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[
"A NOTE ON INDUCED RAMSEY NUMBERS",
"A NOTE ON INDUCED RAMSEY NUMBERS"
]
| [
"David Conlon \nmemory of Jirka Matoušek\n\n",
"Domingos Dellamonica \nmemory of Jirka Matoušek\n\n",
"J R \nmemory of Jirka Matoušek\n\n",
"Steven La Fleur \nmemory of Jirka Matoušek\n\n",
"Vojtěch Rödl \nmemory of Jirka Matoušek\n\n",
"Mathias Schacht \nmemory of Jirka Matoušek\n\n"
]
| [
"memory of Jirka Matoušek\n",
"memory of Jirka Matoušek\n",
"memory of Jirka Matoušek\n",
"memory of Jirka Matoušek\n",
"memory of Jirka Matoušek\n",
"memory of Jirka Matoušek\n"
]
| []
| The induced Ramsey number r ind pF q of a k-uniform hypergraph F is the smallest natural number n for which there exists a k-uniform hypergraph G on n vertices such that every two-coloring of the edges of G contains an induced monochromatic copy of F . We study this function, showing that r ind pF q is bounded above by a reasonable power of rpF q. In particular, our result implies that r ind pF q ď 2 2 ct for any 3-uniform hypergraph F with t vertices, mirroring the best known bound for the usual Ramsey number. The proof relies on an application of the hypergraph container method. | 10.1007/978-3-319-44479-6_13 | [
"https://arxiv.org/pdf/1601.01493v2.pdf"
]
| 119,564,959 | 1601.01493 | 5de9990b57f218c623464f3c966266c1b1c4450b |
A NOTE ON INDUCED RAMSEY NUMBERS
24 Jun 2016
David Conlon
memory of Jirka Matoušek
Domingos Dellamonica
memory of Jirka Matoušek
J R
memory of Jirka Matoušek
Steven La Fleur
memory of Jirka Matoušek
Vojtěch Rödl
memory of Jirka Matoušek
Mathias Schacht
memory of Jirka Matoušek
A NOTE ON INDUCED RAMSEY NUMBERS
24 Jun 2016
The induced Ramsey number r ind pF q of a k-uniform hypergraph F is the smallest natural number n for which there exists a k-uniform hypergraph G on n vertices such that every two-coloring of the edges of G contains an induced monochromatic copy of F . We study this function, showing that r ind pF q is bounded above by a reasonable power of rpF q. In particular, our result implies that r ind pF q ď 2 2 ct for any 3-uniform hypergraph F with t vertices, mirroring the best known bound for the usual Ramsey number. The proof relies on an application of the hypergraph container method.
§1. Introduction
The Ramsey number rpF ; qq of a k-uniform hypergraph F is the smallest natural number n such that every q-coloring of the edges of K pkq n , the complete k-uniform hypergraph on n vertices, contains a monochromatic copy of F . In the particular case when q " 2, we simply write rpF q. The existence of rpF ; qq was established by Ramsey in his foundational paper [17] and there is now a large body of work studying the Ramsey numbers of graphs and hypergraphs. For a recent survey, we refer the interested reader to [5].
In this paper, we will be concerned with a well-known refinement of Ramsey's theorem, the induced Ramsey theorem. We say that a k-uniform hypergraph F is an induced subgraph of another k-uniform hypergraph G if V pF q Ă V pGq and any k vertices in F form an edge if and only if they also form an edge in G. The induced Ramsey number r ind pF ; qq of a k-uniform hypergraph F is then the smallest natural number n for which there exists a k-uniform hypergraph G on n vertices such that that every q-coloring of the edges of G contains an induced monochromatic copy of F . Again, in the particular case when q " 2, we simply write r ind pF q.
For graphs, the existence of induced Ramsey numbers was established independently by Deuber [6], Erdős, Hajnal, and Pósa [8], and Rödl [18], while for k-uniform hypergraphs with k ě 3 their existence was shown independently by Nešetřil and Rödl [16] and Abramson
The first author was supported by a Royal Society University Research Fellowship.
The fourth author was partially supported by NSF grants DMS-1102086 and DMS-1301698.
The fifth author was supported through the Heisenberg-Programme of the DFG. and Harrington [1]. The bounds that these original proofs gave on r ind pF ; qq were enormous.
However, at that time it was noted by Rödl (unpublished) that for bipartite graphs F the induced Ramsey numbers are exponential in the number of vertices. Moreover, it was conjectured by Erdős [7] that there exists a constant c such that every graph F with t vertices satisfies r ind pF q ď 2 ct . If true, the complete graph would show that this is best possible up to the constant c. A result of Conlon, Fox, and Sudakov [3], building on earlier work by Kohayakawa, Prömel, and Rödl [13], comes close to establishing this conjecture, showing that
r ind pF q ď 2 ct log t .
However, the method used to prove this estimate only works in the 2-color case. For q ě 3, the best known bound, due to Fox and Sudakov [11], is r ind pF ; qq ď 2 ct 3 , where c depends only on q.
In this note, we study the analogous question for hypergraphs, showing that the induced Ramsey number is never significantly larger than the usual Ramsey number. Our main result is the following.
r ind pF ; qq ď 2 c 1 k 3 logpqt q R c 2 k 2`c 3 t ,
where R " rpF ; qq is the classical q-color Ramsey number of F .
Define the tower function t k pxq by t 1 pxq " x and, for i ě 1, t i`1 pxq " 2 t i pxq . A seminal result of Erdős and Rado [9] says that
rpK pkq t ; qq ď t k pctq,
where c depends only on k and q. This yields the following immediate corollary of Theorem 1. A result of Erdős and Hajnal (see, for example, Chapter 4.7 in [12] and [4]) says that
rpK pkq t ; 4q ě t k pc 1 tq,
where c 1 depends only on k. Therefore, the Erdős-Rado bound is sharp up to the constant c for q ě 4. By taking F " K pkq t , this also implies that Corollary 1 is tight up to the constant c for q ě 4. Whether it is also sharp for q " 2 and 3 depends on whether rpK pkq t q ě t k pc 1 tq, though determining if this is the case is a famous, and seemingly difficult, open problem.
The proof of Theorem 1 relies on an application of the hypergraph container method of Saxton and Thomason [20] and Balogh, Morris, and Samotij [2]. In Ramsey theory, the use of this method was pioneered by Nenadov and Steger [14] and developed further by Rödl, Ruciński, and Schacht [19] in order to give an exponential-type upper bound for Folkman numbers. Our modest results are simply another manifestation of the power of this beautiful method. §2. Proof of Theorem 1
In order to state the result we first need some definitions. Recall that the degree dpσq of a set of vertices σ in a hypergraph H is the number of edges of H containing σ, while the average degree is the average of dpvq :" dptvuq over all vertices v.
Definition 2.
Let H be an -uniform hypergraph of order N with average degree d. Let
τ ą 0. Given v P V pHq and 2 ď j ď , let d pjq pvq " max dpσq : v P σ Ă V pHq, |σ| " j ( .
If d ą 0, define δ j by the equation
δ j τ j´1 N d " ÿ v d pjq pvq.
The codegree function δpH, τ q is then defined by
δpH, τ q " 2 p 2 q´1 ÿ j"2 2´p j´1 2 q δ j .
If d " 0, define δpH, τ q " 0.
The precise lemma we will need is a slight variant of Corollary 3.6 from Saxton and
Thomason's paper [20]. A similar version was already used in the work of Rödl, Ruciński, and Schacht [19] and we refer the interested reader to that paper for a thorough discussion.
Lemma 3.
Let H be an -uniform hypergraph on N vertices with average degree d. Let 0 ă ε ă 1{2. Suppose that τ satisfies δpH, τ q ď ε{12 ! and τ ď 1{144 ! 2 . Then there exists a collection C of subsets of V pHq such that
(i ) for every set I Ă V pHq such that epHrIsq ď ετ epHq, there is C P C with I Ă C, (ii ) epHrCsq ď εepHq for all C P C, (iii ) log |C| ď 1000 ! 3 logp1{εqN τ logp1{τ q.
Before we give the proof of Theorem 1, we first describe the -uniform hypergraph H to which we will apply Lemma 3.
Construction 4.
Given a k-uniform hypergraph F with edges, we construct an auxiliary hypergraph H by taking
V pHq "ˆr ns k˙a nd EpHq " # E PˆV pHq ˙: E -F + .
In other words, the vertices of H are the k-tuples of rns and the edges of H are copies of F in`r ns k˘.
Proof of Theorem 1. Recall that R " rpF ; qq, the q-color Ramsey number of F , and suppose that F has t vertices and edges. Let us fix the following numbers:
τ " n´1 2 , p " 1000R k qα, α " n´1 2 `1 4 p `1q , ε " 1{p2qR t q, n " 40 2 p `1q p1000qq 8 p `1q R 4k p `1q`4t ˆt k˙4 .
(2.1)
Remark 5.
Note that n is bounded above by an expression of the form
2 c 1 k 3 logpqt q R c 2 k 2`c 3 t ,
as required.
Obviously, R ě t and one can check that p and n satisfy the following conditions, which we will make use of during the course of the proof:
p ď 1, (2.2) n ě p24¨2 p 2 q t t q !R t q 2 , (2.3) n ą p144 ! 2 q 2 , (2.4) n ą 40 2 p `1q , (2.5) n ą p1000qq 8 p `1q R 4k p `1q`4t ˆt k˙4 . (2.6)
We will show that, with positive probability, a random hypergraph G P G pkq pn, pq has the property that every q-coloring of its edges contains an induced monochromatic copy of F . The proof proceeds in two stages. First, we use Lemma 3 to show that, with probability 1´op1q, G has the property that any q-coloring of its edges yields many monochromatic copies of F . Then we show that some of these monochromatic copies must be induced.
More formally, let X be the event that there is a q-coloring of the edges of G which contains at most M :" ετ pnq t autpF q monochromatic copies of F in each color, and let Y be the event that G contains at least M noninduced copies of F . Note that if X X Y happens, then, in any q-coloring, there are more monochromatic copies of F in one of the q colors than there are noninduced copies of F in G. Hence, that color class must contain an induced copy of F .
We now proceed to show that the probability PpXq tends to zero as n tends to infinity.
In order to apply Lemma 3, we need to check that τ and ε satisfy the requisite assumptions with respect to the -uniform hypergraph H defined in Construction 4. Let σ Ă V pHq be arbitrary and define
V σ " ď vPσ v Ă rns.
For an arbitrary set W Ă rns V σ with |W | " t´|V σ |, let emb F pσ, W q denote the number
of copies r F of F with V p r F q " W Y V σ and σ Ă Ep r F q.
Observe that this number does not actually depend on the choice of W , so we will simply use emb F pσq from now on.
Since there are clearly`n´| Vσ| t´|Vσ|˘c hoices for the set W , we arrive at the following claim.
Claim 1. For any
σ Ă V pHq, dpσq "ˆn´| V σ | t´|V σ |˙e mb F pσq.
Let us denote by t j the minimum number of vertices of F which span j edges. From Claim 1, it follows that for any σ Ă V pHq with |σ| " j, we have
dpσq "ˆn´| V σ | t´|V σ |˙e mb F pσq ďˆn´t j t´t j˙e mb F pσq.
On the other hand, for a singleton σ 1 Ă V pHq, we have |V σ 1 | " k and therefore d "
dpσ 1 q is such that dpσq d ď`n´t j t´t jn´k t´k˘e mb F pσq emb F pσ 1 q ď`n´t j t´t jn´k t´k˘ă´n t¯k´t j .
It then follows from Definition 2 and (2.1) that
δ j ă pn{tq k´t j τ j´1 ă t t n k´t j`p j´1q{p2 q . (2.7)
Since t j is increasing with respect to j, t 2 ě k`1, and j ď , we have k´t j`j´1 2 ď´1{2.
Thus, in view of (2.7), we have
δ j ă t t n k´t j`p j´1q{p2 q ď t t n´1 {2 (2.8)
for all 2 ď j ď .
Using Definition 2 and inequality (2.8), we can now bound the codegree function δpH, τ q by δpH, τ q " 2 p 2 q´1
ÿ j"2 2´p j´1 2 q δ j ď 2 p 2 q´1 t t n´1 {2 ÿ j"2 2´p j´1 2 q ď 2 p 2 q t t n´1 {2 . (2.9)
Since n satisfies (2.3), inequality (2.9) implies that
δpH, τ q ď 2 p 2 q t t n´1 {2 ď ε 12 ! .
That is, δpH, τ q satisfies the condition in Lemma 3.
Finally, (2.4) implies that τ satisfies the condition
τ " n´1 {p2 q ă 1 144 ! 2 .
Therefore, the assumptions of Lemma 3 are met and we can let C be the collection of subsets from V pHq obtained from applying Lemma 3. Denote the elements of C by C 1 , C 2 , . . . , C |C| .
For every choice of 1 ď a 1 , . . . , a q ď |C| (not necessarily distinct) let E a 1 ,...,aq be the event that G Ď C a 1 Y¨¨¨Y C aq . Next we will show the following claim. Since each edge in HrI j s corresponds to a copy of F in color j, we have epHrI j sq ď M .
Note that
M " ετ epHq, which means that each I j satisfies the condition ((i )) of Lemma 3. Therefore, for each color class j, there must be a set C a j P C such that C a j Ą I j . Since G " Ť j I j , this implies that G P E a 1 ,...,aq . Since G P X was arbitrary, the bound (2.10) follows and the claim is proved.
Owing to Claim 2, we now bound PpE a 1 ,...,aq q. Recalling the definition of the event E a 1 ,...,aq , we note that PpE a 1 ,...,aq q " p1´pq |V pHq pCa 1 Y¨¨¨YCa q q| .
(2.11)
Hence, we shall estimate |V pHq pC a 1 Y¨¨¨Y C aq q| to derive a bound for PpXq by (2.10). 1 ď a 1 , . . . , a q ď |C| we have |V pHq pC a 1 Y¨¨¨Y C aq q| ě 1 2´n R¯k .
Claim 3. For all choices
Proof. Let a 1 , . . . , a q be fixed and set
A "
" A Pˆr ns R˙:ˆA k˙Ă C a 1 Y¨¨¨Y C aq * . (2.12)
By the definition of R " rpF ; qq, for each set A P A there is an index j " jpAq P rqs such that C a j contains a copy of F with vertices from A. The element e P EpC a j q that corresponds to this copy of F satisfies e Ă`A k˘a nd, thus, Ť xPe x Ă A. We now give an upper bound for |A| by counting the number of pairs in
P " " pe, Aq P q ď i"1 E`C a i˘ˆA with ď xPe x Ă A * .
On
e´5 00qαn k ď e´1 000qαN ,(2.14)
where, in the last step, we used N "`n k˘ď n k 2 . Therefore, (2.10) and (2.14) together with the bound on |C| given by Lemma 3((iii )) imply that PpXq ď ÿ Ca 1 ,...,Ca q PC PpE a 1 ,...,aq q ď |C| q e´1 000qαN
ď exp`1000q ! 3 logp1{εqN τ logp1{τ q´1000qαN" exp`1000qN τ p ! 3 logp1{εq logp1{τ q´α{τ qď exp`1000qN τ p ! 3 log 2 n´n 1{p4 p `1qq q˘ď 1{4,
where we used that n satisfies (2.5). Now, by Markov's inequality, with probability at least 1{2, the number of noninduced copies of F in G will be at most twice the expected number of copies, which is fewer than
2p `1 pnq t autpF qˆt k˙" 2p1000qq `1 R kp `1q n´1 {2´1{p4 q pnq t autpF qˆt kă 1 2qR t pn´1 {p2 q q pnq t autpF q " ετ pnq t autpF q " M,
where the inequality above follows from (2.6). In other words, PpY q ě 1{2 and, therefore, PpX X Y q ě 1{4, so there exists a graph G such that X X Y holds. By our earlier observations, this completes the proof. §3. Concluding remarks
Beginning with Fox and Sudakov [10], much of the recent work on induced Ramsey numbers for graphs has used pseudorandom rather than random graphs for the target graph G. The results of this paper rely very firmly on using random hypergraphs. It would be interesting to know whether comparable bounds could be proved using pseudorandom hypergraphs.
It would also be interesting to prove comparable bounds for the following variant of the induced Ramsey theorem, first proved by Nešetřil and Rödl [15]: for every graph F , there exists a graph G such that every q-coloring of the triangles of G contains an induced copy of F all of whose triangles receive the same color. By taking F " K t and q " 4, we see that |G| may need to be double exponential in |F |. We believe that a matching double-exponential upper bound should also hold.
Theorem 1 .
1Let F be a k-uniform hypergraph with t vertices and edges. Then there are positive constants c 1 , c 2 , and c 3 such that
Corollary 1 .
1For any natural numbers k ě 3 and q ě 2, there exists a constant c such that if F is a k-uniform hypergraph with t vertices, then r ind pF ; qq ď t k pctq.
Pˆł a 1 ,...,aq E a 1 ,...,aq˙ď ÿ a 1 ,...,aq PpE a 1 ,...,aq q. (2.10) Proof. Suppose that G P X. By definition, there exists a q-coloring of the edges of G, say with colors 1, 2, . . . , q, which contains at most M copies of F in each color. For any color class j, let I j denote the set of vertices of H which correspond to edges of color j in G.
the one hand, we have already established that |P| ě |A|. On the other hand, for any fixed e P EpHq, we have | Ť xPe x| " |V pF q| " t and, therefore, there are at most`n´t R´ts ets A Ą Ť xPe x. It follows that By definition, each A P`r ns R˘ A satisfies`A k˘Ć C a 1 Y¨¨¨Y C aq . Hence, V pHq pC a 1 Ÿ¨¨Y C aq q intersects`A k˘. Since an element of V pHq can appear in at most`n´k R´k˘s ets A, it follows from (2.13) that there are at least elements in V pHq pC a 1 Y¨¨¨Y C aq q, as required. In view of Claim 3, our choice of p " 1000R k qα, where α " n´1 {2 `1{4 p `1q , and (2.11), we have, for any C a 1 , . . . , C aq P C, PpE a 1 ,...,aq q ď p1´pq pn{Rq k {2 ď exp`´pn k {2R k˘" exp`´p1000R k qαqn k {2R k"|A| ď |P| ˇˇq
ď
i"1
E`C a i˘ˇˆn´t R´t˙(
(ii ))
ď qεepHqˆn´t
R´t(
2.1)
"
epHq
2R tˆn´t R´t˙ď
pnq t
2R tˆn´t R´t˙ď
1
2ˆn R˙.
(2.13)
1
2ˆn R˙Nˆn´k R´k˙ě
1
2´n R¯k
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| []
|
[
"Generating Adversarial Disturbances for Controller Verification",
"Generating Adversarial Disturbances for Controller Verification"
]
| [
"Udaya Ghai [email protected] \nDepartment of Computer Science\nPrinceton University\n\n",
"David Snyder [email protected] \nDepartment of Mechanical and Aerospace Engineering\nPrinceton University\n\n",
"Anirudha Majumdar [email protected] \nDepartment of Mechanical and Aerospace Engineering\nPrinceton University\n\n\nGoogle AI\nPrinceton\n",
"Elad Hazan \nDepartment of Computer Science\nPrinceton University\n\n\nGoogle AI\nPrinceton\n",
"Ehazan@cs Princeton Edu "
]
| [
"Department of Computer Science\nPrinceton University\n",
"Department of Mechanical and Aerospace Engineering\nPrinceton University\n",
"Department of Mechanical and Aerospace Engineering\nPrinceton University\n",
"Google AI\nPrinceton",
"Department of Computer Science\nPrinceton University\n",
"Google AI\nPrinceton"
]
| []
| We consider the problem of generating maximally adversarial disturbances for a given controller assuming only blackbox access to it. We propose an online learning approach to this problem that adaptively generates disturbances based on control inputs chosen by the controller. The goal of the disturbance generator is to minimize regret versus a benchmark disturbance-generating policy class, i.e., to maximize the cost incurred by the controller as well as possible compared to the best possible disturbance generator in hindsight (chosen from a benchmark policy class). In the setting where the dynamics are linear and the costs are quadratic, we formulate our problem as an online trust region (OTR) problem with memory and present a new online learning algorithm (MOTR) for this problem. We prove that this method competes with the best disturbance generator in hindsight (chosen from a rich class of benchmark policies that includes linear-dynamical disturbance generating policies). We demonstrate our approach on two simulated examples: (i) synthetically generated linear systems, and (ii) generating wind disturbances for the popular PX4 controller in the AirSim simulator. On these examples, we demonstrate that our approach outperforms several baseline approaches, including H ∞ disturbance generation and gradient-based methods. | null | [
"https://arxiv.org/pdf/2012.06695v2.pdf"
]
| 229,157,128 | 2012.06695 | 7c5a61ab85abc70825a62c98acb11ac3fd4c58d3 |
Generating Adversarial Disturbances for Controller Verification
Udaya Ghai [email protected]
Department of Computer Science
Princeton University
David Snyder [email protected]
Department of Mechanical and Aerospace Engineering
Princeton University
Anirudha Majumdar [email protected]
Department of Mechanical and Aerospace Engineering
Princeton University
Google AI
Princeton
Elad Hazan
Department of Computer Science
Princeton University
Google AI
Princeton
Ehazan@cs Princeton Edu
Generating Adversarial Disturbances for Controller Verification
Adversarial DisturbancesController VerificationOnline Learning
We consider the problem of generating maximally adversarial disturbances for a given controller assuming only blackbox access to it. We propose an online learning approach to this problem that adaptively generates disturbances based on control inputs chosen by the controller. The goal of the disturbance generator is to minimize regret versus a benchmark disturbance-generating policy class, i.e., to maximize the cost incurred by the controller as well as possible compared to the best possible disturbance generator in hindsight (chosen from a benchmark policy class). In the setting where the dynamics are linear and the costs are quadratic, we formulate our problem as an online trust region (OTR) problem with memory and present a new online learning algorithm (MOTR) for this problem. We prove that this method competes with the best disturbance generator in hindsight (chosen from a rich class of benchmark policies that includes linear-dynamical disturbance generating policies). We demonstrate our approach on two simulated examples: (i) synthetically generated linear systems, and (ii) generating wind disturbances for the popular PX4 controller in the AirSim simulator. On these examples, we demonstrate that our approach outperforms several baseline approaches, including H ∞ disturbance generation and gradient-based methods.
Introduction
We consider the problem of certifying the safety and correct operation of control algorithms in the context of robotics, as understood by a measure of the worst-case system performance in the presence of uncertainty and disturbances. Motivated by this challenge, we consider the following idealized problem. Consider a control system given by x t+1 = f (x t , u t , w t ), with state x ∈ X ⊆ R dx , control input u ∈ R du , and disturbance w ∈ R dw . Suppose we are provided blackbox access to a controller for this system, i.e., we do not have access to the software that defines the controller, but can observe the closed-loop system's behavior by choosing disturbance values. The controller may be arbitrarily complex (e.g., adaptive, nonlinear, stateful, etc.). Our goal is to generate disturbances w t that maximize a specified cost ∞ t=0 c(x t , u t ) incurred by the controller. Statement of Contributions. We present an online learning approach for tackling the problem of generating disturbances for dynamical systems in order to maximize the cost incurred by a controller arXiv:2012.06695v2 [cs.LG] 31 Jan 2022 GENERATING ADVERSARIAL DISTURBANCES FOR CONTROLLER VERIFICATION Figure 1: Quadrotor in AirSim Mountains Environment. We consider the problem of generating adversarial disturbances (e.g., wind gusts) for a given controller. assuming only blackbox access to it. The key idea behind our approach is to leverage techniques from online learning (see e.g. Hazan (2019)) to adaptively choose disturbances based on control inputs chosen by the controller. Determining the optimal disturbance for a given controller with online blackbox access is computationally infeasible in general. We thus consider regret minimization versus a benchmark disturbance-generating policy class. Since our goal is to maximize the cost of the controller, the natural formulation of our problem in online learning is non-convex. Online non-convex optimization does not admit efficient algorithms in general.
To overcome this challenge, we consider the case when the system is linear and the costs are quadratic. In this case we formulate our problem as a special case of non-convex optimization, namely an online trust region (OTR) problem with memory. We then present a new online trust region with memory algorithm (MOTR) with optimal regret guarantees, which may be of independent interest. Using this technique, we prove that our method competes with the best disturbance-generating policy in hindsight from a reference class. This reference class includes all state-feedback linear-dynamical policies (Def. 3).
We demonstrate our approach on two simulated examples: (i) synthetically generated linear systems, and (ii) generating wind disturbances for the highly-popular PX4 controller (Meier et al., 2015) in the physically-realistic AirSim drone simulator (Shah et al., 2017) (Fig. 1). We compare our approach to several baseline methods, including gradient-based methods and an H ∞ disturbance generator. For linear systems, H ∞ is a Nash equilibrium solution to the offline disturbance problem; however, this does not hold for the case of time-varying costs or when the controller deviates from an H ∞ paradigm. We demonstrate the ability of our method to adaptively generate disturbances that outperform these baselines.
Related Work
Regret minimization for online control. There is a large body of work within the control theory literature on synthesizing robust and adaptive controllers (see, e.g., Stengel (1994); Zhou et al. (1996)). The most relevant work for our purposes is online control with low regret. In classical control theory, the disturbances are assumed to be i.i.d. Gaussian and the cost functions are known ahead of time. In the online LQR setting (Abbasi-Yadkori and Szepesvári, 2011;Dean et al., 2018;Mania et al., 2019;Cohen et al., 2018), a fully-observed linear dynamic system is driven by i.i.d. Gaussian noise and the learner incurs a quadratic state and input cost.
Recent algorithms (Mania et al., 2019;Cohen et al., 2019Cohen et al., , 2018 attain √ T regret for this online setting, and are able to cope with changing loss functions. Agarwal et al. (2019a) consider the more general and challenging setting of non-stochastic control in which the disturbances are adversarially chosen, and the cost functions are arbitrary convex costs. The key insight behind this result is an improper controller parameterization, known as disturbance-action control, coupled with advances in online convex optimization with memory due to Anava et al. (2015). Non-stochastic control was extended to the setting of unknown systems and partial observability Simchowitz et al., 2020).
In contrast to the work mentioned above, we consider the problem of generating adaptive disturbances that maximize the cumulative cost incurred by a given controller. This shift in problem formulation introduces fundamental technical challenges. In particular, the primary challenge is the non-convexity associated with the cost maximization problem. Providing regret guarantees (from the point of view of the disturbance generator) in this non-convex setting constitutes one of they key technical contributions of this work.
Adversarial reinforcement learning. The literature on generating disturbances for control systems is sparse compared to the body of work on synthesizing robust controllers. There has been recent work on generating adversarial policies for agents trained using reinforcement learning (motivated by a long line of work on generating adversarial examples for supervised learning models (Goodfellow et al., 2015)). These results consistently suggest that for high-dimensional problems in RL settings, non-adversarially-trained agents can be directly harmed in training (Behzadan and Munir, 2017) and are highly susceptible to multiple adversarial failure modes (Huang et al., 2017;Gleave et al., 2020). The latter problem motivates Vinitsky et al. (2020) to train agents against an ensemble of adversaries to generate a more robust learned policy. In this vein, Mandlekar et al. (2017) integrates the ideas into a robust training algorithm that allows for noise perturbations in the standard control formulation. However, in contrast to our work, none of the above robust training protocols make theoretical guarantees about the performance of their trained agent, whereas we are able to obtain explicit regret guarantees for the performance of our adversarial agent.
A parallel line of investigation uses sampling-based techniques for probabilistic safety assurance (Sinha et al., 2020a) by actively seeking samples of rare events to estimate the probability of failure modes. This is extended in Sinha et al. (2020b) to include online methods. In particular, they generate models of multiple adversaries (akin to Vinitsky et al. (2020)) and then use online learning to 'decompose' their observed adversary into elements of their set of modeled adversaries, choosing robust actions accordingly. This differs from our work in the optimization paradigm. In particular, they require Monte Carlo sampling of multiple trajectories, repeated over subsequent updates of the environment distribution parameters, in order to obtain guarantees. Our guarantees are 'withintrajectory,' in that we learn and compete with a class of disturbance generators within a single trajectory, rather than by optimizing over many simulations.
Online learning and the trust region problem. We make extensive use of techniques from the field of online learning and regret minimization in games (Cesa-Bianchi and Lugosi, 2006;Hazan et al., 2016). Most relevant to our work is the literature on online non-convex optimization (Agarwal et al., 2019b;Suggala and Netrapalli, 2019), and online convex optimization with memory (Anava et al., 2015). The problem of maximizing a general quadratic function subject to Euclidean norm constraint is known as the Trust Region (TR) problem, which originated in applying Newton's method to non-convex optimization. Despite the non-convex objective, TR is known to be solvable in polynomial time via a semi-definite relaxation (Ben-Tal and Teboulle, 1996), and also allows for accelerated gradient methods (Hazan and Koren, 2015).
Setting and Background
Notation
For vectors, we use the notation x Q = x Qx for a weighted euclidean norm, where Q is a positive definite matrix. For matrices, we use M F to denote the Frobenius norm of matrix M and M to denote the spectral norm.
Setting
We consider a nonstochastic linear time-invariant (LTI) system defined by the following equation:
x t+1 = Ax t + Bu t + Cw t ,
where x t ∈ R dx is the state, u t ∈ R du is the control input, and w t ∈ R dw is the adversarially chosen disturbance. We assume the state and disturbance dimensions are the same and C = I for the remainder of this exposition, but the results still hold in full generality. At time t, a quadratic cost c t (·, ·) is revealed and the controller suffers cost c t (x t , u t ). As the disturbances are adversarially chosen, the trajectory, and thus the costs are determined by this. In this model, a disturbance generator A is a (possibly randomized) mapping from all previous states and actions to a disturbance vector. As the goal is to produce worst-case disturbances, the cost c t for the controller is a reward for A. The states produced by A with controls u t are denoted x A t and the total reward is denoted
J T (A) = T t=1 c t (x A t , u t ) = T t=1 x A t 2 Qt + u t 2 Rt .
For a randomized generator, we consider the expected reward. We use a regret notion of performance, where the goal is to play a disturbance sequence {w t } T t=1 such that the reward is competitive with reward corresponding to the disturbances played by the best fixed disturbance generator π, chosen in hindsight from a comparator class Π.
Regret T (A) = max π∈Π J T (π) − J T (A).
Comparator class
For our comparator class, we consider a bounded set of Control-disturbance Generators, defined as follows.
Definition 1 A Control-disturbance Generator (CDG), π(M ) is specified by parameters M = (M [1] , . . . , M [H] )
, along with a bias 1 w 0 , where the disturbance w t played at state x t is defined as
w t = H i=1 M [i] u t−i + w 0 .
(1)
1. The bias term is not included in the remainder of the theoretical work for simplicity, but equivalent results can be proved including bias. A CDG is the equivalent of a Disturbance-action controller (DAC) (Agarwal et al., 2019a) where the roles of actions and disturbances have been swapped. It has been shown that DACs can approximate Linear Dynamic Controllers (LDCs), a powerful generalization of linear controllers. As such, we analogously define Linear Dynamic Disturbance Generators (LDDGs), which likewise are approximated by CDGs.
Definition 3 (Linear Dynamic Disturbance Generator) A linear dynamic disturbance generator π is a linear dynamical system (A π , B π , C π , D π ) with internal state s t ∈ R dπ , input x t ∈ R dx , and output w t ∈ R dw that satisfies
s t+1 = A π s t + B π x t , w t = C π s t + D π x t .
Assumptions
We make the following assumptions requiring an agent to play bounded controls and requiring bounded system and cost matrices:
Assumption 4 (Bounded controls) The control sequence is bounded so u t ≤ C u . Assumption 5 (Stabilizable Dynamics) Consider the dynamics tuple {A, B, C, u(x, t)}. We as- sume that A = HLH −1 , with L ≤ 1 − γ, matrix A having condition number H H −1 ≤ κ and A , B , C ≤ β.
Note that if A is not open-loop stable, but the pair (A, B) is stabilizable, there exists a matrix K * such thatà = A − BK * satisfies the above criterion, and we can equivalently analyze the system
{Ã, B, u * (x, t)}, where u * (x, t) = u(x, t) + K * x.
This transformation explains the generality of Assumption 5. Importantly, we do not require knowledge of u(x, t) in this transformation.
Assumption 6 (Bounded costs) The cost matrices have bounded spectral norm, Q t , R t ≤ ξ.
Following Chen and Hazan (2020), we use L to denote the complexity of the system and comparator class where L = d x + d u + d w + D + C u + β + κ + ξ + γ −1 . Here, d x , d u , d w are the state, action, and disturbance dimensions respectively. C u bounds the magnitude of controls. The comparator class is Π H,D with H = γ −1 log(κξT ) , in order to capture LDDGs. β and ξ are spectral norm bounds on system matrices and costs respectively. The condition number and decay of dynamics are κ and γ respectively.
Online Trust Region With Memory
This section describes our main building block for the adversarial disturbance generator: an online non-convex learning problem called online trust region (OTR) with memory. In Sec. 3.1 we provide background on the trust region problem. Subsequently, in Sec. 3.2 we formally introduce the online trust region with memory setting.
Trust Region Problem
We show that the cost of a CDG can be approximated closely by a nonconvex quadratic. Optimizing the cost of the policy is then a trust region problem, a well-studied quadratic optimization over a Euclidean ball. The interest in this problem stems from the fact that it is one of the most basic nonconvex optimization problems that admits "hidden-convexity" -a property that allows efficient algorithms that converge to a global solution.
Definition 7 A trust region problem is defined by a tuple (P, p, D) with P ∈ R d×d , p ∈ R d , and D > 0 as the following mathematical optimization problem
max z ≤ D z P z + p z.
We can define a condition number for a trust region problem as follows.
Definition 8 The condition number for a trust region instance (P, p, D) is κ = λ µ , where λ = max(2( P 2 + p 2 ), D, 1) and µ = min(D, 1).
Note that a trust region problem can be solved in polynomial time by conversion to an equivalent convex optimization problem (see e.g. Ben-Tal and Teboulle (1996)).
Theorem 9 Let (P, p, D) be a trust region problem with condition number κ. There exists an algorithm TrustRegion such that TrustRegion(P, p, D, ) produces z a with z a ≤ D such that
z a P z a + p z a ≥ max z ≤ D z P z + p z − ,
and runs in time poly(d, log κ, log 1 ε ).
Online trust region with memory
Consider the setting of online learning, where an algorithm A predicts a point z t with z t ≤ D.
We use the shorthand, z t:H = (z t−H+1 , . . . , z t ) ∈ R dH for the concatenation of the H last points. The algorithm then receives feedback from an adversarially chosen quadratic reward function f t : R dH → R of the last H decisions, parameterized by P t ∈ R dH×dH and p t ∈ R dH . The reward function is defined as
f t (z ) = z P t z + p t z .(2)
The reward function acting on a single point is also useful, so we define g t :
R d → R with g t (z) = f t (z, . . . z) = (z, . . . z) P t (z, . . . z) + p t (z, . . . z) := z C t z + d t z.(3)
The reward earned in round t is f t (z t:H ). The goal of the online player is to minimize the expected regret, compared to playing the single best point in hindsight:
Regret(A) = max z ≤ D T t=H g t (z) − E[ T t=H f t (z t:H )] .(4)
Here the expectation is over randomness of the algorithm. In App. B, we provide a polynomial-time O(H 3 2 T 1 2 ) regret algorithm for the online trust region with memory. Below is an informal statement of this result (See Thm. 15 for the full result).
Theorem 10 Suppose elements of matrices P t and elements of p t bounded. Alg. 3, with suitable parameterization, will incur expected regret at most
Regret(T ) ≤ O(D 2 d 5/2 H 3/2 √ T ) ,
and the runtime of the algorithm for each iteration will be poly(d, H, log D, log T ).
The algorithm works by applying an extension of nonconvex Follow-the-Perturbed-Leader (FPL) (Agarwal et al., 2019b;Suggala and Netrapalli, 2019) to functions with memory (see App. A). In the OTR with memory setting, the perturbed subproblems that need to be solved in each iteration are trust region problems, so they can be solved in polynomial time. (13) and (14) in App. D] Generate random vector σ t ∈ R Hdxdu such that
Algorithm and Main Theorem
(x t , u t ) = x t 2 Qt + u t 2 Rt Generate disturbance w t = H i=1 M [i] t u t−i Observe control u t and update state x t+1 = Ax t + Bu t + w t Define g t (m) = c t (y t (M ), u t ) where y s+1 (M ) = Ay s (M ) + Bu s + C H i=1 M [i] u s−i + w 0 s ≥ t − h 0 otherwise Define S t = S t−1 + (∇ 2 g t )(0) and s t = s t−1 + (∇g t )(0) [seeσ t,i ∼ Exp(η) Update m t+1 ← TrustRegion(S t , s t − σ t , D, 1 T ) Reshape M t+1 ← reshape(m t+1 , R Hdx×du ) end for
In App. C, we show that g t is a quadratic function of m, and that y t is an accurate approximation of x t due to the stabilizability of the dynamics. In App. D, we combine the regret bound for Alg. 3 in Thm. 15 with the approximation guarantee on y t from Lem. 19, yielding the following theorem.
Theorem 11 Suppose Assumptions 4, 5, 6 hold; then Alg. 1 suffers regret at most O(poly(L) √ T ).
Experiments
We evaluate the performance of the disturbance generator MOTR defined in Alg. 1 across two settings. These consist of (1) general (randomly generated) linear systems of varying modal behavior, and (2) a thirteen-dimensional rigid-body model for a quadrotor drone in the AirSim simulation environment (Shah et al., 2017). To evaluate our algorithm in each setting, we compare its performance with the performance of several baseline disturbance generators, against several different controllers. In each setting, the MOTR algorithm is able to outperform the baselines.
Baseline Generators and Controllers
We compare the MOTR algorithm with five baseline generators. Sinusoidal and Gaussian noise are standard within control theory, and form the first two generator classes. A random directional generator (a fixed-norm equivalent of the Gaussian generator) is third. The dynamic game formulation of the H ∞ control problem (Basar and Bernhard, 2008; Bernhard, 1991) yields a Nash equilibrium disturbance generator. The final baseline is a first-order online gradient ascent (OGA) policy, which does not provide theoretical guarantees in this nonconvex setting. OGA produces disturbances via a CDG, with the M learned via gradient ascent on the instantaneous cost.
For the experimental settings in which the true dynamics are linear, the disturbance generators are tested across three controllers: a standard LQR controller, a H ∞ infinite-horizon optimal controller, and an adaptive gradient perturbation controller (GPC) (Agarwal et al., 2019a). In each setting, true, fixed system costs ( x 2 + u 2 ) were provided to the algorithms.
In the AirSim experiment, the two nonlinear controllers tested are the Pixhawk PX4 controller (Meier et al., 2015), which is one of the most popular controllers used by quadrotors in practice, and a pre-tuned PID controller that is defined by the AirSim environment.
Notes on Implementation
In order to ensure fair comparisons across the baselines, the actions chosen by each disturbance generator except the Gaussian are normalized to ensure that the available disturbance 'budget' does not vary across generators. Thus, the sinusoid and random generators are essentially choosing directions within the state space. The Gaussian generator is specified so that its average disturbance norm will be slightly higher than the norm bound in expectation. The frequency and initial phase vector of each sinusoid generator are optimized offline against the open-loop system dynamics. Further details of the implementation of MOTR are deferred to App. E.2
Experiment 1: General Linear Systems
Here, a randomly generated set of 11 linear systems are tested for each controller-generator pair over 10 initial conditions. For each system, A ∈ R 4×4 , B, C ∈ R 4×2 . We define the cost of a trajectory to be equal to the cumulative average cost over the time horizon. For each controller-generator pair, we average the costs over the 10 initial conditions, and then normalize each generator's average cost for a given controller to lie in the range [0, 1], where a higher value indicates stronger performance. These costs are aggregated across the 11 systems and scaled to the best-performing disturbance generator (for the given controller). The results are shown in Tab. 1.
There are several important points to note. First, against an H ∞ controller, the H ∞ disturbance generator is a Nash equilibrium solution, so it is expected that MOTR will recover but not exceed that performance. In addition to MOTR strongly outperforming the Random, Sinusoid, and Gaussian generators in each setting, we see that against adaptive controllers like GPC, MOTR also begins to outperform the H ∞ disturbances. In the presence of model misspecification and cost mismatch, we expect this phenomenon may become more pronounced.
Experiment 2: Rigid-Body Drone in AirSim
Testing MOTR within the AirSim environment allows an empirical test of several key elements of the algorithmic performance, including (1) scaling to higher system dimension, (2) generalizability to nonlinear dynamics about linearized reference conditions, and (3) performance with an accurate but low-dimensional model of the disturbance-to-state transfer function. The model follows the traditional rigid-body, 6 degree-of-freedom (6DOF) model for air vehicle dynamics, but uses quaternions instead of Euler angles, yielding a 13-dimensional state representation. The nonlinear dynamics are propagated about a nominal hover flight condition. This condition was linearized numerically using a least squares regression procedure on simulator data. There are four inputs, corresponding to the motor commands for each propeller of the quadcopter, and three disturbance channels, corresponding to the North-East-Down (NED) coordinates of the inertial wind vector. Results are taken over 14 seeds per generator (b) Results for the popular PX4 controller. Note that the non-adaptive H∞ policy is attenuated, unlike MOTR. Further, the first-order online method struggles in this setting. Results are taken over 15 seeds per generator.
The controllers utilized in the simulator include a 'SimpleFlight' AirSim controller and the PX4 autopilot, which is incorporated into AirSim's software-in-the-loop PX4 stack. Each of these controllers is a nonlinear PID controller, and we note that the PX4 is one of the most commonly used autopilots for quadcopter drones.
We present the results of the simulations in Fig. 2. An important feature of the simulations was the presence of a clear best strategy for large disturbances, corresponding to updrafts and downdrafts. However, because the wind magnitudes are constrained, the PX4 controller is able to adaptively attenuate this behavior. As such, the H ∞ generator, while nearly as strong as MOTR on the SimpleFlight simulations, suffered against PX4. MOTR, however, was able to adapt in the PX4 setting and thus maintain strong disturbance performance.
Conclusions
We have studied the problem of generating the worst possible disturbances for a given controller. This is a challenging non-convex problem, which we pose in the framework of online learning. We describe a novel method based on regret minimization in non-convex games with provable guarantees. Our experimental results demonstrate the ability of our approach to outperform various baselines including gradient-based methods and an H ∞ disturbance generator.
This work raises many intriguing questions: can this approach be generalized to dynamics that are unknown, non-linear, partially observable, admit bandit feedback and/or time-varying? Recent results in non-stochastic control suggest the feasibility of these directions (Chen and Hazan, 2020; Gradu et al., 2020a,b;Simchowitz et al., 2020). Another promising direction is to establish lower bounds on regret for the disturbance generation problem and find algorithms that match these lower bounds. Finally, in the vein of adversarial reinforcement learning, the inclusion of adversarial disturbances may prove a useful tool in synthesizing robust learned controllers, as having access to a an adaptive, online disturbance generation mechanism might enhance robustness and regularize worst-case behavior.
Acknowledgments
We thank Karan Singh for enlightening discussion. Elad Hazan is partially supported by NSF award # 1704860 as well as the Google corporation. Anirudha Majumdar was partially supported by the Office of Naval Research [Award Number: N00014-18-1-2873]. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-2039656. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
References
Yasin Abbasi-Yadkori and Csaba Szepesvári. Regret bounds for the adaptive control of linear quadratic systems.
f t (z t−H , . . . z t )] − min z∈Z H t=H f (z, . . . , z) ≤ (LH 2 + LH) T t=H E[ z t − z t+1 1 ] + dD η + T(7)
We can then follow Thm. 1 of Suggala and Netrapalli (2019) to bound
E[ z t − z t+1 1 ] ≤ 125ηLHd 2 D + 20LH
Substituting into (7), yields the result.
Corollary 14 Let Z ⊆ R d be a convex decision set with ∞ diameter at most D. Suppose the losses are L-Lipschitz wrt. the 1 norm. With η = Θ(L −1 H −3/2 d −1/2 T −1/2 ) and = Θ(T −1/2 ), Alg. 2 achieves expected regret
E[ H t=H f t (z t−H , . . . z t )] − min z∈Z H t=H f (z, . . . , z) ≤ O(LDd 3/2 H 3/2 √ T )
Algorithm 3 Online Trust Region
Input: Rounds T , history length H, noise parameter η, error tolerance , 2 bound D Initialize z 0 , . . . , z H−1 randomly with
z i ≤ D S H−1 = 0 d×d , s H−1 = 0 d for t = H to T do Play z t Observe P t ∈ R dH×dH , p t ∈ R dH Suffer loss f t (z t:H ) = z t:H P t z t:H + p t z t:H Define C t = H i=1 H j=1 P ij t and d t = H i=1 p i t [See (8)] Define S t = S t−1 + C t and s t = s t−1 + d t Generate random vector σ t ∈ R d such that σ t,i ∼ Exp(η) Update z t+1 ← TrustRegion(S t , s t − σ t , D, ) end for
Theorem 15 Suppose elements of matrices P t and elements of p t bounded by R.
Let η = R d 3/2 DH 3/2 · 1 √ T Θ(R(dD + 1) −1 H −3/2 d −1/2 T −1/2 ), ε = Θ(T −1/2 ).
Alg. 3 will incur expected regret at most
Regret(T ) ≤ O(RD 2 d 5/2 H 3/2 √ T ) ,
and the runtime of the algorithm for each iteration will be poly(d, H, log R, log D, log T ).
Proof First note that Alg. 3 is a correct implementation of FTPL with f t and g t as defined in (2) and (8). In particular, TrustRegion(S t , s t − σ t , D, ε) approximately solves
max z ≤ D z S t x + s t z − σ t z = max z ≤ D z t s=H C s z + z t s=H d s − σ t x = max z ≤ D t s=H (z C s z + d s z) − σ t z = max z ≤ D t s=H g s (z) − σ t z .
We first note that the ∞ diameter of the constraint set, D ∞ is 2D. The next step is to bound the 1 , Lipschitz constant of f t (x). We note that ∇f t (x) = (P + P )x + p so
L = max x ∇f t (z) ∞ ≤ max x ( P ∞ + P 1 ) z ∞ + p ∞ ≤ ( P 1 + P ∞ )D + p ∞ ≤ 2dRD + R(9)
The regret bound follows after applying Cor. 14 and (9). To bound the runtime of an iteration, we need to find the condition number for the trust region instances passed to TrustRegion. Because the S matrices sum over time, it can be seen that the condition number can grow at most linearly in T . The runtime then follows from Thm. 9.
Appendix C. Approximating the state and costs with a truncated rollout We return to the control application. We want to analyze the state x A t+1 that results from a noise generator A that plays CDG M t = (M
[1] t . . . M [H]
t ). Unrolling the LDS recursion H times can be done to yield the following transfer matrix Ψ t,i , describing the effect of u t−i on the state x t+1 .
Definition 16 Define the disturbance-noise transfer matrix Ψ t,i to be
Ψ t,i (M t−H . . . M t−1 ) = A i B1 i ≤ H + H j=1 A j−1 M [i−j] t−j 1 i−j∈[1,H](10)
We note that Ψ t,i is linear in (M t−H . . . M t−1 ).
Lemma 17
If w t is chosen using the non-stationary noise policy (ρ 0 . . . ρ T −1 ), then the state sequence satisfies
x A t+1 = A H+1 x A t−H + 2H i=0 Ψ t,i u t−i(11)
As the notion of regret is counter-factual, decisions from all time steps effect the current state and thus the current loss. To mitigate this effect, we approximate the state and cost using only the effect of the last H time steps. Lem. 19 shows that such an approximation is valid, as the current state has a negligible dependence on the long term past.
Definition 18 Define an approximate state y t , which is the state the system would have reached if it played the non-stationary policy (M t−H−1 . . . M t−1 ) at all time steps from t − H to t assuming y t−H = 0. we have
y t (M t−H−1:t−1 ) = 2H i=0 Ψ t−1,i u t−i−1 , where we use the notation M t−H−1:t−1 = [M t−H−1 . . . M t−1 ].
The approximate cost is then given by
f t (M t−H:t ) = c t (y t (M t−H:t ), u t ) = y t (M t−H−1:t−1 ) 2 Qt + u t 2 Rt .
Lemma 19 Suppose A plays according to (M t−H−1 . . . M t−1 ) at all time steps from t − H to t, Assumptions 4, 5, and 6 holds and H = γ −1 log(κξT ) , then we have that |c
t (x A t ) − f t (M t−H:t )| ≤ Cx T , where C x = 2βHDCu γ .
Proof We first show that the states are bounded. We note that w t = M t u t−H:t−1 so w t ≤ HDC u by applying Assumption 4 and the Definition 2. Now applying Assumption 5 and triangle inequality, we have Bu t + Cw t ≤ 2βHDC u . The state x A t−H−1 ≤ 2βHDCu γ = C x via strong stability of A. We first note that
x A t − y M t−H:t t = A H+1 x A t−H−1 [Lem. 17] ≤ A H+1 x A t−H−1 ≤ A H+1 C x ≤ H H −1 L H+1 x A t−H−1 ≤ κ(1 − γ) H+1 C x [Assumption 5] ≤ κC x e −γH [1 + x ≤ e x ]
Now, we use Assumption 6 to complete the result.
|c t (x A t ) − f t (M t−H:t )| = |c t (x A t ) − c t (y t ))| ≤ ξ x A t − y t 2 ≤ ξκC x e −γH ≤ C x T
Appendix D. Applying Alg. 3 to approximate costs
We apply Alg. 3 with the approximate cost f t as defined in Def. 18. To do this, we derive a closed form expression for g t (M ) = f t (M, . . . M ). The transfer matrix can be simplified to
Φ t,i (M . . . M ) = C i M + D i , where C i ∈ R dx×Hdx
Here P t ∈ R Hdxdu×Hdxdu and p t ∈ R Hdxdu are defined by P t,k+Hdx(l−1),m+Hdx(n−1) =
2H i=0 2H j=0 U t ij ln C i Q t C j km(13)p t = 2 2H i=0 2H j=0 vec(C i Q t D j U t ij ) ,(14)
where U t ij = u t−j−1 u t−i−1 . Applying the closed form for g t in Alg. 3 yields Alg. 1. with coefficients of P t bounded by ξC 2 u κ 2 and coefficients of p t bounded by ξC 2 u κ 2 β.
For cases in which the controller deviates from the H ∞ equilibrium, the H ∞ generator is no longer necessarily Nash optimal, and therefore our generator will be able to deviate in order to exploit any arising weaknesses in the controller robustness.
Of course, it is possible to have the bias term include both an H ∞ and learned component, if it is deemed wise to do so. In general, we try to avoid hyperparameter tuning of this nature.
E.2. Computational Notes
In practice the disturbances will need to be bounded, in order to prevent either unbounded growth in the disturbances, or rapid decay (depending on whether the domain is too large or too small). Setting the parameters to be small generally appears to be the better option.
The algorithm runtime is generally short. The AirSim simulation runs in approximately real time on a CPU with a 3.30 GHz Intel i9-9820X processor. For the linear dynamical systems simulations, the four-state, 2-input case runtime is approximately 15 minutes for the simulations comprising 18 controller-generator pairs over 11 systems and 10 seeds.
The AirSim simulation utilizes numerical integration with a timestep dt = 0.001 seconds. In order to analyze our system, the observed rotor speed inputs that are seen by the disturbance generators are averaged over 0.25 second intervals, with 25 data points per interval. The wind policy itself is thus only allowed to update every 0.25 seconds. This limits the wind to oscillations of at most 2 Hz, in line with observed power spectra (Thomas, 1996), which show the presence of pressure fluctuations greater than 1 Hz but general attenuation above 5 Hz.
The key hyperparameter of MOTR (with our bias implementation) is the allowable cumulative norm magnitudes of the matrices M of our learned parameterization [termed D M ]. Because the disturbances are norm-limited, choice of D M affects the relative weighting of the trust region solver (grows with higher D M ) and the bias term. In general, a smaller D M prevents significant projection distances, which keeps the estimate closer to that of the regret minimizing theoretical generator and improves performance. In many cases, even small D M allow significant differentiation between MOTR and the H ∞ generator, which is the limiting case as D M → 0 + .
Definition 2
2Let Π D,H = {π(M ) : M ∈ R Hdx×du , M F ≤ D} be the set of CDGs with history H and size D. We also use the shorthand M ∈ Π D,H .
algorithm (MOTR; Alg. 1) is an application of Alg. 3 in the Appendix for the OTR with memory problem applied to approximations of the costs of playing a CDG, π(M ). Let m = vec(M ) be a flattened version of the CDG matrix M . Our approximate cost, g t (m) = c t (y t (M ), u t ) is the cost of an approximate state from a truncated rollout starting at y t−H (M ) = 0, with y s+1 (M ) = Ay s (M ) + Bu s+ C H i=1 M [i] u s−i + w 0 .(5)Algorithm 1 Memory Online Trust Region (MOTR) GeneratorInput: Rounds T , system parameters (A, B, C), noise parameter η, history H Define u s = 0 for s ≤ 0.Initialize M 0 ∈ Π D,H randomly. S 0 = 0 Hdxdu,Hdxdu , s 0 = 0 Hdxdu . for t = 0 to T do Observe quadratic reward c t and earn c t
Figure 2 :
2(a) Results for the SimpleFlight robust PID controller. Both H∞ and MOTR perform well in this setting.
is a block matrix with blocks either 0 or powers of A, and D i = A i B1 i ≤ H . We can write g t as a quadratic function of M as follows g t (M ) = y t (M . . . M ) (M ) P t vec(M ) + p t vec(M ) + c .
Lemma 20
20Suppose Assumptions 4 and 6 hold, then f t (M t−H:t ) is the quadratic function vec(M t−H:t ) P t vec(M t−H:t ) + p t vec(M t−H:t ) ,
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Appendix A. Follow the Perturbed Leader with MemoryThe core of our algorithm is the (Suggala and Netrapalli, 2019) FPL algorithm applied in the memory setting. Here we let O ,Z be an optimization oracle over a convex set Z.Definition 12 Let z a = O ,Z (f ) be the output of the approximate optimization oracle. Then, we have z a ∈ Z, and sup z∈Z f (z) − f (z a ) ≤Algorithm 2 FPL with memoryInput: Rounds T , history length H, dist. parameter η, approximate opt oracle O Initialize z 0 , . . . , z H−1 randomly for t = H to T do Play z t and suffer loss fTheorem 13 Let Z ⊆ R d be a convex decision set with ∞ diameter at most D. Suppose the losses are L-Lipschitz wrt. the 1 norm. For any η, Alg. 2 achieves expected regretProof We first note that Alg. 2 is the same approximate FPL algorithm as Alg. 1 from Suggala and Netrapalli (2019) with loss functions g t . We note that g t is LH-Lipschitz wrt. the 1 norm as(2019), we have thatNow, by Lemma 4 from Suggala and NetrapalliTo get a regret bound for f t , we need to bound the difference between f t (z t−H , . . . z t ) and g t (z t ).Appendix B. OTR with memory algorithmThe trust region with memory problem can be solved by applying Alg. 2 with a fast approximate trust region optimization algorithm in place of the optimization oracle. The TrustRegion algorithm fromHazan and Koren (2015)can be used here choosing a sufficiently low failure probability.Alg. 3 is a concrete instance of Alg. 2 using a trust region solver. We start with the expansion of g t from Alg. 2. t−j u t−i occur exactly once. Using Assumption 5, we see that y t is κC u Lipshitz in M[i−j]t−j . We can then bound y t Q t y t ≤ ξ y t 2 using Assumption 6, resulting in the coefficients of A t being upper bounded by ξC 2 u κ 2 . Likewise, using the bound B ≤ β, we see the coefficients of b t are bounded by ξC 2 u κ 2 β Theorem 21 Suppose Assumptions 4, 5, and 6 hold, then Alg. 1 suffers regret at most O(poly(L) √ T ).Proof The regret bound follows via the approximation guarantee from Lem. 19 along with applying Thm. 15 for the Online Trust Region algorithm, with bounds on the coefficients for the control setting from Lem. 20.Lem. 20 and Thm. 15]Appendix E. Notes on ImplementationE.1. Dynamics TransformationOur implementation utilizes Assumption 5 to recast the dynamics and disturbance generator. In particular, for any stabilizable system, we apply a transformation to the dynamics using the H ∞ optimal controller as K * . Then we apply the learning algorithm over the residuals, which will be valid so long as the true controller is stabilizing. In particular, define (K H , W H ) as the solution to the H ∞ game acting on dynamics (A, B, C) for arbitrary cost matrices Q, R. This solution exists iff the dynamics are stabilizable. Then we apply the following transformation:Here,Ã is stable and the main analysis can proceed.Ideally, our parameterization of the disturbances would be guaranteed to recover the H ∞ Nash equilibrium against an H ∞ controller. Therefore, we let the bias term of our algorithm be W H x t . Defining r t = u t = K H x t + u t , we see that for u t = −K H x t , we have r t = 0 and recover the desired behavior:
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| []
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[
"Estimating Diffeomorphic Mappings between Templates and Noisy Data: Variance Bounds on the Estimated Canonical Volume Form",
"Estimating Diffeomorphic Mappings between Templates and Noisy Data: Variance Bounds on the Estimated Canonical Volume Form"
]
| [
"Daniel J Tward \nCenter for Imaging Science\nJohns Hopkins University\n21218BaltimoreMD\n",
"Partha P Mitra \nCold Spring Harbor Laboratory\nCold Spring Harbor\n11724NY\n",
"Michael I Miller \nDepartment of Biomedical Engineering\nJohns Hopkins University\n21218BaltimoreMD\n"
]
| [
"Center for Imaging Science\nJohns Hopkins University\n21218BaltimoreMD",
"Cold Spring Harbor Laboratory\nCold Spring Harbor\n11724NY",
"Department of Biomedical Engineering\nJohns Hopkins University\n21218BaltimoreMD"
]
| []
| Anatomy is undergoing a renaissance driven by availability of large digital data sets generated by light microscopy. A central computational task is to map individual data volumes to standardized templates. This is accomplished by regularized estimation of a diffeomorphic transformation between the coordinate systems of the individual data and the template, building the transformation incrementally by integrating a smooth flow field. The canonical volume form of this transformation is used to quantify local growth, atrophy, or cell density. While multiple implementations exist for this estimation, less attention has been paid to the variance of the estimated diffeomorphism for noisy data. Notably, there is an infinite dimensional un-observable space defined by those diffeomorphisms which leave the template invariant. These form the stabilizer subgroup of the diffeomorphic group acting on the template. The corresponding flat directions in the energy landscape are expected to lead to increased estimation variance. Here we show that a least-action principle used to generate geodesics in the space of diffeomorphisms connecting the subject brain to the template removes the stabilizer. This provides reduced-variance estimates of the volume form. Using simulations we demonstrate that the asymmetric large deformation diffeomorphic mapping methods (LDDMM), which explicitly incorporate the asymmetry between idealized template images and noisy empirical images, provide lower variance estimators than their symmetrized counterparts (cf. ANTs). We derive Cramer-Rao bounds for the variances in the limit of small deformations. Analytical results are shown for the Jacobian in terms of perturbations of the vector fields and divergence of the vector field. | 10.1090/qam/1527 | [
"https://arxiv.org/pdf/1807.10834v2.pdf"
]
| 126,344,772 | 1807.10834 | acbb6c7c26023d8efeabe59768363352bbfb51c3 |
Estimating Diffeomorphic Mappings between Templates and Noisy Data: Variance Bounds on the Estimated Canonical Volume Form
Daniel J Tward
Center for Imaging Science
Johns Hopkins University
21218BaltimoreMD
Partha P Mitra
Cold Spring Harbor Laboratory
Cold Spring Harbor
11724NY
Michael I Miller
Department of Biomedical Engineering
Johns Hopkins University
21218BaltimoreMD
Estimating Diffeomorphic Mappings between Templates and Noisy Data: Variance Bounds on the Estimated Canonical Volume Form
(Dated: 19 September 2018)computational anatomymorphometrycell densityHamiltonian dynamics
Anatomy is undergoing a renaissance driven by availability of large digital data sets generated by light microscopy. A central computational task is to map individual data volumes to standardized templates. This is accomplished by regularized estimation of a diffeomorphic transformation between the coordinate systems of the individual data and the template, building the transformation incrementally by integrating a smooth flow field. The canonical volume form of this transformation is used to quantify local growth, atrophy, or cell density. While multiple implementations exist for this estimation, less attention has been paid to the variance of the estimated diffeomorphism for noisy data. Notably, there is an infinite dimensional un-observable space defined by those diffeomorphisms which leave the template invariant. These form the stabilizer subgroup of the diffeomorphic group acting on the template. The corresponding flat directions in the energy landscape are expected to lead to increased estimation variance. Here we show that a least-action principle used to generate geodesics in the space of diffeomorphisms connecting the subject brain to the template removes the stabilizer. This provides reduced-variance estimates of the volume form. Using simulations we demonstrate that the asymmetric large deformation diffeomorphic mapping methods (LDDMM), which explicitly incorporate the asymmetry between idealized template images and noisy empirical images, provide lower variance estimators than their symmetrized counterparts (cf. ANTs). We derive Cramer-Rao bounds for the variances in the limit of small deformations. Analytical results are shown for the Jacobian in terms of perturbations of the vector fields and divergence of the vector field.
I. INTRODUCTION
Computational Anatomy (CA) is a growing discipline. Starting with initial work 1,2 directed towards the the study of transformations between anatomical coordinate systems suitable for volumetric images of the subcompartments of the human brain acquired largely using MRI, contemporary applications have been extended to much larger data volumes acquired using light microscopy. Infinite dimensional diffeomorphisms constitute the central transformation group for studying shape and form. 3,4 The diffeomorphism model underlying this analysis assumes that the space of measured MRI and optical imagery can be generated from exemplars or templates via diffeomorphic changes of coordinates. At the mesoscopic scale the variation of the diffeomorphic change in coordinates from one brain to another can represent transverse individual variation, including pathological conditions, or longitudinal developmental variation.
Of particular interest in mesoscale neuroanatomy are quantities such as the spatial densities of cellular somata or neuronal processes. Mapping estimates of these quantities to a template or reference space requires estimation of the change in the local scale as captured by a) Electronic mail: [email protected] the determinant of the metric tensor (the canonical volume form determined by the Jacobian determinant of the mapping). These applications point to the importance of uncertainty estimation for the diffeomorphic transformations involved. This is the subject area of the current paper.
Dense mapping between coordinate systems began with the low-dimensional matrix Lie groups forming the basis of Kendall's shape theory. 5 Their infinite dimensional analogue, the diffeomorphisms between coordinate systems, were first introduced by Christensen et al. 6 and have occupied a central role in CA. [1][2][3][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] The small deformation methods have been associated with Bookstein's landmark matching [23][24][25] and subsequently for image matching. [26][27][28][29] Large deformations were studied as simply topology preserving transformations without a metric structure. 6,[30][31][32][33] The symmetric approaches for large deformations were variants of these methods. [34][35][36][37] The large deformation diffeomorphic metric mapping algorithms (LDDMM) emerged corresponding to Lagrangian and Hamilton's principles applied to the flow fields incrementally generating the diffeomorphic transformations involved [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57] and provide a metric between images and diffeomorphisms. The survey article by Sotiras, Davatzikos, and Paragios 58 places these works in the greater context of deformable registration.
Uncertainty of nonrigid registration algorithms has been investigated for several applications such as spatially adaptive smoothing of population level data or un-certainty visualization for surgical planning. [59][60][61][62][63][64] These approaches to uncertainty have typically used resampling techniques and linear elastic or spline based deformation models as opposed to diffeomorphisms. Theoretical bounds on uncertainty have been investigated for the case of translation-only registration. 65 In this paper we study problems associated with the estimation of the (local) canonical volume form (ie det(g) where g is the metric tensor of the diffeomorphism) via methods based on large deformation diffeomorphic metric mapping (LDDMM) 41,66 . As an example of an application where knowledge of this local scale change is important, consider mapping estimated cell densities from an individual brain to a template. Investigators may wish to distinguish differential changes in cell density of different types, from an overall change in scale given by the diffeomorphism relating the individual brain to the template. This requires an estimation of the volume form. In human neuroanatomy, this local change in scale is commonly used to quantify patterns of tissue atrophy in neurodegenerative disease 67 .
In the notation followed in the paper, the coordinate transformation φ has Jacobian [Dφ] ij = ∂φi ∂xj and first fundamental form (metric tensor) given by
g(φ) = (Dφ) T (Dφ) .
The canonical volume form is given by the square-root of the determinant | det g|.
We show that the geodesic equations associated with the LDDMM method provide a crucial reduction, removing the nuisance dimensions of the underlying symmetries associated with the stabilizer of the template, i.e. the subgroup of the diffeomorphic transformations that leave the deforming template unchanged. This templatecentered reduction of the stabilizer gives rise to the asymmetric mapping properties central to LDDMM, and provides robustness when the imaging targets suffer from an incomplete and or noisy measurement process. We examine the mean square error of estimation of the canonical volume form. We demonstrate that the asymmetry of LDDMM coupled to geodesic reduction of indeterminate dimensions of the flow leads to favorable performance of this estimate in the presence of noise and variability when compared to symmetric methods for image matching originally proposed by Christensen and H.J.Johnson 68 , and Avants, Grossman, and Gee 69 . We also calculate the Cramer-Rao bound for the variance of the volume form in the case of small deformations.
Explicitly studying the behavior of the canonical volume form under uncertainty, as contrasted to registration accuracy which has been addressed by the community in some situations, is essential for drawing meaningful conclusions about cell or process density and brain morphometry 67 . The current work provides a theoretical basis for the observations presented and for related observations by other authors. It clarifies important differences between symmetric and asymmetric methods as they relate to the uncertainty of the volume form estimates.
The paper is organized as follows:
• We first review the theoretical methods including two optimal control problems that are stated for retrieving the diffeomorphism and the fundamental forms describing changes of coordinate systems between images.
• The necessary condition for the solution to the variational problem is stated in terms of the Euler-Lagrange equation on the conjugate momentum. It is shown that this equation incorporates a Hamiltonian reduction of the infinite-dimensional symmetry group corresponding to the stabilizer of the template.
• We then derive a new analytical result for the Cramer-Rao bound on the variance of the fundamental form in the presence of measurement noise.
• Following this we show results from large deformation simulations showing the decrease in variance for the asymmetric LDDMM which privileges the template coordinate system as ground truth and removes the stabilizer with respect to the template.
• These results are compared to symmetric methods derived by Christensen and Johnson 68 and Avants and Gee et al. 69 It is shown that the Hamiltonianreduced LDDMM and symmetric methods behave similarly at low-noise but LDDMM outperforms with increasing noise.
The significant new results in this paper thus address uncertainty estimation of the canonical volume form for estimated diffeomorphic transformations for both large deformation LDDMM and symmetric methods as well as small deformation methods.
II. METHODS
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A. Theoretical Methods
Geodesic Flows of Diffeomorphisms for Dense Transformation of Coordinate Systems
The diffeomorphism model in Computational Anatomy posits that the diffeomorphism group acts on templates I temp (x), x ∈ X ⊂ R 3 via group action to generate the space of observed anatomical data from individual subjects I,
I = {I = I temp • φ −1 t , φ t ∈ Diff } ,(1)
with diffeomorphisms φ t : X → X, t ∈ [0, 1], φ t ∈ Diff generated via flows:
φ t = v t • φ t , φ 0 = id.(2)
where id is the identity transformation. This is also termed the random orbit model: in group theoretic terms, I is the orbit of I temp under Diff . The Eulerian vector fields v t : X → R 3 are constrained to be spatially smooth, supporting at least 1-continuous derivative in space, ensuring that the flows are well defined and with smooth inverse. 8 They are modeled to be smooth with a finite norm (V, · V ) defined by a differential operator A :
V → V * , v 2 V = X Av · vdx.
It is conventional to use powers of the Laplacian for the differential operator A with a sufficient number of generalized derivatives such that the flow fields are guaranteed to have at least 1 continuous derivative in space. 8 The kernel K of the associated reproducing kernel Hilbert space is at least 1-time continuously differentiable in the spatial variables, and is given by the Green's kernel of A.
Mapping individual brains to reference atlases such as the Allen mouse atlas 70 at the micron scale or the Mori human atlas 71 at millimeter scale is performed via bijective coordinate transformation between anatomical coordinate systems φ. The diffeomorphic change in coordinates is not directly observable and must be inferred from observed brain image data subjected to measurement noise and technical variations. The coordinate system transformations may be estimated by solving the diffeomorphic matching problems as a solution of an optimal control problem. Different optimal control problems are obtained for large and small deformations.
To set up the optimal control problem, we define the transformation φ t as a time dependent state, t → φ t ∈ Diff . The velocity field t → v t is taken to be the control variable. The state satisfies the dynamical equationṡ φ t = v t • φ t , with initial condition φ 0 = id. The goal of the optimal control problem is to drive the state from an initial condition of identity (corresponding to the template), to a state that matches template coordinates to target coordinates. This can be enforced by a cost function for the final state,
U (φ 1 ) = 1 2σ 2 X |J − I • φ −1 1 | 2 dx .(3)
where J is an observed target image. We assume that any differences in position or orientation have already been accounted for by an appropriate similarity transform before deforming the coordinate grid using the diffeomorphic procedure. A scalar parameter σ controls the importance of this data attachment term relative to the regularization term defined below.
Since there are an infinite number of possible flows we use the principle of least-action to minimize a running cost given by the integrated kinetic energy of the vector field v =φ • φ −1 of the flow (LDDMM 11,41 ). This regularization cost term can be interpreted as a kinetic energy Lagrangian:
L(φ t ,φ t ) = 1 2 X A(φ t • φ −1 t ) ·φ t • φ −1 t dx = 1 2 X Av t · v t dx .(4)
The cost function for the optimal control problem is obtained by combining the kinetic energy term (the regularization term) with an end-point cost. Thus, the optimal control problem involves Lagrangian mechanics of the infinite dimensional state defined by the coordinate transformation φ t . We study two versions of this problem, corresponding to large and small deformations (control problems 1 and 2).
Control Problem 1 (Large Deformation). φ t = v t • φ t , φ 0 = id min φ C(φ) . = 1 0 L(φ t ,φ t )dt + U (φ 1 )
For small deformations 26,27,29 there is no time index, and v represents a displacement field rather than a flow field, φ = id + v.
Control Problem 2 (Small Deformation).
φ = id + v min v C(φ) . = 1 2 X Av · vdx + U (φ)
Proposition 1. The solution to the large deformation Control Problem 1 has classical conjugate momentum p t = Av t • φ t |Dφ t | 72,73 which satisfies geodesic equations
p t = α t ∇I, α t = 1 σ 2 (I − J • φ 1 )|Dφ 1 |(Dφ T t ) −1 . (5)
For small deformations, Control Problem 2, p = Av satisfying
p = α∇I , α = 1 σ 2 (I − J • φ)|Dφ|(Dφ T ) −1 .(6)
The conjugate momentum satisfying LDDMM equation (5) is perpendicular to the level lines of the template following the image gradient. This normal condition is satisfied over the entire path of deformation. This is observed most easily by considering the Eulerian momentum, related to the conjugate momentum by a change of coordinates, Av t = 1
σ 2 |Dφ −1 1t |(J(φ −1 1t ) − I(φ −1 t ))∇[I(φ −1 t )], where φ 1t = φ t (φ −1 1 ). Notice that Av t is parallel to ∇[I(ϕ −1 t )] at every point. See Appendix B for a proof.
The fact that the conjugate momentum is in the range space at every point of the gradient of the template is a Hamiltonian reduction that removes the symmetries of the template given by the stabilizer subgroup. The non-identifiable motions that are tangent to level lines are thus removed. This kills off the nuisance dimensions, essentially suppressing components in the null-space of level lines of the template as it flows. 43 For small deformation matching the normal condition of the stabilizer corresponds to a single vector condition that Av is in the span of ∇I the gradient of the image template. Small deformations allow us to both calculate the Cramer-Rao bound (see below) as well as perform a direct perturbation argument. Via a perturbation J → J + δJ we can calculate the accuracy of the divergence of v and how it determines the Jacobian determinant and the canonical volume measure related to the gradient of the template.
Proposition 2. For deformations close to the identity, a perturbation J → J + δJ results in a perturbation to the canonical volume form:
δv = [A + 1 σ 2 (∇I)(∇I) T ] −1 ∇IδJ |Dφ| → |Dφ| + |Dφ|div[δv](7)
where ∇I is a column vector, the outer-product giving a 3 × 3 matrix.
See Appendix D and E for small deformation proofs. Equation (7) clearly shows the crucial matrix involving the gradient of the image and the prior represented via the differential operator A for determining the perturbation on the canonical volume form through the divergence. The Cramer-Rao bound reflects this theme.
Cramer-Rao Bound (CRB) for Small Deformations
We now examine the variational estimator and the variance bound for small deformations in the finite dimensional setting of n-dimensional vector fields v n (x) = n i=1 θ i ψ i (x). Here ψ i is some suitable family of expansion functions. For computing the CRB, we take the observed data to be a conditionally Gaussian random field, conditioned on the mean I • φ n−1 with additive noise
J = I • (φ n ) −1 + noise , φ n = id + v n .(8)
The noise is taken to be zero-mean with non-white inverse covariance Q. The log-likelihood on the n-dimensional cylinder Θ n = (θ 1 , . . . , θ n ) is
n (J; Θ n ) = − 1 2 X X (J(x) − I • φ n−1 (x)) Q(x, y)(J(y) − I • φ n−1 (y))dxdy .(9)
For white-noise the inverse-covariance is Q(x, y) = 1 σ 2 (x) δ(x − y) the n-dimensional log-likelihood (9), for σ 2 (x) a variance at location x. Adding the finitedimensional prior term − 1 2 X Av n (x) · v n (x)dx gives a proper maximum a-posterior estimator (MAP) on ndimensions: max θ 1 ,...,θ n log π n (Θ n |J) = − 1 2 X Av n · v n dx + n (J; θ 1 , . . . , θ n ) .
The Fisher-information is the n × n matrix, i, j = 1, . . . , n:
I ij (Θ n ) = E J|Θ n ∂ n (J; Θ n ) ∂θ i ∂ n (J; Θ n ) ∂θ j .
Taking expectation over Θ n gives the Bayesian version of the Fisher-information:
I B = R n I(J; Θ n )π(dΘ n ) .
Proposition 3. Defining for all (x, y), Q φ (x, y) = Q(φ(x), φ(y)|Dφ(x)||Dφ(y)|,
I ij (Θ n ) = X X ψ iT (x)[Dφ(x)] −T ∇I(x) Q φ (x, y)∇I T (y)[Dφ(y)] −1 ψ j (y)dxdy .(11)
Neglecting quadratic terms using small deformations, noting that the linear terms have an expected value of 0, and taking white noise variance σ 2 gives
I B ij = X ψ T i (x) A + 1 σ 2 (x) ∇I(x)∇I T (x) ψ j (x)dx + E[o(v 2 )] ,(12)
giving the lower bound on the estimator:
Cov[Θ n ] ≥ [I B ] −1
Noteworthy is the CR bound (12) is at the core the same form as direct perturbation (7). We are particularly interested in the variance of div(v), because it directly relates to variance of the Jacobian. This can be written as a linear functional of v, using a test function w which is nonzero in a small neighborhood of the point y (i.e. the sensitivity of an image voxel), namely X w(x)divv(x)dx. In this case the information inequality, which is concerned with estimating functions of the parameter v, becomes
Var[div[v](y)] ≥ n i,j=1 X w(x)divψ i (x)dx [I B ] −1 ij X w(x)divψ j (x)dx(13)
3. The Stabilizer: The normal condition for the nonlinear group action
Unfortunately there is an infinite dimensional nuisance parameter, termed the stabilizer, which is not uniquely determined in estimating the coordinates of the bijections between coordinate systems. The geodesic motion kills the nuisance parameter, suppressing components in the instantaneous null-space of level lines of the template as it flows 43 ; this is Theorem 4 of 73 . Notice there is no momentum tangent to the level lines of the flow of the template image, p t = α t ∇I in (5). This is not true for other diffeomorphic mapping methods. It results from the metric property of the geodesic equation with it's associated conservation law.
To understand flows which are normal to the level lines and in the span of the gradient of the template, examine the stabilizer. Vector fields that are normal to the level lines in the span ∇I do not generate flow through the stabilizer group. The stabilizer is generated from flowsφ t = w • φ t , φ 0 = id for which the vector fields are tangent to level lines of the template:
V I = {w ∈ V : ∇I · w = 0} .
That being normal to the level lines is a necessary condition to null out the stabilizer, examine φ = id + w, then
I • φ = I + ∇I · w + o( ) .
For I • φ = I it must be the case that ∇I · w = 0, implying w must be normal to the level lines. This is the group action version of the pseudo-inverse condition for inverting a matrix with non-zero null-space.
B. Numerical Experiments
Simulating Variance Bounds on Large Deformations
The geodesic properties of LDDMM imply lower variance estimates of the first fundamental form and the fundamental volume measure. To illustrate this we create simulated images by generating large deformation diffeomorphisms under the random orbit model, deforming a template image under these known transformations, and applying Gaussian noise. Our template corresponds to the binary segmentations of the anatomical section in Fig. 2 (left column), a sagittal section of an ex vivo image of the human medial temporal lobe, where hippocampal subfields and surrounding areas are visible. We generate random Gaussian vector fields v 0 (x) = v n (x), x ∈ X as the initial condition of geodesic solutions for deformation. Since geodesics satisfy a conservation law
d dt Dφ T t p t = 0(14)
proved in Appendix C equation C2, we can synthesize the initial condition p 0 = Av n and generate a random spray of deformations φ 1 . The Gaussian vector field is generated from an n-dimensional expansion ψ i : X →
R 3 , i = 1, 2, . . . , v n (x) = n i=1 θ i ψ i (x) ,
coefficients θ i ∈ R, i = 1, . . . , n distributed as multivariate Gaussian, zero-meanθ i = 0 and covariance E[θ i θ j ] = Σ ij . We choose the covariance of vector fields v n to be well defined for large n, sampled at pairs of points x, y ∈ R 3 ,
K n (x, y) . = E[v n (x)v nT (y)] = i,j ψ i (x)E[θ i θ j ]ψ jT (x) .
For large n this tends to covariance specified by the Green's kernel of the operator A originally stated by Beg 41 , the inverse of A = (1 − a 2 ∆) 4 for ∆ the Laplacian at spatial scale of a = 0.25 mm, or 2 pixels). To achieve this, we choose an expansion ψ i that corresponds to a superposition of Green's kernels located on each voxel where the image gradient is nonzero (3432 expansion functions). For the statistical characterization, 100 realizations of random deformation are generated running equation (14) from v n as initial condition. 100 randomly generated images were constructed from these deformations applied to our template (Fig. 2 left), embedded in additive Gaussian noise with standard deviation from 0 to 0.5, spatial correlation of 0 (white noise) or 1.5 pixels (obtained by convolving with a Gaussian of standard deviation 1.5 pixels). Correlated noise is common in medical imaging systems such as radiography 74 , CT 75 , or MRI 76 . Similarly, 100 randomly generated images were constructed using the identity transformation instead of a random transformation.
Applying diffeomorphic mapping for 100 images produced using the identity transformation (θ i = 0∀i), and 100 images produced using these known transformations then gives variance estimates of the diffeomorphic mapping methods. For contrast we compare LDDMM to the symmetrized version discussed in 68 ANTs which does not exploit the reduced representation as symmetry supports momentum on both the template and target:
max v:φ v = 1 0 vt•φtdt+id − 1 2 1 0 v t 2 V dx − 1 2σ 2 I • φ −1 − J 2 L2 − 1 2 1 0 v t 2 V dx − 1 2σ 2 I − J • φ 2 L2 .(15)
Gradient descent is used to determine the optimal v, giving estimates ofφ and the canonical form |Dφ|. We use exactly the same parameters for both algorithms. Parameters were fixed or varied experimentally as summarized in Table I in the Appendix A. In our computa- tional implementation we use two constants in our objective function to be minimized: 1 σ 2 V multiplying the regularization (kinetic energy) term, and 1 σ 2 I multiplying the data attachment term. Since we are estimating the optimizer, but not the optimum, this is equivalent to choosing a single 1
σ 2 = σ 2 V σ 2 I .
Calculating the Cramer Rao Bound
Finite dimensional cylinders were used to illustrate the Cramer Rao lower bound on the divergence ofv n = n i=1 θ i ψ i from (13). Expansion functions are Green's kernels of A located on image boundaries, where the image is downsampled by a factor of (from left to right) 8 (76 expansion functions), 4 (318 expansion functions), or 1 (not downsampled, 3432 expansion functions). Explicitly inverting the ill-conditioned matrix I B was avoided by solving the linear system implied by (13), which was performed using Matlab's linsolve for positive definite symmetric matrices. Figure 2 shows sections (column 1) through the highfield medial temporal lobe MRI phantom used for numerical experiments, with the segmentation into the entorhinal cortex and it substructures subiculum and Cornu Ammonis (CA) compartments. Figure 2 (columns 2,3) show results for simulating known transformations of the template with additive white noise. Column 2 shows the results of solving LDDMM equation (5) to estimate each random deformation φ. The top row shows identity transformation of coordinates; the bottom row shows random deformations φ. LDDMM in homogeneous regions has RMSE error which drops to nearly zero across the entire image, with more error incurred at the contrast boundaries. Column 3 shows the symmetric case which has significantly higher RMSE across the image than LDDMM.
III. RESULTS
A. Variance bounds
To compare homogeneous regions to boundary regions, we show RMSE in Figs. 3, 4 as a function of noise at several regions identified in Fig. 2 (bottom left). Results for each of these regions are shown; Fig. 3 examines white noise; Fig. 4 examines correlated noise.
The symmetric method degrades at higher noise levels, as well as with correlated noise. In the homogeneous regions the LDDMM has no uncertainty, and as one moves away from gradients the uncertainty quickly drops to zero. Location (a) illustrates a place of high gradient on the boundary in which we see at reasonable noise levels equivalent performance. The effect of noise correlation is to increase RMSE, the symmetric method being particularly harshly affected.
B. Cramer-Rao bound
Examples of the Cramer-Rao bound on the divergence ofv n for different n are shown in the top row of Fig. 5. One observers an asymmetry between uncertainty tangent and normal to level lines, which has been noted by other authors 63 and is implied by our consideration of the stabilizer of the diffeomorphism group. In the bottom row of Fig. 5 we show the lower bound as it varies with changing balance between the image gradient (11) term to the quadratic prior term (12). These results have an important implication for image mapping parameter selection in the presence of noise. Note that when both terms are multiplied by the same constant, the bound increases linearly and the displayed image will look the same. The behavior as we transition from "prior dominant" to "image dominant" can be seen as we move from left to right.
IV. DISCUSSION
In this work we demonstrated that image registration algorithms which give comparable accuracy in terms of alignment of observable anatomical boundaries, can lead to widely different estimates of the canonical volume form. In particular we showed that by employing the non-symmetric procedure associated to geodesic matching of LDDMM, we have explicitly accounted for the stabilizer of the diffeomorphism group, reducing the dimensionality of mappings to that of the observable image boundaries, results in favorable performance in the presence of noise. Interestingly the small perturbation argument of Equation (7) also illustrates the importance of the asymmetry, and shows that it is instructive to consider the effect of image variability on estimation of the canonical volume form in 2 scenarios. First, because of the null-space of identifiability of motions along level lines, the homogeneous regions that are distant from image gradients by an amount larger than the spatial scale of K have clear identifiability issues in the symmetric methods. This leads to favorable performance of the asymmetric LDDMM method. Second, boundaries between anatomical structures, such as gray white matter interfaces, have high image gradients. In these cases the δv is approximately related to δJ through the pseudoinverse of (∇I)(∇I) T .
The simulation of noise in the images clearly illustrates that the departure between source and channel in Shannon's classic model is at the heart of the random orbit model, and implies the importance of asymmetry. In this setting, the targets are not in the homogeneous orbit of the template under diffeomorphisms. The prior distribution and the space associated to it is separate from the targets as outputs of the noise channel. Noisy targets have gradients which are nonzero everywhere, which explains the clear superiority of the asymmetric approach of LDDMM in this situation.
This fills an important gap in our knowledge of how image registration algorithms can or should be used to quantify biological properties of tissue. In brain morphomety 67 , the canonical volume form at each voxel in a 3D image is used to quantify structural differences between populations, for example measuring atrophy due to disease or aging. In microscopy it is convenient to idealize cellular locations in terms of spatial point process models 77 . Here the canonical volume used to define a relationship between counts (at the micron scale) and densities (at the mm scale) in a standard atlas coordinate system. During the earliest phases of preclinical Alzheimer's disease for example, when cells have not yet died, cell density changes due to dissolution of the neuropil. When choosing a mapping procedure, one must consider that algorithms or parameters that may be optimal for registration accuracy, may be considerably sub optimal for quantifying local expansion or contraction.
One challenge here is in considering the effect of the many possible parameter choices that define mapping algorithms. The parameters considered here represent realistic choices for neuroimaging applications, and were chosen based on experience in studying brain morphometry. An important parameter not explicitly varied was the spatial scale of the transformation's smoothness. Claims made for "homogeneous regions" or "regions near a boundary", should be understood as related to this spatial scale. The MR image considered for our experiment was chosen because it has features at many different spa-tial scales, from the closely packed structures of the dentate gyrus, to the broadly uniform white matter of the angular bundle. We have observed experimentally that as the spatial scale of the deformation grows very large, every location in the image becomes "near a boundary", and the differences between the two methods considered become less pronounced.
One limitation of this work is that the experiments considered only two dimensional images. This was for reasons of computational speed when considering statistical ensembles of mapping results, as well as for ease of display and interpretation. The theoretical developments here do hold in three dimensions, and we expect our findings to generalize.
This work is important for two important biomedical applications. The first is the effect of data noise on atlas registration procedures. When considering areas such as surgical planning, the quantification of "average performance" may be insufficient. The consequence of deviations from average in this context can incredibly costly. The second is the extension of these atlas mapping technologies to morphometry, and the study of cell and process densities.
In the new era of computational anatomy enabled by large volumes of light microscopic data, quantification of the uncertainty in the coordinate mapping between individual data sets and templates is important both for fundamental science applications, and for applications to the study of pathological conditions or computer aided diagnoses. Integrating by parts gives two equations using zeroboundary δφ 0 = 0: (B3) is the boundary matching term reducing to (p 1 + 1 σ 2 (J • φ 1 − I)(Dφ 1 ) −1T ∇I)|Dφ 1 | · δφ 1 dx = 0 .
giving
p 1 + 1 σ 2 (J • φ 1 − I)(Dφ 1 ) −1T ∇I|Dφ 1 | = 0 .(B4)
The result for p t in terms of p 1 is obtained by applying the result of Appendix C twice. First calculate at time 0: p 0 = Dφ T 1 p 1 , then at time t: p t = Dφ −T t Dφ T 1 p 1 , giving the result
p t + 1 σ 2 (J • φ 1 − I)Dφ −T t ∇I|Dφ 1 | = 0 (B5)
Definition 1 .
1Define the subgroup S ⊂ Diff as the stabilizer of template I if for all φ ∈ S,I • φ = I .
Figure 1
1shows examples of mappings from the stabilizer of an image of a human hippocampus, to a close numerical approximation. The left column shows the identity mapping on the grid; the right two columns show two mappings from the stabilizer.
FIG. 1 .
1Mappings using elements from the stabilizer, leaving the template 2D sections essentially unchanged (top row) but moving the coordinates (bottom row). Column 1 is identity φ = id; columns 2,3 have a tangent component w ∈ VI in the stabilizer.
FIG. 2 .
2Column 1 shows sections through MRI (top) of entorhinal cortex and subiculum and CA partition of the hippocampus (bottom) with three locations depicted for RMSE comparisons. Column 2 shows RMSE's for canonical volume forms for the MTL section with showing LDDMM method and column 3 showing the symmetrized algorithm (15). Top row shows identity transformation; bottom row shows the randomized transformation.
FIG. 3 .
3RMSE of log canonical volume form in white noise; LDDMM solid, symmetric dashed. Top row shows identity transformation; bottom row shows random transformation. Notice panel 1 shows very close performance of two methods; otherwise noteworthy differences.
FIG. 4 .
4RMSE of log canonical volume form in correlated noise; LDDMM solid, symmetric dashed. Top row shows identity transformation; bottom row shows random transformation.
FIG. 5 .
5Top row: Finite dimensional expansions for Cramer-Rao bound on ∇ ·v. Expansion functions are Green's kernels of A on image boundaries, where the image is 76 expansion functions, 318 expansion functions, or 3432 expansion functions. Bottom row: tradeoff between a strong prior term (left) and a strong data term (right).
X
∂L(φ 1 ,φ 1 ) ∂φ 1 · δφ 1 dx + d d U (φ 1 + φ 1 )| =0 = 0 . (B3)To calculate the variation of the endpoint term givesd d | =0 U (φ 1 + δφ 1 ) = X 1 σ 2 (J − I • φ −1 1 )(Dφ 1 ) 2 (J • φ 1 − I)(Dφ 1 ) −1T ∇I|Dφ 1 | · δφ 1 dx .Using the fact that the conjugate momentum is p = ∂L ∂φ = [Av] • φ|Dφ|, then Equation (B2) is the Euler-Lagrange equation; d dt p t + (Dv t ) T • φ t p t = 0 ;
TABLE I .
ISummary of experimental parameters ParameterValues Deformation of atlas identity, random diffeomorphism Spatial scale of deformation, a 0.25mm (2 pixels) Approximate magnitude of deformation 2-3 pixelsImage intensity
Binary 0-1 segmentations
Noise level
0, 0.1, 0.2, 0.3, 0.4, 0.5
Noise correlation
0 pixels, 1.5 pixels
Matching methods
LDDMM, symmetric LDDMM
Matching σI
0.1
Regularization σV
3.33 (0.01, 0.1, ∞ shown in Fig. 5)
Equivalent single parameter σ
0.03
Matching gradient descent step size
0.018
Number of realizations
100
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Appendix A: Experimental ParametersParameters used in our simulations are listed inTable I.Appendix B: Proof of LDDMM Inexact MatchingProof. We will need the perturbation of the inverse. Let φ = φ + δφ, computing the variation uses the fact thatDefine the total cost C(φ) for LDDMM to be minimized:and L(φ,φ) is given by(4). The Euler-Lagrange equation follows from first order perturbation, φ → φ + δφ,φ → φ + d dt δφ:Appendix C: Euler-Lagrange and Conservation are equivalentTo see this, take the derivativeThe last equality is zero by Euler-Lagrange equation (C1).Appendix D: Small deformation momentumProof. Let φ = φ + δφ(φ), then v = v + δφ(φ). Computing the variation requires perturbation of the inverse,Substituting into the second integral x = φ(y) with dx = |dφ|dy gives us(6).Appendix E: Effect of small perturbationsProof. We consider the perturbation J → J + δJ and v → v + δv on (6), seeking a relationship between the two to first order in ∈ R.The left hand side is simply Aδv. The right hand side includes a product of three terms.We consider one of the cases examined experimentally, with I = J and therefore φ = id (identity). Only the first term is nonzero, giving Aδv = 1 σ 2 [DIδv + δJ]∇IRearranging gives the relationship (7).
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| []
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[]
| [
"Francisco Navarro-Lérida ",
"D H Tchrakian ⋆ † \nSchool of Theoretical Physics\nDublin Institute for Advanced Studies\n10 Burlington Road, Dublin 4Ireland\n\nDepartment of Computer Science\nMaynooth University\nMaynoothIreland\n",
"\nDepartamento de Física Atómica\nMolecular y Nuclear\nUniversidad Complutense de Madrid\nCiencias FísicasE-28040MadridSpain\n"
]
| [
"School of Theoretical Physics\nDublin Institute for Advanced Studies\n10 Burlington Road, Dublin 4Ireland",
"Department of Computer Science\nMaynooth University\nMaynoothIreland",
"Departamento de Física Atómica\nMolecular y Nuclear\nUniversidad Complutense de Madrid\nCiencias FísicasE-28040MadridSpain"
]
| []
| We study spherically symmetric finite energy solutions of two Higgs-Chern-Simons-Yang-Mills-Higgs (HCS-YMH) models in 3 + 1 dimensions, one with gauge group SO(5) and the other with SU (3). The Chern-Simons (CS) densities are defined in terms of both the Yang-Mills (YM) and Higgs fields and the choice of the two gauge groups is made so they do not vanish. The solutions of the SO(5) model carry only electric charge and zero magnetic charge, while the solutions of the SU (3) model are dyons carrying both electric and magnetic charges like the Julia-Zee (JZ) dyon. Unlike the latter however, the electric charge in both models receives an important contribution from the CS dynamics. We pay special attention to the relation between the energies and charges of these solutions. In contrast with the electrically charged JZ dyon of the Yang-Mills-Higgs (YMH) system, whose mass is larger than that of the electrically neutral (magnetic monopole) solutions, the masses of the electrically charged solutions of our HCS-YMH models can be smaller than their electrically neutral counterparts in some parts of the parameter space. To establish this is the main task of this work, which is performed by constructing the HCS-YMH solutions numerically. In the case of the SU (3) HCS-YMH, we have considered the question of angular momentum, and it turns out that it vanishes. | 10.1142/s0217751x15500797 | [
"https://arxiv.org/pdf/1412.4654v1.pdf"
]
| 118,602,473 | 1412.4654 | 00705a9fd71c7ac7ace4cc754681136310135728 |
15 Dec 2014 December 16, 2014
Francisco Navarro-Lérida
D H Tchrakian ⋆ †
School of Theoretical Physics
Dublin Institute for Advanced Studies
10 Burlington Road, Dublin 4Ireland
Department of Computer Science
Maynooth University
MaynoothIreland
Departamento de Física Atómica
Molecular y Nuclear
Universidad Complutense de Madrid
Ciencias FísicasE-28040MadridSpain
15 Dec 2014 December 16, 2014Electrically charged finite energy solutions of an SO(5) and an SU (3) Higgs-Chern-Simons-Yang-Mills-Higgs systems in 3 + 1 dimensions
We study spherically symmetric finite energy solutions of two Higgs-Chern-Simons-Yang-Mills-Higgs (HCS-YMH) models in 3 + 1 dimensions, one with gauge group SO(5) and the other with SU (3). The Chern-Simons (CS) densities are defined in terms of both the Yang-Mills (YM) and Higgs fields and the choice of the two gauge groups is made so they do not vanish. The solutions of the SO(5) model carry only electric charge and zero magnetic charge, while the solutions of the SU (3) model are dyons carrying both electric and magnetic charges like the Julia-Zee (JZ) dyon. Unlike the latter however, the electric charge in both models receives an important contribution from the CS dynamics. We pay special attention to the relation between the energies and charges of these solutions. In contrast with the electrically charged JZ dyon of the Yang-Mills-Higgs (YMH) system, whose mass is larger than that of the electrically neutral (magnetic monopole) solutions, the masses of the electrically charged solutions of our HCS-YMH models can be smaller than their electrically neutral counterparts in some parts of the parameter space. To establish this is the main task of this work, which is performed by constructing the HCS-YMH solutions numerically. In the case of the SU (3) HCS-YMH, we have considered the question of angular momentum, and it turns out that it vanishes.
Introduction
The main task of the present work is to establish that introducing CS dynamics to the YMH system can result in the lowering of the energy of the electrically neutral solution, by giving it electric charge. We have tested this with two distinct models, one with gauge group SO(5) and the other SU (3). The main difference between these two models is that, while the solutions of the SU (3) model considered here carry magnetic charge, those of the SO(5) model have zero magnetic charge.
The two CS densities in 3 + 1 dimensions employed here and in the preceding work [1] are the first two in an infinite hierarchy, each resulting from the descent [2,3] from a Chern-Pontryagin density in 2N (N ≥ 3) dimensions. We refer to these as Higgs-CS (HCS) densities. They extend the definition of the usual [4,5] CS densities to all odd and even dimensions, at the cost of importing a Higgs field. In 2 + 1 dimensions, it was found [6,7,8] that the presence of the (usual) CS density in a gauged Higgs system results in finite energy electrically charged solutions.
Here, the corresponding question is considered in 3 + 1 dimensions. It turns out that different choices of the HCS density employed, result in qualitatively quite different solutions.
The SO(5) HCS-YMH model considered here is that employed in a preceding work [1]. In that preliminary work however, the energy of these solutions increased with the electric charge, and the lowest energy solutions turned out to be those with vanishing charge. In this respect, they are qualitatively similar to JZ solitons [9]. There the electrically neutral solutions had non-vanishing electric YM connection A 0 , exhibiting dipole behaviour.
In the present paper we construct more general electrically charged solutions to this SO(5) HCS-YMH model, some of which have lower energy than their neutral counterparts, their energies decreasing with increasing charge. These qualitative features contrast with those of the JZ dyons. As in [1], there are also electrically neutral solutions exhibiting dipole behaviour, namely supporting electrically neutral solutions with non-vanishing electric component of the YM connection A 0 . We have constructed three different families of solutions exhibiting these properties, which we refer to as Types I, II, and III. Types I and II describe electrically charged solutions, while Type III solutions describe electrically neutral solutions with non-vanishing electric component A 0 of the YM potential. None of these three types of solutions carry nonzero magnetic charge.
In addition to the SO(5) HCS-YMH model, we have studied an SU (3) HCS-YMH model. The main difference of the SU (3) model is that its solutions carry nonzero magnetic charge, at the same time supporting nonvanishing HCS terms. The resulting electrically charged solitons are dyons which differ fundamentally from the JZ dyon. The feature of decreasing mass with increasing electrical charge, observed for the solutions of the SO(5) HCS-YMH model, persists also for the SU (3) HCS-YMH model. In addition, we have considered the question of angular momentum in the SU (3) case.
The paper is organised as follows. In Section 2 we define the model, which is formally the same for both the SO(5) and SU (3) HCS-YMH models, except for the Higgs symmetry breaking potentials, which are stated there. Symmetry imposition on the respective SO(5) and SU (3) HCS-YMH models is presented in Sections 3 and 4 respectively. In subsections of Sections 3 and 4, the numerical solutions are presented. Another subsection of Section 4 deals with the question of angular momentum. Finally summary and discussion of our results are given in Section 5.
The models, equations, and charges
The full Lagrangian density is
L = L YMH + κ 1 Ω (1) CS + κ 2 Ω (2) CS ,(1)
with the two HCS densities Ω
CS and Ω
CS given by
Ω (1) CS = i ε µνρσ Tr Φ F µν F ρσ ,(2)Ω (2) CS = i ε µνρσ Tr Φ η 2 F µν F ρσ + 2 9 Φ 2 F µν F ρσ + 1 9 F µν Φ 2 F ρσ − 2 9 (ΦD µ ΦD ν Φ − D µ ΦΦD ν Φ + D µ ΦD ν ΦΦ) F ρσ ,(3)
where ǫ µνρσ is the Levi-Civita tensor in Minkowski spacetime. We do not describe the provenance of the HCS terms Eqs. (2) and (3), since this was given in detail in Appendix A of Ref. [1]. The role the Higgs scalar plays here is somewhat akin to that of the axion [10,11]. The YMH Lagrangian density is 1
L YMH = Tr 1 4 F 2 µν − 1 2 D µ Φ 2 − λ 2 V [η 2 , Φ 2 ] ,(4)
where
D µ = ∂ µ + [A µ , ·].
Here, V [η 2 , Φ 2 ] is the positive definite Higgs selfinteraction potential, with λ its coupling constant, and η denoting the vacuum expectation value of the Higgs field. κ 1 and κ 2 are the coupling strengths of the HCS densities.
The equations of motion resulting from the variations of the Lagrangian with respect to the YM potential and the Higgs field are
D µ F µν + [Φ, D ν Φ] = 2 i κ 1 ε µνρσ {F ρσ , D µ Φ} ,(5)D µ D µ Φ − λ{Φ, (Φ 2 + η 2 1I)} = i κ 1 ε µνρσ F µν F ρσ ,(6)
respectively. { , } denotes the anticommutator. These equations, Eqs. (5) and (6), are written only for the Lagrangian with κ 2 = 0 in Eq. (1). This is because the expressions for the right-hand sides of the corresponding equations for κ 2 = 0 are very cumbersome. There are two types of symmetry breaking potentials consistent with the requirement of finite energy, which we list here for completeness
V 1 = η 2 + a 1 TrΦ 2 2 ,(7)V 2 = 1 4 Tr η 2 1I + a 2 Φ 2 2 ,(8)
where the values of a 1 and a 2 will be chosen according to our convenience when imposing symmetries. As it turns out, we will concentrate mainly on λ = 0 solutions since the presence of the HCS terms, Eqs.
(2)- (3), is sufficient to support nontrivial field configurations outside of SU (2). When we do employ a potential for the purpose of checking that our conclusions are not altered by the presence of one, then our choice is Eq. (7) for both the SO(5) and SU (3) models. The definition of the magnetic monopole charge is
µ = − 1 4π ε ijk S ∞ Tr Φ F ij dS k ,(9)
which presents a lower bound on the energy integral, and the definition of the electric charge is
Q = − 1 4π S ∞ Tr Φ F i0 dS i .(10)
The definitions Eqs. (9) and (10) are valid, both when κ 1 = κ 2 = 0, and when κ 1 = 0 and/or κ 2 = 0.
Solutions of the SO(5) Higgs-Chern-Simons-Yang-Mills-Higgs model
This Section consists of two Subsections. In Subsection 3.1, spherical symmetry is imposed and the boundary values of the solutions sought are stated. The numerical construction of the solutions 2 is presented in Subsection 3.2.
Imposition of symmetry and boundary values
To proceed to the imposition of symmetry, we note that the fields take their values in the 4 × 4 chiral Dirac representation of SO(6)
A µ = A αβ µ Σ αβ , α = i, 4, 5; (i = 1, 2, 3) ,(11)Φ = ψ αβ Σ αβ + φ α Σ α6 ,(12)
where (Σ αβ , Σ α6 ) are the 4 × 4 chiral representation matrices of SO(6) 3 . It is convenient to express our Ansatz using the index notation α = i, M , i = 1, 2, 3 , M = 4, 5 . With this notation, the static spherically symmetric Ansatz for the Higgs field Φ, Eq. (12), and the YM connection
A µ = (A 0 , A i ) , Eq. (11), are Φ = 2η φ M Σ M6 + φ 6x j Σ j6 − (εψ) Mx j Σ jM + ψ 6 Σ 45 ,(13)A 0 = −(εχ) Mx j Σ jM − χ 6 Σ 45 ,(14)A i = ξ 6 + 1 r Σ ijxj + ξ M r (δ ij −x ixj ) + (εA r ) Mx ixj Σ jM + A 6 rx i Σ 45 ,(15)
in which the sum over indices M, N = 4, 5 runs over two values such that we can label the functions (φ M , φ 6 ) ≡ φ,
(χ M , χ 6 ) ≡ χ, (ξ M , ξ 6 ) ≡ ξ, (ψ M , ψ 6 ) ≡ ψ and (A M r , A 6 r ) ≡ A r ,
i.e., in terms of five isotriplets φ, χ, ξ, ψ, and A r , all depending on the 3 dimensional spacelike radial variable r. ε being the two dimensional Levi-Civita symbol.
The full one dimensional subsystems are presented in Appendix A.1. It immediately follows from Eqs. (13) and (A.1) that the magnetic monopole charge Eq. (9) vanishes.
The important quantity for us here is the global electric charge, Eq. (10), which does not vanish. A straightforward calculation yields the electric field E i
E i = Tr Φ F i0 = −2η ψ · D r χ ,(16)
resulting in the electric charge
Q = −1 4 2 π S ∞ Tr Φ F i0 dS i = 1 2 η r 2 ψ · D r χ r=∞ .(17)
For both potentials Eqs. (7) and (8), the finiteness of the energy requires that
lim r→∞ | φ| 2 + | ψ| 2 = 1 ,(18)
so we can introduce an asymptotic angle γ such that
lim r→∞ | φ| 2 = cos 2 γ ,(19)lim r→∞ | ψ| 2 = sin 2 γ .(20)
The SO(3) freedom in this Ansatz results in an invariance at the fixed point of the 2-sphere, due to which only two of the components of each of the five triplets ( A r , ξ, χ, ψ, φ) are independent functions. We thus end up with 10 equations of motion for the functions of r, A r = (ã r , 0, a r ) , ξ = (w, 0, w) , χ = (Ṽ , 0, V ) , ψ = (h, 0, h) , φ = (g, 0, g) .
The equations of motion arising from the variation of A r result in a pair of constraint equations, since there is no non-trivial curvature pertaining to this connection. We will study three types of solutions, for which these constraint equations are identically satisfied, such that A r = 0 effectively. These finite energy solutions may have a non-vanishing electric charge and zero magnetic charge. It is straightforward to check that the magnetic charge density in Eq. (9) vanishes identically for the field configuration parametrised by our spherically symmetric Ansatz, Eqs. (13), (14), and (15).
These three types of zero magnetic charge solutions are described by the following functions
ξ = (0, 0, w) , χ = (Ṽ , 0, 0) , ψ = (h, 0, 0) , φ = (g, 0, 0) ,(22)ξ = (0, 0, w) , χ = (0, 0, V ) , ψ = (0, 0, h) , φ = (0, 0, g) ,(23)ξ = (0, 0, w) , χ = (0, 0, V ) , ψ = (h, 0, 0) , φ = (0, 0, g) ,(24)
to which we refer as Types I, II, and III, respectively. Such solutions exist for models with either of the Higgs potentials, Eqs. (7) and (8).
Types I, II, and III: Numerical results
We have not been able to generate numerically excited solutions when all the components in the multiplets Eq. (21) are present. Only solutions for the restricted cases Eqs. (22)-(24) could be found 4 .
Type I solutions
These solutions are characterized byw = 0, V = 0, h = 0, and g = 0. The expansions at the origin are
w = −1 + w 2 x 2 + 2w 2 x 3 + O(x 4 ) ,(25)V =Ṽ 1 x +Ṽ 1 x 2 + O(x 3 ) ,(26)g =g 0 + O(x 2 ) ,(27)h =h 1 x +h 1 x 2 + O(x 3 ) ,(28)
where x = r/(1 + r). The asymptotic values of the functions are
w = 0 , (29) V =Ṽ 0 ,(30)g = cos γ ,(31)h = sin γ ,(32)
whereṼ 0 and γ are free.Ṽ 0 controls the contribution to the electric charge Eq. (10) of JZ type, while γ gives rise to another contribution to the electric charge, once the HCS terms are present. Our parameters are: λ, κ 1 , κ 2 , V 0 , and γ. The effect of the JZ parameterṼ 0 is exhibited in Fig. 1. For fixed γ and κ 1 = 0 and κ 2 = 0 , when varyingṼ 0 the electric charge Q changes. In this case an increase in |Q| makes the energy of the solutions E increase. This is the behaviour one would expect. In fact, for vanishing λ the theory may be rescaled and the relation between E and Q becomes independent of γ (they both rescale with sin γ).
The situation changes radically when the new CS terms are present. In that case the solution with the lowest energy is not the electrically neutral one, in general. There are regions where the energy is a decreasing function of |Q|. Both types of HCS terms give rise to such an effect, although the first one, Eq. (2), requires the presence of a non-vanishing potential (i.e., λ = 0). This is shown in Fig. 2, where we exhibit the energy E versus the electric charge Q for type I solutions withṼ 0 = 0, κ 1 = 1.0, κ 2 = 0 and λ = 0.0, 0.1, and 1.0. Clearly, the solution with the largest energy corresponds to the electrically uncharged one (excluding the vacuum solution).
When both contributions to the electric charge are present, the structure of the solutions gets more complicated: several solutions may exist for the same value of the electric charge. Moreover, the uncharged solutions may not exist for large enough values ofṼ 0 . This is exemplified in Fig. 3 where no electrically neutral solutions exist for these values of the parameters.
The pattern of solutions may develop a large number of branches in certain regions of the parameter space. In Fig. 4 we present the dependence of the energy E on the electric charge Q for type I solutions withṼ 0 = 0.5, κ 1 = 2.0, κ 2 = −12 and λ = 0.0. We observe that several electrically uncharged solutions exist, none of them having the lowest energy.
Type II solutions
w = −1 + w 2 x 2 + 2w 2 x 3 + O(x 4 ) ,(33)V =V 0 + O(x 2 ) ,(34)g = g 1 x + g 1 x 2 + O(x 3 ) ,(35)h = h 0 + O(x 2 ) ,(36)
where x = r/(1 + r). The asymptotic values of the functions are
w = 0 ,(37)V = V 0 , (38) g = cos γ ,(39)h = sin γ ,(40)
where γ is free. V does not enter the equations directly, but just through its derivatives. So the asymptotic value of V , V 0 , may be given any arbitrary value (gauge freedom). So for this type of solutions we do not have V 0 as a true physical parameter to be varied; that means there is no JZ parameter. Then, only γ allows us to vary the electric charge of the solutions, once the other parameters of the theory, namely, λ, κ 1 , and κ 2 , are given. As opposed to type I solutions, for type II solutions the first HCS term, Eq. (2), can give rise to charged solutions also for λ = 0. When only one type of the HCS term is present, the structure of the solutions is quite simple, as shown in Fig. 5. When both are present, the structure becomes more involved, although the lack of a JZ term prevents the appearance of very complicated structures as in Fig. 4. In Fig. 6 we show the energy E versus the electric charge Q for λ = 0.0, κ 1 = 2.0, and κ 2 = −12.0. Again, the uncharged solutions (excluding the vacuum) do not correspond to the solutions with the lowest energy.
Type III solutions
When we setw = 0,Ṽ = 0,g = 0, and h = 0, type III solutions are obtained. The expansions at the origin now read
w = −1 + w 2 x 2 + 2w 2 x 3 + O(x 4 ) ,(41)V =V 0 + O(x 2 ) , (42) g = g 1 x + g 1 x 2 + O(x 3 ) ,(43)h =h 1 x +h 1 x 2 + O(x 3 ) ,(44)
where x = r/(1 + r). The asymptotic values of the functions are
w = 0 ,(45)V = 0 , (46) g = cos γ ,(47)h = sin γ ,(48)
where γ is free. When the electric charge Q, Eq. (17), is evaluated for these solutions, it is found to be zero. However, the electric potential, A 0 , is not identically zero. This is clearly seen in Fig. 7 where the functions w, V , g, andh are shown for the type III solution with λ = 0.0, γ = 1.2, κ 1 = 1.0, and κ 2 = 2.0.
Since the electric charge vanishes in this case, we may show the structure of branches plotting the energy E versus the asymptotic angle γ. Very intricate patterns appear, as demonstrated in Fig. 8 Solutions of SU (3) monopoles have been studied intensively a long time ago [13]. Here we follow the (some of the) constructions to be found in [14] and [15]. While in the previous example, namely the SO(5) model on IR 3+2 , the dimensional descent from 8 (and resly. 6) over S 3 (and resly. S 1 ) giving rise to HCS(2) (and resly. HCS(1)) was that prescribed in [2], here the corresponding prescription is slightly different. Instead of the gauge field in the bulk being a 8 × 8 (and resly. 4 × 4) anti-Hermitian connection, here it is a 6 × 6 (and resly. 3 × 3) anti-Hermitian connection.
Imposition of symmetry and boundary values
We use the standard SU (3) spherically symmetric Ansatz
A i = 1 − w r λ (1) ijx j ,(49)Φ = 1 2 i η hx j λ (1) j + g λ 8 ,(50)A 0 = 1 2 i ux j λ (1) j + v λ 8 .(51w = −1 + w 2 x 2 + 2w 2 x 3 + O(x 4 ) ,(52)h = h 1 x + h 1 x 2 + O(x 3 ) ,(53)g = g 0 + O(x 2 ) ,(54)u = u 1 x + u 1 x 2 + O(x 3 ) ,(55)v = v 0 + O(x 2 ) .(56)
We seek solutions with the following asymptotic values where γ and u 0 are free parameters, corresponding to an asymptotic angle for the Higgs components and the JZ parameter, respectively. Under these boundary conditions, the magnetic charge, Eq. (9), becomes
lim r→0 w(r) = 1 , lim r→∞ w(r) = 0 ,(57)lim r→0 h(r) = 0 , lim r→∞ h(r) = cos γ ,(58)lim r→0 g ′ (r) = 0 , lim r→∞ g(r) = sin γ ,(59)lim r→0 u(r) = 0 , lim r→∞ u(r) = u 0 ,(60)lim r→0 v ′ (r) = 0 , lim r→∞ v(r) = 0 (gauge choice) ,(61)µ = η cos γ ,(62)
and the electric charge, Eq. (10), results to be
Q = 1 2 η r 2 (hu ′ + gv ′ ) r=∞ .(63)
In the absence of the HCS terms, when λ = 0 the second-order field equations are solved by the first-order selfduality equations. The latter reduce to the BPS equations which have nontrivial solutions only for the functions w(r), h(r) and u(r), while the functions g(r) and v(r) both vanish everywhere. This means that with λ = 0 the only solutions are the SU (2) JZ dyons in that case. However, when the HCS terms are present, nontrivial solutions for the functions g(r) and v(r) are present even in the λ = 0 limit. Since the parameter space is already large enough, we will restrict our attention in this work to the λ = 0 case only, for economy of presentation.
Numerical results
We have generated numerical solutions to this theory. In these numerical ruesults we have set η = 1 to fix the scale.
As happened for SO(5), when κ 1 = 0, κ 2 = 0, and λ = 0 the representation of the scaled energy E/µ versus the scaled electric charge Q/µ shows that E/µ is an increasing function of |Q/µ|; in fact, the figure coincides with Fig. 1 (when rescaled properly). The situation changes, however, when the HCS terms are present. In that case, for a given asymptotic angle γ (i.e., a given magnetic charge µ), the electrically uncharged solution need not be the one with the least energy. We exhibit this fact in Fig. 9 where we represent the energy E versus the magnetic This effect is more clearly observed in Fig. 10, where we represent the energy E of the solutions versus the electric charge Q for 3 asymptotic angles γ = 0, π/6, and π/3 for λ = 0, κ 1 = 1, and κ 2 = 1. For nonvanishing γ the minimal energy occurs for a nonvanishing of the electric charge.
The issue of angular momentum
The issue of angular momentum density can readily be calculated using the Ansatz given in Eqs. (B.8)-(B.11),
4 T ϕ 0 = [(D ρ ξ · D ρ χ) + (D z ξ · D z χ)] + 4η 2 ρ (φεχ)(φεξ) ,(64)
which can be rewritten in the form
4 ρ T ϕ 0 = [∂ ρ (ρ ξ · D ρ χ) + ∂ z (ρ ξ · D z χ)] − [(ξ · D ρ χ) + ρ ξ · (D ρ D ρ χ + D z D z χ)] − 4η 2 ρ (φεχ)(φεξ) ,(65)
where a total divergence term is isolated. Consider now the equation resulting from the variation of Eq. (B.27) with respect to the doublet χ a ,
D ρ χ a + ρ( D ρ D ρ χ a + D z D z χ a ) − 1 ρ (χεξ)(εξ) a + 4η 2 ρ (φεχ)(φε) a = = 1 2 κ 1 η D [ρ (g D z] ξ) a + g f ρz (εξ) a .(66)
Contracting Eq. (66) with ξ a and substituting the result in Eq. (65) The first term on right-hand side in Eq. (67) is a div and its volume integral vanishes by virtue of the asymptotic values of the solutions. The second term is a curl. Using the notation
4 ρ T ϕ 0 = [∂ ρ (ρ ξ · D ρ χ) + ∂ z (ρ ξ · D z χ)] − 1 2 κ 1 η ξ · D [ρ (g D z] ξ) .(67)x A = (ρ, z) ,
the second term in Eq. (67) can be expressed as
ξ a D [ρ (g D z] ξ) a = ε AB ξ a D A (g D B ξ) a = − 1 2 ε AB ∂ A |ξ| 2 ∂ B g ,
which can be evaluated by performing a contour integral, using Stokes' Theorem (like the multi-monopole charge.) On the far hemisphere, |ξ| 2 = 0 so there will be no contribution. On the z-axis ∂ z g changes sign going through the origin, so the line integral on the positive z-axis will cancel against the line integral on the negative z-axis. Thus, the angular momentum of this system vanishes.
Summary, comments and outlook
In this Section, we will summarise our results and comment on their properties. After that we will describe what further questions may arise out of the results. In this paper we have constructed electrically charged solitons in two distinct YMH models in 3 + 1 dimensions, one with gauge group SO(5) and the other SU (3). Both these theories involve two (dynamical) new CS terms which we refer to as HCS terms. The purpose of this investigation is to show that in certain regions of the parameter space, the electrically charged solutions have smaller mass than their electrically neutral counterparts. This property is a consequence of the dynamics of the HCS densities appearing in the respective Langrangian. This is the main result presented here. This investigation is carried out for two distinct models to show that the main result obtained here, is independent of the specific feature of the model chosen, namely of the choice of gauge group. The SO(5) and SU (3) models employed differ in an important respect, namely that the former has zero magnetic charge while the latter has a magnetic charge (in the spherically symmetric case). It is reasonable to treat these two types of solutions separately, to ensure that such a prominent difference does not result in the main feature claimed.
Solutions to the SO(5) and SU (3) models share two properties. First, when the HCS terms are decoupled, i.e. setting κ 1 = κ 2 = 0, the energy of the charged soliton increases with increasing electric charge. This expected result is exhibited in Figure. 1.
Another consequence of setting κ 1 = κ 2 = 0 in these models is, that in the absence of the Higgs symmetry breaking potential (λ = 0) only solutions parametrising the SO(3) subgroup are supported. However, when κ 1 and/or κ 2 are switched on, the gauge fields can take their values outside of SO(3). It is therefore not necessary to consider λ > 0 solutions and for simplicity we have concentrated on the λ = 0. We have nonetheless considered λ > 0 models in a few cases, to ensure that the introduction of the Higgs potential does not alter the qualitative features of our main result.
The SO(5) model
In this case we have only zero magnetic charge solutions. These exhibit the desired property in some regions of the parameter space. To make our investigation complete, we have studied three types of such solutions, Type I, II and III. The numerical construction of these solutions is presented in Section 3.2. parameter free. These results are exhibited in Fig. 5 and 6. As for Type I solutions, these solutions are electrically charged and their mass may be lower than that of the uncharged solution.
• Type III solutions are characterised also by the asymptotic angle γ. Opposite to the previous two type these solutions are electrically uncharged although their electric potential is not identically zero. They describe electric dipoles with zero electric monopole. These results are exhibited in Figs. 7 and 8. The structure of these solutions may get quite complicated as shown in Fig. 8.
Note that in Figs 2 and 5, profiles with λ > 0 appear, which preserve the shapes conformally.
The SU(3) model
The main feature in this case is that the solutions carry both electric and magnetic charge, and are dyons. We see that the qualitative features observed in the SO(5) model, namely our main result, are preseved. While the qualitative result, that the electrically neutral solutions can be more massive than the neutral ones, a specific feature is observed.
• In Fig. 9 we observe that for non-vanishing electric charge, two dyonic solutions are possible for a magnetic charge 0 < µ < 1. The mass of the magnetic monopole (curve in red) is higher than the corresponding value along the lower branch for large ranges in µ. That indicates that the electrically neutral solutions are not necessarily the least energetic ones, in general.
• In Fig. 10 we show this effect more clearly for 3 values of the magnetic charge (including the one chosen in Weinberg's book [15] (green curve)). For 0 < µ < 1 the minimum of the energy occurs for non-vanishing electric charge. In addition in this case we have considered the axially symmetric fields and have constructed the angular momentum density of this SU (3) dyon. It turns out that this vanishes.
Summary and outlook
In this paper we have constructed electrically charged solitons in an SO(5) and SU (3) HCS-YMH theory in 3 + 1 dimensions. These theories contain the new CS terms which were employed in [1] for the SO(5) model. By means of an enlarged spherically symmetric Ansatz, we have been able to endow the solutions of the SO(5) model [1] with an asymptotic angle γ resulting in a larger set of electrically charged solutions, which exhibit the new desired properties. Qualitatively similar results are obtained for the SU (3) model. This way of producing electrically charged solutions differs from the prescription of Julia and Zee [9]. Technically, in the SO(5) model, the obvious difference with the JZ prescription is that the time component of the YM potential A 0 and the Higgs field do not take their values in the same representation of the gauge group. But more importantly, the origin of the electrical fields here is found in the CS dynamics in the case of both the SO(5) and SU (3) models. This is akin to the analogous 2 + 1 dimensional situation in [6] and [7,8].
In the case of the SU (3) model we have calculated the angular momentum of the CS dyon and found that it vanishes. In this respect, the introduction of a new CS term with the attendant enlargement of the gauge group from SU (2) to SU (3), does not change the general result in [16] (and references therein), namely that SU (2) YMH dyons in 3 + 1 dimensions do not rotate. This property contrasts with the analogous 2 + 1 dimensional situation in [6] and [7,8], where the introduction of the CS term results in rotation. In the matter of electric charge the introduction of a CS term plays the same role in gauge-Higgs theories in both 3 + 1 and 2 + 1 dimensions. Thus, the effect of CS dynamics in 3 + 1 and 2 + 1 dimensions is qualitatively different, overlapping in one respect (electric charge) but differing in another (angular momentum). This question is at present under intensive consideration.
Finally, it is natural to inquire what the analogue of the present investigation in the context of gauged Higgs models would be, in the case of gauged Skyrme [17] systems. For this, one would have to employ the Skyrme analogue of the HCS densities used here. This question is at also under intensive consideration.
A The one dimensional quantities subject to spherical symmetry
In this Appendix, we present the curvature field strengths and the covariant derivatives subject to spherical symmetry. The resulting one dimensional static Lagrangian and energy densities used in our computations are then displayed. These quantities are given in the following two subsections, each for the SO(5) and the SU (3) models, respectively.
A.1 SO(5) model
The parametrisation used in the Ansatz, Eqs. (13)- (15), results in a gauge covariant expression for the YM curvature F µν = (F ij , F i0 ) and the covariant derivative of the Higgs
D µ Φ = (D i Φ, D 0 Φ) F ij = 1 r 2 | ξ| 2 − 1 Σ ij + 1 r D r ξ 6 + 1 r | ξ| 2 − 1 x [i Σ j]kxk + 1 r D r ξ Mx [i Σ j]M , (A.1) F i0 = − 1 r ξ M (εχ) M Σ ijxj + 1 r ξ 6 (εχ) M − χ 6 (εξ) M Σ iM − (εD r χ) M + 1 r ξ 6 (εχ) M − χ 6 (εξ) M x ixj Σ jM − D r χ 6x i Σ 45 , (A.2) (2η) −1 D i Φ = − 1 r ( ξ · φ)(δ ij −x ixj ) Σ j6 + D r φ Mx i Σ M6 + D r φ 6x ixj Σ j6 − 1 r ξ M (εψ) M Σ ijxj + 1 r ξ 6 (εψ) M − ψ 6 (εξ) M Σ iM − (εD r ψ) M + 1 r ξ 6 (εψ) M − ψ 6 (εξ) M x ixj Σ jM − D r ψ 6x i Σ 45 , (A.3) (2η) −1 D 0 Φ = φ M (εχ) Mx j Σ j6 − φ 6 (εχ) M − χ 6 (εφ) M Σ M6 +χ M ψ N Σ MN − (ψ 6 χ M − χ 6 ψ M )x j Σ jM , (A.4)
in which we have used the notation
D r φ a = ∂ r φ a + ε abc A b r φ c , ...CS = 8κ 1 η (| ξ| 2 − 1) φ · D r χ − 2( ξ × χ) · ( φ × D r ξ) , (A.6)(1)
which does not receive a contribution from the triplet ψ. The second HCS term, ω
CS , Eq. (3), however does receive a contribution from ψ. The resulting expression being too cumbersome and not instructive, we do not exhibit it here. We have of course verified that its computation using symbolic manipulations is correct.
The reduced one dimensional YM Lagrangian is
− L (1) YM = 2 |D r ξ| 2 + 1 r 2 | ξ| 2 − 1 2 − r 2 |D r χ| 2 + 2 |( ξ × χ)| 2 , (A.7)
the reduced one dimensional Higgs Lagrangian is
L Higgs = 2 η 2 r 2 |( φ × χ)| 2 − |D r φ| 2 + 2 r 2 ( ξ · φ) 2 +|( ψ × χ)| 2 − |D r ψ| 2 + 2 r 2 ( ξ × ψ) 2 , (A.8)
and, finally, the Higgs potentials, Eqs. (7) and (8), reduce (for a 1 = 1/4 and a 2 = 1) to
v 1 = η 4 r 2 1 − | φ| 2 + | ψ| 2 2 , (A.9) v 2 = η 4 r 2 1 − | φ| 2 + | ψ| 2 2 + 4( φ · ψ) 2 , (A.10) with v i def. = r 2 V i , i = 1, 2 .
It is clear that in the case Eq. (A.10) the asymptotic triplet φ must be orthogonal to the asymptotic triplet ψ.
Another quantity we will employ to analyze the solutions is their energy, E, given by
E = ∞ 0 |D r ξ| 2 + 1 2r 2 | ξ| 2 − 1 2 + 1 2 r 2 |D r χ| 2 + |( ξ × χ)| 2 +2 η 2 r 2 |( φ × χ)| 2 + |D r φ| 2 + 2 r 2 ( ξ · φ) 2 + |( ψ × χ)| 2 + |D r ψ| 2 + 2 r 2 ( ξ × ψ) 2 + λ 2 η 4 r 2 1 − | φ| 2 + | ψ| 2 2 dr . (A.11)
Notice that only the first potential Eq. (7) has been included.
A.2 SU(3) model
Subject to the Ansatz Eq. (50), the symmetry breaking potentials, Eqs. (7) and (8), reduce, respectively, to
V 1 = η 4 1 − h 2 + g 2 2 , (A.12) V 2 = 1 4 η 4 3 − a 2 (h 2 + g 2 ) + a 2 2 8 (h 2 + g 2 ) 2 ,(A.F ij = − 1 r 2 (1 − w 2 ) λ (1) ij − w ′ r + 1 r 2 (1 − w 2 ) x [i λ(1)
j]kx k , (A.14)
D i Φ = 1 2 i η wh r λ (1) i + h ′ − wh r x ixj λ (1) j + g ′x i λ 8 , (A.15) F i0 = 1 2 i wu r λ (1) i + u ′ − wu r x ixj λ (1) j + v ′x i λ 8 , (A.16) D 0 Φ = 0 , (A.17)
further resulting in
Tr F 2 ij = − 1 r 2 2 w ′2 + 1 r 2 (1 − w 2 ) 2 , (A.18) Tr F 2 i0 = − 1 2 u ′2 + 2 r 2 w 2 u 2 + v ′2 , (A.19) Tr D i Φ 2 = − 1 2 η 2 h ′2 + 2 r 2 w 2 h 2 + g ′2 . (A.20)
The magnetic charge integral, Eq. (9), reduces to
µ = η[(1 − w 2 )h] r=∞ , (A.21)
and the electric charge integral, Eq. (10), results to be
Q = 1 2 η r 2 (hu ′ + gv ′ ) r=∞ . (A.22)
The energy of the solutions is given by
E = 1 4 ∞ 0 r 2 u ′2 + r 2 v ′2 + 2u 2 w 2 + (1 − w 2 ) 2 r 2 + 2w ′2 + r 2 g ′2 + r 2 h ′2 + 2h 2 w 2 + 2λr 2 (1 − g 2 − h 2 ) 2 dr .
(A.23) Subject to this spherical symmetry, the HCS densities Eqs. (2) and (3), do not identically vanish but yield
Ω (1) CS = − 2 √ 3r 2 η (1 − w 2 ) h v ′ + g[(1 − w 2 )u] ′ , (A.24) Ω (2) CS = − √ 3 54r 2 η 3 −2guw(36 − g 2 − 5h 2 )w ′ + 3h (1 − w 2 )(12 − g 2 − h 2 ) + 2h 2 w 2 v ′ +g (1 − w 2 )(36 − g 2 − 9h 2 ) + 2h 2 w 2 u ′ − h 2 uw 2 g ′ + 4ghuw 2 h ′ .A i = Â i 0 1×2 0 2×1 0 2×2 ; i = α, z ≡ α, 3 ; α = x, y ≡ 1, 2 , (B.1)
The electric component A 0 of the SU (3) connection corresponding to the spherically symmetric Ansatz Eq. (51) and the Higgs field Φ corresponding to Eq. (50), likewise
A 0 = Â 0 0 1×2 0 2×1 0 2×2 + i v(ρ, z) λ 8 , (B.2) and (2η) −2 Φ = Φ 0 1×2 0 2×1 0 2×2 + i g(ρ, z) λ 8 , (B.3)
respectively.
There now remains to impose axial symmetry on the SU (2) algebra valued quantities i = ( α , z ), 0 and Φ. For this, we employ the chiral SO(4) matrices 5 Σ (±) MN representing the SU (±) (2) subalgebra valued quantities in Eqs. (B.1)-(B.3)
In this notation, α = ξ 2 + n ρ (εx) α Σ 12 + ξ 1 ρ (εx) α (εn) γ + a ρxα n γ Σ γ3 , (B.8)
A z = a z n γ Σ γ3 , (B.9)
A 0 = −χ 1 n γ Σ γ4 + χ 2 Σ 34 = χ 1 (εn) γ Σ γ3 + χ 2 Σ 12 , (B.10)
Φ = −φ 1 n γ Σ γ4 + φ 2 Σ 34 = φ 1 (εn) γ Σ γ3 + φ 2 Σ 12 , (B.11)
where n α = (cos nϕ, sin nϕ) is the unit vector in the (x 1 , x 2 ) plane, ϕ is the azimuthal angle and n is the vortex number. The functions (a ρ , a z ), ξ a = (ξ 1 , ξ 2 ), χ a = (χ 1 , χ 2 ) and φ a = (φ 1 , φ 2 ) all depend on the two variables ρ = |x α | 2 and z, and are independant of the time coordinate x 0 . The gauge covariant quantities F µν = (F αβ , F αz , F α0 , F z0 ) and D µ Φ = (D α Φ, D z Φ, D 0 Φ) follow,
F αβ = − 1 ρ ε αβ D ρ ξ 1 (εn) γ Σ γ3 + D ρ ξ 2 Σ 12 , (B.12) F αz = f ρz x α n γ Σ γ3 − 1 ρ (εx) α D z ξ 1 (εn) γ Σ γ3 + D z ξ 2 Σ 12 , (B.13) F α0 = 1 ρ (χεξ)(εx) α n γ Σ γ3 +x α D ρ χ 1 (εn) γ Σ γ3 + D ρ χ 2 Σ 12 , (B.14) F z0 = D z χ 1 (εn) γ Σ γ3 + D z χ 2 Σ 12 , (B.15) and D αΦ =x α D ρ φ 1 (εn) γ Σ γ3 + D ρ φ 2 Σ 12 + 1 ρ (φεξ)(εx) α n γ Σ γ3 ,(f ρz = ∂ ρ a z − ∂ z a ρ ,
the SO(2) covariant derivatives D ρ ξ a = ∂ ρ ξ a + a ρ (εξ) a , D z ξ a = ∂ z ξ a + a z (εξ) a , etc.
and with (f εg) = ε ab f a g b . In particular, we opt for the selfdual case.
The (static) axially symmetric U (1) ≃ SO(2) gauge connection a µ = (a α , a z , a 0 ) can be expressed as a α = u(r, θ) (xε) α , (B.19) a z = 0 , (B.20)
a 0 = a 0 (r, θ) .
(B.21)
In the calcualtion of the angular momentum, the azimuthal component of the Abelian connection a ϕ will be employed, which in the notation of Eq. (B.19) is
a ϕ = ρ u . (B.22)
The components of the Abelian curvature h µν = ∂ µ a ν − ∂ ν a µ follow The reduced two dimensional Lagrangian is
h αβ = 1 ρ (ρ u) ,L (1) = − 1 4 1 ρ |D ρ ξ| 2 + |D z ξ| 2 − ρ |D ρ χ| 2 + |D z χ| 2 + ρ f 2 ρz − 1 ρ (χεξ) 2 − 4ρ ∂ ρ v 2 + ∂ z v 2 −η 2 ρ |D ρ φ| 2 + |D z φ| 2 + 1 ρ (φεξ) 2 − ρ(φεχ) 2 − 4ρ ∂ ρ g 2 + ∂ z g 2 + κ 1 ω (1) , (B.27)
where ω (1) is the reduced two dimensional HCS density Eq. (2),
ω (1) = 8 √ 3 η g (χεξ) f ρz − D [ρ ξ · D z] χ + ∂ [ρ v(φ · D z] ξ) . (B.28)
Figure 1 :
1Energy E versus electric charge Q for type I solutions with λ = 0, κ 1 = 0, and κ 2 = 0;Ṽ 0 is varied and γ is kept fixed.
Figure 2 :
2Energy E versus electric charge Q for type I solutions withṼ 0 = 0, κ 1 = 1.0, and κ 2 = 0 for three values of λ: 0.0, 0.1, and 1.0.
Figure 3 :
3for the type III solutions with λ = 0.0, κ 1 = 1.0, and κ 2 = −12.0. Energy E versus electric charge Q for type I solutions with λ = 0,Ṽ 0 = 0.2, κ 1 = 0.3, and κ 2 = 0.5. 4 Solutions of the SU (3) Higgs-Chern-Simons-Yang-Mills-Higgs model This Section consists of three Subsections. In Subsection 4.1, spherical symmetry is imposed and the boundary values of the solutions sought are stated. The numerical construction of the solutions is presented in Subsection 4.2, and in Subsection 4.3 we impose axial symmetry on this system with a view to show whether the dyon of the SU (3) HCS-YMH model rotates or not.
Figure 4 :(
4Energy E versus electric charge Q for type I solutions with λ = 0,Ṽ 0 = 0.5, κ 1 = 2.0, and κ 2 = −12.0. λ (1) i , i = 1, 2, 3 are the first three su(2) embeddings in su(3), λ 8 is the last diagonal one, We have used anti-Hermitian representations of the su(3) algebra.) Detailed one dimensional reduced quantities used in our computations are given in Appendix A.2. The expansions at the origin read
Figure 5 :
5Energy E versus electric charge Q for type II solutions with λ = 0.0, 1.0, κ 1 = 1.0, and κ 2 = 0.0.
Figure 6 :
6Energy E versus electric charge Q for type II solutions with λ = 0.0, κ 1 = 2.0, and κ 2 = −12.0. charge µ for λ = 0, κ 1 = 1, and κ 2 = 1 and several values of the electric charge Q: 0.0, 0.5, and 1.0. (Notice that in the limit µ = 0 the value of the energy tends to the value of the electric charge.)
Figure 7 :
7Functions w, V , g, andh for type III solutions with λ = 0.0, γ = 1.2, κ 1 = 1.0, and κ 2 = 2.0.
Figure 8 :
8Energy E versus asymptotic angle γ for type III solutions with λ = 0.0, κ 1 = 1.0, and κ 2 = −12.0.
•Figure 9 :
9Type I solutions are characterised by the existence of two parameters: one of them related to the JZ contribution to the electric charge,Ṽ 0 , and one related to the HCS contribution, γ. These solutions posses a non-vanishing electric charge coming from both types of sources. Uncharged solutions may have higher energy than the charged ones. These results are exhibited inFigs. 1-4where we exhibit the dependence of the energy E on the electric charge Q under several circumstances.• Type II solutions are characterised by the presence of the asymptotic angle γ. In this case there is no JZ Energy E versus magnetic charge µ for solutions with λ = 0, κ 1 = 1, and κ 2 = 1 and several values of the electric charge Q: 0.0, 0.5, and 1.0.
Figure 10 :
10Energy E versus electry charge Q for solutions with λ = 0, κ 1 = 1, and κ 2 = 1 and several values of the magnetic charge µ:
as the SO(3) covariant derivatives of the four tripletsξ ≡ ξ a = (ξ M , ξ 6 ), χ ≡ χ a = (χ M , χ 6 ), ψ ≡ ψ a = (ψ M , ψ 6 ), and φ ≡ φ a = (φ M , φ 6 ),with respect to the SO(3) gauge connection A r ≡ A a r . Substituting Eq. (13) and Eqs. (A.2) in the HCS densities, Eqs. (2)-(3), we have the reduced one dimensional HCS densities ω the first HCS term, Eq. (2), we have the reduced one dimensional density ω
B
Imposition of axial symmetry on the SU (3) model In this Appendix, we present the axially symmetric field configurations employed in Section 4.3, in the discussion of the issue of angular momentum in the SU (3) model. We denote the magnetic component A i = (A α , A z ) of the SU (3) connection corresponding to the spherically symmetric Ansatz Eq. (49) as
B.16) D zΦ = D z φ 1 (εn) γ Σ γ3 + D z φ 2 Σ 12 , (B.17) D 0Φ = (φεχ) n γ Σ γ3 , (B.18)which are all expressed in terms of the SO(2) curvature
M Σ N −Σ N Σ M ) , (B.5)where the index M = α, 3, 4, with α = 1, 2. The spin matrices used areΣα = −Σα = i σα , Σ 3 = −Σ 3 = i σ 3 , Σ 4 =Σ 4 = 1I , (B.6)where (σα, σ 3 ) are the usual 2 × 2 Pauli spin matrices.
ρ ε αβ , (B.23) h αz = u ,z (εx) α , (B.24) h α0 = (a 0 ) ,ρxα , (B.25) h z0 = (a 0 ) ,z .(B.26)
13 )
13with a 1 = 2. It is clear that in the case Eq. (A.13), V 2 cannot vanish for any real value of the constant a 2 , i.e. we have only one choice in this case, namely Eq. (A.12). The resulting curvatures and covariant derivative following from Eqs. (49), (50) and (51), are
Since we aspire here to present a 3 + 1 dimensional analogue of the the 2 + 1 dimensional Chern-Simons-Higgs vortices[7,8], it may be relevant to inquire whether we could likewise omit the Yang-Mills term in Eq. (4). This in principle is possible since the system excluding the Yang-Mills term is consistent with the Derrick scaling requirement in the corresponding static Hamiltonian after solving for A 0 using the Gauss-Law equation. However in the non-Abelian system at hand, A 0 cannot be solved for in closed form, rendering such an approach impractical.
We have employed a collocation method for boundary-value ordinary differential equations, equipped with an adaptive mesh selection procedure[12]. A compactified radial coordinate x = r/(1 + r) has been used. Typical mesh sizes include 10 3 − 10 4 points. The solutions have a relative accuracy of 10 −8 .
The chiral Dirac representation matrices Σµν = (Σ αβ , Σ α6 ) used here are defined as Σµν = − 1 4 Σ [µΣν] , in terms of the spin matrices Σ i = −Σ i = iγ i , Σ 4 = −Σ 4 = iγ 4 , Σ 5 = −Σ 5 = iγ 5 , Σ 6 = +Σ 6 = 1I, where (γ i , γ 4 , γ 5 ), i = 1, 2, 3 are the usual Dirac gamma matrices in four dimensions.
We have set η = 1/2 is our numerical schemes. This choice gives rise to a unit energy for type I solutions with λ = 0, κ 1 = 0, κ 2 = 0, and γ = π/2.
Acknowledgments We thank Eugen Radu for fruitful discussions and suggestions on this paper. D.H.Tch. thanks Hermann Nicolai for his hospitality at the Albert-Einstein-Institute, Golm, (Max-Planck-Institut, Potsdam) where parts of this work were carried out. F. N-L. acknowledges financial support of the Spanish Education and Science Ministry under Project No. FIS2011-28013 (MINECO).
. F Navarro-Lérida, E Radu, D H Tchrakian, arXiv:1311.3950Int. J. Mod. Phys. A. 291450149hep-thF. Navarro-Lérida, E. Radu, and D. H. Tchrakian, Int. J. Mod. Phys. A 29 (2014) 1450149 [arXiv:1311.3950 [hep-th]].
. D H Tchrakian, arXiv:1009.3790J. Phys. A. 44343001hep-thD. H. Tchrakian, J. Phys. A 44 (2011) 343001 [arXiv:1009.3790 [hep-th]].
. E Radu, T Tchrakian, arXiv:1101.5068hep-thE. Radu and T. Tchrakian, arXiv:1101.5068 [hep-th].
R See For Example, Jackiw, Chern-Simons terms and cocycles in physics and mathematics. Adam Hilger; BristolE.S. Fradkin F estschrif tsee for example, R. Jackiw, "Chern-Simons terms and cocycles in physics and mathematics", in E.S. Fradkin F estschrif t, Adam Hilger, Bristol (1985).
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. ACM Transactions. 7209ACM Transactions 7 (1981) 209.
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| []
|
[
"Time Dependent Couplings as Observables in de Sitter Space",
"Time Dependent Couplings as Observables in de Sitter Space"
]
| [
"Hiroyuki Kitamoto \nDepartment of Physics and Astronomy\nSeoul National University\n151-747SeoulKorea\n\nIntroduction\n\n",
"Yoshihisa Kitazawa \nKEK Theory Center\n305-0801TsukubaIbarakiJapan\n\nDepartment of Particle and Nuclear Physics\nThe Graduate University for Advanced Studies (Sokendai) Tsukuba\n305-0801IbarakiJapan\n"
]
| [
"Department of Physics and Astronomy\nSeoul National University\n151-747SeoulKorea",
"Introduction\n",
"KEK Theory Center\n305-0801TsukubaIbarakiJapan",
"Department of Particle and Nuclear Physics\nThe Graduate University for Advanced Studies (Sokendai) Tsukuba\n305-0801IbarakiJapan"
]
| []
| We summarize and expand our investigations concerning the soft graviton effects on microscopic matter dynamics in de Sitter space. The physical couplings receive IR logarithmic corrections which are sensitive to the IR cut-off at the one-loop level. The scale invariant spectrum in the gravitational propagator at the super-horizon scale is the source of the de Sitter symmetry breaking. The quartic scalar, Yukawa and gauge couplings become time dependent and diminish with time. In contrast, the Newton's constant increases with time. We clarify the physical mechanism behind these effects in terms of the conformal mode dynamics in analogy with 2d quantum gravity. We show that they are the inevitable consequence of the general covariance and lead to gauge invariant predictions. We construct a simple model in which the cosmological constant is self-tuned to vanish due to UV-IR mixing effect. We also discuss phenomenological implications such as decaying Dark Energy and SUSY breaking at the Inflation era. The quantum effect alters the classical slow roll picture in general if the tensor-to-scalar ratio r is as small as 0.01. | 10.1142/s0217751x14300166 | [
"https://arxiv.org/pdf/1402.2443v2.pdf"
]
| 119,275,496 | 1402.2443 | 28625a24498661fc08a20485860047715117daff |
Time Dependent Couplings as Observables in de Sitter Space
28 Mar 2014 March 2014
Hiroyuki Kitamoto
Department of Physics and Astronomy
Seoul National University
151-747SeoulKorea
Introduction
Yoshihisa Kitazawa
KEK Theory Center
305-0801TsukubaIbarakiJapan
Department of Particle and Nuclear Physics
The Graduate University for Advanced Studies (Sokendai) Tsukuba
305-0801IbarakiJapan
Time Dependent Couplings as Observables in de Sitter Space
28 Mar 2014 March 2014de Sitter spacetime dependent couplingsDark Energy
We summarize and expand our investigations concerning the soft graviton effects on microscopic matter dynamics in de Sitter space. The physical couplings receive IR logarithmic corrections which are sensitive to the IR cut-off at the one-loop level. The scale invariant spectrum in the gravitational propagator at the super-horizon scale is the source of the de Sitter symmetry breaking. The quartic scalar, Yukawa and gauge couplings become time dependent and diminish with time. In contrast, the Newton's constant increases with time. We clarify the physical mechanism behind these effects in terms of the conformal mode dynamics in analogy with 2d quantum gravity. We show that they are the inevitable consequence of the general covariance and lead to gauge invariant predictions. We construct a simple model in which the cosmological constant is self-tuned to vanish due to UV-IR mixing effect. We also discuss phenomenological implications such as decaying Dark Energy and SUSY breaking at the Inflation era. The quantum effect alters the classical slow roll picture in general if the tensor-to-scalar ratio r is as small as 0.01.
Introduction
In the present Universe, the Hubble parameter H 2 is so small in the unit of the Planck mass M 2 P = 1/G: H 2 /M 2 P = 10 −120 .
(1.1)
Its energy scale may be related to that of neutrino mass as
H 2 M 2 P ∼ m 4 ν . (1.2)
It has been long thought to be exactly zero before its recent observations. The present Universe is well described as the space-time with a positive cosmological constant, namely de Sitter (dS) space. However we have not understood why the Hubble parameter (Dark Energy) is so small.
It is likely that we cannot explain its magnitude in the present theoretical framework: relativistic quantum field theory. As is well known, the quantum field theory in the flat space-time is intimately connected to critical phenomena in equilibrium physics. In fact the space-time with a positive cosmological constant is not stationary. The quantum field theory in dS space may belong to nonequilibrium physics. In fact we need to employ the Schwinger-Keldysh formalism instead of the familiar Feynman-Dyson formalism.
In particular, the existence of a cosmological event horizon in dS space may give rise to interesting quantum effects. It generates the fluctuations of the space-time metric with a scale invariant spectrum. Inflation theory explains the origin of large scale structures of space-time by such a mechanism. We may need to take the fluctuations at the super-horizon scale seriously as they may eventually come back into the sub-horizon scale.
In the Poincaré coordinate, the metric in dS space is ds 2 = −dt 2 + a 2 (t)dx 2 , a(t) = e Ht , (1.3) where the dimension of dS space is taken as D = 4 and H is the Hubble constant. In the conformally flat coordinate, (g µν ) dS = a 2 (τ )η µν , a(τ ) = − 1 Hτ . (1.4) Here the conformal time τ (−∞ < τ < 0) is related to the cosmic time t as τ = − 1 H e −Ht . We assume that dS space begins at an initial time t i with a finite spatial extension. After a sufficient exponential expansion, dS space is well described locally by the above metric irrespective of the spatial topology.
The metric is invariant under the scaling transformation τ → Cτ, x i → Cx i .
(1.5)
It is a part of the SO(1, 4) dS symmetry. The central issue is whether the dS symmetry could be broken due to IR effects. Note that this is a large gauge (coordinate) transformation.
As explained shortly, we may need to introduce an IR cut-off in order to regulate the IR divergences of the propagators of the minimally coupled modes. The IR cut-off breaks the scale invariance explicitly. Therefore the equivalent question is to ask whether there exist IR divergences in physical observables.
Let us consider free propagators of a massless and minimally coupled scalar field ϕ and a massless conformally coupled scalar field φ
ϕ(x)ϕ(x ′ ) = H 2 4π 2 1 y − 1 2 log y + 1 2 log a(τ )a(τ ′ ) + 1 − γ , (1.6) φ(x)φ(x ′ ) = H 2 4π 2 1 y , (1.7)
where γ is Euler's constant and y is the dS invariant distance
y = −(τ − τ ′ ) 2 + (x − x ′ ) 2 τ τ ′ . (1.8)
It should be noted that the propagator for a massless and minimally coupled scalar field has the dS symmetry breaking logarithmic term: log a(τ )a(τ ′ ).
To explain what causes the dS symmetry breaking, we recall the wave function for a massless and minimally coupled field
ϕ p (x) = Hτ √ 2p (1 − i 1 pτ )e −ipτ +ip·x . (1.9)
Well inside the cosmological horizon where the physical momentum P ≡ p/a(τ ) ≫ H ⇔ p|τ | ≫ 1, this wave function approaches to that in Minkowski space up to a cosmic scale factor
ϕ p (x) ∼ Hτ √ 2p e −ipτ +ip·x . (1.10)
On the other hand, the behavior outside the cosmological horizon P ≪ H is
ϕ p (x) ∼ H 2p 3 e ip·x .
(1.11)
The IR behavior indicates that the corresponding propagator has a scale invariant spectrum. As a direct consequence of it, the propagator has a logarithmic divergence from the IR contributions in the infinite volume limit.
To regularize the IR divergence, we introduce an IR cut-off ε 0 which fixes the minimum value of the comoving momentum H ε 0 a −1 (τ ) dP. (1.12) With this prescription, more degrees of freedom come out of the cosmological horizon with cosmic evolution. Due to the increase, the propagator acquires the growing time dependence which spoils the dS symmetry [1,2,3]. In tribute to its origin, we call the dS symmetry breaking term the IR logarithm. Physically speaking, 1/ε 0 is recognized as an initial size of universe when the exponential expanding starts. For simplicity, we set ε 0 = H in (1.6). We recall here that the physical momentum P = p/a(τ ) is invariant under the scaling transformation (1.5). Our fundamental hypothesis is to adopt the dS invariant UV cut-off P < Λ UV . In this prescription UV contributions from H < P < Λ UV is time independent. In contrast, the IR contributions from H/a(τ ) < P < H could grow with cosmic expansion.
Actually the metric fluctuations always contain a scale invariant spectrum just like the massless and minimally coupled scalar field. In dealing with the quantum fluctuation of the metric whose background is dS space, we adopt the following parametrization:
g µν = Ω 2 (x)g µν , Ω(x) = a(τ )e κw(x) , (1.13) detg µν = −1,g µν = (e κh(x) ) µν , (1.14)
where κ is defined by the Newton's constant G as κ 2 = 16πG.
The Lagrangian of Einstein gravity on the 4-dimensional dS background is
L gravity = 1 κ 2 √ −g R − 6H 2 (1.15) = 1 κ 2 Ω 2R + 6g µν ∂ µ Ω∂ ν Ω − 6H 2 Ω 4 ,
whereR denotes the Ricci scalar constructed fromg µν . In order to fix the gauge with respect to general coordinate invariance, we adopt the following gauge fixing term [4]:
L GF = − 1 2 a 2 F µ F µ , (1.16) F µ = ∂ ρ h ρ µ − 2∂ µ w + 2h ρ µ ∂ ρ log a + 4w∂ µ log a.
In this paper, the Lorentz indexes are raised and lowered by the flat metric η µν and η µν respectively.
After decomposing the spatial part of the metric and diagonalizing the quadratic action:
h ij =h ij + 1 3 h kk δ ij =h ij + 1 3 h 00 δ ij , (1.17) X = 2 √ 3w − 1 √ 3 h 00 , Y = h 00 − 2w,(1.18)
we find that some modes of gravity behave as the massless and minimally coupled scalar field and the other modes behaves as the massless and conformally coupled mode:
X(x)X(x ′ ) = − ϕ(x)ϕ(x ′ ) ,(1.19)h i j (x)h k l (x ′ ) = (δ ik δ jl + δ i l δ k j − 2 3 δ i j δ k l ) ϕ(x)ϕ(x ′ ) , b i (x)b j (x ′ ) = δ ij ϕ(x)ϕ(x ′ ) , h 0i (x)h 0j (x ′ ) = −δ ij φ(x)φ(x ′ ) , (1.20) Y (x)Y (x ′ ) = φ(x)φ(x ′ ) , b 0 (x)b 0 (x ′ ) = − φ(x)φ(x ′ ) ,
where b,b denote the ghost and anti-ghost fields.
Since we focus on the dS symmetry breaking effects, we may introduce an approximation. We can neglect the conformally coupled modes of gravity (1.20) since they do not induce the IR logarithm. In such an approximation, the following identity holds
h 00 ≃ 2w ≃ √ 3 2 X. (1.21)
Our aim is to investigate quantum IR effects from the minimally coupled modes of gravity (1.19) on microscopic matter dynamics. To do so, we derive the effective equation of motion which takes account of the quantum effects due to soft gravitons. Before investigating specific models, we review the general procedure to derive the effective equation of motion in the next section.
Schwinger-Keldysh formalism
In this section, we review how to derive the effective equation of motion in a time dependent curved space-time. Let us represent the vacuum at t → −∞ as |in , and t → ∞ as |out .
In the Feynman-Dyson formalism on a flat background, it is presumed that |out is equal to |in up to a phase factor. On the other hand, we can't prefix |out in dS space. The correct strategy is to evaluate vacuum expectation values (vev) with respect to |in :
O H (x) = in|T C [U(−∞, ∞)U(∞, −∞)O I (x)]|in ,(2.1)
where O H and O I denote the operators in the Heisenberg and the interaction pictures respectively. U(t 1 , t 2 ) is the time translation operator in the interaction picture
U(t 1 , t 2 ) = exp i t 1 t 2 d 4 x δL I (x) . (2.2)
Here δL denotes the interaction term of the Lagrangian. It is crucial that the operator ordering T C specified by the following path is adopted here
, (2.3) C dt = ∞ −∞ dt + − ∞ −∞ dt − .
We call it the Schwinger-Keldysh formalism [5,6]. Since there are two time indices +, − in this formalism, the propagator has four components
ϕ + (x)ϕ + (x ′ ) ϕ + (x)ϕ − (x ′ ) ϕ − (x)ϕ + (x ′ ) ϕ − (x)ϕ − (x ′ ) = T ϕ(x)ϕ(x ′ ) ϕ(x ′ )ϕ(x) ϕ(x)ϕ(x ′ ) T ϕ(x)ϕ(x ′ ) , (2.4)
where T denotes the time ordering andT denotes the anti-time ordering.
Let us introduce the external source J + , J − for each path and evaluate
Z[J + , J − ] = in|T C [U(−∞, ∞)U(∞, −∞) exp i d 4 x (J + ϕ + − J − ϕ − ) ]|in . (2.5)
The generating functional for the connected Green's functions is
iW [J + , J − ] = log Z[J + , J − ]. (2.6)
We define the classical field aŝ
ϕ A (x) = c AB δW [J + , J − ] δJ B (x) , A, B = +, −, (2.7) c AB = 1 0 0 −1 . (2.8)
By taking the limit J + = J − = J in (2.7), we obtain the vev of ϕ where the action contains the additional Jϕ term
ϕ(x) | Jϕ =φ + (x)| J + =J − =J =φ − (x)| J + =J − =J . (2.9)
Finally, we turn off the source term J = 0
ϕ(x) = δW [J + , J − ] δJ + (x) J + =J − =0 = − δW [J + , J − ] δJ − (x) J + =J − =0 . (2.10)
The effective action is obtained after the Legendre transformation
Γ[φ + ,φ − ] = W [J + , J − ] − d 4 x (J +φ+ − J −φ− ),(2.11)
where J +,− are given byφ +,− as follows
J A (x) = −c AB δΓ[φ + ,φ − ] δφ B (x) . (2.12)
From (2.9) and (2.12), we obtain in the limitφ + =φ − =φ
J(x) = − δΓ[φ + ,φ − ] δφ + (x) φ + =φ − =φ = δΓ[φ + ,φ − ] δφ − (x) φ + =φ − =φ . (2.13)
In the absence of the external source, the exact equation of motion is obtained including quantum effects
δΓ[φ + ,φ − ] δφ + (x) φ + =φ − =φ = − δΓ[φ + ,φ − ] δφ − (x) φ + =φ − =φ = 0. (2.
14)
The quantum equation of motion is an appropriate tool to investigate the non-equilibrium physics and curved space-times [7,8,9] as we do not need to prefix the unknown vacuum state |out . Nevertheless we still need to specify the initial state |in . In this paper we take it to be the Bunch-Davies vacuum.
Free scalar and Dirac fields
Our interests are soft gravitational effects on the local dynamics of matter fields at the subhorizon scale [10,11]. Although we cannot observe the super-horizon modes directly, it is possible that virtual gravitons of the super-horizon scale affect microscopic physics which are directly observable. We propose that observables in dS space are of this type. In some specific models, we have derived the field equation of local matter dynamics corrected by soft gravitons at the one-loop level. As the first example, let us investigate a free massless conformally coupled scalar field:
S = √ −gd 4 x − 1 2 g µν ∂ µ φ∂ ν φ − 1 12 Rφ 2 . (3.1)
We redefine the matter field asφ ≡ Ωφ, (3.2)
S = d 4 x − 1 2g µν ∂ µφ ∂ νφ − 1 12Rφ 2 . (3.3)
This variable corresponds to the canonical quantization for the conformally flat metricg µν = η µν . We believe the freedom of the tensor weight of the matter path integral is uniquely fixed this way by the requirement of unitarity.
Up to the one-loop level, the quantum equation of motion is written as
∂ µ ∂ µφ (x) − i d 4 x ′ Σ 4-pt (x, x ′ ) + Σ ++ 3-pt (x, x ′ ) − Σ +− 3-pt (x, x ′ ) φ (x ′ ) = 0, (3.4)
where the self-energies from the four-point vertices and the three-point vertices are given by
−iΣ 4-pt (x, x ′ ) = 1 2 κ 2 ∂ µ h µ ρ (x)h ρν (x) ∂ ν + 1 12 κ 2 ∂ µ ∂ ν h µ ρ (x)h ρν (x) (3.5) + 1 24 κ 2 ∂ µ h ρα (x)∂ µ h ρα (x) − 1 12 κ 2 ∂ µ h ρα (x)∂ ρ h µα (x) δ (4) (x − x ′ ), −iΣ AB 3-pt (x, x ′ ) = iκ 2 ∂ µ ∂ ′ σ (h µν ) A (x)(h ρσ ) B (x ′ ) ∂ νφA (x)∂ ′ ρφ B (x ′ ) (3.6) + i 6 κ 2 ∂ µ (h µν ) A (x)∂ ′ ρ ∂ ′ σ (h ρσ ) B (x ′ ) ∂ νφA (x)φ B (x ′ ) + i 6 κ 2 ∂ ′ σ ∂ µ ∂ ν (h µν ) A (x)(h ρσ ) B (x ′ ) φ A (x)∂ ′ ρφ B (x ′ ) + i 36 κ 2 ∂ µ ∂ ν (h µν ) A (x)∂ ′ ρ ∂ ′ σ (h ρσ ) B (x ′ ) φ A (x)φ B (x ′ ) .
Here the differential operators are applied after the step functions are assigned. This prescription corresponds with the T * product.
In (3.4), the Lorentz invariance is respected at the tree level. It is because the dS metric is conformally flat and the scalar field is conformally coupled. Even for the minimally coupled case, the Lorentz invariance holds as the effective symmetry in the sub-horizon scale since the mass term of the Hubble scale can be neglected. The Lorentz invariance is a fundamental symmetry of the microscopic physics. In investigating quantum IR effects, we should pay attention whether they respect the Lorentz invariance or not. Its possible violation could be tested stringently by experimental observations.
At the sub-horizon scale, we can neglect the derivative of the scale factor in comparison to the external momentum of the matter as
P ≫ H ⇒ a∂φ ≫ (∂a)φ, (3.7)
where P denotes the external physical momentum scale of the matter. Furthermore the higher derivative terms arise due to UV contributions and as such they are not associated with the IR logarithm. We can focus on the twice differentiate term of the equation (3.4) as the IR logarithms are concerned :
log a(τ )∂∂φ(x). (3.8) It is base on a crucial fact that only the local terms contribute to the dS symmetry breaking. We give a detailed explanation on this fact in the next section. Here we first show the result of the first principle investigation. The quantum equation of motion of a scalar field including the one-loop correction from soft gravitons is evaluated as
1 + 3κ 2 H 2 32π 2 log a(τ ) ∂ µ ∂ µφ (x) ≃ 0. (3.9)
We emphasize that the IR logarithm appears as an overall factor of the kinetic term. In this regard, soft gravitons do not spoil the Lorentz invariance at the sub-horizon scale.
Since the derivative of log a(τ ) is negligible on the local dynamics at the sub-horizon scale, we can eliminate such an overall factor by the following time dependent renormalization of a scalar field: ‡φ
→ Z φφ , Z φ ≃ 1 − 3κ 2 H 2 64π 2 log a(τ ). (3.11)
It is important to understand a physical mechanism behind this result. We may regard the soft metric fluctuations as a slowly varying background a la [18]. In such a view point the propagator forφ in the comoving momentum space behaves as
G(x, p) = 1 g µν p µ p ν = Ω −2 (x) P a P a ,(3.12)
where the physical momentum P a = e µ a (x)p µ involves the vierbein. Our idea is to identify the effective conformal mode dependence e −2κw(x) in Ω −2 (x) as the sole source of the IR logarithms. In other words, we compare the magnitude of the amplitudes for the fixed physical momentum P a . We argue that such a quantum relation is consistent with the effective Lorentz invariance. The conformal mode dependence can arise through the equivalence at the super-horizon scale 2w ∼ h 00 in (1.21). This idea certainly explains the time dependent wave function renormalization factor since e −2κw = Z 2 φ .
(3.12) implies the following UV divergences at the coincident limit ofφ propagator
φ 2 (x) ∼ d 4 p Ω −2 (x) P a P a ∼ d 4 P Ω 2 (x) P a P a ∼ Ω 2 (x)Λ 2 UV , (3.13)
where we fix the UV cut-off of P < Λ U V . Since this behavior is consistent with the general covariance, we argue that the (3.12) is required by it and hence gauge invariant. In fact this point of view holds in more generic situations as we shall explain below.
We have performed the parallel investigation for a free massless Dirac field. The corresponding action is
S = √ −gd 4 x iψe µ a γ a ∇ µ ψ,(3.14)
where e µ a is a vierbein and γ a is the gamma matrix:
γ a γ b + γ b γ a = −2η ab . (3.15)
The vierbein can be parametrized as
e µ a = Ω −1ẽµ a ,ẽ µ a = (e − κ 2 h ) µ a .
(3.16) ‡ For the minimally coupled massless scalar field, the wave function renormalization factor changes due to the additional contribution from soft matter fluctuations
Z ϕ ≃ 1 − κ 2 H 2 16π 2 log a(τ ). (3.10)
In a similar way to (3.3), we redefine the matter field such that it is canonically normalized:
ψ ≡ Ω 3 2 ψ, (3.17) S = d 4 x iψẽ µ a γ a∇ µψ . (3.18)
The quantum equation of motion of a Dirac field including the one-loop correction from soft gravitons is
1 + 3κ 2 H 2 128π 2 log a(τ ) iγ µ ∂ µψ (x) ≃ 0. (3.19)
Just like a scalar field, the IR effect from gravitons to a Dirac field preserves the Lorentz invariance. It can be eliminated by the following time dependent wave function renormalization of a Dirac field:ψ
→ Z ψψ , Z ψ ≃ 1 − 3κ 2 H 2 256π 2 log a(τ ). (3.20)
From (3.11) and (3.20), we can conclude that the IR logarithms due to soft gravitons can be eliminated in the free field theories after the wave function renormalization.
In a similar way to the conformally coupled massless scalar field, we can reproduce the time dependent wave function renormalization ofψ by identifying the conformal mode dependence in the propagator as
Ω −1 (x) iγ a P a . (3.21)
It is because e −κw = Z 2 ψ where we regard P a to be constant. This conformal mode dependence of the propagator ensures that the UV divergences are consistent with the general covariance just like the scalar field case. Thus it also follows from the the fundamental symmetry as we fix the maximum value of the physical momentum P < Λ UV .
We point out that the different choice of quantization schemes with respect to parametrization of the metric and normalization of the matter field leads to different results [15]. We can reproduce the results in the existing literatures by Woodard et al. [16,17]; Giddings and Sloth [18] after taking account of these differences. We believe there is an advantage in our prescription. It is consistent with unitarity since matter fields are canonically normalized. We suspect it is the reason that we can show IR logarithmic effects respect Lorentz invariance of the microscopic physics.
Cancellation of Non-local IR singularities
In investigating the dS symmetry breaking effects, we should distinguish the local and nonlocal IR singularities. When we integrate over the interaction vertices, the amplitudes with soft or collinear particles seem to induce IR singularities. However it is a well-known fact that such non-local IR singularities cancel after summing over degenerate states between real and virtual processes in QED and QCD [12,13].
We have found that the cancellation of the non-local IR singularities holds in quantum gravity on the dS background, at least at the one-loop level [14]. This fact indicates that only the local IR singularities contribute to the dS symmetry breaking.
As seen in (3.5), at the one-loop level, the contribution from the four point vertices consists only of the local terms. Namely the corresponding self-energy is proportional to the delta function
Σ 4-pt (x, x ′ ) ∝ δ (4) (x − x ′ ). (4.1)
In the coefficients of the delta function, the gravitational propagator at the coincident point which is left intact by differential operators h µ ρ (x)h ρν (x) induces the IR logarithm.
In contrast, the contribution from the three point vertices contains the local and the nonlocal terms. In order to extract the local terms, let us recall that the twice differentiated propagator induces the delta function. For example the four-times differentiated propagator of the matter field in Σ ++ 3-pt (x, x ′ ) induces the delta function with two differential operators:
∂ µ ∂ ν ∂ ρ ∂ σ φ + (x)φ + (x ′ ) (4.2) → − i δ 0 µ δ 0 ν ∂ ρ ∂ σ + δ 0 µ δ 0 ρ ∂ ν ∂ σ + δ 0 µ δ 0 σ ∂ ν ∂ ρ + δ 0 ν δ 0 ρ ∂ µ ∂ σ + δ 0 ν δ 0 σ ∂ µ ∂ ρ + δ 0 ρ δ 0 σ ∂ µ ∂ ν − 2(δ 0 µ δ 0 ν δ 0 ρ ∂ σ + δ 0 µ δ 0 ν δ 0 σ ∂ ρ + δ 0 µ δ 0 ρ δ 0 σ ∂ ν + δ 0 ν δ 0 ρ δ 0 σ ∂ µ )∂ 0 + 4δ 0 µ δ 0 ν δ 0 ρ δ 0 σ ∂ 2 0 + δ 0 µ δ 0 ν δ 0 ρ δ 0 σ ∂ α ∂ α δ (4) (x − x ′ )
. We should emphasize that these local terms come from the derivative of the step function and so Σ −+ 3-pt (x, x ′ ) does not contain them. In the coefficients of the local terms (4.2), the gravitational propagator at the coincident point which is left intact by differential operators induces the IR logarithm.
Next, let us investigate the non-local terms in the self-energy from the three point vertices. When we assign p as the external comoving momentum:φ p (x ′ ) ∝ e ipµx ′µ , p µ p µ = 0, the non-local term is written as the following integral:
d 4 x ′ Σ ++ 3-pt (x, x ′ ) − Σ +− 3-pt (x, x ′ ) non-localφ p (x ′ ) (4.3) = d 4 x ′ θ(τ − τ ′ ) Σ −+ 3-pt (x, x ′ ) − Σ +− 3-pt (x, x ′ ) φ p (x ′ ) = e ip·x d 3 p 1 d 3 p 2 (2π) 6 e −iǫτ τ τ i dτ ′ e i(ǫ−p)τ ′ A(p, p 1 , p 2 , τ, τ ′ )(2π) 3 δ (3) (p 1 + p 2 − p) − (h.c.).
Here p 1 and p 2 are respectively the comoving momenta of the intermediate scalar and gravitational fields. Furthermore we have introduced the total energy of intermediate particles as ǫ ≡ p 1 + p 2 . A(p, p 1 , p 2 , τ, τ ′ ) merely denotes the coefficient of the oscillating phase factor.
In the process with a soft or collinear particle, the total energy is close to p:
p 1 ∼ 0 or p 2 ∼ 0 or p 1 , p 2 p ⇒ ǫ ∼ p. (4.4)
In such a process, the contribution from the negatively large conformal time region is dominant since the frequency of the integrand vanishes. In other words, the integral (4.3) is sensitive to the initial time τ i . We call it the non-local IR singularity. Strictly speaking, the non-local IR singularity comes from the integral of
Σ −+ 3-pt (x, x ′ ) but not of Σ +− 3-pt (x, x ′ ).
By considering the off-shell effective equation of motion: p µ p µ = 0, we can avoid the non-local IR singularity. It is because the lower bound of the total energy is given by the virtuality:
ǫ 2 − (p 0 ) 2 > p µ p µ . (4.5)
In the subsequent discussion, we define a physical quantity which gives a proper interpretation into the on-shell limit of the off-shell effective equation of motion.
It is natural to conjecture that the non-local IR singularities cancel after summing over degenerate states between real and virtual processes also in dS space, like QED or QCD. In order to confirm the conjecture, we adopt the Kadanoff-Baym method [14]. The method is valid when the external momentum is at the sub-horizon scale as a particle description holds. In contrast to the investigation by the effective equation of motion, we investigate the Schwinger-Dyson equation of the two point function in this method. So we can systematically obtain the on-shell term and the off-shell term. The two point function depends on two time variables τ 1 , τ 2 . We decompose them as
τ c ≡ (τ 1 + τ 2 )/2 ≫ ∆τ ≡ τ 1 − τ 2 .
We see the following integrals in the investigation of the non-local terms by the Kadanoff-Baym method:
d 4 x ′ θ(τ 1 − τ ′ ) Σ −+ 3-pt (x 1 , x ′ ) − Σ +− 3-pt (x 1 , x ′ ) G −+ p (x ′ , x 2 ) (4.6) − d 4 x ′ θ(τ 2 − τ ′ )Σ −+ 3-pt (x 1 , x ′ ) G −+ p (x ′ , x 2 ) − G +− p (x ′ , x 2 ) .
Here we have introduced the Fourier transformation of the matter propagator:
G −+ p (x 1 , x 2 ) ∝ e −ipµ(x 1 −x 2 ) µ , p µ p µ = 0.
There is a counter part to the first integral in the effective equation of motion (4.3). After performing the integrations other than over the total energy, each integral is written as
d 4 x ′ θ(τ 1 − τ ′ ) Σ −+ 3-pt (x 1 , x ′ ) − Σ +− 3-pt (x 1 , x ′ ) non-local G −+ p (x ′ , x 2 ) (4.7) ∼ − iA ′ e ip·(x 1 −x 2 ) e −ip(τ 1 −τ 2 ) ∞ p dǫ 1 ǫ − p , − d 4 x ′ θ(τ 2 − τ ′ )Σ −+ 3-pt (x 1 , x ′ ) G −+ p (x ′ , x 2 ) − G +− p (x ′ , x 2 ) (4.8) ∼ + iA ′ e ip·(x 1 −x 2 ) ∞ p dǫ e −iǫ(τ 1 −τ 2 ) 1 ǫ − p .
In terms of the characteristic frequency of the oscillating phase factor, we call (4.7) with p the on-shell term and (4.8) with ǫ the off-shell term. The on-shell and off-shell terms have the non-local IR singularities at ǫ ∼ p whose coefficients ∓iA ′ are identical except for opposite signs. If we naively define the on-shell and off-shell terms such as (4.7), (4.8), each term has an IR divergence and the IR cut-off seems to be given by the inverse of the initial time:
p+|1/τ i | dǫ.
Physically speaking, any experiment has a finite energy resolution of observation ∆ǫ. We may divide the integration region of the off-shell term as
∞ p dǫ e −iǫ(τ 1 −τ 2 ) = ∞ p+∆ǫ dǫ e −iǫ(τ 1 −τ 2 ) + p+∆ǫ p dǫ e −iǫ(τ 1 −τ 2 ) . (4.9)
Within the energy resolution, we cannot distinguish the off-shell term from the on-shell term
p+∆ǫ p dǫ e −iǫ(τ 1 −τ 2 ) ∼ e −ip(τ 1 −τ 2 ) p+∆ǫ p dǫ. (4.10)
Thus we need to redefine the on-shell term by transferring the contribution of the off-shell term within the energy resolution p < ǫ < p + ∆ǫ:
− iA ′ e ip·(x 1 −x 2 ) e −ip(τ 1 −τ 2 ) ∞ p dǫ − p+∆ǫ p 1 ǫ − p (4.11) = − iA ′ e ip·(x 1 −x 2 ) e −ip(τ 1 −τ 2 ) ∞ p+∆ǫ dǫ 1 ǫ − p .
The remaining contribution is the well-defined off-shell term:
+iA ′ e ip·(x 1 −x 2 ) ∞ p+∆ǫ dǫ e −iǫ(τ 1 −τ 2 ) 1 ǫ − p .
(4.12)
We have found that there is no IR divergence after the redefinition. Since the energy resolution of observation is at the sub-horizon scale ∆ǫ ∼ 1/∆τ ≫ |1/τ c | > |1/τ i |, the IR cut-off is given by not the inverse of the initial time |1/τ i | but the energy resolution ∆ǫ. The nonlocal IR effect respects the dS symmetry as we fix the physical scale of the energy resolution ∆E ≡ ∆ǫH|τ c |.
Furthermore we can identify the mechanism how the cancellation takes place. Since the zero frequency process is contained in the common integrand
Σ −+ 3-pt (x 1 , x ′ )G −+ p (x ′ , x 2 )
, the integral of the non-local terms (4.6) is evaluated as
τ 1 τ i dτ ′ − τ 2 τ i dτ ′ d 3 x ′ Σ −+ 3-pt (x 1 , x ′ )G −+ p (x ′ , x 2 ). (4.13)
Manifestly the initial time dependences (dS symmetry breaking effects) are canceled.
We summarize this section. As for the non-local IR singularities which originate in the integrations over the vertices including soft or collinear particles, they cancel after summing over degenerate states between real and virtual processes like QED and QCD. Once the nonlocal terms are expressed by physical scales such as P , ∆E, they do not contribute to the dS symmetry breaking. In the coefficients of the local terms, the gravitational propagator at the coincident point breaks the dS symmetry. So we can conclude that only through the local terms, soft gravitons contribute to the dS symmetry breaking.
We have confirmed the cancellation of the non-local IR singularities in the two point function but not in the multi-point functions. In the next section, we investigate soft gravitational effects in interacting field theories. We assume that the cancellation holds also in the multipoint functions.
The assumption is reasonable since the cancellation originates from the fact that the total spectrum weight is preserved. In this regard, it may be a universal phenomenon as far as field theoretic models are consistent with unitarity.
Gauge theory, φ 4 theory and Yukawa theory
In this section, we investigate soft gravitational effects in interacting field theories. For example, let us consider the gauge theory
S gauge = √ −gd 4 x − 1 4e 2 g µρ g νσ F a µν F a ρσ + iψe µ α γ α D µ ψ , (5.1) = d 4 x − 1 4e 2g µρgνσ F a µν F a ρσ + iψẽ µ α γ αD µψ .
We have redefined the Dirac field asψ = Ω 3 2 ψ in the second line of (5.1). Here the gauge group is generic and the Dirac field could be in any representation.
Up to the one-loop level and O(log a(τ )), the bosonic part of the quantum equation of motion is written as
1 e 2 (D µF µν ) a (x) + κ 2 (h µρ ) + (x)(h νσ ) + (x) + 1 2 (h µα ) + (x)(h ρ α ) + (x) η νσ (5.2) + 1 2 η µρ (h να ) + (x)(h σ α ) + (x) 1 e 2 (D µFρσ ) a (x).
Here we have neglected the differentiated gravitational propagator which does not induce the IR logarithm. From (1.19) and (1.21), the bosonic part is evaluated as
1 e 2 1 + 3κ 2 H 2 8π 2 log a(τ ) (D µF µν ) a (x). (5.3)
The result preserves the gauge symmetry manifestly and indicates that the coupling of the gauge interaction decreases with cosmic expansion
e eff ≃ e 1 − 3κ 2 H 2 16π 2 log a(τ ) . (5.4)
It is because there is no wave function renormalization for the classical gauge field in the background gauge. The Lorentz invariance is also preserved and the velocity of light is not renormalized just like the massless scalar and Dirac fields.
We further investigate the fermionic currentψγ ν t aψ in the quantum equation of motion. Up to the one-loop level, this term is evaluated as
1 + 3κ 2 H 2 128π 2 log a(τ ) ψ (x)γ ν t aψ (x). (5.5)
We recall here that the IR logarithm due to soft gravitons modifies the kinetic term of the Dirac field. We have shown that this change of the kinetic term can be absorbed by the wave function renormalization (3.20). It should be noted that the quantum correction in (5.5) can also be absorbed by the identical wave function renormalization. We therefore conclude that the fermionic current is not renormalized by soft gravitons after the wave function renormalization in accord with the gauge invariance.
Let us move on to the investigation of soft gravitational effects in φ 4 and Yukawa theories.
Since the coupling constants are dimensionless, √ −g can be absorbed by the field redefinitioñ
φ = Ωφ,ψ = Ω 3 2 ψ δL 4 = − λ 4 4!φ 4 , (5.6) δL Y = −λ Yφψψ . (5.7)
After the wave function renormalization (3.11), (3.20), the interaction terms are renormalized as
δL 4 = − λ 4 4! Z 4 φφ 4 , (5.8) δL Y = −λ Y Z φ Z 2 ψφψψ . (5.9)
In addition to these wave function renormalization factors, soft gravitons dressing the interaction vertices modify the coupling constants. Considering the both contributions, we have found that the effective couplings of φ 4 and Yukawa interactions decrease with cosmic expansion under the influence of soft gravitons
(λ 4 ) eff ≃ λ 4 1 − 21κ 2 H 2 16π 2 log a(τ ) , (5.10) (λ Y ) eff ≃ λ Y 1 − 39κ 2 H 2 128π 2 log a(τ ) . (5.11)
As mentioned in the previous section, we have assumed that only the local terms contribute to the dS symmetry breaking also in multi-point functions. § § If the minimally coupled massless scalar is involved in the interaction, the couplings evolve as
(λ 4 ) eff ≃ λ 4 1 − 25κ 2 H 2 16π 2 log a(τ ) , (5.12) (λ Y ) eff ≃ λ Y 1 − 41κ 2 H 2 128π 2 log a(τ ) . (5.13)
Although these results are obtained through deriving the quantum equation of motion, we can simply reproduce them by using the effective propagators. Let us investigating the following products of them for the four point Green's function:
G(x, p 1 )G(y, p 2 )G(v, p 3 )G(z, p 4 ) ∼ Ω −2 (x)Ω −2 (y)Ω −2 (v)Ω −2 (z) P 2 1 P 2 2 P 2 3 P 2 4 .
(5.14)
Here we have adopted the postulate (3.12) that the propagators G(x, p) depend on the conformal mode w(x). At short distance when x ∼ y ∼ v ∼ z, the vertex correction where the gravitons are exchanged between different legs can be estimated as
6κ 2 2w(x)2w(x) 1 P 2 1 P 2 2 P 2 3 P 2 4 . (5.15)
We fix the physical momenta P 2 i when we estimate the time evolution of the amplitude. There are 6 ways to pair 2w out of 4 Green's functions. After the wave function renormalization we observe this calculation agrees with (5.10). We can also reproduce (5.11) in this way through the conformal mode dependence of the propagators (3.12) and (3.21). It is our contention that dimensionless couplings acquire time dependence due to soft gravitons. We believe that we have clarified its physical mechanism behind it in terms of the effective conformal mode dynamics of the propagators. We have observed that these behaviors are consistent with the general covariance if we fix the maximum physical momentum as the UV cut-off. As these relations follow from the general covariance, we argue that the IR logarithmic corrections are the inevitable consequence of the fundamental symmetry.
We can also estimate the IR logarithmic corrections to the gravitational couplings at the one-loop level. For a conformally flat metric g µν = a 2 η µν , the Einstein action is
6 κ 2 d 4 x −a∂ µ ∂ µ a − H 2 a 4 . (5.16)
The dS space is the classical solution of this action. Let us assume a more generic time dependent background a here. In dS space, we find minimally coupled gravitational modes. With a generic background a, these modes acquire small mass if we assume a is close to the classical solution. Such a term gives rise to IR logarithms of the one-loop effective action. In this way the IR logarithms in the one-loop effective action are estimated as
6 κ 2 d 4 x −a∂ µ ∂ µ a − 2H 2 a 4 (− 3κ 2 H 2 16π 2 log a).(5.17)
We find that the inverse of G deceases with time at the one-loop level:
1 G eff = 1 G 1 − 3κ 2 H 2 16π 2 log a(τ ) ,(5.18)
Furthermore we find that H 2 /G decreases with time as a soft gravitational effect at the one-loop level:
H 2 G eff = H 2 G 1 − 3κ 2 H 2 8π 2 log a(τ ) . (5.19)
Nevertheless the dimensionless ratio GH 2 is the relevant quantity with respect to the cosmological constant problem. From (5.18) and (5.19), we can conclude that soft graviton effects cancel in this dimensionless ratio at the one-loop level.
In fact the above corrections in (5.18) and (5.19) can be canceled out by shifting the conformal mode of the metric:
Ω → Ω{1 + 3κ 2 H 2 32π 2 log a(τ )}. (5.20)
On the other hand, we can show that the mass operators:
m 2 Ω 2φ2 , mΩψψ. (5.21)
receive no IR logarithmic corrections at the one-loop level in the Schwinger-Keldysh formalism. This fact is consistent with our postulates (3.12) and (3.21). Although the above shift removes time dependence in gravitational couplings, it in turn introduces the IR logarithms into the inertial mass
m eff = m 1 + 3κ 2 H 2 32π 2 log a(τ ) . (5.22)
As the physical Newton constant involves G eff m 2 = Gm 2 eff , we find that it increases with time as (Gm 2 ) eff = 1 + 3κ 2 H 2 16π 2 log a(τ ) , (5.23) irrespective to our renormalization prescription of the conformal mode.
Gauge dependence
It is important to investigate the gauge dependence of our obtained results. In this section, we adopt the following gauge fixing term with a parameter β:
L GF = − 1 2 a 2 F µ F µ , (6.1) F µ = β∂ ρ h ρ µ − 2β∂ µ w + 2 β h ρ µ ∂ ρ log a + 4 β w∂ µ log a.
The gauge condition (1.16) which is adopted in the preceding sections corresponds with the β = 1 case.
For an infinitesimal deformation of the gauge parameter: |β 2 − 1| ≪ 1, the gravitational propagator at the coincident point behaves as
(h µν ) + (x)(h ρσ ) + (x) → (2 − β 2 ) (h µν ) + (x)(h ρσ ) + (x) . (6.2)
Here we have evaluated it perturbatively up to the first order of the gauge deformation:
2 − β 2 = 1 − (β 2 − 1).
From this fact, we can conclude that the IR logarithmic effects respect the Lorentz invariance for a continuous β in scalar and Dirac theory with the wave function renormalization:
Z φ ≃ 1 − (2 − β 2 ) 3κ 2 H 2 64π 2 log a(τ ), (6.3) Z ψ ≃ 1 − (2 − β 2 ) 3κ 2 H 2 256π 2 log a(τ ). (6.4)
It is also the case in gauge theory since there is no wave function renromalization in a background gauge.
We also find that soft gravitons screen the gauge, quartic and Yukawa couplings in this gauge (6.1) as
e eff ≃ e 1 − (2 − β 2 ) 3κ 2 H 2 16π 2 log a(τ ) , (6.5) (λ 4 ) eff ≃ λ 4 1 − (2 − β 2 ) 21κ 2 H 2 16π 2 log a(τ ) , (6.6) (λ Y ) eff ≃ λ Y 1 − (2 − β 2 ) 39κ 2 H 2 128π 2 log a(τ ) . (6.7)
As a consequence, we confirm that these effective couplings depend on the gauge parameter.
Out of these couplings, we can form gauge independent ratios as follows We interpret our findings as follows. The time dependence of each effective coupling is gauge dependent since there is no unique way to specify the time as it depends on an observer. A sensible strategy may be to pick a particular coupling and use its time evolution as a physical time. In (6.8), the coupling of the quartic interaction has been assigned to this role. In this setting the relative scaling exponents measure the time evolution of the couplings in terms of a physical time. Although the choice of time is not unique, the relative scaling exponents are gauge independent and well defined.
(λ Y ) eff /λ Y = (λ 4 ) eff /λ 4
We can draw an analogy with 2-dimensional quantum gravity. In 2-dimensional quantum gravity, the couplings acquire nontrivial scaling dimensions due to quantum fluctuations of metric. Although the scaling dimension of each coupling is gauge dependent, the ratio of them are gauge independent [19,20]. It is because we need to pick a coupling to define a physical scale. In 4-dimensional dS space, we can reproduce the IR logarithmic corrections from the conformal mode dependence in the effective propagators: the scalar (3.12) and Dirac field (3.21). As they follow from the general covariance, we can demonstrate the gauge independence of our such predictions: Lorentz invariance and the relative evolution speed of λ Y , λ 4 . It is because the details of the propagator of the conformal mode does not matter.
In summary we argue that the gauge dependence should be canceled in physical observables such as Lorentz invariance and the ratio of the variation speeds of the couplings.
Self-tuning cosmological constant
In the preceding sections, we have argued that the dimensionless couplings become time dependent and diminish with time in dS space due to soft gravitons. The tree level Lagrangian depends on these couplings as
L(λ 4 , λ Y , e) = √ −g − 1 2 g µν D µ φD ν φ + iψe µ a γ a D µ ψ (7.1) − 1 4! λ 4 φ 4 − λ Y φψψ − 1 4e 2 g µρ g νσ F a µν F a ρσ .
The IR effects are local in such a way to make couplings time dependent:
L(λ 4 , λ Y , e) = √ −g − 1 2 g µν D µ φD ν φ + iψe µ a γ a D µ ψ (7.2) − 1 4! λ 4 (t)φ 4 − λ Y (t)φψψ − 1 4e 2 (t) g µρ g νσ F a µν F a ρσ .
Such effects may have important physical consequences. In an interacting field theory, the cosmological constant is a function of the couplings: f (λ 4 , λ Y , e; Λ UV ) where Λ UV is the ultra-violet (UV) cut-off. As the couplings evolve with time in dS space, the cosmological constant may acquire time dependence: f (λ(t), λ Y (t), e(t); Λ UV ). Here we assume that the UV cut-off Λ UV is kept fixed. Namely the degrees of freedom at the sub-horizon scale are assumed to be constant while the degrees of freedom at the super-horizon scale accumulate with an exponential cosmic expansion.
Let us consider a time evolution trajectory of the cosmological constant and the couplings. Note that the vanishing cosmological constant is the fixed point of the evolution as the couplings stay constant there. Even if we start with a theory with a positive cosmological constant, it may be attracted to the fixed point within the domain of the attraction. In this way, quantum IR effects may provide a self-tuning mechanism for the cosmological constant. We propose it as a simple solution for the cosmological constant problem. Its simplicity may underscore the relevance of non-equilibrium physics to this problem. In order to provide the existence proof of such a mechanism, we first construct a concrete model with a self-tuning cosmological constant.
We consider a conformally coupled scalar field with a quartic coupling λ 4 for simplicity. We have argued that gravitons at the super-horizon scale make λ 4 time dependent:
δL 4 = − √ −g 1 4! λ 4 (t)φ 4 . (7.3)
This interaction term contributes to the cosmological constant as a back-reaction. It is estimated by taking the vacuum expectation value of this potential term
f (λ 4 (t); Λ UV ) = 1 4! λ 4 (t) φ 4 . (7.4)
The leading contribution is quartically UV divergent
f (λ 4 (t); Λ UV ) = 1 8 λ 4 (t)( φ 2 UV ) 2 . (7.5)
By introducing a physical momentum cut-off Λ UV , we estimate the UV divergence as
φ 2 UV = Λ 2 UV 8π 2 , (7.6)
in agreement with (3.13). We stress here that the time evolution of the couplings is the inevitable consequence of this postulate.
We can cancel it by a counter term (bare cosmological constant) at an initial time. However we can no longer do so at late times
∆f (λ 4 (t); Λ UV ) = 1 8 ∆λ 4 (t)( φ 2 UV ) 2 ,(7.7)
where ∆λ(t) = λ(t) − λ. In this way the Hubble parameter becomes time dependent:
∆H 2 = κ 2 48 ( φ 2 UV ) 2 ∆λ 4 (t) = DM 2 P ∆λ 4 (t), (7.8) where the coefficient D is D = π 3 1 8π 2 Λ 2 UV M 2 P 2 . (7.9)
The essential point here is that the bare action is assumed to be generally covariant and hence dS invariant. The symmetry breaking effect comes from IR effects. With this assumption, we can estimate the time evolution of the cosmological constant even if we cannot predict its initial value.
The evolution trajectory of the cosmological constant is a straight line in (λ 4 , H 2 ) plane. This trajectory in (λ 4 , H 2 ) plane is attracted to a fixed point on the horizontal line with H 2 = 0 as λ 4 is decreased by ∆λ 4 = H 2 /(DM 2 P ). We should emphasize that this statement is gauge independent. À ¾ (7.10) Furthermore let us see the implication of this model for Dark Energy. Since H 2 /M 2 P ∼ 10 −120 at present, λ 4 changes little in this process as long as D ∼ (Λ UV /M P ) 4 is not so small. This condition is easily satisfied with Λ UV > m ν .
As seen in (5.10), the effective coupling of the quartic interaction decreases with time as
∆λ 4 = −λ 4 C H 2 M 2 P
log a(t), C = 21 π .
(7.11)
Through the effective coupling, the Hubble parameter acquires the following time dependence ∆H 2 = −λ 4 CDH 2 log a(t). (7.12)
As λ 4 changes little, we can ignore the time dependence of λ 4 in this equation. In this way, we find
H 2 ∼ a(t) −n , n = λ 4 CD. (7.13)
As is well known, n = 3 for dark matter and n = 0 for cosmological constant. The scaling index n is small n < 10 −3 as long as we keep the UV cut-off of this model Λ UV < M P . The important point is that n is non-vanishing unlike a true cosmological constant. The Hubble parameter H 2 is self-tuned to zero at late times in this model.
The theoretical issue we need to clarify here is that the coefficient C depends on a gauge parameter β as
C β = 21 π (2 − β 2 ),(7.14)
where we have evaluated it up to the first order of the gauge deformation in a similar way to (6.2). As seen in (6.6), it originates in the fact that the evolution speed of the coupling is gauge dependent:
∆λ 4 = −λ 4 C β H 2 M 2 P log a(t). (7.15)
We observe that this factor can be absorbed by the reparametrization of the scale factor up to O(β 2 − 1)
(2 − β 2 ) log a(t) ∼ 1 β 2 log a(t) = log a ′ (t ′ ). (7.16)
Of course it is due to the fact that time is an observer dependent quantity. We hence argue that such an ambiguity must be resolved by specifying the coordinates of the observer at the quantum level. In this procedure the symmetry may play an important role. For example the β = 1 case corresponds to the most symmetric space-time since the gravitational propagators possess SO(3) symmetry as seen in (1.19). Namely the scalar, vector and tensor modes of h i j are degenerated in this gauge.
We remark that use of the dimensional regularization to remove power divergences are not allowed in our rule. It is because such a subtraction does not correspond to dS invariant counter terms.
We also mention that supersymmetry (SUSY) does not remove this effect either. SUSY removes quartic divergences of the vacuum energy as long as quartic, Yukawa and gauge coupling are related in a specific way. However as seen in (5.4), (5.10) and (5.11), they evolve differently with time in dS space. So SUSY relations of the couplings are split with time and quartic divergences of the vacuum energy no longer cancel. It is also clear that quadratic divergences also remain and they split mass degeneracy among SUSY multiplets.
Splitting SUSY
In this section, we investigate SUSY splitting effect in dS space in more detail. Let us consider a Wess-Zumino model with a superpotential
W = 1 3 gΦ 3 ,(8.1)
where Φ denotes a chiral super field. In terms of component fields: complex scalar and Weyl spinor ¶ , the interaction term is expressed as
−g 2 4 |φ 2 | 2 − g 3 (φψψ + h.c.),(8.2)
where g 4 = g 3 = g.
As seen in Section 5, each coupling is screened by soft gravitons as
∆g 2 4 = − 21 16π 2 κ 2 H 2 log a(t)g 2 , (8.3) ∆g 2 3 = − 39 64π 2 κ 2 H 2 log a(t)g 2 . (8.4)
As a consequence, the UV-IR mixing effect on the Hubble parameter is evaluated as
∆H 2 = 16DM 2 P (∆g 2 4 − ∆g 2 3 ) (8.5) = − 180 π DH 2 log a(t)g 2 .
The scalar mass term is generated as
∆m 2 φ = 4 φ 2 UV (∆g 2 4 − ∆g 2 3 ) (8.6) = − 45 π EH 2 log a(t)g 2 , where E = Λ 2 UV /8π 2 M 2 P .
It is of order Hubble scale H 2 . The fermion mass term is not perturbatively generated due to the chiral symmetry.
Note that the scalar field becomes tachyonic m 2 φ < 0 if we start with a massless super multiplet. Therefore the scalar field acquires the vacuum expectation value |φ| = √ 2|m φ |/g. Since we have a Mexican hat type potential, we obtain a Nambu-Goldstone boson. The mass of a Higgs type boson is √ 2|m φ |. The fermions acquire the identical mass.
We may draw a lesson from this analysis of a simple SUSY model. In dS space, SUSY degeneracy in the mass spectrum is lifted. The massless particles acquire the mass of order Hubble scale unless they are protected by symmetries. In other words, the gauge bosons ¶ Our convention is ψψ = ǫ αβ ψ α ψ β , ǫ 12 = −ǫ 21 = 1, ǫ 11 = ǫ 22 = 0.
with gauge symmetry and fermions with chiral symmetry remain massless unless the relevant symmetry is spontaneously broken.
The origin of large structure of the Universe is argued to be generated during cosmic inflation with the potential V ∼ (10 16 GeV) 4 (r/0.1). In the inflation era, the Hubble scale is H/M P ∼ 10 −5 unless the tensor to scalar ratio r is much smaller than the current upper bound of 0.1.
It is likely that SUSY is required to make a consistent theory of quantum gravity such as string theory. However our analysis indicates that inflation splits SUSY with SUSY breaking scale of H ∼ 10 −5 M P . A possible way out to have TeV scale SUSY is to make r ratio very small as 10 −20 . We may thus conclude: If Planck observes primordial tensor modes, LHC will find no SUSY and vise versa.
It is certainly desirable to investigate this issue in a UV finite quantum gravity such as string theory. This mechanism could be investigated nonperturbatively by Minkowski version of IIB matrix model [21]. It is interesting to note that an exploratory investigation indicates the beginning of the universe with Inflation. We could also list possibly relevant questions:
• Can we understand Higgs mass m H from this perspective?
• What are the implications to inflation theory?
Diminishing cosmological constant in 2d gravity
In this section we investigate the quantum IR effects in a solvable model: 2-dimensional (2d) quantum gravity. We choose a conformal gauge g µν = e φĝ µν , (9.1) whereĝ µν is a background metric. The effective action for the conformal mode φ is the Liouville action:
−ĝd 2 x − 25 − c 96π (ĝ µν ∂ µ φ∂ ν φ + 2φR) − Λe (1+ γ 2 )φ . (9.2)
Here c denotes the central charge of the matter coupled to 2d quantum gravity. In the free field case, c counts massless scalars and fermionic fields: c = N s + N f /2. We consider the semiclassical regime: c > 25 where the metric for the conformal mode is negative and hence time-like. It is the identical feature with 4d Einstein gravity. In the above expression Λ is the cosmological constant, e (1+ γ 2 )φ is a renormalized cosmological constant operator and γ denotes the anomalous dimension.
The equation of motion with respect to φ is given by
25 − c 48π ∇ 2 φ − Λe φ = 0,(9.3)
where we put γ = 0. Furthermore the equation of motion with respect to h µν is
25 − c 48π (∇ µ φ∇ ν φ − 2∇ µ ∇ ν φ) = ∇ µ f ∇ ν f. (9.4)
where f denotes a free scalar field. The classical solution of the Liouville theory is 2d dS space:
e φc = 1 −Hτ 2 , H 2 = 24π c − 25 Λ. (9.5)
We identify cosmic time with the classical solution for the conformal mode φ c (t) = 2Ht. On the other hand if the cosmological constant can be neglected, we also have a solution with a non-trivial free matter field f :
φ c = Aτ, f c = A 25 − c 48π τ. (9.6)
where A is an arbitrary constant. This is a 2d Friedmann spacetime. This solution should go over to the 2d dS space solution when the cosmological constant becomes dominant.
To renormalize the cosmological constant operator to the leading order in 1/(c − 25), we need to consider the quantum fluctuation of the bare cosmological constant operator:
e φ ∼ e φc(t)+ 1 2 φ 2 . (9.7)
Here φ c (t) denote a classical solution while
φ 2 = − 24 c − 25 Pmax P min dP P . (9.8)
The scalar propagator is both UV and IR divergent in 2-dimension. In this integral with respect to physical momentum, we fix the UV cut-off P max while we identify the IR cut-off as P min = L/a(t). Here a(t) = e φc(t)/2 is the scale factor of the Universe and L is the initial size of the universe. In this way the quantum IR fluctuation grows as the Universe expands:
φ 2 ∼ − 12 c − 25 φ c (t) ⇒ e φ ∼ e (1− 6 c−25 )φc(t) . (9.9)
We have thus found that the effective cosmological constant diminishes as the Universe expands:
H 2 eff ∼ H 2 e − 6 c−25 φc(t) ∼ H 2 a(t) − 12 c−25 . (9.10)
The important point here is that the quantum IR effect is time dependent and hence cannot be subtracted by a dS invariant counter term.
The scaling dimension of the cosmological constant operator is
1 + γ 2 = 2 1 + 1 + 24 c−25 ∼ 1 − 6 c − 25 + · · · . (9.11)
We have reproduced the exact result to the leading order in a simple argument. The exact expression shows that the anomalous dimension γ is negative in the semiclassical regime c > 25. Therefore we can conclude that the quantum IR effects make cosmological constant diminish with time beyond perturbation theory. As the quantum IR effects in 4d dS space can be understood in terms of the conformal mode as well, this model illustrates how the couplings acquire the time dependence.
The effective Newton's constant also decreases with time
1 G eff = 1 G + c − 25 48π φ c (t). (9.12)
In 2-dimension this coupling is topological. For c < 1, (9.11) and (9.12) represent the double scaling relation between these couplings. In such a situation the both couplings grow in the IR limit and we need to fine tune them.
So far we have assumed that the matter system is at the critical point, namely conformally invariant. In a more generic situation, the central charge c is known to be a decreasing function with respect to the IR cut-off and hence time φ c (t). For example a nonlinear sigma model may develop a mass gap. In such a situation, the number of massless scalar fields decreases. This effect may enhance the magnitude of the anomalous dimension and the screening effect of the cosmological constant.
Conclusion
The dS symmetry may be broken due to growing IR effects. Since there exist massless minimally coupled modes in the metric fluctuations, such an effect may be a ubiquitous feature of quantum gravity in dS space. In particular we have found that the couplings acquire time dependence and evolve logarithmically with the scale factor of the Universe. We thus propose that the relative evolution speeds of the couplings are the physical observables in dS space.
We observe these effects can be explained by the nontrivial conformal mode dependence of the operators. We recall here the analogous renormalization takes place in 2d quantum gravity. There the scaling dimensions of the local operators and equivalently couplings to them are renormalized by the scale invariant quantum fluctuation of the metric. We have recalled that the cosmological constant is screened by IR quantum fluctuations in the semiclassical regime. We believe therefore analogous renormalization takes place in 4d dS space due to the scale invariant quantum IR fluctuations of the metric. We argue that the IR logarithmic effects originate from the conformal mode dependence in the effective propagators of the scalar and Dirac fields. In fact we have shown such a quantum effect is the inevitable consequence of the general covariance. Thus we have made a strong case for the validity of our claim when these fields are involved.
The magnitude of this effect is of O(H 2 /M 2 P ). Since this factor is very small (10 −120 ) now, it seems impossible to detect such an effect at first sight. However this factor may be compensated by UV divergences in the case of the cosmological constant which is quartically divergent with respect to the UV cut-off. If so, such an effect could lead to very significant consequences:
• Dark energy may decay with a small but finite n < 10 −3 .
• Inflation may split SUSY in the early universe.
Let us recall BPHZ renormalization procedure. We assume that the theory is renormalized at the n-th order with appropriate counter terms. In the next order, the new divergences occur as over-all divergences since sub-divergences are assumed to be subtracted. If the couplings are time independent, we can renormalize these new local divergences by covariant counter terms. In this way renormalizability implies the absence of UV-IR mixing effect and vise versa. On the other hand, we cannot do so if the couplings become time dependent. The UV-IR mixing effect could arise as the incomplete cancellation of the divergences due to time evolution of the couplings.
So UV-IR mixing effects could occur in non-equilibrium and non-renormalizable field theory. Although our reasoning is different, we agree with Polyakov that the cosmological constant problem may very well be of this nature [22].
Let us consider the density fluctuation in the cosmic microwave background (CMB) due to scalar modes. The standard argument relates it to the two point function of the minimally coupled scalar field (inflaton) at the horizon crossing as it is identified with the curvature perturbation which remains constant after the horizon crossing in the comoving gauge [23,24]. We may be able to estimate the scalar and tensor perturbations from the sub-horizon theory side by following it up to the horizon crossing. We have shown that the leading IR effects can be absorbed into the renormalization of the fields and couplings inside the cosmological horizon. We expect that this picture still holds when the space time is approximately dS such as in the inflation era. We hence believe that our results have an implication for the super-horizon soft graviton loop effect on CMB. If so, it shows that the leading IR effects on the scalar two point functions due to soft gravitons are absent as they can be renormalized away. The time evolution of the couplings modifies slow roll parameters ǫ and η in general.
The curvature and tensor perturbations are shown to stay constant after the horizon crossing in the comoving gauge at the tree level in a single field inflation. It is because they approach pure gauge. The nontrivial potential for them are forbidden due to general covariance. The issues on the higher loop corrections are reviewed in [25]. The IR logarithmic corrections influence the CMB spectrum by making the microscopic parameters of the inflation theory time dependent. We believe that it is important to understand them when we investigate microscopic physics behind inflation.
We illustrate one of such a possibility. The self-tuning effect of the cosmological constant due to UV-IR mixing could alter the classical slow roll picture. The slow roll parameters are
ǫ = 1 16π ( M P V ′ V ) 2 ∼ 4πφ 2 H 2 1 M 2 P , η = 1 8π M 2 p V ′′ V . (10.1)
We note that the slow roll parameter ǫ acts like n/2 in (7.13)
H 2 ∼ a(t) −2ǫ .(10.2)
So the quantum effects could significantly alter the classical slow roll picture when ǫ becomes as small as n. In the case of the single field inflation, n may be small as λ 4 is strongly constrained by the observation. In a generic case, n could be as large as 10 −3 since it could arise from different fields rather than the inflaton. The quantum effect alters the classical slow roll picture in general if the tensor-to-scalar ratio r is as small as 0.01. Such a parameter region will be explored soon.
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| []
|
[
"Filtered-OFDM -Enabler for Flexible Waveform in The 5th Generation Cellular Networks",
"Filtered-OFDM -Enabler for Flexible Waveform in The 5th Generation Cellular Networks"
]
| [
"Xi Zhang \nChengdu Research & Development Centre\nHuawei Technologies Co., Ltd\nPeople's Republic of China\n",
"Ming Jia [email protected] \nOttawa Research & Development Centre\nHuawei Technologies Canada Co., Ltd\nCanada Emails\n",
"Lei Chen \nChengdu Research & Development Centre\nHuawei Technologies Co., Ltd\nPeople's Republic of China\n",
"Jianglei Ma [email protected] \nOttawa Research & Development Centre\nHuawei Technologies Canada Co., Ltd\nCanada Emails\n",
"Jing Qiu \nChengdu Research & Development Centre\nHuawei Technologies Co., Ltd\nPeople's Republic of China\n"
]
| [
"Chengdu Research & Development Centre\nHuawei Technologies Co., Ltd\nPeople's Republic of China",
"Ottawa Research & Development Centre\nHuawei Technologies Canada Co., Ltd\nCanada Emails",
"Chengdu Research & Development Centre\nHuawei Technologies Co., Ltd\nPeople's Republic of China",
"Ottawa Research & Development Centre\nHuawei Technologies Canada Co., Ltd\nCanada Emails",
"Chengdu Research & Development Centre\nHuawei Technologies Co., Ltd\nPeople's Republic of China"
]
| []
| The underlying waveform has always been a shaping factor for each generation of the cellular networks, such as orthogonal frequency division multiplexing (OFDM) for the 4th generation cellular networks (4G). To meet the diversified and pronounced expectations upon the upcoming 5G cellular networks, here we present an enabler for flexible waveform configuration, named as filtered-OFDM (f-OFDM). With the conventional OFDM, a unified numerology is applied across the bandwidth provided, balancing among the channel characteristics and the service requirements, and the spectrum efficiency is limited by the compromise we made. In contrast, with f-OFDM, the assigned bandwidth is split up into several subbands, and different types of services are accommodated in different subbands with the most suitable waveform and numerology, leading to an improved spectrum utilization. After outlining the general framework of f-OFDM, several important design aspects are also discussed, including filter design and guard tone arrangement. In addition, an extensive comparison among the existing 5G waveform candidates is also included to illustrate the advantages of f-OFDM. Our simulations indicate that, in a specific scenario with four distinct types of services, f-OFDM provides up to 46% of throughput gains over the conventional OFDM scheme. | 10.1109/glocom.2014.7417854 | [
"https://arxiv.org/pdf/1508.07387v1.pdf"
]
| 5,397,004 | 1508.07387 | 018e3ba66fd8a252032f1ca9aece450376d34f09 |
Filtered-OFDM -Enabler for Flexible Waveform in The 5th Generation Cellular Networks
Xi Zhang
Chengdu Research & Development Centre
Huawei Technologies Co., Ltd
People's Republic of China
Ming Jia [email protected]
Ottawa Research & Development Centre
Huawei Technologies Canada Co., Ltd
Canada Emails
Lei Chen
Chengdu Research & Development Centre
Huawei Technologies Co., Ltd
People's Republic of China
Jianglei Ma [email protected]
Ottawa Research & Development Centre
Huawei Technologies Canada Co., Ltd
Canada Emails
Jing Qiu
Chengdu Research & Development Centre
Huawei Technologies Co., Ltd
People's Republic of China
Filtered-OFDM -Enabler for Flexible Waveform in The 5th Generation Cellular Networks
Accepted to IEEE Globecom, San Diego, CA, Dec. 2015
The underlying waveform has always been a shaping factor for each generation of the cellular networks, such as orthogonal frequency division multiplexing (OFDM) for the 4th generation cellular networks (4G). To meet the diversified and pronounced expectations upon the upcoming 5G cellular networks, here we present an enabler for flexible waveform configuration, named as filtered-OFDM (f-OFDM). With the conventional OFDM, a unified numerology is applied across the bandwidth provided, balancing among the channel characteristics and the service requirements, and the spectrum efficiency is limited by the compromise we made. In contrast, with f-OFDM, the assigned bandwidth is split up into several subbands, and different types of services are accommodated in different subbands with the most suitable waveform and numerology, leading to an improved spectrum utilization. After outlining the general framework of f-OFDM, several important design aspects are also discussed, including filter design and guard tone arrangement. In addition, an extensive comparison among the existing 5G waveform candidates is also included to illustrate the advantages of f-OFDM. Our simulations indicate that, in a specific scenario with four distinct types of services, f-OFDM provides up to 46% of throughput gains over the conventional OFDM scheme.
I. INTRODUCTION
After years of discussions across the industry and academia, the requirements and expectations for the 5th generation (5G) cellular networks have been made clear [1,2]. Whilst the millimeter wave is expected to deliver short-range high-speed radio access by tens of Gbps [2,3], the lower frequency bands (e.g., those are currently used by the 4G long-term evolution (LTE) networks) will continue to provide ubiquitous and reliable radio access, but with an improved spectrum efficiency. To this end, the air interface, especially the underlying waveform, should be revisited [4,5].
In 4G LTE networks, orthogonal frequency division multiplexing (OFDM) has served as an elegant solution to combat the frequency selectivity and to boost the spectrum efficiency [6]. Recently, it is becoming a consensus that the basic waveform of 5G should be able to offer including but not limited to: 1. Tailored services to different needs and channel characteristics, 2. Reduced out-of-band emission (OOBE), 3. Extra tolerance to time-frequency misalignment [4,5]. In terms of these new requirements, OFDM appears insufficient:
1) The recent development on information society has presented many new types of communication services with diversified performance requirements. For instance, to avoid collision among fast-moving vehicles, the design of vehicle-to-vehicle communication should be aiming at ultra low latency and ultra high reliability [7]. In this case, the OFDM numerology and frame structure of 4G LTE, chosen mainly for mobile broadband (MBB) service, which is not that sensitive to latency or reliability, seems not the best choice. Meanwhile, to provide sufficient coverage with low power consumption thus enabling internet-of-things [1], instead of OFDM, a narrow-band single-carrier waveform could be preferred. Generally speaking, requiring unified numerology across the assigned bandwidth, it is difficult for OFDM to suit the needs of different types of services and the associated channel characteristics simultaneously. 2) Despite the fact that OFDM provides a high spectrum efficiency through orthogonal frequency multiplexing, the OOBE of OFDM is still not very satisfactory. To be specific, in 4G LTE, 10% of the allocated bandwidth was reserved as guard band to give space for the signals to attenuate and thus to meet the spectrum mask. Indubitably, this has been a considerable waste of the frequency resource, which is becoming ever more precious. 3) With OFDM, the time and frequency resources are uniformly split up into many equal-sized elements to carry information. To achieve orthogonality and thus avoid inter-symbol/channel interference, stringent time and frequency alignment is required, resulting in heavy signalling for synchronization, especially for uplink transmission. Failures to establish near-perfect time-frequency alignment will lead to significant performance degradations. That is to say, OFDM requires global synchronization which comes at the price of extra signalling.
To avoid the above-mentioned limitations of OFDM and to meet the new challenges faced by 5G waveform, here, in this paper, we present a new enabler for flexible waveform, named as filtered-OFDM (f-OFDM). With subband-based splitting and filtering, independent OFDM systems (and possibly other waveforms) are closely contained in the assigned bandwidth.
In this way, f-OFDM is capable of overcoming the drawbacks of OFDM whilst retaining the advantages of it. First of all, with subband-based filtering, the requirement on global synchronization is relaxed and inter-subband asynchronous transmission can be supported. Secondly, with suitably designed filters to suppress the OOBE, the guard band consumption can be reduced to a minimum level. Thirdly, within each subband, optimized numerology can be applied to suit the needs of certain type of services. In general, the new performance requirements faced by 5G waveform can be fulfilled by f-OFDM and the overall spectrum efficiency can be improved. Among all the 5G waveform candidates [4,5], to the authors at least, f-OFDM appears as the most promising one, in terms of the overall performance, the associated complexity, and the cost and smoothness on the evolution path from 4G LTE.
II. GENERAL FRAMEWORK
To boost the data rate, it is anticipated that a larger bandwidth will be allocated to 5G, e.g., 100 -200 MHz. With f-OFDM, the assigned bandwidth will be split up into several subbands. In each subband, a conventional OFDM (and possibly other waveform) is tailored to suit the needs of certain type of service and the associated channel characteristics, e.g., with suitable subcarrier (SC) spacing, length of cyclic prefix (CP), and transmission time interval (TTI) [6], etc. Subband-based filtering is then applied to suppress the inter-subband interference, and the time-domain orthogonality between consecutive OFDM symbols in each subband is broken intentionally for a lower OOBE with negligible performance loss in other aspects. Consequently, asynchronous transmission across subbands can now be supported and global synchronization is no longer required, as opposed to the conventional OFDM. In addition, f-OFDM also provides significant reductions on the guard band consumption, leading to a more efficient spectrum utilization.
A. Transceiver Structure
The transceiver structure of f-OFDM is depicted in Fig. 1. As mentioned earlier, different OFDM systems (possibly other waveforms) with different subcarrier spacing, CP length, and TTI duration are to be contained in different subbands. In general, the subbands do not overlap with each other, and between subbands, a small number of guard tones (which is much smaller than the guard band used in 4G LTE) is left to accommodate inter-subband interference and allow for asynchronous transmission. The required number of guard tones depends on the transition region of the filters and will be discussed later. The example in Fig. 1 is mainly for the downlink of f-OFDM. As for the uplink, subband-based filtering can also be combined with DFT-spread-OFDM [8], which was used in 4G LTE networks. Here we spare this part of discussion for the sake of brevity.
B. Flexibility and Coexistence
Fig. 2 illustrates the flexibility and coexistence of waveforms enabled by f-OFDM. As can be seen, instead of a uniform distribution as employed by OFDM in 4G LTE, the time-frequency arrangement/allocation of f-OFDM is much more flexible. For instance, to provide ultra low latency and high reliability for vehicle-to-vehicle communication [7], the TTI duration is shortened while the subcarrier spacing of OFDM is enlarged, as compared with the OFDM numerology used in 4G LTE. Similarly, to enable sufficient coverage with low power consumption for internet-of-things [1], a tailored single-carrier waveform is included, with possibly a small frequency occupation (thus to increase the transmit power density and overcome the penetration loss) and a long TTI duration (exploiting the quasi-static channel for transmission reliability). In general, different waveforms can be incorporated under the framework of f-OFDM, and the time-frequency arrangement may change with time, adapting to the service requirements and channel characteristics of the time.
C. Evolution Path
With independent waveforms embedded in each subband of f-OFDM, a smooth evolution from OFDM to f-OFDM can be expected. Fig. 3 presents a tentative plan for the evolution from OFDM in 4G to f-OFDM in 5G. At the initial stage, the 10% guard band of the 4G networks will be invoked for data transmission through f-OFDM, and no change is required for the legacy 4G mobile devices. In the long run, the bandwidth allocated to 4G will be reduced but will likely continue to exist, while the remaining bandwidth and new spectrums will be allocated to 5G for a more flexible and efficient spectrum utilization enabled by f-OFDM. In this way, f-OFDM provides both backward and forward compatibility.
III. IMPORTANT DESIGN ASPECTS
Here in this section, three important design aspects of f-OFDM will be discussed.
A. Filter Design and Implementation
To enable subband-based filtering and thus enjoy the benefits promised by f-OFDM, properly designed filters are needed. In general, the filter design involves the tradeoff between the time-and frequency-domain characteristics, and is also grounded by the implementation complexity. From our experience, the energy spread in the time domain shall be contained to restrain the inter-symbol interference (ISI), and the sharpness of the transition region in the frequency domain is also worth pursuing. For the sake of complexity, it is recommended to implement the filters in the frequency domain using the overlap-save method [9], with which the benefits of fast Fourier transform (FFT) can be exploited. In addition, to achieve a flexible subband reallocation, a systematic and convenient approach that allows for online generation of filters for any given spectrum requirements is also desirable.
Two types of filters were considered in our investigation. The typical time and frequency response are depicted in Figs. 4 and 5 (with an order of 1024 and 720-KHz passband). A brief discussion about these filters are given as follows:
1) Soft-Truncated Sinc Filters: The impulse response of an ideal low pass filter is a sinc function, which is infinitely long. For practical implementation, the sinc function is soft-truncated with different window functions: 1. Hann window; 2. Root-raised-cosine (RRC) window. In this way, the impulse response of the obtained filters will fade out quickly (see Fig. 4), and thus limiting the ISI introduced between consecutive OFDM symbols. While being easy to generate, the soft-truncated sinc filters will have likely less resolvable frequency taps, e.g., those with a magnitude lower than -30 dB can be excluded from fixed-point frequency-domain implementation [9] (see Fig. 5), which might lead to extra complexity savings. 2) Equiripple Filters: Designed using the Remez exchange algorithm, with equiripple filters, the maximum error between the desired and the actual frequency response is minimized, and thus a sharper transition region can be obtained, as compared with the soft-truncated filters (see Fig. 5), which, as discussed previously, is very desirable for alleviating the inter-subband interference. However, with extremely narrow transition region, the impulse response of the equiripple filters exhibits discontinuities at the head and tail (see Fig. 4), which could be a performance-limiting factor when the operating signal-tonoise ratio (SNR) is relatively high and the ISI becomes the dominant issue, and this does not happen with softtruncated sinc filters. Moreover, the Remez exchange algorithm requires iterative optimization and thus is inconvenient for online filter generation.
While it appears as a good option to use soft-truncated sinc filters, there is still space for improvement and generalization, which requires further investigation.
B. Guard Tone Arrangement
In practice, the designed system will likely support only a finite set of subcarrier spacing, e.g., an integer or fractional copies of the standard subcarrier spacing used in 4G LTE (i.e., 15 KHz). The guard band between subbands, on the other hand, will likely be set as a plurality of the standard subcarrier spacing (i.e., N × 15 KHz), and this is why we named them as guard tones. With properly designed filters, the guard tones between subbands can be minimized to maximize the spectrum utilization. Our simulation results, which will be shown in Section V, has indicated that, with equal transmit power in adjacent subbands: 1. For low to medium modulation orders (e.g., QPSK and 16QAM), no guard tone is required between subbands; 2. For high-order modulation (e.g., 64QAM), up to two guard tones (i.e., two subcarriers in 4G LTE terminology) are needed between subbands. Even if the transmit power in adjacent subbands is lifted by 10 dB, two guard tones are still sufficient for suppressing the inter-subband interference for all the cases. Moreover, if combined with proper scheduling, e.g., putting high-order modulations away from the subband edges, the required number of guard tones can be further reduced.
C. Treatment with Filter Tails
With OFDM, the forward ISI created by multi-path can be suppressed elegantly using CP -if the length of CP exceeds the delay spread. This has been the guideline for choosing the CP duration. Now, with f-OFDM, the filters will lead to long tails in the time domain, both forward and backward. In this case, extending CP to cover the multi-path delay spread and the whole filter tails would result in a high CP overhead and thus is not a good option. Fortunately, if the subband under consideration is of a medium to large bandwidth, the mainlobe of the corresponding filter would be reasonably narrow (see Fig. 4). Hence, in most cases, no special treatment is required for the tails that came with the filters. In extreme cases, if the passband of the filters is extremely narrow (e.g., 180 KHz), the mainlobe will spread out and the tails become non-negligible. For these cases, at the transmitter, one can extend the CP to include the main lobe of the filter and then move the receiver window forward by half of the mainlobe. In this way, both the forward and backward ISI created by filtering can be sufficiently suppressed, as indicated in Fig. 6. However, this type of extended CP and receiver processing is needed only in the cases with extremely narrow subband, which rarely happen in practical systems.
IV. COMPARISON AMONG 5G WAVEFORM CANDIDATES
Except for OFDM and f-OFDM, the most discussed 5G waveform candidates include: 1. Generalized frequency division multiplexing (GFDM) [10], 2. Filter bank multi-carrier (FBMC) [11], 3. Universal filtered multi-carrier (UFMC) [12]. The motivations for these waveforms are similar to f-OFDM, i.e., to reduce OOBE or relax the requirement on synchronization. Filters are applied by all these waveforms, but with different methodology and performance. A detailed comparison among the 5G waveform candidates can be found in Table I and a brief discussion is given as follows.
1) f-OFDM vs. GFDM: The subcarriers of GFDM are arranged in close proximity and are not mutually orthogonal. To suppress inter-subcarrier interference, high-order filtering and tail biting are needed [10]. In addition, precancellation or successive interference cancellation is also required to alleviate the inter-subcarrier interference that still exists after filtering. Opposingly, the subcarriers in each subband of f-OFDM are still quasi-orthogonal, the filter length of f-OFDM is comparatively short, and no complicated pre-/post-processing is required. 2) f-OFDM vs. FBMC: In pursuit of time and frequency localization, the filter length in FBMC is typically very long (e.g., more than 3 times of the symbol duration) and thus is resource-consuming, as compared with the filters in f-OFDM. Moveover, massive antenna transmission has been recognized as the cornerstone of 5G [2]. Hence, the hardship of combining FBMC with multi-antenna transmission has limited its applications [13]. Contrarily, f-OFDM can be combined with multi-antenna transmission without any special processing. 3) f-OFDM vs. UFMC: To avoid the ISI between consecutive OFDM symbols, the filter length of UFMC is typically limited by the length of CP used in OFDM [12], with which the close-in OOBE could be unsatisfactory (see Fig. 7). In sharp contrast, by using a filter length up to half a symbol duration, f-OFDM intentionally gives up the orthogonality between consecutive OFDM symbols, in trade for a lower OOBE, and thus allowing for a minimum number of guard tones to be used. With properly designed filters (e.g., with a limited energy spread), the performance degradation resulted from increasing the filter length is almost negligible, as compared with the savings on guard band consumption. In general, f-OFDM appears as the most promising waveform contender for 5G, providing not only the advantages of OFDM: 1. Flexible frequency multiplexing, 2. Simple channel equalization, 3. Easy combination with multi-antenna transmission, but also many new benefits: 1. Tailored services to different needs, 2. Efficient spectrum utilization, 3. Low OOBE, 4. Affordable computational complexity, 5. Possibility to incorporate other waveforms, and, 6. Backward and forward compatibility.
V. SIMULATION RESULTS
Here we present two parts of simulation results. Following the 4G LTE standard, the first part includes a detailed verification of the OOBE and block error rate (BLER) of f-OFDM, corresponding to the first stage of the evolution path in Fig. 3. In the second part, a preliminary investigation of the throughput performance of f-OFDM is conducted, providing an envision into the last stage of the evolution path in Fig. 3.
A. OOBE and BLER
In this part, the 20-MHz bandwidth of 4G LTE is divided into three subbands. The 1st and 3rd have 1-MHz bandwidth, corresponds to the guard bands in LTE. The 2nd subband has 18-MHz bandwidth, corresponds to the bandwidth loaded in LTE systems. Consistent OFDM numerology is applied across all three subbands, i.e., using uniform subcarrier spacing, normal CP and no special treatment for filter tails. The 1st and 3rd subbands are loaded with 4 resource blocks (RBs) and the 2nd is loaded with 100 RBs, with different numbers of guard tones between them. A half-symbol delay is created between adjacent subbands to simulate inter-subband interference and asynchronous transmission. The modulation and coding scheme (MCS) represents the modulation order and coding rate [6]. The simulation setup is summarized in Table II, and other unlisted issues are set to follow the LTE standard. As depicted in Fig. 8, even with the utilization of the 1-MHz guard bands, the OOBE of f-OFDM is still much lower than that of OFDM. As shown in Fig. 9, with f-OFDM, the 1st subband (i.e., the guard band in LTE systems) is efficiently utilized for data transmission with a BLER that is similar to that of a single OFDM without any interference. As demonstrated in Fig. 10, even if the transmit power in the 2nd subband is lifted by 10 dB, two guard tones are still sufficient for suppressing the inter-subband interference. The BLER in the 2nd subband is similar to that of the 1st subband and thus SNR (dB) is omitted for brevity.
B. Throughput
Now we take an initial step to investigate the throughput gain of f-OFDM over OFDM. Our model employs singleantenna transmission. Four types of services are to be supported: 1. Pedestrian, 2. Urban, 3. Highway, 4. V2V. The numerology of OFDM is limited by the worst case, e.g., extended CP is used to combat rich multi-path scattering in urban environment, and 10% bandwidth is reserved as guard band. With f-OFDM, the number of guard tones is minimized and the bandwidth available is evenly split up into four subbands. In each subband, the OFDM numerology is optimized according to the channel characteristics and service requirements (See Table III). As shown in Fig. 11, significant throughput gains can be achieved in each subband and in total. The throughput gain comes from not only the savings on guard tones, but also the adaptations to the channel characteristics, i.e., reduced CP length for smaller multi-path delay spread and reduced subcarrier spacing for stronger frequency selectivity. In this toy example, the total throughput gain of f-OFDM over OFDM is up to 46%, which is very attractive.
VI. CONCLUSIONS AND FUTURE WORK
In this paper, we presented filtered-OFDM -an enabler for flexible waveform, designed to meet the expectations upon the 5G cellular networks. After outlining the general framework and methodology of f-OFDM, a detailed comparison among the 5G waveform candidates was provided to illustrate the advantages of f-OFDM. Encouraging results were observed in simulations, and to the authors at least, f-OFDM appears as the most promising 5G waveform candidate. Prototyping and field testing of f-OFDM are now in progress. Figure 11. Normalized throughput of OFDM and f-OFDM.
Figure 1 .
1Downlink transceiver structure of f-OFDM.
Figure 2 .
2Flexibility and coexistence of waveforms enabled by f-OFDM.
Figure 3 .
3Possible evolution path from OFDM in 4G LTE to f-OFDM in 5G.
Figure 4 .
4Impulse response of different filters.
Figure 5 .
5Frequency response of different filters.
Figure 6 .
6Treatment with filter tails in f-OFDM.
Figure 7 .
7Typical PSD of OFDM, GFDM, UFMC, f-OFDM and FBMC, under the same level of overhead consumption.
Figure 8 .
8PSD of OFDM and f-OFDM, with distortions from power amplifier.
Figure 9 .Figure 10 .
910BLER of f-OFDM in the 1st subband with equal transmit power in all three subbands, compared with that of a single OFDM in the 1st subband without interference. BLER of f-OFDM in the 1st subband while the transmit power in the 2nd subband is lifted by 10 dB, compared with that of a single OFDM in the 1st subband without interference.
Table I COMPARISON
IAMONG 5G WAVEFORM CANDIDATESWaveform
Filter Granularity
Typical Filter Length
Time Orthogonality
Frequency Orthogonality
OOBE
OFDM
Whole band
≤ CP length
Orthogonal
Orthogonal
Bad
GFDM
Subcarrier
Symbol duration
Non-orthogonal
Non-orthogonal
Good
FBMC
Subcarrier
= (3, 4, 5) × Symbol duration
Orthogonal in real domain
Orthogonal in real domain
Best
UFMC
Subband
= CP length
Orthogonal
Quasi-orthogonal
Good
f-OFDM
Subband
≤ 1/2 × Symbol duration
Non-orthogonal
Quasi-orthogonal
Better
Table II SIMULATION SETUP
IIParameter
Value / Description
Antenna mode
2 × 2 closed-loop beamforming
Channel model
Urban macro
Velocity
3 km/h
Power amplifier input backoff
9.6 dB
Type of filters
Hann windowed sinc filters
Order of filters
1024
Frequency (MHz)
Table III OPTIMIZED
IIIOFDM NUMEROLOGY FOR DIFFERENT SCENARIOSPedestrian Urban Highway
V2V
Channel model
EPA [14] ETU [14] EVA [14] Modified EVA
Velocity (km/h)
3
1
120
240
Subcarrier spacing (KHz)
3.75
3.75
30
60
Symbol duration (µs)
266.67
266.67
33.33
16.67
CP length (µs)
2.6
7.49
2.93
1.95
Total
Highway
OFDM
f-OFDM
+46%
+54%
Pedestrian
Urban
+33%
+34%
V2V
+83%
Throughput
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| []
|
[
"LAYERWISE SYSTEMATIC SCAN: DEEP BOLTZMANN MACHINES AND BEYOND",
"LAYERWISE SYSTEMATIC SCAN: DEEP BOLTZMANN MACHINES AND BEYOND"
]
| [
"Heng Guo ",
"ANDKaan Kara ",
"C E Zhang "
]
| []
| []
| For Markov chain Monte Carlo methods, one of the greatest discrepancies between theory and system is the scan order -while most theoretical development on the mixing time analysis deals with random updates, real-world systems are implemented with systematic scans. We bridge this gap for models that exhibit a bipartite structure, including, most notably, the Restricted/Deep Boltzmann Machine. The de facto implementation for these models scans variables in a layer-wise fashion. We show that the Gibbs sampler with a layerwise alternating scan order has its relaxation time (in terms of epochs) no larger than that of a random-update Gibbs sampler (in terms of variable updates). We also construct examples to show that this bound is asymptotically tight. Through standard inequalities, our result also implies a comparison on the mixing times. | null | [
"https://arxiv.org/pdf/1705.05154v2.pdf"
]
| 4,887,230 | 1705.05154 | 73784387eedac5926fcf7bfed6b28b107ac1f55e |
LAYERWISE SYSTEMATIC SCAN: DEEP BOLTZMANN MACHINES AND BEYOND
Heng Guo
ANDKaan Kara
C E Zhang
LAYERWISE SYSTEMATIC SCAN: DEEP BOLTZMANN MACHINES AND BEYOND
For Markov chain Monte Carlo methods, one of the greatest discrepancies between theory and system is the scan order -while most theoretical development on the mixing time analysis deals with random updates, real-world systems are implemented with systematic scans. We bridge this gap for models that exhibit a bipartite structure, including, most notably, the Restricted/Deep Boltzmann Machine. The de facto implementation for these models scans variables in a layer-wise fashion. We show that the Gibbs sampler with a layerwise alternating scan order has its relaxation time (in terms of epochs) no larger than that of a random-update Gibbs sampler (in terms of variable updates). We also construct examples to show that this bound is asymptotically tight. Through standard inequalities, our result also implies a comparison on the mixing times.
Introduction
Gibbs sampling, or the Markov chain Monte Carlo method in general, plays a central role in machine learning and have been widely implemented as the backbone algorithm for models such as Deep Boltzmann Machines (Salakhutdinov and Hinton, 2009), latent Dirichlet allocations (Blei et al., 2003), and factor graphs in general. Given a set of random variables and a target distribution π, the Gibbs sampler iteratively updates one variable at a time according to the distribution π conditioned on the values of all other variables. If the ergodicity condition is met, then the Gibbs sampler eventually converges to the target distribution.
There are two ways to choose which variable to update at the next iteration: (1) Random Update, where in each epoch (or round) one variable is picked uniformly at random with replacement; and (2) Systematic Scan, where in each epoch all variables are updated using some pre-determined order. Although most theoretical development on analyzing Gibbs sampling deals with random updates (Jerrum, 2003;Levin et al., 2009), systematic scans are prevalent in real-world implementations due to their hardware-friendly nature (cache locality for factor graphs, SIMD for Deep Boltzmann Machines, etc.). It is natural to wonder, whether using systematic scan, rather than random updates, would delay the mixing time, the number of iterations the Gibbs sampler requires to reach the target distribution.
The mixing time of these two update strategies can differ by some high polynomial factors in either directions (He et al., 2016;Roberts and Rosenthal, 2015). Even more pathological examples were constructed for non-Gibbs Markov chains such that systematic scan is not even ergodic whereas the random-update sampler is rapidly mixing (Dyer et al., 2008). Indeed, even for a system as simple as the Ising model, a comparison result remains elusive (Levin et al., 2009, Open problem 5, p. 300). As a consequence, theoretical results on rapidly mixing, such as (Bubley and Dyer, 1997;Mossel and Sly, 2013), do not readily apply to the scan algorithms used in practice.
1.1. Main results. In this paper, we bridge this gap between theory and system. We focus on bipartite distributions, in which variables can be divided into two partitions -conditioned on one of the partitions, variables from the other partition are mutually independent. This bipartite structure arises naturally in practice, including Restricted/Deep Boltzmann Machines. For a bipartite distribution, the de facto implementation is that in each epoch, we scan all variables from one of the partitions first, and then the other. We call this the alternating-scan sampler.
Note that in order to define a valid Markov chain, we have to consider systematic scans in epochs, in which all variables are updated once. Our main theorem is the following.
Theorem 1 (Main Theorem). For any bipartite distribution π, if the random-update Gibbs sampler is ergodic, then so is the alternating-scan sampler. Moreover, the relaxation time of the alternating-scan sampler (in terms of epochs) is no larger than that of the random-update one (in terms of variable updates).
The relaxation time (inverse spectral gap) measures the mixing time from a "warm" start. It is closely related to the (total variation distance) mixing time, and governs mixing times under other metrics as well (Levin et al., 2009). Through standard inequalities, Theorem 1 also implies a comparison result in terms of mixing times, Corollary 10. As we count epochs in Theorem 1, the alternating-scan sampler is implicitly slower by a factor of n, the number of variables. We also show that Theorem 1 is asymptotically tight via Example 11. Thus this implicit factor n slowdown cannot be improved in general.
More specifically, we summarize our contribution as follows.
(1) In Section 4, we establish Theorem 1. By focusing on bipartite systems, we are able to obtain much stronger result than recent studies in the more general setting (He et al., 2016). We note that standard Markov chain comparison results, such as (Diaconis and Saloff-Coste, 1993), do not seem to fit into our setting. Instead, we give a novel analysis via estimates of operator norms of certain carefully defined matrices. One key observation is to consider an artificial but equivalent variant of the alternating-scan sampler, where we insert an extra random update between updating variables from the two partitions. This does not change the algorithm since the extra random update is either redundant with the updates in the first partition or with those in the second.
(2) In Section 5, we discuss bipartite distributions that arise naturally in machine learning.
In particular, our result is a rigorous justification of the popular layer-wise scan sampler for Deep Boltzmann Machines (Salakhutdinov and Hinton, 2009). Our result also applies to other models such as Restricted Boltzmann Machines (Smolensky, 1986) and, more generally, any bipartite factor graph. (3) In Section 6, we conduct experiments to verify our theory and analyze the gap between our worst case theoretical bound and numerical evidences. We observe that in the rapidly mixing regime, the alternating-scan sampler is usually faster than the random-update one, whereas in the slow mixing regime, the alternating-scan sampler can be slower by a factor O(n). We hope these observations shed some light on more fine-grained comparison bounds in the future.
Related Work
Probably the most relevant work is the recent analysis conducted by He et al. (2016) about the impact of the scan order on the mixing time of the Gibbs sampling. They (1) constructed a variety of models in which the scan order can change the mixing time significantly in several different ways and (2) proved comparison results on the mixing time between random updates and a variant of systematic scans where "lazy" moves are allowed. In this paper, we focus on a more specific case, i.e., bipartite systems, and so our bound is stronger -in fact, our bound can be exponentially stronger when the underlying chain is torpidly mixing. Moreover, our result does not modify the standard scan algorithm.
Another related work is the recent analysis by Tosh (2016) considering the mixing time of an alternating sampler for the Restricted Boltzmann Machine (RBM). Tosh showed that, under Dobrushin-like conditions (Dobrushin, 1970), i.e., when the weights in the RBM are sufficiently small, the alternating sampler mixes rapidly. For models other than RBM, mixing time results for systematic scans are relatively rare. Known examples are usually restricted to very specific models (Diaconis and Ram, 2000) or under conditions to ensure that the correlations are sufficiently weak (Dyer et al., 2006;Hayes, 2006;Dyer et al., 2008). Typical conditions of this sort are variants of the classical Dobrushin condition (Dobrushin, 1970). See also (Blanca et al., 2018) for very recent results on analyzing the alternating scan sampler (among others) on the 2D grid under conditions of the Dobrushin-type. In contrast, our work focuses on the relative performance between random updates and systematic scan, and does not rely on Dobrushin-like conditions. In particular, our results extend to the torpid mixing regime as well as the rapid mixing one.
Our primary focus is on discrete state spaces. The scan order question has also been asked and explored in general state spaces. Despite a long line of research (Hastings, 1970;Peskun, 1973;Caracciolo et al., 1990;Liu et al., 1995;Roberts and Sahu, 1997;Roberts and Rosenthal, 1997;Tierney, 1998;Maire et al., 2014;Roberts and Rosenthal, 2015;Andrieu, 2016), to the best of our knowledge, no decisive answer is known.
Another line of related research is about the scan order in stochastic gradient descent (Recht and Ré, 2012;Shamir, 2016;Gürbüzbalaban et al., 2017). Our setting in this paper is very different and the techniques are different as well.
Preliminaries on Markov Chains
Let Ω be a discrete state space and P be a |Ω|-by-|Ω| stochastic matrix describing a (discrete time) Markov chain on Ω. The matrix P is also called the transition matrix or the kernel of the chain. Thus, P t (σ 0 , ·) is the distribution of the chain at time t starting from σ 0 . Let π(·) be a stationary distribution of P . The Markov chain defined by P is reversible (with respect to π(·)) if P satisfies the detailed balance condition:
π(σ)P (σ, τ ) = π(τ )P (τ, σ)(1)
for any σ, τ ∈ Ω. We note that in general the systematic scan sampler is not reversible. The Markov chain is called irreducible if P connects the whole state space Ω, namely, for any σ, τ ∈ Ω, there exists t such that P t (σ, τ ) > 0. It is called aperiodic if gcd{t > 0 : P t (σ, σ) > 0} = 1 for every σ ∈ Ω. We call P ergodic if it is both irreducible and aperiodic. An ergodic Markov chain converges to its unique stationary distribution (Levin et al., 2009).
The total variation distance · T V for two distributions µ and ν on Ω is defined as
µ − ν T V = max A⊂Ω |µ(A) − ν(A)| = 1 2 σ∈Ω |µ(σ) − ν(σ)| .
The mixing time T mix is defined as
T mix (P ) := min t ≥ 0 : max σ∈Ω P t (σ, ·) − π T V ≤ 1 2e ,
where the choice of the constant 1 2e is merely for convenience and is not significant (Levin et al., 2009).
When P is ergodic and reversible, the eigenvalues (ξ i ) i∈ [|Ω|] of P satisfies −1 < ξ i ≤ 1, and additionally, P f = f if and only if f is constant (see (Levin et al., 2009, Lemma 12.1)). The spectral gap of P is defined by λ(P ) := 1 − max{|ξ| : ξ is an eigenvalue of P and ξ = 1}.
The relaxation time for a reversible P is defined as
T rel (P ) := λ(P ) −1 .(3)
The relaxation time and the mixing time differ by at most a factor of log 2e π min where π min = min σ∈Ω π(σ), shown by the following theorem (see, for example, (Levin et al., 2009, Theorem 12.4 and 12.5)). In fact, the relaxation time governs mixing properties with respect to metrics other than the total variation distance as well. See (Levin et al., 2009, Chapter 12) for more details.
Theorem 2. Let P be the transition matrix of a reversible and ergodic Markov chain with the state space Ω and the stationary distribution π. Then
T rel (P ) − 1 ≤ T mix (P ) ≤ T rel (P ) log 2e π min ,
where π min = min σ∈Ω π(σ).
The factor log π −1 min is usually linear in n, the number of variables, in the context of Gibbs sampling which is our primary focus later. Theorem 2 is tight, and there is no good way of avoiding losing this log π −1 min factor in general, with the spectral method. Unfortunately, the systematic-scan sampler is not reversible, and therefore Theorem 2 does not apply. Instead, we use an extension developed by Fill (1991). For a non-reversible transition matrix P , let the multiplicative reversiblization be R(P ) := P P * , where P * is the adjoint of P defined as
P * (σ, τ ) = π(τ )P (τ, σ) π(σ) .(4)
Then R(P ) is reversible. Let the relaxation time for a (not necessarily reversible) P be
T rel (P ) := 1 1 − 1 − λ(R(P )) .(5)
In particular, if P is reversible, then (5) recovers (3) (see Proposition 4). In general, the multiplicative reversibilization mixes similarly to the original non-reversible chain. See (Fill, 1991) for more details.
The following theorem is a simple consequence of (Fill, 1991, Theorem 2.1).
Theorem 3. Let P be the transition matrix of an ergodic Markov chain with the state space Ω and the stationary distribution π. Then
T mix (P ) ≤ log 4e 2 π min T rel (P ),
where π min = min σ∈Ω π(σ).
Note that our definition of relaxation times (5) for non-reversible Markov chains yields asymptotically the same upper bound in Theorem 2.
Proof of Theorem 3. We first restate (Fill, 1991, Theorem 2.1) (note that the norm in (Fill, 1991) is twice the total variation distance):
P t (σ, ·) − π 2 T V ≤ (1 − λ(R(P ))) t π(σ) .(6)
Let λ := λ(R(P )) and T := log 4e 2 π min T rel (P ) =
1 1− √ 1−λ log 4e 2 π min .
Then it is easy to verify that T ≥ 2 λ log 2e √ π min and by (6), we have that
max σ∈Ω P T (σ, ·) − π T V ≤ (1 − λ) T /2 √ π min ≤ (1 − λ) λ −1 log 2e √ π min √ π min ≤ e − log 2e √ π min √ π min = 1 2e .
In other words,
T mix (P ) ≤ T = log 4e 2 π min T rel (P ).
3.1. Operator Norms and the Spectral Gap. We also view the transition matrix P as an operator that mapping functions to functions. More precisely, let f be a function f : Ω → R and P acting on f is defined as
P f (x) := y∈Ω P (x, y)f (y).
This is also called the Markov operator corresponding to P . We will not distinguish the matrix P from the operator P as it will be clear from the context. Note that P f (x) is the expectation of f with respect to the distribution P (x, ·). We can regard a function f as a column vector in R Ω , in which case P f is simply matrix multiplication. Recall (4) and P * is also called the adjoint operator of P . Indeed, P * is the (unique) operator that satisfies f, P g π = P * f, g π . It is easy to verify that if P satisfies the detailed balanced condition (1), then P is self-adjoint, namely P = P * . The Hilbert space L 2 (π) is given by endowing R Ω with the inner product
f, g π := x∈Ω f (x)g(x)π(x),
where f, g ∈ R Ω . In particular, the norm in L 2 (π) is given by
f π := f, f π .
The spectral gap (2) can be rewritten in terms of the operator norm of P , which is defined by
P π := max f π =0 P f π f π .
Indeed, the operator norm equals the largest eigenvalue (which is just 1 for a transition matrix P ), but we are interested in the second largest eigenvalue. Define the following operator
S π (σ, τ ) := π(τ ).(7)
It is easy to verify that S π f (σ) = f, 1 π for any σ. Thus, the only eigenvalues of S π are 0 and 1, and the eigenspace of eigenvalue 0 is {f ∈ L 2 (π) : f, 1 π = 0}. This is exactly the union of eigenspaces of P excluding the eigenvalue 1. Hence, the operator norm of P − S π equals the second largest eigenvalue of P , namely,
λ(P ) = 1 − P − S π π .(8)
The expression in (8) can be found in, for example, (Ullrich, 2014, Eq. (2.8)). In particular, using (8), we show that the definition (5) coincides with (3) when P is reversible.
Algorithm 1 Gibbs sampling with random updates Input: Starting configuration σ = σ 0 for t = 1, . . . , T mix do With probability 1/2, do nothing. Otherwise, select a variable x ∈ V uniformly at random. Set σ ← σ x,s with probability π(σ x,s ) t∈S π(σ x,t ) . end for return σ Proposition 4. Let P be the transition matrix of a reversible matrix with the stationary distribution π. Then 1 λ(P ) = 1
1 − 1 − λ(R(P )) .
Proof. Since P is reversible, P is self-adjoint, namely, P * = P . Hence (P − S π ) * = P * − S π and
(P − S π ) (P − S π ) * = (P − S π ) (P * − S π ) = P P * − P S π − S π P * + S π S π = P P * − S π ,
where we use the fact that P S π = S π P * = S π S π = S π . It implies that
1 − λ(R(P )) = R(P ) − S π π (by (8)) = P P * − S π π = (P − S π ) (P − S π ) * π = P − S π 2 π = (1 − λ(P )) 2 .
Rearranging the terms yields the claim.
Alternating Scan
In this section we describe the random update and the alternating scan sampler, and compare these two. Let V = {x 1 , . . . , x n } be a set of variables where each variable takes values from some finite set S. Let π(·) be a distribution defined on S V .
Let σ ∈ S V be a configuration, namely σ : V → S. Let σ x,s be the configuration that agrees with σ except at x, where σ x,s (x) = s for s ∈ S. In other words, for any y ∈ V ,
σ x,s (y) := σ(y) if y = x; s if y = x.
The lazy 1 Gibbs sampler is defined in Algorithm 1. Let n = |V | be the total number of variables. The transition kernel P RU (where RU stands for "random updates") of the sampler in Algorithm 1 is defined as:
P RU (σ, τ ) = 1 2n · π(σ x,s ) t∈S π(σ x,t ) if τ = σ and there are x ∈ V and s ∈ S such that τ = σ x,s ; 1/2 + x∈V 1 2n · π(σ x,σ(x) ) t∈S π(σ x,t ) if τ = σ; 0 otherwise,
where σ, τ are two configurations. It is not hard to see, for example, by checking the detailed balance condition (1), that π(·) is the stationary distribution of P RU . Note that this Markov
Algorithm 2 Alternating-scan sampler Input: Starting configuration σ = σ 0 for t = 1, . . . , T mix do for i = 1, . . . , n 1 do Set σ ← σ x i ,s with probability π(σ x i ,s ) t∈S π(σ x i ,t ) . end for for j = 1, . . . , n 2 do Set σ ← σ y j ,s with probability π(σ y j ,s ) t∈S π(σ y j ,t )
. end for end for return σ chain is lazy, i.e., it remains at its current state with probability at least 1/2. This self-loop probability is higher than 1/2 because when we update a variable there is positive probability of no change. Lazy chains are often studied in the literature because of its technical conveniences. The self-loop eliminates potential periodicity, and all eigenvalues of a lazy chain are non-negative. In the context of Gibbs sampling, these are merely artifacts of the available techniques and considering the lazy version is not really necessary (Rudolf and Ullrich, 2013;Dyer et al., 2014). Our main result actually applies to both lazy and non-lazy versions. See the remarks after the proof of Theorem 1.
Our main focus is bipartite distributions, defined next. These distributions arise naturally from bipartite factor graphs, including, most notably, Restricted Boltzmann Machines.
Definition 5. The joint distribution π(·) of random variables V = {x 1 , . . . , x n } is bipartite, if V can be partitioned into two sets V 1 and V 2 (namely V 1 ∪ V 2 = V and V 1 ∩ V 2 = ∅), such that conditioned on any assignment of variables in V 2 , all variables in V 1 are mutually independent, and vice versa.
In the following we consider a particular systematic scan sampler for bipartite distributions. For a configuration σ, let σ i := σ| V i be its projection on V i where i = 1, 2. The alternating-scan sampler is given in Algorithm 2, where n 1 = |V 1 | and n 2 = |V 2 |.
In other words, the alternating-scan sampler sequentially resamples all variables in V 1 , and then resamples all variables in V 2 . Note that since we are considering a bipartite distribution, in order to resample x i ∈ V 1 , we only need to condition on σ 2 . In other words, for any i ∈ [n 1 ], the distribution π(σ x i ,s ) t∈S π(σ x i ,t ) s∈S that we draws from depends only on σ 2 . Similarly, resampling y j ∈ V 2 only depends on σ 1 . We will denote the transition kernel of the alternating-scan sampler as P AS , where AS stands for "alternating scan".
An unusual feature of systematic-scan samplers (including the alternating-scan sampler) is that they are not reversible. Namely the detailed balance condition (1) does not in general hold. This is because updating variables x and y in order is in general different from updating y and x in order. This imposes a technical difficulty as most of the theoretical tools of analyzing these chains are not suitable for irreversible chains, such as the Dirichlet form (Diaconis and Saloff-Coste, 1993) or conductance bounds (Jerrum and Sinclair, 1993;Sinclair, 1992).
On the other hand, the scan sampler is aperiodic. Any potential state σ of the chain must be in the state space Ω. Therefore π(σ) > 0 and the probability of staying in σ is strictly positive. Moreover, if the Gibbs sampler is irreducible (namely the state space Ω is connected via single variable flips), then so is the scan sampler. This is because any single variable update can be simulated in the scan sampler, with small but strictly positive probability. Hence if the Gibbs sampler is ergodic, then so is the scan sampler.
We restate our main theorem here in formal terms.
Theorem 1. For any bipartite distribution π, if P RU is ergodic, then so is P AS . Moreover, T rel (P AS ) ≤ T rel (P RU ).
We will prove Theorem 1 next. The transition matrix of updating a particular variable x is the following
T x (σ, τ ) = π(σ x,s ) s∈S π(σ x,s ) if τ = σ x,s for some s ∈ S; 0 otherwise.(9)
Moreover, let I be the identity matrix that I(σ, τ ) = 1(σ, τ ).
Lemma 6. Let π be a bipartite distribution, and P RU , P AS , T x be defined as above. Then we have that
(1) P RU = I 2 + 1 2n x∈V T x ; (2) P AS = n 1 i=1 T x i n 2 j=1 T y j .
Proof. Note that T x is the transition matrix of resampling σ(x). For P RU , the term I 2 comes from the fact that the chain is "lazy". With the other 1/2 probability, we resample σ(x) for a uniformly chosen x ∈ V . This explains the term 1 2n x∈V T x . For P AS , we sequentially resample all variables in V 1 and then all variables in V 2 , which yields the expression.
Lemma 7. Let π be a bipartite distribution and T x be defined as above. Then we have that
(1) For any x ∈ V , T x is a self-adjoint operator and idempotent. Namely, T x = T * x and T x T x = T x .
(2) For any x ∈ V , T x π = 1.
(3) For any x, x ∈ V i where i = 1 or 2, T x and T x commute. In other words T
x T x = T x T x if x, x ∈ V i for i = 1 or 2.
Proof. For Item 1, the fact that T x is self-adjoint follows from the detailed balance condition (1). Idempotence is because updating the same vertex twice is the same as a single update. Item 2 follows from Item 1. This is because
T x π = T x T x π = T x T * x π = T x 2
π . For Item 3, suppose i = 1. Since π is bipartite, resampling x or x only depends on σ 2 . Therefore the ordering of updating x or x does not matter as they are in the same partition.
Define
P GS1 := I 2 + 1 2n 1 n 1 i=1 T x i , and P GS2 := I 2 + 1 2n 2 n 2 j=1 T y j .
Then, since n 1 + n 2 = n,
P RU = 1 n (n 1 P GS1 + n 2 P GS2 ) .(10)
Similarly, define
P AS1 := n 1 i=1 T x i , and P AS2 := n 2 j=1 T y j .
Then P AS = P AS1 P AS2 .
With this notation, Lemma 7 also implies the following.
Corollary 8. The following holds:
(1) P AS1 π ≤ 1 and P AS2 π ≤ 1.
(2) P AS1 P GS1 = P AS1 and P GS2 P AS2 = P AS2 .
Proof. For Item 1, by the submultiplicity of operator norms:
P AS1 π = n 1 i=1 T x i π ≤ n 1 i=1 T x i π = 1. (By Item 2 of Lemma 7)
The claim P AS2 π ≤ 1 follows similarly.
Item 2 follows from Item 1 and 3 of Lemma 7. We verify the first case as follows.
P AS1 P GS1 = n 1 i=1 T x i I 2 + 1 2n 1 n 1 j=1 T x j = 1 2 · n 1 i=1 T x i + 1 2n 1 · n 1 i=1 T x i n 1 j=1 T x j = 1 2 · n 1 i=1 T x i + 1 2n 1 · n 1 j=1 T x j n 1 i=1 T x i = 1 2 · n 1 i=1 T x i + 1 2n 1 · n 1 j=1 T x 1 T x 2 · · · T x j T x j · · · T xn 1 (By Item 3 of Lemma 7) = 1 2 · n 1 i=1 T x i + 1 2n 1 · n 1 j=1 n 1 i=1 T x i (By Item 1 of Lemma 7) = 1 2 · n 1 i=1 T x i + 1 2 · n 1 i=1 T x i = P AS1 .
The other case is similar.
Item 2 of Corollary 8 captures the following intuition: if we sequentially update all variables in V i for i = 1, 2, then an extra individual update either before or after does not change the distribution. Recall Eq. (5).
Lemma 9. Let π be a bipartite distribution and P RU and P AS be defined as above. Then we have that
R(P AS ) − S π π ≤ P RU − S π 2 π .
Proof. Recall (7), the definition of S π , using which it is easy to see that
P AS1 S π = S π P AS2 = S π S π = S π .(12)
Thus, P AS1 (P RU − S π )P AS2 = P AS1 n 1 n P GS1 + n 2 n P GS2 − S π P AS2 (By (10)) = n 1 n P AS1 P GS1 P AS2 + n 2 n P AS1 P GS2 P AS2 − P AS1 S π P AS2 = n 1 n P AS1 P AS2 + n 2 n P AS1 P AS2 − S π (By Item 2 of Cor 8)
= P AS1 P AS2 − S π = P AS − S π ,(13)
where in the last step we use (11). Moreover, we have that
P * AS = n 1 i=1 T x i n 2 j=1 T y j * = n 2 j=1 T * y n 2 +1−j n 1 i=1 T * x n 1 +1−i = n 2 j=1 T y n 2 +1−j n 1 i=1 T x n 1 +1−i (By Item 1 of Lemma 7) = n 2 j=1 T y j n 1 i=1 T x i (By Item 3 of Lemma 7) = P AS2 P AS1 .
Hence, similarly to (13), we have that
P AS2 (P RU − S π )P AS1 = P AS2 P AS1 − S π = P * AS − S π .(14)
Using (12), we further verify that
(P AS − S π ) (P * AS − S π ) = P AS P * AS − P AS S π − S π P * AS + S π S π = P AS P * AS − S π(15)
Combining (13), (14), and (15), we see that R(P AS ) − S π π = P AS P * AS − S π π = (P AS − S π ) (P * AS − S π ) π = P AS1 (P RU − S π ) P AS2 P AS2 (P RU − S π ) P AS1 π ≤ P AS1 π P RU − S π π P AS2 π P AS2 π P RU − S π π P AS1 π
≤ P RU − S π 2 π ,
where the first inequality is due to the submultiplicity of operator norms, and we use Item 1 of Corollary 8 in the last line.
Remark. The last inequality in the proof of Lemma 9 crucially uses the fact that the distribution is bipartite. If there are, say, k partitions, then the corresponding operators P AS1 , . . . , P ASk do not commute and the proof does not generalize.
Proof of Theorem 1. For the first part, notice that the alternating-scan sampler is aperiodic. Any possible state σ of the chain must be in the state space Ω. Therefore π(σ) > 0 and the probability of staying at σ is strictly positive. Moreover, any single variable update can be simulated in the scan sampler, with small but strictly positive probability. Hence if the random-update sampler is irreducible, then so is the scan sampler.
To show that T rel (P AS ) ≤ T rel (P RU ), we have the following T rel (P AS ) = 1 1 − 1 − λ(R(P AS )) (By (5)) = 1 1 − R(P AS ) − S π π (By (8)) ≤ 1 1 − P RU − S π π (By Lemma 9) = 1 λ(P RU ) (By (8)) = T rel (P RU ). (By (3)) This completes the proof.
Remark. It is easy to check that the proof also works if we consider the non-lazy version of P RU . To do so, we just replace I 2 + 1 2n x∈V T x with 1 n x∈V T x and the rest of the proof goes through without changes.
Remark. Our argument can also handle the case of general state spaces, such as Gaussian variables, since the essential property we use is the commutativity of updating variables from the same partition. For general state spaces, in order to apply Theorem 1 on mixing times, we need to replace Theorem 2 and Theorem 3 with their continuous counterparts. See for example (Lawler and Sokal, 1988).
Using Theorem 2 and Theorem 3, we translate Theorem 1 in terms of the mixing time.
Corollary 10. For a Markov random field defined on a bipartite graph, let P RU and P AS be the transition kernels of the random-update Gibbs sampler and the alternating-scan sampler, respectively. Then,
T mix (P AS ) ≤ log 4e 2 π min (T mix (P RU ) + 1) ,
where π min = min σ∈Ω π(σ).
Since n variables are updated in each epoch of P AS , one might hope to strengthen Theorem 1 so that nT rel (P AS ) is also no larger than T rel (P RU ). Unfortunately, this is not the case and we give an example (similar to the "two islands" example due to He et al. (2016)) where T mix (P AS ) T mix (P RU ) and T rel (P AS ) T rel (P RU ). This example implies that Theorem 1 is asymptotically tight. However, it is still possible that Corollary 10 is loose by a factor of log π −1 min . This potential looseness is difficult to circumvent due to the spectral approach we took.
Example 11. Let G = (L ∪ R, E) be a complete bipartite graph K n,n and we want to sample an uniform independent set in G. In other words, each vertex is a Boolean variable and a valid configuration is an independent set I ⊆ L ∪ R. To be an independent set in K n,n , I cannot intersect both L and R. Hence the state space is Ω = {I | I ⊆ L or I ⊆ R} and the measure π is uniform on Ω. Under single-site updates, Ω is composed of two independent copies of the Boolean hypercube {0, 1} n with the two origins identified. The random-update Gibbs sampler has mixing time O(2 n ) because the (maximum) hitting time of the Boolean hypercube is O(2 n ) and the mixing time is upper bounded by the hitting time multiplied by a constant (Levin et al., 2009, Eq. (10.24)). The relaxation time is also O(2 n ) by Theorem 2. In fact, it is not hard to see that both quantities are Θ(2 n ).
On the other hand, the alternating-scan sampler has mixing time Ω(2 n ) and relaxation time Ω(2 n ). For the mixing time, we partition the state space Ω into Ω L = {I | I ⊂ L} and Ω R = {I | I ⊂ R and I = ∅}. Consider the alternating scan projected down to Ω L and Ω R . If the current state is in Ω L , then there is 2 −n probability to go to ∅ after updating all vertices in L, and then with probability 1 − 2 −n the state goes to Ω R after updating all vertices in R. Similarly, going from Ω R to Ω L has also probability O(2 −n ). Thus in each epoch of the alternating scan, the probability to go between Ω L and Ω R is Θ(2 n ) and the mixing time is thus Θ(2 −n ). The relaxation time can be similarly bounded using a standard conductance argument (Sinclair, 1992).
In summary, for this bipartite distribution π, we have that T rel (P AS ) T rel (P RU ) and T mix (P AS ) T mix (P RU ). Therefore, Theorem 1 is asymptotically tight and Corollary 10 is tight up to the factor log π −1 min . We conjecture that the factor log π −1 min should not be in Corollary 10. However, this factor is inherently there with the spectral approach. To get rid of it a new approach is required.
We note that in Example 11, alternating scan is not necessarily the best scan order. Indeed, as shown by He et al. (2016), if we scan vertices alternatingly from the left and right, rather than scanning variables layerwise, the mixing time is smaller by a factor of n. Thus, although Theorem 1 and Corollary 10 provide certain guarantees of the alternating-scan sampler, the layerwise alternating order is not necessarily the best one.
Bipartite Distributions in Machine Learning
The results we developed so far can be applied to probabilistic graphic models with bipartite structures, most notably Restricted Boltzmann Machines (RBM) and Deep Boltzmann Machines (DBM). Although real-world systems for RBM and DBM inference rely on layerwise systematic scans, we are the first to provide a theoretical justification of such implementations.
Markov Random Fields.
A Markov random field (MRF) with binary factors G, S, π is defined on a graph G = (V, E), where each edge describes a "factor" f e and each vertex is a variable drawing from S, a set of possible values. Each factor is a function S 2 → R. A configuration σ ∈ S V is a mapping from V to S. In addition, each vertex is equipped with a factor g v : S → R. Let Ω ⊆ S V be the state space, which is usually defined by a set of hard constraints. When there is no hard constraint, the state space Ω is simply S V . The Hamiltonian of σ ∈ Ω is defined as
H(σ) = e=(u,v)∈E f e (σ(u), σ(v)) + v∈V g v (σ(v)).
The Gibbs distribution π(·) is defined as π(σ) ∝ 1(σ ∈ Ω) exp(H(σ)). These models are popularly used in applications such as image processing (Li, 2009) and natural language processing (Lafferty et al., 2001).
It is easy to check that, when the underlying graph G is bipartite, the Gibbs distribution is bipartite in the sense of Definition 5. Thus Theorem 1 and Corollary 10 apply to this setting. 5.2. Restricted/Deep Boltzmann Machines. Restricted Boltzmann Machines (RBM) was introduced by Smolensky (1986). It is a special case of the general MRF in which all variables are Boolean (i.e., S = {0, 1}) and are partitioned into two disjoint sets, V 1 and V 2 . There is a factor between each variable in V 1 and V 2 , and the Hamiltonian is
H(σ) = u∈V 1 ,v∈V 2 W uv σ(u)σ(v) + v∈V W v σ(v).
where W uv and W v are real-valued weights. Figure 1(a) illustrates the structure of RBMs. We use [f 00 , f 01 , f 10 , f 11 ] to describe a general binary factor defined on Boolean variables. Thus, [0, 0, 0, W ] denotes a standard RBM factor with weight W , and [W, 0, 0, W ] denotes an Ising model with weight W (after some renormalization).
Markov chain Monte Carlo is a common approach to perform inference for RBMs, which involves sampling a configuration from the Gibbs distribution π. The de facto algorithm for this task is Gibbs sampling, in which the conditional probability of each step can be calculated from only the Hamiltonian. In this context, the alternating-scan algorithm we study corresponds to a layerwise scan -first update all variables in V 1 and then all variables in V 2 . This scan (2009), is a Deep Learning model that extends RBM to multiple layers as illustrated in Figure 1(b). This layer structure is indeed bipartite, shown in Figure 1(c). The scan order induced is thus to update odd layers first and even ones after. Like most deep learning models, the scan (evaluation) order of variables has significant impact on the speed and performance of the system. The layerwise implementation is particularly advantageous thanks to dense linear algebra primitives.
Given an RBM or DBM with n variables, it is easy to see that log π −1 min is O(n). Thus, Corollary 10 implies that, comparing to the random-update algorithm, the layerwise systematic scan algorithm incurs at most a O(n 2 ) slowdown in the convergence rate. This comparison result improves exponentially (in the worst case) upon previous result (He et al., 2016).
Experiments
Empirically evaluating the mixing time of Markov chains is notoriously difficult. In general, it is hard under certain complexity assumptions (Bhatnagar et al., 2011) and lower bounds have been established for more concrete settings by Hsu et al. (2015) (see also (Hsu et al., 2015) for a comprehensive survey on this topic). We evaluate the mixing time in either exact and straightforward or approximate but tractable ways, including (1) calculating directly using the transition matrix for small graphs, (2) taking advantage of symmetries in the state space for medium-sized graphs, and (3) using the coupling time (defined later) as a proxy of the mixing time for large graphs.
Mixing Time on Small Graphs. We evaluate the mixing time in a brutal force way, namely, we multiply the transition matrix until the total variation distance to the stationary distribution is below the threshold. Since the state space is exponentially large, such a method is only feasible in small graphs. Figure 2 and Figure 3 contains the comparison of the mixing time for small graphs (RBMs of up to 12 variables and DBMs with 4 layers and 3 variable per layer). We vary (1) number of variables, (2) factor functions (shown as the entries of truth table in the caption), or (3) the weight of factors, in different figures and report the mixing times of random updates and layerwise scan. All solid lines count mixing time in # variable updates and the dotted line in # epochs.
We see that, empirically, alternating scan has comparable, sometimes better, mixing time than random updates, even when counting in the number of variable updates. On one hand, it confirms our result that the mixing time of alternating scan and random updates are similar.
On the other, it shows that our result, although asymptotically tight for the worst case, is not "instance optimal". This observation indicates promising future direction for beyond-worst case analysis.
Medium-sized Graphs. We now turn to Example 11, which has also been studied by He et al. (2016) and is asymptotically the worst case of Theorem 1. Due to certain symmetries, we have a much more succinct representation of the state space, and manage to calculate the mixing and relaxation times for mildly larger graphs (up to 50 variables). As illustrated in Figure 4, the alternating-scan sampler is slower than, but still comparable to the random-update sampler. This is consistent with the discussion in Example 11. Coupling Time on Large Graphs. Lastly, we use the coupling time as a proxy of the mixing time and estimate it on large graphs with 10 4 variables and 5 × 10 4 randomly chosen factors.
We use the grand coupling (Levin et al., 2009, Chapter 5). Let T σ,τ be the first time two copies of the same Markov chain meet, with initial states σ and τ , under certain coupling. Then the coupling time is max (σ,τ )∈Ω 2 T σ,τ . All of the models we tested are monotone (Peres and Winkler, 2013), in which the coupling time under the grand coupling can be easily evaluated by simulating from the top and bottom states. The coupling time is closely related to the mixing time (Levin et al., 2009, Chapter 5). In particular, it is an upper bound of the mixing time regardless of the coupling, and designing a good coupling is an important technique to prove rapid mixing (Bubley and Dyer, 1997). Our experimental findings are summarized in Figure 5.
In these experiments, we choose our parameters to stay within the rapidly mixing regime (Mossel and Sly, 2013) and avoid exponential mixing times. As we can see in Figure 5, alternating scan is faster than random updates (in terms of variable updates). Indeed, numerical evidence suggests that the speedup factor is close to 2.
Concluding Remarks
In summary, we have shown that for a bipartite distribution, the relaxation time of the alternating-scan sampler (in terms of epochs) is no larger than that of the random-update sampler. This is asymptotically tight and implies a (weaker) comparison result on the mixing time. Future directions include more fine-grained comparison results, and going beyond bipartite distributions.
Figure 1 .
1Restricted Boltzmann machines and deep Boltzmann machines as bipartite systems.order allows one to use efficient linear algebra primitives such as dense matrix multiplication implemented with GPUs or SIMD instructions on modern CPUs.Deep Boltzmann Machines (DBM), introduced by Salakhutdinov and Hinton
Figure 2 .Figure 3 .
23Mixing time of Gibbs samplers on Restricted Boltzmann Machines. Mixing time of Gibbs samplers on Deep Boltzmann Machines. See Section 6.
Figure 4 .
4Mixing time comparison on medium-sized graphs.
Figure 5 .
5Coupling time comparison on large and random graphs.
We choose to present the lazy sampler due to its popularity in theoretical analysis. Our arguments later in fact also apply to non-lazy samplers as well. See the remarks after the proof of Theorem 1.
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E-mail address. [email protected] Zurich, Switzerland E-mail address; ETH Zurich, Switzerland E-mailHeng Guo) School of Informatics, University of Edinburgh, United Kingdom ; Kaan Kara) Department of Computer Science ; Ce Zhang) Department of Computer [email protected](Heng Guo) School of Informatics, University of Edinburgh, United Kingdom. E-mail address: [email protected] (Kaan Kara) Department of Computer Science, ETH Zurich, Switzerland E-mail address: [email protected] (Ce Zhang) Department of Computer Science, ETH Zurich, Switzerland E-mail address: [email protected]
| []
|
[
"A steady state quantum classifier",
"A steady state quantum classifier"
]
| [
"Deniz Türkpençe \nDepartment of Electrical Engineering\nİstanbul Technical University\n34469İstanbulTurkey\n",
"Tahir Ç Etin Akıncı \nDepartment of Electrical Engineering\nİstanbul Technical University\n34469İstanbulTurkey\n",
"Serhat Ş Eker \nDepartment of Electrical Engineering\nİstanbul Technical University\n34469İstanbulTurkey\n"
]
| [
"Department of Electrical Engineering\nİstanbul Technical University\n34469İstanbulTurkey",
"Department of Electrical Engineering\nİstanbul Technical University\n34469İstanbulTurkey",
"Department of Electrical Engineering\nİstanbul Technical University\n34469İstanbulTurkey"
]
| []
| We report that under some specific conditions a single qubit model weakly interacting with information environments can be referred to as a quantum classifier. We exploit the additivity and the divisibility properties of the completely positive (CP) quantum dynamical maps in order to obtain an open quantum classifier. The steady state response of the system with respect to the input parameters was numerically investigated and it's found that the response of the open quantum dynamics at steady state acts non-linearly with respect to the input data parameters. We also demonstrate the linear separation of the quantum data instances that reflects the success of the functionality of the proposed model both for ideal and experimental conditions. Superconducting circuits were pointed out as the physical model to implement the theoretical model with possible imperfections. | 10.1016/j.physleta.2019.01.063 | [
"https://arxiv.org/pdf/1810.02261v2.pdf"
]
| 119,214,210 | 1810.02261 | a297f154f10076ed138d1a4173f42632c8667ebc |
A steady state quantum classifier
Deniz Türkpençe
Department of Electrical Engineering
İstanbul Technical University
34469İstanbulTurkey
Tahir Ç Etin Akıncı
Department of Electrical Engineering
İstanbul Technical University
34469İstanbulTurkey
Serhat Ş Eker
Department of Electrical Engineering
İstanbul Technical University
34469İstanbulTurkey
A steady state quantum classifier
Quantum classifierQuantum collision modelInformation reservoirSuperconducting circuits
We report that under some specific conditions a single qubit model weakly interacting with information environments can be referred to as a quantum classifier. We exploit the additivity and the divisibility properties of the completely positive (CP) quantum dynamical maps in order to obtain an open quantum classifier. The steady state response of the system with respect to the input parameters was numerically investigated and it's found that the response of the open quantum dynamics at steady state acts non-linearly with respect to the input data parameters. We also demonstrate the linear separation of the quantum data instances that reflects the success of the functionality of the proposed model both for ideal and experimental conditions. Superconducting circuits were pointed out as the physical model to implement the theoretical model with possible imperfections.
Introduction
Classification of data is of central importance to important implementations such as medical diagnosis, pattern recognition and machine learning. Due to the well-known advantages of quantum computation, studies about the quantum equivalent of machine learning algorithms have been reached to a remarkable level [1,2,3,4,5,6,7]. In contrast to the circuit model of quantum computation in which the system of interest is assumed to be perfectly isolated from the environmental degrees of freedom, one could imagine a quantum classifier as an open quantum system. This model could be referred to as a quantum data driven decision making process, as the environmental states carry information content.
Recent studies underline that the quantum reservoirs are not necessarily the rubbish bins in which the useful information is lost, but they are communication channels that they transmit information [8,9]. Moreover, the proposed quantum equivalent of learning schemes are reported to dynamically violate the unitarity even for a minimal classifier level [4]. These facts motivate us to study the possibility of the basic quantum classifiers in the context of open quantum systems in which the dynamics are non-unitary.
Email address: [email protected] () In this study, we numerically demonstrate that a steady state of a quantum unit subjected to different information environments acts as a quantum data classifier. We investigate a model that represents mixing properties of quantum dynamical maps and demonstrate that the mixture of quantum dynamical maps can be natural data classifiers under some circumstances. The influence of a dissipative environment on the reduced system dynamics is that the evolution of pure states into mixed steady states [10]. Mixed quantum states are mixtures of classical probability distributions carry no quantum signature. Therefore quantum mixed states seem useless for quantum computing implementations. However, it's possible to demonstare most of the quantum apllications by mixed state [11] or dissipative quantum computing [12].
Exploiting the quantum resources exhibit quadratic or non-linear response against linear variation of resource parameters is of importance to quantum thermodynamic or material sciences [13,14,15]. In our model, we focus on a single spin weakly coupled to information reservoirs and calculate the reduced dynamics by tracing out the environmental degrees of freedom in Markov approximation. Single spin magnetization is the figure of merit as the steady state response of the reduced dynamics. In the model, information reservoirs connected to the single spin represents the input data. We find that the steady state response of the model varies non-linearly with respect to the linear variation of input data just . The single spin was initially prepared in |+ state and contacted with a single reservoir spin-down reservoir. Also the time dependent fidelity (inset) between the qubit and the fixed reservoir state has been depicted. (f) The Bloch ball vector trajectory of the single spin during the evolution. The coupling between the each environment unit and the single spin is J = 0.1. The duration of the each unit interaction between the units and the spin is τ = 5 × 10 −2 /J. like the activation functions of the classical classifiers. The underlying physics of our model relies on the complete positivity, additivity [16] and divisibility [17,18] of quantum dynamical maps. We limit our scope with the classification analysis and leave how to perform the training and learning of the model, to an else study.
Framework and system dynamics
The simplest mathematical model for data classification is a perceptron (see Fig. 1 (a) left) that predicts an output for a weighted summation of an input data set depending on an activation function. The input dataset (the features) x 1 , x 2 . . . x N are any measurable individuals with their corresponding adjustable weights w 1 , w 2 , . . . w N and the linear summation y = i x i w i is inserted into an activation function f (y) that returns an output prediction [19]. Figure 1 (b) depicts a few of commonly used activations functions. For instance, a step function yields f (y) = 1 if y = i x i w i ≥ 0 and yields f (y) = −1 else. After these results, if a line correctly separates the data instances (as in Fig. 1 (a) right), this corresponds to a properly functioning perceptron. One can choose activation functions either with linear or non-linear responses but non-linear functions are appealing for multi-layer neural network applications.
There are various reports for quantum models of perceptrons or neural networks [1,3,4,5] relying on the advantages of quantum computing. Generally these schemes require computational resources proportional with the number of input instances to mimic the activation functions [4]. Our scheme is an open quantum system and we transform the dissipative processes into an advantage for data classification. The input data are the quantum information units characterized by qubits which are refered to as information reservoirs [20,21,22]. A qubit is parametrized by polar and azimuthal angles as |ψ (θ, φ) = cos θ 2 | ↑ + e iφ sin θ 2 | ↓ in the well-known Bloch sphere representation. Throughout of our study we take φ = 0 fixed and parametrize 'quantum features' by θ. In the calculations, we use radians and degrees interchangeably. We present our classifier as a model in which a single spin is weakly coupled to different reservoirs carrying information content. We adopt a repeated interaction process to model the open quantum dynamics [23]. Repeated interactions that are also known as collision models have became very popular recently due to their flexibility to choose the associated reservoir states and find applications to model non-Markovian [24] as well as Markovian and quantum correlated reservoirs [25].
As depicted in Fig. 1 (d), initially prepared identi-cal ancillas {R n } sequentially collides with the system S with equal duration τ. It's assumed that initially, system plus reservoir SR state is in a product state (0) = S (0) ⊗ R where S (0) = |+ +| and R = |ψ θ ψ θ |. We choose the initial system states as |+ = (| ↑ + | ↓ )/ √ 2 in order to provide a null magnetization initially. In this study, we use standard collision model in which the ancillas do not interact each other, hence the open system evolution is Markovian. The collisions between the system qubit and the each ancilla are described by unitary propagators U SR n = e −iH SRn τ where the reduced Planck constant was set = 1 throughout the manuscript. H SR is the time-independent reservoir ancilla plus system Hamiltonian where
H SR n = h 2 (σ n z + σ s z ) + J(σ n + σ s − + h.c.).(1)
Here, σ n z and σ n ± are the Pauli matrices acting on the n th ancilla of the reservoir, σ s ± are the Pauli matrices acting on the system qubit, J is the coupling between the system and the n th ancilla and h is the characteristic frequencies of the system and each ancilla.
The defined interactions above, give rise to a dynamical map such that
Φ SR [ ] = U SR 0 SR U † SR (2)
where U SR is composed of cascaded applications of SR. In our Markovian scheme, the system of interest evolves into a state identical to the state of the ancillas after sufficient number of collisions. This discrete dynamical process is called quantum homogenization [26], that is, the system reaches a steady state as n S =Tr n U SR n . . .
Tr 1 [U SR 1 0 S ⊗ R 1 U † SR 1 ] ⊗ . . . . . . ⊗ R n U † SR n(3)
for sufficiently large number of collisions n where Tr i is the partial trace over i th ancilla. The above cascaded dynamical maps can also be presented as
n S = E n • E n−1 • . . . • E 1 ≡ E n [ 0 S ](4)where E i [ S ] = Tr i [U SR i S ⊗ R i U † SR i ].
Note that each map preserves the density matrix properties such as trace unity and complete positivity, that is, each dynamical map written sequentially above is a completely positive trace preserving (CPTP) dynamical map. If a map satisfying Φ t+s = Φ t • Φ s is CP for all t and s ≥ 0 then it is a CP divisible map [17]. Therefore, in this manuscript the standard collision model in which the ancillas are identical and independent, clearly corresponds to CP divisible maps. Moreover it's been reported that a collision model can effectively simulate a Markov master equation ∂ t = L t [ ] as long as it holds the condition of CP divisibility [18].
As a benchmark calculation, we contact the single spin to a data reservoir in the ρ π = | ↓ ↓ | fixed quantum state and apply the above formulation as in Fig. 1 (d). We observe that the time time evolution of spin magnetization converges to σ z (t) = −1 as the spin density matrix approaches to the unit fidelity
F (t) = Tr √ ρ π S (t)
√ ρ π = 1 with the fixed reservoir state monotonically. Fig. 1 (e) illustrates the Bloch vector trajectory during the evolution in terms of the statistics of typical observables. By these numerical results, one concludes that our standard repeated quantum interaction process (collision model) can faithfully simulate the CP divisibility and open quantum dynamics in Markov approximation.
Results
Theoretical model
In this subsection we present the theoretical modelling of the proposed classifier without accounting for the imperfections or the physical decay mechanisms. The objective is to demonstrate that a small quantum system weakly in contact with different quantum environments can be used for classifying the data in which the environments contain. By 'small', it's implied that the system is small enough to be equilibrated toward a steady value in the long term limit [10,27]. To this end, the system of interest is weakly coupled to multiple reservoirs as in Fig. 1 (c). In this scheme, the dynamical evolution can be presented as the mixture of CP divisible dynamical maps
Φ n = q 1 Φ 1 n + q 2 Φ 2 n + . . . + q N Φ N n(5)
by considering their linear convex combinations. Here, q i ≥ 0 and N i q i = 1. Eq. (5) is the mathematical description of the implementation of the proposed open quantum classifier in contact with N reservoirs. It's known that Eq. (5) can also be presented by a master equation composed of weighted combination of effective generators
∂ ∂t = P 1 L (1) t + . . . + P N L (N) t(6)
again depending on the condition that each generator holds the CP divisibility [18] and weak coupling to the reservoirs [16]. Here, P i are the probabilities of the system experiencing from the i th environment. As mentioned above, the steady state magnetization σ z ss = Tr[σ z ss ] is evaluated as the steady state response of the system for the classification process.
Since the system of interest is only a single qubit, the steady state can be defined as a mixed state ss =
i p i Π θ i where Π θ i = |θ i θ i | are rank-one projectors stands for orthogonal basis states |θ i with θ i = 0, π.
Here, the corresponding steady state probabilities of the two-level system can be simply referred to as p e and p g for θ i = 0 and θ i = π respectively. At this final state the classification emerges with class 1 if σ z ss = p e −p g ≥ 0 and with class 2 else, depending on the states of the quantum reservoirs and the weighted couplings of the system to the reservoirs.
Before demonstrating the classification process, we show some results presenting the steady state dynam-ics. In contrast to Fig. 1 (c) for simplicity, we choose only two information reservoirs with states |ψ θ 1 and |ψ θ 2 connected to our single qubit system by dipolar J 1 and J 2 couplings. We consider two cases in our calculations; first the reservoir states are fixed (and orthogonal) and the couplings are varied. Second, the couplings are fixed (and equal) and the reservoir states are varied. Fig. 2 presents the results for these two cases. In Fig. 2 (a) the single qubit with initial |+ state is connected to the two reservoirs with fixed |ψ θ=0 ≡ | ↑ and |ψ θ=π ≡ | ↓ states with corresponding J 1 and J 2 couplings respectively. In this case, the evolution of the qubit magnetization toward steady state is depicted with respect to the variation of the J couplings. In the latter case, the variation of the two reservoir states are parametrized by θ qubit azimuthal angle. As in Fig. 2 (b) at the initial steps of the evolution for θ i 0 or θ i π, highly oscillatory behaviour is evident due to the corresponding nonequilibrium reservoir states. However, in both Fig. 2 (a) and (b) a steady state spin magnetization has been observed after sufficient number of collisions. Note that the couplings are weak and all the conditions for CP divisibility are fulfilled during the evolutions.
After we confirm that our scheme representing the open quantum dynamics is capable of obtaining the steady states, next we examine the steady state response of the system under linear variation of the input parameters. Again, we consider the two distinct cases; one with the fixed reservoir parameters and the other one with the fixed coupling parameters and again choose spin magnetization as a steady state identifier. Fig. 3 (a) presents the first case in which the steady state magnetization depicted against δJ which is a factor governs the variation of the couplings such as J 1 = J/2 + δJ, J 2 = J/2 − δJ to the | ↑ and | ↓ reservoirs respectively. For instance, when δJ = J/2; J 1 = J and J 2 = 0, that is, σ z ss = +1 since the system is coupled only to the first reservoir. As obvious in the figure, the the steady state response of the system is not linear against the overall variation of δJ and exhibits an activation function-like behaviour such as one of the plots of Fig. 1 (b).
In the latter case, the couplings are equal and fixed and the steady response of the system is investigated for different reservoir states defined by the geometrical qubit parameters. In this case, the preferred parameter is the Bloch ball azimuthal angle θ to define to information reservoir states |ψ θ . Fig. 3 (b) depicts the steady response of the system with respect to the variation of one of the two reservoir states while the other one is fixed. Here, the steady response was plotted against φ = π − θ where θ is the sum of the two qubit an- gles θ representing the reservoir states. For instance, according to the calculations, the steady state of our single qubit system with equal dipolar couplings to the two reservoirs with orthogonal |ψ θ=0 and |ψ θ=π states is a maximally mixed qubit state with zero magnetization. Hence, σ z ss = 0 corresponds to φ = π − (0 + π) = 0. Likewise, one obtains the same conditions and maximally mixed qubit state in Fig. 3 (a) when δJ = 0. Fig. 3 (c) is another way of presenting Fig. 3 (b) with more steady state points as explained in the caption. Nonlinear response of open quantum systems was reported for finite temperature quantum reservoirs [28], however temperature is not relevant to our study concerning the information reservoirs.
Finally, we illustrate the classification of the input parameters as an examination of the functionality of the quantum classifier. Figs. 4 (a) and (b) shows that the proposed quantum classifier in the present manuscript is able to linearly separate the input instances composed of the parameters denoting the reservoir states and the couplings to the reservoirs. Bloch sphere representation is very illustrative to represent any two level quantum system. In principle, any point on the sphere is a valid quantum data represented by a pure state. It's well-known that geometrical Bloch sphere representation is parametrized by azimuthal and polar angles θ and φ. The θ parameter governs the variation of the qubit state in terms of two orthogonal states while φ denotes the variation of coherent superposition states. As the classification decision is encoded in the steady state of the classifier in which contains no coherence, we chose to parametrize the initial quantum data of the input channels by θ. As the classical learning algorithms are based on the modification of the weights of the data, visualizing the classification of data in the weight space is quite frequent in classical neuro-computing. Likewise, coupling coefficient (J) of the system qubit to the relevant reservoir is the quantum analogue of the classical weights. Therefore we also choose to present the classification plots in the J space.
By this simple demonstration, it's shown that open dynamics of a single qubit is capable of processing input data in the steady state limit. Though the demonstration is limited to two inputs, extension to arbitrary number of inputs is straightforward due to the convexity of the dynamical maps. As another interesting result, the steady state non-linear response of the system against the linear variation of input parameters encourages one to expand the study toward multi-layer extension of the proposed quantum classifier.
Three input channels
The proposal in which we stress that a single qubit is a binary classifier in the steady state limit was demonstrated for two information reservoirs in the preceding subsection. In the mathematical model, the qubit always returns a binary decision regardless of the number of reservoirs acting as the input information channels as implied in Eqs. (5) and (6). Though generalizing the proposal to larger number of input channels are straightforward due to the additivity of the quantum dynamical maps and the convexity of the density matrix, nevertheless, we give an example for three reservoir states as input information channels for further analysis as depicted in Fig. 5.
Before analysing the three channel character alone, we compare the dynamics with the two channel one. A speed up is obvious on the equilibration dynamics of the classifier for three input channels comparing with the two channel input case. One can observe this qualitatively in Fig. 5 (a) as the spin magnetization curve saturation for three input case takes place before the two input case. Recent reports support this result by analytical expressions [29,30]. That is, unlike the intuitive expectations, the classifier is faster as the new input channels are introduced. This result becomes quite important when the classifier is considered with the realistic parameters.
Figs. 5 (b) and (d) exhibit that the classifier converges to the linear combination of the three reservoir states pointed out in the plot. In this specific example, the coupling strengths of the input channels are equal and the system qubit state reaches the unit fidelity where the target state is the linear combination of the given reservoir states with equal probabilities confirming that Eqs. (5) and (6) applies. More specifically, the steady state of the system qubit is m = p 1 |ψ θ 1 ψ θ 1 | + p 2 |ψ θ 2 ψ θ 2 | + p 3 |ψ θ 3 ψ θ 3 | where N i=1 p i = 1. The probabilities experiencing from each channel are equal p 1 = p 2 = p 3 = 1/3 as the coupling of the system to the channels J 1 = J 2 = J 3 were set equal. This state specifies a steady magnetization, that is, the classifier returns a binary decision for the three channel input case. Moreover, beyond the dynamical analysis of the classification process for two specific three input states, we also performed calculations for random input channel triple states parametrized by geometrical qubit angles θ. The results are visualized as three dimensional parameter space θ i where each triples of θ are generated randomly between 0 and π with mean 0 and variance 1. As clear in Fig. 5 (f) the data instances are linearly separable and the classifier operates properly also in the three input case. In general, for N input channels the steady state of the classifier is m = N i=1 |ψ θ i ψ θ i | where steady state spin polarization σ z ss = Tr[ m σ z ] ss always returns a binary decision.
Physical model
In this subsection, we propose a physical model for the implementation of the quantum classifier. We choose the superconducting circuits [31] as the physical model represents the theoretical example contains a single qubit in contact with two reservoir (ancilla) qubits. Again, the reservoirs are modelled by a repeatedinteraction scheme and the physical qubits are the transmon qubits that interact through a resonator bus [32] in which also serves for qubit readout [33].
In general, the Hamiltonian of N transmon qubits coupled via a coplanar waveguide (CPW) resonator reads
H =ω râ †â + N i=1 E c i (n i − n g i ) 2 − E J i cosφ i + N i=1 g ini (â +â † )(7)
where ω r is the resonator frequency which is in essence, a quantum harmonic oscillator,â andâ † are, respectively, the lowering and raising operators of the oscillator. Transmon qubit is a developed version of a charge qubit (Cooper pair box) based on the Josephson junction tunnelling device [34]. The second term in the Hamiltonian describes the charge qubits wherê n i is the charge quanta number operator, n g i is the offset charge andφ i the quantized flux of qubit i. Here, ϕ i = πΦ i /Φ 0 where Φ i is the tunable magnetic flux of each qubit and Φ 0 is the elementary flux quanta. The capacitive energies E c i and the Josephson energies E J i of the qubits are set E J i E c i so that the qubits operate in the transmon regime in which they capacitively couple to the resonator by g i . As clear in Eq. (7) the desired interaction between qubits does not appear in the form as in Eq. (1). However, this type of interaction can directly be achieved by coupling the transmon qubits to the same resonator dispersively such as |∆ 1,2,3 | = |ω 1,2,3 − ω r | g 1,2,3 . In this scheme, the effective interaction between the qubits are, for instance Q 1 (the system qubit) and one of the ancilla qubits Q 2 , described by [35,36]
J 1,2 = g 1 g 2 2 1 ∆ 1 + 1 ∆ 2(8)
in which the interaction is achieved via virtual exchange of cavity photons. Note that when |ω 1 − ω 2 | J 1,2 the interaction is effectively turned off, that is, the coupling strength can effectively be controlled by tunning the transmon qubit frequencies. Figure 6: (Colour online) Representation of the physical model of the quantum classifier by its Lumped-element circuit diagram. Three transmon qubits (Q 1 stands for the system qubit and Q 2 and Q 3 stand for the reservoir qubits) coupled to the superconducting CPW resonator that serves for both the readout of the qubits (red) and as a coupling bus. Green dots represent the flux tunability of each qubit that allow for the control of coupling to the bus via qubit frequencies. Control fields represented by Microwave lines (blue) acting on reservoir qubits (Q 2 , Q 3 ) are used for resetting and initialization of reservoir qubit states.
There are some specific requirements to implement the proposed classifier by means of the physical system expressed in Fig. 5. First, the qubits that mimic the reservoirs Q 2 and Q 3 , should interact with the system qubit Q 1 and should never interact each other. Second, a successive switch on/off mechanism should be achieved between the interacting qubits in accordance to suitable qubit state preparation and reset scenarios. The first requirement can be easily achieved by tuning the reservoir qubit frequencies |ω 2 − ω 3 | J 2,3 largely dispersive. Therefore, one obtains an effective Hamiltonian
H = ω i 2 3 i=1 σ i z + (ω r + χ i 3 i=1 σ i z )â †â + J 1,i i=2,3 (σ + 1 σ − i + H.c.)(9)
where σ i z and σ ∓ i are the Pauli operators acting on the subspace representing the first two levels of the ith superconducting qubit. Here, χ i are the qubit-dependent resonator frequency shift where there is no energy exchange between dispersively coupled qubit-resonator pairs. However, the second requirement should be evaluated by care, taking the realistic parameters into account. The physical implementation of the repeated interaction model, that is, the realization of the switch on/off mechanism between the reservoir qubits and the resonator can be performed by the externally tunable magnetic flux Φ i as shown in the caption of Fig. 5. [37]. The qubit-CPW coupling can be switched off by detuning the the qubit with the resonator very largely by using the flux bias and the coupling can be reproduced by again tuning Φ i so that the desired dispersive coupling is achieved.
In our scheme, the qubit-CPW coupling is switched on and off by repetitive steps. The time elapsed between two switching instants t i and t i+1 is T ≤ t i+1 − t i where T = τ int + τ r + τ pr . Here, we have several time scales where τ r is the relaxation time of the qubit, τ int is the qubit-CPW interaction time and τ pr represents both, qubit reset and preparation times. In the scenario, the system transmon qubit Q 1 and the reservoir transmon qubits Q 2 , Q 3 couple to a CPW resonator dispersively with strength g that generates effective J 1,2 and J 1,3 couplings between Q 1 − Q 2 and Q 1 − Q 3 as discussed above. The reservoir transmon qubits are initially assumed to be prepared in their reservoir states before the switchon interaction and Q 1 is prepared in any state such that σ z (0) = 0. At time t i = 0 the coupling between Q 1 , Q 2 ,Q 3 and CPW is switched on. After τ int the couplings are switched off and Q 1 and Q 3 are reset to their initial reservoir states after an elapsed τ pr time. Hence, the system qubit is decoupled from the ancilla qubits and ready for the next time in the tensor product state. For reservoir qubits the relaxation time τ r is much longer than the interaction time τ r τ int therefore has no effect on the reservoir qubits between any successive reset times. On the other hand, through a strong field, the qubit reset and preparation time τ pr is much shorter than the interaction time τ pr τ int [37]. Then approximately, the relevant time scale between two successive switch-on operations is t i+1 − t i = T τ int . Many repetitions of the task described above, in principle can successfully simulate the proposed classifier model. However, there are some limitations on the achievement of the physical model. Possible preparation defects of the identical reservoir states should be taken into account. For a better performance analysis of the proposed physical model, one should encounter the experimental parameters to the calculations with the phenomenological decay rates. A comprehensive analysis of the CPW-qubits system can be carried out by a master equation approach with realistic parameters [38].
Here, we repeat the calculations of the proposed classifier with two reservoir qubits (expressed in Section 3.1) taking the reservoir qubit state preparation errors into account in order to see how the system is robust against errors. Typically, for superconductor circuit experiments in the weak coupling regime, the resonator frequency is ω r ∼ 1 − 10 GHz and the qubit resonator coupling is g ∼ 1 − 500 MHz [31,35]. We choose the resonator frequency ω r = 8.625 GHz and the Q 1 , Q 2 , Q 3 qubit frequencies as, respectively, ω 1 /2π = 6.2 GHz, ω 2 /2π = 4.052 GHz and ω 3 /2π = 7.518 GHz where we obtain the effective couplings between the system qubit and the reservoir qubits as J 12 = J 13 = 48.9 MHz by Eq. (8). We also choose the reservoir-system interaction time τ int = 5 ns. The reservoir qubit states were prepared as
= (1 − η )|ψ θ ψ θ | + η 2 1(10)
before any interaction where 1 is the single qubit identity operator, |ψ θ is the perfect reservoir state parametrized by Bloch angle θ and η is the random error parameter. Here, η = ± η and the random parameter η models the errors of the identical reservoir state preparation. The reservoir states can be prepared by single qubit rotations. Current state-of-theart allows for high fidelity logic gates by using single and two qubit rotations. For instance, two-qubit gates with 0.999−0.996 fidelity corresponding to an infidelity = 0.001 − 0.004 were achieved for different types of multi-transmon qubits [39,40].
As shown in Fig. 7 (a)-(b) for relatively large error values = 0.01 or = 0.1 the physical machine can properly classify the data instances. However, for the exaggerated error values of , (0.4 or 0.6 as in Fig. 7 (c)-(d)) the instances become inseparable. In these calculations we observe that the steady states were reached after 1500 − 2000 collisions supporting the results in Fig. 2. Note that, as the interaction time is τ int = 5 ns for each collision, the required time to reach the steady state is ∼ 7.5 − 10µs. As the spin polarization of the system qubit Q 1 is the recognizer of the proposed classification function of our study, energy relaxation time T 1 is relevant to the classifier performance. That is, T 1 of the system qubit should be larger than ∼ 10 µs, the physical classifier response time for the proper functionality of the physical model. Recent studies report that energy relaxation time ranges between T 1 ∼ 20 − 60 µs depending on the coupling and the optimal noise suppressing pulse shape techniques of the transmon qubits [38,40].
Beyond its feasibility, the physical model has advantageous features regarding its speed and the resource requirements. First, depending on the current computer CPU capabilities, one can analyse that just like the typical CPU operations, the classical binary classifiers operate in the ms time range [41]. As stressed above, the quantum classifier can process information in the µs range, that is, the proposed physical model is three orders of magnitude faster than its classical analogues. Second, note that the proposed classifier process quantum information and recently proposed quantum classifiers rely on the circuit model of the universal quantum computing [4,5] in which requires multi-qubit output registers for multi-qubit inputs. However, the proposed quantum perceptron model achieves the binary classification task only by a single qubit output regardless of the number of input channels. Therefore, the proposed quantum classifier in which has the speed superiority in comparison to the classical classifiers, is advantageous also in terms of using the resources in comparison to the other quantum classifiers.
Finally, we would like to mention the possibility of the physical model to implement on the IBM quantum computer. Currently IBM builds a universal quantum computer using a superconductor circuit architecture composed of transmon qubits through IBM Quantum Experience project (IBMQX) [42]. Just like proposed classifier, IBMQX architecture depends on the microwave transmon qubits coupled via CPW resonators. However, the proposed repeated-interaction scheme with frequency tunable transmon qubits, can not be directly applied to IBM architecture as they couple the qubits by exploiting cross-resonance effect in which the qubit frequencies are fixed [43]. But note that universal quantum computers, in principle, can simulate open quantum dynamics through repeated algorithms [44,45]. This type of universal quantum simulation which is the basic motivation of the quantum computers, is called digital quantum simulation [46,47,48]. The limitation of this scheme is that the quantum digi-tal simulation is an approximation of the actual system Hamiltonian and the cost to pay the approximation is the Hamiltonian decomposition errors [49,50].
Conclusion
We propose a general and simple open quantum model for quantum data classification. Repeated interactions were chosen to model open system dynamics for faithful representation of arbitrary states of information reservoirs. It's numerically demonstrated that a single qubit quantum system weakly coupled to quantum reservoirs with quantum information is capable of classifying the input data. Two limit cases with fixed reservoir states and fixed couplings were examined in order to clearly demonstrate the equilibrium state response of the system.
We show that the steady state response of the system has a non-linear activation function-like behaviour for both linear variation of coupling and reservoir state parameters. Three input channel example was also demonstrated. A possible application of the proposed model by superconducting circuits was discussed in detail and the physical performance of the classifier was examined regarding the defects. Considering the experimental parameters, we conjectured that the physical model operates three orders of magnitude faster than the classical counterparts. Thus, we have shown that an open quantum system, which is generally considered to have no useful information in its equilibrated state, is a natural quantum data classifier.
Acknowledgements
Authors acknowledge support from Istanbul Technical University. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Figure 1 :
1(Colour online.) A general view of the proposed methods. (a) A classical perceptron with N inputs. (b) A few of the activation functions for perceptrons. (c) The scheme of the proposed quantum classifier. A single spin is weakly coupled to n number of reservoirs carrying quantum information. (d) The collision model to simulate the open quantum system dynamics. (e) Time evolution of the single spin magnetization depending on the number of collisions (nc)
Figure 2 :
2(Colour online.) Mixing quantum dynamical maps and evolution of single spin towards steady state by two reservoirs depending on the number of collisions (nc). (a) The state of two reservoirs are fixed and | ↑ and | ↓ respectively. The evolution of spin magnetization depending on different couplings to reservoirs depicted. (b) The coupling of the spin to the reservoirs are fixed, equal and J = 0.1. The evolution of spin magnetization depending on different reservoir (qubit) states are depicted. The duration of each interaction between the ancillas and the system qubit is τ = 5 × 10 −2 /J.
Figure 3 :
3(Colour online.) The steady state response of the system depending on the linear variation of the input parameters. (a) The variation of the steady state magnetization of the system qubit depending on the J 1 = J/2 + δJ and J 2 = J/2 − δJ coupling coefficients where δJ is a fraction of J with J = 0.1. The state of the two reservoirs are fixed and | ↑ and | ↓ respectively. (b) The variation of the steady state magnetization of the system qubit coupled to the two environments carrying different information contents parametrized by θ. In the figure, six curves are plotted with each point representing the steady state magnetization during the presence of two |θ 1 and |θ 2 environmental states. Three of the curves (up-right) stand for the three different fixed states of the first environment represented by three azimuthal angles θ 1 = 30 o , 60 o , 90 o (in degrees). Each of these curves are composed of 19 points representing the variation of the state of the second environment parametrized by θ 2 = 0 o , 10 o . . . 180 o . Likewise, the remaining three of the curves (down-left) stand for the three fixed states of the second environment represented by θ 2 = 120 o , 150 o , 180 o which are composed of 19 points representing the variation of the state of the first environment parametrized by θ 1 = 0 o , 10 o . . . 180 o . Coupling of the system to the reservoirs are fixed, equal and J 1 = J 2 = 0.1. (c) The variation of the steady state magnetization of the system qubit coupled to the two environments carrying different information contents parametrized by θ. There are 19 × 19 = 361 plotted dots with each point representing the steady state magnetization during the presence of the first and the second environments, each represented the θ = 0 o , 10 o . . . 180 o azimuthal angles. Coupling of the system to the reservoirs are fixed, equal and J 1 = J 2 = 0.1. The magnetization plotted against φ = π − (θ 1 + θ 2 ) in both (b) and (c) for convenient scaling as explained in the text.
Figure 4 :
4(Colour online) Classification by the response of the steady state magnetization of the system qubit for two dimensional inputs. (a) Classification for the case 1 set up in which the reservoir states are fixed. There are 24 coupling coefficient pairs and the classified instances are linearly separable. (b) Classification for the case 2 set up in which the couplings to the reservoirs are fixed. There are 42 pairs of θ angles denoting the reservoir states and the classified instances are linearly separable.
Figure 5 :
5(Colour online) Analysis of the classifier with two and three input information channels. Evolution of the spin magnetization of the system qubit for (a) two and three reservoirs and only for 3 reservoirs (c),(e) for different input states depending on the number of collisions presented. In (a), the evolution presented with 2 and 3 input reservoirs with | ↑ and | ↓ states. For two input case, the states are | ↑ and | ↓ with corresponding couplings, respectively, J = 0.1 and J = 0.075 and for three input case the reservoir states are | ↑ , | ↑ and | ↓ with corresponding equal couplings J = 0.1. In (b)-(f) three reservoirs were considered with (b), (c); | ↑ , | ↑ , | ↓ states and (d), (e); | ↑ , | ↓ , | ↓ states. The time dependent fidelities of the system qubit was calculated (b), (d) where the target state ρ m denotes the mixture or the linear combination of the reservoir states. The system qubit magnetization and the corresponding diagonal elements were also plotted (c,e) during the evolution. (f) Three dimensional visualization of the classified instances as the steady state response of the system qubit corresponding to three reservoir states denoted by θ. There are 42 triples of θ angles generated randomly between 0 and π with mean 0 and variance 1. (b)-(f) Coupling of the system to the reservoirs are fixed, equal and J 1 = J 2 = J 3 = 0.1
Figure 7 :
7(Colour online) Deformation of the classified instances obtained by the physical model depending on the reservoir state preparation errors. Corresponding error rates are (a) = 0.01, (b) = 0.1, (c) = 0.4 and (d) = 0.6 with corresponding random rates η = /4. Remained parameters are given in the text.
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U L Heras, A Mezzacapo, L Lamata, S Filipp, A Wallraff, E Solano, Digital Quantum Simulation of Spin Systems in Superconducting Circuits. 112U. L. Heras, A. Mezzacapo, L. Lamata, S. Filipp, A. Wallraff, E. Solano, Digital Quantum Simulation of Spin Systems in Su- perconducting Circuits, Physical Review Letters 112 (2014) 200501.
Y Salathé, M Mondal, M Oppliger, J Heinsoo, P Kurpiers, A Potočnik, A Mezzacapo, U Heras, L Lamata, E Solano, S Filipp, A Wallraff, Digital Quantum Simulation of Spin Models with Circuit Quantum Electrodynamics. 210275Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, A. Wallraff, Digital Quantum Simulation of Spin Models with Circuit Quantum Electrodynamics, Physical Review X 5 (2015) 021027.
On the Product of Semi-Groups of Operators. H F Trotter, Proceedings of the American Mathematical Society. 10H. F. Trotter, On the Product of Semi-Groups of Operators, Proceedings of the American Mathematical Society 10 (1959) 545-551.
Optimal Trotterization in universal quantum simulators under faulty control. G C Knee, W J Munro, Physical Review A. 9152327G. C. Knee, W. J. Munro, Optimal Trotterization in universal quantum simulators under faulty control, Physical Review A 91 (2015) 052327.
| []
|
[
"On hyperballeans of bounded geometry",
"On hyperballeans of bounded geometry"
]
| [
"Igor Protasov ",
"Ksenia Protasova "
]
| []
| []
| A ballean (or coarse structure) is a set endowed with some family of subsets, the balls, is such a way that balleans with corresponding morphisms can be considered as asymptotic counterparts of uniform topological spaces. For a ballean B on a set X, the hyperballean B ♭ is a ballean naturally defined on the set X ♭ of all bounded subsets of X. We describe all balleans with hyperballeans of bounded geometry and analyze the structure of these hyperballeans.MSC 54E35, 51F99 | 10.1007/s40879-018-0236-y | [
"https://arxiv.org/pdf/1702.07941v1.pdf"
]
| 119,692,160 | 1702.07941 | c67d651c8de81c8e377c6d9987b9158397e032d2 |
On hyperballeans of bounded geometry
25 Feb 2017 April 25, 2018
Igor Protasov
Ksenia Protasova
On hyperballeans of bounded geometry
25 Feb 2017 April 25, 2018balleanhyperballeancoarse equivalencebounded geometryCantor macrocube
A ballean (or coarse structure) is a set endowed with some family of subsets, the balls, is such a way that balleans with corresponding morphisms can be considered as asymptotic counterparts of uniform topological spaces. For a ballean B on a set X, the hyperballean B ♭ is a ballean naturally defined on the set X ♭ of all bounded subsets of X. We describe all balleans with hyperballeans of bounded geometry and analyze the structure of these hyperballeans.MSC 54E35, 51F99
• for any x, y ∈ X, there exists α ∈ P such that y ∈ B(x, α).
We note that a ballean can be considered as an asymptotic counterpart of a uniform space, and could be defined [9] in terms of entourages of the diagonal ∆ X in X × X. In this case a ballean is called a coarse structure. For categorical look at the ballean and coarse structures as "two faces of the same coin" see [4].
Let B = (X, P, B), B ′ = (X ′ , P ′ , B ′ ) be balleans. A mapping f : X −→ X ′ is called coarse if, for every α ∈ P , there exists α′ ∈ P ′ such that f (B(x, α)) ⊆ B ′ (f (x), α ′ ).
A bijection f : X −→ X ′ is called an asymorphism between B and B ′ if f and f −1 are coarse mappings. In this case B and B ′ are called asymorphic. If X = X ′ and the identity mapping id : X −→ X ′ is an asymorphism, we identify B and B ′ , and write B = B ′ . Given any ballean B = (X, P, B), replacing each ball B(x, α) to B(x, α) ∩ B * (x, α), we get the same ballean, so in what follows we suppose that B(x, α) = B * (x, α).
Let B = (X, P, B) be a ballean. Each non-empty subset Y of X defines a subballean B Y = (X, P, B Y ), where B Y (y, α) = Y ∩ B(y, α). A subset Y is called large if X = B(Y, α)
for some α ∈ P . Two balleans B and B ′ with the support X and X ′ are called coarsely equivalent if there exist large subsets Y ⊆ X and Y ′ ⊆ X ′ such that the balleans B Y and B ′ Y ′ are asymorphic. For a ballean B = (X, P, B), a subset Y of X is called bounded if there exist x ∈ X and α ∈ P such that Y ⊆ B(x, α). A ballean B is called bounded if the support X is bounded. Each bounded ballean is coarsely equivalent to a ballean whose support is a singletone. Now we are ready to introduce the main subject of the note. For a ballean B = (X, P, B), we denote by X ♭ the family of all non-empty bounded subsets of, consider the ballean B ♭ = (X ♭ , P, B ♭ ), where
B ♭ (Y, α) = {Z ∈ X ♭ : Z ⊆ B(Y, α), Y ⊆ B(Z, α)},
and say that B ♭ is the hyperballean of B.
For α ∈ P , a subset S of X is called α-discrete if B(x, α) ∩ S = {x} for each x ∈ S. We say that B is of bounded geometry if there exist α ∈ P and a function f : P −→ N such that if S is an α-discrete subset of a ball B(x, β) then | δ |≤ f (β). A ballean B is called uniformly locally finite if, for every β ∈ P , there is n(β) ∈ N such that | B(x, β) |≤ n(β) for every x ∈ X. By [6], B is of bounded geometry if and only if there exists large subset Y of X such that B Y is uniformly locally finite.
It should be mentioned that the notion of bounded geometry went from asymptotic topology where metric spaces of bounded geometry play the central part [5]. For interrelations between balleans of bounded geometry and G-spaces see [6].
Every metric space (X, d) defines the metric ballean (X,
R + , B α ), where B d (x, r) = {y ∈ X : d(x, y) ≤ r}. A ballean B is called metrizable if B
is asymorphic to some metric ballean. By [ 8, Theorem 2.1.1], for a ballean B, the following statements are equivalent: B is metrizable, B is coarsely equivalent to some metrizable ballean, the set P has a countable confinal subset S. We recall S is confinal if, for every β ∈ P there is α ∈ S such that α > β. Here α > β means that B(x, β) ⊆ B(x, α) for each x ∈ X. Applying this criterion, we conclude that, for every metrizable ballean B, the hyperballean B ♭ is metrizable.
Results
For a non-empty set X and the family F X of all finite subsets of X, we denote by F X the ballean (X,
F X , B F ) where B F (x, F ) = {x} if x / ∈ F ; F if x ∈ F . Then F ♭ X = {F X \{∅}, F X , B ♭ F }, where B ♭ F (H, F ) = {H} if H ∩ F = ∅ and B F (H, F ) = {(H\F ) ∪ Z : Z ⊆ H, Z = ∅} otherwise.
The ballean F ω , ω = {0, 1 . . .} is metrizable (say, by the metric d(m, n) = |2 m − 2 n |), so F ♭ ω is also metrizable (say, by the Hausdorff metric ♭ H ). At the end of the note, we point out some more explicit metrization of F ♭ ω .
Theorem 2.1. For an unbounded ballean B = (X, P, B), the following statements hold:
(i) B ♭ is uniformly locally finite if and only if B = F X ; (ii) B ♭ is of bounded geometry if and only if there exists a large subset Y of X such that B Y = F Y .
For a cardinal κ, we denote by Q κ the ballean with the support
Q κ = {(x α ) α<κ : x α ∈ {0, 1}, x α = 0
for all but finitely many α < κ}, the set of radii F κ and the balls
B Q ((x α ) α<κ , F ) = {(y α ) α<κ : x α = y α for all α ∈ κ \ F }.
The ballean Q ω is known as the Cantor macrocube and sometimes is denoted by 2 <ω or 2 <N . For characterization of balleans coarsely equivalent to the Cantor macrocube see [3]. In [1], 2 <κ denotes the ballean of all {0, 1} κ-sequences (x α ) α<κ such that |{α < κ : x α = 1}| < κ.
A ballean B = (X, P, B) is called asymptotically scattered if, for every unbounded subset Y of X, there is α ∈ P , such that, for every β ∈ P , there exists y ∈ Y such that (B(y, β) \ B(y, α)) Y = ∅.
For asymptotically scattered subbaleans of group balleans see [2].
For a ballean B = (X, P, B), the subset Y, Z of X are called close if there exists α ∈ P such that Y ⊆ B(Z, α), Z ⊆ B(Y, α). (i) the subbalean of F ♭ κ with the support [κ] n is asymptotically scattered;
(ii) the subbalean of F ♭ κ with the support {F ∈ F κ : x ∈ F } is asymorphic to Q κ ;
(iii) F ♭ ω can be partitioned into countably many pairwise close Cantor macrocubes but F ♭ ω is not coarsely equivalent to Q ω .
At the end of the note, we describe some explicit asymorphic embedding of F ♭ ω into Q ω .
Proofs
Proof of Theorem 1.2. (i) By the definition of balls in F ♭ ω , F ♭ ω is uniformly locally finite.
If the identity mapping id : X −→ X is not an asymorphism between B and F X then we can choose α ∈ P and a sequence (x n ) n<ω in X such that |B(x n , α)| > 1 and B(x i , α) B(x j , α) = ∅ for all i < j < ω. For each i < ω, we pick y i ∈ B(x i , α), y i = x i , put X n = {x 0 , . . . , x n }, X n,i = X n {y i }, i ≤ n < ω. Then X n,i ∈ B ♭ (X n , α), so |B ♭ (X n , α)| > n and B is not uniformly locally finite.
(ii) We assume that Y is a large subset of X and choose β ∈ P such that B(Y, β) = X. For each x ∈ X, we pick y x ∈ Y such that
y x ∈ B(x, β). If F ∈ X ♭ then {y x : x ∈ F } ∈ Y ♭ and F ∈ B ♭ ({y κ : x ∈ F }, β). It follows that B ♭ (Y ♭ , β) = X ♭ , Y ♭ is large in X ♭ so B ♭
Y and B ♭ are coarsely equivalent. In particular, if B Y = F Y , we conclude that B is of bounded geometry.
We suppose that B ♭ is of bounded geometry and let α ∈ P and f : P −→ N witness this property. Using Zorn's lemma, we choose a maximal by inclusion subset Y of X such that B(y, α) B(y ′ , α) = ∅ for all distinct y, y ′ ∈ Y . We show that B Y = F Y .
If the identity mapping id : Y −→ Y is not an asymorphism between B Y and F Y then there are β ∈ P and a sequence (y n ) n∈ω in Y such that |B Y (y n , β)| > 1 and B Y (y i , β) B Y (y j , β) = ∅ for all i < j < ω. For each i < ω, we pick z i ∈ B Y (y i , β), z i = y i , put Y n = {y 0 , . . . , y n }, Y n,i = Y n {y i }, i ≤ n < ω. Then Y n,i ∈ B ♭ (Y n , β) and the set {Y n,i : i ≤ n} is α-discrete. Thus, for n > f (β) we get a contradiction with the choice of α and f . ✷ Proof of Theorem 2.2. (i) We say that a subset of a ballean is asymptotically scattered if corresponding subballean has this property. We use the following observation: the union of two asymptotically scattered subsets is asymptotically scattered (see [2]).
We note that every unbounded subset in F ♭ κ is infinite and proceed on induction by n. For n = 1, the statement is evident: given any H ∈ F κ and an infinite subset Y of [κ] 1 , we take {y} ∈ Y , y ∈ H and get B F ({y}, H) = {y}.
Assuming that the statement is true for [κ] n , let Y be an infinite subset of [κ] n+1 . For each F ∈ [κ] n+1 , we denote by min F and max F , the minimal and maximal elements of F with respect to the ordinal ordering of κ and consider two cases.
Case: the set {min F : F ∈ Y } is infinite. We take an arbitrary H ∈ F κ and choose F ∈ Y such that max H < min F . Then B ♭ F (F, H) = {F }. Case: the set {min F : F ∈ Y } is finite, {min F : F ∈ Y } = x 1 , . . . , x n . For each i ∈ {1, . . . , n}, we denote Z i = {F ∈ [κ] n+1 : x i ∈ F }. We note that Z i is asymorphic to [κ] n and, by the inductive assumption, Z i is asymptotically scattered.
Then Z 1 . . . Z n is asymptotically scattered, Y ⊆ Z 1 . . . Z n and we can use definition of asymptotically scattered subsets to choose α ∈ F X suitable for Y .
(ii) We use the standard bijection χ : F κ −→ Q κ defined by χ(K) = (x α ) α<κ , where x α = 1 if and only if α ∈ K. Then the restriction of χ to {F ∈ F κ } : x ∈ F is a asymorphic embedding. Indeed, to verity this property we may use as radii in F ♭ only balls containing x. Clearly, χ{F ∈ F κ : κ ∈ F } is asymorphic to Q κ . ✷ (iii) For every n ∈ ω, let M n = {F ∈ F X : min F = n}. Applying (ii), we see that M n is asymorphic to Q ω . We take arbitrary i, j ∈ ω, denote m = max{i, j}, I m = {0, . . . , m}.
Then M i ⊆ B ♭ F (M j , I m ), M j ⊆ B ♭ F (M i , I m ) so M i , M j are close.
Given H ∈ F ω , we take F ∈ F ω such that max H < min F . Then
B ♭ F (F, H) = {F }.
In terminology of [3], it means that F ♭ ω has an asymptotically isolated balls but every ballean coarsely equivalent to Q ω has no isolated balls. ✷ To embed asymorphically F ♭ ω into Q ω , we use 2N in place of ω. We define a mapping f : F 2N {∅} −→ Q ω by the f (K) = (x n ) n<ω , where x n = 1 if and only if n ∈ {min K − 1} K. We note that the set S = f (F 2N \ {∅}) consists of all sequences (x n ) n<ω with at least two non-zero coordinates and such that the first non-zero coordinate of (x n ) n<ω is odd and all other are even. For each K ∈ F 2N \{∅} and n ∈ N, we have witnessing that f is an asymorphic embedding of F ♭ 2N into Q ω . With this representation, F ♭ ω can be easily metrizable by means of restriction to S of the stadard metric d on Q ω : d((x n ) n∈ω , (y n ) n∈ω ) = min{m : x n = y n for all n ≥ m}.
Theorem 2 . 2 .
22Let κ be an infinite cardinal, n ∈ N, [κ] n = {F ⊂ κ : |F | = n}, x ∈ κ.Then the following statements hold:
f
(B F (K, {2, 4, . . . , 2n}) = S B Q (f (K), {1, 2. . . . , 2n}),
Classifying homogeneous celular ordinal balleans up to coarse equivalence. T Banakh, I Protasov, D Repovs, S Slobodianiuk, preprint (arxiv: 1409.3910v2)T. Banakh, I. Protasov, D. Repovs, S. Slobodianiuk, Classify- ing homogeneous celular ordinal balleans up to coarse equivalence, preprint (arxiv: 1409.3910v2).
Scattered subsets of groups. T Banakh, I Protasov, S Slobodianiuk, Ukr. Math. Zh. 67T. Banakh, I. Protasov, S. Slobodianiuk, Scattered subsets of groups, Ukr. Math. Zh. 67 (2015), P. 304-312.
Characterizing the Cantor bi-cube in asymptotic categories. T Banakh, I Zarichnyi, Groups, Geometry and Dynamics. 5T. Banakh, I. Zarichnyi, Characterizing the Cantor bi-cube in asymp- totic categories, Groups, Geometry and Dynamics 5 (2011), P. 691-728.
Some categorical aspects of coarse spaces and balleans. D Dikranjan, N Zava, Topology Appl. to appearD. Dikranjan, N. Zava, Some categorical aspects of coarse spaces and balleans, Topology Appl., to appear.
. A Dranishnikov, Asymptotic Topology, Russian Math. Surveys. 55A. Dranishnikov, Asymptotic Topology, Russian Math. Surveys, 55 (2000), 1085-1129.
Balleans of bounded geometry and G-spaces. I Protasov, Math. Stud. 30I. Protasov, Balleans of bounded geometry and G-spaces, Math. Stud. 30 (2008), 61-66.
Ball Structures and Colorings of Graphs and Groups. I Protasov, T Banakh, Math. Stud. Monogr. Ser. 11VNTL PublisherI. Protasov, T. Banakh, Ball Structures and Colorings of Graphs and Groups, Math. Stud. Monogr. Ser. 11, Vol. 11, VNTL Publisher, Lviv, 2003.
. I Protasov, I Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser. 12VNTL PublisherI. Protasov, I. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser 12, Vol. 11, VNTL Publisher, Lviv, 2007.
J Roe, Lectures on coarse geometry. Amer31J. Roe,Lectures on coarse geometry, Univ. Lecture Series 31, Amer-
| []
|
[
"Efficient Route Tracing from a Single Source",
"Efficient Route Tracing from a Single Source"
]
| [
"Benoit Donnet \nLaboratoire LIP6-CNRS\nUMR 7606\nUniversité Pierre & Marie Curie\nParisFrance\n",
"Philippe Raoult \nLaboratoire LIP6-CNRS\nUMR 7606\nUniversité Pierre & Marie Curie\nParisFrance\n",
"Timur Friedman \nLaboratoire LIP6-CNRS\nUMR 7606\nUniversité Pierre & Marie Curie\nParisFrance\n"
]
| [
"Laboratoire LIP6-CNRS\nUMR 7606\nUniversité Pierre & Marie Curie\nParisFrance",
"Laboratoire LIP6-CNRS\nUMR 7606\nUniversité Pierre & Marie Curie\nParisFrance",
"Laboratoire LIP6-CNRS\nUMR 7606\nUniversité Pierre & Marie Curie\nParisFrance"
]
| []
| Traceroute is a networking tool that allows one to discover the path that packets take from a source machine, through the network, to a destination machine. It is widely used as an engineering tool, and also as a scientific tool, such as for discovery of the network topology at the IP level. In prior work, authors on this technical report have shown how to improve the efficiency of route tracing from multiple cooperating monitors. However, it is not unusual for a route tracing monitor to operate in isolation. Somewhat different strategies are required for this case, and this report is the first systematic study of those requirements. Standard traceroute is inefficient when used repeatedly towards multiple destinations, as it repeatedly probes the same interfaces close to the source. Others have recognized this inefficiency and have proposed tracing backwards from the destinations and stopping probing upon encounter with a previously-seen interface. One of this technical report's contributions is to quantify for the first time the efficiency of this approach. Another contribution is to describe the effect of non-responding destinations on this efficiency. Since a large portion of destination machines do not reply to probe packets, backwards probing from the destination is often infeasible. We propose an algorithm to tackle non-responding destinations, and we find that our algorithm can strongly decrease probing redundancy at the cost of a small reduction in node and link discovery. | null | [
"https://arxiv.org/pdf/cs/0605133v1.pdf"
]
| 20,540,349 | cs/0605133 | 75fd95f7f9b062bb39062825d6dc6d762b2caeca |
Efficient Route Tracing from a Single Source
29 May 2006
Benoit Donnet
Laboratoire LIP6-CNRS
UMR 7606
Université Pierre & Marie Curie
ParisFrance
Philippe Raoult
Laboratoire LIP6-CNRS
UMR 7606
Université Pierre & Marie Curie
ParisFrance
Timur Friedman
Laboratoire LIP6-CNRS
UMR 7606
Université Pierre & Marie Curie
ParisFrance
Efficient Route Tracing from a Single Source
29 May 2006
Traceroute is a networking tool that allows one to discover the path that packets take from a source machine, through the network, to a destination machine. It is widely used as an engineering tool, and also as a scientific tool, such as for discovery of the network topology at the IP level. In prior work, authors on this technical report have shown how to improve the efficiency of route tracing from multiple cooperating monitors. However, it is not unusual for a route tracing monitor to operate in isolation. Somewhat different strategies are required for this case, and this report is the first systematic study of those requirements. Standard traceroute is inefficient when used repeatedly towards multiple destinations, as it repeatedly probes the same interfaces close to the source. Others have recognized this inefficiency and have proposed tracing backwards from the destinations and stopping probing upon encounter with a previously-seen interface. One of this technical report's contributions is to quantify for the first time the efficiency of this approach. Another contribution is to describe the effect of non-responding destinations on this efficiency. Since a large portion of destination machines do not reply to probe packets, backwards probing from the destination is often infeasible. We propose an algorithm to tackle non-responding destinations, and we find that our algorithm can strongly decrease probing redundancy at the cost of a small reduction in node and link discovery.
Introduction
Traceroute [1] is a networking diagnostic tool natively available on most of the operating systems. It allows one to determine the path followed by a packet. Traceroute allows therefore to draw up the map of router interfaces present along the path between a machine S (the source or the monitor ) and a machine D (the destination). Traceroute has also engineering applications as it can be used, for instance, to detect routers that fail in a network. This report proposes and evaluates improvements to standard traceroute for tracing routes from a single point.
Today's most extensive tracing system at the IP interface level, skitter [2], uses 24 monitors, each targeting on the order of one million destinations. In the fashion of skitter, scamper [3] makes use of several monitors to traceroute IPv6 networks. The Distributed Internet MEasurements & Simulations [4] (Dimes) is a measurement infrastructure somewhat similar to the famous SETI@home [5]. SETI@home's screensaver downloads and analyzes radio-telescope data. The idea behind Dimes is to provide to the research community a publicly downloadable distributed route tracing tool. It was released as a daemon in September 2004. The Dimes agent performs Internet measurements such as traceroute and ping at a low rate, consuming at peak 1KB/sec. At the time of writing this report, Dimes counts more than 8,700 agents scattered over five continents. In the fashion of skitter, scamper [3] makes use of several monitors to traceroute IPv6 networks. Other well known systems, such as Ripe NCC TTM [6] and Nlanr AMP [7], each employs a larger set of monitors, on the order of one-to two-hundred, but they avoid probing outside their own network. Scriptroute [8] is a system that allows an ordinary internet user to perform network measurements from several distributed vantage points. It proposes remote measurement execution on PlanetLab nodes [9], through a daemon that implements ping, hop-by-hop bandwidth measurement, and a number of other utilities in addition to traceroute.
Recently, in the context of large-scale internet topology discovery, we have shown [10] that standard traceroute probing (such as skitter) is particularly inefficient due to duplication of effort at two levels: measurements made by an individual monitor that replicate its own work (intra-monitor redundancy), and measurements made by multiple monitors that replicate each other's work (inter-monitor redundancy). Using skitter data from August 2004, we have quantified both kinds of redundancy. We showed that intra-monitor redundancy is high close to each monitor and, with respect to intermonitor redundancy, we find that most interfaces are visited by all monitors, especially when close to destinations. We further proposed an algorithm, Doubletree, for reducing both forms of redundancy at the same time.
This technical report focuses more deeply on the intra-monitor redundancy problem. Systems that discover internet topology at IP level from a set of isolated vantage points (i.e., there is no cooperation between monitors) have interest to reduce their intra-monitor redundancy. By sending much less probes, monitors can probe the network more frequently. The more frequent snapshots you have, the more accurate should be your view of the topology. This technical report demonstrates how a monitor can act to reduce its intra-monitor redundancy.
The nature of intra-monitor redundancy suggest to start probing far from the traceroute monitor and probe backwards (i.e., decreasing TTLs), as first noticed by Govindan and Tangmunarunkit [11], Spring et al. [8], Moors [12] and Donnet et al. [10]. However, performing backward probing from non-cooperative traceroute monitors in the context of intra-monitor redundancy has never been evaluated previously. Even if backward probing is simple to understand, it is not clear how efficient it is. This report evaluates the redundancy reduction of backward probing as well as the eventual information lost compared to standard traceroute.
Nevertheless, backward probing is based on the assumption that destinations reply to probes in order to estimate path lengths and the distance of the last hop before the destination. Unfortunately, a large set of destinations (40% in our data set) does not reply to probes, probably due to strongly configured firewalls. In this case, backward probing cannot be performed. In this report, we also propose a way to face non-responding destinations. We further propose an efficient algorithm that can handle both cases, i.e., responding and nonresponding destinations. We evaluate these algorithms in terms of intra-monitor redundancy and quantity of information lost.
The remainder of this report is organized as follows: Sec. 2 introduces the data set used throughout this technical report; Sec. 3 gives a key for reading quantile plots; Sec. 4 evaluates standard traceroute; Sec. 5 presents and evaluates separately our backward probing algorithms; Sec. 6 compares the different algorithms; finally, Sec. 7 concludes this report by summarizing its principal contributions.
Data Set
Our study is based on skitter [2] data from August 1 st through 3 rd , 2004. This data set was generated by 24 monitors located in the United States, Canada, the United Kingdom, France, Sweden, the Netherlands, Japan, and New Zealand. The monitors share a common destination set of 971,080 IPv4 addresses. Each monitor cycles through the destination set at its own rate, taking typically three days to complete a cycle. For the purpose of our studies, in order to reduce computing time to a manageable level, we worked from a limited destination set of 50,000, randomly chosen from the original set.
Visits to host and router interfaces are the metric by which we evaluate redundancy. We consider an interface to have been visited if its IP address returned by the host or router appears, at least, at one of the hops in a traceroute. Though it would be of interest to calculate the load at the host and router level, rather than at the individual interface level, we make no attempt to disambiguate interfaces in order to obtain router-level information. The alias resolution techniques described by Pansiot and Grad [13], by Govindan and Tangmunarunkit [11], for Mercator, and applied in the iffinder tool from Caida [14], would require active probing beyond the skitter data, preferably at the same time that the skitter data is collected. The methods used by Spring et al. [15], in Rocketfuel, and by Teixeira et al. [16], apply to routers in the network core, and are untested in stub networks. Despite these limitations, we believe that the load on individual interfaces is a useful measure. As Broido and claffy note [17], "interfaces are individual devices, with their own individual processors, memory, buses, and failure modes. It is reasonable to view them as nodes with their own connections."
How do we account for skitter visits to router and host interfaces? Like many standard traceroute implementations, skitter sends three probe packets for each hop count. An IP address appears thus in a traceroute result if it appears in the replies to, at least, one of the three probes sent (but it may also appear two or three times). For each reply, we account one visit. If none of the three probes are returned, the hop is recorded as non-responding.
Even if an IP address is returned for a given hop count, it might not be valid. Due to the presence of poorly configured routers along traceroute paths, skitter occasionally records anomalies such as private IP addresses that are not globally routable. We account for invalid hops as if they were non-responding hops. The addresses that we consider as invalid are a subset of the special-use IPv4 addresses described in RFC 3330 [18]. Specifically, we eliminate visits to the private IP address blocks 10.0.0.0/8, 172.16.0.0/12, and 192.168.0.0/16. We also remove the loopback address block 127.0.0.0/8. In our data set, we find 4,435 different special addresses, more precisely 4,434 are private addresses and only one is a loopback address. Special addresses account for approximately 3% of the entire set of addresses seen in this trace. Though there were no visits in the data to the following address blocks, they too would be considered invalid: the "this network" block 0. In this report, we plot interface redundancy distributions. Since these distributions are generally skewed, quantile plots give us a better sense of the data than would plots of the mean and variance. There are several possible ways to calculate quantiles. We calculate them in the manner described by Jain [19, p. 194], which is: rounding to the nearest integer value to obtain the index of the element in question, and using the lower integer if the quantile falls exactly halfway between two integers. Fig. 1 provides a key to reading the quantile plots found in subsequent sections of this report.
A dot marks the median (the 2 nd quartile, or 50 th percentile). The vertical line below the dot delineates the range from the minimum to the 1 st quartile, and leaves a space from the 1 st to the 2 nd quartile. The space above the dot runs from the 2 nd to the 3 rd quartile, and the line above that extends from the 3 rd quartile to the maximum. Small tick bars to either side of the lines mark some additional percentiles: bars to the left for the 10 th and 90 th , and bars to the right for the 5 th and 95 th .
In the case of highly skewed distributions, or distributions drawn from small amounts of data, the vertical lines or the spaces between them might not appear. For instance, if there are tick marks but no vertical line above the dot, this means that the 3 rd quartile is identical to the maximum value.
In the figures, each quantile plot sits directly above an accompanying bar chart that indicates the quantity of data upon which the quantiles were based. For each hop count, the bar chart displays the number of interfaces at that distance. For these bar charts, a log scale is used on the vertical axis. This allows us to identify quantiles that are based upon very few interfaces (fewer than twenty, for instance), and so for which the values risk being somewhat arbitrary.
Standard Traceroute
Our basis for comparison is the results from the standard forward tracing algorithm implemented in traceroute. All monitors operate from a set of common destinations, D. Each monitor probes forward starting from TTL=1 and increasing the TTL hop by hop towards each of the destinations in D in turn. As it probes, a monitor i updates the set, S i , initially empty, of interfaces that it has visited.
Evaluating redundancy in the standard traceroute was already published in an authors' SIGMETRICS 2005 paper [10]. For comparison reasons in the next sections of this report, we summarize in this section our redundancy evaluation of standard traceroute. Interested readers might find plots for the 22 other skitter monitors in our technical report [20]. Looking first at the histograms for interface counts (the lower half of each plot), we find data consistent with distributions typically seen in such cases. If we were to look at a plot on a linear scale (not shown here), we would see that these distributions display the familiar bell-shaped curve typical of internet interface distance distributions [21]. If we concentrate on champagne, we see that it discovers 92,354 unique and valid IP addresses. The interface distances are distributed with a mean at 17 hops corresponding to a peak of 9,135 interfaces that are visited at that distance.
The quantile plots show the nature of the redundancy problem. Looking at how the redundancy varies by distance, we see that the problem is worse the closer one is to the monitor. This is what we expect given the treelike structure of routing from a monitor, but here we see how serious the phenomenon is from a quantitative standpoint. For the first two hops from each monitor, the median redundancy is 150,000. A look at the histograms shows that there are very few interfaces at these distances. Just one interface for arin, and two (2 nd hop) or three (3 rd hop) for champagne. These close to the monitor interfaces are only visited three times, as represented by the presence of the 5 th and 10 th percentile marks (since there are only two data points, the lower values point is represented by the entire lower quarter of values on the plot).
Beyond three hops, the median redundancy drops rapidly. By the eleventh hop, in both cases, the median is below ten. However, the distributions remain highly skewed. Even fifteen hops out, some interfaces experience a redundancy on the order of several hundred visits. With small variations, these patterns are repeated for each of the monitors.
Backward Tracing
As seen and discussed in Sec. 4, the most worrisome feature of redundancy in a standard measurement system is the exceptionally high number of visits to the median interfaces close in to the monitor. Also of concern is the heavy tail of the distribution at more distant hop counts, with a certain number of interfaces consistently receiving a high number of visits. Our approach here is to tackle the first problem head-on, and then to see if the second problem remains.
The large number of visits to nodes close in to a monitor is easily explained by the tree-like or conal structure of the graph generated by traceroutes from a single monitor, as described by Broido and claffy [17]. There are typically only a few interfaces close to a monitor, and these interfaces must therefore be visited by a large portion of the traceroutes. The solution to this problem is simple, at least in principle: these close in interfaces must be skipped most of the time.
Traceroute works forward from source to destination. Its first set of probes goes just one hop, its second set goes two, and so forth. It would seem that the best way to reduce intra-monitor redundancy is to start further out and probes backward, i.e., decreasing TTLs. Govindan and Tangmunarunkit [11] do just this in the Mercator system. Using a probing strategy based upon IP address prefixes, Mercator conducts a check before probing the path to a new address that has a prefix P . If paths to an address in P already exist in its database, Mercator starts probing at the highest hop count for a responding router seen on those paths. No results have been published on the performance of this heuristic, though it seems to us an entirely reasonable approach in light of our data.
The Mercator heuristic requires that a guess be made about the relevant prefix length for an address. That guess is based upon the class that the address would have had before the advent of classless inter-domain routing (CIDR) [22]. In this technical report, we have tested a number of simple heuristics that do not require us to hazard such a guess.
Our algorithms work backwards. As illustrated in Fig. 3, a monitor sends its first probes to the destination, its second to one hop short of the destination, and so forth. Now arises the question of when to stop backward probing. Based on the tree-like structure of routes emanating from a single point (i.e., the traceroute source), we choose a stopping rule based on the set of interfaces previously encountered. A monitor will stop backward probing when an already visited interface is encountered. The only redundancy such a strategy should produce would be on interfaces that are branching points in paths between a monitor and its destinations. A backward probing scheme uses the set, S i , of interfaces that a monitor i has visited. In early probing, S i will have few elements, and so paths should be traced from the destination almost all the way back to the monitor. Later probing should terminate further and further out, as more and more interfaces are added to S i .
There are practical problems with a strategy of back-wards probing. They arise because of inherent flaws with methods for establishing the number of hops from a monitor to a destination. These methods rely upon the sending of a ping packet (or a scout packet, following Moors' terminology [12]), and the examination of the time to live (TTL) value in the IP packet that the destination returns. Various heuristics have been described, by Templeton and Levitt [23], Jin et al. [24], Moors [12] and Beverly [25], to guess the original TTL (typically one of a few standard values) based upon the observed value, and thus to guess the hop count from destination to monitor. While these heuristics have been shown to work well, the most serious problem is that they cannot work when the destination does not reply, as is often the case (40.3% in our data). In such a case, a system that takes a backwards probing approach will ideally start from the most distant interface that responds with an ICMP "TTL expired" packet when discarding a hoplimited probe. In practice, this might take some search to locate, adding redundancy. Furthermore, as established by Paxson [26] based upon data from 1995, and confirmed with data from 2002 by Amini et al.
[27], a considerable number of paths in the Internet are asymmetric: most recently almost 70%. This is a less serious problem, however, as the differences in routing often do not translate into considerable differences in hop count. Paxson's work indicated that differences in one or two hops were typical. For the purposes of our simulations we assume that, if a destination does reply to a ping, the system thereby learns the correct number of hops on the forward path.
Pure Backwards
We simulate an algorithm for backwards probing in which the most distant responding interface is assumed to be known a priori. Called pure backward probing, this algorithm is unrealistic because of its assumption. However, its performance sets a benchmark.Against that algorithm, we later compare algorithms that use only information that is actually available to a monitor. Fig. 4 shows redundancy for monitors running the pure backwards algorithm 1 . We notice a significant drop in comparison to the redundancy in straightforward tracerouting shown in Fig. 2. The median drops for the close interfaces, and the distribution tails are significantly shortened overall. However, Figs. 4(a) and 4(b) show still high redundancy for interfaces located one hop from the traceroute source.
We hypothesize that these remanent high redundancies close to the monitors are caused by the existence of firewalls or gateways that either do not permit probes to pass through them, or do not permit replies to return.
1 Plots for others monitors can be found in an appendix at the end of this report. If the destination addresses are invalid, these interfaces could also be default free routers. Under pure backwards probing, a node situated immediately in front of such a device, whatever it might be, will be visited again and again, for each destination that lies beyond, thus resulting in a high visit count for one of its interfaces. Without any further knowledge, the actual cause of such high redundancy under backwards probing remains for us an open question.
However, Figs. 4(a) and 4(b) show that maximum redundancies are in the thousands, rather that the hundreds of thousands as before. Furthermore, median values are a little higher than with standard traceroute. The strong drop in redundancy close to the monitor thus comes at the expense of some increased redundancy further out. The overall effect is one of smoothing the load.
There are costs associated with this drop in redundancy. We measure these in terms of the number of interfaces missed, using the set S i of interfaces visited by the standard algorithm as the reference. For the pure backwards algorithm, the numbers missed are relatively small, as shown in Table 7. Table 7 gives also the cost in term of links missed by the pure backwards algorithm.
Interfaces will necessarily be missed in backwards probing when a hard and fast rule is applied that requires probing to stop once an already visited interface is encountered. Any routing change that might have taken place between the monitor and that interface will go unnoticed. A routing system that adopts a backwards probing algorithm should also adopt a strategy for periodically reprobing certain paths, so as not to miss such changes entirely. So long as the portion of interfaces missed is small, we believe that the development of such a reprobing strategy can be left to future work.
Ordinary Backwards
The ordinary backwards algorithm works in much the same way as perfect backwards, but it is a more realistic algorithm. Just as with pure backwards, when a destination responds, the monitor starts probing backwards from the destination until an already visited interface is met. However, when a destination doesn't reply, the monitor, since it cannot know a priori the most distant responding interface along the path, gives up probing for this particular destination altogether. This is the first of two building blocks that will be used by the algorithm presented in Sec. 5.4, and is not intended to be used in isolation.
Approximately 40% the traceroutes in our data set terminate in a non-responding destination. What does this mean in terms of interfaces that are missed? Table 2 shows the costs of not probing these paths combined with the early stopping that is in any case associated with backwards probing. What is remarkable to note is that, compared to perfect backwards probing, ordinary backwards probing only misses an additional 16% of interfaces. Fig. 5 shows trends very similar to those observed with perfect backwards probing, but some high values are no longer present.
Searching
If we are to use ordinary backwards probing as one element of a larger probing strategy, we need a second element to handle destinations that do not respond. Since the last responding interface on a path to such a destination cannot be known a priori, the monitor must search Our algorithm, labeled searching, now sends its initial probe with a TTL value h. If it receives a response, it continues to probe forwards, to TTLs h + 1, h + 2, and so forth. When the farthest responding interface is found, probing resumes from TTL h − 1, and probes backwards, to TTLs h − 2, h − 3, and so back. If, at any point, a monitor i visits an interface that is in its set S i of interfaces already viewed, probing for that destination stops. The working of the searching algorithm is illustrated in Fig. 6, where h = 3 and R 5 being the last responding interface.
If the algorithm is supposed to start probing from a midpoint h in the network, we have to decide which value give to h. Doubletree [10], proposed by the au-
Figure 6: Searching algorithm with h = 3 ¡ ¢ ¢ ¡ £ £ ¡ ¤ ¤ ¡ ¥ ¦ § ¨ © £ ¥ ¢ ¢ £ ! " # $ % & ' ( # ) 0 # 1 & 2
Figure 7: Incomplete paths distribution thors of this report, is a cooperative and efficient algorithm for large-scale topology discovery. Each Doubletree monitor starts probing at some hop h from itself, performing forwards probing from h and backwards probing from h − 1. The value h is fixed by each monitor according to its probability p of hitting a destination with the very first probe sent. This choice is driven by the risk of probing looking like a distributed denial-ofservice (DDoS) attack. Indeed, when probes sent by multiples monitors converge towards a given destination, the probe traffic might appear, for an end-host, as a DDoS attack. Doubletree aims to minimize this risk and, therefore, each monitor chooses an appropriate h value.
In this report, we are not interested in large-scale distributed probing, i.e., from a large set of monitors that cooperate when probing towards a large set of destinations. We consider that each monitor works in isolation of others. It does not make sense to choose the h value like Doubletree does. Fig. 7 shows the incomplete paths distribution, i.e., the distance distribution of the last responding hop when a traceroute does not terminate by hitting the destination. Such a case occurs in approximately 40% of the traceroutes in our dataset. We propose that each monitor tunes its h value with the mean hop count for its incomplete traceroutes. A monitor can easily evaluates its own h value by performing a small number of standard traceroutes.
In the special case where there is no response at distance h, the distance is halved, and halved again until there is a reply, and probing continues forwards and backwards from that point.
Our results in Fig. 8 indicate that low redundancy can be achieved. We tested the heuristic algorithm using only those traceroutes for which the destination does not respond.
However, we notice that close to the monitor, in the fashion of the pure backwards algorithm (see Fig. 4), the redundancy is still high. We believe that this is caused by very short paths for which the last responding interface is close to the monitor. For those paths, there is a high probability that sending the first probe at h hops to the monitor will corresponds a non-response. The h value will be divided by two, again and again, until reaching a responding interface that will be located close to the monitor, increasing therefore the redundancy of such interfaces.
Searching Ordinary Backwards
In this section, we study a comprehensive strategy for reducing probing redundancy. We employ ordinary backwards probing, along with the heuristic algorithm This algorithm is called searching ordinary backwards. Fig. 9 shows redundancy reduction similar to that obtained with the other algorithms examined so far. Table 3 shows the interfaces and links missed when probing with the searching ordinary backwards algorithm. Table 3 indicates that the numbers of missed interfaces are a bit smaller (this is specially true for arin) than with the supposedly pure backwards algorithm, a surprising fact which will be explained in the next section. Table 4 giving the mean over the 24 monitors.
Algorithms Comparison
Our goal is to avoid both redundancy and missed interfaces as much as possible, however there is a tradeoff between the two. Extremes are represented by the standard traceroute, which by definition misses no interfaces, but has high average redundancy, and by the heuristic algorithm which, having been applied to just those traceroutes for which the destination did not respond, necessarily misses a large number of interfaces.
We see that the ordinary backwards provides the lowest redundancy but, as it is applied on only respond-
Conclusion
This technical report addresses the area of efficient probing in the context of traceroute monitors working in isolation from each others. Prior work stated that standard traceroutes are particularly inefficient by repeatedly reprobing the same interfaces close to the monitor. The solution to this redundancy problem is, at least in principle, simple: those interfaces close to the monitor must be skipped most of the time. It seems that the best way to achieve this solution is to probe backwards from the destinations and stop when encountering a previously seen interface.
In this report, we perform the first careful study of the efficiency of backwards probing, by evaluating it in terms of redundancy reduction and information lost.
Nevertheless, we state that a pure backwards probing algorithm is unrealistic as it is based on the assumption that destinations reply to probes. We therefore propose an algorithm that searches for the last responding interface. The key idea of this algorithm is to start probing at some hop h from the monitor, probe forwards from h until the last responding interface and, then, probe backwards from h − 1 until reaching an already discovered interface.
We finally propose a realistic algorithm that can handle both cases, i.e., responding and non-responding des- tinations. We evaluate this algorithm and state that it is capable of reducing probe traffic by a factor of 10, while only missing 4% of the interfaces discovered by a standard traceroute.
¡ ¡ ¢ ¡ £ ¡ ¤ ¡ ¥ ¡ ¦ ¡ § ¡ ¨ ¡ © ¦ ¢ ¢ ¦ £ £ ¦ ¤ ! " # $ %& % ' ( ) 0 1 2 0 3 2 ( 4 0 3 5 6 7 1 8 ( 4 0 3 5 6 7 1 8 9 3 2 @ A 0 5 B C 0 3 2 ( D E 3 4 A 0 5 B C 0 3 2 ( 9 3 2 7 1 0 3 F A 0 5 B C 0 3 2 ( (a) arin G H G G H I G H P G H Q G H R G H S G H T G H U G H V W X Y W Y X ` a Y b c a d d
As a future work, we aim to provide to the research community an implementation of the algorithms discussed in this report.
References
0.0.0/8, the 6to4 relay any cast address block 192.88.99.0/24, the benchmark testing block 198.18.0.0/15, the multicast address block 224.0.0.0/4, and the reserved address block formerly known as the Class E addresses, 240.0.0.0/4, which includes the lan broadcast address, 255.255.255.255.
Figure 1 :
1Quantiles
Figure 2 :
2Redundancy when probing with the pure forwards algorithm
Fig. 2 Figure 3 :
23Pure backwards algorithm monitors: arin and champagne. The results presented inFig. 2are representative of all the skitter monitors.
Figure 4 :
4Redundancy when probing with the pure backwards algorithm
Figure 5 :
5Redundancy when probing with the ordinary backwards algorithm for it. The search cost is what will make the difference with respect to the pure backwards algorithm.
Figure 8 :
8Redundancy when probing with the searching algorithm for cases in which the destination does not respond.
Fig
. 10 shows the trade-offs between redundancy and missed address interfaces. Redundancy is here repre-
Figure 9 :
9Redundancy when probing with the searching ordinary backwards algorithm sented by the mean number of visits per valid discovered interface, and the missed addresses are expressed as a proportion, using the standard traceroute as the reference. Results shown inFig. 10are representative for all monitors, as demonstrated by
Figure 10 :
10Algorithms comparison
Figure 11 :Figure 12 :Figure 13 :
111213Redundancy when probing with the pure backwards algorithm Redundancy when probing with the pure backwards algorithm -Redundancy when probing with the pure backwards algorithm -
Figure 14 :
14Redundancy when probing with the pure backwards algorithm -4
Figure 15 :
15Redundancy when probing with the ordinary backwards algorithm
Figure 16 :
16Redundancy when probing with the ordinary backwards algorithm -
Figure 17 :
17Redundancy when probing with the ordinary backwards algorithm -
Figure 18 :
18Redundancy when probing with the ordinary backwards algorithm -4
Figure 19 :
19Redundancy when probing with the searching algorithm
Figure 20 :
20Redundancy when probing with the searching algorithm -
Figure 21 :
21Redundancy when probing with the searching algorithm -
Figure 22 :Figure 23 :Figure 24 :Figure 25 :Figure 26 :Figure 27 :Figure 28 :Figure 29 :
2223242526272829Redundancy when probing with the searching algorithm -4 Incomplete Incomplete Incomplete Incomplete Redundancy when probing with the SearchingOB ordinary backwards algorithm Redundancy when probing with the SearchingOB ordinary backwards algorithm -Redundancy when probing with the SearchingOB ordinary backwards algorithm -
Figure 30 :
30Redundancy when probing with the SearchingOB ordinary backwards algorithm
Figure 31 :Figure 32 :Figure 33 :Figure 34 :
31323334Algorithms Algorithms Algorithms Algorithms comparison -4
Table 1: Interfaces missed by the pure backwards algorithmMonitor
Interfaces
Links
total
discovered % missed
total
discovered % missed
arin
92,381
88,204
4.52%
101,850
92,602
9.09%
champagne 92,354
88,012
4.70%
101,652
92,331
9.17%
Monitor
Interfaces
Links
total
discovered % missed
total
discovered % missed
arin
92,381
73,529
20.40%
101,850
75,163
27.21%
champagne 92,354
73,410
20.51%
101,652
74,987
27.24%
Table 2: Interfaces missed by the ordinary backwards algorithm
Monitor
R1
R2
R3
R4
R5 Destination
Probing Sequence
Table 3 :
3Interfaces missed by the searching ordinary backwards algorithmAlgorithms
Mean visit Prop. missed
standard
25.08
0
heuristic
9.16
0.74
search. ord. bwd
6.21
0.03
pure bwd
5.58
0.04
ordinary bwd
4
0.2
Table 4 :
4Algorithms comparison -mean
ing traceroutes, it misses a lot of interfaces. The most
interesting comparison is between the pure backwards
algorithm and searching ordinary backwards. Both are
based upon the full set of traceroutes, and so are strictly
comparable. The searching ordinary backwards algo-
rithm manages to outperform the pure backwards al-
gorithm by paying a slight price in terms of increased
redundancy (in the case of arin).
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Table 5 :
5Interfaces missed by the pure backwards algorithm
Table 6 :
6Interfaces missed by the ordinary backwards algorithm
Table 7 :
7Interfaces missed by the SearchingOB ordinary backwards algorithmE Algorithms Comparison
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A robust classifier for passive TCP/IP fingerprinting. R Beverly, Proc. Passive and Active Measurement Workshop (PAM). Passive and Active Measurement Workshop (PAM)R. Beverly, "A robust classifier for passive TCP/IP fingerprinting," in Proc. Passive and Active Mea- surement Workshop (PAM), Apr. 2004.
End-to-end routing behavior in the internet. V Paxson, Proc. ACM SIGCOMM. ACM SIGCOMMV. Paxson, "End-to-end routing behavior in the in- ternet," in Proc. ACM SIGCOMM, Aug. 1996.
| []
|
[
"arXiv:astro-ph/0208103v1 5 Aug 2002 Principles, Progress and Problems in Inflationary Cosmology",
"arXiv:astro-ph/0208103v1 5 Aug 2002 Principles, Progress and Problems in Inflationary Cosmology"
]
| [
"Robert H Brandenberger \nPhysics Department\nTheory Division\nBrown University\n02912ProvidenceRIUSA\n\nCERN\nCH-1211Genève 23Switzerland\n"
]
| [
"Physics Department\nTheory Division\nBrown University\n02912ProvidenceRIUSA",
"CERN\nCH-1211Genève 23Switzerland"
]
| []
| Inflationary cosmology has become one of the cornerstones of modern cosmology. Inflation was the first theory within which it was possible to make predictions about the structure of the Universe on large scales, based on causal physics. The development of the inflationary Universe scenario has opened up a new and extremely promising avenue for connecting fundamental physics with experiment. This article summarizes the principles of inflationary cosmology, discusses progress in the field, focusing in particular on the mechanism by which initial quantum vacuum fluctuations develop into the seeds for the large-scale structure in the Universe, and highlights the important unsolved problems of the scenario. The case is made that new input from fundamental physics is needed in order to solve these problems, and that thus early Universe cosmology can become the testing ground for trans-Planckian physics. | null | [
"https://arxiv.org/pdf/astro-ph/0208103v1.pdf"
]
| 96,423,720 | astro-ph/0208103 | 6864fbe4294db1c7904746ae1fb860dd076c54a8 |
arXiv:astro-ph/0208103v1 5 Aug 2002 Principles, Progress and Problems in Inflationary Cosmology
(June 1, 2018)
Robert H Brandenberger
Physics Department
Theory Division
Brown University
02912ProvidenceRIUSA
CERN
CH-1211Genève 23Switzerland
arXiv:astro-ph/0208103v1 5 Aug 2002 Principles, Progress and Problems in Inflationary Cosmology
(June 1, 2018)
Inflationary cosmology has become one of the cornerstones of modern cosmology. Inflation was the first theory within which it was possible to make predictions about the structure of the Universe on large scales, based on causal physics. The development of the inflationary Universe scenario has opened up a new and extremely promising avenue for connecting fundamental physics with experiment. This article summarizes the principles of inflationary cosmology, discusses progress in the field, focusing in particular on the mechanism by which initial quantum vacuum fluctuations develop into the seeds for the large-scale structure in the Universe, and highlights the important unsolved problems of the scenario. The case is made that new input from fundamental physics is needed in order to solve these problems, and that thus early Universe cosmology can become the testing ground for trans-Planckian physics.
I. INTRODUCTION
With the recent high-accuracy measurements of the spectrum of the cosmic microwave background (CMB) (see Fig. 1 [1]), cosmology has become a quantitative science. There is now a wealth of new data on the structure of the Universe as deduced from precision maps of the cosmic microwave background anisotropies, from cosmological redshift surveys, from redshift-magnitude diagrams of supernovae, and from many other sources. Standard Big Bang (SBB) cosmology provides the framework for describing the present data. The interpretation and explanation of the existing data, however, requires us to go beyond SBB cosmology and to consider scenarios like the Inflationary Universe in which space-time evolution in the very early Universe differs in crucial ways from what is predicted by the SBB theory. Since inflationary cosmology at later times smoothly connects with the SBB picture, we must begin this article with a short review of the framework of SBB cosmology.
Standard big bang cosmology rests on three theoretical pillars: the cosmological principle, Einstein's general theory of relativity, and the assumption that matter is a classical perfect fluid. The cosmological principle concerns the symmetry of space-time and states that on large distance scales space is homogeneous and isotropic. This implies that the metric of space-time can be written in Friedmann-Robertson-Walker (FRW) form. For simplicity, we consider the case of a spatially flat Universe: ds 2 = dt 2 − a(t) 2 dr 2 + r 2 (dϑ 2 + sin 2 ϑdϕ 2 ) . (1) Here, t, r, θ and ϕ are the space-time coordinates, and ds gives the proper time between events in space-time. The coordinates r, ϑ and ϕ are "comoving" spherical coordinates, and t is the physical time coordinate. Space-time curves with constant comoving coordinates correspond to the trajectories of particles at rest. If the Universe is expanding, i.e. the scale factor a(t) is increasing, then the physical distance ∆x p (t) between two points at rest with fixed comoving distance ∆x c grows as ∆x p = a(t)∆x c .
FIG. 1. Compilation of the spectrum of the CMB. In the region around the peak (the region probed with greatest precision by the COBE satellite) the error bars are smaller than the size of the data points.
The dynamics of an expanding Universe is determined by the Einstein equations, which relate the expansion rate to the matter content, specifically to the energy density ρ and pressure p.
ρ = −3H(ρ + p) ,(3)
where H =ȧ/a is the Hubble expansion rate, and an overdot denotes the derivative with respect to time t. The third key assumption of standard cosmology is that matter is described by a classical ideal gas with some equation of state which is conveniently parametrized in the form p = wρ which some constant w. For cold matter, pressure is negligible and hence w = 0. For radiation we have w = 1/3. In standard cosmology, the Universe is a mixture of cold matter and radiation, the former dominating at late times, the latter dominating in the early Universe. In this case, the FRW equations can be solved exactly, with the result that the Universe had to be born in a "Big Bang" singularity.
SBB cosmology explains Hubble's redshift-distance relationship, and it explains the abundances of the light elements Helium, Deuterium and Lithium. These nucleosynthesis predictions of the SBB model depend on a single free parameter (which is the present ratio of the number densities of baryons to photons), and thus the fact that the abundances of more than one element can be matched by adjusting this ratio is remarkable. Most importantly, SBB cosmology predicts the existence and black-body spectral distribution of a microwave cosmic background radiation, the CMB. The precision measurement [2,3] of the spectrum of the CMB is a triumph for SBB cosmology (see Fig. 1).
However, the triumph of the CMB also leads to one of several major problems for SBB cosmology: within the context of this theory, there is no explanation for the high degree of isotropy of the radiation. As is illustrated in Fig. 2, at the time the microwave radiation last scattered (which occurred when the temperature was about a factor 10 3 of the present temperature), the maximal distance which light could have communicated information starting at the Big Bang (the forward light cone) is much smaller than the distance over which the microwave photons are observed to have the same temperature (the past light cone). This is the famous "horizon problem" of SBB cosmology. Within the context of SBB cosmology it is also a mystery why the Universe today is observed to be approximately spatially flat, since a spatially flat Uni- verse is an unstable fixed point of the FRW equations in an expanding phase. This problem is called the "flatness problem". Finally, as illustrated in Fig. 3, within standard cosmology there is no causal mechanism which can explain the nonrandom distribution of the seed inhomogeneities which develop into the present-day large-scale structure. This constitutes the "formation of structure problem". Under the assumption that only gravity is responsible for the development of inhomogeneities on cosmological scales, the seeds for fluctuations must have been correlated at the time t eq , the time when the energy densities in cold matter and radiation were equal (which occurred when the Universe was about 10 −4 of its present size) when inhomogeneities can first start to grow by gravitational instability. But, at that time, the forward light cone was smaller than the separation between the seeds.
Standard Big Bang cosmology is also internally inconsistent as a theory of the very early Universe. We know that at very high energy densities which the theory predicts in the initial stages, the description of matter as a classical ideal gas is invalid. Since quantum field theory is a better description of matter at high energies, this naturally leads us to consider quantum field theory as the description of matter which must take over in the very early Universe, and this realization led to the discovery of the inflationary scenario. What follows is a brief overview of principles, progress and problems in inflationary cosmology. For more details, the reader is referred to [4].
II. THE INFLATIONARY UNIVERSE SCENARIO
The inflationary Universe scenario [5] is based on the simple hypothesis that there was a time interval in the early Universe beginning at some time t i and ending at a later time t R (the "reheating time") during which the scale factor is exponentially expanding. Such a period is Sketch (physical distance xp vs.time t) of the solution of the homogeneity problem. During inflation, the forward light cone l f (t) is expanded exponentially when measured in physical coordinates. Hence, it does not require many e-foldings of inflation in order that l f (t) becomes larger than the past light cone lp(t) at the time trec of last scattering. The dashed line is the forward light cone without inflation.
called "de Sitter" or "inflationary". The success of Big Bang nucleosynthesis demands that this time interval was long before the time of nucleosynthesis. It turns out that during the inflationary phase, the energy of matter is stored in some new form (see below). At the time t R of reheating, all this energy is released as thermal energy. This is a nonadiabatic process during which the entropy of the Universe increases by a large factor.
Independent of any specific realization, the above simple hypothesis of inflation leads immediately to possible solutions of the horizon, flatness and formation of structure problems. Fig. 4 is a sketch of how a period of inflation can solve the horizon problem. During inflation, the forward light cone increases exponentially compared to a model without inflation, whereas the region over which isotropy is observed is not affected. Hence, provided inflation lasts sufficiently long, the forward light cone at the time of last scattering of CMB photons can be made larger than the region from which the microwave photons are reaching us today.
Inflation also can solve the flatness problem. The key point is that at the time of reheating, the entropy of the Universe increases by a large factor, and this drives the Universe towards spatial flatness, as can be seen from the FRW equations. In fact, one of the key predictions of inflationary cosmology is that the Universe should be spatially flat to great accuracy (although it is possible to construct special models of inflation which produce any given deviation from spatial flatness).
Most importantly, inflation provides a mechanism which in a causal way generates the primordial perturbations required to explain the nonrandom distribution of matter on the scales of galaxies and galaxy clusters, and FIG. 5. A sketch (physical coordinates vs. time) of the solution of the formation of structure problem. Provided that the period of inflation is sufficiently long, the separation dc between two galaxy clusters is at all times smaller than the forward light cone. The dashed line indicates the Hubble radius. Note that dc starts out smaller than the Hubble radius H −1 (t), crosses it during the de Sitter period, and then reenters it at late times.
to produce small-amplitude anisotropies in the CMB. At any given time, microphysics can act coherently on scales up to the Hubble radius H −1 (t). The key point is that during the inflationary period the Hubble radius is constant, whereas the physical length scale associated with fluctuations which have fixed comoving scale increases exponentially. Thus, as is depicted in Fig. 5, provided that the inflationary period is sufficiently long, all comoving scales of cosmological interest today had a physical wavelength smaller than the Hubble radius in the early stages of inflation. Thus, it is possible without violating causality to have a mechanism which generates microscopic-scale fluctuations during the period of inflation whose wavelengths then get stretched exponentially so that they can become the seeds for structure in the present Universe.
As will be discussed in Section 4, the density perturbations produced during inflation are due to quantum fluctuations in the matter and gravitational fields [6,7] . These fluctuations are continously generated during the period of inflation. Once the physical wavelength equals the Hubble radius, the vacuum oscillations freeze out, the quantum state is squeezed, and the highly squeezed state at late time appears as a state with a large number of particles, representing the density fluctuations at late times. The amplitude of these fluctuations is given by the Hubble expansion rate H. Since H is approximately constant during the period of inflation, this mechanism of quantum vacuum fluctuations freezing out and becoming classical density fluctuations leads to the prediction that the spectrum of fluctuations should be scale invariant, i.e. the physical measure of the amplitude of fluctuations at late times should be independent of the wavenumber k:
k 3 |δ k | 2 = const ,(4)
where δ k is the fractional energy density fluctuation in momentum space. Thus, independent of the specific mechanism which drives inflation, as long as the expansion is almost exponential, the spectrum of fluctuations is predicted to be almost scale-invariant. As will be discussed in Section 4, this quantitative prediction of inflationary cosmology is confirmed by observations. However, as will be mentioned in Section 5, this prediction may depend on unstated assumptions about the trans-Planckian physics.
III. HOW TO OBTAIN INFLATION
Because in the year 1980, when inflationary cosmology was being developed, quantum field theory was the best available description of matter at high energies such as must have occurred close to the Big Bang, it in retrospect seems obvious to turn to it for a possible implementation of the inflationary Universe scenario. Since scalar matter field are supposed to play an important role in high energy physics, in particular for implementing the spontaneous breaking of internal gauge symmetries, it is necessary to consider the role of such fields in cosmology.
If we assume that Einstein's equations remain valid in the early Universe, then it follows from the FRW equations (2) that an equation of state with negative pressure p ≃ −ρ is required to obtain exponential inflation. In order to obtain an accelerated expansion ("generalized inflation"), an equation of state with p < −(1/3)ρ is needed. In the context of renormalizable quantum field Lagrangians, it is only scalar fields which provide the possibility to obtain inflation. From the action of a scalar quantum field ϕ (in an expanding space-time) it immediately follows that the energy density and pressure of such a field are given by
ρ = 1 2 (φ) 2 + 1 2a 2 (∇ϕ) 2 + V (ϕ) (5) p = 1 2 (φ) 2 − 1 6a 2 (∇ϕ) 2 − V (ϕ) ,(6)
where V (ϕ) is the potential energy density. It thus follows that if the scalar field is homogeneous and static, but the potential energy positive, then the equation of state p = −ρ necessary for exponential inflation results. This is the idea behind potential-driven inflation. Guth's initial hope was that the same scalar field responsible for the spontaneous symmetry breaking of a unified gauge symmetry could be the inflaton, the field responsible for inflation. However, this hope cannot be realized since the potential of such a scalar field is too steep in order to provide a period of more than a few Hubble times H −1 during which the field kinetic energy remains negligible. This follows almost immediately by considering the variational equation of motion for ϕ, which in the absence of spatial gradients becomes
ϕ + 3Hφ = −V ′ (ϕ) ,(7)
where a prime denotes the derivative with respect to ϕ.
Although at the present time there are many models for inflation, there is no single convincing one. Chaotic inflation [8] is a prototypical model. It assumes the existence of a new scalar field, the "inflaton" with a smooth potential (see Fig. 6). The inflaton is so weakly coupled that it does not start out in thermal equilibrium in the early Universe. Most of the phase space of initial conditions consists of values of ϕ which are large (compared to the Planck scale). It follows from the equation of motion (7) that for such initial conditions the scalar field will roll slowly (in the sense thatφ ≪ 3Hφ) for a long (compared to H −1 ) time, thus yielding inflation. Once ϕ drops much below the Planck scale, the kinetic energy begins to dominate, the field oscillates around its minimum and releases its energy to regular matter via interaction terms in the Lagrangian.
IV. PROGRESS IN INFLATIONARY COSMOLOGY
Simple prototypical models of inflation predict a scaleinvariant spectrum of adiabatic fluctuations (adiabatic in this context means that the energy density fluctuations in all components of matter are proportional), and this will arise naturally in the context of inflation if during the phase of inflationary expansion a single scalar field dominates the dynamics). This prediction of the inflationary scenario has recently been confirmed to high accuracy by the measurements of CMB anisotropies. Microwave anisotropies are usually quantified by expanding the temperature maps into spherical harmonics and calculating the power at different values of the angular quantum number l. Fig. 7 [1] shows a compilation of the recent measurements. The predictions of a theory based on a scale-invariant spectrum of adiabatic fluctuations with a background cosmology which is taken to be dominated today by a remnant cosmological constant which contributes 70% to the density required for a spatially flat Universe (most of the other 30% is believed to be made up of cold dark matter) fits the data very well, whereas a model with a background dominated by cold dark matter does not reproduce the observed details of the spectrum in the region of the oscillations of the spectrum.
Note the oscillations in the spectrum at values of l larger than 100. These "acoustic" oscillations are an imprint of the coherence of the primordial fluctuations. Refer to Fig. 8 for a sketch of the mechanism by which density fluctuations induce CMB anisotropies. If we imagine the inhomogeneities as a superposition of plane waves (which evolve independently in the early Universe since their amplitudes are small), then inflation predicts that these waves start oscillating with the same phase at the time when the wavelength equals the Hubble radius. Thus, when measured at the time of last scattering, the phase of the wave is a periodically varying function of l. Maxima and minima of the waves result in maxima of the temperature anisotropies on the corresponding angular scales, nodes of the waves result in minima (see e.g.
[9] for a detailed discussion of the physics of the acoustic oscillations).
The theoretical understanding of the theory by which quantum vacuum fluctuations on microscopic scales evolve to give rise to classical inhomogeneities on cosmological wavelengths is one of the main areas of progress in inflationary cosmology. Since it is necessary to propagate the fluctuations on scales larger than the Hubble radius, the Newtonian theory of cosmological perturbations obviously is inapplicable, and a general relativistic analysis is needed. On these scales, matter is essentially frozen in comoving coordinates. However, space-time fluctuations can still increase in amplitude. In principle, it is straightforward to work out the general relativistic theory of linear fluctuations. One linearizes the Einstein and scalar matter field equations about an expanding FRW background cosmology. Tedious but straightforward algebra gives a set of coupled linear differential equations for the metric and matter perturbations.
At the level of linear fluctuations, perturbations with different wavelengths decouple. Hence, the analysis becomes simple if we work in momentum space. Furthermore, there are several independent degrees of freedom. First, there are gravitational waves, space-time perturbations which do not couple to matter. Next, there are vector perturbations which correspond to rotational degrees of freedom. Finally, there are the scalar metric fluctuations, space-time inhomogeneities produced by matter perturbations. The scalar metric fluctuations are the most difficult to analyze, in particular since one must be careful to isolate the physical degrees of freedom from gauge artefacts, modes which correspond to space-time coordinate transformations. For matter described by a single scalar field, as is the case in many prototypical inflationary models, there is only one matter fluctuating degree of freedom. According to the Einstein equations, matter fluctuations are coupled to corresponding scalar metric perturbations. There can be no scalar metric fluctuations without matter inhomogeneities. Thus, restricting attention to scalar metric fluctuations of one particular wavelength, there is only one physical degree of freedom. This implies that the analysis of scalar metric fluctuations is reducible to the theory of a single free scalar field (free because we are dealing with linear fluctuations) in a time-dependent background (the time-dependence is set by the background cosmology).
As mentioned earlier, the primordial fluctuations in an inflationary cosmology result from quantum fluctuations. At the linearized level, the equations describing both gravitational and matter perturbations can be quantized in a consistent way (for a detailed review see [10]). The first step of this analysis is to expand the gravitational and matter action to quadratic order in the metric and matter fluctuation variables. Focusing on the scalar metric perturbations, it turns out that one can express the resulting action in terms of a variable v which is the coordinate-invariant scalar matter field fluctuation [11]
v = a δϕ + (ϕ (0) ) ′ H Φ .(8)
Here, ϕ (0) is the background scalar matter field, δϕ is the scalar field fluctuation, a prime denotes the derivative with respect to conformal time η (η being defined by dt = adη), H = a ′ /a, and Φ is the generalized Newtonian gravitational potential (in a coordinate system in which the metric tensor -including fluctuations -is diagonal, 2Φ is the perturbation of the time-time component of the metric). It can then be shown that the action S 2 for the fluctuations reduces to the action of a single gauge invariant free scalar field (namely v) with a time dependent mass [12,13] (the time dependence reflects the expansion of the background space-time)
S 2 = 1 2 dtd 3 x v ′2 − (∇v) 2 + z ′′ z v 2 ,(9)
where the function
z = aϕ ′ 0 H ,(10)
a function of the background fields, determines the effective mass for the field v. The action S 2 has the same form as the action for a free scalar matter field in a time dependent gravitational or electromagnetic background, and we can use standard methods to quantize this theory (see e.g. [14]). Each momentum mode of the field obeys a harmonic oscillator equation with time dependent mass. The time dependence of the mass is reflected in the nontrivial form of the solutions of the mode equations. Let us now follow the evolution of a particular mode from the beginning of inflation until late times. The mode starts out in its vacuum state when its wavelength is smaller than the Hubble radius. While the wavelength remains smaller than the Hubble radius, the mode functions oscillate since the time dependence of the mass is negligible, as can be seen from (9). However, once the mode crosses the Hubble radius, the spatial gradient term becomes negligible and the mass term begins to dominate. The mode function thus no longer oscillates, but its amplitude begins to grow as v ∼ z. This corresponds to squeezing of the vacuum state and corresponds to the generation of fluctuations. This calculation can be followed up to the time the mode re-enters the Hubble radius at late times. It turns out (see Section 5) that the amplitude of the resulting density generically is predicted to be several orders of magnitude larger than what is compatible with current observations, unless some parameter in the quantum field theory Lagrangian is set to a value much smaller than would be inferred from dimensional analysis.
V. PROBLEMS OF INFLATIONARY COSMOLOGY
In spite of the spectacular success of the inflationary scenario in predicting a spectrum of microwave anisotropies and large-scale density fluctuations in excellent agreement with the recent observation, scalar fielddriven inflationary models suffer from some serious conceptual problems.
The main success of inflationary cosmology is that it provides a causal theory for the generation of large-scale cosmological fluctuations. However, this success directly leads to a major problem for most realizations of scalar field-based models of inflation studied up to now. It concerns the amplitude of the density perturbations which are induced by quantum fluctuations during the period of accelerated expansion as discussed in the previous section. Unless a parameter in the scalar field potential is set to have a value several orders of magnitude smaller than what would be given by dimensional analysis, the models predict an amplitude of the fluctuation spectrum several order of magnitude larger than the predicted amplitude. For example, in a model with a single inflaton field with quartic potential, the quartic coupling constant λ must be of the order of 10 −12 in order that the resulting amplitude of fluctuations agrees with observations. This situation is clearly unsatisfactory for a cosmological scenario motivated by the desire to eliminate cosmological fine-tunings. There have been many attempts to justify such small parameters based on specific particle physics models, but no single convincing model has emerged. However, this is probably the least serious of the problems mentioned here.
In many models of inflation, in particular in chaotic inflation, the period of inflation is so long that comoving scales of cosmological interest today correspond to a physical wavelength much smaller than the Planck length at the beginning of inflation. In extrapolating the evolution of cosmological perturbations according to linear theory to these very early times, one is implicitly making the assumption that the theory remains perturbative to arbitrarily high energies, and that the classical theory of general relativity remains the appropriate framework for describing space-time. Both of these assumptions are clearly not justified. It has recently been shown that some (admittedly quite violent) changes to the physics on length scales smaller than the Planck length can lead to a spectrum of density fluctuations totally different from what the usual theory predicts [15]. This can be called the "trans-Planckian problem" for inflationary cosmology. On the other hand it shows that in an inflationary Universe, the spectrum of fluctuations can potentially be used to explore Planck-scale physics, thus turning the "problem" into a "window" of opportunity. Planckscale physics may not only alter the spectrum of fluctuations, it can also dramatically alter the background cosmology, as is seen in "Pre-big-bang Cosmology" [16], a string-motivated dilaton-gravity model which undergoes a dilaton-dominated phase of super-exponential expansion.
Scalar field-driven inflation does not eliminate singularities from cosmology. Although the standard assumptions of the Penrose-Hawking theorems (the theorems which is the context of Einstein gravity coupled to classical fluid matter show that an initial cosmological singularity is inevitable) break down if matter has an equation of state with negative pressure, as is the case during inflation, nevertheless it can be shown that an initial singularity persists in inflationary cosmology [17]. This implies that the theory is incomplete. In particular, the physical initial value problem is not defined.
The Achilles heel of our current inflationary models in without doubt the "cosmological constant problem". There is some as of yet unknown mechanism which prevents the bare cosmological constant, which in theories with quantum fields is predicted to be at least 62 orders of magnitude larger than the observational limit (this number comes from assuming the cancellation of vacuum energies on scales larger than the supersymmetry breaking scale taken to be about 1TeV), from gravitating. How do we know that this unknown mechanism does not also lead the transient "cosmological constant" given by the potential energy of the scalar field to be gravitationally inert, thus eliminating the basis of scalar field-driven inflation?
VI. FUTURE DIRECTIONS
In the light of the problems of scalar field potentialdriven inflation discussed in the previous sections, many cosmologists have begun thinking about new avenues towards early Universe cosmology which, while maintaining (some of) the successes of inflation, address and resolve some of its difficulties. In the same way that inflationary cosmology builds on and transcends standard big bang cosmology by making use of a new theory of matter (quantum field theory in the case of inflation), it is quite likely that a resolution of the problems of inflationary cosmology will once again come from an improved description of matter at high energies. The main candidate at the moment for such a theory is string theory. String theory, in fact, will also lead to a modified picture of space-time at short distances, and may allow a unified quantum description of both the background space-time and of the cosmological fluctuations.
There are at least two ways in which an improved cosmological model could come about. It is possible that a better understanding of string theory will lead to a convincing realization of inflation which does not suffer from the problems mentioned in the previous section, and that many of the cosmological predictions of inflation for present day observations would remain unchanged. However, it is also possible that string theory will provide alternatives to inflationary cosmology, a possibility which would lead to clear observational signatures.
There are several reasons to hope that string theory might lead to an improved realization of inflation. One reason is that the vacuum space of string theory is a complicated moduli space, and the individual directions in this space (e.g. radii of extra dimensions) can be associated with scalar fields with potentials which vanish at the perturbative level. It is hoped that nonperturbative corrections might generate in a natural way the flat potentials required for inflation. The realization that string theory contains higher dimensional fundamental objects called branes has led to speculations that our space-time might be a brane in a higher dimensional space-time. These "brane-world" scenarios have opened new avenues to realizing inflation in the context of string theory (see e.g. [18] for a recent attempt and for references to earlier work). However, it is also possible that string theory will generate an alternative to inflationary cosmology which maintains most of the successes of inflation, but at the same time gives rise to predictions with which this new theory can be distinguished from inflation. One approach which has received a lot of recent attention is pre-big-bang cosmology [16], a theory in which the Universe starts in an empty and flat dilaton-dominated phase which leads to super-exponential expansion. A nice feature of this theory is that the mechanism of acceleration is completely independent of a scalar field potential and thus independent of the cosmological constant issue. Prebig-bang cosmology in its simplest realizations does not give rise to a spectrum of scale-invariant adiabatic fluctuations. Rather, the spectrum is isocurvature (see e.g. [19] for a recent review and original references). Another recent attempt to provide an alternative to inflation in the context of string theory is the "Ekpyrotic Universe" scenario [20], a nonsingular cosmological model designed to solve the problems of SBB cosmology mentioned in Section 1 and to provide a spectrum of scale-invariant adiabatic perturbations without any period of accelera-tion (see, however, [21][22][23] for criticisms of this scenario).
It is, however, also possible that some of the problems of inflationary cosmology mentioned in Section 5 can be addressed within the context of more conventional physics (general relativity plus quantum field theory). In particular, it is possible that some of the problems are consequences of neglecting the intrinsically nonlinear structure of general relativity (see e.g. [24] for some speculations along these lines).
ACKNOWLEDGMENTS
I am grateful to Douglas Scott for providing Figures 1 and 7. This work was supported in part by the US Department of Energy under Contract DE-FG02-91ER40688, Task A.
For a Universe obeying the cosmological principle (and neglecting the possible presence of a cosmological constant) the Einstein equations reduce to the Friedmann-Robertson-Walker (FRW
FIG. 2 .
2A space-time diagram (physical distance xp versus time t) illustrating the homogeneity problem: the past light cone ℓp(t) at the time trec of last scattering is much larger than the forward light cone ℓ f (t) at trec.
FIG. 3 .
3A sketch (conformal separation vs. time) of the formation of structure problem: the comoving separation dc between two clusters is larger than the forward light cone at time teq.
FIG. 4 .
4FIG. 4.
FIG. 6 .
6Sketch of a potential V for a scalar field f in chaotic inflation.
FIG. 7 .
7Compilation of recent data on the angular power spectrum of CMB anisotropies. The green (-) data points are a compilation of the data prior to March 2000; the others represent data from the following experiments: Boomerang (2001 release), blue (diamond); Maxima (2001 release), purple (triangle); DASI, red (x), CBI, black (square).
FIG. 8 .
8Sketch of the geometry which underlies the imprint of density fluctuations as anisotropies in the cosmic microwave background. The surface of last scattering (solid line LS) is a surface of constant temperature. Thus, photons c1 and c2 from different directions in the sky are redshifted by different amounts before reaching us (we are at the point along the t-axis where the lines labelled c1 and c2 hit), and therefore arrive with different temperatures.
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| []
|
[
"Object Instance Mining for Weakly Supervised Object Detection",
"Object Instance Mining for Weakly Supervised Object Detection"
]
| [
"Chenhao Lin \nSenseTime Research\n\n",
"Siwen Wang \nDalian University of Technology\n116024DalianChina\n",
"Dongqi Xu [email protected] \nSenseTime Research\n\n",
"Yu Lu \nSenseTime Research\n\n",
"† Wayne Zhang \nSenseTime Research\n\n"
]
| [
"SenseTime Research\n",
"Dalian University of Technology\n116024DalianChina",
"SenseTime Research\n",
"SenseTime Research\n",
"SenseTime Research\n"
]
| []
| Weakly supervised object detection (WSOD) using only image-level annotations has attracted growing attention over the past few years. Existing approaches using multiple instance learning easily fall into local optima, because such mechanism tends to learn from the most discriminative object in an image for each category. Therefore, these methods suffer from missing object instances which degrade the performance of WSOD. To address this problem, this paper introduces an end-to-end object instance mining (OIM) framework for weakly supervised object detection. OIM attempts to detect all possible object instances existing in each image by introducing information propagation on the spatial and appearance graphs, without any additional annotations. During the iterative learning process, the less discriminative object instances from the same class can be gradually detected and utilized for training. In addition, we design an object instance reweighted loss to learn larger portion of each object instance to further improve the performance. The experimental results on two publicly available databases, VOC 2007 and 2012, demonstrate the efficacy of proposed approach. | 10.1609/aaai.v34i07.6813 | [
"https://ojs.aaai.org/index.php/AAAI/article/download/6813/6667"
]
| 211,020,978 | 2002.01087 | 1a112151151810dd76504aace110be6e95e966ba |
Object Instance Mining for Weakly Supervised Object Detection
Chenhao Lin
SenseTime Research
Siwen Wang
Dalian University of Technology
116024DalianChina
Dongqi Xu [email protected]
SenseTime Research
Yu Lu
SenseTime Research
† Wayne Zhang
SenseTime Research
Object Instance Mining for Weakly Supervised Object Detection
Weakly supervised object detection (WSOD) using only image-level annotations has attracted growing attention over the past few years. Existing approaches using multiple instance learning easily fall into local optima, because such mechanism tends to learn from the most discriminative object in an image for each category. Therefore, these methods suffer from missing object instances which degrade the performance of WSOD. To address this problem, this paper introduces an end-to-end object instance mining (OIM) framework for weakly supervised object detection. OIM attempts to detect all possible object instances existing in each image by introducing information propagation on the spatial and appearance graphs, without any additional annotations. During the iterative learning process, the less discriminative object instances from the same class can be gradually detected and utilized for training. In addition, we design an object instance reweighted loss to learn larger portion of each object instance to further improve the performance. The experimental results on two publicly available databases, VOC 2007 and 2012, demonstrate the efficacy of proposed approach.
Introduction
Object detection has always been one of the most essential technologies in computer vision field. Deep learning techniques introduced in recent years have significantly boosted state-of-the-art approaches for object detection (Girshick 2015;Liu et al. 2016;Redmon et al. 2016;Ren et al. 2015). However, these approaches usually require large-scale manually annotated datasets (Russakovsky et al. 2015). The high-cost of time-consuming accurate bounding box annotations, has impeded the wide deployment of CNN-based object detection technologies in real applications.
To address this limitation, weakly supervised object detection (WSOD) technology, which requires only imagelevel labels for training, has been introduced and explored (Bilen and Vedaldi 2016;Diba et al. 2017;Jie et al. 2017;Oquab et al. 2015;Zhang et al. 2018b;Tang et al. 2017; Figure 1: The original images and corresponding objectness maps to show the evolution of object instance mining during learning process (from left to right). The first to fourth columns represent random initialization, epoch1, epoch3, and final epoch, respectively. Blue or red bounding boxes indicate the detected instances (top-scoring proposals after NMS) with detection scores < 0.5 or ≥ 0.5. Zhang et al. 2018a;Shen et al. 2018;Arun, Jawahar, and Kumar 2019;Pan et al. 2019). Although many approaches have been developed for WSOD and achieved promising results, the lack of object instance level annotations in images leads to huge performance gap between WSOD and fully supervised object detection (FSOD) methods.
Most previous approaches follow the framework of combining multiple instance learning (MIL) with CNN. This framework usually mines the most confident class-specific object proposals for learning CNN-based classifier, regardless of the number of object instances appearing in an image. For the images with multiple object instances from the same class, the object instances (fully annotated with bounding boxes in FSOD) with lower class-specific scores will be probably regarded as background regions. Many images in the challenging VOC datasets contain more than one object instance from the same class. For example, in VOC2007 trainval set the number of image-level object labels and the annotated object instances are 7,913 and 15,662 respectively, which indicates that at least 7,749 instances are NOT selected during training. In this case, the selected object instances with relatively limited scale and appearance variations, may not be sufficient for training a CNN classifier with strong discriminative power. Moreover, the missing instances may be selected as negative samples during training, which may further degrades the discriminative capability of the CNN classifier.
In this paper, an end-to-end object instance mining (OIM) framework is proposed to address the problem of multiple object instances in each image for WSOD. OIM is based on two fundamental assumptions: 1) the highest confidence proposal and its surrounding highly overlapped proposals should probably belong to the same class; 2) the objects from the same class should have high appearance similarity. Formally, spatial and appearance graphs are constructed and utilized to mine all possible object instances present in an image and employ them for training. The spatial graph is designed to model the spatial relationship between the highest confidence proposal and its surrounding proposals, while the appearance graph aims at capturing all possible object instances having high appearance similarities with the most confident proposal. By integrating these two graphs into the iterative training process, an OIM approach that attempts to accurately mine all possible object instances in each image with only image-level supervision is proposed. With more object instances for training, a CNN classifier can have stronger discriminative power and generalization capabilities. The proposed OIM can further prevent the learning process from falling into local optima because more objects per-class with high similarity are employed for training. The original images and the corresponding objectness maps shown in Figure 1 illustrate that with the increasing number of iterations, multiple object instances belonging to the same class can be detected and are employed for training using the proposed approach.
Another observation from existing approaches is that the most confident region proposal is easy to concentrate on the locally distinct part of an object, especially for non-rigid objects such as human and animals. This may lead to the problem of detecting only small part of the object. To alleviate this problem, an object instance reweighted loss using the spatial graph is presented to help the network detect more accurate bounding box. This loss tends to make the network pay less attention on the local distinct parts and focus on learning the larger portion of each object.
Our key contributions can be summarized as follows:
• An object instance mining approach using spatial and appearance graphs is developed to mine all possible object instances with only image-level annotation, and it can significantly improve the discriminative capability of the trained CNN classifier. • An object instance reweighted loss by adjusting the weight of loss function of different instances is proposed to learn more accurate CNN classifier.
Related Work
With only image-level annotations, most existing approaches implement weakly supervised object detection (Bilen and Vedaldi 2016;Tang et al. 2017;Jie et al. 2017;Wan et al. 2019) through multiple instance learning (MIL) framework (Dietterich, Lathrop, and Lozano-Pérez 1997). The training images are firstly divided into bag of proposals (instances) containing positive target objects and negative backgrounds and CNN classifier is trained to classify the proposals into different categories. The most discriminative representation of instances is easy to be distinguished by such classifier that may make network trap into local optima.
Recently, Bilen et al. (Bilen and Vedaldi 2016) proposed a weakly supervised deep detection network (WSDDN) to perform object localization and classification simultaneously. Following this work, Tang et al. (Tang et al. 2017) introduced an online instance classifier refinement (OICR) strategy to learn larger portion of the objects. Such approach improves the performance of WSOD. However, it is also easy to trap into local optima since only the most discriminative instance is selected for refinement. Wan et al. (Wan et al. 2018) developed a min-entropy latent model to classify and locate the objects by minimizing the global and local entropies, which was proved to effectively boost the detection performance. Wan et al. (Wan et al. 2019) also attempted to address the local minima problem in MIL using continuation optimization method. In references Shen et al. 2019;Li et al. 2019), the authors attempted to integrate segmentation task into weakly supervised object detection to obtain more accurate object bounding boxes. However, these methods require complex training framework with high training and test time complexity.
The authors in (Tang et al. 2018) proposed to use proposal cluster to divide all proposals into different small bags and then classifier refinement was applied. This approach attempted to classify and refine all possible objects in each image. However, many proposals containing part of the object might be ignored using proposal cluster during the training. Gao et al. (Gao et al. 2018) introduced a count-guided weakly supervised localization approach to detect per-class objects in each image. A simple count-based region selection algorithm was proposed and integrated into OICR to improve the performance of WSOD. However, the extra count annotations which needs a certain human labor are introduced and their method requires an alternative training process which can be time-consuming. In this paper, the count annotation is replaced by the proposed OIM algorithm without extra labor cost.
Proposed Approach Overall Framework
The overall architecture of the proposed framework illustrated in Figure 2 mainly consists of two parts. The first part is a multiple instance detector (MID) which is similar to the structure presented in (Bilen and Vedaldi 2016). It performs region selection and classification simultaneously using a weighted MIL pooling. The second part is the proposed object instance mining and the proposed instance reweighted Figure 2: Architecture of the proposed object instance mining framework. MID represents multiple instance detector and OIM indicates proposed object instance mining. L CE is multi-class cross entropy loss and L OIR is proposed instance reweighted loss.
loss. During the training phase, we firstly adopt MID to classify the region proposals into different predicted classes. Then the detection outputs and proposal features are integrated to search all possible object instances from the same class in each image using spatial and appearance graphs. In addition, the instance reweighted loss is designed to learn larger portion of each object. As can be seen from the Figure 2, the multiple object instances belonging to the same class can be accurately detected using the proposed method.
Object Instance Mining
Previous methods (Tang et al. 2017;Gao et al. 2018;Zhang et al. 2018b;Wei et al. 2018) often select the most confident proposal from each class as the positive sample to refine the multiple instance detector. The performance improvement can be limited using these methods, since only the top-scoring and surrounding proposals are selected for refinement. While in many conditions, there are multiple object instances belonging to the same class in an image. Those ignored object instances may be regarded as negative samples during the training that may degrade the performance of WSOD. Therefore, we propose an object instance mining (OIM) approach by building spatial graphs and appearance graphs to search all possible object instances in each image and integrate them into the training process. Based on the assumption that the top-scoring and surrounding proposals with large overlaps (spatial similarity) should have the same predicted class, the spatial graphs can be built. We also assume that the objects from the same class should have similar appearance. Based on the similarities between the top-scoring proposal and the other proposals, the appearance graphs are built. Then we search all possible object instances in each image and employ them for training through these graphs.
Given an input image I with class label c, a set of region proposals P = {p 1 , ..., p N } and their corresponding confidence scores X = {x 1 , ..., x N }, the core instance (proposal) p ic with the highest confidence score x ic can be selected.
Here i c donates the index of this core instance (proposal). The core spatial graph can be defined by G s ic = (V s ic , E s ic ), where each node in V s ic represents a selected proposal which has the overlap, i.e. spatial similarity, with the core instance larger than a threshold T . Each edge in E s ic represents such spatial similarity. All the nodes in spatial graph G s ic will be selected and labelled to the same class as p ic .
We define feature vectors of each proposal as F = {f 1 , ..., f N } and it can be generated from the fully connected layer. Each vector encodes a feature representation of a region proposal. Then the appearance graph is defined as G a = (V a , E a ), where each node in V a is a selected proposal which has high appearance similarity with the core instance and each edge in E a represents the appearance similarity. This similarity can be calculated from the feature vectors of core instance and one of the other proposals (e.g. p j ) using the Euclidean distance, denoted as follows,
D ic,j = f ic − f j 2 .
(1) Only if the proposal p j meets the condition that D ic,j < αD avg and p j has no overlap with all the proposals previously selected, such proposal will be added into the nodes in G a . D avg represents the average inter-class similarity of the core spatial graph G s ic using average distance of all the nodes in G s ic and it can be defined as follows,
D avg = 1 M k D ic,k , s.t. IoU(p ic , p k ) > T.(2)
where p k represents the node meet the constraints above and M indicates the number of these nodes in G s ic . α is a hyper parameter which is determined by experiments.
The proposed object instance mining (OIM) approach using spatial and appearance graphs is summarized in Algorithm 1. We also build spatial graph G s for each node in appearance graph G a and then all these nodes will be included for training. If no proposal has high similarity with the core instance, only the core instance and surrounding proposals, i.e. spatial graph G s ic will be employed. In such a way, more instances from the same class with similar appearance and different poses will be employed for training. It results in that not only more object instances can be detected but also more accurate detected boxes can be learned. Figure 3 illustrates the process to detect all possible object instances from to the same class using spatial and appearance graphs. Figure 3 (a) is the core spatial graph and
Instance Reweighted Loss
In addition to exploring all possible object instances in each image, we also design an object instance reweighted loss to learn more accurate detected boxes. During the iterative learning process, the CNN-based classifier is easy to learn the most distinct part of each object instance instead of the whole body, especially for the non-rigid one. We propose to assign different proposal weights to individual proposals to balance the weight of the top-scoring proposal and surrounding less discriminative ones. Thus the larger portion of each instance is expected to be detected. Given an image with label Y and predicted label Y j = [y 0,j , y 1,j , ..., y C,j ] T ∈ R (C+1)×1 for the j-th proposal in a spatial graph G s , where y c,j = 1 or 0 indicates the proposal belonging to class c or not, and c = 0 is index of background class. The loss in Eq. 3 is similar to the loss in (Tang et al. 2017), where w j is the loss weight of j-th proposal. x s c,j with class label c in G s , are the proposals used for training and x s c,ic is center (core) proposal with the highest score.
L = − 1 |P| |P| j=1 C+1 c=1 w j y c,j log x s c,j .(3)
It can be seen from Eq. 3 that proposals in each spatial graph contribute equally. Thus, the non-center proposals with relative low scores in each spatial graph are difficult to be learned during training. To address this problem, an instance reweighted loss function is designed as follows,
L = − 1 |P| |P| j=1 C+1 c=1 w j y c,j (1 + z s j ) log x s c,j ,(4)
where z s j is introduced to balance the proposal weights in spatial graph G s as defined in Eq. 5. β is hyper-parameter.
z s j = β, j = i c β − 1, j = i c(5)
To guide the network to pay more attention on learning the less discriminative regions of the object instance in each graph G s , we balance the weight of the surrounding less discriminative proposals with the center proposal using Eq. 4 and Eq. 5. As a result, gradients of surrounding proposals are scaled up to (1 + β) of its original value, while gradient of the center proposal is scaled to β of its original value during back-propagation. Similar to the implementation in (Gao et al. 2018), we also use the standard multi-class cross entropy loss for the multi-label classification and it is combined with the proposed instance reweighted loss for training.
Experiments Datasets and Evaluation Metrics
Following the previous state-of-the-art methods on WSOD, we also evaluate our approach two datasets, PASCAL VOC2007 (Everingham et al. 2010) and VOC2012 (Everingham et al. 2015), which both contain 20 object categories. For VOC2007, we train the model on the trainval set (5,011 images) and evaluate the performance on the test set (4,952 images). For VOC2012, the trainval set (11,540 images) and the test set (10,991 images) are used for training and evaluation respectively. Additionally, we train our model on the VOC2012 train set (5,717 images) and proceed evaluation on the val set (5,823 images) to further validate the effectiveness of proposed approach. Following previous work, we use mean average precision (mAP) to evaluate the performance of proposed approach. Correct localization (CorLoc) is applied to evaluate the localization accuracy.
Implementation Details
To make a fair comparison, VGG16 model pre-trained on the ImageNet dataset (Russakovsky et al. 2015) is adopted as the backbone network to finetune the CNN classifier. The object proposals are generated using Selective Search (Uijlings et al. 2013). The batch size is set to 2, and the learning rates are set to 0.001 and 0.0001 for the first 40K and the following 50K iterations respectively. During training and test, we take five image scales {480, 576, 688, 864, 1200} along with random horizontal flipping for data augmentation. Following (Tang et al. 2017), the threshold T is set to 0.5. With the increased number of iterations, the network has more stable learning ability, we dynamically set the hyper parameters α as α 1 = 5 for the first 70K and α 2 = 2 for the following 20K iterations. β are empirically set to 0.2 in our experiments. We also analyze the influence of these parameters in the ablation experiments section. 100 top-scoring region proposals are kept and Non-Maximun Suppression with IoU of 0.3 per class is performed to calculate mAP and CorLoc.
Comparison with State-of-the-arts
State-of-the-art WSOD methods are used for comparison to validate the effectiveness of the proposed approach. Table 1 shows performance comparison in terms of mAP on VOC2007 test set. By only using OIM, better or similar results can be achieved compared with previous SOTA methods such as MELM, SDCN, etc. We attribute this improvement to the OIM, which increases the representation capability of the trained CNN by searching more objects from the same class and employing them into training. As the detected bounding boxes and objectness maps shown in Figure 1, the confidence scores of less discriminative objects are gradually improved and more objects from the same class can be detected during the training. It further proves that integrating the less discriminative objects into training improves the performance for WSOD. Further performance improvement can be achieved using the proposed instance reweighted loss. The proposed approach achieves a mAP of 50.1%, which outperforms the PCL,C-WSL * , SDCN, WS-JDS methods, etc, and the performance is similar to the result of C-MIL. We further used the learned objects as pseudo ground-truth to train a Fast-RCNN-based detector, our approach also achieve better or similar performance as compared with previous state-of-the-art methods. In particular, by only using the proposed OIM strategy, our approach outperforms C-WSL method by 1.4 % without introducing extra per-class count supervision. Our work attempts to include all possible object instances from each class for training since many images contain more than one per-class object instance. Figure 5 illustrates that most classes in two datasets have more than one object instance in an image. Specifically, almost half of categories contain more than two object instances in an image. Especially for class "sheep", which the average number of sheep appearing in an image is larger than 3, our OIM method (57.9 % mAP) performs better than all the other methods. In addition, for most non-rigid objects ("cat", "dog", "horse", "person', etc.), as can be seen from Table 1, by applying in- -RPN -------------------- (Tang et al. 2017) 60.6 PCL (Tang et al. 2018) 62.7 C-WSL* (Gao et al. 2018) 63.5 MELM (Wan et al. 2018) 61.4 WS-JDS (Shen et al. 2019) 64.5 C-MIL (Wan et al. 2019) 65.0 OICR+W-RPN (Singh and Lee 2019 CorLoc is also used as the evaluation metric to ascertain the performance of proposed method. Table 2 shows performance comparison in terms of CorLoc on the VOC2007 trainval set. Our result outperforms all existing state-of-theart methods when Fast-RCNN detector is not used. The proposed OIM framework iteratively explores more object instances and larger portion of the instances from the same class with similar appearance and different poses for training, which makes more accurate detected boxes can be learned. Therefore, the proposed approach not only brings the mAP improvements but also makes the detected boxes Table 3: Comparison with the state-of-the-arts in terms of mAP (%) on the VOC2012 test set.
Method
Dataset mAP MELM (Wan et al. 2018) train/val 40.2 C-WSL (Gao et al. 2018) train/val 43.0 OIM+IR train/val 44.4 OICR (Tang et al. 2017) trainval/test 37.9 PCL (Tang et al. 2018) trainval/test 40.6 MELM (Wan et al. 2018) trainval The proposed approach is also evaluated on VOC2012 dataset. Since some approaches (Gao et al. 2018) only use validation set of VOC2012 for evaluation, we use both test and val set to evaluate the proposed approach. In Table 3, the detection results in terms of mAP on test and val set are provided respectively. Table 4 lists the CorLoc results on VOC2012 trainval set. The experimental results in Tables 3 and 4 validate the effectiveness of the proposed approach. Figure 4 visualizes the detection results on the VOC2007 test set. The successful (IoU ≥ 0.5) and failed (IoU < 0.5) detections are marked with red and yellow bounding boxes respectively. The green bounding boxes are the groundtruths. The first two rows indicate our approach can detect tight boxes even multiple objects from the same class cooccur in an image, e.g. "cow", "sheep". The last row shows some failed cases, which are often attribute to localizing the most discriminative parts of non-rigid objects, grouping multiple objects, and background clutter, e.g. "human".
Ablation Experiments
We performed ablation experiments to illustrate the effect of parameters introduced in proposed object instance mining (α) and instance reweighted loss (β). Table 5 indicates when parameter α (α 1 used in the first 70K and α 2 used in the following 20K iterations) becomes smaller or larger, the performance of proposed approach will degrade. If the parameter α is too small, very less instances will be selected in the appearance graph for training. It results in that in many images, only the most discriminative object is selected and used for training. If the parameter α is too large, many false instances (background proposals) will be employed for training and it also leads to performance drop. For the proposed instance reweighted loss, as also can be seen from the Table 5, with the increasing of β the performance decreases. We also studied the WSOD performance by only using appearance graph (AG) or spatial graph (SG) to evaluate their effectiveness separately. The first two columns in Table 6 illustrate the experimental results in terms of mAP on the VOC2007 test set. We can see that the performance can be significantly improved for WSOD by only using appearance or spatial graph.
The effectiveness of the proposed instance reweighted loss is also evaluated. We apply the network structure in OICR but just replace the loss with instance reweighted loss. The performance achieved using the proposed instance reweighted loss in terms of mAP on the VOC2007 test set is shown in Table 6. It can be seen that the mAP can be im- Figure 6: Evolution of object detection during learning process w/o using instance reweighted loss (from left to right). The upper part of each subfigure is the result of OICR (Tang et al. 2017) and the lower part is the result of our method. proved from 41.2% (Tang et al. 2017) to 43.4% by only using instance reweighted loss. The visual comparison shown in Figure 6 also illustrates that larger portion of the object can be gradually detected using the proposed loss. By incorporating the OIM with instance reweighted loss, the best performance (mAP 50.1%) can be achieved.
Conclusion
In this paper, an end-to-end object instance mining framework has been presented to address the limitations of existing approaches for WSOD. Object instance mining algorithm is performed using spatial and appearance graphs to make the network learn less discriminative object instances. Thus more possible objects belonging to the same class can be detected accordingly. Without introducing any extra count information, the proposed approach has achieved improved performance comparable to many state-of-the-art results. The object instance reweighted loss is designed to further help the OIM by learning the larger portion of the target object instances in each image. Experimental results on two publicly available datasets illustrate that the proposed approach achieves competitive or superior performance than state-of-the-art methods for WSOD.
Figure 3 : 9 Compute
39Process to explore all possible object instances from the same class using OIM. (a)-(c) illustrate the spatial and appearance graphs of different epochs and (d) shows all detected instances. Blue bounding boxes represent the detected core instance with the highest confidence score. Red bounding boxes represent the other detected instances which have high appearance similarities with the core instance. Blue and red line represent spatial and appearance graph edge respectively. Red broken line in (b) means the appearance similarities is smaller than the threshold and thus the object instances are not employed in this stage.Algorithm 1: Object Instance Mining Input: Image I, region proposals P = {p1, ..., pN }, image label Y = {y1, y2, ...yc} Output: All the nodes V a in appearance graph 1 Feed Image I and its proposals into the network to produce feature vectors F = {f1, ..., fN } 2 for c in C, C is the list of training data class do 3 if yc == 1 then 4 V s ← ∅, V a ← ∅,D ← 0, Davg ← 0, M ← 0, flag ← 0 5 Choose the top-scoring proposal ic 6 V s ic ← pi c , V a ← pi c 7 for j = 1 to N do 8 Compute the appearance similarity Di c ,j using Eq. 1 IoU(pi c ,pj) 10 if IoU(pi c ,pj) > T then 11 V s ic ← pj 12 D ← Di c ,j + D,M ← M + 1 13 Davg ← D M 14 Sort (ascend) P based on Di c ,j 15 for j = 1 to N do 16 if Di c ,j < αDavg then 17 if ∃ p k ∈ V a , IoU(p k ,pj) > 0 then 18 flag ← 1 19 if flag == 0 then 20 V a ← pj, V s j ← pj figure 3 (b)-(c) describe the spatial and appearance graphs in different epochs. With the increased number of iterations, more instances can be detected using the proposed OIM.
Figure 4 :
4Detections examples on VOC2007 test set. The green bounding boxes represent the ground-truth. The successful detections (IoU ≥ 0.5) are marked with red bounding boxes, and the failed ones are marked with yellow color. We show all detections with scores ≥ 0.5 and NMS is performed to remove duplicate detections.
Figure 5 :
5Objects number of each class divided by the number of images which the corresponding class occurs on VOC2007 and VOC2012.
Table 1 :
1Comparison with the state-of-the-arts in terms of mAP (%) on the VOC2007 test set.Method
aero bike bird boat bottle bus
car
cat chair cow table dog horse mbike person plant sheep sofa train
tv
mAP
OICR
58.0 62.4 31.1 19.4 13.0 65.1 62.2 28.4 24.8 44.7 30.6 25.3 37.8
65.5
15.7
24.1
41.7 46.9 64.3 62.6 41.2
PCL
54.4 69.0 39.3 19.2 15.7 62.9 64.4 30.0 25.1 52.5 44.4 19.6 39.3
67.7
17.8
22.9
46.6 57.5 58.6 63.0 43.5
TS 2 C
59.3 57.5 43.7 27.3 13.5 63.9 61.7 59.9 24.1 46.9 36.7 45.6 39.9
62.6
10.3
23.6
41.7 52.4 58.7 56.6 44.3
C-WSL*
62.9 64.8 39.8 28.1 16.4 69.5 68.2 47.0 27.9 55.8 43.7 31.2 43.8
65.0
10.9
26.1
52.7 55.3 60.2 66.6 46.8
MELM
55.6 66.9 34.2 29.1 16.4 68.8 68.1 43.0 25.0 65.6 45.3 53.2 49.6
68.6
2.0
25.4
52.5 56.8 62.1 57.1 47.3
OICR+W
Table 2 :
2Comparison with the state-of-the-arts in terms of CorLoc (%) on the VOC2007 trainval set.Method
Localization
(CorLoc)
OICR
Table 4 :
4Comparison with the state-of-the-arts in terms of CorLoc (%) on the VOC2012 trainval set. more accurate which results in better CorLoc.Method
Localization
(CorLoc)
OICR(Tang et al. 2017)
62.1
PCL(Tang et al. 2018)
63.2
WS-JDS (Shen et al. 2019)
63.5
OICR+W-RPN (Singh and Lee
2019)
67.5
SDCN (Li et al. 2019)
67.9
OIM+IR
67.1
Pred Net (FRCNN) (Arun et al.
2019)
69.5
WS-JDS + FRCNN (Shen et al.
2019)
69.5
C-MIL + FRCNN (Wan et al.
2019)
67.4
SDCN + FRCNN (Li et al.
2019)
69.5
OIM+IR + FRCNN
69.5
Table 5 :
5Detection performance (mAP%) on the VOC2007 for using different values of parameter α and parameter β.Table 6: Detection performance comparison of proposed approach on the VOC2007 with various configurations.α 1
1
2
5
10
α 2
1
2
2
5
OIM
42.9
48.1
48.2
46.8
OIM+IR
43.4
49.3
50.1
48.4
β
0.2
0.5
0.8
OIM+IR
50.1
48.0
46.3
SG
AG
OIM(SG+AG)
IR
mAP
(%)
34.8
√
42.2
√
46.7
√
√
√
48.2
√
43.4
√
√
√
√
50.1
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| []
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[
"Classification of bi-qutrit positive partial transpose entangled edge states by their ranks",
"Classification of bi-qutrit positive partial transpose entangled edge states by their ranks"
]
| [
"Seung-Hyeok Kye \nDepartment of Mathematics and Institute of Mathematics\nSeoul National University\n151-742SeoulKorea\n",
"Hiroyuki Osaka \nDepartment of Mathematical Sciences\nRitsumeikan University\n525-8577KusatsuShigaJapan\n"
]
| [
"Department of Mathematics and Institute of Mathematics\nSeoul National University\n151-742SeoulKorea",
"Department of Mathematical Sciences\nRitsumeikan University\n525-8577KusatsuShigaJapan"
]
| []
| We construct 3 ⊗ 3 PPT entangled edge states with maximal ranks, to complete the classification of 3 ⊗ 3 PPT entangled edge states by their types. The ranks of the states and their partial transposes are 8 and 6, respectively. These examples also disprove claims in the literature. | 10.1063/1.4712302 | [
"https://arxiv.org/pdf/1202.1699v3.pdf"
]
| 119,708,110 | 1202.1699 | 79e94aefc650b1458e5557881529ee0f1d36c140 |
Classification of bi-qutrit positive partial transpose entangled edge states by their ranks
30 May 2012
Seung-Hyeok Kye
Department of Mathematics and Institute of Mathematics
Seoul National University
151-742SeoulKorea
Hiroyuki Osaka
Department of Mathematical Sciences
Ritsumeikan University
525-8577KusatsuShigaJapan
Classification of bi-qutrit positive partial transpose entangled edge states by their ranks
30 May 2012(Dated: 31 May 2012)AIP/123-QEDnumbers: 0367-a0367Hk0365Fd Keywords: positive partial transposesseparable statesentanglementedge statesproduct vectors 1
We construct 3 ⊗ 3 PPT entangled edge states with maximal ranks, to complete the classification of 3 ⊗ 3 PPT entangled edge states by their types. The ranks of the states and their partial transposes are 8 and 6, respectively. These examples also disprove claims in the literature.
I. INTRODUCTION
Let M n denote the C * -algebra of all n × n matrices over the complex field, with the cone M + n of all positive semi-definite matrices. A positive semi-definite matrix in M m ⊗ M n is said to be separable if it is the convex sum of rank one projectors onto product vectors
x ⊗ y ∈ C m ⊗ C n . A positive semi-definite matrix in M m ⊗ M n is said to be entangled if it is not separable. Since the convex cone of all separable ones coincides with M + m ⊗ M + n , the entanglement consists of (M m ⊗ M n ) + \ M + m ⊗ M + n . The notion of entanglement is a unique phenomenon in non-commutative order structures, and there is no counterpart in classical mechanics. Indeed, it is well-known that the equality (A ⊗ B) + = A + ⊗ B + holds for commutative C * -algebras A and B which are mathematical frameworks for classical mechanics. This notion of quantum entanglement has been one of the key research topics since the nineties, in relation with possible applications to quantum information theory and quantum computation theory.
One of the main research topics in the theory of entanglement is to distinguish entanglement from separability. If we take a rank one projector onto a product vector x ⊗ y, then it is easy to see that its partial transpose is also a rank one projector onto the product vector
x ⊗ y, wherex denotes the vector whose entries are complex conjugates of the corresponding entries of the vector x ∈ C m . Recall that the partial transpose (X ⊗ Y ) τ is given by X t ⊗ Y with the usual transpose X t of X. Therefore, if A ∈ M m ⊗ M n is separable, then its partial transpose A τ is also positive semi-definite, as was observed by Choi 9 and Peres 25 Woronowicz 33 showed that if m = 2 and n ≤ 3 then the notions of separability and PPT coincide, and gave an explicit example of entanglement A ∈ M 2 ⊗ M 4 which is of PPT. This kind of block matrix is called a PPTES (positive partial transpose entangled state) when it is normalized. The first example of PPTES in M 3 ⊗ M 3 was found by Choi 9 . A PPTES A is said to be a PPT entangled edge state, or just an edge state in short, if there exists no nonzero product vector x ⊗ y ∈ RA withx ⊗ y ∈ RA τ as was introduced in Ref. 23, where RA denotes the range space of A. In other words, edge states violate the range criterion for separability 18 in an extreme way.
Since every PPT state is the convex sum of a separable state and an edge state, it is essential to classify edge states to understand the whole structures of PPT states. The first step to classify them is to use the ranks. A PPT state A is said to be of type (p, q) if the rank of A is p and the rank of A τ is q, as was introduced in Ref. 26. Now, we concentrate on the case of 3 ⊗ 3. By the results in Refs. 5, 19, 20, and 28, we have the following possibilities of types for 3 ⊗ 3 PPT entangled edge states: (4,4), (5,5), (6,5), (7,5), (8,5), (6,6), (7,6), (8,6), These examples disprove the above mentioned claim 26 . Our examples also disprove another claim 22 that if D = (RA) ⊥ and E = (RA τ ) ⊥ for a PPT state A ∈ M m ⊗ M n and dim D + dim E = m + n − 2, then there exist finitely many product vectors x ⊗ y ∈ RA with
x ⊗ y ∈ RA τ .
After we explain in the next section the notion of PPT edge states in the context of the whole convex structures of the convex cone generated by PPT states, we present our construction of two parameterized examples of edges of type (8,6) in the Section 3. In the last section, we also exhibit various types of edge states arising from this construction.
II. CONVEX GEOMETRY OF PPT STATES
We denote by V 1 and T the convex cones generated by all separable and PPT states, respectively. The PPT criterion by Choi and Peres tells us that the relation V 1 ⊂ T holds.
One of the best way to understand the whole structures of a given convex set is to characterize The set of all interior point of C will be denoted by int C, which is nothing but the relative interior of C with respect to the affine manifold generated by C. Note that int C is never empty for any convex set C. A point of C which is not an interior point is said to be a boundary point. The set of all boundary points of C will be denoted by ∂C. We recall that From now on, we compare boundary structures of the two convex cones V 1 and T. Basically, we have the following four cases for a given face σ(D, E) of the cone T:
• σ(D, E) ⊆ V 1 . • σ(D, E) V 1 but int σ(D, E) ∩ V 1 = ∅. • int σ(D, E) ∩ V 1 = ∅ but ∂σ(D, E) ∩ V 1 = ∅. • σ(D, E) ∩ V 1 = ∅.
Recall that the range criterion for separability tells us that if a PPT state A is separable with D = RA and E = RA τ then there exist product vectors x ι ⊗ y ι ∈ C m ⊗ C n such that
D = span {x ι ⊗ y ι }, E = span {x ι ⊗ y ι }.
We On the other hand, for a given pair (D, E) of subspaces in C m ⊗ C n , it was shown 20 that there must exist x ⊗ y ∈ D withx ⊗ y ∈ E, whenever either the inequality
dim D + dim E > 2mn − m − n + 2 holds, or dim D + dim E = 2mn − m − n + 2 and r+s=m−1 (−1) r k r ℓ s = 0 hold with k = dim D ⊥ and ℓ = dim E ⊥ .❅ ❅ ❅ ❅ ❅ ❅ p q 8 8 ✉ ✉ ✉ ❜ ❡ ❡ ❡ 2 ⊗ 4 3 ⊗ 3 ❅ ❅ ❅ ❅ ❅ ❅ p q 9 9 ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ❡ ✉ edge states ❡ no edge state ❡ ❜ unknown
III. CONSTRUCTION
We begin with the following 3 × 3 matrix
P [θ] := e iθ + e −iθ −e iθ −e −iθ −e −iθ e iθ + e −iθ −e iθ −e iθ −e −iθ e iθ + e −iθ
which has a kernel vector (1, 1, 1) t . Considering the principal submatrices, we see that P [θ]
is positive semi-definite if and only if cos θ ≥ 0 and 2 cos 2θ ≥ −1 if and only if − π 3 ≤ θ ≤ π 3 . If − π 3 < θ < π 3 then P [θ] is of rank two, and if θ = − π 3 or θ = π 3 then P [θ] is of rank one.
Consider the following matrix
A = e iθ + e −iθ · · · −e iθ · · · −e −iθ · 1 b · · · · · · · · · b · · · · · · · · · b · · · · · −e −iθ · · · e iθ + e −iθ · · · −e iθ · · · · · 1 b · · · · · · · · · 1 b · · · · · · · · · b · −e iθ · · · −e −iθ · · · e iθ + e −iθ (2) in M 3 ⊗ M 3 with the conditions b > 0, − π 3 < θ < π 3 , θ = 0,(3)
where · denote zero. The partial transpose A τ of A is given by
A τ = e iθ + e −iθ · · · · · · · · · 1 b · −e −iθ · · · · · · · b · · · −e iθ · · · −e iθ · b · · · · · · · · · e iθ + e −iθ · · · · · · · · · 1 b · −e −iθ · · · −e −iθ · · · 1 b · · · · · · · −e iθ · b · · · · · · · · · e iθ + e −iθ .
It is clear that A is of PPT under the condition (3), and we have rank A = 8 and rank A τ = 6.
We proceed to show that A is a PPT entangled edge state under the condition (3). First of all, we note that the kernel of A is spanned by Suppose that a product vector z = x ⊗ y ∈ C 3 ⊗ C 3 is in the range of A, andx ⊗ y is in the range of A τ . Then we have
x 1 y 1 + x 2 y 2 + x 3 y 3 = 0,(4)
and bx 1 y 2 + e −iθx 2 y 1 = 0,
bx 2 y 3 + e −iθx 3 y 2 = 0, bx 3 y 1 + e −iθx 1 y 3 = 0.(5)
From (5) we see that at least one of x i , y i is zero. Indeed, we have b 3x 1x2x3 y 1 y 2 y 3 = −e −3iθx 1x2x3 y 1 y 2 y 3 by (5), from whichx 1x2x3 y 1 y 2 y 3 = 0. If x⊗y is nonzero, then we also have x i = 0 ⇐⇒ y i = 0 from (5).
We first consider the case of x 3 = y 3 = 0. Then we have
x 1 y 1 + x 2 y 2 = 0, bx 1 y 2 + e −iθx 2 y 1 = 0,
from which we havex 1 x 1 y 1 +x 1 x 2 y 2 = 0, bx 1 x 2 y 2 + e −iθx 2 x 2 y 1 = 0.
Therefore, we get
|x 1 | 2 y 1 = −x 1 x 2 y 2 = e −iθ b |x 2 | 2 y 1 .
Since θ = 0, we conclude that x 1 = x 2 = 0 or y 1 = 0. If x 1 = x 2 = 0, then x = 0. If y 1 = 0 and either x 1 or x 2 is nonzero, then we have y = 0. Similar arguments for the cases x 1 = y 1 = 0 and x 2 = y 2 = 0 show that if x, y ∈ C 3 satisfy the relations (4) and (5), then x ⊗ y = 0. This shows that there exists no nonzero product vector x ⊗ y ∈ RA with
x ⊗ y ∈ RA τ . Therefore, we conclude that A is a PPT entangled edge state of type (8,6).
Recall 32 that every 5-dimensional subspace of C 3 ⊗ C 3 has a product vector. This is equivalent to say that every system of equations consisting of four homogeneous linear equations with respect to unknowns {x i y j : i, j = 1, 2, 3} must have nontrivial solutions. But, the system of four equations from (4) and (5) between M 3 , as was introduced in Ref. 6. We also recall that the Choi matrix C φ ∈ M m ⊗M n of a linear map φ : M m → M n is given by ], which is a PPT state of type (8,6). On the other hand, we have the following PPT states
C φ := m i,j=1 e ij ⊗ φ(e ij ) ∈ M m ⊗ M n ,A = 1 · · · 1 · · · 1 · 1
b · · · · · · · · · b · · · · · · · · · b · · · · · 1 · · · 1 · · · 1 · · · · · 1 b · · · · · · · · · 1 b · · · · · · · · · b · 1 · · · 1 · · · 1
(6)
of type (7,6) in the literature 13 , which is an edge state whenever b > 0 and b = 1. The key idea of the construction was to parameterized offdiagonals −1 and 1 of these two cases by e iθ . We note that a variant of (6) has been used by Størmer 31 to give a short proof of the indecomposability of the Choi map Φ[1, 0, λ] for λ ≥ 1.
If θ = 0 then A in (2) turns out to be separable. Indeed, if we take product vectors
z 1 (ω) = (0, 1, √ b ω) ⊗ (0, √ b, −ω) = (0, 0, 0 ; 0, √ b, −ω ; 0, b ω, − √ b ) z 2 (ω) = ( √ b ω, 0, 1) ⊗ (−ω, 0, √ b) = (− √ b, 0, b ω ; 0, 0, 0 ; −ω, 0, √ b ) z 3 (ω) = (1, √ b ω, 0) ⊗ ( √ b, −ω, 0) = ( √ b, −ω, 0 ; b ω, − √ b, 0 ; 0, 0, 0) in C 3 ⊗ C 3 then it is straightforward to see that A = 1 3b 3 i=1 ω∈Ω z i (ω)z i (ω) * ,
where Ω = {1, e 2 3 πi , e − 2 3 πi } is the third roots of unity. We note that the Choi matrix of the map Φ[a, b, c] is of PPT if and only if a ≥ 2 and bc ≥ 1, and so it is the sum of a diagonal matrix with nonnegative diagonal entries and a separable one. Therefore, we see that the If we put the following number
a θ = max{e i(θ+ 3 2 π) + e −i(θ+ 3 2 π) , e iθ + e −iθ , e i(θ− 3 2 π) + e −i(θ− 3 2 π) }
in the place of e iθ + e −iθ when we define the matrix A in (2), then we have similar PPT edge states for every θ. Note that a θ is the smallest number so that
a θ −e iθ −e −iθ −e −iθ a θ −e iθ −e iθ −e −iθ a θ
is positive semi-definite.
IV. EDGE STATES OF OTHER TYPES
Let A be the matrix given by (2
ξ i , α i √ b , −γ i √ be iθ ; −α i √ be iθ , η i , β i √ b ; γ i √ b , −β i √ be iθ , ζ i t ,
for scalars ξ i , η i , ζ i , α i , β i and γ i . We denote by P the rank one projector onto the vector
1, 1 √ b , − √ be iθ ; − √ be iθ , 1, 1 √ b ; 1 √ b , − √ be iθ , 1 t ,
and by Q i onto the vector
(ξ i , α i , γ i ; α i , η i , β i ; γ i , β i , ζ i ) t ,
respectively. Here, the projector onto a column vector w means the rank one matrix ww * . Then we see that A τ is the Hadamard product of P and i Q i for suitable choice of ξ i , η i , ζ i , α i , β i and γ i . If we write ξ, η, ζ, α, β and γ the vectors whose i-th entries are ξ i , η i , ζ i , α i , β i and γ i , respectively, then the matrix X = (X τ ) τ is the Hadamard product of the following two matrices:
1 1 √ b − √ be −iθ − √ be iθ −e iθ b 1 √ b 1 b −e −iθ 1 √ b 1 b −e −iθ 1 1 √ b − √ be −iθ − √ be iθ −e iθ b − √ be iθ −e iθ b 1 √ b 1 b −e −iθ 1 1 √ b − √ be −iθ − √ be −iθ 1 1 √ b b − √ be iθ −e iθ −e −iθ 1 √ b 1 b −e −iθ 1 √ b 1 b − √ be −iθ 1 1 √ b b − √ be iθ −e iθ b − √ be iθ −e iθ −e −iθ 1 √ b 1 b − √ be −iθ 1 1 √ b 1 √ b − √ be −iθ 1 −e iθ b − √ be iθ 1 b −e −iθ 1 √ b 1 b −e −iθ 1 √ b 1 √ b − √ be −iθ 1 −e iθ b − √ be iθ −e iθ b − √ be iθ 1 b −e −iθ 1 √ b 1 √ b − √ be −iθ 1 and ( .
Since (1, 0, 0 ; 0, 1, 0 ; 0, 0, 1) t is in the Ker A ⊆ Ker X, we have ξ 2 = e iθ α 2 + e −iθ γ 2 , η 2 = e iθ β 2 + e −iθ α 2 , ζ 2 = e iθ γ 2 + e −iθ β 2 , and so we have α = β = γ||. If α = β = γ|| = 0, then A = 0. So, we may assume that α = β = γ|| = 1.
Then we have
ξ 2 = η 2 = ζ 2 = e iθ + e −iθ .(7)
Considering (2,4), (6,8) and (7,3) principal submatrices, we also have
|(ξ|η)| ≤ 1, |(η|ζ)| ≤ 1, |(ζ|ξ)| ≤ 1.(8)
If we take vectors so that span {ξ, η, ζ} ⊥ span {α, β, γ} with mutually orthonormal vectors α, β, γ then we have
X = e iθ + e −iθ · · · −e iθ · · · −e −iθ · 1 b · (η|ξ) · · · · · · · b · · · (ζ|ξ) · · · (ξ|η) · b · · · · · −e −iθ · · · e iθ + e −iθ · · · −e iθ · · · · · 1 b · (ζ|η) · · · (ξ|ζ) · · · 1 b · · · · · · · (η|ζ) · b · −e iθ · · · −e −iθ · · · e iθ + e −iθ and X τ = e iθ + e −iθ · · · (ξ|η) · · · (ξ|ζ) · 1 b · −e −iθ · · · · · · · b · · · −e iθ · · · −e iθ · b · · · · · (η|ξ) · · · e iθ + e −iθ · · · (η|ζ) · · · · · 1 b · −e −iθ · · · −e −iθ · · · 1 b · · · · · · · −e iθ · b · (ζ|ξ) · · · (ζ|η) · · · e iθ + e −iθ .
It is clear that X is of PPT under the conditions (7) and (8). We note that the rank of X is equal to
2 + rank 1 b (ξ|η) (η|ξ) b + rank 1 b (η|ζ) (ζ|η) b + rank b (ζ|ξ) (ξ|ζ) 1 b
and the rank of X τ is equal to
3 + rank e iθ + e −iθ (ξ|η) (ξ|ζ) (η|ξ) e iθ + e −iθ (η|ζ) (ζ|ξ) (ζ|η) e iθ + e −iθ = 3 + dim span {ξ, η, ζ}.
In the three dimensional space C 3 , it is possible to take linearly independent vectors ξ, η, ζ satisfying (7) and (8)
Then P [ρ, σ, τ ] is a positive semi-definite matrix of rank two. By spectral decomposition, we may get two vectors E 1 = (ξ 1 , η 1 , ζ 1 ) and E 2 = (ξ 2 , η 2 , ζ 2 ) so that P [ρ, σ, τ ] is the sum of rank one projectors onto E 1 and E 2 , respectively. Then we see that
P [ρ, σ, τ ] = |ξ 1 | 2 ξ 1η1 ξ 1ζ1 η 1ξ1 |η 1 | 2 η 1ζ1 ζ 1ξ1 ζ 1η1 |ζ 1 | 2 + |ξ 2 | 2 ξ 2η2 ξ 2ζ2 η 2ξ2 |η 2 | 2 η 2ζ2 ζ 2ξ2 ζ 2η2 |ζ 2 | 2 = (ξ|ξ) (ξ|η) (ξ|ζ) (η|ξ) (η|η) (η|ζ) (ζ|ξ) (ζ|η) (ζ|ζ) .
If we take ρ, σ, τ with (9) so that some of them are of absolute values one and the remainders of them have the absolute values less than one, then we get PPT entangled states of types (8,5), (7,5), (6,5) and (5, 5), as we will now show.
For a given fixed θ with (3), we can take a real number r with −1 < r < 1 so that
P [r, r, r], P [r, −r, 1], P [1, 1, r]
is of rank two, respectively, to get edge states of types (8, 5), (7,5) and (6,5). For example, we see that
P [− cos θ, − cos θ, − cos θ]
is of rank two, and so we get the following natural examples of 3 ⊗ 3 edge states of type (8,5):
e iθ + e −iθ · · · −e iθ · · · −e −iθ · 1 b · − cos θ · · · · · · · b · · · − cos θ · · · − cos θ · b · · · · · −e −iθ · · · e iθ + e −iθ · · · −e iθ · · · · · 1 b · − cos θ · · · − cos θ · · · 1 b · · · · · · · − cos θ · b · −e iθ · · · −e −iθ · · · e iθ + e −iθ .
To get examples of edge states of types (7,5) and (6,5), we put ω = e iθ + e −iθ temporarily.
Note that 1 < ω < 2. We also note that det P [r, −r, 1] = (1 + ω)(ω 2 − ω − 2r 2 ), det P [1, 1, r] = (ω − r)(rω + ω 2 − 2). and zeros r = ω 2 − ω 2 = √ 2 cos 2 θ − cos θ, r = 2 − ω 2 ω = − cos 2θ cos θ of them are in the interval (−1, 1), respectively. In this way, we get edge states of types (7,5) and (6,5). If we consider the rank two matrix P [−e iθ , −e iθ , −e iθ ], which is nothing but P [θ]
at the beginning of the construction, then we have the following parameterized example of edge states of type (5,5):
e iθ + e −iθ · · · −e iθ · · · −e −iθ · 1 b · −e −iθ · · · · · · · b · · · −e iθ · · · −e iθ · b · · · · · −e −iθ · · · e iθ + e −iθ · · · −e iθ · · · · · 1 b · −e −iθ · · · −e −iθ · · · 1 b · · · · · · · −e iθ · b · −e iθ · · · −e −iθ · · · e iθ + e −iθ .
In conclusion, we have constructed 3 ⊗ 3 PPT entangled edge states of type (8, 6) whose existence has been a long-standing question since the claim in Ref. 26 without proof. In this vein, it would be also an interesting question whether there exists a 2 ⊗ 4 edge states of type (6,6) or not, as was explained in Ref. 20. We have shown that there exist edge states of all possible types in the face generated by each PPT state we constructed, except for edge states of (4, 4) types. These include parameterized examples of edge states of types (5,5) and (6,6), for which there have been known very few discrete examples 10,11 . We also have natural parameterized examples of edge states of type (8,5). Compare with Ref. 13. We note that the study of bi-qutrit edge states with minimal ranks was initiated by Ref. 3
. A block matrix A ∈ M m ⊗ M n is said to be of PPT (positive partial transpose) if both of A and A τ are positive semi-definite. The notion of PPT turns out be to very important in quantum physics in relation with bound entanglement. See Ref. 16.
the lattice of all faces. We have very few general information for the facial structures of the convex cone V 1 itself. See Ref. 1 in this direction. On the other hand, we have an easy way to describe faces of the cone T generated by PPT states. Every faces of the cone T is determined 12 by a pair of subspaces of C m ⊗ C n . More precisely, every face of T is of the form σ(D, E) = {A ∈ T : RA ⊆ D, RA τ ⊆ E} for a pair (D, E) of subspaces of C m ⊗ C n . Nevertheless, it is very difficult in general to determine which pairs of subspaces give rise to faces of the convex cone T, and this difficulty is one of the main motivation of this note. In the case of 2 ⊗ 2, all faces of T have been found 12 in terms of pairs of subspaces, using the facial structures 4,30 of the convex cone of all positive linear maps between M 2 . Recall that a point x of a convex set C is said to be an interior point of C if the line segment from any point of C to x may be extended within C.
the interior of σ(D, E) is given by int σ(D, E) = {A ∈ T : RA = D, RA τ = E}.
say that a pair (D, E) satisfies the range criterion if there exist product vectors with the above property. Therefore, we see that if the interior of σ(D, E) has a nonempty intersection with the cone V 1 then (D, E) satisfies the range criterion. The converse of this statement is also true as was shown in Ref. 7, even though the converse of the range criterion itself does not hold. In short, we see that (D, E) satisfies the range criterion if and only if the first two conditions among the above four hold. In terms of a PPT state A itself, we see that (RA, RA τ ) satisfies the range criterion if and only if the smallest face containing A has a separable state in its interior. Recall that every point x of a convex set determines a unique face in which x is an interior point. This is the smallest face containing x. It remains two cases to be considered: A face σ(D, E) either touches the cone V 1 at the boundary or never touches the cone V 1 . It is easy to see that the latter case occurs if and only if every element of the face σ(D, E) is a PPT entangled edge state. If this is the case with dim D = p and dim E = q then every interior point of the face σ(D, E) is an edge state of type (p, q), and every boundary point of σ(D, E) is also an edge state of type (s, t) with s < p or t < q. The first step to characterize the lattice of all faces of the cone T is to find all pairs (p, q) of natural numbers for which there exists a face σ(D, E) with dim D = p and dim E = q. See Ref. 22 for this line of research. This classification is especially important for the cases of separable states and edge states, since every PPT state is the sum of a separable state and an edge state. This task for separable states is nothing but to classify the dimensions of pairs of subspaces satisfying the range criterion. In the case 2 ⊗ n, all pairs (p, q) of natural numbers have been characterized 7 for which there exist pairs (D, E) satisfying the range criterion with dim D = p and dim E = q.As for edge states, there are previous results in the literature in two directions. It was shown 19,24 that if A is supported on C m ⊗ C n and the rank of A ∈ M m ⊗ M n is less than or equal to max{m, n}, then two notions of PPT and separability coincide. This gives a lower bound for the ranks of A and A τ for an edge state A ∈ M m ⊗ M n : If A is an m ⊗ n edge of type (p, q), then we have p, q > max{m, n}.
This gives us an upper bound for the ranks of A and A τ for an edge state A ∈ M m ⊗ M n .In case of m = n = 3, we have 2mn − m − n + 2 = 14. It is easy to see that (k, ℓ) = (2, 2) satisfies the above condition, but (k, ℓ) = (1, 3) does not satisfy. Furthermore, it is now known 5,28 that every PPT entanglement of rank 4 is automatically of type(4,4). All of these arguments give us the possibilities of types as is given in(1). See also Ref.20 for the summary in the case of (m, n) = (2, 4) as well as in the case of m = n = 3. It is unknown whether there exists a 2 ⊗ 4 PPT edge state of type (6, 6) or not. Classifications of possible types of edge states for the 2 ⊗ 4 and 3 ⊗ 3 cases are summarized in the following pictures:
kernel of A τ is spanned by the following three vectors: (0, b, 0 ; e iθ , 0, 0 ; 0, 0, 0) t , (0, 0, 0 ; 0, 0, b ; 0, e iθ , 0) t , (0, 0, e iθ ; 0, 0, 0 ; b, 0, 0) t .
and C φ is of PPT if and only if φ is both completely positive and completely copositive by Ref. 8. We also note that Φ[a, b, c] is both completely positive and completely copositive if and only if a ≥ 2 and bc ≥ 1 by Ref. 6. If θ = 0 then the matrix A in (2) is just the Choi matrix of the map Φ[2, b, 1 b
Choi matrix of the map Φ[a, b, c] is of PPT if and only if it is separable. This shows that the linear map Φ[a, b, c] is super-positive in the sense of Ref. 2, or equivalently an entanglement breaking channel in the sense of Ref. 17 and 21 if and only if it is both completely positive and completely copositive if and only if a ≥ 2 and bc ≥ 1. See Ref. 27 for related topics.
PPT entangled edge state of type (p, q) then p + q ≤ 13.The purpose of this note is to present two parameterized examples of 3⊗3 PPT entangled edge states of type(8,6), to complete the classification of 3 ⊗ 3 edges states by their types.here we list up types (p, q) with p ≥ q by the symmetry. See Ref. 3, 9-11, 13, 14, 18, and
31 for concrete examples of 3 ⊗ 3 edge states of various types. We refer to Ref. 20 for a
summary of examples. All possibilities have been realized in the literature mentioned above,
except for the case of (8, 6). In fact, it has been claimed in Ref. 26 that if there is a 3 ⊗ 3
involve complex conjugates, and may not have nonzero solutions. This seems to be the main point for the wrong statements in Ref.22 and 26.
For nonnegative real numbers a, b and c, we consider the following linear map
Φ[a, b, c](X) =
ax 11 + bx 22 + cx 33
−x 12
−x 13
−x 21
cx 11 + ax 22 + bx 33
−x 23
−x 31
−x 32
bx 11 + cx 22 + ax 33
). Now, we search edge states X in the smallest face containing A by a similar method as in Ref.13. Note that X is in this face if and only if hold. Note that every range vector of A τ is of the formthe relations
RX ⊆ RA,
RX τ ⊆ RA τ
, and have been recently studied in Ref. 5, 15, 28, and 29 very extensively. It is the authors' hope that this is the starting point for the further study of bi-qutrit edge states with maximal ranks. SHK was partially supported by NRFK 2011-0001250. HO was partially supported by the JSPS grant for Scientific Research No.20540220. The first author is grateful to Kil-Chan Ha for helpful discussion.
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| []
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[
"Charmonia Production in W → (cc)D",
"Charmonia Production in W → (cc)D"
]
| [
"Decays A V Luchinsky \n\"Institute for High Energy Physics\" NRC \"Kurchatov Institute\"\n142281ProtvinoRussia\n"
]
| [
"\"Institute for High Energy Physics\" NRC \"Kurchatov Institute\"\n142281ProtvinoRussia"
]
| []
| In the presented paper production of charmonium state Q in exclusive W → QD ( * ) s decays is analyzed in the framework of both leading order Nonrelativistic Quantum Chromodynamics (NRQCD) and light-cone expansion (LC) models. Analytical and numerical predictions for the branching fractions of these decays in both approaches are given. The typical value of the branching fractions is ∼ 10 −11 and it turns out the LC results are about 4 times lager than NRQCD ones, so the effect of internal quark should be taken into account. Some estimates of color-octet contributions are presented and it is shown, that these contributions could be comparable with color-singlet results. | null | [
"https://arxiv.org/pdf/1801.08998v1.pdf"
]
| 119,054,617 | 1801.08998 | f9a71caeb988bf21e4a5b661e2d303c8148a80d9 |
Charmonia Production in W → (cc)D
26 Jan 2018
Decays A V Luchinsky
"Institute for High Energy Physics" NRC "Kurchatov Institute"
142281ProtvinoRussia
Charmonia Production in W → (cc)D
26 Jan 2018numbers: 1440Pq1239St1239Jh338Dg
In the presented paper production of charmonium state Q in exclusive W → QD ( * ) s decays is analyzed in the framework of both leading order Nonrelativistic Quantum Chromodynamics (NRQCD) and light-cone expansion (LC) models. Analytical and numerical predictions for the branching fractions of these decays in both approaches are given. The typical value of the branching fractions is ∼ 10 −11 and it turns out the LC results are about 4 times lager than NRQCD ones, so the effect of internal quark should be taken into account. Some estimates of color-octet contributions are presented and it is shown, that these contributions could be comparable with color-singlet results.
decays is analyzed in the framework of both leading order Nonrelativistic Quantum Chromodynamics (NRQCD) and light-cone expansion (LC) models. Analytical and numerical predictions for the branching fractions of these decays in both approaches are given. The typical value of the branching fractions is ∼ 10 −11 and it turns out the LC results are about 4 times lager than NRQCD ones, so the effect of internal quark should be taken into account. Some estimates of color-octet contributions are presented and it is shown, that these contributions could be comparable with color-singlet results.
I. INTRODUCTION
Heavy quarkonia mesons, i.e. particles that are build from heavy quark-antiquark pair are very interesting states both from theoretical and experimental points of view. Because of the presence of two different mass scales the processes of their production and decays occur in two almost independent steps: production or decay of (QQ) pair and its hadronization into experimentally observed meson. Since the strong coupling constant α s (m Q ) ≪ 1 the first step can be analyzed using perturbative QCD. Final hadronization, on the other hand, is essentially nonperturbative, so some other methods should be used.
One of such methods is Nonrelativistic Quantum Chromodynamics (NRQCD) [1]. In this approach the fact that the velocity of internal quark motion v ∼ α s (m Q ) is small in comparison with the speed of light and the probability of the considered process is written as a series over this small parameter. The hadronization probabilities are parametrized as NRQCD matrix elements, whose numerical values are determined, e.g. from solution of the potential models of analysis or the experimental data. Another interesting NRQCD feature is that production of color-octet (CO) components (when QQ-pair is in color-octet state and total color neutrality of the meson is guaranteed by the presence of additional gluons) can be considered. NRQCD approach was widely used for analysis of various processes and nice agreement with experimental results were achieved.
It should be noted, however, that in the case of charmonium meson production the NRQCD expansion parameter v ∼ α s (m c ) ∼ 0.3 is not really small, so the effect of internal quark motion should be taken into account. Another model for describing charmonium production at high energies is the so called light-cone (LC) expansion model [2], when the amplitude of the reaction is written as a series over small chirality parameter ∼ m c /E, where E is the typical energy scale of the considered reaction. This approach was also highly used in theoretical considerations of various reactions (see, e.g. [3,4]) and often its predictions are more close to experimental data than NRQCD results. Usually the effect of internal quark motion leads to increase of theoretical predictions. For example, in the case of double charmonia production at B-factories Belle and BaBar there is about an order of magnitude difference between LC and NRQCD results and only LC solves the long standing contradiction between theory and experiment [5][6][7][8][9][10] Recently a series of theoretical papers devoted to heavy quarkonia production in exclusive W -, Z-boson decays were published. For example in [11][12][13][14] charmonia Q production in radiative Z-boson decays Z → Qγ was considered. In [15] theoretical analysis of Z → Q 1 Q 2 decay was preformed. It is clear that in these processes the chirality expansion parameter m c /M W ∼ 2 × 10 −2 is small, so LC framework can safely be used for their description. It was shown in mentioned above works that in both cases LC predictions are higher than NRQCD ones. In the presented paper we analyze charmonia production in exclusive W → QD ( * ) s decays. The rest of the paper is organized as follows. In the next section analytical results for the widths of the processes under consideration are given. Numerical predictions for the branching fractions both in color-singlet NRQCD and LC models are presented in section III. In section IV we give some estimates for CO contributions. The last section is reserved for conclusion.
II. ANALYTICAL RESULTS
In our paper we consider charmonium meson Q production in exclusive W -boson decays
W (P ) → Q(p 1 )D ( * ) s (p 2 ).(1)
Typical Feynman diagrams describing this reaction are shown in Fig. 1. In the current section we will restrict ourselves to color-singlet (CS) approximation, so only diagrams shown in Fig. 1 will contribute. Estimates for color-octet (CO) contributions will be given in section IV. A widely used approach for description of heavy quarkonia production is the Non-relativistic Quantum Chromodynamics (NRQCD) formalism [1]. In this model the amplitude of the process is written as a series over small quarks' relative velocity inside the meson. At the leading order over this parameter internal quark motion is neglected completely, so quarks momenta are equal to
p c 1 = pc 1 = m c M Q = P 1 2 , p c,s 2 = m c,s M D ( * ) s P 2 ,(2)
where m c,s are the masses of the corresponding quarks, P 1,2 are the momenta of charmonium and D ( * ) s mesons respectively, and M Q = 2m c , M D ( * ) s = m c + m s are their masses. The projection on physical states is calculated using the technique described, e.g. in paper [6]. It turns out that after straightforward (although rather cumbersome) calculations this approach leads to surprisingly simple expressions for the widths of the considered processes:
Γ NRQCD W → QD ( * ) s = 16πα 2 s λg 2 W 243 m c + m s m c 2 F 2 Q F 2 D ( * ) s M 3 W C QD ( * ) s ,(3)
where α s = α s (M W ) is a strong coupling constant,
g W = eV cs 2 √ 2 sin θ W(4)
is the W → cs vertex coupling constant,
λ = 1 − M Q − M D ( * ) s M W 2 1 − M Q + M D ( * ) s M W 2(5)
is final meson's velocity in W rest frame, and F Q,D ( * ) s are longitudinal constants of the final mesons defined as
F ηc = F J/ψ = O 1 J/ψ m c ,(6)F hc = √ 3F χc0 = 1 √ 2 F χc1 = 3 2 F χc2 = O 1 hC m 3 c ,(7)
where O 1 J/ψ,hc are defined in [6] NRQCD matrix elements for S-and P -wave charmonium mesons. As for dimensionless coefficients C QD ( * ) s , expressions for them are presented in the Appendix. It should be mentioned, however, that in massless quark limit m s,c ≪ m W we have C QD ( * ) s = 1. An alternative way to calculate the widths of the considered processes is so called light-cone (LC) expansions formalism [2]. In the framework of this method the amplitude of the reaction is written as a series over small chirality parameter ∼ m q /M W . According to LC selection rules the total helicity of the hadronic states should be conserved. There are only two hadrons in the reaction, so from this rule it follows that
λ 1 + λ 2 = 0(8)
, where λ 1,2 are the helicities of final mesons. Orbital momentum conservation, in the other hand, requires
−1 ≤ λ W = λ 1 − λ 2 ≤ 1.(9)
It is clear that only λ 1 = λ 2 = 0 satisfy both of these restrictions, so only production of longitudinally polarized mesons is allowed at the leading twist approximation. Thus, in the framework of LC formalism the amplitude of
W → QD ( * ) s decay equals to M(W → QD ( * ) s ) ∼ f Q f D ( * ) s 1 −1 dξ 1 dξ 2 φ Q (ξ 1 )φ D ( * ) s (ξ 2 )A,(10)
where ξ 1,2 = 2x 1,2 − 1 with x 1,2 being the momentum fractions of c-quarks inside Q and D ( * )
s mesons, φ Q,D ( * ) s (ξ 1,2 are light-cone distribution functions, f Q,D ( * )
s are longitudinal mesonic constants, and the amplitude A can be calculated using perturbation QCD on the basis of presented in Fig. 1 diagrams. According to [2] mesonic constants and distribution amplitudes are defined as
Q L (p) c i α (z)[z, −z]c j β (−z) 0 = (p) αβ f Q 4 δ ij 3 1 −1 φ Q (ξ)dξ(11)
for σ-even states Q = J/ψ, χ 0,2 , D ( * ) s , and
Q L (p) c i α (z)[z, −z]c j β (−z) 0 = (pγ 5 ) αβ f Q 4 δ ij 3 1 −1 φ Q (ξ)dξ(12)
for σ-odd states Q = η c , χ c1 , and h c . These definitions for LC constants is consistent with NRQCD mesonic constants (6), (7). In the above expressions α, β and i, j are spinor and colour indices of quark and antiquark respectively. The normalization condition for the distribution amplitudes is
1 −1 φ Q (ξ)dξ = 1, 1 −1 ξφ Q (ξ)dξ = 1(13)
for ξ-even (Q = η c , J/ψ, χ c1 , D ( * ) s ) and ξ-odd (Q = χ c0,2 , h c ) states. In δ approximation, when internal quark motion is neglected, the distribution amplitudes of charmonia states take the form
φ ηc,J/ψ,χc1 (ξ) = δ(ξ), φ χc0,2,hc (ξ) = −δ ′ (ξ),(14)
while for D ( * ) s mesons we have
φ D ( * ) s = δ ξ − m c − m s m c + m s .(15)
The light-cone amplitude corresponding to presented in Fig. 1 diagrams is equal to
M(W → QD ( * ) s ) = 16πα s g W 9 f Q f D ( * ) s M 2 W I QD ( * ) s (P 1 − P 2 ) µ ǫ µ W ,(16)
where ǫ µ W is the polarization vector of the initial W -boson and
I QD ( * ) s = 2 2 −1 dξ 1 dξ 2 φ Q (ξ 1 )φ D ( * ) s (ξ 2 ) (1 − ξ 1 )(1 + ξ 2 ) .(17)
The corresponding width is equal to
Γ LC W → QD ( * ) s = 16πα 2 s g 2 W 243 f 2 Q f 2 D ( * ) s M 2 W I 2 QD ( * ) s .(18)
It is easy to check that in δ-approximation (14), (15) NRQCD result (3) is restored.
III. NUMERICAL RESULTS
Let us first consider NRQCD predictions for the widths of W → QD ( * ) s decays. Numerical values of final meson's masses were taken from PDG tables [16] and quarks' masses were chosen to be equal to
m c = M Q 2 , m s = M D ( * ) s − m c .(19)
The mesonic constants, entering relation (3) can be related to matrix elements O 1 J/ψ,hc using relations (6), (7), where [6]
O 1 J/ψ = 0.22 GeV 3 , O 1 hc = 0.033 GeV 3 .(20)
These values correspond to
F ηc = F J/ψ = 0.38 GeV,(21)
F χc0 = 0.057 GeV, F χc1 = 0.14, GeV, (22) F χc2 = 0.081 GeV, F hc = 0.099 GeV.
The strong coupling constant α s (µ 2 ) is parametrized as
α s (µ 2 ) = 4π b 0 ln(µ 2 /Λ QCD 2 ) , b 0 = 11 − 2 3 n f ,(24)
where Λ QCD ≈ 0.2 GeV and n f = 5 is the number of active flavors. At the scale µ 2 = M 2 W it corresponds to α s (M 2 W ) = 0.14. With presented above values of the parameters it is easy to obtain branching fractions presented in the second columns of tables I, II. Table I. Branching fractions of W → QDs decays. In second, third and fourth columns results of NRQCD formalism, LC approach in δ-approximation (14), (15), and LC results with (28), (29), (30), (32) distribution amplitudes are given. In the last column of the table the effect of internal quark motion is shown. The uncertainties in Br δ LC predictions and first errors in BrLC predictions are caused by mesonic constants uncertainties (25), second errors in fourth and the error in the last column are caused by the variation of distribution amplitudes' parameters (31), (33).
In order to calculate LC predictions (see relations (18), (17)) numerical values of mesonic constants f Q and parametrization for distribution amplitudes φ Q (ξ) is required. It should be noted that both f and φ(ξ) actually depend on the renormalization scale µ (see [17][18][19][20]). According to [21][22][23] the following values of the constants will be used: As for D ( * ) s mesons, in the following we will use [24,25] f Ds (m c ) = 258 MeV,
ff D * s (m c ) = 274 MeV.(27)
The distribution amplitude of charmonium mesons can be written in the form [21][22][23]
φ J/ψ,ηc (ξ, µ 0 ) = c(β S )(1 − ξ 2 ) exp − β S 1 − ξ 2 ,(28)φ χc0,χc2,hc (ξ, µ 0 ) = c 1 (β P )ξ(1 − ξ 2 ) exp − β P 1 − ξ 2 ,(29)φ χc1 (ξ, µ 0 ) = −c 2 (β P ) ξ −1 φ hc (ξ, µ 0 ),(30)
where c(β S ), c 1,2 (β P ) are normalization constants (13) and wave function parameters are equal to
β S = 3.8 ± 0.7, β P = 3.4 +1.5 −0.9 .(31)
Distribution amplitudes of D ( * ) s meson at µ = m c will be parametrized as
φ(ξ) ∼ (1 − ξ) ac (1 + ξ) as ,(32)
where, according to [26,27], a c = 3.1 and 1 ≤ a s < 2 (33) For mean ξ value NRQCD limit
ξ = m c − m s m c + m s(34)
is observed at a s ≈ 1.2. In Figures 2, 3 we show the distributions amplitudes at different scales. From these figures it is clear that with the increase of the scale effective width of the distribution amplitude also increases.
In tables I, II we show LC predictions for branching fractions of the considered decays in δ-approximation (third columns) and using real distribution amplitudes (28), (29), (30), (32) with mentioned above values of the parameters β S,P and a s (fourth columns). In the fifth columns of the tables the effect of internal quark motion is shown. It can be easily seen that as a result of this effect the branching fractions of the decays increase significantly.
IV. COLOR-OCTET CONTRIBUTIONS
As it was shown in the previous section, in spite of the increase caused by internal quark motion the branching fractions of the considered decays are small. This is caused mainly by the large value of W -boson mass, that enters in shown in Fig. 1 gluon propagators. It is clear, on the other hand, that in the case of color-octet (CO) state production the situation is completely different. In the current section we will give rough estimates for CO contributions. In addition to shown in Fig. 1 Feynman diagrams shown in Fig. 4 diagrams also contribute to the process under consideration in color-octet approximation. In order to calculate the corresponding decay width it is convenient simply to change in the projection operator F Q mesonic constant to color-octet parameterF Q and color identity matrix δ ij / √ N c to the corresponding Gell-mann matrix √ 2(T a ) ij , where N c = 3 is the number of colors [28]. With these substitutions the width of W → J/ψD s decay in CO approximation takes the form
Γ CO NRQCD ≈ πα 2 s g 2 W 54 (m 2 c + m 2 c )(m c + m s ) 2 m 4 c m 2 sF 2 J/ψF 2 Ds M W C CO J/ψDs ,(35)
where
C CO J/ψDs = m 2 s (3717m 16 c + 21990m 15 c m s + m 14 c (39548m 2 s + 4435M 2 W ) + m 13 c (39076m s M 2 W − 109006m 3 s )+ m 12 c (−135880m 4 s + 8281m 2 s M 2 W + 3153M 4 W ) + 2m 11 c (64307m 5 s − 41908m 3 s M 2 W − 13555m s M 4 W )+ m 10 c (195276m 6 s + 1123m 4 s M 2 W − 53870m 2 s M 4 W − 5309M 6 W ) − 2m 9 c (14783m 7 s + 514m 5 s M 2 W − 51765m 3 s M 4 W + 5028m s M 6 W ) + m 8 c (−125238m 8 s + 16465m 6 s M 2 W + 38815m 4 s M 4 W + 32583m 2 s M 6 W − 449M 8 W ) − 2m 7 c (19247m 9 s − 12328m 7 s M 2 W + 6678m 5 s M 4 W + 7280m 3 s M 6 W − 2637m s M 8 W )+ m 6 c (28788m 10 s − 623m 8 s M 2 W − 37764m 6 s M 4 W + 7686m 4 s M 6 W − 6664m 2 s M 8 W + 705M 10 W )+ 2m 5 c (9635m 11 s − 5426m 9 s M 2 W − 4030m 7 s M 4 W − 3432m 5 s M 6 W + 3555m 3 s M 8 W − 302m s M 10 W )+ D ( * ) s Q (a) D ( * ) s Q (b)m 4 c (m 2 s − M 2 W ) 2 (2288m 8 s + 2659m 6 s M 2 W + 1653m 4 s M 4 W + 1517m 2 s M 6 W − 117M 8 W )− 2m 3 c m s (m 2 s − M 2 W ) 3 (503m 6 s − 167m 4 s M 2 W − 259m 2 s M 4 W + 51M 6 W ) − m 2 c (m 2 s − M 2 W ) 4 (316m 6 s + 311m 4 s M 2 W + 98m 2 s M 4 W − 9M 6 W ) + 6m c m 3 s (m 2 s − 5M 2 W )(m 2 s − M 2 W ) 5 + 9m 2 s (m 2 s − M 2 W ) 6 (m 2 s + M 2 W )) / 9M 2 W (m 2 c + m 2 s )(m c − m s + M W ) 4 (−m c + m s + M W ) 4 (−4m 3 c + m 2 c m s + 2m c m 2 s + m 3 s − m s M 2 W ) 2(36)
is equal to 1 in large M W limit. It can easily be seen, that the decrease of this decay width with M W is much slower than in CS case:
Γ CO NRQCD Γ NRQCD ∼ M 2 W m 2 c .(37)
This behavior is explained by the fact that gluon virtuality in color-octet diagrams is M 2 J/ψ instead of typical oder M 2 W in the case of color-singlet mechanism. Numerical calculations show that for optimistic assumptions for color-octet constantsF ∼ 10 −1 F the contribution of CO mechanism is comparable with CS one.
V. CONCLUSION
In the presented article production of charmonium Q in exclusive W → QD ( * ) s decays is analyzed using both Non-relativistic Quantum Chromodynamics (NRQCD) and light-cone expansion (LC) approaches.
Presented in the paper theoretical NRQCD predictions show, that the branching fractions of the considered decays are pretty small, although about an order of magnitude higher than obtained in the previous work [15] branching fractions of double charmonium production in exclusive Z-boson decays. The effect of internal quarks' motion, analyzed using LC formalism increases the branching fractions significantly, but they still remains small. The reason for this fact is twofold:
• In contrast to Z → 2Q decay two S-wave mesons can be produced at the leading twist approximation, so no chirality suppression factors occur,
• Production of the lighter system QD ( * ) s instead of Q 1 Q 2 one make the probability of the process larger,
• The widths of the considered decays are, nevertheless, suppressed my large W -boson mass (∼ 1/M 3 W ), so the branching fractions are small.
The last point can be bypassed if production of color-octet (CO) states is considered. In the last section of the article we give rough estimates for CO contributions and show that M W suppression of the resulting width is not so strong and the behavior Γ ∼ 1/M W is observed. In the case of J/ψD s pair production, our calculations show, that with reasonable assumtions on the value of color-octet matrix elements the resulting widths are comparable with color-singlet ones.
In our future work we plan to analyze production of other states (e.g. excited charmonia and P -wave charmonium mesons CO states) in more details.
All calculations in the article were performed with the help of FeynCalc Mathematica package [29,30]. The author would like to thank A. K. Likhoded for fruitful discussions.
PACS numbers: 14.40.Pq, 12.39.St, 12.39.Jh, 3.38.Dg
Figure 1 .
1Typical Feynman diagrams for W → QD ( * ) s decay in color-singlet approximation
ηc (m c ) = (0.35 ± 0.02) GeV, f J/ψ (m c ) = (0.41 ± 0.02) GeV, f χc0 (m c ) = (0.11 ± 0.02) GeV, (25) f χc1 (m c ) = (0.27 ± 0.05) GeV, f χc2 (m c ) = (0.16 ± 0.03) GeV, f hc (m c ) = (0.19 ± 0.03) GeV. (26) Note, the value of c quark in LC model differs from phenomenological choice M J/ψ /2 and is equal to m c = 1.2 GeV.
Figure 2 .
2Distribution amplitudes for S-wave mesons ηc, J/ψ (left figure) and D ( * ) s (right figure). Solid and dashed lines correspond to µ = mc and µ = MW respectively.
Figure 3 .
3Distributions amplitudes for P -wave mesons χc1 (left figure) and χc0,2, hc (right figure). Notations are the same as in Fig.2.
Figure 4 .
4Feynman diagrams for W → QD ( * ) s decays in CO approximation
Table II. Branching fractions of W → QD * s decays. Notations are the same as in table IQ BrNRQCD, 10 −12 Br δ
LC , 10 −12 BrLC, 10 −12
BrLC/Br δ
LC
ηc 3.18
3.38 ± 0.5 14.8 ± 2. +3.
−0.95 4.37 0.9
−0.3
J/ψ 2.97
4.64 ± 0.5 20.3 ± 2. +4.1
−1.3
4.37 0.9
−0.3
hc 0.153
1.02 ± 0.4 2.4 ± 0.9 +0.62
−0.27 2.35 0.6
−0.3
χc0 0.0664
0.341 ± 0.1 0.8 ± 0.3 +0.21
−0.089 2.35 0.6
−0.3
χc1 0.311
2.04 ± 0.8 8.83 ± 3. +1.8
−0.6
4.32 0.9
−0.3
χc2 0.102
0.681 ± 0.3 1.6 ± 0.6 +0.41
−0.18 2.35 0.6
−0.3
Appendix A: NRQCD WidthsBelow we give explicit expressions for C QD ( * ) s coefficients defined in equation(3). It is convenient to introduce dimensionless variablesWith these notations we have X 3 C ηcDs = 1 + r 2 c + 6r c r s + r 2 s − 65r 4 c − 108r 3 c r s − 54r 2 c r 2 s − 12r c r 3 s − r 4 s − 15r 3 c + 23r 2 c r s + 9r c r 2 s + r 3 s 2 ,s + 2 247r 6 c + 774r 5 c r s + 721r 4 c r 2 s + 436r 3 c r 3 s + 121r 2 c r 4 s + 6r c r 5 s − r 6 s + 4 −3r 4 c − 10r 3 c r s + 6r 2 c r 2 s + 6r c r 3 s + r 4 s 2 , 9 X 4 C J/ψDs = 1 − 2 7r 2 c + 6r c r s + 3r 2 s + 53r 4 c + 68r 3 c r s + 86r 2 c r 2 s + 52r c r 3 s + 13r 4 s − 4 7r 6 c − 78r 5 c r s − 83r 4 c r 2 s − 20r 3 c r 3 s + 25r 2 c r 4 s + 18r c r 5 s + 3r 6 s + 4 3r 4 c − 8r 3 c r s + 4r c r 3 s + r 4 s 2 , X 4 C J/ψD * s = 1 + 10(r c + r s ) 2 − 2 87r 4 c + 148r 3 c r s + 118r 2 c r 2 s + 44r c r 3 s + 11r 4 s + 2 53r 6 c − 34r 5 c r s − 21r 4 c r 2 s + 36r 3 c r 3 s + 59r 2 c r 4 s + 30r c r 5 s + 5r 6 s + 17r 2 c + 2r c r s + r 2 It is easy to see that in massless limit r c,s → 0 for all these coefficients we have C QD ( * ) s = 1.
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| []
|
[
"Is the X(3872) a bound state ?",
"Is the X(3872) a bound state ?"
]
| [
"Pablo G Ortega \nGrupo de Física Nuclear\nInstituto Universitario de Física Fundamental y Matemáticas (IUFFyM)\nUniversidad de Salamanca\nE-37008SalamancaSpain\n",
"Enrique Ruiz Arriola \nDepartamento de Física Atómica\nMolecular y Nuclear\nInstituto Carlos I de Física Teórica y Computacional\nUniversidad de Granada\nE-18071GranadaSpain\n"
]
| [
"Grupo de Física Nuclear\nInstituto Universitario de Física Fundamental y Matemáticas (IUFFyM)\nUniversidad de Salamanca\nE-37008SalamancaSpain",
"Departamento de Física Atómica\nMolecular y Nuclear\nInstituto Carlos I de Física Teórica y Computacional\nUniversidad de Granada\nE-18071GranadaSpain"
]
| [
"Chinese Physics C"
]
| All existing experimental evidence of the bound state nature of the X(3872) relies on considering its decay products with a finite experimental spectral mass resolution which is typically ∆m ≥ 2MeV and much larger than its alleged binding energy, B X = 0.00(18)MeV. On the other hand, we have found recently that there is a neat cancelation in the 1 ++ channel for the invariant DD * mass around the threshold between the continuum and bound state contribution. This is very much alike a similar cancelation in the proton-neutron continuum with the deuteron in the 1 ++ channel. Based on comparative fits of experimental cross section deuteron and X(3872) prompt production in pp collisions data with a finite p T to a common Tsallis distribution we find a strong argument questioning the bound state nature of the state but also explaining the large observed production rate likely consistent with a half-bound state. | 10.1088/1674-1137/43/12/124107 | [
"https://arxiv.org/pdf/1907.01441v2.pdf"
]
| 195,776,145 | 1907.01441 | 0a51b8f513b559bb9f137ad4c919e6481cb3f153 |
Is the X(3872) a bound state ?
Pablo G Ortega
Grupo de Física Nuclear
Instituto Universitario de Física Fundamental y Matemáticas (IUFFyM)
Universidad de Salamanca
E-37008SalamancaSpain
Enrique Ruiz Arriola
Departamento de Física Atómica
Molecular y Nuclear
Instituto Carlos I de Física Teórica y Computacional
Universidad de Granada
E-18071GranadaSpain
Is the X(3872) a bound state ?
Chinese Physics C
xxxCharmonium molecular statesParticle ProductionTsallis distribution PACS: 1239Pn1440Lb1440Rt
All existing experimental evidence of the bound state nature of the X(3872) relies on considering its decay products with a finite experimental spectral mass resolution which is typically ∆m ≥ 2MeV and much larger than its alleged binding energy, B X = 0.00(18)MeV. On the other hand, we have found recently that there is a neat cancelation in the 1 ++ channel for the invariant DD * mass around the threshold between the continuum and bound state contribution. This is very much alike a similar cancelation in the proton-neutron continuum with the deuteron in the 1 ++ channel. Based on comparative fits of experimental cross section deuteron and X(3872) prompt production in pp collisions data with a finite p T to a common Tsallis distribution we find a strong argument questioning the bound state nature of the state but also explaining the large observed production rate likely consistent with a half-bound state.
Introduction
The early possibility of loosely bound states near the charm threshold first envisaged in Ref. [1] seems to be confirmed now by the wealth of evidence on the existence of the X(3782) state with binding energy B X = M D + MD * − M X = 0.00 (18)MeV [2] and which has triggered a revolution by the proliferation of the so-called X,Y,Z states (for reviews see e.g. [3,4]). In the absence of electroweak interactions this state has the smallest known hadronic binding energy. However, since this state is unstable, all the detection methods of the X(3872) are based on looking for its decay channels spectra such as X → J/ψπ + π − where the mass resolution never exceeds ∆m ∼ 1 − 2MeV [5][6][7][8] (see e.g. [9] for a pictorial display on spectral experimental resolution). Therefore it is in principle unclear if one can determine the mass of the X(3872) or equivalently its binding energy ∆B X ∆m with such a precision, since we cannot distinguish sharply the initial state.
In most analyses up to now (see however [10]) the bound state nature is assumed rather than deduced. In fact, the molecular interpretation has attracted considerable attention, since for a loosely bound state many properties are mainly determined by its binding energy [4] and characterized by a line shape in production processes [11]. However, we have noticed recently a neat and accurate cancellation between the would-be X(3872) bound state and the DD * continuum which has a sizable impact on the occupation number at finite temperature [12,13]. This reduction stems from a cancellation in the density of states in the 1 ++ channel and potentially blurs any detected signal where a superposition of 1 ++ states is at work. Such a circumstance makes us questioning in the present letter the actual character of the state. We will do so by analyzing the p T distribution of the X(3872) in high energy production experiments and folding the expected distribution with the actual mass distribution corresponding to the 1 ++ spectrum via the level density within the accessible experimental resolution. For our argument a qualitative and quantitative comparison with a truely weakly bound state such as the deuteron, d, will be most enlightening. As a matter of fact, the similarities between d and X(3872) have been inspiring [14][15][16]. Compared to the X(3872) the main difference is that the deuteron is detected directly by analyzing its well defined track and/or stopping power. Actually, the production of loosely bound nuclei and anti-nuclei, including d,d, 3 He Λ , etc. in ultra-high pp collisions is a remarkable and surprissing experimental observation in recent years [17] and so far poorly understood [18].
The cancellation echoes a similar effect on the deuteron pointed out by Dashen and Kane in their discussion on the counting of states in the hadron spectrum in a coarse grained sense [19] which we review in some detail in the next section. In section 3 we analyze the consequences in a production process. Finally, in section 4 we draw our the conclusions and provide an outlook for future work.
Dashen and Kane cancellation mechanism
In order to illustrate the Dashen-Kane mechanism [19] we introduce the cumulative number of states with invariant CM mass √ s below M in a given channel with fixed J PC quantum numbers. This involves the J PC spectrum which contains
N(M) = ∑ i θ (M − M B i ) + 1 π n ∑ α=1 [δ α (M) − δ α (M th )] .(1)
where the index i runs over the M B i bound states and α over the n coupled channels. Here we have separated bound states M B n explicitly from scattering states written in terms of the eigenvalues of the S-matrix, i.e. S = UDiag(e 2iδ 1 , . . . , e 2iδ n )U † , with U a unitary transformation for n-coupled channels and δ i (M) the eigenphaseshifts for the channel i at CM invariant mass √ s = M. This definition fulfills N(0) = 0. In the single channel case, and in the limit of high masses M → ∞ one gets N(∞) = n B + 1 π [δ (∞) − δ (M th )] = 0 due to Levinson's theorem. While the origin of the bound state term is quite obvious, the derivation of the continuum term is a bit subtle but standard and can be found in many textbooks on statistical mechanics dealing with the quantum virial expansion (see e.g. [20,21]). In potential scattering it can be best deduced by confining the system in a large spherical box which quantizes the energy and relates the energy shift due to the interaction to the phase-shift and then letting the volume of the system go to infinity [19].
In the particular case of the deuteron, which is a neutronproton 1 ++ state bound by B d = 2.2MeV, the cancellation between the continuum and discrete parts of the spectrum was pointed out by Dashen and Kane long ago [19]. (see also [22,23] for an explicit picture and further discussion within the resonance gas model framework). The opening of new channels and the impact of confining interactions was discussed in Ref. [24]. In the 1 ++ channel, the presence of tensor force implies a coupling between the 3 S 1 and 3 D 1 channels. While the partial wave analysis of NN scattering data and the determination of the corresponding phase-shifts is a well known subject [25], we note that a similar analysis in the DD * case is at present in its infancy. In our first model determination in Ref. [12,13] the mixing has an influence for larger energies than those considered here. Therefore, in order to illustrate how the cancellation comes about, we consider a simple model which works sufficiently accurately for both the deuteron and the X(3872) by just considering a contact (gaussian) interaction [26] in the 3 S 1 -channel and using efective range parameters to determine the corresponding phase-shift in the d and X(3872) channels [12,27] respectively.
The result for N(M) in both d or X cases depicted in Fig. 1 display a similar pattern for the np or DD * invariant masses respectively. The sharp rise of the cumulative number is followed by a strong decrease generated by the phase-shift. For larger invariant masses M several effects appear, and in particular the nuclear core (see e.g. [23]) or composite nature of the X(3872) and its cc content becomes manifest (see eg. [28]).
O ∆m ≡ m+∆m/2 m−∆m/2 dMρ(M)O(M) .(2)
where ρ(M) is the density of states, defined as
ρ(M) = dN(M) dM = ∑ i δ (M − M B i ) + 1 π n ∑ α=1 δ α (M) ,(3)
where δ α (M) denotes the derivative of the phase shift with respect to the mass.
In the single channel case, with phase shift δ α (M), and if the resolution is much larger than the binding energy ∆m |B| ≡ |M B − M tr | one has
O| M B ±∆m = O(M B ) + 1 π M tr +∆m/2 M tr dMδ α (M)O(M) .(4)
which, on view of Fig. 1 and for a smooth observable O(M), points to the cancellation, anticipated by Dashen and Kane [24]. The effect was explicitly seen in the np virial coefficient at astrophysical temperatures, T ∼ 1 − 10 MeV [29]. We have recently shown [12] how this cancellation can likewise be triggered for the X(3872) ocupation number at quark-gluon crossover temperatures T ∼ 100 − 200MeV. This will be relevant in relativistic heavy ion collisions when X-production yields are measured, because the partition function involves a Boltzmann factor, ∼ e − √ p 2 +m 2 /T with the density of states, Eq. (3) and the measured yields reproduce remarkably the predictions occupation numbers in the hadron resonance gas model [30].
Therefore, given these tantalizing similarities a comparative study of the deuteron and X(3872) production rates to ultra-high energies in colliders provides a suitable callibration xxxxxx-2 tool in order to see the effects of the Dashen-Kane cancellation due to the finite resolution ∆m of the detectors signaling the X(3872) state via its decay products and decide on its bound state character. Here we propose to study the effect in the observed tranverse momentum (p T ) distributions.
X(3872) production abundance
While the theory regarding the shape of transverse momentum distribution is not fully developped (see e.g. Ref. [31] for an early review, Ref. [32] for a historical presentation), we will rest on phenomenological ansatze which describe the data. On the one hand, the asymptotic p T -spectrum [33] provides a production rate 1/p 8 T based on quark-quark scattering. Hagedorn realized that an interpolation between the power correction and a thermal Boltzmann p T -distribution would work [34]. A thermodynamic interpretation for nonextensive systems [35] of the rapidity distribution was proposed by Tsallis [36] and first applied to high energy phenomena in Refs. [37,38], namely the differential occupation number is given by
d 3 N d 3 p = gV (2π) 3 1 + (q − 1) E(p) T − q q−1 q→1 −−→ gV (2π) 3 e − E(p) T (5)
where E(p) = p 2 + m 2 , V is volume of the system, T the temperature and g the degrees of freedom and, as indicated, the limit q → 1 produces the Boltzmann distribution. We use here the form obtained by the maximum Tsallis entropy principle [39].
The invariant differential production rate, d 3 N/(d 2 p T dy) ≡ E p d 3 N/d 3 p with y = tanh −1 (E p /p z ) the rapidity, has the asymptotic matching corresponds to q = 1.25 [40]. While the thermodynamic interpretation is essential to link the degrees of freedom g with the production rate [41], we note that we have checked [42] that a Tsallis distribution describes accurately the results from particle Monte Carlo generators such as PYTHIA [43,44]. This distribution has also been applied recently by the ALICE collaboration to d-production [45] in pp collisions.
We show next that the X(3872), Ψ(2S) and deuteron prompt production cross sections can be described with the same Tsallis distribution:
1 2π p T dσ (m) d p T = N dy E(p T , y) 1 + q − 1 T E(p T , y) q 1−q (6)
with E(p T , y) = p 2 T + m 2 cosh y and N a normalization factor. Obviously, a direct comparison requires similar p T values as possible for both d, Ψ(2S) and X(3872); the closest ones come from ALICE [45] and CMS [46,47] respectively. The ATLAS data for X(3872) [48] confirm a power law behaviour in p T but extend over a much larger range than the available d and hence are not used in this study. The deuteron data is given in invariant differential yields d 2 N/(2π p T d p T dy), hence the inelastic pp cross section at √ s = 7 TeV, σ pp inel = 73.2 ± 1.3 mb, as measured by TOTEM [49] has been used to transform it into differential cross section. Table 1.
Best fit of parameters for Tsallis distribution. The X data used from CMS [47] is multiplied by the branching fraction B X ≡ B(X → J/ψπ + π − ). Correlation between q and T parameters is practically −1 (r = −0.9992).
X(3872) + d X(3872) + Ψ(2S) + d ln(N X B X ) 41.4 ± 0N X B X (9 ± 3) · 10 −6 (9 ± 3) · 10 −6 N Ψ - (2.2 ± 0.3) · 10 −4 p TN Ψ /N d - 1.09 +0.16 −0.17
On a phenomenological level we perform two fits: One including d and X(3872) data and another one adding the Ψ(2S) data. In both cases the N d,X, [Ψ] , q and T are fitted by minimizing the corresponding χ 2 function with Minuit [50]. The experimental error in the x-axis has been incorporated in the χ 2 via a MonteCarlo procedure with 5000 runs, where the p T value of each experimental data has been randomly shifted within the experimental range with an uniform distribution. Due to the scarcity of X data, we assume the production rate is mainly driven by the deuteron. That way, an initial minimization of the q, T and N d is done, and the resulting best-fit values of q and T are employed to fix N X and N Ψ . dσ/dp T [nb/GeV] dσ X(3872) /dp T · B X dσ d /dp T dσ Ψ(2S) /dp T Figure 2.
Comparison between the prompt production cross section in pp collisions of X(3872) (blue), the deuteron (green) and the Ψ(2S) (red). Ψ(2S) data from CMS [46]. The X(3872) data from CMS [47] is multiplied by the branching fraction B(X → J/ψππ). Deuteron data in pp collisions are taken from AL-ICE [45]. The lines are Tsallis distributions fitted to each data set, with the same q and T parameters. We fit both X(3872), Ψ(2S) and deuteron data. The shadowed bands represent the statistical 68% confidence level (CL) obtained from the fit. The results can be found in Tab. 1, and the final production fit at Fig. 2 for the two considered calculations, one with the X(3872) and the deuteron, and another for the X(3872), the Ψ(2S) and the deuteron. They are both compatible, as expected, as the X to Ψ(2S) production ratio, measured at CMS, is almost constant [47]. The X/d production ratio is 0.046 +0.016 −0.013 for X + d fit (and practically the same value for X+Ψ+d fit), dependent on the value of the branching fraction. Note that we do not have the pure cross section for the X, as it is multiplied by the unmeasured branching fraction which has been recently constrained in an analysis of BESIII data by C. Li et al [51] to be B X ≡ B(X → J/ψπ + π − ) = 4.5 +2.3 −1.2 %. This value is consistent with the PDG lower-B X > 3.2% [52], and upper bound B X < 6.6% at 90% C.L. [53]. The uncertainty comes from the most recent value of B X · (B − → K − X(3872)) < 2.6 × 10 −4 at 90% C.L. [54]. We note that in a recent paper, Esposito et al. [55] consider the wider range 8.1 +1.9 −3.1 %. Consequently, we can study the ratio of the X/d occupation numbers as a function of the B X branching fraction. In Fig. 3 we see the results. Considering the error bars, the experimental constrains give ratios between 0.3 and 1.9 for N X /N d .
In our previous fits above we have neglected the role played by the finite resolution of the detectors, ∆m, which we dicuss next. Ref. [47] uses a ±2σ window around the X(3872) mass, with σ = 5 − 6 MeV, to select the X(3872) events in the J/ψππ invariant mass spectrum. That means that the branching fraction B(X → J/ψπ + π − ), as measured by CMS is averaged in the [M X − 2σ , M X + 2σ ] energy window, which includes the continuum. As a consequence of the energy window, there are many decays that can be affected, those involving theD 0 D 0 * channel. In fact, the distribution obtained from Eq. 5 depends on the mass, and hence its observed value undergoes formula 4, reflecting the finite resolution. Similar to the finite temperature case [12] we have checked that the Tsallis p T -shape is basically preserved for p T ∆m, but the occupation number is modified for ∆m B.
For definiteness we use ∆m = 2σ , as CMS measures the X(3872) in a ±2σ region around the central value of the X mass. The net effect is summarized in a ratio, which we find to be practically independent of the transverse momentum p T for the Tsallis distribution shape,
σ m=M X ±∆m σ M X ∼ N ∆m N X .(7)
This formula will allow us to set values for the relative occupation numbers due to the finite resolution. We take M X = M D + MD * − γ 2 X /(2µ D,D * ) as a parameter by looking at the poles of the DD * S-matrix in the 3 S 1 − 3 D 1 channel [12]. Therefore, while in the limit of ∆m → 0 we should expect the ratio N X /N d → 1, 1/2 or 0 for a bound (γ X > 0), half-bound (γ X = 0) or unbound (actually virtual, (γ X < 0)) state, for increasing and finite ∆m the value lies somewhat in between and xxxxxx-4 the different situations can be hardly distinguished. However, as seen in Fig. 4 the numerical value N ∆m /N X ∼ 0.5 − 0.6 is rather stable for a reasonable range of B X and σ values. If we re-interpret N X as N X,∆m this falls remarkably in the bulk of Fig. 3 where N X,∆m /N d ∼ 0.5 implies N X /N d ∼ 1. Thus, unlike expectations, we do not find the production rate to change dramatically due to binding energy effects due to ∆m; the p T shape will likewise not depend on this (unlike expectations [55]). In a recent and insightful paper Kang and Oller have analyzed the character of the X(3872) in terms of bound and virtual states within simple analytical parameterizations [10]. While the Dashen-Kane cancellation has not been explicitly identified, it would be interesting to see if their trends can be reproduced by more microscopic approaches.
Conclusions
Theoretically, it is appealing an scenario where the X(3872) is a half-bound state (zero binding energy) corre-sponding to the so-called unitarity limit, characterized by scale invariance [56]. In this case, the phase-shift becomes δ = π/2 around threshold, and the occupation number becomes a half of that of the bound state. Our analysis shows that the large production rate of the X(3872) at finite p T does not depend strongly on the details of the binding since the experimental bin size is much larger than the binding energy. We also find striking and universal shape similarities with the ψ(2S) and deuteron production data via a common Tsallis distribution. A more direct check of our predicted mild suppression might be undertaken if all production data would be within the same p T values. Finally, we note that in order to envisage a clear fingerprint of the X(3872) binding character a substantial improvement on the current experimental resolution of its decay products would be required.
One of us (E.R.A.) thanks Airton Deppman for discussions on Tsallis distributions. This work is partly supported by the Spanish Ministerio de Economła y Competitividad and European FEDER funds (grants FPA2016-77177-C2-2-P and FIS2017-85053-C2-1-P) and Junta de Andalucía (grant FQM-225).
Figure 1 .
1Cumulative number in the 1 ++ channel for the deuteron (solid) and X(3872) (dashed) as a function of the invariant mass M respect to np and DD * values respectively. We divide by spin degeneracy.An immediate consequence of this effect trivially follows from Eq. (1) for an observable depending on the invariant mass function O(M). The corresponding measured quantity for a bin in the range (m − ∆m/2, m + ∆m/2) becomes
Figure 3 .
3In red, the X(3872) vs deuteron prompt production ratio as a function of the branching fraction B(X → J/ψπ + π − ) when fitting to X(3872), Ψ(2S) and deuteron data. The shadowed band represent the statistical 68% confidence level (CL) obtained from the fit. The green band shows the constrains of the recent analysis of C. Li et al from BESIII data[51] B X ≡ B(X → J/ψπ + π − ) = 4.5 +2.3 −1.2 .
Figure 4 .
4The relative occupation number with and without finite resolution ∆m = 2σ as a function of the X(3872) binding energy. Using M X = M D + MD * − γ 2 X /(2µ D,D * ) we take the positive branch for the virtual state (γ X > 0) and the negative branch for the bound state (γ X < 0) for different σ values.
bound states and continuum states with threshold M th and is given as1) E-mail: [email protected]
2) E-mail: [email protected]
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arXiv:1907.01441v2 [hep-ph] 7 Oct 2019
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| []
|
[
"Encounters of Merger and Accretion Shocks in Galaxy Clusters and their Effects on Intracluster Medium",
"Encounters of Merger and Accretion Shocks in Galaxy Clusters and their Effects on Intracluster Medium"
]
| [
"Congyao Zhang \nDepartment of Astronomy and Astrophysics\nUniversity of Chicago\n60637ChicagoILUSA\n\nMax Planck Institute for Astrophysics\nKarl-Schwarzschild-Str. 1D-85741GarchingGermany\n",
"Eugene Churazov \nMax Planck Institute for Astrophysics\nKarl-Schwarzschild-Str. 1D-85741GarchingGermany\n\nSpace Research Institute (IKI)\nProfsoyuznaya 84/32117997MoscowRussia\n",
"Klaus Dolag \nMax Planck Institute for Astrophysics\nKarl-Schwarzschild-Str. 1D-85741GarchingGermany\n\nUniversity Observatory Munich\nScheinerstr 1D-81679MunichGermany\n",
"William R Forman \nSmithsonian Astrophysical Observatory\nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMA\n",
"Irina Zhuravleva \nDepartment of Astronomy and Astrophysics\nUniversity of Chicago\n60637ChicagoILUSA\n"
]
| [
"Department of Astronomy and Astrophysics\nUniversity of Chicago\n60637ChicagoILUSA",
"Max Planck Institute for Astrophysics\nKarl-Schwarzschild-Str. 1D-85741GarchingGermany",
"Max Planck Institute for Astrophysics\nKarl-Schwarzschild-Str. 1D-85741GarchingGermany",
"Space Research Institute (IKI)\nProfsoyuznaya 84/32117997MoscowRussia",
"Max Planck Institute for Astrophysics\nKarl-Schwarzschild-Str. 1D-85741GarchingGermany",
"University Observatory Munich\nScheinerstr 1D-81679MunichGermany",
"Smithsonian Astrophysical Observatory\nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMA",
"Department of Astronomy and Astrophysics\nUniversity of Chicago\n60637ChicagoILUSA"
]
| []
| Several types/classes of shocks naturally arise during formation and evolution of galaxy clusters. One such class is represented by accretion shocks, associated with deceleration of infalling baryons. Such shocks, characterized by a very high Mach number, are present even in 1D models of cluster evolution. Another class is composed of "runaway merger shocks", which appear when a merger shock, driven by a sufficiently massive infalling subcluster, propagates away from the main-cluster center. We argue that, when the merger shock overtakes the accretion shock, a new long-living shock is formed that propagates to large distances from the main cluster (well beyond its virial radius) affecting the cold gas around the cluster. We refer to these structures as Merger-accelerated Accretion shocks (MA-shocks) in this paper.We show examples of such MA-shocks in 1D and 3D simulations and discuss their characteristic properties. In particular, (1) MA-shocks shape the boundary separating the hot intracluster medium (ICM) from the unshocked gas, giving this boundary a "flower-like" morphology. In 3D, MA-shocks occupy space between the dense accreting filaments.(2) Evolution of MA-shocks highly depends on the Mach number of the runaway merger shock and the mass accretion rate parameter of the cluster. (3) MA-shocks may lead to the misalignment of the ICM boundary and the splashback radius.(racc rsp) if the gas adiabatic index is γ = 5/3 (see e.g. Shi 2016b) 1 .However, the evolution of galaxy clusters is more complicated than those one-dimensional (1D) self-similar solutions. Two major processes tend to break the self-similarity (and also spherical symmetry) of galaxy clusters, i.e. active galactic nucleus (AGN) feedback (see e.g. Werner et al. 2019, for a recent review) and cluster mergers (e.g. Sarazin 2002). The former process perturbs the gas in cluster cores (e.g. 100 kpc); the latter one, 1 Specifically, the alignment of the racc and rsp holds when γ = 5/3 only if the cluster mass accretion rate parameter (Γ) is in the range of 0.5 ≤ Γ ≤ 5 (Shi 2016b; see the definition of Γ in Section 2). | 10.1093/mnras/staa1013 | [
"https://arxiv.org/pdf/2001.10959v2.pdf"
]
| 210,943,080 | 2001.10959 | 92e206acb832cccd98665be4c3172cafed2616b7 |
Encounters of Merger and Accretion Shocks in Galaxy Clusters and their Effects on Intracluster Medium
Congyao Zhang
Department of Astronomy and Astrophysics
University of Chicago
60637ChicagoILUSA
Max Planck Institute for Astrophysics
Karl-Schwarzschild-Str. 1D-85741GarchingGermany
Eugene Churazov
Max Planck Institute for Astrophysics
Karl-Schwarzschild-Str. 1D-85741GarchingGermany
Space Research Institute (IKI)
Profsoyuznaya 84/32117997MoscowRussia
Klaus Dolag
Max Planck Institute for Astrophysics
Karl-Schwarzschild-Str. 1D-85741GarchingGermany
University Observatory Munich
Scheinerstr 1D-81679MunichGermany
William R Forman
Smithsonian Astrophysical Observatory
Harvard-Smithsonian Center for Astrophysics
60 Garden St02138CambridgeMA
Irina Zhuravleva
Department of Astronomy and Astrophysics
University of Chicago
60637ChicagoILUSA
Encounters of Merger and Accretion Shocks in Galaxy Clusters and their Effects on Intracluster Medium
Accepted XXX. Received YYY; in original form ZZZMNRAS 000, 1-10 (2019) Preprint 30 January 2020 Compiled using MNRAS L A T E X style file v3.0hydrodynamics -shock waves -methods: numerical -galaxies: clusters: intracluster medium
Several types/classes of shocks naturally arise during formation and evolution of galaxy clusters. One such class is represented by accretion shocks, associated with deceleration of infalling baryons. Such shocks, characterized by a very high Mach number, are present even in 1D models of cluster evolution. Another class is composed of "runaway merger shocks", which appear when a merger shock, driven by a sufficiently massive infalling subcluster, propagates away from the main-cluster center. We argue that, when the merger shock overtakes the accretion shock, a new long-living shock is formed that propagates to large distances from the main cluster (well beyond its virial radius) affecting the cold gas around the cluster. We refer to these structures as Merger-accelerated Accretion shocks (MA-shocks) in this paper.We show examples of such MA-shocks in 1D and 3D simulations and discuss their characteristic properties. In particular, (1) MA-shocks shape the boundary separating the hot intracluster medium (ICM) from the unshocked gas, giving this boundary a "flower-like" morphology. In 3D, MA-shocks occupy space between the dense accreting filaments.(2) Evolution of MA-shocks highly depends on the Mach number of the runaway merger shock and the mass accretion rate parameter of the cluster. (3) MA-shocks may lead to the misalignment of the ICM boundary and the splashback radius.(racc rsp) if the gas adiabatic index is γ = 5/3 (see e.g. Shi 2016b) 1 .However, the evolution of galaxy clusters is more complicated than those one-dimensional (1D) self-similar solutions. Two major processes tend to break the self-similarity (and also spherical symmetry) of galaxy clusters, i.e. active galactic nucleus (AGN) feedback (see e.g. Werner et al. 2019, for a recent review) and cluster mergers (e.g. Sarazin 2002). The former process perturbs the gas in cluster cores (e.g. 100 kpc); the latter one, 1 Specifically, the alignment of the racc and rsp holds when γ = 5/3 only if the cluster mass accretion rate parameter (Γ) is in the range of 0.5 ≤ Γ ≤ 5 (Shi 2016b; see the definition of Γ in Section 2).
INTRODUCTION
The self-similar spherical collapse model provides an insightful framework for understanding the formation and evolution of galaxy clusters in the expanding Universe (Fillmore & Goldreich 1984;Bertschinger 1985;Adhikari et al. 2014;Shi 2016b). Characteristic sizes of both gaseous and dark matter (DM) components of galaxy clusters naturally co-exist in this model, i.e. the radius of the accretion shock racc (Birnboim & Dekel 2003) and the splashback radius rsp (More et al. 2015). They coincidentally align with each other however, could re-distribute both gas and DM on Mpc scales (e.g. Ricker 1998;Poole et al. 2006). Unlike the prediction of the self-similar model, in cosmological simulations, the accretion shocks are found beyond rsp, and are sometimes even significantly farther outside (e.g. Lau et al. 2015;Schaal et al. 2016; see also Walker et al. 2019 for a review). On average, in simulations racc/rsp is 1.5 throughout the evolution of galaxy clusters (see fig. 1 in Walker et al. 2019). Mergers of galaxy clusters presumably play an important role in this regard.
During the merger process, the cluster splashback radius, as the outermost caustic in the DM density profile, is mainly affected through the change of the gravitational potential of the merging systems. However, due to the collisional nature of the gas, the impact of mergers on accretion shocks is more complicated. One important effect is the encounter of the merger and accretion shocks, which is able to change dramatically the shock radius (Birnboim et al. 2010). Zhang et al. (2019b) have demonstrated that the merger shocks could detach from the infalling subclusters which drive them, and propagate to large distances. They could maintain their shock strength or even get stronger when moving in the diffuse intracluster medium (ICM; say the regions between the high-density filaments), where the radial gas density profile is rather steep. In Zhang et al. (2019b), these shocks were called "runaway merger shocks". On the other hand, galaxy clusters are supposed to frequently experience merger events. For example, the merger rate is a few per halo per unit redshift for the merger mass ratio ξ ≤ 30 (Fakhouri & Ma 2008). Therefore, collisions of the merger and accretion shocks could be very common.
From the theoretical point of view, the collision of two 1D shocks is a Riemann problem. Three discontinuities are subsequently formed after the shock interaction, including forward and reverse shocks/rarefactions and a contact discontinuity (CD) in between (Landau & Lifshitz 1959). More specifically, in our problem, the two colliding shocks are: a runaway merger shock with a moderate Mach number Mrs 3 (Zhang et al. 2019b) and an accretion shock with high Mach number Macc 10 (see Borgani, & Kravtsov 2011), respectively. They both move radially outwards in the rest frame of the cluster. In this case, a strong forward shock is formed with Mach number Mmas MaccMrs, moving away from the cluster center (Birnboim et al. 2010) 2 . In this work, it is referred to as the Merger-accelerated Accretion shock (MA-shock hereafter).
In principle, MA-shocks should naturally appear in all hydrodynamic cosmological simulations if their spatial resolution in the cluster outskirts is high enough to resolve the structures (see e.g. Miniati et al. 2000;Ryu et al. 2003;Skillman et al. 2008;Vazza et al. 2010;Paul et al. 2011;Schaal et al. 2016;Zinger et al. 2016;Ha et al. 2018). These shocks shape atmospheres of galaxy clusters, producing a "blossom-like" morphology (see e.g. fig. 2 in Vazza et al. 2017). By definition, MA-shocks are a subset of the external shocks classified in Ryu et al. (2003), which, however, are expected to behave quite differently from the ordinary accretion shocks. It is therefore worth studying the nature of MA-shocks and their implications in cluster formation. In this work, we argue that most of the time, clusters are encompassed by MA-shocks rather than canonical accretion shocks.
This paper is organized as follows. In Section 2, we illustrate formation of MA-shocks in a 1D cluster model, and explore the evolution of the MA-shock structures and its dependence on the cluster environment. In Section 3, we extend our exploration to full 3-dimensional (3D) cosmological simulations, and make a direct comparison of a 3D cluster with our 1D model. In Section 4, we discuss the impacts of MA-shocks on the ICM, and make conclusions.
MODELLING MA-SHOCKS
For the purposes of demonstrating the formation of MA-shocks, we performed 1D cosmological simulations in this section. These simulations are similar to those used in Birnboim & Dekel (2003) (see also Birnboim et al. 2010), where both gas and DM components of the Universe are simulated in the cosmological comoving background. The 1D model is computationally fast and isolates the formation of MA-shocks from the much more complicated merger/accretion configurations in 3D (see Section 3 for a comparison of the 1D and 3D simulation results).
Here we briefly describe the numerical method and the initial conditions used in the 1D model (see more details in Appendix A). Our simulations employ a hybrid N-body/hydrodynamics method (see e.g. Bryan et al. 1995), where Eulerian scheme is used to solve the gas dynamics while the DM is modelled as Lagrangian shells. All our simulations presented in this section start from the redshift z = 100. The initial gas and DM density profiles are designed so that the cluster grows in a self-similar way with a constant mass accretion parameter Γ defined as
M (t) = M0a(t) Γ ,(1)
where M0 is the cluster mass at present, a(t) is the cosmic scale factor (Fillmore & Goldreich 1984). We stress that, instead of applying more realistic initial conditions (e.g. Birnboim & Dekel 2003), our choice helps to understand the relation of the MA-shock evolution and the state of the cluster growth. Our results show that the trajectory of the MA-shock front strongly depends on the value of Γ (see Figs. 4 and 5 below, and Section 2.3 for more discussions).
We have also used more sophisticated initial conditions in the 1D simulations in Section 3.2, and directly compare the 1D results with the 3D cosmological simulations.
To generate a MA-shock, an additional "merger" shock is initiated at the cluster center at the moment t b by suddenly increasing the gas pressure in the innermost cell by a factor ξ. We vary ξ to obtain "runaway" shocks with different Mach number Mrs when they encounter the cluster accretion shock (see Table 1 for a summary of the main parameters used in our simulations). This method has been used and proved to be robust in Zhang et al. (2019b) when they studied the propagation of runaway merger shocks in cluster outskirts. Fig. 1 shows the evolution of the gas density profile in the simulation S1T2M23. The initial accretion shock forms at the very beginning of the simulations once the collapsing gas decouples from the Hubble flow. Since there is no radiative cooling involved in the simulations, the stable gas atmosphere could exist even when the cluster mass is small (cf. Rees & Ostriker 1977;Birnboim & Dekel 2003). A secondary shock is artificially initiated at the cluster center at t = 2.5 Gyr to mimic a merger shock, which propagates with a high speed in the ICM and rapidly catches up with the accretion shock. As expected, a rarefaction, CD, and MA-shock are formed after the shock collision (marked in the figure; see also Birnboim et al. 2010). Both CD and MA-shock are subsequently decelerated by the inflowing unshocked gas. Their trajectories, however, depend on the environment of the cluster (i.e. mass accretion rate parameter Γ) and the strength of the MA-shock (cf. Fig. A1; see Section 2.2 for more detailed discussions on these dependence). When the Γ is moderate (like the case Γ = 1 shown in Fig. 1), a long time is needed for the MA-shock to re-fall back (even possibly longer than the Hubble time). It is interesting to note that the runaway shock in our model develops an N-shaped wave profile due to its blast-wave nature and the spherical symmetry of the system (Dumond 1946). The front of the N-wave encounters the accretion shock at t 2.7 Gyr. The rear part of the N-wave, however, firstly meets the re-infalling CD at t 6 Gyr. Its velocity increases significantly after crossing the CD from the cold side to the hot side.
Formation of MA-shocks and their structures
To illustrate the structures associated with the MA-shock more clearly, we zoom in on gas profiles where the MA-shock forms, shown in Fig. 2. We can clearly see the aforementioned rarefaction, CD, and MA-shock in the gas density profiles. The CD separates the low-and high-entropy gas on its two sides. On the left (radii smaller than CD) side the gas is successively compressed and heated Figure 1. Evolution of gas density profile in the simulation S1T2M23, where the gas density ρgas is scaled by the cosmic mean density of baryons ρ b . The "runaway" shock is initiated at the cluster center at t = 2.5 Gyr and encounters the accretion shock at tmas 2.7 Gyr. A rarefaction, CD, and MA-shock are subsequently formed after the shock collision. The gaseous structures, including: 1. original accretion shock, 2. front of the runaway shock, 3. rear of the runaway shock, 4. CD, 5. rarefaction, 6. MA-shock, are marked in the figure by their corresponding numbers. As a comparison, the evolution of the accretion shock radius in the run S1 is shown as the black dashed line. This figure shows several new structures form after the collision of the merger and accretion shocks. The boundary of the shock-heated cluster atmosphere is driven much farther outwards by this collision (see Section 2.1).
by the runaway and accretion shocks, while on the right side (outside CD) the gas passes only through the MA-shock, which has velocity comparable to the merger shock velocity. As a result, the gas density is higher but the temperature is lower on the left side of CD. Therefore, a high-entropy gas shell is formed between the CD and MA-shock, and this shell is a robust signature of the past shock collision.
It turns out that, within this high-entropy shell, the entropy is decreasing with radius. The entropy here is defined as Sgas ≡ Tgas/ρ γ−1 gas (Tgas and ρgas are gas temperature and density, respectively). The gas temperature behind the shock is proportional to u 2 mas , where umas is the MA-shock velocity, which is a decreasing function of time/radius for a propagating spherical shock (see Fig. 3, and more discussions in Section 2.2). The upstream cold gas density also decreases with the radius, but rather slowly (approximately ρgas ∝ r −1 before entering the ICM; see also fig. 1 in Shi 2016b). The net result of these competing effects is that the gas entropy profile between the CD and MA-shock is a decreasing function of radius (see bottom panels in Figs. 2 and A1).
Given the radially-decreasing entropy profile, the high-entropy shell is convectively unstable (even though this instability is not captured in the 1D simulations). We note in passing that the temperature is also decreasing with radius and, therefore, the shell is unstable even when the Magneto-Thermal-Instability criterion (Balbus 2000) is used instead of the Schwarzschild one mentioned above. With time, convective motions should kick in and the shell would re-arrange itself to restore the non-decreasing entropy profile. The instability growth rate can be related to the characteristic Keplerian frequency (ΩK) as ∼ ΩK γ −1 d ln Sgas/d ln r, implying that the life-time of MA-shocks is likely shorter than that of the shell. Our estimates show that, given the slope of the entropy profile and its radial extent, the characteristic amplitude of the induced gas motions can be up to ∼ 0.5cs, where cs is the ICM sound speed. This implies that the resulting convection might make an important contribution to the non-thermal pressure in the cluster outskirts (see also Shi, & Komatsu 2014;Shi et al. 2015).
Evolution of MA-shocks and their fate
The evolution of MA-shocks shows a very similar behavior to that of a strong blast wave (see e.g. Ostriker, & McKee 1988). The accreted cold gas on the upstream side strongly decelerates the shocks. The velocity of the MA-shock front in the cluster frame can be represented as a sum of the shock velocity in the rest frame of the upstream gas umas(t) and the infalling velocity of the upstream gas ugas(r, t), i.e.
drmas dt = umas(t) + ugas(rmas, t),(2)
where rmas is the location of the MA-shocks front. Generally, there are three possible fates for a MA-shock, i.e. (1) it survives until z = 0 (like the case shown in Fig. 1); (2) it recedes and is eventually replaced with an ordinary accretion shock (like the case shown in Fig. A1); and (3) it is overrun by a new runaway merger shock and is accelerated again.
To keep the problem simple, we merely address the first two possibilities in this section, and will discuss the last one in Section 3. Fig. 3 shows the evolution of the shock velocities of the radially outermost shocks in our simulations. The MA-shocks behave differently from that of the genuine accretion shocks. The latter one follows uacc ∝ t δ , where δ = (2Γ/3 − 1)/3 (Fillmore & Goldreich 1984;Shi 2016b). The evolution of MA-shocks, however, is close to a Sedov-Taylor solution, i.e. us ∝ (∆t/tmas) −3/5 for not too small ∆t/tmas, where ∆t = t − tmas (see Sedov 1959). This is not surprising, since the MA-shocks are driven from inside, and have negligible upstream gas pressure. Note that the MA-shocks, when viewed as a function of ∆t = t − tmas, go through a transitional stage before approaching the Sedov-Taylor form (i.e. ∆t/tmas 0.1 in our simulations), when the shock velocity is changing slowly. This is because, for small ∆t/tmas, the MA-shocks, which do not start from the cluster center, behave more like plane shock waves rather than spherical ones in their very early phase. In addition, the evolution of MA-shocks is also affected by (1) the gravity, (2) the Hubble expansion, and (3) non-uniform density distribution of the upstream gas, which causes slight deviations of the shock velocity curves from a simple power law. Eventually, some MA-shocks become the accretion shocks again (i.e. the shock velocity curves jump to high values). From the simulations, we find that, the deceleration of the MA-shocks (say the evolution of the shock velocity with time) only mildly depends on its formation time tmas, runaway shock Mach number Mrs, and the mass accretion rate parameter Γ. These allow us to model umas(t) by a very simple form, i.e.
umas(t) = umas(tmas) 0 ≤ ∆t tmas ≤ κ umas(tmas) ∆t κtmas β ∆t tmas > κ ,(3)
where β = −0.7 (slightly steeper than −3/5) and κ is a free parameter to take the shock transitional stage into account. With this approximation, the integration of Eq.
(2) becomes straightforward. The self-similar solution of the spherical collapse model is applied to estimate the gas velocity ugas(r, t) (Bertschinger 1985;Shi 2016b). Fig. 4 shows two groups of solutions of Eq.
(2) (see Table 2 for the parameters used in the calculations). The line color encodes the Mach number of the runaway shocks Mrs just before they encounter the accretion shocks. As a reference, the evolution of the accretion shock radius racc(t) and the turn-around radius rta(t) in the self-similar solutions are shown as the black solid and dashed lines. One can see the modelled shock-front trajectories (color solid lines) show a good match with the simulations (dotted lines) 3 . The MA-shock could survive for a longer time and reach a larger ) t / t mas -1 S1T2M15 S1T2M20 S1T2M23 S1T6M23 S3T2M20 S3T2M25 S3T6M20 Figure 3. Evolution of shock velocities of the radially outermost shocks in the simulations. The results are shown as the solid and dashed lines for the simulations with Γ = 1 and 3, respectively.
The additional thick red line shows ∝ (∆t/tmas) −3/5 for a comparison. This figure shows that the evolution of MA-shocks is close to a Sedov-Taylor solution. The shape of the scaled velocity curves only mildly depends on Γ, tmas, and Mrs (see Section 2.2). radius when Γ is smaller and/or the shock Mach number is higher. All the MA-shocks tend to move back to the radius racc(t) in Fig. 4, even though in some cases the required timescale is longer than the Hubble time. In principle, if strong enough, a MA-shock is able to escape from the halo potential well (e.g. go beyond a few turn-around radii of the halo) and move into a nearly uniform but cosmologically expanding medium. In this case, the shock is described by the cosmological self-similar solution of blast waves found by Bertschinger (1983), where the shock radius evolves as rmas ∝ ∆t 4/5 . However, because the Mach number of merger shocks is usually 3, such a situation must be very rare. Fig. 5 shows the same results as those in Fig. 4 but with rmas and t scaled by the accretion shock radius racc in the self-similar model and the time tmas when forming the MA-shocks 4 . This figure shows that, if Γ = 1, even a moderate runaway merger shock (e.g. Mrs 1.5) could easily expand the size of atmospheres of galaxy clusters by a factor of 2, and the expansion could last for a few tmas. However, when Γ gets larger, it is harder for the MA-shocks to survive for longer time and reach larger cluster radii. The dependence of the upstream gas velocity profile ugas on Γ plays a key role in this result (see Eq. 2). For example, when Γ = 3, the MA-shocks produced by the runaway shocks with Mrs > 2 could only survive for a time period tmas. Nevertheless, those MA-shocks in galaxy clusters with high Γ would still make a significant impact on the ICM at low redshift (e.g. z 1).
Key parameters Γ and Mrs shaping MA-shocks
The two parameters -the mass accretion rate Γ and the Mach number of a runaway shock Mrs play crucial roles in shaping MA-shocks in our simulations, and are also tightly related with the cluster environment and growth history. In this section, we discuss them in detail. The mass accretion rate which quantifies the global growth history of galaxy clusters, has been extensively explored in analytical and numerical studies (e.g. Zhao et al. 2009;Adhikari et al. 2014;Lau et al. 2015). The infalling substructures and filaments contribute a large portion of the accreting mass (see the high peaks in the curves shown in the bottom panel of Fig. 7). However, we have to emphasize that the accretion rate defined there has a different meaning from the one used in our 1D model. Instead of characterizing the mass growth of a whole galaxy cluster, our parameter Γ is used to setup the matter distribution in the model. Zhang et al. (2019b) has argued that prominent runaway merger shocks may only appear in the non-filamentary regions, and so do the MA-shocks. In this sense, the constant Γ applied in our 1D simulations in Section 2 is supposed to represent the non-filamentary environment of the cluster rather than the total mass accretion rate of the cluster integrated over all directions. Fig. 5 implies that runaway merger shocks with moderate Mach number (e.g. Mrs 1.5) are generally required to generate relatively long-lived and significant MA-shocks. However, not all cluster mergers are powerful enough to drive such shocks. For example, one necessary condition is that the infalling subcluster should be sufficiently massive to keep its gas atmosphere after the primary pericentric passage. In this regard, the merger mass ratio ξ is a key factor. We have explored the cluster merger process for a wide range of merger parameters in our previous works (see e.g. Zhang et al. 2014Zhang et al. , 2016) 5 , which provided some intuitions on this question. For example, Zhang et al. (2019a) and Lyskova et al. (2019) presented analysis of two merging systems with ξ = 10 and zero and large impact parameters respectively. In both cases, 5 In those works, we simulated mergers between two idealized galaxy clusters, where each of the merging clusters contains spherical gas and DM halos. The initial gas and DM density profiles both follow r −3 in the cluster outskirts (see eqs. 1-4 in Zhang et al. 2014). A large merger-parameter space (including the cluster mass, mass ratio, initial relative velocity, and impact parameter) has been explored by those simulations.
the merger shocks could reach the cluster outskirts with Mach number larger than 2. Our merger sample with ξ = 60 and zero impact parameter, however, shows that only a weak runaway merger shock (Mach number ∼ 1.4) is formed during the merger process. Overall, these results imply that ξ 50 is a reasonable mass-ratio range for cluster mergers to power significant runaway merger shocks. However, we emphasize that, besides the mass ratio, many other factors (like other merger parameters, initial cluster gas/DM profiles) could also affect the conclusion. To fully characterize this problem, more detailed studies need to be carried out in the future. Nevertheless, we suggest that mergers with mass ratio a few 10s would be able to drive runaway merger shocks with Mrs 1.5, and further excite MA-shocks.
MA-SHOCKS IN 3D COSMOLOGICAL SIMULATIONS
Even though MA-shock formation has been well captured in our 1D models, we extend our study of MA-shocks into the 3D simulations in this section for the following reasons.
(1) Galaxy clusters gradually become asymmetric at large radii. Giant filaments penetrate into the ICM along some directions.
(2) Only a single merger event is considered in our 1D models. But in reality, it is very common for galaxy clusters to experience multiple merger events in a short time period.
We analyzed a galaxy cluster from the COMPASS 6 zoom-in simulations. This simulation is a very high resolution re-simulation of the D.17 region as introduced in Bonafede et al. (2011). The cluster's virial radius and virial mass are R200m = 4.2 Mpc and M200m = 2.2 × 10 15 M 7 , respectively. It is simulated using P-Gadget3, a modernized version of P-Gadget2 (Springel 2005) which implements updated smoothed-particle hydrodynamics (SPH) formulations regarding the treatment of viscosity and the use of kernels (Dolag et al. 2005;Beck et al. 2016), allowing a better treatment of turbulence within the ICM. It also includes a formulation of isotropic, thermal conduction at 1/20th of the classical Spitzer value (Spitzer 1962). The particle mass for DM and gas is 4.7 × 10 6 M and 8.9 × 10 5 M respectively, and the softening for both, DM and gas particles is set to 0.69 kpc. The cluster at redshift z = 0 is therefore resolved with 5.6 × 10 8 particles within the virial radius and hosts almost 10 5 identified sub-structures, making it to one of the most resolved, cosmological, hydrodynamical simulations of massive galaxy clusters. Fig. 7. The white circle shows the shock radius formed in our special 1D simulation (see Section 3.2), which is used to directly compare to the 3D cluster shown here. This figure shows that the ICM in this cluster is mostly covered by MA-shocks along the non-filamentary directions (see Section 3).
MA-shocks in a 3D cluster
We exclude the shocks in the direction of filaments, where the shock structures are complicated and are sometimes even pushed inside the virial radius by the strong inflowing gas streams (Zinger et al. 2016). In this figure, we can clearly see a high-entropy shell lying on the downstream side of the shock predicted by our 1D models, which implies that the cluster gas atmosphere is mostly covered by the MA-shocks but not the genuine accretion shocks along the non-filamentary directions. We further trace the evolution of the averaged radii of the MA-shocks throughout the simulation within three sectors illustrated in Fig. 6. These sectors are selected as they are not affected by the large-scale filaments. The results are shown in the top panel of Fig. 7. One can see that the shock radii are located much beyond the cluster virial radius (black solid line) along all three directions. After t 2 Gyr (or z 3), the cluster experiences three-major merger events (see three major peaks in Γ shown in the bottom panel of Fig. 7). The merger mass ratios are 1, 3, 5, respectively. During the merger, the cluster virial radius shows rapid increases. At the same time, the variations of the MA-shock radii show a temporal correlation with that of the virial radius (also with the merger events). Two rapid increases of the shock radii start at t 5 and 11 Gyr, which are approximately ∼ 1 − 2 Gyr later then the corresponding mergers. This is generally consistent with the timescale of the merger shock crossing the cluster radius, i.e. rmas/(Mrscs). This correlation indicates that the cluster mergers play a dominant role in pushing the shock outwards. . The 3D cluster experiences three-major merger events after t > 2 Gyr. This figure shows that the radially outermost shocks of the cluster reside much beyond the virial radius. The increases of these shock radii associate with the merger events but with a ∼ 1 − 2 Gyr time delay (see Section 3.1).
A direct comparison of 1D and 3D clusters
In this section, we make a direct comparison of the 3D cluster presented in Section 3.1 with our 1D model to further clarify the effect of the mergers on the formation of MA-shocks. Ideally, we need a 1D model which reproduces the growth history of the 3D cluster but suppresses the effect of the merger process on the ICM (i.e. only radial accretion is included). For this purpose, we performed a 1D simulation with a specially designed initial condition, which corresponds to the gas/DM density and velocity profiles of the 3D cluster at redshift z = 160 8 . The bottom panel of Fig. 7 shows a comparison of the parameter Γ in the 1D model and the one from the 3D simulation (here Γ ≡ d ln M200m/d ln a; see also Eq. 1). Both curves show a violent growth of the cluster at high redshift z 3. But at lower redshift, the 1D curve becomes relatively smooth. Two prominent merger events appeared in the 3D simulation are absent in the 1D case. Fig. 8 shows the evolution of the gas density profile in the 1D simulation. Even though only radial accretion is included, the growth of the gas halo is still not smooth. One can see a few high-density shells in the ICM, which are the angularly averaged infalling clumps. They compress the ICM and drive inner shocks inside the halo. As described in Section 2, these inner shocks eventually encounter the accretion shock and expand the cluster shock-heated atmosphere. Nevertheless, this "merger effect" has been much weaker than that in the 3D situation. In other words, mergers and smooth accretion are distinguishable in this regard. As a reference, the black and purple lines in Fig. 8 show the cluster virial radius R200m in the 3D and 1D simulations, respectively. During t = 2 − 9 Gyr, the virial radius of the 1D cluster shows a good match with that of the 3D calculation, but becomes 30% smaller when t > 9 Gyr, because of the absence of the two major mergers (see bottom panel in Fig. 7). The averaged shock radius, along the non-filamentary directions of the 3D cluster, is shown as the white line 9 . The shock radius in the 1D model is found to be much smaller than this curve throughout the simulation. The rmas/R200m (or racc/R200m) is about 2.5 and 1.3 in our 3D and 1D clusters, respectively. The former value is generally consistent with that reported in Walker et al. (2019) (see also Lau et al. 2015); and the latter one agrees with that in the self-similar model (Shi 2016b). The mismatch between the 1D and 3D clusters illustrates the importance of cluster mergers on the expansion of the boundary of the ICM through the shock collisions described in Section 2. Furthermore, the DM splashback radius approximately aligns with the shock radius in our 1D simulation, i.e. rsp racc 1.3R200m 10 . It is interesting to note that both 3D cosmological simulations and the self-similar model show that rsp 0.8 − 1.5R200m depending on the mass accretion rate parameter Γ ( 0 − 5; see More et al. 2015;Mansfield et al. 2017 andalso Shi 2016a). The mergers and radial accretion play similar roles in changing rsp, which are quite different from that for the shock radii. This fact explains the misalignment between the shock and splashback radii in galaxy clusters.
CONCLUSION AND DISCUSSION
In this work, we have explored the encounter of the merger and accretion shocks in galaxy clusters through the 1D and 3D cosmological simulations. As one of the key players, the runaway merger shocks usually exist in the diffuse gas between the high-density filaments (see Zhang et al. 2019b). During the shock collisions, merger-accelerated accretion shocks (MA-shocks) are formed and quickly propagate to larger cluster radii. A notable signature of the shock collisions is that high-entropy shells are generated in 9 Note that this curve is averaged over the entire cluster surface (excluding the filaments) but not only in the x − y plane shown in Fig. 6. 10 However, it is not a trivial task to measure the splashback radius for an individual 3D cluster because of the complicated merger and accretion configurations (Mansfield et al. 2017). Figure 8. Evolution of gas density profile in the 1D simulation (similar to those shown in Figs. 1 and A1 but for the simulation with a more realistic initial condition). The dashed black and purple lines show the virial radii of the 3D and 1D clusters, respectively. As a comparison, the averaged shock radius of the 3D cluster along the non-filamentary directions is shown as the white line. Error bars represent the 1σ scatter. This figure shows that the 3D cluster has a much larger shock radius than that in the 1D cluster. Cluster mergers play a key role in this difference (see Section 3.2).
between the MA-shock fronts and the CDs (see Fig. 2). These entropy structures further excite hydrodynamic instabilities and contribute to the non-thermal pressure in the cluster outskirts. Generally, the same type of MA-shocks should be present for a wide range of halo masses, provided that radiative cooling does not affect the gas and a hot atmosphere forms naturally. Basically, genuine accretion shocks should be present only during relatively quiescent periods of the halo evolution, while, long after each significant merger, the "outer" shocks would be located far outside the virial radius (between the filaments) and would be powered by the merger rather than by accretion. The evolution of the MA-shocks depends on the Mach number of the runaway merger shocks Mrs and the cluster mass accretion rate parameter Γ. We found that, the MA-shock fronts could reach very large cluster radii (i.e. 2 − 3 times larger than those of the ordinary accretion shocks; see Fig. 5) if Γ 3 and Mrs 1.5. These conditions imply that MA-shocks are not rare in galaxy clusters and could make strong impacts on the ICM. As the MA-shock fronts represent the outer boundaries of the ICM, the cluster gas atmospheres are prominently expanded after the shock collision. The formation of MA-shocks thus provides a natural explanation for the misalignment of the shock radii and the splashback radii found in the cosmological simulations (e.g. Lau et al. 2015).
Since the MA-shocks always reside beyond the cluster virial radius R200m, it is a big challenge to detect them through their X-ray signals. The Sunyaev-Zel'dovich (SZ) effect, which linearly scales with the integrated electron pressure, provides a unique opportunity to probe the hot but low density ICM in cluster outskirts (Hurier et al. 2019). However, we have to note that, in the cluster outer region, the electron-ion equilibrium timescale could be very long (e.g. a few Gyr or even longer; see . The non-equilibrium electrons would blur the imprints of MA-shocks on the cosmic microwave background (CMB). Nevertheless, given the nature of collisionless shocks, MA-shocks are ideal targets for the next generations of the X-ray and SZ instruments (e.g. AXIS/Lynx; see Mushotzky et al. 2019;Vikhlinin 2019) to explore the plasma physics of the ICM, e.g. Magneto-Thermal instabilities, acceleration mechanisms of cosmic rays.
Furthermore, MA-shocks are expected to play an important role in boosting radio emissivity of fossil relativistic electrons beyond the cluster virial radius (Enßlin et al. 1998;Enßlin & Gopal-Krishna 2001;Zhang et al. 2019b;Lyskova et al. 2019). Firstly, MA-shocks pass through a very large volume of the intergalactic medium (IGM). The swept gas (also the fossil electrons) tends to accumulate behind the shock fronts (see top panel of Fig. 2). Second, the low-efficiency issue of diffuse shock acceleration (DSA) in merger shocks is not a problem for the MA-shocks any more (see van Weeren et al. 2019, and references therein). The MA-shocks' Mach number could reach a few tens to hundreds depending on the pre-heating process of the IGM.
ACKNOWLEDGMENTS
The simulations performed at the Leibniz-Rechenzentrum are under the project pr86re. KD acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy -EXC-2094 -390783311. EC acknowledges partial support by the Russian Science Foundation grant 19-12-00369. WF acknowledges support from the Smithsonian Institution and the High Resolution Camera program, part of the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060. IZ is partially supported by a Clare Boothe Luce Professorship from the Henry Luce Foundation.
APPENDIX A: 1D COSMOLOGICAL SIMULATIONS
Our 1D cosmological simulation traces the evolution of both gas and DM in the self-gravitational field. A second-order piecewise-parabolic method (PPM) is applied to solve the Euler equations of the gas (Fryxell et al. 2000) on a 1D spherical grid. The DM is however modelled as a series of collisionless shells, which are advanced in time through the leapfrog scheme . In each time step, the DM shells are mapped to the grid to update the gravitational potential felt by the gas.
In all our simulations listed in Table 1, we used 200 gas cells and 80000 DM shells. To simultaneously reach a high spatial resolution for the cluster and suppress the boundary effect on the large-radius side, we arranged 160 cells uniformly to cover the innermost 10 3 comoving kpc (ckpc), and the remaining 40 cells to cover 10 3 − 10 4 ckpc in the logarithmic scale 11 . All cells contain the same number of DM shells in the initial condition. We set the initial gas and DM density profiles so that the cluster grows with a constant mass accretion rate Γ (see Eq. 1). Meanwhile, both DM and gas have zero initial velocity. All these simulations start from redshift z = 100. We assumed a flat ΛCDM cosmology model with Ωm0 = 0.30, Ω b0 = 0.05, ΩΛ0 = 0.70, and H0 = 70 km s −1 Mpc −1 in the calculation. The results of the simulations S1 and S3 show good match with the self-similar solutions (Fillmore & Goldreich 1984). We have also checked the convergence of our simulations in the spatial and mass resolutions. The run with the doubled numbers of gas cells and DM shells shows generally consistent results with those of the low-resolution run, but the excited discontinuities get sharper.
Figure 2 .
2Profiles of the gas density (top panel) and entropy (bottom panel) in the simulation S1T2M23. Note that the x-axis is in comoving coordinates (different from that shown in the bottom panel ofFig. A1). The line colors encode the cosmic time. The numbers marking the gaseous structures correspond to those shown inFig. 1. This figure presents a clear view of the formation of MA-shock. The high-entropy shell between the CD and MA-shock is unambiguous evidence of the past shock collision (see Section 2.1).
Figure 4 .
4Trajectories of the radially outermost shocks in our 1D simulations (dashed lines) and the analytical models (color solid lines). The line color encodes the Mach number of the runaway shocks Mrs right before they encounter the accretion shocks. As a comparison, the black solid and dashed lines show the evolution of the accretion shock radius and the turn-around radius based on the self-similar model. This figure shows a significant dependence of the MA-shock evolution on the value of Γ (see Section 2.2).
Figure 5 .
5The same data shown as the color solid lines inFig. 4, but the radius-and time-axis are scaled by the accretion shock radius racc in the self-similar model and the cosmic time tmas when the MA-shock forms, respectively. This figure shows that the MA-shock could reach a larger cluster radius and survive for a longer time if Γ is smaller and/or the Mach number of the runaway shock is higher (see Section 2.2).
Fig. 6
6shows a gas-entropy slice of the galaxy cluster at the redshift z = 0. The black solid lines mark the positions of the MA-shocks (or accretion shocks) identified in this map.
Figure 6 .
6Gas-entropy slice of a galaxy cluster from the 3D cosmological simulation at redshift z = 0. The black solid lines highlight the MA-shocks (or accretion shocks) identified in this map. Three different sectors are selected (marked by dashed arrows), where evolution of the shock radii are traced and shown in
Figure 7 .
7Top panel: evolution of the shock radii averaged within the three sectors shown and labelled in Fig. 6. The shaded band shows the 1σ scatter. The black line shows the evolution of the cluster virial radius R 200m . Bottom panel: a comparison of the global mass accretion rate Γ of the 3D cluster (purple line) and the 1D model (green line; see Section 3.2)
Figure A1 .
A1Top panel: similar toFig. 1but for the gas density distribution from the simulation S1T2M15. Bottom panel: evolution of the corresponding gas entropy profile. This figure shows complementary results to those presented in Figs. 1 and 2 (where the runaway shock has a larger Mach number). This figure shows that the MA-shock has a shorter lifetime when the runaway shock is weaker (see Section 2.1).
Table 1 .
1Parameters of 1D cosmological simulations.IDs
Γ a
t b (Gyr) b
tmas (Gyr) c
Mrs d
S1
1
-
-
-
S1T2M15
1
2.5
3.0
1.5
S1T2M20
1
2.5
2.8
2.0
S1T2M23
1
2.5
2.8
2.3
S1T6M23
1
6.5
6.7
2.3
S3
3
-
-
-
S3T2M20
3
2.5
2.7
2.0
S3T2M25
3
2.5
2.7
2.5
S3T6M20
3
6.5
7.0
2.0
a The cluster mass accretion rate parameter used to setup
the initial gas/DM density profiles.
b The time a blast wave is initiated at the cluster center. No
blast wave is included in the runs S1 and S3.
c The time the MA-shock is formed. Its uncertainty is
0.05 Gyr, determined by the time interval between two
successive snapshots.
d The Mach number of the "runaway" shock at the moment
just before it encounters the accretion shock.
Table 2 .
2Parameters used in Eq. (2) for calculation of MA-shock trajectories (see Section 2.2).IDs Γ a
tmas (Gyr) b
rmas(tmas) (kpc) c
κ d
G1
1
2.7
140
0.12
G2
3
2.5
120
0.08
a The mass accretion rate.
b The moment MA-shock forms.
c The radius where MA-shock forms at t = tmas.
d MA-shock starts to behave in a self-similar way at t = (1 +
κ)tmas.
Congyao Zhang et al.
At the same time, a reverse rarefaction wave is formed and moves towards the cluster center, which however is quickly diminished when travelling into the denser ICM region.
MNRAS 000, 1-10 (2019)
Though not shown in figures, the analytical model matches the results of the simulation runs S1T6M23 and S3T6M20 as well.
Here we only show the results when tmas 3 Gyr (seeTable 2). But we have compared the models with tmas 3 and 7 Gyr, and found the curves show only weak dependence on tmas.
www.magneticum.org/complements.html#Compass 7 R 200m is referred to as the virial radius of the cluster in this work, which encloses an average matter density 200 times higher than the mean matter density of the Universe. M 200m is the cluster virial mass within R 200m .
Specifically, there is no halo structure at this redshift. We select the position of the peak of the strongest perturbation in the snapshot as the origin of the radial profiles. The main progenitor of the present cluster will form here at a later time.MNRAS 000, 1-10(2019)
In Section 3.2, we present a special 1D simulation, which is used to compare with the 3D cosmological simulation directly. In this simulation, we used 400 gas cells and 80000 DM shells. The size of the computational domain reaches 3 × 10 4 ckpc. The cosmological parameters adopted in this run are the same as those used in the 3D simulation (different from other 1D simulations presented in this paper).MNRAS 000, 1-10(2019)
This paper has been typeset from a T E X/L A T E X file prepared by the author.
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| []
|
[
"Lyapunov function for non-equilibrium transport processes",
"Lyapunov function for non-equilibrium transport processes"
]
| [
"Chuan-Jin Su \nDepartment of Engineering Mechanics\nKey Laboratory for Thermal Science and Power Engineering of Ministry of Education\nTsinghua University\n100084BeijingChina\n",
"Yu-Chao Hua \nDepartment of Engineering Mechanics\nKey Laboratory for Thermal Science and Power Engineering of Ministry of Education\nTsinghua University\n100084BeijingChina\n",
"Zeng-Yuan Guo \nDepartment of Engineering Mechanics\nKey Laboratory for Thermal Science and Power Engineering of Ministry of Education\nTsinghua University\n100084BeijingChina\n"
]
| [
"Department of Engineering Mechanics\nKey Laboratory for Thermal Science and Power Engineering of Ministry of Education\nTsinghua University\n100084BeijingChina",
"Department of Engineering Mechanics\nKey Laboratory for Thermal Science and Power Engineering of Ministry of Education\nTsinghua University\n100084BeijingChina",
"Department of Engineering Mechanics\nKey Laboratory for Thermal Science and Power Engineering of Ministry of Education\nTsinghua University\n100084BeijingChina"
]
| []
| Irreversibility is a critical property of non-equilibrium transport processes. An opinion has long been insisted that the entropy production rate is a Lyapunov function for all kinds of processes, that is, the principle of minimum entropy production. However, such principle is based on some strong assumptions that are rarely valid in practice. Here, the common features of parabolic-like transport processes are discussed. A theorem is then put forward that the dot products of fluxes and corresponding forces serve as Lyapunov function for parabolic-like transport processes. Such fluxes and forces are defined by their actual constitutive relations (e.g., the Fourier's law, the Fick's law, etc.). Then, some typical transport processes are analyzed. Particularly for heat conduction, both the theoretical and numerical analyses demonstrate that its Lyapunov function is the entransy dissipation rather that the entropy production, when the Fourier's law is valid. The present work could be helpful for further understanding on the irreversibility and the mathematical interpretation of non-equilibrium processes. | null | [
"https://arxiv.org/pdf/2202.08001v1.pdf"
]
| 246,867,280 | 2202.08001 | 2816e7a1be41dabcc42c03504479015ba8778050 |
Lyapunov function for non-equilibrium transport processes
16 Feb 2022
Chuan-Jin Su
Department of Engineering Mechanics
Key Laboratory for Thermal Science and Power Engineering of Ministry of Education
Tsinghua University
100084BeijingChina
Yu-Chao Hua
Department of Engineering Mechanics
Key Laboratory for Thermal Science and Power Engineering of Ministry of Education
Tsinghua University
100084BeijingChina
Zeng-Yuan Guo
Department of Engineering Mechanics
Key Laboratory for Thermal Science and Power Engineering of Ministry of Education
Tsinghua University
100084BeijingChina
Lyapunov function for non-equilibrium transport processes
16 Feb 2022
Irreversibility is a critical property of non-equilibrium transport processes. An opinion has long been insisted that the entropy production rate is a Lyapunov function for all kinds of processes, that is, the principle of minimum entropy production. However, such principle is based on some strong assumptions that are rarely valid in practice. Here, the common features of parabolic-like transport processes are discussed. A theorem is then put forward that the dot products of fluxes and corresponding forces serve as Lyapunov function for parabolic-like transport processes. Such fluxes and forces are defined by their actual constitutive relations (e.g., the Fourier's law, the Fick's law, etc.). Then, some typical transport processes are analyzed. Particularly for heat conduction, both the theoretical and numerical analyses demonstrate that its Lyapunov function is the entransy dissipation rather that the entropy production, when the Fourier's law is valid. The present work could be helpful for further understanding on the irreversibility and the mathematical interpretation of non-equilibrium processes.
Introduction
Investigation of transport processes is one crucial topic in non-equilibrium thermodynamics [1]. In practice, linear constitutive relations are generally adopted to handle the non-equilibrium transport problems [2], which lead to parabolic-like governing equations. Thus, the transport processes characterized by paraboliclike partial differential equations (PDEs) can be called parabolic-like transport processes.
The understanding of irreversibility which is a fundamental property of nonequilibrium transport processes, plays an important role in the study of transport processes. For the parabolic-like processes, irreversibility is reflected by governing equations where the partial time derivative terms invalidate the time reversal symmetry. In order to propose a quantity that measures irreversibility, the concept of Lyapunov function in modern stability theory is introduced into non-equilibrium thermodynamics, as irreversibility is the expression of the attraction of stability, quoted by Prigogine [3]. For any system, a scalar function whose time derivative has the opposite is called a Lyapunov function, and the system is stable in the sense of Lyapunov [4]. Classical stability theory formulated by Gibbs for equilibrium states is based on the variation of entropy, where the second term is negative, i.e. 1 2 δ 2 S < 0 [5], however, the similar methodology cannot be applied for the non-equilibrium stationary states because of the limitations of such variational formulations out of equilibrium [6].
Since 1940s, I. Prigogine has proposed and further developed the principle of minimum entropy production (MinEP) [7,8] as the key criterion of stability for non-equilibrium transport processes, characterizing a system at stationary state by producing entropy at the lowest rate compatible with external constraints. This theorem holds only if the following hypotheses [9] are satisfied:
1. Linear relations between generalized forces and fluxes in irreversible phenomenon.
2. Validity of Onsager reciprocal relations.
3. Phenomenological coefficients are considered constant.
When applying to a perturbed system that satisfies them all, we shall see that irreversible processes inside always lower the value of entropy production rate (EPR), and the system returns to the state at which its entropy production is lowest [3], if its boundary conditions are invariant of time. However, questions arise in most practical cases. Firstly, the assumption that constitutive relations between thermodynamic fluxes and forces are entropic (i.e., their dot products should be the entropy production, suggested by previously listed hypotheses) is hardly corresponding to practical situations. As an example, for the heat conduction process, a strong assumption is required that the thermal conductivity must be inversely proportional to the square of the temperature [10,11], i.e. k = l qq /T 2 ∝ 1/T 2 . However, there are few materials of which properties meet such assumption [12]. On the other hand, such assumption is equivalent to the validation of the principle of MinEP, which means that a circular logic is misused, and no additional information is provided by such minimum principle.
In this article, we put forward a theorem stating that for given constitutive relations, the dot products of thermodynamic fluxes and forces serve as Lyapunov functions for parabolic-like transport processes. Lyapunov functions for some typical transport processes are studied in section 3. For the heat conduction process under the Fourier's law, a Lyapunov function provided by the theorem is the entransy dissipation rate which is a measure of the heat conduction process' irreversibility [13], supporting the conclusion of Hua et al. [14,15]. Numerical calculations are also provided to support the theorem in section 4.5.
Applying the theorem, the principle of MinEP and its problematics are discussed in 4. In addition, the attempts to modify the principle of MinEP are reviewed, including Prigogine's approximation that the coefficient l qq should be constant in a small temperature range [5], and the non-total differential [9] which stays non-positive and is claimed to be a general property of entropy production.
2 Lyapunov function for parabolic-like transport processes
As mentioned above, for a system, a scalar function whose time derivative has the opposite sign is a Lyapunov function [4]. We will demonstrate that the dot products of fluxes and forces serve as Lyapunov functions for parabolic-like transport processes.
The local equilibrium assumption of parabolic-like transport processes take the form of the following balance equation [6],
ρ ∂a ∂t = −∇ · J a + σ a ,(1)
where a is an extensive state variable under consideration, J a is the related flux term, σ a is the corresponding source term; and a is conserved when σ a is zero, which is the case that we consider in this paper.
Linear transport processes
For linear transport processes, constitutive relations between driving forces and fluxes are expressed,
J a = KX a = K∇Γ a ,(2)
where K is a constant, Γ a is the corresponding intensive state variable of a. We now define G as the integral of the dot product g of a flux J a and its driving force X a within a system V ,
G a = V g a dV = V X a · J a dV,(3)
whose sign is dependent on the constant K. For example, G e = ∇(−T ) · q = q · q/k ≥ 0 for the heat conduction process, where e is the specific thermal energy, T is the temperature, q is the heat flux, and k is a constant called the thermal conductivity, given the Fourier's law,
q = k∇(−T ).(4)
If the chosen state quantity a is conserved, the time derivative of G a can be immediately expressed using constitutive relation in eq. (2),
dG a dt = d dt V X a · J a dV = d dt V ∇Γ a · K∇Γ a dV = 2 V K∇Γ a · ∇ ∂Γ a ∂t dV Gauss = === = 2 A K ∂Γ a ∂t ∂Γ a ∂n dA − 2 V K ∂Γ a ∂t ∇ 2 Γ a dV,(5)
where the first integral vanishes as the boundary conditions are fixed, i.e. the quantity Γ a is time independent on boundaries. Inserting the local equilibrium assumption in eq. (1) without the source term, i.e. σ a = 0, it becomes,
dG a dt = 2 V ρ ∂Γ a ∂t ∂a ∂t dV = 2 V ρ ∂Γ a ∂a ∂a ∂t 2 dV.(6)
For parabolic-like transport processes (e.g., the heat conduction process), the monotonic functional relation between the state quantities a and Γ a is given as Γ a = Γ a (a), which assures that the sign of ∂Γ a ∂a is opposite to that of the constant K ( ∂Γ a ∂a = − ∂T ∂u = − 1 cv < 0 for the heat conduction process, where c v is the specific heat capacity), governing equations can be expressed in the parabolic-like form,
ρ ∂a ∂t = −K∇ · ∂Γ a ∂a ∇a .(7)
We assert that for parabolic-like transport processes, the sign of such quantity G a is opposite to that of its time derivative dG a dt . Therefore, the quantities defined as G serve as Lyapunov functions in such irreversible processes.
Non-linear transport process
When the constitutive relations between fluxes and corresponding forces are nonlinear,
J a = K(Γ a )∇Γ a ,(8)
where the conductivity term K is Γ a -dependent, such relations can be rewritten in a linear form by applying the methodology in literature [14,16] which was proposed to convert the problem of the temperature dependence of the thermal conductivity. By defining a generalized quantity
F (Γ a ) = Γ a Γ a 0 K(Γ a * ) K 0 dΓ a * ,(9)
where Γ a 0 is the reference point, and K 0 ≡ K(Γ a 0 ). The non-linear relation is converted into a linear problem and can be rewritten as
J a = K 0 ∇F,(10)as ∇F = K(Γ a ) K 0 ∇Γ a .
A Lyapunov function can be given by the previous theorem due to the linear constitutive relation between the flux J a and the generalized driving force X a ge = ∇F , that is
G a ge = V X a ge · J a dV.(11)
Therefore, in the case of G a ge (eq. (11)) serving as the Lyapunov function, the time evolution equation is the parabolic-like transport model in eq. (7), and the approach is to the fix point of the stationary states observed in practice. It should be noted that, in the framework of GENERIC, when considering the reducing rate relation from the arbitrary non-equilibrium level L to the stationary level L, such linearized relation can also be expressed by the GENERIC equation [17][18][19], where the dissipation potential is Ψ = 1 2 K 0 (∇a * ) 2 , and the linearized conjugate of the quantity a is defined as a * = F . A Lyapunov function can be obtained from the rate thermodynamic potential [19] and is in the same form as G a ge in this case.
3 Lyapunov functions for some typical transport processes
The heat conduction process
Given the linear constitutive relation, i.e. the Fourier's law for the heat conduction process,
q = k∇(−T ), (4 revisited)
where q is the heat flux, and X e = ∇(−T ) is the corresponding driving force. Applying our theorem of Lyapunov function for parabolic-like transport processes, a Lyapunov function is expressed as
G e = V X e · qdV = V k∇T · ∇T dV.(12)
Guo et al. [13] introduced entransy to characterize the ability of heat transfer. The local entransy dissipation rate of heat conduction process is the dot product of heat flux and the negative of temperature gradient,
g = −q · ∇T.(13)
The entransy dissipation in solid is a measure of the heat conduction process' irreversibility [13], and a minimum principle can be constructed by the entransy dissipation rate [14,15]. It should be noted that, the Lyapunov function G e defined by the theorem is exactly the entransy dissipation rate,
G e = − V gdV ≥ 0, dG e dt ≤ 0.(14)
where the equality holds if and only if stationary state is reached. Therefore, Lyapunov conditions for the heat conduction process are given by the entransy dissipation rate G e > 0 and its time derivative dG e dt < 0. Least generalized entransy dissipation principle is proposed by Hua and Guo [14], to extend the scope of the entransy concept to the case of temperature dependent thermal conductivity. By defining a generalized temperature,
F (T ) = T T 0 k(T ′ ) k 0 (T 0 ) dT ′ ,(15)
where T 0 is the reference temperature. The generalized entransy dissipation rate is then defined as,
G e ge = − V q · ∇F dV = V k 0 (∇F ) 2 dV,(16)
which is the dot product of the heat flux and its generalized driving force X e ge = ∇(−F ). Indeed, G e ge serves as a Lyapunov function, as pointed out by the nonlinear case of the theorem.
The mass diffusion process
The Fick's law of diffusion describes the mass diffusion process, relating the diffusion flux to the gradient of the concentration,
J α = D α ∇(−φ α ),(17)
where J α is the diffusive flux of component α, D α is the diffusivity, φ is the concentration. A Lyapunov function for the molecular diffusion process is provided by the dot product of the flux and the corresponding driving force X φ = ∇(−φ α ),
G φ = V X φ · J φ dV = V D α ∇φ α · ∇φ α dV(18)
with the conditions G φ ≥ 0 and dG φ dt ≤ 0.
The electrical conduction process
The electrical conduction process in electrical conductors is well described by the Ohm's law, whose vector form is expressed as,
J q = σ∇(−V ),(19)
where J q is the current density, the constant σ is the conductivity, and V is the electric potential. The superscript q represents the charge density.
A Lyapunov function is written as the dot product of the flux J q and its driving force X q = ∇(−V ),
G q = V X q · J q = V σ∇V · ∇V dV,(20)
which is the Joule heating power within the system V , i.e. the rate of electrical energy dissipation. The previous theorem derives the same result as the common sense that the Joule heating power G q decreases to the minimum possible value for given boundary conditions [12].
Thermal conduction q = k∇(−T ) G e = k (∇T ) 2 Mass diffusion J α = D α ∇(−φ α ) G φ = D α (∇φ α ) 2 Electrical conduction J q = σ∇(−V ) G q = σ (∇V ) 2
The defined Lyapunov functions G e , G φ and G q suggest the stability of these typical transport processes, and measure the irreversibility featured by the paraboliclike governing equations. Since the validation of the theorem is verified, it should be pointed out that the fluxes and their corresponding forces should be defined by actual constitutive relations as above. Such G-type Lyapunov function can be also used in the framework of rate thermodynamics [19] to characterize the overall pattern of the preparation process from non-equilibrium state to the non-equilibrium stationary state.
In addition, it is worth mentioning that for the equilibrium states, both the function G a and the entropy S express the approach to the equilibrium, as the final stage can be defined by both equilibrium state and non-equilibrium stationary state. And the Lyapunov function of type G exist even in externally driven systems without local equilibrium for which the entropy S does not exist [19].
4 Misunderstandings on the entropy production and its minimum principle
Limitations of the principle of MinEP
It has been mistakenly believed that the entropy production rate (EPR) is a Lyapunov function for any non-equilibrium transport processes, since the principle of minimum entropy production (MinEP) was proposed by I. Prigogine [7,8], stating that a system at stationary state by producing entropy at the lowest rate compatible with external constraints, and the EPR plays the role of Lyapunov function to characterize the stability of such processes. It is crucial to point out that, the principle of MinEP is valid only when following entropic constitutive relations are given, which invalidate the applicability and the universality of the principle. For the heat conduction process and the mass diffusion process discussed above,
q = l qq ∇ 1 T ,(21)J α = l φφ ∇ − µ α T ,(22)
where l qq and l φφ are phenomenological coefficients in such sense, µ α is the chemical potential of component α. The driving forces are defined as
X u ep = ∇ 1 T , X φ ep = ∇ − µ α T ,(23)
which make the dot products of fluxes and corresponding forces to be the EPR.
Lyapunov functions for such processes are given by our previous theorem,
P u = V J u · X u ep dV = V q · ∇ 1 T dV,(24)P φ = V J φ · X φ ep dV = V J α · ∇ − µ α T dV,(25)
which confirm the conditional validation of the principle of MinEP (see also [1,10,20] for the heat conduction process).
The principle's problematics
However, there are two issues if the EPR is considered to be a Lyapunov function. Firstly, the hypothesis of constant phenomenological coefficients is hardly corresponding to practical situations. As an example, for the heat conduction process, a strong assumption is required that the thermal conductivity must be inversely proportional to the square of the temperature [10,11], i.e., k = lqq T 2 ∝ 1 T 2 . However, there are few materials of which properties meet such assumption [12]. When discussing entropy production in electrical circuit elements, Kondepudi and Prigogine [5] provided the entropic constitutive relation between the voltage and the current as following,
I = L R V R T ,(26)
where L R is a phenomenological coefficient. The same problem arises, that is the resistance R = T L R should be proportional to the temperature, which is not the generally adopted case where the Ohm's law is valid.
It should be noted that the principle of MinEP will be violated if linear phenomenological relations (e.g., the Fourier's law) are valid. Take the heat conduction process as an example, the EPR is written as eq. (24), whose time derivative becomes,
dP u dt = d dt V q · ∇ 1 T dV = d dt V k∇ ln T · ∇ ln T dV = 2 A k ∂ ln T ∂t ∂ ln T ∂n dA − V k ∂ ln T ∂t ∇ 2 ln T dV ,(27)
applying the Gauss's divergence theorem. Given that the boundary conditions are fixed ∂T ∂t A = 0, insertion of the local equilibrium eq. (1) into eq. (27) yields,
dP u dt = −2 V k 2 ρc 1 T ∇ 2 T 1 T ∇ 2 T − ∇T · ∇T T 2 dV,(28)
whose sign is uncertain. Hence, when the initial condition is chosen appropriately, the principle of MinEP can be surely violated (see a counterexample in section 4.5). If the same analysis is applied to the molecular diffusion process given that the Fick's law of diffusion is valid as eq. (17), the same conclusion can be drawn as the time derivative of the EPR is expressed as,
dP φ dt = V RD 2 α ∇φ α · ∇φ α φ 2 α ∇ 2 φ α dV.(29)
The principle of MinEP is violated when the inequality ∇ 2 φ α > 0 holds. As for the case of electrical circuit, the same problematic reappears: the principle of MinEP becomes invalid if the resistance is constant, that is, the Ohm's law rather than the entropic constitutive relation, are valid. A simple counterexample can be given by considering two parallel resistors at different temperatures [12]. Secondly, a circular logic is misused. Given that the theorem holds, the validation of the principle of MinEP is equivalent to the assumption that constitutive relations between fluxes and forces should be entropic, which means that no additional information can be provided by such principle. Such logic of defining fluxes and their corresponding forces is completely different from that we applied above when studying specific transport processes.
Prigogine's approximation on phenomenological coefficients
Prigogine was aware of the issue of constant phenomenological coefficients, an approximation was made when discussing stationary states in the heat conduction process [5],
l qq = kT 2 ≈ kT 2 avg ,(30)
where the coefficient l qq is treated approximately as constant as the average temperature T avg hardly changes too much. However, such approximation can be rejected as the time derivative of the EPR in eq. (28) can be positive no matter how small the temperature range is, as long as we put forward the corresponding initial temperature field. Zullo [10] points out that the assumption of constant average temperature T avg is not valid when the system is not isolated, which is the reason why this approximation is not applicable, and a further assumption should be made to neglect the derivative of T avg . However, it should be clarified that even for an isolated system, the first integral in eq. (27) is still 0 as heat flux on the surface does not exist,
∂T ∂n A = 0.(31)
It is clear that for isolated systems, that is when the time derivative of average temperature T avg vanishes, the EPR is still inapplicable.
A point should be added concerning that approximation, that is in the expression of the EPR
P = V k ∇T T 2 dV ≈ 1 T 2 avg V k (∇T ) 2 dV,(32)
the average temperature T avg is a weighted average of the inverse of the gradient if we extract it outside the integral, which does dot have actual physical meaning. It would be pointless to make further assumptions on T avg . It can be thus asserted that when the thermal conductivity is constant, the principle of MinEP simply does not hold.
The non-total differential
Glansdorff and Prigogine [9] proposed a new methodology to answer the question of what the most general properties of entropy production are, independent of restrictive hypotheses. The non-total differential of the EPR is put forward to play a role of Lyapunov condition. Note P the EPR, for fluxes and driving forces under entropic constitutive relations,
P = V j X j J j dV,(33)
where X j and J j are thermodynamic forces and fluxes, enumerated by j. By decomposing time derivative of eq. (33),
dP dt = V j dX j dt J j dV + V j X j dJ j dt dV ≡ d X P dt + d J P dt ,(34)
it can be obtained that the first component d X P that always stays negative, or zero when reaching stationary state, even at least one of the hypotheses are invalid. Naive proofs of processes of the heat conduction, the chemical reaction, and the isothermal diffusion can be found in literature [5,9,20]. In addition to the EPR itself P ≥ 0, another Lyapunov condition is given by its such non-total differential [1],
d X P dt ≡ V j dX j dt J j dV ≤ 0.(35)
However, the so-called non-total differential is questionable. Firstly, the concept is ambiguous although a new 'kinetic potential' was introduced by Kondepudi and Prigogine [5] such that dW ≡ d X P , as no explicit physical meaning has been interpreted so far. Secondly, the inequality (35) holds not only for entropic constitutive relations. In fact, the properties of such defined non-total differential are universally valid for parabolic-like transport processes discussed previously, regardless of whether the constitutive relations are entropic. For any non-equilibrium process that described by the local equilibrium assumption eq. (1) and a constitutive relation in the form of eq. (2), a Lyapunov function is provided by the dot product G in eq. (3), whose non-total differential is,
d X G a dt ≡ V J a · ∇ dΓ a dt dV = − V dΓ a dt ∇ · J a dV = V ρ dΓ a da ∂a ∂t 2 dV ≤ 0.(36)
The validation of the inequality (36) does not require the constitutive relations to be entropic. As an example, for the heat conduction process under the Fourier's law which is not entropic, the non-total differential of entransy dissipation rate stays negative before the stationary state is reached, i.e., d X G e ≤ 0. Therefore, it is inappropriate to raise such defined quantity to describe properties of entropy production as the feature is commonly shared by non-equilibrium transport processes, whose constitutive relations are not necessarily entropic.
Numerical experiments for the heat conduction process
Given the Fourier's law in the dimensionless form
q * = k * ∇ (−T * ) ,(37)
the dimensionless thermal diffusion equation can be written as,
∂T * ∂t * = α * ∇ * 2 T * ,(38)
where the dimensionless conductivity k * = 1 and diffusivity α * = 1 if characteristic parameters t 0 , x 0 , T 0 are properly chosen accordingly. The dimensionless EPR and entransy dissipation rate (EDR) can be respectively expressed as,
P * u = V * k * ∇ * T * · ∇ * T * T * 2 dV * , G e * = V * k * ∇ * T * · ∇ * T * dV * .(39)
To raise counterexamples that violate the principle of MinEP, initial temperature distribution T * (x * , t * = 0) should be given to satisfy
∀x * ∈ V * , ∇ * 2 T * T * ∇ * 2 T * T * − ∇ * T * · ∇ * T * T * 2 dV * < 0,(40)
in which case the dimensionless EPR increases according to eq. (28). We now study the one-dimensional heat conduction process along x * axis within a unit cube. Such condition can be easily met if we take a parabola T * (x * , t * = 0) = ax * 2 + bx * + c as the initial condition where parameters a, b, c should be properly chosen. Fixing the boundary conditions as,
∀t * > 0, T * (x * = 0, t * ) = T (x * = 0, t * = 0) = c T * (x * = 1, t * ) = T * (x * = 1, t * = 0) = a + b + c(41)
A set of parameters can be given as (a, b, c) = (5, 3, 1). We calculate the evolution of dimensionless temperature field along x * axis following diffusion eq. (38), and the dimensionless EPR and EDR are calculated accordingly. As shown in fig. 1, as the temperature field evolves, the EPR increases until the stationary state is reached, which violates the principle MinEP as discussed. Similar results are reported by Zullo [10], that the EPR may increase or decrease, and cannot be characterized by an extremum principle. Moreover, our numerical calculations show that the EDR decreases and is a Lyapunov function for such process, suggested by the previous theorem.
Conclusion
1. Entropy production has been long regarded as the Lyapunov functions for all kinds of non-equilibrium transport processes as the principle of MinEP claims to characterize the stationary states of irreversible processes. However, the principle of MinEP is questionable. A strong assumption is required that constitutive relations should be entropic, which is hardly corresponding to practical situations. On the other hand, such assumption of the principle is equivalent to the validation of the principle itself, which means that a circular logic is misused, and no additional information can be obtained by the principle.
2. Attempts to modify the principle of MinEP are analyzed. Prigogine's approximation that the phenomenological coefficients of the entropic relations should be constant is inappropriate. The non-total differential, or the kinetic potential, is also inadequate as its property is commonly shard by the dot products of fluxes and forces of non-equilibrium processes, whose constitutive relations are not necessarily entropic.
3. A general model of parabolic-like transport processes is analyzed, and a theorem is derived, demonstrating that the dot products of fluxes and corresponding forces serve as Lyapunov functions. Such fluxes and forces are provided by their actual constitutive relations (e.g., the Fourier's law, the Fick's law, the Ohm's law). Lyapunov functions for the heat conduction process, the mass diffusion process and the electrical conduction process are studied to verify the validation of the theorem. It is worth mentioning that for heat conduction (resp. electrical conduction), the entransy dissipation rate (resp. the Joule heating power), rather than the entropy production rate, serves as a Lyapunov function when the Fourier's law (resp. the Ohm's law) is valid. 4. Numerical experiments for the heat conduction process are effectuated to verify the validation of the theorem. It demonstrates that the principle of MinEP will be violated when the Fourier's law is valid.
Figure 1 :
1The one-dimensional heat conduction process under the Fourier's law within a unit cube, given the initial dimensionless condition T * (x * , t * = 0) = ax * 2 + bx * + c where (a, b, c)
Table 1 :
1Constitutive relations and Lyapunov functions of some typical transportprocesses
Transport process
Constitutive relation
Lyapunov function
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[
"On sound ranging in proper metric spaces",
"On sound ranging in proper metric spaces"
]
| [
"Sergij V Goncharov "
]
| []
| []
| We consider the sound ranging, or source localization, problem -find the source-point from the moments when the wave-sphere of linearly, with time, increasing radius reaches the sensor-points -in the proper metric spaces (any closed ball is compact) and, in particular, in the finite-dimensional normed spaces. We approximate the solution to arbitrary precision by the iterative process with the stopping criterion.MSC2010: Pri 41A65, Sec 54E50, 46B20, 40A05, 68W25 | 10.12697/acutm.2020.24.14 | [
"https://arxiv.org/pdf/1808.03330v1.pdf"
]
| 119,593,405 | 1808.03330 | 761a6449c4bf3a6389ded9395feb3f2d4ac8d864 |
On sound ranging in proper metric spaces
Aug 2018
Sergij V Goncharov
On sound ranging in proper metric spaces
Aug 2018sound ranginglocalizationapproximationalgorithmproper metric spacenormed space
We consider the sound ranging, or source localization, problem -find the source-point from the moments when the wave-sphere of linearly, with time, increasing radius reaches the sensor-points -in the proper metric spaces (any closed ball is compact) and, in particular, in the finite-dimensional normed spaces. We approximate the solution to arbitrary precision by the iterative process with the stopping criterion.MSC2010: Pri 41A65, Sec 54E50, 46B20, 40A05, 68W25
Introduction
Let (X; ρ) be a metric space, i.e. the set X with the metric ρ : X × X → R + . Let s ∈ X be an unknown point, "source". At unknown moment t 0 ∈ R of time the source "emits the (sound) wave", which is the sphere x ∈ X | ρ(x; s) = v(t − t 0 ) for any moment t t 0 . We assume, without loss of generality, that "sound velocity" v = 1 (switch to scaled time t ← vt if v = 1).
Let {r i } i∈I , r i ∈ X, be an indexed set of known "sensors". For each sensor we know the moment t i when it was reached by the expanding wave; that is, t i = t 0 + ρ(r i ; s) are known.
The sound ranging problem (SRP), also called source localization, is to find s and t 0 from known moments when the wave reaches known sensors, ({r i }; {t i }).
SRPs of this and more general forms, usually in Euclidean space, appear in acoustics, geophysics, navigation, sensor networks, tracking among the others; there is an abundant literature of the subject and of the proposed techniques, see e.g. [7,1], [16, 9.1] for further references.
In [13] we investigated noiseless SRP in the infinite-dimensional separable Hilbert space H. The method there is, basically, the "classical" one applied in R m 2 , -we solve the set of implied equations (t i −t 0 ) 2 = ρ 2 (r i ; s) = j (r (i) j −s j ) 2 , where t 0 and coordinates {s j } j∈N of s are unknowns, -with few technicalities related to the countability of coordinates. It is a method of "solving" kind in that we express exact values of s j through known parameters t i and r (i) j , in closed form. This time we look into another generalization of SRP, without Euclidicity in general case. The classical approach doesn't work anymore, because the coordinates, if there are any, are not so easily "extractable" from the equations t i − t 0 = ρ(r i ; s), which become significantly nonlinear.
Instead, we describe the more or less "universal" iterative process that "converges" to the source in certain sense explained further; this is a method of "approximating" kind. In short, we cover the regions of the space by the balls, and repeatedly refine the cover by a) replacing every ball with its cover by the balls of halved radius, then b) removing from the cover each ball such that certain "deviation" at its center is greater than its radius. "Deviation" at the source is 0.
It is presented as an algorithm. How practical such algorithm is depends on its "executor", or, in other words, what (and how many) elementary actions the executor is allowed to perform. If e.g. it were a "computer" U with card(X) cores, we would plainly assign to each core single x ∈ X to verify if x is a solution (t i − ρ(r i ; x) ≡ const, see below). The target executor for our algorithm is far below U, it is closer to the "general purpose" computing devices of nowadays.
Disclaimer. The intent of this paper is not to proclaim the "novelty" of the method being described (to put it mildly, that would be dubious), but to "plant" (develop) the "essence" (approach) of akin methods in more general "soil" (context) and watch how it "blossoms" (works).
We consider "empty" spaces without "physics" such as echoes, varying sound velocity, noisy measurements (except for one remark), focusing on rigour rather than realness and applicability.
Similarly to the H case, the content of this paper has a "folklore" flavor, so -Acknowledgements. We thank everyone who 1) points out where these results or their generalizations have been obtained already (some paper from 1920-30s? something like [18]?), or 2) by means of a time machine, delivers this paper to 1920-30s, when it should've appeared. . .
Root finding vs. Optimization.
Searching for the minimum of f (x) (or the maximum of −f (x)) is the optimization problem that is part of the most of approaches to solving SRPs, especially with noised measurements (f (x) is called "cost" or "plausibility" function there). It is performed either (1-stage) directly in the space of possible source positions to estimate the actual position, or (2-stage) in the space of relative time-delays t i − t j between sensors to estimate these delays, which then allow to obtain the source position in closed form or, alternatively, estimate it as well; see [1], [4], [6], [12], [16]. For example, in [1] the branch & bound technique is applied and compared to other ones. The maximum likelihood estimator is one of common approaches to such optimization as well, though there are issues with local minima when the cost function isn't strictly concave ( [16, 9.4]). The Euclidicity of the space where the wave propagates is important in deriving the closed form solutions and in the least squares localizations ( [4,4], [16, 9.5]).
In our simplified case the exact delays are known and the non-negative function has unique zero; we search for that zero, rather than the extremum, in the (non-Euclidean) space of possible source positions. This is a root finding of "bracketing", or "exclude & enclose", type (see [5], [22]).
The bibliography with somewhat more emphasis on the practice of sound ranging, including historical surveys, was given in [13]; or better, see [2], [7], [16,9]. "♣" indicates the assumptions, or constraints, that we require to hold unless stated otherwise. "•" is for the statements that are considered to be well-known under given assumptions (see e.g. [11], [17], [19], [21], [26]) and included for the sake of completeness, without proofs or references.
SR in proper metric spaces
Preliminaries
• "2nd △-inequality": ∀x, y, z ∈ X ρ(x; z) − ρ(z; y) ρ(x; y).
As usual, x k − −−− → k→∞ y means ρ(x k ; y) − −−− → k→∞ 0.
• Continuity of metric:
x k − −−− → k→∞ y ⇒ ρ(x k ; z) − −−− → k→∞ ρ(y; z). B(c; r) = {x ∈ X | ρ(x; c) < r} and B[c; r] = {x ∈ X | ρ(x; c)
r} denote the open and closed balls with center c and of radius r.
• For any B[c; r] and any point a, ρ(a; c) = d, we have ∀x ∈ B[c; r]: d − r ρ(x; a) d + r
The set A ⊆ X is said to be compact if ∀{x k } k∈N ⊆ A: ∃ {x k l } l∈N : x k l − −− → l→∞ x 0 ∈ A.
• If A is compact, then any closed subset of A is compact too. The family of sets {C j } j∈J , C j ⊆ X, is said to be a cover of
A ⊆ X if A ⊆ j∈J C j .
• The closed A ⊆ X is compact if and only if any open cover of A has finite subcover.
The set A ⊆ X is called bounded if diam A = sup x,y∈A ρ(x; y) < ∞. ♣M1. (X; ρ)
is proper : any closed ball is compact. Such spaces are also called finitely compact or having the Heine-Borel/Bolzano-Weierstrass property ([8, 1.5, p. 43], [23, 1.4, p. 32]; in addition, see [27]).
• In this definition, "any closed ball" can be replaced with "any closed, bounded subset".
• A proper metric space is complete: any fundamental sequence converges. In fact, it would suffice that ∃δ > 0:
∀x ∈ X B[x; δ] is compact. Indeed, if {x k } ∞ k=1 is fundamental, then ∃N : {x k } ∞ k=N ⊆ B[x N ; δ] = B ⇒ ∃{x N +k l } ∞ l=1 : x N +k l − −− → l→∞ y ∈ B, implying x k − −−− → k→∞
y as well. Also, the converse fails: infinite-dimensional H is complete, but not proper.
Now we proceed to the SRP. The source s ∈ X and the emission moment t 0 ∈ R are unknown.
♣1. The set of sensors is finite: {r i } n i=1 , r i ∈ X, and r i = r j , i = j.
These sensors and the moments
t i = t 0 + ρ(r i ; s), i = 1, n (1) define the SRP {r i }; {t i } ; each pair (s ′ ; t ′ ) satisfying the set of equations t i = t ′ + ρ(r i ; s ′ ), i = 1, n(2)
is a solution of this SRP. Since t ′ is defined uniquely from any such equation for given s ′ , the source s ′ itself can be called a solution too.
♣2. The solution s of the SRP {r i }; {t i } is unique.
Obviously, this is not the general case. For n = 1 any y ∈ X is a solution, with t ′ = t 1 −ρ(y; r 1 ). In R 2 2 we can place "true" and "false" sources, s and s ′ respectively, at 2 foci of hyperbola, and place 3 sensors on the same branch of that hyperbola. Then ρ(r i ; s)−ρ(r i ; s ′ ) ≡ d, thus s ′ emitting the wave at the moment t ′ = t 0 + d is another solution.
To ensure the uniqueness of the solution in R m 2 , we can take m + 2 sensors such that {r 2 − r 1 ; . . . ; r m+1 − r 1 } is a basis of R m and r m+2 = 2r 1 − r 2 ([13, Prop. 4]).
Definition 1. For any x ∈ X the backward moments τ i (x) := t i − ρ(x; r i ), i = 1, n τ i (x)
is the moment when the wave must be emitted from x to reach r i at the moment t i .
Definition 2. For any x ∈ X the defect D(x) := 1 n n i=1 τ i (x) − 1 n n j=1 τ j (x)
Cf. e.g. [4,3] or [25, 2.4]. We rewrite
D(x) = 1 n 2 n i=1 nτ i (x) − n j=1 τ j (x) = 1 n 2 n i=1 n j=1 τ i (x) − τ j (x) 1 n 2 n i=1 n j=1 τ i (x) − τ j (x)
The elementary properties of D(·) follow (Props. [1][2][3][4].
Proposition 1. s ′ ∈ X is the solution of the SRP if and only if D(s ′ ) = 0. Proof. If s ′ is such solution, then t i = t ′ + ρ(r i ; s ′ ), τ i (s ′ ) ≡ t ′ ⇒ τ i (s ′ ) − τ j (s ′ ) ≡ 0, so D(s ′ ) = 0. Contrariwise, D(s ′ ) = 0 implies τ i (s ′ ) ≡ t ′ = 1 n n j=1
τ j (s ′ ), and (s ′ ; t ′ ) is the solution.
Corollary 1. D(x) has exactly one zero in X, at x = s.
Proposition 2. ∀x ∈ X ∃i, j: |τ i (x) − τ j (x)| D(x).
Proof. Assuming the contrary, we have D(x) < 1
n 2 n i=1 n j=1 D(x) = D(x) -a contradiction. Proposition 3. ∀x, y ∈ X: D(x) − D(y)
2ρ(x; y).
Proof. D(x) − D(y) = 1 n 2 i | . . . | − i | . . . | = 1 n 2 i | j (. . .)| − | j (. . .)| 1 n 2 i j (. . .) − j (. . .) = 1 n 2 n i=1 n j=1 τ i (x) − τ j (x) − {τ i (y) − τ j (y)} 1 n 2 n i=1 n j=1 τ i (x) − τ i (y) − τ j (x) − τ j (y) 1 n 2 i,j τ i (x) − τ i (y) + τ j (x) − τ j (y) = = 1 n 2 i,j ρ(x; r i ) − ρ(y; r i ) + ρ(x; r j ) − ρ(y; r j ) 1 n 2 i,j ρ(x; y) + ρ(x; y) = 2ρ(x; y) -D(·)
is a Lipschitz function (see [15,6], [26, 9.4]).
Corollary 2. D(x) is uniformly continuous on X. Proposition 4. If s ∈ B = B[c; r], then ∀δ > 0 ∃ε > 0: x ∈ B, D(x) < ε ⇒ ρ(x; s) < δ. Proof. Let S = x ∈ B | ρ(x; s) δ and ε = inf x∈S D(x). We claim that S is compact: indeed,
S ⊂ B and S is closed due to continuity of metric. Since ∀x ∈ S: D(x) > 0 due to Cor. 1, we have ε > 0 (otherwise ∀k ∈ N ∃x k ∈ S: D(x k ) < 1 k , and it follows from compactness of S that ∃{x k l } l∈N :
x k l − −− → l→∞ x 0 ∈ S; D(x) is continuous, so D(x k l ) − −− → l→∞ D(x 0 ), but 0 D(x k l ) < 1 k l 1 l ⇒ D(x 0 ) = lim l→∞ D(x k l ) = 0 -a contradiction). Now, if x ∈ B and D(x) < ε, then x / ∈ S, which means ρ(x; s) < δ.
Test for a ball. Consider arbitrary closed ball (3) and the condition "2r < D(c)" divides the family of all closed balls in X into 2 families: 1) N (egative) -the balls that satisfy this condition and thus do not contain s, 2) S (uspicious) -the balls that do not satisfy it. Of course, even if B ∈ S, more "advanced" tests may prove that s / ∈ B. For the method at hand we can weaken ρ(c; z) r to ρ(c; z) K 1 r, K 1 1
B = B[c; r] ⊆ X. If s ∈ B, then ρ(r i ; c) − r ρ(r i ; s) ρ(r i ; c) + r, i = 1, n ⇔ ⇔ t 0 = t i − ρ(r i ; s) ∈ t i − ρ(r i ; c) − r; t i − ρ(r i ; c) + r , i = 1, n hence t 0 ∈ C =Proposition 5. Suppose D(x) > 0. If x ∈ B = B[y; r] and r < 1 4 D(x), then B ∈ N . Proof. By Prop. 3, ρ(y; x) r < 1 4 D(x) ⇒ |D(y) − D(x)| < 1 2 D(x) ⇒ D(y) > 1 2 D(x) > 0. Since r < 1 4 D(x) < 1 2 D(y), (3) implies B ∈ N . -if D(x) > 0,
When (X; ρ) has some additional properties, e.g. its points can be provided with coordinates, we are able to make B more "constructively"; in particular, next section describes this procedure in the finite-dimensional normed spaces. Another example is the Riemannian manifolds with intrinsic metric (see [8, 7.
Proof. By construction, ∀B
′ = B[z k ; r 2 k ] ∈ B k ∃B ′′ = B[z k−1 ; r 2 k−1 ] ∈ B k−1 : B ′ ∈ B(B ′′ ), so by (4): ρ(z k ; z k−1 ) K 1 · r 2 k−1 = 2K 1 r · 1 2 k . Therefore ρ(z k ; z) ρ(z k ; z k−1 ) + ρ(z k−1 ; z) 2K 1 r · 1 2 k + ρ(z k−1 ; z) 2K 1 r · 1 2 k + 2K 1 r · 1 2 k−1 + ρ(z k−2 ; z) . . . 2K 1 r k i=1 2 −i + ρ(z; z) 2K 1 r Now, ∀x ∈ B∈B∞ B: ∃B[z k ; r 2 k ] ∋ x for some k, hence ρ(x; z) ρ(x; z k ) + ρ(z k ; z) r 2 k + 2K 1 r (2K 1 + 1)r
Method
We add one more assumption for the sake of simplicity: That means the ball being "big enough" to contain all possible positions of the source.
Step 0. Let k = 0, C 0 = {B 0,1 }, and r 0 = r.
Step 1. Let C k+1 = ∅. For each ball B = B[y; r k ] ∈ C k , r k = r 2 k , there is the cover B of B, which consists of the balls B ′ = B[c; r k+1 ], r k+1 = 1 2 r k = r 2 k+1 . Consider each B ′ in turn and apply test (3) to B ′ . If B ′ ∈ S, then add B ′ to C k+1 .
Step 2. Let z k+1 be the center of the arbitrarily chosen ball from C k+1 .
Step 3. k := k + 1, goto Step 1. Take any δ > 0. By Prop. 4, ∃ε > 0: x ∈ B, D(x) < ε ⇒ ρ(x; s) < δ.
Since r k = r 2 k − −−− → k→∞ 0, we see that ∃k 0 : ∀k k 0 r k < 1 2 ε. By construction, ∀B = B[c; r k ] ∈ C k : B ∈ S, so D(c) 2r k < ε. Hence ρ(c; s) < δ; in particular, ρ(z k ; s) < δ.
In practice, however, we would like to know when to halt this process. We need a discernible "sign" that z k is close enough to s, ρ(z k ; s) < δ for the preselected precision δ. We do not rely on the condition r k < 1 2 ε, because ε is, in a sense, unknown -defined "not constructively enough"; put differently, the convergence rate is unknown.
To attain this goal, we add the stopping criterion to Step 3 and replace it by
Step 3'. k := k + 1. If 1) ρ(c ′ ; c ′′ ) < 2 3 δ for any B[c ′ ; r k ], B[c ′′ ; r k ] ∈ C k and 2) r k < 1 3 δ, then halt; else goto Step 1.
By Prop. 4, ∃ε > 0: x ∈ B, D(x) < ε ⇒ ρ(x; s) < 1 3 δ. When r k = r 2 k < 1 2 ε, for two balls B[c ′ ; r k ], B[c ′′ ; r k ] to be in C k ⊆ S it is necessary that D(c ′ ), D(c ′′ ) 2r k < ε, thus ρ(c ′ ; c ′′ ) ρ(c ′ ; s) + ρ(s; c ′′ ) < 2
3 δ, -for big enough k the condition (1) holds. Obviously, the condition (2) holds when r 2 k < 1 3 δ ⇔ k > log 2 3r δ . As soon as the process reaches k such that both conditions hold and halts, z k is the sought approximation of s: suppose s ∈ B[c; r k ] ∈ C k , then ρ(z k ; s) ρ(z k ; c) + ρ(c; s) < 2 3 δ + 1 3 δ = δ
SR in finite-dimensional normed spaces
Now we denote by (X; · ) the normed space over the field R of real numbers. θ is the zero of X as linear vector space.
We apply the same "refining cover by defect" (RCD) method to approximate s, only the RC itself becomes more "constructible" due to the usage of bases and coordinates. Most of the reasonings above for metric spaces remain though, with usual ρ(x; y) = x − y .
We keep the constraints ♣1-3. As for ♣M1, it is provided by
♣N1. X is finite-dimensional: dim X = m ∈ N.
• X is a complete (Banach) space.
• If A ⊆ X is closed and bounded (A ⊆ B[θ; R]), then A is compact. In particular, any closed ball is compact -♣M1.
• If L is a (linear) subspace of X, L < X, then L is a closed subspace.
• If L < X and x / ∈ L, then ρ(x; L) = inf u∈L x − u > 0 and ∃h ∈ L: x − h = ρ(x; L).
In principle, we could take any normalized basis E = {e j } m j=1 of X (that is, e j ≡ 1, E is linearly independent, and X = L(E) = m j=1
x j e j | x j ∈ R, j = 1, m ). However, for the sake of optimization of the refining cover we prefer the so-called Auerbach bases.
We denote by X * the dual, or adjoint, space of X, that is, the space of all linear bounded functionals f : X → R. · * is the norm of X * .
• ∀f ∈ X * , ∀x ∈ X: |f (x)| f * · x .
Auerbach theorem ( [3], [17, 20.12]). There exist {e j } m j=1 ⊂ X and {f j } m j=1 ⊂ X * such that e j = f j * = 1, j = 1, m (normality), and f i (e j ) = δ ij , i, j = 1, m (biorthogonality).
Let E = {e j } m j=1 from Auerbach theorem. ∀e j , ∀u = i =j u i e i ∈ L −j = L(E\{e j }) we have 1 = |δ jj − 0| = |f j (e j ) − i =j u i f j (e i )| = |f j (e j − u)| f j * · e j − u = e j − u
thus ρ(e j ; L −j ) 1. On the other hand, θ ∈ L −j and e j − θ = 1, so ρ(e j ; L −j ) = e j = 1, j = 1, m (5) -in addition to being the normalized basis of X (E is linearly independent and |E| = dim X) this Auerbach basis E has the "orthogonality" property.
See also [21, 11.1, p. 517-519]. If we have some non-Auerbach basis E of X and want to "construct", or approximate, the Auerbach one E (i.e. calculate the coordinates of e j ∈ E in E) using the referenced "canonical" proof, which involves the maximization of the determinant, then we can search for that maximum in the m 2 -dimensional space of the coordinates of the m-tuples of the points on {x ∈ X : x = 1}, -a complicated task as m increases; on the other hand, we perform this search only once for given (X; · ).
Refining cover. We describe (or just recall) the cover of B = B[θ; 1] by the "lattice" of the closed balls of radius 1 2 ; cf. [10, 2.2, 6.3]. Let x ∈ B and x = m j=1
x j e j . For x j = 0
1 x = |x j | · e j + i =j xi xj e i Since − i =j] such that c > 3 2 , because B ′ ∩ B = ∅ then (∃x ∈ B ′ ∩ B ⇒ c c − x + x 1 2 + 1). We claim that ∀x ∈ B ∃c: x ∈ B[c; 1 2 ] ∈ B.
To obtain such c, we take c ij that is closest to x j (i j = rnd(m(1 + x j )), where rnd(x) = ⌊x⌋ + ⌊2{x}⌋ = ⌊2x⌋ − ⌊x⌋), then |x j − c ij |
Remarks
Defects. Other defect functions
D 1 (x) = 2 n 2 1 i<j n τ i (x) − τ j (x) , D 2 (x) = 1 n n i=1 τ i (x) − 1 n n j=1 τ j (x) 2 , D I (x) = I(x) = max i τ i (x) − min i τ i (x) = max 1 i<j n τ i (x) − τ j (x) , . . .
have the properties similar to those of D(·). For instance,
D 2 (x) − D 2 (y) 8M ρ(x; y) where M = max i,j ρ(r i ; r j ).
2. Issues with gradient method (GM) of searching for the minima of the defect function f D (x), which starts at the initial point x 0 and "moves" in the direction of the steepest descent (another name of this method) of f D (x); see [5, 6.6], [22,25].
Obviously, s is a local (and global) minimum, locmin, of f D (·). However, in general case there can be more than one locmin, even without disturbances caused by noise (cf. [7, 5.2], [16, 9. 3. Towards Noise. When, instead of exact t i , we know only "shifted" t i = t i + ξ i and τ i (x) = t i − ρ(x; r i ) (noises ξ i are supposed to be random variables with certain properties), D = 1 Maybe D has a continuum of zeros, maybe none.
n 2 i j τ i − τ j is "distorted";
If we opt to keep using the root finding approach rather than the optimization one, then the following crude, vaguely described "trick" may be applied, assuming |ξ i | γ for small enough γ:
consider D = | D − 2γ|.
It is easy to see that | D − D| 2γ, thus D(s) − 2γ 0, while at some distant x presumably D(x) − 2γ > 0; D has zeros. Like D, D is a "distortion" of D, but due to γ ≈ 0 this distortion should be small enough for the (continuum of) zeros of D to be "not too far" from s. These zeros form the closed (topologically and geometrically) "surface(s)" Z = {x ∈ X | D(x) = 0} around or near s; in a sense, the distortion "inflates" single zero, turning it into surface(s).
Prop. 3 remains valid for D, and instead of Prop. 4 we'll have its analogue with ρ(x; s) replaced by ρ(x; Z). We use the test "2r k < D(c)", and the refining cover {C k } ∞ k=0 constisting of B[c; r k ] that do not satisfy this inequality "converges" to Z, which stays mostly within the union of the balls from C k . At some iteration we halt and take some point "between" the centers of these balls (their mean in normed space, for instance). . . and rely on this point being close enough to s.
The locmins cause one of evident drawbacks of this trick: false, or "ghost", solutions can appear near such minima, relatively far from the true source s. At least they should not appear, and C k shouldn't break into the disjointed groups of the balls as k → ∞, if 2γ < µ, where µ is the minimal value of D(b) at the locmins b that are not the "descendants" of s.
Meta-refinement: we run, in parallel, several instances of | D − λ|-trick with different λ (e.g. λ ij = ± i 2 j−1 γ), compare how the respective covers behave, and spawn new instances if needed.
(c) − r; τ i (c) + r = ∅. It is easy to see that C = ∅ if and only (c) − τ j (c) =: I(c) Thus we have the inference: if s ∈ B[c; r], then 2r I(c). Conversely, 2r < I(c) ⇒ s / ∈ B[c; r]. From |τ i (c) − τ j (c)| I(c) it follows that D(c) = I(c), so 2r < D(c) ⇒ s / ∈ B[c; r]
then the suspicious balls that are small enough do not contain x. Refining cover. Consider any B = B[z; r]. A 0 = B(c; r 2 ) | c ∈ B is a cover of B by open sets. From ♣M1 it follows that there exists a finite subcover A 1= B(c i ; r 2 ) | i = 1, N ⊆ A 0 , which also covers B. Clearly, B = B[c i ; r 2 ] | i = 1, Nis a finite cover of B too. Thus we obtained Proposition 6. For any B = B[z; r] there exists a finite cover B = B(B) = B[c i ; r 2 ] | i = 1, N of B, and ∀B[c; r 2 ] ∈ B: ρ(c; z) r. Naturally, we want N to be as small as possible; however, the time spent in the (intricate) positioning of less balls can exceed the time gained by not testing more balls. This topic is omitted here; cf. [9], [14, 2]. If ∃N d ∈ N such that any B[z; r] can be covered by at most N d closed balls of radius r 2 , then (X; ρ) is called doubling, and N d is its doubling constant ([15, 10.13, p. 81]).
Let B 0 =B
0{B[z; r]} and B 1 = B(B[z; r]). We then denote by B 2 the union of the covers of all balls from B 1 , . . . , by B k the union of the covers of all balls from B k−1 , . . . At that, the balls in B k are of radius r ⊆ B[z; Kr], where K = 2K 1 + 1.
♣ 3 .
3At least one ball B 0,1 = B[c 0,1 ; r] ∋ s is known.
B
that at least one ball from B(B 0,1 ) contains s, this ball belongs to S and therefore C 1 = ∅. Similarly, at least one ball from B∈C1 B(B) contains s, implying C 2 = ∅, etc.: ∀k ∈ Z + C k = ∅. Hence these steps define the infinite sequence of the covers {C k } ∞ k=0 and the infinite sequence of the centers {z k } ∞ k=1 . Proposition 8. z k − −−− → k→∞ s. Proof. By Prop. 7, {z k } k∈N ⊆ B = B[c 0,1 ; Kr] and s ∈ B 0,1 ⊆ B.
xi xj e i ∈ L −j = L {e 1 ; . . . ; e j−1 ; e j+1 ; . . . ; e m } , we obtain 1 |x j | · ρ(e j ; L −j ). By construction of E, we have(5): ρ(e j ; L −j ) = 1, therefore |x j | 1. Let c i = −1 + i m , i = 0,2m: we break [−1; 1] into the segments [c i ; c i+1 ] of length 1 m . Consider the set of the balls B = B[c; 1 2 ] | c = m j=1 c ij e j , i j = 0, 2m, j = 1, m . There are (2m + 1) m of them, and we instantly remove from B the balls B ′ = B[c; 1 2
B is the cover of B. Analogously, scaled and translatedB = z + rB = B[z + c; r 2 ] : B[c; 1 2 ] ∈ B is the sought cover of B[z; r], and ∀B[z ′ ; r 2 ] ∈B: z ′ − z 3 2 r (∼ Prop. 6).
.
Consider the defect D 2 (·), which is also the variance of the random variable with equiprobable values τ i (x). Let (X; ρ) = R 2 2 , s = (0; 0), and the sensors r 1 = (8; 6), r 2 = (5; 5), r 3 = (−2; 6), r 4 = (−6; 4), r 5 = (−10; 2) ⊳ "Numerical experiments" show that D 2 (x) has locmin at b ≈ (−3.6901; 21.5627), at that D 2 (b) ≈ 0.69044. Therefore we cannot start GM at arbitrary initial point to search for the solution (r 5 = 2r 4 − r 3 , so [13, Prop. 4] implies the uniqueness of the solution s). ⊲ Similar configurations exist for higher dimensionalities. Moreover, GM that starts at the sensor nearest to s can converge to the locmin b = s: Example 2. Let (X; ρ) = R 2 2 , s = (0; 0), and the sensors r 1 (1.885; 0.014), r 2 (2.523; −0.76), r 3 (2.552; −0.756), r 4 (2.94; −0.78), r 5 (2.081; 0.986) ⊳ GM with the initial point r 1 , which is nearest to s, converges to the locmin of D 2 (x) at b ≈ (2.039; 0.253), D 2 (b) ≈ 0.00318. ⊲ Again, similar behaviour can occur in R m 2 for m > 2.
we lose Prop. 1, Prop. 4, and what is built on top of them.
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| []
|
[
"LINEAR RESPONSE THEORY FOR RANDOM SCHRÖDINGER OPERATORS AND NONCOMMUTATIVE INTEGRATION",
"LINEAR RESPONSE THEORY FOR RANDOM SCHRÖDINGER OPERATORS AND NONCOMMUTATIVE INTEGRATION"
]
| [
"Nicolas Dombrowski ",
"Franç Ois Germinet "
]
| []
| []
| We consider an ergodic Schrödinger operator with magnetic field within the non-interacting particle approximation. Justifying the linear response theory, a rigorous derivation of a Kubo formula for the electric conductivity tensor within this context can be found in a recent work of Bouclet, Germinet, Klein and Schenker [BoGKS]. If the Fermi level falls into a region of localization, the well-known Kubo-Stȓeda formula for the quantum Hall conductivity at zero temperature is recovered. In this review we go along the lines of [BoGKS] but make a more systematic use of noncommutative L p -spaces, leading to a somewhat more transparent proof. | null | [
"https://arxiv.org/pdf/1103.5498v1.pdf"
]
| 119,313,399 | 1103.5498 | f36915c7b37edb69abebec8e34ce0a6002ab6194 |
LINEAR RESPONSE THEORY FOR RANDOM SCHRÖDINGER OPERATORS AND NONCOMMUTATIVE INTEGRATION
28 Mar 2011
Nicolas Dombrowski
Franç Ois Germinet
LINEAR RESPONSE THEORY FOR RANDOM SCHRÖDINGER OPERATORS AND NONCOMMUTATIVE INTEGRATION
28 Mar 2011arXiv:1103.5498v1 [math-ph]
We consider an ergodic Schrödinger operator with magnetic field within the non-interacting particle approximation. Justifying the linear response theory, a rigorous derivation of a Kubo formula for the electric conductivity tensor within this context can be found in a recent work of Bouclet, Germinet, Klein and Schenker [BoGKS]. If the Fermi level falls into a region of localization, the well-known Kubo-Stȓeda formula for the quantum Hall conductivity at zero temperature is recovered. In this review we go along the lines of [BoGKS] but make a more systematic use of noncommutative L p -spaces, leading to a somewhat more transparent proof.
Introduction
In [BoGKS] the authors consider an ergodic Schrödinger operator with magnetic field, and give a controlled derivation of a Kubo formula for the electric conductivity tensor, validating the linear response theory within the noninteracting particle approximation. For an adiabatically switched electric field, they then recover the expected expression for the quantum Hall conductivity whenever the Fermi energy lies either in a region of localization of the reference Hamiltonian or in a gap of the spectrum.
The aim of this paper is to provide a pedestrian derivation of [BoGKS]'s result and to simplify their "mathematical apparatus" by resorting more systematically to noncommutative L p -spaces. We also state results for more general time dependent electric fields, so that AC-conductivity is covered as well. That von Neumann algebra and noncommutative integration play an important rôle in the context of random Schrödinger operators involved with the quantum Hall effect goes back to Bellissard, e.g. [B,BES,SB1,SB2].
The electric conductivity tensor is usually expressed in terms of a "Kubo formula," derived via formal linear response theory. In the context of disordered media, where Anderson localization is expected (or proved), the electric conductivity has driven a lot of interest coming for several perspective. For time reversal systems and at zero temperature, the vanishing of the direct conductivity is a physically meaningful evidence of a localized regime [FS, AG]. Another direction of interest is the connection between direct conductivity and the quantum Hall effect [ThKNN, St, B, Ku, BES, AvSS, AG]. On an other hand the behaviour of the alternative conductivity at small frequencies within the region of localization is dictated by the celebrated Mott formula [MD] (see [KLP, KLM, KM] for recent important developments). Connected to conductivities, current-current correlations functions have recently droven a lot of attention as well (see [BH, CGH] and references therein).
During the past two decades a few papers managed to shed some light on these derivations from the mathematical point of view, e.g., [P,Ku,B,NB,AvSS,BES,SB1,SB2,AG,Na,ES,CoJM,CoNP]. While a great amount of attention has been brought to the derivation from a Kubo formula of conductivities (in particular of the quantum Hall conductivity), and to the study of these conductivities, not much has been done concerning a controlled derivation of the linear response and the Kubo formula itself; only the recent papers [SB2,ES,BoGKS,CoJM,CoNP] deal with this question.
In this note, the accent is put on the derivation of the linear response for which we shall present the main elements of proof, along the lines of [BoGKS] but using noncommutative integration. The required material is briefly presented or recalled from [BoGKS]. Further details and extended proofs will be found in [Do]. We start by describing the noncommutative L p -spaces that are relevant in our context, and we state the properties that we shall need (Section 2). In Section 3 we define magnetic random Schrödinger operators and perturbations of these by time dependent electric fields, but in a suitable gauge where the electric field is given by a time-dependent vector potential. We review from [BoGKS] the main tools that enter the derivation of the linear response, in particular the time evolution propagators. In Section 4 we compute rigorously the linear response of the system forced by a time dependent electric field. We do it along the lines of [BoGKS] but within the framework of the noncommutative L p -spaces presented in Section 2. The derivation is achieved in several steps. First we set up the Liouville equation which describes the time evolution of the density matrix under the action of a timedependent electric field (Theorem 4.1). In a standard way, this evolution equation can be written as an integral equation, the so-called Duhamel formula. Second, we compute the net current per unit volume induced by the electric field and prove that it is differentiable with respect to the electric field at zero field. This yields in fair generality the desired Kubo formula for the electric conductivity tensor, for any non zero adiabatic parameter (Theorem 4.6 and Corollary 4.7). The adiabatic limit is then performed in order to compute the direct / ac conductivity at zero temperature (Theorem 4.8, Corollary 4.9 and Remark 4.11). In particular we recover the expected expression for the quantum Hall conductivity, the Kubo-Stȓeda formula, as in [B, BES]. At positive temperature, we note that, while the existence of the electric conductivity measure can easily be derived from that Kubo formula [KM], proving that the conductivity itself, i.e. its density, exists and is finite remains out of reach.
Acknowledgement. We thank warmly Vladimir Georgescu for enlightening discussions on noncommutative integration, as well as A. Klein for useful comments.
Covariant measurable operators and noncommutative L p -spaces
In this section we construct the noncommutative L p -spaces that are relevant for our analysis. We first recall the underlying Von Neumann alegbra of observables and we equip it with the so called "trace per unit volume". We refer to [D, Te] for the material. We shall skip some details and proofs for which we also refer to [Do].
2.1. Observables. Let H be a separable Hilbert space (in our context H = L 2 (R d )). Let Z be an abelian locally compact group and U = {U a } a∈Z a unitary projective representation of Z on H, i.e.
• U a U b = ξ(a, b)U a+b , where ξ(a, b) ∈ C, |ξ(a, b)| = 1;
• U e = 1; Now we take a set of orthogonal projections on H , χ := {χ a } a∈Z , Z → B(H). Such that if a = b ⇒ χ a χ b = 0 and a∈Z χ a =1. Furthermore one requires a covariance relation or a stability relation of χ under U i.e U a χ b U * a = χ a+b . Next to this Hilbertian structure (representing the "physical" space), we consider a probability space (Ω, F , P) (representing the presence of the disorder) that is ergodic under the action of a group τ = {τ a } a∈Z , that is,
• ∀a ∈ Z, τ a : Ω → Ω is a measure preserving isomorphism;
• ∀a, b ∈ Z, τ a • τ b = τ a+b ;
• τ e = 1 where e is the neutral element of Z and thus τ −1 a = τ −a , ∀a ∈ Z; • If A ∈ F is invariant under τ , then P(A) = 0 or 1. With these two structures at hand we define the Hilbert spacẽ
H = ⊕ Ω H dP(ω) := L 2 (Ω, P, H) ≃ H ⊗ L 2 (Ω, P),
(2.1) equipped with the inner product
ϕ, ψ H = Ω ϕ(ω), ψ(ω) H dP(ω), ∀ϕ, ψ ∈H 2 . (2.2)
We are interested in bounded operators onH that are decomposable elements A = (A ω ) ω∈Ω , in the sense that they commute with the diagonal ones. Measurable operators are defined as decomposable operators such that for all measurable vector's field {ϕ(ω), ω ∈ Ω}, the set {A(ω)ϕ(ω), ω ∈ Ω} is measurable too. Measurable decomposable operators are called essentially bounded if ω → A ω L(H) is a element of L ∞ (Ω, P). We set, for such A's,
A L(H) = A ∞ = ess − sup Ω A(ω) ,(2.3)
and define the following von Neumann algebra
K = L ∞ (Ω, P, L(H)) = {A : Ω → L(H), measurable A ∞ < ∞}. (2.4)
There exists an isometric isomorphism betwen K and decomposable operators on L(H).
We shall work with observables of K that satisfy the so-called covariant property.
Definition 2.1. A ∈ K is covariant iff U a A(ω)U ⋆ a = A(τ a ω) , ∀a ∈ Z, ∀ω ∈ Ω. (2.5) We set K ∞ = {A ∈ K, A is covariant}. (2.6)
IfŨ a := U a ⊗τ (−a),with the slight notation abuse where we note τ for the action induct by τ on L 2 andŨ = (Ũ a ) a∈Z , we note that
K ∞ = {A ∈ K, ∀a ∈ Z, [A,Ũ a ] = 0} (2.7) = K ∩ (Ũ ) ′ , (2.8)
so that K ∞ is again a von Neumann algebra.
2.2. Noncommutative integration. The von Neumann algebra K ∞ is equipped with the faithfull, normal and semi-finite trace
T (A) := E{tr(χ e A(ω)χ e )},(2.9)
where "tr" denotes the usual trace on the Hilbert space H.In the usual context of the Anderson model this is nothing but the trace per unit volume, by the Birkhoff Ergodic Theorem, whenever T (|A|) < ∞, one has
T (A) = lim |ΛL|→∞ 1 |Λ L | tr(χ ΛL A ω χ ΛL ), (2.10)
where Λ L ⊂ Z and χ ΛL = a∈ΛL χ a . There is a natural topology associated to von Neumann algebras equipped with such a trace. It is defined by the following basis of neighborhoods:
N (ǫ, δ) = {A ∈ K ∞ , ∃P ∈ K proj ∞ , AP ∞ < ε , T (P ⊥ ) < δ}, (2.11)
where K proj ∞ denotes the set of projectors of K ∞ . It is a well known fact that
A ∈ N (ε, δ) ⇐⇒ T (χ ]ε,∞[ (|A|)) ≤ δ. (2.12)
As pointed out to us by V. Georgescu, this topology can also be generated by the following Frechet-norm on K ∞ [Geo]:
A T = inf P ∈K proj ∞ max{ AP ∞ , T (P ⊥ )}. (2.13)
Let us denote by M(K ∞ ) the set of all T -measurable operators, namely the completion of K ∞ with respect to this topology. It is a well established fact from noncommutative integration that Theorem 2.2. M(K ∞ ) is a Hausdorff topological * -algebra. , in the sens that all the algebraic operations are continuous for the τ -measure topology.
Definition 2.3. A linear subspace E ⊆ H is called T -dense if, ∀δ ∈ R + , there exists a projection P ∈ K ∞ such that P H ⊆ E and T (P ⊥ ) ≤ δ.
It turns out that any element A of M(K ∞ ) can be represented as an (possibly unbounded) operator, that we shall still denote by A, acting onH with a domain D(A) = {ϕ ∈H, Aϕ ∈H} that is T -densily defined. Then, adjoints, sums and products of elements of M(K ∞ ) are defined as usual adjoints, sums and products of unbounded operators.
For any 0 < p < ∞, we set
L p (K ∞ ) := {x ∈ K ∞ , T (|x| p ) < ∞} · p = {x ∈ M(K ∞ ), T (|x| p ) < ∞}, (2.14)
where the second equality is actually a theorem. For p ≥ 1, the spaces
L p (K ∞ ) are Banach spaces in which L p,o (K ∞ ) := L p (K ∞ ) ∩ K ∞ are dense by definition. For p = ∞, in analogy with the commutative case, we set L ∞ (K ∞ ) = K ∞ .
Noncommutative Hölder inequalities hold: for any 0 <, p, q, r ≤ ∞ so that
p −1 + q −1 = r −1 , if A ∈ L p (K ∞ ) and B ∈ L q (K ∞ ), then the product AB ∈ M(K ∞ ) belongs to L r (K ∞ ) with AB r ≤ A p B q . (2.15) In particular, for all A ∈ L ∞ (K ∞ ) and B ∈ L p (K ∞ ), AB p ≤ A ∞ B p and BA p ≤ A ∞ B p , (2.16) so that L p (K ∞ )-spaces are K ∞ two-sided submodules of M(K ∞ ). As another con- sequence, bilinear forms L p,o (K ∞ )×L q,o (K ∞ ) ∋ (A, B) → T (AB) ∈ C continuously extends to bilinear maps defined on L p (K ∞ ) × L q (K ∞ ). Lemma 2.4. Let A ∈ L p (K ∞ ), p ∈ [1, ∞[ be given, and suppose T (AB) = 0 for all B ∈ L q (K ∞ ), p −1 + q −1 = 1. Then A = 0.
The case p = 2 is of particular interest since L 2 (K ∞ ) equipped with the sesquilinear form A, B L 2 = T (A * B) is a Hilbert space. The corresponding norm reads
A 2 2 = Ω tr(χ e A * ω A ω χ e )dP(ω) = Ω A ω χ e 2 2 dP(ω).
(2.17) (Where · 2 denotes the Hilbert-Schmidt norm.) From the case p = 2, we can derive the centraliy of the trace. Indeed, by covariance and using the centrality of the usual trace, it follows that T (AB) = T (BA) whenever A, B ∈ K ∞ . By density we get the following lemma.
Lemma 2.5. Let A ∈ L p (K ∞ ) and B ∈ L q (K ∞ ), p −1 + q −1 = 1 be given. Then T (AB) = T (BA).
We shall also make use of the following observation.
Lemma 2.6. Let A ∈ L p (K ∞ ) and (B n ) a sequence of elements of K ∞ that converges strongly to B ∈ K ∞ . Then AB n converges to AB in L p (K ∞ ).
Commutators of measurable covariant operators.
Let H be a decomposable (unbounded) operator affiliated to K ∞ with domain D, and A ∈ M(K ∞ ).
In particular H need not be T -measurable, i.e. in M(K ∞ ). If there exists a Tdense domain D ′ such that AD ′ ⊂ D, then HA is well defined, and if in addition the product is T -measurable then we write HA ∈ M(K ∞ ). Similarly, if D is Tdense and the range of HD ⊂ D(A), then AH is well defined, and if in addition the product is T -measurable then we write AH ∈ M(K ∞ ).
Remark 2.7. We define the following (generalized) commutators:
(i): If A ∈ M (K ∞ ) and B ∈ K ∞ , [A, B] := AB − BA ∈ M(K ∞ ), [B, A] := −[A, B]. (2.18) (ii): If A ∈ L p (K ∞ ), B ∈ L q (K ∞ ), p, q ≥ 1 such that p −1 + q −1 = 1 , then [A, B] := AB − BA ∈ L 1 (K ∞ ). (2.19) Definition 2.8. Let HηK ∞ (i.e H affiliated to K ∞ ). If A ∈ M(K ∞ ) is such that HA and AH are in M(K ∞ ), then [H, A] := HA − AH ∈ M(K ∞ ) .
(2.20)
We shall need the following observations.
Lemma 2.9. 1) For any A ∈ L p (K ∞ ), B ∈ L q (K ∞ ), p, q ≥ 1, p −1 + q −1 = 1, and C ω ∈ K ∞ , we have T {[C, A]B} = T {C[A, B]} . (2.21) 2) For any A, B ∈ K ∞ and C ∈ L 1 (K ∞ ), we have T {A[B, C]} = T {[A, B]C} . (2.22) 3) Let p, q ≥ 1 be such that p −1 +q −1 = 1. For any A ∈ L p (K ∞ ), resp. B ∈ L q (K ∞ ), such that [H, A] ∈ L p (K ∞ ), resp. [H, B] ∈ L q (K ∞ ), we have T {[H, A]B} = −T {A[H, B]} . (2.23) 2.4. Differentiation. A * -derivation ∂ is a * -map defined on a dense sub-algreba of K ∞ and such that: • ∂(AB) = ∂(A)B + A∂(B) • ∂(A + λB) = ∂(A) + λ∂(B) • ∂(A ⋆ ) = ∂(A) ⋆ • [α a , ∂] = 0 in the sense that α a • ∂(A) = ∂ • α a (A) ∀a ∈ Z , ∀A ∈ K ∞ . If ∂ 1 , .
.., ∂ d are * -derivations we define a non-commutative gradient by ∇ := (∂ 1 , ..., ∂ d ), densily defined on K ∞ . We define a non-commutative Sobolev space
W 1,p (K ∞ ) := {A ∈ L p (K ∞ ), ∇A ∈ L p (K ∞ )}.
(2.24) and a second space for HηK ∞ ,
D (0) p (H) = {A ∈ L p (K ∞ ), HA, AH ∈ L p (K ∞ )} .
(2.25)
The setting: Schrödinger operators and dynamics
In this section we describe our background operators and recall from [BoGKS] the main properties we shall need in order to establish the Kubo formula, but within the framework of noncommutative integration when relevant (i.e. in Subsection 3.2).
Magnetic Schrödinger operators and time-dependent operators.
Throughout this paper we shall consider Schrödinger operators of general form
H = H(A, V ) = (−i∇ − A) 2 + V on L 2 (R d ), (3.1)
where the magnetic potential A and the electric potential V satisfy the Leinfelder-Simader conditions:
• A(x) ∈ L 4 loc (R d ; R d ) with ∇ · A(x) ∈ L 2 loc (R d ). • V (x) = V + (x) − V − (x) with V ± (x) ∈ L 2 loc (R d ), V ± (x)
≥ 0, and V − (x) relatively bounded with respect to ∆ with relative bound < 1, i.e., there are 0 ≤ α < 1 and β ≥ 0 such that
V − ψ ≤ α ∆ψ + β ψ for all ψ ∈ D(∆). (3.2)
Leinfelder and Simader have shown that [LS,Theorem 3]. It has been checked in [BoGKS] that under these hypotheses H(A, V ) is bounded from below:
H(A, V ) is essentially self-adjoint on C ∞ c (R d )H(A, V ) ≥ − β (1 − α) =: −γ + 1, so that H + γ ≥ 1. (3.3)
We denote by x j the multiplication operator in L 2 (R d ) by the j th coordinate x j , and x := (x 1 , · · · x d ). We want to implement the adiabatic switching of a time dependent spatially uniform electric field E η (t) · x = e ηt E(t) · x between time t = −∞ and time t = t 0 . Here η > 0 is the adiabatic parameter and we assume that
t0 −∞ e ηt |E(t)|dt < ∞.
( 3.4) To do so we consider the time-dependent magnetic potential
A(t) = A + F η (t), with F η (t) = t −∞ E η (s)ds.
In other terms, the dynamics is generated by the time-dependent magnetic operator
H(t) = (−i∇ − A − F η (t)) 2 + V (x) = H(A(t), V ) , (3.5) which is essentially self-adjoint on C ∞ c (R d ) with domain D := D(H) = D(H(t)
) for all t ∈ R. One has (see [BoGKS,Proposition 2.5])
H(t) = H − 2F η (t) · D(A) + F η (t) 2 on D(H), (3.6) where D = D(A) is the closure of (−i∇ − A) as an operator from L 2 (R d ) to L 2 (R d ; C d ) with domain C ∞ c (R d ). Each of its components D j = D j (A) = (−i ∂ ∂xj − A j ), j = 1, . . . , d, is essentially self-adjoint on C ∞ c (R d ).
To see that such a family of operators generates the dynamics a quantum particle in the presence of the time dependent spatially uniform electric field E η (t) · x, consider the gauge transformation
[G(t)ψ](x) := e iFη (t)·x ψ(x) ,(3.7)
so that
H(t) = G(t) (−i∇ − A) 2 + V G(t) * . (3.8) Then if ψ(t) obeys Schrödinger equation i∂ t ψ(t) = H(t)ψ(t),(3.9)
one has, formally,
i∂ t G(t) * ψ(t) = (−i∇ − A) 2 + V + E η (t) · x G(t) * ψ(t).
(3.10)
To summarize the action of the gauge transformation we recall the The key observation is that the general theory of propagators with a time dependent generator [Y,Theorem XIV.3.1] applies to H(t). It thus yields the existence of a two parameters family U (t, s) of unitary operators, jointly strongly continuous in t and s, that solves the Schrödinger equation.
U (t, r)U (r, s) = U (t, s) (3.14) U (t, t) = I (3.15) U (t, s)D = D , (3.16) i∂ t U (t, s)ψ = H(t)U (t, s)ψ for all ψ ∈ D , (3.17) i∂ s U (t, s)ψ = − U (t, s)H(s)ψ for all ψ ∈ D .
(3.18)
We refer to [BoGKS,Theorem 2.7] for other relevant properties.
To compute the linear response, we shall make use of the following "Duhamel formula". Let U (0) (t) = e −itH . For all ψ ∈ D and t, s ∈ R we have [BoGKS,Lemma 2
.8] U (t, s)ψ = U (0) (t−s)ψ+i t s U (0) (t−r)(2F η (r)·D(A)−F η (r) 2 )U (r, s)ψ dr . (3.19) Moreover, lim |E|→0 U (t, s) = U (0) (t − s) strongly .
(3.20)
3.2. Adding the randomness. Let (Ω, P) be a probability space equipped with an ergodic group {τ a ; a ∈ Z d } of measure preserving transformations. We study operator-valued maps A : Ω ∋ ω → A ω . Throughout the rest of this paper we shall use the material of Section 2 with H = L 2 (R d ) and Z = Z d . The projective representation of Z d on H is given by magnetic translations (U (a)ψ)(x) = e ia·Sx ψ(x − a), S being a given d × d real matrix. The projection χ a is the characteristic function of the unit cube of R d centered at a ∈ Z d .
In our context natural * -derivations arise:
∂ j A := i[x j , A], j = 1, · · · , d, ∇A = i[x, A]. (3.21)
We shall now recall the material from [BoGKS,Section 4.3]. Proofs of assertions are extensions of the arguments of [BoGKS] to the setting of L p (K ∞ )-spaces. We refer to [Do] for details.
We state the technical assumptions on our Hamiltonian of reference H ω .
Assumption 3.2. The ergodic Hamiltonian ω → H ω is a measurable map from the probability space (Ω, P) to self-adjoint operators on H such that
H ω = H(A ω , V ω ) = (−i∇ − A ω ) 2 + V ω ,(3.
22)
almost surely, where A ω (V ω ) are vector (scalar) potential valued random variables which satisfy the Leinfelder-Simader conditions (see Subsection 3.1) almost surely. It is furthermore assumed that H ω is covariant as in (2.5). We denote by H the operator (H ω ) ω∈Ω acting onH.
As a consequence f (H ω ) ≤ f ∞ and f (H) ∈ K ∞ for every bounded Borel function f on the real line. In particular H is affiliated to K ∞ . For P-a.e. ω, let U ω (t, s) be the unitary propagator associated to H ω and described in Subsection 3.1. Note that (U ω (t, s)) ω∈Ω ∈ K ∞ (measurability in ω follows by construction of U ω (t, s), see [BoGKS]). For A ∈ M(K ∞ ) decomposable, let
U(t, s)(A) := ⊕ Ω U ω (t, s)A ω U ω (s, t) dP(ω).(K ∞ ), p ∈ 1, ∞]. Pick p > 0. Let A ∈ L p (K ∞ ) be such that H(r 0 )A and AH(r 0 ) are in L p (K ∞ ) for some r 0 ∈ [−∞, ∞). Then both maps r → U(t, r)(A) ∈ L p (K ∞ ) and t → U(t, r)(A) ∈ L p (K ∞ ) are differentiable in L p (K ∞ ), with (recalling Definition 2.8) i∂ r U(t, r)(A) = −U(t, r)([H(r), A]). (3.27) i∂ t U(t, r)(A) = [H(t), U(t, r)(A)].
(3.28)
Moreover, for t 0 < ∞ given, there exists C = C(t 0 ) < ∞ such that for all t, r ≤ t 0 ,
(H(t) + γ) U(t, r)(A) p ≤ C (H(r) + γ)A p ,(3.29)
[H(t), U(t, r)(A)] p ≤ C ( (H(r) + γ)A p + A(H(r) + γ) p ) .
(3.30)
We note that in order to apply the above formula and in particular (3.27) and (3.28), it is actually enough to check that (H(r 0 ) + γ)A and A(H(r 0 ) + γ) are in L p (K ∞ ). Whenever we want to keep track of the dependence of U ω (t, s) on the electric field E = E η (t), we shall write U ω (E, t, s). When E = 0, note that
U ω (E = 0, t, s) = U (0) ω (t − s) := e −i(t−s)Hω . (3.31) For A ∈ M(K ∞ ) decomposable, we let U (0) (r)(A) := ⊕ Ω U (0) ω (r)A ω U (0) ω (−r) dP(ω). (3.32)
We still denote by U (0) (r)(A) its extension to M(K ∞ ).
Proposition 3.3. Let p ≥ 1 be given. U (0) (t) is a one-parameter group of operators on M(K ∞ ), leaving L p (K ∞ ) invariant. U (0) (t) is an isometry on L p (K ∞ ), and unitary if p = 2. It is strongly continuous on L p (K ∞ ). We further denote by L p the infinitesimal generator of U (0) (t) in L p (K ∞ ):
U (0) (t) = e −itLp for all t ∈ R .
(3.33)
The operator L p is usually called the Liouvillian. Let
D (0) p = {A ∈ L p (K ∞ ), HA, AH ∈ L p (K ∞ )} . (3.34) Then D (0) p
is an operator core for L p (note that L 2 is essentially self-adjoint on D Moreover, for every B ω ∈ K ∞ there exists a sequence B n,ω ∈ D
∞ such that B n,ω → B ω as a bounded and P-a.e.-strong limit.
We finish this list of properties with the following lemma about the Gauge transformations in spaces of measurable operators. The map
G(t)(A) = G(t)AG(t) * ,(3.36)
with G(t) = e i t −∞ Eη (s)·x as in (3.7), is an isometry on L p (K ∞ ), for p ∈]0, ∞].
Lemma 3.4. For any p ∈]0, ∞], the map G(t) is strongly continuous on L p (K ∞ ), and lim
t→−∞ G(t) = I strongly (3.37) on L p (K ∞ ). Moreover, if A ∈ W 1,p (K ∞ ), then G(t)(A) is continuously differen- tiable in L p (K ∞ ) with ∂ t G(t)(A) = E η (t) · ∇(G(t)(A)).
(3.38)
Linear response theory and Kubo formula
4.1. Adiabatic switching of the electric field. We now fix an initial equilibrium state of the system, i.e., we specify a density matrix ζ ω which is in equilibrium, so [H ω , ζ ω ] = 0. For physical applications, we would generally take ζ ω = f (H ω ) with f the Fermi-Dirac distribution at inverse temperature β ∈ (0, ∞] and Fermi energy
E F ∈ R, i.e., f (E) = 1 1+e β(E−E F ) if β < ∞ and f (E) = χ (−∞,EF ] (E) if β = ∞; explicitly ζ ω = F (β,EF ) ω := 1 1+e β(Hω −E F ) , β < ∞ , P (EF ) ω := χ (−∞,EF ] (H ω ) , β = ∞ .
(4.1)
However we note that our analysis allows for fairly general functions f [BoGKS]. We set ζ = (ζ ω ) ω∈Ω ∈ K ∞ but shall also write ζ ω instead of ζ. That f is the Fermi-Dirac distribution plays no role in the derivation of the linear response. However computing the Hall conductivity itself (once the linear response performed) we shall restrict our attention to the zero temperature case with the Fermi projection P (EF ) ω . The system is described by the ergodic time dependent Hamiltonian H ω (t), as in (3.5). Assuming the system was in equilibrium at t = −∞ with the density matrix ̺ ω (−∞) = ζ ω , the time dependent density matrix ̺ ω (t) is the solution of the Cauchy problem for the Liouville equation. Since we shall solve the evolution equation in L p (K ∞ ), we work with H(t) = (H ω (t)) ω∈Ω , as in Assumption 3.2.
The electric field E η (t) · x = e ηt E(t) · x is swichted on adiabatically between t = −∞ and t = t 0 (typically t 0 = 0). Depending on which conductivity on is interested, one may consider different forms for E(t). In particular E(t) = E leads to the direct conductivity, while E(t) = cos(νt)E leads to the AC-conductivity at frequency ν 1 . The first one is relevant for studying the Quantum Hall effect (see subsection 4.4), while the second enters the Mott's formula [KLP, KLM].
We write
ζ(t) = G(t)ζG(t) * = G(t)(ζ), i.e., ζ(t) = f (H(t)).
(4.2)
Theorem 4.1. Let η > 0 and assume that
t −∞ e ηr |E(r)|dr < ∞ for all t ∈ R. Let p ∈ [1, ∞[. Assume that ζ ∈ W 1,p (K ∞ ) and that ∇ζ ∈ D o p . The Cauchy problem i∂ t ̺(t) = [H(t), ̺(t))] lim t→−∞ ̺(t) = ζ , (4.3)
has a unique solution in L p (K ∞ ), that is given by
̺(t) = lim s→−∞ U(t, s) (ζ) (4.4) = lim s→−∞ U(t, s) (ζ(s)) (4.5) = ζ(t) − t −∞
dr e ηr U(t, r)(E(r) · ∇ζ(r)).
(4.6)
We also have
̺(t) = U(t, s)(̺(s)) , ̺(t) p = ζ p , (4.7)
for all t, s. Furthermore, ̺(t) is non-negative, and if ζ is a projection, then so is ̺(t) for all t.
Remark 4.2. If the initial state ζ is of the form (4.1), then the hypotheses of Theorem 4.1 hold for any p > 0, provided ζ ω = P (EF ) ω that E F lies in a region of localization. This is true for suitable A ω , V ω and E F , by the methods of, for example, [CH,W,GK1,GK2,GK3,BoGK,AENSS,U,GrHK] and for the models studied therein as well as in [CH,GK3]. The bound E |x|ζ ω χ 0 2 < ∞ or equivalently ∇ζ ∈ L 2 (K ∞ ) is actually sufficient for our applications. For p = 1, 2, we refer to [BoGKS,Proposition 4.2] and [BoGKS,Lemma 5.4] for the derivation of these hypotheses from known results.
Proof of Theorem 4.1. Let us first define ̺(t, s) := U(t, s)(ζ(s)).
(4.8)
We get, as operators in M(K ∞ ),
∂ s ̺(t, s) = iU(t, s) ([H(s), ζ(s)]) + U(t, s) (E η (s) · ∇ζ(s)) = U(t, s) (E η (s) · ∇ζ(s)) ,(4.9)
where we used (3.27) and Lemma 3.4. As a consequence, with E η (r) = e ηr E(r),
̺(t, t) − ̺(t, s) = t s
dr e ηr U(t, r)(E(r) · ∇ζ(r)).
(4.10)
Since U(t, r)(E(r) · ∇(ζ(r)) p ≤ c d |E(r)| ∇ζ p < ∞, the integral is absolutely convergent by hypothesis on E η (t), and the limit as s → −∞ can be performed in L p (K ∞ ). It yields the equality between (4.5) and (4.6). Equality of (4.4) and (4.5) follows from Lemma 3.4 which gives
ζ = lim s→−∞ ζ(s) in L p (K ∞ ). (4.11)
Since U(t, s) are isometries on L p (K ∞ ), it follows from (4.4) that ̺(t) p = ζ p . We also get ̺(t) = ̺(t) * . Moreover, (4.4) with the limit in L p (K ∞ ) implies that ̺(t) is nonnegative.
Furthermore, if ζ = ζ 2 then ̺(t) can be seen to be a projection as follows. Note that convergence in L p implies convergence in M(K ∞ ), so that,
̺(t) = lim (τ ) s→−∞ U(t, s) (ζ) = lim (τ ) s→−∞ U(t, s) (ζ) U(t, s) (ζ) = lim (τ ) s→−∞ U(t, s) (ζ) lim (τ ) s→−∞ U(t, s) (ζ) = ̺(t) 2 . (4.12)
where we note lim (τ ) the limit in the topological algebra M(K ∞ ). To see that ̺(t) is a solution of (4.3) in L p (K ∞ ), we differentiate the expression (4.6) using (3.28) and Lemma 3.4. We get
i∂ t ̺(t) = − t −∞ dr e ηr [H(t), U(t, r) (E(r) · ∇ζ(r))]
(4.13)
= − H(t), t −∞
dr e ηr U(t, r) (E(r) · ∇ζ(r)) (4.14)
=
H(t), ζ(t) − t −∞ dr e ηr U(t, r) (E(r) · ∇ζ(r)) = [H(t), ̺(t)] . (4.15)
The integral in (4.13) converges since by (3.30) ,
[H(t), U(t, r) (E(r) · ∇ζ(r))] p ≤ 2C (H + γ)(E(r) · ∇ζ) p . (4.16)
Then we justify going from (4.13) to (4.14) by inserting a resolvent (H(t) + γ) −1 and making use of (3.29). It remains to show that the solution of (4.3) is unique in L p (K ∞ ). It suffices to show that if ν(t) is a solution of (4.3) with ζ = 0 then ν(t) = 0 for all t. We defineν (s) (t) = U(s, t)(ν(t)) and proceed by duality. Since p ≥ 1, with pick q s.t. p −1 + q −1 = 1. If A ∈ D (0) q , we have, using Lemma 2.5,
i∂ t T Aν (s) (t) = i∂ t T {U(t, s)(A)ν(t)} (4.17) = T {[H(t), U(t, s)(A)] ν(t)} + T {U(t, s)(A)L q (t)(ν(t))} = −T {U(t, s)(A)L q (t)(ν(t))} + T {U(t, s)(A) L q (t)(ν(t))} = 0 .
We conclude that for all t and A ∈ D
(0) q we have T Aν (s) (t) = T Aν (s) (s) = T {Aν(s)} . (4.18)
Thusν (s) (t) = ν(s) by Lemma 2.4, that is, ν(t) = U(t, s)(ν(s)). Since by hypothesis lim s→−∞ ν(s) = 0, we obtain that ν(t) = 0 for all t.
The current and the conductivity. The velocity operator v is defined as
v = v(A) = 2D(A), (4.19) where D = D(A) is defined below (3.6). Recall that v = 2(−i∇ − A) = i[H, x] on C ∞ c (R d ).
We also set D(t) = D(A + F η (t)) as in (3.13), and v(t) = 2D(t). From now on ̺(t) will denote the unique solution to (4.3), given explicitly in (4.6). If H(t)̺(t) ∈ L p (K ∞ ) then clearly D j (t)̺(t) can be defined as well by
D j (t)̺(t) = D j (t)(H(t) + γ) −1 ((H(t) + γ)̺(t)) ,(4.20)
since D j (t)(H(t) + γ) −1 ∈ K ∞ , and thus D j (t)̺(t) ∈ L p (K ∞ ).
Definition 4.3. Starting with a system in equilibrium in state ζ, the net current (per unit volume), J(η, E; ζ, t 0 ) ∈ R d , generated by switching on an electric field E adiabatically at rate η > 0 between time −∞ and time t 0 , is defined as
J(η, E; ζ, t 0 ) = T (v(t 0 )̺(t 0 )) − T (vζ) . (4.21)
As it is well known, the current is null at equilibrium:
Lemma 4.4. One has T (D j ζ) = 0 for all j = 1, · · · , d, and thus T (vζ) = 0.
Throughout the rest of this paper, we shall assume that the electric field has the form (4.22) where E ∈ C d gives the intensity of the electric in each direction while |E(t)| = O(1) modulates this intensity as time varies. As pointed out above, the two cases of particular interest are E(t) = 1 and E(t) = cos(νt). We may however, as in [KLM], use the more general form (4.23) for suitableÊ(ν) (see [KLM]). It is useful to rewrite the current (4.21), using (4.6) and Lemma 4.4, as
E(t) = E(t)E,E(t) = R cos(νt)Ê(ν)dν,J(η, E; ζ, t 0 ) = T {2D(0) (̺(t 0 ) − ζ(t 0 ))} (4.24) = −T 2 t0 −∞ dr e ηr D(0) U(t 0 , r) (E(r) · ∇ζ(r)) . = −T 2 t0 −∞
dr e ηr E(r)D(0) U(t 0 , r) (E · ∇ζ(r)) .
The conductivity tensor σ(η; ζ, t 0 ) is defined as the derivative of the function J(η, E; ζ, t 0 ) : R d → R d at E = 0. Note that σ(η; ζ, t 0 ) is a d×d matrix {σ jk (η; ζ, t 0 )}.
Definition 4.5. For η > 0 and t 0 ∈ R, the conductivity tensor σ(η; ζ, t 0 ) is defined as σ(η; ζ, t 0 ) = ∂ E (J(η, E; ζ, t 0 )) |E=0 , (4.25)
if it exists. The conductivity tensor σ(ζ, t 0 ) is defined by σ(ζ, t 0 ) := lim η↓0 σ(η; ζ, t 0 ) , (4.26)
whenever the limit exists.
4.3.
Computing the linear response: a Kubo formula for the conductivity. The next theorem gives a "Kubo formula" for the conductivity at positive adiabatic parameter.
Theorem 4.6. Let η > 0. Under the hypotheses of Theorem 4.1 for p = 1, the current J(η, E; ζ, t 0 ) is differentiable with respect to E at E = 0 and the derivative σ(η; ζ) is given by
σ jk (η; ζ, t 0 ) = −T 2 t0 −∞ dr e ηr E(r)D j U (0) (t 0 − r) (∂ k (ζ)) . (4.27)
The analogue of [BES,Eq. (41)] and [SB2, Theorem 1] then holds:
Corollary 4.7. Assume that E(t) = ℜe iνt , ν ∈ R, then the conductivity σ jk (η; ζ; ν) at frequency ν is given by
σ jk (η; ζ; ν; 0) = −T 2D j (iL 1 + η + iν) −1 (∂ k ζ)} , (4.28)
Proof of corollary 4.7. Recall (4.11) , in particular ζ = ζ 1 2 ζ 1 2 . It follows that σ(η; ν; ζ; 0) in (4.27) is real (for arbitrary ζ = f (H) write f = f + − f − ). As a consequence ,
σ(η; ν; ζ; 0) = −ℜT 2 t0 −∞ dr e ηr e iνr D j U (0) (t 0 − r) (∂ k (ζ)) .
(4.29)
Integrating over r yields the result.
Proof of Theorem 4.6. For clarity, in this proof we display the argument E in all functions which depend on E. From (4.24) and J j (η, 0; ζ, t 0 ) = 0 (Lemma 4.4), we have
σ jk (η; ζ, t 0 ) = − lim E→0 2T t0 −∞ dr e ηr E(r)D E,j (0)U(E, 0, r) (∂ k ζ(E, r)) . (4.30)
First understand we can interchange integration and the limit E → 0, and get
σ jk (η; ζ, t 0 ) = −2 t0 −∞ dr e ηr E(r) lim E→0 T {D j (E, 0)U(E, 0, r) (∂ k ζ(E, r))} . (4.31)
The latter can easily be seen by inserting a resolvent (H(t) + γ) −1 and making use of (3.29), the fact that H∇ζ ∈ L 1 (K ∞ ), the inequality : |T (A)| ≤ T (|A|) and dominated convergence. Next, we note that for any s we have Recalling that D j,ω (E, 0) = D j,ω − F j (0) and that ∂ k ζ(E, r) 1 = ∂ k ζ 1 < ∞, using Lemma 2.6,
lim E→0 T {D j (E, 0)U(E, 0, r) (∂ k ζ(E, r))} = lim E→0 T {D j U (E, 0, r)(∂ k ζ)U (E, r, 0)} = lim E→0 T D j U (E, 0, r)(∂ k ζ)U (0) (r) ,(4.34)
where we have inserted (and removed) the resolvents (H(E, r)+γ) −1 and (H +γ) −1 .
To proceed it is convenient to introduce a cutoff so that we can deal with D j as if it were in K ∞ . Thus we pick f n ∈ C ∞ c (R), real valued, |f n | ≤ 1, f n = 1 on [−n, n], so that f n (H) converges strongly to 1. Using Lemma 2.6, we have
T D j U (E, 0, r)(∂ k ζ)U (0) (r) = lim n→∞ T f n (H)D j U (E, 0, r)(∂ k ζ)U (0) (r) = lim n→∞ T U (E, 0, r) ((∂ k ζ)(H + γ)) U (0) (r)(H + γ) −1 f n (H)D j = T U (E, 0, r) ((∂ k ζ)(H + γ)) U (0) (r)(H + γ) −1 D j ,(4.35)
where we used the centrality of the trace, the fact that (H + γ) −1 commutes with U (0) and then that (H + γ) −1 D j ∈ K ∞ in order to remove to limit n → ∞. Finally, combining (4.34) and (4.35), we get
lim E→0 T {D j (E, 0)U(E, 0, r) (∂ k ζ(E, r))} (4.36) = T U (0) (−r) ((∂ k ζ)(H + γ)) U (0) (r)(H + γ) −1 D j = T D j U (0) (−r)(∂ k ζ) .
(4.37)
The Kubo formula (4.27) now follows from (4.31) and (4.37).
4.4. The Kubo-Stȓeda formula for the Hall conductivity. Following [BES, AG], we now recover the well-known Kubo-Stȓeda formula for the Hall conductivity at zero temperature (see however Remark 4.11 for AC-conductivity). To that aim we consider the case E(t) = 1 and t 0 = 0. Recall Definition 4.5. We write
σ (E f ) j,k = σ j,k (P (EF ) , 0) , and σ (E f ) j,k (η) = σ j,k (η; P (EF ) , 0) . (4.38)
Theorem 4.8. Take E(t) = 1 and t 0 = 0. If ζ = P (EF ) is a Fermi projection satisfying the hypotheses of Theorem 4.1 with p = 2, we have
σ (EF ) j,k = −iT P (EF ) ∂ j P (EF ) , ∂ k P (EF ) , (4.39)
for all j, k = 1, 2, . . . , d. As a consequence, the conductivity tensor is antisymmetric; in particular σ (EF ) j,j = 0 for j = 1, 2, . . . , d.
Clearly the direct conductivity vanishes, σ (EF ) jj = 0. Note that, if the system is time-reversible the off diagonal elements are zero in the region of localization, as expected.
Corollary 4.9. Under the assumptions of Theorem 4.8, if A = 0 (no magnetic field), we have σ (EF ) j,k = 0 for all j, k = 1, 2, . . . , d.
We have the crucial following lemma for computing the Kubo-Stȓeda formula, which already appears in [BES] (and then in [AG]).
Lemma 4.10. Let P ∈ K ∞ be a projection such that ∂ k P ∈ L p (K ∞ ), then as operators in M(K ∞ ) (and thus in L p (K ∞ )),
∂ k P = [P, [P, ∂ k P ]] .
(4.40)
Proof. Note that ∂ k P = ∂ k P 2 = P ∂ k P + (∂ k P )P so that multiplying left and right both sides by P implies that P (∂ k P )P = 0. We then have, in L p (K ∞ ),
∂ k P = P ∂ k P + (∂ k P )P = P ∂ k P + (∂ k P )P − 2P (∂ k P )P = P (∂ k P )(1 − P ) + (1 − P )(∂ k P )P = [P, [P, ∂ k P ]] .
Remark that Lemma (4.10) heavily relies on the fact P is a projection. We shall apply it to the situation of zero temperature, i.e. when the initial density matrix is the orthogonal projection P (EF ) . The argument would not go through at positive temperature.
Proof of Theorem 4.8. We again regularize the velocity D j,ω with a smooth function f n ∈ C ∞ c (R), |f n | ≤ 1, f n = 1 on [−n, n], but this time we also require that f n = 0 outside [−n − 1, n + 1], so that f n χ [−n−1,n+1] = f n . Thus D j f n (H) ∈ L p,o (K ∞ ), 0 < p ≤ ∞. Moreover f n (H)(2D j )f n (H) = f n (H)P n (2D j )P n f n (H) = −f n (H)∂ j (P n H)f n (H) (4.41) where P n = P 2 n = χ [−n−1,n+1] (H) so that HP n is a bounded operator. We have, using the centrality of the trace T , that It follows from the spectral theorem (applied to L 2 ) that lim η→0 (L 2 + iη) −1 L 2 = P (Ker L2) ⊥ strongly in L 2 (K ∞ ) , (4.50)
where P (Ker L2) ⊥ is the orthogonal projection onto (Ker L 2 ) ⊥ . Moreover, as in [BoGKS] one can prove that P (EF ) , ∂ j P (EF ) ∈ (Ker L 2 ) ⊥ .
(4.51)
Combining (4.49), (4.50), (4.51), and Lemma 2.9, we get σ (EF ) j,k = i P (EF ) , ∂ j P (EF ) , ∂ k P (EF )
L 2 = −iT P (EF ) ∂ j P (EF ) , ∂ k P (EF ) ,
which is just (4.39).
Remark 4.11. If one is interested in the AC-conductivity, then the proof above is valid up to (4.49). In particular, with E(t) = ℜe iνt , one obtains σ (EF ) jk (η) = −ℜ i (L 2 + ν + iη) −1 L 2 P (EF ) , ∂ j P (EF ) , ∂ k P (EF ) L 2
.
(4.52)
The limit η → 0 can still be performed as in [KLM,Corollary 3.4]. It is the main achievement of [KLM] to be able to investigate the behaviour of this limit as ν → 0 in connection with Mott's formula.
Lemma 3. 1 .
1[BoGKS, Lemma 2.6] Let G(t) be as in (3.7). ThenG(t)D = D ,(3.11)H(t) = G(t)HG(t) * , (3.12) D(A + F η (t)) = D(A) − F η (t) = G(t)D(A)G(t) * . (3.13)Moreover, i[H(t), x j ] = 2D(A + F η (t)) as quadratic forms on D ∩ D(x j ), j = 1, 2, . . . , d.
(t, s) extends a linear operator on M(K ∞ ), leaving invariant M(K ∞ ) andL p (K ∞ ), p ∈]0, ∞], with U(t, r)U(r, s) = U(t, s) , (3.24) U(t, t) = I , (3.25) {U(t, s)(A)} * = U(t, s)(A * ) .(3.26)Moreover, U(t, s) is to a unitary on L 2 (K ∞ ) and an isometry in L p (K ∞ ), p ∈ [1, ∞].In addition, U(t, s) is jointly strongly continuous in t and s on L p
lim E→0 G
E→0(E, s) = I strongly in L 1 (K ∞ ) , (4.32) which can be proven by a argument similar to the one used to prove Lemma 3.4. Along the same lines, for B ∈ K ∞ we have lim E→0 G(E, s)(B ω ) = B ω strongly in H, with G(E, s)(B) ∞ = B ∞ . (4.33)
( 0 )P
0(r)(f n (H)2D j,ω f n (H))∂ k P(EF ) . (4.43)Using Lemma 2.9 and applying Lemma 4.10 applied to P = P (EF ) , it follows thatT U (0) (r)(f n (H)2D j f n (H))∂ k P (EF ) (4.44) = T U (0) (r)(f n (H)2D j f n (H)) P (EF ) , P (EF ) , ∂ k P (EF ) = T U (0) (r) P (EF ) , P (EF ) , f n (H)2D j f n (H) ∂ k P (EF ) , = −T U (0) (r) P (EF ) , f n (H) P (EF ) , ∂ j (HP n ) f n (H) ∂ k P (EF ) ,where we used that P (EF ) commutes with U (0) and f n (H), and (4.41). Now, as elements in M(K ∞ ), P (EF ) , ∂ j HP n = HP n , ∂ j P (EF ) .(4.45)Since [H, ∂ j P (EF ) ]] is well defined by hypothesis, f n (H) HP n , ∂ j P (EF ) f n (H) converges in L p to the latter as n goes to infinity. Combining (4.43), (4.44), and (4.45), we get after taking n → ∞,σ (EF ) jk (r) = −T U (0) (r) P (EF ) , H, ∂ j P (EF ) ∂ k P (EF ) . (EF ) , H, ∂ j P (EF ) = H, P (EF ) , ∂ j P (EF ) , ) = −T U (0) (r) H, P (EF ) , ∂ j P (EF ) ∂ k P (EF ) = − e −irL L 2 P (EF ) , ∂ j P (EF ) , ∂ k P (EF ) L 2 ,(4.48)where A, B L 2 = T (A * B). Combining (4.27), (4.42), and (4.48), we get σ (EF ) jk (η) = − i (L 2 + iη) −1 L 2 P (EF ) , ∂ j P (EF ) , ∂ k P (EF ) L 2 .(4.49)
The AC-conductivity may be better defined using the from (4.23) as argued in[KLM].
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Site de Saint-Martin, 2 avenue Adolphe Chauvin, F-95302 Cergy-Pontoise, France E-mail address: [email protected] (Germinet). CNRS UMR. 8088Université de Cergy-Pontoise ; CNRS UMR 8088, Laboratoire AGM, Département de Mathématiques ; Université de Cergy-PontoiseUniversité de Cergy-Pontoise, CNRS UMR 8088, Laboratoire AGM, Département de Mathématiques, Site de Saint-Martin, 2 avenue Adolphe Chauvin, F- 95302 Cergy-Pontoise, France E-mail address: [email protected] (Germinet) Université de Cergy-Pontoise, CNRS UMR 8088, Laboratoire AGM, Département de Mathématiques, Site de Saint-Martin, 2 avenue Adolphe Chauvin, F-95302 Cergy- Pontoise, France E-mail address: [email protected]
| []
|
[
"Novel Mechanism to Defend DDoS Attacks Caused by Spam",
"Novel Mechanism to Defend DDoS Attacks Caused by Spam"
]
| [
"Dhinaharan Nagamalai \nDepartment of Computer Engineering\nWireilla Net Solutions Inc\nChennaiIndia\n",
"Cynthia Dhinakaran \nHannam University\nSouth Korea\n",
"Jae Kwang Lee \nHannam University\nSouth Korea\n"
]
| [
"Department of Computer Engineering\nWireilla Net Solutions Inc\nChennaiIndia",
"Hannam University\nSouth Korea",
"Hannam University\nSouth Korea"
]
| [
"International Journal of Smart Home"
]
| Corporate mail services are designed to perform better than public mail services. Fast mail delivery, large size file transfer as an attachments, high level spam and virus protection, commercial advertisement free environment are some of the advantages worth to mention. But these mail services are frequent target of hackers and spammers. Distributed Denial of service attacks are becoming more common and sophisticated. The researchers have proposed various solutions to the DDOS attacks. Can we stop these kinds of attacks with available technology? These days the DDoS attack through spam has increased and disturbed the mail services of various organizations. Spam penetrates through all the filters to establish DDoS attacks, which causes serious problems to users and the data. In this paper we propose a novel approach to defend DDoS attack caused by spam mails. This approach is a combination of fine tuning of source filters, content filters, strictly implementing mail policies, educating user, network monitoring and logical solutions to the ongoing attack. We have conducted several experiments in corporate mail services; the results show that this approach is highly effective to prevent DDoS attack caused by spam. The novel defense mechanism reduced 60% of the incoming spam traffic and repelled many DDoS attacks caused by spam. | null | [
"https://arxiv.org/pdf/1012.0610v1.pdf"
]
| 6,894,480 | 1012.0610 | d15698916652e8928f5fcc9ff718f13773ca3685 |
Novel Mechanism to Defend DDoS Attacks Caused by Spam
July, 2007
Dhinaharan Nagamalai
Department of Computer Engineering
Wireilla Net Solutions Inc
ChennaiIndia
Cynthia Dhinakaran
Hannam University
South Korea
Jae Kwang Lee
Hannam University
South Korea
Novel Mechanism to Defend DDoS Attacks Caused by Spam
International Journal of Smart Home
12July, 200783
Corporate mail services are designed to perform better than public mail services. Fast mail delivery, large size file transfer as an attachments, high level spam and virus protection, commercial advertisement free environment are some of the advantages worth to mention. But these mail services are frequent target of hackers and spammers. Distributed Denial of service attacks are becoming more common and sophisticated. The researchers have proposed various solutions to the DDOS attacks. Can we stop these kinds of attacks with available technology? These days the DDoS attack through spam has increased and disturbed the mail services of various organizations. Spam penetrates through all the filters to establish DDoS attacks, which causes serious problems to users and the data. In this paper we propose a novel approach to defend DDoS attack caused by spam mails. This approach is a combination of fine tuning of source filters, content filters, strictly implementing mail policies, educating user, network monitoring and logical solutions to the ongoing attack. We have conducted several experiments in corporate mail services; the results show that this approach is highly effective to prevent DDoS attack caused by spam. The novel defense mechanism reduced 60% of the incoming spam traffic and repelled many DDoS attacks caused by spam.
Introduction
Email is a source of communication for millions of people world wide [8]. But spam is abruptly disturbing the email users by eating their resource, time & money. In Internet community the spam has always been considered as bulk and unsolicited. Spam mails accounts for 80% of the entire mail traffic. Many researchers have proposed different solutions to stop the spam. But the effort has become a drop of water in the ocean. No matter how hard, spammers always find new ways to deliver spam mail to the user's inbox. Of late the spammers target the mail servers to disturb the activities of organizations which results in economic and reputation loss. The DDoS attack is a common mode of attack to cripple the particular server. The spammers take DDoS attack in their arms to disturb the mail servers. This paper is going to study the DDoS attacks through spam mails. We proposed a multi layer approach to defend the DDoS attack caused by spam mails. We implemented this methodology in our mail system and monitored the results. The result shows that our approach is very effective to defend DDoS attack caused by spam.
E-mail life cycle : The composed mail in the source machine will be handover to the Message Transfer Agent (MTA). The MTA will find the destination machine with the help of DNS server and relay the mail to the destination systems MTA [14]. The MTA at the destination machine delivers the mail to the destination user's mail box. The machines between source and destination will act as intermediate machines for the data transfer called relay. MTA relay mail between each other using the Simple Mail Transfer Protocol.
Most of the corporate mail services are pretty faster and sophisticated than other free mail services. The corporate mail services deliver mails faster and it provides a facility to attach bigger size files and unlimited storage facilities. To deliver mails faster, the server generally jumps most of the time consuming spam protection tests. To attach the big size files it has to bypass the content filter settings. This makes the corporate mail servers vulnerable to spam mail which ultimately causing DDOS attacks. We proposed a multi layer approach to defend the DDoS attack caused by spam mails. We implemented this methodology in our mail system and monitored the results. The result shows that our approach is very effective to defend DDoS attack caused by spam.
The rest of the paper is organized as follows. Section 2 discusses related work. Section 3 provides background on mechanism of DDOS attack through spam and the effects. In section 4, we describe our mechanism to defend the attack. Section 5 provides data Collection and experimental results. We conclude in section 6.
Related work
In [1] luiz, Crsitino presented an extensive study on characteristics of spam traffic in terms of email arrival process, size distribution, the distributions of popularity and temporal locality of email recipients etc., compared with legitimate mail traffic. Their study reveals major differences between spam and non spam mails. In [2] examines the use of DNS black lists. They have examined seven popular DNSBLs and found that 80% of the spam sources are listed in some DNSBL. [4] Presented a comprehensive study of clustering behavior of spammers and group based anti spam strategies. Their study exposed that the spammers has demonstrated clustering structures. They have proposed a group based anti spam frame work to block organized spammers. [5] Presented a network level behavior of spammers. They have analyzed spammers IP address ranges, modes & characteristics of botnet. Their study reveals that blacklists were remarkably ineffective at detecting spamming relays. Their study states that to trace senders the internet routing structure should be secured. [9] Presented a comprehensive study of spam and spammers technology. His study reveals that few work email accounts suffer from spam than private email. To the best of our knowledge our study is a first paper, comprehensively studying the DDoS attacks by spam.
Mechanism of DDoS attacks through spam
Distributed Denial of Service (DDoS) attack is a large scale, coordinated attack on the availability of services at a victim system or network resource [3]. DDOS attack through spam mail is one of the new versions of common DDoS attack. In this type, the attacker penetrates the network by a small program attached to the spam mail. After the execution of the attached file, the mail server resources will be eaten up by mass mails from other machines in the domain results denial of services. The working scenario of this attack is explained in fig 1. The attackers take maximum effort to pass through the spam filters and deliver the spam mail to the user's inbox. Here the hackers are doing enough to make the mail recipient to believe that the spam mail is from the legitimate user. The attackers use fake email ids from victim domains to penetrate the network. The spam mail had sent in the name of Network administrator/well wisher of the victim or boss of the organization. Note that the spam mail does not have the signature.
The spam contains small size of .exe file as an attachment (for example update.exe). The attackers used double file extension to confuse the filter (Update_KB2546_*86.BAK.exe (140k)) and user. The attachment size ranges from 140 to 180 KB. Mostly the spam mail asks the recipient to execute the .exe file to update anti virus software. Upon execution of the attachment, it will drop new files in windows folder and change the registry file, link to the attacker's website to download big programs to harm the network further. The infected machine collected email addresses through windows address book and automatically send mails to others in the same domain. Even if the users don't use mail service programs like Outlook express and others, it will send mails by using its own SMTP. Mostly this kind of spam mail attracts the group mail ids, and will send mails to groups. By sending mails to the group, it will spread the attack vigorously. If any of the users forward this mail to others it will worsen the situation. Ultimately the server will receive enormous request from others beyond its processing capacity. In this way it will spread the attack and results in a DDoS attack. After the first mail, for every minute it will send same kind of mail with different subject name & different contents to the group email ids. With in a day it will eat up server resources and end up in distributed denial of service attack. The names of the worms used in these kind of DDoS attacks are WORM_start.Bt,WORM_STRAT.BG,WORM_STRAT.BR, TROJ_PDROPPER.Q. Upon execution, these worms dropped files namely serv.exe, serv.dll, serv.s, serv.wax, E1.dll, rasaw32t.dll etc.
DDoS malware cause direct and indirect damage by flooding specific targets [14]. Mass mailers and network worms cause indirect damage when they clog mail servers and network bandwidth. In Network, It will consume the network bandwidth and resources, causing slow mail delivery further resulting Denial of service. The server will be down due to enormous request from clients and bulk mail processing.
Proposed Defense Mechanism
We proposed a multi layer approach to defend the DDoS attack caused by spam mails [ Figure 2]. We implemented this approach in our mail system and monitored the results. The result shows that our approach is very effective. The approach has six layers as shown in Figure4. This approach is a combination of fine tuning of source filters, content filters, network monitoring policy, general email policies, educating the user & timely logical solutions of the network administrator. Fine tuning of source filters reject the incoming connections before the spam mail delivery. The content filters analyses the contents of the mails and blocks the incoming unwanted mails. Network monitoring approach provides general solution to identify the attacks prior to the attack and also during the attack. Business houses should educate the user about possible attack scenarios & reacting ways to it. The logical solutions of the network administrator play an important role during the attack period and even post attack period. The combination of these layers provides best methodology to stop the DDoS attacks established though spam mails.
Source Filters
Currently, There is a prediction that the spam will be 70% of the email traffic in 2007 [1]. There are lot of source filters are available in real time. But by simply enabling all the filters will not help to prevent the attacks. It will slow down the mail delivery process. So the fine tuning of filter is an important to handle the attacks. The figure .3 shows the structure of the filters.
Bayesian Filter: Bayesian filtering is one of the effective filtering technologies used by most of the antispam software developers [9]. This filter works based on the mathematical theorem of Bayes a British mathematician. Anti spam developers have developed various algorithms by modifying the Bayes theorem to effectively filter the spam. Anti spam companies have developed various algorithms by modifying the Bayes theorem to effectively filter the spam. In Bayes methodology, the system develops two tables from the contents of incoming spam mail & out bound legitimate mails. The tables referred as a dictionary. Each word from an incoming new mail will be compared to the spam mail table and legitimate mail table or dictionary. For incoming mail words, the probability value is calculated based on the number of occurrences of particular word in spam mail table & legitimate mail table. For example the word "Viagra" occurs 400 times in 3000 spam mails and in 5 out of 300 legitimate mails the probability would be .889 [400/3000]/ [5/300+400/3000]. It will perform the same operation for all words in incoming mails. The mail is classified as spam, if the calculated probability is higher than a given threshold mostly above 0.5. The normal threshold value ranges from .7 and above for a corporate mail server. Based on the threat level the administrator can change the threshold value. After the classification of the mail as a spam, the contents of the spam will be added to the dictionary. It will be useful for the future calculation. In this way the system will learn the latest technologies used by the spammers. The administrator can enable or disable the learning from the spam and outbound mails. Most of the filters offer an administrator to select the number of words for dictionary. We recommend 50000 words are recommended for small & medium size mail server. If you increase the dictionary size, the lookup time or processing time will increase causing the delay in mail delivery. The system will take two weeks to build a spam word table but some of the filters use static tables. Bayesian filter is the most important and successful antispam method [9].The spammers learned how to pass this filter to deliver spam into inbox even it is powered with learning. Enhancements may be useful [10] but an ever increasing the quality of filter is impossible since the dictionary size & the content of the mail is limited. "Mail, server, report, firewall, virus, windows, customer, support" are some of the common tokens listed in legitimate mail table and also used by the attacker in a given sample mail. The server will send a regular report titled "server report" to the network administrator. So the probability of being a spam for this mail is very less results safe delivery to the user's inbox. After the attack the system will add these tokens to the spam tokens table. Meantime the spammers will come with new techniques. The legitimate mail word table is a standard one not enables for learning. The hackers can get the data and use it to bypass the filter. Bayesian filtering is effective but it can not filter 100% of the spam. With the combination of various filters Bayesian Filter will work more effective and overall performance of spam filtering will be increased.
DNSBL:
Even though the spam generation is not accepted widely as a legal actively, 80% of the spam mail is generated by particular users. If we have the list of these spam generators IP addresses, we can effectively block the spam messages. DNSBL is based on the concept above said. DNS black hole list or black list is a well defined source filtering technology it works before delivering the mail to the user's inbox. The list publishes the list of IP addresses through DNS of massive spam generators. Lot of DNSBLs offers various list of IP addresses based on open relay, spam or virus source. The most widely used DNSBLs are spamhaus, spamcop,sorbs, abuseat,dsbl, rfc-ignorant etc., these DNSBLs list out thousands of IP addresses of spam generators. By blocking this well known IP addresses to deliver message we can effectively block the incoming spam traffic. There might be a overlapping of IP addresses in various Black lists. Legitimate Mail delivery will be delayed, if all the available DNS black lists included in the DNSBL settings to filter the spam. The IP address of the black listed IP addresses will change frequently based on their spam generating behaviors. Some DNSBLs will check the particular IP address regularly; if they stop the spamming activity, it will remove the particular IP address and add the new IP addresses of spammers [2].
If you enable "immediately reject the connections" from blacklisted server option, the connection will not be established to the particular spammer. Most of the mail servers provide the option to include more lists or delete the DNSBLs from the list. But the remaining 20% of the spam mail generators are not listed by any DNSBLs, still you have to depend on other filters or methodologies to stop the spam. Still there is no single Blacklist with all the complete spam generating IP addresses since the spammers will change their IP addresses frequently. Moreover these list providers are frequent target to hackers. The spammers used Mimail.E worm to perform Dos attack on spamhaus site. In 2003, Spamhaus servers came under distributed Denial of Service (DDoS) attacks by thousands of virus-infected computers throughout the Internet [20]. In 2006 also the spamhaus servers are out of service due to DDoS attacks [25]. It is clear that the angry spammers are trying to stop the services of DNSBLs. These attacks clearly show the use more than one DNSBLs in the List. Even if one DNS black list is out of service the mail server can manage with other lists. In recent days the DNSBL lookups are increased tremendously of total DNS lookups compared to 5 years before [1]. Nearly 80% of the spam generated by relays that appear in one at least one of eight major blacklists [4]. Fine tuning of multiple black lists is more effective than simply using all lists. The DNSBLs is not effective when the spam is being sent from larger set of IP addresses [2]. SURBL: SURBL Searches for URLs in incoming mails. SURBL is a collection of spam supported websites, domains, web servers. If there is any URL or IP address in the message, the system will contact the SURBL list to check whether the URL is listed. If the URL is listed in SURBLs, it blocks the messages. The available SURBL lists are sc.surbl.org, ws.surbl.org, ob.surbl.org, ab.surbl.org. multi.surbl.org is a combination of all the lists. If the system uses other SURBLs with multi.surbl.org, it will take long time to process the mail. If use only multi.surbl.org for SURBL check, and if the service is not available, no checks will be performed. We recommend using other four surbls rather than multi.surbl.org. The administrator can edit the list whenever a high rate of false positive is present [17].
In the mentioned DDoS attack through spam [in section 2], the worm downloaded malicious code from the following websites.
If SURBL was enabled, there was less possibility of the attack. This kind of URL based filter is very effective against the DDoS attack since these references are faked websites. Some attacker includes multi URLs to confuse the filters. For multi domain messages, it is hard to determine the real spam domain among all the domains [10]. The combination of checking SURBL database with other filters is a best way to defend the DDoS attacks. [23]. Sender address forgery is a big threat to the users as well as the entire network. In the attack mentioned in the section 2, all the users received mails from the unknown person within their organization. The attacker's mail id is a fake, it has victims domain name. That is why most of the users obeyed the instruction and executed the file attachment leading to the DDoS attack. We can stop this kind of forgery by SPF (Sender Policy Framework).The current version of SPF -called SPFv1 or SPF Classic [13]. The Sender Policy Framework allows you to check whether a particular email sender is forged or not. Most of today's spammers use forged email addresses to hide their identity. SPF requires that the organization of the sender has published its mail server in an SPF record. If you receive a mail from a user, you can check that mail is coming from the particular organization by the sender's IP address. The SPF record will inform the receiver whether the user is allowed to use their network or not [15]. If the organization recognizes particular machine, it passes the test. Otherwise it is an attacker or a spammer. There are to kind of fails like fail and soft fail.
Sender Policy Framework: Sender Policy Framework reject message if SPF test is fail or soft fail
Grey Listing: Grey listing is a simple technique to fight against spam [18]. It will reject all incoming mails from unfamiliar IP addresses with an error code. The mail server records the combination of sender, recipient id & IP address. If the same sender is trying to send the mail after 10 seconds to 12 hours, the server will check for the combination in its record, if it matches, it will allow the sender to deliver the message. This is based on assumption that the spammers will not try again but legitimate users. But spammers learned this technology & how to bypass. But results show that there is substantial reduction of spam after the implementation of grey listing. The old version of Grey list used to accept the second mail after 4 hours [19]. But the legitimate user faces delay in mail delivery.
Reverse DNS: The incoming system should have rDNS ie. The sending system should give domain name and IP address to prove that is from the legitimate user. Most of the spam doesn't have reverse DNS [12]. Rejecting all incoming mails without rDNS is an effective way to filter the spam. "Reject message if sending server IP does not have a reverse DNS entry", "Reject message if the reverse DNS entry does not match Helo host" are two options supported by most mail services. SPF & rDNS are useful to filter the spam into some extends.
Content Filters
Once cleared from the SMTP server, the sender is allowed to deliver the message headers and body of the mail [12]. By carefully checking each and every word of the header and contents still we can block the spam. Most spam headers try to confuse the filters. Spammers will use recognizable words as a subject and clear from address. If the incoming mail has particular content or subject, the content filter will stop the mail delivery. Most of the spam caused DDOS attack has subjects like test, server report, status, helo etc; In this case the attacker carefully selected the words to avoid the content filtering. "Server report" is a word used by servers to send report to the administrator. The content filter blocks the mail which has some specific words like Viagra, ViAgRa, install updates, customer support service etc., Multiple words separated by comma, space are allowed in content filters to search the mail contents .
Most of the manuals say that the blocking of particular id & IP address is not useful [26]. But such activity highly reduces the spam mail delivery. Even the entire domain blocking is highly advisable to stop further spam delivery or attack. But in this DDoS attack, further attacks will be from its own domain mail ids except the first mail. Blocking own mail id is highly impossible. Another option of content filter is if you know your regular contacts, you can block all other mail senders. We can block all mails not send by particular user. If the administrator doesn't want to receive from other mail ids rather than own domain ids, he can block all the incoming and outgoing mails by enabling option "Block incoming message not sent by ". Content filters allow the admin to block all the mails with big size attachments. For example he can block all incoming mails with the attachment size is more than 100 KB. In this case the attacker can not deliver worms which are bigger in size. This will completely block the DDOS attack through this way. Moreover chances of DDoS attack caused by dumping of larger size files will be eliminated at greater extend. Denial of service will be delayed from starting time of the attack; meantime the admin can take other steps to defend the DDoS attack.
The content filters can block the mails with particular type of files as an attachment [12]. Most of these worms have .exe as an extension. If the filters block all the incoming mails with attachment of exe files, can stop the incoming DDOS attackers spam. But these days the spammers learnt this and started to send the attachments with double extension like update.doc.exe, update.txt.exe etc. It will confuse the filter and will be delivered to mailbox. To avoid this, content filter is providing the options like *.*.exe. By this kind of technique the content filters are helpful to defend the DDoS attacks through spam. Before the attack, the content filters can not identify the contents, headers, subject, and the attachment size & file extension of the incoming mails. Anyway this spam will take some time to result the DDoS attack. After the first one or two mails the administrator can identify these item and block by using content filters. The experience of the administrators can play a wide role in this filter. While all of the filters are effective to some extends, a combination of these filters will effectively stop the DDoS attacks through spam mails. Even if it is pass through one filter, it will be blocked by another one. After the initial attack the combination of content filters play an important role to defend the DDoS attacks resulted by spammer..
Policies
Mail is the primary source of communication between all employees at an organization. Therefore it is appropriate that an email-etiquette be established to distinguish between what is Push vs. Pull information. As any organization of any size, it needs an agreed upon system of sending, sorting and utilizing files in their mail server. The type and number of emails / files sent via mail has increased exponentially over the past few years. If the server reaches its capacity levels that cause significant delay in email ultimately results DOS. The policy helps to avoid the DDoS attack kind of situations.
The users required to use signatures or set default signatures to all ids in the domain. So the users can easily differentiate spam and legitimate mail. Mostly the attackers' mail will not contain any signatures. Encourage user to use specific words in certain place. For example if the organization is religious organization, force all the users to use particular word to identify the mail is from legitimate user. For example if all the users are using "In His Service" instead of "regards" at the end of the mail. The attackers can not identify such words to make the user to believe the mail is from legitimate user. Since most of the DDOS attacks through the spam mail takes an extra effort to make the user to believe the mails from legitimate user. Mostly the attack mails sent in the name of network administrator or head of the Institute. These unique code words will separate the spam from legitimate mails. The user will not open the mail or execute the attached file. It will help the network to fight against DDoS attacks through spam effectively.
Educate the user
The user's action during the attack or before the attack plays an important role to defend the DDoS attacks. So the users need to be educated how to behave generally and during the attack. The users have to be educated about spam mails and DDOS attacks. The users should be asked not to open or reply or forward or any kind of activities to the mails from unknown users. The user should inform the network administrator, if they responded to the spam in any method. The user can choose to flag spam so that the server knows to block it. The users should be asked not to use their work email addresses when registering in news groups and others. They should also be asked not to run any exe file or any file sent by email. Automatically deleting spam after particular day should be implemented. If not user should be advised to clean up their spam regularly. After the attack if spam mail exists with DDoS attack weapon, by error it can reappear and results DDoS attack. So the users should clear their old mail and spam regularly.
Monitoring the Network
To defend network against DDoS attack through spam requires real time monitoring of network wide traffic to obtain timely and important information. Monitoring the performance of network plays an important role to avert the DDOS attack [14]. Unusual activities can be detected, if the network is monitored by 24*7. If the speed of the mail service is down, we can assume that the server is processing a bulk data. Even the heavy regular network traffic causes the congestion; the administrator can regulate the data flow by his regular procedures to increase the speed. But during the attack, the net admin can not ease the data flow by his regular practices. It indicates that there is something wrong in the network. If the DDoS attack takes place automatically the mail server's speed will go down. Continuously monitoring the network performance is a useful practice to defend the attack. Two mail servers were monitored simultaneously with minimum of 200 mail users. In one domain the attacker launched DDoS attack through spam, because of continuous monitoring the net administrator marked the mail as a spam and deleted before spread to others. Absolutely the domain was escaped from the attack. In another domain for experiment, the spam was not marked and allowed to user ' s inbox. Some of the users responded to the spam by forwarding and executing the mail. Since this attack targeted only group ids, the mail server was out of service with in a day. Maintaining the history of network activities and network problems are useful to handle the current situation. Experiences are highly useful to design a solution to DDoS attacks through spam.
Logical solutions
Any attack can be handled with minimum impact by the network administrator's skills. After the attack, shutting down the server is not useful. The ways should be identified to change the path of the data dumping. The DDoS attack through spam mail targeted only group ids. So the mail service will become out of service very soon. But the wise net administrator can change all the group ids to new ids. For example [email protected] can be changed in to [email protected]. These group mail ids are converted into private users and not for public users. So the attacker is not allowed to send more mails. Since the incoming spam has diverted, all the spam mails stopped immediately. But already infected machines will give trouble to the particular users. The infected machines need to be removed from the network. In order to view the impact of the attack, these machines have to be analyzed. After the removal of worms from these machines, they can be allowed to join the network. There will be a logical solution to every attack, no need to be panic.
Data Collection and Results
We have conducted several experiments to measure the effectiveness of DNSBL, SURBL and the proposed defense mechanism. The test was conducted on client computers connected through local area network. The web server provides service to 200 users with 20 group email IDs and 200 individual mail IDs. The speed of the Internet connection is 100 Mpbs for the LAN, with 20 Mbps upload and download speed (Due to security and privacy concerns we are not able to disclose the real domain name). Our dataset consists of the spam mails collected at a large spam trap. The trap is a collection of spam mails filtered by source, content filters, and other settings mentioned in this paper.
Figure3. Effectiveness of DNSBL
Figure6. Defense Mechanism effects
Several experiments were conducted to measure the effectiveness of the proposed defense mechanism. We observed the system for six months continuously. Our dataset consists of the spam mails collected at a large spam trap. The graph shows the number of spam received before and after implementing the defense mechanism. We have selected five sessions of data to display. As shown in Figure6, a session holds good for three hours. The graph shows that after implementing the defense mechanism the incoming spam has reduced by 50% to 60%. Our corporate mail service did not face any DDoS attack for past six months. We have observed that the individual users are not receiving more spam like before implementing the defense mechanism.
Conclusion
In this paper we have proposed a multi layer defense mechanism to defend the mail services from DDoS attacks caused by spam. Experimental results show that this system is highly effective and the mail service experiencing strong protection against DDoS attacks caused by spam. There is no single step solution to the DDoS attacks established through spam mails. Simply using various filters doesn't help to stop the possible DDoS attacks caused by spam. But fine tuning of filters mentioned in our mechanism prevented DDoS attacks through spam. The content filters clogged the attack by filtering the spam with unwanted contents and programs. Continuous monitoring of the network averted possible attacks and gave enough time to defend the attacks. Since the educated users are responding well to this kind of attacks, the attacks avoided in an efficient way. The policies prevented the spam mails by utilizing policies of using signatures, no bulk mails, and the limitations of usage of group ids. Last but certainly not the least, the logical solutions to these attacks plays an important role to stop the attacks. The experiments show the effectiveness of SURBL to filter the spam. The experimental results show that there is 60% of reduction in spam traffic after implementing the defense mechanism. Also we didn't face DDoS attack through spam for past six months.
Figure 1 .Figure 2 .
12Defense Combination of Source Filters
Several experiments were conducted to measure the effectiveness of DNSBL. Relays.ordb.org, bl.spamcop.net, sbl.spamhaus.org is a good combination to effectively filter the spam. For continues seven days we have included relays.ordb.org, bl.spamcop.net, sbl.spamhaus.org in DNS black lists and collected the spam. Then we have removed relays.ordb.org from DNSBL list and measured the spam trap. When we removed relays.ordb.org from the list, we observed 40~50% increase of spam traffic as shown in Figure3. Apart from this each user received 2~3 false negatives per day. The standard deviation of false negatives received by an end user for 7 days is shown in Figure4. Figure4. Standard Deviation of false negative We conducted several experiments to measure the effectiveness of SURBL. The test was conducted in the same network. To test the SURBL, we observed mail delivery for particular period of time (sessions). Each session is about 3 hours period of time. The experiment result shows that the effectiveness of the SURBL test. Our dataset consists of the spam mails collected at a large spam trap. Figure5. SURBL test-Spam delivery The number of spam had increased to the user's inbox when the SURBL test is not conducted to check the spam; at the same time the number spam has decreased to the spam trap. SURBL test was unchecked for five sessions. Most of the users received spam in their inbox during this test. The results are shown in the Figure5.
http://www2.{BLOCKED}tinmdesachlion.com http://www3.{BLOCKED}tinmdesachlion.com http://www4.{BLOCKED}tinmdesachlion.com http://www6.{BLOCKED}tinmdesachlion.com
Emil sit: An empirical study of spam traffic and the use of DNS Black lists. Jae Yeon , Jung , ACM SIGCOMM Internet measurement conferences. Jae Yeon Jung, Emil sit: An empirical study of spam traffic and the use of DNS Black lists, ACM SIGCOMM Internet measurement conferences, pp 370-75, 2004.
Nick Feamster: Can DNS-based Blacklists keep up with Bots. Anirudh Ramachandran, David Dagon, CEAS. Anirudh Ramachandran, David Dagon, Nick Feamster: Can DNS-based Blacklists keep up with Bots, CEAS 2006, July 27-28, 2006.
Cooperative mechanism against DDOS attacks. Guangsen Zhang, Manish Parashar, SAM 2005. Guangsen Zhang, Manish Parashar: Cooperative mechanism against DDOS attacks, , SAM 2005, pg 86-96, June, 2005.
Understanding the network level behaviour of spammers. Anirudh Ramachandran, Nick Feamster, SIGCOMM 06. Anirudh Ramachandran, Nick Feamster : Understanding the network level behaviour of spammers, SIGCOMM 06, September 11-16,2006.
Proof of work proves not to work. Ben Laurie, Richard Clayton, 04Minneapolis MNBen Laurie, Richard Clayton: Proof of work proves not to work, WEIS04, Minneapolis MN, May 13-14, 2004.
Ben Adida, David Chau, Susan Hohenberger, L Ronald, Rivest, Lightweight Signatures for Email, DIMACS. Ben Adida, David Chau, susan Hohenberger, Ronald L rivest: Lightweight Signatures for Email, DIMACS, 2006.
Controlling spam through Lightweight currency. A David, Turner, M Daniel, Havey, 4David A Turner, Daniel M Havey: Controlling spam through Lightweight currency, ICCS-04, 2004.
Carl Eklund, Spam -from nuisance to Internet Infestation, Peer to Peer and SPAM in the Internet Raimo Kantola's technical report. Carl Eklund: Spam -from nuisance to Internet Infestation, Peer to Peer and SPAM in the Internet Raimo Kantola's technical report, 126-134, 2004.
Mechanism for detecting and prevention of email spamming, Peer to Peer and SPAM in the Internet Raimo Kantola's technical report. Vladimir Mijatovic, Vladimir Mijatovic: Mechanism for detecting and prevention of email spamming, Peer to Peer and SPAM in the Internet Raimo Kantola's technical report, 135-145, 2004.
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Sender Policy Framework. Sender Policy Framework, http://new.openspf.org/
Spamhaus survives DDoS. Spamhaus survives DDoS attack http://www.virus.org/news, September 2006.
. Faq Greylisting, Greylisting FAQ https://hdc.tamu.edu/
The Spamhous Project. The Spamhous Project. www.spamhaus.org
SpamAssassin ww.spamassassin.apache.org. SpamAssassin ww.spamassassin.apache.org
SpamCop. SpamCop http://spamcop.net
Cloudmark. Cloudmark, http://www.cloudmark.com
Sender Policy Framework. Sender Policy Framework, http://new.openspf.org/
. Email Bombing, Spamming, Email Bombing and Spamming: www.cert.org/tech_tips/email_bombing_spamming.html
| []
|
[
"Cosmoparticle Physics - the Challenge for the Millenium",
"Cosmoparticle Physics - the Challenge for the Millenium"
]
| [
"Maxim Yu Khlopov \nCosmoparticle Physics \"Cosmion\"\n125047MoscowRussia\n\nPhysics Department\nUniversity \"LaSapienza\"\nPle A.Moro,200185RomeItaly\n"
]
| [
"Cosmoparticle Physics \"Cosmion\"\n125047MoscowRussia",
"Physics Department\nUniversity \"LaSapienza\"\nPle A.Moro,200185RomeItaly"
]
| []
| Cosmoparticle physics is the natural result of development of mutual relationship between cosmology and particle physics. Its prospects offer the way to study the theory of everything and the true history of the Universe, based on it, in the proper combination of their indirect physical, astrophysical and cosmological signatures. We may be near the first positive results in this direction. The basic ideas of cosmoparticle physics are briefly reviewed. | 10.1007/978-3-642-18534-2_15 | [
"https://arxiv.org/pdf/astro-ph/0309704v1.pdf"
]
| 174,135 | astro-ph/0309704 | 32dc575619895c5cd74da441d1822760d3471f33 |
Cosmoparticle Physics - the Challenge for the Millenium
25 Sep 2003
Maxim Yu Khlopov
Cosmoparticle Physics "Cosmion"
125047MoscowRussia
Physics Department
University "LaSapienza"
Ple A.Moro,200185RomeItaly
Cosmoparticle Physics - the Challenge for the Millenium
25 Sep 2003
Cosmoparticle physics is the natural result of development of mutual relationship between cosmology and particle physics. Its prospects offer the way to study the theory of everything and the true history of the Universe, based on it, in the proper combination of their indirect physical, astrophysical and cosmological signatures. We may be near the first positive results in this direction. The basic ideas of cosmoparticle physics are briefly reviewed.
Cosmoparticle physics originates from the well established relationship between microscopic and macroscopic descriptions in theoretical physics. Remind the links between statistical physics and thermodynamics, or between electrodynamics and theory of electron. To the end of the XX Century the new level of this relationship was realized. It followed both from the cosmological necessity to go beyond the world of known elementary particles in the physical grounds for inflationary cosmology with baryosynthesis and dark matter as well as from the necessity for particle theory to use cosmological tests as the important and in many cases unique way to probe its predictions.
The convergence of the frontiers of our knowledge in micro-and macro worlds leads to the wrong circle of problems, illustrated by the mystical Uhroboros (selfeating-snake). The Uhroboros puzzle may be formulated as follows: The theory of the Universe is based on the predictions of particle theory, that need cosmology for their test. Cosmoparticle physics [1], [2], [3] offers the way our of this wrong circle. It studies the fundamental basis and mutual relationship between micro-and macro-worlds in the proper combination of physical, astrophysical and cosmological signatures.
Let's specify in more details the set of links between fundamental particle properties and their cosmological effects.
The role of particle content in the Einstein equations is reduced to its contribution into energy-momentum tensor. So, the set of relativistic species, dominating in the Universe, realizes the relativistic equation of state p = ε/3 and the relativistic stage of expansion. The difference between relativistic bosons and fermions or various bosonic (or fermionic) species is accounted by the statistic weight of respective degree of freedom. The very treatment of different species of particles as equivalent degrees of freedom physically assumes strict symmetry between them.
Such strict symmetry is not realized in Nature. There is no exact symmetry between bosons and fermions (e.g. supersymmetry). There is no exact symme-try between various quarks and leptons. The symmetry breaking implies the difference in particle masses. The particle mass pattern reflects the hierarchy of symmetry breaking.
Noether's theorem relates the exact symmetry to conservation of respective charge. The lightest particle, bearing the strictly conserved charge, is absolutely stable. So, electron is absolutely stable, what reflects the conservation of electric charge. In the same manner the stability of proton is conditioned by the conservation of baryon charge. The stability of ordinary matter is thus protected by the conservation of electric and baryon charges, and its properties reflect the fundamental physical scales of electroweak and strong interactions. Indeed, the mass of electron is related to the scale of the electroweak symmetry breaking, whereas the mass of proton reflects the scale of QCD confinement.
Extensions of the standard model imply new symmetries and new particle states. The respective symmetry breaking induces new fundamental physical scales in particle theory. If the symmetry is strict, its existence implies new conserved charge. The lightest particle, bearing this charge, is stable. The set of new fundamental particles, corresponding to the new strict symmetry, is then reflected in the existence of new stable particles, which should be present in the Universe and taken into account in the total energy-momentum tensor.
Most of the known particles are unstable. For a particle with the mass m the particle physics time scale is t ∼ 1/m, so in particle world we refer to particles with lifetime τ ≫ 1/m as to metastable. To be of cosmological significance metastable particle should survive after the temperature of the Universe T fell down below T ∼ m, what means that the particle lifetime should exceed t ∼ (m P l /m) · (1/m). Such a long lifetime should find reason in the existence of an (approximate) symmetry. From this viewpoint, cosmology is sensitive to the most fundamental properties of microworld, to the conservation laws reflecting strict or nearly strict symmetries of particle theory.
However, the mechanism of particle symmetry breaking can also have the cosmological impact. Heating of condensed matter leads to restoration of its symmetry. When the heated matter cools down, phase transition to the phase of broken symmetry takes place. In the course of the phase transitions, corresponding to given type of symmetry breaking, topological defects can form. One can directly observe formation of such defects in liquid crystals or in superfluids. In the same manner the mechanism of spontaneous breaking of particle symmetry implies restoration of the underlying symmetry. When temperature decreases in the course of cosmological expansion, transitions to the phase of broken symmetry can lead, depending on the symmetry breaking pattern, to formation of topological defects in very early Universe. The defects can represent the new form of stable particles (as it is in the case of magnetic monopoles), or the form of extended structures, such as cosmic strings or cosmic walls.
In the old Big bang scenario the cosmological expansion and its initial conditions was given a priori. In the modern cosmology the expansion of the Universe and its initial conditions is related to the process of inflation. The global properties of the Universe as well as the origin of its large scale structure are the result of this process. The matter content of the modern Universe is also originated from the physical processes: the baryon density is the result of baryosynthesis and the nonbaryonic dark matter represents the relic species of physics of the hidden sector of particle theory. Physics, underlying inflation, baryosynthesis and dark matter, is referred to the extensions of the standard model, and the variety of such extensions makes the whole picture in general ambiguous. However, in the framework of each particular physical realization of inflationary model with baryosynthesis and dark matter the corresponding model dependent cosmological scenario can be specified in all the details. In such scenario the main stages of cosmological evolution, the structure and the physical content of the Universe reflect the structure of the underlying physical model. The latter should include with necessity the standard model, describing the properties of baryonic matter, and its extensions, responsible for inflation, baryosynthesis and dark matter. In no case the cosmological impact of such extensions is reduced to reproduction of these three phenomena only. The nontrivial path of cosmological evolution, specific for each particular realization of inflational model with baryosynthesis and nonbaryonic dark matter, always contains some additional model dependent cosmologically viable predictions, which can be confronted with astrophysical data. The part of cosmoparticle physics, called cosmoarcheology, offers the set of methods and tools probing such predictions.
Cosmoarcheology considers the results of observational cosmology as the sample of the experimental data on the possible existence and features of hypothetical phenomena predicted by particle theory. To undertake the Gedanken Experiment with these phenomena some theoretical framework to treat their origin and evolution in the Universe should be assumed. As it was pointed out in [4] the choice of such framework is a nontrivial problem in the modern cosmology.
Indeed, in the old Big bang scenario any new phenomenon, predicted by particle theory was considered in the course of the thermal history of the Universe, starting from Planck times. The problem is that the bedrock of the modern cosmology, namely, inflation, baryosynthesis and dark matter, is also based on experimentally unproven part of particle theory, so that the test for possible effects of new physics is accomplished by the necessity to choose the physical basis for such test. There are two possible solutions for this problem: a) a crude model independent comparison of the predicted effect with the observational data and b) the model dependent treatment of considered effect, provided that the model, predicting it, contains physical mechanism of inflation, baryosynthesis and dark matter.
The basis for the approach (a) is that whatever happened in the early Universe its results should not contradict the observed properties of the modern Universe. The set of observational data and, especially, the light element abundance and thermal spectrum of microwave background radiation put severe constraint on the deviation from thermal evolution after 1 s of expansion, what strengthens the model independent conjectures of approach (a).
One can specify the new phenomena by their net contribution into the cosmological density and by forms of their possible influence on parameters of matter and radiation. In the first aspect we can consider strong and weak phenomena. Strong phenomena can put dominant contribution into the density of the Universe, thus defining the dynamics of expansion in that period, whereas the contribution of weak phenomena into the total density is always subdominant. The phenomena are time dependent, being characterized by their time-scale, so that permanent (stable) and temporary (unstable) phenomena can take place. They can have homogeneous and inhomogeneous distribution in space. The amplitude of density fluctuations δ ≡ δ̺/̺ measures the level of inhomogeneity relative to the total density, ̺. The partial amplitude δ i ≡ δ̺ i /̺ i measures the level of fluctuations within a particular component with density ̺ i , contributing into the total density ̺ = i ̺ i . The case δ i ≥ 1 within the considered i-th component corresponds to its strong inhomogeneity. Strong inhomogeneity is compatible with the smallness of total density fluctuations, if the contribution of inhomogeneous component into the total density is small:
̺ i ≪ ̺, so that δ ≪ 1.
The phenomena can influence the properties of matter and radiation either indirectly, say, changing of the cosmological equation of state, or via direct interaction with matter and radiation. In the first case only strong phenomena are relevant, in the second case even weak phenomena are accessible to observational data. The detailed analysis of sensitivity of cosmological data to various phenomena of new physics are presented in [3].
The basis for the approach (b) is provided by a particle model, in which inflation, baryosynthesis and nonbaryonic dark matter is reproduced. Any realization of such physically complete basis for models of the modern cosmology contains with necessity additional model dependent predictions, accessible to cosmoarcheological means. Here the scenario should contain all the details, specific to the considered model, and the confrontation with the observational data should be undertaken in its framework. In this approach complete cosmoparticle physics models may be realized, where all the parameters of particle model can be fixed from the set of astrophysical, cosmological and physical constraints. Even the details, related to cosmologically irrelevant predictions, such as the parameters of unstable particles, can find the cosmologically important meaning in these models. So, in the model of horizontal unification [5], [6], [7], the top quark or B-meson physics fixes the parameters, describing the dark matter, forming the large scale structure of the Universe.
To study the imprints of new physics in astrophysical data cosmoarcheology implies the forms and means in which new physics leaves such imprints. So, the important tool of cosmoarcheology in linking the cosmological predictions of particle theory to observational data is the Cosmophenomenology of new physics. It studies the possible hypothetical forms of new physics, which may appear as cosmological consequences of particle theory, and their properties, which can result in observable effects.
The simplest primordial form of new physics is the gas of new stable massive particles, originated from early Universe. For particles with the mass m, at high temperature T > m the equilibrium condition, n · σv · t > 1 is valid, if their annihilation cross section σ > 1/(mm P l ) is sufficiently large to establish the equilibrium. At T < m such particles go out of equilibrium and their relative concentration freezes out. More weakly interacting species decouple from plasma and radiation at T > m, when n · σv · t ∼ 1, i.e. at T dec ∼ (σm P l ) −1 . The maximal temperature, which is reached in inflationary Universe, is the reheating temperature, T r , after inflation. So, the very weakly interacting particles with the annihilation cross section σ < 1/(T r m P l ), as well as very heavy particles with the mass m ≫ T r can not be in thermal equilibrium, and the detailed mechanism of their production should be considered to calculate their primordial abundance.
Decaying particles with the lifetime τ , exceeding the age of the Universe, t U , τ > t U , can be treated as stable. By definition, primordial stable particles survive to the present time and should be present in the modern Universe. The net effect of their existence is given by their contribution into the total cosmological density. They can dominate in the total density being the dominant form of cosmological dark matter, or they can represent its subdominant fraction. In the latter case more detailed analysis of their distribution in space, of their condensation in galaxies, of their capture by stars, Sun and Earth, as well as of the effects of their interaction with matter and of their annihilation provides more sensitive probes for their existence. In particular, hypothetical stable neutrinos of the 4th generation with the mass about 50 GeV are predicted to form the subdominant form of the modern dark matter, contributing less than 0,1 % to the total density. However, direct experimental search for cosmic fluxes of weakly interacting massive particles (WIMPs) may be sensitive to the existence of such component [8], [9], and may be even favors it [9]. It was shown in [10], [11], [12] that annihilation of 4th neutrinos and their antineutrinos in the Galaxy can explain the galactic gamma-background, measured by EGRET in the range above 1 GeV, and that it can give some clue to explanation of cosmic positron anomaly, claimed to be found by HEAT. 4th neutrino annihilation inside the Earth should lead to the flux of underground monochromatic neutrinos of known types, which can be traced in the analysis of the already existing and future data of underground neutrino detectors [12].
New particles with electric charge and/or strong interaction can form anomalous atoms and contain in the ordinary matter as anomalous isotopes. For example, if the lightest quark of 4th generation is stable, it can form stable charged hadrons, serving as nuclei of anomalous atoms of e.g. crazy helium [13].
Primordial unstable particles with the lifetime, less than the age of the Universe, τ < t U , can not survive to the present time. But, if their lifetime is sufficiently large to satisfy the condition τ ≫ (m P l /m) · (1/m), their existence in early Universe can lead to direct or indirect traces. Cosmological flux of decay products contributing into the cosmic and gamma ray backgrounds represents the direct trace of unstable particles. If the decay products do not survive to the present time their interaction with matter and radiation can cause indirect trace in the light element abundance or in the fluctuations of thermal radiation. If the particle lifetime is much less than 1s the multi-step indirect traces are possible, provided that particles dominate in the Universe before their de-cay. On the dust-like stage of their dominance black hole formation takes place, and the spectrum of such primordial black holes traces the particle properties (mass, frozen concentration, lifetime) [14]. The particle decay in the end of dust like stage influences the baryon asymmetry of the Universe. In any way cos-mophenomenoLOGICAL chains link the predicted properties of even unstable new particles to the effects accessible in astronomical observations. Such effects may be important in the analysis of the observational data.
So, the only direct evidence for the accelerated expansion of the modern Universe comes from the distant SN I data. The data on the cosmic microwave background (CMB) radiation and large scale structure (LSS) evolution (see e.g.
[15]) prove in fact the existence of homogeneously distributed dark energy and the slowing down of LSS evolution at z ≤ 3. Homogeneous negative pressure medium (Λ-term or quintessence) leads to relative slowing down of LSS evolution due to acceleration of cosmological expansion. However, both homogeneous component of dark matter and slowing down of LSS evolution naturally follow from the models of Unstable Dark Matter (UDM) (see [3] for review), in which the structure is formed by unstable weakly interacting particles. The weakly interacting decay products are distributed homogeneously. The loss of the most part of dark matter after decay slows down the LSS evolution. The dominantly invisible decay products can contain small ionizing component [6]. Thus, UDM effects will deserve attention, even if the accelerated expansion is proved.
The parameters of new stable and metastable particles are also determined by the pattern of particle symmetry breaking. This pattern is reflected in the succession of phase transitions in the early Universe. The phase transitions of the first order proceed through the bubble nucleation, which can result in black hole formation. The phase transitions of the second order can lead to formation of topological defects, such as walls, string or monopoles. The observational data put severe constraints on magnetic monopole and cosmic wall production, as well as on the parameters of cosmic strings. The succession of phase transitions can change the structure of cosmological defects. The more complicated forms, such as walls-surrounded-by-strings can appear. Such structures can be unstable, but their existence can lead the trace in the nonhomogeneous distribution of dark matter and in large scale correlations in the nonhomogeneous dark matter structures, such as archioles [16]. The large scale correlations in topological defects and their imprints in primordial inhomogeneities is the indirect effect of inflation, if phase transitions take place after reheating of the Universe. Inflation provides in this case the equal conditions of phase transition, taking place in causally disconnected regions.
If the phase transitions take place on inflational stage new forms of primordial large scale correlations appear. The example of global U(1) symmetry, broken spontaneously in the period of inflation and successively broken explicitly after reheating, was recently considered in [17]. In this model, spontaneous U(1) symmetry breaking at inflational stage is induced by the vacuum expectation value ψ = f of a complex scalar field Ψ = ψ exp (iθ), having also explicit symmetry breaking term in its potential V eb = Λ 4 (1 − cos θ). The latter is negligible in the period of inflation, if f ≫ Λ, so that there appears a valley relative to values of phase in the field potential in this period. Fluctuations of the phase θ along this valley, being of the order of ∆θ ∼ H/(2πf ) (here H is the Hubble parameter at inflational stage) change in the course of inflation its initial value within the regions of smaller size. Owing to such fluctuations, for the fixed value of θ 60 in the period of inflation with e-folding N = 60 corresponding to the part of the Universe within the modern cosmological horizon, strong deviations from this value appear at smaller scales, corresponding to later periods of inflation with N < 60. If θ 60 < π, the fluctuations can move the value of θ N to θ N > π in some regions of the Universe. After reheating, when the Universe cools down to temperature T = Λ the phase transition to the true vacuum states, corresponding to the minima of V eb takes place. For θ N < π the minimum of V eb is reached at θ vac = 0, whereas in the regions with θ N > π the true vacuum state corresponds to θ vac = 2π. For θ 60 < π in the bulk of the volume within the modern cosmological horizon θ vac = 0. However, within this volume there appear regions with θ vac = 2π. These regions are surrounded by massive domain walls, formed at the border between the two vacua. Since regions with θ vac = 2π are confined, the domain walls are closed. After their size equals the horizon, closed walls can collapse into black holes. The minimal mass of such black hole is determined by the condition that it's Schwarzschild radius, r g = 2GM/c 2 exceeds the width of the wall, l ∼ f /Λ 2 , and it is given by M min ∼ f (m P l /Λ) 2 . The maximal mass is determined by the mass of the wall, corresponding to the earliest region θ N > π, appeared at inflational stage. This mechanism can lead to formation of primordial black holes of a whatever large mass (up to the mass of AGNs [18]). Such black holes appear in the form of primordial black hole clusters, exhibiting fractal distribution in space [17]. It can shed new light on the problem of galaxy formation.
Primordial strong inhomogeneities can also appear in the baryon charge distribution. The appearance of antibaryon domains in the baryon asymmetrical Universe, reflecting the inhomogeneity of baryosynthesis, is the profound signature of such strong inhomogeneity [19]. On the example of the model of spontaneous baryosynthesis (see [23] for review) the possibility for existence of antimatter domains, surviving to the present time in inflationary Universe with inhomogeneous baryosynthesis was revealed in [24]. Evolution of sufficiently dense antimatter domains can lead to formation of antimatter globular clusters [25]. The existence of such cluster in the halo of our Galaxy should lead to the pollution of the galactic halo by antiprotons. Their annihilation can reproduce [26] the observed galactic gamma background in the range tens-hundreds MeV. The prediction of antihelium component of cosmic rays [27], accessible to future searches for cosmic ray antinuclei in PAMELA and AMS II experiments, as well as of antimatter meteorites [28] provides the direct experimental test for this hypothesis.
So the primordial strong inhomogeneities in the distribution of total, dark matter and baryon density in the Universe is the new important phenomenon of cosmological models, based on particle models with hierarchy of symmetry breaking.
The new physics follows from the necessity to extend the Standard model. The white spots in the representations of symmetry groups, considered in the extensions of the Standard model, correspond to new unknown particles. The extension of the symmetry of gauge group puts into consideration new gauge fields, mediating new interactions. Global symmetry breaking results in the existence of Goldstone boson fields.
For a long time the necessity to extend the Standard model had purely theoretical reasons. Aesthetically, because full unification is not achieved in the Standard model; practically, because it contains some internal inconsistencies. It does not seem complete for cosmology. One has to go beyond the Standard model to explain inflation, baryosynthesis and nonbaryonic dark matter. Recently there has appeared a set of experimental evidences for the existence of neutrino oscillations (see for recent review e.g. [20], [21], [22]), of cosmic WIMPs [9], and of double neutrinoless beta decay [29]. Whatever is the accepted status of these evidences, they indicate that the experimental searches may have already crossed the border of new physics.
In particle physics direct experimental probes for the predictions of particle theory are most attractive. The predictions of new charged particles, such as supersymmetric particles or quarks and leptons of new generation, are accessible to experimental search at accelerators of new generation, if their masses are in 100GeV-1TeV range. However, the predictions related to higher energy scale need non-accelerator or indirect means for their test.
The search for rare processes, such as proton decay, neutrino oscillations, neutrinoless beta decay, precise measurements of parameters of known particles, experimental searches for dark matter represent the widely known forms of such means. Cosmoparticle physics offers the nontrivial extensions of indirect and non-accelerator searches for new physics and its possible properties. In experimental cosmoarcheology the data is to be obtained, necessary to link the cosmophenomenology of new physics with astrophysical observations (See [4]). In experimental cosmoparticle physics the parameters, fixed from the consitency of cosmological models and observations, define the level, at which the new types of particle processes should be searched for (see [30]).
The theories of everything should provide the complete physical basis for cosmology. The problem is that the string theory [31] is now in the form of "theoretical theory", for which the experimental probes are widely doubted to exist. The development of cosmoparticle physics can remove these doubts. In its framework there are two directions to approach the test of theories of everything.
One of them is related with the search for the experimentally accessible effects of heterotic string phenomenology. The mechanism of compactification and symmetry breaking leads to the prediction of homotopically stable objects [32] and shadow matter [33], accessible to cosmoarcheological means of cosmoparticle physics. The condition to reproduce the Standard model naturally leads in the heterotic string phenomenology to the prediction of fourth generation of quarks and leptons [34] with a stable massive 4th neutrino [10], what can be the subject of complete experimental test in the near future. The comparison between the rank of the unifying group E 6 (r = 6) and the rank of the Standard model (r = 4) implies the existence of new conserved charges and new (possibly strict) gauge symmetries. New strict gauge U(1) symmetry (similar to U(1) symmetry of electrodynamics) is possible, if it is ascribed to the fermions of 4th generation. This hypothesis explains the difference between the three known types of neutrinos and neutrino of 4th generation. The latter possesses new gauge charge and, being Dirac particle, can not have small Majorana mass due to sea saw mechanism. If the 4th neutrino is the lightest particle of the 4th quark-lepton family, strict conservation of the new charge makes massive 4th neutrino to be absolutely stable. Following this hypothesis [34] quarks and leptons of 4th generation are the source of new long range interaction (y-electromagnetism), similar to the electromagnetic interaction of ordinary charged particles. New strictly conserved local U(1) gauge symmetries can also arise in the development of D-brane phenomenology [35], [36]. If proved, the practical importance of this property could be hardly overestimated.
It is interesting, that heterotic string phenomenology embeds even in its simplest realisation both supersymmetric particles and the 4th family of quarks and leptons, in particular, the two types of WIMP candidates: neutralinos and massive stable 4th neutrinos. So in the framework of this phenomenology the multicomponent analysis of WIMP effects is favorable.
In the above approach some particular phenomenological features of simplest variants of string theory are studied. The other direction is to elaborate the extensive phenomenology of theories of everything by adding to the symmetry of the Standard model the (broken) symmetries, which have serious reasons to exist. The existence of (broken) symmetry between quark-lepton families, the necessity in the solution of strong CP-violation problem with the use of broken Peccei-Quinn symmetry, as well as the practical necessity in supersymmetry to eliminate the quadratic divergence of Higgs boson mass in electroweak theory is the example of appealing additions to the symmetry of the Standard model. The horizontal unification and its cosmology represent the first step on this way, illustrating the approach of cosmoparticle physics to the elaboration of the proper phenomenology for theories of everything [7].
We can conclude that from the very beginning to the modern stage, the evolution of Universe is governed by the forms of matter, different from those we are built of and observe around us. From the very beginning to the present time, the evolution of the Universe was governed by physical laws, which we still don't know. Observational cosmology offers strong evidences favoring the existence of processes, determined by new physics, and the experimental physics approaches to their investigation.
Cosmoparticle physics[1] [2], studying the physical, astrophysical and cosmological impact of new laws of Nature, explores the new forms of matter and their physical properties, what opens the way to use the corresponding new sources of energy and new means of energy transfer. It offers the great challenge for the new Millennium.The work was performed in the framework of the State Contract 40.022.1.1.1106 and was partially supported by the RFBR grant 02-02-17490 and grant UR.02.01.026.
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[
"Néel and Valence-Bond Crystal phases of the Two-Dimensional Heisenberg Model on the Checkerboard Lattice",
"Néel and Valence-Bond Crystal phases of the Two-Dimensional Heisenberg Model on the Checkerboard Lattice"
]
| [
"S Moukouri \nDepartment of Physics\nMichigan Center for Theoretical Physics\nUniversity of Michigan\n2477 Randall Laboratory48109Ann ArborMI\n"
]
| [
"Department of Physics\nMichigan Center for Theoretical Physics\nUniversity of Michigan\n2477 Randall Laboratory48109Ann ArborMI"
]
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| I use an improved version of the two-step density matrix renormalization group method to study ground-state properties of the 2D Heisenberg model on the checkerboard lattice. In this version the Hamiltonian is projected on a tensor product of two-leg ladders instead of chains. This allows investigations of 2D isotropic models. I show that this method can describe both the magnetically disordered and ordered phases. The ground-state phases of the checkerboard model as J2 increases are: (i) Néel with Q = (π, π), (ii) a valence bond crystal (VBC) of plaquettes, (iii) Néel with Q = (π/2, π), and (iv) a VBC of crossed dimers. In agreement with previous results, I find that at the isotropic point J2 = J1, the ground state is made of weakly interacting plaquettes with a large gap ∆ ≈ 0.67J1 to triplet excitations. | null | [
"https://arxiv.org/pdf/0709.2688v1.pdf"
]
| 117,045,274 | 0709.2688 | ce44df35633ca7686f2601edebc6a51ccde1754e |
Néel and Valence-Bond Crystal phases of the Two-Dimensional Heisenberg Model on the Checkerboard Lattice
17 Sep 2007
S Moukouri
Department of Physics
Michigan Center for Theoretical Physics
University of Michigan
2477 Randall Laboratory48109Ann ArborMI
Néel and Valence-Bond Crystal phases of the Two-Dimensional Heisenberg Model on the Checkerboard Lattice
17 Sep 2007
I use an improved version of the two-step density matrix renormalization group method to study ground-state properties of the 2D Heisenberg model on the checkerboard lattice. In this version the Hamiltonian is projected on a tensor product of two-leg ladders instead of chains. This allows investigations of 2D isotropic models. I show that this method can describe both the magnetically disordered and ordered phases. The ground-state phases of the checkerboard model as J2 increases are: (i) Néel with Q = (π, π), (ii) a valence bond crystal (VBC) of plaquettes, (iii) Néel with Q = (π/2, π), and (iv) a VBC of crossed dimers. In agreement with previous results, I find that at the isotropic point J2 = J1, the ground state is made of weakly interacting plaquettes with a large gap ∆ ≈ 0.67J1 to triplet excitations.
Frustration-induced magnetically disordered phases in two dimensions (2D) recently have attracted substantial interest [1]. Frustrated magnets are known to display unconventional ground states with, in some cases, a large set of low-lying degenerate singlet excitations that are still not well understood. Among models of frustrated systems, the Heisenberg model on the checkerboard lattice (HMCL) has recently been intensively studied by various techniques [2,3,4,5,6,7]. This model is seen as a first step in the investigation of the 3D pyrochlore model. The emerging picture is that at the isotropic point (J 1 = J 2 ), the HMCL spontaneously breaks the lattice's translational symmetry. The ground state is a singlet made of a collection of weakly coupled plaquettes with a large gap, ∆ ≈ 0.7J 1 , to triplet excitations. Away from the isotropic point, the situation is less clear. There is no single method which can capture the full phase diagram.
In this letter, I introduce an improved version of the two-step density-matrix renormalization group (TS-DMRG) [8,9] which, as I will show, is very convenient in the study of the HMCL and other 2D frustrated models. This new version is based on using the two-leg ladder, instead of chains, as the starting point to build the 2D lattice. The main insight in using the two-leg ladder to construct the 2D lattice comes from large N predictions [11] that frustration often induces ground states in which the translationaly symmetry is broken. In the strongcoupling regime of the disordered phase of S = 1/2 systems, the system is made of a collection of singlets or plaquettes. This strong coupling regime cannot be described starting from independent chains which are gapless. Starting from a single chain, small transverse perturbations can yield a gap within the TSDMRG. But this gap is often small and it is difficult to obtain reliable extrapolations. The two-leg ladder does not present this problem. It does already present a large gap ∆ ≈ 0.5 even in absence of frustration. Coupled ladders naturally evolve toward the 2D Néel state as the number of legs increases. Hence, in principle, disordered and or-dered phases could be described within a two-leg ladder version of the TSDMRG. This suggests that the two-leg ladder is a more natural starting point to describe ground state phases of 2D antiferromagnets than the single chain.
Additional insights into this idea came from my comparative study of coupled chains with half-integer and integer spins [10]. In Ref. [10], when starting from single chains, I found that although chains with S = 1 display the Haldane gap, ∆ ≈ 0.4, they converge much faster to the Néel state than those with S = 1/2. Furthermore, when a frustration induced disordered phase is present, it can be much more easily found in the case S = 1. Hence, following the equivalence between the two-leg ladder and the Haldane spin chain, suggested by the Affleck-Kennedy-Lieb-Tasaki construction [12], it would be better to adopt the two-leg ladder as the building block for two-dimensional lattices.
I will now illustrate this idea in the case of the HMCL. Following the usual notation, the HMCL is given by:
H = J 1 <i,j> S i S j + J 2 [i,j] S i S j ,(1)
where < i, j > represents nearest-neighbor sites and [i, j] stand for next-nearest neighbors on every other plaquette. J 1 is set as the unit energy. The TSDMRG with ladders is similar to the method with chains. So I refer the reader to Ref. [8,9] for a complete exposition of the algorithm. Here, I will discuss only briefly the main points of the algorithm. I start by dividing the 2D lattice into two-leg ladders; the Hamiltonian (1) is written as,
H = ladders H ladder + H int ,(2)
where H ladder is the Hamiltonian of a single two-leg ladder, H int contains the inter-ladder part. In the first step of the method, the usual DMRG method is applied to generate a low energy Hamiltonian of an isolated ladder of N x sites keeping m 1 states. Then m 2 low-lying states of the superblock states, the corresponding energies, and all the local spin operators are kept. These energies represent the renormalized low energy Hamiltonian of a single ladder. The Hamiltonian (2) is then projected onto the tensor product basis of independent ladders,
Ψ = ladders Φ ladder ,(3)
where Φ ladder is an eigenfunction of H 0,ladder . This yield an effective Hamiltonian,
H ef f = ladders H 0,ladder +H int .(4)
The resulting effective coupled ladder problem which is 1D is studied again by the DMRG method in the transverse direction. The TSDMRG is variational, as the original DMRG method, the subspace spanned by the wavefunctions of the form Ψ is a subspace of the full Hilbert space of Hamiltonian (1). Its convergence depends on m 1 and m 2 , the error is given by max(ρ 1 , ρ 2 ), where ρ 1 and ρ 2 are the truncation errors in the first and second steps respectively. m 2 fixes the energy band-width δE. The method is accurate only when the inter-ladder couplings are small with respect to δE. In the present simulations δE ≈ 4. Since for the HMCL the inter-ladder and intra-ladder are of the same magnitude, in principle this approach would be plagued by the same deficiencies the block RG method. But if the starting point is chosen so that the essential physics is already contained at the level of the ladder, the effective strength of the interladder couplings will be small even if the bare couplings are not. This is particularly the case of models with frustration in which the competing interactions largely cancel each other in the strong frustration regime, yielding weakly coupled sub-clusters. The ground state properties of an isolated ladder can readily be obtained. I keep up to m 1 = 96 and N x = 16 and I target spin sectors from S z = 0 to S z = ±4 and used open boundary conditions (OBC). The maximum error is ρ 1 = 1 × 10 −4 . There is a gap ∆ for all values of J 2 investigated between 0 and 2. The finite size behavior of gaps for some typical values of J 2 are shown in Fig.1. The case J 2 = 0 reduces to the usual two-leg ladder which has been widely studied in the literature [13]. For J 2 = 0, ∆ ≈ 0.5 . As J 2 increases, ∆ has a non-monotonous behavior. This suggests a rich structure which is revealed more clearly by the analysis of the correlation functions. I computed the following short-range correlation functions: the bond strength along a leg C lu,c = S i,1 S i+1,1 u,c for uncrossed (u) and crossed (c) plaquettes, the diagonal correlation C du,c = S i,1 S i+1,2 u,c , and bond strength along the rungs C r = S i,1 S i,2 . Note that I have introduced a second index to the local spin. These correlations are shown in Fig.2. Four regions can be identified: (i) region I (rung dimers): 0 < ∼ J 2 < ∼ 0.6, C lu,c < 0, C lu ≈ C lc , C du,c > 0, C r < 0, and |C r | > |C lu |; the dominant spinspin correlations are along the rungs. The ground state properties of the ladder in this region are identical to those of the unfrustrated ladder (J 2 = 0). (ii) Region II (plaquettes I): 0.6 < ∼ J 2 < ∼ 1, C lu,c < 0, |C lu | > |C lc |, C du,c > 0, C r < 0, and |C r | < |C lu |; the physics is dominated by that of the isotropic point. At this point, the ground state is a collection of weakly interacting uncrossed plaquettes. Both C dc and C lc vanish at J 2 = 1. In this region, the local spin configuration is the same on all the uncrossed plaquettes as shown in Fig.3(b). (iii) region III (plaquettes II): 1 < ∼ J 2 < ∼ 1.3, C lu < 0, C lc > 0, C du > 0, C dc < 0, C r < 0, and |C r | < |C lu |; in this region, the ground state is again dominated by uncrossed plaquettes. But now the local spin configura- tions on two consecutive uncrossed plaquettes are images of one another by reflection with respect to a plane passing through the middle of the crossed plaquette between them. Region IV (crossed dimers): 1.3 < ∼ J 2 , C lu < 0, C dc < 0, C lc = C r = C du ≈ 0, and |C r | < |C lu |; the ground state is dominated by the crossed dimers on crossed plaquettes as shown in Fig.3(d). The sketch of the spin structure corresponding to each region is summarized in Fig.3. Since I applied OBC, for a given size, there are two possible ground states depending on the plaquette pattern: (a) ucu...ucu or (b) cuc...cuc. In region I, the configurations (a) and (b) have nearly the same energy. This is consistent with the fact that the translational symmetry is not broken. But in Region II and III, (a) has the lowest energy, since it has a larger number of uncrossed plaquettes. By contrast, in Region IV where dimer order is dominant, it is (b) that has the lowest energy.
The 2D systems are obtained by applying the DMRG on H ef f in the transverse direction. I studied systems of size N x × N y = 4 × 6, 8 × 10, 12 × 14 and 16 × 18. I kept up to m 2 = 96 and used OBC. Inter-ladder interactions will have very different effects depending on whether they correspond to a magnetic regime or a disordered regime. I will first consider their effects on region II, which includes the isotropic point. Recently, there have been a number of studies which strongly suggest that the physics of the 2D systems is identical to that displayed by the two-leg ladder. In other words, the ground state is essentially made of weakly interacting plaquettes. If this is the case, it means that the inter-ladder interactions will not strongly modify the ground state wave function of decoupled ladders. Fig.1 shows that the ground state energy and ∆ remain very close to that of an isolated plaquette. Thus in the vicinity of J 2 = 1, inter-ladder interactions do not strongly renormalize the properties of an isolated ladder which themselves are close to those of an isolated plaquette. The extrapolated gap is found to be ∆ = 0.67J 1 which is in good agreement with the prediction from exact diagonalization [5]. The same conclusion is seen in Fig.1 for region IV where the crossed- dimer ground state found for the ladder is also the ground state of the 2D lattice. In both cases, the wave function made of the tensor product of the wave function of single two-leg ladders is a good variational wave function for the 2D system. In each case, the ground state energy of the 2D system remains very close to that of individual plaquettes or crossed dimers. This can be explained as follows: when ladders are brought together to build the 2D lattice, the dominant local correlations are C lu in region II and C dc in region IV; during this process, magnetic energy cannot efficiently be gained. For region II, this is because the two neighboring plaquettes of an uncrossed plaquette in the direction of the rungs involve frustrated bonds. Hence the system prefers the original configuration to avoid increasing its energy. For region IV, C r is very small. The system cannot increase it when the ladders are coupled, because the spins are already involved in strong diagonal dimers. There is, however, the possibility to gain magnetic energy by forming Néel order along the direction of the diagonal bonds (J 2 direction) as suggested in Ref. [7]. This is unlikely, however, because once such a phase is reached, I do not see how the system could go to crossed dimers at larger J 2 . The action of J 1 which act as frustration in this regime decreases as J 2 increases. Hence once this hypothetical Néel phase along the J 2 bonds is reached, there is no obvious mechanism that could destroy it as J 2 increases to yield the crossed-dimer phase as suggested in Ref. [7]. Such a Néel phase would be favored only when J 2 ≫ J 1 . I made rough calculations with J 2 = 4, 8 and I found that the system remains in the crossed-dimer phase. The situation is apparently identical to the J 1 −J 2 chain where the independent chains regime is only reached in the infinite J 2 limit.
The situation is very different for regions I and III. In region I, the dominant local correlation is C r ; when the ladders are brought together, magnetic energy can be gained by an antiferromagnetic arrangement along the rungs. This enhances the local antiferromagnetic order which exists along the legs and ultimately leads to a Néel order with Q = (π, π). This is seen in the vanishing of the spin gap for J 2 = 0 and J 2 = 0.5 shown in Fig.4(a). This is in agreement with results for J 2 = 0 from quantum Monte Carlo (QMC) simulations [14] and large S analysis [4]. I find that the TSDMRG ground state energy −0.6011 at J 2 = 0. is not in very good agreement with the QMC result −0.6699 of Ref. [14]. Despite this discrepancy, the TSDMRG is nevertheless able to reproduce the low-energy behavior of the ordered phase. This is not in fact surprising. In the Resonating valence bond picture, the Néel state and its low energy excitations can be written as linear combination of a tensor product of dimers. The TSDMRG variational solution of Hamiltonian(1) which is a linear combination of the wave functions Ψ has exactly this form. A similar analysis also applies for region III. C lu is dominant in region II. But as seen in Fig.2, C lu has a minimum at J 2 = 1 and then increases. It becomes very close to C r when J 2 enters region III. Hence magnetic energy can be gained again through the rungs. Since the structure along the legs is not modified from Fig.3(d), the resulting wave vector will be Q = (π/2, π). This is seen in Fig.4 in the behavior of the spin-spin correlation function C l (i) along the legs. C l (i) displays a period of 4. The correlations between the rungs (not shown) oscillate with q y = π. Fig.5 presents a sketch of the different ground state phases of the HMCL as function of J 2 . I note that in Ref. [7], a very similar phase diagram was suggested; the only difference with the TSDMRG phase diagram is the wave vector of the Néel phase between the plaquette and crossed-dimer phases.
In summary, I have shown that the TSDMRG method can reliably be used to study the disordered phases with short correlation lengths of isotropic 2D models. In these phases, the system is a collection of dimers or plaquettes. This makes the two-leg ladder a very good starting point for a variational calculation. I showed that the basic physics of 2D systems could already be read through short-range correlations of the two-leg ladders. This variational calculation is able to predict reasonably magnetically ordered phases as well. In this work, I did not discuss the question of low-lying singlet excitations within the gap. Targeting them will lead to large truncation errors and the calculations will become impractical. These excitations are naturally truncated out when they are not needed to form a target state. Finally, The same method could be applied to the Sutherland-Shastry, J 1 − J 2 or the Kagomé models in 2D. It could also be applied to the pyrochlore lattice, provided that the Hamiltonian could be written in some form involving 1D subsystems with a large gap.
This work started during a visit at the Weizmann Institute. The author thanks E. Altman for hospitality. This work was supported by the NSF Grant No. DMR-0426775.
gap of the two-leg ladder for J2 = 0 (circles), 0.5 (squares), 1 (diamonds), 1.1 (triangles up), and 2 (triangles down). (b) Ground-state energies as function of the system size for a two-leg ladder for J2 = 1 (circles), J2 = 2 (diamonds), and for the 2D lattice for J2 = 1 (squares), J2 = 2 (triangle up).
FIG. 2 :
2Short-range correlations C l (circles), Cr (squares), C d (diamonds) for the two-leg ladder for uncrossed (a) and crossed (b) plaquettes as function of J2
FIG. 3 :
3The four phases of the two-leg ladder: rung dimers (a), plaquette I (b), plaquette II (c), and crossed dimers (d).
gaps as function of the system size J2 = 0 (circles), 0.5 (squares), 1 (diamonds), 1.1 (triangles up), and 2 (triangles down). (b) Correlation function along the legs as function of the distance for J2 = 1.25.
FIG. 5 :
5Ground-state phases of the 2D checkerboard model as function of J2. Note that the phase boundaries are rough estimates taken from the phases of the two-leg ladder.
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| []
|
[
"Lower Bounds for Buchsbaum* Complexes",
"Lower Bounds for Buchsbaum* Complexes"
]
| [
"Jonathan Browder [[email protected] \nDepartment of Mathematics\nUniversity of Washington\nBox 35435098195-4350SeattleWAUSA\n",
"Steven Klee klees]@math.washington.edu \nDepartment of Mathematics\nUniversity of Washington\nBox 35435098195-4350SeattleWAUSA\n"
]
| [
"Department of Mathematics\nUniversity of Washington\nBox 35435098195-4350SeattleWAUSA",
"Department of Mathematics\nUniversity of Washington\nBox 35435098195-4350SeattleWAUSA"
]
| []
| The class of (d − 1)-dimensional Buchsbaum* simplicial complexes is studied. It is shown that the rank-selected subcomplexes of a (completely) balanced Buchsbaum* simplicial complex are also Buchsbaum*. Using this result, lower bounds on the h-numbers of balanced Buchsbaum* simplicial complexes are established. In addition, sharp lower bounds on the h-numbers of flag m-Buchsbaum* simplicial complexes are derived, and the case of equality is treated. | 10.1016/j.ejc.2010.07.007 | [
"https://arxiv.org/pdf/1002.1256v1.pdf"
]
| 31,857,019 | 1002.1256 | bfef0800ea18e5dffeff48d746d9cd8e0e6c7f88 |
Lower Bounds for Buchsbaum* Complexes
5 Feb 2010 February 5, 2010
Jonathan Browder [[email protected]
Department of Mathematics
University of Washington
Box 35435098195-4350SeattleWAUSA
Steven Klee klees]@math.washington.edu
Department of Mathematics
University of Washington
Box 35435098195-4350SeattleWAUSA
Lower Bounds for Buchsbaum* Complexes
5 Feb 2010 February 5, 2010
The class of (d − 1)-dimensional Buchsbaum* simplicial complexes is studied. It is shown that the rank-selected subcomplexes of a (completely) balanced Buchsbaum* simplicial complex are also Buchsbaum*. Using this result, lower bounds on the h-numbers of balanced Buchsbaum* simplicial complexes are established. In addition, sharp lower bounds on the h-numbers of flag m-Buchsbaum* simplicial complexes are derived, and the case of equality is treated.
Introduction
One commonly studied invariant of a finite, (d − 1)-dimensional simplicial complex ∆ is its f -vector f (∆) = (f −1 , f 0 , . . . , f d−1 ), where f i denotes the number of i-dimensional faces in ∆. It is equivalent, and oftentimes more convenient, to study the h-vector h(∆) = (h 0 , . . . , h d ) defined by the relation
d j=0 h j λ d−j = d i=0 f i−1 (λ − 1) d−i .
When studying the h-numbers of simplicial complexes, it is natural to study the class of simplicial complexes that are Cohen-Macaulay over a fixed field k. This includes the classes of k-homology balls and k-homology spheres. A more specialized class of simplicial complexes is the class of complexes that are doubly Cohen-Macaulay (2-CM) over k. This class was introduced and studied by Baclawski [3]. Any complex that is 2-CM over k is Cohen-Macaulay over k with non-vanishing top dimensional homology (computed with coefficients in k). For example, k-homology spheres are 2-CM over k, but k-homology balls are not 2-CM over k.
The advantage of studying topological manifolds is that they are locally homeomorphic to topological balls. Using this local property as motivation, it is natural to define the class of Buchsbaum simplicial complexes. We say that a simplicial complex ∆ is Buchsbuam over k if the link of each vertex of ∆ is Cohen-Macaulay over k. Hence k-homology manifolds are Buchsbaum simplicial complexes over k. If ∆ is a Buchsbaum simplicial complex, it is convenient to study the h ′ -numbers of ∆, denoted by h ′ j (∆), (defined in Section 2) which encode both the h-numbers of ∆ and the underlying geometric structure of ∆.
Athanasiadis and Welker [2] define the class of Buchsbaum* complexes that specialize Buchsbaum complexes in the same way that 2-CM complexes specialize Cohen-Macaulay complexes. They show that if ∆ is Buchsbaum* over k, then the link of each vertex of ∆ is 2-CM over k and that a homology manifold is Buchsbaum* over k if and only if it is orientable over k.
Stanley [11] introduces the class of balanced simplicial complexes and shows that the rank selected subcomplexes of a balanced Cohen-Macaulay complex are Cohen-Macaulay. This result easily generalizes to the classes of 2-CM and Buchsbaum simplicial complexes. Our first goal in this paper is to show (Theorem 3.1) that the rank-selected subcomplexes of a Buchsbaum* simplicial complex are Buchsbaum*. This result answers a question posed in [2] in the affirmative.
Barnette's Lower Bound Theorem [4] says that if ∆ is a (d−1)-dimensional homology manifold without boundary and d ≥ 3, then h 2 (∆) ≥ h 1 (∆). This gives a sharp lower bound on all the f -numbers of ∆ in terms of d and n, the number of vertices in ∆. Equality in this lower bound is achieved by a stacked simplicial (d − 1)-sphere on n vertices. Kalai [7] used the theory of rigidity frameworks to prove that h 2 ≥ h 1 for simplicial homology manifolds without boundary and d ≥ 3. Nevo [9] extended this result to the class of 2-CM complexes, and Athanasiadis and Welker [2] further extend this result to the class of connected Buchsbaum* complexes. Using these results, together with Theorem 3.1, we extend a result of Goff, Klee, and Novik [5], showing that 2h 2 ≥ (d − 1)h 1 for all balanced, connected Buchsbaum* simplicial complexes with d ≥ 3. As in the case of Barnette's LBT, this bound is sharp, with equality achieved by a so-called stacked cross-polytopal sphere.
Athanasiadis and Welker [2] show that h ′ j (∆) ≥ d j when ∆ is a flag, Buchsbaum* simplicial complex of dimension d − 1. Equality is achieved in this bound by P × d , the boundary complex of a d-dimensional cross polytope.
We generalize this result, showing that h ′ j (∆) ≥ d j m j when ∆ is a flag, m-Buchsbaum* simplicial complex of dimension d−1 in Theorem 4.4. Moreover, we show that equality holds here only for the d-fold join of m + 1 disjoint vertices, answering a question posed in [2] in the case that m = 1.
The paper is structured as follows. In Section 2, we review the definitions and background information that will be necessary for the remainder of the paper. In Section 3, we study balanced Buchsbaum* complexes, proving that the rank selected subcomplexes of Buchsbaum* complexes are Buchsbaum* (Theorem 3.1). In Section 4, we prove lower bounds on balanced Buchsbaum* complexes (Theorem 4.1) and flag Buchsbaum* complexes (Theorem 4.4).
Notation and Definitions
We begin by reviewing some basic definitions on simplicial complexes. A simplicial complex ∆ on vertex set V = V (∆) is a collection of subsets τ ⊆ V, called faces, that is closed under inclusion. We say that a simplicial complex ∆ is pure if all of its facets (maximal faces under inclusion) have the same dimension.
The dimension of a face τ ∈ ∆ is dim τ = |τ | − 1, and the dimension of ∆ is dim ∆ = max{dim τ : τ ∈ ∆}. The f -vector of ∆ is the vector f (∆) = (f −1 , f 0 , . . . , f d−1 ) where dim ∆ = d − 1 and the f -numbers f i = f i (∆) denote the number of i-dimensional faces of ∆. It is oftentimes more convenient to study the h-numbers h j (∆) defined by the relation
d j=0 h j λ d−j = d i=0 f i−1 (λ − 1) d−i .
It is easy to see that the h-numbers of ∆ can be written as integer linear combinations of its f -numbers and that the f -numbers of ∆ can be written as nonnegative integer linear combinations of its h-numbers. In particular, bounds on the h-numbers of ∆ immediately yield bounds on the f -numbers of ∆.
If ∆ is a simplicial complex and τ is a face of ∆, the contrastar of τ in
∆ is cost ∆ (τ ) := {σ ∈ ∆ : σ τ }, and the link of τ in ∆ is lk ∆ (τ ) := {σ ∈ ∆ : σ ∩ τ = ∅, σ ∪ τ ∈ ∆}. If A ⊂ V (∆) is a collection of vertices in ∆, then ∆ − A is the restriction of ∆ to V (∆) \ A. When A consists of a single vertex v ∈ V (∆), we simply write ∆ − v instead of ∆ − {v}.
If Γ and ∆ are simplicial complexes on disjoint vertex sets, their simplicial join is the
(dim Γ + dim ∆ + 1)-dimensional simplicial complex Γ * ∆ := {σ ∪ τ : σ ∈ Γ, τ ∈ ∆}.
We are particularly interested in studying the class of balanced simplicial complexes, introduced by Stanley [11].
Definition 2.1 A (d − 1)-dimensional simplicial complex ∆ is balanced if there is a coloring κ : V (∆) → [d] with the property that κ(u) = κ(v) for all edges {u, v} ∈ ∆.
We assume that a balanced complex comes equipped with such a coloring κ.
The order complex of a graded poset of rank d is one example of a balanced simplicial complex. If ∆ is a balanced simplicial complex and S ⊆ [d], it is often important to study the S-rank selected subcomplex of ∆, which is defined as the collection of faces in ∆ whose vertices are colored by S.
Specifically, ∆ S = {τ ∈ ∆ : κ(τ ) ⊆ S}.
In [11] Stanley showed that
h i (∆) = |S|=i h i (∆ S );(1)
and that if ∆ is Cohen-Macaulay, then so are its rank-selected subcomplexes. A more specialized class of Cohen-Macaulay complexes is the class of doubly Cohen-Macaulay [14] showed that double Cohen-Macaulayness is a topological property, meaning that it only depends on the homeomorphism type of the geometric realization |∆| of ∆. In particular, if a (d − 1)-dimensional simplicial complex ∆ is 2-CM, then cost ∆ τ is Cohen-Macaulay of dimension d − 1 for all nonempty faces τ ∈ ∆.
(2-CM) complexes. A (d − 1)-dimensional simplicial complex ∆ is 2-CM (over k) if it is Cohen-Macaulay and ∆ − v is Cohen- Macaulay of dimension d − 1 for all vertices v ∈ ∆. Walker
Athanasiadis and Welker [2] define Buchsbaum* complexes as specializations of Buchsbaum complexes, much in the same sense that 2-CM complexes are specializations of Cohen-Macaulay complexes. For the purposes of this paper, we will use the following definition of a Buchsbaum* complex.
Definition 2.2 A (d − 1)-dimensional simplicial complex ∆ that is Buchs- baum over k is Buchsbaum* over k if, for all p ∈ |∆|, the canonical map ρ * : H d−1 (|∆|; k) → H d−1 (|∆|, |∆| − p; k) is surjective.
Henceforth we will fix a field k. When we say that a simplicial complex ∆ is Buchsbaum* without qualification, we implicitly mean that ∆ is Buchsbaum* over k. Moreover, we will implicitly compute all homology groups with coefficients in k, and we will suppress this from our notation for convenience.
First we will give an equivalent definition of the Buchsbaum* property in a combinatorial language.
Lemma 2.3 Let ∆ be a (d − 1)-dimensional Buchsbaum simplicial complex.
The following are equivalent.
(a) ∆ is Buchsbaum*;
(b) For all faces σ ⊆ τ of ∆, the map j * : H d−1 (∆, cost ∆ (σ)) → H d−1 (∆, cost ∆ (τ )),
induced by inclusion, is surjective;
(c) For all faces τ ∈ ∆, the map
ρ * : H d−1 (∆) → H d−1 (∆, cost ∆ (τ )),
induced by inclusion, is surjective.
Proof:
The equivalence of (a) and (b) is Proposition 2.8 in [2]. Taking σ = ∅ shows that (b) implies (c). Next, consider any point p ∈ |∆|, and let τ be the unique minimal face of |∆| whose relative interior contains p. Then |∆| − p retracts onto cost ∆ (τ ) and (c) implies (a).
The h ′ -vector of a Buchsbaum complex ∆ encodes both the underlying geometry of ∆ and the combinatorial data of ∆. Let β i (∆) := dim k H i (∆; k) denote the (reduced) k-Betti numbers of ∆. The h ′ -numbers of ∆ are defined by
h ′ j (∆) := h j (∆) + d j j−1 i=0 (−1) j−i−1 β i−1 (∆). We encode the h ′ -numbers of ∆ into the h ′ -polynomial h ′ ∆ (t) := d j=0 h ′ j (∆)t j .
Athanasiadis and Welker [2] prove a number of very nice properties about Buchsbaum* complexes, which we summarize here.
Theorem 2.4 Let ∆ be a (d − 1)-dimensional Buchsbaum* complex with d ≥ 2. Then 1. H d−1 (∆) = 0; 2. lk ∆ v is 2-CM for all vertices v ∈ ∆; and 3. ∆ is doubly-Buchsbaum, meaning that ∆ − v is a Buchsbaum complex of dimension d − 1, for all vertices v ∈ ∆.
Balanced Buchsbaum* Complexes
It is well known (see, for example, [11]) that if ∆ is a balanced, Cohen-Macaulay complex, then ∆ S is Cohen-Macaulay for any S ⊆ [d]. It is easy to see that this result generalizes to the classes of 2-CM complexes and Buchsbaum complexes. The purpose of this section is to generalize this result to the class of Buchsbaum* complexes.
Theorem 3.1 Let ∆ be a (d − 1)-dimensional balanced, doubly-Buchsbaum complex. For any S ⊂ [d]
, the rank selected subcomplex ∆ S is Buchsbaum*.
In particular, since a Buchsbaum* complex is doubly-Buchsbaum, Theorem 3.1 implies that the rank selected subcomplexes of a Buchsbaum* complex are Buchsbaum*.
We begin with a series of lemmas, the first of which is well-known (see, for example, [8]).
Lemma 3.2 Let Γ be a simplicial complex, and let τ be a nonempty face in
Γ. Then H i (Γ, cost Γ (τ )) ∼ = H i−|τ | (lk Γ τ ).H d−2 (cost ∆ i−1 (τ ), cost ∆ i (τ )) = 0.
Proof: By Lemma 3.2, Let τ be a nonempty face in ∆ S ⊂ ∆ k−1 ⊂ · · · ⊂ ∆ 1 ⊂ ∆. We claim that for any 0 ≤ i ≤ k, the canonical map
H d−2 (cost ∆ i−1 (τ ), cost ∆ i (τ )) ∼ = H d−3 (lk cost ∆ i−1 (τ ) v i ). Since ∆ is balanced, lk cost ∆ i−1 (τ ) v i = lk cost ∆ (τ ) v i , and lk cost ∆ (τ ) v i = {σ ∈ ∆ : σ τ, v i / ∈ σ, σ ∪ v i ∈ ∆} = cost lk v i (τ ). Since lk ∆ v i is 2-CM of dimension d − 2, it follows that H d−3 (cost lk v i (τ )) = 0.ρ i * : H d−2 (∆ i ) → H d−2 (∆ i , cost ∆ i (τ ))
is a surjection. We proceed by induction on i. When i = 0, ∆ 0 = ∆ is Buchsbaum so H d−2 (∆, cost ∆ (τ )) = 0, and the map ρ 0 * is surely surjective. Suppose now that i > 0 and consider the long exact sequence for the pair (cost ∆ i−1 (τ ), cost ∆ i (τ )):
→ H d−2 (cost ∆ i−1 (τ ), cost ∆ i (τ )) → H d−3 (cost ∆ i (τ )) ι * → H d−3 (cost ∆ i−1 (τ )) → . By Lemma 3.3, H d−2 (cost ∆ i−1 (τ ), cost ∆ i (τ )
) ∼ = 0, and hence the map ι * is an injection.
Next, we consider the inclusion map (∆ i , cost ∆ i (τ )) ֒→ (∆ i−1 , cost ∆ i−1 (τ )), which induces the following commutative diagram of long exact sequences.
H d−2 (∆ i ) H d−2 (∆ i , cost ∆ i (τ )) H d−3 (cost ∆ i (τ )) H d−2 (∆ i−1 ) H d−2 (∆ i−1 , cost ∆ i−1 (τ )) H d−3 (cost ∆ i−1 (τ )) - ρ i * ? ? - ∂ i ? ι * - ρ i−1 * - ∂ i−1
By the inductive hypothesis, the map ρ i−1 * is surjective and so the map ∂ i−1 is the zero map. By commutativity of the above diagram, ι * • ∂ i is the zero map, and since ι * is an injection, the map ∂ i is also the zero map. Thus by exactness, ρ i * :
H d−2 (∆ i ) → H d−2 (∆ i , cost ∆ i (τ )
) is a surjection. This establishes the claim. In particular, when i = k, we have shown that the canonical map ρ * : H d−2 (∆ S ) → H d−2 (∆ S , cost ∆ S (τ )) is surjective, and hence ∆ S is Buchsbaum*.
∆ is m- Buchsbaum* if ∆ is Buchsbaum and ∆−A is Buchsbaum* of dimension d−1 for any subset A ⊂ V (∆) with |A| < m. Corollary 3.5 Let ∆ be a (d − 1)-dimensional balanced simplicial complex that is m-Buchsbaum* over k. For any S ⊆ [d]
, the rank selected subcomplex ∆ S is m-Buchsbaum* over k.
Proof:
By Lemma 5.6 in [2], ∆ is (m + 1)-Buchsbaum over k. For any
subset A ⊆ V (∆ S ) with |A| < m, the complex ∆ − A is doubly Buchsbaum. Thus (∆ − A) S = ∆ S − A is Buchsbaum* by Theorem 3.1.
Lower Bounds
Fix integers n and d such that d divides n. Let P × d denote the boundary complex of the d-dimensional cross polytope. Following [5], define a stacked cross-polytopal sphere ST × (n, d − 1) by taking the connected sum of n d − 1 copies of P × d . In each connected sum, we identify vertices of the same colors so that ST × (n, d − 1) is a balanced (d − 1)-sphere on n vertices.
Athanasiadis and Welker ( [2], Theorem 4.1) prove that if ∆ is a connected, (d − 1)-dimensional Buchsbaum* complex with d ≥ 3, then the graph of ∆ is generically d-rigid. This generalizes Nevo's result that h 2 (∆) ≥ h 1 (∆) when ∆ is 2-CM ( [9], Theorem 1.3). Using Theorem 4.1 from [2] in place of Nevo's result and the conclusion of Theorem 3.1, the techniques used to prove Theorem 5.3 in [5] give the following Lower Bound Theorem for balanced Buchsbaum* complexes.
Theorem 4.1 Let ∆ be a connected, balanced, Buchsbaum* complex of di- mension d − 1 with d ≥ 3. Then d · h 2 (∆) ≥ d 2 h 1 (∆). In particular, if d divides n = f 0 (∆), then f j (∆) ≥ f j (ST × (n, d − 1) for all j.
Hersh and Novik [6] define the short simplicial h-numbers of a simplicial complex ∆ as h j (∆) := v∈∆ h j (lk ∆ v) for 0 ≤ j ≤ d − 1. Swartz [12] proves that the short simplicial h-numbers satisfy
h j−1 (∆) = j · h j (∆) + (d − j + 1)h j−1 (∆).(2)
We use this formula, together with Theorem 4.1 to prove the following theorem.
Theorem 4.2 Let ∆ be a balanced Buchsbaum* complex of dimension
d − 1 with d ≥ 4. Then d · h 3 (∆) ≥ d 3 h 1 (∆).
Proof: The link of each vertex v ∈ ∆ is 2-CM, and hence by Theorem 5.3 in [5],
2h 2 (lk ∆ v) ≥ (d − 2)h 1 (lk ∆ v) for all v ∈ ∆. Thus 2 · (3h 3 (∆) + (d − 2)h 2 (∆)) = 2 · h 2 (∆) ≥ (d − 2) h 1 (∆) = (d − 2)(2h 2 (∆) + (d − 1)h 1 (∆)),
and the desired result follows.
Björner and Swartz [13] have conjectured that the inequality h d−1 ≥ h 1 holds for all 2-CM complexes with d ≥ 3. When d = 4, the conclusion of Theorem 4.2 continues to hold without the assumption that ∆ is balanced, establishing that h 3 ≥ h 1 for all 3-dimensional Buchsbaum* (and hence 2-CM) complexes.
Athanasiadis and Welker [2] prove that if a (d − 1)-dimensional Buchsbaum* simplicial complex ∆ is flag, then h ′ ∆ (t) ≥ (1 + t) d , where the inequality is interpreted coefficient-wise. Motivated by a question in [1], they pose the following question.
h ′ j = d j for some 1 ≤ j ≤ d − 1, is ∆ necessarily isomorphic to P × d ?
We will generalize the result of Athanasiadis and Welker to the class of m-Buchsbaum* simplicial complexes and answer Question 4.3 for this class. For fixed positive integers m and d, we define the simplicial complex P(m + 1, d) to be the d-fold join of m+1 disjoint vertices. We remark first that P(m+1, d) is (m+1)-CM and hence m-Buchsbaum* by Proposition 5.6 in [2], and second that P(2, d) = P × d .
Theorem 4.4 Let ∆ be a (d − 1)-dimensional flag simplicial complex that is m-Buchsbaum* over the field k. Then h ′ ∆ (t) ≥ (1 + mt) d . Moreover, if h ′ j (∆) = d j m j for some 1 ≤ j ≤ d − 1, then ∆ is isomorphic to P(m + 1, d).
Proof: We prove the claim by induction on d. When d = 1, it is clear that f 0 (∆) ≥ m + 1, so suppose that d ≥ 2.
Let F be a (d − 2)-dimensional face of ∆. Since lk ∆ F is m-Buchsbaum*, there are at least m + 1 vertices v 1 , . . . , v m+1 in lk ∆ F . Since ∆ is flag, no two of these vertices v i are connected by an edge in ∆. In particular, this means that lk ∆−{v 1 ,...,v i } (v i+1 ) = lk ∆ v i+1 . Following the techniques of Theorem 1.3 in [1] or Corollary 3.3 in [2],
h ′ ∆ (t) = h ′ ∆−v 1 (t) + t · h lk v 1 (t) = h ′ ∆−{v 1 ,v 2 } (t) + t · h lk v 2 (t) + t · h lk v 1 (t) = · · · = h ′ ∆−{v 1 ,v 2 ,...,vm} (t) + t · h lk vm (t) + · · · + t · h lk v 1 (t) ≥ h lk v m+1 (t) + t · h lk vm (t) + · · · + t · h lk v 1 (t) ( †) ≥ (1 + mt) d−1 + mt(1 + mt) d−1 = (1 + mt) d .
To obtain line ( †), we use the fact that ∆ is m-Buchsbaum* and hence (m + 1)-Buchsbaum.
Thus ∆ − {v 1 , . . . , v m } is a (d − 1)-dimensional Buchs- baum complex and h ′ i (∆ − {v 1 , . . . , v m }) ≥ h i (lk ∆ v m+1 ) for all 0 ≤ i ≤ d. Suppose next that h ′ j (∆) = d j m j for some 1 ≤ j ≤ d − 1.
When d = 2, the only case to consider is j = 1. It is easy to see that the complete bipartite graph on two disjoint vertex sets of size m + 1 is the only flag (i.e. trianglefree) m-Buchsbaum* graph with exactly 2(m + 1) vertices.
Suppose now that d ≥ 3. From the argument used above to show that h ′ j ≥ d j m j , it follows that h ′ j (∆) = d j m j if and only if h j (lk ∆ u) = d−1 j m j and h j−1 (lk ∆ u) = d−1 j−1 m j−1 for all vertices u ∈ ∆. In particular, one of the numbers j and j −1 lies in the set {1, 2, . . . , d−2}, and so lk ∆ u is isomorphic to P(m + 1, d − 1) for all vertices u ∈ ∆ by our inductive hypothesis.
Choose a vertex u 1 ∈ ∆, and let Γ := lk ∆ u 1 . Let v be a vertex of Γ. Then v has (m + 1)(d − 1) neighbors in ∆, and (m + 1)(d − 2) of these neighbors lie in Γ. Let u 1 , u 2 , . . . , u m+1 be the neighbors of v in ∆ that do not lie in Γ. Now consider a vertex v ′ ∈ Γ that is adjacent to v. Then lk ∆ {v, v ′ } has (m + 1)(d − 2) vertices, and (m + 1)(d − 3) of these vertices lie in Γ. The remaining m + 1 vertices of lk ∆ {v, v ′ } are adjacent to v, and the only such vertices are {u 1 , . . . , u m+1 }. Thus u i v ′ is an edge for all i. Since Γ is connected, it follows that u i w is an edge for all w ∈ Γ. We claim that ∆ is connected, and since ∆ is flag, it follows that ∆ is isomorphic to P(m + 1, d).
Finally, we show that ∆ is connected. When j = 1, this is obvious as each connected component of ∆ requires (m + 1) · d vertices. When j ≥ 2, it is relatively easy to see that h ′ j (∆) = h ′ j (∆ t ), where the sum is taken over all connected components ∆ t of ∆. Each connected component of ∆ is m-Buchsbaum*, and the claim follows.
Taking m = 1 answers Question 4.3. We note that the assumption that ∆ is flag in Theorem 4.4 can easily be replaced by the assumption that ∆ is balanced to yield the same conclusion. The following corollary is immediate and very interesting, especially when m is large.
Concluding Remarks
Recall that a set of vertices τ in a simplicial complex ∆ is called a missing face if τ / ∈ ∆ but σ ∈ ∆ for all σ τ . A flag simplicial complex, for example, only has missing faces of size two. For any simplicial complex ∆, let ∆ * q denote the simplicial join of q disjoint copies of ∆. Let Σ j denote the j-dimensional simplex and ∂Σ j its boundary complex.
In [10], Nevo studies the class of (d − 1)-dimensional simplicial complexes with no missing faces of dimension larger than i. For fixed integers d and i, write d = qi + r where q and r are integers with 1 ≤ r ≤ i. Nevo defines a certain (d − 1)-dimensional sphere S(i, d − 1) with no missing faces of dimension larger than i by S(i, d − 1) := (∂Σ i ) * q * ∂Σ r .
Goff, Klee, and Novik ( [5], Theorem 3.1(2)) prove that h ∆ (t) ≥ h S(i,d−1) (t) for all (d−1)-dimensional simplicial complexes ∆ that are 2-CM with no missing faces of dimension larger than i. This result and Theorem 4.4 motivate the following question.
Fix integers m, i, and d and consider the class of (d − 1)-dimensional simplicial complexes that are m-CM with no missing faces of dimension larger than i. As before, write d = qi+ r with 1 ≤ r ≤ i, and consider the simplicial complex S(m, i, d − 1) := (Skel i−1 (Σ m+i−2 )) * q * Skel r−1 (Σ m+r−2 ).
Notice that S(2, i, d − 1) is Nevo's S(i, d − 1), and S(m, 1, d − 1) is P(m, d). Each join-summand Skel i−1 (Σ m+i−2 ) is m-CM, and hence S(m, i, d − 1) is a (d − 1)-dimensional simplicial complex that is m-CM with no missing faces of dimension larger than i. It seems natural, therefore, to pose the following question.
Question 5.1 Let ∆ be a (d − 1)-dimensional simplicial complex that is m-CM with no missing faces of dimension larger than i. Is it necessarily true that h ∆ (t) ≥ h S(m,i,d−1) (t)?
Lemma 3. 3
3Let ∆ be a balanced, doubly Buchsbaum simplicial complex of dimension d − 1, and let c ∈ [d]. Choose vertices v 1 , . . . , v i of color c, and let ∆ i−1 = ∆ \ {v 1 , . . . , v i−1 } and ∆ i = ∆ \ {v 1 , . . . , v i }. Then for S = [d] − c and any nonempty face τ ∈ ∆ S ,
Now we proceed with the proof of Theorem 3.1. Proof: (Theorem 3.1) Fix a coloring κ : V (∆) → [d]. We need only consider those S ⊂ [d] with |S| = d − 1. Inductively, this is sufficient as any Buchsbaum* complex is doubly Buchsbaum. Suppose S = [d] − {c} and consider the vertices {v 1 , . . . , v k } ∈ ∆ with κ(v i ) = c. For 1 ≤ i ≤ k, let ∆ i := ∆ − {v 1 , . . . , v i }, and when i = 0, set ∆ 0 := ∆.
)
Let ∆ be a (d − 1)-dimensional simplicial complex, and let m be a nonnegative integer. We say that
i)) Let ∆ be a (d − 1)-dimensional flag, Buchsbaum* simplicial complex. If
Corollary 4. 5
5Let ∆ be a (d − 1)-dimensional flag (or balanced) simplicial complex that is m-Buchsbaum* over k. Then (−1) d−1 χ(∆) ≥ m d . In particular, if ∆ is Cohen-Macaulay over k, then β d−1 (∆) ≥ m d .
AcknowledgementsWe are grateful to Christos Athanasiadis and Volkmar Welker for introducing us to the problems discussed in this paper. Our thanks also go to Christos Athanasiadis and Isabella Novik for a number of helpful conversations during the development of this paper. Jonathan Browder's research is partially supported by VIGRE NSF Grant DMS-0354131.
Some combinatorial properties of flag simplicial pseudomanifolds and spheres. Ark. Mat. C A Athanasiadis, to appearC.A. Athanasiadis. Some combinatorial properties of flag simplicial pseudomanifolds and spheres. Ark. Mat. to appear.
C A Athanasiadis, V Welker, arXiv:0909.1931Buchsbaum* complexes. C.A. Athanasiadis and V. Welker. Buchsbaum* complexes. arXiv: 0909.1931.
Cohen-Macaulay connectivity and geometric lattices. K Baclawski, European J. Combin. 34K. Baclawski. Cohen-Macaulay connectivity and geometric lattices. Eu- ropean J. Combin., 3(4):293-305, 1982.
A proof of the lower bound conjecture for convex polytopes. D Barnette, Pacific J. Math. 46D. Barnette. A proof of the lower bound conjecture for convex polytopes. Pacific J. Math., 46:349-354, 1973.
M Goff, S Klee, I Novik, arXiv: 0907:1669Balanced complexes and complexes without large missing faces. to appear in Arkiv för MatematikM. Goff, S. Klee, and I. Novik. Balanced complexes and complexes without large missing faces. arXiv: 0907:1669, to appear in Arkiv för Matematik.
A short simplicial h-vector and the upper bound theorem. P Hersh, I Novik, Discrete Comput. Geom. 283P. Hersh and I. Novik. A short simplicial h-vector and the upper bound theorem. Discrete Comput. Geom., 28(3):283-289, 2002.
Rigidity and the lower bound theorem. G Kalai, I. Invent. Math. 881G. Kalai. Rigidity and the lower bound theorem. I. Invent. Math., 88(1):125-151, 1987.
Topological results in combinatorics. J Munkres, Michigan Math. J. 15J. Munkres. Topological results in combinatorics. Michigan Math. J., 15:113-128, 1984.
Rigidity and the lower bound theorem for doubly Cohen-Macaulay complexes. E Nevo, Discrete Comput. Geom. 391-3E. Nevo. Rigidity and the lower bound theorem for doubly Cohen- Macaulay complexes. Discrete Comput. Geom., 39(1-3):411-418, 2008.
Remarks on missing faces and lower bounds on face numbers. E Nevo, Electronic J. Combin. 16211Research Paper 8E. Nevo. Remarks on missing faces and lower bounds on face numbers. Electronic J. Combin., 16(2), 2009-10. Research Paper 8, 11pp.
Balanced Cohen-Macaulay complexes. R Stanley, Trans. Amer. Math. Soc. 2491R. Stanley. Balanced Cohen-Macaulay complexes. Trans. Amer. Math. Soc., 249(1):139-157, 1979.
Lower bounds for h-vectors of k-CM, independence, and broken circuit complexes. E Swartz, SIAM J. Discrete Math. 1835E. Swartz. Lower bounds for h-vectors of k-CM, independence, and bro- ken circuit complexes. SIAM J. Discrete Math., 18(3):647-661, 2004/05.
g-elements, finite buildings and higher Cohen-Macaulay connectivity. E Swartz, J. Combin. Theory Ser. A. 1137E. Swartz. g-elements, finite buildings and higher Cohen-Macaulay con- nectivity. J. Combin. Theory Ser. A, 113(7):1305-1320, 2006.
Topology and combinatorics of ordered sets. J W Walker, PhD thesisJ.W. Walker. Topology and combinatorics of ordered sets. PhD thesis, M.I.T., 1981.
| []
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[
"Melting Scenario for Coulomb-interacting Classical Particles in Two-dimensional Irregular Confinements",
"Melting Scenario for Coulomb-interacting Classical Particles in Two-dimensional Irregular Confinements"
]
| [
"Dyuti Bhattacharya \nIndian Institute of Science Education and Research-Kolkata\nMohanpur Campus741252India\n",
"Amit Ghosal \nIndian Institute of Science Education and Research-Kolkata\nMohanpur Campus741252India\n"
]
| [
"Indian Institute of Science Education and Research-Kolkata\nMohanpur Campus741252India",
"Indian Institute of Science Education and Research-Kolkata\nMohanpur Campus741252India"
]
| []
| The "melting" of self-formed rigid structures made of a small number of interacting classical particles confined in an irregular two-dimensional space is investigated using Monte Carlo simulations. It is shown that the interplay of long-range Coulomb repulsions between these particles and the irregular confinement yields a solid-like phase at low temperatures that possesses a bond-orientational order, however, the positional order is depleted even at the lowest temperatures. Upon including thermal fluctuations, this solid-like phase smoothly crosses over to a liquid-like phase by destroying the bond-orientational order. The collapse of solidity is shown to be defect mediated, and aided by the proliferation of free disclinations. The behavior of different physical observables across the crossover region are obtained. Our results will help quantifying melting found in experiments on systems with confined geometries. | null | [
"https://arxiv.org/pdf/1302.4575v2.pdf"
]
| 119,105,278 | 1302.4575 | ad328de8bf79989d43bd8619d782c54dba4b8827 |
Melting Scenario for Coulomb-interacting Classical Particles in Two-dimensional Irregular Confinements
12 Jun 2013
Dyuti Bhattacharya
Indian Institute of Science Education and Research-Kolkata
Mohanpur Campus741252India
Amit Ghosal
Indian Institute of Science Education and Research-Kolkata
Mohanpur Campus741252India
Melting Scenario for Coulomb-interacting Classical Particles in Two-dimensional Irregular Confinements
12 Jun 2013arXiv:1302.4575v2 [cond-mat.dis-nn]
The "melting" of self-formed rigid structures made of a small number of interacting classical particles confined in an irregular two-dimensional space is investigated using Monte Carlo simulations. It is shown that the interplay of long-range Coulomb repulsions between these particles and the irregular confinement yields a solid-like phase at low temperatures that possesses a bond-orientational order, however, the positional order is depleted even at the lowest temperatures. Upon including thermal fluctuations, this solid-like phase smoothly crosses over to a liquid-like phase by destroying the bond-orientational order. The collapse of solidity is shown to be defect mediated, and aided by the proliferation of free disclinations. The behavior of different physical observables across the crossover region are obtained. Our results will help quantifying melting found in experiments on systems with confined geometries.
I. INTRODUCTION
The phenomenon of melting [1][2][3][4][5] has always fascinated physicists from the early dawn of study of the condensed phases, because it is encountered in everyday life. The order (or quasi-order) characterizing a solid, as well as the mechanism through which a solid melts, depends crucially on the spatial dimensionality. 6 For example, a two-dimensional (2D) system presents additional challenges in comprehending melting than three-dimensional one because of the increased significance of fluctuations. 7 While comprehensive research effort has enriched a coherent understanding of 2D melting in bulk systems, 5 the physics of melting in confined systems made of small number of particles has not been studied as much. One cannot expect a sharp phase-transition 8 in a confined system unlike in the bulk -a thermodynamic phase itself is ill-defined with a finite number of particles. However, some signatures of the "melting", say, a Crossover (CO) from a solid-like to a liquid-like phase have been found. 9,10 Such 2D-clusters find significant experimental relevance and are realized in a variety of experiments, such as, radiofrequency ion traps, 11 electrons on the surface of liquid He, 12 electrons in quantum dots in semiconductor heterostructure, 13 and in dusty plasmas. 14 The interactions between the constituent charged particles are expected to maintain the bare long range (∼ r −1 ) Coulomb repulsion due to the lack of screening in small clusters compared to the bulk. The resulting solid-like phase in a cluster has been termed as a Wigner Molecule (WM), 13 due to its analogy with the Wigner Crystal (WC) in bulk systems. 15 Incidentally, a 2D WC undergoes melting to a Fermi liquid even at zero temperature (T = 0), 16 driven by quantum fluctuations, and so does a WM in circularly symmetric traps. [17][18][19] However, for a variety of experimental clusters at low T , the quantum zeropoint motion, inherent to particles in a bound states, and the associated quantum fluctuations are not so important: either due to the finite operating T , or because of the heavy mass of the particles or their low density, or a combination of various experimental reasons. The rich physics of the CO is thus contributed entirely by the classical thermal fluctuations. Temperature driven melting of classical Coulomb-interacting 2D plasma was studied numerically 20 in the past, and the corresponding Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory of 2D melting has also been developed. [21][22][23][24] Similar thermal melting of clusters in 2D harmonic trap has recently been studied extensively, 17,31 and also been realized experimentally, 32 leading to a detailed insight through the measurements of various observables including relative positional fluctuations and associated Lindemann ratio, 32 pair-correlation function, 33 static structure factor 34 and addition spectra. 35 Progresses in the field of high accuracy fabrication made the clusters ultra-tunable and the shape of confinement is controlled at will by electrostatic and magnetic methods. 14 These systems naturally have become a hotbed for systematic study of a complex interplay of Coulomb-repulsion, quantum interference effects of the confinement, level quantization due to their smallness, and finally, disorder in the form of irregularities in the confinement. Existence of irregularities, at least in large quantum dots has been established in Coulomb blockade experiments, 36,37 and corresponding theories based on "Universal Hamiltonian" 38,39 have been put forward. It is this last point -the effect of disorder or irregularities on the melting mechanism, that has drawn a significant attention in the recent past 40 in the context of the quantum melting in 2D WC. 41 While the melting scenario is still unsettled even for the disordered bulk 2D systems, with proposals of intermediate exotic phases, 42,43 the role of irregularities on the melting of WM has received relatively little attention (See however, [44][45][46][47] and is addressed in this work in details.
There are relevant questions with respect to the CO from a solid-like to a liquid-like phase in an Irregular Wigner Molecule (IWM): Is there a low-temperature solid-like phase at all, in spite of the irregularities? Does the CO occur more or less abruptly or gradually with T ? What are the reasonable criteria for identifying the possible CO in an IWM? And finally, what is the physical mechanism governing the CO, if any? While addressing these important questions within a framework of a systematic calculation, we first show that there indeed exists a phase that is solid-like, where the solidity is arising primarily from the orientational order at low T . Upon inclusion of thermal fluctuations, the IWM crosses over to a liquid-like phase. From a detailed study of the behavior of different observables, we estimate the temperature width ∆T X for the crossover. We also present compelling numerical evidences illuminating the mechanism for this crossover, which turns out to be associated with formation and proliferation of free disclinations on top of the irregular geometry that diminishes the positional order in the IWM.
II. MODEL, PARAMETERS AND METHOD
We consider a system of N classical particles interacting via a Coulomb repulsion
V Coul = i<j ( r i − r j ) −1 (The standard
Coulomb energy factor C = q 2 /ε is assumed to be unity). The effect of irregularity is introduced through the following 2D irregular quartic confinement potential V c :
V c (x, y) = a x 4 b + by 4 − 2λx 2 y 2 + γ(x − y)xyr ,(1)
where r = x 2 + y 2 . The parameter a makes the confinement narrow or shallow and thus controls the average density of particles in the system. We present all our results for a = 1.
Effect of a change in the value of a will be commented on later in this report. We choose b = π/4 49 that breaks x-y symmetry of the quartic oscillator. λ in Eq. (1) controls the chaoticity, and γ breaks the reflection symmetry. Our interests lie primarily in unfolding the universal features of disordered systems, it is thus essential that the chosen V c respects those. Fortunately, there exists a large body of literature confirming the universal behavior of chaotic quantum dots with our choice of V c in Eq. (1). This is particularly important because a 'soft' potential (with smooth boundary like ours, and relevant for experiments where confinement is set up by electrostatic and magnetic means) rarely shows a wide range of chaotic behavior, unlike the billiards. 48 The classical dynamics of non-interacting particles in the above confinement has been studied in details 48,49 with the identification of the integrable and the chaotic regimes. The interplay of Coulomb-repulsion between spin-1/2 fermions and the irregularities of V c have also been looked at. 50,51 With the aim of generating data on self-similar copies of irregularities for better statistics, we choose λ ∈ [0.565, 0.635] and γ ∈ [0.1, 0.2] and generate 5 different realizations of confinement, each for a given combination of (λ, γ). Finally, we study the behavior of N = 10 to 151 particles and the thermal evolution of several observables described in the next section.
While the particles have no dynamics at T = 0 due to the lack of zero-point motion, many low-energy configurations with updated location of particles contribute to the partition function for T > 0. As a result, the thermal contribution to the physical observables is weighted with appropriate Boltzmann factors. These observables thus carry in their thermal evolution the complex interplay of thermal kicks on these classical particles and their inter-particle Coulomb potential energy, as well as the effect of the external irregular confinement. The thermal evolution of the system is studied using standard Monte Carlo (MC) simulated annealing 52 aided by Metropolis algorithm. 53 Simulated annealing is expected to track the appropriate low-energy states consistent with the Boltzmann probability at a given T (T includes k B , the Boltzmann factor for all our results), and can obtain the Ground state configuration in the limit T → 0, for an appropriate choice of a slow annealing schedule. 54
III. RESULTS
We describe our results by showing evidences for the 'solidity' in the proposed IWM at T = 0. An example of a groundstate configuration (GSC) is shown in Fig. 1(a) for a fixed set of parameters (See the caption). Such a spatial configuration of particles (for all different realizations) was obtained by running the simulated annealing to a very low T ∼ 10 −5 , and then letting the system relax to T = 0 configuration following the downhill moves alone in the energy landscape. While it is not possible to access the true GSC for each individual run, we ensure that the GSC is indeed achieved by the following statistical analysis: We start the T = 0 calculation from ∼ 200 different configurations generated in the equilibrium MC runs at the lowest T , for a given realization. Starting with these as initial configurations, not all of the 200 runs converge at T = 0 to the same final GSC. However, for all the cases we study in the entire parameter space, more than 60% of them do, with exactly the same energy E 0 . Furthermore, the runs for which the final ground-state at T = 0 are different from the true GSC (with energy E 0 ) always had an energy, E > E 0 , giving us the confidence of identifying the true GSC. We also ensure that all the forces (originating from the inter-particle potential and confinement) on each particle in the true GSC add up to zero, modulo a numerical tolerance, by using first principle Newtonian mechanics.
Snapshots of the MC configurations, tracing the paths traversed by individual particles in space during MC evolution, are presented in Fig. 1(b) and (c) for T = 0.015 and 0.065 respectively, in the unit of energy. These snapshots play important role in visualizing the crossover from a solid-like to a liquid-like behavior in the IWM in the following manner: In Fig. 1(b), where the "melting" has just started, the thermal motion of some particles becomes correlated leading to the incipient melting through the formation of a tortuous path. While such paths (for a given N and realization) always occur at similar T , signaling the commencement of melting, their locations are completely random in space without any preference to the bulk or to the boundary. We will bring back our attention to these interesting structures when we discuss the possible mechanism of the crossover. Fig. 1(c) illustrates the "melted" state, where every particle travels almost everywhere in the system. Fig. 1(d) presents a similar snapshot in the same confinement depicting the incipient melting for hard-core particles, 55 with the average core size same as the Coulomb-interacting particles in the IWM as inferred from the pair distribution function (defined below). Comparison of Fig. 1(b) and Fig. 1(d) shows that the melting starts predominantly on the boundary for the hard-core particles, unlike the Coulomb-interacting ones. While pre-melting on the boundary in solids with short-range interactions has been discussed for a long time, 56,57 our results with long-range Coulomb re- pulsion show that it can occur anywhere in the system, based on the statistics from different realizations of the confinement.
Having established that we indeed track the true GSC, the next question is: Are these IWM equivalent to the solid-like phase in confined systems? Even the bulk 2D WC leads to very interesting physics. 16,58 Extensive research 59 confirmed that an ideal 2D WC phase is characterized by both positional and orientational orders, and the collapse of both are necessary to cause its final melting, and thus it is likely to occur in two stages. 5,23 Fig. 1(a) clearly shows that the Positional Order (PO) is already depleted even at T = 0, because of the breaking of translational symmetry by the confinement, yielding an amorphous solid-like phase. Lack of positional order is also verified by calculating ρ k = N −1 i exp(i k. r i ), r i being position of i-th particle. ρ k shows only broad humps for IWM even at T = 0 instead of sharp Bragg-peaks. This points towards the absence of PO down to T = 0. We do not see any significant thermal evolution of ρ k either.
The orientational order, on the other hand, is evident from the nearly perfect 6-coordinated environment for all the particles, except obviously for those on the boundary. We reiterate that the orientational order, together with the positional order, characterizes solidity in the 2D bulk systems. Our results also indicate that a self-formed IWM makes approximately a triangular lattice (modulo the irregular lattice lines in the absence of PO) -typical 2D Bravais lattice minimizing the total energy with long-range interactions. A quantitative estimate of the orientational order is measured through Bond-Orientational Order (BOO), 24 ψ 6 ( r) (See Fig. 3), and the Bond-Orientational Correlation Function (BOCF), g 6 (r), to be defined in the next section (See Fig. 4). The other signature of solidity, even for the amorphous solid-like phase in an IWM, lies in its rigidity prohibiting any significant root mean square fluctuations of the constituent particles from their equilibrium positions at low T . Such fluctuations measured in terms of the average inter-particle spacing are known as Lindemann Ratio (LR), L. 25 Our results in Fig. 2 demonstrate that the average LR is essentially zero for all T < 0.01, and thus provide further evidence for the solidlike IWM at T = 0.
How does an IWM melt with increasing T ? Insight could be derived from the LR given as:
L = N −1 i L i , where L i = a −1 i | u i | 2 for i-th particle, and u i = r i − r (0) i . Here r (0) i
is the position of the i-th particle in the initial configuration (i.e., the one at the end of the equilibration MC steps), and a i is the average distance of the i-th particle with its neighbors. We track the thermal evolution of L (averaged over all realizations) in Fig. 2. While L(T < 0.01) ≈ 0, LR increases dramatically for 0.01 ≤ T ≤ 0.05, beyond which the growth of L becomes far more gradual. This range of temperature thus identifies the crossover width, ∆T X , between the IWM and its melted state. Distribution of L i , P (L i ), collected over all particles and also over realizations of confinement is shown in the top-left inset of Fig. 2 for 0.025 ≤ T ≤ 0.065. It indicates that a peak in P (L i ) starts appearing for non-zero values of L i for all T ≥ 0.03. The variance, σ L , of P (L i ) follows a similar T -evolution as the L itself, as presented in the main panel of Fig. 2, giving further confidence in identifying ∆T X as the crossover width.
The above definition of L is difficult to implement for the bulk 2D systems, because, | u| 2 can show logarithmic divergence 26 with system size. Therefore a modified Lindemann parameter, Γ, has been proposed 27 in terms of the relative inter-particle distance fluctuations:
Γ = 1 N N i=1 1 a 2 i N b N b j=1 ( u i − u j ) 2 1/2 ,(2)
where, the summation on j runs over N b number of nearest neighbors of i-th particle. Γ is free from any divergence and it has been used extensively to track 2D melting. 28,29 However, particles are confined in an IWM within a finite length-scale, and thus, even L remains free of divergence in an IWM in contrast to the bulk. We, nevertheless, calculated the evolution of Γ and the variance σ Γ of P (Γ i ), defined exactly in the same way as those for L i . The resulting Γ and σ Γ are shown as a function of T in the bottom-right inset of Fig. 2 for N = 141.
The similarity of their T -evolution with those for L serves as a consistency check for our results. While the thermal evolution of Γ shows a sharp transition (i.e., ∆T X → 0) for the bulk 2D system, 27 a confined (and hence finite) system would always have a finite ∆T X We found that its value for IWM is typically larger than that for a circularly symmetric WM. 31 This can be crudely taken as the smearing of ∆T X by irregularities. Our study also indicates that ∆T X does not change much for N > 35, and does not seem to depend on the model parameters [γ, λ]. Beyond T > 0.05 the thermal evolution of L follows a weaker linear trend, as expected for a liquid with particles moving around diffusively. 60 T -dependence of LR had been used to identify melting in the bulk (respecting translational symmetry in all directions), 27 as well as in circular dots. 17,31 In the latter case, the translational symmetry in the azimuthal direction and the lack of it in the radial direction helps defining the radial and the azimuthal LR separately, however, we don't see such separation because our confinement breaks all symmetries.
As mentioned already, GSC in an IWM seems to possess 6coordinated bond-orientational order. It is important that we study the temperature dependence of the local 6-fold bondorientational order parameter, 61 defined as: , for N = 125 shows a strong peak at ψ6 ≈ 1 signifying nearly regular 6-coordinated IWM at low T . P (ψ6) broadens with T across ∆TX , finally producing a very broad distribution ranging between 0 and 1 beyond the crossover from solid-like to liquid-like phase. The boundary-particles, not having 6 neighbors surrounding them, were not considered in P (ψ6).
ψ 6 ( r) = 1 6 6 k=1 exp(6iθ k )(3)
Here r is the position of the particle in question, and θ k is the angle that the particle at r makes with its six closest neighbors k, relative to any arbitrary reference axis. ψ 6 ( r) measures local orientational order in the IWM based on the principle that all bonds in a perfect triangular lattice should have the same θ k , modulo π/3, implying ψ 6 = 1 for all r. We present the distribution P (ψ 6 ) in Fig. 3, for N = 125 collecting data from all realizations. The sharp peak in P (ψ 6 ) at ψ 6 = 1 demonstrates the persistence of strong BOO in the GSC of an IWM at low T , particularly for large N . In a liquid, on the other hand, lack of BOO forces random θ k ∈ [0, 2π], resulting in a broad distribution of P (ψ 6 ) ranging between 0 and 1. Our result of T -dependence of P (ψ 6 ) indicates that while the change from solid-like to liquid-like crossover takes place continuously with T , the most dramatic change in which the remnance of the low-T peak of P (ψ 6 ) gets washed out, occur in same range of ∆T X where L and σ L changes most rapidly (See Fig. 2). This provides further support for the identification of ∆T X as the CO width. For clusters with N ≤ 35, P (ψ 6 ) develops qualitatively non-universal features, including multiple peaks for some values of N , which prohibits reliable identification of ∆T X . The ratio of the number of particles at the boundary to those in the bulk increases resulting in a fewer data-points for constructing P (ψ 6 ). Nevertheless, 6-coordinated bulk particles can be broadly identified down to N ∼ 20, at the lowest temperature.
While ψ 6 ( r) describes strength of local BOO, the longrange nature of this order is best discussed in terms of the associated bond-orientational correlation function (BOCF), de- These peaks weaken with increasing T , finally producing only couple of initial humps like a liquid. Data for different T are shifted vertically for clarity. The boundary particles are not taken into account for g6(r), because they don't have 6 surrounding neighbors. Inset shows the T -evolution of PVR (defined in the text). The smooth decrease of PVR over ∆TX is indicative of a smooth CO. fined as: g 6 (r) ≡ g 6 (| r|) = ψ * 6 (| r ′ |)ψ 6 (| r ′ − r|) ,
which measures the distance up to which local ψ 6 are correlated. It is exactly the same way the pair distribution function (PDF), g(r), defined as:
g(r) ≡ g(| r|) = δ(| r ′ |)δ(| r ′ − r|) ,(5)
which measures the positional correlation, describing the probability of finding another particle at a distance r from a given one. Our results for BOCF in Fig. 4 at low T show well defined Bragg-peaks at specific values of r for the largest inter-particle distances in IWM indicating that the BOO is long-ranged in the scale of linear dimension of the IWM. It is well known that g 6 (r) (and g(r) as well) in a bulk system tends to a constant value for r → ∞ indicating a uniform density. In IWM the distance between any two particles is limited by the linear dimension of IWM and as a result g 6 (r) ( and g(r) too) must vanish beyond the system size. With increasing T , the long-range nature diminishes by washing out the peaks progressively at large r, and beyond T ≥ 0.05, only a liquid like behavior persists featuring only the first couple of humps, as presented in Fig. 4. In the liquid-like phase, g 6 (r) (and g(r) as well) is expected to follow the profile of the average radial density for large r, larger than the positions of the initial humps. This is the origin of a smooth r-dependent background, we obtain in IWM at large T . The window of T , in which the peaks at large r of g 6 (r) starts disappearing, leaving only liquid-like features, is consistent with the ∆T X reported from the Lindemann analysis. An obvious quantitative measure of how fast these Bragg peaks are depleting is the peak-to-valley ratio (PVR) -the ratio of the value of g 6 at its first peak at the lowest r to the same at the valley immediately after, is also presented as an inset in the same figure.
We have also studied the thermal evolution of g(r), which is presented in Fig. 5. The qualitative features are similar to those of g 6 (r), except that the peaks in g(r) are less sharply defined. It is known that for 2D bulk system the long-range nature of g 6 (r) survives up to larger T 5 than for g(r), leading to a hexatic phase 22 at intermediate temperatures. A reliable identification of such a phase from numerics 30 is difficult in the IWM owing to its small linear dimension. More insights on the CO, however, are obtained from the PDF of individual particle, g i (r), in the following manner. The pair-distribution function of i-th particle, g i (r), is defined as the distribution of distances of the i-th particle with all the other (N − 1) particles averaged over all the MC steps. This definition, in fact, ensures that g(r) = N −1 N i g i (r). The nearest neighbor distance of the i-th particle, a i , averaged over all the neighbors and also over the MC steps is defined as the value of r, where the first peak of g i (r) appears. This same a i has been used in the Lindemann analysis as the inter-particle spacing of the i-th particle. Similarly, the location of the first peak of g(r) defines the mean inter-particle spacing for the system.
While every particle travels through the whole system for large T ensuring g(r) ≈ g i (r), such an independence of g i (r) on i does not hold at small T . For T → 0, g i (r) is determined by the environment of the i-th particle, which could differ significantly from that of any other particle due to irregular con- (defined in the text). Traces of individual gi(r) for two particles (1 and 7 in our nomenclature) are shown on the left-insets at low T , showing significant particle-to-particle fluctuations in it, which are attributed by the irregular confinement. At large T , on the other hand, such fluctuations vanish, producing gi(r) similar to g(r) for all i, as shown on the right-inset for the same two particles. The resulting P (∆g), thus becomes narrow and symmetric with T in the same range ∆TX . finement. Thus, the low-temperature distribution, P (∆g), of ∆g(r) ≡ g i (r) − g(r) collected for all r, and also over the realizations of irregularity, will be rather broad, and need not even be symmetric. The asymmetry in P (∆g) for a given realization of V c depends on its parameters, and is washed out in the ensemble averaging over many realizations. On the contrary, P (∆g) is narrow and symmetric at larger T , because all particles are equally likely to be everywhere in the system. It is then expected that the thermal evolution of P (∆g) must shed some light on the crossover. Note that a similar argument for the bulk system would lead to a symmetric and possibly narrow P (∆g) for both large and small T . We present the T -dependence of P (∆g) in Fig. 6 illuminating the gradual progression towards the melting.
Having seen a consistent ∆T X for solid-like to liquid-like behavior in an IWM, we turn to the T -dependence specific heat, c V , of the IWM,. 62 With E = E MC -the total MC energy, andÊ = E , c V is defined as,
c V = dÊ dT = ( E 2 − E 2 )/T 2 .(6)
The temperature evolution of c V is presented in Fig. 7, showing a distinct hump as expected in a CO, which gets sharper as N increases. It is important to note that the position of the hump, T X ≈ 0.03, is fairly insensitive to N , and falls close to the midway of ∆T X . We also emphasize that it is the same T X , at which the distribution P (L i ) developed a peak at a non-zero value of L i for the first time as T was increased , showing a hump characterizing a crossover in critical phenomena. The hump gets sharper as N increases. Also, it is interesting to note that the hump occurs at TX = 0.03, and is insensitive to N . The inset compares the N -dependence of TX (obtained from the location of the hump in cV ) and Tc, the transition temperature for a bulk system with the same average density of particles, as in our irregularly confined system for each N . Modulo the large uncertainty in TX (Note ∆TX ∼ TX ), their behavior indicates that the thermal fluctuations could destabilize an IWM more than an equivalent bulk system. from zero (See top left inset of Fig. 2). The estimate of the T X from the peak-value of smoothed c V is presented in the inset of Fig. 7 as a function of N . We also present for comparison the corresponding T c for bulk using √ πn/T c ≈ 137 from Ref. 20, where the value of average density, n, is obtained from our results on IWM. Interestingly, T X is found to lie lower than the T c for N ≥ 35. This is qualitatively consistent with what happens in a circular Wigner molecule. 31 However, serious significance might not be associated with this comparison, because the uncertainty in T X is large (∆T X ∼ T X ). Based on different criteria for T X and more broadly for ∆T X that we report here, we find that the melting is strongly smeared, with the width of transition comparable to the melting temperature.
IV. MECHANISM FOR THE CROSSOVER
Our results evidently raise the next fundamental question: What's the mechanism driving the crossover found in our IWM? This is particularly important in comparison with the established mechanism of melting in the bulk 2D system, and circularly confined systems. As discussed before, the thermal melting of 2D WC, described by KTHNY theory, is a two-step process mediated by production of crystal defects, e.g., dislocations and disclinations leading to the breaking of positional and orientation orders respectively. Melting in circular con-finements, also a two-step melting process, is enforced by the symmetry, delocalizing particles along the azimuthal direction in one step and melting along radial direction in the other. As confirmed already -radial and azimuthal melting loose relevance in our V c due to its complete lack of symmetry. Can we still identify any crystal defects? Can their thermal evolution help in understanding the CO encountered?
In absence of a firm analytical theory describing 'melting' in irregular confinements, we turn to numerical evidences from our calculations and we find the following: The positional order is largely depleted in the IWM even at the lowest temperature, and this is consistent with our occasional finding of signatures of dislocations, that resemble an extra row of particles stuck partly, in the solid-like phase. However, we note that rigorous identification of dislocations is difficult due to: (a) smallness of the system itself, and (b) presence of irregularities resulting into inhomogeneous and irregular lattice structures. An example of a dislocation found in a realization of the confinement is shown in Fig. 8(a). The thermal evolution of such dislocations is less clear for the reasons above, but to the extent we can infer, we do not see any trend for T -dependent proliferation of them.
Disclinations, on the other hand, characterized by a mismatch in the orientation as one circumnavigates it, signal the loss of BOO and are indeed found to proliferate for all T ≥ 0.01 in our data. They are best seen as a particle having the "incorrect" number of nearest neighbors as we discuss below. The number of nearest neighbors of a particle, also called the coordination number (CN) of that particle, in a given configuration is best measured using the Voronoi diagram (VD). 63 The VD assigns a 2D polygon around each particle in a given configuration, such that, any point within that polygon will be closer to that given particle than from all other particles. Thus, the VD determines the coordination number for all the N particles, and identifies their closest neighbors in an unbiased manner. Obviously, the VD corresponding to a perfect triangular lattice would be regular hexagons of same size touching each other with lattice points lying at the centers, and each particle on such a lattice will have 6 nearest neighbors (CN = 6).
We present three VDs on an arbitrary equilibrium MC configuration particles in Fig. 8 for the same realization of confinement as in Fig. 1 (a-c). The result at T = 0 shows CN= 6 for nearly all particles, except for those on the boundary and for the bound pair of disclinations contained within the dislocation. The VDs at T = 0.015, and T = 0.065 illustrate progressive proliferation of the disclinations (i.e., particles with CN= 5 or 7) with T .
We now come back to the question: Why does the Lindemann ratio show rapid increase for T ≥ 0.01? Analysis of our numerical data indicates that for T ≥ 0.01 disclinations start growing causing re-adjustments of particles in space, which expectedly raises L. Such re-adjustments obviously affect the particles close by, leading naturally to spatially correlated movements of particles in certain regions in space. This not only explains the incipient melting through the tortuous paths found in Fig. 1b, but also the large value of σ L (Fig. 2) for T in the range of ∆T X as well. Such a physical picture for melt- ing was found consistent with our data obtained for N ≥ 40, and also across all the realizations of confinement.
In order to emphasize the last point, we present the evolution of the disclinations for two other realizations of the confinement in Fig. 9. The top panels show the snapshot of particles over 100 independent MC steps at T = 0.02, where the melting has just commenced. These are thus similar to Fig. 1(b), but for different realizations, as well as for different N . We show the corresponding VDs for two profiles for independent MC steps in the middle and bottom panels. These figures demonstrate that the tortuous path of the melting is rife with disclinations. We further observe that the particles with CN= 6 close to such tortuous paths are more likely to have distorted hexagons as the Voronoi-plaquettes surrounding them than the others away from such paths. Our results, therefore, suggest strongly that the crossover in an IWM from solid-like to a liquid-like phase is associated with disclinations destroying BOO. The emergence of such a mechanism constitute the key finding of our study. FIG. 9. The configuration for two different realizations of (λ, γ) with N = 148 and the Voronoi diagrams corresponding to two different equilibrium MC configurations. Filled squares for CN = 7 and filled circles for CN= 5. The disclination mainly has started along the line of the tortuous paths and also the Voronoi plaquettes corresponding to the particles along such line is rather distorted than those of other particles.
V. DISCUSSION
We found that the IWMs with N ≥ 35, present qualitatively similar physics discussed so far. But, for smaller values of N , the crossover from a solid-like to a liquid like behavior is lot more smeared, and is rife with larger fluctuations in all physical observables. Further,a higher ratio of boundary to bulk particles being for smaller N makes it harder to look for 'universal' features.
All the results discussed here are for a = 1 in V c (Eq. 1), which essentially fixes the average density of particles in the system, for example, n = 6.857 for N = 100. We have repeated the same calculations for lower densities on a few realizations, namely for a = 0.1, 0.01 resulting into n = 2.73 and n = 1.06, respectively for N = 100, and find qualitatively similar results. However, T X , as well as ∆T X were found to decrease with n. In any case, such inferences are expected to change at very low densities where quantum effects become significant, particularly at lowest temperatures. Those effects, while significant, are beyond the scope of the current work and remain as an important future direction.
In conclusion, we have studied the thermal crossover from a Wigner-type solid-like to a liquid-like phase in an irregular confinement containing classical particles. Our results demonstrate that such crossover takes place gradually without any sharp changes, and hence the width ∆T X is large. Thermal evolution of different observables points towards a unique ∆T X . Interestingly, the mechanism for melting appears to be the proliferation of disclinations that destroys the quasi-long range orientational order. Breaking of all symmetries does not, as such, stabilize the quasi-order state in IWM than in the bulk. Experiments study some of the dynamical properties of melting, such as, diffusion constant, frequency dependence of structure factor, the dynamics of the defects and their role in melting etc. While our method constrains us to study only the static properties, an extension to include the dynamics of the defects seems to be a bright direction. We hope that our finding will help understanding the physics of chaotic quantum dots in the experimental regime.
FIG. 1 .
1(a) The GSC (at T = 0) with N = 148 particles in one realization of the confinement Vc of Eq. (1) with λ = 0.2, γ = 0.635, showing the amorphous solid-like behavior. Snapshots of particles in the same confinement are shown in (b) and (c) at larger temperatures. (b) At T = 0.015, where the melting has just started, the spatially correlated 'movements' of several particles in the MC configuration space have produced a tortuous path connecting these particles. (c)At T = 0.065 the melting is nearly complete. 100 independent MC configurations were used for generating the snapshots, and these MC configurations were taken from the equilibrium MC runs separated by the intervals in which the particles would have moved the same distance diffusively. For comparison, similar snapshot is presented in (d) for hard-core particles (core size ∼ 0.285) in the same confinement, showing the 'pre-melting' primarily occurring near the boundaries.
FIG. 2 .
2The main panel shows the evolution of L (defined in the text) with T for different N . Each trace for a given N is averaged over 5 realizations of confinement. While L(T < 0.01) ≈ 0, its smooth rise in the range of 0.01 ≤ T ≤ 0.05 indicates 'melting'. For T ≥ 0.06, L still rises but with a much weaker slope, as expected from diffusive motion of particles in a 'fluid' phase. The top-left inset presents the T -evolution of the probability distribution of Li, P (Li) for N = 141. The variance of P (Li), σL, (scaled by a factor 155 for clarity) is also presented in the main panel, showing similar rise as L itself. The bottom-right inset describes the same crossover as in main panel, but in terms of modified Lindemann parameter Γ (defined in the text), and its variance σΓ (scaled by 1200) for N = 141.
FIG. 3 .
3The distribution, P (ψ6)
FIG. 4 .
4Evolution of BOCF, g6(r) for N = 125, with T , showing clear Bragg-type peaks at low T signifying bond-orientational order.
FIG. 5 .
5Evolution of PDF, g(r) for N = 125, with T , showing much weaker Bragg peaks at low T compared to those in g6(r) ofFig. (4), indicating weaker positional order than bond-orientational order. The other features follow the same trend of g6(r). The inset showing a smooth decrease in the PVR of g(r), and this decrease is less drastic than that of g6(r).
FIG. 6 .
6The main panel shows the distribution P (∆g) for N = 141
FIG. 7 .
7The specific heat, cV , as calculated from energy fluctuations in MC (see text)
FIG. 8 .
8The Voronoi diagram corresponding to single equilibrium MC configuration with N = 148 corresponding toFig. (1). The location of actual particles are shown by thin dots for those with CN = 6, filled squares for CN = 7, and filled circles for CN = 5. Note that CN = 5, 7 implies local disclinations. (a) VD for GSC at T = 0 shows predominantly regular hexagonal area surrounding each particle, resulting strong BOO. Note the identification of a dislocation even at T = 0 (as discussed in text) by the region bounded by thin dashed line that contains a bound pair of disclinations, as expected. (b) The configuration at T = 0.015 illustrates the beginning of the formation of a correlated path of free disclinations. (c) The proliferation of disclinations is demonstrated on the VD at T = 0.065 (d) The VD for hard-core particles implies that disclinations are limited primarily near the boundary leaving sizable BOO in the bulk.
ACKNOWLEDGMENTSWe would like to thank D. Dhar, S. Lal, H. U. Baranger, D. Sen and J. Chakrabarti for valuable conversations.
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| []
|
[
"Spitzer Observations of Supernova Remnant IC443",
"Spitzer Observations of Supernova Remnant IC443"
]
| [
"A Noriega-Crespo ",
"D C Hines ",
"K Gordon ",
"F R Marleau ",
"G H Rieke ",
"J Rho ",
"& W B Latter "
]
| []
| []
| We present Spitzer observations of IC 443 obtained with MIPS and IRS as part of our GTO program on the astrophysics of ejecta from evolved stars. We find that the overall morphology at mid/far IR wavelengths resembles even more closely a loop or a shell than the ground based optical and/or near IR images.The dust temperature map, based on the 70/160µm ratio, shows a range from 18 to 30 K degrees. The IRS spectra confirm the findings from previous near+mid IR spectroscopic observations of a collisionally excited gas, atomic and molecular, rich in fine structure atomic and pure H 2 rotational emission lines, respectively. The spectroscopic shock indicator, [Ne II] 12.8µm, suggests shock velocities ranging from 60-90 km s −1 , consistent with the values derived from other indicators.Subject headings: infrared: ISM -ISM: individual (IC 443) -supernova remnants | null | [
"https://arxiv.org/pdf/0804.4665v1.pdf"
]
| 18,317,672 | 0804.4665 | 9e8d8879d952235ad040e0fb38124db67dd39504 |
Spitzer Observations of Supernova Remnant IC443
29 Apr 2008
A Noriega-Crespo
D C Hines
K Gordon
F R Marleau
G H Rieke
J Rho
& W B Latter
Spitzer Observations of Supernova Remnant IC443
29 Apr 2008arXiv:0804.4665v1 [astro-ph] The Evolving ISM in the Milky Way & Nearby Galaxies
We present Spitzer observations of IC 443 obtained with MIPS and IRS as part of our GTO program on the astrophysics of ejecta from evolved stars. We find that the overall morphology at mid/far IR wavelengths resembles even more closely a loop or a shell than the ground based optical and/or near IR images.The dust temperature map, based on the 70/160µm ratio, shows a range from 18 to 30 K degrees. The IRS spectra confirm the findings from previous near+mid IR spectroscopic observations of a collisionally excited gas, atomic and molecular, rich in fine structure atomic and pure H 2 rotational emission lines, respectively. The spectroscopic shock indicator, [Ne II] 12.8µm, suggests shock velocities ranging from 60-90 km s −1 , consistent with the values derived from other indicators.Subject headings: infrared: ISM -ISM: individual (IC 443) -supernova remnants
Introduction
As one of the best examples of a supernova remnant (SNR) interacting with a molecular cloud IC 443 has been studied over all possible wavelength ranges, from the radio (see e.g. Leahy 2004), through the sub-mm (van Dishoeck et al. 1993) to the X-rays (see e.g. Troja et al. 2006Troja et al. , 2008, including TEV γ emission that is thought to be associated with pulsars (Albert et al. 2007;Humesky et al. 2007). At an estimated distance of 1.5 Kpc (Welsh & Sallmen 2003), IC 443 covers approximately a square degree over the sky. Until recently because of its relatively large size, most of IC 443 imaging data was a by-product of large sky surveys (IRAS,2MASS,ROSAT,MSX,etc), and to this date the spectroscopic data only samples a handful of specific regions. The spectroscopic data, nevertheless, do confirm that the emission arising from IC 443 carries the signature of collisionally excited (atomic & molecular) gas, the result of a shock wave impinging on a nearby molecular cloud. (see e.g. Shull et al 1982;Graham et al. 1987, Burton 1987, van Dishoeck et al. 1993Cesarsky et al. 1999, Oliva et al. 1999Rho et al. 2001, Neufeld et al. 2007, Rosado et al. 2007. Thus IC 443 continues to provide a excellent laboratory to study the evolution and interaction of a SNR with its surrounding medium.
In this communication we present the images obtained with the far infrared (FIR) photometer MIPS (Rieke et al. 2004), complemented with mid infrared (MIR) spectroscopy data obtained with IRS (Houck et al. 2004), both instruments on board of the Spitzer Space Telescope (Werner et al. 2004).
Observations
The MIPS & IRS observations are part of our GTO program (Rieke PID 77 and Houck PID 18) to study the physical characteristics of the ejecta from evolved stars. Although the observations were taken very early in the Spitzer mission, we have learned a handful of new things on data reduction as to provide the best possible images and spectra. The MIPS observations were obtained at three different epochs using fast scan mapping (3 sec per frame, 5 pointings per pixel), with scan legs offset of 148" to sample completely the 70 and 160um arrays. One of the remarkable features of the MIPS instrument is its capability to map large areas of the sky in a very efficient way, and therefore the new MIPS images at 24, 70 & 160µm capture the SNR in its entirety (Fig. 1). The IRS observations were carried out using both short and long high resolution modules at 5 fixed cluster positions (including an off-position) using 6 and 14sec ramps (one cycle), respectively. The off-position was used to remove the background from the on-target spectra.
Preliminary Analysis and Summary
The morphology of the IC 443 is shown in superb detail in the high angular resolution MIPS images (standard beam sizes of 6 ′′ , 18 ′′ and 40 ′′ at 24, 70 & 160µm respectively). Nevertheless the overall shell morphology can be seen already in the IRAS images (Fig. 2 top, HiRes fresco first iteration; see also Braun & Strom 1986). The comparison with Hα (a tracer of the ionized gas) and 24µm confirms that a significant fraction of the emission at 24µm is due to fine structure atomic and H 2 molecular emission lines, and not necessarily to dust continuum emission from small dust grains. his conclusion is further supported by the IRS spectra (Fig. 3), which show strong [Fe II] 26µm and H 2 0-0 S(0) 28.2µm emission lines at the four observed positions, but no detected continuum emission.Indeed, except for the South Rim of the shell, the 160µm emission (a tracer of cold dust) does not match the morphology of the HI 1.4GHz emission (Fig. 2, bottom left), suggesting that a large fraction of the emission is not due to dust continuum. The 2MASS Ks observations at 2µm were interpreted as due to H 2 excitation from shocks (Rho et al. 2001), if this is the case, then is possible that [C II] 158µm contribute to the 160µm emission. Certainly [O I] 63µm has been detected in several positions across the shell (Rho et al. 2001), and is very likely to contribute significantly to the 70µm emission band. Even so, one can use the 70 to 160µm ratio to estimate the dust temperature, and at first approximation, we found a range of 18−30 K, with higher dust temperature at the NE, where the Hα and 24µm emission are brighter.
The IRS spectra, as expected from previous work in the NIR+MIR, contains a handful of atomic fine structure lines from Fe, Ne and Si, plus the H2 pure rotational lines (Fig. 3, bottom). The most interesting aspect is the obvious differences as a function position in the excitation along the shell. The standard shock indicator of [Ne II] 12.8µm suggests shock velocities ranging from 60-90 km/s, and consistent with some previous estimates to account for the emission of the atomic/ionic lines (Rho et al.2001).
Finally, the excitation diagrams derived from the three H2 lines covered by the IRS observations (12.23, 17.03 and 28.22µm) do also show differences in column densities and temperatures as a function of position, ranging from T ex ∼ 300 − 600 K and N H2 ∼ 6.6 × 10 19 − 1.4 × 10 21 cm −1 (Fig. 4) suggesting that the interaction between the shock wave and its environment is non-symmetric.
Fig. 1 .
1-From left to right: MIPS maps of IC 443 at 24, 70 & 160µm. FOV∼0.9 • ×1.9 • . The bright source at the top of the image is IC 444 or IRAS 0655+2319. North is up and East is left Fig. 2.-Top; IRAS HiRes fresco of IC 443 with a similar FOV as Fig 1. Bottom Left: Hα (false color) and MIPS 24µm (contours). Rigft: MIPS 160µm (false color) and HI 1.4 GHz (contours). The color scales are in MJy/sr and the FOV ∼ 0.9 • radius -5 -Fig. 3.-Top: A schematic view of the 5 IRS observed positions. Bottom: Sample spectra obtained with the IRS short & high resolution modules.
Fig. 4 .
4-Excitation diagrams at positions one (left) and four (right) based on the three H 2 0-0 lines (12.23, 17.03 and 28.22µm) present within the wavelength range of our IRS spectra.
. J Albert, ApJ. 66487Albert, J. et al. 2007, ApJ, 664, L87
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| []
|
[
"Reverse Engineering Code Dependencies: Converting Integer-Based Variability to Propositional Logic",
"Reverse Engineering Code Dependencies: Converting Integer-Based Variability to Propositional Logic"
]
| [
"Adam Krafczyk \nUniversity of Hildesheim\n31141HildesheimGermany\n",
"Sascha El-Sharkawy \nUniversity of Hildesheim\n31141HildesheimGermany\n",
"Klaus Schmid [email protected] \nUniversity of Hildesheim\n31141HildesheimGermany\n"
]
| [
"University of Hildesheim\n31141HildesheimGermany",
"University of Hildesheim\n31141HildesheimGermany",
"University of Hildesheim\n31141HildesheimGermany"
]
| [
"Sweden Proceedings of 22nd International Conference on Software Product Line (SPLC'18)"
]
| A number of SAT-based analysis concepts and tools for software product lines exist, that extract code dependencies in propositional logic from the source code assets of the product line. On these extracted conditions, SAT-solvers are used to reason about the variability. However, in practice, a lot of software product lines use integer-based variability. The variability variables hold integer values, and integer operators are used in the conditions. Most existing analysis tools can not handle this kind of variability; they expect pure Boolean conditions. This paper introduces an approach to convert integer-based variability conditions to propositional logic. Running this approach as a preparation on an integer-based product line allows the existing SAT-based analyses to work without any modifications. The pure Boolean formulas, that our approach builds as a replacement for the integer-based conditions, are mostly equivalent to the original conditions with respect to satisfiability. Our approach was motivated by and implemented in the context of a real-world industrial case-study, where such a preparation was necessary to analyze the variability.Our contribution is an approach to convert conditions, that use integer variables, into propositional formulas, to enable easy usage of SAT-solvers on the result. It works well on restricted variables (i.e. variables with a small range of allowed values); unrestricted integer variables are handled less exact, but still retain useful variability information.CCS CONCEPTS• Theory of computation → Equational logic and rewriting; • Software and its engineering → Software product lines; Software reverse engineering; | 10.1145/3236405.3237202 | [
"https://arxiv.org/pdf/2110.05875v1.pdf"
]
| 52,152,170 | 2110.05875 | 1d09b2a123fc42ac515bb2f4f97181dc82a0bd31 |
Reverse Engineering Code Dependencies: Converting Integer-Based Variability to Propositional Logic
September 10-14. 2018
Adam Krafczyk
University of Hildesheim
31141HildesheimGermany
Sascha El-Sharkawy
University of Hildesheim
31141HildesheimGermany
Klaus Schmid [email protected]
University of Hildesheim
31141HildesheimGermany
Reverse Engineering Code Dependencies: Converting Integer-Based Variability to Propositional Logic
Sweden Proceedings of 22nd International Conference on Software Product Line (SPLC'18)
GothenburgSeptember 10-14. 201810.1145/3236405.3237202ACM Reference Format: Adam Krafczyk, Sascha El-Sharkawy, and Klaus Schmid. 2018. Reverse Engineering Code Dependencies: Converting Integer-Based Variability to Propositional Logic. In Proceedings of 22nd International Conference on Software Product Line (SPLC'18). ACM, New York, NY, USA, 8 pages. This is the author's version of the work. It is posted here for your personal use. Not for redistribution. The definitive Version of Record was published inSoftware product linesSatisfiabilityInteger-based expressionsPropositional logicVariability managementReverse engineering
A number of SAT-based analysis concepts and tools for software product lines exist, that extract code dependencies in propositional logic from the source code assets of the product line. On these extracted conditions, SAT-solvers are used to reason about the variability. However, in practice, a lot of software product lines use integer-based variability. The variability variables hold integer values, and integer operators are used in the conditions. Most existing analysis tools can not handle this kind of variability; they expect pure Boolean conditions. This paper introduces an approach to convert integer-based variability conditions to propositional logic. Running this approach as a preparation on an integer-based product line allows the existing SAT-based analyses to work without any modifications. The pure Boolean formulas, that our approach builds as a replacement for the integer-based conditions, are mostly equivalent to the original conditions with respect to satisfiability. Our approach was motivated by and implemented in the context of a real-world industrial case-study, where such a preparation was necessary to analyze the variability.Our contribution is an approach to convert conditions, that use integer variables, into propositional formulas, to enable easy usage of SAT-solvers on the result. It works well on restricted variables (i.e. variables with a small range of allowed values); unrestricted integer variables are handled less exact, but still retain useful variability information.CCS CONCEPTS• Theory of computation → Equational logic and rewriting; • Software and its engineering → Software product lines; Software reverse engineering;
INTRODUCTION
Many software product lines use the C-preprocessor for implementing variability in their source code [6,11]. The #if-statement is used to conditionally compile source code parts. The expressions use variability variables to determine whether the following code lines should be included in the product for a given configuration. Typically, such product lines use mostly Boolean variables. A well known example for this kind of code variability is the Linux Kernel. Most of its variability variables are Boolean or tristate. Tristates allow three different values, and are implemented using pairs of Boolean variables in the C-preprocessor.
There exist a lot of family-based analysis [15] approaches and tools for these kinds of product lines. Undertaker finds code blocks that can never be (un-)selected [14,17]. TypeChef, among other things, finds type errors across all configurations at once [7,16]. The feature-effect approach defines Boolean formulas that represent under which condition a given variability variable has an effect on the final product [12]. The configuration mismatch analysis builds on that and finds constraint mismatches between the code and the variability model [5]. All of these approaches have in common that they use SAT-solvers (or techniques similar to SAT) to reason about the variability in the source code. Because of this, they only work with code dependencies in propositional logic; all variability variables are expected to be Booleans, and the #if-conditions in the code may only contain Boolean operations.
However, there are also software product lines that are not limited to propositional logic in their variability. For example, an industrial product line we analyzed uses integer-based variability [4]. Each variability variable holds an integer value, out of a defined (typically small) range of allowed values. The #if-conditions in the code then use the integer arithmetic and comparison operators that the C-preprocessor defines. None of the existing analysis tools mentioned above can handle such variability conditions, because they are built on pure SAT-based approaches. However, we still wanted to use some of the analysis approaches that they offer on the integer-based product line.
To deal with this incompatibility, we developed and implemented an approach to convert the variability in the source code of the product line to a format that is suitable for the existing analysis tools. Our tool creates a copy of the source files in the product line, and replaces all integer-based #if-conditions with propositional formulas. This allows easy usage of SAT-solvers on the converted code conditions. As a result, all existing analysis tools are able to extract variability information from the modified source code files, without any modifications of the tools. Our tool is implemented in the context of the ITEA3 project REVaMP 2 , as an open-source plugin for the KernelHaven analysis framework [8][9][10].
This paper introduces the approach we developed to transform integer-based variability conditions to propositional logic. It reverse engineers the code dependencies from an existing product line, and transforms them into a format that is understood by existing, SATbased analysis tools. This is based on the industrial use case we analyzed. The variability variables all hold integer values, and most of them have a (typically small) range of allowed values. These ranges are an input to the transformation process and are, for example, defined in a variability model of the product line. The execution time of the transformation process scales linearly with the size of the product line to transform. The overhead is generally insignificant, compared to the following analysis steps.
Some corner cases can not be covered by our approach; thus the resulting propositional formulas are sometimes not completely equisatisfiable, compared to the original integer-based ones. However, we implemented a fallback strategy that produces less accurate results, but still retains useful variability information in these cases. In practice, the resulting formulas are good enough for the analysis tools mentioned above to produce reasonable results.
The remainder of this paper is structured as follows: Section 2 presents work related to integer-based variability. Section 3 introduces the basic concept of our transformation approach. Based on this, Section 4 describes the implementation of our approach in more detail and walks through a full example transformation. Section 5 discusses the limitations of our approach, while Section 6 evaluates the implementation. Finally, Section 7 summarizes this paper and shows possible future work.
RELATED WORK
Existing research shows that integer-based (or more broadly: non-Boolean) variability is used in real-world software product lines. Passos et al. studied the non-Boolean variability used in the eCos product line [13]. They specifically study the kinds of constraints that non-Boolean variables are used in, and describe the challenges that this kind of variability poses for analyses. Berger et al. studied real-world variability modeling techniques [2]. All the product lines they studied use integers as variability variables, although the percentage of variability that is integer-based varies. They found that academic analyses often make simplifying assumptions about the structure of variability. In contrast to that, our approach presented in this paper was driven by a real-world industrial use-case, that uses integer-based variability [4]. Thus, it does not include the common simplifications found in some academic concepts.
Analyses that run on product lines that use integer-based variability have to use strategies to deal with the integer variables and constraints. This could be done by employing a solver that can directly work with such constraints. Barrett et al. developed a first-order logic solver that incrementally translates parts of the expression to SAT, while solving it [1]. Satisfiability modulo theory (SMT) solvers, for example the Z3 solver by Microsoft [3], handle first-order logic conditions directly. Such a solver does not require a translation to SAT. Xiong et al. use Z3 for their approach to fix ill-formed configurations that contain integer variables [18]. An ill-formed configuration violates one or more of the constraints imposed by the variability model. Their approach calculates valid ranges for integer variables, that adhere to these violated constraints. It works directly on the integer-based constraint, without a translation to SAT.
The existing SAT-based analysis approaches and tools, introduced in Section 1, could use such a solver to handle integer-based variability. However, this would require to modify the tools; they need to be extended to parse and represent the integer-based conditions, and their SAT-solver needs to be swapped for one of the solvers mentioned above. In contrast, our approach works as a preparation for the product line, and no modification is required to the SAT-based analysis tools. Our approach takes advantage of the fact that in the analyzed product line, most of the integer variables have only a limited range of allowed values. Additionally, we expect the runtime of a more powerful solver to be much higher than the runtime of our preparation approach plus the simple SAT-solvers.
CONCEPT
This section introduces our concept for converting integer-based conditions to propositional formulas. First, we describe the structure of the integer-based conditions of the product line. Then we explain why existing analysis tools can not work with these conditions directly. Finally, we introduce our approach that solves the incompatibility between the integer-based conditions and the existing, propositional logic-based analysis tools.
The source code in product lines that we want to analyze contains integer-based variability conditions in the C-preprocessor. The variables used in the conditions hold integer values, and have a (typically small) range of allowed values. All integer arithmetic operations (+, −, * , /, %, &, |,^, ∼) and integer comparison operators (==, ! =, >, >=, <, <=) that the C-preprocessor defines are used in the conditions. Additionally, the Boolean operators (&&, ||, !) and the defined function are used. The defined(VAR) function returns whether any value has been set for the variable. For instance, a condition from such a product line may look like this:
#if (VAR_A * 2 > VAR_B) || defined(VAR_C)(1)
with the variability model specifying the possible ranges for the integer variables: VAR_A ∈ {1, 2, 3} and VAR_B ∈ {5, 6}. The existing tools for analyzing C-preprocessor based variability, introduced in Section 1, can only handle conditions in propositional logic. This is because they use SAT-based approaches to reason about variability. Only Boolean operators (&&, ||, !) and Boolean variables are allowed in the variability conditions that they extract. The convention for Boolean variables in the C-preprocessor is to use the defined function on a variable, and denoting the two possible states by either defining or not defining the variable. An example of such a product line is the Linux Kernel, which is also a common target for research analysis tools [14]. The existing analysis tools expect the analyzed code to follow this convention for pure Boolean conditions; they can not handle integer-based conditions, where the value that a variable holds is important. Additionally, they often simply can not parse the integer operations that are used in the conditions.
The goal of our approach is to construct conditions in propositional logic, that are, with respect to satisfiability, mostly equal to the integer-based conditions in the source code. Thus, after preparing the code artifacts with our approach, all the existing tools can be used on the previously integer-based product line. The propositional replacements hold mostly the same variability information as the integer-based conditions, with respect to the satisfiability of the code dependencies. This ensures that the results produced by the SAT-based analysis approaches are reasonable and useful for the analyzed product line.
Our approach introduces Boolean variables for each possible value of the integer variables. This is feasible, because most of the integer variables have only a small range of allowed values. For each integer-based condition, we calculate which combinations of values satisfy this condition. From this, a propositional formula using the introduced Boolean variables is created, that reflects these combinations. Additionally, we introduce another Boolean variable for each integer variable, that denotes whether the variable is set to any value, or whether it is undefined. This is used for resolving defined calls, and as a fallback for cases that our approach can not calculate exact solutions for (see Section 5).
More formally, let be the set of all integer-based variability variables, :
→ P (Z) a function that defines the range of allowed values for each variable, and a set of Boolean variables. We then introduce a function :
× (Z ∪ { }) →
which injectively maps a variable and one possible value of it to a Boolean variable. in place of a value maps to the Boolean variable that denotes whether the integer variable is defined or not (i.e. whether it is set to any value). For instance, consider the variable VAR_A with (VAR_A) = {1, 2, 3}. VAR_A may be set to either 1, 2, or 3. (VAR_A, ) returns the Boolean variable that denotes whether VAR_A is defined. (VAR_A, 1) returns the Boolean variable that denotes that VAR_A is set to 1. In practice, such variable names may look like this: (VAR_A, 1) = VAR_A_eq_1, (VAR_A, ) = VAR_A.
The exact naming scheme depends on the context that the approach is applied in; it has to ensure that no name collisions occur, and that the names are valid identifiers for the C-preprocessor. There are two constraints for the introduced Boolean variables, which are not explicitly modeled. These have to be manually considered when interpreting the result of any analysis done on the propositional formulas:
(1) The Boolean variables for the possible values of an integer variable are mutually exclusive:
∀ ∈ , ∀ , ∈ ( ), ≠ | ( , ) =⇒ ¬ ( , )(2)
The Boolean variable is true if and only if a value is set for the integer variable, that is if any of the Boolean variables denoting the possible values is true:
∀ ∈ | ( , ) ⇐⇒ ∈ ( ) ( , )
These newly introduced Boolean variables are used to replace the integer-based (sub-)expressions in the conditions. We calculate which combinations of allowed values fulfill the integer-based (sub-)expression, and build a Boolean formula that is equally satisfiable to this. For the example condition shown in Formula 1, the propositional replacement would look like this:
#if (VAR_A_eq_3 && VAR_B_eq_5) || VAR_C (the defined calls around each of these variables has been left out for brevity). The integer sub-expression VAR_A * 2 > VAR_B is only fulfilled by the combination VAR_A=3 (with (VAR_A, 3) = _ _ _3) and VAR_B=5 (with (VAR_B, 5) = _ _ _5). Thus, the replacement for this sub-expression is a Boolean formula for this combination of values. The sub-expression defined(VAR_C) is replaced with (VAR_C, ) = _ . In many cases, the resulting propositional formula is much larger, compared to the original integer-based one. The goal, however, is not to produce small or readable conditions, but to provide input data for analysis tools. Thus, it is not a primary concern to keep the resulting propositional formulas concise.
A special case are integer variables that have no restriction on the allowed values. The "allowed values" for such a variable are for example the whole range of 32 bit integer variables. It is not possible (or at least not feasible) to introduce Boolean variables for each of the possible values. In this case, we take a less exact approach to still be able to handle these variables: we only introduce the single Boolean variable ( , ). Then, in each condition where the unrestricted variable appears, we use this Boolean variable, no matter which actual value of the variable would fulfill the condition. This way, we still have the variability information that the given condition somehow depends on the unrestricted variable; however, we lose the information which specific values it depends on. For example, the condition VAR_A+2 > 5, with VAR_A as an unrestricted integer variable, is converted to (VAR_A, ).
Another special case are variables with only one possible allowed value. These variables are not real variability, since they can not be set to any other value. Thus, we treat them as constants and replace each occurrence of them with their literal value. This makes it easier to calculate the combinations of the other integer variables, that satisfy a constraint. It also removes unnecessary variables from the propositional formulas, and thus reduces the unnecessary complexity of the output.
IMPLEMENTATION
This section describes the implementation of our approach. First, the general steps for converting a condition are explained. Then, the transformation of the integer (sub-)expressions is described in detail in Section 4.1. This is followed by Section 4.2 with an example of a complete conversion of a code condition.
Our tool reads through all source files and replaces the C-preprocessor conditions in them with a propositional formula. The result is a copy of the source files with all the C-preprocessor conditions transformed into a purely Boolean form. This allows existing analysis tools, which are based on propositional logic, to work with these files. Each of the C-preprocessor conditions found in the source files is converted in the following steps:
(1) Parse the condition into an abstract syntax tree (AST) (2) Replace constants with their literal value (3) Walk bottom-up through the AST to replace integer-parts (4) Convert AST back to a C-preprocessor condition string
The first step is straightforward parsing of the C-preprocessor condition. All the subsequent steps will work on the AST that is produced by this. In the second step, each constant that has only one allowed value is replaced by its literal value. This removes constant values, which do not need to be considered when analyzing variability.
The third step is the main part of the conversion to a propositional form. The goal is to convert all integer-parts of the conditions to pure Boolean ones, that have equivalent satisfiability. The strategy used here is to walk bottom-up through the AST and apply a number of rules on the AST that define how the integer operations are converted. The following section will explain these rules in detail.
Finally, after the integer parts are eliminated, the AST is converted back into a C-preprocessor condition string. This string is then used as the replacement for the original condition. It will only contain Boolean variables and operators, so that existing tools for propositional logic can handle it.
Transformation of Integer Expressions
This section describes how the integer-based (sub-)expressions are converted to Boolean formulas. In the AST, the highest operator of an integer (sub-)expression is a comparison operator 1 . On the left and right side of this comparison, there are literals, variables, or arithmetic operations combining both. The general idea is to find all possible combinations of allowed values for the variables on the left and right side that fulfill the comparison operator. These values are then transformed into Boolean variables using the function. A propositional formula is constructed from them, that is satisfiable for all combinations that fulfill the comparison.
Integer arithmetic operations are evaluated bottom-up, so that eventually the comparison operation can be evaluated on the resulting values. When resolving the arithmetic operations, they are applied on all possible values of a variable. This results in a set of values, instead of a single result value for the operation. This is needed, because we want to find all possible values that fulfill the comparison at once.
It is also not enough to just compute the results of integer arithmetic operations on variables. When a result value that fulfills the comparison operator is found, the original value of the variable that led to this result value is required to construct the Boolean variable using the function. We call the result of arithmetic operations the current value and the initial value of the allowed range of the variable that this result stems from the original value. For instance, consider the integer expression VAR_A + 1 == 2. When evaluating the + operation, the original value 1 of VAR_A led to the current value 2. When resolving the equality operator, the current value 2 for the left side fulfills this comparison. Thus, the Boolean variable of the original value that led to this is constructed: (VAR_A, 1).
The following sub-sections describe the different evaluation rules in detail. The rules are applied on the AST, based on which integer operator is used on which input types.
Comparison Operator on Variable and Literal.
A comparison of a variable and an integer literal is resolved to a propositional formula, which contains all possible original values that satisfy the comparison. The comparison is computed on all the current values stored in the variable. For each current value that satisfies the comparison, the corresponding original value of the variable is turned into a Boolean variable via the function. All these Boolean variables that fulfill the comparison are then combined with a Boolean disjunction operator.
||
In this figure, VAR_A is compared with the literal 4 with a "greater than" comparison operator. In this example, the current and original values in the tuples are different; this is because some previous arithmetic operation on VAR_A has modified them. This is not always the case (a variable can also be compared without doing arithmetic on it first), but we chose this for illustration purposes, to make it clear that the current and original values have to be treated differently.
Full Example
This section shows an example of a full transformation from an integer-based C-preprocessor condition to a propositional formula. The original condition to convert is: Then the comparison operator can be resolved. Only one pair of tuples from VAR_A and VAR_B fulfills this: the third tuple of VAR_A and the first tuple of VAR_B (6 > 5). This is then converted into a propositional formula, that specifies that the original values 3 from VAR_A and 5 from VAR_B fulfill this comparison: The right side of the disjunction operator, defined(VAR_C), is replaced with the variable for VAR_C: 3) σ(VAR_B,5)
σ(VAR_C,ϵ) && σ(VAR_A,
||
The AST now only contains Boolean variables and operators. The fourth and final step is to convert it back into a C-preprocessor condition string (the defined calls around each of the variables have been left out for brevity):
#if (VAR_A_eq_3 && VAR_B_eq_5) || VAR_C
LIMITATIONS
This section will discuss the limitations of the approach presented in this paper. Both, the conceptual problems of our approach, and the technical issues of our implementation are discussed.
An important conceptual problem is that unrestricted integer variables are handled not exact. For integer variables that have an infinite (or very large) range of allowed values, our approach does not differentiate between the different values that this variable holds. Only the fact that a condition depends on the variable having a defined value is represented in the replacement propositional formula. For example, the condition VAR_A * 2 > 5, with VAR_A as an unrestricted integer variable, leads to the replacement condition defined(VAR_A). This will also lead to some conditions, such as VAR_A > 0 && VAR_A < 0, to appear satisfiable in the propositional formula.
The impact of this inexact strategy depends on the concrete use-case where our approach is applied. The more unrestricted integer variables are present in the variability model, the more of a problem this becomes. Also the usage of these variables in the conditions needs to be examined: if the unrestricted variables are mixed together with the other integer variables in the conditions, then the inexact results may influence the other variables, too. In contrast, if they are mostly used in separate conditions, then the results for the restricted variables are unaffected and remain exact.
In our industrial use-case, we found that this inexact strategy does not have a large effect on calculating feature-effects [4]. The feature-effect analysis builds formulas from the variability conditions in the code, which specify when a certain feature has an effect on the final product. If a condition contains an unrestricted integer variable, the propositional formula for it created by our approach retains the information that the condition somehow depends on this unrestricted variable. This is because our approach adds a (VAR, ) variable in place of the unrestricted variable. This dependency then also appears in the calculation of the feature effects. It is not as exact as the per-value analysis of the restricted variables, however the general dependency is still included.
Another conceptual limitation of our approach is that resulting propositional formulas may be much larger than the original integer-based ones. This is because our approach considers all possible values for integer variables when resolving integer operations on them. When both sides of an operation are integer variables, a pair-wise combination of their possible values is necessary. This leads to a quadratic growth of combinations when multiple variables are combined in a series of arithmetic operations.
Generally, creating long replacement conditions is not a problem, since they are only used as input for further analysis tools. The formulas are not meant to be human-readable. However, in practice, we encountered runtime and memory problems with too large conditions in a very small number of cases.
To circumvent the runtime and memory problems of too large conditions, our tool defines a fixed upper limit for the number of value combinations to consider. When a series of arithmetic operations on variables exceeds this limit, we drop the per-value analysis for this sub-expression. Instead, we fall back to a similar approach used for the infinite integer variables: only (VAR, ) variables are created for all involved variables. This retains the information, that the condition somehow depends on these variables, but the information which concrete values it depends on is lost.
Finally, there are a few minor technical issues with our implementation. These stem from the specifics of the C-preprocessor, and are thus not inherent to our approach itself.
• The C-preprocessor has no well-defined data types. This leads to a problem when evaluating the bit-wise negation operator ∼, where the concrete type of an integer (bit size, and whether it is signed or unsigned) is important for correct results. However, in practice, this operator is not used much. • Based on the underlying industrial use-case, our tool is only designed to handle integer and Boolean variables in the Cpreprocessor conditions. It can not handle string variables, or the string concatenation operator (##). When encountering this, our tool will print a warning and skip replacing the condition. • The C-preprocessor allows defining functions (with the #define statement) that can be used in #if-statements. Our tool can not interpret these functions; when such a function appears in a condition, our tool will print a warning and skip replacing the condition.
EVALUATION
We have evaluated the implementation of the approach presented in this paper both in practice and with generated test cases. The generated test cases were used to evaluate the performance of the implementation. Each test case consists of 100 generated C source files with 10 #if-conditions each. The #if-conditions are generated in a way that each of the transformation rules described in Section 4.1 are covered. 5 integer variables are used throughout these conditions, each with (VAR) = {1, 2, 3, 4} as the range of allowed values. The test cases were executed on a machine with an Intel Core i7-6700 CPU with 3.40 GHz and 16 GiB RAM. The execution time of the file preparation, that is copying the files and converting all #if-conditions, was measured. Figure 1 shows that the execution time grows linearly with the number of conditions to process. This means, that when analyzing a whole product line, the overhead of the preparation will also grow linearly with the size of the product line. For this test series, the size of the range of allowed values per variable was gradually increased, starting from 2 and up to 18. Additionally, the upper limit for the number of value combinations to consider (see Section 5) was removed. When an integer operator has variables on both sides, our approach considers all combinations of the allowed values for both variables. This leads to a quadratic growth of combinations, that is visible in Figure 2. However, when the upper limit for combinations to consider is not removed, the execution time stays below 500 ms for all test cases shown in this Figure. Limiting the number of combinations to consider produces less accurate results (see Section 5), but it mitigates the (potential) performance problem visible in Figure 2.
The implementation of the approach presented in this paper has also been used in practice in the analysis of the Bosch PS-EC product line [4]. Our tool converted the integer-based C-preprocessor conditions to propositional logic. This allowed existing SAT-based analyses to be used on the product line, without any modification to the existing analysis tools. The execution time of our preparation tool was not significant, compared to the execution time required for the following analysis steps. The resulting replacement conditions created by our approach allowed the following analysis steps to create meaningful results.
SUMMARY
In this paper, we developed an approach to convert integer-based variability conditions to propositional logic. The original conditions are defined using the C-preprocessor in source code files. The variability variables used in the conditions hold integer values and are restricted to a (usually small) range of allowed values. Our approach converts all the conditions found in the source files, and replaces them with the created propositional formulas.
The goal of our approach is to easily use SAT-solvers even on integer-based product lines. This enables the usage of a number of existing, SAT-based tools and approaches without any modification to them. Thus, our approach ensures that the propositional formulas, that replace the integer-based one, are mostly equal to the original conditions with respect to satisfiability. Section 3 described the general concept of our approach: Boolean variables are introduced for each allowed value of the integer variables. This is viable, because the range of allowed values of a variable is usually small. Our approach then converts the integer-based conditions using these Boolean variables. It calculates for each integer (sub-)expression, which combination of allowed values fulfills it. A propositional formula, which reflects this combination of values that fulfill the expression, is used as the replacement.
Section 4 explained how this is implemented. A set of rules is applied on the abstract syntax tree, to evaluate the integer-parts bottom-up. This evaluation keeps track of all allowed values of the integer variables at once. When a comparison operator is reached, a propositional formula can be constructured that specifies which combinations fulfill this comparison.
The approach also has a few limitations, as described in Section 5. Most importantly, the handling of unrestricted integer variables is not exact. Unrestricted integer variables are variables that have no restrictions on the allowed values, or have a very large range of allowed values. For these variables, our approach does not evaluate each possible value individually. Instead, only a single Boolean variable is used, that specifies whether the variable is set to any value, or whether it is left undefined. This retains some useful variability information, although it is not as exact as the per-value analysis. In practice, this inexact approach still leads to reasonable results in the further analyses.
Our approach is implemented as an open-source plugin for the KernelHaven analysis framework [8,9]. This implementation has been used in practice to analyze the Bosch PS-EC product line [4].
Future work on integer-based product lines can utilize the approach presented in this paper to efficiently re-use existing, SATbased analyses. Additionally, the approach presented here can be refined in the future. We already experiment with a heuristic to find intervals of unrestricted integer variables, that can be treated as a single value. These equivalence classes would allow a per-value analysis of unrestricted variables in the approach presented here, and would thus improve the quality of the results.
• comparison operator refers to integer comparison operators (==, ! =, <, <=, >, >=) • arithmetic operator refers to integer arithmetic operators (+, −, * , /, %, &, |,^, ∼) • literal refers to literal integer values 1 Sometimes there are no explicit comparison operators to convert integer expressions to Boolean values. In this case, a != 0 comparison can be assumed, since all integer values except 0 are defined to be true in the C-preprocessor. • variable refers to integer variables, figure, the expression 2 * 4 results in the literal value 8.4.1.2 Comparison Operator on two Literals. For comparison operations on two literal values, the resulting Boolean constant is simply calculated. Neither side of the comparison contains any integer variables, thus a single Boolean literal can express the satisfiability of this (sub-)expression. 4.1.3 Arithmetic Operator on Literal and Variable. For integer arithmetic operations on variables, the operation is calculated on each of the allowed values. A set of tuples is stored in the variable, which contains for each original value, the currently computed value. All arithmetic operations on this variable will always update the current value. When resolving the variable to a Boolean formula later on, it is important which original value led to the currently computed one. figure, the literal value 2 is added to VAR_A. VAR_A has three possible values: 1, 2, and 3. For each of these possible values, the operation is computed and the result stored in the first component in the tuple. The second component is not modified; it contains the original value of VAR_A that led to the currently computed one. For example, in the second tuple of the result, the original value 2 of VAR_A led to the current value of 4 (via the addition of 2).
Two of the current values (the first component of the tuples) of VAR_A fulfill the comparison: the second and the third tuple. From both these tuples, the original values (the second component) are transformed into Boolean variables and combined with a logical disjunction. 4.1.5 Comparison Operator on two Variables. A comparison of two variables is resolved to a propositional formula that contains all possible combinations of original values that satisfy the comparison. For each pair of the current values of the two variables it is checked if they fulfill the comparison operator. For each pair that does fulfill it, the two original values of the variables are turned into Boolean variables (via the function) and combined with a logical conjunction operator. All of these conjunction terms are then combined with a logical disjunction operator. figure, there are two pairs of tuples that have the same current value and thus fulfill the equality operator: the second one from VAR_A and the first one of VAR_B both have the value 4, the third one from VAR_A and the second one from VAR_B both have the value 5. For the first pair, the original value of VAR_A that led to the current value is 2 (the second component in the tuple), while the original value of VAR_B is 1. Thus, the Boolean representation of this combination is (VAR_A, 2) ∧ (VAR_B, 1).Similarly, the Boolean representation of the second matching pair is (VAR_A, 3) ∧ (VAR_B, 2). Since both of these pairs fulfill the equality operator, they are combined with a disjunction operator.
Operator on two Variables. For integer arithmetic operations on two variables, the operation is done on each combination of the current values of both variables. For each of these calculated values, both of the original values of the variables that led to this current value are stored. When turning this tuple into a Boolean formula, not a single variable is created (e.g. (VAR_A, 2)), but a logical disjunction of the two variables with the original values stored in the tuple (e.g. (VAR_A, 2) ∧ (VAR_B, figure, the two variables VAR_A and VAR_B, both with two possible values, are added together. Combining the first tuple of both, results in the first tuple of the result: the current values (the first components of the tuples: 3 and 4) are added together, resulting in the new current value 7. Then, both of the original values (the second components in the tuples) are stored in the result, to indicate which original values of VAR_A and VAR_B led to the current value of 7. When turning this tuple into a Boolean formula (if the current value of this tuple fulfills a comparison later on), then the original values of both variables have to be considered: the resulting formula is (VAR_A, 1) ∧ (VAR_B, 1).
#if (VAR_A * CONST_A > VAR_B) || defined(VAR_C) with (VAR_A) = {1, 2, 3}, (VAR_B) = {5, 6}, (VAR_C) = {0, 1}, and (CONST_A) = {2}.The first step is to parse the condition into an abstract syntax tree (AST): second step replaces the constant CONST_A with its literal value 2: step is the main part of the conversion. The integer sub-expression on the left side, VAR_A * 2 > VAR_B, is converted into a propositional formula. First, the multiplication operation is resolved, by multiplying each of the possible values of VAR_A with the literal value 2:
Figure 1 :
1Runtime with varying number of conditions
Figure 1
1shows how the total number of #if-conditions in the C source files relates to the execution time of the tool. For this test series, the number of #if-conditions per C source file is increased in steps of 50, starting from 50 and up to 1000 conditions per file.
Figure 2 :
2Runtime with varying ranges per variableFigure 2 shows how the number of allowed values per variable impacts the execution time.
ACKNOWLEDGMENTSThis work is partially supported by the ITEA3 project REVaMP 2 , funded by the BMBF (German Ministry of Research and Education) under grant 01IS16042H. Any opinions expressed herein are solely by the authors and not by the BMBF.
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Feature Consistency in Compile-time-configurable System Software: Facing the Linux 10,000 Feature Problem. Reinhard Tartler, Daniel Lohmann, Julio Sincero, Wolfgang Schröder-Preikschat, 10.1145/1966445.1966451Proceedings of the Sixth Conference on Computer Systems (EuroSys '11). the Sixth Conference on Computer Systems (EuroSys '11)New York, NY, USAACMReinhard Tartler, Daniel Lohmann, Julio Sincero, and Wolfgang Schröder- Preikschat. 2011. Feature Consistency in Compile-time-configurable System Software: Facing the Linux 10,000 Feature Problem. In Proceedings of the Sixth Conference on Computer Systems (EuroSys '11). ACM, New York, NY, USA, 47-60. https://doi.org/10.1145/1966445.1966451
A Classification and Survey of Analysis Strategies for Software Product Lines. Thomas Thüm, Sven Apel, Christian Kästner, Ina Schaefer, Gunter Saake, 10.1145/2580950ACM Comput. Surv. 476Thomas Thüm, Sven Apel, Christian Kästner, Ina Schaefer, and Gunter Saake. 2014. A Classification and Survey of Analysis Strategies for Software Product Lines. ACM Comput. Surv. 47, 1, Article 6 (June 2014), 45 pages. https://doi.org/ 10.1145/2580950
Undertaker. visited: 22.05Undertaker 2015. https://vamos.informatik.uni-erlangen.de/trac/undertaker Last visited: 22.05.2018.
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"P Sekatski \nInstitut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstr. 25A-6020InnsbruckAustria\n",
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"W Dür \nInstitut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstr. 25A-6020InnsbruckAustria\n",
"P Sekatski \nInstitut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstr. 25A-6020InnsbruckAustria\n",
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"Institut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstr. 25A-6020InnsbruckAustria",
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| We show that one can deterministically generate out of N copies of an unknown unitary operation up to N 2 almost perfect copies. The result holds for all operations generated by a Hamiltonian with an unknown interaction strength. This generalizes a similar result in the context of phase covariant cloning where, however, super-replication comes at the price of an exponentially reduced probability of success. We also show that multiple copies of unitary operations can be emulated by operations acting on a much smaller space, e.g., a magnetic field acting on a single n-level system allows one to emulate the action of the field on n 2 qubits.Introduction.-Quantum information can not be cloned. This simple statement, first manifested in [1], has far reaching consequences particularly in the context of quantum cryptography where the no-cloning principle ensures security[2]. A violation of the no-cloning principle would allow for super-luminal communication or the violation of Heisenberg's uncertainty principle, illustrating its fundamental character.However, imperfect replication of quantum information is possible and various works have derived optimal cloning devices under different circumstances [3] (see also[4]). Given N copies of a system in some pure state |ψ , one can deterministically produce M > N copies with a nonunit fidelity that depends on N and M . Moreover, it was shown in [5] that for phase covariant states, i.e., states generated by a Hamiltonian with unknown interaction strength, up to N 2 almost perfect copies can be generated using a probabilistic replication processes. This super-replication of states comes at the price of a success probability that drops exponentially with N .In this letter we show that a similar super-replication can be achieved for the cloning of unitary operations. Here the goal is to produce out of N copies of an unknown unitary operation (given in the form of a black box that can be applied to arbitrary states) M ≥ N copies. This is in general a harder task than cloning of states, as one needs to replicate the action of the operation on all possible input states[6]. Nevertheless, we find that deterministic super-replication of unitary transformations of the form U = e iϑH , where H is the Hamiltonian generating the unitary evolution and ϑ is an unknown interaction strength, is possible in contrast to super-replication of phase covariant states. We demonstrate this result by providing an explicit protocol that makes use of likely sequences as in Schumacher's compression theorem[7].We also consider the emulation of multiple copies of unitary operations by operations acting on a smaller space. Specifically, we find that a single unitary operation with a given interaction strength, ϑ, acting on an n-dimensional system is sufficient to emulate n 2 copies of an operation with the same ϑ acting on a two-dimensional system. In addition, we show that if one can also interject the evolution generated by the Hamiltonian with additional control operations, then one can generate an arbitrary number, M , of perfect copies from single instance of the unitary operation at the cost of a √ M reduction in interaction strength.Background.-We start by specifying the set-up. We | 10.1103/physrevlett.114.120503 | [
"https://arxiv.org/pdf/1410.6008v2.pdf"
]
| 801,460 | 1410.6008 | 1f5a80b90fecc2d960d6c21524f2c582072a63c9 |
Deterministic super-replication of unitary operations
W Dür
Institut für Theoretische Physik
Universität Innsbruck
Technikerstr. 25A-6020InnsbruckAustria
P Sekatski
Institut für Theoretische Physik
Universität Innsbruck
Technikerstr. 25A-6020InnsbruckAustria
M Skotiniotis
Institut für Theoretische Physik
Universität Innsbruck
Technikerstr. 25A-6020InnsbruckAustria
Deterministic super-replication of unitary operations
(Dated: October 23, 2014)numbers: 0367-a0365Ud0367Hk0365Ta
We show that one can deterministically generate out of N copies of an unknown unitary operation up to N 2 almost perfect copies. The result holds for all operations generated by a Hamiltonian with an unknown interaction strength. This generalizes a similar result in the context of phase covariant cloning where, however, super-replication comes at the price of an exponentially reduced probability of success. We also show that multiple copies of unitary operations can be emulated by operations acting on a much smaller space, e.g., a magnetic field acting on a single n-level system allows one to emulate the action of the field on n 2 qubits.Introduction.-Quantum information can not be cloned. This simple statement, first manifested in [1], has far reaching consequences particularly in the context of quantum cryptography where the no-cloning principle ensures security[2]. A violation of the no-cloning principle would allow for super-luminal communication or the violation of Heisenberg's uncertainty principle, illustrating its fundamental character.However, imperfect replication of quantum information is possible and various works have derived optimal cloning devices under different circumstances [3] (see also[4]). Given N copies of a system in some pure state |ψ , one can deterministically produce M > N copies with a nonunit fidelity that depends on N and M . Moreover, it was shown in [5] that for phase covariant states, i.e., states generated by a Hamiltonian with unknown interaction strength, up to N 2 almost perfect copies can be generated using a probabilistic replication processes. This super-replication of states comes at the price of a success probability that drops exponentially with N .In this letter we show that a similar super-replication can be achieved for the cloning of unitary operations. Here the goal is to produce out of N copies of an unknown unitary operation (given in the form of a black box that can be applied to arbitrary states) M ≥ N copies. This is in general a harder task than cloning of states, as one needs to replicate the action of the operation on all possible input states[6]. Nevertheless, we find that deterministic super-replication of unitary transformations of the form U = e iϑH , where H is the Hamiltonian generating the unitary evolution and ϑ is an unknown interaction strength, is possible in contrast to super-replication of phase covariant states. We demonstrate this result by providing an explicit protocol that makes use of likely sequences as in Schumacher's compression theorem[7].We also consider the emulation of multiple copies of unitary operations by operations acting on a smaller space. Specifically, we find that a single unitary operation with a given interaction strength, ϑ, acting on an n-dimensional system is sufficient to emulate n 2 copies of an operation with the same ϑ acting on a two-dimensional system. In addition, we show that if one can also interject the evolution generated by the Hamiltonian with additional control operations, then one can generate an arbitrary number, M , of perfect copies from single instance of the unitary operation at the cost of a √ M reduction in interaction strength.Background.-We start by specifying the set-up. We
k=0 λ k |ϕ k ϕ k | is known but ϑ is not. For instance, this may correspond to a situation where ϑ specifies the unknown interaction strength and the time for applying H is fixed. For simplicity we will consider d = 2 in the following where H = |1 1| and ϑ ∈ [0, 2π), i.e., U (ϑ) is equivalent, up to an irrelevant global phase factor, to a rotation around the z-axis by an angle ϑ. Generalization of the results to arbitrary d and arbitrary Hamiltonians are straightforward.
The goal is to generate M approximate copies of U (ϑ), i.e.,Ṽ (ϑ) ≈ U (ϑ) ⊗M , given only N copies, where M ≥ N . To achieve this task we make use of a suitable number of auxiliary qubits and appropriate unitary operation A-to be applied before and after the application of U (ϑ) ⊗N -that yield an approximation of U (ϑ) ⊗M on an arbitrary input state, see Fig. 1.
FIG. 1. Illustration of the overall procedure to obtain M approximate copies from N applications of an (unknown) unitary operation U (ϑ). By applying the unitary basis change A † , A, before and after U (ϑ) ⊗N we can obtain the operatioñ V (ϑ) of Eq. (5) which is a good approximation of U (ϑ) ⊗M , i.e., A † (1 l ⊗M ⊗ U ⊗N )A |ψ ⊗ |0 ⊗N ≈ U ⊗M |ψ ⊗ |0 ⊗N . Notice that N auxiliary systems are used that are not affected by the transformation.
We quantify the performance ofṼ (ϑ) resulting from our protocol by the global Jamio lkowski Fidelity (process fidelity), F E , averaged over all possible input operations U (ϑ) [6,8]. For an n-dimensional unitary operation, X, the process fidelity of a completely positive map E is defined as where |ψ X , ρ E are the Choi-Jamio lkowski states associated to X, E respectively via the Choi-Jamio lkowski isomorphism [9]. The latter associates to the operations X, E the states |ψ X = 1l ⊗ X|Φ and ρ E =1l ⊗ E (|Φ Φ|) respectively, where |Φ = 1/ √ n n j=1 |j ⊗ |j is a maximally entangled n-level state.
F E (E, X) = ψ X |ρ E |ψ X(
The process fidelity is closely related to the average fidelity,F (E, X) = dψ ψ|U † E(|ψ ψ|)U |ψ , where the average is taken over all input states |ψ . It is known that F (E, X) = (F E (E, X)n + 1)/(n + 1) [10], meaning that a sufficiently large process fidelity ensures that the map E provides a good approximation, on average, for all input states. Throughout this article we consider only unitary operations, where the process fidelity reduces to the overlap of the corresponding pure Jamio lkowski states.
Faithful approximation of U (ϑ) ⊗M .-Consider M copies of an operation
U (ϑ) = e −iϑ|1 1| = |0 0| + e −iϑ |1 1|.(2)
We have that
U (ϑ) ⊗M = k e −i|k|ϑ |k k|,(3)
where we denote by |k ∈ (C 2 ) ⊗M the basis vectors of the M -qubit systems using binary notation, i.e., |0 = |00 . . . 0 , and |k| denotes the Hamming weight of the vector k-the number of ones in binary notation. The corresponding Jamio lkowski state, 1l ⊗ U (ϑ) ⊗M |Φ , with |Φ = 2 −M/2 k |k ⊗ |k is given by
|ψ U (ϑ) ⊗M = 2 −M/2 k e −i|k|ϑ |k ⊗ |k ,(4)
and all basis vectors with the same Hamming weight pick up the same phase factor. Our goal is to approximate the action of U (ϑ) ⊗M . To this aim, consider an operationṼ (ϑ) acting on M qubits that only produces the appropriate phases for the majority of basis vectors. The underlying distribution of the basis vectors in U (ϑ) ⊗M is binomial, centered at k = |k| = M/2, and in the limit of large M approaches the Gaussian distribution of the same mean and standard deviation σ = √ M /2 [11]. Hence, it suffices to reproduce phases for k ∈ (k − , k + ) with k ± = M/2 ± αM β for some α > 0 and 1/2 < β < 1. The operatioñ
V (ϑ) = |k|∈(k−,k+) e −i|k|ϑ |k k| + |k| ∈(k−,k+) e −iγ k |k k|, (5) with arbitrary γ k approximates U (ϑ) ⊗M , where the process fidelity F E (Ṽ (ϑ), U (ϑ) ⊗M ) = | ψṼ (ϑ) |ψ U (ϑ) ⊗M | 2 is bounded from below by Φ(2αM β−1/2 ) = 1/ √ 2π αM β−1/2
−αM β−1/2 e y 2 /2 dy for any value of ϑ. For our choice of α, β, we have that F E → 1 for large M . Notice that also for finite, moderate values of of M one obtains a faithful approximation, which can be checked by directly evaluating the sum of binomial coefficients. Using Stirling's formula, one can approximate the binomial coefficients directly instead of invoking the Gaussian approximation, and arrives at the same conclusion, i.e. for our choice of α, β, F E → 1 in the limit of large M .
Cloning protocol.-We now show how to obtaiñ V (ϑ) ≈ U (ϑ) ⊗M from U (ϑ) ⊗N whenever N = M β , ∀β > 1/2. As mentioned above it is sufficient to obtain the proper phases on all basis states |k with |k| ∈ (k − , k + ). The latter set contains 2αM β + 1 different phases, with values k − + mϑ where 0 ≤ m ≤ 2αM β . Furthermore, we need only reproduce the phases mϑ as the resulting operation is equivalent up to an irrelevant global phase factor e −ik−ϑ . As U (ϑ) ⊗N contains N +1 distinct phases, e i|k|ϑ , 0 ≤ |k| ≤ N (see Eq. (3)), choosing N = 2αM β is sufficient to reproduce all the required phases ofṼ (ϑ) in the interval (k − , k + ) (see Eq. (5)).
To properly approximate U (ϑ) ⊗M each phase e −i|k|ϑ has to be reproduced on all the M |k| levels that lay in the multiplicity space for each |k| ∈ (k − , k + ). To do so, we attach M additional auxiliary systems and consider the operation 1l ⊗M ⊗ U (ϑ) ⊗N (see Fig. 2). As the largest multiplicity inṼ (ϑ) is M M/2 , M auxiliary systems are sufficient as each eigenstate in 1l
⊗M ⊗ U (ϑ) ⊗N is 2 M N |k| - degenerate.
To obtainṼ (ϑ) from 1l ⊗M ⊗ U (ϑ) ⊗N , all we need is to establish a basis change that maps the eigenstates with the appropriate phases onto each other. This is done as follows. Consider the M + N qubit state |k ⊗ |0 where |k is an M -qubit state and |0 is the state of N auxiliary qubits. We use the mapping
|k ⊗ |0 → |k ⊗ |0 if |k| ∈ (k − , k + ) |k ⊗ |0 → |k ⊗ ||k| − k − if |k| ∈ (k − , k + ), (6) where ||k| − k − = |0 ⊗N −(|k|−k−) ⊗ |1 ⊗|k|−k−
is a specific N -qubit state upon which U (ϑ) ⊗N acts and |k is an M -qubit state upon which the identity acts (See Fig: 2). Notice that for |k| ∈ (k − , k + ), the state |k ⊗ ||k| − k − picks up the phase e −iϑ(|k|−k−) , which is the correct phase up to an overall phase factor e ik−ϑ . Moreover, the number of states with this phase factor corresponds to all M -bit strings |k with Hamming weight |k|, which is precisely the multiplicity of e −i|k|ϑ for |k| ∈ (k − , k + ) in Eq. (5). All other states outside the bulk do not obtain a phase [12]. For all other states {|k ⊗ |l } we can choose an arbitrary mapping to one of the other basis states such that the overall operator, A, is unitary [13]. After application of U ⊗N to the last N qubits, one only needs to undo the basis change by applying A † , see Fig. 1. The choice of N = 2αM β for β > 1/2 ensures that the Jamio lkowski fidelity is close to 1 in the limit of large N, M , and hence super-replication with a rate of O(N 2 ) is achieved. Note that one can indeed show that this rate is optimal. It is known that in state super-replication, the Heisenberg limit, i.e., a replication rate of N 2 , is optimal [5]. As this also applies to the Choi-Jamio lkowski state-which can be obtained deterministically from the unitary-any higher replication rate for unitaries would imply a corresponding higher rate for the state which is impossible [14].
Notice that in contrast to state super-replication our protocol works deterministically. This also holds when we apply the protocol to input states |+ ⊗M , which corresponds to the case of phase-covariant state cloning. The difference is that in our case the information on the unknown parameter, ϑ, is encoded in the unitary operation and not in a particular state as is the case in [5]. Whereas standard cloning protocols deal with input states that are of tensor product structure, here it is possible to apply the unitary operations to general (entangled) states, which is effectively achieved by the mapping A. One can also directly adapt the protocol of [5] to accomplish deterministic state super-replication if we incorporate the filter into the state preparation procedure, prior to the application of the unitaries -which imprint the state informationand the cloning protocol.
We remark that our result can be generalized to arbitrary d-dimensional unitary operations generated by a Hamiltonian with unknown interaction strength, W (ϑ) = exp(−iϑH) where H = j λ j |ϕ j ϕ j |. For d > 2, the relevant, likely subspace of W (ϑ) ⊗N follows a multinomial, rather than a binomial, distribution that converges to a multivariable Gaussian distribution centered at p k = λ k N . As long as the Gaussian has a width of O( √ N ) in each dimension, the approximation is faithful. It follows that one can generate an approximation of W (ϑ) ⊗N 2 from W (ϑ) ⊗N in this case as well, where the required protocol is a direct generalization of the one presented for d = 2. The key ingredient is again the unitary operation A where now the tensor product of eigenstates |ϕ k , belonging to the likely subspace, are appropriately mapped so that they pick up the correct phase factor when W (ϑ) ⊗N is applied. As the spectral properties of the Hamiltonian have no bearing in our argument, superreplication is possible for arbitrary Hamiltonians as well.
Emulation of multi-qubit operations.-In a similar way one can also consider emulation of operations that depend on the same (unknown) parameter, ϑ, but act on different systems. For example, consider the operation V (ϑ) = exp(−iϑH V ), where H V = n−1 j=0 j|j j| is the Hamiltonian acting on an n-level system, and the unitary operation U (ϑ) of Eq. (2) acting on a qubit. The above operations describe a spin-(n − 1)/2 and a spin-1/2 particle coupled to the same magnetic field of unknown strength ϑ. Using the techniques established in the previous section it is straightforward to show that a single use of V (ϑ) is sufficient to approximate M uses of U (ϑ) whenever n = 2αM β and α > 0, β > 1/2. To see this first note that V (ϑ) and U (ϑ) ⊗n have the same spectrum; only the multiplicities of the various eigenvalues differ. By attaching M auxiliary qubits, on which the identity acts, one can construct a similar unitary operator to A above and obtain U (ϑ) ⊗n exactly [15]. Using the scheme described in the previous section we can now obtain an approximation of U (ϑ) ⊗n 2 from n uses of U (ϑ).
The above result highlights an important equivalence between higher dimensional systems and the number of uses of a unitary operator on a two-level system. One can trade a single use of a unitary acting on an n-level system for an approximate n 2 uses of a unitary operator acting on qubits.
So far we have considered that additional control is available only before and after the application of the unitary operations. However, in many physically relevant situations, where U (ϑ) is generated by a Hamiltonian with unknown interaction strength that is applied for a fixed time, additional control is available. In these cases one can interject the Hamiltonian evolution with ultrafast control pulses thus modifying the effective evolution [16]. This technique, also known as "bang-bang control", allows one to generate an effective Hamiltonian with a modified spectrum. The use of bang-bang control techniques allows for more advanced emulation schemes. For example, consider the n-fold degenerate Hamiltonian with eigenvalues 0, 1. Such a Hamiltonian describes, for example, the spin and motional degrees of freedom of an electron, where the spin degrees of freedom are acted upon by the Hamiltonian H = ϑ|1 1|-the same Hamiltonian that generates U (ϑ) in Eq. (2)-and n motional degrees of freedom are acted on by the identity. Intermediate control pulses allow one to modify the spectrum of the effective Hamiltonian such that it contains n eigenstates, whose eigenvalues are evenly gapped, and all but the ground state level are non-degenerate. Up to a multiplicative factor of n, this is the same spectrum as for the Hamiltonian H V above where the multiplicative factor leads to an evolution V (ϑ/n) instead of V (ϑ). Hence, one can use the same technique as before to obtain multiple single-qubit operations. In fact, as n can be freely chosen, we have that from a single application of H for time t = 1, a single qubit operation U (ϑ), one can generate up to n 2 copies of an operation with reduced strength ϑ/n, i.e., U (ϑ) → U (ϑ/n) ⊗n 2 .
Links to quantum metrology.-We now discuss connections between the super-replication of unitary operations established above and quantum metrology. The latter deals with optimally estimating an unknown parameter, ϑ, by choosing an optimal input state on which ϑ is imprinted, and reading out the desired information by means of an optimal measurement [17]. When the input state is a product state of N qubits and the parameter, ϑ, is imprinted by applying the operation U (ϑ) of Eq. (2) on each qubit, then the achievable precision, δϑ, in the estimation of ϑ is bounded by δϑ ≥ O(1/N ), the standard quantum limit. When the N qubits are prepared in an entangled state, however, an accuracy of δϑ = O(1/N 2 ) can be achieved.
Our super-replication procedure establishes an equivalence between different resources namely, N uses of U (ϑ) on an entangled input state of N qubits, N 2 uses of U (ϑ) on the optimal product state of N 2 qubits, and a single use of
V (ϑ) = exp(−iϑH V ), where H V = N −1 j=0 j|j j|
acts on a single N -dimensional spin.
In particular, consider the case of quantum metrology where the input state comprises multiple qubits in the state |0 x = 1/ √ 2(|0 + |1 ). This set-up corresponds precisely to phase covariant cloning [5], where the information on the phase is contained in the unitary U (ϑ), and, in this case, the global fidelity of the cloned states is equivalent to the process fidelity. In [5] it was shown that the optimal super-replication strategy for states can produce at most M = N 2 copies, and saturates the Heisenberg limit. Using our super-replication procedure for unitary operations, and applying it to the product input state |0 x ⊗N , one achieves the same, optimal, precision in quantum metrology as when U (ϑ) ⊗N act directly on the optimal entangled input state of N qubits.
However, this does not guarantee that a high fidelity is achieved for all input states. The figure of merit for the super-replication procedure is the process fidelity, which provides a bound on the average state fidelity averaged over all possible input states. We stress that the process fidelity is the standard way of measuring the accuracy of operations and processes, and a high process fidelity implies a good approximation of the process [6,8]. On the one hand, there exist states where the fidelity exceeds the process fidelity, e.g. for any state of the form |ψ = |k|∈(k−,k+) α k |k the fidelity is one. On the other hand, there are also several input states for which the achievable fidelity is smaller than the process fidelity.
In fact, it turns out that the action of U (ϑ) ⊗M is not appropriately mimicked for input states that are themselves useful for parameter estimation, i.e., have a quantum Fisher information that is of O(M 2 ). This is to be expected, as otherwise the Heisenberg limit for metrology would be violated by combining the super-replication of unitary operations as established here, and letting the protocol act on entangled input states.
Indeed, the protocol will not work for the optimal input state for quantum metrology, (|0 ⊗M + |1 ⊗M )/ √ 2. Only random phases will be imprinted on both, |0 ⊗M and |1 ⊗M , rather than the required phases 0 and M ϑ. By construction, our super-replication protocol yields a faithful approximation only for the bulk of states where the Hamming weight is approximately M/2, i.e., only for energy eigenstates with energy approximately M/2 ± √ M . All states with a large support on this subspace have quantum Fisher information that scales only as O(M ). States with a quantum Fisher information scaling as O(M 2 ) are superpositions of eigenstates where the eigenvalues differ by O(M ) [18]. For all these states the proper phases are not reproduced by our superreplication protocol, however the relative volume of those states goes to zero with increasing M .
Finally, a single use of V (ϑ) on a N -dimensional spin also allows to mimic the action of U (ϑ) on N 2 product states |0 x , and hence to achieve the same precision in the estimation of ϑ. One can trade between the number of levels and the number of copies of a two-level system.
consider a class of d-dimensional unitary operations U (ϑ) = exp(−iϑH) generated by a Hamiltonian, H, and parametrized by ϑ. The operations are provided in the form of an unknown black box, where H = d−1
AFIG. 2 .
2Illustration of the unitary mapping A. Eigenstates |k of U ⊗M with corresponding degeneracies are depicted on the right. Eigenstates |j of U (ϑ) ⊗N are depicted on the left, where the degeneracy are achieved by adding M ancillay systems on which an identity operation acts. The relevant part of the spectrum of U (ϑ) ⊗M is mapped to the appropriate eigenstates of U ⊗N ⊗ I ⊗M .
Conclusion and outlook.-We have demonstrated the deterministic super-replication of unknown unitary operations. For all operations generated by a Hamiltonian with unknown interaction strength, one can produce up to N 2 copies of the operation using the operation only N times. This surprising result is in perfect agreement with similar effects in state super-replication and quantum metrology. Whether a similar improvement can be obtained for arbitrary unitary operations of the group SU (2) remains an open question.
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. H Fan, Y.-N Wang, L Jing, J.-D Yue, H.-D Shi, Y.-L Zhang, L.-Z Mu, 10.1016/j.physrep.2014.06.004accepted in Physics ReportsH. Fan, Y.-N. Wang, L. Jing, J.- D. Yue, H.-D. Shi, Y.-L. Zhang, and L.-Z. Mu, (2014), 10.1016/j.physrep.2014.06.004, accepted in Physics Re- ports.
. G Chiribella, Y Yang, A C , -C Yao, Nature communications. 4G. Chiribella, Y. Yang, and A. C.-C. Yao, Nature com- munications 4 (2013).
. G Chiribella, G M D'ariano, P Perinotti, 10.1103/PhysRevLett.101.180504Phys. Rev. Lett. 101180504G. Chiribella, G. M. D'Ariano, and P. Perinotti, Phys. Rev. Lett. 101, 180504 (2008).
. B Schumacher, 10.1103/PhysRevA.51.2738Phys. Rev. A. 512738B. Schumacher, Phys. Rev. A 51, 2738 (1995).
. A Gilchrist, N K Langford, M A Nielsen, 10.1103/PhysRevA.71.062310Phys. Rev. A. 7162310A. Gilchrist, N. K. Langford, and M. A. Nielsen, Phys. Rev. A 71, 062310 (2005);
. W Dür, M Hein, J I Cirac, H.-J Briegel, 10.1103/PhysRevA.72.052326Phys. Rev. A. 7252326W. Dür, M. Hein, J. I. Cirac, and H.-J. Briegel, Phys. Rev. A 72, 052326 (2005).
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We note that, whereas the binomial distribution is strictly defined over the positive real line, in the limit of large M it can be shown that the Gaussian distribution over the negative real numbers only incurs an error in the approximation. that scales as O(M −1 ) [19We note that, whereas the binomial distribution is strictly defined over the positive real line, in the limit of large M it can be shown that the Gaussian distribution over the negative real numbers only incurs an error in the approx- imation that scales as O(M −1 ) [19].
One may introduce an additional random phase e −iγ|k| with γ ∈ (0, 2π] whenever |k| ∈ (k−, k+). which guarantees that the protocol works equally well for all ϑOne may introduce an additional random phase e −iγ|k| with γ ∈ (0, 2π] whenever |k| ∈ (k−, k+), which guaran- tees that the protocol works equally well for all ϑ.
Note that it is sufficient for the map to perform the right action only when the N auxiliary systems are prepared in a given state. Note that it is sufficient for the map to perform the right action only when the N auxiliary systems are prepared in a given state.
. G Chiribella, private communicationG. Chiribella, private communication.
We stress that there is no approximation of U (ϑ) ⊗n here as the entire spectrum of the latter can be obtained. not just the typical subspaceWe stress that there is no approximation of U (ϑ) ⊗n here as the entire spectrum of the latter can be obtained, not just the typical subspace.
. L Viola, S Lloyd, 10.1103/PhysRevA.58.2733Phys. Rev. A. 582733L. Viola and S. Lloyd, Phys. Rev. A 58, 2733 (1998);
. L Viola, E Knill, S Lloyd, 10.1103/PhysRevLett.82.2417Phys. Rev. Lett. 822417L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. 82, 2417 (1999).
. V Giovannetti, S Lloyd, L Maccone, Science. 3061330V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330 (2004).
. M Skotiniotis, F Fröwis, W Dür, B Kraus, arXiv:1409.2316arXiv preprintM. Skotiniotis, F. Fröwis, W. Dür, and B. Kraus, arXiv preprint arXiv:1409.2316 (2014).
. M Skotiniotis, G Gour, New Journal of Physics. 1473022M. Skotiniotis and G. Gour, New Journal of Physics 14, 073022 (2012).
| []
|
[
"Sun's Three Conjectures on Apéry-like Sums Involving Harmonic Numbers",
"Sun's Three Conjectures on Apéry-like Sums Involving Harmonic Numbers"
]
| [
"Ce Xu \nSchool of Mathematics and Statistics\nAnhui Normal University\n241002WuhuPRC\n",
"Jianqiang Zhao \nDepartment of Mathematics\nThe Bishop's School\nLa Jolla92037CAUSA\n"
]
| [
"School of Mathematics and Statistics\nAnhui Normal University\n241002WuhuPRC",
"Department of Mathematics\nThe Bishop's School\nLa Jolla92037CAUSA"
]
| []
| In this paper, we will give another proof of Zhi-Wei Sun's three conjectures on Apéry-like sums involving harmonic numbers by proving some identities among special values of multiple polylogarithms. | null | [
"https://arxiv.org/pdf/2203.04184v2.pdf"
]
| 247,315,564 | 2203.04184 | e00dcc00e33c5be55e5bc96d5dfd699494af8ffb |
Sun's Three Conjectures on Apéry-like Sums Involving Harmonic Numbers
13 Mar 2022
Ce Xu
School of Mathematics and Statistics
Anhui Normal University
241002WuhuPRC
Jianqiang Zhao
Department of Mathematics
The Bishop's School
La Jolla92037CAUSA
Sun's Three Conjectures on Apéry-like Sums Involving Harmonic Numbers
13 Mar 2022arXiv:2203.04184v2 [math.NT]Apéry-like sumsharmonic numbersmultiple polylogarithm functioniterated integrals AMS Subject Classifications (2020): 11M3211B65
In this paper, we will give another proof of Zhi-Wei Sun's three conjectures on Apéry-like sums involving harmonic numbers by proving some identities among special values of multiple polylogarithms.
Introduction
In his new book [12], Prof. Zhi-Wei Sun listed 820 mathematical conjectures, including ten concerning the Apéry-like sums involving harmonic numbers, some of which had appeared in his previous paper [11]. In this paper, we will prove the following three conjectures by using a few results of Akhilesh [1] and , together with Au's package [3,4].
Proof of conjectures
The Riemann zeta values ζ(k) (2 ≤ k ∈ N) are defined by
ζ(k) := ∞ n=1 1 n k . (2.1)
In particular, Euler determined the explicit values of zeta values function at even integers in 1775:
ζ(2k) = − B 2k 2(2k)! (2π √ −1) 2k ,
where B n are Bernoulli numbers defined by the generating function
t e t − 1 = ∞ n=0 B n t n n! .
These can be regarded as special values of the polylogarithm function Li k (x). More generally, for any k 1 , . . . , k r ∈ N, the classical multiple polylogarithm function is defined by Li k 1 ,...,kr (x 1 , . . . , x r ) := n 1 >n 2 >···>nr>0
x n 1 1 · · · x nr r n k 1 1 · · · n kr r which converges if |x 1 · · · x j | < 1 for all 1 ≤ j ≤ r. In particular, for |x| ≤ 1 and (x, Remark 2.2. The formula (2.2) has also already appeared in Hessami Pilehroods [9], page 220, between (7) and (8).
ln 4 (ϕ) − 7 50 π 2 ln 2 (ϕ) + π 4 50 , (2.4) ∞ n=1 (−1) n n 4 2n n = 8 Li 3 (ϕ) ln(ϕ) + 1 2 Li 4 (ϕ 2 ) − 8 Li 4 (ϕ) − 4 5 ζ(3) ln(ϕ) + 13 6 ln 4 (ϕ) − 7 15 π 2 ln 2 (ϕ) + 7π 4 90 . (2.5) Lemma 2.4. ([7, (3.12)-(3.13)]) For u ∈ (−∞, 0) ∪ (4, +∞), define y := 1 − u u−4 1 + u u−4 , H −1,0,0,1 (−y) := −y 0 Li 3 (x) 1 + x dx.
Then we have Remark 2.5. In [7], Li 2,1 (y) and Li 3,1 (y) are denoted by S 1,2 (y) and S 2,2 (y), respectively. More general, for a, b ∈ N and z ∈ [0, 1] the function S a,b (z) is defined by
∞ n=1 u n n 3 2n n H n−1 = 4H −1,0,0,1 (−y) + Li 3,1 (y 2 ) − 4 Li 3,1 (y) − 4 Li 3,1 (−y) − 6 Li 4 (−y) −2 Li 4 (y) + 4 Li 2,1 (−y) ln y + 4 Li 2,1 (y) ln y − 2 Li 2,1 (y 2 ) ln(y) + 4 Li 3 (−y) ln(1 − y) +2 Li 3 (−y) ln y + 2 Li 3 (y) ln y − Li 2 (y) ln 2 y − 4 Li 2 (−y) ln y ln(1 − y) − 1 3 ln 3 y ln(1 − y) + 1 24 ln 4 y + 2ζ(2) Li 2 (y) − 1 2 ζ(2) ln 2 y + 2ζ(2) ln y ln(1 − y) +6ζ(3) ln(1 − y) − 3ζ(3) ln y − 4ζ(4) , (2.6) ∞ j=1 u n n 3 2n n H 2n−1 = 4H −1,0,0,1 (−y) + Li 3,1 (y 2 ) − 8 Li 3,1 (y) − 4 Li 3,1 (−y) − 6 Li 4 (−y)S a,b (z) := (−1) a−1+b (a − 1)!b! 1 0 ln a−1 (t) ln b (1 − zt) t dt = 1 (a − 1)!b! 1 0 1 t dt t a−1 dt t t 0 z dt 1 − zt b = 1 0 dt t a z dt 1 − zt b = z 0 dt t a dt 1 − t b = Li a+1,1 b−1 (z),
where we have used Chen's iterated integrals above to represent the single-variable multiple polylogarithm, see [15,Ch. 2].
Theorem 2.6. We have Hence, the (2.9) can be rewritten as the form
n (H 2n + 4H n ) = −8 Li 3 (ϕ) ln(ϕ) − 9 2 Li 4 (ϕ 2 ) + 8 Li 4 (ϕ) + 4 Li 3,1 (ϕ 2 ) + Li 2 2 (ϕ 2 ) − 8 Li 2,1 (ϕ 2 ) ln(ϕ) + 8 Li 3 (ϕ 2 ) ln(ϕ) − 6 Li 2 (ϕ 2 ) ln 2 (ϕ) − 2ζ(2) Li 2 (ϕ 2 ) + 4 5 ζ(3) ln(ϕ) − 19 6 ln 4 (ϕ) + 2 15 π 2 ln 2 (ϕ).
(2.14)
By applying Au's package [4] (also see Remark 2.7 below), we have
Li 2,1 (ϕ 2 ) = MZIteratedIntegral[0, ϕ −2 , ϕ −2 ] = ζ(3) + π 2 10 ln(ϕ) − Li 3 (ϕ), (2.15) Li 3,1 (ϕ 2 ) = MZIteratedIntegral[0, 0, ϕ −2 , ϕ −2 ] = π 4 90 − π 2 20 ln 2 (ϕ) + 3 8 ln 4 (ϕ) + 9 8 Li 4 (ϕ 2 ) − 2 Li 4 (ϕ) + 1 5 ζ(3) ln(ϕ).
(2.16)
Further, from [10, (1.20) and (6.13)] we have
Li 2 (ϕ 2 ) = π 2 15 − ln 2 (ϕ), (2.17) Li 3 (ϕ 2 ) = 4 5 ζ(3) − 2 3 ln 3 (ϕ) + 2 15 π 2 ln(ϕ(y, x) = Li * 3 (x) − Li * 3 x − xy 1 − xy + Li * 3 (xy) − Li 3 y − xy 1 − xy + Li 3 (y) − Li 3 (xy) − ln(1 − xy)(Li 2 (x) + Li 2 (y)) − 1 2 ln 2 1 − x 1 − xy ln 1 − y 1 − xy , where Li * 3 (x) = Li 3 (1) − Li 3 (1 − x) + Li 2 (1) ln(1 − x) − 1 2 ln(x) ln 2 (1 − x).
Taking y = ϕ 2 and using 1 − y = ϕ we see that
lim x→1 − Li * 3 (x) − Li * 3 x − xy 1 − xy = Li 2 (1) ln(1 − y) = π 2 6 ln(ϕ).
Hence Letting x = ϕ 2 and noting the fact that (see [10, (1.20
Li 2,1 (ϕ 2 ) = Li 3 (1) − Li 3 (ϕ) − ln(ϕ) Li 2 (ϕ 2 ) − ln 3 (ϕ) + π 2 6 ln(ϕ) = ζ(3) +
)])
Li 2 (ϕ) = π 2 10 − ln 2 (ϕ)
gives Li 2,1 (ϕ 2 ) = ζ(3) − Li 3 (ϕ) + ln(ϕ) Li 2 (ϕ) + ln 3 (ϕ) = ζ(3) + π 2 10 ln(ϕ) − Li 3 (ϕ).
A human proof of (2.16) is possible although it is conceivably much more complicated than (2.15). In particular, setting k = 3 and r = 2 [13, Thm.
where H n is the classical harmonic number defined by H 0 := 0 and H n
Remark
1 ,...,kr (x) := Li k 1 ,...,kr (x, 1, . . . , classical single-variable multiple polylogarithm function. In particular, if x = 1 then Li k 1 ,...,kr (1) become the multiple zeta values ζ(k 1 , . . . , k r ), namely, ζ(k 1 , . . . , k r ) := Li k 1 ,...,kr (1). Lemma 2.1. [1, Eq. (122)]
Lemma 2. 3 .
3([4, Example 6.16 and 6.18]) Let ϕ := 3 (ϕ) ln(ϕ) + 3 20 Li 4 (ϕ 2 ) − 12 5 Li 4 (ϕ) − 6 25 ζ(3) ln(
+2
Li 4 (y) − [Li 2 (y)] 2 + 4 Li 2,1 (−y) ln y + 8 Li 2,1 (y) ln y − 2 Li 2,1 (y 2 ) ln y + 1 48 ln 4 y +4 Li 3 (−y) ln(1 − y) − 4 Li 3 (y) ln(1 − y) + 2 Li 3 (−y) ln y − 4 Li 2 (−y) ln y ln(1 − y) +2 Li 2 (y) ln y ln(1 − y) − 1 2 Li 2 (y) ln 2 y − 1 6 ln 3 y ln(1 − y) + 4ζ(2) ln y ln(1 − y) −ζ(2) ln 2 y + 10ζ(3) ln(1 − y) − 5ζ(3) ln y + 4ζ(2) Li 2 (y) − 19 2 ζ(4). (2.7)
First, applying (2.4) and (2.5) yields equation(2.8). To prove (2.9), by applying the stuffle relations (or quasi-shuffle relations, see[8])
( 2 )(
2n−1 − H 2n−1 ) = 4 Li 3,1 (y) − 4 Li 4 (y) + Li 2 2 (y) − 4 Li 2,1 (y) ln y + 2 Li 3 (y) ln y − 1 2 Li 2 (y) ln 2 y + 4 Li 3 (y) ln(1 − y) − 2 Li 2 (y) ln y ln(1 − y) ln 2 y − 2ζ(2) ln y ln(1 − y) − 4ζ(3) ln(1 − y) H n−1 − H 2n−1 ) = 4 Li 3,1 (ϕ 2 ) − 4 Li 4 (ϕ 2 ) + Li 2 2 (ϕ 2 ) − 8 Li 2,1 (ϕ 2 ) ln(ϕ) + 8 Li 3 (ϕ 2 ) ln(ϕ) − 6 Li 2 (ϕ 2 ) ln 2 (ϕ) − ln 4 (ϕ)− 2ζ(2) Li 2 (ϕ 2 ) − 2ζ(2) ln 2 (ϕ)
π 2
210 ln(ϕ) − Li 3 (ϕ) by (2.17). This proves (2.15). Second, setting k = r = 2 in [2, Thm. 8] or [13, Thm. 2.1] yields Li 2,1 (x) = ζ(3) − Li 3 (1 − x) + ln(1 − x) Li 2 (1 − x) + 1 2 ln(x) ln 2 (1 − x).
3,1 (1 − x) = ln(x) (ζ(3) − Li 3 (1 − x) + ln(1 − x) Li 2 (1 − x))
Multiple zeta values and multiple Apéry-like sums. P Akhilesh, J. Number Theory. 226P. Akhilesh, Multiple zeta values and multiple Apéry-like sums, J. Number Theory. 226(2021), pp. 72-138.
Multiple zeta values, poly-Bernoulli numbers, and related zeta functions. T Arakawa, M Kaneko, Nagoya Math. J. 153T. Arakawa and M. Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J. 153 (1999), pp. 189-209.
Evaluation of one-dimensional polylogarithmic integral. K C Au, arXiv:2007.03957with applications to infinite series. K.C. Au, Evaluation of one-dimensional polylogarithmic integral, with applications to infi- nite series, arXiv:2007.03957.
K C Au, arXiv:2201.01676Iterated integrals and special values of multiple polylogarithm at algebraic arguments. K.C. Au, Iterated integrals and special values of multiple polylogarithm at algebraic argu- ments, arXiv:2201.01676.
Alternating series of Apeŕy-type for the Riemann zeta function. W Chu, Contribut. Discrete Math. 15W. Chu, Alternating series of Apeŕy-type for the Riemann zeta function, Contribut. Discrete Math. 15(2020), pp. 108-116.
Further Apeŕy-like series for Riemann zeta function. W Chu, Math. Notes. 109W. Chu, Further Apeŕy-like series for Riemann zeta function, Math. Notes 109(2021), pp. 136-146.
Massive Feynman diagrams and inverse binomial sums. A I Davydychev, M Yu, Kalmykov, arXiv:hep-th/0303162v4Nuclear Phys. B. 699A.I. Davydychev and M. Yu. Kalmykov, Massive Feynman diagrams and inverse binomial sums, Nuclear Phys. B 699 (2004), pp. 3-64. arXiv:hep-th/0303162v4.
Quasi-shuffle products. M E Hoffman, J. Algebraic Combin. 11M.E. Hoffman, Quasi-shuffle products, J. Algebraic Combin. 11(2000), 49-68.
Congruences arising from Apeŕy-type series for zeta values. Kh, T. Hessami Hessami Pilehrood, Pilehrood, Adv. Appl. Math. 49Kh. Hessami Pilehrood and T. Hessami Pilehrood, Congruences arising from Apeŕy-type series for zeta values, Adv. Appl. Math. 49(2012), 218-238.
Polylogarithms and Associated Functions. L Lewin, Elsevier Sci. PublishersNew York, New YorkL. Lewin, Polylogarithms and Associated Functions, Elsevier Sci. Publishers, New York, New York, 1981.
New series for some special values of L-functions. Z.-W Sun, Nanjing Univ. J. Math. Biquarterly. 322Z.-W. Sun, New series for some special values of L-functions, Nanjing Univ. J. Math. Biquarterly 32(2015), no.2, 189-218.
Z.-W Sun, New Conjectures in Number Theory and Combinatorics. HarbinHarbin Institute of Technology Press2021in ChineseZ.-W. Sun, New Conjectures in Number Theory and Combinatorics (in Chinese), Harbin Institute of Technology Press, Harbin, 2021.
Explicit relations between multiple zeta values and related variants. C Xu, Adv. Appl. Math. 130102245C. Xu, Explicit relations between multiple zeta values and related variants, Adv. Appl. Math. 130(2021), 102245.
Motivic complexes of weight three and pairs of simplices in projective 3-space. J Zhao, Adv. Math. 161J. Zhao, Motivic complexes of weight three and pairs of simplices in projective 3-space, Adv. Math. 161(2001), pp. 141-208.
Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values. J Zhao, Series on Number Theory and its Applications. Hackensack, NJWorld Scientific Publishing Co. Pte. Ltd12J. Zhao, Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values, Series on Number Theory and its Applications, Vol. 12, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016.
| []
|
[
"A Private and Unlinkable Message Exchange Using a Public bulletin board in Opportunistic Networks",
"A Private and Unlinkable Message Exchange Using a Public bulletin board in Opportunistic Networks"
]
| [
"Ardalan Farkhondeh [email protected] \nDepartment of Information and Communications Engineering Autonomous\nUniversity of Barcelona\n\n"
]
| [
"Department of Information and Communications Engineering Autonomous\nUniversity of Barcelona\n"
]
| []
| We plan to simulate a private and unlinkable exchange of messages by using a Public bulletin board and Mix networks in Opportunistic networks. This Opportunistic network uses a secure and privacy-friendly asynchronous unidirectional message transmission protocol. By using this protocol, we create a Public bulletin board in a network that makes individuals send or receive events unlinkable to one another [18]. With the design of a Public bulletin board in an Opportunistic network, the clients can use the benefits of this Public bulletin board in a safe environment. When this Opportunistic network uses the protocol, it can guarantee an unlinkable communication based on the Mix networks. The protocol can work with the Public bulletin board exclusively with acceptable performance. Also, this simulation can be used for hiding metadata in the bidirectional message exchange in some messengers such as WhatsApp. As we know, one of the main goals of a messenger like WhatsApp is to protect the social graph. By using this protocol, a messenger can protect social graph and a central Public bulletin board. | null | [
"https://arxiv.org/pdf/1909.02380v1.pdf"
]
| 202,537,713 | 1909.02380 | 8622399ea7fa0cfe05c6fca9a2a5ee36d36204ca |
A Private and Unlinkable Message Exchange Using a Public bulletin board in Opportunistic Networks
Ardalan Farkhondeh [email protected]
Department of Information and Communications Engineering Autonomous
University of Barcelona
A Private and Unlinkable Message Exchange Using a Public bulletin board in Opportunistic Networks
1
We plan to simulate a private and unlinkable exchange of messages by using a Public bulletin board and Mix networks in Opportunistic networks. This Opportunistic network uses a secure and privacy-friendly asynchronous unidirectional message transmission protocol. By using this protocol, we create a Public bulletin board in a network that makes individuals send or receive events unlinkable to one another [18]. With the design of a Public bulletin board in an Opportunistic network, the clients can use the benefits of this Public bulletin board in a safe environment. When this Opportunistic network uses the protocol, it can guarantee an unlinkable communication based on the Mix networks. The protocol can work with the Public bulletin board exclusively with acceptable performance. Also, this simulation can be used for hiding metadata in the bidirectional message exchange in some messengers such as WhatsApp. As we know, one of the main goals of a messenger like WhatsApp is to protect the social graph. By using this protocol, a messenger can protect social graph and a central Public bulletin board.
INTRODUCTION
General Objective
We plan to simulate a private and unlinkable exchange of messages by using a Public bulletin board and Mix networks in Opportunistic networks. To do this, we have designed an Opportunistic network that uses a secure and private integrated protocol. We can use this project in mail providers or messengers. In our proposal, clients can send their messages in a safe and secure environment. Users can use the advantage of a public bulletins board and save their messages in a database securely. Two nodes allow for communication in an Opportunistic network so that data transference takes place by using Mix networks and Public bulletin boards in a way that the contact address is not revealed to the participants in the communication [18].
By using Mixer nodes in our proposal, we can make a nontracing Opportunistic network, and a client can send and receive a message anonymously. In this Opportunistic network, malicious users cannot rebuild social structure networks.
In other words, with the simulation of this proposal, we designed indirect and non-tracing communication. In a nutshell, we can say that our primary goal of simulation of this proposal is creating Opportunistic networks by using Public bulletin boards and Mix networks in a non-tracing space.
Document Structure
This document consists of the Abstract and Introduction sections and the Related works section. The Related works section includes three main subsections: Definition of Opportunistic networks, Anonymity in Opportunistic networks and Mix networks.
In the first subsection, the document offers a definition for Opportunistic networks and discusses five routing methods in Opportunistic networks as well as the Architecture of Opportunistic networks. This section tries to familiarize the reader with the basic concepts of Opportunistic networks.
In Anonymity in Opportunistic networks section have two subsections. In the first part, the documents introduced a definition of anonymity in the opportunistic network. Also, this part tries to familiarization the readers with some basic concept of Anonymity in Opportunistic networks. In the second subsection the document introducing some anonymous routing protocols and explain all of them briefly.
The third subsection explains about Mix networks. Mix networks subsection has three parts that including a definition of Mix networks, the description of mixer nodes and privacy issues in Mix networks. Since our simulation model has used Mix networks, this document needs to explain the structure and general concepts of Mix networks. In this subsection, we review Mix networks and explain about the node's behaviour in them. The section illustrates how Mix networks can create paths by using mixer nodes. This subsection also discusses privacy issues in Mix network. Furthermore, we offer some suggestion for improving privacy in Mix networks in the end.
The fourth section is the Proposal section. It includes two main subsections. The first one discusses the structure of Public bulletin boards and their advantages. It offers a detailed explanation of different parts of the Public bulletin board structure. We clarify that a Public bulletin board can have that include Write method, Read method and response method. The second subsection of the proposal discusses client protocol for the transmission of packets between nodes and a Public bulletin board by Mix networks.
The fifth section of the document is the Experimentation section. It includes four subsections. The first subsection describes the simulation environment. In general, in this subsection, we talk about the ONE simulator and describe the differences between java functions in the ONE simulator. The next subsection is Scenario in which we explain about two scenarios run in our simulation. Also, we discuss the differences and similarities of those scenarios. The next subsection is the ONE modification in which we explain all the changes that we made on some functions of the ONE simulator. At the end of Experimentation section, we review the results of both scenarios and discuss all the metrics we have used. Next, we compare both scenarios and present a conclusion on the results.
The final two sections of the document are the conclusion of the whole document and the bibliography.
RELATED WORK
Opportunistic networks
Definition and characteristics
One subclass of the Delay-Tolerant Networks is Opportunistic networks. An Opportunistic network has intermittent communication opportunities in a way that there ever exists an end-to-end path between the sources and the destinations. In an Opportunistic network, link performance is too variable. We cannot use TCP/IP protocol in an Opportunistic network, because of an end-toend path created between the source and the destination for a short period of time and its unpredictability. An Opportunistic network can use node mobility to exploits and local forwarding for data transference, which can be an excellent suggestion to forfeit the lack of TCP/IP protocols [1]. Each node can store and carry data until these nodes have opportunistic contact. The nodes can forward the data when they have opportunistic contact with other nodes. In general, entire messages transferred from a node's buffer to another node's buffer until the messages reach their destination by using a path.
In general, we can mention three main characteristics of Opportunistic networks. These characteristics include flexibility for different environments, low budget and easy implementation. These characteristics help us develop Opportunistic networks quickly. In what follows, we explain each characteristic in more details.
The first characteristic of Opportunistic networks is their flexibility. Opportunistic networks can include a large number of nodes that are always on the move. The movement of nodes can be used to convey messages. The nodes can share their messages with their neighbour nodes. Since nodes are moving, they can transfer the messages from one node to other groups of nodes until the messages arrive at the destination node. The movement of nodes eliminates the need to install the specific infrastructure for transmission of messages in Opportunistic networks. Network designer can design different protocols on movement devices easily, and these protocols can smoothly adapt to the needs of different environments in order to convey messages. Advantages of using these networks include their utilizing a variety of protocols, high versatility with different environments and their use of the movement of nodes for transmission which Opportunistic networks highly flexible. Therefore, we do not need to spend much money to install specific infrastructures. For this reason, we can implement Opportunistic networks without any government funding most of the time. To do so, we only need to use some cheap or free applications on local devices which are capable of sending and receiving messages in one area. Due to the affordability of Opportunistic networks, using them in poor or remote areas has been welcomed.
The last characteristic of Opportunistic networks is related to their easy implementation. Network designer needs to define some protocols and some programming in the form of an application, after that, this application will be installed on the local devices in one area. Thus, this process can be done in a short time. We can imagine an area which has lost all of the network infrastructures by flooding or war and this area needs implementation of a network in a short period of time. Obviously, in such situations, one of the most reasonable choices are Opportunistic networks because of their quick implementation.
Two Examples of Opportunistic network
In Figure 1, a simple example of the operation of an Opportunistic network is displayed. In this figure, we can see different nodes of people and vehicles. These nodes include peoples, cars and satellites that form networks based on the different infrastructures on earth and sky. Some nodes use wired communication to form the networks; for instance, these nodes connect to a central telephone office with wired connections. Also, we can see that some nodes create networks by using a wireless connection. We also see that some sensors or some nodes have established networks based on internet routing. This figure well shows a tolerant network based on a set of heterogeneous networks. In this figure, all subset networks eventually connect into DTN nodes. A DTN node is a node which has the potential to tolerate the delay in message transmission paths.
As shown in Figure 2, this Opportunistic network consists of three groups (A, B, and C) of nodes located in separate locations. Each group consists of two types of node. The first type of node merely consists of group members and plays a role in transmitting messages inside their location. These normal nodes cannot transfer the messages to the outside, but these nodes can transfer ordinary messages between members. Other types of nodes are intermediate nodes which can send obfuscated messages between different groups of nodes in different locations [3].
As shown in this figure, the node U1 generates the message and sends it to the node U2, which is an intermediate node. Node U2 transfers the message from the group location A to the group location B. Node U2 moves to the group location B, and then node U2 publishes the messages among the second-nodal group members. After being published, the message arrives at the node U4, which is an intermediate node. Node U4 moves to the next location and publishes the message to the group C. After the message reaches U5, we have established a perfect connection between the nodes in different places. Now, the node U1 can send the message through the intermediate nodes to the node U5. The node U5 can send the message through antenna and route through the various routers to a service provider centre. Thus, this figure shows an implementation that a node that can by using intermediate nodes in different locations send a message to a long-distance service provider. Figure 2 shows that we can make a path between two longdistance nodes. Moreover, due to the heterogeneity of the path, the Opportunistic network may be associated with many disturbances. Some disturbances may cause the Opportunistic network to fail the message along the transferring path. Network designers need to plan nodes to solve such kinds of problems in different situations. It illustrates a condition in which the node may not have enough space to buffer a message. This node may not have enough space to buffer a message. Therefore, we propose that buffers be shared between neighbouring nodes. In this case, if a node encounters a space shortage, it can use the neighbouring node buffers. This suggestion can be an example of network flexibility in node planning to solve problems.
Classification of routing in Opportunistic networks
We can classify routing algorithms based on the data forwarding behaviour in the Opportunistic networks in five main categories. These categories include Direct Transmission, Flooding Based, Prediction Based, Coding Based, and Context-based routing [2].
A. Direct Transmission
In this algorithm of routing, each node generates a packet for sending to the destination. To do so, they store the in their buffer at first. The packets remain in the buffers until the nodes meet up the node destination.
The nodes can forward the message directly to their desired destination, and they just forward a single copy of the message. Therefore, nodes must wait for an opportunity to meet their desired destination and then send their message directly. The disadvantage of this method is that it is unclear when a sender and receiver can meet each other; thus, this routing algorithm has an unbounded delivery delay. This algorithm has an advantage since it needs one transmission to forward a single copy of the packets.
B. Flooding Based routing
In Flooding Based algorithm of routing, the nodes generate a multi-copy of packets and inject them to the Opportunistic network. This process continues until the packets arrive at the destinationflooding at source nodes classified into two categories, which includes controlled and uncontrolled categories. In the controlled category, replication of the packets is unlimited to reduce the long delivery delay. In the controlled category, the opportunistic networks set a limit for replication of the packets to reduce the network contention.
C. Prediction Based routing
It should be noted that the Flooding Based routing sometimes receives the amount of traffic overhead, and Prediction Based algorithm introduces a way to reduce the traffic. In Prediction Based algorithm, the Opportunistic network predicts and calculates the nodes' behaviour according to their contact history, and the Opportunistic network can decide to predict the best probabilistic routing. If a neighbour node has higher delivery predictability value, a sender node forwards its packet to this node.
D. Coding Based routing
In Coding Based algorithm of routing, the Opportunistic network changes the format of all the packets before transmission. Coding based routing embeds additional information in blocks of the packets. If the receivers want to reconstruct the original packets, they must open a certain number of blocks successfully.
E. Context based routing
The Context-Based algorithm of routing tries to improve the Prediction based algorithm by utilizing the context information. Sometimes, Prediction Based algorithm fails to predict the right paths; thus, the Opportunistic network uses the context information that can refine the wrong prediction and improve the delivery ratio.
Architecture of opportunistic networks
It is important to know than an Opportunistic network typically includes separate partitions. These partitions are called regions. By using devices in different regions, Opportunistic networks try to store-carry-forward the packet and make connections between heterogeneous regions. By using a bundle layer, the intermediate nodes implement the store-carry forward message switching mechanism. The act of nodes is defined by the bundle layer. It should be noted that that nodes can perform three acts, as a host, a router and a gateway. The bundle layer of the router can store, carry and forward the packets between nodes in the same region. The bundle layer of the gateway can store, carry and forward the packets between different regions. The gateway can also be a host, optionally [3].
Anonymity in Opportunistic networks
Introduction to Anonymous communication
The purpose of anonymous communication is to communicate between senders and receivers who know each other, while observers and network entities do not recognize the identities of the senders and receivers [6]. Malicious users cannot trace the traffics, nor can they discover the identities of the sender and receiver.
As this document discussed, the Opportunistic networks can be used in mobile communications, network communications in remote areas, intelligent transportation and many other types. However, in each different implementation of Opportunistic networks, we must always protect the privacy. The information potential of each node can be a threat and cause serious security problems. We should find a way to protect the information and prevent from tracing it by dangerous people who can cause irreparable problems. One of the best ways to protect the nodes' privacy is to keep their anonymity. Therefore, discussing the anonymity of nodes and protecting personal information is always a severe challenge in Opportunistic networks.
Opportunistic networks must strive to preserve critical information, including names, messages, places and other types of information. The formation of social graphs and social relationships that are created by the nodes meeting must be preserved. The anonymity of the spheres on the Opportunistic network guarantees that it can prevent from the restructuring of social graphs from malicious users [7].
Keeping the node's anonymity by the Opportunistic networks does not mean that any of the node's information cannot be shared because in many cases, they should also share some of the node's information with other nodes. Then Opportunistic networks should be able to balance between the disclosure of the node's identity and the necessary information. The Opportunistic network must try to prevent malicious users that try to use confidential information. For this purpose, an Opportunistic network needs an integration mechanism to protect privacy. An Opportunistic network needs an integrated security mechanism across the entire network environment, and the network must increase the level of trust between nodes. With an integrated security system, the network can better preserve the social graph.
An Opportunistic network can do two general possibilities for authentication. Before the Opportunistic network publishes all identifying information of nodes, there is no social graph. The first possibility is to allow the nodes to place all of their identity information on the Opportunistic network. In this case, the Opportunistic network is very vulnerable. Therefore it must consider a policy among the nodes that the network can protect the form of the social graph and keep important information at the same time. If an Opportunistic network publishes all the information about all nodes, creating a social graph is meaningless. Another possibility is that the network hides all the nodes' information. Moreover, this policy cannot be realistic. It does not allow for the formation of social graphs, and this network cannot be very efficient [9].
This paragraph suggests implementing an integrated security mechanism. One of the solutions for authentication with anonymity is the use of digital signatures on packets. The use of a unique signature in the router enables the Opportunistic network to identify identities from packets and send messages in a more efficient way. Also, the network can better use resources. Also, after the security authorization was granted by routers, the routers add individual licenses to the packets. All this requires additional computing on the Opportunistic network; instead, it makes for integrated security on the network [8,11].
Introducing some anonymous routing protocols
A. Onion routing
One famous DTN routing protocol for anonymity is onion routing. Onion routing with layered encryption on packets by using different secret keys preserves anonymity. The peeling off of each layer in DTNs is done by the corresponding secret key. This protocol remains the anonymous connection between the sender, receivers and the final destination, so the connection between them becomes untraceable [10]. The paths in onion routing are made by some onion routers. Each onion router only recognizes the previous and the next router, and none of them knows the first sender and the final destination.
B. ALAR
ALAR is an anonymity protocol for improving level privacy by combining the ideas of on-demand routing, identity-free routing, and neighbourhood traffic. The main idea of ALAR is to fragment a message to K segment and to send each segment to at least N neighbours [12]. In ALAR protocol, the malicious user receives many copies of a packet from different routers at a different time so that if they want to trace the packets of the traffic probably cannot find the sender and receivers.
C.MASK
MASK is an anonymous on-demand protocol. Neighbouring nodes share a pairwise secret key between each other to establish a neighbourhood authentication without revealing their identities. With node's secret key, MASK can forward and route packets without disclosing of identities of nodes [12]. However, we should know that malicious users can localize the sender's position, which is a weakness.
D. ASR
ASR is a secure routing protocol that only encrypts a small piece of a packet instead of encrypting the whole packet. ASR only encrypts a small part of a packet which contains the information on the identity of the sender and the receiver. It should also be noted that each relay node only verifies that small encrypted piece of a packet. The forwarding process functions by sharing a key between any two consecutive nodes until the message arrives at the destination [12].
E. Privhab
Privhab uses homomorphic encryption with a protocol that functions according to the location of the nodes (Homomorphic encryption can encrypt data with additional evaluation capability, and Homomorphic encryption can use that additional evaluation value for computing over encrypted data without using the secret key). This method seeks nodes that are within the boundary range and use them to select the probability of intermediate nodes. Also, this method uses long-term predictions to select the intermediate nodes, which increases network efficiency. Privhab uses an innovative way to protect privacy. Instead of using the actual user name, this method uses aliases to keep the identities of the nodes intact [13].
F. Eprivo
Eprivo uses Homomorphic encryption for keeping privacy and anonymization. The Homomorphic encryption had used for avoiding reveals information corresponding to the node's neighbouring graph in Eprivo. When two nodes meet each other, they do not share their routing metrics. They only compare routing metrics by using Homomorphic encryption. The Eprivo have defined two main methods to keep anonymous of the nodes. These two methods include Neighborhood-randomization and Binary-anonymization [14].
Mix networks
Defination of Mix networks
Mix networks are introduced as anonymous communication protocols that can prevent the tracing of senders and recipients. A Mix network uses the mixer nodes to prevent malicious users from tracking messages. As we know, Opportunistic networks can build paths to transmit messages by using intermediate nodes. Mix networks can use mixer nodes between intermediate nodes to construct the paths. Moreover, Mix networks have a high degree of flexibility, the same as Opportunistic networks. For this reason, we can consider Mix networks for different locations and environments. However, the term "Mix networks" is often referred to as non-traceable secure networks. Mix networks are very suitable for particular environments where information preservation is of great importance. For example, a government can use a Mix network to hold an important election in a country, where the disclosure of information or information tracking can lead to many problems.
Defination of Mixer nodes
Mixer nodes store all the received messages sent by the surrounding nodes in their buffers. As we see in figure 3, when the mixer node receives messages, they do not send messages immediately to other nodes. The mixer nodes change the order of all the messages before sending them to the next destination. These changes randomly occur in the mixer node buffer. We should know that the volume of Mix network traffic directly depends on the number of mixer nodes. The higher the number of the mixer nodes grows, the higher the amount of traffic becomes equally [15]. After the messages shuffled in a mixed column in a mixer node, they are sent to the nodes around the mixer nodes.
All messages in Mix networks are encrypted by the public key for the mixer nodes. We can imagine this encryption in the form of Russian dolls with the message in the innermost layer. The message includes many layers and the mixer nodes by opening each layer, understand which node is the next destination. It is important to know that the mixer nodes open the layers by using their private keys. In the end, mixer nodes change the order of messages and send them to the next destination.
Privacy in Mix network
Mixer nodes can be vulnerable targets because they are attractive for malicious users. If malicious users penetrate in mixer nodes (for example, by injection of repackaged messages), they can damage the whole Mix network operation. In this section, we present several suggestions for improving privacy by introducing four dangerous threats to the privacy of Mixed Networks. We know that mixer nodes destroy links between senders and receivers. Therefore, malicious threats cannot identify users by analyzing the traffics of Mix networks. In case malicious users succeed in controlling some of the mixer nodes, they are not aware of previous message links, and they cannot identify the source and destination of the message. Mix networks can maintain anonymity in this way. The messages in the Mix networks contain unique information about the sender or receiver nodes. By this information, the nodes identify each other better, which is an advantage of Mix networks to improve sending and receiving in their next forward messeges. Due to this, the networks know their nodes better in the next update. There is a negative aspect to this feature because if there is more information about nodes, the malicious users know more about the nodes' contacts, and they can analyze traffics of the Mix networks more easily and can break the anonymity and plan for more convenient attacks. To prevent this from happening, we need to limit the amount of the information we place in the Mix networks to only what is needed.
Each message in each mixer node is encryptedthis encryption made by a public key. Before each proxy is encrypted, this layer indicates which node is the next or which one is the final destination [16]. If a network does these actions poorly or does repeated actions in the proxy, malicious users can make predictions and compromise the privacy of the network users by analyzing the Mix networks' traffic and will find out about the connections between the network nodes. In such situations, malicious users monitor the arrival and departure of messages for a long time, or they test nodes that send messages from duplicate actions. In this way, they can predict the behaviour of nodes. The weakness in the transmitting process can also provide conditions for network penetration by malicious users. In this situation, malicious users can quickly create a template for prediction. They can use these weaknesses and, after penetration, break the privacy of the nodes and reach social graphs. To protect the networks, we must program proxies in a way that they do not allow duplicate actions or poor performance on encryption and decryption.
THE PROPOSAL
We plan to simulate a private and unlinkable exchange of messages by using a Public bulletin board and Mix network in Opportunistic networks. The goal is that the nodes can send and receive messages in an anonymous manner. In this simulation, the nodes can use paths which include some mixer nodes, and these mixer nodes can protect paths from being traced. This proposal can make sure that the security of the client runs in the end-user. Also, we simulate a Public bulletin board in which nodes can store their messages in a protected form. Senders can store their messages to the Public bulletin board, and the receivers can receive their messages by the Public bulletin board [18].
The Public bulletin board
As sketched in figure 4, we assume a Public bulletin board with n cells. The client can access to the Public bulletin board by using the Mix network. All cells in a Public bulletin board can contain a set of value/tag pairs. All cells are known to have an index that they have.
Users can operate two functions on a Public bulletin board. In the first function, the user can send an encrypted message with a preimage of the tag and an index to the Public bulletin board. A Public bulletin board can write encryption of a message in the value part of a cell. Also, a Public bulletin board can store preimage of the tag in the tag's part of a cell. The index of the cell in the Public bulletin board is chosen by the sender, and the Public bulletin board writes the message in the same index mentioned by the sender. It should be noted that the encrypted message must include a tag and an index for the next message from the sender.
In the second function, a receiver sends the preimage of a corresponding tag and an indexthe preimage of the corresponding tag can open the Public bulletin board cells. We know that any node which is to be read in a Public bulletin board must know the corresponding tag of value. If a Public bulletin board can open the cell by the preimage of the corresponding tag, the value of the cell will return to the receiver; otherwise, it returns null. It should also be noted that only the intended recipients can delete the value of the cell by using the preimage of the corresponding tag.
We can adjust the size of a Public bulletin board in proportion to what a network needs. The size of the Public bulletin board can dynamically adjust with the network size so that we can reduce the amount of network activity and server activity.
Public bulletin boards can be attractive targets for malicious users. If malicious users can control Public bulletin boards, they can learn about all vital network information. It should be noted that the Public bulletin board does not know about the identity of senders and recipients. In a sense, the proposal will increase the level of security and privacy for Public bulletin boards.
Client protocols
As figure 4 shows, node A and B want to send a message to each other. They have met and reached an agreement before exchanging the first message. We suppose that nodes A and B shared a secret symmetric key in that meeting. After sharing the secret symmetric key, node B can use this key for opening the message from node A. It should be noted that both node A and B create a fresh key for each message. Each fresh key is made by a key derivation function. After receiving each message to node B, both node A and B update the symmetric key by key derivation function and delete the old copy of the key. We know that there has been an agreement between nodes A and B for the first index of a cell on the Public bulletin board and that node A gives the preimage of the first message.
As it was mentioned earlier, each cell has a number index in the Public bulletin board. Both node A and B can access a specific cell by the tag and index they share. Node B can read the value from a specific Public bulletin board cell because it recognizes the preimage of the corresponding tag and the first number index of that specific cell. Figure 4 is an example of our simulation. There is no direct connection between nodes A and B when they send a message to each other. Instead, there are some usual and mixer nodes which can be used to create a path between node A and B.
Writing and reading from a Public bulletin board by nodes include two events during the process of simulation. During the first event, we assume that node A sends a message in a path that includes some mixer nodes and usual nodes. The message was sent along with a tag and index. The first mixer node receives the message and delivers it to the next mixer node, as long as the message and the rest of the content arrive in the Public bulletin board. Then the message will be stored in the same Public bulletin board cell according to the index number determined by node A.
For security reasons, it is better not to save messages in continuous cells in the Public bulletin board. Therefore, node A chooses a random index for the next message that refers to a new place in the Public bulletin board. This process can continue to work as long as the Public bulletin board is not filled.
In the second event, we suppose that the Public bulletin board receives a request message from node B. We know that nodes A and B have agreed about the first index and preimage of the corresponding tag. This message, along with the preimage of the corresponding tag and the first number index, will be read by the specific cell. The request message passes from a path with some mixer nodes, which makes it impossible to trace the path. After the request message arrives at the Public bulletin board, it checks the request message with the corresponding index and tag in the Public bulletin board cell. If the index of the tag was not empty, and the preimage of the corresponding tag can open the cell, the value of the cells returns to node B; otherwise, returns null. The Public bulletin board form the value of the cell as a new message and publish it in a path which includes some mixer nodes again.
Figure 4. An example of a Public bulletin board
As we said earlier, a path with mixer nodes is protected from being traced. After the message passes along the path, it will arrive in node B along with the value.
After the arrival of the message (with a value) to node B, node B can decrypt the value of the message by the secret symmetric key shared before between node A and B. The information about the next index and tag mentioned in the encrypted value that node B can decrypt it now and know it. Node B decrypts the value and recognizes the index and tag of the next cell for a new message. Therefore, node B knows where the next message in Public bulletin board cells can be found. Also, node B can open the next cell by the tag it receives from the encrypted message of the value. Despite all the advantages, this proposal cannot say that this Public bulletin board is safe from any attack.
For example, we can consider a situation in which a malicious user sends many messages and fills all the cells in the Public bulletin board in his/her favour. A simple solution is that the Public bulletin board has registration for each node. It can prevent the Public bulletin board from attacks by a malicious node. By using this method, we can protect our public bulletin board form potential threats. Also, we suppose a value for each node, and the value decreases each time that a node stores a message in the Public bulletin board. Such actions could prevent these types of attacks from blocking the Public bulletin board.
Experimentation
Simulation environment
This proposal uses the ONE simulator for simulation-the ONE simulator is a java based simulator which is used for research in delay tolerant networks [19]. The ONE simulator can define different nodes with different behaviours and engage them in different scenarios.
The ONE simulator has some default functions which can be used to make, send, receive and route messages between nodes. In this simulator, we can create different scenarios by explaining different occurrences. We can also change the default functions of ONE simulator for specific routing protocols and design new protocols. We can change in the structure of messages and nodes too.
The ONE simulator can display outputs of simulation in two ways: graphically and textual. Our proposal has not used any graphics outputs; we have used textual outputs. We can customize outputs of the ONE simulator according to our proposal.
Also, the ONE simulator uses a function for making reports of simulation. The ONE simulator considers a Report file to gather all reports. These reports can provide enough information on how to work the simulation.
Scenarios
We define two scenarios in this proposal by using the ONE simulator. These two scenarios have some differences in speed movement of nodes and spatial dimensions of simulation. It should be noted that both scenarios have the same execution time.
In the first scenario, we assume a park in the dimension of one kilometer square and that forty-one nodes exist in this park. Each of these nodes is a human that can slowly walk with the speed of one to two kilometers per hour in the park and each node moves in a random direction. The number zero node is a Public bulletin board, and the size of it is one hundred cells. Other nodes can be normal nodes or mixer nodes. The range of the node's transmission is twenty-five meter, and the speed of the transmission is one hundred megabytes per second. Each node can use two prefixes for sending a message to the Public bulletin board: one prefix to write in the Public bulletin board and the second one to read from it. Each message can use three mixer nodes at maximum to reach the Public bulletin board.
In the second scenario, the Setting file is the same, except for the dimension of the park and the node's speed of movement. We assume a park as big as five hundred squares. Also, the node's speed of the movement is double time faster, and it is between two to three kilometres per hours.
The ONE modification
The ONE simulator contains various functions, each containing a set of different compiled java functionseach one designed for a specific purpose in the simulator. This proposal had the most changes in a series of javafunctions in Core and Routing functions. Also, we created a new function in the core file which we called "BB". The BB function used to create a Public bulletin board. We also needed to make changes to the Routing functions so that we could define a protocol between senders and receivers with a public bulletins board. Also, this simulation considers Mix network function, which can change each node to become a mixer node. This section only discusses changes in the functions and the creation of BB java function.
We should know that after all changes on the ONE simulator (which will be explained), we can compile all the functions to record the changes. After running the simulator on our desired setting on Setting files according to our proposal, the ONE simulator can execute the simulation. Also, in the terminal, we can see outputs of each step of sending, receiving, and storing messages from the Public bulletin board. After the occurrence of each step, the status of the Public bulletin board will be shown in the terminal.
Public bulletin board function
In the first step, we write a java function for creating a Public bulletin board structure. We create this function in core functions.
The structure of the Public bulletin board contains two arrays, one array for storing the preimage of tags and another for storing values. Also, each cell of arrays has a number index by which we can access the cell.
The Public bulletin board function has two main methods. One of the methods is the WRITE method, which is used to write the value of messages and the preimage of tags from senders, and another one is the READ method to return values from a Public bulletin board to receivers.
The WRITE method is the first method that we considered in the java Public bulletin board function (BB). At first, when a message arrives in the Public bulletin board, BB function checks the message's prefix. If the message's prefix is equal to the WRITE prefix, the WRITE method becomes active. Then, the WRITE method starts to store the message in the Public bulletin board arrays. As we know, a message includes a value, the preimage of tag and an id. The message's id refers to the numbered cell of arrays. The WRITE method receives the message's value and stores it in the value array. The cell's number of the array is equal to the message's id. The WRITE method does the same with the preimage of the message's tag and stores it in tags array.
The READ method is the second method we considered in the java Public bulletin board function (BB). After checking the prefix message by the Public bulletin board function, if the message's prefix is equal to the READ prefix, the READ method becomes active. It should be noted that a message to get value from the Public bulletin board needs an id and a corresponding tag. The READ method checks the cell's number of the tags array according to the message's id. The READ method checks the value array with the same number of id to see if the cell's tag is equal to the corresponding message's tag. Then the READ method returns the value in the form of a new message. -The READ method sends the new message in a path that including mixer nodes to reach the leading destination by using routing function.
The DTNHost function
In the ONE simulator, all nodes are created by DTNHOST function. The structure of a node created by the DTNHOST function consists of the name, location, type of movement, range of transmission, speed of transition, etc. Even we can by call BB function in a host structure and use a Public bulletin board structure in it. Also, we add three values, including a secret key, an agreed-id and an agreed-tag in DTNHOST function. It should be noted that we wrote these three functions to generate the secret key, the agreed-id and the agreed-tag that it takes amount randomly for each one. Also, we write methods to generate their value.
The SIM Scenario function
Scenario function can create a scenario that is supposed to happen according to Setting files. We have modified the Scenario function. In a part of this function that takes the number of the host in simulation from the Setting function, we can have constructed the first node as the Public bulletin board to create nodes in the simulation when it uses DTNHOST function.
The Message function
The Message function is used for creating message structures in the ONE simulator. The message structure has values including sender hosts, receiver hosts, the id of the message, the size of the message, etc. We add some values for carrying the agreed-id, the agreed-tag and the encrypted value. Also, we have created a method that works automatically when a message is generated. This method simulates the agreement between sender hosts and receiver hosts. When a message is created automatically, the value of the secret key, the agreed-tag and the agreed-id between a sender host and a receiver host becomes equal. Also, the id and tag in the message structure become equal with the agreed-id and the agreed-tag values from the sender.
The Mix network function
We create a java function in the routing functions named Mix network function which generates the mixer nodes. As mentioned in the previous section, using mixer nodes in the simulation can create an untraceable message which has been one of the goals of the present proposal. At each mixer node, messages become mixed without any change to their context.
The Message routing function
The message routing function introduces a protocol to route a message from senders to receivers. To do so, we should have some changes on message routing function to adapt
Results
Metrics to be studied
We review the result of the simulation by considering three metrics that include Total packet delivery ratio, Total packet latency average and Total packet latency median.
The Total packet delivery ratio is the ratio of the total packets successfully sent to the Public bulletin board. We calculate the Total packet delivery ratio by the sum of the ratio delivery of all packets that arrive at the Public bulletin board (all packets that use one, two, or three mixers or do not use any mixers). Then, we divided the total delivery ratio by four. We show the Total packet delivery ratio in Figure 5 and Figure 8 for each scenario by two columns in blue and red. The blue column refers to nodes which store messages in the Public bulletin board, and the red column refers to the nodes which receive messages from the Public bulletin board.
The total packet latency average refers to the total delay of packets that arrive at the Public bulletin board. We calculate the total packet latency average by the sum of the latency of all packets that arrive at the Public bulletin board, packets that use one, two, or three mixers or do not use any mixers. Also, in Figure 6 and Figure 9, we consider two columns with red and blue colours.
The total packet latency median refers to the total delay of packets to arrive at the Public bulletin board for both methods. We calculate Total packet latency median by the sum of the median of all packets that arrive at Public bulletin board, all packets that use one mixer or two mixers or three mixers or do not use any mixer. Also, in Figure 7 and Figure 10, we consider two columns with different colours as previous figures.
Result
For the first scenario, we provided three charts. Figure 5 refers to the Total packet delivery ratio. In both methods (READ with the red column, WRITE with the blue column), the results of the Total packet delivery ratio have an acceptable ratio. Therefore, we can understand that if a node sends a message for storing in the Public bulletin board, this message reaches the destination with a high rate. Also, in Figure 6 and Figure 7, we can see the Total packet latency average and the Total packet latency median for the total number of messages for each method. Unfortunately, if we look at these figures, we notice that the simulation has not worked very well in terms of network latency and median. A message should stay quite a few minutes in intermediate nodes to reach the destination, which is a long time for the delivery of a packet, as we know. If our Setting file includes the slow movement of the nodes and the size of the simulated environment, we can justify these results. In general, these results are not acceptable.
As we said, we had some changes in the Setting file for the second scenario. In the following next three charts, we obtained results by doubling the speed of node movement and half of the simulation park dimension but with the same execution time. Figure 8 shows the Total packet delivery ratio for the second scenario. At first, the rate for the second scenario is very acceptable as the first scenario. It can be a reasonable rate for simulation to deliver each message. Figure 9 and Figure 10 show the Total packet latency average and the Total packet latency median. An important point in Figure 9 and Figure 10 is the Mix network condition (because of the messages stored in the mixer nodes many times), which makes the latency average, and latency median acceptable. In general, these results are good results for the second scenario.
Conclusions on the results
By considering all the charts, we can compare two scenarios. Based on the Total packet delivery ratio of both scenarios, we can see that our simulation has a high performance in terms of delivery rates. In general, we can conclude that the simulation has a good Total packet delivery ratio.
As we have seen in the first scenario, the latency time rate was very high; it is not an acceptable rate for our simulation. Moreover, with the changes we made in the second scenario, we achieved acceptable results in terms of network latency and median. We know that we had a small modification in the second scenario, but we can see a massive difference between the charts. Thus, the simulation performance is related directly to the node's speed of movement and the size of the simulation environment.
By doubling the speed of the nodes, we reduce the delay of the message several times. We can say that the simulation in the second scenario has a good result by considering the Total packet latency average and the Total packet latency median. If we look at our simulation performance by considering all the charts, it can be concluded that our simulation can work well with a central Public bulletin board for environments where nodes have a faster speed of movement. Given the overall results, it can be concluded that the simulation has a reasonably good performance.
CONCLUSIONS
As we discussed in the document, we intended to simulate a private and unlinkable exchange of messages by using a Public bulletin board and Mix networks in opportunistic networks. We used the Mix network, where the nodes could send their messages through intermediate nodes anonymously. By considering the materials from the related work section, we provided a protocol in which nodes could send their packets to the public bulletin in a secure environment. Also, we considered two different scenarios, and we analyzed and compared them.
If we want to draw a general conclusion about the results of the simulation, we can say that our simulation has a satisfactory performance. Nodes can operate and use the benefits of the public bulletins board in our simulation. Also, the nodes can send their packets in anonymity by using Mix network. At all stages, we succeeded in preserving the anonymity of the nodes. In general, we can say that this proposal was able to achieve overall goals at an acceptable level.
Figure 1 .
1An example of an Opportunistic network. Adapted from[4]
Figure 2 .
2An example of an Opportunistic network in real life. Adapted from[5]
Figure 3 .
3An example of Mix network. Adapted from [17]
Figure 5 . 1 Figure 6 . 1 Figure 7 .Figure 8 . 2 Figure 9 . 2 Figure 10 .
51617829210Total delivery probability ratio for scenario Total latency average for scenario Total latency median for scenario 1 Total delivery probability ratio for scenario Total latency average for scenario Total latency median for scenario 2
simulation with the Public bulletin board. Firstly, in Message routing function, we check the message destination to make sure whether or not it exists in the Public bulletin board. If the destination exists in the Public bulletin board, we activate the Public bulletin board function to check the message. Also, as we know, a Public bulletin board uses Message routing function to send new messages it generates to return the values. After generating a new message by the Public bulletin board, we change the address of the host who wants to receive a message of the Public bulletin board with zero node address. Then we add the new message in the list of the message for routing in simulation. New messages will use mixer nodes to reach the leading destination. Also, after the routing of all new messages, we can access the information about all new messages in reports files.
Total delivery probability ratio for scenario1 Total delivery probability ratio for scenario 2Total latency average for scenario 2Total median for scenario 1 Total median for scenario 2Total delivery ratio
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Per second
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An example of an Opportunistic network. An example of an Opportunistic network. Adapted from https://groups.geni.net/geni/rawattachment/wiki/GENIExperiment er/Tutorials/Graphics/DelayDisruption.png
An example of an Opportunistic network in real life. An example of an Opportunistic network in real life. Adapted from https://ars.els-cdn.com/content/image/1-s2.0- S1084804517303557-gr1.jpg
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Improving Podcast Distribution on Gwanda using PrivHab: a Multiagent Secure Georouting Protocol. Adrián Sánchez-Carmona, Sergi Robles, Carlos Borrego, ADCAIJ: Advances in Distributed Computing and Artificial Intelligence Journal. Salamanca 4.1Sánchez-Carmona, Adrián, Sergi Robles, and Carlos Borrego. "Improving Podcast Distribution on Gwanda using PrivHab: a Multiagent Secure Georouting Protocol. ADCAIJ: Advances in Distributed Computing and Artificial Intelligence Journal (ISSN: 2255-2863)." Salamanca 4.1 (2015).
Efficient broadcast in opportunistic networks using optimal stopping theory. Carlos Borrego, Joan Borrell, Sergi Robles, Ad Hoc Networks. 88Borrego, Carlos, Joan Borrell, and Sergi Robles. "Efficient broadcast in opportunistic networks using optimal stopping theory." Ad Hoc Networks 88 (2019): 5-17.
| []
|
[
"Harmonic Analysis in Phase Space and Finite Weyl-Heisenberg Ensembles",
"Harmonic Analysis in Phase Space and Finite Weyl-Heisenberg Ensembles"
]
| [
"Daniel Luís ",
"Abreu [email protected] \nAcoustics Research Institute\nAustrian Academy of Sciences\nWohllebengasse 12-141040ViennaAustria\n",
"Karlheinz Gröchenig [email protected] \nFaculty of Mathematics\nUniversity of Vienna\nOskar-Morgenstern-Platz 11090ViennaAustria\n",
"José Luis Romero [email protected] \nAcoustics Research Institute\nAustrian Academy of Sciences\nWohllebengasse 12-141040ViennaAustria\n\nFaculty of Mathematics\nUniversity of Vienna\nOskar-Morgenstern-Platz 11090ViennaAustria\n",
"B José ",
"Luis Romero [email protected] ",
"Daniel Luís ",
"Abreu ",
"Karlheinz Gröchenig "
]
| [
"Acoustics Research Institute\nAustrian Academy of Sciences\nWohllebengasse 12-141040ViennaAustria",
"Faculty of Mathematics\nUniversity of Vienna\nOskar-Morgenstern-Platz 11090ViennaAustria",
"Acoustics Research Institute\nAustrian Academy of Sciences\nWohllebengasse 12-141040ViennaAustria",
"Faculty of Mathematics\nUniversity of Vienna\nOskar-Morgenstern-Platz 11090ViennaAustria"
]
| [
"Journal of Statistical Physics"
]
| Weyl-Heisenberg ensembles are translation-invariant determinantal point processes on R 2d associated with the Schrödinger representation of the Heisenberg group, and include as examples the Ginibre ensemble and the polyanalytic ensembles, which model the higher Landau levels in physics. We introduce finite versions of the Weyl-Heisenberg ensembles and show that they behave analogously to the finite Ginibre ensembles. More specifically, guided by the observation that the Ginibre ensemble with N points is asymptotically close to the restriction of the infinite Ginibre ensemble to the disk of area N , we define finite WH ensembles as adequate finite approximations of the restriction of infinite WH ensembles to a given domain . We provide a precise rate for the convergence of the corresponding one-point intensities to the indicator function of , as is dilated and the process is rescaled proportionally (thermodynamic regime). The construction and analysis rely neither on explicit formulas nor on the asymptotics for orthogonal polynomials, but rather on phase-space methods. Second, we apply our construction to study the pure finite Ginibre-type polyanalytic ensembles, which model finite particle systems in a single Landau level, and are defined in terms of complex Hermite polynomials. On a technical level, we show that finite WH ensembles provide an approximate model for finite polyanalytic Ginibre ensembles, and we quantify the corresponding deviation. By means of this asymptotic description, we derive estimates for the rate of convergence of the one-point intensity of polyanalytic Ginibre ensembles in the thermodynamic limit. | 10.1007/s10955-019-02226-2 | null | 85,530,534 | 1704.03042 | 1e2cf8f4765c5d672146604919cf426afadf0e2e |
Harmonic Analysis in Phase Space and Finite Weyl-Heisenberg Ensembles
2019
Daniel Luís
Abreu [email protected]
Acoustics Research Institute
Austrian Academy of Sciences
Wohllebengasse 12-141040ViennaAustria
Karlheinz Gröchenig [email protected]
Faculty of Mathematics
University of Vienna
Oskar-Morgenstern-Platz 11090ViennaAustria
José Luis Romero [email protected]
Acoustics Research Institute
Austrian Academy of Sciences
Wohllebengasse 12-141040ViennaAustria
Faculty of Mathematics
University of Vienna
Oskar-Morgenstern-Platz 11090ViennaAustria
B José
Luis Romero [email protected]
Daniel Luís
Abreu
Karlheinz Gröchenig
Harmonic Analysis in Phase Space and Finite Weyl-Heisenberg Ensembles
Journal of Statistical Physics
174201910.1007/s10955-019-02226-2Received: 3 March 2018 / Accepted: 8 January 2019 / Published online: 22 January 2019L. D. A. was supported by the Austrian Science Fund (FWF): START-project FLAME (Frames and Linear Operators for Acoustical Modeling and Parameter Estimation, Y 551-N13), and by FWF P. 31225-N32. K. G. was supported in part by the project P31887-N32 of the Austrian Science Fund (FWF), J. L. R. gratefully acknowledges support from the Austrian Science Fund (FWF):P 29462-N35, and from the WWTF grant INSIGHT (MA16-053). 123Landau level · Polyanalytic Ginibre ensemble · Hyperuniformity · Weyl-Heisenberg ensemble · Phase-space · Time-frequency analysis
Weyl-Heisenberg ensembles are translation-invariant determinantal point processes on R 2d associated with the Schrödinger representation of the Heisenberg group, and include as examples the Ginibre ensemble and the polyanalytic ensembles, which model the higher Landau levels in physics. We introduce finite versions of the Weyl-Heisenberg ensembles and show that they behave analogously to the finite Ginibre ensembles. More specifically, guided by the observation that the Ginibre ensemble with N points is asymptotically close to the restriction of the infinite Ginibre ensemble to the disk of area N , we define finite WH ensembles as adequate finite approximations of the restriction of infinite WH ensembles to a given domain . We provide a precise rate for the convergence of the corresponding one-point intensities to the indicator function of , as is dilated and the process is rescaled proportionally (thermodynamic regime). The construction and analysis rely neither on explicit formulas nor on the asymptotics for orthogonal polynomials, but rather on phase-space methods. Second, we apply our construction to study the pure finite Ginibre-type polyanalytic ensembles, which model finite particle systems in a single Landau level, and are defined in terms of complex Hermite polynomials. On a technical level, we show that finite WH ensembles provide an approximate model for finite polyanalytic Ginibre ensembles, and we quantify the corresponding deviation. By means of this asymptotic description, we derive estimates for the rate of convergence of the one-point intensity of polyanalytic Ginibre ensembles in the thermodynamic limit.
Introduction
Weyl-Heisenberg Ensembles
We study the class of determinantal point processes on R 2d whose correlation kernel is given as
K g ((x, ξ), (x , ξ )) = R d e 2πi(ξ −ξ)t g(t − x )g(t − x)dt (1.1)
for some non-zero (normalized) function g ∈ L 2 (R d ) and (x, ξ), (x , ξ ) ∈ R 2d . These determinantal point processes are called Weyl-Heisenberg ensembles (WH ensembles) and have been introduced recently in [8]. They form a large class of translation-invariant hyperuniform point processes [36,55,61]. The prototype of a Weyl-Heisenberg ensemble is the complex Ginibre ensemble. Choosing g in (1.1) to be the Gaussian g(t) = 2 1/4 e −π t 2 and writing z = x + iξ, z = x + iξ , the resulting kernel is then
K g (z, z ) = e iπ(x ξ −xξ) e − π 2 (|z| 2 +|z | 2 ) e π zz , z = x + iξ, z = x + iξ . (1.2)
Modulo conjugation with a phase factor, this is essentially the kernel of the infinite Ginibre ensemble K ∞ (z, z ) = e − π 2 (|z| 2 +|z | 2 ) e π zz . Another important class of examples arises by choosing g to be a Hermite function. In this case one obtains a pure polyanalytic Ginibre ensemble [8,57], which models the electron density in a single (pure) higher Landau level (see Sect. A.5 for some background).
The Ginibre ensemble with kernel K ∞ arises as limit of corresponding processes with N points, whose kernels
K N (z, z ) = e − π 2 (|z| 2 +|z | 2 ) N −1 j=0 π zz j j! ,(1.3)
are obtained simply by truncating the expansion of the exponential e π zz . It is not obvious how to obtain the analogous finite-dimensional process for a general Weyl-Heisenberg ensemble (1.1), because for most choices of g ∈ L 2 (R d ) there is no treatable explicit formula available for K g . We present a canonical construction of finite Weyl-Heisenberg ensembles and show that they enjoy properties similar to the finite Ginibre ensemble. The construction and analysis is based on spectral theory of Toeplitz-like operators and harmonic analysis of phase space. The abstract construction is instrumental to study the asymptotic properties of a particularly important class of finite-dimensional determinantal point processes, namely the finite pure polyanalytic Ginibre ensembles, which model the electron density in higher Landau levels. This is an example where the Plancherel-Rotach asymptotics of the basis functions are not available. Moreover, the relevant polynomials do not satisfy the classical three-term recurrence relations which are used in Riemann-Hilbert type methods [25,27]. We develop a new approach based on spectral methods and harmonic analysis in phase space and show that the finite WH ensembles associated with a Hermite function are asymptotically close to finite polyanalytic ensembles. Thus, our analysis of the finite polyanalytic ensembles has two steps: (i) the abstract construction of finite WH ensembles and their thermodynamic limits; (ii) the comparison of the finite WH ensembles associated with Hermite functions and the finite pure polyanalytic ensembles.
Planar Hermite Ensembles
The complex Hermite polynomials are given by
H j,r (z, z) = ⎧ ⎨ ⎩ r ! j! π j−r 2 z j−r L j−r r π |z| 2 , j > r ≥ 0, (−1) r − j j! r ! π r− j 2 z r − j L r − j j π |z| 2 , 0 ≤ j ≤ r ,(1.4)
where L α r denotes the Laguerre polynomial
L α j (x) = j i=0 (−1) i j + α j − i x i i! , x ∈ R, j ≥ 0, j + α ≥ 0. (1.5)
Complex Hermite polynomials satisfy the doubly-indexed orthogonality relation C H j,r (z, z)H j ,r (z, z)e −π |z| 2 dz = δ j j δ rr , and form an orthonormal basis of L 2 C, e −π |z| 2 [4]. 1 The complex Hermite polynomials form a complete set of eigenfunctions of the Landau operator L z := −∂ z ∂ z + π z∂ z (1.6) acting on the Hilbert space L 2 (C, e −π |z| 2 ). The Landau operator is the Schrödinger operator that models the behavior of an electron in R 2 in a constant magnetic field perpendicular to the C-plane. The spectrum of L z , i.e., the set of possible energy levels, is given by σ (L z ) = {r π : r = 0, 1, 2, . . .} and the eigenspace associated with the eigenvalue r π is called the Landau level of order r . For the minimal energy r = 0, i.e., the ground state, the eigenspace is the classical Fock space, for r > 0, the eigenspaces are spanned by the orthonormal basis {H j,r : j ∈ N}. The Landau levels are key for the mathematical formulation of the integer quantum Hall effect discovered by von Klitzing [64]. We will consider a variety of ensembles associated with the complex Hermite polynomials. Complex Hermite polynomials are an example of polyanalytic functions-that is, polynomials in z with analytic coefficients (see Sect. A.4). While most classes of orthogonal polynomials satisfy a three-term recurrence relation-which puts them in the scope of Riemann-Hilbert type techniques [25,27]-the complex Hermite polynomials satisfy instead a system of doubly-indexed recurrence relations [34,45].
Several important determinantal point processes arise as special cases of (1.7). First, since H j,0 (z, z) = (π j / j!) 1 2 z j , the set J = {0, . . . , N − 1} × {0} in (1.7) leads to the kernel of the Ginibre ensemble (1.3). A second important example arises for J := {( j, r ) : 0 ≤ j ≤ n − 1, r = m − n + j} with n, m ∈ N. The corresponding one-point intensity is a radial version of the marginal probability density function of the unordered eigenvalues of a complex Gaussian Wishart matrix after the change of variables t → π |z| 2 , see, e.g. [62,Theorem 2.17]. Thirdly, choosing J = {0, . . . , N − 1} × {0, . . . , q − 1} one obtains the polyanalytic Ginibre ensemble introduced by Haimi and Hedenmalm [40]. The polyanalytic Ginibre ensemble gives the probability distribution of a system composed by several Landau levels. The case of more general interaction potentials has been investigated in [40,41], by considering polyanalytic Ginibre ensembles with general weights. These investigations parallel the ones of weighted Ginibre ensembles [9][10][11].
We are particularly interested in finite versions of the infinite pure polyanalytic ensembles defined by Shirai [57]. The infinite ensembles are defined by the reproducing kernels of an eigenspace of the Landau operator (1.6) which is given by
K r (z, z ) = L 0 r (π z − z 2 )e π zw− π 2 (|z| 2 +|z | 2 ) = e − π 2 (|z| 2 +|z | 2 ) ∞ j=0 H j,r (z, z) H j,r z , z .
Here the second identity follows from the fact that H j,r (z, z) j∈N spans the rth eigenspace of the Landau operator. The corresponding finite pure polyanalytic ensembles can now be defined as planar Hermite ensembles with J = {0, . . . , N − 1} × {r }. In analogy to (1.3), the finite (r , N )-pure polyanalytic ensemble is the determinantal point process with correlation kernel
K r ,N (z, z ) = e − π 2 (|z| 2 +|z | 2 ) N −1 j=0 H j,r (z, z) H j,r z , z . (1.8)
While pure polyanalytic ensembles describe individual Landau levels, their finite counterparts model a finite number of particles confined to a certain disk (for example, as the result of a radial potential). In this article, we prove the following theorem, which supports this interpretation, and provides a rate of convergence for the one-point intensity related to each Landau level.
Theorem 1.2 Let ρ r ,N (z) = K r ,N (z, z) be the one-point intensity of the finite (r , N )-pure polyanalytic Ginibre ensemble. Then, for each r > 0,
ρ r ,N N π · −→ 1 D , (1.9) in L 1 (R 2 ), as N −→ +∞. Moreover, ρ r ,N − 1 D √ N /π 1 ≤ C r √ N . (1.10)
The convergence rate in Theorem 1.2 is independent of the energy level r of the Landau operator. It is known to be sharp for the first Landau level r = 0, and we believe that (1.10) is also sharp for all Landau levels r ∈ N. 2,3 In statistical terms, (1.10) means that the number of points of the (r , N )-pure polyanalytic Ginibre ensemble that belong to a certain domain A ⊆ C, n r ,N (A), satisfies
E{n r ,N (A)} = D √ N /π ∩ A + O √ N .
(1.11) Theorem 1.2 supports and validates the interpretation of finite pure polyanalytic ensembles as models for N particles confined to a disk by giving asymptotics for the first order statistics (1.11) that indeed show concentration on the disk area N , up to an error comparable to the perimeter of that disk. In addition, (1.11) implies that, after proper rescaling, the particles are, in expectation, asymptotically equidistributed on the disk. This statistical description is consistent with the notion of a filling factor of each Landau level-that is, a certain limit to the number of particles that each level can accommodate. The incremental saturation of each individual Landau level, corresponding to incremental energy levels, is part of the mathematical description of the integer quantum Hall effect discovered by von Klitzing [64].
(The integer quantum Hall effect is not to be confused with the fractional quantum Hall effect, whose mathematical formulation is related to the Laughlin's wave function [48] and the so-called beta-ensembles [18,19].) As a first step towards a description of finite pure polyanalytic ensembles, we introduce a general construction of finite versions of Weyl-Heisenberg ensembles that may be of independent interest.
Finite Weyl-Heisenberg Ensembles
The construction of finite WH ensembles relies on methods from harmonic analysis on phase space [32,33], and on the spectral analysis of phase-space Toeplitz operators. Write z = (x, ξ) ∈ R 2d , z = (x , ξ ) ∈ R 2d for a point in phase space and
π(z) f (t) := e 2πiξ t f (t − x) (1.12)
for the phase-space shift by z. Then the kernel in (1.1) is given by
K g (z, z ) = π(z )g, π(z)g . (1.13)
Let us now describe the construction of the finite point processes associated with the kernel K g . For normalized g ∈ L 2 (R d ), g 2 = 1, the integral operator with kernel K g , i.e., F → R 2d K g (z, z )F(z )dz , is an orthogonal projection (see for example [32,Chapter 1], [38,Chapter 9]). Consequently, the range of this projection is a reproducing kernel Hilbert
space V g ⊆ L 2 (R 2d ) with the explicit description V g = F ∈ L 2 (R 2d ) : F(z) = f , π(z)g , for f ∈ L 2 (R d ) ⊆ L 2 (R 2d ).
Thus every F ∈ V g is a phase-space representation of a function f defined on the configuration space R d .
Step 1: Concentration as a smooth restriction Let X g be a WH ensemble (with correlation kernel K g ) and let ⊆ R 2d be a measurable set. The restriction of X g to is a determinantal point process (DPP) X g | with correlation kernel
K g | (z, z ) = 1 (z)K g (z, z )1 (z ).
(1.14)
An expansion of the kernel K g | can be obtained as follows. We consider the Toeplitz operator on V g defined by
M g F(z) = F(z )K g (z, z ) dz . (1.15) Since F(z ) = R 2d F(z )K g (z , z ) dz for F ∈ V g , M g can be expressed as an integral operator M g F(z) = R 2d F(z ) 1 (z )K g (z, z ) dz (1.16) = R 2d F(z ) R 2d K g (z, z )1 (z )K g (z , z ) dz dz . (1.17)
By definition (1.15), M g acts on a function F ∈ V g by multiplication by 1 , followed by
projection onto V g . On the other hand, if F ∈ V ⊥ g ,to L 2 (R 2d ) that is 0 on L 2 (R 2d ) V g .
For ⊆ R 2d of finite measure, M g is a compact positive (self-adjoint) operator on L 2 (R 2d ); see for example [21,54]. By the spectral theorem, M g is diagonalized by an orthonormal set { p g, j : j ∈ N} ⊆ V g of eigenfunctions, with corresponding eigenvalues λ j = λ j (ordered non-increasingly):
M g = j≥1 λ j p g, j ⊗ p g, j . (1.18)
The key property is that the eigenfunctions p g, j are doubly-orthogonal: since
M g F, F = |F(z)| 2 dz, F ∈ L 2 (R 2d ),
it follows that p g, j , p g, j L 2 ( ) = M g p g, j , p g, j L 2 (R 2d ) = λ j δ j, j , and consequently the restricted kernel has the orthogonal expansion This process is thus a smoother variant of the restricted process X g | , because it involves the (smooth) functions p g, j (z) instead of their truncations p g, j (z)1 (z), which may have discontinuities along ∂ . The construction of DPPs from the spectrum of selfadjoint operators has been suggested in [16,17] as an analogue of the construction of DPPs from the spectral measure of a group. In a related work [52], a combination of methods from operator theory and representation theory has been used to show that a DPP is the spectral measure for an explicit commutative group of Gaussian operators in the fermionic Fock space.
K g | (z, z ) = j≥1 p g, j (z)1 (z) · p g, j (z )1 (z ) ;(1.
Step 2: Spectral truncation Since M g F, F = |F| 2 , by the min-max principle,
λ j = max |F(z)| 2 dz : F 2 = 1, F ∈ V g , F ⊥ p g,1 , . . . , p g, j−1 . (1.21)
Thus, the eigenvalues λ j describe the best possible simultaneous phase-space concentration of waveforms within . In particular, (1.21) implies that
0 ≤ λ j ≤ 1, j ≥ 1.
It is well-known that there are ≈ | | eigenvalues λ j that are close to 1. As a precise statement we cite the following Weyl-type law: for any δ ∈ (0, 1),
#{ j : λ j > 1 − δ} − | | ≤ max 1 δ , 1 1 − δ C g |∂ | 2d−1 ,(1.22)
where |∂ | 2d−1 is the perimeter of (the surface measure of its boundary), and C g is a constant depending explicitly on g. See for instance [6,Proposition 3.4] or [24]. The dependence of the constant C g on g is made explicit below in (1.27).
We now look into the concentrated process X g,con introduced in Step 1. The Toeplitz operator M g is not a projection. However, the corresponding DPP can be realized as a random mixture of DPP's associated with projection kernels [44,Theorem 4.5.3]. Indeed, if I j ∼ Bernoulli(λ j ) are independent (taking the value 1 or 0 with probabilities λ j and 1 − λ j respectively), then X g,con is generated by the kernel corresponding to the random operator M g,ran = j≥1 I j · p g, j ⊗ p g, j .
(1.23)
Precisely, this means that one first chooses a realization of the I j 's and then a realization of the DPP with the kernel above. Because of (1.22), the first eigenvalues λ j are close to 1 and thus the corresponding I j will most likely be 1. Similarly, for j | |, the corresponding I j will most likely be 0. As a finite-dimensional model for WH ensembles, we propose replacing the random Bernoulli mixing coefficients with
1, for j ≤ | | , 0, for j > | | . (1.24) Definition 1.3 Let g ∈ L 2 (R d )
be of norm 1-called the window function, let ⊆ R 2d with non-empty interior and finite measure and perimeter, and let N = | | the least integer greater than or equal to the Lebesgue measure of . The finite Weyl-Heisenberg ensemble is the determinantal point process X g with correlation kernel 4 K g, (z, z ) = N j=1 p g, j (z) p g, j (z ). 4 We do not denote this kernel by K g in order to avoid a possible confusion with the restricted kernel K g | .
Note also the notational difference between the finite ensemble X g and the restriction of the infinite ensemble
X g | .
To illustrate the construction, consider g(t) = 2 1/4 e −π t 2 and = D R = {z ∈ C : |z| ≤ R}. The eigenfunctions of M g D R are explicitly given as p D R g, j (z) = e πi xξ (π j / j!) 1 2 z j e −π |z| 2 /2 , z = x + iξ . They are independent of the radius R of the disk, and choosing R such that |D R | = N , the corresponding finite WH ensemble is precisely the finite Ginibre ensemble given by (1.3). This well known fact also follows as a special case from Corollary 4.6.
Scaled Limits and Rates of Convergence
We now discuss how finite WH ensembles behave when the number of points tends to infinity. Let
ρ g, (z) = K g, (z, z) = N j=1 | p g, j (z)| 2
be the one-point intensity of a finite Weyl-Heisenberg ensemble, so that
D ρ g, (z)dz = E X g (D)
is the expected number of points to be found in D ⊆ R 2d (see Sect. 1). The following describes the scaled limit of the one-point intensities.
Theorem 1.4 Let ⊂ R 2d be compact. Then the 1 -point intensity of the finite Weyl- Heisenberg ensemble satisfies ρ g,m (m·) −→ 1 , (1.25) in L 1 (R 2d ), as m −→ +∞.
In statistical terms, the convergence in Theorem 1.4 means that, as m −→ ∞,
1 m 2d E X g m (m D) = 1 m 2d m D ρ g,m (z)dz = D ρ g,m (mz)dz −→ D 1 (z)dz = |D ∩ | .
(1.26) Theorem 1.4 follows immediately from [6, Theorem 1.3], once the one-point intensity ρ g, is recognized as the accumulated spectrogram studied in [6, Definition 1.2]. We make a few remarks as a companion to the illustrations in Figs. 2 and 3.
(i) When g(t) = 2 1/4 e −π t 2 and is a disk of area N , Theorem 1.4 follows from the circular law of the Ginibre ensemble. (ii) The asymptotics are not restricted to disks, but hold for arbitrary sets with finite measure and also hold in arbitrary dimension, not just for planar determinantal point processes. (iii) The limit distribution in (1.25) is independent of the parameterizing function g. This can be seen as an another instance of a universality phenomenon [26,50,59].
In view of Theorem 1.4 we will quantify the deviation of the finite WH ensemble from its limit distribution in the L 1 -norm, using the results in [7], where the sharp version of the main result in [6] has been obtained. Assume that g satisfies the condition
g 2 M * := R 2d |z| | g, π(z)g | 2 dz < +∞. (1.27)
If has finite perimeter and |∂ | 2d−1 ≥ 1, then
ρ g, − 1 1 ≤ C g |∂ | 2d−1 (1.28)
with a constant depending only on g M * .
The condition on the window g in (1.27) amounts to mild decay in the time and frequency variables, and is satisfied by every Schwartz function. See Sects. 5.1 and A.3 for a discussion on closely-related function classes. The error rate in Theorem 1.5 is sharp-see [7, Theorem 1.6]. Intuitively, in (1.28) we compare the continuous function ρ g, with the characteristic function 1 . Thus, along every point of the boundary of (of surface measure |∂ | 2d−1 ) we accumulate a pointwise error of O(1), leading to a total L 1 -error at least of order |∂ | 2d−1 .
Approximation of Finite Polyanalytic Ensembles by WH Ensembles
The second ingredient towards the proof of Theorem 1.2 is a comparison result that bounds the deviation between finite pure polyanalytic ensembles and finite WH ensembles with Hermite window functions. Before stating the result, some preparation is required. We consider the following transformation, which is usually called a gauge transformation, and the change of
variables f * (z) := f (z), z ∈ C d . Given an operator T : L 2 (R 2d ) → L 2 (R 2d ) we denote: T f * := m T ( f * m), m(x, ξ) := e −πi xξ . (1.29)
Hence, if T has the integral kernel K , then T has the integral kernel
K (z, z ) = e πi(x ξ −xξ) K z, z , z = x + iξ, z = x + iξ . (1.30)
(See Sect. 1 for details). We call the operation K → K a renormalization of the kernel K . With this notation, if K g is the kernel in (1.2) and g is the Gaussian window, then K g is the kernel of the infinite Ginibre ensemble. In addition, the DPP's on C d associated with the kernels K and K are related by the transformation z → z. Now, let the window g be a Hermite function
h r (t) = 2 1/4 √ r ! −1 2 √ π r e π t 2 d r dt r e −2π t 2 , r ≥ 0. (1.31)
The corresponding kernel K h r describes (after the renormalization above) the orthogonal projection onto the Bargmann-Fock space of pure polyanalytic functions of type r (see Sect.
A.4). Let us consider a Toeplitz operator on L 2 (R 2 ) with a circular domain = D R . By means of an argument based on phase-space symmetries (more precisely, the symplectic covariance of Weyl's quantization) we show in Sect. 4 that the eigenfunctions
{ p D R h r , j : j ≥ 1} of M h r D R
are the normalized complex Hermite polynomials H j,r (z,z)e − π 2 |z| 2 . In particular, as with the Ginibre ensemble, the eigenfunctions are independent of the radius R. Choosing R such that N D R = N , and recalling that we order the eigenvalues of M h r D R by magnitude, we obtain a map σ : N 0 → N 0 , such that
p D R h r , j = H σ ( j),r (z,z)e − π 2 |z| 2 .
Thus, the finite WH ensemble associated with h r and D R is a planar Hermite ensemble, with correlation kernel For small values of R > 0, the eigenvalue corresponding to H 1,1 is bigger than the one corresponding to H 1,0 , and thus for small N , the kernels in (1.8) and (1.32) do not coincide. The following result shows that this difference is asymptotically negligible. Theorem 1.6 Let N ∈ N and R > 0 be such that N D R = |D R | = N . Let K h r ,D R be the correlation kernel of the finite Weyl-Heisenberg ensemble associated with the Hermite window h r and the disk D R , and K r ,N the correlation kernel of the (r , N )-pure polyanalytic ensemble given by (1.8). Then
K h r ,D R (z, z ) = e − π 2 (|z| 2 +|z | 2 ) N D R j=1 H σ ( j),r (z, z)H σ ( j),r (z , zK h r ,D R − K r ,N S 1 |∂ D R | 1 √ N ,
where · S 1 denotes the trace-norm of the corresponding integral operators.
Since K h r ,D R S 1 = K r ,N S 1 = N , the finite pure polyanalytic ensemble-defined by a lexicographic criterion-is asymptotically equivalent to a finite WH ensemble -defined by optimizing phase-space concentration. To derive Theorem 1.6, we resort to methods from harmonic analysis on phase space. More precisely, we will use Weyl's correspondence and account for the difference between (1.32) and (1.8) as the error introduced by using two different variants of Berezin's quantization rule (anti-Wick calculus).
Finally, Theorem 1.2 follows by combining the comparison result in Theorem 1.6 with the asymptotics in Theorem 1.5 applied to Hermite windows-see Sect. 5.4. This argument is reminiscent of Lubinsky's localization principle [50] that concerns deviations between kernels of orthogonal polynomials. In the present context, the difference between the two kernels does not stem from an order relation between two measures, but from a permutation of the basis functions.
Simultaneous Observability
The independence of the eigenfunctions of M h r D R of the radius R yields another property of the (finite and infinite) r -pure polyanalytic ensembles.
Theorem 1.7 The restrictions { p h r , j D R : j ∈ N} are orthogonal on L 2 (D R ) for all R > 0.
In the terminology of determinantal point processes this means that the family of disks {D R : R > 0} is simultaneously observable for all r -pure polyanalytic ensembles.
This recovers and slightly extends a result of Shirai [57]. As an application, we obtain an extension of Kostlan's theorem [47] on the absolute values of the points of the Ginibre ensemble of dimension N .
f Y j (x) := 2 π j−r +1 r ! j! x 2( j−r )+1 L j−r r (π x 2 ) 2 e −π x 2 , where L α j are the Laguerre polynomials of (1.5). (Hence, Y 2 j is distributed according to a generalized Gamma function with density f Y 2 j (x) = π j−r+1 r ! j! x j−r L j−r r (π x) 2 e −π x ).
Organization
Section 2 presents tools from phase-space analysis, including the short-time Fourier transform and Weyl's correspondence. Section 3 studies finite WH ensembles and more technical variants required for the identification of finite polyanalytic ensembles as WH ensembles with Hermite windows. This identification is carried out in Sect. 4 by means of symmetry arguments. The approximate identification of finite polyanalytic ensembles with finite WH ensembles is finished in Sect. 5 and gives a comparison of the processes defined by truncating the complex Hermite expansion on the one hand, and by the abstract concentration and spectral truncation method on the other. We explain the deviation between the two ensembles as stemming from two different quantization rules. The proof resorts to a Sobolev embedding for certain symbol classes known as modulation spaces. Some of the technical details are postponed to the appendix. Theorem 1.2 is proved in Sect. 5. In Sect. 6 we apply the symmetry argument from Sect. 4 to rederive the so-called simultaneous observability of polyanalytic ensembles. We also clarify the relation between the spectral expansions of the restriction and Toeplitz kernels. Finally, the appendix provides some background material on determinantal point processes, a certain symbol class for pseudo-differential operators, functions of bounded variation, and polyanalytic spaces.
Harmonic Analysis on Phase Space
In this section we briefly discuss our tools. These methods from harmonic analysis are new in the study of determinantal point processes.
The Short-Time Fourier Transform
Given a window function g ∈ L 2 (R d ), the short-time Fourier transform of f ∈ L 2 (R d ) is
V g f (x, ξ) = R d f (t)g(t − x)e −2πiξ t dt, (x, ξ) ∈ R 2d . (2.1)
The short-time Fourier transform is closely related to the Schrödinger representation of the Heisenberg group, which is implemented by the operators
T (x, ξ, τ )g(t) = e 2πiτ e −πi xξ e 2πiξ t g(t − x), (x, ξ) ∈ R d , τ ∈ R.
The corresponding representation coefficients are
f , T (x, ξ, τ )g = e −2πiτ e πi xξ f , e 2πiξ · g(· − x) = e −2πiτ e πi xξ V g f (x, ξ).
As the variable τ occuring in the Schrödinger representation is unnecessary for DPPs, we will only use the short-time Fourier transform. We identify a pair (x, ξ) ∈ R 2d with the complex vector z = x + iξ ∈ C d . In terms of the phase-space shifts in (1.12), the short-time Fourier transform is V g f (z) := f , π(z)g . The phase-space shifts satisfy the commutation relations
π(x, ξ)π(x , ξ ) = e −2πiξ x π(x + x , ξ + ξ ), (x, ξ), (x , ξ ) ∈ R d × R d ,(2.V g 1 f 1 , V g 2 f 2 L 2 (R 2d ) = f 1 , f 2 L 2 (R d ) g 1 , g 2 L 2 (R d ) . (2.3)
In particular, when g 2 = 1, the map V g is an isometry between L 2 (R d ) and a closed subspace of L 2 (R 2d ):
V g f L 2 (R 2d ) = f L 2 (R d ) , f ∈ L 2 (R d ). (2.4)
The commutation rule (2.2) implies the following formula for the short-time Fourier transform:
V g (π(x, ξ) f )(x , ξ ) = e −2πi x(ξ −ξ) V g f (x − x, ξ − ξ), (x, ξ), (x , ξ ) ∈ R d × R d .
Since the phase-space shift of f on R d corresponds to a phase-space shift of V g f on R 2d , this formula is usually called the covariance property of the short-time Fourier transform.
Special Windows
If we choose the Gaussian function h 0 (t) = 2 1 4 e −π t 2 , t ∈ R, as a window in (2.1), then a simple calculation shows that
e −iπ xξ + π 2 |z| 2 V h 0 f (x, −ξ) = 2 1/4 R f (t)e 2π tz−π t 2 − π 2 z 2 dt = B f (z),(2.F 2 F (C) = C |F(z)| 2 e −π |z| 2 dz < ∞. (2.6)
We now explain the relation between polyanalytic Fock spaces and phase-space analysis with Hermite windows {h r : r ≥ 0}. The r -pure polyanalytic Bargmann transform [2] is the map B r :
L 2 (R) → L 2 (C, e −π |z| 2 ) B r f (z) := e −iπ xξ + π 2 |z| 2 V h r f (x, −ξ), z = x + iξ. (2.7)
This map defines an isometric isomorphism between L 2 (R) and the pure polyanalytic-Fock space F r (C) (see Sect. A.5). The orthogonality relations (2.3) show that for r = r , V h r f 1 is orthogonal to V h r f 2 for all f 1 , f 2 ∈ L 2 (R). The relation between phase-space analysis and polyanalytic functions discovered in [2] can be understood in terms of the Laguerre connection [32, Chapter 1.9]:
V h r h j (x, −ξ) = e iπ xξ − π 2 |z| 2 H j,r (z,z),(2.8)
which, in terms of the polyanalytic Bargmann transform reads as
B r h j (z) = H j,r (z,z),(2.9)
see also [2].
The Range of the Short-Time Fourier Transform
For g 2 = 1, the short-time Fourier transform V g defines an isometric map V g :
L 2 (R d ) → L 2 (R 2d ) with range V g := V g f : f ∈ L 2 (R d ) ⊆ L 2 (R 2d ).
The adjoint of V g can be written formally as V * g :
L 2 (R 2d ) → L 2 (R d ), V * g F = R 2d F(z)π(z)g dz, t ∈ R d ,
where the integral is to be taken as a vector-valued integral. The orthogonal projection P V g :
L 2 (R 2d ) → V g is then P V g = V g V * g . Explicitly, P V g is the integral operator P V g F(z) = R 2d K g (z, z )F(z )dz , z = (x, ξ) ∈ R 2d ,
where the reproducing kernel K g is given by (1.1). Every function F ∈ V g is continuous and satisfies the reproducing formula F(z) = R 2d F(z )K g (z, z )dz .
Metaplectic Rotation
We will make use of a rotational symmetry argument in phase space. Let R θ :=
cos(θ ) − sin(θ ) sin(θ ) cos(θ )
denote the rotation by the angle θ ∈ R. The metaplectic rotation is the operator given in the Hermite basis {h r : r ≥ 0} by
μ(R θ ) f = r ≥0 e irθ f , h r h r , f ∈ L 2 (R) ,(2.10)
in particular, μ(R θ )h r = e irθ h r . The standard and metaplectic rotations are related by
V g f (R θ (x, ξ)) = e πi(xξ −x ξ ) V μ(R −θ )g μ(R −θ ) f (x, ξ), where (x , ξ ) = R θ (x, ξ).(2.
Time-Frequency Localization and Toeplitz Operators
Let us consider g with
g 2 = 1. For m ∈ L ∞ (R 2d ), the Toeplitz operator M g m : V g → V g is M g m F := P V g (m · F), F ∈ V g ,
and its integral kernel at a point (z, z ) is given by 5 The situation is depicted in the following diagram.
K m (z, z ) = R 2d K g (z, z )m(z )K g (z , z ) dz .g m ) = R 2d K m (z, z) dz = R 2d R 2d |K g (z, z )| 2 m(z ) dzdz = R 2d m(z ) dz , (2.13) because the isometry property (2.4) implies that R 2d |K g (z, z )| 2 dz = R 2d | π(z )g, π(z)g | 2 dz = 1 .L 2 (R d ) V g H g m L 2 (R d ) V g V g m· M g m V g L 2 (R 2d ) P Vg (2.14)
Explicitly, the time-frequency localization operator applies a mask to the short-time Fourier transform:
H g m f := R 2d m(z)V g f (z)π(z)g dz, f ∈ L 2 (R 2d ).
As we will use the connection between time-frequency localization on R d and Toeplitz operators on R 2d in a crucial argument, we write (2.14) as a formula
H g m f , u = V g (V * g M g m V g f ), V g u = P V g (m V g f ), V g u = m V g f , V g u . (2.15)
This formula makes sense for f , u ∈ L 2 (R d ) and m ∈ L ∞ (R 2d ), but also for many other assumptions [21]. Time-frequency localization operators are useful in signal processing because they model time-varying filters. For Gaussian windows, they have been studied in signal processing by Daubechies [22] and as Toeplitz operators on spaces of analytic functions by Seip [56]; see also [6,Section 1.4]. When m ∈ L 1 (R 2d ), H g m is trace-class by (2.13) and
trace(H g m ) = R 2d m(z)dz .
(2.16)
For more details see [21,42,43]. When m = 1 , the indicator function of a set , we write M g and H g . In this case, the positivity property implies that 0 ≤ M g ≤ I .
The Weyl Correspondence
The Weyl transform of a distribution σ ∈ S (R d × R d ) is an operator σ w that is formally defined on functions f :
R d → C as σ w f (x) := R d ×R d σ x + y 2 , ξ e 2πi(x−y)ξ f (y)dydξ, x ∈ R d .
Every continuous linear operator T : S(R d ) → S (R d ) can be represented in a unique way as T = σ w , and σ is called its Weyl symbol (see [32,Chapter 2]). The Wigner distribution of a test function g ∈ S(R d ) and a distribution f ∈ S (R d ) is
W ( f , g)(x, ξ) = R 2d f (x + t 2 )g(x − t 2 )e −2πitξ dt.
The integral has to be understood distributionally. The map ( f , g) → W ( f , g) extends to other function classes, for example, for f ,
g ∈ L 2 (R d ), W ( f , g) is well-defined and W ( f , g) 2 = f 2 g 2 . (2.17)
The Wigner distribution is closely related to the short-time Fourier transform:
W ( f , g)(x, ξ) = 2 d e 4πi x·ξ Vg f (2x, 2ξ),
whereg(x) = g(−x). The action of σ w on a distribution can be easily described in terms of the Wigner distribution:
σ w f , g = σ, W (g, f ) .
Finite Weyl-Heisenberg Ensembles
Definitions
To define finite Weyl-Heisenberg processes, we consider a domain ⊆ R 2d with nonempty interior, finite measure and finite perimeter, i.e., the characteristic function of has bounded variation (see Sect. A.1). Since M g is trace-class, the Toeplitz operator M g can be diagonalized as We remark that the eigenvalues λ j do depend on the window function g. When we need to stress this dependence we write λ j ( , g). The finite Weyl-Heisenberg ensemble X g is given by Definition 1.3. For technical reasons, we will also consider a more general class of WH ensembles depending on an extra ingredient. Given a subset I ⊆ N, we let X g ,I be the determinantal point process with correlation kernel K g, ,I (z, z ) = j∈I p g, j (z) p g, j (z ).
M g = j≥1 λ j p g, j ⊗ p g, j , f ∈ L 2 (R 2d ),(3.
When I = {1, . . . , N } we obtain the finite WH ensemble X g , while for I = N we obtain the infinite ensemble. (In the latter case, the resulting point-process is independent of domain .) Later we need to analyze the properties of the ensemble X g ,I with respect to variations of the index set I . When no subset I is specified, we always refer to the ensemble X g associated with I = {1, . . . , N }.
Remark 3.1
The process X g ,I is well-defined due to the Macchi-Soshnikov theorem (see Sect. 1). Indeed, since the kernel K g, ,I represents an orthogonal projection, we only need to verify that it is locally trace-class. This follows easily from the facts that 0 ≤ K g, ,I (z, z) ≤ K g (z, z) = 1 and that the restriction operators are positive (see Sect. 6.1).
Universality and Rates of Convergence
The one-point intensity associated with a Weyl-Heisenberg ensemble X For X g , the intensity ρ g, has been studied in the realm of signal analysis, where it is known as the accumulated spectrogram [6,7]. (Another interesting connection between DPP's and signal analysis is the completeness results of Ghosh [35].) The results in [6,7] imply Theorems
ρ g, ,I − 1 L 1 (R 2d ) = #I − | | + 2 j / ∈I λ j .
Proof Using that 0 ≤ ρ g, ,I ≤ 1 and (1.20) and (3.2), we first calculate
ρ g, ,I − 1 L 1 ( ) = 1 − ρ g, ,I (z) dz = | | − j∈I λ j = j / ∈I λ j .
Second, since the eigenfunctions are normalized and | p g, j (z)| 2 dz = λ j , we have
ρ g, ,I − 1 L 1 (R 2d \ ) = R 2d \ ρ g, ,I (z) dz = j∈I R 2d | p g, j (z)| 2 dz − | p g, j (z)| 2 dz = j∈I 1 − λ j = #I − j∈I λ j = #I − | | + j / ∈I λ j .
The conclusion follows by adding both estimates.
Hermite Windows and Polyanalytic Ensembles
Eigenfunctions of Toeplitz Operators
We first investigate the eigenfunctions of Toeplitz operators with Hermite windows {h r : r ≥ 0} and circular domains. Proof Consider the metaplectic rotation R θ with angle θ ∈ R defined in (2.10). For f , u ∈ L 2 (R), we use first (2.15) and then the covariance property in (2.11) and the rotational invariance of D R to compute:
μ(R θ ) * H h r D R μ(R θ ) f , u = H h r D R μ(R θ ) f , μ(R θ )u = 1 D R V h r μ(R θ ) f , V h r μ(R θ )u = 1 D R V μ(R θ )h r μ(R θ ) f , V μ(R θ )h r μ(R θ )u = 1 D R V h r f (R −θ ·), V h r u(R −θ ·) = D R V h r f (z)V h r u(z)dz = H h r D R f , u .
We conclude that μ(R θ ) * H h r D R μ(R θ ) = H h r D R , for all θ ∈ R. Applying this identity to a Hermite function gives
μ(R θ ) * H h r D R h j = μ(R θ ) * H h r D R μ(R θ ) e −i jθ h j = e −i jθ μ(R θ ) * H h r D R μ(R θ )h j = e −i jθ H h r D R h j .
Thus, H h r D R h j is an eigenfunction of μ(R θ ) * with eigenvalue e −i jθ . For irrational θ , the numbers {e −i jθ : j ≥ 0} are all different, and, therefore, the eigenspaces of μ(R θ ) * are one-dimensional. Hence, H h r D R h j must be a multiple of h j . Thus, we have shown that each Hermite function is an eigenfunction of H h r D R . Since the family of Hermite functions is complete, the conclusion follows. The statement about the complex Hermite polynomials follows from (2.8) and (2.14); the extra phase-factors and conjugation bars disappear due to the renormalization M h r D R → M h r D R .
Eigenvalues of Toeplitz Operators
As a second step to identify polyanalytic ensembles as WH ensembles, we inspect the eigenvalues of Toeplitz operators.
Lemma 4.2 Let R > 0. Then the eigenvalue of H h r D R corresponding to h j and the eigenvalue of M h r D R corresponding to H j,r
(z, z)e −π |z| 2 /2 are μ r j,R := H h r D R h j , h j = D R H r , j (z,z) 2 e −π |z| 2 dz. (4.1)
In particular, μ r j,R = 0 for all j, r ≥ 0 and R > 0, and
H h r D R = j≥0 μ r j,R h j ⊗ h j . (4.2)
Proof (4.1) follows immediately from the definitions. According to (1.4), H r , j vanishes only on a set of measure zero, thus we conclude that μ r j,R = 0. The diagonalization follows from Proposition 4.1. Figure 4 shows a plot of μ 1 0,R (solid, blue) and μ 1 1,R (dashed, red) as a function of R. Note that for a certain value of R, the eigenvalue μ 1 0,R = μ 1 1,R is multiple.
Remark 4.3
Identification as a WH Ensemble
We can now identify finite pure polyanalytic ensembles as WH ensembles. As a consequence, we obtain the following. and #I r ,N = N , whose existence is granted by Proposition 4.4. Then K h r ,D R N ,I r,N = K r ,N , and the corresponding point processes coincide. In particular
ρ r ,N (z) = ρ h r ,D R N ,I r,N (z), z ∈ C. (4.5)
Proof Since #I r ,N = N , we can write
K h r ,D R N (z, z ) = j∈I r,N p D R N h r , j (z) p D R N h r , j (z ) = N −1 j=0 V h r h j (z)V h r h j (z ).
Using (1.30) and (2.8) we conclude that
K h r ,D R N (z, z ) = N −1 j=0 H j,r (z, z)e −π |z| 2 /2 H j,r (z , z )e −π |z 2 /2 = K r ,N (z, z ),
as desired. This implies that the point processes corresponding to K h r ,D R N and K r ,N are related by transformation z → z. Since H j,r (z, z) = H j,r (z, z), the intensities of the pure
ρ 0,N (z) = ρ h 0 ,D R N (z), z ∈ C. (4.6)
Proof The claim amounts to saying that the eigenvalues μ 0 j,R in (4.1) are decreasing for all R > 0, so that the ordering of the eigenfunctions in (3.1) coincides with the indexation of the complex Hermite polynomials. The explicit formula in (4.1) in the case r = 0 gives the sequence of incomplete Gamma functions:
μ 0 j,R = 1 j! π R 2 0 t j e −t dt = 1 − e −π R 2 j k=0 π k k! R 2k ,
which is decreasing in j (see for example [1, Eq. 6.5.13]).
Comparison Between Finite WH and Polyanalytic Ensembles
Having identified finite pure polyanalytic ensembles as WH ensembles associated with a certain subset of eigenfunctions I , we now investigate how much this choice deviates from the standard one I = {1, . . . , N }. Thus, we compare finite pure polyanalytic ensembles to the finite WH ensembles of Definition 1.3.
Change of Quantization
As a main technical step, we show that the change of the window of a time-frequency localization operator affects the distribution of the corresponding eigenvalues in a way that is controlled by the perimeter of the localization domain. When g is a Gaussian, the map m → H g m is called Berezin's quantization or anti-Wick calculus [32,Chapter 2] or [49]. The results in this section show that if Berezin's quantization is considered with respect to more general windows and in R 2d , the resulting calculus enjoys similar asymptotic spectral properties. We consider the function class
M 1 (R d ) := f ∈ L 2 (R d ) : f M 1 := V φ f L 1 (R 2d ) < +∞ ,(5.1)
where φ(x) = 2 d/4 e −π |x| 2 . The class M 1 is one of the modulation spaces used in signal processing. It is also important as a symbol-class for pseudo-differential operators. Indeed, the following lemma, whose proof can be found in [37], gives a trace-class estimate in terms of the M 1 -norm of the Weyl symbol (see also [21,42,43]).
Proposition 5.1 Let σ ∈ M 1 (R 2d ).
Then σ w is a trace-class operator and
σ w S 1 σ M 1 ,
where · S 1 denotes the trace-norm.
The next lemma will allow us to exploit cancellation properties in the M 1 -norm. Its proof is postponed to Sect. A.3.
Lemma 5.2 (A Sobolev embedding for
M 1 ) Let f ∈ L 1 (R d ) be such that ∂ x k f ∈ M 1 (R d ), for k = 1, . . . , d. Then f ∈ M 1 (R d ) and f M 1 f L 1 + d k=1 ∂ x i f M 1 .
We can now derive the main technical result. Its statement uses the space of BV(R 2d ) of (integrable) functions of bounded variation; see Sect. A.1 for some background.
Theorem 5.3 Let g 1 , g 2 ∈ S(R d ) with g i 2 = 1 and m ∈ BV(R 2d ). Then H g 1 m − H g 2 m S 1 ≤ C g 1 ,g 2 var(m),
where C g 1 ,g 2 is a constant that only depends on g 1 and g 2 . In particular, when m = 1 we obtain that
H g 1 − H g 2 S 1 ≤ C g 1 ,g 2 |∂ | 2d−1 .
Proof of Theorem 5. 3 Let us assume first that m is smooth and compactly supported. We use the description of time-frequency localization operators as Weyl operators. By (2.18), H g i m = (m * W (g i , g i )) w . Now, let h := W (g 1 , g 1 ) − W (g 2 , g 2 ). Then h ∈ S-see, e.g., [32,Proposition1.92]-and h = g 1 2 2 − g 2 2 2 = 0 by (2.17). Hence, by Proposition 5.1, H
g 1 m − H g 2 m S 1 = (m * h) w S 1 m * h M 1 ,
Therefore, it suffices to prove that m * h M 1 var(m). We apply Lemma 5.2 to this end. First note that ∂ x i (m * h) = ∂ x i m * h and, consequently,
∂ x i (m * h) M 1 ∂ x i m L 1 h M 1 var(m).
Second, we exploit the fact that h = 0 to get
(m * h)(z) = R d m(z )h(z − z )dz = R d (m(z ) − m(z))h(z − z )dz = R d 1 0 ∇(m)(tz + (1 − t)z), z − z dt h(z − z )dz ,
and consequently
R d |m * h(z)| dz ≤ 1 0 R d R d ∇(m)(tz + (1 − t)z) z − z h(z − z ) dz dzdt = 1 0 R d R d |∇(m)(tw + z)| |w| |h(−w)| dwdzdt = ∇m L 1 1 0 R d |w| |h(w)| dwdt = ∇m L 1 R d |w| |h(w)| dw.
Since h ∈ S the last integral is finite. We conclude that m * h L 1 ∇m
Comparison of Correlation Kernels
We now state and prove the main result on the comparison between finite WH ensembles associated with different subsets of eigenfunctions.
K h r ,D R N − K h r ,D R N ,I r,N S 1 ∂ D R N 1 √ N ,(5.
2)
where · S 1 denotes the trace-norm of the corresponding integral operators.
D R N = j≥1 λ j (D R N , h r ) p D R N h r , j ⊗ p D R N h r , j ,(5.
3)
M h r D R N = j≥0 μ r j,R N V h r h j ⊗ V h r h j . (5.4)
Recall that, while the eigenvalues in (5.4) are ordered non-increasingly, the eigenvalues in (5.3) follow the indexation of Hermite functions. When r = 0, according to Corollary 4.6, the two expansions coincide: the sequence μ 0 j,R N is decreasing, and
λ j+1 (D R N , h 0 ) = μ 0 j,R N , j ≥ 0. (5.5)
We now quantify the deviation between the two eigen-expansions for general r .
μ 0 ·,R N − μ r ·,R N 1 = H h 0 D R N − H h r D R N S 1 ≤ C r ∂ D R N 1 R N √ N . (5.6)
Step 2. Estimates for the spectral truncations. According to Proposition 4.5,
K h r ,D R N ,I r,N = N −1 j=0 V h r h j ⊗ V h r h j . (5.7)
For clarity, in what follows we denote by T K the operator with integral kernel K . Let L j := 1 for 1 ≤ j ≤ N and L j := 0, for j > N . Using the expansion in (5.4) and (3.1), we estimate the trace-norm:
T K hr ,D R N − M h r D R N S 1 = j≥1 L j − λ j (D R N , h r ) p D R N h r , j ⊗ p D R N h r , j S 1 ≤ j≥1 L j − λ j (D R N , h r ) = N j=1 1 − λ j (D R N , h r ) + j>N λ j (D R N , h r ) = N − j≥1 λ j (D R N , h r ) + 2 j>N λ j (D R N , h r ) = 2 j>N λ j (D R N , h r ) ,as j λ j = |D R N | = N by (3.2). Since μ r j,R N is a rearrangement of λ j (D R N , h r )
, we can use (5.3) and (5.7) to mimic the argument. Thus, a similar calculation gives
T K hr ,D R N ,I r,N − M h r D R N S 1 ≤ 2 j>N −1 μ r j,R N ,
and consequently,
T K hr ,D R N − T K hr ,D R N ,I r,N S 1 j>N λ j (D R N , h r ) + j>N −1 μ r j,R N . (5.8)
Step 3. Final estimates Combining (5.8) with (5.5) and (5.6) we obtain
T K hr ,D R N − T K hr ,D R N ,I r,N S 1 j>N λ j (D R N , h r ) + j>N λ j (D R N , h 0 ) + √ N . (5.9)
We now invoke Lemma 3.2 and Theorem 1.5 to estimate
j>N λ j (D R N , h r ) ρ h r ,D R N − 1 D R N L 1 ∂ D R N 1 √ N . (5.10)
Finally, (5.2) follows by combining (5.9) and (5.10).
Transference to Finite Pure Polyanalytic Ensembles
Proof of Theorem 1. 6 We use Proposition 4.5 to identify the (r , N )-polyanalytic ensemble with a Weyl-Heisenberg ensemble with parameters (h r , D R N , I r ,N ), with correlation K h r ,D R N ,I r,N as in Theorem 5.4. By Proposition 4.5, K h r ,D R N ,I r,N = K r ,N . Therefore, the conclusion follows from (5.2).
The One-Point Intensity of Finite Polyanalytic Ensembles
Proof of Theorem 1. 2 We use the notation of Theorem 5.4; in particular R N = N π . By (4.5), ρ r ,N = ρ h r ,D R N ,I r,N , and we can estimate
ρ r ,N − 1 D R N 1 ≤ ρ h r ,D R N ,I r,N − ρ h r ,D R N 1 + ρ h r ,D R N − 1 D R N 1 . By Theorem 1.5, ρ h r ,D R N − 1 D R N 1 √ N .
In addition, by Lemma A.1 in the appendix,
ρ h r ,D R N ,I r,N − ρ h r ,D R N 1 = R 2d K h r ,D R N ,I r,N (z, z) − K h r ,D R N (z, z) dz ≤ K h r ,D R N ,I r,N − K h r ,D R N S 1 .
Hence, the conclusion follows from Theorem 5.4.
Note that the proofs of Theorems 5.4 and 1.2 combine our main insights: the identification of the finite polyanalytic ensembles with certain WH ensembles, the analysis of the spectrum of time-frequency localization operators and Toeplitz operators, and the non-asymptotic estimates of the accumulated spectrum.
Double Orthogonality
Restriction Versus Localization
Let X g be an infinite WH ensemble on R 2d and ⊆ R 2d of finite measure and non-empty interior. We consider the restriction operator T g : L 2 (R 2d ) → L 2 (R 2d ),
T g F := 1 P V g (1 · F),
and the inflated Toeplitz operator S g : L 2 (R 2d ) → L 2 (R 2d ),
S g F := P V g (1 · P V g F).
In view of the decomposition L 2 (R 2d ) = V g ⊕ V ⊥ g , S g and M g are related by
S g = M g 0 0 0 ,
and therefore share the same non-zero eigenvalues, and the corresponding eigenspaces coincide. The integral representation of S g is given by (1.17). Since P V g and F → F · 1 are orthogonal projections, both T g and S g are self-adjoint operators with spectrum contained in [0, 1]. The integral kernel of T g is given by (1.14) and K g | (z, z)dz = | | < +∞.
Therefore, T g is trace-class (see e.g. [58, Theorems 2.12 and 2.14]). It is an elementary fact that T g and S g have the same non-zero eigenvalues with the same multiplicities (this is true for P Q P and Q P Q whenever P and Q are orthogonal projections). Morever, for λ = 0, the map
F −→ 1 √ λ 1 F
is an isometry between the eigenspaces
F ∈ L 2 (R 2d ) : S g F = λF −→ F ∈ L 2 (R 2d ) : T g F = λF .
Therefore, if M g is diagonalized as in (1.18), then T g can be expanded as in (1.19). This justifies the discussion in Sect. 1.3.
Simultaneous Observability
Let X be a determinantal point process (with a Hermitian locally trace-class correlation kernel). We say that a family of sets γ : γ ∈ is simultaneously observable for X , if the following happens. Let = γ ∈ γ . There is an orthogonal basis {ϕ j : j ∈ J } of the closure of the range of the restriction operator T consisting of eigenfunctions of T such that for each γ ∈ , the set {ϕ j | γ : j ∈ J } of the restricted functions is orthogonal. This is a slightly relaxed version of the notion in [44, p. 69]: in the situation of the definition, the functions {ϕ j | γ : j ∈ J } \ {0} form an orthogonal basis of the closure of the range of T γ , but we avoid making claims about the kernel of T . As explained in [44, p. 69], the motivation for this terminology comes from quantum mechanics, where two physical quantities can be measured simultaneously if the corresponding operators commute (or, more concretely, if they have a basis of common eigenfunctions). : V h r → V h r corresponding to the indices in I :
K h r D R 0 ,I (z, z ) = j∈I p D R 0 h r , j (z) p D R 0 h r , j (z ).
Since, by part (i), the functions p g, j are orthogonal when restricted to disks, the conclusion follows.
As a consequence, we obtain Theorem 1.7, which we restate for convenience.
Theorem 1.7
The family D = D R : r ∈ R + of all disks of C centered at the origin is simultaneously observable for every finite and infinite pure-type polyanalytic ensemble.
Proof This follows immediately from Proposition 4.5 and Theorem 6.1. (This slightly extends a result originally derived by Shirai [57].)
An Extension of Kostlan's Theorem
Theorem 1.8 is a consequence of the following slightly more general result.
f Y j (x) := 2 π j−r +1 r ! j! x 2( j−r )+1 L j−r r (π x 2 ) 2 e −π x 2 . (Hence, Y 2 j is distributed according to f Y 2 j (x) = π j−r+1 r ! j! x j−r L j−r r (π x) 2 e −π x .)
Proof We want to show that the point processes |X | := x∈X δ |x| on R and Y := j∈J δ Y j on C have the same distribution. Let I k = [r k , R k ], k = 1, . . . N , be a disjoint family of subintervals of [0, +∞). Then
(Y(I 1 ), . . . , Y(I N )) d = j∈J ζ j ,
where the ζ j are independent, P(ζ j = e k ) =
R k r k f Y j (x)dx, and P(ζ j = 0) = R\∪ k [r k ,R k ] f Y j (x)dx.
On the other hand, Theorem 1.7 implies that the annuli A k := {z ∈ C : r k ≤ |z| ≤ R k } are simultaneously observable for X . Hence, by [44,Proposition 4.5.9]-which is still applicable for the slightly more general definition of simultaneous observability in Sect. 6. where the ζ j are independent, P(ζ j = e k ) = A k H j,r (z, z) 2 e −π |z| 2 dz, and P(ζ j = 0) = C\∪ k A k H j,r (z, z) 2 e −π |z| 2 dz. A direct calculation, together with the identity
(−x) k k! L k−r r (x) = (−x) r r ! L r −k k (x)
shows that ζ j : j ∈ J d = ζ j : j ∈ J and the conclusion follows.
Remark 6.3
Let n(R) denote the number of points of a point process in the disk of radius R centered at the origin. An immediate consequence of Theorem 6.2 is the following formula for the probability of finding such a disk void of points, when the points are distributed according to the a polyanalytic Ginibre ensemble of the pure type:
P [n(R) = 0] = j P Y j ≥ R
This is known as the hole probability (see [44,Section 7.2] for applications in the case of the Ginibre ensemble).
A DPP can be represented by different kernels. If m : R d → C is unimodular (i.e., |m(z)| = 1), then the kernel
K m (x, x ) = m(x)K (x, x )m(x ),
produces the same intensities in (A.1) as K does. (This is a so-called gauge transformation). The integral operator with kernel K m is related to the one with kernel K by
m(x)T K (m f )(x) = R d m(x)K (x, x )m(x ) f (x )dx = T K m f (x).
Similarly, a linear transformation of a DPP corresponds to a linear change of variables in the kernel K .
A.1 Functions of Bounded Variation
A real-valued function f ∈ L 1 (R d ) is said to have bounded variation, f ∈ BV(R d ), if its distributional partial derivatives are finite Radon measures. The variation of f is defined as
var( f ) := sup R d f (x) div φ(x)dx : φ ∈ C 1 c (R d , R d ), |φ(x)| 2 ≤ 1 ,
where C 1 c (R d , R d ) denotes the class of compactly supported C 1 -vector fields and div is the divergence operator. If f is continuously differentiable, then f ∈ BV(R d ) simply means that ∂ x 1 f , . . . , ∂ x d f ∈ L 1 (R d ), and var( f ) = R d |∇ f (x)| 2 dx = ∇ f L 1 . A set ⊆ R d is said to have finite perimeter if its characteristic function 1 is of bounded variation, and the perimeter of is defined as |∂ | d−1 := var(1 ). If has a smooth boundary, then |∂ | d−1 is just the (d − 1)-Hausdorff measure of the topological boundary. See [30,Chapter 5] for an extensive discussion of BV.
A.2 Trace-Class Operators
Lemma A.1 Let K : R d × R d → C be a continuous function and assume that the integral operator
T K f (x) = R d K (x, y) f (y)dy, f ∈ L 2 (R d ),
is well-defined, bounded, and trace-class. Then R d |K (x, x)| dx ≤ T K S 1 , where · S 1 denotes the trace-norm.
Proof Let T K = j μ j ϕ j ⊗ ψ j , with μ j ≥ 0 and {ϕ j : j ≥ 1}, {ψ j : j ≥ 1} orthonormal. Then K (x, y) = j μ j ϕ j (x)ψ j (y) for almost every (x, y), and we can formally compute
R d |K (x, x)| dx ≤ j μ j R d ϕ j (x) ψ j (x) dx ≤ j μ j R d ϕ j (x) 2 dx 1/2 R d ψ j (x) 2 dx 1/2 = j μ j = T K S 1 .
Polyanalytic Bargmann-Fock spaces appear naturally in vector-valued time-frequency analysis [2,39] and signal multiplexing [12,13]. Within F q (C) we distinguish the polynomial subspace Pol π,q,N = span{z j z l : 0 ≤ j ≤ N − 1, 0 ≤ l ≤ q − 1}, with the Hilbert space structure of L 2 (C, e −π |z| 2 ). The polyanalytic Ginibre ensemble, introduced in [40], is the DPP with correlation kernel corresponding to the orthogonal projection onto Pol π,q,N (weighted with the Gaussian measure). In [40,Proposition 2.1] it is shown that Pol π,q,N = span{H j,r (z, z) :
0 ≤ j ≤ N − 1, 0 ≤ r ≤ q − 1},
where H j,r are the complex Hermite polynomials (1.4). Thus, the reproducing kernel of Pol π,q,N can be written as
A.5 Pure Polyanalytic-Fock Spaces
The pure polyanalytic Fock spaces F r (C) have been introduced by Vasilevski [63], under the name of true polyanalytic spaces. They are spanned by the complex Hermite polynomials of fixed order r and can be defined as the set of polyanalytic functions F integrable in L 2 (C, e −π |z| 2 ) and such that, for some entire function H [2], F(z) = π r r ! 1 2 e π |z| 2 (∂ z ) r e −π |z| 2 H (z) .
Vasilevski [63] obtained the following decomposition of the polyanalytic Fock space F q (C) into pure components F q (C) = F 0 (C) ⊕ · · · ⊕ F q−1 (C). (A.5)
Pure polyanalytic spaces are important in signal analysis [2] and in connection to theoretical physics [5,40]. Indeed, they parameterize the so-called Landau levels, which are the eigenspaces of the Landau Hamiltonian and model the distribution of electrons in high energy states (see e.g. [57, Section 2], [8, Section 4.1]). The complex Hermite polynomials (1.4) provide a natural way of defining a polynomial subspace of the true polyanalytic space:
Pol π,r ,N = span{H j,r (z, z) : 0 ≤ j ≤ N − 1}.
Thus,
Pol π,q,N = Pol π,0,N ⊕ · · · ⊕ Pol π,q−1,N .
The reproducing kernel of Pol π,r ,N is therefore
Definition 1. 1
1Let J ⊆ N 0 × N 0 . The planar Hermite ensemble based on J is the determinantal point process with the correlation kernel K (z, z ) = e − π 2 (|z| 2 +|z | 2 ) j,r ∈J H j,r (z, z) H j,r z , z . (1.7)
Fig. 1 18 Fig. 2
1182A plot of the eigenvalues of the Toeplitz operator M g , with g a Gaussian window and of area ≈ The eigenfunctions # 1, 7, 18 corresponding to the operator inFig. 1
Fig. 3 1 Theorem 1. 5
315The one-point intensity of a WH ensemble plotted over the domain in Fig. Let ρ g, be the one-point intensity of the finite Weyl-Heisenberg ensemble.
Fig. 4
4A plot of the eigenvalues λ = M h 1 D R H j,1 (z, z)e − π 2 |z| 2 , as a function of R, corresponding to j = 0 (blue, solid) and j = 1 (red, dashed) (Color figure online) ordered according to their Hermite index, while in the other they are ordered according to the magnitude of their eigenvalues.Figure 4 shows the eigenvalues of M h 1 D R , as a function of R, corresponding to the eigenfunctions H 0,1 (z, z)e − π 2 |z| 2 and H 1,1 (z, z)e − π 2 |z| 2 .
Theorem 1. 8
8The set of absolute values of the points distributed according to the r -pure polyanalytic Ginibre ensemble has the same distribution as {Y 1,r , . . . , Y n,r }, where the Y j 's are independent and have density
m = 1 , the last expression coincides with (1.17). (The operator M g m is defined on V g ; the kernel in (2.12) represents the extension of M g m to L 2 (R 2d ) that is 0 on V ⊥ g .) Clearly, M g m V g →V g ≤ m ∞ . In addition, it is easy to see that if m ≥ 0, then M g m is a positive operator. If m ∈ L 1 (R 2d ), then M g m is trace-class. By (2.12) the trace of M
The time-frequency localization operator with window g and symbol m is Hg m := V * g M g m V g : L 2 (R d ) → L 2 (R d ). Hence M g m and H g m are unitarily equivalent.
ρ g, ,I (z) := j∈I p g, j (z) 2 .
Proposition 4. 4
4Let J ⊆ N 0 and R > 0, then there exist a set I ⊆ N with #I = # J such thatV h r h j : j ∈ J = p D R h r , j : j ∈ I .(4.3) Proof By Proposition 4.1 every Hermite function h j is an eigenfunction of H h r D R . In addition, by Lemma 4.2, the corresponding eigenvalue μ r j,R is non-zero. Hence V h r h j is one of the functions p D R h r , j in the diagonalization (3.1). The set I := { j : j ∈ J } satisfies (4.3).
(r , N )-polyanalytic ensemble are invariant under the map z → z and the conclusion follows. While Proposition 4.5 identifies finite pure polyanalytic ensembles with WH ensembles in the generalized sense of Sect. 3 , this is just a technical step. Our final goal is to compare finite polyanalytic ensembles with finite WH ensembles in the sense of Definition 1.3, where the index set is I r ,N = {1, . . . , N }. Before proceeding we note that for the Gaussian h 0 such comparison is in fact an exact identification.
Corollary 4. 6
6For r = 0, the set I 0,N from Proposition 4.5 is I 0,N = {0, . . . , N − 1}. Thus, the N -dimensional Ginibre ensemble has the same distribution as the finite WH ensemble
L 1 =
1var(m), providing the argument for smooth, compactly supported m. For general m ∈ BV(R d ), there exists a sequence of smooth, compactly supported functions {m k : k ≥ 0} such that m k → m in L 1 , and var(m k ) → var(m), as k → +∞ (see for example[30, Sec. trace norm, and the conclusion follows by a continuity argument.
Theorem 5. 4
4Consider the identification of the (r , N )-pure polyanalytic ensemble as a finite WH ensemble with parameters (h r , D R N , I r ,N ) given by Proposition 4.5. Let K h r ,D R N ,I r,N be the corresponding correlation kernel, and let K h r ,D R N be the correlation kernel of the finite Weyl-Heisenberg ensemble associated with the Hermite window h r and the disk D R N . Then
2, we have (|X | (I 1 ), . . . , | X | (I N )) = (X (A 1 ), . . . , X (A N )) d = j∈J ζ j ,
,r (z, z)H j,r (z , z ). (A.4)
K
r ,π,N (z, z ) = N −1 j=0 H j,r (z, z)H j,r (z , z ), and the corresponding determinantal point processes have been introduced in [40].
Comparing the correlation kernels of the finite pure polyanalytic ensemble (1.8) with the finite (renormalized) WH ensemble with a Hermite window (1.32), we see that in each case different subsets of the complex Hermite basis intervene: in one case functions are).
(1.32)
Proposition 4.1 Let D R ⊆ R 2 be a disk centered at the origin. Then the family of Hermite functions is a complete set of eigenfunctions for H h r D R . As a consequence, the set {H j,r (z, z)e −π |z| 2 /2 : j ≥ 0} forms a complete set of eigenfunctions for M h rD R (where M h r
D R
is related to M h r
D R by (1.29).
Theorem 6.1 Let D = D R : R ∈ R + be the family of all disks of R 2 centered at the origin and r ∈ N. Then (i) D is simultaneously observable for the infinite Weyl-Heisenberg ensemble with window h r . (ii) Let D R 0 be a disk and I ⊆ N. Then D is simultaneously observable for the Weyl-Heisenberg ensemble X h r D R 0 ,I . Proof Let us prove (i). Since the definition of simultaneous observability involves the orthogonal complement of the kernels of the restriction operators T ) ⊥ , the discussion in Sect. 6.1 implies that it suffices to show that the Toeplitz operators M h r D R have a common basis of eigenfunctions. Since V * h r M h r D R V h r = H h r D R , and, by Proposition 4.1, the Hermite basis diagonalizes H h r D R for all R > 0, the conclusion follows. Let us now prove (ii). The ensemble X h r D R 0 ,I is constructed by selecting the eigenfunctions of the Toeplitz operator M h r D R 0g
D R , ran(T
g
D R ) = (ker T
g
D R
Theorem 6.2 Let X be the determinantal point process associated with the (r , J )-pure polyanalytic ensemble, with J ⊆ N 0 finite. Then the point process on [0, +∞) of absolute values |X | has the same distribution as the process generated by {Y j : j ∈ J } where the Y j 's are independent random variables with density
Perelomov[53] mentions that (1.4) has been used by Feynman and Schwinger as the explicit expression for the matrix elements of the displacement operator in Bargmann-Fock space.
The first Landau level is also called ground level because it corresponds to the lowest energy.3 See also[51, Proposition 14] and[20], where it is pointed out that the sharp rate for the ground level also follows from pointwise estimates for Bergman kernels[60].
The operator H g m should not be confused with the complex Hermite polynomial H j,r .
Acknowledgements Open access funding provided by Austrian Science Fund (FWF).Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.Appendix A: Additional Background MaterialDeterminantal Point Processes and IntensitiesWe follow the presentation of[15,44]. Let K : R d × R d → C be a locally trace-class Hermitian kernel with spectrum contained in [0, 1], and consider the functions ρ n (x 1 , . . . , x n ) := det K (x j , x k ) j,k=1,...,d ,The Macchi-Soshnikov theorem implies that there exists a point process X on R d such that for every family of disjoint measurable sets 1 , . . . n ⊆ R d ,where X ( ) denotes the number of points of X to be found in . The functions ρ n are known as correlation functions or intensities and X is called a determinantal point process. The one-point intensity ρ is simply the diagonal of the correlation kerneland allows one to compute the expected number of points to be found on a domain :The one-point intensity can also be used to evaluate expectations of linear statistics:1An approximation argument using the continuity of K is needed to justify the computations with the restriction of K to the diagonal-see [58, Chapters 1,2,3] for related arguments.A.3 Properties of Modulation SpacesRecall the definition of the modulation space M 1 in (5.1). It is well-known that, instead of the Gaussian function φ, any non-zero Schwartz function can be used to define M 1 , giving an equivalent norm[31],[38,Chapter 9]. Using this fact, the following lemma follows easily.Then:where C m is a constant that depends on m.We now prove the Sobolev embedding lemma that was used in Sect. 5.1.Proof of Lemma 5.2Let g be such thatĝ = f . By Lemma A.2, it suffices to show that g ∈ M 1 (R) and satisfies a suitable norm estimate. Let η ∈ C ∞ (R) be such that η(ξ ) ≡ 0 for |ξ | ≤ 1/2 and η(ξ ) ≡ 1 for |ξ | > 1. We write η(ξ ) = d k=1 ξ k η k (ξ ), where η k ∈ C ∞ (R) has bounded derivatives of all orders. We set g 1 := η · g and g 2 :and η k has bounded derivatives of all orders, we conclude from Lemma A.2 (iii) that g 1 ∈ M 1 (R) and thatOn the other hand, since g has an integrable Fourier transform, so does g 2 = (1 − η) · g and g 2 L 1 f L 1 . In addition, g 2 is supported on D 1 (0). Therefore, by Lemma A.2, g 2 ∈ M 1 and g 2 M 1 f L 1 . Hence g = g 1 + g 2 ∈ M 1 , and it satisfies the stated estimate.A.4 Polyanalytic Bargmann-Fock SpacesA complex-valued function F(z, z) defined on a subset of C is said to be polyanalytic of order q − 1, if it satisfies the generalized Cauchy-Riemann equationsEquivalently, F is a polyanalytic function of order q − 1 if it can be written aswhere the coefficients {ϕ k (z)} q−1 k=0 are analytic functions. The polyanalytic Fock space F q (C) consists of all the polyanalytic functions of order q − 1 contained in the Hilbert space L 2 (C, e −π |z| 2 ). The reproducing kernel of the polyanalytic Fock space F q (C) is K q (z, z ) = L 1 q (π z − z 2 )e π zz .
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"Manifestaion of SUSY in B decays a",
"Manifestaion of SUSY in B decays a"
]
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"Yasuhiro Okada [email protected] \nInstitute for Particle and Nuclear Studies\nKEK\nOho 1-1305-0801TsukubaJapan\n"
]
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"Institute for Particle and Nuclear Studies\nKEK\nOho 1-1305-0801TsukubaJapan"
]
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| SUSY effects on various flavor changing neutral current processes are discussed in the minimal supergravity model and the SU(5) grand unified theory with righthanded neutrino supermultiplets. In particular, in the latter case the neutrino Yukawa coupling constants can be a source of the flavor mixing in the right-handeddown-type-squark sector. It is shown that due to this mixing the time-dependent CP asymmetry of radiative B decay can be as large as 30% and the ratio of Bs-Bs mixing and B d -B d mixing deviates from the prediction in the standard model and the minimal supergravity model without the neutrino interaction. | 10.1142/9789812791870_0068 | [
"https://arxiv.org/pdf/hep-ph/0002296v1.pdf"
]
| 18,666,170 | hep-ph/0002296 | eeb2af5c2c83a2ae44808a4dcc1c5a5d75828340 |
Manifestaion of SUSY in B decays a
arXiv:hep-ph/0002296v1 29 Feb 2000
Yasuhiro Okada [email protected]
Institute for Particle and Nuclear Studies
KEK
Oho 1-1305-0801TsukubaJapan
Manifestaion of SUSY in B decays a
arXiv:hep-ph/0002296v1 29 Feb 2000
SUSY effects on various flavor changing neutral current processes are discussed in the minimal supergravity model and the SU(5) grand unified theory with righthanded neutrino supermultiplets. In particular, in the latter case the neutrino Yukawa coupling constants can be a source of the flavor mixing in the right-handeddown-type-squark sector. It is shown that due to this mixing the time-dependent CP asymmetry of radiative B decay can be as large as 30% and the ratio of Bs-Bs mixing and B d -B d mixing deviates from the prediction in the standard model and the minimal supergravity model without the neutrino interaction.
Introduction
In order to explore supersymetry (SUSY) indirect search experiments can play a complementary role to direct search for SUSY particles at collider experiments. Since SUSY particles may affect flavor changing neutral current (FCNC) processes and CP violation in B and K meson decays, it is possible that new experiments in B decay at both e + e − colliders and hadron machines reveal new physics signals which can be interpreted as indirect evidence of SUSY.
In the context of SUSY models flavor physics has important implications. Because the squark and the slepton mass matrices become new sources of flavor mixings generic mass matrices would induce too large FCNC and lepton flavor violation (LFV) effects if the superpartners' masses are in a few-hundred-GeV region. For example, if we assume that the SUSY contribution to the K 0 −K 0 mixing is suppressed because of the cancellation among the squark contributions of different generations, the squarks with the same gauge quantum numbers must be highly degenerate in masses at least for the first two generations.
There are several scenarios to solve this flavor problem. In the minimal supergravity model flavor problem are avoided by taking SUSY soft-breaking terms as flavor-blind structure. The scalar mass terms are assumed to be common for all scalar fields at the Planck scale and therefore there are no FCNC effects nor LFV from the squark and slepton sectors at this scale. The physical a Talk given at the Third International Conference on B Physics and CP Violation, December 3 -7, 1999, Taipei. squark and slepton masses are determined taking account of renormalization effects from the Planck to the weak scale. This induces sizable SUSY contributions to various FCNC and LFV processes.
In this talk we consider two types of SUSY models and discuss FCNC processes. The first one is the minimal supersymmetric standard model (MSSM) with a universal SUSY breaking terms at the Planck scale which is realized in the minimal supergravity model. The other is the SU(5) grand unified theory with right-handed neutrino supermultiplets. This model incorporates the seesaw mechanism for neutrino mass generation. In the latter case the neutrino Yukawa coupling constants can be a source of the flavor mixing in the righthanded-down-type-squark sector and due to this mixing the time-dependent CP asymmetry of radiative B decay can be as large as 30% and the ratio of B s -B s mixing and B d -B d mixing deviates from the prediction for the standard model (SM) and the MSSM without the neutrino interaction.
Update of FCNC Processes in the Supergravity Model
In the minimal SM various FCNC processes and CP violation in B and K decays are determined by the Cabibbo-Kobayashi-Maskawa (CKM) matrix. Constraints on the parameters of the CKM matrix can be conveniently expressed in terms of the unitarity triangle. With CP violation at B factory as well as rare K decay experiments we will be able to check consistency of the unitarity triangle and at the same time search for effects of physics beyond the SM. In order to distinguish possible new physics effects it is important to identify how various models can modify the SM predictions.
Although general SUSY models can change the lengths and the angles of the unitarity triangle in variety ways, the supergravity model predicts a specific pattern of the deviation from the SM. 1 Namely, we can show that the SUSY loop contributions to FCNC amplitudes approximately have the same dependence on the CKM elements as the SM contributions. In particular, if we assume that there are no CP violating phases from SUSY breaking sectors, the complex phase of the B 0 −B 0 mixing amplitude does not change even if we take into account the SUSY contributions. The case with supersymmetric CP phases was also studied within the minimal supergravity model and it was shown that effects of new CP phases on the B 0 −B 0 mixing amplitude and the direct CP asymmetry in the b → s γ process are small once constraints from neutron and electron EDMs are included. 2 We calculate various FCNC processes in the supergravity model with universal soft breaking terms at a high energy scale. The results are summarized as follows.
1. The amplitude for b → sγ can receive a large contribution from the SUSY and the charged-Higgs-top-quark loop diagrams. The experimental branching ratio puts a strong constraint on SUSY parameter space.
Since the SUSY contribution can interfere with other contributions either constructively or destructively we cannot exclude the light charged Higgs boson region unlike the non-SUSY type II two Higgs doublet model. 3 2. When the sign of the b → sγ amplitude is opposite to that of the SM, B(b → sl + l − ) can be twice larger than the SM prediction. This can occur for a large tan β region where tan β is the ratio of two vacuum expectation values of Higgs fields. The deviation is also evident in the differential branching ratio and the lepton forward-backward asymmetry. 4 3. In terms of consistency check of the unitarity triangle the supergravity model has the following features. 5 (i)∆M B d and ǫ K are enhanced by the SUSY and charged-Higgs loop effects. When these quantities are normalized by the corresponding quantities in the SM they are almost independent of the CKM matrix element, and the enhancement factors for ∆M B d and ǫ K are almost equal. (ii) The branching ratios for K + → π + ν ν and K L → π 0 ν ν processes are suppressed compared to the SM prediction. Again the suppression factor are almost the same for two branching ratios and does not depend strongly on the CKM matrix element. (iii) CP asymmetries in various B decay modes such as B → J/ψK S and the ratio of ∆M Bs and ∆M B d are the same as the SM prediction.
In Fig. 1 we present the correlation between ∆M B d and B(K L → π 0 ν ν) normalized by the corresponding quantities in the SM for tan β = 3. The constraint on the SUSY parameter space from the recent improved SUSY Higgs search is implemented. 5 We have calculated the SUSY particle spectrum based on two different assumptions on the initial conditions of renormalization group equations. The minimal case corresponds to the minimal supergravity where all scalar fields have a common SUSY breaking mass at the GUT scale. For "nonminimal" we enlarge the SUSY parameter space by relaxing the initial conditions for the SUSY breaking parameters, namely all squarks and sleptons have a common SUSY breaking mass whereas an independent SUSY breaking parameter is assigned for Higgs fields. The square(dot) points correspond to the minimal (enlarged) parameter space of the supergravity model. We can see that the ∆M B d (and ǫ K ) can deviated from the SM by 20% whereas the deviation in K L → π 0 ν ν and K + → π + ν ν processes are small. These deviations may be evident in future when B factory experiments provide additional information on the CKM parameters.
FCNC in SUSY GUT with Right-handed Neutrino
In this section we consider FCNC and LFV of charged lepton decays in the model of a SU(5) SUSY GUT which incorporates the see-saw mechanism for the neutrino mass generation. 6 In this model sources of the flavor mixing are Yukawa coupling constant matrices for quarks and leptons as well as that for the right-handed neutrinos. Because the quark and lepton sectors are related by GUT interactions, the flavor mixing relevant to the CKM matrix can generate LFV such as µ → e γ and τ → µ γ processes 7 in addition to FCNC in hadronic observables. 8 In the SUSY model with right-handed neutrinos, branching ratios of the LFV processes can become large enough to be measured in near-future experiments. 9 When we consider the right-handed neutrinos in the context of GUT, the flavor mixing related to the neutrino oscillation can be a source of the flavor mixing in the squark sector. We show that due to the large mixing of the second and third generations suggested by the atmospheric neutrino anomaly, B s -B s mixing, the time-dependent CP asymmetry of the B → M s γ process, where M s is a CP eigenstate including the strange quark, can have a large deviation from the SM prediction. 6 The Yukawa coupling part and the Majorana mass term of the superpotential for the SU(5) SUSY GUT with right-handed neutrino supermultiplets is The renormalization effects due to the Yukawa coupling constants induce various FCNC and LFV effects from the mismatch between the quark/lepton and squark/slepton diagonalization matrices. In particular the large top Yukawa coupling constant is responsible for the renormalization of theq L andũ R mass matrices. At the same time theẽ R mass matrix receives sizable corrections between the Planck and the GUT scales and various LFV processes are induced. In a similar way, if the neutrino Yukawa coupling constant f ij N is large enough, thel L mass matrix and thed R mass matrix receive sizable flavor changing effects due to renormalization between the Planck and the right-handed neutrino mass scales and the Planck and the GUT scales, respectively. These are sources of extra contributions to LFV processes and various FCNC processes such as b → s γ, the B 0 -B 0 mixing and the K 0 -K 0 mixing.
given by W = 1 8 f ij U Ψ i Ψ j H 5 + f ij D Ψ i Φ j H5 + f ij N N i Φ j H 5 + 1 2 M ij ν N i N j ,where Ψ i , Φ i
The flavor mixing in thed R sector can induce large time-dependent CP asymmetry in the B → M s γ process. Using the Wilson coefficients c 7 and c ′ 7 in the effective Lagrangian for the b → s γ decay L = c 7s σ µν b R F µν + c ′ 7s σ µν b L F µν +H.c., the asymmetry is written as We also take M ν to be proportional to a unit matrix with a diagonal element of M R = 4 × 10 14 GeV. We fix CKM parameters as V cb = 0.04, |V ub /V cb | = 0.08 and δ 13 = 60 • . We take tan β = 5 and vary other SUSY parameters. We take account of various phenomenological constraints on SUSY parameters including B(b → s γ). We also calculated B(µ → e γ) and ǫ K and imposed constraints from these quantities. The upper part of Fig.2 shows a correlation between ∆m s /∆m d (ratio of B s -B s and B d -B d mass splittings) and B(τ → µ γ). We can see that ∆m s /∆m d can be enhanced up to 30% compared to the SM prediction. This feature is quite different from the minimal supergravity model without the GUT and right-handed neutrino interactions 5 where ∆m s /∆m d is almost the same as the SM value. A t for the same parameter set is shown as a function of B(τ → µ γ) in the lower part of Fig. 2. We can see that |A t | can be close to 30% when B(τ → µ γ)is larger than 10 −8 . The large asymmetry arises because the renormalization effect due to f N induces sizable contribution to c ′ 7 through gluino-d R loop diagrams. In this example possible new physics signals in B(τ → µ γ), B s -B s mixing and A t all come from the renormalization effect on squark and slepton mass matrices from the large neutrino Yukawa coupling constant. Because these signals provide quite different signatures compared to the SM and the minimal supergravity model without GUT and right-handed neutrino interactions, future experiments in B physics and LFV can provide us important clues on the interactions at very high energy scale. The work was supported in part by the Grant-in-Aid of the Ministry of Education, Science, Sports and Culture, Government of Japan (No.09640381), Priority area "Supersymmetry and Unified Theory of Elementary Particles" (No. 707), and "Physics of CP Violation" (No.09246105).
Γ(t)−Γ(t) Γ(t)+Γ(t) = ξA t sin ∆m d t, A t = 2Im(e −iθ B c7c ′ 7 ) |c7| 2 +|c ′ 7 | 2 , where Γ(t) (Γ(t))
Figure 1 :
1Correlation between ∆M B d and ǫ K normalized by the SM value for tan β = 3. The square(dot) points correspond to the minimal (enlarged) parameter space of the supergravity model.
and N i are 10,5 and 1 representations of SU(5) gauge group. i, j = 1, 2, 3 are the generation indices. H 5 and H5 are Higgs superfields with 5 and5 representations.
V
is the decay width of B 0 (t) → M s γ (B 0 (t) → M s γ) and M s is some CP eigenstate (ξ = +1(−1) for a CP even (odd) state) such as K S π 0 . 10 ∆m d = 2|M 12 (B d )| and θ B = arg M 12 (B d ) where M 12 (B d ) is the B d -B d mixing amplitude. Because the asymmetry can be only a few percent in the SM, a sizable asymmetry is a clear signal of new physics beyond the SM.We calculated various FCNC and LFV observables in this model under the assumption of the universal soft breaking terms at the Planck scale. As typical examples of the neutrino parameters, we consider the following parameter set corresponding to the Mikheyev-Smirnov-Wolfenstein (MSW) small mixing case. m ν = 2.236 × 10 −3 , 3.16 × 10 −3 , 5.92 × 10 −2 eV and the Maki-Nakagawa-Sakata (MNS) matrix is given by MNS
Figure 2 :
2The ratio of Bs-Bs and B d -B d mass splittings ∆ms/∆m d and the magnitude factor At of the time-dependent CP asymmetry in the B → Ms γ process as a function of B(τ → µ γ) for the small mixing.
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| []
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[
"Learning Domain Invariant Representations in Goal-conditioned Block MDPs",
"Learning Domain Invariant Representations in Goal-conditioned Block MDPs"
]
| [
"Beining Han \nIIIS\nTsinghua University\nCarnegie Mellon University\nUniversity of Toronto & Vector Institute\n\n",
"Chongyi Zheng [email protected] \nIIIS\nTsinghua University\nCarnegie Mellon University\nUniversity of Toronto & Vector Institute\n\n",
"Harris Chan [email protected] \nIIIS\nTsinghua University\nCarnegie Mellon University\nUniversity of Toronto & Vector Institute\n\n",
"Keiran Paster \nIIIS\nTsinghua University\nCarnegie Mellon University\nUniversity of Toronto & Vector Institute\n\n",
"Michael R Zhang [email protected] \nIIIS\nTsinghua University\nCarnegie Mellon University\nUniversity of Toronto & Vector Institute\n\n",
"Jimmy Ba \nIIIS\nTsinghua University\nCarnegie Mellon University\nUniversity of Toronto & Vector Institute\n\n"
]
| [
"IIIS\nTsinghua University\nCarnegie Mellon University\nUniversity of Toronto & Vector Institute\n",
"IIIS\nTsinghua University\nCarnegie Mellon University\nUniversity of Toronto & Vector Institute\n",
"IIIS\nTsinghua University\nCarnegie Mellon University\nUniversity of Toronto & Vector Institute\n",
"IIIS\nTsinghua University\nCarnegie Mellon University\nUniversity of Toronto & Vector Institute\n",
"IIIS\nTsinghua University\nCarnegie Mellon University\nUniversity of Toronto & Vector Institute\n",
"IIIS\nTsinghua University\nCarnegie Mellon University\nUniversity of Toronto & Vector Institute\n"
]
| []
| Deep Reinforcement Learning (RL) is successful in solving many complex Markov Decision Processes (MDPs) problems. However, agents often face unanticipated environmental changes after deployment in the real world. These changes are often spurious and unrelated to the underlying problem, such as background shifts for visual input agents. Unfortunately, deep RL policies are usually sensitive to these changes and fail to act robustly against them. This resembles the problem of domain generalization in supervised learning. In this work, we study this problem for goalconditioned RL agents. We propose a theoretical framework in the Block MDP setting that characterizes the generalizability of goal-conditioned policies to new environments. Under this framework, we develop a practical method PA-SkewFit that enhances domain generalization. The empirical evaluation shows that our goal-conditioned RL agent can perform well in various unseen test environments, improving by 50% over baselines. | null | [
"https://arxiv.org/pdf/2110.14248v2.pdf"
]
| 239,998,383 | 2110.14248 | 1889f93a37a707975ddb04ab4423ad07484d0d41 |
Learning Domain Invariant Representations in Goal-conditioned Block MDPs
Beining Han
IIIS
Tsinghua University
Carnegie Mellon University
University of Toronto & Vector Institute
Chongyi Zheng [email protected]
IIIS
Tsinghua University
Carnegie Mellon University
University of Toronto & Vector Institute
Harris Chan [email protected]
IIIS
Tsinghua University
Carnegie Mellon University
University of Toronto & Vector Institute
Keiran Paster
IIIS
Tsinghua University
Carnegie Mellon University
University of Toronto & Vector Institute
Michael R Zhang [email protected]
IIIS
Tsinghua University
Carnegie Mellon University
University of Toronto & Vector Institute
Jimmy Ba
IIIS
Tsinghua University
Carnegie Mellon University
University of Toronto & Vector Institute
Learning Domain Invariant Representations in Goal-conditioned Block MDPs
Deep Reinforcement Learning (RL) is successful in solving many complex Markov Decision Processes (MDPs) problems. However, agents often face unanticipated environmental changes after deployment in the real world. These changes are often spurious and unrelated to the underlying problem, such as background shifts for visual input agents. Unfortunately, deep RL policies are usually sensitive to these changes and fail to act robustly against them. This resembles the problem of domain generalization in supervised learning. In this work, we study this problem for goalconditioned RL agents. We propose a theoretical framework in the Block MDP setting that characterizes the generalizability of goal-conditioned policies to new environments. Under this framework, we develop a practical method PA-SkewFit that enhances domain generalization. The empirical evaluation shows that our goal-conditioned RL agent can perform well in various unseen test environments, improving by 50% over baselines.
Introduction
Deep Reinforcement Learning (RL) has achieved remarkable success in solving high-dimensional Markov Decision Processes (MDPs) problems, e.g., Alpha Zero [1] for Go, DQN [2] for Atari games and SAC [3] for locomotion control. However, current RL algorithms requires massive amounts of trial and error to learn [1][2][3]. They also tend to overfit to specific environments and often fail to generalize beyond the environment they were trained on [4]. Unfortunately, this characteristic limits the applicability of RL algorithms for many real world applications. Deployed RL agents, e.g. robots in the field, will often face environment changes in their input such as different backgrounds, lighting conditions or object shapes [5]. Many of these changes are often spurious and unrelated to the underlying task, e.g. control. However, RL agents trained without experiencing these changes are sensitive to the changes and often perform poorly in practice [5][6][7].
In our work, we seek to tackle changing, diverse problems with goal-conditioned RL agents. Goalconditioned Reinforcement Learning is a popular research topic as its formulation and method is practical for many robot learning problems [8,9]. In goal-conditioned MDPs, the agent has to achieve a desired goal state g which is sampled from a prior distribution. The agent should be able to achieve not only the training goals but also new test-time goals. Moreover, in practice, goal-conditioned RL agents often receive high-dimensional inputs for both observations and goals [10,11]. Thus, it is important to ensure that the behaviour of goal-conditioned RL agents is invariant to any irrelevant environmental changes in the input at test time. Previous work [6] tries to address these problems via model bisimulation metric [12]. These methods aim to acquire a minimal representation which 35th Conference on Neural Information Processing Systems (NeurIPS 2021), Sydney, Australia. is invariant to irrelevant environment factors. However, as goal-conditioned MDPs are a family of MDPs indexed by the goals, it is inefficient for these methods to acquire the model bisimulation representation for every possible goal, especially in high-dimensional continuous goal spaces (such as images).
In our work, we instead choose to optimize a surrogate objective to learn the invariant policy. Our main contributions are:
1. We formulate the Goal-conditioned Block MDPs (GBMDPs) to study domain generalization in the goal-conditioned reinforcement learning setting (Section 2), and propose a general theory characterizing how well a policy generalizes to unseen environments (Section 3.1). 2. We propose a theoretically-motivated algorithm based on optimizing a surrogate objective, perfect alignment, with aligned data (Section 3.2). We then describe a practical implementation based on Skew-Fit [13] to achieve the objective (Section 3.3). 3. Empirically, our experiments for a sawyer arm robot simulation with visual observations and goals demonstrates that our proposed method achieves state-of-the-art performance compared to data augmentation and bisimulation baselines at generalizing to unseen test environments in goal-conditioned tasks (Section 4). The agent takes in the goal g and observation x t , which is produced by the domain invariant state s t and environmental state b t , and acts with action a t . Note that b t may have temporal dependence indicated by the dashed edge.
Problem Formulation
In this section, we formulate the domain invariant learning problem as solving Goal-conditioned Block MDPs (GBMDPs). This extends previous work on learning invariances [6,14] to the goal-conditioned setting [15,16,8].
We consider a family of Goal-conditioned Block MDP environments M E = {(S, A, X e , T e , G, γ)|e ∈ E} where e stands for the environment index. Each environment consists of shared state space S, shared action space A, observation space X e , transition dynamic T e , shared goal space G ⊂ S and the discount factor γ.
Moreover, we assume that M E follows the generalized Block structure [6]. The observation x e ∈ X e is determined by state s ∈ S and the environmental factor b e ∈ B e , i.e., x e (s, b e ) ( Figure 6(c)). For brevity, we use x e t (s) to denote the observation for domain e at state s and step t. We may also omit t as x e (s) if we do not emphasize on the step t or the exact environmental factor b e t . The transition function is thus consists of state transition p(s t+1 |s t , a t ) (also p(s 0 )), environmental factor transition q e (b e t+1 |b e t ). In our work, we assume the state transition is nearly deterministic, i.e., ∀s, a, entropy H(p(s t+1 |s t , a t )), H(p(s 0 )) 1, which is quite common in most RL benchmarks and applications [2,17,18]. Most importantly, X E = ∪ e∈E X e satisfies the disjoint property [14], i.e., each observation x ∈ X E uniquely determines its underlying state s. Thus, the observation space X E can be partitioned into disjoint blocks X (s), s ∈ S. This assumption prevents the partial observation problem.
The objective function in GBMDP is to learn a goal-conditioned policy π(a|x e , g) that maximizes the discounted state density function J(π) [9] across all domains e ∈ E. In our theoretical analysis, we do not assume the exact form of g to the policy. One can regard π(·|x e , g) as a group of RL policies indexed by the goal state g.
J(π) = E e∼E,g∼G,π (1 − γ) ∞ t=0 γ t p e π (s t = g|g) = E e∼E [J e (π)](1)
p e π (s t = g|g) denotes the probability of achieving goal g under policy π(·|x e , g) at step t in domain e. Besides, e ∼ E and g ∼ G refers to uniform samples from each set. As p e π is defined over state space, it may differs among environments since policy π takes x e as input. Fortunately, in a GBMDP, there exist optimal policies π G (·|x e , g) which are invariant over all environments, i.e., π G (a|x e (s), g) = π G (a|x e (s), g), ∀a ∈ A, s ∈ S, e, e ∈ E.
During training, the agent has access to training environments {e i } N i=1 = E train ⊂ E with their environment indices. However, we do not assume that E train is i.i.d sampled from E. Thus, we want the goal-conditioned RL agent to acquire the ability to neglect the spurious and unrelated environmental factor b e and capture the underlying invariant state information. This setup is adopted in many recent works such as in [6] and in domain generalization [19,20] for supervised learning.
Method
In this section, we propose a novel learning algorithm to solve GBMDPs. First, we propose a general theory to characterize how well a policy π generalizes to unseen test environments after training on E train . Then, we introduce perfect alignment as a surrogate objective for learning. This objective is supported by the generalization theory. Finally, we propose a practical method to acquire perfect alignment.
Domain Generalization Theory for GBMDP
In a seminal work, Ben-David et al. [21] shows it is possible to bound the error of a classifier trained on a source domain on a target domain with a different data distribution. Follow-up work extends the theory to the domain generalization setting [22,23]. In GBMDP, we can also derive similar theory to characterize the generalization from training environments E train to target test environment t. The theory relies on the Total Variation Distance D TV [24] of two policies π 1 , π 2 with input (x e , g), which is defined as follows.
D TV (π 1 (·|x e , g) π 2 (·|x e , g)) = sup A ∈σ(A) |π 1 (A |x e , g) − π 2 (A |x e , g)|
In the following statements, we denote ρ(x, g) as some joint distributions of goals and observations that g ∼ G and x is determined by ρ(x|g). Additionally, we use ρ e π (x e |g) to denote the discounted occupancy measure of x e in environment e under policy π(·|x e , g) and refer ρ e π (x e ) as the marginal distribution. Furthermore, we denote ρ(x,g) (π 1 π 2 ) as the average D TV between π 1 and π 2 , i.e., ρ(x,g) (π 1 π 2 ) = E ρ(x,g) [D TV (π 1 (·|x, g) π 2 (·|x, g))]. This quantity is crucial in our theory as it can characterize the performance gap between two policies (see Appendix C).
Then, similar to the famous H∆H-divergence [21,22] in domain adaptation theory, we define Π∆Π-divergence of two joint distributions ρ(x, g) and ρ(x, g) in terms of the policy class Π:
d Π∆Π (ρ(x, g), ρ(x, g) ) = sup π,π ∈Π | ρ(x,g) (π π ) − ρ(x,g) (π π )|
On one hand, d Π∆Π is a distance metric which reflects the distance between two distributions w.r.t function class Π. On the other hand, if we fix these two distributions, it also reveals the quality of the function class Π, i.e., smaller d Π∆Π means more invariance to the distribution change. Finally, we state the following Proposition in which π G is some optimal and invariant policy. Proposition 1 (Informal). For any π ∈ Π, we consider the occupancy measure {ρ ei π (x ei , g)} N i=1 for training environments and ρ t π G (x t , g) for the target environment. For simplicity, we use ei as the abbreviation of ρ e i π (x e i ,g) , t as ρ t π G (x t ,g) and δ = max ei,e i ∈Etrain d Π∆Π (ρ ei π (x ei , g), ρ e i π (x e i , g)). Let
λ = 1 N N i=1 ei (π * π G ) + t (π * π G ), π * = arg min π ∈Π N i=1 ei (π π G )
Then, we have
J t (π G ) − J t (π) ≤ 1 N N i=1 ei (π π G ) + λ + δ + min ρ(x,g)∈B d Π∆Π (ρ(x, g), ρ t π G (x t , g))(2)
where B is a characteristic set of joint distributions determined by E train and policy class Π.
The formal statement and the proof are shown in Appendix C.2. Generally speaking, the first term of the right hand side in Eq. (2) quantifies the performance of π in the N training environments. λ quantifies the optimality of the policy class Π over all environments. δ reflects how the policy class Π can reflect the difference among {ρ ei π (x ei , g), e i ∈ E train }, which should be small if the policy class is invariant. The last term characterizes the distance between training environment and target environment and will be small if the training environments are diversely distributed. Different from their perspectives, in GBMDPs, we propose a simple but effective criteria to minimize the bound. From now on, we only consider the policy class Π = Π Φ = {w(Φ(x), g), ∀w}. Usually, Φ will be referred as an encoder which maps x ∈ X E to some latent representation z = Φ(x). We will also use the notation z(s) = Φ(x(s)) if we do not emphasize on the specific environment. Definition 1 (Perfect Alignment). An encoder is called a perfect alignment encoder Φ w.r.t environment set E if ∀e, e ∈ E and ∀s, s ∈ S, Φ(x e (s)) = Φ(x e (s )) if and only if s = s .
As illustrated in Figure 5, an encoder is in perfect alignment if it maps two observations of the same underlying state s to the same latent encoding z(s) while also preventing meaningless embedding, i.e., mapping observations of different states to the same z. We believe perfect alignment plays an important role in domain generalization for goal-conditioned RL agents. Specifically, it can minimize the bound of Eq. (2) as follows. Proposition 2 (Informal). If the encoder Φ is a perfect alignment over E train , then
J t (π G ) − J t (π) ≤ 1 N N i=1 ei (π π G ) (E) + t (π * π G ) + d ΠΦ∆ΠΦ (ρ(x, g), ρ t π G (x t , g)) (t)(3)
whereρ(x, g) and π * are defined in Proposition 1 (also Appendix C).
In Appendix C.3, we formally prove Proposition 2 when Φ is a (η, ψ)-perfect alignment, i.e., Φ is only near perfect alignment. The proof shows that the generalization error bound is minimized on the R.H.S of Eq. (3) when Φ asymptotically becomes an exact perfect alignment encoder. Therefore, in our following method, we aim to learn a perfect alignment encoder via aligned sampling (Section 3.2).
For the remaining terms in the R.H.S of Eq. (3), we find it hard to quantify them task agnostically, as similar difficulties also exist in the domain generalization theory of supervised learning [22]. Fortunately, we can derive upper bounds for the remaining terms under certain assumptions and we observe that these upper bounds are significantly reduced via our method in the experiments (Section 4). The (E) term represents how well the learnt policy π approximates the optimal invariant policy on the training environments and is reduced to almost zero via RL (Table 1). For the (t) term, we show that an upper bound of (t) is proportion to the invariant quality of Φ on the target environment. Moreover, we find that learning a perfect alignment encoder over E train empirically improves the invariant quality over other unseen environments (t) ( Figure 4). Thus, this (t) term upperbound is reduced by learning perfect alignment. Please refer to Appendix C.4 for more details.
Based on the theory we derived in this subsection, we adopt perfect alignment as the heuristic to address GBMDPs in our work. In the following subsections, we propose a practical method to acquire a perfect alignment encoder over the training environments.
Learning Domain Invariant via Aligned Sampling
First, we discuss about the if condition on perfect alignment encoder Φ, i.e., ∀s, Φ(x e (s)) = Φ(x e (s)). The proposed method is based on aligned sampling. In contrast, most RL algorithms use observation-dependent sampling from the environment, e.g., -greedy or Gaussian distribution policies [3,30,13,2]. However, with observation-dependent sampling, occupancy measures ρ e π (s), ∀e ∈ E train will be different. Thus, simply aligning the latent representation of these observations will fail to produce a perfect alignment encoder Φ.
Thus, we propose a novel strategy for data collection called aligned sampling. First, we randomly select a trajectory (e.g., from replay buffer etc.), denoted as Second, we take the same action sequence a 0:T in another domain e to get another trajectory {x e 0 , a 0 , x e 1 , a 1 , . . . , x e T } (so as {s e t (a 0:t )} T t=0 ). We refer to the data collected by aligned sampling from all training environments as aligned data. These aligned observations {x ei t (a 0:t )}, ∀e i ∈ E train are stored in an aligned buffer R align corresponding to the aligned action sequence a 0:t .
Under the definition of GBMDP, we have ∀t ∈ [0 : T ], s ∈ S, ρ(s e t (a 0:t )) = ρ(s e t (a 0:t )), i.e., the same state distribution. Therefore, we can use MMD loss [31] to match distribution of Φ(x e (s)) for the aligned data. More specifically, in each iteration, we sample a mini-batch of B aligned observations of every training environment e i ∈ E train from R align , i.e., B align = {x ei (s ei t (a b 0:t )), ∀e i ∈ E train } B b=1 . Then we use the following loss as a computationally efficient approximation of the MMD metric [32,27].
L MMD (Φ) = E e,e ∼Etrain,Balign∼Ralign [ 1 B B b=1 ψ(Φ(x e (s e t (a b 0:t ))) − 1 B B b=1 ψ(Φ(x e (s e t (a b 0:t )))) 2 2 ]
where ψ is a random expansion function.
In Figure 2(a), we illustrate the intuition of the above approach. When the transition is nearly deterministic, the entropy for ρ(s e t (a 0:t )) is much smaller, i.e., H(ρ(s e t (a 0:t ))) H(ρ e π (s t )). Thus, ρ(s e t (a 0:t )) can be regarded as small patches in S. We use the MMD loss L MMD to match the latent representation {Φ(x e (s)), s ∼ ρ(s e t (a 0:t ))}, ∀e ∈ E train together. As a consequence, we should achieve an encoder Φ that is more aligned. We discuss the theoretical property of L MMD in detail in Appendix C.5.
However, simply minimizing L MMD may violate the only if condition for perfect alignment. For example, a trivial solution for L MMD = 0 is mapping all observations to some constant latent. To ensure that Φ(x e (s)) = Φ(x e (s )) only if s = s , we additionally use the difference loss L DIFF as follows.
L DIFF (Φ) = −E e∼Etrain,x e ,x e ∈R e Φ(x e ) − Φ(x e ) 2 2
where R e refers to the replay buffer of environment e. Clearly, minimizing L DIFF encourages dispersed latent representations over all states s ∈ S.
We refer to the combination α MMD L MMD + α DIFF L DIFF as our perfect alignment loss L PA . Note that L PA resembles contrastive learning [33,34]. Namely, observations of aligned data from R align are positive pairs while observations sampled randomly from a big replay buffer are negative pairs. We match the latent embedding of positive pairs via the MMD loss while separating negative pairs via the difference loss. As discussed in Section 3.1, we believe this latent representation will improve generalization to unseen target environments.
Perfect Alignment for Skew-Fit
In Section 4, we will train goal-conditioned RL agents with perfect alignment encoder using the Skew-Fit algorithm [13]. Skew-Fit is typically designed for visual-input agents which learn a goal-conditioned policy via purely self-supervised learning.
First, Skew-Fit trains a β-VAE with observations collected online to acquire a compact and meaningful latent representation for each state, i.e., z(s) from the image observations x(s). Then, Skew-Fit optimizes a SAC [3] agent in the goal-conditioned setting over the latent embedding of the image observation and goal, π(a|z, g). The reward function is the negative of l 2 distance between the two latent representation z(s) and z(g), i.e., r(s, g) = − z(s) − z(g) 2 . Furthermore, to improve sample efficiency, Skew-Fit proposes skewed sampling for goal-conditioned exploration.
In our algorithm, Perfect Alignment for Skew-Fit (PA-SF), the encoder Φ is optimized via both β-VAE losses as [13,35] and L PA loss to ensure meaningful and perfectly aligned latent representation.
L(Φ, D) = L RECON (x e ,x e ) + βD KL (q Φ (z|x e ) p(z)) + α MMD L MMD + α DIFF L DIFF(4)
In addition, we use both aligned sampling and observation-dependent sampling. Aligned sampling provides aligned data but hurts sample-efficiency while observation-dependent sampling is exploration-efficient but fails to ensure alignment. In practice, we find that collecting a small portion (15% of all data collected) of aligned data in R align is enough for perfect alignment via L PA .
Additionally, inspired by [27], we also change the β-VAE structure to what is shown in Figure 2(b), since in GBMDP data are collected from N training environments and thus, the identity Gaussian distribution is no longer a proper fit for prior. The encoder Φ maps x e (s) to some latent representation z(s) while the decoder D takes both z(s) and the environment index e as input to reconstruct x e (s). Note that by using both L PA and L RECON , we require static environmental factors in E train (unnecessary for testing environments) for a stable optimization. In future work, we will address the limit from β-VAE by training two latent representations simultaneously to stabilize the optimization for generality.
Experiments
In this section, we conduct experiments to evaluate our PA-SF algorithms. The experiments are based on multiworld [18]. Our empirical analysis tries to answer the following questions: (1) How well does PA-SF perform in solving GBMDP problems? (2) How does each component proposed in Section 3 contribute to the performance?
Comparative Evaluation
In this subsection, we aim to answer the question (1) by comparing our proposed PA-SF method with vanilla Skew-Fit and several other baselines that attempt to acquire invariant policies for RL agents.
Baselines Current methods for obtaining robust policies can be characterized into two categories:
(1) data augmentation and (2) model bisimulation. . These methods utilize bisimulation metrics to learn a minimal but sufficient representation which will neglect irrelevant features of Block MDPs. We include MISA [6] and DBC [7] in our comparison as they are the most successful implementations for high-dimensional tasks. Moreover, in the goal-conditioned setting, we use an oracle state-goal distance − s − g 2 as rewards for these two algorithms in GBMDP. In contrast, our PA-SF method does not have such information.
Environments We evaluate PA-SF and all baselines on a set of GBMDP tasks based on multiworld benchmark [18], which is widely used to evaluate the performance of visual input goal-conditioned algorithms. We use the following four basic tasks [35, 13]: Reach, Door, Pickup and Push. In GBMDP, we create different environments with various backgrounds, desk surfaces, and object appearances. During testing, we also create environments with unseen video backgrounds to mimic environmental factor transitions q e (b e t+1 |b e t ). This makes policy generalization more challenging. Please refer to Appendix E for a full description of our experiment setup and implementation details of the baselines and our algorithm.
Results
In Table 1, we show the final average performance of each algorithm on unseen test environments E test . The corresponding learning curves are shown in Figure 3. This metric shows the generalizability of each RL agent. All these agents are allowed to collect data from E train (N = 3) with static environmental factors. Our PA-SF achieves SOTA performance on all tasks. On testing environments, we achieve a relative reduction around 40% to 65% of the corresponding metrics over vanilla Skew-Fit w.r.t the optimal metric possible (Oracle Skew-Fit). Oracle Skew-Fit refers to the performance of a Skew-Fit algorithm trained directly on the single environment (and not simultaneously on all E train ).
Other invariant policy learning methods perform sluggishly on all tasks. For DBC and MISA, we hypothesize that they struggle for goal-conditioned problems since the model bisimulation metric is defined for a single MDP. In GBMDPs, this means acquiring a set of encoders Φ g that achieves model bisimulation for every possible g and is thus inefficient for learning. By design, our method is not susceptible to this issue as the perfect alignment is a universal invariant representation for all goals. Data augmentation via RAD provides marginal improvement over the vanilla Skew-Fit. Nevertheless, we believe developing adequate data augmentation techniques for GBMDPs is an important research problem and is orthogonal with our method.
Additionally, we also show the performance of PA-SF on the training environments in Table 1. PA-SF is still comparable and as sample-efficient as Skew-Fit in the training environments. This supports the claim that the (E) term in the R.H.S of Eq. (3) is reduced to almost zero via RL training in practice.
Design Evaluation
In this subsection, we conduct comprehensive analysis on the design of PA-SF to interpret how well it carries out the theoretical framework discussed in Section 3.1 and Section 3.2.
To begin with, we show the learning curves in Figure Additionally, we also quantify the quality of the latent representation Φ(x e (s)) in Figure 4 via the metric Latent Error Rate (LER). LER is defined as the average over environment set E ∈ {E train , E test } as follows:
Err(Φ) = E e∼E,s∼S Φ(x e (s)) − Φ(x e0 (s)) 2 Φ(x e (s)) 2
In general, the smaller Err(Φ) is, the closer the encoder Φ is to perfect alignment over the environments E. We first focus on the discussion about training performance. 2. The only if condition, i.e., Φ(x e (s)) = Φ(x e (s )) only if s = s , is also achieved empirically by visualizing the reconstruction of the VAE ( Figure 9 in Appendix D) and we believe this is satisfied by both the difference loss and the reconstruction loss. Under the only if condition, the SAC [3] trained on the latent space achieves the optimal performance. In contrast, PA-SF (w/o AS) fails to learn well on the training environments as its latent representation is mixed over different states.
Second, we focus on the generalization performance on target domains t, i.e., term (t) in Eq. (3). We observe the following: . This indicates that the increased test performance indeed comes from the improved representation quality of the encoder Φ, i.e., more aligned. This supports our claim at the end of Section 3.1 and the upper bound analysis on the (t) term in Appendix C.4, that the increased invariant property of Φ produces better domain generalization performance. 2. In test environment ablations, the LER is reduced significantly on methods with L MMD . This supports our claim that a perfect alignment encoder on training environments also improves the encoder's invariant property on unseen environments. In addition, by encouraging dispersed latent representation, the difference loss L DIFF also plays a role in reducing LER during testing. This supports the necessity of both losses for generalization. We observe the similar results in other tasks as well (Appendix D). Here, we also visualize the latent space by t-SNE plot to illustrate the perfect alignment on task Push. Dots in training environments are matched perfectly and the corresponding test environment dot is approximately near as expected. Learning Invariants in RL: Robustness to domain shifts is crucial for real-world applications of RL. [6,7,50] implement the model-bisimulation framework [12] to acquire a minimal but sufficient representation for solving the MDP problem. However, model-bisimulation for high-dimension problems typically requires domain-invariant and dense rewards. These assumptions do not hold in GBMDPs. Contrastive Metric Embeddings (CME) [51] instead uses π * -bisimulation metric but it also requires extra information of the optimal policy. Another line of work tries to address these issues via self-supervised learning.
[36] tests multiple data augmentation methods including RAD [37] and DrQ [52] to boost the robustness of the representation as well as the policy. Our work can also apply data augmentation in practice. However, we find that RAD is not very helpful in the Skew-Fit framework. Additionally, [53, 54] use self-supervised correction during real-world adaptation like sim2real transfer but these methods are incompatible for domain generalization.
Conclusion
In this paper, we study the problem of learning invariant policies in Goal-conditioned RL agents. The problem is formulated as a GBMDP, which is an extension of Goal-conditioned MDPs and Block MDPs where we want the agent's policy to generalize to unseen test environments after training on several training environments.
As supported by the generalization bound for GBMDP, we propose a simple but effective heuristic, i.e., perfect alignment which we can minimize the bound asymptotically and benefit the generalization. To learn a perfect alignment encoder, we propose a practical method based on aligned sampling. The method resembles contrastive learning: matching latent representation of aligned data via MMD loss and dispersing the whole latent representations via the DIFF loss. Finally, we propose a practical implementation Perfect Alignment for Skew-Fit (PA-SF) by adding the perfect alignment loss to Skew-Fit and changing the VAE structure to handle GBMDPs.
The empirical evaluation shows that our method is the SOTA algorithm and achieves a remarkable increase in test environments' performance over other methods. We also compare our algorithm with several ablations and analyze the representation quantitatively. The results support our claims in the theoretical analysis that perfect alignment criteria is effective and that we can effectively optimize the criteria with our proposed method. We believe the perfect alignment criteria will enable applications in diverse problem settings and offers interesting directions for future work, such as extensions to other goal-conditioned learning frameworks [9,10].
[10] Keiran Paster, Sheila A McIlraith, and Jimmy Ba. Planning from pixels using inverse dynamics models. arXiv preprint arXiv:2012.02419, 2020.
[
x e t = x e (s t , b e t )
Observation determined by state s and environmental factor b e for environment e at timestep t p(s t+1 |s t , a t ) State transition shared among environments q e (b e t+1 |b e t ) Environmental factor transition for environment e X E = ∪ e∈E X e Joint set of observation spaces π(a|x e , g)
Goal-conditioned policy shared among environments J(π)
Objective function for policy π J e (π)
Objective function for policy π in environment e p e π (s t = g|g)
Probability of achieving goal g under policy π(·|x e , g) at timestep t in environment e π G (·|x e , g) Optimal policies which are invariant over all environments
{e i } N i=1 = E train
Training environments ρ(x, g) = ρ(g)ρ(x|g) Joint distributions of goals and observations ρ e π (x e |g)
Occupancy measure of x e in environment e under policy π(·|x e , g) ρ e π (x e ) Marginal distribution of ρ e π (x e , g) over goals ρ(x,g) (π 1 π 2 ) Averaged Total Variation between policy π 1 and π 2 Π Policy class (i.e. space for all possible policies) d Π∆Π (ρ(x, g), ρ(x, g) ) Π∆Π-divergence of two joint distributions ρ(x, g) and ρ(x, g) in terms of the policy class Π ei (π 1 π 2 ) = ρ e i π (x e i ,g) (π 1 π 2 )
Total Variation between policy π 1 and π 2 averaged over joint occupancy measure under policy π in training environment e i t (π 1 π 2 ) = ρ t π G (x t ,g) (π 1 π 2 )
Total Variation between policy π 1 and π 2 averaged over joint occupancy measure under policy π G in testing environment t π *
The closest policy for training environments in policy class Π w.r.t.optimal invariant policy π G measured by averaged Total Variation Continued on next page
B align = {x ei (s ei t (a b 0:t )), ∀e i ∈ E train } B b=1
Batch of aligned observations from all the training environments L MMD (Φ)
MMD loss for encoder Φ ψ(z)
Random expansion function for latent representation z L DIFF (Φ)
Difference loss for encoder Φ R e Replay buffer for transitions from environment e L PA Perfect alignement loss
L RECON Reconstruction loss β KL divergence coefficient α MMD MMD loss coefficient α DIFF Difference loss coefficient Err(Φ)
Latent error rate for encoder Φ
B Algorithm
The main difference between the PA-SF and Skew-Fit are (i) separate replay buffer for each training environments R = {R e , e ∈ E train }, (ii) an additional aligned buffer for the aligned data R align = {R e align , e ∈ E train } and a corresponding aligned sampling procedure, (iii) VAE training uses Eq. (4) with mini-batches from both replay buffer and aligned buffer. The overall algorithm is described in Algorithm 1 and implementation details are listed in Appendix E.
Algorithm 1 Perfect Alignment for Skew-Fit (PA-SF).
Require: β-VAE decoder, encoder q φ , goal-conditioned policy π θ , goal-conditioned value function Q w , skew parameter α, VAE training schedule, training environments E train , replay buffer R = {R e , e ∈ E train }, aligned buffer R align = {R e align , e ∈ E train }, coefficients in Eq. (4). Sample goal observation x e (g) ∼ p e,m skewed and encode as z e (g) = q φ (x e (g)).
4:
Sample initial observation x e 0 from the environment e.
5:
for t = 0, . . . , H − 1 steps do 6: Get action a t ∼ π θ (q φ (x e t ), g).
7:
Get next state x e t+1 ∼ p(· | x e t , a t ).
8:
Store (x e t , a t , x e t+1 , x e (g)) into replay buffer R e . Sample transition (x e t , a t , x e t +1 , z e (g)) ∼ R e for all e ∈ E train .
19:
Encode z e t = q φ (x e t ), z e t +1 = q φ (x e t +1 ).
20:
(Probability 0.5) replace z e (g) with q φ (x (g)) where x (g) ∼ p e,m skewed .
21:
Compute new reward r = −||z e t +1 − z e (g)|| 2 .
22:
Update π θ and Q w via SAC on (z e t , a t , z e t +1 , z e (g), r).
C Proofs and Discussions
In this section, we provide detailed proofs and statements omitted in the main text. In addition, we also discuss the assumptions we make in the analysis in detail.
C.1 Illustration of Different MDP Problems
Here, we illustrate different graphical models of related MDPs including Block MDPs (Figure 6(a)), Goal-conditioned MDPs (Figure 6(b)), and ours Goal-conditioned Block MDPs (GBMDP) ( Figure 6(c)). We use the indicator funtion in the Goal-conditioned and GBMDP settings to emphasize that the reward is sparse. In practice, the goal g may only be indirectly observed as x e (g), such as future state in pixel space for a particular domain.
C.2 Proof of Proposition 1
Recall that Proposition 1 bounds the generalization performance by 4 terms: (1) average training environments' performance, (2) optimality of the policy class, (3) d Π∆Π over all training environments, and (4) the discrepancy measure between training environments and the target environment.
We begin our analysis by proving the following two Lemmas. For simplicity, we denote p ∆,e π (s|g) as the discounted state density as follows.
p ∆,e π (s|g) = (1 − γ) ∞ t=0 γ t p e π (s t = s|g)
where p e π (s t = s|g) is the probability of state s under goal-conditioned policy π(·|x e , g) at step t in domain e (marginalized over the initial state s 0 ∼ p(s 0 ), previous actions a i ∼ π(a i |x e i , g), i = 0, . . . , t − 1, and previous states s i ∼ p(s i |s i−1 , a i−1 ), i = 0, . . . , t − 1). Lemma 1. ∀e ∈ E all , let ρ e π (x e , g) denote joint distributions of g ∼ G and x e under policy π(·|x e , g), then ∀π 1 , π 2 , we have
|J e (π 1 ) − J e (π 2 )| ≤ 2γ 1 − γ E ρ e π 1
(x e ,g) [D T V (π 1 (·|x e , g) π 2 (·|x e , g))]
Proof. By the definition of J e (π), we have
|J e (π 1 ) − J e (π 2 )| = |E g∼G [p ∆,e π1 (g|g) − p ∆,e π2 (g|g)]| ≤ E g∼G [|p ∆,e π1 (g|g) − p ∆,e π2 (g|g)|]
Thus, it suffices to prove ∀g ∈ G,
|p ∆,e π1 (g|g) − p ∆,e π2 (g|g)| ≤ 2γ 1 − γ E ρ e π 1
(x e |g) [D T V (π 1 (·|x e , g) π 2 (·|x e , g))]
First, we consider |p e π1 (s T = s|g) − p e π2 (s T = s|g)| for some fixed step T . Denote {π 1 < t, π 2 ≥ t} as another policy which imitates policy π 1 for first t steps and then imitates π 2 for the rest. By the telescoping operation, we have ∀s ∈ S, g ∈ G
|p e π1 (s T = s|g) − p e π2 (s T = s|g)| ≤ T −1 t=0 |p e π1<t,π2≥t (s T = s|g) − p e π1<t+1,π2≥t+1 (s T = s|g)| = T −1 t=0 P e t (s|π 1 , π 2 , g, T )
where we use P e t (s|π 1 , π 2 , g, T ) to denote each term for brevity.
P e t (s|π 1 , π 2 , g, T ) = | st,a p e π1 (s t |g)π 2 (a|x e (s t ), g)p e π2 (s T = s|s t , a, g)ds t da− − st,a p e π1 (s t |g)π 1 (a|x e t (s t ), g)p e π2 (s T = s|s t , a, g)ds t da| ≤ st,a p e π1 (s t |g)|π 1 (a|x e (s t ), g) − π 2 (a|x e (s t ), g)|p e π2 (s T = s|s t , a, g)ds t da ≤ st p e π1 (s t |g) a |π 1 (a|x e (s t ), g) − π 2 (a|x e (s t ), g)|da ds t ≤ 2 st p e π1 (s t |g)D T V (π 1 (·|x e (s t ), g) π 2 (·|x e (s t ), g))ds t
Here, p e π2 (s T = s|s t , a, g) is the probability of achieving state s at step T under policy π 2 (·|x e , g) when it takes action a at s t . Noticing that, the upper bound for P e t (s|π 1 , π 2 , g, T ) is not dependent on T . Thus, we have ∀s ∈ S, g ∈ G,
|p ∆,e π1 (s|g) − p ∆,e π2 (s|g)| ≤(1 − γ) ∞ T =0 γ T T −1
t=0 P e t (s|π 1 , π 2 , g, T )
=(1 − γ) ∞ t=0 γ t+1 1 − γ P e t (s|π 1 , π 2 , g, t + 1) ≤2 ∞ t=0 γ t+1 st p e π1 (s t |g)D T V (π 1 (·|x e (s t ), g) π 2 (·|x e (s t ), g))ds t =2γ st ∞ t=0 γ t p e π1 (s t |g)D T V (π 1 (·|x e (s t ), g) π 2 (·|x e (s t ), g))ds t = 2γ 1 − γ E ρ e π 1 (x e |g) [D T V (π 1 (·|x e , g) π 2 (·|x e , g)]
The Lemma holds by averaging over g ∼ G.
Lemma 1 bounds the objective function between two policies π 1 and π 2 with the Total Variation distance. Recall that we use ρ(x,g) (π 1 π 2 ) to denote the average D TV between the two policies under the joint distribution. We refer to π G as some optimal and invariant policy. Then, we have the following Lemma. Lemma 2. For any policy class Π and two joint distributions ρ s (x, g) and ρ t (x, g), suppose π * s,t = arg min π ∈Π ρ s (x,g) (π π G ) + ρ t (x,g) (π π G ) then we have for any π ∈ Π ρ t (x,g) (π π G ) ≤ ρ s (x,g) (π π G ) + sup
π,π ∈Π ρ s (x,g) (π π ) − ρ t (x,g) (π π ) dΠ∆Π(ρ s (x,g),ρ t (x,g)) +λ s,t where λ s,t = ρ s (x,g) (π * s,t π G ) + ρ t (x,g) (π * s,t π G ).
Proof. Noticing that D TV is a distance metric that satisfies the triangular inequality and is symmetric. Thus, we have ∀π 1 , π 2 , π 3 and any joint distribution ρ(x, g).
ρ(x,g) (π 1 π 2 ) = E ρ(x,g) [D TV (π 1 (·|x, g) π 2 (·|x, g))]
≤ E ρ(x,g) [D TV (π 1 (·|x, g) π 3 (·|x, g)) + D TV (π 3 (·|x, g) π 2 (·|x, g))]
= ρ(x,g) (π 1 π 3 ) + ρ(x,g) (π 3 π 2 )
Based on this property, we have
ρ t (x,g) (π π G ) ≤ ρ t (x,g) (π * s,t π G ) + ρ t (x,g) (π π * s,t ) ≤ ρ t (x,g) (π * s,t π G ) + ρ s (x,g) (π π * s,t ) + ρ t (x,g) (π π * s,t ) − ρ s (x,g) (π π * s,t ) ≤ ρ s (x,g) (π π G ) + ρ s (x,g) (π * s,t π G ) + ρ t (x,g) (π * s,t π G ) + ρ t (x,g) (π π * s,t ) − ρ s (x,g) (π π * s,t ) ≤ ρ s (x,g) (π π G ) + d Π∆Π (ρ s (x, g), ρ t (x, g)) + λ s,t
Lemma 2 resembles the seminal bound in domain adaptation theory [21]. Then, we extend the generalization bound to the GBMDP setting. We consider the joint distributions
ρ E = {ρ ei (x, g)} N i=1
and define the characteristic set as follows. 2 Definition 2. The characteristic set B(δ, E|Π) is a set of joint distributions ρ(x, g) which ∀e i ∈ E with ρ ei (x, g),
d Π∆Π (ρ(x, g), ρ ei (x, g)) ≤ δ
In other words, the characteristic set is a set of joint distributions that is close to the training environments' distributions in terms of the d Π∆Π divergence. Notice that if we define δ E = max e,e ∈E d Π∆Π (ρ e (x, g), ρ e (x, g)), then we have that the convex hull Λ({ρ ei (x, g)} N i=1 ) ⊂ B(δ E , E|Π). Namely, the characteristic set contains all the distributions of convex combinations of training environments' distributions [22]. Proposition 3. For any policy class Π, a set of source joint distributions ρ E = {ρ ei (x, g)} N i=1 and the target distribution ρ t (x, g), suppose for any unit sum weights
{α i } N i=1 , i.e., 0 ≤ α i ≤ 1, i α i = 1. λ α = N i=1 α i ei (π * α π G ) + t (π * α π G ), π * α = arg min π ∈Π N i=1 α i ei (π π G ) + t (π π G )
where ei and t are short of ρ e i and ρ t respectively. Let
ρ(x, g) = arg min ρ∈B(δ E ,E|Π) d Π∆Π (ρ(x, g), ρ t (x, g)) Then, ∀π ∈ Π t (π π G ) ≤ min α N i=1 α i ei (π π G ) + λ α + max e,e ∈E d Π∆Π (ρ e (x, g), ρ e (x, g)) δ E + d Π∆Π (ρ(x, g), ρ t (x, g))
2 ei is only used as an index here.
Proof. By Lemma 2, we have ∀e i ∈ E t (π π G ) ≤ ei (π π G ) + d Π∆Π (ρ ei (x, g), ρ t (x, g)) + λ ei,t Then, we have for any unit sum weights α
t (π π G ) ≤ N i=1 α i ei (π π G ) + N i=1 α i d Π∆Π (ρ ei (x, g), ρ t (x, g)) + N i=1 α i λ ei,t Noticing that N i=1 α i λ ei,t = N i=1 α i min π ∈Π ei (π π G ) + t (π π G ) ≤ min π ∈Π N i=1 α i ei (π π G ) + t (π π G ) = λ α Thus, we have t (π π G ) ≤ N i=1 α i ei (π π G ) + N i=1 α i d Π∆Π (ρ ei (x, g), ρ t (x, g)) + λ α
Since d Π∆Π divergence also follows the triangular inequality [21], we have
t (π π G ) ≤ N i=1 α i ei (π π G ) + N i=1 α i d Π∆Π (ρ ei (x, g),ρ(x, g)) + d Π∆Π (ρ(x, g), ρ t (x, g)) + λ α ≤ N i=1 α i ei (π π G ) + λ α + δ E + d Π∆Π (ρ(x, g), ρ t (x, g))
The proposition holds by taking the minimum over α.
Finally, we are able to provide the formal statements and proofs for Proposition 1 as follows. Proposition 1 (Formal). For any π ∈ Π, we consider the occupancy measure ρ Etrain = {ρ ei π (x ei , g)} N i=1 for training environments and ρ t π G (x t , g) for the target environment. For simplicity, we use ei and t as the abbreviations. Considering
λ = 1 N N i=1 ei (π * π G ) + t (π * π G ), π * = arg min π ∈Π N i=1 ei (π π G )
and δ = max e,e ∈Etrain d Π∆Π (ρ e π (x e , g), ρ e π (x e , g)), the characteristic set B(δ, E train |Π). Definẽ ρ(x, g) = arg min
ρ∈B(δ,Etrain|Π) d Π∆Π (ρ(x, g), ρ t π G (x t , g))
Then, we have
J t (π G ) − J t (π) ≤ 2γ 1 − γ 1 N N i=1 ei (π π G ) + λ + δ + d Π∆Π (ρ(x, g), ρ t π G (x t , g))
Proof. By Lemma 1 and Proposition 3, we have
J t (π G ) − J t (π) ≤ 2γ 1 − γ t (π π G ) ≤ 2γ 1 − γ min α N i=1 α i ei (π π G ) + λ α + δ + d Π∆Π (ρ(x, g), ρ t π G (x, g)) ≤ 2γ 1 − γ 1 N N i=1 ei (π π G ) + λ α= 1 N + δ + d Π∆Π (ρ(x, g), ρ t π G (x, g))
Noticing that λ and λ α= 1 N have different definitions. But, we have
λ α= 1 N = min π ∈Π 1 N N i=1 ei (π π G ) + t (π π G ) ≤ 1 N N i=1 ei (π * π G ) + t (π * π G ) = λ where π * = arg min π ∈Π N i=1
ei (π π G ). Thus, we have
J t (π G ) − J t (π) ≤ 2γ 1 − γ 1 N N i=1 ei (π π G ) + λ + δ + d Π∆Π (ρ(x, g), ρ t π G (x t , g))
This completes the proof.
Remark 1. The informal version Proposition 1 omits unessential constant part and the definition of the characteristic set.
C.3 Proof of Proposition 2
Here, we provide the formal proof and statement of Proposition 2. To begin with, we prove the following Lemma. Lemma 3. For two goal-conditioned policies π, π of the Goal-conditioned MDP S, G, A, p, γ , suppose that max s,g D TV (π(·|s, g) π (·|s, g)) ≤ , then we have D TV (ρ π (s, g) ρ π (s, g)) ≤ γ 1−γ .
Proof. The proof follows the perturbation theory in Appendix B of [55]. By the definition of total variation distance, it suffices to prove that ∀g, D TV (ρ π (s|g) ρ π (s|g)) ≤ γ 1−γ , where ρ π (s|g) denotes the discounted occupancy measure of s under policy π(·|s, g). Consequently, in the following notations, we may omit specifying g if unambiguous.
First, we refer P π as the state transition matrix under policy π(·|s, g), i.e., (P π ) xy = a p(s = x|s = y, a)π(a|s = y, g)da and subsequently, G π = I + γP π + γ 2 P 2 π + · · · = (I − γP π ) −1 . Then, the transition discrepancy matrix is defined as ∆ = P π − P π . Observing that
G −1 π − G −1 π = γ(P π − P π ) = γ∆ ⇒ G π − G π = γG π ∆G π
Thus, for any initial state distribution ρ 0 , we have
D TV (ρ π (s|g) ρ π (s|g)) = 1 2 s |ρ π (s|g) − ρ π (s|g)| = 1 − γ 2 (G π − G π )ρ 0 1 = (1 − γ)γ 2 G π ∆G π ρ 0 1 ≤ (1 − γ)γ 2 G π 1 ∆ 1 G π 1 ρ 0 1
Noticing that P π , P π are matrices whose columns have sum 1. Thus, G π 1 = G π 1 = 1 1−γ . Furthermore,
∆ 1 = max y
x a p(s = x|s = y, a)(π (a|s = y, g) − π(a|s = y, g))da dx = max y a x p(s = x|s = y, a) |π (a|s = y, g) − π(a|s = y, g)| dxda = max y 2D TV (π(·|s = y, g) π (·|s = y, g)) ≤ 2
In all, we have
D TV (ρ π (s|g) ρ π (s|g)) ≤ γ 1 − γ
To state Proposition 2 formally, we define the L-lipschitz policy class Π E Φ,L , whose Φ : X E → Z maps the input x e s to latent vector zs. 3
Π E Φ,L = {w(Φ(x e ), g), ∀w|∀g ∈ G, z, z ∈ Φ(X E ), D TV (w(z, g) w(z , g)) ≤ L z − z 2 }
Namely, the nonlinear function w is L-smooth over the latent space Φ(X E ) for each g. Furthermore, we extend the definition of perfect alignment encoder to the (η, ψ)-perfect alignment as follows. Definition 3 ((η, ψ)-Perfect Alignment). An encoder Φ is a (η, ψ)-perfect alignment encoder over the environments E, if it satisfies the following two properties:
1. ∀s ∈ S, ∀e, e ∈ E, Φ(x e (s)) − Φ(x e (s)) 2 ≤ η.
2. ∀s, s ∈ S, ∀e, e ∈ E, Φ(x e (s)) − Φ(x e (s )) 2 ≥ ψ s − s 2
Essentially, η quantifies the if condition of perfect alignment (Definition 1), i.e., how aligned the representation Φ(x e (s)), e ∈ Es are. Moreover, ψ quantifies the only if condition, i.e., how the representations of different states s are separated. Based on the definition of (η, ψ)-perfect alignment, we prove the formal statement of Proposition 2 as follows.
Proposition 2 (Formal). ∀π ∈ Π Etrain Φ,L and occupancy measure ρ Etrain = {ρ ei π (x ei , g)} N i=1 for training environments and ρ t π G (x t , g) for the target environment. For simplicity, we use ei and t as the abbreviations. Considering π * = arg min π ∈Π E train Φ,L N i=1 ei (π π G ) and ρ(x, g) = arg min
ρ∈B(δ,Etrain|Π E train Φ,L ) d Π E train Φ,L ∆Π E train Φ,L (ρ(x, g), ρ t π G (x t , g))
Then, if the encoder Φ is a (η, ψ)-perfect alignment over E train and π G is a u-smooth invariant policy with u = Lψ, i.e., ∀s, s , ∀g, D TV (π G (·|x e (s), g) π G (·|x e (s ), g)) ≤ u s − s 2 , we have
J t (π G ) − J t (π) ≤ 2γ 1 − γ 1 N N i=1 ei (π π G ) (E) +(3 + γ 1 − γ )ηL + 2γ 1 − γ t (π * π G ) + d Π E train Φ,L ∆Π E train Φ,L (ρ(x, g), ρ t π G (x t , g)) (t) .
Proof. It suffices to prove the following two statements with Proposition 1.
(1) max ei,e i ∈Etrain d Π E train
Φ,L ∆Π E train Φ,L (ρ ei π (x ei , g), ρ e i
π (x e i , g)) ≤ (2 + γ 1−γ )ηL.
(2) 1 N N i=1 ei (π * π G ) ≤ ηL. 3 X E = ∪e∈EX e
Proof of (1): By the definition of Π Etrain Φ,L , we have ∀s, g and e, e ∈ E train , D TV (π(·|x e (s), g) π(·|x e (s), g)) ≤ ηL. Without loss of generality, we have ∀e, e ∈ E train ,
d Π E train Φ,L ∆Π E train Φ,L
(ρ e π (x e , g), ρ e π (x e , g)) = sup π,π ∈Π E train Φ,L |E ρ e π (s,g) [D TV (π(·|x e (s), g) π (·|x e (s), g))]
− E ρ e π (s,g) [D TV (π(·|x e (s), g) π (·|x e (s), g))]| ≤ sup π,π ∈Π E train Φ,L E ρ e π (s,g) [D TV (π(·|x e (s), g) π(·|x e (s), g))]
+ sup π,π ∈Π E train Φ,L
E ρ e π (s,g) [D TV (π (·|x e (s), g) π (·|x e (s), g))]
+ sup π,π ∈Π E train Φ,L
|E ρ e π (s,g) [D TV (π(·|x e (s), g) π (·|x e (s), g))]
− E ρ e π (s,g) [D TV (π(·|x e (s), g) π (·|x e (s), g))]| ≤ 2ηL + sup A∈σ(S,G) |ρ e π (A) − ρ e π (A)| = 2ηL + D TV (ρ e π (s, g) ρ e π (s, g))
Then, by Lemma 3, we have D TV (ρ e π (s, g) ρ e π (s, g)) ≤ γηL 1−γ . Consequently, we have ∀e, e ∈ E train ,
d Π E train Φ,L ∆Π E train Φ,L (ρ e π (x e , g), ρ e π (x e , g)) ≤ (2 + γ 1 − γ )ηL
Proof of (2): First, for each z ∈ Φ(X Etrain ), we assign one s(z) such that ∃e(z) ∈ E train , s.t. Φ(x e(z) (s(z))) = z. Then, we choose thew thatw(z, g) = π G (·|s(z), g), ∀z ∈ Φ(X Etrain ).
Clearly,w is a mapping of Φ(X Etrain ) → Π Etrain Φ,∞ . Besides, ∀z 1 , z 2 ∈ Φ(X Etrain ), g ∈ G, we have
w(z 1 , g) −w(z 2 , g) 2 ≤ u s(z 1 ) − s(z 2 ) 2 ≤ u ψ ψ s(z 1 ) − s(z 2 ) 2 ≤ u ψ Φ(x e(z1) (s(z 1 ))) − Φ(x e(z2) (s(z 2 ))) 2 ≤ L z 1 − z 2 2
Thus, the policyπ =w(Φ(x e ), g) ∈ Π Etrain Φ,L . Furthermore, by the definition of (η, ψ)-perfect alignment, we have
1 N N i=1 ei (π π G ) = 1 N N i=1
E ρ e i π (s,g) [D TV (π(·|x ei (s), g) π G (·|s, g)))]
≤ 1 N N i=1 E ρ e i π (s,g) [D TV (π(·|x ei (s), g) w(Φ(x e(z) (s)), g))]
≤ ηL
This completes the proof.
C.4 Discussions on Eq. (3)
Here, we discuss the remaining terms (E), (t) in the R.H.S of Eq. (3), i.e., upper bound of J t (π G ) − J t (π). Together with the empirical analysis, we show how these terms are reduced by our perfect alignment criterion.
Discussion on (E). Theoretically speaking, for a (η, ψ)-perfect alignment encoder, we have min(E) ≤ ηL, as proved in Appendix C.3. Thus, when Φ is an ideal perfect alignment over E train , i.e., η = 0, ψ > 0 and L → ∞, the optimal invariant policy π G ∈ Π Etrain Φ,∞ . In Section 4.2, empirical results demonstrate that η is minimized to almost 0 and the reconstruction (Figure 9) demonstrates that the only if condition is also satisfied.
Moreover, in GBMDPs with finite states, the perfect alignment encoder Φ over E train maps all training environments to the same goal-conditioned MDP (Figure 6(b)) with state space {z(s), s ∼ S}.
Noticing that the mapping is bijective and share the same actions and rewards with the original problem. Thus, a RL algorithm (e.g. Q-learning [56]) on the equivalent MDP will converge to an optimal policy π G in the original MDP, i.e., invariant and maximize J e , e ∈ E train .
Empirically, in Table 1, we find that ours PA-SF achieves the near-optimal performance on all E train s, i.e., the same performance as a policy trained on a single environment. Therefore, we conclude that (E) term is reduced to almost zero.
Discussion on (t). In general, it is hard to conduct task-agnostic analysis on (t) term, as discussed in [22]. Moreover, owing to the intractable sup operators, it is almost impossible to measure the (t) term directly in the experiments. Here, we analyze the generalization error term (t) with an upper bound and we find evidence that this upper bound is reduced significantly under our perfect alignment criteria.
Here, we analyze the generalization performance when the policy converges to the optimal policy over E train , which is the case empirically. Furthermore, we assume that ∀s ∈ S, e ∈ E train , p e π (s) ≥ . Since it has been proven that the relabeled goal distribution of Skew-Fit will converge to U (S) (uniform over the bounded state space) [13], it is reasonable to assume that each state has non-zero occupancy measure under the well-trained policy π(·|s, g). We derive the following proposition with the same notation as in Proposition 2 except that Π Etrain Φ,L is replaced by Π E Φ,L . Proposition 4. Suppose that Φ is a (η t , ψ t )-perfect alignment encoder over E and π G , π ∈ Π E Φ,L . Besides, ∀s ∈ S, e ∈ E train , p e π (s) ≥ . Then, ∀t ∈ E/E train , we have
(t) ≤ 2γ 1 − γ N (E) + 4η t L + (E)
Consequently, we have
J t (π G ) − J t (π) ≤ 4γ 1 − γ (1 + γN (1 − γ) )(E) + (14 − 12γ)γ (1 − γ) 2 η t L
Proof. It is the straight forward to check that the statement and the proofs in Proposition 2 also hold under the policy class Π E Φ,L ⊂ Π Etrain Φ,L when Φ is (η t , ψ t )-perfect alignment over E. Namely, ∀π, π G ∈ Π E Φ,L , training environment set E train and the target environment t ∈ E/E train , we have
J t (π G ) − J t (π) ≤ 2γ 1 − γ 1 N N i=1 ei (π π G ) (E) +(3 + γ 1 − γ )η t L + 2γ 1 − γ t (π * π G ) + d Π E Φ,L ∆Π E Φ,L (ρ(x, g), ρ t π G (x t , g)) (t) .
By definition, ∀e ∈ E train , we have ρ e π ∈ B(δ, E train |Π E Φ,L ). As a consequence, we have g), ρ e π G (x e , g))
(t) = 1 N N i=1 ( t (π * π G ) − ei (π * π G )) + 1 N N i=1 ei (π * π G ) + d Π E Φ,L ∆Π E Φ,L (ρ(x, g), ρ t π G (x t , g)) ≤ max e∈Etrain t (π * π G ) − e (π * π G ) + d Π E Φ,L ∆Π E Φ,L (ρ e π (x, g), ρ t π G (x t , g)) + λ ≤ 2 max e∈Etrain d Π E Φ,L ∆Π E Φ,L (ρ e π (x e , g), ρ t π G (x t , g)) + λ ≤ λ + 2 max e∈Etrain d Π E Φ,L ∆Π E Φ,L (ρ e π (x e ,+ 2 max e∈Etrain d Π E Φ,L ∆Π E Φ,L (ρ e π G (x e , g), ρ t π G (x t , g))
Without loss of generality, we have
d Π E Φ,L ∆Π E Φ,L (ρ e π G (x e , g), ρ t π G (x t , g)) = sup π,π ∈Π E Φ,L |E ρ e π G (s,g) [D TV (π(·|x e (s), g) π (·|x e (s), g))] − E ρ t π G (s,g) [D TV (π(·|x t (s), g) π (·|x t (s), g))]| ≤ E ρπ G (s,g) [ sup π,π ∈Π E Φ,L |D TV (π(·|x e (s), g) π (·|x e (s), g)) − D TV (π(·|x t (s), g) π (·|x t (s), g))|] ≤ E ρπ G (s,g) [ sup π,π ∈Π E Φ,L
|D TV (π(·|x e (s), g) π(·|x t (s), g)) + D TV (π(·|x t (s), g) π (·|x t (s), g))
+ D TV (π (·|x t (s), g) π (·|x e (s), g)) − D TV (π(·|x t (s), g) π (·|x t (s), g))|] ≤ E ρπ G (s,g) [ sup π,π ∈Π E Φ,L |D TV (π(·|x e (s), g) π(·|x t (s), g)) + D TV (π (·|x t (s), g) π (·|x e (s), g))|]
Clearly, we have ∀e, e ∈ E, D TV (π(·|x e (s), g) π(·|x e (s), g)) ≤ η t L and ∀e, e ∈ E train , D TV (π(·|x e (s), g) π G (·|x e (s), g)) ≤ N (E) . Thus, by Lemma 3, we have ∀e ∈ E train , D TV (ρ e π (s, g) ρ e π G (s, g)) ≤ γ 1−γ N (E) . By the fact that λ ≤ (E), we have
(t) ≤ 2γ 1 − γ N (E) + 4η t L + (E)
Consequently, we have
J t (π G ) − J t (π) ≤ 4γ 1 − γ (1 + γN (1 − γ) )(E) + (14 − 12γ)γ (1 − γ) 2 η t L
Intuitively speaking, when the RL policy converges to nearly optimal on E train , the generalization regret over target environment can be bounded by the sum of two components: training environment performance regret and the level of invariant over the target environment. The former is relative small as the policy is near optimal. The latter can be reduced by learning perfect alignment encoder on the training environments. As shown in Figure 4, the LER of different ablations over test environments suggests that η t is reduced empirically in PA-SF. Moreover, both losses L MMD and L DIFF contribute to the reduction. Furthermore, we notice that PA-SF < PA-SF (w/o D) < PA-SF (w/o MD) < Skew-Fit in both LER and generalization performance, which coincides with our theoretical analysis.
C.5 Discussions on L MMD
Noticing that the (η, ψ)-perfect alignment requires the l 2 distance of z e (s)−z e (s) is bounded for any state s. In practice, we adopt the convention [55] to minimize the average l 2 distance over the state space as a surrogate objective, i.e., E ρ(s) [ Φ(x e (s)) − Φ(x e (s)) 2 2 ]. It prevents the unstable and intractable training of robust optimization in our setting and we find it works well empirically. Here, we introduce a theoretical property of L MMD , which justifies its validity to ensure the if condition of perfect alignment. To begin with, we prove the following Lemma.
Proof. We assume that ∀z, z , ψ(z) − ψ(z ) 2 ≥ u z − z 2 for some u > 0. For brevity, we denote z e (s) = Φ(x e (s)) and s e b = s e t (a b 0:t ) (the b th sample: state s sampled in environment e after executing some action a b 0:t ). Since s e b s are sampled from R align independently, we have
L MMD (Φ) = E e,e ∼Etrain,Balign∼Ralign [ 1 B B b=1 ψ(z e (s e b )) − 1 B B b=1 ψ(z e (s e b )) 2 2 ] ≥ 1 B 2 E e,L MMD (Φ) ≥ 1 B E e,0:t )) [ z e (s e b ) − z e (s e b ) 2 2 ] ≥ u 2B E e,e ∼Etrain,a b 0:t ∼Ralign,s b ∼ρ(st(a b 0:t )) [ z e (s b ) − z e (s b ) 2 2 ] (Lemma 4) = u 2B E e,e ∼Etrain,s b ∼Ralign [ z e (s b ) − z e (s b ) 2 2 ]
This completes the proof.
D Additional Results
Here, we show the ablation study of PA-SF on other tasks: Reach, Push, and Pickup. large in Pickup, perhaps owing to the relative large stochasity in this environment. However, the training and test performance are still satisfactory.
Additionally, we compare t-SNE plots of PA-SF with that of vanilla Skew-Fit in Figure 8. Clearly, the t-SNE plot generated from PA-SF are more aligned than that in Skew-Fit. Noticing that Skew-Fit also encodes the irrelevant environmental factors into the latent embedding.
Finally, we visualize how well the VAE trained with L PA satisfies the perfect alignment condition. In Figure 9, we visualize the reconstruction (middle line) of the original input (bottom line) as well as the shuffled reconstruction (top line), i.e., reconstruction with the same latent representation z but a different environment index e (Figure 2(b)). Clearly, in all tasks, the VAE successfully acquires the perfectly aligned latent space w.r.t the training environments as the shuffled reconstruction shares the almost same ground truth state s with the reconstruction and the original input. Noticing that the original input images are sampled uniformly from the state space S. Figure 10 illustrates some of the environments and we provide brief descriptions as follows.
Reach: A 7-DoF sawyer arm task in which the goal is to reach a desired target position. We construct multiple training and test environments by altering the backgrounds with various images and dynamic videos and the foregrounds with diverse textures.
Door: A 7-DoF sawyer arm task with a box on the table. The goal is to open the door to a target angle. We construct multiple environments in the same way as Reach but with different ingredients. Additionally, we use the task with reset at the end of each episode.
Push: A 7-DoF sawyer arm task and a small puck on the table. The goal is to push the puck to a target position. We only change table textures as the camera has almost no background as input.
Pickup: The task setting is the same as Push. The goal is to pick up the object and place it in the desired position. We construct different environments with different backgrounds.
E.2 Implementation Details of PA-SF
In our experiments, we use the same VAE architecture as Skew-Fit except for the environment index as an extra input for the decoder. Most of the hyper-parameters in VAE including the training schedule is the same as that in Skew-Fit except for the components we added in our algorithm. In L MMD , we use the same random expansion function as in [27] as it works well in practice. Namely,
ψ(z) = 2 D ψ cos 2 γ ψ W z + b
where z ∈ R d denotes latent embedding of observation, and W ∈ R D ψ ×d and b ∈ R D ψ are random weight and bias. L MMD and L DIFF are computed with respect to samples from the replay buffer and the aligned buffer. Table 3 lists the hyper-parameters that are shared across four tasks. Table 4 lists hyper-parameters specified to each task. Notice that we fine-tune the hyper-parameters on some validation environments and test on other environments.
E.3 Implementation of Baselines
Skew-Fit [13]: Skew-Fit is designed to learn a goal-conditioned policy by self-learning in Goalconditioned MDPs. We extend it to Goal-conditioned Block MDPs by training the β-VAE with observations sampled from replay buffers of each training environments and constructing skewed distribution respectively. We modified some of the hyper-parameters of Skew-Fit in Table 5, did a grid search over latent dimension size, β, VAE training schedule, and number of training per train loop (Table 6).
Skew-Fit + RAD [37]: Previous work [52] and [37] have found that data augmentation is a simple yet powerful technique to enhance performance for visual input agents. We compare to the most well known method, i.e., Reinforcement Learning with Augmented Data (RAD) and re-implement 4 it upon Skew-Fit for Goal-conditioned Block MDPs. Note that RAD is originally designed to augment states for agents directly. In our setting, the augmentation is added to β-VAE training phase, which increases the robustness of latent space against irrelevant noise. Specifically, we augmented observations for training β-VAE as well as constructing skewed data distributions. At the beginning of each episode, we sample goals from the skewed distribution and encode augmented goal as latent goal. The training of SAC algorithm depends on the latent code of augmented current and next state. We also incorporate data augmentation with the skewed distribution and hindsight relabeling steps. To investigate the performance of different augmentation methods, we chose to experiment with crop, cutout-color, color-jitter, and grayscale and found that crop worked best among the four augmentations as reported in the RAD paper. Other augmentation methods such as cutout-color and color-jitter either replace a patch of images with single color, which may include the end-effector of Sawyer arm, or alter the color of the whole image, which may include target object (i.e., puck in Push) and thus hurt performance. We use the same hyper-parameters as in Skew-Fit.
MISA [6] and DBC [7]: Bisimulation metrics have been used to learn minimal yet sufficient representations in Block MDPs. We compare with two SOTA methods: Model-irrelevance State Abstraction (MISA) and Deep Bisimulation for Control (DBC), and modify the code for Goalconditioned Block MDPs. In particular, we add goals' inputs into the reward predictor and use oracle ground truth distance between the current state and goal state (i.e., end-effector's position and object's position) as rewards. Our code is built upon the publicly available codes 5 6 .For fair comparison, we also fine-tuned some hyper-parameters on each task respectively. For MISA, we did a grid search over the encoder and decoder learning rates ∈ {10 −3 , 10 −5 } and reward predictor coefficient ∈ {0.5, 1.0, 2.0}. For DBC, we did a grid search over the encoder and decoder learning rates ∈ {10 −3 , 10 −4 } and bisimulation coefficients ∈ {0.25, 0.5, 1}. Please refer to Table 7 and Table 8 for our final choices.
Figure 1 :
1Graphical model for Goal-conditioned Block MDPs (GBMDPs) setting.
Figure 2 :
2{x e 0 , a 0 , x e 1 , a 1 , . . . , x e T } from environment e. The set of corresponding states along this trajectory are denoted as {s e t (a 0:t )} (a): Illustration of Aligned Sampling. Square represents the whole state space S, gray area represents the distribution ρ e π (s) in two different environments. Small colored areas are the aligned state distribution generated by aligned sampling in Section 3.2. (b): Overall VAE structure in our PA-SF. Encoder maps x e to the latent embedding z and decoder D reconstructs the observations with z and index e. L MMD and L DIFF denote the two losses in Section 3.2.
Figure 3 :
3Learning curve of all algorithms on average across test environments for each task. All curves show the mean and one standard deviation (a half for Pickup to show clearly) of 7 seeds.
Figure 2 Figure 4 :
244 of different ablations of PA-SF in the Door environment during both training and testing. To understand the roles of L DIFF and L MMD , PA-SF (w/o D) excludes L DIFF and PA-SF (w/o MD) excludes both losses 1 . Noticing that PA-SF (w/o MD) is equivalent to the Skew-Fit algorithm with our proposed VAE structure (Ablation of PA-SF and visualization of the latent representation via LER metric. All curves represent the mean and one standard deviation across 7 seeds.
1 .
1Φ achieves the if condition of perfect alignment over E train via L MMD as the LER value of PA-SF and PA-SF (w/o D) is almost 0. While without MMD loss, PA-SF (w/o MD) and Skew-Fit struggle with large LER value despite achieving good training performance. Furthermore, the comparison between PA-SF and PA-SF (w/o AS) demonstrates the importance of using aligned data in the MMD loss (Otherwise, the matching is inherently erroneous).
1 .
1As shown by the learning curve of test environments, the target domain performance of different ablations match that of the LER metric: SkewFit, PA-SF (w/o AS) > PA-SF (w/o MD) > PA-SF (w/o D) > PA-SF. During training, these ablations have almost the same performance, except PA-SF (w/o AS)
Figure 5 :
5t-SNE visualization of the latent space Φ(x e ) trained with PA-SF for three environments: 2 training and 1 testing of Push as well as instances visualization.
B
Characteristic set of joint distributions determined by E train and policy class Π Φ Observation encoder z e (s) = Φ(x e (s)) Latent representation of observation x e with state s Π Φ = {w • (Φ(x), g)}, ∀w} Policy class induced by encoder Φ with any function w ρ(x, g) The closest occupancy measure in characteristic set B w.r.t. occupancy measure in testing environment under Π∆Π-divergence {s e t (a 0:t )} T t=0 Set of states along trajectory {x e 0 , a 0 , x e 1 , a 1 , . . . , x e T } {x ei t (a 0:t )} Aligned observations in environment e i for action sequence {a 0 , . . . , a t−1 } (numpy style indexing) R align Replay buffer for aligned transitions
1 :
1for m = 0, . . . , M − 1 episodes do 2: for e = 0, . . . , N − 1 training environments do Exploration Rollout 3:
sequences {a 0:T } by executing the policy in a random training environment. Aligned Sampling 12: for e = 0, . . . , N − 1 training environments do 13:Sample initial state x e 0 (s e 0 ) from the environment e.14:Rollout action sequence {a 0:T } to get {x e t (s e t (a 0:t ))e t (s e t (a 0:t ))} T t=0 in aligned buffer R e align indexed by a 0:t for e ∈ E train .17:for i = 0, . . . , I − 1 training iterations do Policy Gradient 18:
Figure 6 :
6Comparison of graphical models of (a) a Block MDP [14, 6], (b) a Goal-Conditioned MDP [15, 16, 8], and (c) our proposed Goal-Conditioned Block MDP. The agent takes in the goal g and observation x t , which is produced by the domain invariant state s t and environmental state b t , and acts with action a t . 1 t denotes the indicator function on whether the inputs are the same state. Note that b t may have temporal dependence indicated by the dashed edge.
Remark 2 .
2If we choose ψ = 1 √ L , η = 1 L 2 and u = √ L, the informal version of Proposition 2 describes the case when L → ∞ and omits unessential constants.
z e (s e i )) − ψ(z e (s e i )), ψ(z e (s e j )) − ψ(z e (s e j )) ]
[ z e (s e b ) − z e (s e
[ ψ(z e (s e b )) − ψ(z e (s e b )) [ ψ(z e (s e b )) − ψ(z e (s e b ))
Figure 7 Figure 7 :
77demonstrates that similar results in Section 4.2 also holds on Reach and Push while on Pickup, PA-SF (w/o D) outperforms PA-SF on test environments marginally. We also find that the LER is Ablations of our algorithm PA-SF and visualization of the latent representation via LER metric on Reach, Push, and Pickup. All curves represent the mean and one standard deviation (except Pickup with half standard deviation) across 7 seeds.
Figure 8 Figure 9 :Figure 10 :
8910: t-SNE visualization of the latent space Φ(x e ) trained with PA-SF and Skew-Fit for three environments, i.e., 2 training and 1 testing on different tasks. Visualization of the VAE on all four tasks. For each figure, the bottom line shows the original input images (sampled uniformly from the state space). The middle line is the reconstruction of the input image. The top line show the shuffled reconstruction image, i.e., reconstruction with the same latent space z but a shuffled environment index e. Task setups for training (left) and test (right) environments Our base environments are first used in [35].
Many works on domain generalization of supervised learning [21, 25, 22, 23, 26] spend much effort in discussing the trade-offs among different terms similar to the ones in Eq. (2), e.g., minimizing δ may increase λ [26], and in developing sophisticated techniques to optimize the bound, e.g. distribution matching [27-29] or adversarial learning [25].
Table 1 :
1Evaluation of PA-SF and baselines on four control tasks. We report the mean and one standard deviation on each task (lower metric is better).Algorithm
Reach
Door
Push
Pickup
Hand distance Angle difference Puck distance Object distance
(35K)
(150K)
(400K)
(280K)
Test Avg
Skew-Fit
0.111 ± 0.010 0.194 ± 0.018
0.086 ± 0.004 0.037 ± 0.006
Skew-Fit + RAD 0.105 ± 0.010 0.162 ± 0.030
0.082 ± 0.008 0.040 ± 0.004
MISA
0.239 ± 0.0142 0.255 ± 0.027
0.099 ± 0.006 0.043 ± 0.004
DBC
0.185 ± 0.037 0.320 ± 0.033
0.095 ± 0.006 0.045 ± 0.002
PA-SF(Ours)
0.076 ± 0.005 0.106 ± 0.015 0.069 ± 0.005 0.028 ± 0.004
Train Avg
PA-SF(Ours)
0.067 ± 0.005 0.058 ± 0.074
0.060 ± 0.005 0.020 ± 0.008
Oracle Skew-Fit 0.055 ± 0.010 0.057 ± 0.012
0.054 ± 0.006 0.020 ± 0.006
5 Related Work
RelatedGoal-conditioned RL: Goal-conditioned RL [15, 16] removes the need for complicated reward shaping by only rewarding agents for reaching a desired set of goal states. RIG [35] is the seminal work for visual-input, reward-free learning in goal-conditioned MDPs. Skew-Fit [13] improves over RIG [35] in training efficiency by ensuring the behavioural goals used to explore are diverse and have wide state coverage. However, Skew-Fit has its own limitation in understanding the semantic meaning of the goal-conditioned task. To acquire more meaningful goal's and observation's latent representation, several approaches apply inductive biases or seek human feedback. ROLL [38] applies object extraction methods under strong assumptions, while WSC [39] uses weak binary labeled data as the reward function. Others explore the same goal-conditioned RL problem via hindsight experience replay [8, 40, 41], unsupervised reward learning [11], inverse dynamics models [10], C-learning [9], goal generation [42-44], goal-conditioned forward models [45], and hierarchical RL [46-49]. Our study focus on learning goal-conditioned policies that is invariant of spurious environmental factors. We aim to learn a policy that can generalize to visual goals in unseen test environments.
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13
A Notation
Table 2 :
2Description for symbolsSymbol
Description
M E
A family of Goal-conditioned Block MDPs
e
Environment index
S
Shared state space among environments e ∈ E
A
Shared action space among environments e ∈ E
X e
Specific observation space for environment e
T e
Specific transition dynamic for environment e
G
Shared goal space among environments e ∈ E
γ
Shared discount factor among environments e ∈ E
b e
Environmental factor for environment e
(e.g. background)
B e
Specific environmental factor space for environment e
(i.e. video backgrounds are allowed)
Table 2 -
2continued from previous pageSymbol
Description
δ
Maximum Π∆Π-divergence between occupancy
measure for two different training environments under
given policy π
λ
Performance of π * in both training and testing
environments in terms of average TV distance
Sample future state x ehj , t < h j ≤ H − 1 for all e ∈ E train .23:
end for
24:
for t = 0, ..., H − 1 steps do
Hindsight Relabeling
25:
for j = 0, ..., J − 1 steps do
26:
27:
Store (x e
t , a t , x e
t+1 , q φ x e
hj ) into R e .
28:
end for
29:
end for
30:
Construct skewed replay buffer distribution p e,m+1
skewed using data
from R e for all e ∈ E train .
Skewing Replay Buffers
31:
Fine-tune β-VAE on {x e } B
b=1 ∼ p e,m+1
skewed and {x e (s e
t (a b
0:t ))} B
b=1 ∼ R e
aligned
for all e ∈ E train according to the VAE training schedule and via Eq. (4).
VAE Training
32: end for
Table 3 :
3Shared hyper-parameters for PA-SF Hyper-parameter Value Aligned Path Length 50 VAE Relay Buffer Batch Size 32 VAE Aligned Buffer Batch Size 32 Random Expansion Function Dimension D ψ 1024 Random Expansion Function Scalar γ ψ 1.0 Number of Training per Train Loop 1500 Number of Total Exploration Steps per Epoch 900
Table 4 :
4Task specific hyper-parameters for PA-SF Hyper-parameter Reach Door Push Pickup Proportion of Aligned Sampling in Exploration 1MMD Coefficient α MMD
1000 1000 200 100
Difference Coefficient α DIFF
0.1
1.0
0.04 0.04
β for β-VAE
20
20
20
10
Skew Coefficient α
-0.1
-0.5 -1
-1
6
1
3
1
6
1
6
Table 5 :
5Modified general hyper-parameters for Skew-Fit Hyper-parameter Value Replay Buffer Size for each e ∈ E train 50000Exploration Noise
None
RL Batch Size
1200
VAE Batch Size
96
Table 6 :
6Task specific hyper-parameters for Skew-Fit p skewed p skewed p skewed Number of Training per Train Loop 1200 2000 2000 2000Hyper-parameter
Reach Door Push Pickup
Path Length
50
100
50
50
β for β-VAE
20
20
40
15
Latent Dimension Size
8
25
15
20
α for Skew-Fit
0.1
0.5
1.0
1.0
VAE Training Schedule
2
1
2
1
Sample Goals From
q G
φ
Table 7 :
7Task specific hyper-parameters for MISA Hyper-parameter Reach Door Push Pickup Encoder and Decoder Learning Rate 10 −5 10 −3 10 −3 10 −5Reward Predictor Coefficient
0.5
1.0
2.0
0.5
Table 8 :
8Task specific hyper-parameters for DBC Hyper-parameter Reach Door Push Pickup Encoder and Decoder Learning Rate 10 −4 10 −4 10 −4 10 −3Bisimulation Coefficient
0.5
1.0
0.25 1.0
A single L DIFF is not useful here.
https://github.com/MishaLaskin/rad
https://github.com/facebookresearch/icp-block-mdp 6 https://github.com/facebookresearch/deep_bisim4control
AcknowledgementsWe are grateful for the feedback from anonymous reviewers. Resources used in preparing this research were provided, in part, by the Province of Ontario, the Government of Canada through CIFAR, and companies sponsoring the Vector Institute <http://www.vectorinstitute.ai/partners>.
George van den Driessche, Thore Graepel, and Demis Hassabis. David Silver, Julian Schrittwieser, Karen Simonyan, Aj Antonoglou, Ioannis, Arthur Huang, Thomas Guez, Lucas Hubert, Matthew Baker, Adrian Lai, Yutian Bolton, Timothy Chen, Fan Lillicrap, Laurent Hui, Sifre, Nature. 550Mastering the game of go without human knowledgeDavid Silver, Julian Schrittwieser, Karen Simonyan, Aj Antonoglou, Ioannis abd Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, Yutian Chen, Timothy Lillicrap, Fan Hui, Laurent Sifre, George van den Driessche, Thore Graepel, and Demis Hassabis. Mastering the game of go without human knowledge. Nature, 550, 2017.
Dharshan umaran, Daan Wierstra, Shane Legg, and Demis Hassabis. Human-level control through deep reinforcement learning. Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc Bellemare, Alex Graves, Martin Riedmiller, Andreas Fidjeland, Georg Strovski , Stig Petersen, Nature. 518Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc Bellemare, Alex Graves, Martin Riedmiller, Andreas Fidjeland, Georg strovski, Stig Petersen, Charles Beattie, Amir Sadik, Ioannis Antonoglou, Helen King, Dharshan umaran, Daan Wierstra, Shane Legg, and Demis Hassabis. Human-level control through deep reinforcement learning. Nature, 518, 2015.
Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor. Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, Sergey Levine, PMLRProceedings of the 35th International Conference on Machine Learning. the 35th International Conference on Machine LearningStockholmsmässan, Stockholm Sweden80Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, and Sergey Levine. Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor. In Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pages 1861-1870, Stockholmsmässan, Stockholm Sweden, 10-15 Jul 2018. PMLR.
Assessing generalization in deep reinforcement learning. CoRR, abs/1810.12282. Charles Packer, Katelyn Gao, Jernej Kos, Philipp Krähenbühl, Vladlen Koltun, and Dawn SongCharles Packer, Katelyn Gao, Jernej Kos, Philipp Krähenbühl, Vladlen Koltun, and Dawn Song. Assessing generalization in deep reinforcement learning. CoRR, abs/1810.12282, 2018. URL http://arxiv.org/abs/1810.12282.
Efficient adaptation for end-to-end vision-based robotic manipulation. Ryan Julian, Benjamin Swanson, S Gaurav, Sergey Sukhatme, Chelsea Levine, Karol Finn, Hausman, arXiv:2004.10190arXiv preprintRyan Julian, Benjamin Swanson, Gaurav S Sukhatme, Sergey Levine, Chelsea Finn, and Karol Hausman. Efficient adaptation for end-to-end vision-based robotic manipulation. arXiv preprint arXiv:2004.10190, 2020.
Invariant causal prediction for block MDPs. Amy Zhang, Clare Lyle, Shagun Sodhani, Angelos Filos, Marta Kwiatkowska, Joelle Pineau, Yarin Gal, Doina Precup, International Conference on Machine Learning. PMLRAmy Zhang, Clare Lyle, Shagun Sodhani, Angelos Filos, Marta Kwiatkowska, Joelle Pineau, Yarin Gal, and Doina Precup. Invariant causal prediction for block MDPs. In International Conference on Machine Learning, pages 11214-11224. PMLR, 2020.
Learning invariant representations for reinforcement learning without reconstruction. Amy Zhang, Rowan Mcallister, Roberto Calandra, Yarin Gal, Sergey Levine, arXiv:2006.10742arXiv preprintAmy Zhang, Rowan McAllister, Roberto Calandra, Yarin Gal, and Sergey Levine. Learning invariant representations for reinforcement learning without reconstruction. arXiv preprint arXiv:2006.10742, 2020.
Abbeel Pieter, and Zaremba Wojciech. Hindsight experience replay. Andrychowicz Marcin, Wolski Filip, Ray Alex, Schneider Jonas, Fong Rachel, Welinder Peter, Mcgrew Bob, Tobin Josh, Advances in Neural Information Processing Systems. Andrychowicz Marcin, Wolski Filip, Ray Alex, Schneider Jonas, Fong Rachel, Welinder Peter, McGrew Bob, Tobin Josh, Abbeel Pieter, and Zaremba Wojciech. Hindsight experience replay. In Advances in Neural Information Processing Systems, 2017.
Benjamin Eysenbach, Ruslan Salakhutdinov, Sergey Levine. C-Learning, arXiv:2011.08909Learning to achieve goals via recursive classification. arXiv preprintBenjamin Eysenbach, Ruslan Salakhutdinov, and Sergey Levine. C-learning: Learning to achieve goals via recursive classification. arXiv preprint arXiv:2011.08909, 2020.
| [
"https://github.com/MishaLaskin/rad",
"https://github.com/facebookresearch/icp-block-mdp",
"https://github.com/facebookresearch/deep_bisim4control"
]
|
[
"GEM: Glare or Gloom, I Can Still See You - End-to-end Multi-modal Object Detection",
"GEM: Glare or Gloom, I Can Still See You - End-to-end Multi-modal Object Detection"
]
| [
"Osama Mazhar ",
"Robert Babuška ",
"Jens Kober "
]
| []
| []
| Deep neural networks designed for vision tasks are often prone to failure when they encounter environmental conditions not covered by the training data. Single-modal strategies are insufficient when the sensor fails to acquire information due to malfunction or its design limitations. Multi-sensor configurations are known to provide redundancy, increase reliability, and are crucial in achieving robustness against asymmetric sensor failures. To address the issue of changing lighting conditions and asymmetric sensor degradation in object detection, we develop a multi-modal 2D object detector, and propose deterministic and stochastic sensor-aware feature fusion strategies. The proposed fusion mechanisms are driven by the estimated sensor measurement reliability values/weights. Reliable object detection in harsh lighting conditions is essential for applications such as self-driving vehicles and human-robot interaction. We also propose a new "r-blended" hybrid depth modality for RGB-D sensors. Through extensive experimentation, we show that the proposed strategies outperform the existing state-of-the-art methods on the FLIR-Thermal dataset, and obtain promising results on the SUNRGB-D dataset. We additionally record a new RGB-Infra indoor dataset, namely L515-Indoors, and demonstrate that the proposed object detection methodologies are highly effective for a variety of lighting conditions. | 10.1109/lra.2021.3093871 | [
"https://arxiv.org/pdf/2102.12319v3.pdf"
]
| 232,035,590 | 2102.12319 | 1ee13d53b7427c3d383515b3107db31a0074e00e |
GEM: Glare or Gloom, I Can Still See You - End-to-end Multi-modal Object Detection
Osama Mazhar
Robert Babuška
Jens Kober
GEM: Glare or Gloom, I Can Still See You - End-to-end Multi-modal Object Detection
Deep neural networks designed for vision tasks are often prone to failure when they encounter environmental conditions not covered by the training data. Single-modal strategies are insufficient when the sensor fails to acquire information due to malfunction or its design limitations. Multi-sensor configurations are known to provide redundancy, increase reliability, and are crucial in achieving robustness against asymmetric sensor failures. To address the issue of changing lighting conditions and asymmetric sensor degradation in object detection, we develop a multi-modal 2D object detector, and propose deterministic and stochastic sensor-aware feature fusion strategies. The proposed fusion mechanisms are driven by the estimated sensor measurement reliability values/weights. Reliable object detection in harsh lighting conditions is essential for applications such as self-driving vehicles and human-robot interaction. We also propose a new "r-blended" hybrid depth modality for RGB-D sensors. Through extensive experimentation, we show that the proposed strategies outperform the existing state-of-the-art methods on the FLIR-Thermal dataset, and obtain promising results on the SUNRGB-D dataset. We additionally record a new RGB-Infra indoor dataset, namely L515-Indoors, and demonstrate that the proposed object detection methodologies are highly effective for a variety of lighting conditions.
I. INTRODUCTION
Modern intelligent systems such as autonomous vehicles or assistive robots should have the ability to reliably detect objects in challenging real-world scenarios. Object detection is one of the widely studied problems in computer vision. It has been addressed lately by employing deep convolutional neural networks where the state-of-the-art methods have achieved fairly accurate detection performances on the existing datasets [1][2][3][4]. However, these vision models are fragile and do not generalize across realistic unconstrained scenarios, such as changing lighting conditions or other environmental circumstances which were not covered by the training data [5]. The failure of the detection algorithms in such conditions could lead to potentially catastrophic results, as in the case of self-driving vehicles.
One way of addressing this problem is to employ a data-augmentation strategy [6]. It refers to the technique of perturbing data without altering class labels, and it has been proven to greatly improve robustness and generalization performance [7]. Nevertheless, this is insufficient for the The research presented in this article was carried out as part of the OpenDR project, which has received funding from the European Union's Horizon 2020 research and innovation programme under Grant Agreement No. 871449. 1 cases where the sensor fails to acquire information due to malfunction or its technical limitations. For example, the output of standard passive cameras degenerates with reduced ambient light, while thermal cameras or LiDARs are less affected by illumination changes. Multi-sensor configurations are known to provide redundancy and often enhance the performance of the detection algorithms. Moreover, efficient sensor fusion strategies minimize uncertainties, increase reliability, and are crucial in achieving robustness against asymmetric sensor failures [8]. Although, increasing the number of sensors might enhance the performance of detection algorithms, this comes with a considerable computational and energy cost. This is not desirable in mobile robotic systems, which typically have constraints in terms of computational power and battery consumption. In such cases, intelligent choice and combination of sensors are crucial.
Furthermore, multi-modal data fusion often requires an estimate of the sensor signal uncertainty to guarantee efficient fusion and reliable prediction without a priory knowledge of the sensor characteristics [9]. The existing multi-modal object detection methods fuse the sensor data streams without explicitly modeling the measurement reliability. This may have severe consequences when the data from an individual sensor degrades or is missing due to sheer sensor failure.
To address the above problems, we propose sensor-aware multi-modal fusion strategies for object detection in harsh lighting conditions, thus the title " GEM: Glare or gloom, I can still see you -End-to-end Multimodal object detection". The output samples of GEM are shown in Figure 1. Two fusion methods are proposed: deterministic weighted fusion and stochastic feature fusion. In the deterministic weighted fusion, the measurement certainty of each sensor is estimated either by learning scalar weights or masks through separate neural networks. The learned weights are then assigned to the feature maps extracted from the feature extractor backbones for each sensor modality. The weighted feature maps can be fused either by averaging or concatenation. Moreover, we can visualize and interpret the measurement certainty of each sensor in the execution phase, which provides deeper insights into the relative strengths of each data stream. The stochastic feature fusion creates a one-hot encoding of the feature maps of each sensor, which can be assumed as a discrete switch that allows only the dominant/relevant features to pass. The obtained selected features are then concatenated before they are passed to the object detection and classification head. The proposed sensor-aware multi-modal object detector, referred to simply as GEM in the rest of the paper, is trained in an end-to-end fashion along with the fusion mechanism.
Most modern object detectors, including YOLO, Faster-RCNN and SSD employ many hand-crafted features such as anchor generation, rule-based assignment of classification and regression targets as well as weights to each anchor, and non-maximum suppression postprocessing. The overall performance of these methods often relies on careful tuning of the above-mentioned hyper-parameters. Following their success in sequence/language modeling, transformers have lately emerged in vision applications, outperforming competitive baselines and demonstrating a strong potential in this field. Therefore, we employ transformers in our work as in [10], which thanks to their powerful relational modeling capability eliminates the need of hand-crafted components in object detection. Our main contributions in this paper are:
• Evaluation of feature fusion in two configurations, i.e., deterministic weighted fusion and stochastic feature fusion for multi-modal object detection. • Estimation of measurement reliability of each sensor as scalar or mask multipliers through separate neural networks for each modality to efficiently drive the deterministic weighted fusion. • Use of transformers for multi-modal object detection to harness the efficacy of self-attention in sensor fusion.
II. RELATED WORK
In this section, we first review deep learning-based object detection strategies, followed by a discussion on existing methods for multi-modal fusion methods in relevant tasks.
A. Deep learning-based Object Detection
Detailed literature surveys for deep learning-based object detectors have been published in [11,12]. Here we briefly discuss some of the well-known object detection strategies. Typically, object detectors can be classified into two types, namely two-stage and singe-stage object detectors.
1) Two-stage object detection: Two-stage object detectors exploit a region proposal network (RPN) in their first stage. RPN ranks region boxes, alias anchors, and proposes the ones that most likely contain objects as candidate boxes. The features are extracted by region-of-interest pooling (RoIPool) operation from each candidate box in the second stage. These features are then utilized for bounding-box regression and classification tasks.
2) Single-stage object detection: Single-stage detectors propose predicted boxes from input images in one forward pass directly, without the region proposal step. Thus, this type of object detectors are time efficient and can be utilized for real-time operations. Lately, an end-to-end object detection strategy has been proposed in [10] that eliminates the need for hand-crafted components like anchor boxes and nonmaximum suppression. The authors employ transformers in an encoder-decoder fashion, which have been extremely successful and become a de facto standard for natural language processing tasks. The transformer implicitly performed region proposals instead of using an R-CNN. The multi-head attention module in transformers jointly attended to different semantic regions of an image/feature maps and linearly aggregates the outputs through learnable weights. The learned attention maps can be visualized without requiring dedicated methods, as in the case of convolutional neural networks. The inherent non-sequential architecture of transformers allows parallelization of models. Thus, we opted to build upon the methodology of [10] for our multi-modal object detector for harsh lighting conditions.
B. Sensor Fusion
Sensor fusion strategies can be roughly divided into three types according to the level of abstraction where fusion is performed or in which order transformations are applied compared to feature combinations, namely low-level, midlevel, and high-level fusion [13]. In low-level or early fusion, raw information from each sensor is fused at pixel level, e.g., disparity maps in stereovision cameras [14]. In midlevel fusion, a set of features is extracted for each modality in a pre-processing stage, while multiple approaches [15] are exploited to fuse the extracted features. Late-fusion often employs a combination of two fusing methods, e.g., convolution of stacked feature maps followed by several fully connected layers with dropout regularization [16]. In highlevel fusion or ensemble learning methods, predictions are obtained individually for each modality and the learnt scores or hypotheses are subsequently combined via strategies such as weighted majority votes [17]. Deep fusion or cross fusion [18] is another type of fusion strategy which repeatedly combines inputs, then transforms them individually. In each repetition, the transformation learns different features. For example in [8], features from the layers of VGG network are exchanged among all modalities driven by sensor entropy after each pooling operation.
C. Multi-sensor Object Detection
Most of the efforts on multi-modal object detection in the literature are focused on pedestrian or vehicle detection in the automotive context. Sensor fusion strategies are typically proposed for camera-LiDAR, camera-radar, and cameraradar-LiDAR setups. Here, we briefly go through the relevant state-of-the-art methods. The authors in [8] proposed an entropy-steered multimodal deep fusion architecture for adverse weather conditions. The sensor modalities exploited in their method include RGB camera, gated camera (NIR band), LiDAR, and radar. Instead of employing BeV projection or point cloud representation for LiDAR, the authors encoded depth, height, and pulse intensity on an image plane. Moreover, the radar output was also projected onto an image plane parallel to the image horizontal dimension. Considering the radar output invariant along the vertical image axis, the scan was replicated across the horizontal image axis. They utilized a modified VGG architecture for feature extraction, while features were exchanged among all modalities driven by sensor entropy after each pooling operation. Fused feature maps from the last 6 layers of the feature extractors were passed to the SSD object detection head. In [19], the authors proposed a pseudo multi-modal object detector from thermal IR images in a Faster-RCNN setting. The features from ResNet-50 backbones for the two modalities are concatenated and a 1 × 1 convolution is applied to the concatenated features before they are passed to the rest of Faster-RCNN network. They exploited I2I translation networks, namely CycleGAN [20] and UNIT [21] to transform thermal images from the FLIR Thermal [22] and KAIST [23] datasets to the RGB domain, thus the names MM-CG and MM-UNIT. Here, we also discuss some fusion strategies which were originally proposed for applications other than object detection but are relevant to our work. In [24], the authors proposed a sensor fusion methodology for RGB and depth images to steer a self-driving vehicle. The latent semantic vector from an encoder-decoder segmentation network trained on RGB images was fused with the depth features. The fusion architecture proposed by [24] is similar to the gating mechanism driven by the learned scalar weights presented in [25]. The method proposed in [26] is closest to our work. The authors proposed two sensor fusion strategies for Visual-Inertial Odometry (VIO), namely soft fusion and hard fusion. In soft fusion, they learned soft masks which were subsequently assigned to each element in the feature vector. Hard fusion employed a variant of the Gumbel-max trick, which is often used to sample discrete data from categorical distributions. Learning masks equal to the size of feature vectors might introduce computational overhead. Therefore, we learn dynamic scalar weights for each sensor modality, which adapt to the environmental/lighting conditions. These scalar weights represent the reliability or relevance of the sensor signals. Moreover, we also learn single-channel masks with a spatial size equal to that of the feature maps obtained from the feature extractor backbone. Nevertheless, we also implement the Gumbel-Softmax trick for comparison, as a stochastic feature fusion for multi-modal object detectors.
III. SENSOR-AWARE MULTI-MODAL FUSION
In this work, we propose a new method for sensoraware feature selection and multi-modal fusion for object detection. We actually evaluate feature fusion in two configurations, i.e., deterministic weighted fusion with scalar and mask multipliers, and stochastic feature fusion driven by the Gumbel-Softmax trick that enables sampling from a discrete distribution. The overall pipeline of the proposed multi-modal object detector is illustrated in Figure 2. The proposed methodologies are trained and evaluated on datasets with RGB and thermal or depth images. However, it can be extended to include data from other sensors like LiDAR or radar, either by projecting the sensor output onto an image plane as proposed in [8] or by employing sensor-specific feature extractors such as [27].
A. Deterministic Weighted Fusion
The proposed deterministic weighted fusion scheme is conditioned on the measurement certainty of each sensor. These values are obtained either by learning scalar weights or masks through separate neural networks. Subsequently, the weights are assigned to the feature maps extracted from the backbones as (scalar or mask) multipliers for each sensor modality. Given the output of the backbone feature extractors s for a single modality, the neural network f optimizes parameters θ to obtain measurement certainty w of the corresponding sensor as described as follows:
w = f (s, θ) × 1 rows × cols × k rows l=1 cols m=1 k n=1 s(l, m, n) (1)
where k is the selected number of channels. The network f learns the parameters θ in an end-to-end fashion. In the case of sensor degradation, the output of the neurons in the early layers of the corresponding backbone will remain close to zero. Thus, we multiply the output of the network f by the mean of first k feature maps, 16 in our case, from s in a feedforward setting to obtain w. This allows f to dynamically condition its output to changing lighting/sensor degradation scenarios, which subsequently guides the transformers to focus on the dominant sensor modality for object detection. Furthermore, the multiplication of the raw output of f with the mean of the selected feature maps is performed without gradient calculation to prevent the distortion of the feature maps in the back-propagation phase.
The weighted feature maps are fused by either taking an average of the two feature sets, or by concatenating them. The fused features are then passed to the transformer for object detection and localization. Our scalar fusion functions g sa (averaging) and g sc (concatenating) are represented as:
g sa (s RGB , s IR ) = φ(w RGB s RGB , w IR s IR ) g sc (s RGB , s IR ) = [w RGB s RGB ; w IR s IR ](2)
where φ denotes the mean operation, s RGB and s IR are feature maps obtained from the backbone feature extractor for RGB and thermal/IR imagers respectively, while w RGB and w IR are the sensor measurement certainty weights obtained through Equation (1). Similar to the scalar fusion method, feature selection is also modelled by learning masks for each modality, in this case m RGB and m IR /m depth , with a spatial size equal to that of the features maps. The fusion scheme with mask multipliers is represented as:
g ma (s RGB , s IR ) = φ(m RGB s RGB , m IR s IR ) g mc (s RGB , s IR ) = [m RGB s RGB ; m IR s IR ](3)
B. Stochastic Feature Fusion
In addition to the weighted fusion schemes, we exploit a variant of the Gumbel-max trick to learn a one-hot encoding that either propagates or blocks each component of the feature maps for intelligent fusion. The Gumbel-max resampling strategy allows to draw discrete samples from a categorical distribution during the forward pass through a neural network. It exploits the reparametrization trick to separate out the deterministic and stochastic parts of the sampling process. However, it adds Gumbel noise instead of that from a normal distribution, which is actually used to model the distribution of the maximums for samples taken from other distributions. Gumbel-max then employs the arg max function to find the class that has the maximum value for each sample.
Considering α be the n-dimensional probability variable conditioned for every row on each channel of the feature volume such that α = [π 1 , . . . , π n ], representing the probability of each feature at location n, the Gumbel-max trick can be represented by the following equation:
Q = arg max i (log π i + G i )(4)
where, Q is a categorical variable with class probabilities π 1 , π 2 , . . . , π n and {G i } i≤n is an i.i.d. sequence of standard Gumbel random variables which is given by:
G = − log(− log(U )), U ∼ Uniform[0, 1](5)
The use of arg max makes the Gumbel-max trick nondifferentiable. However, it can be replaced by Softmax with a temperature factor τ , thus making it a fully-differentiable resampling method [28]. Softmax with temperature parameter τ can be represented as where τ determines how closely the Gumbel-Softmax distribution matches the categorical distribution. With low temperatures, e.g., τ = 0.1 to τ = 0.5, the expected value of a Gumbel-Softmax random variable approaches the expected value of a categorical random variable [28]. The Gumbel-Softmax resampling function can therefore be written as
f τ (x) i = exp(x i /τ ) Σ n j=1 exp(x j /τ )(6)Q τ i = f τ (log π i + G i ) = exp((log π i + G i )/τ ) Σ n j=1 exp((log π j + G j )/τ )(7)
with i = 1, . . . , n.
We set τ = 1 and obtain feature volume approximate onehot categorical encodings for each modality e RGB and e IR . Then a Hadamard product is taken between the encodings and the feature volumes and the resultants are subsequently concatenated and passed on to the bounding box regressor and classification head. We illustrate our selective fusion process developed for multi-modal object detector in Figure 3, while the selective fusion function g sf is given as follows
g sf (s RGB , s IR ) = [e RGB s RGB ; e IR s IR ].(8)
IV. EXPERIMENTS
A. Datasets
Three datasets are utilized in the training and evaluation of GEM, including the FLIR Thermal, SUNRGB-D [29] and a new L515-Indoor dataset that we recorded for this research. The FLIR Thermal dataset provides 8,862 training and 1,366 test samples of thermal and RGB images recorded in the streets and highways in Santa Barbara, California, USA. Only the thermal images in the dataset are annotated with four classes, i.e., People, Bicycle, Car and Dog. The given RGB images in the dataset are neither annotated nor aligned with their thermal counterparts, while the camera matrices are also not provided. Thus, to utilize this dataset in a multimodal setting, the given RGB images must be annotated or aligned with their corresponding thermal images. One way to address this problem is to create artificial RGB images from input thermal images through GANs or similar neural networks as performed in [19]. However, we opted to employ the concept of homography by manually selecting matching features in multiple RGB and thermal images. The selected feature points are then employed to estimate a transformation matrix between the two camera modalities. RGB images are subsequently transformed with the estimated homography matrix such that they approximately align with their thermal equivalents. The Dog class constitutes only 0.29% of all the annotations in the aligned FLIR-Thermal dataset, thus it is not included in our experiments.
The SUNRGB-D dataset contains 10,335 RGB-D images taken by Kinect v1, Kinect v2, Intel RealSense, and Asus Xtion cameras. The annotations provided consist of 146,617 2D polygons and 64,595 3D bounding boxes, while 2D bounding boxes are obtained by projecting the coordinates of 3D bounding boxes onto the image plane. Although the dataset contains labels for approximately 800 objects, we evaluate our method on the selected 19 objects similar to [29]. We first divide the dataset into three subsets such that the training set consists of 4,255 images, the validation set has 5,050 images, while the test set contains 1,059 images.
The L515-Indoor dataset provides 482 training and 207 validation RGB and IR images recorded with Intel RealSense L515 camera with various ambient light conditions in an indoor scene. It contains annotations of 1,819 2D bounding boxes of 6 object categories in total. The IR images are aligned with their RGB counterparts through a homography matrix which is computed in a similar fashion as explained for the FLIR-Thermal dataset. The population distributions of the datasets are illustrated in Figure 4.
B. Pre-processing Sensor Outputs
For the FLIR-Thermal and L515-Indoor datasets, aligned RGB and thermal/IR images are fed into our feature extractor backbones without any pre-processing. However, techniques that exploit datasets with depth images including [29] often apply HHA encoding [30] on the depth sensor modality for early feature extraction prior to being fed into the neural networks. HHA is a geocentric embedding for depth images that encodes horizontal disparity, height above ground, and angle with gravity for each pixel. In a multi-threading setup on a 12-Core Intel ® Core TM i7-9750H CPU, HHA encoding of a batch of 32 images takes approximately 119 seconds, which is far from its application in real-time object detection or segmentation tasks.
To address this problem, assuming that we are working with RGB and depth modalities, we create a new hybrid image that introduces scene texture in a depth image. As the red light is scattered the least by air molecules, we blend the depth images and the red image channels through a blend weight α. Thus, we name our hybrid depth image as "rblended" depth image. img r-blended = α img depth + (1 − α) img red (9) The value of α is set to 0.9 for depth images while the weight value for the red channels becomes 0.1. This is to make sure that when the neural network is trained with "r-blended depth" image, it should focus on learning the depth features while information from the red channel only complements the raw depth map. The idea to blend the red channel is also supported by the fact that CMOS cameras are often more sensitive to green and red light. We first train our multi-modal object detector on RGB and HHA encoded depth images. Later, we fine-tune the trained model by replacing HHA encoded images with r-blended depth images and achieve comparable results in terms of detection accuracy, while the fine-tuned model can indeed be used for real-time multimodal object detection.
C. Training
GEM is trained with scalar fusion and mask fusion methods, i.e., g sa , g sc , g ma and g mc for deterministic weighted fusion driven by Equations (2) and (3), while it is also trained with g sf for stochastic feature fusion. The backbone feature extractors for both sensor streams and the transformer block are pre-trained on MSCOCO dataset on RGB images as in [10]. For the FLIR thermal dataset, each model is trained on a cluster with 2 GPUs for 100 epochs while the models for the SUNRGB-D dataset are trained with 4 GPUs for 300 epochs. Similarly, the models for L515-Indoor are trained on a cluster with 2 GPUs for 300 epochs with a batch size of 1. The batch size for the FLIR-Thermal and SUNRGBD datasets is set to 2, while the learning rate for the feature extractor backbones, fusion networks, and transformer block is set to 8 × 10 −6 for all datasets. We employ ResNet-50 as the feature extractor, while we also train g sc on MobileNet v2 [31] for the FLIR thermal dataset. To guide the fusion process and mimic harsh lighting conditions for the RGB sensor, we also employ Random Shadows and Highlights (RSH) data augmentation as proposed in [7]. RSH develops immunity against lighting perturbations in the convolutional neural networks, which is desirable for real world applications. We additionally implement SSD512 object detector with VGG16 backbones in a multi-modal setting in two configurations, i.e., a simple averaging fusion as the baseline method (SSD-BL) and a weighted averaging fusion scheme (SSD-WA) similar to g sa . The anchor/default boxes are configured for both SSD-BL and SSD-WA in a fashion similar to that for the MS-COCO dataset. These models are trained for 800 epochs in a single GPU setup with a batch size 1 and a learning rate 1×10 −4 which decays with a decaying factor 0.2 after the first 520,000 iterations.
D. Evaluation
FLIR-Thermal: performance evaluation of the proposed networks on the FLIR-Thermal dataset is shown in Table I. We show Average Precision (AP) values at Intersection over Union (IoU) of 0.5 for each dominant class, while the mean Average Precision (mAP) is also estimated with and without lighting perturbations. These lighting corruptions are introduced by creating Random Shadows and Highlights The results are compared with the single modality object detector, the multi-modal baseline fusion networks, and the existing state-of-the-art methods on this dataset. In the baselines, the features from the backbones are fused in two configurations: averaged and concatenated, without any weighing or re-sampling mechanism. Additionally, we compare the performance of SSD-BL and SSD-WA on the FLIR-Thermal dataset. It is clear from the evaluation results, that our proposed methodologies, i.e., g sa , g sc , g ma , g mc , and g sf , outperform the previously reported results on this dataset. Our methods also demonstrate robustness against lighting perturbation, while a significant performance drop of single modality and baseline methods can be seen when tested with RSH. The "avg-baseline" obtained comparable results, but as it is only a blind fusion, hence no sensor contribution or reliability measure can be obtained with this methodology. Additionally, its performance can significantly drop in the case of asymmetric sensor failure. This can partially be observed in Table I where the baselines are tested with RSH perturbations. Concerning the evaluation of multi-modal SSD on the FLIR-Thermal dataset, SSD-WA certainly improves the results compared to SSD-BL, specifically in terms of robustness against lighting perturbations introduced by RSH. The overall performance of SSD-based detectors turned out to be inferior to that of our transformer-based multi-modal object detection methods. Among our proposed fusion methods, g sa obtained the best overall performance on the FLIR-Thermal dataset. Scalar multiplication amplifies the information in the feature maps by retaining the learned structure. Nevertheless, mask multiplication may amplify a certain spatial portion of the feature maps in some channels, but it could also potentially distort the learned information depth-wise. Concatenation might be useful when the feature spaces of the utilized sensor modalities differ, e.g., image versus point cloud. However, in our case of image modalities, the averaging features g sa performed better than concatenation g sc . Similarly, switching off the features with selective fusion g sf has affected the performance of the model adversely. We plan to explore this method further in our future research, especially in the cases when information from the sensor modalities of dissimilar domains are fused, e.g., camera versus LiDAR/radar. SUNRGB-D: The evaluation results on SUNRGB-D dataset are shown in Table II. We not only present a comparison of single vs. multi-modal settings on the selected 19 categories of the SUNRGB-D dataset, but also between raw vs. processed depth images. The table only shows the results for eight categories due to limited space. Two single modality networks are trained, one with RGB images and the other with HHA-encoded depth images. We also evaluate the performance of "conc-baseline" and "avg-baseline" with RGB and HHA-encoded depth modalities. Motivated by the performance of g sa and g sc on the FLIR-Thermal dataset, we chose to evaluate their performance on SUNRGB-D dataset exclusively. Since HHA-encoding introduces a significant computational burden inhibiting the possibility of real-time object detection, we first train g sa and g sc with on RGB and HHA-encoded depth images, later we fine-tune these models on raw-depth images as well as on our "r-blended" hybrid depth images. It is evident in Table II that both g sa and g sc obtain promising results on this dataset with RGB and "r-blended" depth images. Further analysing the results of Table I, we observe that the comparative performance of the models on the Bicycle class is not stable. Looking at the distribution of the datasets in Figure 4, we realize that the Bicycle class only constitutes 8.23% of the dataset. This indicates its comparative inconsistent performance on various models. However, analysing the results in Table II, we realize this performance instability might also be related to the object size. The proposed networks are able to distinguish large sized objects even if their contribution in the dataset is relatively small e.g., Baththub and Bed classes. This problem can be traced back to [10] which itself struggles to perform equally on detecting small sized objects. L515-Indoor: Table III presents the evaluation results of L515-Indoor dataset. We tested the performance RGB-only and IR-only networks, as well as the g sa variant of GEM on this dataset. Evidently, g sa outperformed both single modality detectors providing an additional functionality of switching between the dominant sensors in changing lighting conditions. The performance of g sa with MobileNet v2 backbone is also presented in the table. The qualitative results on all three datasets are shown in Figures 5 and 6.
MobileNet v2: On a mobile platform having a 12-Core Intel ® Core TM i7-9750H CPU, and Nvidia GeForce RTX 2080 GPU, with ResNet-50 backbones, it takes approx 106.0 ms for a single forward pass on the proposed multi-modal object detector. However, with MobileNet v2 backbone feature extractors, the time for a single forward pass reduces to 49.7 ms obtaining approximately 20.1 fps. The drop in prediction accuracy of the deep models with the decrease in the number of network parameters for faster detection speed, is a well-known dilemma (e.g., in our case 23 million parameters for ResNet-50 to 3.4 million parameters for MobileNet v2). A compromise on prediction accuracy should only be made in non-critical cases where human safety is not at stake. Otherwise, the use of lightweight backbones should be avoided V. CONCLUSION In this paper, we propose GEM, a novel sensor-aware multi-modal object detector, with immunity against adverse lighting scenarios. Among the proposed sensor fusion configurations, the scalar averaging variant of the deterministic weighted fusion outscored the state-of-the-art and other fusion methods. The mask multipliers may amplify a certain spatial portion of the feature maps, but could also potentially distort the learned features depth-wise. Concatenation might be useful in cases where the feature spaces of the utilized sensor modalities differ. Regarding RGB-D data, the proposed "r-blended" hybrid depth modality has proven to be a promising and lightweight alternative to the commonly employed HHA-encoded depth images. However, instead of employing a fixed blend weight α, dynamic adaptation driven by ambient light intensity could demonstrate a more realistic use of the proposed hybrid image. GEM brings along the shortcomings of [10] in multi-modal object detection setting as well, e.g., it struggles to detect small objects and suffers from the computational complexity of the attention layers. These issues will be addressed in the future work.
Cognitive Robotics Department, Delft University of Technology, Delft, The Netherlands {O.Mazhar, R.Babuska, J.Kober}@tudelft.nl
Fig. 1 .
1Output samples of the proposed multi-modal object detector. The blue/green bar at the top illustrates the contribution/reliability of each sensor modality in obtaining the final output. Images from two modalities are merged diagonally only for illustration purposes. (a) Shows the results on the FLIR-Thermal dataset with RGB and thermal sensor modalities, (b) Shows the output on the SUNRGB-D dataset with RGB and our proposed "r-blended" hybrid depth modality.
Fig. 2 .
2Our proposed pipeline for a multi-modal object detector with transformers. The features from each backbone are fused and passed to the transformer encoder-decoder network. The decoder output is subsequently exploited by Multilayer Perceptrons (MLPs) for bounding box regression and object classification.
Fig. 3 .
3Illustration of our stochastic feature fusion strategy that employs the Gumbel-Softmax sampling trick.
Fig. 4 .
4Class distributions of the datasets (a) L515-Indoor (b) FLIR-Thermal (c) SUNRGB-D. The number of annotations in (c) are presented in the logarithmic scale.
Fig. 5 .
5Qualitative analysis of our multi-modal object detector, gsa in this case. Columns (a), (b) and (d) are the outputs of gsa in various asymmetric sensor failure conditions imitated artificially, which are mentioned on the upper-right corner of each image in row I. The top blue/green bar represents the contribution of each sensor modality in obtaining the final results (RGB: blue and Thermal/Infra: green). (c) and (e) are the outputs from single modal baselines. (f) is the ground-truth. Rows I and II are from FLIR-Thermal dataset while III and IV are from L515-Indoor dataset. Row IV represents a true sensor failure case when IR camera gets saturated due to sun-light even in indoors.
Fig. 6 .
6(a) Sample output of GEM (gsc) on the SUNRGB-D dataset with RGB images and "r-blended" depth modality. In (b), the output of single modal object detector trained only on RGB images is shown, while (c) is the ground truth.
TABLE I PERFORMANCE
IEVALUATION ON FLIR-THERMAL DATASET.Model
w/ RSH
w/o Random Shadows and Highlights
w/ RSH
AP@IoU=0.5
mAP@
IoU=0.5
mAP@
IoU=0.5
Person Bicycle
Car
FLIR Baseline
0.794
0.580
0.856
0.743
-
rgb-only
0.383
0.168
0.638
0.395
0.376
thermal-only
0.683
0.499
0.783
0.655
0.316
MM-UNIT [19]
0.644
0.494
0.707
0.615
-
MM-CG [19]
0.633
0.502
0.706
0.614
-
SSD-BL
0.450
0.341
0.719
0.503
0.478
SSD-WA
0.526
0.314
0.718
0.519
0.516
avg-baseline
0.801
0.562
0.879
0.747
0.731
conc-baseline
0.533
0.417
0.675
0.541
0.492
gsa
0.828
0.593
0.891
0.770
0.769
gsc
0.809
0.637
0.862
0.769
0.764
gma
0.803
0.575
0.862
0.746
0.744
gmc
0.800
0.611
0.857
0.755
0.755
g sf
0.790
0.584
0.874
0.749
0.756
gsc m-net
0.696
0.472
0.823
0.663
0.664
(RSH) on the test RGB images. The evaluation with lighting
perturbation is performed for 10 trials in all experiments,
while the average of the obtained mAP is shown in the
table.
TABLE II PERFORMANCE
IIEVALUATION ON SUNRGB-D DATASET.Models
Tested without Random Shadows and Highlights (RSH)
w/ RSH
AP@IoU=0.5
mAP@
IoU=0.5
mAP@
IoU=0.5
bathtub
bed
bookshelf
box
chair
. . .
door
dresser
lamp
night stand
RGB-only
0.116
0.461
0.038
0.084 0.457 . . .
0.370
0.085
0.185
0.095
0.224
0.169
HHA-only
0.355
0.409
0.002
0.020 0.413 . . .
0.113
0.024
0.199
0.057
0.165
0.093
conc-baseline
0.333
0.440
0.002
0.068 0.456 . . .
0.367
0.056
0.195
0.041
0.211
0.155
avg-baseline
0.174
0.461
0.032
0.062 0.470
0.339
0.030
0.220
0.049
0.207
0.166
gsc(raw-depth)
0.404
0.411
0.008
0.073 0.487 . . .
0.360
0.051
0.225
0.044
0.226
0.209
gsc(r-blended)
0.350
0.457
0.040
0.085
0.490
. . .
0.381
0.107
0.226
0.108
0.242
0.230
gsa(raw-depth)
0.204
0.399
0.033
0.087
0.478
0.344
0.102
0.220
0.025
0.221
0.214
gsa(r-blended)
0.253
0.423
0.106
0.080 0.474
0.379
0.035
0.219
0.079
0.239
0.236
TABLE III
IIIPERFORMANCE EVALUATION ON L515-INDOOR DATASET.Model
w/ RSH
w/o Random Shadows and Highlights
w/ RSH
AP@IoU=0.5
mAP@
IoU=0.5
mAP@
IoU=0.5
Chair
Cycle
Bin
Laptop
rgb-only
0.909
0.912
0.920 0.911
0.912
0.769
ir-only
0.141
0.557
0.012 0.690
0.386
0.311
gsa m-net 0.851
0.895
0.705 0.740
0.811
0.685
gsa
0.968
0.998
0.997 0.979
0.982
0.945
Czech Institute of Informatics, Robotics, and Cybernetics, Czech Technical University in Prague, Czech Republic
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|
[
"Cones with a Mapping Cone Symmetry in the Finite-Dimensional Case",
"Cones with a Mapping Cone Symmetry in the Finite-Dimensional Case"
]
| [
"Lukasz Skowronek \nInstytut Fizyki im. Smoluchowskiego\nUniwersytet Jagielloński\nReymonta 430-059KrakówPoland\n"
]
| [
"Instytut Fizyki im. Smoluchowskiego\nUniwersytet Jagielloński\nReymonta 430-059KrakówPoland"
]
| []
| In the finite-dimensional case, we present a new approach to the theory of cones with a mapping cone symmetry, first introduced by Størmer. Our method is based on a definition of an inner product in the space of linear maps between two algebras of operators and the fact that the Jamio lkowski-Choi isomorphism is an isometry. We consider a slightly modified class of cones, although not substantially different from the original mapping cones by Størmer. Using the new approach, several known results are proved faster and often in more generality than before. For example, the dual of a mapping cone turns out to be a mapping cone as well, without any additional assumptions. The main result of the paper is a characterization of cones with a mapping cone symmetry, saying that a given map is an element of such cone if and only if the composition of the map with the conjugate of an arbitrary element in the dual cone is completely positive. A similar result was known in the case where the map goes from an algebra of operators into itself and the cone is a symmetric mapping cone. Our result is proved without the additional assumptions of symmetry and equality between the domain and the target space. We show how it gives a number of older results as a corollary, including an exemplary application. | 10.1016/j.laa.2011.01.019 | [
"https://arxiv.org/pdf/1008.3237v2.pdf"
]
| 119,253,583 | 1008.3237 | 2c2e03699f3395b58b25225aa463b19c2bf89eb5 |
Cones with a Mapping Cone Symmetry in the Finite-Dimensional Case
Lukasz Skowronek
Instytut Fizyki im. Smoluchowskiego
Uniwersytet Jagielloński
Reymonta 430-059KrakówPoland
Cones with a Mapping Cone Symmetry in the Finite-Dimensional Case
mapping conespositive mapsconvex geometry
In the finite-dimensional case, we present a new approach to the theory of cones with a mapping cone symmetry, first introduced by Størmer. Our method is based on a definition of an inner product in the space of linear maps between two algebras of operators and the fact that the Jamio lkowski-Choi isomorphism is an isometry. We consider a slightly modified class of cones, although not substantially different from the original mapping cones by Størmer. Using the new approach, several known results are proved faster and often in more generality than before. For example, the dual of a mapping cone turns out to be a mapping cone as well, without any additional assumptions. The main result of the paper is a characterization of cones with a mapping cone symmetry, saying that a given map is an element of such cone if and only if the composition of the map with the conjugate of an arbitrary element in the dual cone is completely positive. A similar result was known in the case where the map goes from an algebra of operators into itself and the cone is a symmetric mapping cone. Our result is proved without the additional assumptions of symmetry and equality between the domain and the target space. We show how it gives a number of older results as a corollary, including an exemplary application.
Introduction
Mapping cones were introduced by Størmer in [1] as a way to better understand the structure of positive maps. They are an abstract notion mimicking a well-known property of the cones of positive and more generally, k-positive maps. Namely, for any k-positive map Φ and a pair of completely positive maps Υ and Ω, the map Υ • Φ • Ω is k-positive again. Størmer called a closed cone C, different from 0, in the space of positive linear maps from an algebra B (K) of bounded operators on a Hilbert space K into itself a mapping cone [1] if and only if for all Φ ∈ C and a, b ∈ B (K), the map x → aΦ (bxb * ) a * is an element of C again. Equivalently in the case of finite-dimensional K, Υ • Φ • Ω is an element of C for arbitrary completely positive maps Υ, Ω of B (K). In contrast to the original paper [1], in the following we do not only consider cones of maps from B (K) into itself, but also into B (H) for another Hilbert space H. Most of the time, we use the convexity assumption, which was absent in [1], but is actually very much in line of Størmer's later work on mapping cones. Even though the mentioned differences are not substantial, we shall stick to the term "cones with a mapping cone symmetry" (mcs-cones for short) in order to give sufficient credit to [1].
In a number of recent papers [2][3][4][5], Størmer and coauthors proved various characterization theorems for mapping cones. Some of them are of special interest because they relate purely geometrical properties to properties of algebraic nature. In particular, they reveal an intrinsic link between the condition that a product of two maps is completely positive and the fact that the two maps belong to a pair of dual mapping cones (cf. [3] and [5]). In the present paper, we aim at a similar, and very general characterization for mcs-cones. We also use a new approach that allows us to prove results more quickly and to directly exploit the mapping cone symmetry. Our methods work well in the finite-dimensional setting, whereas their applicability to the infinite-dimensional case is not obvious.
Basic notions
Let K and H be two Hilbert spaces. We denote with ., . the inner product in K or H. In the following, we shall assume that K and H are finite-dimensional and thus equivalent to C m and C n for some m, n ∈ N, dim K = m, dim H = n. We also fix orthonormal bases {f j } m j=1 and {e i } n i=1 of K and H, resp. Thus we have a very specific setting for our discussion, but we shall keep the abstract notation of Hilbert spaces, hoping to bring the attention of the reader to possible generalizations to the infinite-dimensional case. Let us denote with B (K) and B (H) the spaces of bounded operators on K and H resp. and choose their canonical bases
{f kl } m k,l=1 , {e ij } n i,j=1 .
That is, f kl (e j ) = δ lj f k and similarly for the e ij . Positive elements of B (K) are
operators A ∈ B (K) such that v, A (v) 0 ∀ v∈H .
Similarly for elements of B (H). The sets of positive elements of B (K) and B (H) will be denoted by B (K) + and B (H) + . In the finite-dimensional case, there exists a natural inner product in B (K), given by the formula
A, B := Tr (AB * )(1)
for A, B ∈ B (K). An identical definition works for A, B ∈ B (H) and we do not distinguish notationally between the inner products in B (H) and B (K). Note that the bases {f kl } m k,l=1 and {e ij } n i,j=1 are orthonormal with respect to ., . .
In the following, we will be mostly dealing with linear maps from B (K) to B (H). Because of the finitedimensionality assumption, they are all elements of B (B (K) , B (H)), the space of bounded operators from B (K) to B (H). Given a map Φ ∈ B (B (K) , B (H)), we define its conjugate Φ * as a map from B (H) into B (K) satisfying A, Φ (B) = Φ * (A) , B for all A ∈ B (H) and B ∈ B (K). In our setting, there also exists a natural inner product in B (B (K) , B (H)), given by the formula
Φ, Ψ := m k,l=1 Φ (f kl ) , Ψ (f kl ) .
(
Note that the spaces B (B (H) , B (K)), B (B (K)) and B (B (H)) can be endowed with analogous inner products and we shall not notationally distinguish between them. The following proposition summarizes a few elementary facts about ., . that will be useful for our later discussion.
Proposition 1. For all Φ, Ψ ∈ B (B (K) , B (H)) and α ∈ B (B (H)), β ∈ B (B (K)), one has the following equalities
1. Φ • β, Ψ = β, Φ * • Ψ = Ψ * • Φ, β * , 2. α • Φ, Ψ = α, Ψ • Φ * = Φ • Ψ * , α * , 3. α • Φ • β, Ψ = Φ, α * • Ψ • β * .
Proof. The first equality in point one follows directly from Φ
• β (f kl ) , Ψ (f kl ) = β (f kl ) , Φ * • Ψ (f kl )
and the definition of ., . , eq. (2). To prove the other equalities, we can use a simple lemma.
Φ, Ψ = Ψ * , Φ *(3)
Proof. Starting from the definition of ., . , we get
Φ, Ψ = m k,l=1 Φ (f kl ) , Ψ (f kl ) = n i,Ψ ij,rs f r,s , Φ ij,kl f kl = n i,j=1 Ψ * (e ij ) , Φ * (e ij ) ,(4)
where the last equality follows because Φ * (e ij ) = m k,l=1 Φ ij,kl f kl as a consequence of f kl , Φ * (e ij ) = Φ (f kl ) , e ij = m r,s=1 Φ rs,kl e rs , e ij = Φ ij,kl . Similarly, Ψ * (e ij ) = m r,s=1 Φ ij,rs f rs holds. The final expression in (4) clearly equals Ψ * , Φ * .
Note that the assertion of Lemma 1 holds for any choice of K and H, and thus also when the two finitedimensional Hilbert spaces are different from the K and H referred to in the statement of the proposition. Using the lemma, we get β, Φ * • Ψ = Ψ * • Φ, β * , which proves the second equality in point one. Furthermore,
α • Φ, Ψ = Ψ * , Φ * • α * = Φ * • α * , Ψ * = α * , Φ • Ψ * = Φ • Ψ * , α * = α, Ψ • Φ * ,(5)
where we successively used Lemma 1, the conjugate symmetry of ., . , the first equation in point one, the conjugate symmetry again, and finally Lemma 1 for the second time. Obviously, the first, the fifth and the sixth term in equation (5) are the same as in point two of the proposition. Hence the only remaining thing to prove is point three. We have
α • Φ • β, Ψ = α, Ψ • β * • Φ * = β • Ψ * • α, Φ * = Φ, α * • Ψ • β * ,(6)
where we used the properties α
• Φ, Ψ = α, Ψ • Φ * with Φ → Φ • β, β, Φ * • Ψ = Φ • β, Ψ with β → α, Φ → β
• Ψ * and Ψ → Φ * , and finally Lemma 1.
Consider the tensor product K ⊗ H. This space has a natural inner product, inherited from K and H, and an orthonormal basis {f kl ⊗ e ij } n;m i,j=1;k,l=1 . Similarly to B (K) and B (H), the space B (K ⊗ H) of bounded operators on K ⊗ H is endowed with a natural Hilbert-Schmidt product, defined by formula (1) with A, B ∈ B (K ⊗ H). We shall again denote the inner product with ., . to avoid excess notation. There exists a one-to-one correspondence between linear maps Φ of B (K) into B (H) and elements of B (K ⊗ H), given by
Φ → C Φ := m k,l=1 f kl ⊗ Φ (f kl ) .(7)
The symbol C Φ denotes the Choi matrix of Φ [6] and the mapping J : Φ → C Φ is sometimes called the Jamio lkowski-Choi isomorphism [7]. In fact, J is not only an isomorphism, but also an isometry between B (B (K) , B (H)) and B (K ⊗ H) in the sense of Hilbert-Schmidt type inner products. One has the following Property 1. The Jamio lkowski-Choi isomorphism is an isometry. One has
Φ, Ψ = C Φ , C Ψ (8) for all Φ, Ψ ∈ B (B (K) , B (H)) (with C Φ , C Ψ ∈ B (K ⊗ H)).
Proof. By the definition of C Φ and C Ψ ,
C Φ , C Ψ = m k,l=1 f kl ⊗ Φ (f kl ) , m r,s=1 f rs ⊗ Ψ (f rs ) = . . .(9)
Since
Tr (A ⊗ A ) (B ⊗ B ) * = Tr (AB * ) Tr (A B * ) for arbitrary A, B ∈ B (K) and A , B ∈ B (H), by formula (1) we have . . . = m k,l=1 m r,s=1 f kl , f rs Φ (f kl ) , Ψ (f rs ) = m k,l=1 Φ (f kl ) , Ψ (f kl ) ,(10)
where we used orthonormality of {f kl } m k,l=1 . The last expression equals Φ, Ψ by definition (2).
A linear map Φ from B (K) to B (H) is called positive if it preserves positivity of operators, which means Φ B (K) + ⊂ B (H) + . Moreover, Φ is called k-positive if Φ ⊗ id M k (C) is positive as a map from B (K) ⊗ M k (C) into B (H) ⊗ M k (C), where M k (C) denotes the space of k × k matrices with complex entries
and id refers to the identity map. A map Φ is called completely positive if it is k-positive for all k ∈ N.
From the Choi's theorem on completely positive maps [6] (cf. also Lemma 2) it follows that every such map has a representation Φ = i Ad Vi as a sum of conjugation maps,
Ad Vi : ρ → V i ρV * i with V i ∈ B (K, H)
. Conversely, every map Φ of the form i Ad Vi is completely positive. If all the V i 's can be chosen of rank k for some k ∈ N, Φ is said to be k-superpositive [5]. One-superpositive maps are simply called superpositive [8]. The sets of positive, k-positive, completely positive, k-superpositive and superpositive maps from
B (K) to B (H) will be denoted with P (B (K) , B (H)), P k (B (K) , B (H)), CP (B (K) , B (H)), SP k (B (K) , B (H)), SP (B (K) , B (H)) or P, P k , CP, SP k , SP for short.
It is clear that all of them are closed convex cones contained in P (B (K) , B (H)). They also share a more special property that the product Υ • Φ • Ω of Φ ∈ C, Υ ∈ CP (B (H)) and Ω ∈ CP (B (K)) is an element of C again, where C stands for one of the sets P, P k , CP, SP k and SP (cf. e.g. [5]). Thus, following rather closely the original definition by Størmer [1], a cone with a mapping cone symmetry, or an mcs-cone for short, is defined as a closed convex cone C in P (B (K) , B (H)), different from 0, such that Υ • Φ • Ω ∈ C for all Φ ∈ C, Υ ∈ CP (B (H)) and Ω ∈ CP (B (K)). In the following, the convexity assumption could sometimes be skept, and we do include appropriate comments.
Note that the set of positive maps from B [9]. Therefore ., . induces a symmetric inner product on HP (B (K) , B (H)) (cf. Property 1). By definition, all mapping cones are subsets of P and thus of HP. Since HP is a finitedimensional space over R with a symmetric inner product ., . , one can easily apply to it tools of convex analysis. In particular, given any cone C ⊂ HP, one defines its dual C • as the cone of elements Ψ ∈ HP such that Ψ, Φ 0 for all Φ ∈ C,
(K) into B (H) is contained in the real-linear subspace HP (B (K) , B (H)) ⊂ B (B (K) , B (H)) (HP for short) consisting of all Hermiticity-preserving maps, i.e. Φ such that Φ (X * ) = Φ (X) * . Moreover, the image of HP (B (K) , B (H)) by J : Φ → C Φ equals the set of self- adjoint elements of B (K ⊗ H)C • := Ψ ∈ HP (B (K) , B (H)) Ψ, Φ 0 ∀ Φ∈C .(11)
Obviously, C • is closed and convex. It has a clear geometrical interpretation as the convex cone spanned by the normals to the supporting hyperplanes for C. The dual cone has a well-known counterpart in convex analysis [10], C = −C • , which is called the polar of C. We have the following Property 2. Let C be a closed convex cone. Then
C = C •• .(12)
Proof. Formula (2) is equivalent to C = C for a closed convex cone C. The latter equality is a known fact in convex analysis. A proof can be found e.g. in [10] (Theorem 14.1).
It can be shown (cf. e.g. [5]) that a duality relation P • k = SP k holds for all k ∈ N. The converse relation SP • k = P k is also true, as a consequence of Property 2. In particular, for k = 1 we get SP • = P and P • = SP. Taking k = min {m, n}, one obtains CP • = CP, which is in accordance with Choi's theorem on completely postive maps [6] and with Property 1.
In the following, we shall be interested in duality relations between mcs-cones. This is in general a well-posed problem, because the operation C → C • acts within the "mcs" class. We have Proposition 2. Let C ⊂ P (B (K) , B (H)) be an arbitrary mcs-cone. Then C • , defined as in (11), is an mcs-cone as well.
Proof. Let Ψ be an element of C • . First we prove that Υ • Ψ • Ω ∈ C • for all Υ ∈ CP (B (H)) and Ω ∈ CP (B (K)). We have Υ * ∈ CP (B (H)) and Ω * ∈ CP (B (K)) because the sets of completely positive maps are * -invariant. Therefore Υ * • Φ • Ω * ∈ C for an arbitrary element Φ of the cone C. By the definition (11) of C • , we have Ψ, Υ * • Φ • Ω * 0 ∀ Φ∈C . Using Proposition 1, point three, we can rewrite this as
Υ • Ψ • Ω, Φ 0 ∀ Φ∈C .(13)
According to definition (11), condition (13) means that Υ • Ψ • Ω ∈ C • . This holds for arbitrary Υ ∈ CP (B (H)) and Ω ∈ CP (B (K)). The only thing which is left to prove is C • ⊂ P (B (K) , B (H)). The inclusion holds because every mcs-cone C contains all the conjugation maps Ad V with rk V = 1. Consequently,
C • ⊂ convhull {Ad V rk V = 1} • = SP • = P.
To show that indeed {Ad V rk V = 1} ⊂ C for any mcscone C, take an arbitrary nonzero Φ ∈ C. There must exist normalized vectors υ ∈ K and ω ∈ H such that p ω , Φ (p υ ) 0, where p υ and p ω are orthogonal projections onto the one-dimensional subspaces spanned by υ and ω. Denote χ := p ω , Φ (p υ ) . Consider a pair of maps, U : K a → a, υ υ ∈ K and W : H b → b, ω ω ∈ H, where υ and ω are arbitrary normalized vectors in K and H. A map Φ , defined as λ/χ (Ad W • Φ • Ad U ) acts in the following way, Φ : ρ → λ p υ , ρ p ω or Φ = Ad V with V : K c → λ υ , c ω . Any rank one operator V can be written in the latter form for some υ and ω . But Φ is an element of C because of the assumption that C is an mcs-cone. Thus indeed Ad V ∈ C for all V ∈ B (K, H) such that rk V = 1. In the case of K = H and mapping cones C as in the original definition by Størmer, the inclusion Ad V ∈ C follows from Lemma 2.4 in [1]. It should be kept in mind that we never used convexity of C in the proof.
Note that a version of Proposition 2 was proved in [11] using different methods, with the additional assumption of H = K and C being a symmetric mapping cone. It is instructive to see how that result of [11] follows using our method. First, note that a mapping cone C ⊂ P (B (K)) is called symmetric [11] if C = C * = C t , with C * := {Φ * |Φ ∈ C} and C t := {t • Φ • t|Φ ∈ C}, where t stands for the transposition map, t : B (K) f kl → f lk ∈ B (K). We have the following Proposition 3. Consider the case H = K. Let C be a symmetric mapping cone of maps from B (K) into itself, i.e. C ∈ P (B (K)). The dual C • is also a symmetric mapping cone.
Proof. By Proposition 2 and the redundancy of the convexity assumption, we know that C • is an mcs-cone of maps from B (K) into itself, and thus a mapping cone in the sense of [1]. We only need to show that it is symmetric. Let Ψ be an arbitrary element of C • . By the symmetry C t = C, we know that the condition Ψ, Φ 0 ∀ Φ∈C is equivalent to Ψ, t • Φ • t 0 ∀ Φ∈C . By Proposition 1, point three, this is the same as
t • Ψ • t, Φ 0 ∀ Φ∈C , or t • Ψ • t ∈ C • . Thus (C • ) t = C • .
To show (C • ) * = C • , one only needs to note that Ψ, Φ = Φ * , Ψ * by Lemma 1. Now, the property C * = C can be used.
The main theorem
Using the properties discussed in the previous section, we can almost immediately prove a surprising characterization theorem for mcs-cones, which was strongly suggested by earlier results on the subject [3,5,11]. It holds without any additional assumptions about the cone, and is noteworthy as it links the condition that two maps Φ, Ψ lay in a pair of dual mcs-cones to the fact that the product Ψ * • Φ is a CP map. Thus it reveals a connection between convex geometry and a fact which is more likely to be called algebraic than geometrical. Before we proceed with the proof, let us show a simple lemma, which is a version of [12, Lemma 1(i)] for K = H.
Lemma 2. Let V : K a → n i=1 m j=1
V ij a, f j e i ∈ H be an arbitrary operator in B (B (K) , B (H)) and consider the map Ad V : ρ → V ρV * . Then
C Ad V = |υ υ| ,(14)
where υ = n i=1 m j=1 V ij f j ⊗ e i is a vector in K ⊗ H and |υ υ| : w → w, υ υ is proportional to an orthogonal projection onto the subspace spanned by υ.
Proof. Obviously, the map V * acts in the following way,
V * : H b → n i=1 m j=1 V ij b, e i f j ∈ K. Thus V f kl V * : H b → n i,r=1 m j,s=1 V rs f kl (f j ) , f s V ij b, e i e r ∈ H,(15)
where the last expression is easily verified to be equal to n i,r=1 V rk V il b, e i e r . Thus we have V f kl V * = n i,r=1 V rk V il e ri and by the definition (7) of the Choi matrix,
C Ad V = m k,l=1 n i,r=1 V rk V il f kl ⊗ e ri = |υ υ| ,(16)with υ = n i=1 m j=1 V ij f j ⊗ e i .
A proof of the last equality in (16) is left as an elementary exercise for the reader.
We are ready to prove the following (cf. Theorem 1 in [3]). B (H)) be an mcs-cone. The following conditions are equivalent,
Theorem 1. Let C ⊂ P (B (K) ,1. Φ ∈ C, 2. Ψ * • Φ ∈ CP (B (K)) for all Ψ ∈ C • , 3. Φ • Ψ * ∈ CP (B (H)) for all Ψ ∈ C • .
Proof. We first show 1 ⇔ 2. Let us start with 2 ⇒ 1. Since Ψ * • Φ ∈ CP ∀ Ψ∈C • , we can use the facts that CP • = CP and id ∈ CP to get
Ψ * • Φ, id 0 ∀ Ψ∈C • .(17)
By using point one of Proposition 1 with the identity map id substituted for β, we get Φ, Ψ 0 ∀ Ψ∈C • , which means that Φ ∈ C •• . But C •• = C because C is a closed convex cone and Property 2 holds. Hence Φ ∈ C. The proof of 1 ⇒ 2 strongly builds on the assumption that C has the mapping cone symmetry. By Proposition 2, we know that C • is an mcs-cone as well. Therefore Ψ • Ad V ∈ C • for an arbitrary Ψ ∈ C • and V ∈ B (K). We have Ψ
• Ad V , Φ 0 ∀ V ∈B(K) ∀ Ψ∈C • . By Proposition 1, point one, we get Ψ • Ad V , Φ = Ad V , Ψ * • Φ .
Using Property 1 and Lemma 2 with H = K, the last term can be rewritten as
Ad V , Ψ * • Φ = C Ad V , C Ψ * •Φ = |v v| , C Ψ * •Φ = υ, C Ψ * •Φ (υ) ,(18)where υ = m i,j=1 V ij f j ⊗f i for V : K a → m i,j=1 V ij a, f j f i ∈ K.
The vector υ ∈ K⊗K can be arbitrary, since we do not assume anything about the operator V . Consequently, the condition Ψ
• Ad V , Φ 0 ∀ V ∈B(K) ∀ Ψ∈C • is equivalent to υ, C Ψ * •Φ (υ) 0 ∀ υ∈K⊗K ∀ Ψ∈C • ,(19)
which means that C Ψ * •Φ ∈ B (K ⊗ K) + for all Ψ ∈ C • . By the Choi theorem on completely positive maps [6], Ψ * • Φ ∈ CP (B (K)) for all Ψ ∈ C • . Thus we have finished proving that 1 ⇔ 2. The proof of the equivalence 1 ⇔ 3 only needs a minor modification of the above argument. Instead of using point one of Proposition 1, point two of the same proposition has to be used. Other details are practically the same as above and we shall not give them explicitly.
In case of H = K and a * -invariant mcs-cone C ∈ P (B (K)), Theorem 1 can be further simplified.
Theorem 2. Let C ⊂ P (B (K)) be a * -invariant mcs-cone. Then the following conditions are equivalent,
1. Φ ∈ C, 2. Ψ • Φ ∈ CP (B (K)) for all Ψ ∈ C • , 3. Φ • Ψ ∈ CP (B (K)) for all Ψ ∈ C • .
Proof. Obvious from Theorem 1.
This result was earlier known for P k (B (K)) and SP k (B (K)) [5], and inexplicitly for all symmetric (and convex) mapping cones [11]. As it was pointed to the author by Erling Størmer, in the case of k-positive maps, not necessarily from B (K) into itself, an even stronger characterization of the type of Theorems 1 and 2 is valid. First, we have the simple Theorem 3. The following conditons are equivalent
1. Φ ∈ P k (B (K) , B (H)), 2. Ad V * • Φ ∈ CP (B (K)) for all V ∈ B (K, H) s.t. rk V k, 3. Φ • Ad V * ∈ CP (B (H)) for all V ∈ B (K, H) s.t. rk V k.
Proof. Obvious from Theorem 1. The duality relation
P k (B (K) , B (H)) • = SP k (B (K) , B (H)) = convhull {Ad V |V ∈ B (K, H) , rk V k}(20)
holds (cf. [5]) and we can substitute Ψ in Theorem 1 with Ad V , rk V k. We also use the elementary fact that Ad * V = Ad V * . The next result on k-positive maps seems to be less obvious.
Theorem 4. Denote with Π k (K) and Π k (H) the sets of k-dimensional projections in K and H, resp. The following conditons are equivalent
1. Φ ∈ P k (B (K) , B (H)), 2. Ad E • Φ ∈ CP (B (K) , B (H)) for all E ∈ Π k (H), 3. Φ • Ad F ∈ CP (B (K) , B (H)) for all F ∈ Π k (K), 4. Ad E • Φ • Ad F ∈ CP (B (K) , B (H)) for all E ∈ Π k (H), F ∈ Π k (K).
Proof. We shall prove the equivalence 1 ⇔ 4. The other ones follow analogously. Since CP • = CP and any CP map can be written as i Ad Vi with V i arbitrary, the condition Ad
E • Φ • Ad F ∈ CP (B (K) , B (H)) is equivalent to Ad E • Φ • Ad F , Ad V 0 ∀ E∈Π k (H),F ∈Π k (K) ∀ V ∈B(K,H) .(21)
By Proposition 1, point three, equation (21) can be rewritten as
Φ, Ad EV F 0 ∀ E∈Π k (H),F ∈Π k (K) ∀ V ∈B(K,H) ,(22)
where we used the fact that Ad E • Ad V • Ad F = Ad EV F and the self-adjointness of E and F . Note that U = EV F is an element of B (K, H) of rank k. Conversely, every map in U ∈ B (K, H) of rank k can be written in the form EV F for some V ∈ B (K, H), E ∈ Π k (H) and F ∈ Π k (K). It is sufficient to take V = U and E, F as the range and rank projections for U , resp. Therefore the condition (22) is equivalent to Φ, Ad U 0 for all U ∈ B (K, H) s.t. rk U 0. But this is the same as Φ, Ψ 0 for all Ψ ∈ SP k (B (K) , B (H)), or Φ ∈ SP k (B (K) , B (H)) • = P k (B (K) , B (H)). Thus 1 ⇔ 4.
Example 1.
A very instructive application of Theorem 4, due to Størmer, has recently been given in [12]. It concerns maps of the form Φ λ : ρ → Tr ρ · id −λ Ad V (ρ), or Φ λ = Tr −λ Ad V for short, where V ∈ B (K, H) and λ > 0. Consider first the question of complete positivity of such maps. It is not difficult to check that C Tr = 1. Thus by Lemma 2,
C Φ λ = 1 − λ |υ υ|, where υ = n i=1 m j=1 V ij f j ⊗ e i , V : K a → n i=1 m j=1
V ij a, f j e i ∈ H and 1 denotes the identity operator in K ⊗ H. According to the Choi theorem on completely positive maps [6], complete positivity of Φ λ is equivalent to positivity of C Φ λ . For an arbitrary vector w ∈ K ⊗ H, the product w, C Φ λ (w) equals w, w − λ | w, υ | 2 . Minimizing over vectors w of unit norm, we get
1 − λ | υ, υ |. But | υ, υ | = n i=1 m j=1 V ij V ij ,
which is the same as Tr (V V * ). Thus C Φ λ is positive, or equivalently, Φ λ completely positive if and only if λ Tr (V V * ) 1. Now we turn to the question about k-positivity of Φ λ . According to Theorem 4, condition 2., we need to check complete positivity of the map Ad E • Φ λ for all projections E of rank k. Interestingly, Ad E • Φ λ = E Tr −λ Ad EV = E (Tr −λ Ad EV ) E, which is the same as Φ λ save the E at both ends and EV in place of V . Actually, if we consider Ad E • Φ λ as a map from B (K) into B (EH), it is of the same form as Φ λ , with EV instead of V . Hence, using the result obtained above, Ad E • Φ λ is a CP map if and only if λ Tr EV (EV ) * = λ Tr (EV V * ) 1, where we skip a few technical details of the argument (cf. [12]). According to Theorem 4, we should maximize the expression on the left over all possible choices of E. The maximum turns out to be equal to λ times the square of the k-th Ky Fan norm of V (cf. e.g. [13]). Thus we rederive a result by Chruściński and Kossakowski concerning k-positivity of maps of the form Φ λ [14].
Conclusion
In the finite-dimensional setting, we presented a number of results concerning convex cones with a mapping cone symmetry, or "mcs-cones". Our focus was on convex cone duality. The use of a slightly modified class of cones, but much in the spirit of the original definition of a mapping cone by Størmer [1], allowed us to make very general statements. In Proposition 2, we showed that the dual of an mcscone is an mcs-cone. Our main result, which is the characterization included in Theorem 1, can be very loosely described as saying that the surface of mcs-cones has an additional structure, which makes them more "smooth". There is a stronger relation between a pair of dual cones with a mapping cone symmetry than the relation between a mere pair of dual convex cones. There also exist stronger versions of the main characterization theorem, valid for * -invariant cones (Theorem 2) and specifically for k-positive maps (Theorems 3 and 4). Example 1 shows a practical application.
It is natural to ask how large the class of mcs-cones is. We know that k-positive and k-superpositive maps (for k = 1, 2, . . .) provide examples of such cones, including completely positive maps. Another class includes the same cones multiplied by the transposition map, i.e. cones C • t := {Φ • t Φ ∈ C}, where C stands for any of P, P k , CP, SP k , SP. For example, elements of CP • t are called completely co-positive. We can also provide a variety of new examples by taking the intersection and the convex sum of any two known mapping cones C 1 and C 2 , one not included in another. Figure 1 shows roughly how it works for C 1 = CP and C 2 = CP • t. The key remaining question is Question 1. Are there any "untypical" mcs-cones?
By a typical mcs-cone we mean a cone which is obtained from P, P k , CP, SP k or SP using the mentioned operations C → C • t, (C 1 , C 2 ) → C 1 ∩ C 2 and (C 1 , C 2 ) → C 1 ∨ C 2 . So far, no answer to the question is known. Note that all typical mcs-cones are symmetric.
At this place, it seems desirable to shortly mention a connection between mcs-cones and matrix ordered * -vector spaces [15], or operator systems. For any mcs-cone C ∈ B (B (K)), one has a matrix ordering of B (K), given by the cones C C n := (X ij ) ∈ M n (B (K)) (Φ (X ij )) ∈ M n (B (K)) + ∀ Φ∈C for n = 1, 2, . . . . A similar definition could also be used in the case when C does not have a mapping cone symmetry, however mcs-cones seem to be a preferred choice, as they provide a definition of positivity of (X ij ) ∈ M n (B (K)) which is invariant under any conjugation map (X ij ) → (Ad V (X ij )) = (V X ij V * ). This is particularly natural if we think about the case n = 1. The mapping cone symmetry also takes care of a redundancy in the condition (Φ (X ij )) ∈ M n (B (K)) + ∀ Φ∈C allowing a conjugation of Φ, (Ad U •Φ (X ij )) ∈ M n (B (K)) + ∀ Φ∈C ∀ U ∈B(K) , to have no effect on C C n . Thus a left-invariance of C under conjugation can always be assumed when a matrix ordering given by the cones C C n for some C ⊂ B (B (K)) is considered. In many respects, orderings of this type provide a generalization of the OMIN k and OMAX k structures recently investigated in [16]. In that broader context, Question 1 alludes to the subject of possible operator system structures for B (K).
Throughout the paper, we used the definition (2) of an inner product in the space of linear maps from an algebra B (K) into B (H), where K, H stand for two Hilbert spaces and we assumed K and H to be finitedimensional. We also exploited the property that the Jamio lkowski-Choi isomorphism (cf. eq. (7)) is an isometry. Because of the finite-dimensionality assumption, the definition (2) was certainly correct and quite natural. The same for the Jamio lkowski-Choi isomorphism. If the dimension of K or H was not assumed to be finite, there will be serious problems with both definitions. Nevertheless, it seems that similar methods may work at least for some cones in the infinite-dimensional case, e.g. assuming that their elements are trace class in a proper sense.
In the end, let us mention that k-positive maps are of special interest in the theory of quantum information and computation, a very active branch of contemporary physics and information science (cf. e.g. [17]). The case k = 1 corresponds to entanglement witnesses [18] and k = 2 to undistillable quantum states, with the fundamental question about the existence of NPT bound entanglement [19].
Acknowledgement
Lemma 1 .
1For any finite-dimensional Hilbert spaces K, H and maps Φ, Ψ ∈ B (B (K) , B (H)), we have
Figure 1 :
1A family of four mapping cones, constructed out of the cone of completely positive maps. The set CP ∨ CP • t of decomposable maps is included.
The author would like to thank Erling Størmer, Marcin Marciniak and KarolŻyczkowski for collaboration and for comments on the manuscript. The project was operated within the Foundation for Polish Science International Ph.D. Projects Programme co-financed by the European Regional Development Fund covering, under the agreement no. MPD/2009/6, the Jagiellonian University International Ph.D. Studies in Physics of Complex Systems. Some of the results presented in the paper came to existence during a visit of the author to the University of Oslo, where he enjoyed hospitality at the Mathematics Institute and was financially supported by Scholarschip and Training Fund, operated by Foundation for the Development of the Education System.
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Cones of positive maps and their duality relations. L Skowronek, E Størmer, K Życzkowski, J. Math. Phys. 5062106L. Skowronek, E. Størmer, K.Życzkowski, Cones of positive maps and their duality relations, J. Math. Phys. 50 (2009) 062106.
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| []
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[
"Fast Interpretable Greedy-Tree Sums (FIGS)",
"Fast Interpretable Greedy-Tree Sums (FIGS)"
]
| [
"Yan Shuo Tan \nEqual contribution\n\n\nDepartment of Statistics\nUC Berkeley\nBerkeleyCaliforniaUSA\n",
"Chandan Singh \nEqual contribution\n\n\nEECS Department\nUC Berkeley\nBerkeleyCaliforniaUSA\n",
"Keyan Nasseri \nEECS Department\nUC Berkeley\nBerkeleyCaliforniaUSA\n",
"Abhineet Agarwal \nDepartment of Physics\nUC Berkeley\nBerkeleyCaliforniaUSA\n",
"Bin Yu \nDepartment of Statistics\nUC Berkeley\nBerkeleyCaliforniaUSA\n\nEECS Department\nUC Berkeley\nBerkeleyCaliforniaUSA\n"
]
| [
"Equal contribution\n",
"Department of Statistics\nUC Berkeley\nBerkeleyCaliforniaUSA",
"Equal contribution\n",
"EECS Department\nUC Berkeley\nBerkeleyCaliforniaUSA",
"EECS Department\nUC Berkeley\nBerkeleyCaliforniaUSA",
"Department of Physics\nUC Berkeley\nBerkeleyCaliforniaUSA",
"Department of Statistics\nUC Berkeley\nBerkeleyCaliforniaUSA",
"EECS Department\nUC Berkeley\nBerkeleyCaliforniaUSA"
]
| []
| Equal contribution | null | [
"https://arxiv.org/pdf/2201.11931v2.pdf"
]
| 246,411,594 | 2201.11931 | 054c685469088dc7f55194ff4dff51d163c55672 |
Fast Interpretable Greedy-Tree Sums (FIGS)
Yan Shuo Tan
Equal contribution
Department of Statistics
UC Berkeley
BerkeleyCaliforniaUSA
Chandan Singh
Equal contribution
EECS Department
UC Berkeley
BerkeleyCaliforniaUSA
Keyan Nasseri
EECS Department
UC Berkeley
BerkeleyCaliforniaUSA
Abhineet Agarwal
Department of Physics
UC Berkeley
BerkeleyCaliforniaUSA
Bin Yu
Department of Statistics
UC Berkeley
BerkeleyCaliforniaUSA
EECS Department
UC Berkeley
BerkeleyCaliforniaUSA
Fast Interpretable Greedy-Tree Sums (FIGS)
Equal contribution
Modern machine learning has achieved impressive prediction performance, but often sacrifices interpretability, a critical consideration in many problems. Here, we propose Fast Interpretable Greedy-Tree Sums (FIGS), an algorithm for fitting concise rule-based models. Specifically, FIGS generalizes the CART algorithm to simultaneously grow a flexible number of trees in a summation. The total number of splits across all the trees can be restricted by a pre-specified threshold, thereby keeping both the size and number of its trees under control. When both are small, the fitted tree-sum can be easily visualized and written out by hand, making it highly interpretable. A partially oracle theoretical result hints at the potential for FIGS to overcome a key weakness of single-tree models by disentangling additive components of generative additive models, thereby reducing redundancy from repeated splits on the same feature. Furthermore, given oracle access to optimal tree structures, we obtain 2 generalization bounds for such generative models in the case of C 1 component functions, matching known minimax rates in some cases. Extensive experiments across a wide array of real-world datasets show that FIGS achieves state-of-the-art prediction performance (among all popular rule-based methods) when restricted to just a few splits (e.g. less than 20). We find empirically that FIGS is able to avoid repeated splits, and often provides more concise decision rules than fitted decision trees, without sacrificing predictive performance. All code and models are released in a full-fledged package on Github. 1
Introduction
Modern machine-learning methods such as random forests [2], gradient boosting [3,4], and deep learning [5] display impressive predictive performance, but are complex and opaque, leading many to call them "black-box" models. This is unfortunate, as model interpretability is critical in many applications [6,7], particularly in highstakes settings such as medicine, biology, and policymaking. Interpretability allows models to be audited for general validation, errors, biases, and therefore also more amenable to improvement by domain experts. It facilitates counterfactual reasoning, which is the bedrock of scientific insight, and it instills trust/distrust in a model when warranted. As an added benefit, interpretable models tend to be faster and more computationally efficient than black-box models.
Decision trees are a prime example of interpretable models [1,3,[8][9][10]. They can be easily visualized and simulated even by non-experts, and thus fit naturally into the operating human-in-the-loop AI workflow of many organizations. While they are flexible, and thus have the potential to adapt to complex data, they often tend to be outperformed by black-box models in terms of prediction accuracy. However, this performance gap is not intrinsic to interpretable models, e.g. see examples in [1,[10][11][12]. Indeed, in this paper, we will show how this gap can be partially bridged by carefully examining how and why decision trees fall short, and then directly targeting these weaknesses.
Our starting point is the observation that decision trees can be statistically inefficient at fitting regression functions with additive components [13]. To illustrate this, consider the following toy example: y = 1 X1>0 +1 X2>0 ·1 X3>0 . 2 The two components of this function can be individually implemented by trees with 1 split and 2 splits respectively. However, implementing their sum with a single tree requires at least 5 splits, as we are forced to combine their tree structures in a fractal manner: a copy of the second tree has to be grown out of every leaf node of the first tree (see Fig 1). Indeed, it is easy to see that a single tree f implementing Iteration 1 Iteration 1 2 3 4 2 3 6 Potential splits Figure 1. FIGS algorithm overview for learning the toy function y = 1X 1 >0 + 1X 2 >0 · 1X 3 >0. FIGS greedily adds one node at a time, considering splits not just in an individual tree but within an ensemble of trees. This can lead to much more compact models, as it avoids repeated splits (e.g. in the final CART model shown in the top-right).
FIGS X 1 > 0 + X 1 > 0 + X 2 > 0 X 3 > 0 Best split CART X 1 > 0 X 1 > 0 + X 2 > 0 + X 1 > 0 X 2 > 0 X 1 > 0 X 2 > 0 X 3 > 0 X 2 > 0 X 3 > 0 . . .
the sum of independent tree functions f 1 , . . . , f k satisfies
#leaves(f ) ≥ K k=1 #leaves(f k ),
and so is much more complicated than simply encoding the function in terms of the original trees in the summation.
This need to grow a deep tree implies two statistical weaknesses of decision trees when fitting them to additive generative models. First, growing a deep tree greatly increases the probability of splitting on noisy features. Second, leaves in a deep tree contain fewer samples, which means that the tree predictions have higher variance. These two weaknesses could be avoided if we could fit a separate decision tree to each additive component of the generative model and present their sum as our model estimate. Existing ensemble methods are unable to disentangle the separate additive components, because they fit each tree either individually (random forests), or sequentially (gradient boosting).
To address these weaknesses, we propose Fast Interpretable Greedy-Tree Sums (FIGS), a novel yet natural algorithm which is able to grow a flexible number of trees simultaneously. This procedure is based on a simple modification to Classification and Regression Trees (CART) [8], allowing it to adapt to additive structure if present by starting new trees, while still maintaining the ability of CART to adapt to higher-order interaction terms. Meanwhile, the running time of FIGS remains largely similar to CART due to the similarity of the two algorithms. FIGS also remains interpretable by keeping the total number of splits in the model limited, allowing for the model to be easily visualized and simulated by hand.
While CART cannot achieve the minimax rate for fitting (generalized) additive generative models with C 1 component functions even with oracle access to the optimal tree structure [13], we show that FIGS can do so under this setting (Thm 1). In a population setting, we also show that FIGS is able to disentangle separate additive components (Thm 2) without any constraints on the component functions. We verify both theorems in finite-sample simulations, showing situations where FIGS even outperforms random forests. Meanwhile, extensive experiments across a wide array of real-world datasets show that FIGS achieves state-of-the-art performance while maintaining a concise, interpretable model (e.g. having less than 20 total splits).
In particular, they greatly improve upon the predictive performance of CART, and this improvement hints at the presence of approximate additive structure in many of these datasets.
In what follows, Sec 2 introduces FIGS, Sec 3 covers related work, Sec 4 establishes two theoretical results underpinning its performance, Sec 5 shows simulations supporting these two results, and Sec 6 shows extensive experiments suggesting FIGS predicts well with very few splits on real-world datasets.
FIGS: Algorithm description and runtime
FIGS proposes a natural but powerful extension to CART which forms a sum of trees rather than a single tree. The total number of splits in the model is restricted by a threshold (chosen either by a user or cross-validation). Given this threshold, the greedy algorithm flexibly determines how to allocate these splits among a variable number of trees.
Formally, suppose we are given training data
D n = {(x i , y i )} n i=1 .
When growing a tree, CART chooses for each node t the split s that maximizes the (weighted) im-purity decrease in the responses y. This has the formulâ
∆(s, t, y) := xi∈t (y i −ȳ t ) 2 − xi∈t L (y i −ȳ t L ) 2 − xi∈t R (y i −ȳ t R ) 2 ,
where t L and t R denote the left and right child nodes of t respectively. We call such a split s a potential split, and note that for each step in the algorithm, CART actualizes the potential split with the largest impurity decrease value.
FIGS extends CART to greedily grow a small tree-sum (see Algorithm 1). That is, at each iteration of FIGS, the algorithm chooses either to make a split on one of the current K treesf 1 , . . . ,f K in the sum, or to add a new stump to the sum. To make this decision, it still applies the CART splitting rule detailed above to identify potential splits, but instead of using the original response vector, it makes use of the leave-f k -out residual vector r
(−k) i = y i − l =kf l (x i )
to compute the impurity decrease for each treef k . FIGS makes only one split among the K + 1 potential splits: The one corresponding to the largest impurity decrease. The prediction over each of the new leaf nodes is defined to be the mean of the r for tree in all trees: 7:
y residuals = y -predict(all trees except tree) 8:
for leaf in tree: 9:
potential split = split(X, y residuals, leaf ) 10:
potential splits.append(potential split) 11:
best split = split with min impurity(potential splits) 12:
trees.insert(best split)
FIGS is related to backfitting [15], but differs from it in important ways: FIGS neither assumes a fixed number of component predictors, nor updates them in a cyclic manner; in fact, FIGS coordinates a competition among the trees being fitted at each iteration, thus mitigating backfitting's potential to overfit to residuals.
Due to its similarity to CART, FIGS supports many natural modifications that are used in CART trees. For example, different impurity measures can be used; here we use Gini impurity for classification and mean-squared-error for regression. Additionally, FIGS could benefit from pruning or by being used as part of an ensemble model.
Run-time analysis
The run-time complexity for FIGS to grow a model with m splits in total is O(dm 2 n 2 ), where d the number of features, and n the number of samples (see derivation in Appendix S1). In contrast, CART has a run-time of O(dmn 2 ). Both of these worst-case run-times given above are quite fast, and the gap between them is relatively benign as we usually make a small number of splits for the sake of interpretability.
Selecting the model's stopping threshold. Choosing a threshold on the total number of splits can be done similar to CART: using a combination of the model's predictive performance and domain knowledge on how interpretable the model needs to be. Alternatively, the threshold can be selected using an impurity decrease threshold [8] rather than a hard threshold on the number of splits. We discuss potential data-driven choices of the threshold in the Discussion (Sec 7).
Background and related work
There is a long history of greedy methods for learning individual trees, e.g. C4.5 [9], CART [8], and ID3 [16]. Recent work has proposed global optimization problems rather than greedy algorithms for trees, which can incur a high computational cost but improve performance given a fixed rule budget [17][18][19]. However, due to the limitations of a single tree, all these methods suffer from the problem of having repeated splits or repeated subtrees [20], a failure we will quantify in the results section.
Besides trees, there are a variety of other interpretable methods such as rule lists [21,22] or rule sets [23,24]; for an overview and python implementation, see [1].
Also similar to the work here are methods that learn an additive model of rules, where a rule is defined to be an axisaligned, rectangular region in the input space. RuleFit [25] is a popular method that learns a model by first extracting rules from multiple greedy decision trees fit to the data and then learning a linear model using those rules as features. FIGS is able to improve upon RuleFit by greedily selecting higher-order interactions when needed, rather than simply using all rules from some pre-specified tree depth. MARS [26] greedily learns an additive model of splines in a manner similar to FIGS, but loses interpretability as a result of using splines rather than rules.
Loosely related to this work are additive models of trees, such as Random Forest [2], gradient-boosted trees [27], BART [28] and AddTree [29], which use tree ensembles as a way to boost predictive accuracy without focusing on finding an interpretable model. Also loosely related are posthoc methods which aim to help understand a black-box model, but ultimately cannot be as interpretable as an individual interpretable model [3,30,31].
Theoretical evidence that FIGS adapts to additive structure
Tight generalization upper bounds have proved elusive for CART due to the complexity of analyzing the tree growing procedure, and are difficult for FIGS for the same reason. However, even if we knew the optimal tree structure for CART, having to use empirical averages instead of population means for the prediction over each leaf leads to an 2 generalization lower bound of Ω(n −2/(d+2) ) when the data is generated from an additive model with C 1 component functions, which is much worse than the minimax rate ofÕ dn −2/3 for this problem [13]. In comparison, assuming that we know the optimal tree structures, but not the optimal tree predictions, we are able to derive a much faster rate for models comprising sums of trees.
To formalize our theorem, consider a collection of trees C = {T 1 , T 2 , . . . , T L }, We define a tree-sum model on C to be a functionf that is a sum of component functionsf 1 , . . . ,f L , withf l implementable by T l , for l = 1, . . . , L. Now suppose we are given training data
D n = {(x i , y i )} n i=1
. Define a tree-sum model on C to be best-fit with respect to D n if it is an empirical 2 risk minimizer in this class of models. 3 For each query point x, this must satisfy the best-fit property that
f l (x) = 1 N (t l (x)) xi∈t l (x) y i − k =lf k (x i ) ,(1)
where t l (x) is the leaf in T l containing x, and that this property is satisfied approximately by FIGS because of the update formula.
Our generative model: Let x be a random variable with distribution π on [0, 1] d . Suppose that we have independent blocks of features I 1 , . . . ,
I K , of sizes d 1 , . . . , d K . For each k, let P k : [0, 1] d → [0, 1] I k denote the projection onto the coordinates in I k . Let y = f (x)+ where E{ | x} = 0 and f (x) = K k=1 f k (P k (x)) + f 0 .(2)
Theorem 1 (Generalization upper bounds using oracle tree structure). Given the generative model described above, further suppose the distribution π k of each independent block x I k has a continuous density, each f k in (2) is C 1 , with ∇f k 2 ≤ β k , and that is homoskedastic with variance E 2 | x = σ 2 . Then there exists an oracle collection of K trees C = {T 1 , . . . , T K }, with T k splitting only on features in I k for each k, and a best-fit tree-sum model on C with respect to D n ,f = K k=1f k , for which we have 3 Under some regularity conditions on the trees, it is possible to show this to be unique (see Appendix S5.4).
the following 2 upper bound on the complement of a vanishing event E:
E Dn,x (f (x) − f (x)) 2 1{E c } ≤ K k=1 c k σ 2 n 2 d k +2 .(3)
Here, c k :
= 8 d k β 2 k π k ∞ d k d k +2 , while P{E} = O(n −2/(dmax+2) ) where d max = max k d k is the size of the largest feature block in (2).
It is instructive to consider two extreme cases: If d k = 1 for each k, then we have an upper bound of O dn −2/3 . If on the other hand K = 1, we have an upper bound of O n −2/(d+2) . Both (partially oracle) bounds match the well-known minimax rates for their respective inference problems [32], hinting that FIGS might be able adapt to both additive structure as well as higher-order interactions. We also believe that (3) is the minimax rate in general for any block structure.
We note that the error event E is due to the query point possibly landing in leaf nodes containing very few or even zero training samples, which can be thus be detected and avoided in practice by imputing a default value.
The proof is deferred to Appendix S5, and builds on recent work [33] which shows how to interpret CART as a "local orthogonal greedy procedure": Growing a CART tree corresponds to greedily adding to a set of engineered linear predictors. This interpretation has a natural extension to FIGS, but at the cost of orthogonality.
Our next result shows that FIGS is able to disentangle the different additive components of f into distinct trees as intended, if the algorithm is run in the large sample limit.
Theorem 2 (Oracle disentanglement). Suppose we run Algorithm 1 with the following oracle modifications:
1. Split impurities are defined via:
∆(s, t, r) := π(t)Var{r | x ∈ t} (4) − π(t L )Var{r | x ∈ t L } − π(t R )Var{r | x ∈ t R } 2.
The prediction over each new leaf node is defined to be the population mean of the residual function r (−k) over the leaf.
At any number of iterations, letf = K k=1f k denote the working model. Then for each treef k , the set of features split upon is contained within a single index set I k for some k.
The number of terms K in the fitted model need not be equal to K. The proof for this theorem is again deferred to Appendix S5. Note that the two modifications are equivalent to running FIGS in the large sample limit, as for any function h(x, y), we have n −1∆ (s, t, h) → ∆(s, t, h) and h t → E{h | t} as n → ∞. Appendix S2.2 shows further comparisons of the prediction performance of FIGS against that of four other algorithms: CART, RF, XGBoost, and penalized iteratively reweighted least squares (PIRLS) on the log-likelihood of a generative additive model. We generate data from fully additive generative models and generative models with interactions. FIGS is able to adapt reasonably well to both generative models, whereas the other models cannot (e.g. tree-based methods perform poorly on additive data whereas PIRLS performs poorly on data with interactions).
Simulations support theoretical results
FIGS disentangles additive components of additive models
To investigate disentanglement (Thm 2), we add interactions into our generating model, and set
y = 4 i=0 x 3i+1 x 3i+2 x 3i+3 + ,
while keeping the other parameters as before and using 2,500 training samples to fit FIGS. When training FIGS on this data, we hope that each tree learned by the algorithm will contain splits only from a single interaction.
FIGS results on real-world datasets
This section gives a brief overview of the datasets analyzed here before Sec 6.1 shows FIGS's predictive performance and Sec 6.2 shows its ability to identify additive structures in real-world data.
For classification, we study four large datasets previously used to evaluate rule-based models [34] along with the two largest UCI binary classification datasets used in the classic Random-Forest paper [2,35] (overview in Table 1). For regression, we study all datasets used in the Random-Forest paper with at least 200 samples along with three of the largest non-redundant datasets from the PMLB benchmark [36] (more data details in Appendix S3). 80% of the data is used for training/3-fold cross-validation and 20% of the data is used for testing. For both classification and regression, FIGS is compared to CART, RuleFit, and Boosted Stumps (CART stumps learned via gradient-boosting). For classification, we additionally compare against C4.5 and for regression we additionally compare against CART using the mean-absoluteerror (MAE) splitting-criterion. We finally also add a Random Forest black-box baseline with 100 trees, which uses many more splits than all the other models. 5 The top two rows of Fig 4 show results for classification (measured using the ROC area under the curve, i.e. AUC), and the bottom three rows show results for regression (measured using R 2 . On average, FIGS outperforms baseline models when the number of splits is very low. The performance gain from FIGS over other baselines is larger for the datasets with more samples (e.g. the top row of Fig 4), matching the intuition that FIGS performs better because of its increased flexibility. For two of the large datasets (Credit and Recidivism), FIGS even outperforms the black-box Random Forest baseline, despite using less than 15 rules. For the smallest classification dataset (Diabetes), FIGS performs extremely well with very few (less than 10) rules, but starts to overfit as more rules are added. [39,43]. In this dataset, eight risk factors were collected and used to predict the onset of diabetes within five years. The dataset consists of 768 female subjects from the Pima Native American population near Phoenix, AZ, USA 268 of the subjects developed diabetes, which is treated as a binary label. Fig 5 shows two models, one learned by FIGS and one learned by CART. In both models, a higher prediction corresponds to a higher risk of developing diabetes. Both achieve roughly the same performance (FIGS yields an AUC of 0.820 whereas CART yields an AUC of 0.817), but the models have some key differences. The FIGS model includes fewer features and fewer total rules than the CART model, making it easier to understand in its entirety. Moreover, the FIGS model completely decouples interactions between features, making it clear that each of the features contributes independently of one another, something which any single-tree model is unable to do.
FIGS predicts well with few splits on real-world datasets
FIGS diagnoses possible additive structures in real-world datasets
The FIGS model makes its prediction by summing the contribution for the leaf-node of each tree in the model (where some trees consist of only one split). For example, if a subject's plasma glucose is greater than 166, their BMI (bodymass index) is greater than 29, and their age is less than 29, then their final risk score is 0.55 + 0.26 + 0 = 0.81. To make this prediction, the CART model must instead use an interaction between plasma glucose and BMI.
Next, Fig 6 investigates whether FIGS avoids the issue of repeated rules. It shows the fraction of rules which are repeated within a learned model as a function of the total number of rules in the model. We define a rule to be repeated if the model contains another rule using the same feature and a threshold whose value is within 0.01 of the original rule's threshold. 6 FIGS consistently learns fewer repeated rules than CART, one signal that it is avoiding learning redundant subtrees by separately modeling additive components. For clarity, Fig 6 shows only the largest three datasets studied here, but other datasets demonstrate the same relationship (see Fig S2).
Discussion
FIGS is a powerful and natural extension to CART which achieves improved predictive performance over popular baseline tree-based methods across a wide array of datasets while maintaining interpretability by using very few splits.
FIGS has many natural extensions. It is a greedy algorithm, but could be extended by using a global optimization algorithm over the class of tree-sum models. Alternatively, a FIGS model could be distilled into a simpler model (e.g. a single tree or rule-list). Additionally, the class of FIGS models could be further extended to include linear terms or allow for summations of trees to be present at split nodes, rather than just at the root. Future work could also explore using FIGS (or a randomized version of FIGS), for interaction detection, building off of Thm 2 and Fig 3. In this work, we vary the total number of splits in the model and analyze the performance. As mentioned earlier, this regularization parameter in FIGS can be tuned as done in CART. In some situations, a data-driven choice of threshold may be desirable. As seen in Sec 6, using crossvalidation (CV) to select the threshold almost always leads to the largest allowed value for the total number of splits for the datasets and parameter ranges that we considered. This is not surprising as CV doesn't consider stability or interpretability when selecting a model. Future work can use criteria related to BIC [44] or stability in combination with CV [45] for selecting this threshold based on data. In future work, one could also vary the total number of splits and number of trees separately, helping to build prior knowledge into the fitting process.
FIGS as proposed has some potential limitations. It is more flexible than CART, and as such could potentially overfit to small data faster than CART. To mitigate overfitting, FIGS's flexibility could be penalized via novel regularization techniques, such as regularizing individual leaves or regularizing a linear model formed from the rules extracted by FIGS. Alternatively, FIGS might be distilled into an even simpler rule-based model to impose more regularization. We note however, that the potential for overfitting does not materialize in our experiments (e.g. Fig 4), perhaps since starting a new tree helps combat the problem of estimating the mean value of a leaf node with very few points. We hope FIGS can pave the way towards more transparent and interpretable modeling that can improve machine-learning practice going forward, particularly in high-stakes domains such as medicine and policy making.
Acknowledgements
We gratefully acknowledge partial support from NSF Proof. Each iteration of the outer loop adds exactly one split, so it suffices to bound the running time for each iteration, where it is clear that the cost is dominated by the operation split in Algorithm 1 line 9, which takes O(n 2 d), since there are at most nd possible splits, and it takes O(n) time to compute the impurity decrease for each of these. Consider iteration s, in which we have a FIGS model f with s splits. Suppose f comprises k trees in total, with tree i having s i splits, and so that s = s 1 + . . . + s k . The total number of potential splits is equal to l + 1, where l is the total number of leaves in the model. The number of leaves on each tree is s i + 1, so the total number of leaves in f is
l = k i=1 (s i + 1) = s + k.
Since each tree has at least one split, we have k ≤ s, so that the number of potential splits is at most 2s + 1 The total time complexity is therefore
(x) = 8 i=1 1{x i > 0.1} (C) Sum of polynomial interactions model: f (x) = 4 i=0 x 3i+1 x 3i+2 x 3i+3 (B) Sum of Boolean interactions model: f (x) = 4 i=0 3 j=1 1{x 3i+j > 0.5}
We ran FIGS with a minimum impurity decrease threshold of 5σ 2 . We used the implementation of PIRLS in pygam [46], with 20 splines term for each feature. All other algorithms were fitted using default settings, except that we set min samples leaf=5 in CART. We computed the noiseless test MSE for all five algorithms on each of the generative models for a range of sample sizes n, averaging the results over 10 runs.
The results, plotted in Fig S2, show that while all other models suffer from weaknesses (PIRLS performs poorly whenever there are interactions present, i.e. for (B), (C) and (D), and tree-based methods perform poorly when there is additive structure in (A)), FIGS is able to adapt well to all scenarios, usually outperforming all other methods in moderate sample sizes. Proof of Thm 1. We may assume WLOG that E π k {f k } = 0 for each k. We first define the feature mappings Ψ k for each set of indices I k and concatenate them to form our feature map Ψ. To define Ψ k , consider a tree T k that partitions [0, 1] d k into cubes of side length h k , where h k is a parameter to be determined later. Let p k denote the number of internal nodes in T k . Let g k be defined by
g k (x I k ) := E f k (x I k ) | x I k ∈ t k (x I k )
where t k (x I k ) is the leaf in T k containing x I k , and x I k an independent copy of x I k . Set
θ * (t) := N (t L )N (t R ) N (t) (E{y | t L } − E{y | t R }),(5)
for each node t to form a vector θ * ∈ R p , where p = K k=1 p k . One can check that
g k (x I k ) = θ * T I k Φ k (x I k ). Now define g(x) = K k=1 g k (x I k ) = θ T Φ(x).
For any event E, we may apply Cauchy-Schwarz to get
E Dn,x∼π f (x) − f (x) 2 1{E c } ≤ 2E (f (x) − g(x)) 2 + 2E g(x) −f (x) 2 1{E c } .(6)
By independence, and the fact that E{g k (x I k )} = 0 for each k, we can decompose the first term as
E (f (x) − g(x)) 2 = K k=1 E (f k (x) − g k (x)) 2 .
Meanwhile, note that we have the equation
y = θ * T Ψ(x) + η + where η := f (x) − g(x) satisfies E{η | Ψ(x)} = E K k=1 (f k (x I k ) − g k (x I k )) | Ψ(x) = K k=1 E{f k (x I k ) − g k (x I k ) | Ψ k (x I k )} = 0.
As such, we may apply Theorem 8 with the event E given in the statement of the theorem to get
E g(x) −f (x) 2 1{E c } ≤ 2 pσ 2 n + 1 + 2E (f (x) − g(x)) 2 . E Dn,x x T θ n − θ 2 1{E c } ≤ 2 pσ 2 n + 1 + 2σ 2 η .(7)
Plugging this into (6), we get
E Dn,x∼π f (x) − f (x) 2 1{E c } ≤ 10 K k=1 E (f k (x I k ) − g k (x I k )) 2 + 4pσ 2 n + 1 = K k=1 10E (f k (x I k ) − g k (x I k )) 2 + 4p k σ 2 n + 1 .(8)
We reduce to the case of uniform distribution µ, via the inequality
E π k (f k (x I k ) − g k (x I k )) 2 ≤ π k ∞ E µ (f k (x I k ) − g k (x I k )) 2 ,
and from now work with this distribution, dropping the subscript for conciseness. Next, observe that
E (f k (x I k ) − g k (x I k )) 2 = E{Var{f k (x I k ) | t k (x I k )}}.
Using Lemma 5, we have that
Var{f k (x I k ) | t k (x I k )} ≤ β 2 k d k h 2 k 6 .
Meanwhile, a volumetric argument gives
p k ≤ h −d k k .
We use these to bound each term of (8) as
10 π k ∞ E (f k (x I k ) − g k (x I k )) 2 + 4p k σ 2 n + 1 ≤ 2 π k ∞ β 2 k d k h 2 k + 4h −d k k σ 2 n + 1 .(9)
Pick h k = 2σ 2 π k ∞ β 2 k d k (n + 1)
1 d k +2
, which sets both terms on the right hand side to be equal, in which case the right hand of (9) has the value
4 2 π k ∞ β 2 k d k d k d k +2 σ 2 n + 1 2 d k +2
. Summing these quantities up over all k gives the bound (3), with the error probability obtained by computing 2p/n.
Var µ {f (x) | x ∈ C} ≤ β 2 6 d j=1 (b j − a j ) 2 .(10)
Proof. For any x, x ∈ C, we may write
(f (x) − f (x )) 2 = ∇f (x ), x − x 2 ≤ β 2 x − x 2 2 .
Next, note that
E x − x 2 2 | x, x ∈ C 2 = 1 3 d j=1 (b j − a j ) 2 .
As such, we have
Var µ {f (x) | x ∈ C} = 1 2 E (f (x) − f (x )) 2 | x, x ∈ c ≤ β 2 6 d j=1 (b j − a j ) 2 .
S5.2. Proof of Thm 2
Proof of Thm 2. We prove this by induction on the total number of splits, with the base case being trivial. By the induction hypothesis, we may assume WLOG thatf 1 only has splits on features in I 1 . Consider a candidate split s on a leaf t ∈f 1 based on a feature m ∈ I 2 . Let t = P 1 (t). As sets in R d , we may then write
t = t × R [d]\I1 ,(11)t L = t × (−∞, τ ] × R [d]\I1∪{m} ,(12)
and
t R = t × (τ, ∞) × R [d]\I1∪{m} .(13)
Recall that we work with the residual r (−1) = f (x) − k>1f k . Now using the law of total variance, we can rewrite the weighted impurity decrease in a more convenient form:
∆(s, t, r (−1) ) = π(t L )π(t R ) π(t) E r (−1) | x ∈ t L − E r (−1) | x ∈ t R 2 .(14)
We may assume WLOG that this quantity is strictly positive. By the induction hypothesis, we can divide the set of component trees into two collections, one of which only splits on features in I 2 , and those which only split on features in [d]\I 2 . Denoting the function associated with the second collection of trees by g 2 , we observe that
E r (−1) | x ∈ t L − E r (−1) | x ∈ t R = E{f 2 − g | x ∈ t L } − E{f 2 − g | x ∈ t R }.
Since f 2 and g do not depend on features in I 1 , we can then further rewrite this quantity as
E{f 2 − g | x m ≤ τ } − E{f 2 − g | x m > τ }.(15)
Meanwhile, using (11), (12), and (13), we may rewrite π(t L )π(t R ) π(t) = π 1 (t )π 2 (x m ≤ τ )π 2 (x m > τ ).
Plugging (15) and (16) back into (14), we get ∆(s, t, r (−1) ) = π 1 (t )π 2 (x m ≤ τ )π 2 (x m > τ )(
E{f 2 − g | x m ≤ τ } − E{f 2 − g | x m > τ }) 2 .(17)
In contrast, if we split a new root node t 0 on m at the same threshold and call this split s , we can run through a similar set of calculations to get ∆(s , t 0 , r) = π 2 (x m ≤ τ )π 2 (x m > τ )(
E{f 2 − g | x m ≤ τ } − E{f 2 − g | x m > τ }) 2 .(18)
Comparing (17) and (18), we see that ∆(s, t, r (−1) ) = π 1 (t )∆(s , t 0 , r), and as such, split s will be chosen in favor of s.
S5.3. CART as a local orthogonal greedy procedure
In this section, we build on recent work which shows that CART can be thought of as a "local orthogonal greedy procedure" [33]. To see this, consider a tree modelf , and a leaf node t in the tree. Given a potential split s of t into children t L and t R , we may associate the normalized decision stump
samples it contains. At inference time, the prediction is made by summing the predictions of each tree. Algorithm 1 FIGS fitting algorithm. 1: FIGS(X: features, y: outcomes, stopping threshold) 2: trees = [] 3: while count total splits(trees) < stopping threshold: 4:all trees = join(trees, build new tree()) # add new tree 5:potential splits = [] 6:
achieves fast rates for 2 generalization error for additive modelsFig 2 investigates the 2 generalization error for FIGS as a function of the number of training samples used. As predicted by Thm 1, FIGS error decreases at a faster rate than that of either CART or Random Forest (RF). We simulated data via a sparse sum of squares model y = 20 j=1 x 2 j + with x ∼ Unif [0, 1] 50 , and ∼ N (0, 0.01).
FIGS slope : 1.38
Figure 2 .
2FIGS test error rate for additive data decreases faster than CART and random forest (RF), as predicted by Thm 1. Averaged over 4 runs (errors bars are standard error of the mean and are often within the points).
Fig 3 shows that this is largely what happens. Let T l be the number of trees learned by FIGS on dataset l, and set T = 10 l=1 T l . Given the collection of all trees a single index, we construct the T by 15 matrix M , whose (i, j)th entry is the number of splits in tree i on feature j. We then compute the pairwise cosine similarities between the columns of M , displaying the results in Fig 3. Note that pairs of features that never get split upon on in the same tree have a similarity value of 0, while pairs of features that always have the same number of splits in each tree have a value of 1. Fig 3 shows that the empirically observed similarity values are remarkably close to this ideal. Ground truth interactions FIGS learned interactions
Figure 3 .
3FIGS disentangles interactions into different additive components, as predicted by Thm 2. When fitted to a sum of three-way interactions, FIGS correctly places interacting terms into the same tree (dark blocks).
Fig 4
4shows the models' performance results (on test data) as a function of the number of splits in the fitted model 4 .
Fig 5
5shows an example comparing individual models learned by FIGS and CART on the Diabetes classification dataset
FIGS CART Rulefit Boosted Stumps C4.5
Figure 4 .
4FIGS performs extremely well using very few splits, particularly when the dataset is large. Top two rows show results for classification datasets (measured by AUC of the ROC curve) and the bottom three rows show results for regression datasets (measured by R 2 ). Errors bars show standard error of the mean, computed over 6 random data splits.
FIGS
FIGS
Figure 5 .
5Comparison between FIGS and CART on the diabetes dataset. FIGS learns a simpler model, which disentangles interactions between features. Both models achieve the same generalization performance (FIGS yields an AUC of 0.820 whereas CART yields 0.817.)
Figure 6 .
6FIGS learns less redundant models than CART. As a function of the number of rules in the learned model, we plot the fraction of rules, repeated for three different datasets. Error bars show standard error of the mean, computed over 6 random splits.
TRIPODS Grant 1740855, DMS-1613002, 1953191, 2015341, IIS 1741340, ONR grant N00014-17-1-2176, the Center for Science of Information (CSoI), an NSF Science and Technology Center, under grant agreement CCF-0939370, NSF grant 2023505 on Collaborative Research: Foundations of Data Science Institute (FODSI), the NSF and the Simons Foundation for the Collaboration on the Theoretical Foundations of Deep Learning through awards DMS-2031883 and 814639, and a Weill Neurohub grant. Supplement S1. FIGS run-time analysis Proposition 3. The run time complexity for FIGS to grow a model with m splits in total is O(dm 2 n 2 ), where d the number of features, and n the number of samples.
Figure S1 .
S1· O(n 2 d) = O(m 2 n 2 d).S2. Simulations S2.1. Error rate for FIGS for two generative models. FIGS test error rate is faster than CART and random forest. In both (A) and (B), the generative model for X is uniform with 50 features. Noise is Gaussian with mean zero and standard deviation 0.1 for training but no noise for testing. (A) Y is generated as sum of squares of Xi for sparsity 20 with coefficient 1. Averaged over 4 runs. (B) Y is generated from a linear model where Xi for sparsity 10 with coefficient 1. Averaged over 4 runs.S2.2. Comparison of FIGS performance with those of other algorithms over more generative modelsWe compare the prediction performance of FIGS against that of four other algorithms: CART, RF, XGBoost, and penalized iteratively reweighted least squares (PIRLS) on the log-likelihood of a generative additive model. We simulated data via y = f (x) + with x ∼ Unif [0, 1] 50 , and ∼ N (0, 0.01), where f is one of the four regression functions:
Figure S2 .
S2FIGS is able to adapt to generative models, handling both additive structure and interactions gracefully. In both (A) and (B), the generative model for X is uniform with 50 features. Noise is Gaussian with mean zero and standard deviation 0.1 for training but no noise for testing. (A) Y is generated as linear model with sparsity 20 and coefficient 1. (B) Y is generated from a single Boolean interaction model of order 8. (C) Y is generated from a sum of 5 three-way polynomial interactions. (D) Y is generated from a sum of 5 three-way Boolean interactions. All results are averaged over 10 runs.
Figure S1 .Figure S2 .
S1S2Number of trees learned as a function of the total number of rules in FIGS for different classification datasets. Fraction of repeated splits for all datasets. Corresponds to Fig 6.
Table S2. Regression datasets (extended).S3. Data details
Name
Samples Features Class 0 Class 1 Majority class %
Diabetes [39]
768
8
500
268
65.1
German credit
1000
20
300
700
70.0
Juvenile [38]
3640
286
3153
487
86.6
Recidivism
6172
20
3182
2990
51.6
Credit [37]
30000
33
23364
6636
77.9
Readmission
101763
150
54861
46902
53.9
Table S1. Classification datasets (extended).
Name
Samples Features Mean
Std Min
Max
Breast tumor [36]
116640
9
24.7
10.3 -8.5
62.0
California housing [40]
20640
8
2.1
1.2
0.1
5.0
Echo months [36]
17496
9
22.0
15.8 -4.4
74.6
Satellite image [36]
6435
36
3.7
2.2
1.0
7.0
Abalone [41]
4177
8
9.9
3.2
1.0
29.0
Diabetes [42]
442
10 152.1
77.0 25.0
346.0
Friedman1 [26]
200
10
14.7
5.2
2.5
26.5
Friedman2 [26]
200
4 462.9 373.4 10.1 1657.0
Friedman3 [26]
200
4
1.3
0.3
0.0
1.6
S4. Experiment results
0
10
20
30
40
Number of rules
Fast Interpretable Greedy-Tree Sums (FIGS) Corollary 4. Assume a sparse additive model, i.e. in(2), assume I k = {k} for k = 1, . . . , K. Then we haveE Dn,x∼π f (x) − f (x) 1{E c } ≤ 8K maxLemma 5 (Variance and side lengths). Let µ be the uniform measure on [0, 1] d . Let C ⊂ [0, 1] d be a cell. Let f be any differentiable function such that ∇f (x) 2 2 ≤ β 2 . Then we have2
k
π k ∞ β 2
k
1/3 σ 2
n
2
3
.
This toy model is an instance of a Local Spiky and Sparse (LSS) model[14], which is potentially grounded in real biological mechanisms whereby an outcome is related to interactions of inputs which display thresholding behavior.
For RuleFit, each term in the linear model is counted as one split5 We also compare against Gradient-boosting with decision trees of depth 2, but find that it is outperformed by CART in this limited-rule regime, so we omit these results for clarity. We also attempt to compare to optimal tree methods, such as GOSDT[17], but find that they are unable to fit the dataset sizes here.
This result is stable to reasonable variation in the choice of this threshold.
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| []
|
[
"Q-CURVATURE FLOW WITH INDEFINITE NONLINEARITY",
"Q-CURVATURE FLOW WITH INDEFINITE NONLINEARITY"
]
| [
"M A Li "
]
| []
| [
"Mathematics Subject Classification"
]
| In this note, we study Q-curvature flow on S 4 with indefinite nonlinearity. Our result is that the prescribed Q-curvature problem on S 4 has a solution provided the prescribed Q-curvature f has its positive part, which possesses non-degenerate critical points such that ∆ S 4 f = 0 at the saddle points and an extra condition such as a nontrivial degree counting condition. | 10.1016/j.crma.2010.02.014 | [
"https://arxiv.org/pdf/0809.4826v1.pdf"
]
| 18,489,514 | 0809.4826 | 5411b9ecadeda1512cb1698eca4d93a8589487c6 |
Q-CURVATURE FLOW WITH INDEFINITE NONLINEARITY
2000
M A Li
Q-CURVATURE FLOW WITH INDEFINITE NONLINEARITY
Mathematics Subject Classification
53352000Q-curvature flowindefinite nonlinearityblow-upcon- formal class
In this note, we study Q-curvature flow on S 4 with indefinite nonlinearity. Our result is that the prescribed Q-curvature problem on S 4 has a solution provided the prescribed Q-curvature f has its positive part, which possesses non-degenerate critical points such that ∆ S 4 f = 0 at the saddle points and an extra condition such as a nontrivial degree counting condition.
Introduction
Following the works of A.Chang-P.Yang [4], M.Brendle [3], Malchiodi and M.Struwe [6], we study a heat flow method to the prescribed Q-curvature problem on S 4 . Given the Riemannian metric g in the conformal class of standard metric c on S 4 with Q-curvature Q g . Then it is the well-known that Q g = − 1 12 (∆ g R g − R 2 g + 3|Rc(g)| 2 ) := Q, where R g , Rc(g), ∆ g are the scalar curvature, Ricci curvature tensor, the Laplacian operator of the metric g respectively.
Recall the Chern-Gaussian-Bonnet formula on S 4 is
S 4 Q g dv g = 8π 2 ,
Hence,we know that Q g has to be positive somewhere. This gives a necessary condition for the prescribed Q-curvature problem on S 4 . Assuming the prescribed curvature function f being positive on S 4 , the heat flow for the Q-curvature problem is a family of metrics of the form g = e 2u(x,t) c satisfying
(1) u t = αf − Q, x ∈ S 4 , t > 0, where u : S 4 × (0, T ) → R, and α = α(t) is defined by (2) α S 4 f dv g = 8π 2 .
Date: July 12th, 2008. * The research is partially supported by the National Natural Science Foundation of China 10631020 and SRFDP 20060003002 .
Here dv g is the area element with respect to the metric g. It is easy to see that
α t S 4 f dv g = 2α S 4 (Q − αf )f dv g .
Malchiodi and M.Struwe [6] can show that the flow exists globally, furthermore, the flow converges at infinity provided f possesses non-degenerate critical points such that ∆ S 4 f = 0 at the saddle points with the condition
{p:∇f (p)=0;∆ S 4 f (p)<0} (−1) ind(f,p) = 0.
Here ∆ S 4 := ∆ is the Analyst's Laplacian on the standard 4-sphere (S 4 , c). Recall that S 4 dv c = 8 3 π 2 . The purpose of this paper is to relax their assumption by allowing the function f to have sign-changing or to have zeros.
Since we have
Q = 1 2 e −4u (∆ 2 u − div(( 2 3 R(c)c − 2Rc(c))du) + 6),
the equation (1) define a nonlinear parabolic equation for u, and the flow exists at least locally for any initial data u| t=0 = u 0 . Clearly, we have
∂ t S 4 dv g = 2 S 4 u t dv g = 0.
We shall assume that the initial data u 0 satisfies the condition
(3) f e 4u dv c > 0.
We shall show that this property is preserved along the flow. It is easy to compute that
(4) Q t = −4u t Q − 1 2 P u t = 4Q(Q − αf ) + P (αf − Q),
where P = P g = e −4u P c and P c is the Paneitz operator in the metric c on S 4 [4]. Using (4), we can compute the growth rate of the Calabi-type energy
S 4 |Q − αf | 2 dv g .
Our main result is following
m 0 = 1 + k 0 , m i = k i−1 + k i , 1 ≤ i ≤ 4, k 4 = 0.
Then f is the Q curvature of the conformal metric g = e 2u c on S 4 .
Note that this result is an extension of the famous result of Malchiodi-Struwe [6] where only positive f has been considered. A similar result for Curvature flow to Nirenberg problem on S 2 has been obtained in [8]. See also J.Wei and X.Xu's work [9].
For simplifying notations, we shall use the conventions that dc = dvc 8 3 π 2 and u =ū(t) defined by
S 4 (u −ū)dv c = 0.
Basic properties of the flow
Recall the following result of Beckner [2] that
(5) S 4 (|∆u| 2 + 2|∇u| 2 + 12u)dc ≥ log( S 4 e 4u dc) = 0,
where |∇u| 2 is the norm of the gradient of the function u with respect to the standard metric c. Here we have used the fact that S 4 e 4u dc = 1 along the flow (1).
We show that this condition is preserved along the flow (1). In fact, letting
E(u) = S 4 (uP u + 4Q c u)dc = S 4 (|∆u| 2 c + 2|∇u| 2 c + 12u)dc
be the Liouville energy of u and letting
E f (u) = E(u) − 3 log( S 4 f e 4u dc)
be the energy function for the flow (1), we then compute that
(6) ∂ t E f (u) = − 3 2π 2 S 4 |αf − Q| 2 dv g ≤ 0.
One may see Lemma 2.1 in [6] for a proof. Hence
E f (u(t)) ≤ E f (u 0 ), t > 0.
After using the inequality (5) we have
(7) log(1/ S 4 f e 4u dc) ≤ E f (u 0 ), which implies that S 4 f e 4u dv c > 0 and e E f (u 0 ) S 4 e 4u dc ≤ S 4 f e 4u dc.
Note also that S 4 f e 4u dc = 1/α(t). Hence,
α(t) ≤ 1 e E f (u 0 ) .
Using the definition of α(t) we have
α(t) ≥ 1 max S 4 f .
We then conclude that α(t) is uniformly bounded along the flow, i.e.,
1 max S 4 f ≤ α(t) ≤ 1 e E f (u 0 ) .(8)
We shall use this inequality to replace (26) in [6] in the study of the normalized flow, which will be defined soon following the work of Machiodi and M.Struwe [6]. If we have a global flow, then using (6) we have
2 ∞ 0 dt S 4 |αf − Q| 2 dv g ≤ 4π(E f (u 0 ) + log max S 4 f ).
Hence we have a suitable sequence t l → ∞ with associated metrics g l = g(t l ) and α(t l ) → α > 0, and letting Q l = Q(g l ) be the Q-curvature of the metric g l , such that
S 4 |Q l − αf | 2 → 0, (t l → ∞).
Therefore, once we have a limiting metric g ∞ of the sequence of the metrics g l , it follows that Q(g ∞ ) = αf . After a re-scaling, we see that f is the Gaussian curvature of the metric βg ∞ for some β > 0, which implies our Theorem 1.
Normalized flow and the proof of Theorem 1
We now introduce a normalized flow. For the given flow g(t) = e 2u(t) c on S 4 , there exists a family of conformal diffeomorphisms φ = φ(t) : S 4 → S 4 , which depends smoothly on the time variable t, such that for the metrics h = φ * g, we have Using this we can obtain the uniform H 1 norm bounds of v for all t ≥ 0 that sup
t |v(t)| H 1 (S 2 ) ≤ C.
See the proof of Lemma 3.2 in [6]. Using the Aubin-Moser-Trudinger inequality [1] we further have
4 sup {0≤t<T } S 4 |u(t)|dc ≤ sup t S 4 e 4|u(t)| dc ≤ C < ∞. Note that v t = u t • φ + 1 4 e −4v div S 4 (ξe 4v )
where ξ = (dφ) −1 φ t is the vector field on S 2 generating the flow (φ(t)), t ≥ 0, as in [6], formula (17), with the uniform bound
|ξ| 2 L ∞ (S 4 ) ≤ C S 4 |αf − K| 2 dv g .
With the help of this bound, we can show (see Lemma 3.3 in [6]) that for any T > 0, it holds sup 0≤t<T S 2 e 4|u(t)| dc < +∞.
Following the method of Malchiodi and M.Struwe [6] (see also Lemma 3.4 in [5]) and using the bound (8) and the growth rate of α, we can show that
S 4 |αf − Q| 2 dv g → 0
as t → ∞. Once getting this curvature decay estimate, we can come to consider the concentration behavior of the metrics g(t). Following [5], we show that Lemma 2. Let (u l ) be a sequence of smooth functions on S 4 with associated metrics g l = e 2u l c with vol(S 4 , g l ) = 8 3 π 2 , l = 1, 2, ... as constructed above. Suppose that there is a smooth function Q ∞ , which is positive somewhere in
S 4 such that |Q(g l ) − Q ∞ | L 2 (S 4 ,g l ) → 0
as l → ∞. Let h l = φ * l g l = e 2v l c be defined as before. Then we have either 1) for a subsequence l → ∞ we have u l → u ∞ in H 4 (S 4 , c), where g ∞ = e 2u∞ c has Q-curvature Q ∞ , or 2) there exists a subsequence, still denoted by (u l ) and a point q ∈ S 4 with Q ∞ (q) > 0, such that the metrics g l has a measure concentration that dv g l → 8 3 π 2 δ q weakly in the sense of measures, while h l → c in H 4 (S 4 , c) and in particular, Q(h l ) → 3 in L 2 (S 4 ). Moreover, in the latter case the conformal diffeomorphisms φ l weakly converges in H 2 (S 4 ) to the constant map φ ∞ = q.
Proof. The case 1) can be proved as Lemma 3.6 in [6]. So we need only to prove the case 2). As in [6], we choose q l ∈ S 4 and radii r l > 0 such that sup q∈S 4 B(q,r l )
|K(g l )|dv g l ≤ B(q l ,r l ) |K(g l )|dv g l = 2π 2 ,
where B(q, r l ) is the geodesic ball in (S 4 , g l ). Then we have r l → 0 and we may assume that q l → q as l → ∞. For each l, we introduce φ l as in Lemma 3.6 in [6] so that the functionŝ
u l = u l • φ l + 1 4 log(det(dφ l ))
satisfy the conformal Q-curvature equation −P R 4û l = 2Q l e 4û l , on R 4 , whereQ l = Q(g l ) • φ and P R 4 is the Paneitz operator of the standard Euclidean metric g R 4 . Note that forĝ l = φ * g l = e 2û l g R 4 , we have
V ol(R 4 ,ĝ l ) = V ol(S 4 , g l ) = 8 3 π 2 .
Arguing as in [6], we can conclude a convergent subsequenceû l →û ∞ in H 4 loc (R 4 ) whereû ∞ satisfies the Liouville type equation
−∆ 2 R 4û∞ =Q ∞ (q)e 4û∞ , on R 4 , with R 4 K ∞ (q)e 4û∞ dz ≤ 8 3 π 2 .
We only need to exclude the case when Q ∞ (q) ≤ 0. Just note that by (7) we have
log(1/ R 4 f • φ l e 4û l ) ≤ E f (u 0 ). Hence, sending l → ∞, we always have f • φ l → f • φ(q) > 0 uniformly on any compact domains of R 4 .
The remaining part is the same as in the proof of Lemma 3.6 in [6]. We confer to [6] for the full proof.
We remark that some other argument can also exclude the case Q ∞ (q) < 0. It can not occur since there is no such a solution on the whole space R 4 (see also the argument in [7]). If Q ∞ (q) = 0, then ∆ R 4û := ∆ R 4û ∞ is a harmonic function in R 4 . Letū(r) be the average of u on the circle ∂B r (0) ⊂ R 4 . Then we have
∆ 2 R 4ū = 0.
Hence ∆ R 4ū = A + Br −2 for some constants A and B, where r = |x|. Since ∆ R 4ū is a continuous function on [0, ∞), we have ∆ R 4ū = A, which gives us thatū = A + Br 2 + Cr −2 for some constants A, B, and C. Again, usingū is regular, we have C = 0 andū = A + Br 2 with B < 0. However, it seems hard to exclude this case without the use of the fact (7).
With this understanding, we can do the same finite-dimensional dynamics analysis as in section 5 in [6]. Then arguing as in section 5 in [6] we can prove Theorem 1. By now the argument is well-known, so we omit the detail and refer to [6] for full discussion.
Theorem 1 .
1Let f be a positive somewhere, smooth function on S 4 with only non-degenerate critical points on the its positive part f + with its Morse index ind(f + , p). Suppose that at each critical point p of f + , we have ∆f = 0. Let m i be the number of critical points with f (p) > 0, ∆ S 4 f (p) < 0 and ind(f, p) = 4 − i. Suppose that there is no solutions with coefficients k i ≥ 0 to the system of equations
= 0, f or all t ≥ 0.Here x = (x 1 , x 2 , x 3 , x 4 , x 5 ) ∈ S 4 ⊂ R 5 is a position vector of the standard 4-sphere. Let v = u • φ + 1 4 log(det(dφ)).Then we have h = e 2v c. Using the conformal invariance of the Liouville energy[4], we have E(v) = E(u), and furthermore,V ol(S 4 , h) = V ol(S 4 , g) = 8 3 π 2 , f or all t ≥ 0.Assume u(t) satisfies (1) and(2). Then we have the uniform energy bounds0 ≤ E(v) ≤ E(u) = E f (u) + log( S 4f e 4u dc) ≤ E f (u 0 ) + log(max
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| []
|
[
"EVIDENCE FOR SOLAR-LIKE OSCILLATIONS IN BETA HYDRI",
"EVIDENCE FOR SOLAR-LIKE OSCILLATIONS IN BETA HYDRI"
]
| [
"Timothy R Bedding ",
"R Paul Butler ",
"Hans Kjeldsen ",
"Ivan K Baldry ",
"Simon J O'toole ",
"Christopher G Tinney ",
"Geoffrey W Marcy ",
"Francesco Kienzle ",
"Fabien Carrier "
]
| []
| []
| We have made a clear detection of excess power, providing strong evidence for solar-like oscillations in the G2 subgiant β Hyi. We observed this star over five nights with the UCLES echelle spectrograph on the 3.9-m Anglo-Australian Telescope, using an iodine absorption cell as a velocity reference. The time series of 1196 velocity measurements shows an rms scatter of 3.30 m s −1 , and the mean noise level in the amplitude spectrum at frequencies above 0.5 mHz is 0.11 m s −1 . We see a clear excess of power centred at 1.0 mHz, with peak amplitudes of about 0.5 m s −1 , in agreement with expectations for this star. Fitting the asymptotic relation to the power spectrum indicates the most likely value for the large separation is 56.2 µHz, also in good agreement with the known properties of β Hyi. | 10.1086/319139 | [
"https://arxiv.org/pdf/astro-ph/0012417v1.pdf"
]
| 119,481,438 | astro-ph/0012417 | 3ba52e4737a797de4196884a31695f1b85b5ba2d |
EVIDENCE FOR SOLAR-LIKE OSCILLATIONS IN BETA HYDRI
20 Dec 2000
Timothy R Bedding
R Paul Butler
Hans Kjeldsen
Ivan K Baldry
Simon J O'toole
Christopher G Tinney
Geoffrey W Marcy
Francesco Kienzle
Fabien Carrier
EVIDENCE FOR SOLAR-LIKE OSCILLATIONS IN BETA HYDRI
20 Dec 2000Submitted to ApJ LettersSUBMITTED TO APJ LETTERS Preprint typeset using L A T E X style emulateapj v. 14/09/00Subject headings: stars: individual (β Hyi) -stars: oscillations-techniques: radial velocities
We have made a clear detection of excess power, providing strong evidence for solar-like oscillations in the G2 subgiant β Hyi. We observed this star over five nights with the UCLES echelle spectrograph on the 3.9-m Anglo-Australian Telescope, using an iodine absorption cell as a velocity reference. The time series of 1196 velocity measurements shows an rms scatter of 3.30 m s −1 , and the mean noise level in the amplitude spectrum at frequencies above 0.5 mHz is 0.11 m s −1 . We see a clear excess of power centred at 1.0 mHz, with peak amplitudes of about 0.5 m s −1 , in agreement with expectations for this star. Fitting the asymptotic relation to the power spectrum indicates the most likely value for the large separation is 56.2 µHz, also in good agreement with the known properties of β Hyi.
INTRODUCTION
The search for solar-like oscillations in other stars has been long and difficult. Observers have mostly concentrated on three stars: Procyon (α CMi), η Boo and α Cen A. Reviews of those efforts have been given by Brown & Gilliland (1994), Kjeldsen & Bedding (1995, hereafter KB95), Heasley et al. (1996) and . More recently, measured Balmer-line equivalent widths in α Cen A and set an upper limit on oscillation amplitudes of only 1.4 times solar, with tentative evidence for p-mode structure. More recently still, measurements of velocity variations in Procyon by Martic et al. (1999, see also Barban et al. 1999), showed very good evidence for oscillations with peak amplitudes of about 0.5 m s −1 and frequencies centred at about 1 mHz.
Here, we report the clear detection of excess power, providing evidence for oscillations in the G2 subgiant β Hydri (HR 98, V = 2.80, G2 IV). This star is the closest G-type subgiant, with luminosity 3.5 L ⊙ , mass 1.1 M ⊙ and age about 6.7 Gy (Dravins, Lindegren, & VandenBerg 1998). It has received less attention than the three stars mentioned above, presumably due to its extreme southerly declination (−77 • ). An attempt to measure oscillations in β Hyi in radial velocity was made by Edmonds & Cram (1995) and gave upper limits on the strongest modes of 1.5 to 2.0 m s −1 , consistent with our detection.
OBSERVATIONS AND DATA REDUCTION
The observations were made over five nights (2000 June 11-15). We used the University College London Echelle Spectrograph (UCLES) at the coudé focus of the 3.9-m Anglo-Australian Telescope (AAT) at Siding Spring Observatory, Australia. To produce high-precision velocity measurements, the star was observed through an iodine absorption cell mounted directly in the telescope beam, immediately behind the spectrograph entrance slit. The cell is temperature-stabilized at 55±0.1 • C and imprints a rich forest of molecular iodine absorption lines from 500 nm to 600 nm directly on the incident starlight. Echelle spectra were recorded with the MITLL 2k×4k 15µm pixel CCD, denoted MITLL2a, which covered the wavelength range 470-880 nm.
Exposure times were typically 60 s, with a dead-time of 55 s between exposures (using "FAST" readout). The signal-tonoise ratio for most spectra was in the range 200 to 400, depending on the seeing and extinction. In total, 1196 spectra were collected, with the following distribution over the five nights: 59, 166, 301, 325 and 345. The first two nights were affected by poor weather and technical problems.
Extraction of radial velocities from the echelle spectra followed the method described by Butler et al. (1996). This involved using the embedded iodine lines both as a wavelength reference and also to recover the spectrograph point-spreadfunction. Essential to this process were template spectra taken of β Hyi with the iodine cell removed from the beam, and of the iodine cell itself superimposed on a rapidly rotating B-type star.
The resulting velocity measurements for β Hyi are shown in the lower panel of Fig. 1. They have been corrected to the solar system barycentre, as described by Butler et al. (1996). No other corrections, decorrelation or high-pass filtering have been applied: the measurements are exactly as they emerged from the pipeline processing. Note that the velocity measurements are relative to the velocity of the star when the template was taken. The slow variations in velocity within and between nights are due to uncorrected instrumental drifts. The rms scatter of these measurements is 3.30 m s −1 .
Uncertainties for the velocity measurements were estimated from residuals in the fitting procedure and are shown in the upper panel of Fig. 1. Most lie in the range 2.5-4 m s −1 , and fall gradually during each night as the target rises in the sky. Observations of β Hyi were also made using the CORALIE spectrograph on the 1.2-m Leonard Euler Swiss telescope at La Silla Observatory in Chile (Queloz et al. 2000). The precision of those measurements was poorer than those from UCLES, presumably due -at least in part -to the smaller telescope aperture. The CORALIE data are not included in this Letter; their analysis is postponed to a future paper.
TIME SERIES ANALYSIS
The amplitude spectrum of the velocity time series was calculated as a weighted least-squares fit of sinusoids Arentoft et al. 1998), with a weight being assigned to each point according its uncertainty estimate. Figure 2 shows the resulting power spectrum. There is a striking excess of power around 1 mHz which is the clear signature of solar-like oscillations. As discussed below, the frequency and amplitude of this excess power are in excellent agreement with expectations. We also note that the excess is apparent in the power spectra of individual nights.
Typically for such a power spectrum, the noise has two components:
1. At high frequencies it is flat (i.e., white), indicative of the Poisson statistics of photon noise. The mean noise level in the amplitude spectrum in the range 3-5 mHz is 0.11 m s −1 . Since this is based on 1196 measurements, we can calculate (e.g., KB95) that the velocity precision on the corresponding timescales is 2.2 m s −1 .
2. Towards the lowest frequencies, we see rising power that arises from the slow drifts mentioned above. A logarithmic plot shows that, as expected for instrumental drift, this noise goes inversely with frequency in the amplitude spectrum (and inversely with frequency squared in power). It has a value at 0.1 mHz of 0.3 m s −1 .
The white line in Fig. 2 shows the combined noise level, and we see that the contribution from 1/ f noise is neglible above about 0.5 mHz.
DISCUSSION
Oscillation frequencies
Mode frequencies for low-degree oscillations in the Sun are reasonably well approximated by the asymptotic relation:
ν(n, l) = ∆ν(n + 1 2 l + ε) − l(l + 1)D 0 .
(1) Here n and l are integers which define the radial order and angular degree of the mode, respectively; ∆ν (the so-called large separation) reflects the average stellar density, D 0 is sensitive to the sound speed near the core and ε is sensitive to the surface layers. Note that D 0 is δν 0 /6, where δν 0 is the so-called small frequency separation between adjacent modes with l = 0 and l = 2.
A similar relation to equation (1) is expected for other solar-like stars, although there may be significant deviations in more evolved stars. Models of the G subgiant η Boo suggest that modes with l = 1 undergo 'avoided crossings,' in which their frequencies are shifted from their usual regular spacing by effects of gravity modes in the stellar core (Christensen-Dalsgaard, . Some evidence for this effect was seen in the proposed detection of oscillations in η Boo by .
In attempting to find peaks in our power spectrum matching the asymptotic relation, we were severely hampered by the single-site window function. As is well known, daily gaps in a time series produce aliases in the power spectrum at spacings ±11.57 µHz which are difficult to disentangle from the genuine peaks. Various methods have been discussed in the literature for searching for a regular series of peaks, such as autocorrelation, comb response and histograms of frequencies (e.g., Gilliland et al. 1993;Mosser et al. 1998;Barban et al. 1999;). Here we use a type of comb analysis, in which we calculated a response function for all sensible values of ∆ν, D 0 and ε.
Since we are searching for mode structure in the region of excess power, it is convenient to rewrite equation (1) as ν(n ′ , l) = ν 0 + ∆ν(n ′ + 1 2 l) − l(l + 1)D 0 .
Here, ν 0 is the frequency of a radial (l = 0) mode in the region of maximum power, for which n ′ = 0. Thus, ν 0 replaces ε as the parameter for the absolute position of the comb. The comb response function was obtained as follows. We first thresholded the power spectrum at the noise level. In other words, any points less than the noise level were set at this level. For each triplet (∆ν, D 0 , ν 0 ) we then measured the (modified) power spectrum at each of the frequencies predicted by equation (2). We summed these numbers to produce a response value which is a measure of the goodness-of-fit of that triplet. We limited the sum to l = 0, 1, 2, and n ′ = −4, −3, . . . , 4, with ν 0 centred at 1000 µHz. The restriction on n ′ was made for two reasons: as in the Sun, we only see modes excited to observable amplitudes over a restricted range of frequencies and, also as in the Sun, we expect deviations from equation (2) over large ranges in frequency.
From this analysis we identified the parameters that maximised the comb response, searching over the ranges ∆ν = 45-80 µHz and D 0 = 0.5-2.0 µHz. These ranges encompass the values that would be expected for this star. For ν 0 , we examined a range of width ∆ν centred on 1000 µHz. The following two triplets of (∆ν, D 0 , ν 0 ) gave the best responses: (56. calculated and observed peaks is not the same for the two solutions. The matches in Solution B occur mostly at the lower frequencies, while those for Solution A cover the full range of excess power. This suggests that Solution A is more likely to be the correct one.
How do these two solutions agree with expectations for β Hyi? The large separation ∆ν of a star scales approximately as the square root of density. Extrapolating from the solar case (∆ν = 135.0 µHz) using parameters for β Hyi of L = 3.5 L ⊙ , T eff = 5800 K and M = 1.1 M ⊙ (see Dravins et al. 1998, and references therein) gives ∆ν = 56.0 µHz. Given that we searched over the entire range 45-80 µHz, the excellent agreement between observation and theory is very encouraging, especially for solution A.
There is no simple way to estimate D 0 for a star as evolved as β Hyi, since this parameter is very sensitive to evolution. Extrapolating the grid calculated by Christensen-Dalsgaard (1993) leads us to expect D 0 in the range 0.5 to 1.0, which is consistent with both solutions. The value of ν 0 (or, equivalently, ε) depends on details of the surface layers, which also requires construction of models specifically for β Hyi.
We note that simulations in which input frequencies obeyed the asymptotic relation precisely showed a more definite (and unambiguous) peak in the comb response than did the real data. We conclude that the oscillation frequencies in β Hyi have significant departures from the asymptotic relation. This is of great astrophysical interest and not unexpected, but it does make it very difficult to extract the correct frequencies from our singlesite data.
At the suggestion of the referee, we examined the comb response of random power spectra that were generated by multiplying white noise by an envelope similar to the observed power excess. For these spectra, the comb response sometimes showed peaks similar to those from the real data, at various values of ∆ν. The comb analysis therefore does not prove that the excess power is due to individual modes. However, given (i) the agreement between observations and expectations in all respects -position and amplitude of excess power (see below), and the best-fit value for ∆ν -and (ii) the absence of any theoretical or observational reason to attribute the power excess to other sources, we consider that solar-like oscillations are the most likely explanation for the observed power excess.
Oscillation amplitudes
The strongest peaks in the amplitude spectrum of β Hyi (square root of power) reach about 0.6 m s −1 . However, these are likely to have been strengthened significantly by constructive interference with noise peaks. As stressed by KB95 (Appendix A.2), the effects of the noise must be taken into account when estimating the amplitude of the underlying signal. To do this, we have generated simulated time series consisting of artificial signal plus noise. We conclude that the underlying oscillations have peaks of about 0.5 m s −1 .
Solar-like oscillations are excited by convection and the ex-pected amplitudes have been estimated using theoretical models. Based on models by Christensen-Dalsgaard & Frandsen (1983), KB95 suggested that amplitudes in velocity should scale as L/M. More recent calculations by Houdek et al. (1999) confirm this scaling relation, at least for stars with near-solar effective temperatures. For β Hyi, the implied amplitude is about 3.2 times solar, which is 0.7 to 0.8 m s −1 (based on the strongest few peaks -see KB95). The observed amplitudes are therefore consistent with expectations. We also note that the frequency of excess power (1 mHz) is in excellent agreement with the value expected from scaling the acoustic cutoff from the solar case (Brown et al. 1991, KB95).
CONCLUSION
Our observations of β Hyi show an obvious excess of power, clearly separated from the 1/ f noise, and with a position and amplitude that are in agreement with expectations. Although hampered by the single-site window, a comb analysis shows evidence for approximate regularity in the peaks at the spacing expected from asymptotic theory. There seem to be significant departures from regularity, perhaps indicating mode shifts from avoided crossings. We hope that further analysis along one or more of the following lines will allow us to explore further the oscillation spectrum of β Hyi: including the CORALIE observations which, despite their lower precision, could usefully improve the spectral window; reducing the noise of the velocity measurements (UCLES and CORALIE) by decorrelating against external parameters (e.g., Gilliland et al. 1991); combining these results with an analysis of equivalent-width variations of strong lines in the spectral region uncontaminated by iodine lines (e.g., Hα); and using theoretical models to calculate expected shifts due to avoided crossings, to help identify the affected modes.
The clear strength of the technique used in these observations is its ability to precisely calibrate spectrograph variations at the timescales of primary interest. Although the best radial velocity precisions achieved in the long term from this technique are 3-4 m s −1 , our data clearly demonstrate that the precision at frequencies around 1 mHz is significantly better (2.2 m s −1 ). Our results provide valuable confirmation that oscillations in solar-like stars really do have the amplitudes that we have been led to expect by extrapolating from the Sun. This bodes extremely well for success of space missions such as MOST (Matthews et al. 2000), MONS (Kjeldsen, Bedding, & Christensen-Dalsgaard 2000) and COROT (Baglin et al. 1998), which will provide photometric data of high quality for a wide sample of stars.
This work was supported financially by: the Australian Research Council (TRB and SJOT), NSF grant AST-9988087 (RPB), SUN Microsystems, and the Danish Natural Science Research Council and the Danish National Research Foundation through its establishment of the Theoretical Astrophysics Center (HK).
1 FIG. 1 .
11-Velocity measurements of β Hyi obtained with the AAT (lower panel) and the corresponding uncertainties (upper panel).
2, 0.83, 1030.1) µHz (solution A) and (60.3, 0.75, 1005.3) µHz (solution B). The frequencies corresponding to these solutions are shown in Fig. 3, overlaid on the measured power spectrum. The distribution as a function of frequency of matches between FIG. 2.-Power spectrum of the AAT velocity measurement of β Hyi. The white line shows a two-component noise model (see Sec. 3).
FIG. 3 .
3-Close-up of the power spectrum of β Hyi, with dashed lines showing the frequencies given by equation(2)for solutions A (upper panel) and B (lower panel). The dotted lines show the ±1/day aliases of those frequencies.
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| []
|
[
"Bi-phasic vesicle: instability induced by adsorption of proteins",
"Bi-phasic vesicle: instability induced by adsorption of proteins"
]
| [
"Jean-Marc Allain \nLaboratoire de Physique Statistique\nEcole Normale Supérieure\n24 rue Lhomond75231, Cedex 05ParisFrance\n",
"Martine Ben Amar [email protected] \nLaboratoire de Physique Statistique\nEcole Normale Supérieure\n24 rue Lhomond75231, Cedex 05ParisFrance\n"
]
| [
"Laboratoire de Physique Statistique\nEcole Normale Supérieure\n24 rue Lhomond75231, Cedex 05ParisFrance",
"Laboratoire de Physique Statistique\nEcole Normale Supérieure\n24 rue Lhomond75231, Cedex 05ParisFrance"
]
| []
| The recent discovery of a lateral organization in cell membranes due to small structures called 'rafts' has motivated a lot of biological and physico-chemical studies. A new experiment on a model system has shown a spectacular budding process with the expulsion of one or two rafts when one introduces proteins on the membrane. In this paper, we give a physical interpretation of the budding of the raft phase. An approach based on the energy of the system including the presence of proteins is used to derive a shape equation and to study possible instabilities. This model shows two different situations which are strongly dependent on the nature of the proteins: a regime of easy budding when the proteins are strongly coupled to the membrane and a regime of difficult budding. | 10.1016/j.physa.2003.12.058 | [
"https://arxiv.org/pdf/q-bio/0312028v2.pdf"
]
| 15,397,151 | q-bio/0312028 | 13c2ef6eede034693415cc6b889905341c2af342 |
Bi-phasic vesicle: instability induced by adsorption of proteins
19 Apr 2004
Jean-Marc Allain
Laboratoire de Physique Statistique
Ecole Normale Supérieure
24 rue Lhomond75231, Cedex 05ParisFrance
Martine Ben Amar [email protected]
Laboratoire de Physique Statistique
Ecole Normale Supérieure
24 rue Lhomond75231, Cedex 05ParisFrance
Bi-phasic vesicle: instability induced by adsorption of proteins
19 Apr 2004Preprint submitted to Elsevier Science 7 February 2008arXiv:q-bio/0312028v2 [q-bio.SC]RaftBuddingProteinsMembranesVesicles shapeSpherical cap harmonics
The recent discovery of a lateral organization in cell membranes due to small structures called 'rafts' has motivated a lot of biological and physico-chemical studies. A new experiment on a model system has shown a spectacular budding process with the expulsion of one or two rafts when one introduces proteins on the membrane. In this paper, we give a physical interpretation of the budding of the raft phase. An approach based on the energy of the system including the presence of proteins is used to derive a shape equation and to study possible instabilities. This model shows two different situations which are strongly dependent on the nature of the proteins: a regime of easy budding when the proteins are strongly coupled to the membrane and a regime of difficult budding.
Introduction
Classical and over-simplified models of the cell reduces the membrane to a bilayer of lipids in a fluid state which is a solvent for the proteins of the membrane (1). But the cell membrane is a much more complex and inhomogeneous system. The inhomogeneities come from a phase separation between small structures called 'rafts' (2) and the surrounding liquid phase. These rafts have been discovered a decade ago and remain an important issue of cell biology but also immunology, virology, etc (3). A lot of biological studies concern the rafts and examine their composition (4), their in-vivo size (5), their role in signaling (6) or in lipid traffic (7) for example. The raft is roughly a mixture of cholesterol and sphingolipid but the exact nature of the sphingolipid and its concentration can vary between different rafts. In any case and whatever its composition, the raft has different physical or chemical properties than the rest of the membrane. In this paper, we focus on this specificity which is at the origin of an elastic instability that we want to explain.
Experimentally, the raft in vivo cannot be easily studied and artificial systems like GUV (giant unilamellar vesicle) (8) appear more appropriate. GUV consist in a membrane of lipids with the possibility of a raft inclusion. On these artificial systems, a better control of the experimental parameters can be obtained and explored. For example, they have been used to study the coupled effects of both the membrane composition and the temperature on the nucleation of rafts (9). Recently, a new experiment on GUV with rafts has shown a spectacular budding process (10) induced by injection of proteins called P LA 2 (phospholipase A 2 ). Before injection, the GUV membrane is in a stable, nearly spherical state. But, more precisely, high-quality pictures of vesicles reveal two spherical caps, one for each phase: the raft and the fluid phase (11). These two caps have a radius of the same order of magnitude (about 5 micrometers, depending on the experimental conditions) and are separated by a discontinuous interface. Few seconds after injection of P LA 2 with a micro-pipette in the vicinity of the raft, one observes a rather strong destabilization of the initial shape: the raft tries to rise. The discontinuity of slope at the interface between the two caps becomes more and more pronounced. This lifting can be strong enough to expel the raft from the vesicle. When two or three rafts are present initially, successive expulsions can be observed. We present here a theoretical treatment showing that the driving force of the deformation is the absorption of proteins which locally deforms the membrane. We neglect chemical reactions since we focus here on the early stages of the instability: the time-scale of the instability is small compared to the characteristic time of chemical effects. We restrict ourselves to the simplest model relevant for the experiment we want to describe. It involves standard physical concepts of membrane mechanics. The initial shape of the system is given by a minimum of the energy of the whole system (that is the inhomogeneous vesicle including proteins). A linear perturbation treatment allows to examine the existence of another solution which may lead to a new minimum of energy. This approach is sufficient to predict the experimental observation of destabilization and to derive a concentration threshold for an elastic instability of the vesicle. The calculation presented in this paper concerns only the first stages of the instability. Intermediate stages require at least a dynamical nonlinear calculation including possible chemical effects of proteins. The final stage can be achieved by a non-linear calculation or, in case of fission, by the energy evaluation of two separated homogeneous spheres following the strategy described in (12).
Models of vesicles have been widely described in previous papers (14; 15; 16; 17; 18; 19). They vary depending on the physical interactions involved taken into account. The backbone of all models is based on the minimization of the average curvature energy of the bilayer, with the introduction of a possible local membrane asymmetry (13; 14). A large number of shapes have been predicted in the past by this model (16). They suitably describe experimental results such as the various shapes of red blood cells. Other physical effects can be introduced, such as the difference of area between the two layers of the membrane (20), suggesting a differential compressive stress in the bilayer. These effects are visible under suitable experimental conditions (19). Here, our scope is to study quantitatively the protein-membrane interaction using a generalization of the Leibler's model (21) to an inhomogeneous system. It turns out that this model, which describes the proteins as defects on the membrane, leads to a spatially inhomogeneous spontaneous curvature which is shown to be responsible for the destabilization of the whole system. Going back to the microscopic level, we derive a threshold for the protein concentration, which appears as a control parameter. Moreover, depending on the shape of the proteins, we are able to select two different regimes: a protein-stocking regime and a destabilization regime with possible raft-ejection. The idea of a nonhomogeneous spontaneous curvature is not new since it has been used for mono-phasic vesicles to explain a possible thermal budding (22). This does not concern the experiment described in (10) since the temperature is not the relevant control parameter. Another scenario for the budding process of a raft has been proposed by (18): the increase of line tension by the proteins leads to an apparent slope discontinuity and to a neck. Again, this approach, which is different from ours, is not quantitatively related to the amount of proteins. It is why we suggest a different treatment as an interpretation of the raft ejection. This paper is organized as follows. Section 2 is devoted to a detailed description of the model defining precisely the elastic energy plus the energy of interaction combined to the constraints. Section 3 determines an obvious solution of the minimization of energy in terms of two joined spherical caps. A linear perturbation is performed which gives the threshold of proteins when a destabilization occurs. In section 4, the results are analyzed and discussed taking into account known or estimated orders of magnitude of physical parameters.
The model
Membrane description
The energetic model of the membrane is well established nowadays. It can incorporate many different interactions, constraints or restrictions. Here, we focus on a precise experiment and we think the model suitable for this experiment (10). Nevertheless, it can be modified easily for another experimental set-up.
We consider a slightly stretched vesicle made of amphiphilic molecules difficult to solubilize in water. The raft will be denoted by phase 1, it is usually considered as an ordered liquid. The remaining part is denoted by phase 2 and is considered as a disordered liquid. As the two phases are liquid, we describe them by two similar free energies, each of them having its own set of physical constants. Quantities which remain fixed in the experiment are constraints expressed via Lagrange multipliers in the free energy. So we define the energy of the bilayer in the phase (i):
F b i = S i κ i 2 H 2 + κ G i K + Σ i dS(1)
with H the mean curvature and K the Gaussian curvature. The square of H is the classical elastic energy (14) when we get rid of the spontaneous curvature. Here, there is no physical reason to introduce a spontaneous curvature, sign of asymmetry between the two layers. The membrane contains enough cholesterol, which has a fast rate of flip-flop and which relaxes the constraints inside the bilayer. As for the Gaussian curvature, when a bi-phasic system without topological changes is concerned, it gives (Gauss-Bonnet theorem) a mathematical contribution only at the boundary (18). Σ i means the surface tension: it is the combination between the stretching energy of the membrane and an entropic effect due to invisible fluctuations (23). In addition to this energy of the bare membrane, we need to introduce the protein-membrane interactions.
Protein-membrane interactions
Both phases absorb the proteins, as soon as they are introduced, but probably with a different affinity. These proteins are not soluble in water, so we think that they remain localized on the membrane and neglect possible exchange with the surrounding bath. As a consequence, the number of these molecules remains constant. Moreover, we assume that the proteins can not cross the interface (24). As suggested by S. Leibler (21), the average curvature is coupled to the protein concentration, for two possible reasons. First, this can be due to the conical shape of the proteins which locally make a deformation of the membrane. Second, an osmotic pressure on the membrane results from the part of the protein in the water (25). Whatever the microscopic effect, the proteins force the membrane to tilt nearby and thus induce a local curvature. We define the energy due to proteins in the phase (i):
F p i = S i Λ i Hφ + λ i φ + α i 2 (φ − φ eq i ) 2 + β i 2 (∇φ) 2 dS(2)
with φ the concentration of proteins on the surface. The coupling constant is Λ i . In Eq. (2), λ i is a Lagrange multiplier which allows to maintain the number of proteins constant in each phase. The model can be easily changed by considering λ i as the chemical potential of the proteins. In this case, the proteins are free to move everywhere on the membrane, to cross the interfaces or to go in the surrounding water. The last term in Eq.(2) is a Landau's expansion of the energy needed to absorb proteins on the surface nearby the equilibrium concentration φ eq . The φ gradient indicates a cost in energy to pay for a spatially inhomogeneous concentration. Since the two phases are coupled together to make a unique membrane, let us describe now the interaction between them.
Two phases in interaction
The total energy of this inhomogeneous system is the sum of these two individual energies for each phase plus at least two coupling terms. First, a more or less sharp interface exists between the raft and the phospholipidic part of the membrane. The interface is a line, the cost of energy of the transition being given by a line tension σ equivalent to the surface tension in a vapor-liquid mixing. Second, the surface of the vesicle is lightly porous to the water but not to the ions or big molecules present in the solution. So, the membrane is a semi-permeable surface and an osmotic pressure appears. As a small variation of the composition of the medium surrounding the vesicle induces a large variation of the size of the vesicle, the volume does not change when the proteins are injected, however the membrane will break down or transient pores will appear (26), which is not observed here. So, one needs to introduce a Lagrange's parameter −P to express the constraint on the volume. Physically, P is the difference of osmotic pressure between the two sides of the membrane. Then, the free energy of the bilayer becomes:
F T OT = i=1,2 (F b i + F p i ) + σ C dl − P dV(3)
with C the boundary between the two phases. A variational approach is used to find the initial state and to study its stability to small perturbations.
Static Solution and Stability analysis
Initial state
First, we look for the simplest realistic solution with an homogeneous concentration of proteins in each phase. It can be found by minimizing the energy F T OT . This minimization gives the Euler-Lagrange equations (E-L equations) plus the boundary conditions in an arbitrary set of coordinates. The surface has initially an axis of symmetry. Thus, the cylindrical coordinates seem to be the best choice. The parameterization of the surface is done by the arc-length s alone (see Fig.1). The energy becomes:
F = 2π s 1 s 0 L 1 ds + s 2 s 1 L 2 ds + σr(s 1 ) (4) with L i = κ i 2 sin(ψ) 2 r + ψ ′2 r + 2ψ ′ sin(ψ) + κ G i sin(ψ)ψ ′ −Λ i φ (sin(ψ) + ψ ′ r) + α i 2 φ 2 r + β i 2 φ ′2 r + Σ ′ i r + λ ′ i φr − P 2 r 2 sin(ψ) + γ(r ′ − cos(ψ))
with the new parameters:
Σ ′ i = Σ i + α i 2 φ 2 eq i and λ ′ i = λ i − α i φ eq i .(5)
Minimization with respect to small perturbations of the spatial coordinates (r and ψ) and of the protein concentration φ leads to the E-L equations:
ψ ′′ = sin(ψ) cos(ψ) r 2 − P r 2κ i cos(ψ) − ψ ′ r cos(ψ) + Λ i κ i φ ′ + γ κ i r sin(ψ), (6a) γ ′ = κ i 2 ψ ′2 − κ 2r 2 sin(ψ) 2 + Σ ′ i − P r sin(ψ) − Λ i φψ ′ + α i 2 φ 2 + β i 2 φ ′2 + λ ′ i φ, (6b) φ ′′ = − Λ i β i r (sin(ψ) + ψ ′ r) + α i β i φ − φ ′ cos(ψ) r + λ ′ i β i ,(6c)r ′ = cos(ψ). (6d)
The boundary conditions deduced from Eq.(4) at the junction (defined by s 1 ) between the two phases and the continuity of the radius r give:
κ 1 ψ ′ (s 1 − ǫ)r(s 1 ) + (κ 1 + κ G 1 ) sin(ψ(s 1 − ǫ)) − Λ 1 φ(s 1 − ǫ)r(s 1 ) (7a) −κ 2 ψ ′ (s 1 + ǫ)r(s 1 ) − (κ 2 + κ G 2 ) sin(ψ(s 1 + ǫ)) + Λ 2 φ(s 1 + ǫ)r(s 1 ) = 0, γ(s 1 − ǫ) − γ(s 1 + ǫ) + σ = 0, (7b) β 1 φ ′ (s 1 − ǫ)r(s 1 ) − β 2 φ ′ (s 1 + ǫ)r(s 1 ) = 0.(7c)
The boundary conditions are deduced from the bounds in the variational process. The observation of a shape discontinuity at the boundary between the two phases (11) suggests a solution which exhibits such a discontinuity of the slope at s = s 1 . The angle ψ between the surface and the radius axis is chosen discontinuous at the interface (see Fig.2). This allows a tilt of the surface at the boundary C. The simplest solution, strongly suggested by the experiment, is two spherical caps of radius R 1 and R 2 , one for each phase with a constant concentration φ i (see Fig.2, for clarity, the deformation of the raft is stronger than the real one in the initial state). The minimization shows that one must satisfy the following "bulk" conditions for each phase in order to have this solution:
λ ′ i R i = 2Λ i − α i R i φ i so R i λ i = 2Λ i (8a) 2Σ ′ i R 2 i − P R 3 i + 2Λ i φ i R i − α i φ 2 i R 2 i = 0.(8b)
The boundary conditions give two other relations: Fig. 2. Parameterization of the vesicle near its initial state. The deformation is larger than the real one in the initial state.
(2) 2 θ 1 ϕ (1) θ2κ 1 + κ G 1 R 1 − Λ 1 φ 1 = 2κ 2 + κ G 2 R 2 − Λ 2 φ 2 ,(9a)R 2 cos(θ 2 ) − R 1 cos(θ 1 ) = 2σ P R 1 sin(θ 1 ) . (9b)
where θ 1 and θ 2 are the polar angles in each phase at the boundary (see Fig.2). Note that, for each phase, the couple of equations given by Eq.(8) derive the Lagrange parameters like λ i (equivalent to a chemical potential) and Σ i (the tension) which are quantities not easy to measure experimentally. On the contrary, Eq.(9) give geometrical informations. These informations with the other constraints such as the continuity of the radius and the ratio between area of both phases are enough to fix completely the values of R 1 , R 2 , θ 1 and θ 2 . Both conditions have to be satisfied for all protein concentrations in order to ensure the existence of the initial homogeneous spherical caps, whether they are stable or not. If there is no protein, the conditions (8) reduce to λ ′ = 0 and 2Σ ′ i R 2 0 = P R 3 0 , which is the classical Laplace equation for an interface with a surface tension. Contrary to the law of capillarity, where the surface tension is a physical parameter dependent on the chemical phases involved, the tension here is not a constant characteristic of the lipids of the vesicle. It is a stress (times a length) which varies with the pressure.
Linear perturbation analysis
Now, we examine the stability of this solution. Due to the geometry, it turns out that the spherical coordinate system is more appropriate here and make the calculations easier. A perturbation of the spherical cap (i) is described by: R(θ; ϕ) = R i (1 + u(θ; ϕ)) with R i the initial radius; in a similar way, a perturbation of the protein concentration is φ = φ i (1 + v(θ; ϕ)) with φ i the initial and homogeneous concentration of proteins on the surface. We assume that the line tension is not modified by the addition of proteins, at least linearly. One can expand the free energy Eq.(4) to second order in u and v in the phase i:
F i = 2π s 1 i s 0 i L i (u, ∇u, ∆u, v, ∇v) sin θdθ with (10) L i = 2κ i −∆u + 1 4 ∆u 2 + u∆u + ∇u 2 2 + Σ ′ i R 2 i 2u + u 2 + ∇u 2 2 −2Λ i φ i R i u + v − ∆u 2 + ∇u 2 2 + uv − v∆u 2 − P R 3 i 3 (3u + 3u 2 ) + α i 2 (φ i R i ) 2 2u + 2v + u 2 + 4uv + v 2 + ∇u 2 2 + β i 2 φ 2 i (∇v) 2 +λ ′ i φ i R 2 i 2u + v + u 2 + 2uv + ∇u 2 2
The total free energy is then a function of u and v, which allows a variational approach to find the Euler-Lagrange's equations (E-L equations) and the boundary conditions. The E-L equations give shapes which are extrema of the free energy. Two sets of equations are derived: one for the zeroth order in u and v and one for the first order, the energy being calculated up to the second order of the perturbation. The zero-order equations gives the same results as Eq.(9). The first order equations are:
Λ i φ i R i (2u + ∆u) + φ 2 i R 2 i α i v − β i R 2 i ∆v = 0 (11a) −2Λ i φ i R i + 2α i φ 2 i R 2 i + 2λ ′ i φ i R 2 i v + Λ i φ i R i ∆v + 2Σ ′ i R 2 i + α i φ 2 i R 2 i + 2λ ′ i φ i R 2 i − 2P R 3 i u (11b) + 2κ i − Σ ′ i R 2 i + 2Λ i R i φ i − α i 2 φ 2 i R 2 i − λ ′ i φ i R 2 i ∆u + κ i ∆∆u = 0 equivalent to Λ i φ i R i (2v + ∆v) + Λ i φ i R i − P R 3 i 2 (2u + ∆u) + κ i ∆ (2u + ∆u) = 0 (11c)
This coupling imposes boundary conditions which must be treated at the zeroorder and the first order. We have already studied the zero-order which gives relations (9). As usual for linear perturbation analysis, the boundary conditions for the perturbation are homogeneous: u = ∆u = v = 0 at the boundary between the two phases. Contrary to first intuition and usual procedures, although Eq.(11a) and (11c) are linear, we cannot use the Legendre polynomial basis, due to the specific boundary conditions in this problem. The convenient angular basis in this case turns out to be the spherical cap harmonics, following standard techniques in geophysics (27) (see Appendix A where we recall some mathematical useful relations). These spherical cap harmonics are Legendre functions P x l (cos θ). The regular function at the pole of the cap is of the first kind and since we restrict on axisymmetric perturbations, these Legendre functions are simply hypergeometric function
2 F 1 −x l , x l + 1, 1, 1 − cos θ 2 .
Notice that x l is not an integer. In the case where it is, we recover the Legendre polynomial basis. We select the spherical cap harmonics which vanish at the boundary angle (θ = θ 1 or θ = θ 2 , see Fig.2). This condition at the boundary gives a discrete infinite set of non-integer x l + 1). We define u(θ) = Σ l u l P x l (cos θ) and v(θ) = Σ l v l P x l (cos θ). From the first E-L equation (11a), one can deduce the amplitude v i,l the protein concentration from u i,l in the phase (i):
v i,l = Λ i x (i) l (x (i) l + 1) − 2 φ i R i α i + β i (x (i) l + 1)x (i) l /R 2 i u i,l .(12)
We introduce
q 2 = x (i) l (x (i) l +1)/R 2 i ,
which is similar to the spatial period of the perturbation. Then, from the second E-L equation (11b) and after elimination using Eq.(12), we derive
Λ 2 i (q 2 − 2/R 2 i ) = Σ ′ i − α i 2 φ 2 i + κ i q 2 (α i + β i q 2 ).(13)
Our result can be compared to previous analysis made in two different asymptotic limits in the homogeneous case. In these cases, the cap is a complete sphere and x l is an integer. First, for β = 0, we recover the result of (28) for an homogeneous vesicle without diffusion. Second, when R i goes to infinity, we recover the result for an homogeneous flat membrane (21).
Notice that, in Eq.(13), the protein concentration has a similar significance as a negative surface tension: one can make the change of variable
Σ ′′ i = Σ ′ i − α i φ 2 i /2 = Σ i + α i (φ 2 eq i − φ 2 i )/2.
The principal effect of the proteins is to decrease the surface tension which is an obvious sign of instability.
Discussion
We will use the protein concentration φ i as our control parameter. Eq. (13) gives for each mode x (i) l a threshold concentration Φ i such that for φ i ≤ Φ i the initial state is stable and for φ i ≥ Φ i , one of the two phases is unstable, leading to a complete instability. The threshold concentration Φ i strongly depends on the physical properties of each phase. Then, the two parts of the initial system have no reason to be unstable simultaneously. The deformation of the other phase (not unstable to linear order) will be induced by the non-linear effects not included in this analysis.
Since the thermal energy kT is the only external energy and the typical length of the phase (i) is R i , one can introduce dimensionless parameters:
q = x l (x l + 1), κ i = κ i /kT ,Σ ′ i = Σ i R 2 i /kT ,Λ i = Λ i /(kT R i ),α i = α i /(R 2 i kT ) andβ i = β i /(R 2 i L 2 c kT )
with L c a characteristic length for the gradient of protein concentration. Then, the protein concentration is replaced by R 4 i φ 2 i which is proportional to the square of the number of proteins in the phase (i).
Rewriting the threshold (13), we find for the threshold concentration, in dimensionless parameters:
R 4 i Φ 2 i = 2 α i Σ ′ i +κ iq 2 −Λ 2 i (q 2 − 2) α i +β i (L c /R i ) 2q2 .(14)
From Eq. (14), the search of the smallest threshold concentration gives two different regimes depending on the value of the dimensionless constant:
c =Λ i 2 α i + 2β i L 2 c R 2 i κ iαi 2 = Λ 2 i (α i + 2β i /R 2 i ) κ i α 2 i .(15)
This constant describes the strength of the coupling of the protein with the membrane (Λ i ) to the resistance of the membrane (κ i ) and to the absorption power (α i ). In the weak interaction regime (c ≤ 1), the protein concentration is an increasing function ofq (see fig. 3). So, the threshold is obtained for the smallest possible x l : x 0 . The direction of the deformation (inside or outside the initial cap) would be deduced from a third order calculation or from a numerical simulation. According to the definition of Σ ′ i (Eq.(5)), the concentration required to destabilize the membrane is found bigger than the equilibrium concentration. But in this case, one expects that this threshold Φ i is difficult or impossible to reach since it requests the absorption of a concentration of proteins larger than φ eq i : probably, the excess of proteins would prefer to dissolve in the surrounding water then forming aggregates. So, the weak regime of instability is not observable experimentally, our basic state made of two spherical caps is stable and proteins are stocked only.
In the strong interaction regime, the limiting concentration shows a minimum not necessary for the smallest x l (see fig.3). This has two consequences. First, the limiting concentration is less than the equilibrium concentration and it is easier to induce the instability. Second, the first unstable mode could be modified:q ≈ 10 for the chosen numerical values. So, we have something more complex than the simple oblate/prolate (x 0 ) deformation. This regime is observable and probably corresponds to the observed shape instability. In any case, our basic state cannot be seen in the experiment except as a transient.
The existence of these two regimes, depending on the nature of the proteins, allows two possible and distinct scenarios for the cell: there is no doubt that this property is useful and probably used for biological purpose. The main difficulty of this study is the quantitative determination of the parameters since experimental values are not available even for this minimal model. Let us estimate c. The curvature of the membrane is of order 1/R i , its surface is close to R 2 i . κ is of order 10kT for an unstretched vesicle, soκ ≈ 10. α, the cost of energy needed to increase the concentration of proteins, can be deduced from the energy required to remove a protein from the surface which is is about 100kT . It changes the concentration of proteins of 1/R 2 i so 100kT = α i /R 2 i andα ≈ 100. β is deduced from L c which should be a small fraction of the radius of the sphere. We will take hereafter L c ≈ R i /10. The proteins are moving at the surface of the membrane due to the Brownian motion. So the energy to move one protein by the length L c is about kT . Then, β i = kT L 2 c R 2 i andβ i ≈ 1. The value of Λ i is more difficult to determine. Λ i is the coupling constant between the membrane and the proteins. This is expressed by the spontaneous curvature radius R P . If R P is small, the coupling effect is strong and, on the opposite, if R P is large, Λ i is small, which suggests that Λ i is proportional to 1/R P . But Λ i is an energy multiplied by a length. Then, a good order of magnitude for Λ i is kT R 2 i /R P , soΛ i = R i /R P . Finally, we get c ≈ R P /R i . If R P is smaller than R i , the system is in the strong coupling regime and in the other case, the interaction between membrane and proteins is weak. When the two phases have approximately the same physical constants (11), the instability occurs first in the largest phase, as shown by Eq. (14) in the previous conditions. Nevertheless, the ejection of one part of the membrane requires a complete nonlinear dynamical treatment which will be derived from this energy formulation.
Conclusions
We have proposed a model of instability for an inhomogeneous vesicle which absorbs proteins. This instability is at the origin of a separation into two vesicles, one for each phase as seen experimentally. Our model rests on a "bulk" effect and assumes that the proteins are distributed everywhere on the membrane contrary to the "line tension "model which assumes a high concentration of the proteins at the raft boundary. To validate (or invalidate) our model, an experimental test could be the use of phosphorescent proteins with the same properties. It would be a way to follow the place where the proteins prefer to diffuse and stay on the membrane. Although we ignore the feasibility of such an experiment, it would provide a very useful information.
We thank G. Staneva, M. Angelova and K. Koumanov for communicating their results prior to publication. We acknowledge enlightening discussions with J.B. Fournier.
A Spherical cap harmonics
The spherical cap harmonics are eigenvalues of the Laplace's equation in spherical coordinates. The Laplace's problem can then be rewritten as a Legendre's equation. The general solution is:
U m n = f (φ)L m x l (cos θ)
with θ the colatitude, φ the longitude and L m x l an associated Legendre function. The eigenvalues associated to this solution are m 2 and −x l (x l + 1). So the solutions are symmetric with respect to x l = −1/2. So we can restrict to x l ≥ −1/2 in all cases.
Generally speaking, m and x l can be integer, real or even complex and are determined by the boundary conditions. For a sphere, the solution must be periodic in the φ angle. This implies m real. In the particular case of an axisymmetric solution, which is the case in this paper, m = 0.
The boundary condition on θ for θ = 0 is a condition of regularity:
∂U 0 x l ∂θ = 0 for m = 0 (A.1) U m x l = 0 for m = 0 (A.2)
It is satisfied by the Legendre functions of the first kind and excludes those of the second kind. Notice that this condition is required both for a complete sphere and for a spherical cap.
In the case of the sphere, the boundary condition θ = π is similar to Eq.(A.1). The values of x l are then integer and the solutions are the classical associated Legendre polynomials.
For a spherical cap whose ends are given by θ = ±θ 0 , the boundary conditions at θ 0 are given by standard physical requirements. These boundary conditions can be satisfied by using two kinds of solutions such that either: Functions in one set are orthogonal to each other but are not orthogonal to those of the other set. It is easy to show that: .4), the set of solutions y l or z l is enough to form a basis of solution of the problem. In the other case, one have to combine both of them and the resolution of the complete problem becomes more harder. In the case of this paper, we focus on the case of axisymmetric solutions (m = 0). The boundary conditions are given by A.4, so the good set of parameters are the y l (0). The table A.1 presents the first values of y l , calculated for two angles θ 0 (π/6 and 5π/6), chosen as example.
The figure A.1 shows the three first P s l for θ 0 = π/6. The figure A.2 shows the three first P x l for θ 0 = 5π/6. The figure A.3 shows the deformation of the spherical cap in the case of a perturbation by the three first P x l for θ 0 = π/6.
Fig. 1 .
1Parameterization of an axisymmetric vesicle with two phases in cylindrical coordinates.
for the phase (i). l is an integer index used to order the allowed values x
increasing values. It is also the number of zero of the function P x (i) l on the cap. These harmonics have the properties to be an orthogonal basis and to be eigenfunctions of the Legendre equation with eigenvalues: x
concentration versus the reduced modeq for two different values of (14) are taken asα i = 1,κ i = 1,Σ ′ i = 60,β i (L c /R i ) 2 = 0.01. The solid line is the case of large tilt due to each protein:Λ i = 1.7. The dashed curve is the case of a small tilt:Λ i = 0.01.
= 0 for θ = ±θ 0 (A.4) These conditions are satisfied by Legendre functions P m x l (cos θ) with x l not necessary integer. No function can satisfy simultaneously the conditions (A.3) and (A.4) and there is two sets of x l which depend on the m value. We call y l (m) the values of x l such as (A.3) is satisfied and z l (m) the values of x l such as (A.4) is satisfied.
1 (m) (cos θ)P m y l 2 (m) (cos θ) sin θdθ = 0 for l 1 1 (m) (cos θ)P m z l 2 (m) (cos θ) sin θdθ = 0 for l 1 = l 2 θ 0 0 P m y l 1 (m) (cos θ)P m z l 2 (m) (cos θ) sin θdθ = − sin θ 0 P m y l 1 (m) (cos θ){[P m z l 2 (m) (cos θ)]/dθ} (y l 1 − z l 2 )(y l 1 + z l 2 + 1) If the physics requires the boundary condition (A.3) or (A
FigFig
. A.1. Legendre functions P y l versus the angle θ for θ 0 = π/6 and for l = 0 (solid curve), l = 1 (large dashing) and l = 2 (small dashing) . A.2. Legendre functions P y l versus the angle θ for θ 0 = 5π/6 and for l = 0 (solid curve), l = 1 (large dashing) and l = 2 (small dashing)Fig. A.3. Deformations of a spherical cap of radius 1 and half-angle π/6. The deformations are dues to Legendre functions P y l . For clarity, the maximum amplitude of the deformation is fixed to 0.1. This value is too large in principle for the linear analysis. The solid curve is the initial state. The largest dashing is the case l = 0. The intermediate dashing is the case l = 1. The smallest dashing is the case l = 2.
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[
"Quantum-geometric perspective on spin-orbit-coupled Bose superfluids",
"Quantum-geometric perspective on spin-orbit-coupled Bose superfluids"
]
| [
"A L Subaşı \nDepartment of Physics\nFaculty of Science and Letters\nIstanbul Technical University\n34469Maslak, IstanbulTurkey\n",
"M Iskin \nDepartment of Physics\nKoç University\nRumelifeneri Yolu34450Sarıyer, IstanbulTurkey\n"
]
| [
"Department of Physics\nFaculty of Science and Letters\nIstanbul Technical University\n34469Maslak, IstanbulTurkey",
"Department of Physics\nKoç University\nRumelifeneri Yolu34450Sarıyer, IstanbulTurkey"
]
| []
| We employ the Bogoliubov approximation to study how the quantum geometry of the helicity states affects the superfluid properties of a spin-orbit-coupled Bose gas in continuum. In particular we derive the low-energy Bogoliubov spectrum for a plane-wave condensate in the lower helicity band and show that the geometric contributions to the sound velocity are distinguished by their linear dependences on the interaction strength, i.e., they are in sharp contrast to the conventional contribution which has a square-root dependence. We also discuss the roton instability of the plane-wave condensate against the stripe phase and determine their phase transition boundary. In addition we derive the superfluid density tensor by imposing a phase-twist on the condensate order parameter and study the relative importance of its contribution from the interband processes that is related to the quantum geometry. arXiv:2110.01385v2 [cond-mat.quant-gas] | 10.1103/physreva.105.023301 | [
"https://arxiv.org/pdf/2110.01385v2.pdf"
]
| 238,259,828 | 2110.01385 | d37c8c7c4aec07efbcdfbfafc058ff96f8d9bc17 |
Quantum-geometric perspective on spin-orbit-coupled Bose superfluids
A L Subaşı
Department of Physics
Faculty of Science and Letters
Istanbul Technical University
34469Maslak, IstanbulTurkey
M Iskin
Department of Physics
Koç University
Rumelifeneri Yolu34450Sarıyer, IstanbulTurkey
Quantum-geometric perspective on spin-orbit-coupled Bose superfluids
(Dated: January 21, 2022)
We employ the Bogoliubov approximation to study how the quantum geometry of the helicity states affects the superfluid properties of a spin-orbit-coupled Bose gas in continuum. In particular we derive the low-energy Bogoliubov spectrum for a plane-wave condensate in the lower helicity band and show that the geometric contributions to the sound velocity are distinguished by their linear dependences on the interaction strength, i.e., they are in sharp contrast to the conventional contribution which has a square-root dependence. We also discuss the roton instability of the plane-wave condensate against the stripe phase and determine their phase transition boundary. In addition we derive the superfluid density tensor by imposing a phase-twist on the condensate order parameter and study the relative importance of its contribution from the interband processes that is related to the quantum geometry. arXiv:2110.01385v2 [cond-mat.quant-gas]
I. INTRODUCTION
Recent studies have shown that the quantum geometry of the Bloch states can play important roles in characterizing some of the fundamental properties of Fermi superfluids (SFs) [1,2]. The physical mechanism is quite clear in a multiband lattice: the geometric effects originate from the dressing of the effective mass of the SF carriers by the interband processes, which in return controls those SF properties that depend on the carrier mass. Besides the SF density/weight, the list includes the velocity of the low-energy Goldstone modes and the critical BKT temperature [1][2][3][4][5][6][7][8]. On the other hand the intraband processes give rise to the conventional effects. Depending on the band structure and the strength of the interparticle interactions, it has been established that the geometric effects can become sizeable and may even dominate in an isolated flat band [1]. Furthermore such geometric effects on Fermi SFs can be traced all the way back to the twobody problem in a multiband lattice in vacuum [9,10].
Despite the growing number of recent works exposing the role of quantum geometry for the Fermi SFs, there is a lack of understanding in the bosonic counterparts which are much less studied [11][12][13]. For instance Julku et al. have considered a weakly-interacting BEC in a flat band, and showed that the speed of sound has a linear dependence on the interaction strength and a square-root dependence on the quantum metric of the condensed Bloch state [11,12]. They have also showed that the quantum depletion is dictated solely by the quantum geometry and the SF weight has a quantum-geometric origin.
Motivated by the success of analogous works on spinorbit-coupled Fermi SFs [3][4][5]7], here we investigate the SF properties of a spin-orbit-coupled Bose gas from a quantum-geometric perspective. Our work differs from the existing literature in several ways [15][16][17]. In particular we derive the low-energy Bogoliubov spectrum for a plane-wave condensate in the lower helicity band and identify the geometric contributions to the sound velocity. The geometric effects survive only when the single-particle Hamiltonian has a a σ z term in the pseudospin basis that is coupled with a σ x (and/or equivalently a σ y ) term. In contrast to the conventional contribution that has a square-root dependence on the interaction strength, we find that the geometric ones are distinguished by a linear dependence. Similar to the Fermion problem where the geometric effects dress the effective mass of the Goldstone modes, here one can also interpret the geometric terms in terms of a dressed effective mass for the Bogoliubov modes. We also discuss the roton instability of the plane-wave ground state against the stripe phase and determine the phase transition boundary. All of these results are achieved analytically by reducing the 4 × 4 Bogoliubov Hamiltonian (that involves both lower and upper helicity bands) down to 2 × 2 through projecting the system onto the lower helicity band. The projected Hamiltonian works extremely well except for a tiny region in momentum-space around the point where the helicity bands are degenerate. In addition we derive the SF density tensor by imposing a phase-twist on the condensate order parameter and analyze the relative importance of its contribution from the interband processes [13].
The rest of the paper is organized as follows. We begin with the theoretical model in Sec. II: the many-body Hamiltonian is introduced in Sec. II A and the noninteracting helicity spectrum is reviewed in Sec. II B. Then we present the Bogoliubov mean-field theory for a planewave condensate in Sec. III: the four branches of the full Bogoliubov spectrum are discussed in Sec. III A and the two branches of the projected (i.e., to the lower-helicity band) Bogoliubov spectrum are derived in Sec. III B. Furthermore, by analyzing the resultant Bogoliubov spectrum in the low-energy regime, we find closed-form analytic expressions for the Bogoliubov modes in Sec. III C and for the roton instability of the plane-wave condensate against the stripe phase in Sec. III D. Finally we derive and analyze the SF density tensor and condensate density in Sec. IV. The paper ends with a summary of our conclusions given in Sec. V.
II. THEORETICAL MODEL
In order to study the interplay between a BEC and SOC, and having cold-atom systems in mind, here we consider a two-component atomic Bose gas that is characterized by a weakly-repulsive zero-ranged (contact) interactions in continuum. It is customary to refer to such a two-component bosonic system as the pseudospin-1/2 Bose gas.
A. Pseudospin-1/2 Bose Gas In particular, by making use of the momentum-space representation, we express the single-particle Hamiltonian in the usual form
H 0 = k Λ † k ε k + ε k0 σ 0 + d k · σ m Λ k ,(1)
where k = (k x , k y , k z ) is the momentum vector with = 1 and Λ † k = a † ↑k a † ↓k is a two-component spinor with the creation operator a † σk for a pseudospin-σ particle in state |σk = a † σk |0 . Here σ = {↑, ↓} labels the two components of the Bose gas and |0 is the vacuum state. The first term ε k = k 2 /(2m) is the kinetic energy of a particle where ε k0 is a convenient choice of an energy offset (k 0 is defined below) and σ 0 is an identity matrix. The second term is the so-called SOC where σ = (σ x , σ y , σ z ) is a vector of Pauli spin matrices and d k = d x k , d y k , d z k is the SOC field with linearly dispersing components d i k = α i k i . Here we choose α i ≥ 0 and α x ≥ {α y , α z } without the loss of generality.
Similarly a compact way to express the intraspin and interspin interaction terms is
H U = 1 2V σσ k1+k2=k3+k4 U σσ a † σk1 a † σ k2 a σ k3 a σk4 ,(2)
where V is the volume and U σσ ≥ 0 is the strength of the interactions. Here we consider a sufficiently weak U ↑↓ in order to prevent competing phases that are beyond the scope of this paper. See Sec. III D for a detailed account of the stability analysis. In addition we include a chemical potential term H µ = − σk µ σ a † σk a σk to the total Hamiltonian H = H 0 + H U + H µ of the system, and determine µ σ in a self-consistent fashion.
B. Helicity Bands
Let us first discuss the single-particle ground state. The eigenvalues of the Hamiltonian matrix shown in Eq. (1) can be written as
ξ sk = ε k + ε k0 + s d k m ,(3)
where s = ± labels, respectively, the upper and lower band and d k = |d k | is the magnitude of the SOC field. Therefore the single-particle (helicity) spectrum exhibits two branches due to SOC. In the pseudospin basis |σk , the corresponding eigenvectors (i.e., helicity basis) |sk = a † sk |0 can be represented as |+, k = u k v k e iϕ k T for the upper and |−, k = −v k e −iϕ k u k T for the lower
helicity band, where u k = (d k + d z k )/(2d k ), v k = (d k − d z k )/(2d k ), ϕ k = arg(d x k + id y k )
, and T denotes the transpose. Alternatively,
a ↑k a ↓k = u k −v k e −iϕ k v k e iϕ k u k a +,k a −,k
is the transformation between the annihilation operators for the pseudospin and helicity states. For notational convenience, the lower helicity state |−, k is denoted as |φ k in the rest of the paper. Then the single-particle ground state |φ k0 is determined by setting ∂ξ −,k /∂k i = 0, leading to either k i = 0 or
k x / α x −2 −1 0 1 2 kz / αx −2 −1 0 1 2 ξ sk/(α 2 x /2m)α 2 i = d k .
Here we choose k 0 = (α x , 0, 0) without the loss of generality [18][19][20], for which case the singleparticle ground-state energy ξ −,k0 = 0 vanishes (see Fig. 1) and the single-particle ground state |φ k0 = −1/ √ 2 1/ √ 2 T admits a real representation. Note that the ground state is at least two-fold degenerate with the opposite-momentum state
|φ −k0 = 1/ √ 2 1/ √ 2 T ,
and we highlight its competing role in Sec. III D. Having introduced the theoretical model, and discussed its single-particle ground state, next we analyze the manybody ground state within the Bogoliubov mean-field approximation.
III. BOGOLIUBOV THEORY
Under the Bogoliubov mean-field approximation, the many-body ground state is known to be either a planewave condensate or a stripe phase depending on the relative strengths between the intraspin and interspin interactions [18][19][20][21][22]. See Sec. III D for a detailed account of the stability analysis. Assuming that U ↑↓ is sufficiently weak, here we concentrate only on the former phase.
A. Bogoliubov Spectrum
In order to describe the many-body ground state |φ k0 that is macroscopically occupied by N 0 particles, we replace the annihilation and creation operators in accordance with a σk = ∆ σ √ V δ kk0 +ã σk . Here the complex field ∆ σ = √ n 0 σ|φ k0 corresponds to the meanfield order parameter for the condensate with condensate density n 0 = N 0 /V , δ ij is a Kronecker-delta, and the operatorã σk denotes the fluctuations on top of the ground state. Following the usual recipe, we neglect the third-and fourth-order fluctuation terms in the interaction Hamiltonian. Then the excitations are described by the so-called Bogoliubov Hamiltonian
H B = 1 2 q Ψ † q H pp q H ph q H hp q H hh q Ψ q ,(4)H pp q = K ↑q U ↑↓ ∆ ↑ ∆ * ↓ U ↑↓ ∆ * ↑ ∆ ↓ K ↓q + d k0+q · σ m ,(5)H ph q = U ↑↑ ∆ 2 ↑ U ↑↓ ∆ ↑ ∆ ↓ U ↑↓ ∆ ↑ ∆ ↓ U ↓↓ ∆ 2 ↓ ,(6)
where Ψ † q = ã † ↑,k0+qã † ↓,k0+qã ↑,k0−qã↓,k0−q is a four-component spinor and K σq = ε k0+q + ε k0 − µ σ + 2U σσ |∆ σ | 2 + U ↑↓ |∆ −σ | 2 with the index −σ denoting the opposite component of the spin. The other terms are simply related via H hh q = (H pp −q ) * and H hp q = (H ph q ) † . The prime symbol indicates that the summation is over all of the non-condensed states. In this approximation, µ σ is determined by setting the first-order fluctuation terms to 0, leading to
µ σ = U σσ |∆ σ | 2 + U ↑↓ |∆ −σ | 2 . Note that ∆ ↑ = −∆ ↓ = − n 0 /2 are real for our particular choice for the ground state |φ k0 .
The Bogoliubov spectrum E n sq is determined by the eigenvalues of τ z H q [11,12], i.e.,
τ z H q |χ n sq = E n sq |χ n sq ,(7)
where τ z is a Pauli matrix acting only on the particlehole sector, H q is the 4 × 4 Hamiltonian matrix shown in Eq. (4), and |χ n sq is the corresponding Bogoliubov state. Here n = ± labels, respectively, the upper and lower Bogoliubov band, and s = ± labels, respectively, the quasiparticle and quasihole branch for a given band n, leading to four Bogoliubov modes for a given q. The Bogoliubov states are normalized in the usual way, i.e., if we denote |χ n sq = |χ n sq 1 |χ n sq 2 then 1 χ n sq |χ n sq 1 − 2 χ n sq |χ n sq 2 = s. While the Bogoliubov spectrum exhibits E n sq = −E n −s,−q as a manifestation of the quasiparticle-quasihole symmetry, Eq. (7) does not allow for a closed-form analytic solution in general, and its characterization requires a fully numerical procedure.
In order to gain some analytical insight into the lowenergy Bogoliubov modes, we assume that the energy gap between the lower and upper helicity bands nearby the ground state |φ k0 is much larger than the interaction energy. This occurs when the SOC energy scale is much stronger than the interaction energy scale. In this case the occupation of the upper band is negligible, and the system can be projected solely to the lower band as discussed next.
B. Projected System
The total Hamiltonian H of the system can be projected to the lower helicity band as follows [18]. Using the identity operator σ 0 = s |sk sk| for a given k, we first reexpress a σk = s σ|sk a sk , and discard those terms that involve the upper band, i.e., a σk → σ|φ k a −,k . This procedure leads to
h 0 + h µ = k ξ −,k − µ a † −,k a −,k ,(8)h U = 1 2V k1+k2=k3+k4 f k3k4 k1k2 a † −,k1 a † −,k2 a −,k3 a −,k4 , (9) f k3k4 k1k2 = σσ U σσ φ k1 |σ φ k2 |σ σ |φ k3 σ|φ k4 ,(10)
where µ = (µ ↑ + µ ↓ )/2 is the effective chemical potential and f k3k4
k1k2 = U ↑↑ v k1 v k2 v k3 v k4 e i(ϕ k 1 +ϕ k 2 −ϕ k 3 −ϕ k 4 ) + U ↓↓ u k1 u k2 u k3 u k4 + U ↑↓ v k1 u k2 u k3 v k4 e i(ϕ k 1 −ϕ k 4 )
is the effective long-range interaction for the projected system. We note that the long-range nature of the effective interaction plays a crucial role in the Bogoliubov spectrum as discussed in Sec. III D.
Under the Bogoliubov mean-field approximation that is used in Sec. III A, we replace the creation and annihilation operators in accordance with a −,k = √ N 0 δ kk0 +ã −,k and set the first-order fluctuation terms to 0. This leads to µ = n 0 f k0k0 k0k0 = (n 0 /4) σσ U σσ , which is consistent with µ σ that is found in Sec. III A. The zeroth-order fluctuation terms give −µN 0 + n 0 f k0k0 k0k0 N 0 /2. Then the excitations above the ground state are described by the Bogoliubov Hamiltonian
h B = 1 2 q ψ † q h pp q h ph q h hp q h hh q ψ q ,(11)h pp q = ξ −,k0+q − µ + n 0 2 f k0,k0+q k0,k0+q + f k0+q,k0 k0+q,k0 +f k0,k0+q k0+q,k0 + f k0+q,k0 k0,k0+q ,(12)h ph q = n 0 2 f k0,k0 k0+q,k0−q + f k0,k0 k0−q,k0+q ,(13)
where
ψ † q = ã † −,k0+qã −,k0−q
is a two-component spinor, and the other terms are simply related via h hh q = h pp −q and h hp q = (h ph q ) * . The Bogoliubov spectrum sq is determined by the eigenvalues of τ z h q , leading to two Bogoliubov modes for a given q, i.e.,
sq = h pp q − h hh q 2 + s h pp q + h hh q 2 2 − |h ph q | 2 , (14) h pp q = ξ −,k0+q − µ + n 0 2 σσ U σσ | φ k0+q |σ | 2 +n 0 σσ U σσ φ k0+q |σ σ |φ k0 φ k0 |σ σ|φ k0+q , (15) h ph q = n 0 σσ U σσ φ k0+q |σ σ|φ k0 φ * k0 |σ σ |φ * k0−q .(16)
Here s = ± labels, respectively, the quasiparticle and quasihole branch of the lower Bogoliubov band (i.e., n = −) that is discussed in Sec. III A. See Fig. 2 for their excellent numerical benchmark except for the spurious jumps at q = ∓k 0 that are discussed in Sec. III E. The Bogolibov spectrum exhibits +,q = − −,−q as a manifestation of the quasiparticle-quasihole symmetry. Note that when U σσ = U δ σσ , these expressions reduce exactly to those of Ref. [11,12] with M = 2, where our h pp q = ξ −,k0+q + U n 0 /2 and h ph q correspond, respectively, to their q 2 /(2m ef f ) + µ and µα(q) provided that µ = U n 0 /2 in this particular case. Such a reduction may not be surprising since the intraspin interactions U ↑↑ and U ↓↓ play the roles of sublattice-dependent onsite interactions U AA and U BB , and the interspin interaction U ↑↓ plays the role of a (long-range) inter-sublattice interaction U AB . Thus our U σσ = U δ σσ limit corresponds precisely to the U = U AA = U BB and U AB = 0 case that is considered in Ref. [11,12].
We can make further analytical progress through a lowq expansion around the ground state, and use the fact that | σ|φ k0 | 2 = 1/2 for both pseudospin components, i.e., the z component of k 0 vanishes for the ground state.
−3 −2 −1 0 1 2 3 q x /α x −10 −5 0 5 10 E/(n 0 U ) −0.5 0.0 0.5 −1.1 −1.0 −0.9 E + sq E − sq sq FIG. 2:
Bogoliubov spectrum is shown as a function of qx when qy = 0 = qz, U = U ↑↑ = 2U ↓↓ = 4U ↑↓ , αx = αy = 2/ξ with the healing length ξ = 1/ √ 2mnU , and αz = 0. Here the total particle density n ≈ n0 is set to na 3 = 10 −6 where a = mU/(4π) is the scattering length. The full spectrum (solid lines) is shown together with the projected one (dotted lines) that is given by Eq. (14). In addition the low-q expansion Eq. (20) is shown as dashed black lines in the right inset. If one sets U ↑↑ = U ↓↓ then the band gap shown in the left inset disappears, i.e., see Sec. III E for the analysis of the spurious jumps at qx = ∓αx. If one sets U ↑↓ = U ↑↑ = U ↓↓ then two additional zero-energy modes appear at qx = ∓2αx, i.e., see Sec. III D for the analysis of the roton instability.
C. Low-Momentum Expansion
Up to second order in q, the low-energy expansions around the ground state |φ k0 can be written as
h pp q = 1 2 ij q i q j M −1 ij − µ + n 0 2 σσ U σσ + 2n 0 iσσ q i U σσ × Re ∂ i φ k |σ σ|φ k0 + n 0 ijσσ q i q j U σσ × Re ∂ i ∂ j φ k |σ σ|φ k0 + ∂ i φ k |σ σ|∂ j φ k /2 + ∂ i φ k |σ σ |φ k0 φ k0 |σ σ|∂ j φ k ,(17)h ph q = n 0 4 σσ U σσ + n 0 2 ijσσ q i q j U σσ ∂ i ∂ j φ k |σ σ|φ k0 − 2 ∂ i φ k |σ σ|φ k0 φ * k0 |σ σ |∂ j φ * k ,(18)
where the spectrum of the lower helicity band is expanded as ξ −,k0+q = (1/2) ij q i q j M −1 ij . Here M −1 is the inverse of the effective-mass tensor whose elements are given by M −1 xx = 1/m, M −1 yy = 1/m − α 2 y /(mα 2 x ), M −1 zz = 1/m − α 2 z /(mα 2 x ), and 0 otherwise. In addition Re denotes the real part of an expression and |∂ i φ k stands for ∂|φ k /∂k i in the k → k 0 limit. By plugging these expansions in Eq. (14), and keeping up to second-order terms in q, we obtain
sq = 2n 0 iσσ q i U σσ Re ∂ i φ k |σ σ|φ k0 + s X q , (19) X q = µ ij q i q j M −1 ij + n 0 σσ U σσ ∂ i φ k |σ σ|∂ j φ k +Re ∂ i ∂ j φ k |σ σ|φ k0 + 2 ∂ i φ k |σ σ |φ k0 φ k0 |σ × σ|∂ j φ k + 2Re ∂ i φ k |σ σ |φ k0 φ * k0 |σ σ|∂ j φ * k ,
for the low-energy Bogoliubov spectrum of the projected system. In addition to the conventional effective-mass term that depends only on the helicity spectrum, here we have the so-called geometric terms that depend also on the helicity states. The quantum geometry of the underlying Hilbert space is masked behind those terms that depend on |∂ i φ k and |∂ i ∂ j φ k [11,12]. While most of these terms cancel one another, they lead to
sq = n 0 U ↓↓ − U ↑↑ α z q z 2α 2 x + s 2 n 0 (U ↑↑ + U ↓↓ + 2U ↑↓ ) × ij q i q j M −1 ij + n 0 U ↑↑ + U ↓↓ − 2U ↑↓ α 2 z q 2 z 4α 4 x ,(20)
manifesting explicitly the quasiparticle-quasihole symmetry. When U ↑↑ = U ↓↓ , Eq. (20) is in full agreement with the recent literature for the reported parameters [18]. In addition see the right inset in Fig. 2 for its numerical benchmark with Eq. (14). Our work reveals that the linear term in α z q z that is outside of the square root as well as the quadratic term in α z q z that is in the inside have a quantum-geometric origin. Note that the geometric terms that depend on α x and α y vanish all together. Thus we conclude that the geometric effects survive only in the presence of a finite σ z coupling assuming a σ x (and/or equivalently a σ y ) coupling to begin with. See Sec. II B for our initial assumption in choosing k 0 . Although we choose a k 0 that is symmetric in y and z directions, the condition | σ|φ k0 | 2 = 1/2 breaks this symmetry in general for other k 0 values as it requires k 0z = 0. The remaining geometric terms can be isolated from the conventional effective-mass term in the q → (0, 0, q z ) limit when α z ≈ α x , leading to q i q j M −1 ij = 0. Therefore this particular limit can be used to distinguish the geometric origin of sound velocity from the conventional one, i.e., unlike the conventional term that has ∝ √ U dependence on the interaction strength, the geometric ones have ∝ U dependence. The square root vs. linear dependence is consistent with the recent results on multi-band Bloch systems [11,12]. We note that the geometric term that is inside the square root can be incorporated into the conventional effective mass term, leading to a 'dressed' effective mass M −1
zz → M −1 zz + n 0 (U ↑↑ + U ↓↓ − 2U ↑↓ )α 2 z /(4α 4
x ) for the Bogoliubov modes [11,12]. While this geometric dressing shares some similarities with the dressing of the effective-mass tensor of the Cooper pairs or the Goldstone modes in spin-orbit-coupled Fermi SFs, their mathematical structure is entirely different [4,5]. The latter involves a k-space sum over the quantum-metric tensor of the helicity bands that is weighted by a function of other quantities including the excitation spectrum.
We note in passing that when U σσ = U δ σσ , our Eq. (19) reduce exactly to that of Ref. [11,12] with M = 2, for which case we obtain sq = s q with
q = (U n 0 /2) ij q i q j M −1 ij + U n 0 ∂ i φ k |∂ j φ k + 2U n 0 σ Re ∂ i φ k |σ σ|φ k0 φ * k0 |σ σ|∂ j φ * k 1/2 . Fur-
thermore, using the fact that |φ k0 is real for the ground state, we find q = (
U n 0 /2) ij q i q j M −1 ij + U n 0 ∂ i φ k |∂ j φ k + U n 0 Re ∂ i φ k |∂ j φ * k 1 2 .
In comparison the quantum metric of the lower helicity band is defined by
g k ij = Re ∂ i φ k | σ 0 − |φ k φ k | |∂ j φ k , and it reduces to g k ij = ∂ i φ k |∂ j φ k only when |φ k is real for all k. This is because ∂ i φ k |φ k = − φ k |∂ i φ k = − ∂ i φ * k |φ * k
must vanish when φ k is real. Thus we conclude that the geometric dressing of the effective mass of the Bogoliubov modes can be written in terms of g k ij when |φ k is real for all k. This is clearly the case when d y k = 0 in twoband lattices and when α y = 0 in spin-orbit-coupled Bose SFs.
Furthermore, when α z = 0, we find that the competition between the linear term in q z that is outside of the square root and the quadratic terms within the square root in Eq. (20) causes an energetic instability (i.e., sq changes sign and becomes ±,q≶0 ) in the q → 0 limit unless
4U 2 ↑↓ − (3U ↑↑ − U ↓↓ )(3U ↓↓ − U ↑↑ ) U ↑↑ + U ↓↓ + 2U ↑↓ ≤ 4α 2 x mn 0 α 2 x α 2 z − 1(21)
is satisfied. For instance this condition reduces to 3U ↓↓ ≥ U ↑↑ ≥ U ↓↓ /3 when α z = α x in the U ↑↓ → 0 limit, revealing a peculiar constraint on the strength of the interactions. Our calculation suggests that the physical origin of this instability is related to the quantum geometry of the underlying space without a deeper insight. In addition, when α z = 0, Eq. (20) further suggests that there is a dynamical instability (i.e., sq becomes complex) unless the quadratic terms within the square root are positive, i.e., 1 − α 2
z /α 2 x + mn 0 α 2 z (U ↑↑ + U ↓↓ − 2U ↑↓ )/(4α 4
x ≥ 0. This condition is most restrictive when α z → α x , giving rise to (U ↑↑ + U ↓↓ )/2 ≥ U ↑↓ for the dynamical stability of the system. Next we show that the dynamical instability never takes place because it is preceded by the so-called roton instability, given that the geometric mean of U ↑↑ and U ↓↓ is guaranteed to be less than or equal to the arithmetic mean.
D. Roton Instability at q = ∓2k0
The zero-energy Bogoliubov mode that is found at q = 0 is a special example of the Goldstone mode that is associated with the spontaneous breaking of a continuous symmetry in SF systems. In addition to this phonon mode, the Bogoliubov spectrum also exhibits the so-called roton mode at finite q. This peculiar spectrum clearly originates from the long-range interaction characterized by Eq. (10), and it is a remarkable feature given the surge of recent interest in roton-like spectra in various other cold-atom contexts [23][24][25][26][27] that paved the way for the creation of dipolar Bose supersolids [26,27]. Furthermore the roton spectrum [28,29] along with some supersolid properties [30,31] have also been measured with Raman SOC. As a consequence of these outstanding progress, the roton spectrum is nowadays considered as a possible route and precursor to the solidification of Bose SFs.
Depending on the interaction parameters, our numerics show that there may appear an additional pair of zero-energy modes at finite q when the roton gap vanishes. See also Refs. [15,17,19,20,22,32] for related observations. It turns out they always appear precisely at opposite momentum q = ∓2k 0 when the local minimum (maximum) of ±,q touches the zero-energy axis with a quadratic dispersion away from it. For instance the roton minimum and its gap is clearly visible in Fig. 2 at q x = ∓2α x .
Given this numerical observation, we evaluate Eqs. (15) and (16) at q = ∓2k 0 , leading to, e.g., the quasiparticle-quasiparticle element h pp −2k0 = (U ↑↑ + U ↓↓ − 2U ↑↓ )n 0 /4, quasihole-quasihole element h hh −2k0 = ε 3k0 +(U ↑↑ +U ↓↓ +2U ↑↓ )n 0 /4, and quasiparticle-quasihole element h ph −2k0 = (U ↓↓ − U ↑↑ )n 0 /4. Then, by plugging them into Eq. (14), and noting that the stability of the Bogoliubov theory requires the local minimum (maximum) of the quasiparticle (quasihole) spectrum to satisfy ±,∓2k0 ≷ 0, we obtain the following condition
2α 2 x mn 0 + U ↑↑ 2α 2 x mn 0 + U ↓↓ > 2α 2 x mn 0 + U ↑↓ 2 .(22)
This condition guarantees the energetic stability of the many-body ground state that is presumed in Sec. II B to begin with, and it is in full agreement with the previously known results. For instance it reduces to U ↑↑ U ↓↓ > U ↑↓ in the absence of a SOC when α x = 0, and it reduces to U > U ↑↓ for equal intraspin interactions U ↑↑ = U ↓↓ = U when α x = 0 [18,21]. In general Eq. (22) suggests that while the ground state is energetically stable for all α x values when U ↑↑ U ↓↓ > U ↑↓ , it is stable for sufficiently strong SOC strengths α x > α c when
U ↑↑ U ↓↓ < U ↑↓ < (U ↑↑ + U ↓↓ )/2. Here α c = [mn 0 (U 2 ↑↓ − U ↑↑ U ↓↓ )/(2U ↑↑ + 2U ↓↓ − 4U ↑↓ )] 1/2 is the critical threshold.
Both the appearance of an additional pair of zeroenergy modes at q = ∓2k 0 and the associated instability of the many-body ground state that is caused by ±,q ≶ 0 can be traced back to the degeneracy of the lower-helicity band ξ −,k that is discussed in Sec. II B. For instance, when α x ≥ {α y , α z }, our single-particle ground state |φ k0 is at least two-fold degenerate with the opposite-momentum state |φ −k0 . Note that the relative momentum between these two particle (hole) states is exactly ∓2k 0 . Then Eq. (22) suggests that while our initial choice for a plane-wave condensate that is described purely by the state |φ k0 is energetically stable for sufficiently weak U ↑↓ , it eventually becomes unstable against competing states with increasing U ↑↓ . Since this instability also occurs precisely at q = ∓2k 0 , it clearly signals the possibility of an additional condensate that is described by the state |φ −k0 . Thus, when Eq. (22) is not satisfied, we conclude that the many-body ground state corresponds to the so-called stripe phase that is described by a superposition of two states with opposite momentum, i.e., |φ k0 and |φ −k0 [15-17, 21, 22, 32]. Indeed some supersolid properties of the stripe phase have already been observed with Raman SOC [30,31].
We would like to emphasize that this conclusion is immune to the increased degeneracy of the helicity states when the SOC field is isotropic in momentum space. For instance, despite the circular degeneracy caused by a Rashba SOC when α x = α y , the zero-energy modes still appear at q = ∓2k 0 , and therefore, the stripe phase again involves a superposition of two states with opposite momentum.
E. Spurious Jumps at q = ∓k0
As shown in Fig. 2, there is an almost perfect agreement between the Bogoliubov spectrum of the 4 × 4 Hamiltonian and that of the 2 × 2 projected one except for a tiny region in the vicinity of a peculiar jump at q = ∓k 0 . In order to reveal its physical origin, here we set α z = 0 for its simplicity, and expand the Hamiltonian matrix at q = −k 0 + δ for a small δ = (δ, 0, 0). We find that
h pp δ = ξ −,δ + n 0 4 U ↑↑ + U ↓↓ + 2U ↑↓ cos(ϕ δ ) , h ph δ = n 0 4 U ↑↑ − U ↓↓ + 2U ↑↓ cos(ϕ δ ) ,
where the phase angle ϕ k is defined in Sec.II B leading to cos(ϕ δ ) = sgn(δ). This analysis shows it is those coupling terms U ↑↓ cos(ϕ δ ) between the ↑ and ↓ sectors in the Bogoliubov Hamiltonian that is responsible for the spurious jump at δ = 0 upon the change of sign of δ. Note that our initial motivation in deriving the projected Hamiltonian in Sec. III A is the assumption that the energy gap between the lower and upper helicity bands nearby the single-particle ground state |φ k0 is much larger than the interaction energy. While the validity region of this assumption in k space is not limited with the ground state, it clearly breaks down in the vicinity of k = 0 where the s = ± helicity bands are degenerate (see Fig. 1). For this reason our projected Hamiltonian becomes unphysical and fails to capture the actual result in a tiny region around q = −k 0 . Having presented a detailed analysis of the Bogoliubov spectrum, next we determine the SF density tensor and compare it to the condensate density of the system.
IV. SUPERFLUID vs. CONDENSATE DENSITY
In this paper we define the SF density ρ s by imposing a so-called phase twist on the mean-field order parameter [34][35][36]. When the SF flows uniformly with the momentum Q, the SF order parameter transforms as ∆ σ → ∆ σ e iQ·r , and the SF density tensor ρ ij is defined as the response of the thermodynamic potential Ω Q to an infinitesimal flow, i.e.,
ρ ij = m V lim Q→0 ∂ 2 Ω Q ∂Q i ∂Q j .(23)
Here the derivatives are taken for a constant ∆ σ and µ σ , i.e., the mean-field parameters do not depend on Q in the Q → 0 limit. We note that the SF mass density tensor mρ ij is a related quantity, and it corresponds to the total mass involved in the flow.
Let us now calculate Ω Q in the low-Q limit. In the absence of an SF flow when Q = 0, the thermodynamic potential Ω 0 can be written as Ω 0 = Ω zp + (T /2) q Tr ln G −1 0 q , where Ω zp = −µN 0 /2 − q ε q + µ /2 is the zero-point contribution, T is the temperature with the Boltzmann constant k B = 1, Tr is the trace, and G −1 0 q = iω σ 0 τ z − H q is the inverse of the Green's function for the Bogoliubov Hamiltonian that is given in Eq. (4). Here ω = 2π T is the bosonic Matsubara frequency with an integer. In order to make some analytical progress, we make use of the Bogoliubov states and spectrum determined by Eq. (7), and define [11,12]
G 0 q = ns s|χ n sq χ n sq | iω − E n sq .(24)
This expression clearly satisfies G −1 0 q G 0 q = σ 0 τ 0 . In the presence of an SF flow when Q = 0, the thermodynamic potential Ω Q can be obtained through a gauge transformation of the bosonic field operatorsã σq →ã σq e iQ·r . This transformation removes the phases of the SF order parameters, and we obtain the inverse Green's function G −1 Q q = G −1 0 q − Σ Q of the twisted system. Its Qdependent part has three terms Σ Q = Σ Q,1 + Σ Q,2 + Σ Q,3 [36]: while the SOC-independent terms Σ Q,1 =
Q 2 2m σ 0 τ 0 and Σ Q,2 = σ0 m (k 0 + q) · Q 0 0 (k 0 − q) · Q are
diagonal both in the spin and particle-hole sectors, the SOC-induced term Σ Q,3 = 1 m d Q · σ 0 0 d Q · σ * is diagonal only in the particle-hole sector. These terms can be conveniently reexpressed as Σ Q,1 =
(1/2) ij Q i Q j ∂ i ∂ j H q and Σ Q,2+3 = i Q i τ z ∂ i H q , where ∂ i H q stands for ∂H q /∂q i .
Since we are interested only in the low-Q limit of Ω Q , we can use the Taylor expansion ln det G −1 Q q = Tr ln G −1 0 q − Tr ∞ l=1 (G 0 q Σ Q ) l /l, and keep up to second-order terms in Σ Q . This calculation leads to
ρ ij =mn 0 M −1 ij − mT 2V q Tr G 0 q ∂ i ∂ j H q + Tr G 0 q τ z ∂ i H q G 0 q τ z ∂ j H q .(25)
Here the first term is due to the kinetic energy of the condensate in the presence of an SF flow, i.e., there is an additional quadratic contribution
(N 0 /2) ij Q i Q j M −1 ij
to Ω zp coming from the low-Q expansion of k ξ −,k+Q a † −,k a −,k around k 0 . Thus, when the inverse of the effective-mass tensor M −1 ij vanishes, ρ ij is determined entirely by the Bogoliubov Hamiltonian, i.e., the quantum fluctuations above the condensate. The trace of the Green's function in the second term is related to the density of excited (noncondensate) particles n e since its diagonal elements yield n e = −(T /V ) q G 11 0 q +G 22 0 q e −iω 0 + , or alternatively, n e = −(T /V ) q G 33 0 q + G 44 0 q e iω 0 + . Thus, by performing the summation over the Matsubara frequencies, we eventually obtain
ρ ij = n t δ ij − n 0 α 2 y δ iy + α 2 z δ iz α 2 x + m 2V nn ss q ss χ n sq |τ z ∂ i H q |χ n s q χ n s q |τ z ∂ j H q |χ n sq f B (E n sq ) − f B (E n s q ) E n sq − E n s q ,(26)
where n t = n 0 + n e is the total density of particles in the system and f B (x) is the Bose-Einstein distribution function. Here a partial derivative ∂f B (E n sq )/∂E n sq = −[1/(4T )]cosech 2 E n sq /(2T ) is implied when the summation indices coincide simultaneously (n = n and s = s ). In comparison to the SF density, the noncondensate density can be written as n e = 1 2V nsq s χ n sq |χ n sq f B (E n sq ) + 2 χ n sq |χ n sq 2 ,
= 1 2V nsq s χ n sq |χ n sq f B (E n sq ) − 1/2 .(27)
We checked that both expressions yield the same numerical result. Note that n 0 e = [1/(2V )] nq (−1 + χ n −,q |χ n −,q ) is the so-called quantum depletion of the condensate at T = 0.
As an illustration, in the case of a single-component Bose gas, there is a single Bogoliubov band with the usual quasiparticle-quasihole symmetric spectrum E sq = sE q where E q = ε q (ε q + 2U n 0 ), and by plugging χ sq |τ z ∂ i H q |χ s q = (sq i /m)δ ss into Eq. (26), we recover the textbook definition ρ ij = n t δ ij + [1/(mV )] q q i q j ∂f B (E q )/∂E q of ρ s [37]. This shows that ρ ij = n t δ ij at zero temperature and that the entire gas is SF. Similarly, by plugging χ sq |χ s q = (ε q + U n 0 )/E q into Eq. (28), we recover the textbook definition of n e = n 0 e + n T e , where n 0 e = [1/(2V )] q [−1 + (ε q + U n 0 )/E q ] is the quantum depletion and n T e = (1/V ) q (ε q + U n 0 )f B (E q )/E q is the thermal one [37].
We note in passing that Eq. (26) is consistent with the so-called SF weight that is derived in Ref. [11,12] for a multi-band Bloch Hamiltonian. See also [13]. Unlike our phase-twist method, they define the SF weight as the long-wavelength and zero-frequency limit of the currentcurrent linear response. In particular our expression for a continuum model is formally equivalent to their D s 1,µν + D s 2,µν + D s 3,µν with the caveat that D s 2,µν is cancelled by the interband contribution of D s 1,µν . This is similar to the cancellation that they observed for the Kagome lattice. Furthermore Eq. (26) can also be split into two parts ρ ij = ρ intra ij +ρ inter ij depending on the physical origin of the terms [11,12]: the intraband (interband) processes give rise to the conventional (geometric) contribution. This division is motivated by the success of a similar description with Fermi SFs [3,7].
In order to provide further evidence for its geometric origin, in Fig. 3 we compare the interband contribution with that of the intraband one coming from the summation term in Eq. (26). Here we set T → 0. First of all this figure shows that the total contribution from the summation term decreases with the increased strength and isotropy of the SOC fields, i.e., when α y → α x in Figs. 3(a,b,c) and when α z → α in Figs. 4(d,e,f). Thus ρ xx always decreases from n t with SOC. However, depending on the value of α y and α z , the remaining contribution n e + n 0 (α 2
x − α 2 y δ iy − α 2 z δ iz )/α 2 x to ρ yy and ρ zz in Eq. (26) may compete with or favor the contribution from the summation term. More importantly Fig. 3 shows that not only ρ zz has the largest interband contribution but also its relative weight is predominantly controlled by α z = 0. These findings are in support of our Bogoliubov dispersion given in Eq. (20) whose quantum-geometric contributions are fully controlled by α z = 0. For completeness, in Fig. 4 we present the quantum depletion n 0 e as a function of SOC parameters when U = U ↑↑ = U ↓↓ = 10U ↑↓ /9. This figure shows that n 0 e increases with the increased strength and isotropy of the SOC fields [18], i.e., when α y → α x in Fig. 4(a) and when α z → α in Fig. 4(b). This is clearly a direct consequence of the increased degeneracy of the single-particle spectrum. However, since n 0 e n even for moderately strong SOC fields, the Bogoliubov approximation is expected to work well in general.
V. CONCLUSION
To summarize here we considered the plane-wave BEC phase of a spin-orbit-coupled Bose gas and reexamined its SF properties from a quantum-geometric perspective. In order to achieve this task analytically, we first reduce the 4 × 4 Bogoliubov Hamiltonian (that involves both lower Quantum depletion of the condensate density is shown as a function of SOC parameters when T = 0 and U = U ↑↑ = U ↓↓ = 10U ↑↓ /9. The fraction of the depletion is plotted (a) as a function of αz for three values of αy when αx = 2/ξ is fixed, and (b) as a function of the SOC strength α = αx = αy for three values of the αz/α ratio. and upper helicity bands) down to 2 × 2 through projecting the system onto the lower helicity band. This is motivated by the assumption that the energy gap between the lower and upper helicity bands nearby the single-particle ground state is much larger than the interaction energy. Then, given our numerical verification that the projected Hamiltonian provides an almost perfect description for the lower (higher) quasiparticle (quasihole) branch in the Bogoliubov spectrum, we exploited the low-momentum Bogoliubov spectrum analytically and identified the geometric contributions to the sound velocity. In contrast to the conventional contribution that has a square-root dependence on the interaction strength, we found that the geometric ones are distinguished by a linear dependence. It may be important to emphasize that these geometric effects are not caused by the negligence of the upper helicity band. Similar to the Fermion problem where the geometric effects dress the effective mass of the Goldstone modes [5,6], here one can also interpret the geometric terms in terms of a dressed effective mass for the Bogoliubov modes. We also discussed the roton instability of the plane-wave ground state against the stripe phase and determined the phase transition boundary. In addition we derived the SF density tensor by imposing a phase-twist on the condensate order parameter and analyzed the relative importance of its contribution from the interband processes that is related to the quantum geometry. As an outlook we believe it is worthwhile to do a similar analysis for the stripe phase.
FIG. 1 :
1Helicity bands ξ sk ( in units of α 2 x /2m) are shown for αx = 2αz and αy = 0 at ky = 0. The upper (red) and lower (blue) bands touch at k = 0. The single-particle ground state is doubly degenerate at k = (±αx, 0, 0).
FIG. 3 :
3The intraband (solid lines) and interband (dashed lines) contributions to the summation term in the superfluid-density tensor ρij are shown when T = 0 and U = U ↑↑ = U ↓↓ = 10U ↑↓ /9. Here the left, middle and right columns correspond, respectively, to the diagonal elements ρxx, ρyy and ρzz (in units of n0 ≈ n), and all of the off-diagonal elements vanish. In the top panel the diagonal elements are shown as a function of αz for three values of αy when αx = 2/ξ is fixed. In the bottom panel the diagonal elements are shown as a function of the SOC strength α = αx = αy for three values of the αz/α ratio.
FIG. 4: Quantum depletion of the condensate density is shown as a function of SOC parameters when T = 0 and U = U ↑↑ = U ↓↓ = 10U ↑↓ /9. The fraction of the depletion is plotted (a) as a function of αz for three values of αy when αx = 2/ξ is fixed, and (b) as a function of the SOC strength α = αx = αy for three values of the αz/α ratio.
AcknowledgmentsWe thank Aleksi Julku for email correspondence and M. I. acknowledges funding from TÜBİTAK Grant No. 1001-118F359.
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A L Fetter, J D Walecka, Quantum Theory of Many-Particle Systems. McGraw-HillA. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, 1971).
| []
|
[
"Stunted accretion growth of black holes by combined effect of the flow angular momentum and radiation feedback",
"Stunted accretion growth of black holes by combined effect of the flow angular momentum and radiation feedback"
]
| [
"Kazuyuki Sugimura \nAstronomical Institute\nTohoku University\n980-8578AobaSendaiJapan\n",
"Takashi Hosokawa \nDepartment of Physics\nKyoto University\n606-8502SakyoKyotoJapan\n",
"Hidenobu Yajima \nAstronomical Institute\nTohoku University\n980-8578AobaSendaiJapan\n\nFrontier Research Institute for Interdisciplinary Sciences\nTohoku University\n980-8578AobaSendaiJapan\n",
"Kohei Inayoshi \nDepartment of Astronomy\nColumbia University\n550 W. 120th Street10027New YorkNYUSA\n",
"Kazuyuki Omukai \nAstronomical Institute\nTohoku University\n980-8578AobaSendaiJapan\n"
]
| [
"Astronomical Institute\nTohoku University\n980-8578AobaSendaiJapan",
"Department of Physics\nKyoto University\n606-8502SakyoKyotoJapan",
"Astronomical Institute\nTohoku University\n980-8578AobaSendaiJapan",
"Frontier Research Institute for Interdisciplinary Sciences\nTohoku University\n980-8578AobaSendaiJapan",
"Department of Astronomy\nColumbia University\n550 W. 120th Street10027New YorkNYUSA",
"Astronomical Institute\nTohoku University\n980-8578AobaSendaiJapan"
]
| [
"Mon. Not. R. Astron. Soc"
]
| Accretion on to seed black holes (BHs) is believed to play a crucial role in formation of supermassive BHs observed at high-redshift (z > 6). Here, we investigate the combined effect of gas angular momentum and radiation feedback on the accretion flow, by performing 2D axially symmetric radiation hydrodynamics simulations that solve the flow structure across the Bondi radius and the outer part of the accretion disc simultaneously. The accreting gas with finite angular momentum forms a rotationallysupported disc inside the Bondi radius, where the accretion proceeds by the angular momentum transport due to assumed α-type viscosity. We find that the interplay of radiation and angular momentum significantly suppresses accretion even if the radiative feedback is weakened in an equatorial shadowing region. The accretion rate is O(α) ∼ O(0.01 -0.1) times the Bondi value, where α is the viscosity parameter. By developing an analytical model, we show that such a great reduction of the accretion rate persists unless the angular momentum is so small that the corresponding centrifugal radius is 0.04 times the Bondi radius. We argue that BHs are hard to grow quickly via rapid mass accretion considering the angular momentum barrier presented in this paper. | 10.1093/mnras/sty1298 | [
"https://arxiv.org/pdf/1802.07264v2.pdf"
]
| 119,332,139 | 1802.07264 | d06a5f336dfc0987a1855b7eb046b258cc9bd112 |
Stunted accretion growth of black holes by combined effect of the flow angular momentum and radiation feedback
September 2018
Kazuyuki Sugimura
Astronomical Institute
Tohoku University
980-8578AobaSendaiJapan
Takashi Hosokawa
Department of Physics
Kyoto University
606-8502SakyoKyotoJapan
Hidenobu Yajima
Astronomical Institute
Tohoku University
980-8578AobaSendaiJapan
Frontier Research Institute for Interdisciplinary Sciences
Tohoku University
980-8578AobaSendaiJapan
Kohei Inayoshi
Department of Astronomy
Columbia University
550 W. 120th Street10027New YorkNYUSA
Kazuyuki Omukai
Astronomical Institute
Tohoku University
980-8578AobaSendaiJapan
Stunted accretion growth of black holes by combined effect of the flow angular momentum and radiation feedback
Mon. Not. R. Astron. Soc
0000000September 2018(MN L A T E X style file v2.2)quasars: supermassive black holes-cosmology: theory
Accretion on to seed black holes (BHs) is believed to play a crucial role in formation of supermassive BHs observed at high-redshift (z > 6). Here, we investigate the combined effect of gas angular momentum and radiation feedback on the accretion flow, by performing 2D axially symmetric radiation hydrodynamics simulations that solve the flow structure across the Bondi radius and the outer part of the accretion disc simultaneously. The accreting gas with finite angular momentum forms a rotationallysupported disc inside the Bondi radius, where the accretion proceeds by the angular momentum transport due to assumed α-type viscosity. We find that the interplay of radiation and angular momentum significantly suppresses accretion even if the radiative feedback is weakened in an equatorial shadowing region. The accretion rate is O(α) ∼ O(0.01 -0.1) times the Bondi value, where α is the viscosity parameter. By developing an analytical model, we show that such a great reduction of the accretion rate persists unless the angular momentum is so small that the corresponding centrifugal radius is 0.04 times the Bondi radius. We argue that BHs are hard to grow quickly via rapid mass accretion considering the angular momentum barrier presented in this paper.
INTRODUCTION
Observations of supermassive black holes (SMBHs) with mass ∼ 10 9 M⊙ at redshift z 6, or 1 Gyr after the big bang, severely constrain their formation mechanism (e.g., Fan et al. 2001;Willott et al. 2010;Mortlock et al. 2011;Venemans et al. 2013;Wu et al. 2015; see also Gallerani et al. 2017 for review). Heavy seed BHs have been invoked in several scenarios to reconcile the short available time with their hugeness (see, e.g., Volonteri 2012; Haiman 2013, for a review), which includes (1) Pop III remnant BHs with mass of MBH 10 3 M⊙ (e.g., Yoshida et al. 2008;Hosokawa et al. 2011Hosokawa et al. , 2016Susa et al. 2014;Hirano et al. 2015;Stacy et al. 2016); (2) direct collapse BHs with MBH ∼ 10 5 M⊙ formed via the collapse of supermassive stars (e.g., Omukai 2001;Bromm & Loeb 2003;Hosokawa et al. 2012;Sugimura et al. 2014Sugimura et al. , 2016Inayoshi et al. 2014;Chon et al. 2016;Umeda et al. 2016); and (3) massive BHs with MBH ∼ 10 3 M⊙ ⋆ E-mail: [email protected] formed as a consequence of stellar mergers in dense clusters (e.g., Omukai et al. 2008;Devecchi & Volonteri 2009;Katz et al. 2015;Tagawa et al. 2015;Yajima & Khochfar 2016;Sakurai et al. 2017).
In all the cases, the seeds have to increase their mass further by several orders of magnitude. Although Pop III remnants with relatively small mass have been claimed to become SMBHs by z 6 by very rapid (super-Eddington) gas accretion (Volonteri & Rees 2005;Madau et al. 2014;Alexander & Natarajan 2014;Volonteri et al. 2015), whether this actually occurs or not is still very uncertain. This is because our knowledge on the realistic accretion process is still limited. Here, λB is the non-dimensional factor depending on the polytropic index γ (λB = 1.12 for the isothermal, i.e., γ = 1, case) and cs,∞ = 8 km s −1 for a neutral primordial gas with temperature T = 10 4 K. In the second expression of Eq. (1), we use the relation between the number density of hydrogen atoms nH,∞ and ρ∞ for the primordial gas.
Although the Bondi rate provides a rough estimate of an accretion rate onto a BH, various effects reduce the rate in realistic situations. For instance, radiation feedback from circum-BH discs can significantly disturb the accretion flow so that the rate decreases down to 0.01ṀB (e.g., Milosavljević et al. 2009;Park & Ricotti 2011Park et al. 2017). Recent studies have proposed pathways to alleviate the feedback, in such cases as where the radiation is highly obscured around the equatorial plane by disc winds (Sugimura et al. 2017;Takeo et al. 2018) or where the radiation is trapped in a dense (super-Eddington) accretion flow Sakurai et al. 2016). These studies, however, do not consider the effect of angular momentum on the large-scale accretion flow, assuming that the size of the circum-BH disc is much smaller than the Bondi radius. This assumption may not be always the case. We need to know the condition that the effect of angular momentum becomes significant.
Accretion to active galactic nuclei (AGNs) has been investigated in the last few decades (e.g., Ciotti & Ostriker 2001;Wada & Norman 2002;Kawakatu & Wada 2008;Kurosawa & Proga 2009;Novak et al. 2011;Kawaguchi & Mori 2010;Barai et al. 2012;Yuan et al. 2012). In such a context, the effect of gas angular momentum has partly been investigated. Even without radiation feedback, moderate gas angular momentum significantly suppresses the accretion in lowluminosity AGN (Proga & Begelman 2003b,a;Cuadra et al. 2006;Li et al. 2013;Gaspari et al. 2015;Inayoshi et al. 2018). This implies that the effect of angular momentum, as well as radiation feedback, needs to be considered in studying the accretion on to the seed BHs. In Begelman & Volonteri (2017), they have studied the properties of super-Eddington accretion flows for cases where the disc size is smaller than the photon-trapping radius. In contrast, we are interested in cases where the disc is much larger than the photon-trapping radius, which is far smaller than the Bondi radius.
In this work, to see the combined effect of the angular momentum and the radiation feedback on seed BH accretion, we perform a set of 2D axisymmetric radiation hydrodynamics simulations, considering both finite gas angular momentum and radiation from the circum-BH discs. We follow formation of rotationally-supported discs and subsequent viscous accretion through them. We find that angular momentum of the gas, in cooperation with radiation feedback, suppresses the accretion on to seed BHs. To understand its mechanism, we also develop an analytical model for accretion from a rotating medium.
The paper is organized as follows. In Sec. 2, we describe the numerical method and the parameter sets explored. In Sec. 3, we present the results of our simulations. In Sec. 4, we develop an analytical model, which allows us to obtain the condition for suppression of accretion. In Sec. 5, we discuss the possible growth history of Pop III remnant BHs based on our findings, as well as the caveats of our simulations. Finally, we summarize in Sec. 6.
METHOD
We perform axisymmetric 2D radiation hydrodynamics (RHD) simulations, by using a modified version of a public grid-based multidimensional magnetohydrodynamics (MHD) code PLUTO 4.1 (Mignone et al. 2007), which is mostly the same as used in our previous work (Sugimura et al. 2017; see also Kuiper et al. 2010aKuiper et al. ,b, 2011Kuiper & Klessen 2013;Hosokawa et al. 2016). Major update here is implementation of physics related to the rotation of gas. In the rest of this paper, we use both spherical (r, θ, φ) and cylindrical (R, z, φ) coordinates interchangeably, although the spherical one was actually used in the calculation.
Modelling accretion on to a BH under radiation feedback
We briefly describe the basic properties of the code that are common to our previous work (see Sugimura et al. 2017, for details). With this code, we follow BH accretion from a homogeneous surrounding medium under radiation feedback. We put a BH at the centre of computational domain and treat it as a sink. Through the sink surface, the gas is allowed to flow in, while the radiation is emitted according to a semi-analytical model described below. We solve the coupled equations of hydrodynamics, radial multi-frequency radiation transport and primordial gas chemistry. In the current version of the code, helium ionization is considered, hydrogen molecules are assumed to be completely destroyed, and the self-gravity of gas is ignored.
We consider ionizing photons emitted from the circum-BH discs with a semi-analytical prescription. Using the accretion rateṀ evaluated at the sink surface, the luminosity L is given by (Watarai et al. 2000)
L = 2 LE [1 + ln (ṁ/20)]ṁ > 20 0.1 LEṁṁ < 20 ,(2)
withṁ ≡Ṁ /ṀE. Here, LE is the Eddington luminosity,
LE = 4πGMBHcmp σT = 3.3 × 10 7 MBH 10 3 M⊙ L⊙ ,(3)
andṀE the (efficiency-independent) Eddington accretion rate,Ṁ
E = LE c 2 = 2.2 × 10 −6 MBH 10 3 M⊙ M⊙ yr −1 .(4)
Note that the decrease of radiative efficiency atṀ ≫ṀE is caused by photon trapping in the slim discs (Abramowicz et al. 1988). We assume that the spectral energy distribution is given by the power law Lν ∝ ν −1.5 with the frequency minimum at hνmin = 13.6 eV and that L = ∞ ν min
Lν dν (e.g., Park & Ricotti 2011).
Radiation from the sink is supposed to be anisotropic, partly because the photons from the hot inner part of the circm-BH disc is obscured by some outer materials. For example, Proga et al. (2000) suggested that line-driven winds from an SMBH accretion disc make a shadowing region with opening angle from the equatorial plane ∼ 10 • . However, especially in the case with smaller BH mass or lower metallicity, the anisotropy is highly uncertain due to the lack of knowledge on obscuring materials, which are presumably (failed) winds or coronae above the disc 1 (see, e.g., Begelman et al. 1983;Hollenbach et al. 1994;Woods et al. 1996;Proga et al. 2000;Wada 2012;Suzuki & Inutsuka 2014;Nomura et al. 2016). We thus do not attempt to realistically model the anisotropy, but instead, consider the two limiting cases: isotropic radiation and anisotropic radiation with a ∼ 45 • shadowing region ( Fig. 1; Equation 9 of Sugimura et al. 2017). We expect the reality lies somewhere between the two.
Modelling gas rotation
We assume that the surrounding gas has the initial profile of specific angular momentum That is, the gas has constant j throughout the computational domain, except near the rotation axis with R < rB, where the angular velocity Ω = j/R 2 is constant instead. Below, we interchangeably use j and the centrifugal radius Rc, where the centrifugal force and the BH gravity balances, to indicate the angular momentum of the accreted gas, as Rc is related to j as
j(x) = R rB 2 j∞ R < rB j∞ rB < R ,(5)Rc = j 2 GMBH .(7)
Angular momentum must be transported for a gas with finite j to reach the vicinity of the BH. We assume the α-type viscous stress (Shakura & Sunyaev 1973; see also Igumenshchev & Abramowicz 1999;Stone et al. 1999;Li et al. 2013), which is possibly due to turbulence driven by the magnetorotational instability (MRI; Balbus & Hawley 1998). The gravitational torque is insignificant in the cases studied here, because the disc is highly gravitationally stable (see Appendix E). Note that while the value of α corresponding to the actual MRI turbulence is yet to be fully understood, Bai & Stone (2013), for example, have reported that α is ∼ 0.01-0.02 in the case with a weak vertical magnetic field but can be even larger than unity in the strong field case.
To assure that the viscosity works only inside the disc (e.g., Stone et al. 1999), we introduce a confinement factor f and use the following expression for viscosity: 2
ν = f Ω ΩK α γ c 2 s ΩK ,(8)
with isothermal sound speed cs, adiabatic index γ = 5/3 and Keplerian angular velocity ΩK = GMBH/R 3 . Here, we identify the disc region based on the degree of rotational support against the BH gravity: we set f = 0 for Ω/ΩK <Ω th and f = 1 forΩ th + ∆Ω < Ω/ΩK, and linearly interpolate between them, with the threshold valueΩ th = 0.8 and the transition width ∆Ω = 0.1 in most of our calculations. We check the effect of changingΩ th in Sec. A2. Note that we have seen in test runs that if the viscosity is not limited to the disc, the disc will disappear and the flow becomes spherical due to angular momentum transport in the entire computational domain.
Cases considered
Our runs are summarized in Table 1. In all runs, we set MBH = 10 3 M⊙ and fix it constant during the calculations. The surrounding medium is assumed to be the neutral primordial gas with density nH,∞ = 10 5 cm −3 and temperature T∞ = 10 4 K. In the fiducial run, we assume anisotropic radiation field with α = 0.01 and Rc,∞ ≡ j 2 ∞ /GMBH = 0.1 rB. In order to examine the parameter dependence, we study cases with different parameters of (α, Rc,∞/rB) = (0.01, 0.03), (0.1, 0.1) and (0.01, 0.3). For comparison, we also solve the flow structure under the isotropic radiation field with the similar parameter sets.
We start from the homogeneous and quasi-static (vr = v θ = 0) initial condition. We turn the radiation off for the first 2 × 10 5 yr to allow the flow to settle in a steady state with a rotationally-supported disc. Note that this period is longer than either the dynamical time at rB, tB = Ω −1 K (rB) ∼ 10 4 yr, or the viscous time at Rc,∞, tvisc,c = R 2 c,∞ /ν(Rc,∞) ∼ 10 5 (α/0.01) −1 (Rc,∞/0.1rB) 1/2 yr. We then turn on the radiation and follow the evolution until t end = 1.2 × 10 6 yr in the anisotropic radiation runs and 4 × 10 5 yr in the isotropic radiation runs. We impose the reflection symmetry with respect to the equatorial plane, as well as axisymmetry around the rotation axis, and thus θ extends from 0 • to 90 • . In the r-direction, our computational domain ranges from rin = 10 −2 rB to rout = 10 2 rB. Note that rin is smaller than either Rc,∞ ∼ 10 −1 rB (see above) or the Bondi radius for H ii gas rB,HII ∼ 10 −1 rB with ∼ 7 × 10 4 K, while rout is larger than the size of H ii region. We use logarithmic grids in the r direction and homogeneous ones in the θ directions, with Nr × N θ = 512 × 144. We discuss the dependence of our results on the numerical configuration in Appendix D.
At the outer boundary, we let the flow go out from the computational domain but not come into it. At the inner boundary, we impose the same boundary condition in most cases, but when the gas is judged to belong to the Keplerian accretion disc (specifically, when vr,in < 0 and Ωin > 0.95 ΩK,in, where quantities with subscripts "in" are evaluated at rin), we determine the physical quantities in the ghost cells according to the radial profile of the isothermal Keplerian disc: ρ = (r/rin) −3 ρin, p = (r/rin) −3 pin, vr = (r/rin) 1/2 vr,in, v φ = (r/rin) −1/2 v φ,in and Q = Qin for the other variables (see Appendix A2). We set a temperature floor at Tmin = 10 4 K for simplicity (e.g., Sugimura et al. 2017), as well as minimum density nmin = 10 −1 cm −3 and maximum velocity vmax = 150 km/s for numerical reasons. For the lowest angular momentum (Rc,∞/rB = 0.03) run, we additionally limit j to below j∞, because otherwise the disc accretes the gas with j > j∞, receiving additional angular momentum transported from the inner part of the Both fiducial anisotropic radiation case (blue) and isotropic radiation case (orange) are shown. In these runs, the radiation is turned off for the initial 2 × 10 5 years. The bottom thin-dashed lines represent the averaged values for 3 × 10 5 yr < t < 4 × 10 5 yr. The results for the non-rotating case (Sugimura et al. 2017) are also shown by arrows for comparison.
disc. Recall that our aim here is to study how the accretion flow structure varies with different j∞ in a well controlled manner. We discuss how such a situation may be realized in Sec. 5.2 later. For the other higher angular momentum (Rc,∞/rB 0.1) cases, we do not put the same upper limit on j because it hardly changes the result.
RESULTS
Results of our simulations are summarized in Table 1, along with our previous cases for the non-rotating gas (Sugimura et al. 2017), for comparison. Below, we first describe the fiducial anisotropic radiation run in Sec. 3.1. We then present the isotropic radiation run with the same parameter set in Sec. 3.2 and finally see the cases with different parameter sets in Sec. 3.3.
The fiducial run
We first describe the fiducial run, where the parameter set is given as follows: MBH = 10 3 M⊙, nH,∞ = 10 5 cm −3 , T∞ = 10 4 K, α = 0.01 and Rc,∞ = 0.1 rB. The radiation field is assumed to be anisotropic with ∼ 45 • shadow (see Fig. 1). Such a wide obscuration can be regarded as an extreme of strong anisotropy. Starting from the homogeneous initial condition, we follow the evolution of flow with the radiation turned off until 2 × 10 5 yr. We then turn on the radiation and follow the evolution until t end = 1.2 × 10 6 yr. Figs. 2(a) and (b) show the time evolution of the accretion rateṀ and the luminosity L, respectively, along with . The gas distribution on the scales of (a) 10 4 au and (b) 10 3 au just before turning on the radiation in the fiducial anisotropic radiation run. In each panel, the four quadrants (clockwise from top left) represent number density n H [cm 3 ], temperature T [K], neutral fraction of hydrogen x H and specific angular momentum shown by the corresponding centrifugal radius Rc (= j 2 /GM BH ) [au]. The arrows represent the velocity vector v, shown only when |v| > 1 kms −1 . The contours in the bottom left panel represent Ω/Ω K = 0.5 (white), 0.6 (pink), 0.7 (orange), 0.8 (red) and 0.9 (dark red). The Bondi radii for neutral and ionized gases are shown as dashed black and white circles, respectively.
the result for the case with isotropic radiation, which will be described in the next section. Fig. 2(a) shows that during the early period without radiation the accretion rateṀ once converges to the constant value 7.2×10 −5 M⊙ yr −1 , only less than 1/10 of the Bondi rateṀB (= 1.7 × 10 −3 M⊙ yr −1 ). This remarkable reduction inṀ is totally attributable to the effect of angular momentum. Fig. 3 shows the gas distribution just before turning on the radiation. A flared disc formed inside rB can be seen. In the disc, the gravity is balanced with the centrifugal force inside Rc,∞ and with the pressure gradient outside Rc,∞, so that the dynamical equilibrium (vr = v θ = 0) is approximately maintained throughout. In the bipolar regions, a gas with low angular momentum directly flows into the sink without hitting on the disc.
Before turning on the radiation, the gas is almost isothermal at ∼ 10 4 K due to the efficient Lyα cooling throughout the computational domain (see Fig. 3). As an experiment, we have rerun the isothermal simulation setting T = 10 4 K and have confirmed that the result does not change. Therefore, in the rest of this paper, we adopt the isothermal equation of state with T = 10 4 K in cases without radiation, to save computational costs.
After turning on the radiation, the accretion rate decreases further, as seen in Fig. 2(a). It reaches the smaller constant valueṀ = 5.0 × 10 −6 M⊙ yr −1 at t end = 1.2 × 10 6 yr, which is about 0.1 of the value before turning on the radiation and even less than 0.01 of the Bondi rate. As shown in Sugimura et al. (2017), the accretion rate with this parameter set but without rotation is very high witḣ M ∼ 0.6ṀB. Therefore, this result demonstrates that the accretion rate is largely reduced by the interplay of angular momentum and radiation. The amount of the reduction will be analytically understood in Sec. 4. In Fig. 2(b), we see that the luminosity behaves in the same way asṀ following Eq. (2). The luminosity L is generally sub-Eddington, with L ≈ 0.2 LE at t end . Fig. 4 shows the flow structure at t end . The accretion occurs through the neutral disc remaining inside the shadow. The boundary between the ionized and neutral regions is determined by the shadow angle, ∼ 45 • from the equatorial plane. Outside the Bondi radius for the ionized gas, rB,HII, material on the surface of the neutral gas is photoevaporated and flows out in the vertical direction, as the sound speed, cs,HII, is larger than the escape velocity, vesc = (2GMBH/r) 1/2 . Conversely, inside rB,HII, where cs,HII < vesc, the photoionized gas falls back to the disc again. The outflows in the polar directions blow away the low-j gases initially locating near the poles. Recall that accretion of such material boostedṀ before the radiation is turned on.
In Fig. 4(c), a low-density region appears near the poles as a result of the centrifugal barrier. 3 However, structures along the z-axis are partly artifacts of assumed axisymmetry (e.g., Sugimura et al. 2017). In addition, jets or outflows launched near the BH, if included, would largely change such features. Since these structures hardly affectṀ anyway because of the small solid angle, we do not attempt to study them in more detail below.
Although the ionization boundary looks similar in shape to that in the non-rotating case, the flow structure is significantly altered by the angular momentum. Whereas the gas in the shadowed region falls freely in the non-rotating case, it has to pass through the accretion disc slowly in a viscous timescale otherwise. In the non-rotating case with large MBH, Takeo et al. (2018) found that, for a massive enough (∼ 10 5 M⊙) BH surrounded by a non-rotating medium, the radiation is obscured by the gas which is pushed by the ram pressure and intrudes into the polar regions, thereby mitigating the radiation feedback. In the rotating case, however, the ram pressure of the flow is so weak that we do not see such a phenomenon. We have confirmed this by performing an additional simulation with MBH enhanced to 10 5 M⊙.
Isotropic radiation run
Next, we describe the isotropic radiation run. The set-up is the same as in the previous run, except that the radiation is isotropic. Figs. 2(a) and (b) show the time evolution ofṀ and L, respectively. Fig. 2(a) shows that after the radiation is turned on,Ṁ oscillates violently repeating burst and quiescent phases. This behaviour is similar to what is found in the non-rotating case (e.g., Sugimura et al. 2017), but the averaged accretion rate in the last 10 5 years,Ṁ = 1.9 × 10 −7 M⊙ yr −1 , is about 10 times smaller. Here, we see again the reduction ofṀ by the interplay of angular momentum and radiation. Again, Fig. 2(b) shows that L oscillates in the same way asṀ following Eq. (2). Its absolute value is generally small and L ≪ LE even at the burst phases.
Figs. 5 (a, b) and (c, d) show the gas distribution before and after an accretion burst, respectively. The H ii region contracts in a quiescent phase, and expands in a burst phase (e.g., Park & Ricotti 2011). The neutral disc initially formed (Fig. 3) is completely ionized and vanishes soon after the radiation is turned on. The H ii region is elongated along the z-axis (Figs. 5a and c), because the density decreases near the axis.
Cases with different sets of the parameters
As described in Sec. 2.3, we also study a number of cases with different parameters of (α, Rc,∞/rB) = (0.01, 0.03), (0.1, 0.1) and (0.01, 0.3), in addition to the fiducial case presented above. We here only study the evolution of the accretion rates for these cases. More detailed analyses are provided later in Sec. 4, where we develop an analytical model that well explains the numerical results for a wide range of the parameters. Fig. 6 shows the time variation ofṀ in the anisotropic radiation runs with the various parameter sets (see Table 1). By the time radiation turns on (t 2 × 10 5 yr), the accretion flows reach steady sates. At that time,Ṁ is larger with larger α, while it is larger with smaller Rc,∞ with weak de-pendence for Rc,∞/rB 0.1. After the radiation turns on, the flows reach different steady states by the end of simulations (t = 1.2 × 10 6 yr). The final accretion rates depend on α and Rc,∞ in a similar way, but with the greater variation in this case.
In the isotropic radiation runs,Ṁ oscillates violently as shown in Sec. 3.2, with the average accretion rates for 3 × 10 5 yr < t < 4 × 10 5 yr given in Table 1. They are very small in all three runs, with the dependence on α and Rc,∞ similar to those obtained above. In this case, the dependence is as weak as before radiation turns on.
ANALYTICAL FORMULATION
In the previous section, we found that the angular momentum can largely suppress the accretion in the case with anisotropic radiation, where the accretion rate would be high without angular momentum. In order to understand such suppression, we here develop an analytical model describing accretion through a neutral disc connected to a medium. This model is motivated by the fact that the accretion occurs through the neutral disc remaining inside the shadow in the anisotropic radiation runs (Fig. 4). Note that this model is applicable to neither the case with isotropic radiation where the disc is completely photoionized (Fig. 5), nor that without radiation where polar inflows of low angular momentum gas contribute to the accretion (Fig. 3). In the latter case, however, such polar inflows might be prevented by jets or outflows launched near the BH.
After developing the model, we derive the critical angular momentum of the medium needed to suppress the accretion, as the model predicts that the accretion rate goes back to the Bondi rate in the case with low angular momentum. Using the critical value, we can roughly estimate the impact of the angular momentum on the accretion rate from the property of the medium (e.g., Sec. 5.1).
Below, we first develop the analytical model in Sec. 4.1. Then, in Sec. 4.2, we interpret the simulation results obtained in Sec. 3 with the model. Furthermore, we derive the condition for accretion suppression in Sec. 4.3.
Analytical model of accretion through a disc connected to a medium
Here, to understand the mechanism of the suppression of accretion by the angular momentum, as well as to analytically estimate the extent of the suppression, we develop an analytical model for the accretion through a neutral disc connected to a rotating medium with constant specific angular momentum j = j∞. In this model, we suppose that the accretion is suppressed because the gas stagnates at the centrifugal radius and accumulates between the centrifugal and the Bondi radii, and thus the pressure within the Bondi radius is enhanced. We develop the model by connecting the following two types of solutions at the centrifugal radius Rc,∞. Outside Rc,∞, where the dynamical equilibrium is held between the gravity, pressure gradient and centrifugal force, the accretion is associated with the inward gas supply that compensates the depletion to the inner disc at Rc,∞. Inside Rc,∞, where the Keplerian disc is formed, the accretion is caused by the viscous angular momentum loss, which transports the inward mass flux supplied at Rc,∞. Below, we assume that the gas is isothermal.
We start by obtaining the surface density Σ for the outer dynamically equilibrium distribution that is connected to the medium with ρ = ρ∞ (Papaloizou & Pringle 1984). Here, we assume that the gas has j = j∞ everywhere. Then, the equation for the force balance can be analytically solved, as shown in Appendix A1. Inside rB, where the scale height Hs = cs/ΩK is smaller than the radius R and the distribution is disc-like, Σ is given by (Appendix A1)
Σ = √ 2πcs ΩK ρ∞ exp rB R − Rc,∞rB 2R 2 .(9)
Next, we obtain the relation between the accretion ratė M and Σ for the inner viscous Keplerian disc (e.g., Shakura & Sunyaev 1973;Kato et al. 1998;Frank et al. 2002). In Appendix A2, assuming that the disc is thin and adopting the α-type viscosity, ν = α γ c 2 s /ΩK, we obtain (e.g., Eq. 5.19 of Frank et al. 2002
)Ṁ = 3πνΣ .(10)
In a steady accretion discṀ becomes radially constant by adjusting Σ. We then connect the two solutions at Rc,∞ by substituting Eq. (9) into Eq. (10). We obtaiṅ
Msuppr = √ 18π 3 α γ cs R 3 c,∞ rB ρ∞ exp rB 2Rc,∞ ,(11)
where we have used Ω 2 K = c 2 s rB/R 3 c,∞ at Rc,∞. Here, if Rc,∞ is so small that Eq. (11) yieldsṀsuppr >ṀB, the assumption of the outer dynamically-equilibrium distribution breaks down, because the gas cannot be supplied to the disc at a rate exceedingṀB. Thus, imposing thatṀ does not exceedṀB, we modify Eq. (11) aṡ
M = Ṁ suppr (Ṁsuppr <ṀB) MB (Ṁsuppr >ṀB) ,(12)
which gives an analytical estimate for the accretion rate from a rotating medium. In Fig. 7, the parameter dependence of the analytical accretion rateṀ is shown, along with the numerical results obtained in the equivalent settings (see Appendix B). In the figure, we normalizeṀ byṀB. Note that how mucḣ M is suppressed with respect toṀB depends only on the combination of α and Rc,∞/rB (not on either MBH, ρ∞ or cs), because, with Eqs. (1) and (11),Ṁsuppr/ṀB can be rewritten aṡ
Msuppṙ MB = 2.8 α Rc,∞ rB 3 exp rB 2Rc,∞ ,(13)
where we have used γ = 5/3. The agreement of the analytical and numerical results is remarkable considering the simplicity of the model. In the case that the accretion is suppressed by the angular momentum, i.e.,Ṁsuppr <ṀB, the α dependence is thatṀ is proportional to α (Eq. 11). As for the Rc,∞ dependence,Ṁ rapidly increases with decreasing Rc,∞ for Rc,∞ 0.1 rB, mainly because of the exponential increase of Σ(Rc,∞) under the assumption of the dynamical equilibrium (see Eq. 9). WhenṀsuppr reachesṀB, however, the dynamical equilibrium is broken and the increase ofṀ stops. For lower Rc,∞,Ṁ is equal toṀB, with little effect of the angular momentum. The disagreement between the numerical and analytical results can be partly attributed to the considerable thickness of disc at Rc,∞ (see Fig. B1c), which is not consistent with our assumption of thin inner disc. As the thickness at Rc,∞ increases with Rc,∞ and the aspect ratio reaches unity when Rc,∞ ∼ rB, our model is reliable only when Rc,∞ is sufficiently smaller than rB.
Interpreting the simulation results with the analytical model
In this section, we understand the simulation results in Sec. 3 with the analytical model developed in the previous section, focusing mainly on the case with anisotropic radiation. We begin with overplotting the simulation results (star symbols) in Fig. 7, where the parameter dependence ofṀ in the model (lines), as well as that for the isothermal accretion (dot symbols), is shown. For all the four cases with (α, Rc,∞/rB) = (0.01, 0.1), (0.1, 0.01), (0.01, 0.03) and (0.01, 0.3) examined in this paper, the simulation results are well reproduced by the model. Especially, the transition between low and high-Ṁ regimes occurring at Rc,∞/rB 0.1 is clearly seen both in the simulation results and the model. The accretion rate in the run with (α, Rc,∞/rB) = (0.01, 0.03) is ∼ 0.5ṀB, which is close to the value obtained with the assumption of negligible angular momentum in the run with the same anisotropic radiation field (Sugimura et al. 2017). The downward shifts ofṀ by at most a factor of three compared to the isothermal case (dot symbols) are partly attributable to the photoevaporation mass-loss from the surface of the disc. Note that we have also confirmed that the equatorial gas profile in the anisotropic radiation run with α = 0.01 and Rc,∞ = 0.1 rB is consistent with that of the model (Appendix C). The agreement both in the accretion rate and the gas profile suggests that the accretion is suppressed by the same mechanism as in the model, i.e., it is due to the stagnation of the gas at the centrifugal radius and the consequent pressure enhancement within the Bondi radius. Below, we make some remarks on the simulation results in the case without radiation, as well as in the case with isotropic radiation (see Table 1). In the former case, the accretion rates for the runs with (α, Rc,∞/rB) = (0.01, 0.1), (0.1, 0.1) and (0.01, 0.3) are larger than in the model, because, in addition to the accretion through the disc, the polar low-j gas directly flows into the sink and contributes to the accretion rate (see Fig. 3). Recall that in the anisotropic radiation runs, polar photoevaporative outflows prevent such inflows. For the run with (α, Rc,∞/rB) = (0.01, 0.03), however,Ṁ at t = 2 × 10 5 yr is 0.8ṀB and slightly lower thanṀB as predicted by the model, because at that point of timeṀ has yet to reach the asymptotic value and is still on the rise. Conversely, in the latter case, the accretion rates are smaller than in the model, because the accretion disc is totally photoevaporated (see Fig. 5). In the anisotropic radiation runs, however, the disc survives without being photoevaporated inside the shadow.
Condition for accretion suppression by the angular momentum
Next, we obtain the critical angular momentum above which the accretion is significantly suppressed. As mentioned in Sec. 4.1, we suppose that the effect of angular momentum becomes negligible when Eq. (13) yieldsṀsuppr/ṀB > 1. Thus, we define the critical centrifugal radius R cr c by the conditionṀsuppr/ṀB = 1. In Fig. 8, we plot R cr c as a function of α. The former slowly increases with the latter, with R cr c = 0.04 rB at α = 0.01 (R cr c = 0.05 rB at α = 0.1). For a wide range of α with 10 −6 < α < 1, the suppression of accretion becomes important as Rc,∞ exceeds O(10 −2 rB), withṀ decreased to O(αṀB) when Rc,∞ ∼ 10 −1 rB (see Fig. 7). Recall that the case with Rc,∞ rB is beyond the scope of our analytical model (Sec. 4.1).
As expected with our analyses in Sec. 4.2, Fig. 8 agrees well with the numerical results overall. Most of the anisotropic radiation runs examined are located in the shaded region, showing the significant suppression of the ac- cretion rate. Only an exception is the case with (Rc,∞, α) = (0.03, 0.01), which is really below the critical line in Fig. 8.
DISCUSSION
Growth of Pop III remnant BHs
Rapid growth of BHs is critically important for the formation of SMBHs in the early Universe, especially if their growth starts from "light seeds", i.e., the Pop III remnant BHs. Reflecting the expected diversity of the stellar mass of the Pop III stars (e.g., Hirano et al. 2014Hirano et al. , 2015, their remnant BHs would have a variety of masses, ∼ 100 − 10 3 M⊙. The subsequent growth of such BHs should depend on the environments surrounding them (Fig. 9). In the standard bottom-up structure formation paradigm in the ΛCDM Universe, Pop III stars normally form in mini-halos with masses of M h ≃ 10 5−6 M⊙ at z > 20 (e.g, Yoshida et al. 2006). In addition, Pop III stars also form in atomic-cooling halos with M h 3 × 10 7 M⊙, if prior star formation is hampered by Lyman-Werner radiation and/or baryonic streaming motions (e.g., Wise & Abel 2007b;Visbal et al. 2014;Tanaka & Li 2014;Hirano et al. 2017;Schauer et al. 2017). In the following, we discuss the possibility of rapid growth of Pop III remnant BHs, separately considering their different formation sites of mini-halos and atomic-cooling halos. We argue that the BH growth via rapid mass accretion is not easily realized for either case. In particular, the angular momentum effect studied in the previous sections is a critical obstacle for the growth of BHs formed in atomic-cooling halos.
Pop III remnants formed in mini-halos
Let us first consider the growth of Pop III remnant BHs in a mini-halo with M h ≃ 10 5−6 M⊙. Massive Pop III stars with 10 2 − 10 3 M⊙ produce intense ionizing radiation before collapsing into BHs. This ionizing radiation feedback evacuates a large fraction of gas from the mini-halo, because of its shallow gravitational potential (Kitayama et al. 2004). Rapid mass accretion on to the remnant BH is thus not expected, at least when the BH is still harbored in the same mini-halo (Johnson & Bromm 2007). Through the assembly of DM halos, a significant fraction of Pop III remnant BHs fall into an atomic-cooling halo with M h 3 × 10 7 M⊙ at z 15. In such a massive and gas-rich halo, the BHs could grow via accretion if they could sink to the galactic disc due to dynamical friction. We assume that a fraction f⋆(≃ 0.3) of the gas in the halo forms stars, i.e., M⋆ = f⋆(Ω b /Ωm)M h , and those stars exert friction on the remnant BHs. Here, we adopt Ω b /Ωm = 0.16 (Planck Collaboration et al. 2016). If the density distribution of the stars is approximated by a singular isothermal sphere, the dynamical friction timescale for a BH 4 is estimated as (Binney & Tremaine 1987) , (14) where the crossing time is tcross = r 3 /(GM⋆) and the Coulomb logarithm is set to ln Λ ∼ ln[M⋆/MBH] ∼ 6. In the above expression, we have defined M h,7.5 ≡ M h /3 × 10 7 M⊙ and MBH,3 ≡ MBH/10 3 M⊙. We assume that the BH is located at the outskirts of the galactic disc, i.e., r ≃ λ Rvir (Mo et al. 1998), where λ ≃ 0.05 is the dimensionless spin parameter and Rvir ≃ 1 kpc M h,7.5 (1 + z) −1 16 is the virial radius of the halo (Barkana & Loeb 2001), with (1 + z)16 ≡ (1 + z)/16. Using this relation, the ratio of the dynamical friction timescale to the Hubble timescale is estimated as
tDF tH ≃ 2 M 2 h,7.5 M −1 BH,3 λ 3/2 0.05 f⋆ 0.3 1/2 ,(15)
with λ0.05 = λ/0.05. This suggests that dynamical friction is inefficient for BHs with MBH 10 3 M⊙. As a result, Pop III remnant BHs coming originally from mini-halos will just continue to wander in the outskirts of the galactic disc. Such BHs will hardly grow via accretion, simply because the density of the surrounding gas is low (nH ≪ 10 4 cm −3 , also see §5.1.2 below). In this case, the rapid growth of BHs is unlikely to occur regardless of whether the angular momentum of the gas affects the BH feeding.
Pop III remnants formed in atomic-cooling halos
Pop III stars can be also formed in a galactic disc in an atomic-cooling halo with M h 3×10 7 M⊙ (or Tvir 10 4 K). Such a halo can hold the gas against the stellar feedback (Kitayama et al. 2004). The remnant BHs are initially embedded in the gas-rich disc. Let us suppose that the BH is embedded in an isothermal, exponential disc with a gas temperature Tgas ≃ 8000 K.
The gas density at the mid-plane within the disc radius (r λRvir) is estimated as (Oh & Haiman 2002) nH ≃ 1 × 10 4 cm −3 Tvir,4λ −4 0.05 (1 + z) 3
16 f d 0.3 2 .(16)
Assuming that a remnant BH has a peculiar velocity comparable to the circular velocity of the gas disc,
V = G(M d + M⋆) λRvir ≃ 20 km s −1 T 1/2 vir,4 λ −1/2 0.05 f d + f⋆ 0.6 1/2 ,(17)
which is higher than the sound speed of the gas cs ≃ 7 km s −1 (Tgas/8000 K) 1/2 . Thus, the Bondi-Hoyle-Lyttleton (BHL) accretion rate is reduced by a factor of ≃ [1 + (V /cs) 2 ] 3/2 ≃ 20 from the value in Eq. (1),
MBHL ≃ 1 × 10 −5 M⊙ yr −1 f d 0.3 1/2 f d + f⋆ 2f d −3/2 × T −1/2 vir,4 M 2 BH,3 λ −5/2 0.05 (1 + z) 3 16 .(18)
Note that Eq. (18) , which means that Pop III remnant BHs with MBH 3 × 10 2 M⊙ formed inside an atomiccooling halo may undergo rapid accretion. However, the angular momentum of the accreting gas here comes into play to prevent the BH growth.
According to cosmological simulations of the first galaxy, the gas flow in an atomic-cooling halo is turbulent in general (e.g., Wise & Abel 2007a). We consider the same density fluctuation of the turbulent medium as in our Galaxy, δρ/ρ ∼ (L/2 pc) 1/3 , where L is the characteristic spacial length of the fluctuation (Armstrong et al. 1995;Draine 2011). Accreting gas with such density fluctuation brings a net angular momentum, which is estimated as ℓ ∼ (V rB/4) · (δρ/ρ)|L=2r B (Ipser & Price 1977;Ioka et al. 2017;Matsumoto et al. 2018). Thus, the ratio of the centrifugal radius to the Bondi radius is given by
Rc rB ≃ 3 × 10 −2 M 2/3 BH,3 V 20 km s −1 −4/3 .(19)
This ratio is comparable to the critical value R cr c /rB ∼ 0.04 obtained in Sec. 4.3, above which the accretion rate is suppressed by a factor of α ∼ O(0.01 -0.1), but the former exceeds the latter as MBH becomes larger than ∼ 10 3 M⊙. Thereafter, even if the BH is embedded in the gas disc, its growth timescale becomes even longer than the age of the Universe at z = 6,
t ang grow ∼ MBḢ Msuppr ∼ 10 Gyr M −1 BH,3 (1 + z) −3 16 α−2 ,(20)
where α−2 ≡ α/10 −2 . Therefore, it seems that Pop III remnant BHs are hard to grow to high-z SMBHs via rapid accretion. Tagawa et al. (2015) and Ryu et al. (2016) considered the evolution of Pop III remnant BHs in an atomic-cooing halo, performing N-body simulations. In fact, they concluded that some remnant BHs can fall into the central region much faster than we estimated. This is mainly because the gas density is assumed to be higher than ∼ 10 4 cm −3 or to follow an isothermal singular profile (nH ∝ r −2 ). Because of the higher gas densities (at smaller scales), dynamical friction allows the BHs to quickly sink into the galactic center. When the remnant BH reaches the central region with nH 10 6 cm −3 M −1 BH,3 , the Bondi accretion rate on to it becomes high enough (∼ 5000 LE/c 2 ) to realize hyper-Eddington accretion without impeded by radiation feedback Sakurai et al. 2016;Sugimura et al. 2017;Takeo et al. 2018). This process may quickly form intermediate massive BHs with ∼ 10 5 M⊙ at the centre of the protogalaxy.
Obviously, a key uncertainty is the density structure of the gas containing the BHs. Tagawa et al. (2015) and Ryu et al. (2016) assume that the density profile does not change during the orbital evolution of BHs. In reality, however, the star formation will easily occur in the high-density regions, so that the original density structure could be modified by feedback effects such as supernova explosions (e.g., Dubois et al. 2015;Yajima et al. 2017). In future work, we will study the evolution of the BH orbital motions and ambient density structure, self-consistently incorporating the feedback effects. Such treatment will also allow us to accurately estimate how much angular momentum is brought by the accreting gas, and to assess whether super-Eddington accretion is possible circumventing the angular momentum barrier presented in this paper.
Caveats
We have made a number of simplifications and approximations in this work, and now discuss their significances.
First, we have examined only the two limiting cases of the anisotropy of radiation field. In the case of a nonrotating medium, Sugimura et al. (2017) have found that there is a critical shadowing angle (∼ 10 • from the equatorial plane) above which the efficient accretion is realized by the neutral Bondi-like inflows through the equatorial layer that exceeds the photoevaporative mass-loss from the surfaces. In the rotating case, however, the above condition needs to be modified, because the photoevaporative mass-loss from the surfaces of the rotationally-supported disc can be significant. We have seen that the disc is completely ionized by the isotropic radiation, while it is not photoevaporated inside the ∼ 45 • shadow in the case of the anisotropic radiation. We will study the dependence ofṀ on the anisotropy more in the future.
Second, the actual anisotropy created inside the sink is highly uncertain, althoughṀ strongly depends on it. In the literature, generation of (failed) winds or coronae above the disc have been investigated (e.g., Hollenbach et al. 1994;Begelman et al. 1983;Woods et al. 1996;Proga et al. 2000;Wada 2012;Suzuki & Inutsuka 2014;Nomura et al. 2016). We expect that the materials associated with such structures obscure the outward radiation and create its anisotropy. In a future work, we plan to study such process, considering its dependence on MBH,Ṁ , the metallicity of gas, etc.. Third, we have studied the idealized system of a static BH embedded in a homogeneous medium, in order to understand how angular momentum and radiation feedback affect the accretion flow. In considering more realistic BH accretion systems, however, we need to take into account the effects of turbulence (e.g., Krumholz et al. 2006;Hobbs et al. 2011) and/or galactic-scale inflows (e.g., Hobbs et al. 2012;Park et al. 2016). In a highly symmetric system as studied by our axisymmetric 2D simulations, the angular momentum transported outward through a disc may accumulate near the disc outer edge, resulting in the reduction of the accretion rate. We in fact confirm this effect for the cases where the accreting gas has small angular momentum (Rc,∞ ≪ 0.1 rB; see Appendix B). In the case of accretion from a turbulent medium, however, such accumulation may not occur because the disc can change its rotational axis before the accumulation proceeds, as the angular momentum vector of the accreting gas varies in time. Recall that in this work, we have artificially removed the accumulated angular momentum by imposing upper bound on specific angular momentum. This procedure may qualitatively mimic the above mechanism that works in a turbulent medium.
Fourth, we have adopted the α-type viscosity to mimic the angular momentum transport via the turbulence driven by the MRI. Although our results depend on the value of α, as well as where the viscosity works, i.e., the confinement factor f in Eq. (8), it is computationally too expensive to perform 3D magnetohydrodynamics simulations of the same problem. The unstable non-axisymmetric modes can also affect the flow in the 3D simulations (Papaloizou & Pringle 1984). In the cases studied here, the Toomre Q parameter is above unity and the disc is gravitationally stable. In a case with different parameter set, however, the gravitational instability can play a role in transporting the angular momentum depending on MBH and nH,∞ (see Appendix E). It is also likely that the Rayleigh-Taylor instability of the HII bubble, as seen in our 2D simulations (e.g., Fig. 5), grows differently in 3D. The former 3D simulations (Park et al. 2017), however, suggest that such difference does not significantly change the accretion rate. In addition, the disc is known to be Rayleigh-unstable when the angular momentum decreases outward, so that the accumulated angular momentum would be transported in 3D simulations (see, e.g., Inayoshi et al. 2018, and reference therein).
Finally, although we assume that the dominant cooling process in the neutral gas is the Lyα cooling, the H − freebound cooling becomes dominant and cools the gas to ∼ 4 × 10 3 K when nH ≫ 10 6 cm −3 (Omukai 2001). In addition, the temperature might drop even to ∼ 2 × 10 2 K if H2 molecules somehow form in spite of the UV irradiation from the BH neighbourhood. Consideration of these processes could lead to modification ofṀ .
SUMMARY
We have investigated the combined effect of gas angular momentum and radiation feedback on seed BH accretion, by preforming a suit of 2D axisymmetric simulations considering both finite gas angular momentum and radiation from the circum-BH disc. The BH is located at the center of a rotating medium, whose centrifugal radius is typically a tenth of the Bondi radius. We follow the formation of the rotationally-supported disc, through which the accretion proceeds by the angular momentum transport due to the assumed α-type viscosity.
We have found that the accretion is strongly suppressed by the gas angular momentum. Except for the case with very low angular momentum, the accretion rate is reduced by one order of magnitude even without radiation feedback and becomes even smaller with the feedback. In particular, the accretion rate in the case with anisotropic radiation field, which would be in the same order as the Bondi rate without gas rotation (Sugimura et al. 2017), is reduced by a factor of α ∼ O(0.01 -0.1). Our results clearly indicate the importance of the interplay of the angular momentum and radiation feedback.
We have also developed an analytical model that describes accretion through a neutral disc connected to a medium. This model is capable of reproducing the accretion rate obtained in the simulations with anisotropic radiation. Furthermore, the model suggests the presence of the critical angular momentum above which the accretion is significantly suppressed. The corresponding critical centrifugal radius normalized by the Bondi radius is a weakly increasing function of α and equal to 0.04 for α = 0.01, suggesting that even such small angular momentum is enough to reduce the accretion rate. This provides a useful estimate for the impact of the angular momentum on the accretion rate, for example, in future cosmological simulations of SMBH formation.
Finally, we have discussed the implications of our findings on the growth of Pop III remnant BHs. In the literature, those BHs are claimed to grow to high-z SMBHs by very rapid (super-Eddington) accretion. However, the angular momentum effect studied in this paper can be a crucial obstacle for such rapid mass accretion. Whilst the condition for the formation of direct-collapse BHs is known to be difficult to achieve (e.g., Sugimura et al. 2014), it should be equally challenging for the Pop III remnant BHs to rapidly grow via the super-Eddington accretion. Clearly, further studies are needed to unveil the nature.
321, 669
Yoshida, N., Omukai, K., Hernquist, L., & Abel, T. 2006, ApJ, 652, 6 Yuan, F., Bu, D., & Wu, M. 2012 APPENDIX A: INNER AND OUTER SOLUTIONS OF ANALYTICAL MODEL Our model consists of an outer dynamically equilibrium distribution (Sec. A1) and an inner viscous Keplerian disc (Sec. A2), which are connected at the centrifugal radius. Below, we describe the outer and inner solutions in this order.
A1 Outer dynamically equilibrium distribution
Here, we describe an outer dynamically equilibrium distribution connected to a homogeneous medium (Papaloizou & Pringle 1984). We assume that the gas is isothermal and that the angular momentum j = j∞ everywhere.
We start by describing the equation for the dynamical equilibrium between the gravity, pressure gradient and centrifugal force,
− 1 ρ ∇p − GMBH r 2 r + j 2 ∞ R 3 R = 0 ,(A1)
wherer andR are the unit vectors in the r and R directions, respectively. Using the isothermal equation of state, p = c 2 s ρ, Eq. (A1) can be rewritten as
∇ c 2 s ln ρ − GMBH r + j 2 ∞ 2R 2 = 0 .(A2)
Then, imposing the outer boundary condition of constant density except near the pole, i.e., ρ → ρ∞ as r → ∞ with finite θ, we obtain
ρ = ρ∞ exp GMBH rc 2 s − j 2 ∞ 2R 2 c 2 s = ρ∞ exp rB r − Rc,∞rB 2R 2 ,(A3)
where we have used rB = GMBH/c 2 s and Rc,∞ = j 2 ∞ /GMBH in the second equality. The density ρ rapidly decreases towards the pole, i.e., as R → 0, due to the assumed constant angular momentum.
The density profile corresponds to that of a thin disc at R ≪ rB. Near the equatorial plane, where rB/r = rB/(
√ R 2 + z 2 ) ≈ (rB/R)(1 − z 2 /2R 2 ), we can approximate Eq. (A3) as ρ ≈ ρeq(R) exp − z 2 2H 2 s ,(A4)
with the equatorial density,
ρeq = ρ∞ exp rB R − Rc,∞rB 2R 2 ,(A5)
and the scale height,
Hs = cs ΩK .(A6)
At R ≪ rB, where the aspect ratio Hs/R = (R/rB) 1/2 is small, the majority of gas is confined to a layer with |z| O(Hs). In such layer, the approximate form of Eq. (A4) is valid and the integration in the z-direction yields the expression for surface density,
Σ ≈ √ 2πHsρeq(R) ,(A7)
which can be rewritten as Eq. (9) with Eq. (A4). Note that ρeq has its maximum at R = Rc,∞, as the pressure gradient is balanced with the gravity at R > Rc,∞ but with the centrifugal force at R < Rc,∞.
A2 Inner viscous Keplerian disc
Next, we describe a thin Keplerian disc extended towards the BH (e.g., Shakura & Sunyaev 1973;Kato et al. 1998;Frank et al. 2002). Through the disc, the steady accretion is driven by the angular momentum transport via the α-type viscosity. Again, we assume the isothermal gas. The gas distribution is governed by the following equations. Since the disc is vertically hydrostatic, the zdependence of density is the same as Eq. (A4) and thus the relation between ρeq and Σ is given by Eq. (A7) (though ρeq is different from Eq. A5). Radially, the gravity is balanced with the centrifugal force with angular velocity
Ω = ΩK .(A8)
The vertically-integrated mass conservation equation can be written as
2π R vR Σ = −Ṁ ,(A9)
with the accretion rateṀ (> 0). Finally, the verticallyintegrated angular momentum conservation equation is given by
−Ṁ R 2 ΩK + 3πR 2 νΣΩK =J ,(A10)
with the net angular momentum fluxJ (= const). Note that we have used Eqs. (A8) and (A9) to obtain Eq. (A10). Finally, we adopt the α-type viscosity
ν = α γ c 2 s ΩK ,(A11)
which is the same as Eq. (8) with f = 1. The structure of the disc is uniquely determined once cs, α,Ṁ andJ are given. Here,J can be determined from the inner boundary condition: we impose the torque-free condition, i.e., the second viscous stress term in Eq. (A10) is set to zero, at the inner boundary where the first advection term, which is proportional to R 1/2 , is also small. 5 Thus, neglect-ingJ in the right-hand side of Eq. (A10), we finally obtaiṅ M = 3πνΣ (Eq. 10). For givenṀ , with Eq. (10), as well as Eqs. (A7), (A8), (A9) and (A11), we can straightforwardly derive Figure B1. The radial profiles of the gas in the isothermal accretion from a medium with constant angular momentum. Each panel represents (a) equatorial density, (b) equatorial specific angular momentum, (c) disc height and (d) net accretion rate. The case with α = 0.01 and Rc,∞ = 0.1 r B is shown.
ρeq = GMBHṀ 3 √ 2π 3 αγc 3 s R 3 ∝ R −3 ,(A12)vR = − 3αγc 2 s √ R 2 √ GM ∝ R 1/2 ,(A13)10 −3 M [M /yr] r B R c,∞ M B isothermal 3πνΣ disc (a) (b) (c) (d)
and
v φ = GM R ∝ R −1/2 ,(A14)
which motivate the inner boundary conditions adopted in our simulations (see Sec. 2).
APPENDIX B: NUMERICAL CONFIRMATION OF ANALYTICAL MODEL
To confirm the validity of our analytical model, we compare the analytical results with the numerical ones, by performing simulations in the equivalent settings.
Here, we start the simulation from the dynamical equilibrium distribution with j = j∞ everywhere (Eq. A3). For the computational reason, if the density ρ is initially below the floor density, we set ρ at the floor density and j = 0. We assume the isothermal gas at T = 10 4 K. Here, we adopt α = 0.01 or 0.1 and Rc,∞/rB = 0.03, 0.04, 0.05, 0.06, 0.08, 0.14, 0.2 or 0.3. We stop the calculation when the time variation ofṀ becomes sufficiently small. For the simulations presented here, we limit j to below j∞, as in the lowest angular momentum (Rc,∞/rB = 0.03 ) run in Sec. 3. The rest of the numerical method is the same as that described in Sec. 2.
To begin with, let us investigate the gas distribution, focusing on the case with α = 0.01 and Rc,∞ = 0.1 rB. In Fig. B1, the profiles of several quantities obtained at the end of the simulation are compared with the analytical model. First, Fig. B1(a) shows the equatorial density neq. It agrees excellently with that of the dynamically equilibrium distribution (Eq. A5) at R > Rc,∞, while its R dependence of ∝ R −3 is the same as the viscous Keplerian disc at R < Rc,∞. Second, in Fig. B1(b), the equatorial specific angular momentum jeq is presented. It is equal to the asymptotic value of j∞ at R ≫ Rc,∞, while it is close to the Keplerian profile jK = R 2 ΩK at R ≪ Rc,∞. However, the agreement is not perfect around Rc,∞, because the nonnegligible pressure gradient reduces j compared with jK. Recall that we here impose the upper limit at j∞ by removing j otherwise accumulated just outside the outer edge of the disc. Third, in Fig. B1(c), we show the disc height H disc at which ρ drops to one hundredth of ρeq. The disc is thin, i.e., H disc /R 1, at R Rc,∞, but it rapidly swells up at R > Rc,∞. For the thin part, the relation H disc ≈ 3Hs holds, consistent with the vertical dependence of ρ in Eq. (A4). Finally, in Fig. B1(d), we plot the net accretion rate over the entire solid angle,
M = 2 π/2 0 ρ(r, θ)vr(r, θ)2πr sin θ dθ .(B1)
It is constant with R, consistent with the steady accretion. For the range where the disc is thin (H disc /R < 1), we also plot 3πνΣ disc (see Eq. 10), using the disc surface density, Σ disc = H disc −H disc ρ(R, z) dz, obtained from the simulation. It agrees surprisingly well withṀ , implying that the accretion is indeed driven by the angular momentum loss. In summary, the numerical result is fully consistent with the analytical model. As expected from the above agreement in the gas distribution, the analytical model can nicely reproduce the numerical accretion rates for the various sets of the parameters, as shown in Fig. 7. In all cases, the difference between the analytical and numerical results is less than a factor of three, ensuring the qualitative validity of the analytical model.
In order to check the effect of imposing the upper limit on j, we have made test calculations without the limit. In such a case, we have found that j is accumulated outside the outer edge of the disc, which gradually expands as the accumulation proceeds. The effective enhancement of j makeṡ M smaller, especially in the case with smaller Rc,∞. As a result, the rapid increase ofṀ with decreasing Rc,∞, as seen in Fig. 7, disappears. The effect of imposing the limit is insignificant for the case with Rc,∞ 0.1 rB, where its impact onṀ is at most 30 percent.
Finally, we briefly mention the effect of changingΩ, which regulates the region where the viscosity works (Eq. 8). In the test runs withΩ different from the fiducial value of 0.8 (with α = 0.01 and Rc,∞ = 0.1 rB), we have seen thatṀ increases with decreasingΩ and becomes three times larger in the case ofΩ = 0.6. While the quantitative determination ofṀ is affected by the specific prescription of viscosity, we do not expect it qualitatively changes the conclusion of this work.
APPENDIX C: EQUATORIAL GAS PROFILE IN ANISOTROPIC RADIATION RUN
Here, we show the equatorial gas profile in the anisotropic radiation run in Sec. 3.1, where α = 0.01 and Rc,∞ = 0.1 rB.
We average the profiles of neq, jeq, H disc andṀ for the last 3 × 10 5 yr with the time intervals of 2 × 10 3 yr (using 150 snapshots in total) and plot them in Fig. C1 in the same way as Fig. B1. In each panel, the profile in the anisotropic radiation run generally agrees with that considered in the analytical model. Let us make several remarks on Fig. C1. First, in Fig. C1(c), while H disc is smaller than the assumed 45 • shadow and ionizing photons do not reach the disc surfaces at R 0.5 Rc,∞, H disc /R ∼ 1 at R ∼ Rc,∞ because H disc is determined by the shadow angle there. Second, in Fig. C1(b), jeq exceeds j∞ at R Rc,∞ due to the accumulation of j transported from the inside. Note that, for simplicity, we do not impose the upper limit on j in this run. Recall, however, that we have seen in Appendix B that the imposition of the upper limit does not significantly affecṫ M for the case with Rc,∞ 0.1 rB. Finally, in Fig. C1(d), the net accretion rateṀ is roughly constant at R rB but largely varies outside, corresponding to the small time variation remaining even near the end the simulation, as seen in Fig. 2(a).
APPENDIX D: DEPENDENCE ON NUMERICAL CONFIGURATIONS
To examine the dependence of our results on the numerical configuration, we rerun the simulation of the fiducial case in Appendix B with different numerical configurations. Until now, we adopt the following configurations: the inner boundary is at rin = 10 −2 rB; the number of grids is Nr × N θ = 512 × 144; and the range of θ is 0 < θ < π/2 under mid-plane symmetry. However, we here replace one of them as: rin = 5 × 10 −3 rB (r0005 run); rin = 2 × 10 −2 rB (r002); Nr × N θ = 1024 × 288 (N1024); Nr × N θ = 256 × 72 (N256); and 0 < θ < π without mid-plane symmetry (no-Sym). Fig. D1 shows the time evolution ofṀ . While there are some differences in the early stage (t 5 × 10 4 yr), all the runs give practically the same value in the end. Note thaṫ M is not affected by the assumption of mid-plane symmetry, while the non-symmetric modes are known to be evident in the former simulations of the accretion to active galactic nuclei (e.g., Stone et al. 1999). Note also that we find considerable deviations between the cell-centered and cell-boundary mass fluxes in the run with the reduced grids (N256), al-thoughṀ is not affected by such deviations. In summary, our simulation results ofṀ are apparently independent of the numerical configuration in the case without radiation feedback.
In this work, we consider the radiation feedback, as well as the angular momentum. For the non-rotating case with radiation feedback, Sugimura et al. (2017) have shown that the resolution dependence is insignificant with the current numerical configuration. Thus, we expect that the dependence of our results on the numerical configuration in the case with both angular momentum and radiation feedback is also modest.
Figure 1 .
1Directional dependence of ionizing flux. The radial extent represents the strength of the flux compared to the isotropic case.
Figure 2 .
2Time evolution of the (a) accretion rate and (b) luminosity in the runs with α = 0.01 and Rc,∞ = 0.1 r B .
Figure 3
3Figure 3. The gas distribution on the scales of (a) 10 4 au and (b) 10 3 au just before turning on the radiation in the fiducial anisotropic radiation run. In each panel, the four quadrants (clockwise from top left) represent number density n H [cm 3 ], temperature T [K], neutral fraction of hydrogen x H and specific angular momentum shown by the corresponding centrifugal radius Rc (= j 2 /GM BH ) [au]. The arrows represent the velocity vector v, shown only when |v| > 1 kms −1 . The contours in the bottom left panel represent Ω/Ω K = 0.5 (white), 0.6 (pink), 0.7 (orange), 0.8 (red) and 0.9 (dark red). The Bondi radii for neutral and ionized gases are shown as dashed black and white circles, respectively.
Figure 4 .
4Same asFig. 3but at the end of the simulation, when the gas in the polar regions are photoionized by the anisotropic radiation. Here, the gas distribution is plotted on the scales of (a) 10 5 au, (b) 10 4 au and (c) 10 3 au.
Figure 5 .
5Same as Fig. 3 but (a, b) before and (c, d) after an accretion burst in the isotropic radiation run.
Figure 6 .
6Same asFig. 2(a) but for the anisotropic radiation runs with different parameter sets. The colours represent Rc,∞/r B = 0.03 (red), 0.1 (blue) and 0.3 (green), while the line types represent α = 0.01 (solid) and 0.1 (dashed). The run with (α, Rc,∞/r B ) = (0.01, 0.1) is also presented inFig. 2(a).
Figure 7 .
7The parameter dependence ofṀ for the accretion from an isothermal medium with j = j∞. We normalizeṀ byṀ B . The horizontal axis represents Rc,∞ (≡ j 2 ∞ /GM BH ), while the colors correspond to α = 0.1 (orange) and 0.01 (blue). The lines are the analytical estimates given by Eq.(12), whereas the filled dots are the numerical values obtained in Appendix B. We overplot the results of anisotropic radiation runs in Sec. 3 with open stars.
Figure 8 .
8The critical centrifugal radius R cr c normalized by r B as a function of α. The accretion is significantly suppressed by the angular momentum in the shaded region (Rc,∞ > R cr c ).
Figure 9 .
9Two evolutionary paths of a Pop III remnant BH and its environment. A Pop III remnant BH can form either in a minihalo (Sec. 5.1.1) or in an atomic-cooling (AC) halo (Sec. 5.1.2). See text for details.
Figure C1 .
C1Same as Fig. B1 but for the anisotropic radiation run with α = 0.01 and Rc,∞ = 0.1 r B . In panel (d), solid (dot-dashed) lines show a positive (negative) value.
Figure D1 .
D1Same as Fig. 2(a) but for the fiducial case in Sec. B with different numerical configurations (see text). The lines for the fiducial, no-Sym and N1024 runs are overlapped with each other.
Table 1 .
1Summary of runs. The accretion rate can also be normalized byṀ E using the relationṀ /Ṁ E ≈ 800Ṁ /Ṁ B (SeeEqs. 1 and 4).Rc,∞/r B
α
radiation
t end [yr]Ṁ /Ṁ B
c,d
0.1
0.01
anisotropic a
1.2 × 10 6
2.9 × 10 −3
0.1
0.1
anisotropic a
1.2 × 10 6
5.2 × 10 −2
0.3
0.01
anisotropic a
1.2 × 10 6
1.2 × 10 −3
0.03
0.01
anisotropic a
1.2 × 10 6
4.3 × 10 −1
0.1
0.01
isotropic
4 × 10 5
1.1 × 10 −4
0.1
0.1
isotropic
4 × 10 5
2.6 × 10 −4
0.3
0.01
isotropic
4 × 10 5
9.7 × 10 −5
0.1
0.01
no
2 × 10 5
4.1 × 10 −2
0.1
0.1
no
2 × 10 5
1.2 × 10 −1
0.3
0.01
no
2 × 10 5
3.4 × 10 −2
0.03
0.01
no
2 × 10 5
8.1 × 10 −1
Non-rotating case b
0
-
anisotropic a
2 × 10 6
5.9 × 10 −1
0
-
isotropic
5 × 10 5
1.7 × 10 −3
0
-
no
-
1
NOTES.-We set M BH = 10 3 M ⊙ , n H,∞ = 10 5 cm −3 and
T∞ = 10 4 K in all runs.
a Anisotropic radiation with ∼ 45 • shadow.
b Results from analytical estimate and the simulations in
Sugimura et al. (2017).
c Accretion rate at the end of simulation (averaged near the end
of simulation if oscillating).
d
c 0000 RAS, MNRAS 000, 000-000
Note that the anisotropy generated in this way exists independently of the self-shielding effect in slim discs.
Equation(8)can also be written as ν = f α c 2 s,ad /Ω K , with adiabatic sound speed c s,ad = √ γ cs.c 0000 RAS, MNRAS 000, 000-000
The temperature is also low due to the inefficient photoionization heating, whose heating rate is proportional to the recombination rate and hence the square of the density. c 0000 RAS, MNRAS 000, 000-000
Note that the dynamical friction timescale could be shortened by a factor of two if the friction is exerted by a gas instead of stars(Ostriker 1999;Escala et al. 2004)
17H04827 (HY) and 17H01102 and 17H06360 (KO) and by the Simons Foundation through the Simons Society of Fellows (KI).
In the literature, the torque-free boundary condition is often imposed at the innermost stable circular orbit (ISCO), which is located at 6GM BH /c 2 in the case of Schwarzschild BHs. c 0000 RAS, MNRAS 000, 000-000
ACKNOWLEDGEMENTSThe authors would like to thank Ken Ohsuga, Sanemichi Takahashi and Kenji Toma for fruitful discussions. The numerical simulations were performed on the Cray XC30 at CfCA of the National Astronomical Observatory of Japan, as well as on the computer cluster, Draco, at Frontier Research Institute for Interdisciplinary Sciences of Tohoku University and on the Cray XC40 at Yukawa Institute for Theoretical Physics in Kyoto University. This work is supported in part by MEXT/JSPS KAKENHI Grant Number 15J03873 (KS), 25800102, 15H00776 and 16H05996 (TH),APPENDIX E: GRAVITATIONAL STABILITY OF DISCSIn the cases studied in this work, the disc is always gravitationally stable according to the Toomre criterion, i.e., the Toomre Q defined aswith the epicyclic frequency κΩ = ((2Ω/R)d/dR(R 2 Ω)) 1/2 , is larger than unity. In the following, to see under what conditions the disc becomes gravitationally unstable, we derive the parameter dependence of Q based on the analytical model in Sec. 4.1.Using Σ of the analytical model obtained from Eqs.(10)and(12), we can rewrite Eq. (E1) aswhere the R and α dependences are canceled out. This yields Q ∼ 10 3 when MBH = 10 3 M⊙, nH,∞ = 10 5 cm −3 and Rc,∞ = 0.1 rB, consistent with the simulation results. With the Jeans length λJ = πc 2 s /Gρ, Eq. (E2) can be further rewritten as Q ≈ 0.3 (rB/Rc,∞) 3 (λJ/rB) 2 exp[−rB/2Rc,∞]. This suggests that Q could be less than unity, i.e., the disc could be gravitationally unstable, in the case with smaller λJ/rB (∝ M −1 BH n −1/2 H,∞ ) and/or Rc,∞/rB.
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| []
|
[
"Anomaly Inflow and Membrane Dynamics in the QCD Vacuum",
"Anomaly Inflow and Membrane Dynamics in the QCD Vacuum"
]
| [
"H B Thacker \nDepartment of Physics\nUniversity of Virginia\nP.O. Box 40071422901-4714CharlottesvilleVAUSA\n",
"Chi Xiong \nInstitute of Advanced Studies\nNanyang Technological University\n639673Singapore\n"
]
| [
"Department of Physics\nUniversity of Virginia\nP.O. Box 40071422901-4714CharlottesvilleVAUSA",
"Institute of Advanced Studies\nNanyang Technological University\n639673Singapore"
]
| []
| Large N c and holographic arguments, as well as Monte Carlo results, suggest that the topological structure of the QCD vacuum is dominated by codimension-one membranes which appear as thin dipole layers of topological charge. Such membranes arise naturally as D6 branes in the holographic formulation of QCD based on IIA string theory. The polarizability of these membranes leads to a vacuum energy ∝ θ 2 , providing the origin of nonzero topological susceptibility. Here we show that the axial U (1) anomaly can be formulated as anomaly inflow on the brane surfaces. A 4D gauge transformation at the brane surface separates into a 3D gauge transformation of components within the brane and the transformation of the transverse component. The in-brane gauge transformation induces currents of an effective Chern-Simons theory on the brane surface, while the transformation of the transverse component describes the transverse motion of the brane and is related to the Ramond-Ramond closed string field in the holographic formulation of QCD. The relation between the surface currents and the transverse motion of the brane is dictated by the descent equations of Yang-Mills theory. * | 10.1103/physrevd.86.105020 | [
"https://arxiv.org/pdf/1208.4784v1.pdf"
]
| 58,892,873 | 1208.4784 | d9ccc01396684387d0871551c17df747d3c7efa2 |
Anomaly Inflow and Membrane Dynamics in the QCD Vacuum
23 Aug 2012
H B Thacker
Department of Physics
University of Virginia
P.O. Box 40071422901-4714CharlottesvilleVAUSA
Chi Xiong
Institute of Advanced Studies
Nanyang Technological University
639673Singapore
Anomaly Inflow and Membrane Dynamics in the QCD Vacuum
23 Aug 2012
Large N c and holographic arguments, as well as Monte Carlo results, suggest that the topological structure of the QCD vacuum is dominated by codimension-one membranes which appear as thin dipole layers of topological charge. Such membranes arise naturally as D6 branes in the holographic formulation of QCD based on IIA string theory. The polarizability of these membranes leads to a vacuum energy ∝ θ 2 , providing the origin of nonzero topological susceptibility. Here we show that the axial U (1) anomaly can be formulated as anomaly inflow on the brane surfaces. A 4D gauge transformation at the brane surface separates into a 3D gauge transformation of components within the brane and the transformation of the transverse component. The in-brane gauge transformation induces currents of an effective Chern-Simons theory on the brane surface, while the transformation of the transverse component describes the transverse motion of the brane and is related to the Ramond-Ramond closed string field in the holographic formulation of QCD. The relation between the surface currents and the transverse motion of the brane is dictated by the descent equations of Yang-Mills theory. *
I. INTRODUCTION
The possible importance of codimension one membrane-like topological charge structures in the QCD vacuum is suggested by both theoretical considerations [1,2] and by Monte Carlo studies [3,4]. Theoretically, the suggestion of topological domain wall structures in the vacuum emerged from large-N c chiral Lagrangian arguments. These arguments showed that, in the large-N c limit the multivaluedness of the effective η ′ mass term induced by the chiral U (1) anomaly implies the existence of multiple, discrete, quasistable, and nearly degenerate "k-vacua". These vacua are labeled by effective local values of the QCD θ parameter which differ by integer multiples of 2π, and are separated by domain walls where the value of θ jumps by ±2π. With the emergence of the holographic string theory framework for QCD-like gauge theories [1,5,6], it was shown that the role of the domain wall predicted by large N c was played by the D6 brane of IIA string theory. (More precisely, by an "I2 brane" which is the intersection of the D6 brane with the D4 color branes [1].)
In the holographic framework the local θ parameter is given by the Wilson line of the closed string Ramond-Ramond U(1) gauge field around the compactified direction of the D4 branes. In 9+1 dimensions, a D6 brane plays the role of a magnetic monopole source for the RR field, and is dual to an instanton, which is represented by a D0 brane in the holographic model. The incompatibility of the instanton model with large N c chiral dynamics [2] indicated that, at least for sufficiently large N c , instantons should be replaced by codimension one membranes or domain walls.. From the 9+1dimensional string viewpoint, the membrane-dominated vacuum is in a precise sense dual to the instanton vacuum, with instantons (D0 branes) and domain walls (D6 branes) being, respectively, electric and magnetic sources of Ramond-Ramond field. The success of large N c phenomenology, combined with the Monte Carlo evidence for topological charge membranes in SU (3) gauge theory [3,4] strongly indicates that N c = 3 is large enough that the membrane-dominated vacuum is the correct qualitative picture for real QCD. In 4-dimensional spacetime, a D6 brane appears as a 2+1 dimensional intersection with the color branes, with the other 4 spatial dimensions of the D6 brane compactified on an S 4 . As a color excitation, the D6 brane appears as a codimension one dipole layer of topological charge in the gauge field. The quantized jump in the value of θ across the membrane is just the Dirac quantization condition for RR monopoles. In this paper, we show that the existence and dynamics of these topological membranes in the QCD vacuum can be studied without reference to the higher-dimensional string theory framework, using the descent equations and cohomology structure of 4-dimensional Yang-Mills theory [7][8][9]. This provides a "bottom-up" perspective on the holographic framework. It also identifies the exact mechanism by which the RR U(1) gauge field remains in 4-dimensional QCD as an auxiliary field representing singular membrane-like excitations of the Yang-Mills field.
We consider 4D SU(N) Yang-Mills theory and, in order to construct a domain wall, we add an external source coupled to topological charge,
S = S Y M + d 4 x θ(x)Q(x)(1)
where
Q(x) = 1 16π 2 N c ε µνστ T rF µν F στ(2)
is the Yang-Mills topological charge density, and θ(x) represents the local value of the theta parameter. We construct a straight flat membrane or domain wall by taking θ(x) to be a step function along one spatial direction, which we label x 1 , with discontinuity θ 0 ,
θ(x) = θ 0 x 1 > 0 (3) = 0 x 1 < 0(4)
This produces a codimension one membrane occupying the dimensions transverse to x 1 . Integrating by parts, we can write the source term as an integral on the surface of the brane, where the integrand is proportional to the Chern-Simons current,
K µ = ε µαβγ Tr A α ∂ β A γ + 2 3 A α A β A γ ≡ ε µαβγ K αβγ 3(5)
whose divergence is proportional to the topological charge density,
∂ µ K µ = 32π 2 N c Q(x)(6)
The term in the action is thus expressed in terms of Yang-Mills fields localized to the brane surface, but it is no longer explicitly gauge invariant (because we have discarded a surface term at x 1 = ∞).
The gauge variation of the CS 3-form in (5) is dictated by the descent equations [7][8][9], and can be written formally as the exterior derivative of a 2-form. However, one of the terms in the resulting 2-dimensional surface integral has a topological ambiguity. It has the form of a Wess-Zumino-Witten term and is only defined modulo 2π times an integer which identifies the winding number of the gauge transformation in the 3-volume enclosed by the 2D surface. Gauge invariance of the exponentiated WZW term requires the integer quantization of the coupling constant, which in this case is θ 0 /2π. In this way we find that the quantization of the step in θ across the domain wall follows from the requirement of invariance under "large" gauge transformations, in a manner similar to the quantization of the WZW coupling constant in 2D sigma models.
δ(∂ µ θ) = −δK µ(7)
The gauge invariance constraint (7) is an "anomaly inflow" condition that specifies the gauge variation of the RR field (i.e. of θ(x)) that must accompany a 4D Yang Mills gauge transformation.
In a string theory framework, the idea of anomaly inflow is important for understanding the relation between string theory in the bulk, where only closed strings propagate, and open-string gauge theories defined on lower dimensional brane surfaces. Anomalous, fermionic currents of the gauge theory are seen as currents which are conserved overall, but which can flow onto and off of the brane surface, so only the combination of brane and bulk current is conserved. From the reverse perspective of the bulk theory, anomaly inflow is a generalization of the Dirac monopole construction, with the Bianchi identity being preserved by a cancellation between the bulk magnetic flux and that carried away by the Dirac string. Normally one would expect that the axial U (1)
anomaly in QCD could only be interpreted in terms of anomaly inflow by embedding it in a higher dimensional "bulk" theory, e.g. type IIA string theory in 10 dimensions. However, the role of codimension one membranes in the vacuum of 4-dimensional QCD allows the anomaly inflow mechanism to be operative in a strictly 4-dimensional context, with the spatial direction transverse to the membrane playing the role of a bulk coordinate.
In the case of the axial U (1) anomaly in QCD the physical gauge invariant flavor singlet axial currentĵ µ 5 is not conserved, ∂ µĵ µ 5 = Q(x), but it is sometimes convenient to introduce a conserved, non-gauge invariant axial current j µ 5 by subtracting off the Chern-Simons current of the gauge field,
j µ 5 =ĵ µ 5 − 1 32π 2 N c K µ(8)
which is conserved by Eq. (6). Here we interpret this construction as an anomaly inflow constraint.
If only the in-brane components of the gauge field are nonzero, the Chern-Simons current K µ is a vector transverse to the brane whose support is localized on the membrane surface. Away from the brane surface, in the 4-dimensional bulk, the axial current is both conserved and gauge invariant.
The nonconservation of the currentĵ µ 5 occurs only on the brane surface, where it can be carried away in the form of surface currents associated with the gauge variation of the Chern-Simons 3-form.
As shown in Ref. [10], the anomaly inflow condition (7) enforces a Kogut-Susskind cancellation [11] between massless poles coupled to the (separately non gauge invariant) operators ∂ µ θ and K µ , so that there are no massless poles coupled to the gauge invariant combination ∂ µ θ + K µ . In the holographic description, this invokes the anomaly inflow requirement that the gauge variation of the CS 3-form on the I2 brane should be cancelled by a gauge variation of the bulk RR field on the brane surface [10,12]. The Kogut-Susskind cancellation of massless poles in gauge invariant amplitudes is a manifestation of the gauge invariance that connects the in-brane components of the gauge field to the transverse component. As we will discuss in detail for the 2-dimensional case considered in Section II, this relates the Chern-Simons current on the brane to the local spacetime orientation of the brane surface.
In 4D Yang-Mills theory, the gauge cancellation (7) specifies a relation between a 3D Yang-Mills transformation within the brane and a change of the local orientation of the brane surface, as determined by ∂ µ θ. In this way, we relate the variation of ∂ µ θ to the 1-cocycle of the SU (N ) gauge transformation g ≡ e iω , as constructed from the descent equations [7][8][9]. For our purposes, a 1-cocycle can be thought of as a "Berry phase" associated with transporting a representation of the gauge group around a closed orbit in the parameter space of gauge transformations.
The cocycle is a functional of the gauge transformation and is given by the integral of a local 2-form density over the surface of the brane at fixed time.. The 1-cocycle that is attached to a 2-dimensional brane surface is constructed from the topological charge by the descent procedure [9]. The gauge variation of the Chern-Simons current is the sum of two terms,
δK µ = ε µαβγ δK αβγ 3A + δK αβγ 3B(9)
where one term is the Maurer-Cartan form
δK αβγ 3A = 1 3 Tr g −1 ∂ α gg −1 ∂ β gg −1 ∂ γ g(10)
Since topological charge is gauge invariant,
δ(∂ µ K µ ) = ∂ µ δK µ = 0(11)
we expect that, locally, we can write the 3-form in (9) as an exterior derivative,
ε µαβγ δK αβγ 3A + δK αβγ 3B = ε µαβγ ∂ α δK βγ 2A + δK βγ 2B(12)
The Maurer-Cartan term can be written formally as the exterior derivative of a 2-form. However, K 2A is not single-valued because it depends on the multivalued gauge phase ω = −i ln g. Up to terms of order (ω) 4 , it is given by
δK βγ 2A = i 3 Tr ωg −1 ∂ β gg −1 ∂ γ g + O(ω 4 )(13)
As in the 2D WZW sigma model, the integral of this term over a closed 2D surface is ambiguous mod 2π. It depends not only on the values of g on the 2-dimensional boundary, but on its winding number in the enclosed 3-dimensional volume. The second term in (9) describes an interaction between the WZW current and the Yang-Mills gauge potential,
δK βγ 2B = Tr ∂ β g g −1 A γ(14)
This is a single-valued, nontopological contribution to the 1-cocycle. It describes the emission of gluons which accompanies brane fluctuations.
II. WILSON LINES AS MEMBRANES IN 2-DIMENSIONAL U(1) GAUGE THEORY
The basic idea of our formulation of brane dynamics in gauge theory is illustrated in a particularly simple context by the case of 2-dimensional U (1) gauge theory. For this case, the Chern-Simons current is K µ = ε µν A ν , and the analog of the descent equation (9) is simply related to the gauge transformation g = e iω itself,
δK µ = −iε µν g −1 ∂ ν g = ε µν ∂ ν ω(15)
For definiteness, we consider the Schwinger model (2-dimensional QED), but most of the discussion applies equally well to the 2-dimensional CP N −1 sigma model. As described in the Introduction, we construct a codimension one membrane by including a topological source term
S θ = 1 2π d 2 x θ(x)ǫ µν F µν(16)
For notational simplicity, we denote the coordinates by x 1 ≡ x, x 2 ≡ y and take the source field to be a spatial step function at x = 0,
θ(x) = θ 0 x > 0 (17) = 0 x < 0(18)
Integrating by parts, we see that the source term is equivalent to an ordinary Wilson line operator of the gauge field,
S θ = −θ 0 A y dy(19)
We consider the variation of this term under a U(1) gauge transformation A µ → A µ + ∂ µ ω. If we compactify the y coordinate over a finite range from 0 to L, the variation is given by
δS θ = − θ 0 2π L 0 ∂ y ω dy = − θ 0 2π [ω(L) − ω(0)] = −θ 0 n (20)
Imposing periodic boundary conditions on g requires n to be an integer. Thus the gauge variation of The vector ∂ µ θ is a vector normal to the brane surface and thus specifies its local orientation. In the Schwinger model, this can be identified with the conserved, non-gauge invariant axial vector current ∂ µ δθ = 2πj µ 5 . Here j µ 5 is an auxiliary free fermion current which couples to an unphysical massless Goldstone boson in the covariant gauge formulation of the model [11,13]. Its introduction explicitly separates the fermionic component of the axial current from the gauge anomaly. The gauge invariant currentĵ µ 5 is related to the conserved current bŷ
j µ 5 = j µ 5 + 1 2π K µ(21)
where K µ = ε µν A ν . Here,ĵ µ 5 is the physical axial vector current which includes the anomaly. Note that if we interpret the Wilson line in the usual way as a charged particle world line representing the flow of vector current j µ , then j µ 5 = ε µν j ν is always normal to the Wilson line. The Kogut-Susskind mechanism [11] has a simple physical interpretation as the separation of a physical charged particle into the bare particle and its comoving gauge field. A proper gauge invariant particle state must include both the particle and its surrounding field. But in order to quantize in a covariant gauge, the KS pole cancellation mechanism must be employed. This introduces two massless scalar fields associated with the two terms on the right hand side of (21). The field representing j µ 5 is an ordinary massless boson field, but the one representing the gauge anomaly term in (21) is a massless ghost field. Physical, gauge invariant amplitudes are constructed with operators that only contain the gauge invariant sum of the two fields, and massless poles cancel. The physical spectrum has a mass gap given by the mass of the Schwinger boson µ 2 = e 2 /π, which is the analog of the η ′ in QCD.
The KS mechanism is thus a cancellation between the long range effects which would be induced by separately varying the position of a particle and that of its surrounding gauge field. Varying either one separately would induce long range effects, but if the particle and its surrounding field are varied together, as required physically, the effects are short range, and there are no massless particles in gauge invariant amplitudes.
When expressed in terms of branes in the gauge field, the Kogut-Susskind mechanism generalizes straightforwardly to the case of 4-dimensional QCD. In the 2D case the charge carrying object is a pointlike bare fermion, while in 4D QCD it is the codimension-one θ membrane. The Kogut-Susskind mechanism is a cancellation between the massless fluctuations of the membrane surface, described by ∂ µ θ, and the wrong-sign massless pole in the Chern-Simons current correlator [10].
III. ANOMALY INFLOW, TRANSVERSE BRANE FUZZ, AND THE
RAMOND-RAMOND FIELD
In order to calculate the contribution of a brane to the topological susceptibility in 2D U (1) gauge theory, consider the calculation of a Wilson loop around a contour C that cuts across a membrane, as depicted in Fig. 1. For simplicity, we first consider the case of a straight brane along the y-axis. As in the case of a Dirac-Wu-Yang monopole, a description of such a field configuration with no unphysical singularities requires that the A µ field (as well as the Ramond-Ramond field θ)
to the left and right of the brane must be written in different gauges A µ L and A µ R . At the location of the brane along the y-axis, we must match the field description to the left and right of the brane by a gauge transformation g = e iω defined on the surface of the brane,
A y R = A y L − ig −1 ∂ y g ≡ A y L + ∂ y ω for x = 0 (22) θ R = θ L + 2π(23)
We can now identify the contribution of the membrane to the Wilson loop integral around the contour C by writing it in terms of the two closed subcontours C L and C R , which do not cross the brane. The Wilson loop integral is given by the sum of the contributions from the two subcontours C L and C R , plus a contribution from the membrane surface coming from the gauge mismatch between the two contours,
C A · dl = C L A L · dl + C R A R · dl + y 2 y 1 (A R − A L ) y dy(24)
where y 1 and y 2 are the two points where the contour C punctures the membrane. The effect of the membrane on the Wilson loop is to add a phase
y 2 y 1 (A R − A L ) µ dx µ = −i y 2 y 1 g −1 ∂ µ g dx µ = ω(y 2 ) − ω(y 1 )(25)
This is just the Wu-Yang prescription for the phase of a charged particle propagating in the field of a magnetic monopole: In addition to the A µ phase, if the particle passes from one coordinate patch to another one in a different gauge, the Wilson loop phase receives a contribution from the gauge transformation that matches the fields along the interface between sections.
In (25)
G y : δA x = 0 , δA y = −iG −1 ∂ y G = ∂ y Ω (27) G x : δA x = −iG −1 ∂ x G = ∂ x Ω , δA y = 0(28)
where we choose x to be transverse and y parallel to the brane. For example, a straight, uniform brane along the y-axis at x = 0 is represented by G y for the gauge function
Ω(x, y) = 2πy × δ(x)(29)
giving the transformations
G x : δA x = 2πyδ ′ (x), δA y = 0 (30) G y : δA x = 0, δA y = 2πδ(x)(31)
While the combined effect of G x and G y is a gauge transformation, the transformation G y by itself inserts a physical membrane into the system. When we generalize this to 4D Yang-Mills, the phase ω(y) in (26) will be replaced by a WZW 2-form on the 2+1-dimensional brane.
For the 2D case, we can write any gauge field in the 2D plane in a transverse/longitudinal decomposition,
A µ = ε µν ∂ ν σ + ∂ µ Ω(32)
A uniform brane along the y-axis corresponds to a gauge configuration
A x = 0, A y = 2πδ(x)(33)
which is obtained from (32) with σ = 2πΘ(x), Ω = 0 (34) (Here we use upper case Θ(x) to denote a unit step function.) The field strength for this configuration is a dipole layer of topological charge,
F = 2πδ ′ (x)(35)
A general variation of the Chern-Simons current includes both a physical variation δσ and a gauge variation δΩ,
δK µ = ε µν ∂ ν δΩ − ∂ µ δσ(36)
Let's now consider the effect of an infinitesimal transformation applied only to the in-brane component A y of the brane configuration (33),
δA x = 0, δA y = 2πεδ(x)(37)
corresponding to the physical variation of the discontinuity of the gauge field across the brane,
δσ = 2πǫΘ(x), δΩ = 0(38)
This excitation also varies the density of the topological charge dipole layer,
δF = 2πǫδ ′ (x)(39)
On the other hand, the A y field in (37) can also be obtained by an in-brane gauge transformation of the form (26),
δΩ = −2πǫyδ(x)(40)
This gives the same variation of the in-brane component as in (37),
δA y = 2πǫδ(x)(41)
but also introduces transverse "brane fuzz"
δA x = 2πǫyδ ′ (x)(42)
which cancels the variation (39) from δA y to give a net δF = 0.
We can now combine the variation (38) with the gauge transformation (40) to show that the inbrane gauge variation of A y (37) is gauge equivalent to a uniform infinitesimal spacetime rotation of the membrane in the x-y plane. We perform physical and gauge variations whose combined effect leaves the A y component of the field unchanged,
δσ = 2πǫΘ(x), δΩ = 2πǫyδ(x)(43)
This leads to a variation of the Chern-Simons vector that can be interpreted as an infinitesimal spacetime rotation of the original brane configuration (33), as depicted in Fig. (2),
δK x = 2πǫδ(x) − 2πǫδ(x) = 0 (44) δK y = −2πǫyδ ′ (x) ≈ 2π [δ(x − ǫy) − δ(x)](45)
This is just the field configuration that would be obtained by an infinitesimal rotation of the original configuration (33). By the anomaly inflow constraint, the gauge variation of the RR field is specified by δ(∂ µ θ) = −δK µ , so (44)-(45) describes an infinitesimal rotation of the θ domain wall boundary.
To summarize, if we start with a straight brane along the y-axis, the gauge variation of the in-brane component A y , Eq. (37) when accompanied by a gauge transformation δΩ, Eq. (40), is just an infinitesimal rotation of the brane. The variation (37) appears as a uniform variation of the topological charge dipole density,
δF = 2πǫδ ′ (x)(46)
Similarly, when we generalize this construction to 4D Yang-Mills theory, the codimension one dipole that results from this mapping is not quantization of localized topological charge, but rather, the quantization of the step-function discontinuity of the θ field across the brane.
Since the identification between the in-brane gauge transformation and the orientation of the brane surface can be made locally along the brane, the previous argument can be extended to describe any infinitesimal fluctuations of the brane. As we will discuss in the next Section It is interesting to consider the role of anomaly inflow on the brane in the conservation of axial vector current. As discussed in Section II, in the presence of quarks the θ field becomes the U (1) chiral field, and the gauge variation of ∂ µ θ is related to the conserved, gauge noninvariant axial vector current j µ 5 . The flow of axial current near the brane is dictated by the anomaly inflow constraint. Taking the y direction along the brane as Euclidean time, we equate the conserved axial current j 5 µ to the variation δ(∂ µ θ) under a gauge transformation of the form (26),
j x 5 = ω ′ (y) δ(x) (47) j y 5 = −ω(y) δ ′ (x)(48)
The flow of spatial current j x 5 into the brane is balanced by the accumulation of chiral charge j y Recall that in that model, the axial U (1) anomaly can be obtained by a simple point splitting method [11]. We take the y-direction in Fig. 1 to be Euclidean time, and the axial vector chargeψγ 0 γ 5 ψ = ψ † γ 5 ψ may be constructed as a gauge invariant operator by point
splitting,ĵ 5 y = ψ † (x + ǫ)γ 5 exp i ǫ −ǫ A x dx ψ(x − ǫ) → j 5 y + A x(49)
Here the anomaly A x arises from the O(1/ǫ) singularity in the short distance expansion of the quark bilinear. In Fig. 1 the brane fluctuation term induces quark-antiquark annihilation into a pure gauge excitation of the Chern-Simons tensor as depicted in Fig. 3. As discussed in Ref. [10], this is the origin of the 4-quark contact term that is responsible for the η ′ mass.
Note that, although the 2D gauge function Ω(x, y) in (29)
IV. MEMBRANES IN 4D YANG-MILLS THEORY
As in the 2D example, the anomaly inflow constraint allows one to reduce the gauge dynamics of 4D Yang-Mills theory at the codimension one surface of a brane to a lower-dimensional theory on the brane surface coupled to a bulk θ field. In the 2D U(1) case, the matching of gauge fields across the brane depicted in Fig. 1 gives a contribution to the Wilson loop phase proportional to the length of the membrane,
y 2 y 1 δA dy = ω(y 2 ) − ω(y 1 ) = 2π(y 2 − y 1 )(50)
We interpret this as the action associated with the membrane world line between y 1 and y 2 . We may think of the 1-cocycle ω(y) as a phase attached to the pointlike brane at a fixed time y. In 4D
Yang-Mills theory, the world volume action of the brane is the gauge variation of the 3D Chern-Simons tensor, and the 1-cocycle obtained from the descent equations (9)- (14) plays the role of the Hamiltonian density for the 2-dimensional brane at a fixed time. The gauge transformation g becomes a local field on the world sheet of the brane describing its fluctuations in the bulk space.
The approach we use for constructing a topological charge membrane in 4D Yang-Mills theory is the same as in 2D U(1). We add a brane which spans three of the four Euclidean dimensions by including a theta term which is a step function in the transverse coordinate x 4 ≡ x and independent of the other coordinates, as in Eq. (3). Again, integrating by parts, we write the action S θ as an integral localized to the brane surface,
S θ = − d 4 x∂ µ θ K µ = −θ 0 R 3 K αβγ 3 dx α ∧ dx β ∧ dx γ(51)
where the Chern-Simons current K µ and the dual CS tensor K αβγ If we keep the brane flat by holding the discontinuity of θ(x) fixed at x = 0, the action S θ is not gauge invariant. The gauge variation of K µ is given by the sum of two terms, Eqns. (13) and (14). The term (13) depends only on g, and is proportional to the winding number density,
w(x) = 1 24π 2 ǫ αβγ Tr g −1 ∂ α g g −1 ∂ β g g −1 ∂ γ g(52)= 1 8π 2 ǫ αβγ ∂ α K βγ 2A(53)
where K αβ 2A is the WZW term (13). The mod 2π ambiguity of the surface integral of K αβ 2A again leads to the requirement that the coefficient θ 0 /2π be integer quantized to maintain gauge invariance of the exponentiated WZW term.
Invoking the anomaly inflow constraint, we again require that ∂ µ θ transform under a Yang-Mills gauge transformation g = e iω , so that it cancels the variation of the Chern-Simons current given by (9),
δ(∂ µ θ) = −δK µ = ǫ µαβγ ∂ α K βγ 2A + K βγ 2B(54)
The WZW term K βγ 2A , Eq. (13) is analogous to the gauge phase ω in (26) for 2D U(1). As in that case, a gauge transformation on K µ induces a fluctuation of the vector ∂ µ θ, which represents a fluctuation of the brane surface. To study this further, we consider a truncated brane with a finite boundary by taking the source field θ(x) to be a constant inside a 3-dimensional ball of radius R carved out of the Euclidean brane, representing the propagation of a 2-dimensional disk over a finite time interval:
θ(x) = θ 0 x 1 > 0, x 2 2 + x 2 3 + x 2 4 < R (55) = 0 otherwise(56)
Then the gauge variation of the action S θ can be written as a 2-dimensional action on the surface of the ball,
δS θ = S 2 dx α ∧ dx β 1 3 Tr ω g −1 ∂ α g g −1 ∂ β g + Tr ∂ α g g −1 A β + O(ω 4 )(57)
Note that δS θ = S θ (g) − S θ (1), so that the expression (57) is the g-dependent part of the action.. For simplicity we will discuss SU (2) gauge theory, but generalization to N c > 2 is straightforward. To construct a brane at x = 0 we perform a gauge transformation on the 3 in-brane components of the Yang-Mills field by an SU (2) phase ω = −i log g = π y · σ ℓ (58)
The topological WZW term of the corresponding 1-cocycle, Eq. (13), has the form
K βγ 2A = π 3 ǫ αβγ y α(59)
This is embedded in 4-dimensional space by restricting it to the 3D surface of the brane at x = 0 with a delta-function. Then the gauge variation of the Chern-Simons current is given by
δK µ = π 3 ǫ µαβγ ∂ α K βγ 2 × δ(x)(60)
The quantity ∂ α K βγ 2 consists of the two terms in the gauge variation of the Chern-Simons 3-form on the membrane surface, Eq. (12). For the topological term K 2A , this gives
δK µ = π 3 ǫ µαβγ ∂ α ǫ βγi y i δ(x)(61)
Anomaly inflow for the CS current K x transverse to the brane shows that the in-brane gauge transformation (58) creates a uniform codimension one brane transverse to the x-axis. From Eq.
(61) we find
∂ x θ = −δK x = −2πδ(x)(62)
Once again, as in the 2D discussion, we consider an additional infinitesimal in-brane transformation of the same form,
δω = πǫ y · σ ℓ(63)
Transforming only the in-brane components, this varies the θ discontinuity across the brane
δ(∂ x θ) = −δK x = −2πǫδ(x) (64) δ(∂ y i θ) = 0 (65)
Following the same argument applied to the 2D Wilson line excitation, we can apply a 4D Yang-Mills gauge transformation to write this in a form where the discontinuity of θ across the brane remains 2π (to first order in ǫ), but the local orientation of the brane surface has rotated slightly, For the 2-dimensional U(1) case, the brane action depends only on the gauge phase ω. The analogous term in 4D Yang-Mills is the topological WZW term, K 2A integrated over the spatial surface of the brane. This also depends only on the gauge transformation, but unlike the 2-dimensional case, the WZW action includes self-interactions for the membrane fluctuations. Another new feature of the 4D Yang-Mills case is the additional, nontopological term in the action, K 2B , which defines a coupling between the color Kac-Moody current associated with the WZW field g and the color gauge field A µ . This term describes the emission of a gluon from a fluctuating brane.
δ(∂ x θ) = 0 (66) δ(∂ y i θ) = 2π [δ(x + εy i ) − δ(x)] ≈ 2πεy i δ ′ (x)(67)
V. DISCUSSION
Large N c chiral lagrangian arguments, gauge-string holography, and Monte Carlo results all indicate that the topological structure of the QCD vacuum is dominated by codimension one membranes which appear as dipole layers of topological charge, i.e. juxtaposed positively and negatively charge sheets. In this paper we have discussed an approach to the dynamics of these membranes based on their interpretation as Wilson bags [14], i.e. singular, sheet-like excitations of the Chern-Simons tensor on codimension one surfaces. A Wilson bag plays the role of a domain wall between local quasi-vacua with discrete values of the QCD θ parameter differing by ±2π.
Holographically, the Wilson bag is interpreted as a D6 brane in IIA string theory, carrying Ramond-Ramond charge. The analogy with Wilson line excitations in 2-dimensional U(1) gauge theory is very instructive. In Coleman's original discovery of the topological θ parameter in the massive Schwinger model [13], he showed that θ could be interpreted as a background electric field. A domain wall between vacua with different values of θ is just a charged particle world line. The associated Wilson line integral of the A field can be reinterpreted as a surface integral of the Chern-Simons flux, which is the 2D analog of a Wilson bag. As discussed in [10], the Ramond-Ramond field θ plays the role of the background electric field in the 4D Yang-Mills generalization of Coleman's discussion.
The approach we have pursued in this paper avoids any direct use of the holographic framework to introduce branes into QCD. Instead, a membrane is constructed from its 4-dimensional definition as a discrete step in the QCD θ parameter, or equivalently, a surface integral of the Chern-Simons tensor. This approach allows us to address questions of brane dynamics in the powerful mathematical framework of gauge group cohomology, anomaly inflow, and the descent equations of Yang-Mills theory [7][8][9]15]. The anomaly inflow constraint at the brane surface defines the connection between the θ field and the gauge field. It can be thought of as "Gauss's law" for the θ field, with the source term given by the Chern-Simons tensor on the brane. This implies a nontrivial transformation of the θ field under Yang-Mills gauge transformations. This transformation is specified by the gauge variation of the Chern-Simons tensor, as expressed by the descent equations of Yang-Mills theory [9]. The sequence of arguments is simplest in the 2-dimensional U (1) case, where the relevant descent equation is just the gauge transformation itself δA µ = ∂ µ ω. In 4D Yang-Mills, the analog of A µ is the 3-index Chern-Simons tensor, and its gauge variation defines a 1-cocycle ω µν , which is a Wess-Zumino-Witten 2-form K µν 2 integrated over the 2-dimensional spatial surface of the brane. It is a functional of the gauge group variation g on the brane which appears as the WZW field. In the same sense that the gauge phase ω along the timelike Wilson line in 2D can be thought of as the phase attached to the wave function of a pointlike charged particle, the 1-cocycle ω µν can be thought of as the gauge phase attached to a 2-dimensional membrane in the µ-ν plane.
The results presented here suggest a very appealing model for the chiral condensate. In the membrane vacuum, the near-zero Dirac eigenmodes which are needed to form a condensate appear as surface modes on the topological charge membranes. The fact that the membrane consists of opposite-sign topological charge sheets on opposite sides of the brane implies that left-and right-handed chiral densities q(1 ± γ 5 )q will appear on opposite sides of the same brane. We saw in Section III that conservation of axial vector current near the brane surface arises from a balance between the current impinging on the brane from the transverse direction and the current flowing along the brane. The axial U (1) anomaly arises by the following physical mechanism: when a membrane fluctuates the quark and antiquark states on opposite sides of the brane will overlap and thus can annihilate, as in Fig. 3, if the quark and antiquark are of the same flavor. This is the origin of the η ′ mass insertion and the nonconservation of axial U (1) current. If we suppress thē qq annihilation process (either by taking the large N c limit, or by introducing two flavors of quark and considering flavor nonsinglet pions), this picture also provides an understanding of massless Goldstone boson propagation.
It was argued in Ref. [10] that the Ramond-Ramond field in QCD gives rise to effective 4-quark contact terms responsible for both the η ′ mass insertion and a Nambu-Jona Lasinio-type interaction that provides the attractive interaction between chiral pairs that produces theqq condensate. To see the connection between the Ramond-Ramond field and Goldstone bosons, we recall the equivalence between a rotation of the QCD θ parameter and a variation of the flavor singlet chiral field η ′ . In the usual discussion this equivalence follows from the index theorem. Our discussion exhibits a physical mechanism for this connection by identifying the quark near-zero modes as surface modes of the topological charge membranes. The anomaly inflow formalism defines a spacetime dependent θ(x) which is sourced by singular Chern-Simons excitations of the gauge field. The discontinuities of the θ field define the location of the membranes. The connection between θ and the chiral field follows from the assumption that the condensate lives on the brane surfaces. This leads to the identification of ∂ µ θ as the axial vector current. In a vacuum filled with a "topological sandwich" of membranes [16][17][18], long wavelength Goldstone bosons propagate masslessly via chiral quark pairs occupying delocalized surface modes on the branes combined with a collective transverse oscillation of the branes. The bulk oscillation and surface mode propagation are locked together by 4D gauge invariance and the anomaly inflow constraint, which balances the bulk and surface currents to give massless Goldstone boson propagation.
S
θ depends on the winding number of the U (1) gauge phase around the compactified y-axis. Gauge invariance of exp iS θ requires that the coefficient θ 0 /2π in (20) is an integer. This is the simplest example of gauge group cohomology, where the 1-cocycle associated with the group element g = e iω is just ω, the gauge phase itself. Since δA µ = ∂ µ ω, the phase (20) is given by the gauge variation of the Chern-Simons 1-form integrated over the Wilson line. In the case of 4D Yang-Mills theory, a similar argument applies, where the 1-cocycle is obtained from the descent equations, and is given by the gauge variation of the Chern-Simons 3-form [7-9] integrated over the brane surface. Now let us allow the brane defined by (17) to fluctuate around its flat starting position at x = 0.
FIG. 1 :
1Calculating the contribution of a brane along the y-axis to a Wilson loop.
the gauge transformation g = e iω which specifies the matching across the brane depends only on the in-brane y-coordinate and appears only in the difference of the A y components of the gauge field. The gauge component A x transverse to the brane does not enter into the matching.However, the definition of g as the transformation which matches the gauge on the two sides of the brane indicates that it should be regarded as localized to the brane at x = 0. For this reason, we define a gauge transformation G ≡ e iΩ , whereΩ(x, y) = ω(y) × δ(x)(26)On a lattice, this would be a gauge transformation that is applied only on a single row of sites along the brane at x = 0. In the continuum description employed here, the gauge transformationG always appears in the form of either Ω or G −1 ∂ µ G = i∂ µ Ω, so (26) always leads to well-defined expressions in terms of delta functions and derivatives thereof. The separation of a singular bulk gauge transformation at a brane surface into a transformation depending only on the in-brane coordinates multiplied by a delta function in the transverse coordinate will play a central role in the following discussion. Small non-topological variations of the gauge transformation g are related by anomaly inflow to local fluctuations of the surface orientation vector ∂ µ θ. In this way, the gauge transformation g which matches the fields on the two sides of the brane is promoted to a dynamical field describing fluctuations on the surface of the brane. This seems surprising at first, since a gauge transformation should not produce a physical excitation. The key point is that the transformation of the inbrane component(s) of the A µ field is not a gauge transformation in the bulk theory unless it is accompanied by a transformation of the transverse component A x . The cohomology of the Wilson line and the brane action (25) depends only on the component within the brane A y , and not on the transverse component A x . In general, we can distinguish between transformation of the in-brane components of the gauge field, which determines the gauge cohomology, and transformation of the transverse component, which, by a certain choice of gauge, can be interpreted as transverse motion of the brane surface. Let G ≡ e iΩ be a gauge transformation defined in 2D space-time, δA µ = −iG −1 ∂ µ G, and define separately the transformation of the x and y components of the A field,
FIG. 2 :
2layers of topological charge (which are in fact observed in Monte Carlo studies[3,4]) represent the brane fuzz associated with the transverse component of the gauge field at the brane surface.By repeatedly applying infinitesimal in-brane transformations of the form (41) alternated with bulk gauge transformations, we can extend this identification to the case of finite rotations and finite in-brane gauge transformations. This provides a novel view of the topological connection between spacetime and the gauge group. In the usual discussion of gauge group topology associated with instantons, one assumes that the topological charge is localized in spacetime, and that the A field For a flat uniform brane, the Chern-Simons current is a vector transverse to the brane surface. Gauge variation of the dual Chern-Simons form on the brane surface is equivalent to an infinitesimal rotation of the surface orientation.on a circle at infinity is everywhere gauge equivalent to A µ = 0. The global topology of the gauge field then reduces to the winding number of the mapping of the spacetime circle to the group phase e iω , where A µ = ∂ µ ω. In our discussion of topological charge membranes, topological charge is delocalized, and the mapping between the group phase and the spacetime direction arises in a different context. By the anomaly inflow argument, we have related the gauge phase δω on the brane to the local orientation angle of the brane surface in the x-y plane. The quantization
, the relation imposed by the anomaly inflow constraint between transformation of the in-brane component(s) of A and the fluctuations of the brane surface represented by the θ field generalizes to the 4D Yang-Mills case. Combined with the descent equations, this provides a mathematical framework for studying the dynamics of branes and their role in QCD vacuum structure.
5 on the brane, giving ∂ µ j µ 5 = 0. For example, the gauge function ω(y) = εy describes a constant flow of current into the brane, j x 5 = εδ(x), and a linearly increasing chiral charge with time y, j y 5 ≡ j 0 5 = −εyδ ′ (x). The axial anomaly and η ′ mass arise from the possibility of quark-antiquark annihilation between chiral surface modes on opposite sides of the brane. This causes some of the axial charge to disappear from the brane, leaving a Chern-Simons excitation in the form of a transverse brane fluctuation. The 2D Schwinger model is an instructive example which suggests the role of transverse brane fluctuations in inducing the quark-antiquark annihilation that gives the U(1) Goldstone boson a mass.
we can interpret the two vertical Wilson lines on opposite sides of the brane as representing a quark bilinear which straddles the brane. The point splitting procedure (49) suggests that the matching at the brane surface, Eq. (25), is sensitive to not only the inbrane gauge field component A y , but also to the transverse A x component. The form of the line integral (25) indicates that such an effect would arise from a fluctuating brane whose world line deviates from the y-axis, picking up a contribution from the x-component of the gauge field. In the picture where the Wilson lines in Fig. 1 are the fermions (quarks) of the Schwinger model, FIG. 3: Brane puncture induced by the axial anomaly corresponding to quark-antiquark annihilation across the brane. The Wilson line contribution from the puncture is proportional to the x-component of the A field, as in Eq. (49)
is dimensionless, we are implicitly taking x and y to be given in units of the physical length scale. For example, on the lattice, the observed membranes are of finite thickness in lattice units, but become singular delta-functions in the continuum limit. In the 2D Schwinger model, the length scale is determined by the gauge coupling constant e which has dimensions of mass, while in the CP N −1 models and in QCD, it is determined dynamically in terms of the cutoff scale by asymptotic freedom. In QCD, the relevant length scale is set by the pseudoscalar decay constant ℓ ∼ f −1 π .
over the 3-dimensional brane world volume.. The integral of the 3-index CS tensor K 3 over a 3-dimensional surface has been referred to by Luscher [14] as a "Wilson bag" operator. In the discussion of topological charge structure of 4D Yang-Mills theory, it plays a role analagous to the Wilson line in the 2D U(1) case. Just as the Wilson line can be interpreted as the gauge phase attached to the world line of a charged particle, the Wilson bag integral is the gauge phase associated with the world volume of a 2+1 dimensional membrane. Note also that the value of a closed Wilson loop in 2D U(1) theory is equal to the total topological charge contained inside the loop. Similarly, the integral of K 3 over a closed Wilson bag is equal to the amount of Yang-Mills topological charge contained in the 4-volume enclosed by the bag.
For a given Yang-Mills potential A γ the equation (54) allows us to translate a color gauge transformation g into a fluctuation of the brane orientation vector ∂ µ θ. Thus the first term in the action(57) describes the self interaction of the fluctuations of the brane inside the 3-volume of the ball in terms of a 2D WZW model on the surface of the ball. In a Hamiltonian framework, the 3D ball represents at fixed time a 2-dimensional spatial disk of maximum radius R, with the Kac-Moody currents of the WZW model flowing around the boundary of the disk.Just as we did in the 2D case, we may study the contribution to topological susceptibility of a Yang-Mills membrane in the vacuum by determining its effect on a probe Wilson bag operator that is cut into two sections by the 3D plane of the membrane. As before, the gauge choice on the two sides of the membrane must differ by a relative gauge transformation g. The analog of the Wu-Yang phase (25) is the 1-cocycle of the gauge transformation g, given by Eq. (57). As in the 2D case, a straight flat brane can be introduced by a gauge transformation which transforms the 3 in-brane components of the gauge field by a topologically nontrivial gauge transformation. Of the four Euclidean coordinates x µ , µ = 1, . . . 4, we denote the 3 coordinates within the brane byx i ≡ y i , i = 1, . . . 3, and the transverse coordinate by x 4 ≡ x.
We have shown that a small fluctuation of the 3-dimensional in-brane gauge transformation g = e iω is equivalent to a fluctuation of the surface in the transverse space. Note that g is defined entirelyon the 3-dimensional brane without any reference to the transverse coordinate. The intepretation of g as describing a transverse fluctuation of the brane arises when we embed the 3-dimensional gauge transformation g in the 4D gauge configuration with a transverse delta function. The relation between gauge variations on the brane and transverse fluctuations is reminiscent of similar connections in string theory. In the case of gauge theory, this connection is a direct consequence of the descent equations and cohomology structure of Yang-Mills theory, which describes the interplay between gauge variations and spacetime derivatives. The gauge variation of the Chern-Simons 3form as a functional of g plays the role of a world sheet action for the brane. The descent equations express this 3-dimensional action as the exterior derivative of a WZW 2-form K αβ 2 , Eq. (13)and(14).
Next we show that by considering small, nontopological gauge transformations we obtain information about the dynamics of fluctuating branes. This occurs because invariance under a 4D Yang-Mills gauge transformation provides a relation between the gauge variation of the Chern-Simons 3-form on the brane surface and the transformation of the gauge field component transverse to the brane surface. The Chern-Simons form that appears in the surface integral at x 1 = 0 depends only on the three "in brane" components. The transverse component of A µ is related to membrane fluctuations in the transverse coordinate. A central role in this discussion is played by the descent equations of 4D Yang-Mills theory [7-9], which describes the intertwining of gauge cohomology with spacetime de Rahm cohomology by relating gauge variations to exterior derivatives. For a curved or fluctuating membrane ∂ µ θ is a vector normal to the surface. This vector thus specifies the local orientation of the brane surface. In order to construct gauge invariant amplitudes in the presence of a fluctuating brane, we define the auxiliary field ∂ µ θ to transform under a Yang-Mills gauge transformation to cancel the variation of the Chern-Simons current on the brane surface:
AcknowledgmentsThis work was supported by the Department of Energy under grant DE-FG02-97ER41027. CX is supported by the research funds from the Institute of Advanced Studies, Nanyang Technological
. Singapore University, University, Singapore.
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| []
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[
"Anomalies for Galilean fields",
"Anomalies for Galilean fields"
]
| [
"Kristan Jensen \nC.N. Yang Institute for Theoretical Physics SUNY Stony Brook\n11794-3840Stony BrookNY\n"
]
| [
"C.N. Yang Institute for Theoretical Physics SUNY Stony Brook\n11794-3840Stony BrookNY"
]
| []
| We initiate a systematic study of 't Hooft anomalies in Galilean field theories, focusing on two questions therein. In the first, we consider the non-relativistic theories obtained from a discrete light-cone quantization (DLCQ) of a relativistic theory with flavor or gravitational anomalies. We find that these anomalies survive the DLCQ, becoming mixed flavor/boost or gravitational/boost anomalies. We also classify the pure Weyl anomalies of Schrödinger theories, which are Galilean conformal field theories (CFTs) with z = 2. There are no pure Weyl anomalies in even spacetime dimension, and the lowest-derivative anomalies in odd dimension are in one-to-one correspondence with those of a relativistic CFT in one dimension higher. These results classify many of the anomalies that arise in the field theories dual to string theory on Schrödinger spacetimes. | 10.21468/scipostphys.5.1.005 | [
"https://arxiv.org/pdf/1412.7750v2.pdf"
]
| 55,322,257 | 1412.7750 | 03011c63c0ebebfb867200a316fa8a54444b2703 |
Anomalies for Galilean fields
December 25, 2014 24 Dec 2014
Kristan Jensen
C.N. Yang Institute for Theoretical Physics SUNY Stony Brook
11794-3840Stony BrookNY
Anomalies for Galilean fields
December 25, 2014 24 Dec 2014
We initiate a systematic study of 't Hooft anomalies in Galilean field theories, focusing on two questions therein. In the first, we consider the non-relativistic theories obtained from a discrete light-cone quantization (DLCQ) of a relativistic theory with flavor or gravitational anomalies. We find that these anomalies survive the DLCQ, becoming mixed flavor/boost or gravitational/boost anomalies. We also classify the pure Weyl anomalies of Schrödinger theories, which are Galilean conformal field theories (CFTs) with z = 2. There are no pure Weyl anomalies in even spacetime dimension, and the lowest-derivative anomalies in odd dimension are in one-to-one correspondence with those of a relativistic CFT in one dimension higher. These results classify many of the anomalies that arise in the field theories dual to string theory on Schrödinger spacetimes.
Introduction
Anomalous global symmetries provide one of the most useful handles on non-perturbative field theory. Their utility stems largely from being simultaneously exact, calculable, and a universal feature across the space of field theories. Anomalies must be matched across scales, and so give stringent checks on renormalization group flows and dualities. Furthermore, the language of anomalies and anomaly inflow is the natural one to discuss and classify topologically non-trivial phases of matter.
For all of these reasons, we would like to better understand anomalies in nonrelativistic (NR) field theories, for which little is presently known. 1 Indeed, the role of anomalies in topologically non-trivial phases is almost entirely discussed in terms of the anomalies of relativistic field theory. This seems at best dubious to us.
In this note we begin a proper classification of anomalies in Galilean field theories. 2 We focus entirely on Galilean theories, as they possess more symmetry than a generic NR system and moreover the potentially anomalous symmetries are completely understood. The end result [5] after much history [6][7][8][9][10][11] is that Galilean theories couple to a version of Newton-Cartan (NC) geometry, and the symmetries one demands are invariance under coordinate reparameterizations, gauge transformations for the particle number and any other global symmetries, and a shift known in the NC literature (see e.g. [6]) as a Milne boost. The latter is the difference between Galilean theories and NR theories without a boost symmetry. 3 This suite of background geometry and symmetries satisfies a number of checks as summarized in [5], and moreover can be obtained by carefully taking the NR limit of relativistic theories [16].
Rather than performing a complete analysis, we elect to answer two basic questions. First, one natural route to an anomalous Galilean theory is to start with a relativistic theory with flavor and/or gravitational anomalies and perform a discrete light-cone quantization (DLCQ), i.e. to put the relativistic theory on a background with a lightlike circle. The dimensionally reduced theory is Galilean-invariant. Does it also have an anomaly? We find that the answer is "yes," insofar as there is no local counterterm which can render the NR theory invariant under all symmetries. Moreover, we show that the anomalies descend to mixed flavor/Milne or gravitational/Milne anomalies. They are "mixed" in the sense that one can arrange for the NR theory to be invariant under one symmetry or the other, but not both simultaneously.
Second, we consider Schrödinger theories, that is Galilean CFTs whose global symmetries in flat space comprise the Schrödinger group. We classify the pure Weyl anomalies of these theories, in analogy with the Weyl or trace anomaly of relativistic CFT, using consistency properties of field theory. The easiest way to perform this analysis is to lift the NC data to an ordinary Lorentzian metric in one more dimension with a null circle. We find that the NR Weyl anomaly takes the same form as the ordinary relativistic Weyl anomaly built from this higher-dimensional metric. For example, 2 + 1-dimensional Schrödinger theories have two central charges, one which is formally analogous to a in four-dimensional CFT, and the other to c.
Our analysis has one caveat: we only classify anomalous variations with at most d + 1 derivatives. We expect that there are Weyl anomalies with more derivatives, and we conjecture below that they are all Weyl-covariant.
A corollary of this result is the following. The DLCQ of a relativistic CFT is a Schrödinger theory, and the Weyl anomaly of the relativistic parent survives the DLCQ to become the NR Weyl anomaly. This in turn gives a prediction for the Weyl anomaly of Schrödinger theories holographically dual to string theory on so-called Schrödinger spacetimes [17,18] with z = 2. It would be nice to see this directly in holography using the dictionary (see [19,20]) which allows for the field theory to couple to a curved geometry.
Similarly, our results for flavor and gravitational anomalies hold for the string theory embeddings of Schrödinger holography [21][22][23], wherein the NR field theory is obtained by DLCQ together with a holonomy for a global symmetry around the null circle.
The rest of this note is organized as follows. In the next Section, we summarize the details of Newton-Cartan geometry we require, and its relation to null reductions. We go on in Section 3 to show that flavor and gravitational anomalies survive DLCQ. Finally, we classify pure Weyl anomalies in Schrödinger theories in Section 4.
Preliminaries
We presently review the machinery which we require for the rest of this note.
Newton-Cartan geometry
The version of Newton-Cartan geometry which will be useful for us is the following. A Newton-Cartan structure (see e.g. [7,24]) on a d-dimensional spacetime M d is comprised of a one-form n µ , a symmetric, positive-semi-definite rank d − 1 tensor h µν , and a U(1) connection A µ . The tensors (n µ , h µν ) are almost arbitrary: we require that
γ µν = n µ n ν + h µν , (2.1)
is positive-definite. These tensors algebraically determine the upper-index data v µ and h µν satisfying
v µ n µ = 1 , h µν v ν = 0 , h µν n ν = 0 , h µρ h νρ = δ µ ν − v µ n ν . (2.2)
The "velocity vector" v µ defines a local time direction, and h µν gives a metric on spatial slices. (NC geometry with general n not closed was only studied recently [5,25].) We can define a covariant derivative using the tensors that make up the NC structure. Unlike in Riemannian geometry where there is essentially one derivative that can be defined with a metric, there are many possible derivatives that can be defined from the NC data. One choice of connection is [5]
Γ µ νρ = v µ ∂ ρ n ν + 1 2 h µσ ∂ ν h ρσ + ∂ σ h νρ − ∂ σ h νρ + h µσ n (ν F ρ)σ , (2.3)
where the brackets indicate symmetrization with weight 1/2 and F is the field strength of A µ . The corresponding derivative D µ has the nice feature that
D µ n ν = 0 , D µ h νρ = 0 . (2.4)
Galilean field theories naturally couple to this sort of NC geometry [5,10,11], where A µ is the gauge field which couples to particle number. For instance, the action of a free Galilean scalar ϕ carrying charge m under particle number is
S f ree = d d x √ γ iv µ 2 ϕ † D µ ϕ − (D µ ϕ † )ϕ − h µν 2m ϕ † ϕ , (2.5) with D µ ϕ = (∂ µ − imA µ )ϕ. Note(v µ , h µν , A µ ) shift as v µ → v µ + ψ µ , h µν → h µν − (n µ ψ ν + n ν ψ µ ) + n µ n ν ψ 2 , A µ → A µ + ψ µ − 1 2 n µ ψ 2 , (2.6)
Here, ψ µ is spatial, meaning v µ ψ µ = 0, and we have used the shorthand ψ µ = h µν ψ ν , ψ 2 = ψ µ ψ µ . One can easily verify that (2.5) is invariant under the Milne boosts. The boosts are crucial: they impose a covariant version of the Galilean boost invariance. However, it is troublesome to obtain tensors which are invariant under both the boosts and U(1) gauge invariance. For example, the connection we defined above (2.3) is gaugeinvariant, but not Milne-invariant [5]. One can define another connection which is Milneinvariant, but the resulting derivative is not gauge-invariant. There is no connection which is invariant under both symmetries. 4 As a result, the covariant derivative of an boost and gauge-invariant tensor is not a boost and gauge-invariant tensor. C'est la vie, but this fact complicates the classification of potential anomalies.
In the absence of any anomalies, W is invariant under infinitesimal coordinate transformations, gauge transformations, and Milne boosts. These symmetries lead to (potentially anomalous) Ward identities [5,11]. From the generating functional W of correlation functions, one defines a sort of stress tensor complex. Letting W depend on an overcomplete parameterization of the background, W = W[n µ , v µ , h µν , A µ ], we define the number current J µ , momentum current P µ , energy current E µ , and spatial stress tensor T µν via
δW = d d x √ γ δA µ J µ − δv µ P µ − δn µ E µ − δh µν 2 T µν . (2.7)
Here we let the variations of n µ be completely arbitrary, in which case some of the variations of v µ and h µν are fixed in terms of δn µ , e.g.
δv µ = −v µ v ν δn ν + P µ ν δv ν , (2.8)
where δv µ is arbitrary, and similarly for δh µν . The U(1) gauge invariance implies that J µ is conserved, the Milne invariance equates momentum with the spatial part of the particle number current, P µ = h µν J ν , and the coordinate reparameterization invariance leads to conservation equations for the energy current and spatial stress tensor. When the theory has "local" anomalies, W has an infinitesimal variation under at least one of these transformations, and the anomaly may be characterized by the variation δ χ W = 0. Equivalently, the anomaly may be characterized by the modified, anomalous Ward identities.
A Galilean CFT is also invariant (up to anomalies) under "Weyl" rescalings which are characterized by a critical exponent z, under which the background fields rescale as
n µ → e zΩ n µ , h µν → e 2Ω h µν , A µ → e (2−z)Ω A µ . (2.9)
The case z = 2 is special, and in that case the CFT is called a "Schrödinger" theory, as its global symmetries in flat space form the Schrödinger group. In Section 4 we will consider Weyl anomalies for Schrödinger theories. In that case, denoting the Weyl variation of W as
δ Ω W = d d x √ γ δΩ A ,(2.10)
the anomalous Weyl Ward identity is
2n µ E µ − h µν T µν = A . (2.11)
Null reduction
This geometric structure could have been (and perhaps should have been) anticipated from DLCQ. Recall that one way to construct a d-dimensional Galilean-invariant theory is to start with a relativistic theory in d + 1 dimensional Minkowski space in light-cone coordinates,
g = 2dx 0 dx − + d x 2 ,
and compactify the null coordinate x − with some radius R. On purely algebraic groundsthe Poincaré generators which commute with P − generate the Galilean algebra, with P − playing the role of particle number -the d-dimensional theory one obtains on (x 0 , x) is Galilean-invariant. Similarly, if one starts with a relativistic CFT, the lower-dimensional theory is invariant under the Schrödinger group [21], which is generated by the Galilean algebra in addition to a dilatation and special conformal generator. It is well known that DLCQ is subtle (see e.g. [21,22] and references therein). The zeromodes of the null reduction have to be treated carefully, and this is not always understood. However we are only interested in the symmetries of the problem, rather than a careful definition of the ensuing Galilean theory, and so these subtleties are irrelevant for us.
It is worth noting that the string theoretic realizations of Schrödinger field theories in holography either arise from DLCQ, or from DLCQ in addition to a holonomy for a global symmetry around the null circle [21][22][23].
Of course, we could consider the most general DLCQ. That is, we could put a relativistic theory on the most general d + 1-dimensional spacetime with a null isometry,
g = 2n µ dx µ (dx − + A) + h µν dx µ dx ν , (2.12)
where the component functions depend on the x µ but not x − and h µν is a rank d − 1 positive-semi-definite tensor. The null isometry is
n M ∂ M = ∂ − . (2.13)
After reducing on x − , the lower-dimensional theory is a Galilean theory which couples to (n µ , h µν , A µ ), which we recognize as the defining data of a NC structure. Moreover, all of the symmetries we outlined in the previous Subsection are manifest here. The gauge field A µ is just the graviphoton of the reduction, and its U(1) gauge invariance just corresponds to additive reparameterizations of x − . The Milne boosts correspond to an ambiguity in the extraction of (h µν , A µ ) from g, as discussed in [5]. One can redefine A and h as
A µ → A µ + Ψ µ , h µν → h µν − (n µ Ψ ν + n ν Ψ µ ) ,(2.14)
for any one-form Ψ µ (x ν ), which leaves g unchanged. In order for h µν to remain rank d − 1, however, we must have
Ψ µ = ψ µ − 1 2 n µ ψ 2 , v µ ψ µ = 0 . (2.15)
But these are just the Milne boosts (2.6). And of course d-dimensional reparameterizations are just d + 1-dimensional reparameterizations along fibers of constant x − .
So DLCQ automatically gives NR theories coupled to NC geometry, invariant under the symmetries above.
Later, it will be important that there is no one-form dx − + V µ (x ν )dx µ which is invariant under both the U(1) particle number symmetry and the Milne boosts:
dx − is boost- invariant, but not U(1)-invariant, while dx − + A is U(1)-invariant but not boost-invariant.
Historically, some of these results were known some time ago, although from a very different perspective. It was first recognized in [7] (see also [27]) that a NC structure can be obtained via null reduction of a Lorentzian d + 1 dimensional manifold, although these authors restricted n to be closed. Of course this reduction still works if dn = 0 [5,25]. The role of the Milne boosts in d + 1 dimensions was only realized in [5].
Aside from its obvious importance in DLCQ, the null reduction (2.12) will be very useful in what follows, even for NR theories which do not follow from DLCQ. Its primary virtue for us is that all of the NC symmetries are manifest therein. So we can efficiently construct tensors under the NC symmetries by constructing tensors on an auxiliary d + 1dimensional spacetime with metric (2.12). For instance, using the Levi-Civita connection
(Γ g ) M NP = 1 2 g MQ (∂ N g PQ + ∂ P g NQ − ∂ Q g NP ) ,(2.R M NPQ = ∂ P (Γ g ) M NQ − ∂ Q (Γ g ) M NP + (Γ g ) M SP (Γ g ) S NQ − (Γ g ) M SQ (Γ g ) S NP ,(2.17)
and so R = g MN R P MPN . Because R is invariant under all of the higher-dimensional symmetries, it gives an invariant d-dimensional scalar under all of the NC symmetries.
As we mentioned above, the flat-space DLCQ of a relativistic CFT gives a Schrödinger theory. So it should not be a surprise that the Weyl symmetry of a Schrödinger theory nicely fits into the null reduction (2.12). Note that the Weyl rescaling of g g → e 2Ω g , (2.18) is equivalent to the NR z = 2 Weyl rescaling of n µ and h µν in (2.9). This will be useful for us when we consider pure Weyl anomalies in Schrödinger theories in Section 4. As a simple example, consider the DLCQ of a conformally coupled complex scalar Φ
S R = − 1 4π d d+1 x −g g MN ∂ M Φ † ∂ N Φ + ξRΦ † Φ , ξ = d − 1 4d ,(2.19)
and compactify x − with periodicity 2π. This theory is
Weyl-invariant provided that Φ transforms with weight 1−d 2 . Expanding Φ in Fourier modes of x − , Φ = ∑ n ϕ n (x µ )e inx − , the relativistic action S R becomes (using that √ −g = √ γ) S R = ∑ n d d x √ γ inv µ 2 ϕ † n D µ ϕ n − (D µ ϕ † n )ϕ n − h µν 2 D µ ϕ † n D ν ϕ n − ξR 2 ϕ † n ϕ n , (2.20)
with D µ ϕ n = (∂ µ − inA µ )ϕ n . The term for each n = 0 is just n times the action of the NR analogue of a conformally coupled scalar ϕ n carrying charge n under particle number [5,28]. The zero-mode action is also invariant under the local Galilean and Weyl symmetries, but it is obvious that it must be treated carefully when computing observables.
Flavor and/or gravitational anomalies from DLCQ
Suppose we study the NR theory that arises from the DLCQ of a relativistic theory with flavor and/or gravitational anomalies. 5 Does the NR theory have anomalies?
We proceed in two steps. First, we efficiently represent the anomalies of the underlying relativistic theory with anomaly inflow. Then we put the relativistic theory on a background with a null isometry (2.12), whereby we express the anomalies in terms of the NC data to which the NR theory couples. This variation can be cancelled off by adding a counterterm built from the NC data. However said counterterm is not boost-invariant.
It is easiest to work with this boost-non-invariant description. We find that this boost variation cannot be removed by a further counterterm, and so is a genuine anomaly which we then write in terms of an anomaly inflow. So the anomalies of the relativistic parent descend to mixed flavor/boost or gravitational/boost anomalies in the NR theory, and the counterterm we construct is a "Bardeen counterterm" which shifts the anomaly from the flavor or gravitational sector to being a boost anomaly.
Anomaly inflow
Perhaps the simplest way to describe flavor and gravitational anomalies in relativistic QFT is via the anomaly inflow mechanism [29]. Given a field theory on a D-dimensional spacetime M D coupled to a flavor gauge field A M and Riemannian metric g MN , the local anomalies are encoded in a D + 2-form P known as the anomaly polynomial. P is built from the Chern classes of the field strength F MN of A M and Pontryagin classes of the Riemann curvature R M NPQ , and so dP = 0. P determines the variation of the field theory generating functional W through the descent equations. P is the exterior derivative of a Chern-Simons d + 1-form I, P = dI, whose gauge and/or coordinate variation is a derivative of the d-form G χ
δ χ I = dG χ . (3.1)
This determines the anomalous variation of W via
δ χ W = − M D G χ ,(3.2)
or equivalently, one constructs 3) neatly encodes the idea of anomaly inflow: we imagine that our field theory lives on the boundary of a "Hall" system whose action is the Chern-Simons term I, and the anomalies of the boundary theory arise because currents or energy-momentum can flow from the "bulk theory" on M D+1 into the boundary. The anomaly inflow mechanism is also at play in holography. When an anomalous field theory has a gravity dual, the metric and gauge field in the field theory are the asymptotic values of a dynamical bulk metric and gauge field, and the gravitational theory has a Chern-Simons term I and the field theory anomaly polynomial is P = dI.
W cov = W + M D+1 I , (3.3) M d reduce on x − ←−−−−−−−− M d+1 = ∂M d+2 inflow ←−−−−−−−− M d+2 descent: I←P ←−−−−−−−− M d+3
Constructing the counterterm
Now consider the d-dimensional Galilean theory on M d obtained by performing the most general DLCQ of a relativistic theory on M d+1 with anomaly polynomial P. That is, the relativistic theory is coupled to a d + 1-dimensional metric g MN and (not necessarily abelian) flavor gauge field A M with a null symmetry along x − ,
g = 2n µ dx µ (dx − + A) + h µν dx µ dx ν , A =Â µ dx µ + A − (dx − + A) ,(3.4)
where the component fields depend on x µ but not x − , and implicitly x − is compactified with some radius. The (n µ , h µν , A µ ) become the NC structure to which the d-dimensional NR theory couples, µ becomes a background flavor gauge field which couples to the flavor symmetry of the NR theory, and A − becomes a scalar source. The anomalies of the d + 1-dimensional theory are most efficiently described in terms of the Chern-Simons form I on a d + 2-dimensional spacetime M d+2 which has M d+1 as its boundary. Reducing on the null circle leads to the d-dimensional spacetime M d on which the NR theory lives. This gives the short sequence represented in Fig. 1.
When one instead reduces a relativistic theory with anomalies on a spatial or thermal circle, the d-dimensional description is also relativistic. In that case one can construct a local counterterm built out of the d-dimensional background which cancels the anomalous variation. That such a counterterm exists is a corollary of the fact that there are no flavor or gravitational anomalies in odd-dimensional relativistic theories. It turns out that we can construct such a counterterm in the null case, although we lack an a priori argument that this should be so. However, said counterterm is not invariant under Milne boosts.
If one is interested just in the lower-dimensional theory obtained from a spatial circle reduction, the precise form of this counterterm is uninteresting. However, for thermal circles, this counterterm is physical: it is intimately related to non-renormalized anomaly-induced transport. See [30] (as well as [31]) for details. For an arbitrary anomaly polynomial, the requisite counterterm was constructed in [32] (see also [33]) using the technology of transgression forms, which will also be useful here.
The basic idea can be illustrated with a pure flavor anomaly. From the point of view of the NR theory, A − is a scalar source which transforms in the adjoint representation of the flavor symmetry, rather than as a component of a connection. So we can define a new flavor connectionĀ by simply subtracting off the A − component. To do this in a U(1)-invariant way, we subtract off a term proportional to dx − + A, 6
A ≡ A − A − (dx − + A) =Â µ dx µ ,(3.A →Â − A − Ψ , Ψ µ = ψ µ − 1 2 n µ ψ 2 (3.6)
which still has no leg along dx − . In any case, it is clear that both and the field strength which is where transgression forms become useful. We refer the reader who is unfamiliar with the transgression machinery to the Appendix of [32] for a concise and modern discussion. Essentially, this technology is the natural way to describe the way characteristic classes, and objects like Chern-Simons forms constructed from them, depend on the connection. The result we require here is that a Chern-Simons form I evaluated for two different connections A 1 and A 2 , which we denote as I n = I[A n , F n ], obeys
I 1 − I 2 = V 12 + dW 12 , (3.8) 6
There is a fully d + 1-dimensionally covariant version of this construction along the lines of that presented in [32] for thermal circles. The generalization to the null case is straightforward: the covariant version of A − is n M A M + Λ K where Λ K is the flavor gauge transformation which together with n M generates the symmetry, and the covariant version of dx − + A is any one-form u which obeys n M u M = 1. However any such choice breaks the Milne redundancy, and so the redefined connection varies under the Milne boosts.
where
A(τ) = A 2 + τ(A 1 − A 2 ) , F (τ) = dA(τ) + A(τ) ∧ A(τ) , V 12 = 1 0 dτ (A 1 − A 2 ) ∧ · ∂P (τ) ∂F (τ) , W 12 = 1 0 dτ (A 1 − A 2 ) ∧ · ∂I(τ) ∂F (τ) . (3.9)
Here A(τ) interpolates between A 2 and A 1 , P (τ) and I(τ) refer to the anomaly polynomial and Chern-Simons form evaluated on it, and · refers to a trace over flavor indices. When A 1 and A 2 differ by an adjoint tensor, V 12 is a gauge-invariant form and the variations of I 1 and I 2 are carried by the boundary term W 12 .
Combining this with (3.3) and (3.7), taking A 2 =Â and A 1 = A, and denoting
I −Î =V + dŴ ,(3.10)
then we can rewrite the covariant field theory generating functional (3.3) as
W cov = W + M d+2 I = W + M d+1Ŵ + M d+2V = W + M d+2V ,(3.11)
where we have redefined W by the local countertermŴ as Before going on, we observe that (3.11) now looks like anomaly inflow for boosts, whereV plays the role of a Chern-Simons form for Milne boosts. That is, dV = P −P , (3.12) which is boost-invariant, but the boost variation ofV is a total derivative, 7 δ ψV = dG ψ , (3.13) so that W varies under Milne boosts as
W → W + M d+1Ŵ . Because A −Â = A − (dx − + A) isδ ψ W = − M d+1 G ψ . (3.14)
In fact, this demonstrates that the Milne variation is a genuine anomaly. That is, there is no local d + 1-dimensional counterterm which can remove (3.14). 8 To see this, suppose that such a counterterm exists, which we denote as M d+1 C. ThenV ≡V + dC is invariant under all symmetries, and dV = P −P. So we must be able to "transgress" the difference P −P in a flavor/gravitational/boost-invariant way. The most general such transgression fromP to P is to consider the most general flow A(τ) which interpolates 7 We will justify both of these assertions and compute the Milne variation ofV in the next Subsection. 8 In this statement we implicitly require d > 1 in order to have a Milne symmetry in the first place.
from at τ = 0 and flows to A at τ = 1. In terms of that flow one has
P −P = dV ,V = 1 0 dτ ∂ τ A(τ) ∧ · ∂P (τ) ∂F (τ) . (3.15)
In order forV to be invariant, we require that each of the terms in the integral expression (3.15) forV to be covariant under the symmetries. However, it is easy to see that no such A(τ) exists so that ∂ τ A(τ) is covariant: the only zero-derivative covariant one-form available is n = n M dx M , and one cannot reach by adding a scalar times n to A. So far, we have shown that the DLCQ of a theory with a pure flavor anomaly leads to a NR theory with a mixed flavor/boost anomaly. The term "mixed" refers to the fact that one can use a local counterterm to make the NR theory invariant under one symmetry or the other, but not both simultaneously. Now we consider the DLCQ of a theory with arbitrary anomalies. This is only an upgraded version of the analysis above, so we just mention the highlights.
In addition to extending the flavor connection A and symmetry data to higher dimensions, we also extend the metric g and so the Levi-Civita connection Γ g . 9 It is convenient to represent Γ g as a matrix-valued connection one-form, (3.16) and the Riemann curvature R M NPQ as a curvature two-form,
(Γ g ) M N = (Γ g ) M NP dx P ,R M N = 1 2 R M NPQ dx P ∧ dx Q = d(Γ g ) M N + (Γ g ) M P ∧ (Γ g ) P N . (3.17)
In the coordinates in which we expressed g MN (3.4), the − component of Γ g is a tensor
(Γ g ) M N− = D N n M , (3.18)
which we can use to define a "hatted" connectionΓ g along the same lines as in (3.5),
(Γ g ) M N = (Γ g ) M N − D N n M (dx − + A) . (3.19)
BothΓ g and the corresponding curvatureR have no leg along dx − . (The covariant statement is that dx − + A is a one-form u obeying n M u M = 1,R M NPQ n Q = 0, and (Γ g ) M NP n P can be set to zero by a coordinate reparameterization.)Γ g transforms as a connection under coordinate reparameterizations, but inherits a non-trivial transformation law under Milne boosts owing to the transformation of dx − + A,
(Γ g ) M N → (Γ g ) M N − D N n M Ψ , Ψ µ = ψ µ − 1 2 n µ ψ 2 .
(3.20)
As above, the Chern-Simons form I evaluated for the hatted connections,Î = I[Â,F ,Γ g ,R], has no leg along dx − and so its integral vanishes on M d+2 , giving
M d+2 I = M d+2 I − M d+2Î .
In analogy with (3.9), the difference between I evaluated for two sets of connections {A 1 , Γ 1 } and {A 2 , Γ 2 } may be represented as
I 1 − I 2 = V 12 + dW 12 ,
where V 12 and W 12 are transgression forms. Flowing in the space of connections as
A(τ) = A 2 + τ(A 1 − A 2 ) , Γ M N (τ) = (Γ 2 ) M N + τ (Γ 1 ) M N − (Γ 2 ) M N ,(3.21)
we have
V 12 = 1 0 dτ (A 1 − A 2 ) ∧ · ∂P (τ) ∂F (τ) + (Γ 1 ) M N − (Γ 2 ) M N ∧ ∂P (τ) ∂R M N , W 12 = 1 0 dτ (A 1 − A 2 ) ∧ · ∂I(τ) ∂F (τ) + (Γ 1 ) M N − (Γ 2 ) M N ∧ ∂I(τ) ∂R M N . (3.22)
When A 1 − A 2 and Γ 1 − Γ 2 are covariant tensors, V 12 is a gauge and reparameterizationinvariant form and W 12 carries the variations of I 1 and I 2 . This is indeed the case when
A 2 =Â , A 1 = A , Γ 2 =Γ g , Γ 1 = Γ g ,
and as before we denote I −Î =V + dŴ .
Putting this together with (3.3) we can rewrite W cov in the same form as (3.11),
W cov = W + M d+2V , W = W + M d+1Ŵ .
As beforeŴ is a local counterterm which renders the NR theory invariant under flavor gauge transformations and coordinate reparameterizations, but leaves it anomalous under Milne boosts. We conclude this Subsection with some simplified formulae forV andŴ. First, we simplify the hatted curvatures. We denote u ≡ dx − + A so du = F, and let D be the exterior covariant derivative. We also decompose the ordinary curvatures into components which are longitudinal and transverse to n M ,
F = E ∧ u + B , R M N = (E R ) M N ∧ u + (B R ) M N , (3.23) where E M = F MN n N , B MN n N = 0 , (E R ) M NP = R M NPQ n Q , (B R ) M NPQ n Q = 0 . (3.24)
The fact that n generates a symmetry fixes the longitudinal parts E and E R as
DA − = E , D D N n M = (E R ) M N . (3.25)
Then the hatted curvatures arê
F = B − A − F ,R M N = (B R ) M N − D N n M F . (3.26)
As a result, u ∧F and u ∧R differ from u ∧ F and u ∧ R by a term proportional to F. Consequently, the formal d + 4 form
u ∧ P −P ,
can be expanded as a formal power series in F where each term has at least one F, 27) and the V i are d + 2(1 − i) forms. Similarly, the formal d + 3 form u ∧ (I −Î) can be written as a power series in F,
u ∧ P −P = ∑ i=0 V i ∧ F i+1 ,(3.u ∧ I −Î = ∑ i=0 W i ∧ F i+1 ,(3.28)
where the W i are d + 1 − 2i forms. Then we define the formal objects 29) and in terms of these we find the compact expressionŝ
u F ∧ P −P ≡ ∑ i=0 V i ∧ F i , u F ∧ I −Î ≡ ∑ i=0 W i ∧ F i ,(3.V = u F ∧ P −P ,Ŵ = u F ∧ I −Î . (3.30)
One can also obtain these from the direct integration of (3.22).
The anomalous boost Ward identity
We presently justify the assertions we made in (3.12) and (
δ ψ u = ψ , n M ψ M = 0 . (3.31)
Using (3.23) and that F and R are boost-invariant, this in turn induces
δ ψ = −A − ψ , δ ψ (Γ g ) M N = −(D N n M )ψ , δ ψ F = dψ , (3.32) δ ψF = −D(A − ψ) , δ ψR M N = −D D N n M ψ ,
whereD is the exterior covariant derivative defined using the hatted connections.
Although it is not obvious at this stage, these ingredients tell us that the computation of δ ψV has already appeared in the literature. In the Appendix of [32], we and other authors computed the variations of a transgression form much likeV where the difference between two connections was a one-form, just as is the case here. Rather than walk through that computation, let us simply quote the result, are essentially the Hodge duals of the "Hall currents" that one gets by varyingÎ with respect to the hatted connections andΓ g . However, both δ ψ and δ ψΓg have no leg along dx − , nor do the Hall currents in (3.34). As a result, the boost variation of the integral ofV, which is what actually appears in the anomaly inflow (3.11), only picks up the boundary term,
δ ψV = d ψ ∧ ∂V ∂F − δ ψ ∧ · ∂P ∂F − δ ψ (Γ g ) M N ∧ ∂P ∂R M N ,(3.δ ψ M d+2V = M d+1 ψ ∧ ∂V ∂F . (3.35)
This is what we meant when we made our earlier assertion that the boost variation ofV is a boundary term (3.13). This also gives
G ψ = ψ ∧ ∂V ∂F , δ ψ W = − M d+1 G ψ . (3.36)
Using the epsilon tensor to dualize Q ≡ ∂V ∂F into a vector
q M ≡ 1 d! ε MP 1 ...P d Q P 1 ...P d ,(3.δ ψ W = − d d x √ γ ψ µ h µν q ν ,(3.38)
where ψ µ = h µν ψ ν . The anomalous boost Ward identity then reads
P µ − h µν J ν = h µν q ν . (3.39)
This has all been rather abstract. Let us see how this machinery works for pure U(1) anomalies when the relativistic parent is two or four dimensional. In the first case, there is no Milne symmetry in the first place as there are no spatial directions, but we nevertheless proceed to illustrate how the machinery works. We have P = c A F ∧ F , (3.40) and we readily find the formal objectŝ
V = c A A − u ∧ (2B − A − F) , G ψ = −c A A 2 − ψ ∧ u . (3.41)
But in this case ψ = 0 and so G ψ = 0, so there is no Milne anomaly for the non-existent Milne symmetry.
In the four-dimensional case, we have
P = c A F ∧ F ∧ F , (3.42) so thatV = c A A − u ∧ 3B ∧ B − 3A − F ∧ B + A 2 − F ∧ F , G ψ = −c A A 2 − ψ ∧ u ∧ (3B − 2A − F) .
(3.43)
The anomalous boost Ward identity is
P µ − h µν J ν = − c A A − 2 h µν ε νρσ 3F ρσ − 2A − F ρσ . (3.44) 4
Weyl anomalies for z = 2
The basic idea
In what follows we will classify pure Weyl anomalies for Schrödinger theories. We do not presume or employ a Lagrangian description, but will simply use consistency conditions that all quantum field theories satisfy. We proceed in three steps.
1. First, we parameterize the most general local Weyl variation,
δ Ω W = d d x √ γ δΩ A ,
where A is a boost and gauge-invariant scalar built from the NC data (n µ , h µν , A µ ) and derivatives.
2.
Write down the most general set of local gauge/boost-invariant counterterms which may be added to W. Then compute the Weyl variation of these counterterms, and deduce which terms in A can be removed by a judicious choice of counterterm.
3. Impose Wess-Zumino (WZ) consistency [34], which in the present instance means that we demand
[δ Ω 1 , δ Ω 2 ] = 0 . (4.1)
It should be clear that we can only perform this algorithm once we know the symmetries to consider and the background fields out of which we can build A and counterterms.
After performing this analysis, there are three classes of terms which appear in A. We refer to terms which can be generated by local counterterms as class C, terms which are Weyl-covariant with weight −(d + 1) as class B, and terms which are not Weyl-covariant yet are WZ consistent as class A. (This is a different convention than that in [35].)
Because we consider pure Weyl anomalies, we need to efficiently classify boost and gauge-invariant scalars. As we mentioned in Subsection 2.2, the simplest way to do this is via the null reduction of a Lorentzian manifold with metric (2.12). In what follows we write down tensors in terms of the metric, isometry, and covariant derivative on this auxiliary higher-dimensional spacetime.
Two spatial dimensions
Let us warm up with the simplest non-trivial case, namely theories in 2 + 1 dimensions. The 0 + 1-dimensional case is trivial: in that case there are no tensors that may be formed from the NC structure, and so no potential Weyl anomaly.
For simplicity, we consider parity-preserving theories. Then the scalars that may contribute to A are built from the following basis of tensors: the inverse metric g MN , the isometry n M , the Riemann curvature R MNPQ , the derivative of n, D M n N , and D M . Because n M generates a null isometry, we have
D (M n N) = 0 , n N D M n N = 0 , R MNPQ n Q = D P D N n M . (4.2)
Under constant Weyl transformations e Ω = λ, the basic building blocks transform as
g MN → λ −2 g MN , n M → n M , R MNPQ → λ 2 R MNPQ , D M n N → λ 2 D M n N , (4.3)
and D M does not rescale. The putative Weyl anomaly transforms as A → λ −4 A under constant Weyl rescalings, and the local counterterms which can remove terms in A are the spacetime integrals of scalars C which also transform as C → λ −4 C. Scalars with weight −4 containing p Riemann's, q factors of the "twist" D M n N , and r additional covariant derivatives D M necessarily contain 2 + p + q factors of the inverse metric and 2(p − 2) + r factors of n M . In fact, we can only build such scalars if either p ≥ 2 or p = 1 and r ≥ 2. Such a scalar contains 2p + q + r derivatives. So the lowest number of derivatives possible is four, which can be done with either p = 2, q = r = 0 or p = 1, q = 0, and r = 2. In either case n M does not appear. That is, the scalars of weight −4 with the lowest number of derivatives are just the ordinary four-derivative scalars which can be built from the Riemann, the covariant derivative, and the metric. We parameterize them using the basis
E 4 = R MNPQ R MNPQ − 4R MN R MN + R 2 , W MNPQ W MNPQ , R 2 , D M D M R ,R 2 R MN n M n N .
At this point, it should be clear that the problem of classifying the four-derivative terms in the Weyl anomaly is just the same problem as for 3 + 1-dimensional relativistic CFT. The Euler and W 2 counterterms are Weyl-invariant up to a boundary term, the R 2 counterterm can be used to remove the D M D M R term in A, and the D M D M R counterterm integrates to a boundary term. The remaining E 4 and W 2 terms in A are WZ consistent, but the R 2 term is not. So we have
A = aE 4 − cW 2 + O(∂ 6 ) .
(4.5)
The four-derivative part is just the usual Weyl anomaly of a 3 + 1-dimensional CFT on the background (2.12). The Euler term is a class A anomaly, the W 2 term is class B, and the D M D M R term is class C. It is worth noting that even though there is an isometry both E 4 and W MNPQ can be nonzero. However, if the isometry has no twist, D M n N = 0, then (4.2) implies that R MNPQ is effectively the Riemann tensor of a three-dimensional space, in which case E 4 and W MNPQ vanish. In the coordinates (2.12), the nonzero components of the twist are
D µ n ν = 1 2 (∂ µ n ν − ∂ ν n µ ) . (4.6)
So the Weyl anomaly is only visible when dn = 0. What of the potential higher-derivative terms in A? While we do not rigorously classify them, we offer some observations which lead to a conjecture for A.
First, by WZ consistency, any scalar which transforms covariantly with weight −4 under inhomogeneous Weyl transformations may appear in A. If n was everywhere timelike or spacelike rather than null, then we could use it to redefine the covariant derivative in a Weyl-invariant way. Then the Weyl-covariant tensors would be built from this Weyl-covariant derivative. However, because n is null no such redefinition of D M exists. So the manifestly Weyl-covariant tensors are built from W MNPQ along with g MN and n N . We also expect there to be non-manifest Weyl-covariant tensors which cannot be expressed this way. It seems that one can build weight −4 scalars with arbitrarily many derivatives from this data by a similar counting argument to that above. The manifestly Weyl-covariant scalars can be counted as follows. A 2p-derivative scalar possesses p Weyl tensors along with 2 + p factors of the inverse metric and 2(p − 2) factors of n M (with p ≥ 2). For example, a six-derivative scalar is
W MNPQ W MNP S W Q A S B n A n B .(4.7)
We have not found an argument that there are a finite number of such scalars. Second, the Weyl variation of any local weight −4 counterterm is always of the form (we can represent the counterterm equivalently as a 4-dimensional integral or as a 3-dimensional one, since no sources depend on the null circle)
δ Ω d 3 x √ γ C = d 3 x
√ γ δΩ D M C M + (boundary terms) . (4.8) With this in mind, experience has taught us that counterterms can be used to remove all terms in A of the form D M A M , although we lack a proof that this is always the case. Third, we conjecture that E 4 is the unique "exceptional" scalar built from g MN and n M and derivatives which is not Weyl-covariant, yet is WZ consistent when it appears in A.
Putting all three of these ingredients together, we can make a conjecture for A, namely that it is of the form
A = aE 4 − cW 2 + ∑ i d i W n i ,(4.9)
where the W n i are Weyl-covariant scalars with at least six derivatives, built using n M .
The general result
The general argument proceeds in the same way as that above. First, consider a Schrödinger theory in even spacetime dimension d. We build an auxiliary odd-dimensional metric and isometry from the NC structure. The Weyl anomaly A must transform with weight d + 1 under constant Weyl transformations, which in this instance means it transforms with odd weight. However, there is simply no way to assemble an odd weight scalar out of the building blocks (4.3). So A = 0 in this case.
In odd spacetime dimension d = 2m − 1, we can build weight −(d + 1) scalars with p Riemanns, q factors of the "twist" D M n N , and r derivatives, provided that we contract indices with 2 + p + q factors of the inverse metric and 2(p − m) + r factors of n M . Such a scalar possess 2p + q + r derivatives. The terms with the lowest number of derivatives have 2p + r = m and q = 0, in which case no factors of n M appear. So, as in the 2 + 1dimensional case, the lowest-derivative analysis reduces to the relativistic one, for which we can appeal to the literature [35], A = aE d+1 + ∑ n c n W n + O(∂ d+3 ) , (4.10) where the W n are weight −2m Weyl-covariant scalars, and E d+1 is the d + 1 dimensional Euler density. We conjecture that all of the higher-derivative terms are Weyl-covariant scalars W n i which depend on n M , giving
A = aE d+1 + ∑ n c n W n + ∑ i d i W n i .(4.11)
A BRST-inspired analysis [36] should be able to verify or disprove our conjecture.
Figure 1 :
1The short sequence which relates the anomalies of a relativistic theory to those of the d-dimensional NR theory realized by DLCQ. The relativistic parent lives on M d+1 , and its null reduction leads to the NR theory on M d . The anomalies of the parent are described via inflow by letting M d+1 be the boundary of a d + 2-manifold M d+2 which we equip with a Chern-Simons form I. The anomaly polynomial P is a formal d + 3 form which may be thought of as living on a formal M d+3 .which is invariant under all symmetries, where we have extended the gauge field and metric on M D to a gauge field and metric on the D + 1-dimensional manifold M D+1 which has M D as its boundary.(3.
ofÂ,F , have no legs along dx − . (The covariant statement is thatF MN n N = 0 and M n M can be set to zero by a gauge transformation.) Consequently, the Chern-Simons form I evaluated on the hatted connection and field strength has no leg along dx − , and so its integral on M d+2 vanishes. Denoting I[Â,F ] =Î,
3.13) by explicitly computing the Milne variation ofV. From (3.26) and (3.30) it is clear that we can regardV as u wedge a functional of B, (B R ) M N , F, A − , D N n M . Under an infinitesimal boost u shifts as
33) where the partial derivative ofV is taken at fixed u, A − , D M n N , B, B R , and the objects
37) (here we have chosen an orientation such that ε 0...d− = 1 √ γ ) the boost variation of the field theory generating functional becomes
In coupling a Galilean theory to NC geometry, one demands invariance under (i.) reparameterization of coordinates, (ii.) U(1) gauge transformations, and (iii.) shift transformations known in the NC literature[6] as Milne boosts. Under the boost, the tensors (n µ , h µν ) are invariant andthat we have used γ µν defined in (2.1) to define an
invariant measure d d x
√
γ.
16 )
16obtained from g MN to define the d + 1-dimensional covariant derivative D M , we can define the Riemann tensor
dx − + A varies under the Milne boosts, and so varies aŝ5)
which we recognize as just the hatted connection from (3.4). This hatted connection
is almost a good flavor connection under the d-dimensional symmetries: while A is
Milne-invariant,
an adjoint tensor,V is gauge-invariant and so W is too. That is, the local counterterm Ŵ renders the NR theory invariant under flavor gauge transformations. However, because A −Â = A − (dx − + A) varies under Milne boosts, the redefined W is now non-invariant under Milne boosts.
(4.4) where R MN = R P MPN is the Ricci curvature, E 4 is the four-dimensional Euler density built from R MNPQ , and W MNPQ the Weyl tensors. We have also used that D M D N R MN =1 2 D M D M R by the Bianchi identity. Due to (4.2) there are no weight −4 scalars with five derivatives. There are a number of six-derivative scalars, like
For example, consider a Hall system in two spatial dimensions for which a gravitational Chern-Simons term for a SO(2) spin connection appears in the low-energy effective action. Despite much ink spilled, it is not known if the boundary field theory has a corresponding anomaly, or if a boundary counterterm cancels the variation of the Chern-Simons term.2 There has been some work classifying pure Weyl[1][2][3] and axial anomalies[4] in Lifshitz theories.3 See[12][13][14][15] for other perspectives on the local symmetries of Galilean theories.
If one has a gauge and Milne-invariant vector v µ satisfying v µ n µ > 0 everywhere, then one can use this vector to build a boost and gauge-invariant derivative[26].
This includes those theories which are dual to string theory on asymptotically Schrödinger backgrounds, where there are bulk Chern-Simons terms.
To ensure that the gravitational anomaly inflow is completely intrinsic, we require that the extrinsic curvature of M d+1 = ∂M d+2 vanishes. In these coordinates that means (Γ g ) ⊥ M = (Γ g ) M ⊥ = 0. Alternatively, one can allow for extrinsic curvature and add a local counterterm which depends on it in such a way as to obtain the correct gravitational anomaly on M d+1 .
AcknowledgmentsWe are pleased to thank J. Hartong for useful discussion. The author would like to thank the organizers of the "2014 Simons Summer Workshop in Mathematics and Physics" at the Simons Center for Geometry and Physics for their hospitality during which most of this work was completed. The author was supported in part by National Science Foundation under grant PHY-0969739.
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| []
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[
"Born-Infeld magnetars: larger than classical toroidal magnetic fields and implications for gravitational-wave astronomy",
"Born-Infeld magnetars: larger than classical toroidal magnetic fields and implications for gravitational-wave astronomy"
]
| [
"Jonas P Pereira \nCentro de Ciências Naturais e Humanas\nUniversidade Federal do ABC\nAvenida dos Estados 5001, Santo André09210-170SPBrazil\n\nMathematical Sciences and STAG Research Centre\nUniversity of Southampton\nSO17 1BJSouthamptonUnited Kingdom\n",
"Jaziel G Coelho \nDivisão de Astrofísica\nInstituto Nacional de Pesquisas Espaciais\nAvenida dos Astronautas 175812227-010São José dos CamposSPBrazil\n\nDepartamento de Física\nUniversidade Tecnológica Federal do Paraná\n85884-000MedianeiraPRBrazil\n",
"Rafael C R De Lima \nUniversidade do Estado de Santa Catarina\nAv. Madre Benvenuta, 88.035-9012007Itacorubi, FlorianópolisBrazil\n"
]
| [
"Centro de Ciências Naturais e Humanas\nUniversidade Federal do ABC\nAvenida dos Estados 5001, Santo André09210-170SPBrazil",
"Mathematical Sciences and STAG Research Centre\nUniversity of Southampton\nSO17 1BJSouthamptonUnited Kingdom",
"Divisão de Astrofísica\nInstituto Nacional de Pesquisas Espaciais\nAvenida dos Astronautas 175812227-010São José dos CamposSPBrazil",
"Departamento de Física\nUniversidade Tecnológica Federal do Paraná\n85884-000MedianeiraPRBrazil",
"Universidade do Estado de Santa Catarina\nAv. Madre Benvenuta, 88.035-9012007Itacorubi, FlorianópolisBrazil"
]
| []
| Magnetars are neutron stars presenting bursts and outbursts of X-and soft-gamma rays that can be understood with the presence of very large magnetic fields. Thus, nonlinear electrodynamics should be taken into account for a more accurate description of such compact systems. We study that in the context of ideal magnetohydrodynamics and make a realization of our analysis to the case of the well known Born-Infeld (BI) electromagnetism in order to come up with some of its astrophysical consequences. We focus here on toroidal magnetic fields as motivated by already known magnetars with low dipolar magnetic fields and their expected relevance in highly magnetized stars. We show that BI electrodynamics leads to larger toroidal magnetic fields when compared to Maxwell's electrodynamics. Hence, one should expect higher production of gravitational waves (GWs) and even more energetic giant flares from nonlinear stars. Given current constraints on BI's scale field, giant flare energetics and magnetic fields in magnetars, we also find that the maximum magnitude of magnetar ellipticities should be 10 −6 − 10 −5 . Besides, BI electrodynamics may lead to a maximum increase of order 10% − 20% of the GW energy radiated from a magnetar when compared to Maxwell's, while much larger percentages may arise for other physically motivated scenarios. Thus, nonlinear theories of the electromagnetism might also be probed in the near future with the improvement of GW detectors. | 10.1140/epjc/s10052-018-5849-2 | [
"https://arxiv.org/pdf/1804.10182v1.pdf"
]
| 54,633,603 | 1804.10182 | e445dba4203b1e076ffaff366bd227cac1c8bacc |
Born-Infeld magnetars: larger than classical toroidal magnetic fields and implications for gravitational-wave astronomy
Jonas P Pereira
Centro de Ciências Naturais e Humanas
Universidade Federal do ABC
Avenida dos Estados 5001, Santo André09210-170SPBrazil
Mathematical Sciences and STAG Research Centre
University of Southampton
SO17 1BJSouthamptonUnited Kingdom
Jaziel G Coelho
Divisão de Astrofísica
Instituto Nacional de Pesquisas Espaciais
Avenida dos Astronautas 175812227-010São José dos CamposSPBrazil
Departamento de Física
Universidade Tecnológica Federal do Paraná
85884-000MedianeiraPRBrazil
Rafael C R De Lima
Universidade do Estado de Santa Catarina
Av. Madre Benvenuta, 88.035-9012007Itacorubi, FlorianópolisBrazil
Born-Infeld magnetars: larger than classical toroidal magnetic fields and implications for gravitational-wave astronomy
Received: date / Revised version: dateEur. Phys. J. C manuscript No. (will be inserted by the editor)
Magnetars are neutron stars presenting bursts and outbursts of X-and soft-gamma rays that can be understood with the presence of very large magnetic fields. Thus, nonlinear electrodynamics should be taken into account for a more accurate description of such compact systems. We study that in the context of ideal magnetohydrodynamics and make a realization of our analysis to the case of the well known Born-Infeld (BI) electromagnetism in order to come up with some of its astrophysical consequences. We focus here on toroidal magnetic fields as motivated by already known magnetars with low dipolar magnetic fields and their expected relevance in highly magnetized stars. We show that BI electrodynamics leads to larger toroidal magnetic fields when compared to Maxwell's electrodynamics. Hence, one should expect higher production of gravitational waves (GWs) and even more energetic giant flares from nonlinear stars. Given current constraints on BI's scale field, giant flare energetics and magnetic fields in magnetars, we also find that the maximum magnitude of magnetar ellipticities should be 10 −6 − 10 −5 . Besides, BI electrodynamics may lead to a maximum increase of order 10% − 20% of the GW energy radiated from a magnetar when compared to Maxwell's, while much larger percentages may arise for other physically motivated scenarios. Thus, nonlinear theories of the electromagnetism might also be probed in the near future with the improvement of GW detectors.
Abstract Magnetars are neutron stars presenting bursts and outbursts of X-and soft-gamma rays that can be understood with the presence of very large magnetic fields. Thus, nonlinear electrodynamics should be taken into account for a more accurate description of such compact systems. We study that in the context of ideal magnetohydrodynamics and make a realization of our analysis to the case of the well known Born-Infeld (BI) electromagnetism in order to come up with some of its astrophysical consequences. We focus here on toroidal magnetic fields as motivated by already known magnetars with low dipolar magnetic fields and their expected relevance in highly magnetized stars. We show that BI electrodynamics leads to larger toroidal magnetic fields when compared to Maxwell's electrodynamics. Hence, one should expect higher production of gravitational waves (GWs) and even more energetic giant flares from nonlinear stars. Given current constraints on BI's scale field, giant flare energetics and magnetic fields in magnetars, we also find that the maximum magnitude of magnetar ellipticities should be 10 −6 − 10 −5 . Besides, BI electrodynamics may lead to a maximum increase of order 10% − 20% of the GW energy radiated from a magnetar when compared to Maxwell's, while much larger percentages may arise for other physically motivated scenarios. Thus, nonlinear theories of the electromagnetism might also be probed in the near future with the improvement of GW detectors.
Introduction
Soft Gamma Repeaters (SGRs) and Anomalous X-Ray pulsars (AXPs) (also known as magnetars [see, e.g., 1, and references therein]) are spectacular astrophysical systems which a e-mail: [email protected] b e-mail: [email protected] c e-mail: [email protected] hold the unique possibility of probing yet unknown particle physics under extremely high magnetic fields. Believed to be "lonely wolves", among other properties such objects are observationally characterized by outbursts of X-ray and soft gamma-ray flares and have rotational periods P ∼ (2-12) s and slowing down ratesṖ ∼ (10 −15 −10 −10 ) s/s [2]. Duncan and Thompson [3,4] proposed that they would be neutron stars (NSs) with huge magnetic fields (magnetars), of the order of 10 14 − 10 15 G. One of the reasons for this would be their transient activities in the form of giant flares, whose typical luminosities are 10 44 -10 47 erg s −1 [2], not possible in general to be powered by their rotational energy. 1 Notwithstanding, three counterexamples of magnetars with low surface magnetic fields are already known [6][7][8]. This apparent "glitch" of the magnetar model has motivated different scenarios for the explanation of SGRs/AXPs, e.g.: the possibility of a fallback disk slowing down a neutron star pulsar up to its current spin period [see, e.g., 9,10]; drift waves near the light-cylinder of NSs [see 11, and references therein]; exotic scenarios involving quark stars [12]; and massive, fast rotating, highly magnetized white dwarfs to explain these types of sources [13][14][15]. Low-B values are related to surface poloidal magnetic fields, obtained by assuming that the spinning down of a star is due to its magnetic dipolar radiation [16], namely B pol 3 × 10 19 (PṖ) 1/2 G [2]. Thus, low-B observations rendered the issue of magnetars much more complex than just outbursts and high dipolar magnetic fields and strengthened the relevance of toroidal magnetic fields in stars.
Numerical simulations of ordinary twisted magnetic fields in magnetars suggest they should be poloidal field dominated, which is at odds with some inferences from observa-tions [see details in 17, 18, and references therein], such as the three above mentioned low-poloidal-B magnetars. This has been addressed in the context of a modified twisted magnetic field, where ratios of the toroidal energy to the total magnetic energy in stars could be up to 90% [18]. This has been done in the context of Maxwellian electromagnetism and toroidal fields as high as 10 15 G could be obtained. However, for such large magnetic fields, nonlinearities of the electromagnetism should play a significant role in the observational properties of the systems bearing them. Motivations for nonlinear electrodynamics in the context of high fields naturally come from QED [19] but in this paper we focus on an older approach, due to Born and Infeld [20], interesting even nowadays due to the reasons that follow.
Born-Infeld's electromagnetism was conceived with the purpose of healing the energy singularities of point-like charged particles such as the electron and is motivated by the finiteness of physical observables in special relativity [20]. It was showed in the 1980s that it is consistent with the low-energy limit of string theory and this has, since then, drawn very much attention to it [see 21, and references therein]. It already has an exact black hole solution [see for instance 22, and references therein] though in this context singularities are unavoidable [23]. When applied to the hydrogen atom, Born-Infeld theory as the description for electromagnetic interactions meets the observational spectrum of this atom only if its scale factor b is larger than the one inferred by Born and Infeld themselves within the unitarian viewpoint [24][25][26], approximately 10 15 statvolt cm −1 (or also 10 15 G) [20] (around 100 times QED's critical fields [19]). There are also constraints on b coming from particle accelerators. It has been recently shown that LHC light-by-light scattering in Pb-Pb collisions would restrict b to be larger than (100 GeV ) 2 ≈ 10 23 statvolt cm −1 ≈ 10 23 G [1eV 2 ≈ 14.4 G ≈ 14.4 statvolt cm −1 ] [27]. However, it is important to point out that the kinematic cuts made in ATLAS and the selfconsistency of the linear analysis used for cross section calculations are such that this experiment does not allow for the test of smaller values of b [27]. All the above means that the window 10 15 G b 10 23 G cannot be assessed by LHC experiments so far and hence they leave a "hole" in the probe of the b parameter. Given that magnetars are believed to have surface fields as high as 10 15 G (even larger fields in their interiors), they could be the ideal systems for assessing scale fields to the Born-Infeld theory in the region the LHC cannot. This is important and is a motivation our analysis because the Born-Infeld electrodynamics can only be disregarded when its whole space of parameters is excluded. Besides, magnetars could also be ideal testbeds for other nonlinear theories of the electromagnetism. Impressively enough, perhaps due to the intrinsic difficulties already present in the Maxwell theory, such analysis seems scarce in the literature of magnetars. We try to partially fill this gap in here.
We structure this work in the following way. Next section is devoted to the deduction of the field equations in nonlinear electromagnetism for the case of ideal magnetohydrodynamics. In Sec. 3.1 we work out a realization of nonlinear toroidal fields, by focusing on some consequences of it for the case of Born-Infeld electrodynamics. Some astrophysical consequences thereof are investigated in Sec. 4, especially related to the increase of the magnetic ellipticity and gravitational-wave energy budget when compared to their Maxwellian counterparts. Finally, discussions and conclusions are given in Sec. 5. We work with Gaussian and geometric units.
Nonlinear electrodynamics in ideal magnetohydrodynamics
For a Lagrangian density L depending upon both invariants of the electromagnetism, F . [28], the appropriate electromagnetic action is
= F µν F µν and G . = F µν * F µν , * F µν the dual of F µνS em = a d 4 x √ −gL(F, G) − d 4 x √ −g j µ A µ ,(1)
where a is a numerical factor that depends upon the system of units used, g is the determinant of the spacetime metric g µν (assumed in this work to be given), j µ is the current four-vector of the system and A µ is the four-potential of the electromagnetic fields. We work with Gaussian units, which means we take a = −1/16π. By assuming that F µν = A µ;ν − A ν;µ = A µ,ν − A ν,µ and varying Eq. (1) with respect to A µ , we obtain the field equations
∂ µ ( √ −gL F F µν + √ −gL G * F µν ) = 4π √ −g j ν or (L F F µν + L G * F µν ) ;µ = 4π j ν ,(2)
complemented with (a natural consequence of the definition of F µν ) [28] ∂ µ (
√ −g * F µν ) = 0 or ( * F µν ) ;µ = 0 (3) where L X . = ∂ L/∂ X, X = (F, G), * F µν . = η µναβ F αβ /(2 √ −g), η 0123 .
= +1, is a totally antisymmetric tensor. Note that since the Lagrangian must be an even function of the invariant G (due to symmetry requirements from electrodynamics), L G is an odd function of G, which means it is zero when the fields are orthogonal.
In order to simplify our description and evidence the physical picture involved in our analysis, we assume now the case where the electromagnetic fields are orthogonal and take the spacetime metric to be the Minkowski metric. Crossed fields naturally arise in the context of ideal magnetohydrodynamics (MHD) [see e.g., [16,29]], which we will consider throughout this work, and the use of a flat spacetime metric, though being an idealized case, will allow us to find the maximal changes of physical quantities regarding Maxwell's electrodynamics. 2 As commented previously, L G = 0 for the case under interest and Eqs. (2) and (3) can be cast as (we restore Gaussian units here)
∇ · (L F E) = 4πρ,(4)∇ · B = 0,(5)∇ × E = − 1 c ∂ B ∂t(6)
and
∇ × (L F B) = 1 c ∂ L F E ∂t + 4π c j,(7)
where we have defined ρ as the system's charge density and j its current vector.
Since ρ and j are given aspects of a system, one could take as reference Maxwell's electromagnetism. There, 4πρ = ∇ · E Ma and 4π
c j = ∇ × (B Ma ) − 1 c ∂ E Ma
∂t , which allows us to recast Eqs. (4) and (7) as
∇ · (L F E − E Ma ) = 0(8)
and
∇ × (L F B − B Ma ) = 1 c ∂ ∂t (∂ L F E − E Ma ).(9)
Care must be taken at this point due to Eqs. (5) and (6), which are formally identical to their Maxwellian counterparts but are related to the fields E and B, instead of B Ma and E Ma . This means, for instance, that a dipolar magnetic field in Maxwell's electromagnetism cannot in general have the same functional form in nonlinear electrodynamics. Nonetheless, for a very good conductive region of a rigidly rotating star with an angular frequency ω in the regime of small velocities v/c 1 (v . = ||v||) , one has that (ideal MHD [16,29])
E = − v c × B = − ω × r c × B,(10)
which comes from the assumption that Ohm's law also holds for nonlinear electrodynamics, taken in this work as a first approach, and the consideration the current density is finite in the limit of infinite conductivity. 3 If one assumes that the magnetic field lines are dragged along the motion of the star (frozen), then it follows that (Alfvén's theorem) [29]
∂ B ∂t = ∇ × (v × B) = ∇ × (ω × r × B).(11)
However, in nonlinear electrodynamics this point might be subtler in principle. If one inserts E = j/σ − v × B/c (from Ohm's law in first order in v/c; see footnote) into Eq. (6), takes σ to be constant, assumes that ∂ L F E/∂t is negligible when compared to the current density (which could be justified due to the fact that in a given limit the equations of nonlinear electrodynamics tend to the Maxwell ones where that holds true for large σ [32]), one has that
∂ B ∂t = ∇ × (v × B) − c 2 4πσ ∇ × ∇ × (L F B).(12)
One sees from the above equation that Eq. (11) is recovered when σ → ∞ (ideal MHD) and that Eq. (10) arises in this limit. The above means that the magnetic flux through any closed loop in a perfectly conducting medium is constant also in nonlinear electrodynamics if the conditions leading to Eq. (12) hold. We leave analyses of other scenarios (for instance finite σ , violation of Ohm's law, etc.) to be investigated elsewhere. From Eq. (8), one has
L F E = E Ma + ∇ × C,(13)
where C is an arbitrary vector. If one assumes that the time derivative of C is negligible [which should be justified by the assumption of large conductivity and would be equivalent to the disregard of the time derivative term of Eq. (7) w.r.t. j, as happens in Maxwell's ideal magnetohydrodynamics [29]], then it follows from Eq. (9) that
L F B = B Ma + ∇ f ,(14)
where f is also an arbitrary function. One can solve Eq. (14) for B (given a nonlinear electrodynamics), which, after replaced in Eq. (5), will fix the function f . 3 Let us elaborate on this point. If one assumes the validity of Ohm's law also in nonlinear electrodynamics, then in a locally comoving reference frame (or rest-frame) K (with respect to a conducting fluid), one has that j = σ E , where j and E are the current density and the (nonlinear) electric field there, respectively, and σ is the conductivity of the fluid. Take now the limit σ → ∞ (ideal conductor). If one assumes that j is finite (could be any), then, necessarily, E = 0. Make use now of another inertial coordinate system K (usually called the laboratory frame) such that K moves with respect to it with velocity v. When v/c 1, it follows that E = E + v × B/c [28]. (Due to the smallness of the charge density in a conductor after a small characteristic time [32], from the transformation laws of four-vectors in the limit of v/c 1, it also follows that j = j.) Thus, in the limit of ideal MHD, Eq. (10) follows as a kinematic constraint, and hence would be valid for any theory of the electromagnetism. From Eqs. (13), (14) and (10), it immediately follows that
E Ma + ∇ × C = − ω × r c × (B Ma + ∇ f ),(15)
which from the Maxwellian relationship of the fields
[E Ma = −(ω × r) × B Ma /c] implies ∇ × C = − ω × r c × ∇ f .(16)
Thus, any non-null f induces a non-null C and this is done through the assumption of the system's large conductivity, exactly as suggested by previous intuitive arguments. We stress that Eqs. (13), (14) and (16) are simple just because they relate fields in different theories. Without the solutions for B Ma and ∇ f , which are not easy in general, they are just indicative. The electromagnetic energy density in nonlinear electrodynamics can be easily obtained with the mixed 00 component of the electromagnetic energy momentum tensor, which to the case of Eq. (1) for orthogonal fields is given by [33] 16πT
ν µ = −4L F F µα F γβ g νγ g αβ + L δ ν µ ,(17)
where δ ν µ is the Kronecker delta function [28]. When the hypothesis of very conductive media is taken into account (|E| |B|), Eq. (17) implies that
16πT 0 0 ≈ L.(18)
Nonlinear toroidal fields
In order to have some insights into the influence of nonlinear electrodynamics on magnetars and motivated by the expected dominance of azimuthal fields inside such systems [18], here we focus on the nonlinear field equations for purely toroidal fields with axial symmetry (thought of as a rough approximation to the real scenario), such that B t = B φ (r, θ )φ , whereφ is the azimuthal unit vector. This field is such that Eq. (5) is automatically satisfied. From Eq. (10), the case of toroidal fields in rigidly rotating stars (we define the z-axis such that ω = ωẑ, which means v = vφ , v being any) result in E = 0, which automatically satisfies Eq. (6) [as well as Eq. (11)] and from Eqs. (13) and (16)
L F B φ = B Ma φ .(19)
Since in the small field regime of nonlinear electrodynamics L F = 1 − |something small| [34], it follows generically from the above equation that the magnitudes of nonlinear toroidal fields in ideal MHD are in general larger than their Maxwellian counterparts. Given that, we investigate an interesting case with an exact solution in the next subsection.
Born-Infeld toroidal field analysis
Born-Infeld's Lagrangian is defined as [see for instance 33,35]
L B.I . = 4b 2 1 + F 2b 2 − G 2 16b 4 − 1 ,(20)
where b is the scale field of the theory and from the definition of the invariants of the electromagnetism, F = 2(B 2 − E 2 ) and G 2 = 16(E · B) 2 . Thus, for orthogonal fields, our main interest in this work, the only relevant derivative to the Lagrangian is
L F = b √ b 2 + B 2 − E 2 ≈ b √ b 2 + B 2 ,(21)
since for very good conducting stars it follows that |E| |B|. To the case of Born-Infeld toroidal magnetic fields, Eqs. (19) and (21) lead to a very simple solution, namely,
B φ = bB Ma φ b 2 − (B Ma φ ) 2 ,(22)
which clearly evidences the increase nonlinear electrodynamics impinges on B φ . For toroidal-dominated fields, it follows from Eqs. (18), (20) and (22) that the energy densities in Born-Infeld and Maxwell's theories are related by
(T 0 0 ) BI (T 0 0 ) Ma = 2 b B Ma φ 2 1 − B Ma φ b 2 − 1 2 − 1 ,(23)
which is always larger than the unit for B Ma φ < b, needed for the consistency of Born-Infeld toroidal fields. This means that toroidal fields in nonlinear electrodynamics imply larger energy reservoirs when compared to Maxwell's predictions.
Ellipticities for Born-Infeld toroidal fields
Strong toroidal magnetic fields also have important implications for the ellipticity, as we discuss now. Assume that the magnetic ellipticity of a star is proportional to the mean value of B 2 φ , i.e.,
ε = c 1 B 2 φ ,(24)
where c 1 is a constant that depends upon aspects of the star such as its radius, compactness, etc. Indeed, when toroidal magnetic fields are dominant, one has that [36] ε
∝ − 1 E g dV B 2 φ ,(25)
where E g ∝ M 2 /R is the magnitude of the gravitational energy of the star and dV is a volume element. Thus, by defining B 2 φ . = dV B 2 φ /V , Eq. (24) ensues with c 1 depending on the compactness of the star and its radius, all byproducts of its microphysics.
From Eq. (22), it follows that the Born-Infeld to the Maxwellian ellipticity ratio is
ε BI ε Ma = Ė BI GẆ E Ma GW 1 2 ≈ ∆ E BI GW ∆ E Ma GW 1 2 ≈ 1 − B Ma φ b 2 −1 ,(26)
whereĖ GW is the energy loss due to gravitational waves. We have assumed that B Ma φ is slowly varying within the star (where it takes place), which might be taken for maximal estimates. Since B Ma φ < b from consistency of Eq. (22), one has from the above equation that the production of gravitational waves is larger in the Born-Infeld theory than in Maxwell's.
Assume now that c 1 is given, so as an upper limit to the norm of the magnetic ellipticity of a star and the Maxwellian toroidal magnetic field. (One could easily estimate B Ma φ by means of magnetar's observables such as flare luminosities and upper limits to the ellipticities could be inferred from gravitational-wave analysis.) In this case, from Eq. (24), it follows that Maxwell's ellipticity is known. If one takes the true theory of the electromagnetism as Born-Infeld's, thus
b ≥ B Ma φ 1 − 1 C − 1 2 , C . = |ε ul | |ε Ma | ,(27)
where ε ul stands for the observational upper limit of the magnetic ellipticity. From Eqs. (22) and (24) one sees that when the measured ellipticity is the one connected with the Born-Infeld theory, then it follows that C > 1 automatically. Besides, Eq. (27) is totally equivalent to the consistency of Eq. (22) since the minimum value of b is B Ma φ (1−1/C ) −1/2 , which is larger than B Ma φ . Therefore, Born-Infeld electrodynamics is self-consistent.
Possible Astrophysical Implications of Born-Infeld magnetars
LIGO measurements already constrain ellipticities in ordinary pulsars and their magnitude should be smaller than approximately 10 −6 [37]. The values of the ellipticities they found should be taken as upper limits since no continuous gravitational-wave signals have been detected from neutron stars. In what follows we do not assume the above value holds true for magnetars, but we rather find an upper limit to it by means of outcomes of Born-Infeld theory applied to the hydrogen atom. We will focus on the systems supposed to have dominant toroidal fields such that the analysis of the previous section could be taken into account. This is actually believed to be the case in all magnetars and even possibly in several pulsars [18].
In this context, mean toroidal Maxwellian fields could be estimated with giant flare events by dint of
(B Ma φ ) 2 = 6E f l R 3 ,(28)
where R is the radius of the magnetar and E f l is the total energy released in the giant flare. Precise |c 1 | calculations for the case of a homogeneous ellipsoid show that [36]
|c 1 | ≈ R 4 GM 2 ,(29)
where G is Newton's constant and M is the magnetar's mass. From Eq. (27) [or Eq. (26)], |ε ul | could be constrained if a minimum value for b, b min , was given, implying that
|ε ul | = 1 − B Ma φ b min 2 −1 |ε Ma |.(30)
Maximum values for B Ma φ are estimated by taking the maximum energy released during giant flares, around 10 47 erg to the magnetar SGR 1806-20 [2,38]. In this case, toroidal magnetic fields of order 7.7×10 14 G arise and from Eqs. (24) and (29), by making use of fiducial magnetar parameters [R = 10 km and M = 1.4 M ], it follows that |ε Ma | = 1.20× 10 −6 . Taking into account hydrogen experiment outcomes [26], one learns that the absolute minimum value for Born-Infeld's scale field is b min ≈ 3.96 × 10 15 statvolt cm −1 (or b min ≈ 3.96 × 10 15 G) [20] [most recent data for the mass and charge of the electron have been used]. Thus, from the above and Eqs. (30) and (26), we have that |ε ul | ≈ 1.24 × 10 −6 and ∆ E BI GW /∆ E Ma GW 1.08. Naturally, if larger values of E f l are the case (for instance associated with uncertainties in distance measurements or even newly-born magnetars), then larger ellipticities would emerge. If E f l = 10 47 − 10 48 erg, say E f l = 5 × 10 47 erg, which might be possible to giant flare events when they are associated with nonlinear electrodynamics [which increases the magnetic energy reservoir of highly magnetized stars, see Eq. (23)], or are associated with smaller than usual short GRBs 4 [typical isotropic energy of ordinary short-GRBs are in the range 10 49 −10 52 erg [40,43] which in turn would imply in |ε ul | ≈ 2 × 10 −6 . Thus, upper limits to the magnitude of magnetic ellipticities of magnetars would be around 10 −6 (10 −5 ) and the maximum increase of gravitational-wave energy released in Born-Infeld theory with respect to Maxwell's theory would be around 15% (even larger than 50%) for flare energy up to 10 47 (10 48 ) erg.
Discussions and conclusions
In order to explain magnetars' observational properties, it seems accepted today in the literature that very large resultant magnetic fields should take place in the vicinities of their surfaces and also interiors. Fields as high as 10 15 G should be present in order to function as energy reservoirs to the flare activities magnetars display. With such high fields, it seems reasonable to assume nonlinear electrodynamics could give a more precise description of magnetars and concomitantly they could be natural candidates for probing nonlinear electrodynamics astrophysically. Given that there are known magnetars with low-poloidal magnetic fields, toroidal fields should be particularly relevant for highly magnetized stars. Fields of order 10 15 G are a natural scale for magnetic fields in magnetars and this value is of the same order as the scale field in Born-Infeld's original theory. This was one of our motivations to our analysis. Axially symmetric toroidal nonlinear fields have a very simple solution in the context of magnetars' ideal magnetohydrodynamics because electric fields are not induced. Note this would be the case only where toroidal fields would exist and be much larger than poloidal ones, believed to be the case just in the interior of magnetars [18]. Besides, nonlinear electrodynamics leads to toroidal magnetic field increase when compared to their Maxwellian counterparts. This would imply the increase of the magnetic energy of a magnetar, exactly as we have showed, which could lead to more energetic flare activities. Besides, more magnetic energy could be converted into gravitational-wave energy, as also explicitly showed. The kinematic reason for so is the increase in magnitude of the Born-Infeld magnetic ellipticity when compared to its Maxwellian counterpart.
Hydrogen atom byproducts show that Born-Infeld's scale field should be larger than the one obtained by Born and Infeld themselves. Actually, such a value defines an absolutely lower limit to the scale field and has to be taken into account for physically relevant constraints. (For the upper limit on b which could not have been assessed by the ATLAS experiment, the changes introduced by the Born-Infeld Lagrangian are negligible when compared to the Maxwell theory.) When that is done, Born-Infeld theory predicts an upper limit to the magnetic ellipticity of a magnetar, which also lead to an upper limit to the gravitational-wave energy it may emit. Giant flares are important in this case because they result in the largest energy budgets of the system, which according to the magnetar theory should have exclusive magnetic origins. We have showed that the upper limit to the magnitude of the ellipticity should be within the range 10 −5 − 10 −6 for current observations of the magnetar SRG 1806-20. Born-Infeld's gravitational-wave energy for giant flare events of around 10 47 erg would be at most 10% − 20% larger than their classical counterparts, while for flare energy up to 10 48 erg the maximum percentage could be much higher than 50%. Naturally, the previous incredibly large Born-Infeld gravitational-wave energy increase should be seen only as an indicative value, given that b should be larger than b min . When LIGO/VIRGO are able to constrain magnetar ellipticities, we see from Eq. (27) that minimum values for the Born-Infeld's scale field could be inferred astrophysically. This seems very interesting since it would work as a possible cross-check to the already available constraints on b and be able to probe the region of parameters current experiments cannot.
It is argued that a fiducial upper limit to the norm of the ellipticity should be around 10 −6 when asymmetries supported by anisotropic stresses built up during the crystallization period of the crust are taken into account [44]. It is worth mentioning that Refs. [45,46] have predicted GW amplitudes for all known pulsars and, when an extremely optimistic case is considered, ellipticities should be at most 10 −5 (for PSR J1846-0258). Thus, since the predicted GW amplitudes are extremely small, observation times of thousands of years would be needed even for advanced detectors such as aLIGO and AdVirgo, and the planned Einstein Telescope might not be able to detect these pulsars. All of the above actually evidences the relevance of finding the maximum increase of gravitational-wave energy in Born-Infeld theory with respect to Maxwell's theory. The conclusion in the context of Born-Infeld theory is that magnetars with giant flare energy 10 47 erg would be the most promising candidates for gravitational-wave detections and by consequence potential tests of nonlinear electrodynamics.
In a realistic stellar model, magnetic fields should present both poloidal and toroidal components and should be dependent upon the azimuthal coordinate and time too (besides r and θ ). However, for slowly rotating highly magnetized nonlinear stars it seems reasonable to start with simplified stationary and axially symmetric models, as suggested by some MHD simulations [18]. Therefore, one could conceive models in the form B = B pol (r, θ )+B tor (r, θ ). Notwithstanding, in the context of nonlinear theories, finding B pol and B tor is expected to be even harder than in Maxwell's electrodynamics due to the natural coupling of these components. When toroidal fields are very large, though, one could uncouple the system of equations. Besides, it is also pending stability analysis for magnetic fields in the context of nonlinear electrodynamics and when Ohm's law does not hold. It is already known that Maxwell's theory in the context of ideal MHD leads to instabilities of purely poloidal or toroidal fields (see e.g., [47,48] and references therein). These analyses are interesting on their own and are left for future works.
Additionally, for describing more realistically nonlinear electrodynamics in magnetars and the effects thereof, curved geometries should also be taken into account. However, in a first approach, it seems justifiable to neglect them such that maximal field strength (and derived quantities) and a better understanding of the physics taking place in the above scenario could be obtained. This would allow one to find which aspects are more important to be focused on in more precise analyses. We have seen that magnetic ellipticities are clear candidates. Curved geometries may significantly affect them by changing the spatial distributions of magnetic fields. Besides, the presence of both poloidal and toroidal magnetic fields and their couplings might also cancel out one another effects on the magnetic ellipticity. Due to the connections of the above effects with magnetar GWs, we plan to investigate these issues elsewhere.
Summing up, we have showed that when ideal magnetohydrodynamics and nonlinear electrodynamics are taken into account in magnetars, axially symmetric toroidal fields should be larger than their Maxwellian counterparts. This implies larger magnetic energy and ellipticities, which could thus increase the emission of gravitational waves from nonlinear highly magnetized stars. Current constraints on the Born-Infeld theory, giant-flare energetics and magnetic fields in magnetars point to a maximum increase of 10% − 20% of the energy emitted in the form of gravitational waves by Born-Infeld magnetars when compared to Maxwellian ones. When larger giant-flare energy are taken into account, in principle plausible due to the nonlinearities of the electromagnetism which could increase the magnetic energy reservoirs of magnetars, much higher percentages may appear. Thus, the possibility may arise for also probing nonlinear electrodynamics with the advancement of gravitational-wave detectors.
Recently, though, the authors of Ref.[5] have discussed the possibility of some SGRs/AXPs being rotation-powered NSs and explored the entire range of NS parameters allowed by the conditions of stability of the star, not only fiducial parameters.
Indeed, when general relativistic corrections are taken into account, quantities such as the magnetic fields should decrease with respect to their flat spacetime counterparts by a function depending on the compactness factor[5,30,31]. We plan to investigate the case of nonlinear electrodynamics in curved geometries as due to magnetars in a forthcoming work.
The authors of Ref.[39] have estimated that an appreciable quantity of short GRBs might be related to extragalactic unstable magnetars. However, Ref.[40] has cast doubt on whether or not the recent subluminous short gamma-ray burst GRB 170817 A[41], associated with GW 170817[42], might be related to a magnetar flare.
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| []
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[
"NON-WANDERING FATOU COMPONENTS FOR STRONGLY ATTRACTING POLYNOMIAL SKEW PRODUCTS",
"NON-WANDERING FATOU COMPONENTS FOR STRONGLY ATTRACTING POLYNOMIAL SKEW PRODUCTS"
]
| [
"J I Zhuchao "
]
| []
| []
| We show a partial generalization of Sullivan's non-wandering domain theorem in complex dimension two. More precisely, we show the non-existence of wandering Fatou components for polynomial skew products of C 2 with an invariant attracting fiber, under the assumption that the multiplier λ is small. We actually show a stronger result, namely that every forward orbit of any vertical Fatou disk intersects a bulging Fatou component. | 10.1007/s12220-018-00127-6 | [
"https://arxiv.org/pdf/1802.05972v2.pdf"
]
| 119,608,617 | 1802.05972 | ce1e34e5971213a845074dd47b81591350f0c361 |
NON-WANDERING FATOU COMPONENTS FOR STRONGLY ATTRACTING POLYNOMIAL SKEW PRODUCTS
12 Dec 2018
J I Zhuchao
NON-WANDERING FATOU COMPONENTS FOR STRONGLY ATTRACTING POLYNOMIAL SKEW PRODUCTS
12 Dec 2018
We show a partial generalization of Sullivan's non-wandering domain theorem in complex dimension two. More precisely, we show the non-existence of wandering Fatou components for polynomial skew products of C 2 with an invariant attracting fiber, under the assumption that the multiplier λ is small. We actually show a stronger result, namely that every forward orbit of any vertical Fatou disk intersects a bulging Fatou component.
Introduction
Complex dynamics, also known as Fatou-Julia theory, is naturally subdivided according to these two terms. One is focused on the Julia set. This is the set where chaotic dynamics occurs. The other direction of investigation is concerned with the dynamically stable partthe Fatou set. In this paper we will concentrate on the Fatou theory.
In a general setting, let M be a complex manifold, and let f : M → M be a holomorphic self map. We consider f as a dynamical system, that is, we study the long-time behavior of the sequence of iterates {f n } n≥0 . The Fatou set F (f ) is classically defined as the largest open subset of M in which the sequence of iterates is normal. Its complement is the Julia set J(f ). A Fatou component is a connected component of F (f ).
In one-dimensional case, we study the dynamics of iterated holomorphic self map on a Riemann surface. The classical case of rational functions on Riemann sphere P 1 occupies an important place and produces a fruitful theory. The non-wandering domain theorem due to Sullivan [11] asserts that every Fatou component of a rational map is eventually periodic. This result is fundamental in the Fatou theory since it leads to a complete classification of the dynamics in the Fatou set: the orbit of any point in the Fatou set eventually lands in an attracting basin, a parabolic basin, or a rotation domain.
The same question arises in higher dimensions, i.e. to investigate the non-wandering domain theorem for higher dimensional holomorphic endomorphisms on P k . A good test class is that of polynomial skew products hence one-dimensional tools can be used.
A polynomial skew product is a map P : C 2 −→ C 2 of the form
P (t, z) = (g(t), f (t, z)),
where g is an one variable polynomial and f is a two variable polynomial. See Jonsson [4] for a systematic study of such polynomial skew products, see also Dujardin [3] for an application of polynomial skew products.
To investigate the Fatou set of P , let π 1 be the projection to the t-coordinate, i.e.
π 1 : C 2 → C, π 1 (t, z) = t.
We first notice that π 1 (F (P )) ⊂ F (g), pass to some iterates of P , we may assume that the points in F (g) will eventually land into an immediate basin or a Siegel disk (no Herman rings for polynomials), thus we only need to study the following semi-local case:
P = (g, f ) : ∆ × C → ∆ × C,
where g(0) = 0 which means the line {t = 0} is invariant and ∆ is the immediate basin or the Siegel disk of g. P is called an attracting, parabolic or elliptic local polynomial skew product when g ′ (0) is attracting, parabolic or elliptic respectively.
The first positive result is due to Lilov. Under the assumption that 0 ≤ |g ′ (0)| < 1, Koenigs' Theorem and Böttcher's Theorem tell us that the dynamical system is locally conjugated to P (t, z) = (λt, f (t, z)), when g ′ (0) = λ = 0, or P (t, z) = (t m , f (t, z)), m ≥ 2, when g ′ (0) = 0. In the first case the invariant fiber is called attracting and in the second case the invariant fiber is called super-attracting. Now f is no longer a polynomial, and f can be written as a polynomial in z, f (t, z) = a 0 (t) + a 1 (t)z + · · · + a d (t)z d , with coefficients a i (t) holomorphic in t in a neighborhood of 0, we further assume that a d (0) = 0 (and we make this assumption in the rest of the paper). In this case the dynamics in {t = 0} is given by the polynomial p(z) = f (0, z) and is very well understood. In his unpublished PhD thesis [5], Lilov first showed that every Fatou component of p in the super-attracting invariant fiber is contained in a two-dimensional Fatou component, which is called a bulging Fatou component. We will show that this bulging property of Fatou component of p also holds in attracting case.
Lilov's main result is the non-existence of wandering Fatou components for local polynomial skew products in the basin of a super-attracting invariant fiber. Since this is a local result, it can be stated as follows.
Theorem (Lilov). For a local polynomial skew product P with a super-attracting invariant fiber, P (t, z) = (t m , f (t, z)), m ≥ 2, every forward orbit of a vertical Fatou disk intersects a bulging Fatou component. This implies that every Fatou component iterate to a bulging Fatou component. In particular there are no wandering Fatou components. On the other hand, recently Astorg, Buff, Dujardin, Peters and Raissy [1] constructed a holomorphic endomorphism h : P 2 −→ P 2 , induced by a polynomial skew product P = (g(t), f (t, z)) : C 2 −→ C 2 with parabolic invariant fiber, processing a wandering Fatou component, thus the non-wandering domain theorem does not hold for general polynomial skew products.
At this stage it remains an interesting problem to investigate the existence of wandering Fatou components for local polynomial skew products with attracting but not super-attracting invariant fiber. As it is clear from Lilov's theorem, Lilov actually showed a stronger result, namely that every forward orbit of a vertical Fatou disk intersects a bulging Fatou component. Peters and Vivas showed in [8] that there is an attracting local polynomial skew product with a wandering vertical Fatou disk, which shows that Lilov's proof breaks down in the general attracting case. Note that this result does not answer the existence question of wandering Fatou components, but shows that the question is considerably more complicated than in the super-attracting case. On the other hand, by using a different strategy from Lilov's, Peters and Smit in [7] showed that the non-wandering domain theorem holds in the attracting case, under the assumption that the dynamics on the invariant fiber is sub-hyperbolic. The elliptic case was studied by Peters and Raissy in [6]. See also Raissy [9] for a survey of the history of the investigation of wandering domains for polynomial skew products.
In this paper we prove a non-wandering domain theorem in the attracting local polynomial skew product case without any assumption of the dynamics on the invariant fiber. Actually we show that Lilov's stronger result holds in the attracting case when the multiplier λ is small.
Theorem (Main Theorem). For a local polynomial skew product P with an attracting invariant fiber, We can also apply this local result to globally defined polynomial skew products, see Theorem 6.4 for the precise statement.
P (t, z) = (λt, f (t, z)), for any fixed f , there is a constant λ 0 = λ 0 (f ) > 0 such that if λ satisfies 0 < |λ| < λ 0 ,
The proof of the main theorem basically follows Lilov's strategy. The difficulty is that Lilov's argument highly depends on the super-attracting condition and breaks down in the attracting case by [8]. The main idea of this paper are to use and adapt an one-dimensional lemma due to Denker-Przytycki-Urbanski(the DPU Lemma for short) to our case. This will give estimates of the horizontal size of bulging Fatou components and of the size of forward images of a wandering vertical Fatou disk (these concepts will be explained later). We note that some results in our paper already appear in Lilov's thesis [5] (Theorem 3.4, Lemma 4.1, Lemma 5.1 and Lemma 5.2). Since his paper is not easily available, we choose to present the whole proof with all details. On the other hand we believe that the introduction of the DPU Lemma makes the argument conceptually simpler even in the super-attracting case.
The outline of the paper is as follows. In section 2 we start with some definitions, then we present the DPU Lemma and some corollaries. In section 3 we show that every Fatou component of p in the invariant fiber bulges, i.e. is contained in a two-dimensional bulging Fatou component. This result follows classical ideas from normal hyperbolicity theory.
In section 4 we give an estimate of the horizontal size of the bulging Fatou components by applying the one-dimensional DPU Lemma. Let z ∈ F (p) be a point in a Fatou component of the invariant fiber and denote by r(z) the supremum radius of a horizontal holomorphic disk (see Definition 2.1 for the precise definition) centered at z that is contained in the bulging Fatou component. We have the following key estimate. Theorem 4.3. If λ is chosen sufficiently small, then there are constants k > 0, l > 0, R > 0 such that for any point z ∈ F (p) ∩ {|z| < R},
r(z) ≥ k d(z, J(p)) l ,
where J(p) is the Julia set in the invariant fiber.
In section 5 we adapt the DPU Lemma to the attracting local polynomial skew product case, to show that the size of forward images of a wandering vertical Fatou disk shrinks slowly, which is also important in the proof of the main theorem.
Proposition 5.5. Let ∆ 0 ⊂ {t = t 0 } be a wandering vertical Fatou disk centered at x 0 = (t 0 , z 0 ). Let x n = (t n , z n ) = P n (x 0 ). Define a function ρ as follows: for a domain U ⊂ C, for every z ∈ U ⊂ C, define ρ(z, U ) = sup {r > 0| D(z, r) ⊂ U } .
Set ∆ n = P n (∆ 0 ) for every n ≥ 1 and let ρ n = ρ(z n , π 2 (∆ n )), here π 2 is the projection π 2 : (t, z) → z. If λ is chosen sufficiently small, we have lim n→∞ |λ| n ρ n = 0.
The proof of the main theorem is given in section 6. The main point are to combine Theorem 4.3 and Proposition 5.5 to show wandering vertical Fatou disk can not exist. We finish section 6 with some remarks around the main theorem. We also show how our main theorem can be applied to globally defined polynomial skew products in theorem 6.4.
Acknowledgements. I would like to thank Romain Dujardin for drawing my attention to the subject and for his invaluable help. The work is partially supported by ANR-LAMBDA, ANR-13-BS01-0002. I also would like to thank the referee for the nice suggestions on the structure of the paper. Recall that after a local coordinate change our map has the form P : ∆ × C → ∆ × C, here ∆ ⊂ C is a disk centered at 0, such that
P (t, z) = (λt, f (t, z)),
here f is a polynomial in z with coefficients a i (t) holomorphic in ∆, and a d (0) = 0, λ satisfies 0 < |λ| < 1.
Definition 2.1. • A horizontal holomorphic disk is a subset of the form {(t, z) ∈ ∆ × C | z = φ(t), |t| < δ}
where φ(t) is holomorphic in {|t| < δ} for some δ > 0. δ is called the size of the horizontal holomorphic disk.
• Let π 2 denote the projection to the z-axis, that is
π 2 : ∆ × C −→ C, (t, z) −→ z. A subset ∆ 0 lying in some {t = t 0 } is called a vertical disk if π 2 (∆ 0 )
is a disk in the complex plane. A vertical disk centered at x 0 with radius r is denoted by ∆(x 0 , r). ∆ 0 is called a vertical Fatou disk if the restriction of {P n } n≥0 to ∆ 0 is a normal family.
In the rest of the paper, for a disk on the complex plane centered at z with radius r, we denote it by D(z, r) to distinguish.
Remark 2.2. A vertical disk contained in a Fatou component of P is a vertical Fatou disk.
We define a positive real valued function r(z), which measures the horizontal size of the bulging Fatou components. Definition 2.3. For z ∈ C satisfying |z| < R and z lying in the Fatou set of p, we define r(z) to be the supremum of all positive real numbers r such that there exist a horizontal holomorphic disk passing through z with size 2r, contained in F (P ) ∩ {|z| ≤ R}.
2.2.
Denker-Przytycki-Urbanski's Lemma. In this subsection we introduce the work of Denker-Przytycki-Urbanski in [2], and give some corollaries. Denker, Przytycki and Urbanski consider rational maps on P 1 , and study the local dynamical behavior of some neighborhood of a critical point lying in Julia set. As a consequence they deduce an upper bound of the size of the pre-images of a ball centered at a point in Julia set.
In the following let f be a rational map on P 1 , denote by C(f ) the set of critical points of f lying in Julia set. Assume that #C(f ) = q. We begin with a definition,
k c (x) = − log d(x, c), if x = c ∞, if x = c. Define a function k(x) by k(x) = max c∈C(f ) k c (x).
Here the distance is relative to the spherical metric on P 1 .
Let x 0 be arbitrary and consider the forward orbit {x 0 , x 1 , · · · , x n , · · · }, where x n = f n (x 0 ). We let the function k(x) acts on this orbit and the following DPU Lemma gives an asymptotic description of the sum of k(x) on this orbit. Recall that q denotes the number of critical points lying in J.
Lemma 2.5 (Denker, Przytycki, Urbanski). There exist a constant Q > 0 such that for every x ∈ P 1 , and n ≥ 0, there exists a subset j 1 , · · · , j q ′ ⊂ {0, 1, · · · , n}, such that
n j=0 k(x j ) − q α=1 k(x jα ) ≤ Qn,
here q ′ ≤ q is an integer.
Lemma 2.5 implies that in a sense the orbit of a point can not come close to C(f ) very frequently. As a consequence Denker, Przytycki, Urbanski deduce an upper bound of the size of the pre-images of a ball centered at a point in J(f ). Corollary 2.6 (Denker, Przytycki, Urbanski). There exist s ≥ 1 and ρ > 0 such that for every x ∈ J(f ), for every ǫ > 0, n ≥ 0, and for every connected component V of f −n (B(x, ǫ)), one has diam V ≤ s n ǫ ρ . Corollary 2.7. Let f be a polynomial map in C. For fixed R > 0, there exist s ≥ 1 and ρ > 0 such that for any n ≥ 0 and any z ∈ C satisfying f n (z) ∈ {|z| < R}, we have
d(z, J(f )) ≤ s n d(f n (z), J(f )) ρ ,
where the diameter is relative to the Euclidean metric.
Proof. Since the Euclidean metric and the spherical metric are equivalent on a compact subset of C, by Corollary 2.6 for fixed R > 0, there exist s ≥ 1 and ρ > 0 such that for every z satisfying z ∈ J(f ), 0 < ǫ ≤ R, n ≥ 0, and for every connected
component V of f −n (D(z, ǫ)), one has diam V ≤ s n ǫ ρ .
For any z and n satisfy f n (z)
∈ {|z| < R}, let y ∈ J satisfy d(f n (z), J(f )) = d(y, f n (z)) = ǫ. For every connected component V of f −n (D(y, 2ǫ)), one has diam V ≤ s n 1 (2ǫ) ρ , so that d(z, J(f )) ≤ d(z, f −n (y)) ≤ diam V ≤ s n 1 2 ρ ǫ ρ .
Set s = 2 ρ s 1 and the proof is complete.
Remark 2.8. The existence of such a result is intuitive since the Julia set is expected to be repelling in some sense -however the presence of critical points on J makes it non-trivial.
Structure of bulging Fatou components
In this section we show that every Fatou component of p in the invariant fiber is actually contained in a Fatou component of P , which is called a bulging Fatou component, and in this case we call the Fatou component of p bulges. By Sullivan's theorem every Fatou component of p is pre-periodic, it is sufficient to show that every periodic Fatou component of p is contained in a Fatou component of P . There are three kinds of periodic Fatou components of p, i.e. attracting basin, parabolic basin and Siegel disk. For all these three kinds we study the structure of the associated bulging Fatou components.
We may iterate P many times to ensure that all periodic Fatou components of p are actually fixed, and all parabolic fixed points have multiplier equals to 1. In the following of this paper the metric referred to is the Euclidean metric.
3.1. Attracting basin case. In the attracting basin case, assume that we have an attracting basin B of p in the invariant fiber. Without loss generality we may assume 0 is the fixed point in B, so that (0, 0) becomes a fixed point of P , and p ′ (0) = λ ′ with |λ ′ | < 1. We have the following well-known theorem [10]. In our case we have
DP (0, 0) = λ 0 ∂f ∂t (0, 0) λ ′ ,
so that all the all eigenvalues of the derivative DP (0, 0) are less than 1 in absolute value. As a consequence B is contained in a two dimensional attracting basin of (0, 0), say U , so that B bulges.
Parabolic basin case.
In the parabolic basin case suppose 0 is a parabolic fixed point of p. Assume that p is locally conjugated to z → z + az s + O(z s+1 ) for some s ≥ 2, a = 0. We first prove that near the fixed point (0, 0), P is locally conjugated to
(t, z) → (λt, z + az s + O(z s+1 )).
where O(z s+1 ) means there are constant C such that the error term ≤ C|z| s+1 , for all (t, z) in a neighborhood of the origin. Then we prove in this coordinate every parabolic basin of p bulges.
Lemma 3.2. Assume (0, 0) is a fixed point of P , and |p ′ (0)| = 1, then there exist a stable manifold through the origin in the horizontal direction. More precisely, there is a holomorphic function z = φ(t) defined on a small disk {|t| < δ} such that
φ(0) = 0, and f (t, φ(t)) = φ(λt).
Proof. This is related to the two dimensional Poincaré's theorem. See [12] Theorem 3.1 for the proof.
We have the following theorem which is a special case of [12, §7.2]. Theorem 3.3. We assume that the local skew product is given by
P (t, z) = (λt, z + az s + O(z s+1 )),
then there exist a constant δ > 0 and s-1 pairwise disjoint simply connected open sets U j ⊂ {|t| < δ} × C, referred to as two dimensional attracting petals, with the following properties:
(1) P (U j ) ⊂ U j , points in U j converge to (0, 0) locally uniformly.
(2) For any point x 0 = (t 0 , z 0 ) such that P n (x 0 ) → (0, 0), there exist integer N and j such that for all n ≥ N either P n (x 0 ) ∈ U j or z n = 0.
(3) U j = {|t| < δ} × (U j ∩ {t = 0}).
Thus by Theorem 3.3, for fixed j, all the points x 0 whose orbit finally lands on U j form an open subset Ω j , which is contained in the Fatou set of P . It is obvious that every parabolic basin of p is contained in one of such Ω j , this implies all parabolic basins of p bulge.
3.3. Siegel disk case. In the Siegel disk case, we assume that 0 is a Siegel point with a Siegel disk D ⊂ {t = 0}. We are going to prove that D is contained in a two dimensional Fatou component.
Theorem 3.4. Assume that p is locally conjugated to z → e iθ z with θ an irrational multiple of π 2 , then there is a neighborhood Ω of D such that D ⊂ Ω ⊂ C 2 , and there exists a biholomorphic map ψ defined on Ω such that
ψ • P • ψ −1 (t, z) = (λt, e iθ z).
Proof. We may assume that p is conjugated to z → e iθ z, then by Lemma 3.2 there is a stable manifold z = φ(t). A change of variables z → z + φ(t) straightens the stable manifold so that P is conjugated to (t, z) → (λt, e iθ z + tg(t, z)),
where g(t, z) is a holomorphic function. By an abuse of notation we rename this map by P . Let U be a relatively compact neighborhood of D in C 2 . Set C = sup |g(t, z)| on U . Let δ be so small that Cδ 1−δ < dist(D, ∂U ), and then Ω = {|t| < δ} × D is an open subset of U . Let (t 0 , z 0 ) be an arbitrary initial point in Ω, and denote P n (t 0 , z 0 ) by (t n , z n ), then
||z n+1 | − |z n || ≤ |t n g(t n , z n )| ≤ C|λ| n δ.
Then we have
||z n | − |z 0 || ≤ Cδ 1 − δ + |z 0 | ≤ dist(z 0 , ∂U ) + |z 0 |,
so that (t n , z n ) still lies in U . Thus {P n } is a normal family on Ω, for the reason that P n (Ω) is uniformly bounded.
Thus we can select a sub-sequence {n j } for which the sequence
φ t 0 (z 0 ) = lim j→∞ e −in j θ f tn j • f tn j −1 • · · · • f t 0 (z 0 )
uniformly converges on compact subset of Ω. Thus φ t (z) is a holomorphic function on Ω, and we have
φ λt 0 • f t 0 (z 0 ) = e iθ φ t 0 (z 0 )
for every (t 0 , z 0 ) ∈ Ω. Thus if we let ψ(t, z) = (t, φ t (z)), since φ 0 (z) = z we can shrink Ω if necessary to make sure that ψ is invertible on Ω, and we have
ψ • P • ψ −1 (t, z) = (λt, e iθ z).
For every (t, z) ∈ Ω.
It is obvious that Ω is contained in the Fatou set of P . Since D ⊂ Ω, this implies that every Siegel disk of p bulges.
3.4. Wandering vertical Fatou disks. We finish section 3 with a definition. We note that "wandering" has special meaning in our definition. The definition of wandering vertical Fatou disk we made here is not equivalent to vertical Fatou disks containing wandering points. Proof. This is simply because for every x = (t, z) ∈ ∆, if P n (x) tends to (0, z 0 ) ∈ F (p) then eventually P n ((t, z)) lands in the bulging Fatou component that contains (0, z 0 ). This contradicts the fact x lying in a wandering Fatou disk.
Estimate of horizontal size of bulging Fatou components
In this section we deduce an estimate of the horizontal size of the bulging Fatou components, by applying the one-dimensional DPU Lemma.
In the following we choose R > 0 such that if (t, z) satisfies t ∈ ∆, |z| > R, then |f (t, z)| ≥ 2|z|. This follows that the line at infinity is super-attracting. Thus for any holomorphic function φ(t) defined on {|t| < r} such that |φ(t)| ≤ R, we have for all |t| < r,
(4.1) |φ(t) − φ(0)| ≤ 2R |t| r ,
this follows from the classical Schwarz Lemma.
We begin with a lemma.
Lemma 4.1. Let Crit(P ) = (t, z)| ∂f ∂z (t, z) = 0 , then there exist constants 0 < δ 1 < 1 and K > K 1 > 0 such that any connected component C k of Crit(P ) ∩ {|t| < δ 1 } intersects the line {t = 0}
in a unique point, say c k , and for any point x = (t, z) ∈ Crit(P ), say x ∈ C k , we have
(4.2) |z − c k | ≤ K 1 |t| 1 d 1 .
and
(4.3) |f (t, z) − p(c k )| ≤ K|t| 1 d 1 ,
where c k = C k ∩ {t = 0}, and d 1 is the maximal multiplicity of critical points of p.
Proof. Since Crit(P ) is an analytic variety, by Weirstrass preparation theorem we can let
δ 1 < 1 small enough so that Crit(P ) ∩ {|t| < δ 1 } = ∪ l k=1 C k where C k , 1 ≤ k ≤ l are local connected analytic sets, C k ∩ {t = 0} = {c k }. For each fixed component C intersect {t = 0}
at c, C is given by the zero set of a Weirstrass polynomial,
C = {(t, z) ∈ {|t| < δ 1 } × C, g(t, z) = 0} , where g(t, z) = (z − c) m + a m−1 (t)(z − c) m−1 + · · · + a 0 (t) is a Weirstrass polynomial, m ≤ d 1 is an integer, a i (t) are holomorphic functions in t satisfying |a i (t)| ≤ M |t| for some constant M > 0 .
We show that |z − c| ≤ mM |t|
|a m−1 (t 0 )(z 0 − c) m−1 + · · · + a 0 (t 0 )| ≤ mM a m−1 |t 0 |. (4.4) Thus we have |z 0 − c| m > |a m−1 (t 0 )(z 0 − c) m−1 + · · · + a 0 (t 0 )|, which contradicts to (t 0 , z 0 ) ∈ C . Setting K 1 = 2d 1 M we get (4.2).
Let Ω be a relatively compact open set that contains Crit(P ) ∩ {|t| < δ 1 }. Let
M ′ = max ∂f ∂z , ∂f ∂t : (t, z) ∈ Ω .
Then for (t, z) ∈ C k we have
|f (t, z) − p(c k )| ≤ M ′ |t| + M ′ |z − c k | ≤ M ′ (1 + K 1 )|t| 1 d 1 .
To get (4.3) we set K = 2 max {M ′ (1 + K 1 ), 2R}. Thus the proof is complete.
Remark 4.2. We note that K 1 and K are invariant under a local coordinate change of the form t → φ(t) with φ(0) = 0 and φ ′ (0) = 1. To see this let a i (t) be the coefficients of the Weirstrass polynomial, the coordinate change t → φ(t) with φ ′ (0) = 1 takes a i (t) become a i (φ(t)). We have |a i (φ(t)| ≤ 2M |t| by shrinking δ 1 (φ) if necessary , then we get (4.2) with the same constant K 1 (this is the reason for the constant 2 in definition of K 1 ). By shrinking δ 1 (φ) we see that Ω and R are invariant, and
max ∂f (φ(t), z) ∂z , ∂f (φ(t), z) ∂t : (t, z) ∈ Ω ≤ 2M ′ .
By the same reason we get (4.3) with the same constant K.
We are going to prove the following estimate of r(z) under the assumption that the multiplier λ is sufficiently small. We would like to give an outline of the proof of Theorem 4.3 first. Since there are only finitely many invariant Fatou components of p, and every Fatou component is pre-periodic to one of them, it is enough to prove Theorem 4.3 holds for z in the basin of an invariant Fatou component. To do this, we first fix an invariant Fatou component U , and we prove Theorem 4.3 holds for a subset W satisfying ∪ ∞ i=0 p −i (W )= the basin of U , this is the first step. In step 2, we use the following Pull Back Lemma to get the relation between r(z) and r(p(z)), together with the DPU Lemma we are able to give the estimate for the points in p −i (W ), for every i. We start with the Pull Back Lemma.
V = {z ∈ F (p), d(z, J(p)) < ǫ} ,
then for any z 0 ∈ F (p) ∩ {|z| < R} such that p(z 0 ) ∈ V , at least one of the following holds:
(4.5) r(z 0 ) ≥ α |λ| r(p(z 0 ))d(z 0 , C(p)) d 1 (d 1 +1) ; or (4.6) r(z 0 ) ≥ β d(z 0 , J(p)) d 1 (d 1 +1) .
Here α, β are positive constants only depending on p and the constant K from Lemma 4.1, and C(p) is the set of critical points lying in J(p).
Proof. Let Crit(p) be the set of critical points of p, We choose ǫ small such that p(z) ∈ V implies d(z, p(C(p))) = d(z, p(Crit(p)). Let φ be the associated holomorphic function with respect to p(z 0 ) with size r(p(z 0 )). We are going to show that the critical value set of P does not intersect the graph of φ when the domain of φ is small . Suppose x ′ = (t ′ , z ′ ) lies in Crit(P ) satisfying t ′ < r(z 0 ) and P (x ′ ) lying in the graph of φ. then by Lemma 4.1 the connected component of Crit(P ) containing x ′ intersects {t = 0} at a unique point c. Then we have
d(p(z 0 ), p(C(p)) ≤ |p(z 0 ) − p(c)| = |φ(0) − φ(λt ′ ) + f (t ′ , z ′ ) − p(c)| ≤ |φ(0) − φ(λt ′ )| + |f (t ′ , z ′ ) − p(c)| ≤ K |λt ′ | r(p(z 0 )) + K|t ′ | 1/d 1 .|t ′ | ≥ r(p(z 0 )) |λ| . (b) If |λt ′ | r(p(z 0 )) < 1, then |λt ′ | r(p(z 0 )) ≤ |λt ′ | 1/d 1 r(p(z 0 )) 1/d 1 ,
so that by (4.7) we have
d(p(z 0 ), p(C(p)) ≤ K |λt ′ | 1/d 1 r(p(z 0 )) 1/d 1 + K|t ′ | 1/d 1 .
For case (b), there are two subcases, (b1) If r(p(z 0 )) ≤ |λ|, then
d(p(z 0 ), p(C(p)) ≤ 2K |λt ′ | 1/d 1 r(p(z 0 )) 1/d 1 ,
by applying the fact that there is a constant c = c(p) > 0 such that d(p(z 0 ), p(C(p)) ≥ c d(z 0 , C(p)) d 1 +1 we have
|t ′ | ≥ α |λ| r(p(z 0 ))d(z 0 , C(p)) d 1 (d 1 +1) , where α = c 2K d 1 .
(b2) If r(p(z 0 )) > |λ|, then
d(p(z 0 ), p(C(p)) < 2K|t ′ | 1/d 1 .
Thus we have
|t ′ | > 1 2K d 1 d(p(z 0 ), J(p)) d 1 .
By applying the fact that there is a constant
c = c(p) > 0 such that d(p(z 0 ), J(p)) ≥ cd(z 0 , J(p) d 1 +1 , we get |t ′ | ≥ β d(z 0 , J(p)) d 1 (d 1 +1) . where β = c c 2K d 1 .
We can let α small enough such that actually αd(z 0 , C(p)) d 1 (d 1 +1) < 1, thus for case (b1) we have
|t ′ | ≥ α |λ| r(p(z 0 ))d(z 0 , C(p)) d 1 (d 1 +1) ≥ r(p(z 0 )) |λ| ,
thus case (a) is actually contained in case (b1). In either case (b1) or (b2) we get a lower bound on t ′ . Thus for any t which does not exceed that lower bound, φ(λt) is not a critical value of f t and so all branches of f −1 t are well defined and holomorphic in a neighborhood of the graph of φ. Therefore, choose g t to be the branch of f −1 t for which g 0 (f 0 (z)) = z, then the function ψ(t) = g t (φ(λt)) is well defined from t = 0 up to |t| < η satisfying ψ(0) = z 0 and the graph of ψ containing in the Fatou set, where η is the lower bound from (4.5) and (4.6). We know that ψ is also bounded by R, since otherwise φ would not be bounded by R. To avoid the case |t ′ | ≥ δ 1 , we can shrink β such that β d(z 0 , J(p)) d 1 (d 1 +1) < δ 1 for all z 0 . Thus |t ′ | ≥ δ 1 implies |t ′ | ≥ β d(z 0 , J(p)) d 1 (d 1 +1) . Thus at least one of (4.5) and (4.6) holds.
Proof of Theorem 4.3. In the following we fix an invariant Fatou component U of p, denote the basin of U by B (If B is the basin of infinity we let B be contained in {|z| < R} ). We can shrink ǫ to ensure that the set {z ∈ B, d(z, J(p)) < 2ǫ} is contained in {|z| < R}. In either case we first construct a subset W of B, satisfies the following conditions, (1) W eventually traps the forward orbit of any point in B.
(2) W contains the compact subset {z ∈ B, d(z, J(p)) ≥ ǫ}.
(3) Theorem 4.3 holds for z ∈ W .
Finally we use the Pull Back Lemma to prove Theorem 4.3 holds for z ∈ B.
Step 1: Construction of W . We split the argument in several cases.
• U is an immediate attracting basin. Let ω ⊂ U be a compact neighborhood of the attracting fixed point. We set W = {z ∈ B, d(z, J(p)) ≥ ǫ}∪ω, then W automatically satisfies (1) and (2). Since W is also compact and contained in F (P ), there is a lower bound a > 0 such that r(z) ≥ a for every z ∈ W . So there exist k > 0 such that r(z) ≥ k d(z, J(p)) for z ∈ W .
• U is the attracting basin of ∞. We set W = {z ∈ B, d(z, J(p)) ≥ ǫ}, then W automatically satisfies (1) and (2). There is a lower bound a > 0 such that r(z) ≥ a for every z ∈ W . So there exist k > 0 such that r(z) ≥ k d(z, J(p)).
• U is an immediate parabolic basin. Let Q be the associated attracting petal of Theorem 3.3. We set W = {z ∈ B, d(z, J(p)) ≥ ǫ} ∪ Q, then W automatically satisfies (1) and (2). By Theorem 3.3 there is a lower bound a > 0 such that r(z) ≥ a for every z ∈ P . Thus there is a lower bound b > 0 such that r(z) ≥ b for every z ∈ W . So there exist k > 0 such that r(z) ≥ k d(z, J(p)).
• U is a Siegel disk. We set W = U ∪{z ∈ B, d(z, J(p)) ≥ ǫ}, then W automatically satisfies (1) and (2). To prove (3), it is enough to prove (3) for z ∈ U .
Lemma 4.5. Let U be a Siegel disk, then there are constants k > 0, l > 0 such that for any point z ∈ U , r(z) ≥ k d(z, J(p)) l .
Further more l only depends on p.
Proof. Since the technique of the proof is similar to that of Theorem 4.3, we postpone the proof to the end of this subsection.
Step 2: Pull back argument.
We already have the estimate for z ∈ W . For every z 0 ∈ B\W , let {z i } i≥0 be its forward orbit, and let n be the smallest integer such that z n lies in W . Let m be the smallest integer such that case (4.5) does not happen, if this m dose not exist, let m = n, in either case we have r(z m ) ≥ k d(z m , J(p)) l , for some k, l > 0, and for all z i , 0 ≤ i ≤ m − 1, we have
(4.8) r(z i ) ≥ α |λ| r(z i+1 ))d(z i , C(p)) d 1 (d 1 +1) . By (4.8) we have log r(z i ) ≥ log r(z i+1 ) + log d(z i , C(p)) d 1 (d 1 +1) + log α |λ| = log r(z i+1 ) − d 1 (d 1 + 1)k(z i ) + log α |λ| , for all 0 ≤ i ≤ m − 1, where k(z i ) is as in Lemma 2.5. Thus we have log r(z 0 ) ≥ log r(z m ) − d 1 (d 1 + 1) m−1 i=0 k(z i ) + m log α |λ| .
By Lemma 2.5 there exist a subset i 1 , · · · , i q ′ ⊂ {0, 1, · · · , m − 1} such that
m−1 i=0 k(z i ) − q ′ α=1 k(z iα ) ≤ Qm.
Therefore we have
log r(z 0 ) ≥ log r(z m ) − d 1 (d 1 + 1) q ′ α=1 k(z iα ) − d 1 (d 1 + 1)Qm + m log α |λ| ≥ log r(z m ) + d 1 (d 1 + 1) q ′ α=1 log d(z iα , J(p)) − d 1 (d 1 + 1)Qm + m log α |λ| . (4.9)
By Corollary 2.7 we have for each i α ,
log d(z iα , J(p)) ≥ 1 ρ log d(z 0 , J(p)) − 1 ρ i α log s ≥ 1 ρ log d(z 0 , J(p)) − 1 ρ m log s. Likewise we have, log r(z m ) ≥ log k + l log d(z m , J(p)) ≥ log k + l ρ log d(z 0 , J(p)) − l ρ m log s.
Thus applying the estimates of log d(z iα , J(p)) and log r(z m ) to (4.9) gives
log r(z 0 ) ≥ log r(z m ) + d 1 (d 1 + 1) q ′ α=1 log d(z iα , J(p)) − d 1 (d 1 + 1)Qm + m log α |λ| ≥ log k + l + qd 1 (d 1 + 1) ρ log d(z 0 , J(p)) − l + qd 1 (d 1 + 1) ρ m log s − d 1 (d 1 + 1)Qm + m log α |λ| .
Let us now fix λ 1 so small such that
(4.10) log α λ 1 ≥ l + qd 1 (d 1 + 1) ρ log s + d 1 (d 1 + 1)Q,
then for every |λ| < λ 1 we have
log r(z 0 ) ≥ log k + l + qd 1 (d 1 + 1) ρ log d(z 0 , J(p)),
which is equivalent to
r(z 0 ) ≥ k d(z 0 , J(p)) l ′ , where l ′ = l+qd 1 (d 1 +1) ρ .
We have shown that there are constants k > 0, l ′ > 0 such that r(z) ≥ k d(z, J(p)) l ′ for z ∈ B, and l ′ only depends on p, this finishes the proof of Theorem 4.3.
Proof of Lemma 4.5 It is enough to prove the estimate for an invariant subset U ǫ ⊂ U \ {z ∈ B, d(z, J(p)) ≥ ǫ}. First note that the conclusion of Lemma 4.4 holds for all z 0 ∈ U ǫ , since for all z 0 ∈ U ǫ the condition p(z 0 ) ∈ V holds. Since U is a Siegel disk, the forward orbit {z n } n≥0 lies in a compact subset S of U , where z n = p n (z 0 ). Thus there is a lower bound a > 0 such that r(z) ≥ a for z ∈ S, a depending on S. By Lemma 4.4 there are two cases, (1) There is no such integer n that r(z n ) ≥ β d(z n , J(p)) (d 1 +1)d 1 , thus all z n satisfy r(z n ) ≥ α |λ| r(z n+1 )d(z n , C(p)) d 1 (d 1 +1) .
(2) There is an integer n such that r(z n ) ≥ β d(z n , J(p)) d 1 (d 1 +1) .
In case (1) for every i ≥ 0
log r(z i ) ≥ log r(z i+1 ) + log d(z i , C(p)) d 1 (d 1 +1) + log α |λ| = log r(z i+1 ) − d 1 (d 1 + 1)k(z i ) + log α |λ| .
Thus we have for every n ≥ 0,
log r(z 0 ) ≥ log r(z n ) − d 1 (d 1 + 1) n−1 i=0 k(z i ) + n log α |λ| ≥ log a + qd 1 (d 1 + 1) ρ log d(z 0 , J(p)) − qd 1 (d 1 + 1) ρ n log s − d 1 (d 1 + 1)Qn + n log α |λ| .
Let us now fix |λ 1 | so small such that
(4.11) log α λ 1 ≥ l + qd 1 (d 1 + 1) ρ log s + d 1 (d 1 + 1)Q + 1,
thus for every |λ| < λ 1 we have log r(z 0 ) ≥ log a + qd 1 (d 1 + 1) ρ log d(z 0 , J(p)) + n.
Let n → ∞ then r(z 0 ) can be arbitrary large, which is a contradiction, thus actually case (1) can not happen. For the case (2), the proof is same as the proof of Theorem 4.3, thus the proof is complete.
Estimate of size of forward images of vertical Fatou disks
In this section we adapt the DPU Lemma to the attracting local polynomial skew product case, to show that the size of forward images of a wandering vertical Fatou disk shrinks slowly. We begin with two classical lemmas. We follow Lilov's presentation.
Lemma 5.1. There exist c 0 > 0 depending only on p and δ 2 > 0 such that when |t 0 | < δ 2 , let ∆(x, r) ⊂ {t = t 0 } be an arbitrary vertical disk, then P (∆(x, r)) contains a disk ∆(P (x), r ′ ) ⊂ {t = λt 0 } of radius ≥ c 0 r d .
Proof. For fixed x = (t, z) satisfying |t| < δ 2 , z ∈ C, and for fixed r > 0, define a function
f t,z,r (w) = 1 rM t,z.r (f t (z − rw) − f t (z))
which is a polynomial defined on the closed unit disk D(0, 1). The positive number M t,z,r is defined by M t,z,r = sup
w∈π 2 (∆(x,r)) |f ′ t (w)|.
Let A be the finite dimensional normed space containing all polynomials with degree ≤ d on D(0, 1), equipped with the uniform norm. Since |f ′ t,z,r (w)| ≤ 1 on D (0, 1), the family f ′ t,z,r is bounded in A. Notice that f t,z,r (0) = 0, so that {f t,z,r } is also bounded in A. The closure of {f t,z,r } contains no constant map since the derivative of constant map vanishes. but sup D(0,1) |f ′ t,z,r (w)| = 1. Now suppose that there is a sequence {f tn,zn,rn } such that f tn,zn,rn (D(0, 1)) does not contains D(0, δ n ), with δ n → 0. We can take a sub-sequence f tn,zn,rn → g , where g is a nonconstant polynomial map with g(0) = 0. Therefore by open mapping Theorem g(D(0, 1 2 )) contains D(0, δ) for some δ > 0. Then for n large enough f tn,zn,rn (D(0, 1)) also contains D(0, δ), which is a contradiction. Therefore for all parameter {t, z, r}, f t,z,r (D(0, 1)) contain a ball D(0, δ), which is equivalent to say that (5.1) ∆(P (x), δrM t,z,r ) ⊂ P (∆(x, r)).
Next we estimate M t,z,r from below. Let z 1 (t), z 2 (t), ..., z d−1 (t) be all zeroes of f ′ t (z). Then
f ′ t (z) = da d (t)(z −z 1 (t)) · · · (z −z d−1 (t)). We choose δ 2 small such that c 0 = inf |t|≤δ 2 |da d (t)| > 0. Then we have M t,z,r = sup w∈π 2 (∆(x,r)) |f ′ t (w)| = sup w∈π 2 (∆(x,r)) |da d (t)(z − z 1 (t)) · · · (z − z d−1 (t))| ≥ c 0 r d−1 sin d−1 π 2 d − 1 ,
this with (5.1) finishes the proof.
Lemma 5.2. There exist 0 < c < c 0 , δ 2 > 0 such that if a vertical disk ∆(x, r) ⊂ {t = t 0 } satisfies ∆(x, r) ⊂ {|z| < R}, |t 0 | < δ 2 and η = d(∆(x, r), {t = t 0 } ∩ Crit(P )) > 0, then P (∆(z, r)) contains a disk ∆(P (x), r ′ ) ⊂ {t = λt 0 } of radius ≥ cη 2d−2 r.
Proof. Let V = {x 0 = (t 0 , z 0 ) : |t 0 | < δ 2 , |z 0 | < R, d(x 0 , {t = t 0 } ∩ Crit(P )) > η}, and set
M 1 = inf V ∂f ∂z > 0, M 2 = sup V ∂ 2 f ∂z 2 < ∞,
here M 1 depends on η but M 2 does not.
Thus for ∆(x 0 , r) ⊂ {t = t 0 } satisfying ∆(x 0 , r) ⊂ {|z| < R} and η = d(∆(x 0 , r), {t = t 0 } ∩ Crit(P )) > 0, we have ∆ = ∆(x 0 , r) ⊂ V ∩ {t = t 0 }.
Pick an arbitrary a in the interior of π 2 (∆). Then for all z ∈ ∂π 2 (∆), we let
h(z) = f t 0 (z) − f t 0 (a) = f ′ t 0 (z)(z − a) + 1 2 (z − a) 2 g(z).
We know g(z) satisfies |g(z)| ≤ M 2 , so that
|f ′ t 0 (z)(z − a)| ≥ M 1 |z − a| ≥ M 1 |z − a| 2 2r .
In the case r ≤ M 1 2M 2 we have
|f ′ t 0 (z)(z − a)| ≥ M 2 |z − a| 2 > 1 2 |(z − a) 2 g(z)|.
Thus by Rouché's Theorem the function h(z) has the same number of zero points as f ′ t 0 (z)(z − a), thus h(z) has exactly one zero point {z = a} . Since a ∈ π 2 (∆) is arbitrary we have f t 0 is injective on ∆. The classical Koebe's one-quarter Theorem shows that P (∆(x 0 , r)) contains a disk with radius at least
(5.2) 1 4 ∂f t 0 ∂z (z 0 ) r. Now we estimate ∂ft 0 ∂z (z 0 ) from below. Let z 1 (t), z 2 (t), ..., z d−1 (t) be all zeroes of f ′ t (z). Then f ′ t (z) = da d (t)(z − z 1 (t)) · · · (z − z d−1 (t)). We choose δ 2 such that c 0 = inf |t|≤δ 2 |da d (t)| > 0. We have for every 1 ≤ i ≤ d − 1, |z 0 − a i (t 0 )| ≥ η. Thus we have ∂f t 0 ∂z (z 0 ) = |da d (t 0 )(z 0 − z 1 (t 0 )) · · · (z 0 − z d−1 (t 0 ))| ≥ c 0 η d−1 ,
this with (5.2) gives
r ′ ≥ 1 4 c 0 η d−1 r.
In the case r ≥ M 1 2M 2 , by the same argument we have
(5.3) r ′ ≥ 1 4 c 0 η d−1 M 1 2M 2 ≥ 1 8M 2 c 2 0 η 2d−2 . Setting c = 1 2 min c 0 4R d−1 , 1 8RM 2 c 2 0
we get the conclusion. here D(z, r) is a disk centered at z with radius r.
Proposition 5.5. Let ∆ 0 ⊂ {t = t 0 } be a wandering vertical Fatou disk centered at x 0 = (t 0 , z 0 ). Let x n = (t n , z n ) = P n (x 0 ). Set ∆ n = P n (∆ 0 ) for every n ≥ 1 and let ρ n = ρ(z n , π 2 (∆ n )). There is a constant λ 2 (f ) such that for fixed |λ| < λ 2 , we have lim n→∞ |λ| n ρ n = 0.
Proof. Let λ 3 be a positive constant to be determined. It is sufficient to prove the result by replacing ∆ n by ∆ n ∩ ∆(x n , λ n+1
3 ). In the following we let ∆ n always be contained in ∆(x n , λ n+1 3 ). Without loss generality we can assume that |t 0 | < min {δ 1 , δ 2 , λ 3 }, where δ 1 is the constant in Lemma 4.1 and δ 2 is the constant in Lemma 5.1 and Lemma 5.2 . Let N be a fixed integer such that N > d q + 1, where q is the number of critical points lying in J(p). Let K = {|t| < min {δ 1 , δ 2 , λ 3 }} × {|z| < R} be a relatively compact subset of C 2 such that for (t, z) / ∈ K, |f (t, z)| ≥ 2|z|. Since the orbits of points in ∆ 0 cluster only on J(p),we have ∆ n ∈ K for every n . We need the following lemma:
Lemma 5.6. There is a constant M > 0 such that if |λ| < λ 3 , for every n and for every x ′ = (t n , w n ) ∈ ∆ n , for every integer m, letting (t n+m , w n+m ) = P m (x ′ ) we have,
(5.4) |w n+m − p m (z n )| ≤ M m λ n+1 3 .
Proof. We prove it by induction. Let M satisfying for (t, z) ∈ K, ∂f (t,z)
∂t ≤ M 2 and ∂f (t,z) ∂z ≤ M 2 .
We can also assume M is larger than the constant K in Lemma 4.1. Thus For m = 0 it is obviously true. Assume that when m = k − 1 is true, we have
|w n+k − p k (z n )| = |f (t n+k−1 , w n+k−1 ) − f (0, p k−1 (z n ))| ≤ M 2 |t n+k−1 | + M 2 |w n+k−1 − p k−1 (z n )| ≤ M 2 |λ| n+1 + M k 2 λ n+1 3 ≤ M k λ n+1 3 .
Thus for every m, (5.4) holds.
Remark that when w n = z n , the same argument gives
|z n+m − p m (z n )| ≤ M m |λ| n+1 .
Let C(P ) be the union of components of Crit(P ) such that meet C(p) = Crit(p) ∩ J(p) in the invariant fiber. For every point x ∈ ∆ n , we define k(x) = − log d(x, C(P ) ∩ {t = t n }), and k n = sup x∈∆n k(x). (This definition allows k n = +∞.) Recall that N is a fixed integer such that N > d q + 1. We are going to prove a two dimensional DPU Lemma for attracting polynomial skew products: Lemma 5.7 (Two Dimensional DPU Lemma). Let |λ| < λ 3 , then for every N k ≤ n < N k+1 , there is a subset α 1 , · · · , α q ′ ⊂ N k − 1, N k , · · · , n − 1 and a constant Q > 0 such that
(5.5) n−1 i=N k −1 k i − q ′ i=1 k α i ≤ Q(n − N k + 1),
here k is an arbitrary integer, q ′ ≤ q is an integer. Recall that q is the number of critical points lying in J(p).
Proof. Recall that the DPU Lemma implies that there is a subset α 1 , · · · , α q ′ ⊂ N k − 1, N k , · · · , n − 1 and a constant Q > 0 such that
(5.6) n−1 i=N k −1 k(p i−N k +1 (z N k −1 )) − q ′ j=1 k(p α j −N k +1 (z N k −1 )) ≤ Q 2 (n − N k + 1).
So it is sufficient to prove k i ≤ 2k(p i−N k +1 (z N k −1 )) for every i not appearing in α 1 , . . . , α q ′ . This is equivalent to
(5.7) d(∆ i , C(P ) ∩ {t = t i }) ≥ d(p i−N k +1 (z N k −1 ), C(p)) 2 .
To prove (5.7), assume that d(
p i−N k +1 (z N k −1 ), C(p)) = d(p i−N k +1 (z N k −1 ), c k ) for some point c k ∈ C(p),d(∆ i , C(P ) ∩ {t = t i }) ≥ d(p i−N k +1 (z N k −1 ), C(p)) − sup x ′ ∈∆ i |π 2 (x ′ ) − p i−N k +1 (z N k −1 )| − |w i − c k | ≥ d(p i−N k +1 (z N k −1 ), C(p)) − M i−N k +1 λ N k 3 − M |λ| N k d 1 , where w i is π 2 (C k ∩ {t = t i }). By |λ| < λ 3 we have M i−N k +1 λ N k 3 + M |λ| N k d 1 = M i−N k +1 + M λ N k d 1 3 . Thus we have d(∆ i , C(P ) ∩ {t = t i }) ≥ d(p i−N k +1 (z N k −1 ), C(p)) − M i−N k +1 + M λ N k d 1 3 .
To prove (5.7) it is sufficient to prove
(5.8) M i−N k +1 + M λ N k d 1 3 ≤ d(p i−N k +1 (z N k −1 ), C(p)) − d(p i−N k +1 (z N k −1 ), C(p)) 2 . By (5.6) we have d(p i−N k +1 z N k −1 , C(p)) ≥ e − Q 2 (n−N k +1) , thus it is sufficient to prove (5.9) (M n−N k +1 + M )λ N k d 1 3 ≤ e − Q 2 (n−N k +1) − e −Q(n−N k +1)
. We can always choose λ 3 sufficiently small to make (5.9) holds for all k ≥ 0. This ends the proof of the two dimensional DPU Lemma (5.5).
By Lemma 5.1 and Lemma 5.2 there is a constant c > 0 such that (5.10) ρ n+1 ≥ ce −(2d−2)kn ρ n . and (5.11) ρ n+1 ≥ cρ d n . From the above we can now give some estimates of ρ n . Recall that ρ n is assumed smaller than |λ 3 | n+1 otherwise we replace it by min ρ n , λ n+1 3 . Lemma 5.8. There is a constant c 1 > 0 such that for N k ≤ n < N k+1 , we have 1 , · · · , α q } we apply inequality (5.10), if i / ∈ α 1 , · · · , α q ′ we apply inequality (5.11). Thus we have
ρ n ≥ c N k 1 ρ d q N k −1 . Proof. For N k ≤ i ≤ n, if i − 1 ∈ {αρ n ≥ c n−α q ′ +1 exp −(2d − 2) n j=α q ′ +1 k j · · · c α 1 −N k +1 exp −(2d − 2) α 1 −1 j=N k −1 k j ρ N k −1 d · · · d ≥ c (n−N k +1)d q exp −d q n−1 j=N k −1 k i + d q q ′ j=1 k α j ρ d q N k −1 (because q ′ ≤ q) ≥ c (n−N k +1)d q exp −Qd q n − N k + 1 ρ d q N k −1 (by Lemma 5.7) ≥ c N k+1 d q exp −Qd q N k+1 ρ d q N k −1 . Setting c 1 = min c N d q e −QN d q , λ 3 we get the desired conclusion. Lemma 5.9. For N k ≤ n < N k+1 , ρ 0 ≤ λ 3 we have ρ n ≥ c N k+1 1 ρ d q(k+1) 0 .
Proof. By iterating Lemma 5.8 we get that
ρ N k −1 ≥ c N k −d qk N−d q 1 ρ d qk 0 ≥ c N k 1 ρ d qk 0 , so that ρ n ≥ c N k 1 ρ d q N k −1 ≥ c N k+1 1 ρ d q(k+1) 0
, this finishes the proof. Now we can conclude the proof of Proposition 5.5. For N k ≤ n < N k+1 we get
|λ| n ρ n ≤ |λ| N k c N k+1 1 ρ d q(k+1) 0 .
Choosing λ 2 small such that (5.12) λ 2 < c N 1 , since N > d q + 1 we deduce that for every |λ| < λ 0 ,
lim k→∞ |λ| N k c N k+1 1 ρ d q(k+1) 0 = 0,
finally lim n→∞ |λ| n ρn = 0, which finishes the proof.
Remark 5.10. The constant λ 2 appearing in Proposition 5.5 is invariant under a local coordinate change of the form t → φ(t) with φ(0) = 0 and φ ′ (0) = 1. To see this we know that by (5.9) λ 3 depends only on M and p, M can be dealt with by replacing it everywhere by 2M (see Remark 4.2), so that λ 3 is invariant. By putting c 1 = min c N d q e −QN d q , λ 3 we get that c 1 is invariant. Then by (5.12) we get that λ 2 is invariant. Proof. By Proposition 5.5 if |λ| < λ 2 , then lim n→∞ |λ| n ρn = 0 holds. For any l > 0, we then let |λ| smaller than λ l 2 to make the conclusion holds.
Proof of the non-wandering domain theorem
In this section we prove the non-existence of wandering Fatou components. Let us recall the statement Theorem 6.1 (No wandering Fatou components). Let P be a local polynomial skew product with an attracting invariant fiber,
P (t, z) = (λt, f (t, z)).
Then for any fixed f , there is a constant λ 0 (f ) > 0 such that if λ satisfies 0 < |λ| < λ 0 , every forward orbit of a vertical Fatou disk intersects a bulging Fatou component. In particular every Fatou component iterates to a bulging Fatou component, and there are no wandering Fatou components.
Proof. We argue by contradiction. Suppose ∆ 0 ⊂ {t = t 0 } is a vertical disk lying in a Fatou component which does not iterate to a bulging Fatou component. Without loss generality we may assume |t 0 | < min {1, δ 1 , δ 2 , λ 3 }. By Remark 2.2, ∆ 0 is a vertical Fatou disk. Let x 0 = (t 0 , z 0 ) ∈ ∆ 0 be the center of ∆ 0 and set x n = (t n , z n ) = P n (t 0 , z 0 ) and ∆ n = P n (∆ 0 ). We divide the proof into several steps, We set ρ n = ρ(z n , π 2 (∆ n )) as before and assume that ρ 0 ≤ λ 3 . Notice that ∆ 0 can not be contained in the basin of infinity, thus ∆ n is uniformly bounded. Let λ 0 < min λ 1 , λ l 2 , where λ 1 and λ 2 come from Theorem 4.3 and Proposition 5.5. In the course of the proof we will have to shrink λ 0 one more time.
• Step 1. By Remark 3.6, the orbits of points in ∆ 0 cluster only on J(p).
• Step 2. We show that there exist N 0 > 0 such that when n ≥ N 0 , the projection π 2 ∆(x n , ρn 4 ) intersects J(p). We determine N 0 in the following. Suppose π 2 ∆(x n , ρn 4 ) does not intersect J(p). Thus z n ∈ F (p) and Theorem 4.3 implies r(z n ) ≥ k d(z n , J(p)) l , then we have |t n | r(z n ) ≤ |t n | k d(z n , J(p)) l ≤ 4 l |t n | k ρ l n . By Corollary 5.11 we can let N 0 large enough so that for all n ≥ N 0 , 4 l |tn| k ρ l n < 1. From the definition of r(z n ) we get a horizontal holomorphic disk defined by φ(t), |t| < r(z n ) contained in the bulging Fatou components, with φ(0) = z n , and t n is in the domain of φ. Then we have
|φ(t n ) − z n | = |φ(t n ) − φ(0)| ≤ 2R |t n | r(z n ) ≤ 2R |t n | k d(z n , J(p)) l ≤ 2R 4 l |t n | k ρ l n .
Again by Corollary 5.11, we can let N 0 large enough that for all n ≥ N 0 , 2R 4 l |tn| k ρ l n < ρn 4 . Thus φ(t n ) ∈ ∆(x n , ρn 4 ) ⊂ ∆ n . Since φ(t n ) is contained in the bulging Fatou components that contains z n , this implies ∆ n intersects the bulging Fatou component so it can not be wandering. This contradiction shows that π 2 ∆(x n , ρn 4 ) intersects J(p). Let y n ∈ ∆ n satisfies π 2 (y n ) ∈ π 2 ∆(x n , ρn 4 ) ∩ J(p), then for all x ∈ ∆(y n , ρn 4 ) we have ρ(π 2 (x), π 2 (∆ n )) ≥ ρn 2 . • Step 3. We show that there is an integer N 1 > N 0 such that for every x ∈ ∆(y N 1 , ρ N 1 4 ), for every m ≥ 0, p m (π 2 (x)) ∈ π 2 (∆ m+N 1 ), here π 2 (y N 1 ) ∈ π 2 ∆(x N 1 ,
ρ N 1 4 ) ∩ J(p)
. This means that the orbit of π 2 (x) is always shadowed by the orbit of ∆ N 1 , which will contradict the fact that π 2 (∆ m+N 1 ) intersects J(p). To show this, we inductively prove the more precise statement that for fixed N > d q + 1, there exist a large N 1 = N k 0 − 1 > N 0 , such that for every k ≥ k 0 , N k ≤ n < N k+1 , we have (6.1) p n−N 1 (π 2 (x)) ∈ π 2 (∆ n ) and
(6.2) ρ ′ n ≥ c N k+1 2 ρ d q(k+1) 0 ,
where ρ ′ n = ρ(p n−N 1 (π 2 (x)), π 2 (∆ n ), c 2 = c 1 2 comes from Lemma 5.8 and Lemma 5.9. We will determine k 0 in the following.
From Lemma 5.9 we know that (6.1) and (6.2) hold for n = N 1 . Assume that for some k ≥ k 0 , for all n ≤ N k − 1, (6.1) and (6.2) holds. Then for N k ≤ n < N k+1 , let y = P n−N k +1 (t N k −1 , p N k −1−N 1 (π 2 (x))), by Lemma 5.8 we have
(6.3) ρ(y, ∆ n ) ≥ c N k 1 ρ ′ N k −1 d q .
To estimate the distance between π 2 (y) and p n−N 1 (π 2 (x)), by Lemma 5.6 we have |π 2 (y) − p n−N 1 (π 2 (x))| ≤ M n−N k +1 |λ| N k .
ρ ′ n ≥ c N k 1 ρ ′ N k −1 d q − M n−N k +1 |λ| N k ≥ c N k 1 (c N k 2 ρ d qk 0 ) d q − M n−N k +1 |λ| N k (By the induction hypothesis (6.2)) ≥ c N k 1 c N k d q 2 ρ d q(k+1) 0 − M n−N k +1 |λ| N k .
By the choice c 2 = c 1 2 we have
ρ ′ n ≥ 2c N k+1 2 ρ d q(k+1) 0 − M n−N k +1 |λ| N k .
To get (6.2) it is sufficient to prove
c N k+1 2 ρ d q(K+1) 0 ≥ M n−N k +1 |λ| N k .
We take λ 0 sufficiently small such that
(6.5) λ 0 ≤ ( c 2 M ) 2N .
Thus to prove (6.2) it is sufficient to prove that when |λ| < λ 0 , (6.6) ρ d q(k+1) 0 ≥ |λ| N k 2 . Since N > d q + 1, we can choose k 0 large enough such that for every k > k 0 (6.6) holds. This finishes the induction.
This shows that (6.1) and (6.2) are true for all n ≥ N 1 .
• Step 4. Since for every x ∈ ∆(y N 1 , ρ N 1 4 ), for every m ≥ 0, p m (π 2 (x)) ∈ π 2 (∆ m+N 1 ), and ∆ n is uniformly bounded, the family {p m } m≥0 restricts on D(π 2 (y N 1 ), ρ N 1 4 )) is a normal family. Thus π 2 (y N 1 ) belongs to the Fatou set F (p), this contradicts to π 2 (y N 1 ) ∈ J(p). Thus the proof is complete.
Remark 6.2. The constant λ 0 appearing in Theorem 6.1 is invariant under a local coordinate change of the form t → φ(t) with φ(0) = 0 and φ ′ (0) = 1. To see this we know that the constants c 2 = c 1 2 , M and N are invariant under a local coordinate change of the form t → φ(t) with φ(0) = 0 and φ ′ (0) = 1 (M can be dealt with by replacing it everywhere by 2M , see Remark 4.2). Then by (6.5) λ 0 only depends on c 2 , M and N , thus λ 0 is invariant. Remark 6.3. Lilov's Theorem can be seen as a consequence of Theorem 6.1. In fact, for the super-attracting case, the Fatou components of p bulge for a similar reason. Since when |t| is very small, the contraction to the invariant fiber is stronger than any geometric contraction t → λt, Theorem 4.3 and Proposition 5.5 follows easily. Thus following the argument of Theorem 6.1 gives the result.
In the following theorem we show how the main theorem can be applied to globally defined polynomial skew products.
Theorem 6.4. Let P (t, z) = (g(t), f (t, z)) : C 2 → C 2 be a globally defined polynomial skew product, where g, f are polynomials. Assume deg f = d and the coefficient of the term z d of f is non-vanishing, then there exist a constant λ 0 (t 0 , f ) > 0 depending only on f and t 0 such that if g(t 0 ) = t 0 and |g ′ (t 0 )| < λ 0 then there are no wandering Fatou components in B(t 0 ) × C, where B(t 0 ) is the attracting basin of g at t 0 in the t-coordinate.
Proof. First by a coordinate change φ 0 : t → t + t 0 , P is conjugated to P 0 : (t, z) → (g 0 (t), f 0 (t, z)),
where g 0 (t) = g(t + t 0 ) − t 0 , and f 0 (t, z) = f (t + t 0 , z). It is clear that {t = 0} becomes an invariant fiber.
By Koenig's Theorem we can introduce a local coordinate change φ : t → φ(t) with φ(0) = 0 and φ ′ (0) = 1 such that P 0 is locally conjugated to (6.7) (t, z) → (λt, f 0 (φ(t), z)),
where λ = g ′ (t 0 ). We have seen in Remark 6.2 that the constant λ 0 (f ) is invariant under a local coordinate change of the form t → φ(t) with φ(0) = 0 and φ ′ (0) = 1. This means that for fixed f , for every such φ, P φ : (t, z) → (λt, f (φ(t), z)) has no wandering Fatou components when |λ| < λ 0 (f ). Thus applying this to (6.7) when |λ| = |g ′ (t 0 )| < λ 0 (f 0 ) we get the local skew product (t, z) → (λt, f 0 (φ(t), z)) has no wandering Fatou components. Thus by conjugation P has no wandering Fatou components in a neighborhood of {t = t 0 } , thus actually P has no wandering Fatou components in B(t 0 ) × C, where B(t 0 ) is the attracting basin of g at t 0 in the t-coordinate.
See
Definition 2.1 for the definition of the vertical Fatou disk.
Horizontal holomorphic disk and vertical Fatou disk. In this subsection we make the precise definitions appearing in the statement of Theorem 4.3 and Proposition 5.5.
Definition 2 . 4 .
24For a critical point c ∈ C(f ), define a positive valued function k c (x) by
Theorem 3 . 1 .
31If P : Ω → Ω is a holomorphic self map, where Ω is an open set of C 2 and (0, 0) ∈ Ω is a fixed point. If all eigenvalues of the derivative DP (0, 0) are less than 1 in absolute value then P has an open attracting basin at the origin.
Definition 3.5. A vertical Fatou disk ∆ is called wandering if the forward images of ∆ do not intersect any bulging Fatou component.
Remark 3 . 6 .
36The forward orbit of a wandering vertical Fatou disk clusters only on J(p).
We argue by contradiction. Suppose there exist a point (t 0 , z 0 ) ∈ C such that |z 0 then we have |z 0 − c| m = a m |t 0 |, and
Theorem 4. 3 .
3There exist a constant λ 1 = λ 1 (f ) > 0 such that for fixed |λ| < λ 1 , there are constants k > 0, l > 0 such that for any point z ∈ F (p) ∩ {|z| < R}, r(z) ≥ k d(z, J(p)) l , here J(p) is the Julia set of p in the invariant fiber. Furthermore l depends only on p.
Lemma 4. 4 (
4Pull Back lemma). There exist a constant 0 < ǫ < 1, such that if we let
holds by applying Lemma 4.1 and inequality (4.1). Now there are two possibilities, (a) If |λt ′ | r(p(z 0 )) ≥ 1, then
Remark 4. 6 .
6The constant λ 1 appearing in Theorem 4.3 is invariant under local coordinate change t → φ(t) with φ(0) = 0 and φ ′ (0) = 1. To see this from Lemma 4.4 and Remark 4.2 we know that α is invariant since it only depends on p and K. By (4.10) and (4.11) λ 1 only depends on α and p, hence λ 1 (f ) is invariant.
Remark 5. 3 .
3We note that c is invariant under a local coordinate change of the form t → φ(t) with φ(0) = 0 and φ ′ (0) = 1. To see this, we know c 0 and R are invariant under a local coordinate change of the form t → φ(t) with φ(0) = 0 and φ ′ (0) = 1, and by shrinking δ 2 get that c is invariant. Now we show that the size of forward images of a wandering vertical Fatou disk shrinks slowly. We begin with a definition.Definition 5.4. Define the inradius ρ as follows: for a domain U ⊂ C, for every z ∈ U ⊂ C, define ρ(z, U ) = sup {r > 0| D(z, r) ⊂ U } ,
let C k be the component of C(P ) which meats c k at invariant fiber, by (5.4) and Lemma 4.1 we have
every Fatou component iterates to a bulging Fatou component. In particular there are no wandering Fatou components.
A two-dimensional polynomial mapping with a wandering Fatou component. Matthieu Astorg, Xavier Buff, Romain Dujardin, Han Peters, Jasmin Raissy, Annals of mathematics. 1841Matthieu Astorg, Xavier Buff, Romain Dujardin, Han Peters, and Jasmin Raissy. A two-dimensional polynomial mapping with a wandering Fatou component. Annals of mathematics, 184(1):263-313, 2016.
On the transfer operator for rational functions on the Riemann sphere. Ergodic Theory and Dynamical Systems. Manfred Denker, Feliks Przytycki, Mariusz Urbański, 16Manfred Denker, Feliks Przytycki, and Mariusz Urbański. On the transfer operator for rational functions on the Riemann sphere. Ergodic Theory and Dynamical Systems, 16(02):255-266, 1996.
A non-laminar dynamical green current. Romain Dujardin, Mathematische Annalen. 3651-2Romain Dujardin. A non-laminar dynamical green current. Mathematische Annalen, 365(1-2):77-91, 2016.
Dynamics of polynomial skew products on C 2. Mattias Jonsson, Mathematische Annalen. 3143Mattias Jonsson. Dynamics of polynomial skew products on C 2 . Mathematische Annalen, 314(3):403-447, 1999.
Fatou theory in two dimensions. Krastio Lilov, University of MichiganPhD ThesisKrastio Lilov. Fatou theory in two dimensions. University of Michigan PhD Thesis, 2004.
Fatou components of elliptic polynomial skew products. Ergodic Theory and Dynamical Systems. Han Peters, Jasmin Raissy, Han Peters and Jasmin Raissy. Fatou components of elliptic polynomial skew products. Ergodic Theory and Dynamical Systems, pages 1-13, 2017.
Fatou components of attracting skew-products. The Journal of Geometric Analysis. Han Peters, Iris Marjan Smit, Han Peters and Iris Marjan Smit. Fatou components of attracting skew-products. The Journal of Geo- metric Analysis, pages 1-27, 2017.
Polynomial skew-products with wandering Fatou-disks. Han Peters, Liz Raquel Vivas, Mathematische Zeitschrift. 2831-2Han Peters and Liz Raquel Vivas. Polynomial skew-products with wandering Fatou-disks. Mathematische Zeitschrift, 283(1-2):349-366, 2016.
Polynomial skew-products in dimension 2: Bulging and wandering fatou components. Bollettino dell'Unione Matematica Italiana. Jasmin Raissy, 10Jasmin Raissy. Polynomial skew-products in dimension 2: Bulging and wandering fatou components. Bollettino dell'Unione Matematica Italiana, 10(3):441-450, 2017.
Holomorphic maps from C n to C n. Jean- , Pierre Rosay, Walter Rudin, Transactions of the American Mathematical Society. 3101Jean-Pierre Rosay and Walter Rudin. Holomorphic maps from C n to C n . Transactions of the American Mathematical Society, 310(1):47-86, 1988.
Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou-Julia problem on wandering domains. Dennis Sullivan, Annals of mathematics. 1222Dennis Sullivan. Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou-Julia problem on wandering domains. Annals of mathematics, 122(2):401-418, 1985.
Local structure of analytic transformations of two complex variables. Tetsuo Ueda, I. Journal of Mathematics of Kyoto University. 262Tetsuo Ueda. Local structure of analytic transformations of two complex variables, I. Journal of Mathe- matics of Kyoto University, 26(2):233-261, 1986.
Statistique et Modélisation (LPSM, UMR 8001. Sorbonne Universités, Laboratoire de Probabilités4 place Jussieu, 75252 Paris Cedex 05, France E-mail address: [email protected] Universités, Laboratoire de Probabilités, Statistique et Modélisation (LPSM, UMR 8001), 4 place Jussieu, 75252 Paris Cedex 05, France E-mail address: [email protected]
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"Topological phase transiton of anisotropic XY model with Dzyaloshinskii-Moriya interaction",
"Topological phase transiton of anisotropic XY model with Dzyaloshinskii-Moriya interaction",
"Topological phase transiton of anisotropic XY model with Dzyaloshinskii-Moriya interaction",
"Topological phase transiton of anisotropic XY model with Dzyaloshinskii-Moriya interaction"
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| [
"T Farajollahpour \nDepartment of Physics\nSharif University of Technology\n11155-9161TehranIran\n",
"S A Jafari \nDepartment of Physics\nSharif University of Technology\n11155-9161TehranIran\n\nCenter of excellence for Complex Systems and Condensed Matter (CSCM)\nSharif University of Technology\n1458889694TehranIran\n",
"T Farajollahpour \nDepartment of Physics\nSharif University of Technology\n11155-9161TehranIran\n",
"S A Jafari \nDepartment of Physics\nSharif University of Technology\n11155-9161TehranIran\n\nCenter of excellence for Complex Systems and Condensed Matter (CSCM)\nSharif University of Technology\n1458889694TehranIran\n"
]
| [
"Department of Physics\nSharif University of Technology\n11155-9161TehranIran",
"Department of Physics\nSharif University of Technology\n11155-9161TehranIran",
"Center of excellence for Complex Systems and Condensed Matter (CSCM)\nSharif University of Technology\n1458889694TehranIran",
"Department of Physics\nSharif University of Technology\n11155-9161TehranIran",
"Department of Physics\nSharif University of Technology\n11155-9161TehranIran",
"Center of excellence for Complex Systems and Condensed Matter (CSCM)\nSharif University of Technology\n1458889694TehranIran"
]
| []
| Within the real space renormalization group we obtain the phase portrait of the anisotropic quantum XY model on square lattice in presence of Dzyaloshinskii-Moriya (DM) interaction. The model is characterized by two parameters, λ corresponding to XY anisotropy, and D corresponding to the strength of DM interaction. The flow portrait of the model is governed by two global Ising-Kitaev attractors at (λ = ±1, D = 0) and a repeller line, λ = 0. Renormalization flow of concurrence suggests that the λ = 0 line corresponds to a topological phase transition. The gap starts at zero on this repeller line corresponding to super-fluid phase of underlying bosons; and flows towards a finite value at the Ising-Kitaev points. At these two fixed points the spin fields become purely classical, and hence the resulting Ising degeneracy can be interpreted as topological degeneracy of dual degrees of freedom. The state of affairs at the Ising-Kitaev fixed point is consistent with the picture of a p-wave pairing of strength λ of Jordan-Wigner fermions coupled with Chern-Simons gauge fields. | 10.1103/physrevb.98.085136 | [
"https://arxiv.org/pdf/1803.08665v1.pdf"
]
| 119,106,711 | 1803.08665 | 334ac23a59dfd6707591460d402d3fcb1e06995a |
Topological phase transiton of anisotropic XY model with Dzyaloshinskii-Moriya interaction
T Farajollahpour
Department of Physics
Sharif University of Technology
11155-9161TehranIran
S A Jafari
Department of Physics
Sharif University of Technology
11155-9161TehranIran
Center of excellence for Complex Systems and Condensed Matter (CSCM)
Sharif University of Technology
1458889694TehranIran
Topological phase transiton of anisotropic XY model with Dzyaloshinskii-Moriya interaction
numbers: 7510Jm0510Cc0367Mn7343Nq
Within the real space renormalization group we obtain the phase portrait of the anisotropic quantum XY model on square lattice in presence of Dzyaloshinskii-Moriya (DM) interaction. The model is characterized by two parameters, λ corresponding to XY anisotropy, and D corresponding to the strength of DM interaction. The flow portrait of the model is governed by two global Ising-Kitaev attractors at (λ = ±1, D = 0) and a repeller line, λ = 0. Renormalization flow of concurrence suggests that the λ = 0 line corresponds to a topological phase transition. The gap starts at zero on this repeller line corresponding to super-fluid phase of underlying bosons; and flows towards a finite value at the Ising-Kitaev points. At these two fixed points the spin fields become purely classical, and hence the resulting Ising degeneracy can be interpreted as topological degeneracy of dual degrees of freedom. The state of affairs at the Ising-Kitaev fixed point is consistent with the picture of a p-wave pairing of strength λ of Jordan-Wigner fermions coupled with Chern-Simons gauge fields.
I. INTRODUCTION
The two-dimensional classical (vector) XY model is a paradigm for the celebrated Berezinskii-Kosterlitz-Thouless (BKT) transition upon which the phase coherence of an underlying super-fluid is lost by the proliferation of topological excitations known as vortices [1][2][3] . Quantum version of this model was initially proposed by Matsubara and Matsuda as a lattice model to understand the liquid helium 4 . Since then there has been tremendous studies of the the two-dimensional quantum XY (2DQXY) model. Berezinskii used the term "anisotropic planar magnetic substances" to refer to the quantum XY model 5 . The isotropic limit of the XY model refers to the situation where σ x σ x and σ y σ y couplings have equal strength. This is the isotropic limit of the XY model. Oitma and Betts found that the ground state of this model has finite transverse magnetization 6 . The exact diagonalization study of Tang on the anti-ferromagnetic XY model found isotropic staggered magnetization in the XY plane 7 . Drzewinsky and Sznajd used a block-spin renormalization group at finite temperatures to find a BKT transition temperature in this system 8 . The BKT transition for the 2DQXY was confirmed in quantum Monte Carlo studies [9][10][11][12] . The critical exponents extracted from the quantum Monte Carlo study of Ding and coworkers suggested that 2DQXY belongs to the same universality class as the classical (vector) XY model 10 .
An equivalent way of thinking about 2DQXY model is in terms of hardcore bosons 13 . This bosonic language is particularly convenient for the study of super-fluid transition measured by super-fluid density, ρ s , which in the spin language corresponds to spin-stiffness 14 . In the bosonic language for a system with filling fraction n at the classical level the zero temperature super-fluid-density is given by ρ cl s ∝ n(1 − n). Quantum fluctuations enhance the above stiffness by few percent 15 . The emerging picture is that the zero temperature phase of the isotropic 2DQXY is that of a super-fluid. Indeed in a remarkable paper a much stronger version of this for all spins and for all dimensions higher than one was proven by Kennedy, Lieb and Shastry 16 .
Extensions of the isotropic 2DQXY model are also very interesting. Dekeyser and coworkers employed the quantum renormalization group method to suggest that extending the 2DQXY by an σ z σ z Ising term gives a very simple picture that the greater of Ising and XY exchange interaction dominates the low-energy phase 17 . Such an Ising exchange is equivalent to interaction among bosons. Placing this model on triangular lattice 18 sets a very interesting competition between the Mott localization, geometric frustration and super-fluidity of hardcore bosons where a diagonal solid order emerges at strong interactions 19 and remains stable for arbitrary large values of interaction 18 . This can be a possible explanation for the super-solid phase of helium 20 . Another possible extension is by plaquette interactions in presence of an external field where the four-site terms encourage valence bond solid 21 . Allowing for bond-disorder in the 2DQXY model enhances the amplitude of zero-point phase fluctuations giving rise to vanishing of the spin-stiffness which then turns the ground state into spin liquid 22 .
In addition to the above bosonic picture of the 2DQXY model and its extensions, there is also fermionic picture which is based on a Jordan-Wigner transformation. In this approach the spin system is mapped into a Chern-Simons (CS) theory coupled with spin-1/2 fermions 3,23,24 . This mapping is quite general and applies to larger family of spin systems than the 2DQXY on any bipartite lattice 25 . This approach is quite powerful, and is used to relate the 1/3 magnetization plateau of the regime of XY anisotropy to a bosonic fractional Laughlin state with filling fraction 1/2 26 .
In this paper we extend the anisotropic 2DQXY model by adding a Dzyaloshinskii-Moriya (DM) interaction of strength D. We consider a planar anisotropy λ that makes the exchange in x and y directions different. On top of that we add a DM interaction between the planar components of the spin. We employ the block-spin renormalization group (BSRG) to study the phase transitions of this model. We construct a phase portrait of the model from our BSRG equations. We find that there are two global attractors that attract the flow to gapped states which correspond to Ising phases polarized along x or y directions 7 . These two are separated by a gap-closing and hence should correspond to topologically non-trivial phases, similar to its one-dimensional counterpart 27 . We corroborate the topological nature of this quantum phase transition with the calculation of concurrence. The whole λ = 0 line in the plane of λ and D will be a gapless repeller which is unstable with respect to smallest anisotropy λ (irrespective of the sign of λ). This is reminiscent of the pairing instability in a gapless system of Jordan-Winger fermions 27 which from the exact solution of the one-dimensional problem can be interpreted as the p-wave pairing interaction. Indeed such a p-wave pairing resulting from the anisotropy λ can be obtained from the study of equivalent fermions coupled to Chern-Simons gauge fields on the honeycomb lattice 28 .
The paper is organized as follows. In Sec. II the XY model in the presence of DM interaction has been considered. The effective Hamiltonian of the system for the renormalized coupling constant and anisotropic parameters is obtained. In the Sec. III we present the details of the phase diagram. The discussions and results are presented in Sec. IV.
II. MODEL AND METHOD
The Hamiltonian of XY model on a 2D square lattice in the presence of DM interaction with N×N sites can be written as,
H(J, λ, D) = J N i=1 N j=1 [(1 + λ) σ x i,j σ x i+1,j + σ x i,j σ x i,j+1 + (1 − λ)(σ y i,j σ y i+1,j + σ y i,j σ y i,j+1 ) + D(σ x i,j σ y i+1,j − σ y i,j σ x i+1,j ) + D(σ x i,j σ y i,j+1 − σ y i,j σ x i,j+1 )](1)
where J > 0 is the exchange coupling, λ is anisotropy parameter, D is the DM interaction term and σ n i (n = x, y, z) are Pauli matrices at site i. The basic idea of block-spin renormalization method is to partition the lattice into clusters. Then if the cluster allows for a Kramers doublet ground states, the fluctuations between such doublet can be captured with an effective (coarse grained) spin variable 29,30 .
A. Block spin RG equations
To study the ground state phases of the above Hamiltonian, we partition the square lattice into blocks of five sites as depicted in Fig. 1. Out of the five sites in the cluster, four are from one sub-lattice and one is from the other sub-lattice. For interactions involving the Ising term of the form σ z σ z such a sub-lattice imbalance erroneously biases the ground state towards the wrong ground state total spin. This is due to Lieb-Mattis theorem for the Hubbard and Heisenberg family of models. However for XY family where the only conserved charge is ζ = j σ z j 27 where j runts over the whole lattice, this sub-lattice asymmetry does not destroy the doublet structure of the ground state and we still get a doublet of ground states each belonging to ζ = ±1 sectors. The conserved charge ζ already breaks the 2 5 dimensional Hilbert space into two sectors, each of dimension 16. States in each sector are in one-to-one correspondence in the above two sectors. These two sectors are mapped to each other by replacing the role of ↑ and ↓ spins. The clusters in Fig. 1 have further four-fold rotational symmetry. This allows to use standard methods of group theory 31 to further reduce the 16 dimensional space corresponding to a given ζ. The details of the straightforward but lengthily algebra is given in the appendix. The sector that contains the ground state is a 6 × 6 dimensional space which can be diagonalized to give the set of eigenvalues depicted in Fig. 2 in the parameter space of D, λ.
The ground state energy in both ζ = ±1 sectors is
e 0 = −2J 5(1 + D 2 ) + 5λ 2 + η,(2)
where
η = λ 4 + 34λ 2 (1 + D 2 ) + (1 + D 2 ) 2(3)
and the ground state eigen-vector in the ζ = +1 sector is,
|φ + =γ 1 | ↓↓↓↓↓ + γ 2 | ↑↑↑↑↓ + γ 3 (| ↑↑↑↓↑ + | ↑↑↓↑↑ + | ↑↓↑↑↑ + | ↓↑↑↑↑ ) + γ 4 (| ↑↓↓↓↑ + | ↓↓↓↑↑ + | ↓↑↓↓↑ + | ↓↓↑↓↑ ) + √ 2 2 (| ↑↓↓↑↓ + | ↑↑↓↓↓ + | ↓↓↑↑↓ + | ↓↑↑↓↓ + | ↑↓↑↓↓ + | ↓↑↓↑↓ ). (4)
The ground state in ζ = −1 sector is simply obtained by the spin-flip transformation of the above ground state, ↑↔↓.
|φ − =γ 1 | ↑↑↑↑↑ + γ 2 | ↓↓↓↓↑ + γ 3 (| ↓↓↓↑↓ + | ↓↓↑↓↓ + | ↓↑↓↓↓ + | ↑↓↓↓↓ ) + γ 4 (| ↓↑↑↑↓ + | ↑↑↑↓↓ + | ↑↓↑↑↓ + | ↑↑↓↑↓ ) + √ 2 2 (| ↓↑↑↓↑ + | ↓↓↑↑↑ + | ↑↑↓↓↑ + | ↑↓↓↑↑ + | ↓↑↓↑↑ + | ↑↓↑↓↑ ) (5)
where the coefficients are obtained as,
γ 1 = 6 √ 2λ(1 + iD) 5(1 + D 2 ) + λ 2 + η γ 2 = √ 2(1 + iD)(−1 + λ 2 − D 2 + η) λ(5(1 + D 2 ) + λ 2 + η) γ 3 = (−1 − 5λ 2 − D 2 + η) 5(1 + D 2 ) + 5λ 2 + η 4 √ 2λ(−1 − D 2 + λ 2 ) γ 4 = − 3i(−i + D) 5(1 + D 2 ) + 5λ 2 + η 5(1 + D 2 ) + λ 2 + η .(6)
As illustrated in Fig. 2 the presence of DM interaction will not generate any band crossing and the ground state remains stable with respect to change of anisotropy parameters and DM interactions. The relation between Hamiltonian (1) and the coarse-grained effective Hamiltonian is formally given as,
H eff = T † 0 HT 0 ,(7)
where the projection operator T 0 basically assigns a new coarse grained spins ⇑, ⇓ to the two degenerate (Kramers double) ground states |φ + and |φ − in the ζ = ±1 sectors:
T 0 = |φ + ⇑ | + |φ − ⇓ |.(8)
The Pauli matrix σ z j can not change the charge ζ = ±1 and therefore in the space composed of doublet of |φ ± , the action of each σ z j contributes to the formation of a coarse grained σ z for the whole cluster. Similarly each σ x(y) j flips one of the spins, thereby flipping the sign of ζ and can be interpreted as flipping the coarse grained spins | ⇑ and | ⇓ which can then be represented by σ x(y) in the space of coarse grained spins. In this process some coefficients from the ground state wave function will be collected. For the coupling of neighboring coarse-grained spins one needs to collect interactions on the bonds connecting a cluster to its neighbors 27 . The effective Hamiltonian in the n'th step of the above process will be
H(J n , λ n , D n ) = J n N/5 r=1 N/5 s=1 [(1 + λ n )(σ x r,s σ x r+1,s + σ x r,s σ x r,s+1 ) + (1 − λ n )(σ y r,s σ y r+1,s + σ y r,s σ y r,s+1 ) + D n (σ x r,s σ y r+1,s − σ y r,s σ x r+1,s ) + D n (σ x r,s σ y r,s+1 − σ y r,s σ x r,s+1 )],(9)
where the BSRG transformation connecting two consecutive steps becomes,
J n+1 = α 2 + ξ 2 + α 2 − ξ 2 λ n 2 J n λ n+1 = α 2 − ξ 2 + α 2 + ξ 2 λ n α 2 + ξ 2 + (α 2 − ξ 2 ) λ n D n+1 = 2αξ α 2 + ξ 2 + (α 2 − ξ 2 ) λ n D n .(10)
The coefficients are given by,
α = 1 N 2 [γ 1 γ * 3 + γ 3 γ * 1 + γ 2 γ * 4 + γ 4 γ * 2 3 √ 2 2 (γ 3 + γ * 3 + γ 4 + γ * 4 )] ξ = 1 N 2 [γ 1 γ * 3 + γ 3 γ * 1 − γ 2 γ * 4 − γ 4 γ * 2 3 √ 2 2 (−γ 3 − γ * 3 + γ 4 + γ * 4 )] with N = |γ 1 | 2 + |γ 2 | 2 + 4|γ 3 | 2 + 4|γ 4 | 2 + 3.
The γ * i is the conjugate of γ i and |γ i | 2 = γ i γ * i .
B. Entanglement analysis
Among various tools, entanglement is standard tool for the diagnosis of the phase transitions in general [32][33][34][35][36][37][38][39] and change of topology [40][41][42] in particular. Therefore let us use this tool in our five-site problem and to study its evolution under the RG transformations obtained above. Let us consider the entanglement between two spins in the corners of each blocks. For this purpose we first calculate the reduced density matrix between every two pairs of neighboring sites, namely, ρ 12 , ρ 13 , ρ 14 , ρ 23 , ρ 24 and ρ 34 which involves tracing out all the rest of degrees of freedom 43,44 . To construct the 4 × 4 matrix ρ ij (i, j are site indices, not matrix indices) one first constructs the full density matrix,
ρ = |ψ 0 ψ 0 |(11)
where the |ψ 0 can be any of the Kramers doublet degenerate ground states |φ ± and then traces all sites except for sites i, j. We then form a matrix ρ ijρij whereρ ij = σ y i ⊗ σ y j ρ * ij σ y i ⊗ σ y j and number its four eigenvalues such that λ ij,m , (m = 1, 2, 3, 4) such that λ ij,4 > λ ij,3 > λ ij,2 > λ ij,1 . From these eigenvalues we then evaluate the bipartite concurrence defined by 44 ,
C ij = max λ ij,4 − λ ij,3 − λ ij,2 − λ ij,1 , 0 .
Then we construct a geometric mean of the concurrence between the above 6 pairs of sites as 44 ,
C g = 6 C 12 × C 13 × C 14 × C 23 × C 24 × C 34 .(12)
In the present case, by rotational symmetry all of the six density matrices are equal and given by
ρ 12 = ρ 23 = ρ 34 =ρ 13 = ρ 14 = ρ 24 = = 1 N 2 γ 1 γ * 1 + 2γ 4 γ * 4 + 1 2 0 0 √ 2 2 (γ 1 + γ * 2 ) + 2γ 4 γ * 3 0 γ 3 γ * 3 + γ 4 γ * 4 + 1 γ 3 γ * 3 + γ 4 γ * 4 + 1 0 0 γ 3 γ * 3 + γ 4 γ * 4 + 1 γ 3 γ * 3 + γ 4 γ * 4 + 1 0 √ 2 2 (γ * 1 + γ 2 ) + 2γ * 4 γ 3 0 0 γ 2 γ * 2 + 2γ 3 γ * 3 + 1 2 (13)
The above matrix gives,
C g = Γ 2 2 + 2Γ 1 Γ 3 + Γ * 2 2 + 2 √ Λ Γ 2 √ 2 − Γ 2 2 + 2Γ 1 Γ 3 + Γ * 2 2 − 2 √ Λ Γ 2 √ 2 − 2 Γ 4(14)
where
Λ = Γ 2 2 + 4Γ 1 Γ 3 − 2Γ 2 Γ * 2 + Γ * 2 2 , Γ 1 = 1 N 2 γ 1 γ * 1 + 2γ 4 γ * 4 + 1 2 , Γ 2 = 1 N 2 √ 2 2 (γ 1 + γ * 2 ) + 2γ 4 γ * 3 , Γ 3 = 1 N 2 (γ 3 γ * 3 + γ 4 γ * 4 + 1) and Γ 4 = 1 N 2 γ 2 γ * 2 + 2γ 3 γ * 3 + 1 2 .
We will use the above formula in our analysis of the phase transitions of the 2DQXY with planar anisotropy λ and DM interaction D.
III. PHASE DIAGRAM OF THE MODEL
A. Analysis of the phase portrait
The standard method for analysis of the phase diagram of a model that depends on set of parameters R is to study ∆R n ≡ R n+1 − R n and its dependence to the initial values R 0 ≡ R 27, 45 . In our problem the parameters are given by R = (λ, D). In Fig. 3 we have presented two such cuts. In the first row, for two fixed values of D = 0 (left) and D = 4 (right) we plot how ∆λ depends on the initial value λ. As can be seen there are two fixed points. Repulsive fixed point at λ 0 * = 0 and two attractors at λ ± * = ±1 27 . These values do not change by replacing D = 0 with D = 4. In the second row of Fig. 3, for two fixed values of λ = 0 and λ = 1 we have plotted how ∆D depends on the initial value D. As can be seen, independent of value of λ, there is always an attractor at D * = 0: Slightly moving to the right (left) of D * = 0, gives a negative (positive) ∆D that returns D to the attractor D * = 0. Therefore the coupling D is irrelevant and any Hamiltonian of the form (1) with a non-zero D in the long wave-length limit behaves similar to the D * = 0 and the DM interaction is renormalized away in the infrared limit.
Let us put the above picture in a global perspective in a plane composed of λ and D. In Fig. 4 we have provided a stream plot of the vectors ∆R = (∆λ, ∆D) as a function of the initial value R = (λ, D). As can be seen the fact that in Fig. 3 the fixed point at λ ± * does not depend on D is reflected in Fig. 4 as the fact that the two attractors at (λ ± * , D * ) = (±1, 0) are globally attractive fixed points. However the fact that the repulsive fixed point λ 0 * in Fig. 3 does not depend on λ is reflected in Fig. 4 as a repulsive line. The symmetry of the above phase portrait under λ → −λ is the direct manifestation of the fact that Hamiltonian is invariant under σ x j → σ y j , σ y j → −σ x j (π/2 rotation around z axis), D → D and λ → −λ. The gap between the ground state and the first excited state is given by,
E C g =(15)2J 5(1 + D 2 ) + 5λ 2 + η − 5(1 + D 2 ) + 5λ 2 − η
The effect of RG flow on this quantity when it is iterated up to large enough RG steps to ensure machine precision convergence is plotted in Fig. 5 for various values of the DM interaction D indicated in the legend. In this figure we plot the gap at the 8-th RG step (converged within 10 −5 ). As can be seen for every value of D, the point λ 0 * = 0 is the only gapless point, and any non-zero value of λ, either positive or negative gives rise to a non-zero gap. The gap is normalized per lattice site, and the natural unit of the gap is J. The fact that for every value of D we have a non-zero gap for λ = 0 agrees with the existence of a line of fixed points λ = λ 0 * in the (λ, D) plane of Fig. 4. As can be seen in Fig. 5 although for all values of D the gap is a function of λ that vanishes at λ = 0, but the way it vanishes depends on D and is not universal. To extract these information, in Fig. 6 we produce a log-log plot of the gap versus λ for D = 0. Note that very small values of λ ∼ 10 −3 are needed to extract the dependence of gap on λ. The linear dependence of the log-log plot suggests a perfect power-law dependence of the gap, E g ∝ λ m , where the non-universal exponent m actually does depend on D. This is analogous to the behavior of the corresponding 1D system 46 where in the absence of DM term one has E g ≈ λ. The BSRG for threesite problem in 1D with D = 0 gives E g ∝ λ 0.63 . In 2D square lattice Fig. 6 suggests that this exponent for D = 0 is given as E g ≈ λ 0.4869 . Note that the value of the exponent (0.4869) has finite size errors.
By turning on the DM interaction D as can be seen in Fig. 5 still the gap vanishes as λ approaches zero. To quantify this, we repeat the above log-log analysis for various values of D, and extracting the corresponding exponent m as a function of D, we obtain the set of data points in Fig. 7. As can be seen Fig. 7 for larger D the exponent becomes smaller. Using the following ansatz for the fit,
m = exp(αD 2 + βD + γ),(16)
gives, α = −0.02042 ± 0.00106, β = −0.1828 ± 0.00523 and γ = −0.6732 ± 0.00499.
C. Analysis of the concurrence
So far we have established that for any D, the λ = 0 repulsive line is a gapless line. This is consistent with a picture of underlying phase coherent super-fluid, albeit not limited to D = 0, but also valid for nonzero values of D. The value of D only affects the exponent m that determines how fast the gap vanishes. Its repulsive nature indicates some form of instability towards a gapped state. Both positive and negative λ sides are gapped states. Is the gap closing at λ = 0 line a topological phase transition? In the λ = 0 (isotropic) XY model, the non-analytic value of GMC is suggested as and indicator of the spin fluid phase in the 2D system 47 . In Fig. 8 we have plotted the GMC versus anisotropy parameter λ for the D = 0 case. As can be seen by repeating the RG steps, the convergence can be attained very quickly, and the GMC at λ = 0 becomes non-analytic. This suggest that the gap closing at λ = 0 line is a topological phase transition 42 . A nice feature of the above plot is the vanishing of GMS at λ = ±1 which corresponds to Ising-Kitaev limit polarized alongx or y directions. For such a product state the entanglement must be zero.
To put the above picture in a global perspective, in Fig. 9 we plot intensity profile of GMC at first two steps of the RG process. This figure suggests that at the λ = 0 line the gapclosing is accompanied by a change of topology 42 .
IV. SUMMARY AND DISCUSSION
The phase portrait of anisotropy 2DQXY model with DM interaction in Fig. 4 indicates that the DM interaction is irrele- vant in the infrared limit. The λ = 0 line is a gapless line that separates two gapped states for positive and negative λ. The analysis of concurrence in Fig. 9 suggest that the gap-closing transition at λ = 0 is a topological phase transition 42 . In the bosonic language, the gapless state at λ = 0 corresponds to a super-fluid phase of underlying bosons 13,14 , and vanishing of the gap can be attributed to the soft phase fluctuations of a super-fluid 15 . There are two ways to destroy the long range order in the phase variable: The well known way is by the BKT mechanism, i.e. the proliferation of vortices at elevated temperatures.
The second way to gap the super-fluid state is to stay at zero temperature but turn on the anisotropy λ. According to present study, as long as anisotropy λ stays at zero, the DM interaction does not help with gapping the state. Having established that λ = 0 generates a gapped state for any D, the question is, what kind of gapped state is it? Is it topologically trivial or non-trivial? Fermionic representation of the problem in terms of Jordan-Wigner fermions coupled with the Chern-Simons gauge fields 3,23,24,26 suggests that the gapped state is a topological superconductor 28 . The super-fluid picture at λ = 0 (in the bosonic language) corresponds to a liquid of JW fermions coupled with CS gauge fields in the fermionic picture. In the fermionic language, the anisotropy parameter λ triggers a superconducting pairing instability in the Fermi sear of JW fermions leading to a topologically non-trivial superconducting state of JW fermions 28 .
In our RG picture this can be understood as follows: Deep in the gapped phase, at the Ising-Kitaev fixed points, λ = +(−)1 the long distance behavior of the system is equivalent to a simple 2D Ising model polarized alongx (ŷ) direction. The ground state at these fixed points is factorizable and this explains why in Fig. 8 the entanglement indicator at all RG steps gives zero. This means that at the Ising-Kitaev fixed point the Hamiltonian is given in terms of entirely commuting variables, and hence it has become purely classical (hence zero entanglement). The fact that entanglement at every RG step (i.e. for every system size) in Fig. 8 is zero, already in- dicates that it has been protected by some sort of topology, and therefore the resulting Ising degeneracy can be interpreted in a dual picture as topological degeneracy 48 . At these fixed points the resulting classical 2D Ising model translates via celebrated Lieb-Schultz-Mattis mapping 49 to a one-dimensional p-wave superconductor in modern terms. This is nothing but the well known Kitaev model of a topological superconductor. Therefore the ground states at the fixed points λ = ±1 is entitled to a winding number. Now moving slightly away from these fixed points and deforming the Hamiltonian in such a way that it ultimately returns to the fixed points upon enlarging the length scale, the topological number does not change, as there is no gap-closing as long as one does not hit the λ = 0 repeller line. Therefore our real space RG is consistent with a non-trivial topological charge for the gapped states at λ = 0.
To summarize, we have considered the quantum XY model in 2D square lattice in the presence of DM interaction. The symmetry of problem allows us to obtain analytical expressions for the ground state doublet of this system which then enables us to set up a real space block spin RG. The DM interaction turns out to be irrelevant at long wave-lengths. The RG flow consists in a gapless repulsive λ = 0 line, and two attractive (λ = ±1, D = 0) points corresponding to Ising-Kitaev limit. Non-analyticity of concurrence shows that the phase transition at λ = 0 is of topological nature 42 . The Ising Kitaev-limit enables us to assign a topological charge to the gapped phases at λ = 0. These features are very similar to corresponding 1D system 27 In this appendix, details of the exact diagonalization for selected cluster in square lattice are presented. To reduce the dimension of ensuing matrix we employ group theory method.
To obtain the eigenvalues (Eq. 2) and eigen-states (Eq. 4 and 5) first we consider the possible states of spin-1/2 system 2 5 in cluster P . Each state of the cluster is in the following form,
|α i = |σ 4 , σ 3 , σ 2 , σ 1 , σ 0 (A1)
where i = 1 ... 32 and σ present the two possible values ↑↓ in Fig. 1
Now we proceed calculations by employing symmetry consideration to reduce 32 dimensional Hilbert space to smaller blocks in matrix representation. The + shape of cluster in Fig. 1. is invariant under rotations by π 2 which is denoted by C and then the rotation group is given by {C 0 , C 1 , C 2 , C 3 }. The C operates on the site labels as,
C = 1 → 2 2 → 3 3 → 4 4 → 1 (A3)
By successive operation of C on a one state for e.i. |α 3 , the following pattern is obtained,
|3 C − → |4 C − → |5 C − → |6 C − → |3 (A4)
which is the concise representation of
C 0 |3 = |3 , C 1 |3 = |4 , C 2 |3 = |5 , C 3 |3 = |6 ,(A5)
According to projection theorem in group theory we construct the symmetry adopted state in representation which is labeled by n from an arbitrary state |φ
|ψ (n) ∼ g gΓ n [g] |φ (A6)
where g interprets the member of group and Γ n [g] denotes the n-th irreducible representation for element g in the group. Our case is a rotation group and the irreducible representations of the cyclic group are tagged by means of three (angular momentum) n = 0, ± 1. These are presented by {ω 0 , ω n , ω 2n , ω 3n } where ω = exp(iπ/2). The Γ n (C p ) = ω pn is the well-set representation of above cyclic group. A symmetry adopted state build from e.i. |3 is,
C 0 ω 0 + C 1 ω n + C 2 ω 2n + C 3 ω 3n |3 (A7)
where by applying Eq. A5, the obtained state is as,
|3 + ω n |4 + ω 2n |5 + ω 3n |6 (A8)
with n is the angular momentum. By applying the same symmetry to every other states we obtain,
|7 C − → |11 C − → |14 C − → |16 C − → |7 |8 C − → |10 C − → |12 C − → |13 C − → |8 |18 C − → |19 C − → |20 C − → |21 C − → |18 |22 C − → |26 C − → |29 C − → |31 C − → |22 |23 C − → |25 C − → |27 C − → |28 C − → |23 |9 C − → |15 C − → |9 |24 C − → |30 C − → |24 |1 C − → |1 , |2 C − → |2 |17 C − → |17 , |32 C − → |32 (A9)
The normalized states are as, |φ 1 = |α 1 , |φ 2 = |α 2 , |φ 3 = 1 2 (|α 3 + |α 4 + |α 5 + |α 6 ) , |φ 4 = 1 2 (|α 7 + |α 11 + |α 14 + |α 16 ) , |φ 5 = 1 2 (|α 8 + |α 10 + |α 12 + |α 13 ) , |φ 6 = 1 √ 2 (|α 9 + |α 15 ) , |φ 7 = 1 2 (|α 18 + |α 19 + |α 20 + |α 21 ) , |φ 8 = 1 2 (|α 22 + |α 26 + |α 29 + |α 31 ) , |φ 9 = 1 2 (|α 23 + |α 25 + |α 27 + |α 28 ) , |φ 10 = 1 √ 2 (|α 24 + |α 30 ) , |φ 11 = |α 17 , |φ 12 = |α 32 .
(A10) The same approach will lead to normalized state at n = +1 sector. Due to the time reversal symmetry the n = −1 sector has identical spectrum. The n = +1 sector normalized states are, |χ 1 = 1 2 |α 3 + ω|α 4 + ω 2 |α 5 + ω 3 |α 6 , |χ 2 = 1 2 |α 7 + ω|α 11 + ω 2 |α 14 + ω 3 |α 16 , |χ 3 = 1 2 |α 8 + ω|α 10 + ω 2 |α 12 + ω 3 |α 13 , |χ 4 = 1 2 |α 18 + ω|α 19 + ω 2 |α 20 + ω 3 |α 21 , |χ 5 = 1 2 |α 22 + ω|α 26 + ω 2 |α 29 + ω 3 |α 31 , |χ 6 = 1 2 |α 23 + ω|α 25 + ω 2 |α 27 + ω 3 |α 28 ,
It should be noted that the other symmetry such as parity symmetry in the selected cluster is in the heart of the rotation symmetry. The other operator that we introduced is
ζ = i σ z i (A12)
which operates as a constant of motion. Consider one arbitrary state with arrangements of spins of up and down. The operation of XY Hamiltonian on a selected arrangements does not change the value of q. The reason is that in the presence of the two consecutive σ x or σ y operator the total number of spin flip is even. This operator acts on the 32 basis of cluster and breaks it in two family with ζ = +1 which consist of |α 1 , |α 7 , |α 8 , |α 9 |α 10 , |α 11 , |α 12 , |α 13 |α 14 , |α 15 , |α 16 , |α 17 |α 18 , |α 19 , |α 20 , |α 21 (A13) and ζ = −1 |α 2 , |α 3 , |α 4 , |α 5 |α 6 , |α 22 , |α 23 , |α 24 |α 25 , |α 26 , |α 27 , |α 28 |α 29 , |α 30 , |α 31 , |α 32 .
By considering all the symmetries and constant of motion, it is possible to diagonalize Hamiltonian analytically for obtain the ground state and energy bands. For e.i. in n = 0 sector the Hamiltonian of the system by considering above symmetries in ζ = 1 reduced to
H = 0 0 0 4Jλ 0 0 0 0 4J(1 + iD) 0 0 0 0 4J(1 − iD) 0 0 4Jλ 2 √ 2Jλ 4Jλ0 0 0 0 4J(1 + iD) 2 √ 2J(1 + iD) 0 0 4Jλ 4J(1 − iD) 0 0 0 0 2 √ 2Jλ 2 √ 2J(1 − iD) 0 0
where the eigenvalues of above matrix of Hamiltonian are e 0 = −2J 5(1 + D 2 ) + 5λ 2 + η, e 1 = −2J 5(1 + D 2 ) + 5λ 2 − η, e 2 = e 3 = 0, e 4 = 2J 5(1 + D 2 ) + 5λ 2 − η, e 5 = 2J 5(1 + D 2 ) + 5λ 2 + η,
in which η = λ 4 + 34λ 2 (1 + D 2 ) + (1 + D 2 ) 2 (A16)
FIG. 1 .
1(Color online) The selected cluster in a square lattice where the dashed lines shows the block-block interactions.
FIG. 2 .
2(Color online) (a) The bands plots of selected five-site cluster in terms of λ and J when D = 0. (b) The plots of bands in nonzero DM interaction at J = 1.
FIG. 3 .
3(Color online) Fixed points of the 2DQXY model with anisotropy (λ) and DM interaction (D). In top (bottom) row we have fixed D (λ) to study flow of λ (D). There are two attractors at λ = ±1 and a repulsive fixed point at λ = 0. The DM interaction has only one attractor at D = 0.
FIG. 4 .
4(Color online) Phase portrait of the 2DQXY model with anisotropy λ and DM interaction D. Two global attractors at (λ ± * , D * ) = (±1, 0) along with repeller at infinity and a repulsive line (λ 0 * , D) for every D, completely characterize the above RG flow profile.
B. Analysis of the gap So far our phase portraits in Figs. 3 and 4 indicate the irrelevance of D and a possible phase transition at λ = 0 line. Let us see how does this manifest itself in the spectral gap.
FIG. 5 .
5(Color online) Dependence of the gap on anisotropy λ for various values of the DM interaction D indicated in the legend.
FIG. 6 .
6(Color online) The power law behavior of gap in terms of anisotropy parameter λ = 0 for D = 0. Different colors correspond to various RG steps as indicated in the legend.
FIG. 7 .
7(Color online) Exponent m of Eg ∼ λ m which determines how does the gap vanish as a function of λ. This exponent variers with D. These values are extracted from 8th level RG step which converges within the precision of 10 −5 from
FIG. 8 .
8(Color online) GMC as a function of anisotropy parameter λ in the absence of DM interaction in different RG steps. The non-vanishing GMC indicates that the gap closing at λ = 0 is a topological phase transition.
FIG. 9 .
9(Color online) Intensity map of GMC in the (λ, D) plane for various RG steps: (a) 0-th RG step, (b) 1-st RG step and (c) 2-nd RG step. In the all values of DM interaction at the nontrivial point of λ = 0 the GMC shows non-analytic behavior.
and in agreement with results of studies based on JW fermions coupled with CS gauge fields 28 . ACKNOWLEDGMENT SAJ appreciates financial supports by Alexander von Humboldt fellowship for experienced researchers Appendix A: Details of exact diagonalization of selected cluster in square lattice
. The basis in this 32 dimensional Hilbert space are as (for brevity in representation of basis states we drop | ),|α 1 =↑↑↑↑↑, |α 2 =↑↑↑↑↓, |α 3 =↑↑↑↓↑, |α 4 =↑↑↓↑↑, |α 5 =↑↓↑↑↑, |α 6 =↓↑↑↑↑, |α 7 =↓↑↑↑↓, |α 8 =↓↑↑↓↓, |α 9 =↓↑↓↑↓,|α 10 =↓↓↑↑↓, |α 11 =↑↑↑↓↓, |α 12 =↑↑↓↓↑, |α 13 =↑↓↓↑↑, |α 14 =↑↓↑↑↓, |α 15 =↑↓↑↓↑, |α 16 =↑↑↓↑↓, |α 17 =↓↓↓↓↑, |α 18 =↓↓↓↑↓, |α 19 =↓↓↑↓↓, |α 20 =↓↑↓↓↓, |α 21 =↑↓↓↓↓, |α 22 =↑↓↓↓↑, |α 23 =↑↓↓↑↓, |α 24 =↑↓↑↓↓, |α 25 =↑↑↓↓↓, |α 26 =↓↓↓↑↑, |α 27 =↓↓↑↑↓, |α 28 =↓↑↑↓↓, |α 29 =↓↑↓↓↑, |α 30 =↓↑↓↑↓, |α 31 =↓↓↑↓↑, |α 32 =↓↓↓↓↓,
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| []
|
[
"De-projection of radio observations of axi-symmetric expanding circumstellar envelopes",
"De-projection of radio observations of axi-symmetric expanding circumstellar envelopes"
]
| [
"1⋆P T Nhung \nDepartment of Astrophysics\nAcademy of Science and Technology (VAST)\nUMR 8112\nNational Space Center (VNSC)\n18 Hoang Quoc Viet, Ha Noi, Viet Nam 2 LERMAVietnam, Vietnam\n\nCNRS and Observatoire de Paris\nPSL Research University\n61 av. de l'ObservatoireF-75014ParisFrance\n",
"D T Hoai \nDepartment of Astrophysics\nAcademy of Science and Technology (VAST)\nUMR 8112\nNational Space Center (VNSC)\n18 Hoang Quoc Viet, Ha Noi, Viet Nam 2 LERMAVietnam, Vietnam\n\nCNRS and Observatoire de Paris\nPSL Research University\n61 av. de l'ObservatoireF-75014ParisFrance\n",
"P Tuan-Anh \nDepartment of Astrophysics\nAcademy of Science and Technology (VAST)\nUMR 8112\nNational Space Center (VNSC)\n18 Hoang Quoc Viet, Ha Noi, Viet Nam 2 LERMAVietnam, Vietnam\n\nCNRS and Observatoire de Paris\nPSL Research University\n61 av. de l'ObservatoireF-75014ParisFrance\n",
"P Darriulat \nDepartment of Astrophysics\nAcademy of Science and Technology (VAST)\nUMR 8112\nNational Space Center (VNSC)\n18 Hoang Quoc Viet, Ha Noi, Viet Nam 2 LERMAVietnam, Vietnam\n\nCNRS and Observatoire de Paris\nPSL Research University\n61 av. de l'ObservatoireF-75014ParisFrance\n",
"T Le Bertre ",
"J M Winters \nIRAM\n300 rue de la Piscine, Domaine UniversitaireF-38406St. Martin d'HéresFrance\n",
"P N Diep \nDepartment of Astrophysics\nAcademy of Science and Technology (VAST)\nUMR 8112\nNational Space Center (VNSC)\n18 Hoang Quoc Viet, Ha Noi, Viet Nam 2 LERMAVietnam, Vietnam\n\nCNRS and Observatoire de Paris\nPSL Research University\n61 av. de l'ObservatoireF-75014ParisFrance\n",
"N T Phuong \nDepartment of Astrophysics\nAcademy of Science and Technology (VAST)\nUMR 8112\nNational Space Center (VNSC)\n18 Hoang Quoc Viet, Ha Noi, Viet Nam 2 LERMAVietnam, Vietnam\n\nCNRS and Observatoire de Paris\nPSL Research University\n61 av. de l'ObservatoireF-75014ParisFrance\n"
]
| [
"Department of Astrophysics\nAcademy of Science and Technology (VAST)\nUMR 8112\nNational Space Center (VNSC)\n18 Hoang Quoc Viet, Ha Noi, Viet Nam 2 LERMAVietnam, Vietnam",
"CNRS and Observatoire de Paris\nPSL Research University\n61 av. de l'ObservatoireF-75014ParisFrance",
"Department of Astrophysics\nAcademy of Science and Technology (VAST)\nUMR 8112\nNational Space Center (VNSC)\n18 Hoang Quoc Viet, Ha Noi, Viet Nam 2 LERMAVietnam, Vietnam",
"CNRS and Observatoire de Paris\nPSL Research University\n61 av. de l'ObservatoireF-75014ParisFrance",
"Department of Astrophysics\nAcademy of Science and Technology (VAST)\nUMR 8112\nNational Space Center (VNSC)\n18 Hoang Quoc Viet, Ha Noi, Viet Nam 2 LERMAVietnam, Vietnam",
"CNRS and Observatoire de Paris\nPSL Research University\n61 av. de l'ObservatoireF-75014ParisFrance",
"Department of Astrophysics\nAcademy of Science and Technology (VAST)\nUMR 8112\nNational Space Center (VNSC)\n18 Hoang Quoc Viet, Ha Noi, Viet Nam 2 LERMAVietnam, Vietnam",
"CNRS and Observatoire de Paris\nPSL Research University\n61 av. de l'ObservatoireF-75014ParisFrance",
"IRAM\n300 rue de la Piscine, Domaine UniversitaireF-38406St. Martin d'HéresFrance",
"Department of Astrophysics\nAcademy of Science and Technology (VAST)\nUMR 8112\nNational Space Center (VNSC)\n18 Hoang Quoc Viet, Ha Noi, Viet Nam 2 LERMAVietnam, Vietnam",
"CNRS and Observatoire de Paris\nPSL Research University\n61 av. de l'ObservatoireF-75014ParisFrance",
"Department of Astrophysics\nAcademy of Science and Technology (VAST)\nUMR 8112\nNational Space Center (VNSC)\n18 Hoang Quoc Viet, Ha Noi, Viet Nam 2 LERMAVietnam, Vietnam",
"CNRS and Observatoire de Paris\nPSL Research University\n61 av. de l'ObservatoireF-75014ParisFrance"
]
| [
"MNRAS"
]
| The problem of de-projection of radio line observations of axi-symmetric expanding circumstellar envelopes is studied with the aim of easing their analysis in terms of physics models. The arguments developed rest on the remark that, in principle, when the wind velocity distribution is known, the effective emissivity can be calculated at any point in space. The paper provides a detailed study of how much this is true in practice. The wind velocity distribution assumed to be axi-symmetric and in expansion, is described by four parameters: the angles defining the orientation of the symmetry axis, an overall velocity scale and a parameter measuring the elongation (prolateness) of the distribution. Tools are developed that allow for measuring, or at least constraining, each of the four parameters. The use of effective emissivity as relevant quantity, rather than temperature and density being considered separately, implies important assumptions and simplifications meaning that the approach being considered here is only a preliminary to, and by no means a replacement for, a physics analysis accounting for radiative transfer and hydrodynamics arguments. While most considerations are developed using simulated observations as examples, two case studies (EP Aqr, observed with ALMA, and RS Cnc, with NOEMA) are presented that illustrate their usefulness in practical cases. | 10.1093/mnras/sty2005 | [
"https://arxiv.org/pdf/1807.10205v1.pdf"
]
| 119,484,372 | 1807.10205 | 6675921e0db87d1cf8afb55927c3f82b14a13480 |
De-projection of radio observations of axi-symmetric expanding circumstellar envelopes
Jul 2018. 2018
1⋆P T Nhung
Department of Astrophysics
Academy of Science and Technology (VAST)
UMR 8112
National Space Center (VNSC)
18 Hoang Quoc Viet, Ha Noi, Viet Nam 2 LERMAVietnam, Vietnam
CNRS and Observatoire de Paris
PSL Research University
61 av. de l'ObservatoireF-75014ParisFrance
D T Hoai
Department of Astrophysics
Academy of Science and Technology (VAST)
UMR 8112
National Space Center (VNSC)
18 Hoang Quoc Viet, Ha Noi, Viet Nam 2 LERMAVietnam, Vietnam
CNRS and Observatoire de Paris
PSL Research University
61 av. de l'ObservatoireF-75014ParisFrance
P Tuan-Anh
Department of Astrophysics
Academy of Science and Technology (VAST)
UMR 8112
National Space Center (VNSC)
18 Hoang Quoc Viet, Ha Noi, Viet Nam 2 LERMAVietnam, Vietnam
CNRS and Observatoire de Paris
PSL Research University
61 av. de l'ObservatoireF-75014ParisFrance
P Darriulat
Department of Astrophysics
Academy of Science and Technology (VAST)
UMR 8112
National Space Center (VNSC)
18 Hoang Quoc Viet, Ha Noi, Viet Nam 2 LERMAVietnam, Vietnam
CNRS and Observatoire de Paris
PSL Research University
61 av. de l'ObservatoireF-75014ParisFrance
T Le Bertre
J M Winters
IRAM
300 rue de la Piscine, Domaine UniversitaireF-38406St. Martin d'HéresFrance
P N Diep
Department of Astrophysics
Academy of Science and Technology (VAST)
UMR 8112
National Space Center (VNSC)
18 Hoang Quoc Viet, Ha Noi, Viet Nam 2 LERMAVietnam, Vietnam
CNRS and Observatoire de Paris
PSL Research University
61 av. de l'ObservatoireF-75014ParisFrance
N T Phuong
Department of Astrophysics
Academy of Science and Technology (VAST)
UMR 8112
National Space Center (VNSC)
18 Hoang Quoc Viet, Ha Noi, Viet Nam 2 LERMAVietnam, Vietnam
CNRS and Observatoire de Paris
PSL Research University
61 av. de l'ObservatoireF-75014ParisFrance
De-projection of radio observations of axi-symmetric expanding circumstellar envelopes
MNRAS
000Jul 2018. 2018Accepted XXX. Received YYY; in original form ZZZPreprint 27 July 2018 Compiled using MNRAS L A T E X style file v3.0stars: circumstellar matter, AGB and post-AGB; methods: data analysis
The problem of de-projection of radio line observations of axi-symmetric expanding circumstellar envelopes is studied with the aim of easing their analysis in terms of physics models. The arguments developed rest on the remark that, in principle, when the wind velocity distribution is known, the effective emissivity can be calculated at any point in space. The paper provides a detailed study of how much this is true in practice. The wind velocity distribution assumed to be axi-symmetric and in expansion, is described by four parameters: the angles defining the orientation of the symmetry axis, an overall velocity scale and a parameter measuring the elongation (prolateness) of the distribution. Tools are developed that allow for measuring, or at least constraining, each of the four parameters. The use of effective emissivity as relevant quantity, rather than temperature and density being considered separately, implies important assumptions and simplifications meaning that the approach being considered here is only a preliminary to, and by no means a replacement for, a physics analysis accounting for radiative transfer and hydrodynamics arguments. While most considerations are developed using simulated observations as examples, two case studies (EP Aqr, observed with ALMA, and RS Cnc, with NOEMA) are presented that illustrate their usefulness in practical cases.
INTRODUCTION
Recent years have seen high quality observations of molecular line emissions from evolved stars become available. Such observations, in particular from NOEMA (NOrthern Extended Millimeter Array) and from the Atacama Large Millimeter/submillimeter Array (ALMA), offer a spatial and spectral resolution calling for analysis methods making the best possible use of it. The present work addresses the case of expanding circumstellar envelopes of evolved stars, particularly stars populating the Asymptotic Giant Branch (AGB, for a recent review see Höfner & Olofsson 2018, and references therein). Such envelopes have shapes that often evolve from spherical to axi-symmetric morphology, providing the seed for possibly more irregular configurations later observed in post-AGB stars and Planetary Nebulae. The physics governing the breaking of spherical symmetry is currently the subject of active research. In the case of binaries, attraction from the companion has been shown to play an important role. Yet, many unanswered questions, such as the role played by magnetic fields, remain to be elucidated.
The analysis of radio observations of molecular line emissions requires, as a preliminary, a de-projection in space. This is a largely under-determined problem: only two out of three position coordinates are measured, those in the sky plane; the position along the line of sight is unknown. And only one out of three velocity components is measured, that along the line of sight, from the observed Doppler shift. In a recent paper (Diep et al. 2016) we made general considerations on the problem of de-projection, with particular emphasis on the differentiation between expansion and rotation and on the use of Position-Velocity (PV) diagrams. We were then addressing issues related to both the physics of proto-stars and of evolved stars. In the present work, we concentrate instead on expanding circumstellar envelopes. The aim is to shed new light on the problem of de-projection in as simple terms as possible and to understand in depth the difficulties that its solution needs to face. To do so, we deliberately ignore complications such as arising from optical thickness or from the possible presence of rotation, and more generally from any form of departure from exact axi-symmetry and exact radial expansion. We mostly exploit the simplification offered by the constraint of axi-symmetry (one relation) and from the hypothesis of radial expansion (two relations), helping with the solution of the problem of de-projection.
The considerations that follow have no ambition at replacing the physics analysis required by an in-depth understanding of the physics mechanisms at play. They are simply meant as a useful preliminary step, providing helpful tools and possibly inspiring considerations on the issue of de-projection.
THE FRAMEWORK
We use coordinates (x, y, z) attached to the sky plane and (x ′ , y ′ , z ′ ) attached to the star (Figure 1). The z axis is parallel to the line of sight, pointing away from Earth, while x is pointing east and y north. The z ′ axis is the symmetry axis of the star morphology and kinematics, making an angle ϕ with the line of sight. Its projection on the sky plane makes an angle θ (position angle) with the y axis, where θ is the angle between the x axis and the x ′ axis, the latter taken to be in the (x, y) plane. The transformation relations between the two systems of coordinates read
x ′ = x cos θ + y sin θ y ′ = (−x sin θ + y cos θ) cos ϕ + z sin ϕ z ′ = −(−x sin θ + y cos θ) sin ϕ + z cos ϕ x = x ′ cos θ − (y ′ cos ϕ − z ′ sin ϕ) sin θ y = (y ′ cos ϕ − z ′ sin ϕ) cos θ + x ′ sin θ z = y ′ sin ϕ + z ′ cos ϕ
(1) However, in much of what follows, we redefine the y axis as the projection of the star axis on the sky plane, which is equivalent to setting θ = 0. In this case x = x ′ and the transformation relations between (y, z) and (y ′ , z ′ ) read: y ′ = y cos ϕ + z sin ϕ y = y ′ cos ϕ − z ′ sin ϕ z ′ = −y sin ϕ + z cos ϕ z = y ′ sin ϕ + z ′ cos ϕ
In order to illustrate our arguments as simply as possible we use a model (in the remainder of the article we refer to it as "the simple model") in which the wind velocity is radial and independent of the distance from the star with a dependence on star latitude α of the form V = V pole sin 2 α + Veq cos 2 α = V0(1 − λ cos 2α)
with the polar velocity V pole = V0(1 + λ) and the equatorial velocity Veq = V0(1 − λ) (Figure 2 left). For λ between 0 and 1 one obtains prolate velocity distributions (bipolar outflow) and for λ between −1 and 0 oblate velocity distributions (equatorial outflow), λ=0 corresponding to an isotropic (spherical) velocity distribution. Here V0 defines the velocity scale but its precise value is irrelevant; in practice, we set it at 5 km s −1 for the purpose of illustration. The Doppler velocity at space point (x, y, z) is
Vz = V sin ζ(4)
where ζ is the angle between the (x, y, z) direction and the plane of the sky. We also define R = x 2 + y 2 = z/tan ζ and r = √ R 2 + z 2 = z/sin ζ = R/cos ζ. From Relations 2, 3 and 4, using sin α = z ′ /r and replacing z ′ by its expression in Relation 2 we can express the Doppler velocity as a function of ζ, ϕ and ψ, defined as the position angle measured with respect to the projection of the star axis on the sky plane, y = R cos ψ:
Vz = V0(z/r)[1 − λ(1 − 2 sin 2 α)] = V0(z/r)[1 − λ + 2λ(z cos ϕ − y sin ϕ) 2 /r 2 ] (5) Vz =V0 sin ζ[1 − λ + 2λ(sin ζ cos ϕ − cos ζ cos ψ sin ϕ) 2 ]
=V0[(1 − λ) sin ζ + 2λ sin 3 ζ cos 2 ϕ − 4λ sin 2 ζ cos ζ sin ϕ cos ϕ cos ψ + 2λ sin ζ cos 2 ζ sin 2 ϕ cos 2 ψ] (5a)
Taking the derivative along the line of sight (R and ψ being fixed) and using the identities d/dz = [d/d(sin ζ)][d(sin ζ)/dz] = r −1 (cos 2 ζ)d/d(sin ζ) and d(cos ζ)/d(sin ζ) = − tan ζ we obtain:
rdVz/dz = V0[(1 − λ) cos 2 ζ + 2λP ](5b)
with P =3 cos 2 ϕ sin 2 ζ cos 2 ζ − cos ψ sin 2ϕ cos ζ sin ζ(2 cos 2 ζ − sin 2 ζ) + cos 2 ψ sin 2 ϕ cos 2 ζ(cos 2 ζ − 2 sin 2 ζ) (5c) Radio astronomy measurements are in the form of a data-cube with elements f (x, y, Vz) measuring the brightness along the line (x, y) normal to the sky plane at Doppler velocity Vz. It is convenient to define the effective emissivity at space point (x, y, z) as ρ(x, y, z) = f (x, y, Vz)dVz/dz (6) and the measured intensity as
F (x, y) = f (x, y, Vz)dVz = ρ(x, y, z)dz(7)
As mentioned in the introduction, the assumption of pure radial velocity and of axi-symmetry should help with the de-projection of the effective emissivity using Relation 6. How much this is true in practice is the subject of the present article. . Left: sin ζ (abscissa) distribution of the extrema of the Vz for a million of uniformly distributed (λ, ϕ, ψ) triplets. Dashed line is for λ<0 (oblate) and solid line for λ>0 (prolate). Right: region (in blue) of the λ (ordinate) vs ϕ (abscissa) plane where de-projection is unambiguous, z increasing monotonically with Vz in the region | sin ζ| < 0.9.
A number of symmetries are apparent in Relations 5. Changing ϕ in −ϕ and ψ in 180 • − ψ leaves Vz invariant, so does also changing ψ in −ψ; changing ϕ in 180 • − ϕ and ζ in −ζ changes Vz in −Vz. Accordingly, it is sufficient to limit the ranges of ϕ and ψ to respectively [0 • , 90 • ] and [0 • , 180 • ].
A remarkable consequence of the above relations is the ability to estimate the r-dependence of the effective emissivity independently from the form chosen for the wind velocity (V0, λ, ϕ) as long as V does not depend on r (Figure 2 right). Indeed, consider a line passing by the star and having direction (ψ, ζ), namely making an angle ζ with the plane of the sky and projecting on it at position angle ψ; this line corresponds to a single value of Vz, independent of both R and r as long as the radial expansion velocity is constant on it, namely independent of r. Therefore, as long as both the real wind velocity, V (ψ, ζ), and the wind velocity used for de-projection, V ′ (ψ, ζ) are both independent of r (but depend on ψ and ζ, generally in different ways) the de-projected line is also a straight line, having the same projection on the plane of the sky; its direction (ψ ′ , ζ ′ ) is related to the direction (ψ, ζ) by the relations ψ ′ = ψ and V ′ (ψ, ζ ′ ) sin ζ ′ = V (ψ, ζ) sin ζ. De-projection simply transforms ζ into ζ ′ and the ratio z ′ /z is independent of R. As a result, the true effective emissivity ρ and the de-projected effective emissivity ρ ′ have the same dependence on R on each of these two lines, (ψ, ζ) for the former and (ψ, ζ ′ ) for the latter: they are proportional to a same data-cube element f (x, y, Vz), independently from R . To the extent that the true wind velocity and the wind velocity used for deprojection are not too different, the r-dependence of the deprojected effective emissivity is similar to the r-dependence of the true effective emissivity, both being dominated by the R dependence of the Doppler velocity spectrum.
λ=-0.9 ϕ=0°λ =-0.3 ϕ=0°λ =0.3 ϕ=0°λ =0.9 ϕ=0°λ =-0.9 ϕ=22.5°λ =-0.3 ϕ=22.5°λ =0.3 ϕ=22.5°λ =0.9 ϕ=22.5°λ =-0.9 ϕ=45°λ =-0.3 ϕ=45°λ =0.3 ϕ=45°λ =0.9 ϕ=45°λ =-0.9 ϕ=67.5°λ =-0.3 ϕ=67.5°λ =0.3 ϕ=67.5°λ =0.9 ϕ=67.5°λ =-0.9 ϕ=90°λ =-0.3 ϕ=90°λ =0.3 ϕ=90°λ =0.9 ϕ=90°F
igure 4. Dependence of sin ζ (ordinate) on Vz (abscissa) for different values of ψ (from 0 • to 180 • in steps of 20 • ). Panels are in five rows of ϕ (up down from 0 • to 90 • in steps of 22.5 • ) and in columns of λ (left to right from −0.9 to +0.9 in steps of 0.6).
GENERAL CONSIDERATIONS
De-projection implies using Relation 6 to calculate the effective emissivity ρ(x, y, z) from the measured brightness f (x, y, Vz) by associating to each value Vz of the measured Doppler velocity spectrum a point (x, y, z) in space. This is only possible if the relation giving Vz as a function of z can be inverted into a relation giving z as a function of Vz. In general, an extremum of the dependence of Vz on z will generate in its vicinity two values of z for a same value of Vz. In principle, this should prevent de-projection as one does not know how to share the brightness measured at Doppler velocity Vz between the two corresponding space points; this issue will be discussed in some detail in Section 6. As the relation between z and sin ζ is one-to-one, z = R tan ζ = R sin ζ/ 1 − sin 2 ζ, the extrema of Vz vs z are the same as of Vz vs sin ζ.
In order to obtain some insight into this question, θ-θ 0 (degree) we consider a sample of uniformly distributed (λ, ϕ, ψ) triplets. Figure 3 (left) displays the values of sin ζ at which an extremum of Vz vs z is found; they concentrate near sin ζ = ±1. To see it more directly, we display in Figure 4 a set of representative functions sin ζ vs Vz. For sin ζ = ±1, Vz = ±V0(1 + λ cos 2ϕ) and when sin ζ departs from ±1, Vz increases or decreases depending on the sign of λ sin 2ϕ cos ψ. Globally, Vz must increase with z; if it starts in the wrong direction, it needs to turn back and one obtains an extremum. When such extrema are in the vicinity of sin ζ = ±1, they are not too harmful for de-projection: they simply mean that each end region of the Doppler velocity spectrum has to be assigned a broad range of sin ζ values in the vicinity of the line of sight. However, the farther away they are from sin ζ = ±1, the larger the fraction of the Doppler velocity spectrum that becomes unsuitable for de-projection. In a majority of cases (68%) there is no "harmful" extremum, defined as having | sin ζ| > 0.9.
The right panel of Figure 3 displays the region of the (λ, ϕ) plane where no "harmful" extremum occurs, namely where dVz/dz does not cancel in the | sin ζ| < 0.9 region. In this region, as clearly illustrated in Figure 4, large values of |Vz| are associated with large values of | sin ζ| and deprojection can be unambiguously performed as long as V0 is large enough. In the complementary ambiguous region, on the contrary, the larger values of |Vz| are associated with intermediate values of | sin ζ|, the larger values of | sin ζ| being now associated with lower values of |Vz|: the Doppler velocity spectrum is folded on itself, the larger values of |Vz| being associated with two values of | sin ζ|, making de-projection ambiguous and unreliable. The strong qualitative difference between the two regions will be seen to play a major role in the arguments developed in the present study.
POSITION ANGLE OF THE PROJECTION OF THE STAR AXIS ON THE SKY PLANE
The invariance of Relations 5 when ψ changes sign implies an exact symmetry of the data-cube with respect to the plane of position angle θ, perpendicular to the sky plane and containing the star axis and the line of sight. It is indeed a consequence of axi-symmetry and is expected to apply for any axi-symmetric model, not just the simple model used here for illustration. In principle, the symmetry plane, and therefore the value of θ, can be simply found by minimizing the quantity
χ 2 rot = [(f (x, y, Vz) − f (x * , y * , Vz)] 2 /(∆f ) 2(8)
where (x * , y * ) is the symmetric of (x, y) with respect to direction θ:
x * = −(x cos 2θ+y sin 2θ) y * = −(x sin 2θ−y cos 2θ) (9) and where ∆f is the uncertainty on the f measurement. In the present section we use a simulation having a wind of the form given in Relation 3 and we simply take as ∆f the quadratic sum of the rms deviations of f from its mean in the vicinity of each of (x, y, Vz) and (x * , y * , Vz).
As an illustration of the procedure, Figure 5 (left) shows the dependence of χ 2 rot on θ − θ0 for λ = 0.5 and various values of ϕ, with θ0 being the value of θ used to produce the simulated effective emissivity. The angle θ is undefined in two obvious cases: for an isotropic velocity distribution, λ = 0, and for a star axis parallel to the line of sight, ϕ = 0; the latter case is a trivial effect of geometry, the elementary solid angle being dΩ = sin ϕdθdϕ rather than simply dθdϕ. Indeed, the dependence of χ 2 rot on θ − θ0 is observed to display a steep minimum at 0 as long as λ and ϕ are not too close from zero.
In order to assess the accuracy of the θ measurement, we calculate the relative increase Srot of χ 2 rot associated with a shift of ±1 • from θ = θ0:
Srot = 1/2[χ 2 rot (θ = ϕ(degree) λ Figure 6. Maps of V zmin /V 0 (upper panels) and Vzmax/V 0 (lower panels) in the λ (ordinate) vs ϕ (abscissa) plane for 0 < ψ < 30 • (left), 30 • < ψ < 60 • (middle) and 60 • < ψ < 90 • (right). θ0 − 1 • ) + χ 2 rot (θ = θ0 + 1 • )]/χ 2 rot (θ = θ0).
The larger Srot and the farther away from unity, the better defined is the value of θ. Figure 5 (right) displays the dependence of Srot on λ and ϕ. Its actual value when applied to real observations depends on the relevant uncertainties, usually caused by the lumpiness of the effective emissivity rather than by noise; but its behaviour in the (λ, ϕ) plane remains essentially the same as found here. However, a systematic rather than random deviation from axi-symmetry may affect the measurement of θ, as was already discussed in the case of rotation by Diep et al. (2016). However, a systematic violation of axi-symmetry by the effective emissivity or by the wind radial expansion velocity of the form 1 + ǫ cos(ω − ω0) does not shift the value of θ but simply broadens the minimum of χ 2 rot , because, contrary to rotation, it distorts in a same way the red-shifted and blue-shifted hemispheres.
MAGNITUDE OF THE WIND VELOCITIES: WIDTH AND OFFSET OF THE DOPPLER VELOCITY SPECTRUM
In general, the radial wind velocity V is confined between two finite values; in the case of the simple model, these are Veq and V pole , respectively V0(1 − λ) and V0(1 + λ). However, at a given point (x, y) in the sky plane, the measured Doppler velocity varies between two values Vzmin and Vzmax that are not simply related to the above. Yet, when choosing a velocity distribution with which to de-project the effective emissivity, it is essential to have some idea of its scale, meaning the value of V0 in the case of the simple model. In principle, having chosen a pair (λ, ϕ)
for de-projection, the values of Vzmin/V0 and Vzmax/V0 that they generate in each pixel are known. One should then choose for V0 the ratio between the measured values of Vzmin and Vzmax and the model values of Vzmin/V0 and Vzmax/V0. The values of Vzmin/V0 and Vzmax/V0 depend on the pixel. In general, different pixels produce different values of V0 and their mean, or better their maximum, should be retained for de-projection. Note that changing z in −z and cos ψ in − cos ψ changes Vz in −Vz but leaves dVz/dz invariant; therefore it changes Vzmin in −Vzmax and Vzmax in −Vzmin: it is sufficient to confine ψ to the [0 • , 90 • ] interval. Figure 6 displays the maps of Vzmin/V0 and Vzmax/V0 in the (λ, ϕ) plane for three intervals of ψ, each 30 • wide, covering between 0 • and 90 • . Qualitatively, the dependence on ψ is weak. The main features reflect the effect of the width of the Doppler velocity spectrum, which is large on the descending and small on the ascending diagonal. On the former, from (λ, ϕ)= (+1, 0 • ) to (−1, 90 • ), namely from a bipolar outflow parallel to the line of sight to an equatorial outflow having its axis in the sky plane, Vzmin/V0 varies between −1 and −2 and Vzmax/V0 between 1 and 2. On the latter, from (λ, ϕ)=(−1, 0 • ) to (+1, 90 • ), namely from a bipolar to an equatorial outflow, both in the sky plane, Vzmin/V0 varies between −1 and 0 and Vzmax/V0 between 0 and 1.
When de-projecting the effective emissivity using a (λ, ϕ) pair of parameters, agreement between the values obtained in each pixel for V0 as a function of ψ and R is a useful indicator of the suitability of the particular (λ, ϕ) pair to describe the observations. The ratio Q = Vrms/Vmean between ϕ(degree) λ Figure 7. Dependence of Q on the values of λ (ordinate) and ϕ (abscissa) used in de-projection for a few simulated wind configurations of the form V (km s −1 )= 5(1 − λ cos 2α) indicated as a cross. From left to right, the values of (λ, ϕ) used in the simulation are (±0.8, 10 • ), (±0.4, 45 • ) and (±0.8, 80 • ) respectively. their rms value Vrms and their mean value Vmean, calculated over the whole image, when too large, can be used to reject unsuitable wind configurations: the study of the dependence on position angle of the width and offset of the Doppler velocity spectrum is not only a tool to obtain an evaluation of the scale of the space velocity (here V0) but also, in principle, to reject unsuitable values of the (λ, ϕ) pair. This is illustrated in Figure 7, which displays the dependence of Q on (λ, ϕ) for a few typical simulated wind configurations. In all cases a steep minimum of Q is obtained in a narrow region of the (λ, ϕ) plane containing the values used in the simulation, suggesting that an important fraction of the (λ, ϕ) plane could be eliminated by simply requiring Q not to exceed some threshold. In practice, however, as will be seen in Section 7, the minimum of Q is much less steep for real than for simulated data.
We remark that if V0 is slightly overestimated, its largest values will de-project in a region of the observed Doppler velocity spectrum where there are no data and will accordingly set the de-projected effective emissivity to zero. On the contrary, if V0 is slightly underestimated, the larger values of the observed Doppler velocity spectrum will be ignored. It is therefore important, in practice, to make sure that the obtained value of V0 is optimal and, if necessary, to fine-tune it.
MEASURING THE PROLATENESS PARAMETER λ AND THE INCLINATION ϕ OF THE STAR AXIS WITH RESPECT TO THE LINE OF SIGHT
The results obtained in the preceding two sections are largely independent from the particular form of the dependence of the effective emissivity on stellar latitude. They provide reliable estimates of the scale V0 of the space velocity distribution and of the position angle θ of the projection of the star axis on the sky plane. Moreover, they eliminate regions of the (λ, ϕ) plane that are unsuitable for de-projection. We are then left with two parameters, λ and ϕ, to be measured in the case of the simple model. In the general case, several parameters will be necessary to describe the wind velocity in place of the pair (V0, λ) and λ must be seen as a measure of the effective prolateness of the wind velocity distribution.
In the present section, we exploit the constraint resulting from the requirement of axi-symmetry of the effective emissivity to help with the measurement of the (λ, ϕ) pair. This constraint is less useful in some cases than in others. For example, in the case of a wind velocity having its axis parallel to the line of sight any pair (λ, ϕ=0) used in deprojection will produce an axi-symmetric effective emissivity, independently from the value assumed for λ.
In order to understand under which conditions the constraint of axi-symmetry is strong, we use a simple measure of the amount of axi-symmetry of the de-projected effective ϕ(degree) λ Figure 8. Dependence of χ 2 axi on the values of λ (ordinate) and ϕ(abscissa) used in de-projection for the same simulated wind configurations indicated as a cross as in Figure 7. In each panel, the colour scale extends from minimum to ten times minimum. emissivity about axis (θ, ϕ):
χ 2 axi = [ρ(r, α, ω)− < ρ(r, α) >] 2 /(∆ρ) 2(10)
where ω is the stellar longitude (x ′ = r cos α cos ω, y ′ = r cos α sin ω and tan ω = y ′ /x ′ ); ρ(r, α, ω) is the effective emissivity de-projected using a wind configuration of axis (θ, ϕ), effective prolateness λ and velocity scale V0 estimated from the procedure described in the preceding section; < ρ(r, α) > is its mean value at (r, α), averaged over longitude ω; ∆ρ is the uncertainty attached to the evaluation of ρ.
The sum extends over the whole space over which measurements are available. The evaluation of ∆ρ = ∆f |dVz/dz|, where ∆f is the uncertainty on the de-projected data-cube element, uses here the same estimate of ∆f as used for χ 2 rot in Section 4, namely the rms deviation of f from its mean in the vicinity of the de-projected space point. In real cases, however, the definition of ∆f is delicate and needs to account for both experimental uncertainties and uncertainties attached to the de-projection; we discuss this point in more detail in Section 7. In practice the calculation of χ 2 axi proceeds as follows: one chooses a circle having the star axis as axis and defined by its position (R ′ , z ′ ) in the star frame and scans over it by varying the stellar longitude ω, each time calculating the space position (x, y, z) and the associated value of Vz for the values of (V0, λ, ϕ) of the axi-symmetric wind used in de-projection.
The main weakness of this method is its mishandling of cases where two different values of z/r are associated with a same bin of Doppler velocity. Such bins contain contributions from each of the two regions but the de-projection algorithm wrongly assigns their total content to each of the two regions, generating double-counting. As was already remarked in Section 3, one does not know how to share it between the two regions and such bins are unsuitable for de-projection. We have seen in Section 3 that for z/r = ±1, Vz = ±V0(1 + λ cos 2ϕ) and that when z/r departs from ±1, Vz increases or decreases depending on the sign of λ sin 2ϕ cos ψ. Rather than simply ignoring the ambiguous intervals of Doppler velocity between Vzmin and −V0(1 + λ cos 2ϕ) > Vzmin and between V0(1 + λ cos 2ϕ) < Vzmax and Vzmax, we simply assign to each of the two associated z values half the de-projected emissivity, thereby avoiding double-counting. Figure 8 illustrates the dependence of χ 2 axi on parameters λ and ϕ for the same simulated wind configurations as displayed in Figure 7. The effective emissivity used to produce the simulated data-cube is of the form 1/r 2 for 0.5 < r < 5 arcsec with no dependence on stellar latitude in order not to bias the evaluation of χ 2 axi or, more precisely, not to complicate its interpretation. For r > 5 arcsec, the effective emissivity is taken to cancel and for r < 0.5 arcsec it is taken to be constant.
In all cases χ 2 axi is minimal at the simulated value of (λ, ϕ) but in some cases a broader region of the (λ, ϕ) plane is observed to be equally acceptable. The minimum is better behaved along the descending diagonal (wind velocities out of the sky plane) than along the ascending diagonal (wind velocities near the sky plane): in the former case χ 2 axi is more efficient than Q to constrain the (λ, ϕ) pair while in the latter case Q is more efficient than χ 2 axi .
Using ϕ = 0 for de-projection deserves a special comment; in this case, a given circle (R ′ , z ′ ) is made of points having all the same value of z = z ′ and the same value of R = R ′ , therefore a same value of Vz and a same value of dVz/dz: χ 2 axi is simply a measure of the axi-symmetry of the data-cube about the z axis for Doppler velocities |Vz| < (1 + λ)V0. In particular, when the real wind and the wind used for de-projection are both axi-symmetric with respect to the z axis, with respective prolateness λtrue and λ deproj , χ 2 axi = 0 is of no help to constrain the value of λtrue. The same is true of the study of Vzmin, Vzmax and Q, in the approximation where the wind velocity and mass loss rate do not depend on r: whatever the value of λ the extrema of the velocity spectrum are reached in all pixels at a same value of Vzmax and Vzmin = −Vzmax. The result of the de-projection depends on the location of λtrue and λ deproj with respect to the value −0.2 that separates ambiguous from unambiguous cases. When both λtrue and λ deproj are larger than −0.2, the maximum of |Vz| is reached at the poles and the value of V0 used for de-projection is the true value multiplied by (1+λtrue)/(1+λ deproj ). However, when one of the λ parameters takes values that vary between −0.2 and −1, the maximum of |Vz| is reached at values of | sin ζ| that vary between 1 and 1/ √ 3, the corresponding values of |Vz| varying between 0.8V0 and 4V0/[3 √ 3] ∼ 0.77V0 with a minimum at λ = −1/2 where | sin ζ| = 1/ √ 2 and |Vz| = 1/ √ 2V0 ∼ 0.71V0. This near independence on λ (∼ ±3V0/4) of Vzmin and Vzmax when ϕ = 0 and λ < −0.2 could already be seen in Figure 6. In general, when using ϕ = 0 and λ < −0.2 for de-projection of a (λ0, ϕ0) data-cube located in the unambiguous region will require the use of a V0 value significantly larger than the true value and will deproject the large values of |Vz|, associated with large values of | sin ζ|, to intermediate values of | sin ζ| resulting in relatively low values of χ 2 axi .
CASE STUDIES: RS Cnc AND EP Aqr
In the preceding sections, arguments were developed using simulated rather than real observations. In practice, the morphology of circumstellar envelopes of evolved stars are far from being as smooth and well-behaved as those simulated in the present study. The question of the practicability of using the arguments and the tools developed in the preceding sections remains therefore open at this stage. While each particular case must be considered separately, we find it useful to devote the present section to two case studies in order to get some idea of the nature and magnitude of the difficulties that one may have to face.
RS Cnc
The first star selected for this purpose is RS Cnc, an S-type AGB star (CSS 589 in Stephenson's 1984 catalogue) in the thermally-pulsing phase (Lebzelter & Hron 1999), which can be considered as representative of its family, the morphology of its envelope being quite clumpy but not excessively. Analyses of Plateau de Bure observations of its CO(1-0) and CO(2-1) emissions have been published earlier (Libert et al. 2010;Hoai et al. 2014;Nhung et al. 2015a;Le Bertre et al. 2016). Here, we use CO(2-1) observations from the upgraded Plateau de Bure interferometer (NOEMA) having a beam size (FWHM) of 0.44 × 0.28 arcsec 2 and a spectral resolution of ∼ 0.2 km s −1 . The noise level per data cube element is 1.6 mJy beam −1 for pixels of 0.07 × 0.07 arcsec 2 . In order to improve the signal to noise ratio, we limit the study to the region R < 4 arcsec and we group pixels by 3 × 3 = 9, meaning 0.21 × 0.21 arcsec 2 ; we use as experimental uncertainty on the brightness the quadratic sum of the noise and of 20% of the measured value. Figure 9 displays sky maps of the measured intensity multiplied by R and of the mean Doppler velocity, together with the integrated Doppler velocity spectrum.
As already apparent from the map of the mean Doppler velocity, the projection of the star axis on the sky plane is nearly north-south oriented. Indeed we find that χ 2 rot is minimal for θ0 ∼ 7 • and we display in Figure 10 (left) the dependence on θ − θ0 of its normalized value (divided by the number of degrees of freedom). The value of χ 2 rot at minimum, ∼ 4, is the result of the relative lumpiness of the measured brightness. The uncertainty on θ is accordingly poorly defined; as an indication of how well θ is measured we quote as effective uncertainty the value associated with a 10% increase of χ 2 rot with respect to minimum, namely θ =7 • ±10 • . Indeed, masking measured brightness values below noise level (one sigma) minimizes χ 2 rot at θ ∼ 10 • instead of ∼ 7 • .
Having obtained an evaluation of θ we rotate the datacube by 7 • about the line of sight in order to have effectively θ = 0 and proceed with the evaluation of the scale V0 of the wind velocity and the exclusion from the (λ, ϕ) plane of regions unsuited for de-projection as was done in Section 5 using simulated data. The dependence of Q on λ and ϕ is displayed in the central panel of Figure 10. At first sight, the resemblance with the upper-middle panel of Figure 7 (λ = 0.4, ϕ = 45 • ) is striking; however, the minimum is now considerably less steep than it was for simulated data and extends over a broad region that excludes wind velocities closely confined to the sky plane, whether bipolar or equatorial outflows. The resulting V0 values are displayed in the right panel of Figure 10 and range between ∼ 4 and ∼ 15 km s −1 , the lower values being associated with prolate bipolar outflows near the line of sight or oblate equatorial outflows having their axis near the sky plane, namely wind configurations in the unambiguous de-projection region. The larger values are instead confined to the ambiguous de-projection region of the (λ, ϕ) plane.
A major cause of error in the evaluation of V0 is the difficulty to measure accurately Vzmin and/or Vzmax in pixels where the brightness is close to noise level at the extremities of the Doppler velocity spectrum. In calculating in each pixel the values of Vzmin and Vzmax we used brightness in excess of two noise σ's and checked that the result was essentially unaffected when using instead one or three noise σ's. The left panels of Figure 11 display distributions of the values obtained for V0 in each pixel (two values per pixel) for some representative values of the (λ, ϕ) pair. Ideally, if the selected wind configuration describes the data well, the V0 distribution must have the shape of a narrow peak centered at the proper V0 value. Such a peak is indeed visible for some values of the (λ, ϕ) pairs, but it is broad and accompanied by a low V0 tail: exploiting the information contained in the dependence over the sky plane of Vzmin and Vzmax is clearly more difficult when dealing with real data than it is with simulated data. In pixels where the brightness is close to noise level at the extremities of the Doppler velocity spectrum |Vzmin| and/or |Vzmax| are under-evaluated and the same is therefore true for V0. Pixels at larger values of R, being associated with lower intensities, are more likely to be of that kind. Indeed, as can be seen in Figure 11, the distribution of V0 obtained inside the circle R < 3 arcsec is narrower and less contaminated by a low V0 tail than inside the circle R < 4 arcsec. The distributions displayed in the left panels of Figure 11 suggest that a figure of merit revealing the presence of a peak in the V0 distribution might be more efficient than Q in exploiting the information carried by Vzmin and Vzmax. We use a simple algorithm to evaluate, for each value of the (λ, ϕ) pair the peak to tail ratio of the V0 distribution, P/T . The distribution of P/T over the (λ, ϕ) plane is displayed in Figure 11 (right). It is consistent with the information carried by Q but is much more selective: a reasonable cut Q > 0.36 is only a factor 1.06 above minimum while an equivalent cut P/T < 2 is a factor ∼ 4 above minimum and a factor 2.5 below maximum, allowing for safely rejecting regions of the (λ, ϕ) plane having P/T < 2.
In order to further restrict the acceptable region of the (λ, ϕ) plane, we still need to exploit the constraint of axisymmetry as was done in Section 6 using simulated observations. This, however, cannot be done reliably in the regions of ambiguous de-projection (right panel of Figure 3). As these have a large overlap with the regions disfavoured by the analysis of the (λ, ϕ)-dependence of Vzmin and Vzmax, we may exclude them from the analysis by rejecting regions containing a "harmful" extremum of the Vz vs z relation as defined in the right panel of Figure 3, namely regions where dVz/dz cancels in the | sin ζ| < 0.9 interval. When an extremum occurs at | sin ζ| > 0.9, we share the de-projected emissivity equally between the two associated values of sin ζ. The distribution of χ 2 axi in the (λ, ϕ) plane is displayed in Figure 12 together with the boundaries associated with regions containing a "harmful" extremum and with the con- tours associated with Q = 0.36 and P/T = 2. The main contribution of χ 2 axi is to disfavour wind configurations close to spherical with an axis at intermediate inclination with respect to the sky plane, leaving a relatively narrow region of acceptable bipolar outflows in the upper-left quadrant of the (λ, ϕ) plane with λ between ∼ 0.3 and ∼ 0.8 and ϕ between ∼ 15 • and ∼ 45 • .
The joint analysis of CO(1-0) and CO(2-1) emissions presented in Nhung et al. (2015a) gave (V0, λ, θ, ϕ) = (5.0 ± 0.2 km s −1 , 0.50±0.02, 9 • ± 6 • , 38 • ± 2 • ) , however allowing for some r-dependence of the expansion velocity. This corresponds indeed to values of Q < 0.36, P/T > 2 and χ 2 axi < 10 favoured by the present study. We display in Figure 12 (right) the dependence of the de-projected emissivity, multiplied by r 2 on respectively r, ω and z ′ /r = sin α for three different values of the (λ, ϕ) pair in the favoured region, (λ, ϕ)= (0.5, 40 • ), (0.6, 30 • ) and (0.7, 20 • ) respectively. All three distributions are averaged in the sphere r < 4 arcsec. A remarkable result is the independence of the r-dependence of the de-projected effective emissivity on the value of (V0, λ, ϕ) used for de-projection, a result that had been anticipated and explained in Section 2 using Relations 5. However, the longitudinal dependence differs significantly from uniform, with an excess in the 180 • < ω < 360 • hemisphere compared with 0 < ω < 180 • . The latitudinal dependence displays a clear asymmetry with respect to the star equator. Table 1 summarizes the results.
Commenting further on these results goes beyond the scope of the present study. Physics arguments need now to be used for interpreting the observed behaviour of the deprojected emissivity and its relation with the probably much too simple form assumed for the wind configuration. Yet, the present results are of considerable help in constructing a physics model of the morphology and kinematics of the circumstellar envelope and have provided a deep insight in the constraints that such a model has to obey.
EP Aqr
The second star used for illustration is EP Aqr, an oxygenrich M type AGB star that is probably at the beginning of its evolution on the thermally pulsing phase (Lebzelter & Hron 1999;Cami et al. 2000) in spite of observations of trailing gas (Cox et al. 2012;Le Bertre & Gérard 2004) suggesting a mass loss episode at the scale of 10 4 to 10 5 years. Recently, observations of 12 CO(1-0) and 12 CO(2-1) emissions using the IRAM 30-m telescope and the Plateau de Bure Interferometer have been reported (Winters et al. 2003(Winters et al. , 2007Nhung et al. 2015b;Le Bertre et al. 2016). Here, we use CO(1-0) and CO(2-1) observations made in Cycle 4 of ALMA operation (Nhung et al. 2018(Nhung et al. , 2016.S). The beam size (FWHM) is respectively 0.78×0.70 and 0.33×0.30 arcsec 2 and the noise respectively 8 and 7 mJy beam −1 for pixels of respectively 0.2 × 0.2 and 0.1 × 0.1 arcsec 2 and Doppler velocity bins of 0.2 km s −1 . Similar but different ALMA observations, including also SiO and SO2 emission, have recently been presented by Homan et al. (2018).
We restrict the analysis to the sky plane region having R < 8 arcsec. Figure 13 displays sky maps of the deviation from unity of the ratio ∆(x, y) = F (x, y)/ < F (x, y) > between the measured intensity F (x, y) and its average < F (x, y) > over position angle. It gives evidence for approximate isotropy in the sky plane, suggesting that the wind is either spherical or axi-symmetric with the symmetry axis near the line of sight. Indeed, χ 2 rot is found to display a very broad minimum, centered at θ0 ∼ −20 • for CO(1-0) emission and at θ0 ∼ −31 • for CO(2-1) emission; its dependence on θ − θ0 is shown in Figure 14. Using the same criterion as for RS Cnc to measure the uncertainty on its measurement, namely a 10% increase of χ 2 rot , gives θ = −20 • ± 37 • and −31 • ± 43 • respectively, consistent with a common value of −25 • ± 28 • . The agreement between the CO(1-0) and CO(2-1) results suggests that the deviation from sphericity is real ϕ (degree) λ Figure 12. RS Cnc. Left: distribution of χ 2 axi in the (λ, ϕ) plane. The contours are for ambiguous de-projection (green), Q = 0.36 (black) and P/T = 2 (red). Rightmost panels (from left to right): dependence of the de-projected emissivity multiplied by r 2 (Jy beam −1 km s −1 arcsec) on r (arcsec), ω (degree) and sin α respectively. The values of (λ, ϕ) are, from up down, (0.5, 40 • ), (0.6, 30 • ) and (0.7, 20 • ), corresponding to the crosses in the left panel. and the broad distribution of χ 2 rot as a function of θ − θ0 indicates that the star axis must make a small angle with the line of sight. Also shown in Figure 14 (middle panels) is the dependence of Q on λ and ϕ. It remains nearly constant over a very large region of the (λ, ϕ) plane. The reason is that Vzmin and Vzmax are nearly invariant over the sky plane. In order to quantify better this statement, we integrate the Doppler velocity spectrum between R = 1 arcsec and R = 8 arcsec and display it in 18 bins of position angle ψ, each 20 • wide. The high sensitivity that results allows for measuring in each bin Vzmin and Vzmax with a precision of ∼ 0.1 km s −1 . We obtain this way measurements of the width Wvz and the offset ∆vz of the Doppler velocity spectrum in each bin of ψ. Their mean and rms values for CO(1-0) and CO(2-1) emission respectively, relative to the spectrum width, are < Wvz >= 21.30 km s −1 and 21.61 km s −1 , Rms(Wvz)/< Wvz >= 2.1 10 −2 and 2.9 10 −2 and Rms(∆vz)/< Wvz >= 1.2 10 −2 and 2.0 10 −2 respectively. The latter two quantities are compared with the predictions of the simple model in the right panels of Figure 14. Compared with Q they are more reliable discriminants in rejecting regions of the (λ, ϕ) plane but leave a very broad acceptable region along the descending diagonal. The situation here is very different from what was found for RS Cnc: it would not have helped to use a discriminant such as P/T because the V0 distribution is extremely narrow in the low Q region of the (λ, ϕ) plane. Figure 15 displays the dependence of χ 2 axi on λ and ϕ. We use as uncertainty on the measured brightness the quadratic sum of 20% of its value and of the noise. For both CO(1-0) and CO(2-1) emissions, large values of ϕ are excluded; taking into account the constraint imposed by the study of Vzmin and Vzmax leaves a large region in the upper left quadrant of the map acceptable for de-projection. The result obtained earlier by Nhung et al. (2015b), (V0, λ, θ, ϕ)=(6.0 km s −1 , 0.67, −36 • , 13 • ) is contained in this region. We select three representative (λ, ϕ) pairs for purpose of illustration: (0, 0 • ), (0.35, 6 • ) and (0.7, 12 • ). The corresponding parameters are listed in Table 2 and the dependence on r, ω and sin α of the associated de-projected effective emissivity multiplied by r 2 inside the sphere r < 8 arcsec is displayed in Figure 16. The difference between the CO(1-0) and CO(2-1) radial distributions is understood as being caused by the different temperature dependence (Nhung et al. 2018). The smaller λ, the larger V0 and the stronger the concentration of the effective emissivity near the equatorial plane. Indeed one evolves from a bipolar outflow for λ = 0.7 with no significant equatorial enhancement of effective emissivity to a spherical wind producing a strong enhancement of effective emissivity at equator for λ = 0. Only physics arguments can help choosing between the possible wind configurations. In particular, the presence of a companion star or massive planet in the equatorial plane would probably favour an enhancement of the effective emissivity while not making the wind velocity much deviate from spherical. A bipolar wind associated with an isotropic effective emissivity would be more difficult to explain. However, such arguments are well beyond the scope of the present article.
SUMMARY AND CONCLUSION
The above analysis of the problem of de-projection of radially expanding axi-symmetric circumstellar envelopes has provided a deep insight into its main features and has devised tools that help with its solution. However, Figure 13. EP Aqr CO(1-0) (upper panels) and CO(2-1) (lower panels) emission. Sky maps of ∆(x, y) − 1 for R < 10 arcsec. Left panels are for Vz < −2 km s −1 , central panels for |Vz| < 2 km s −1 and right panels for Vz > 2 km s −1 as indicated in the Doppler velocity spectra displayed above the sky maps. From Nhung et al. (2018). it does not offer a substitute to a detailed analysis that takes into account the physics of the mechanisms at play, such as hydrodynamical constraints and radiative transfer considerations. It is only meant to shed light on some of its intricacies and to set the frame for preliminary considerations that can help the construction of a realistic physical model. The effective emissivity used throughout the paper to describe observations is a convenient quantity that offers simplicity but hides the difficulty of disentangling the effects of temperature from those of density. Its assumed axi-symmetry is often a good approximation but is only valid for envelopes that are sufficiently optically thin and the assumption of radial expansion, implying the absence of rotation, is often violated in the later stages of the star evolution on their way to planetary nebulae. It is therefore essential to keep in mind the limited scope of the results of the present work.
With these caveats in mind, we summarize below the main results that have been obtained:
i) The position angle θ of the projection of the star axis on the sky plane has been shown to minimize a quantity, χ 2 rot , independently from the particular form taken by the effective emissivity; its minimization, as long as it is made over the whole data-cube and not simply on its projection on the sky plane, makes optimal use of the available information. Only in cases where the problem has no solution, either because the star axis is close to the line of sight or because the wind velocity and the effective axi on λ and ϕ. On both panels the colour scale runs from minimum to 2.5 times minimum. The black contours indicate the regions rejected by Rms(Wvz)/< Wvz > and Rms(∆vz)/< Wvz >. The green contours show the limits of ambiguous de-projection. Crosses indicate values of the (λ, ϕ) pair listed in Table 2. emissivity are nearly isotropic, is θ ill-defined.
ii) A good evaluation of the scale of the wind velocity −V0 in the simple model−has been obtained from a study, in each pixel, of the dependence on position angle ψ of the end points Vzmin and Vzmax of the Doppler velocity spectra. To this effect, the rms deviation of the velocity scale relative to its mean over the sky plane, Q, was found to be sufficient, in the case of simulated data, to quantify the agreement with a model. Its minimization helps in eliminating regions of the prolateness (λ) versus inclination (ϕ) plane which are unsuitable for de-projection. The method is particularly efficient when the wind velocity is confined near the sky plane, either as strongly prolate with axis close the sky plane or as Figure 16. Dependence of the de-projected emissivity for EP Aqr multiplied by r 2 (Jy beam −1 km s −1 arcsec) on r (arcsec, left), ω (degrees, middle) and sin α (right) respectively. The values of (λ, ϕ) are, from up down, (0, 0 • ), (0.35, 6 • ) and (0.7, 12 • ), corresponding to the crosses in Figure 15. Upper panels are for CO(1-0) emission and lower panels for CO(2-1) emission.
strongly oblate with axis close to the line of sight. However, in the case of real data, a more careful analysis is needed in order to exploit the richness of the information contained in the dependence of Vzmin and Vzmax on λ and ϕ. A careful evaluation of Vzmin and Vzmax, taking the noise level in proper account, is mandatory and algorithms allowing for separating the V0 peak from a background of improper values may be helpful.
iii) Having obtained sensible estimates of θ and V0, one can further constrain the (λ, ϕ) pair by imposing axisymmetry on the de-projected effective emissivity. To this effect, a quantity χ 2 axi has been constructed, which is minimal for maximal axi-symmetry. Contrary to the minimization of Q, the minimization of χ 2 axi is inefficient when the wind velocity is confined near the sky plane. In such cases, the evaluation of the optimal value of the (λ, ϕ) pair relies more on the constraints imposed by the distribution on the sky plane of the width and offset of the Doppler velocity spectra than on the constraints imposed by the requirement of axi-symmetry. iv) Having obtained estimates of the orientation of the star axis, θ and ϕ, of the scale of the wind velocity, V0, and of the effective prolateness of its distribution, λ, we are then in a position to de-project the effective emissivity as long as a single value of z/r is associated with each bin of measured Doppler velocity. However, the effective emissivity cannot be directly de-projected in regions of z/r that are associated with ambiguous velocity bins. Such regions are particularly important when the wind velocity is confined in the vicinity of the sky plane, implying that the Doppler velocity distribution is folded on itself and confined to lower values. In such cases, the observations measure mostly the projec-tion of the effective emissivity on the sky plane, namely the integrated brightness (or intensity). v) To a good approximation, the r-dependence of the de-projected effective emissivity is obtained independently from the choice of the wind configuration used for deprojection as long as the wind velocity does not depend on r.
vi) As an illustration, we have presented two case studies of CO emission of AGB stars. They were chosen to be representative of typical observations rather than of the best space and spectral resolutions and sensitivity available today. In both cases, the method and tools developed in the present article have been shown to select efficiently wind configurations suitable for de-projection. Published analyses of the relevant observations have proposed models that are indeed favoured by these results. However, they also suggest exploring other wind configurations, of different inclination and prolateness, which may deserve being considered in the framework of a physics analysis. More importantly, they provide a deep understanding of the constraints imposed on a physics model and of how unique is the region of the (λ, ϕ) plane ultimately selected.
vii) The main contribution of the present work may be the insight it has provided on the issue of the underdetermination of the problem of de-projection. It has underlined the importance of being conscious that a broad family of wind configurations, illustrated in blue in the right panel of Figure 3, can be used in principle to de-project the observed brightness data-cube into an effective emissivity at each point in space. In practice, however, constraints imposed by the requirement of axi-symmetry of the deprojected effective emissivity and of the need to populate in each pixel the totality of the observed Doppler velocity spectrum (but no more) have been found to complement each other; taken together, they are efficient in restricting the domain of the (λ, ϕ) plane acceptable for de-projection. A good approximation to the r-distribution of the de-projected emissivity has been obtained under the hypothesis of constant velocity but physics considerations must then be used to decide how best to combine a possible velocity gradient with such r-dependence. More generally, the constraints imposed by hydrodynamics on the relation between density, temperature and velocity will be determinant in deciding on the "best" physics model.
Figure 1 .
1Coordinate systems for θ = 0. Relations 5 transform from the sky frame (left) to the star frame (right).
Figure 2 .Figure 3
23Left: polar diagrams of V in the upper meridian quadrant (R ′ in abscissa and z ′ in ordinate) for different values of the prolateness parameter λ. Right: Deprojection in the (R, z) plane (see text).
Figure 5 .
5Left: dependence of χ 2 rot (ordinate) on θ − θ 0 (degrees, abscissa) for λ=0.5 and ϕ=5 • , 10 • , 15 • , 30 • , 45 • (from down up). Right: dependence of Srot on λ (ordinate) and ϕ (abscissa).
Figure 9 .Figure 10 .
910NOEMA observations of the CO(2-1) emission of RS Cnc in the region R < 4 arcsec. Left: sky map of the intensity (Jy beam −1 ) multiplied by R (arcsec). Middle: sky map of < Vz > (km s −1 ). Right: Doppler velocity Vz spectrum (km s −1 ) integrated over the region R < 4 arcsec. In the first two panels north is up and east is left. RS Cnc CO(2-1) emission. Left: dependence of χ 2 rot on θ − 7 • ; the curve is a parabolic fit. Middle: (λ, ϕ) map of Q for the region R < 4 arcsec. The colour scale runs from minimum (∼ 0.34) to 1.2 times minimum. Right: (λ, ϕ) map of the velocity scale V 0 (km s −1 ).
Figure 11 .
11RS Cnc. Left: V 0 distributions obtained for four different values of the (λ, ϕ) pair and for R < 4 arcsec (solid) or R < 3 arcsec (dashed). Right: distribution of P/T in the (λ, ϕ) plane for R < 3 arcsec. The contours correspond to Q = 0.345, 0.360 and 0.375 respectively.
Figure 14 .Figure 15 .
1415EP Aqr CO(1-0) and CO(2-1) emission. Left: dependence of χ 2 rot on θ − θ 0 ; the curves are cosine square fits. Middle: (λ, ϕ) maps of Q for the region R < 8 arcsec. The contours show Q = 0.31 for CO(1-0) and Q = 0.39 for CO(2-1). Right: (λ, ϕ) maps of Rms(Wvz)/< Wvz > and Rms(∆vz)/< Wvz > as predicted by the simple model. The black contours display the values measured for CO(1-0) and CO(2-1) emission. The white contours are the same as shown in black in the middle panels. EP Aqr CO(1-0) (left) and CO(2-1) (right) emission. Dependence of χ 2
Table 1 .
1De-projection of RS Cnc CO(2-1) observations: summary of resultsCase
V 0 (km /s)
λ
ϕ( • )
θ( • )
Q
P/T
χ 2
axi
Nhung et al. (2015a)
5.0±0.2
0.50±0.02
38±2
9±6
−
−
−
1
6.0
0.5
40
7
0.344
3.6
9.5
2
5.0
0.6
30
7
0.345
2.0
7.5
3
4.5
0.7
20
7
0.356
2.2
8.7
Table 2 .
2De-projection of EP Aqr CO(1-0) observations: summary of resultsEmission
Case
V 0 (km s −1 )
λ
ϕ ( • )
θ ( • )
Q
χ 2
axi
CO(1-0)&(2-1)
Nhung et al. (2015b)
6.0
0.67
13
−36
−
−
CO(1-0)
1
10.6
0
0
−20
0.29
0.44
2
8.0
0.35
6
0.57
3
6.5
0.7
12
0.66
CO(2-1)
1
10.8
0
0
−31
0.37
0.68
2
8.1
0.35
6
0.89
3
6.6
0.7
12
1.00
MNRAS 000, 1-16(2018)
ACKNOWLEDGEMENTSWe thank Professor Pierre Lesaffre for a careful reading of the manuscript and very useful and pertinent comments that helped improving significantly its content. This paper makes use of the following ALMA data: 2016.1.00026.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. It also makes use of observations carried out with the IRAM NOEMA Interferometer and the IRAM 30 m telescope. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain). This research is funded by Graduate University of Science and Technology under grant number GUST.STS.DT2017-VL01. Financial and/or material support from the Vietnam National Space Center, the National Foundation for Science and Technology Development (NAFOSTED), the World Laboratory and Odon Vallet fellowships is gratefully acknowledged.
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| []
|
[
"THE NONLINEAR 2D SUPERCRITICAL INVISCID SHALLOW WATER EQUATIONS IN A RECTANGLE",
"THE NONLINEAR 2D SUPERCRITICAL INVISCID SHALLOW WATER EQUATIONS IN A RECTANGLE"
]
| [
"Aimin Huang ",
"Madalina Petcu ",
"Roger Temam "
]
| []
| []
| In this article we consider the inviscid two-dimensional shallow water equations in a rectangle. The flow occurs near a stationary solution in the so called supercritical regime and we establish short term existence of smooth solutions for the corresponding initial and boundary value problem.X m 1 (Ω) = {θ ∈ H m (Ω) : T θ = λθ x + θ y ∈ H m (Ω)}. | 10.1007/s10884-015-9507-1 | [
"https://arxiv.org/pdf/1503.00283v1.pdf"
]
| 119,322,271 | 1503.00283 | 62a0929edc0f4d9bfdf24a287af79152474d9e8e |
THE NONLINEAR 2D SUPERCRITICAL INVISCID SHALLOW WATER EQUATIONS IN A RECTANGLE
1 Mar 2015
Aimin Huang
Madalina Petcu
Roger Temam
THE NONLINEAR 2D SUPERCRITICAL INVISCID SHALLOW WATER EQUATIONS IN A RECTANGLE
1 Mar 2015
In this article we consider the inviscid two-dimensional shallow water equations in a rectangle. The flow occurs near a stationary solution in the so called supercritical regime and we establish short term existence of smooth solutions for the corresponding initial and boundary value problem.X m 1 (Ω) = {θ ∈ H m (Ω) : T θ = λθ x + θ y ∈ H m (Ω)}.
Introduction
Motivated by the study of the inviscid primitive equations, we consider in this article the inviscid two-dimensional shallow water equations in a rectangle in the so-called supercritical regime. It has been shown that a certain vertical expansion of the inviscid primitive equations leads to a system of coupled nonlinear equations similar to the inviscid shallow water equations; see [RTT08b] and [HT14a]. Hence beside their intrinsic interest, the nonlinear shallow water equations can be seen as one mode of the vertical expansion of the primitive equations.
The issue of the boundary conditions to be associated with the primitive or shallow water equations has been emphasized as a major problem and limitation for the so-called Local Area Models for which weather predictions are sought and simulations are performed within a domain for which the boundary has no physical significance, so that there are no physical laws prescribing the boundary conditions (see [WPT97] and e.g. [RTT08a,RTT08b], [CSTT12,SLTT]). The choice of the boundary conditions relies then on mathematical considerations (derivation of a well-posed mathematical problem), and on general computational considerations and physical intuition. The boundary conditions suitable for the one-dimensional shallow water equations were derived in an intuitive context in the book of Whitham [Whi99] and in [NHF08]; see [PT13] for a rigorous study. For general results on boundary value problems for quasilinear hyperbolic system in space dimension one see [LY85]; for initial and boundary value problems for hyperbolic equations in smooth domain see the thorough book [BS07]. The present article follows the study of the one-dimensional inviscid shallow water equations in [PT13,HPT11] and the study of the linearized shallow water equations in [HT14a]. In the study of the linearized inviscid shallow water equations in [HT14a] we have shown that five cases can occur depending on the respective values of the velocity and the height (not counting the non-generic cases and the symmetries). The nonlinear case that we consider in this article relates to what was called the supercritical case in [HT14a]; see [HT14b] for the study of a subcritical case. In this article, we consider the inviscid fully nonlinear 2D shallow water equations (SWE)
(1.1) u t + uu x + vu y + gφ x − f v = 0, v t + uv x + vv y + gφ y + f u = 0, φ t + uφ x + vφ y + φ(u x + v y ) = 0;
here U = (u, v, φ) t , (x, y) ∈ Ω := (0, L 1 ) × (0, L 2 ), t ∈ (0, T ), u and v are the two horizontal components of the velocity, φ is the height of the water, and g is the gravitational acceleration, f is the Coriolis parameter. The first and second equations (1.1) are derived from the equations of conservation of horizontal momentum, and the third one expresses the conservation of mass. We consider equations (1.1) for certain values of u, v, φ as described below, corresponding to a "supercritical" flow and we associate with (1.1), initial conditions for u, v, φ and boundary conditions at x = 0 and y = 0, u, v, φ vanishing on that part of the boundary.
This article is organized as follows. After this introductory section, we derive in Section 2 suitable density theorems, density of certain smooth functions in certain function spaces of Sobolev type. Section 3 is devoted to the modified (symmetrized) SWE operator for the time-independent and the time-dependent cases. It prepares Section 4 in which we deal with the linear SWE, linearized around a non-constant time-dependent flow unlike in [HT14a] where the background flow is time-independent. In Section 4 we prove the well-posedness of the linearized SWE at the price of a loss of derivatives (see Theorem 4.1), and then the wellposedness of the linearized SWE in a short time without a loss of derivatives (see Theorem 4.2). Section 5 considers the fully nonlinear SWE, for which the local well-posedness result is obtained. In the Appendices A and B, we collect some useful theorems about semigroup and evolution systems, and several classical estimates about functions in Sobolev spaces.
The density theorems
In this section, we establish general density theorems for certain Sobolev spaces; the results supplement and complement those of Section 3 in [HT14a] which we recall when needed. These theorems have independent interest, and also will be needed for proving later on that −A generates a quasi-contraction semigroup on certain Sobolev spaces, where A is the 2D modified SWE operator associated with suitable boundary conditions. Throughout this section, let m be a non-negative integer and let λ = λ(x, y) satisfy
(2.1) c 0 ≤ λ(x, y) ≤ c 1 ,
where c 0 , c 1 are positive constants. Furthermore, we say that λ = λ(x, y) satisfies the positive m-condition (m integer ≥ 0) if λ satisfies (2.1) and
(2.2) ∇λ ∈ L ∞ (Ω), for m = 0, 1, λ ∈ H 3∨m (Ω), for m ≥ 2,
where a ∨ b = max(a, b). It is easy to see that if m 1 ≥ m 2 and λ satisfies the positive m 1 -condition, then λ also satisfies the positive m 2 -condition. We now set for any function θ = θ(x, y), T θ = λ(x, y)θ x + θ y , where λ is assumed to satisfy the positive m-condition, and introduce the function space We observe that X m 1 (Ω) is a space of local type, that is (2.3)
If θ ∈ X m 1 (Ω), ψ ∈ C ∞ (Ω), then θψ ∈ X m 1 (Ω). This property follows from T (ψθ) = ψT θ + (λψ x + ψ y )θ, and λθ (and hence (λψ x + ψ y )θ) is in H m (Ω) because of Lemma B.1 i).
We give an equivalent characterization of the space X m 1 (Ω). In the following and throughout this article, we let ∂ α = ∂ α 1
x ∂ α 2 y with α = (α 1 , α 2 ) and set |α| = α 1 + α 2 . We also denote by
[∂ α , f ] the commutator [∂ α , f ]g = ∂ α (f g) − f ∂ α g.
Proposition 2.1. We assume that λ satisfies the positive m-condition. Then
(2.4) X m 1 (Ω) = {θ ∈ H m (Ω) : T ∂ α θ ∈ L 2 (Ω), ∀ |α| = m}. Proof.
It is clear that (2.4) holds when m = 0. For m = 1, we observe that
(2.5) T ∂ α θ = ∂ α (T θ) − (∂ α λ)θ x ,
where ∂ α = ∂ x or ∂ y . Then if θ ∈ H 1 (Ω), (∂ α λ)θ x belongs to L 2 (Ω) since ∇λ is bounded, and by (2.5), T ∂ α θ belongs to L 2 (Ω) if and only if ∂ α (T θ) belongs to L 2 (Ω); (2.4) follows for m = 1. For m ≥ 2, we observe that
(2.6) T ∂ α θ = ∂ α (T θ) − [∂ α , λ]θ x ,
holds for all |α| = m. We note that for θ ∈ H m (Ω), [∂ α , λ]θ x belongs to L 2 (Ω) from Lemma B.1 iii) with d = 2, a = λ and k = 3 ∨ m. Hence by (2.6), for |α| = m, T ∂ α θ belongs to L 2 (Ω) if and only if ∂ α (T θ) belongs to L 2 (Ω), and (2.18) follows as well for m ≥ 2. Now, we need to show that the smooth functions are dense in X m 1 (Ω). Later on we will prove more involved density theorems, showing that if u ∈ X m 1 (Ω) vanishes on certain parts of ∂Ω, then u can be approximated in X m 1 (Ω) by smooth functions, vanishing on the same parts of the boundary. For the moment, we prove the following:
Proposition 2.2. C ∞ (Ω) ∩ X m 1 (Ω) is dense in X m 1 (Ω). Proof.
Using a proper covering of Ω by sets O 0 , O 1 , · · · , O N , we consider a partition of unity subordinated to this covering, 1 = N i=0 ψ i . Here and again in this section we will use a covering of Ω consisting of O 0 , a relatively compact subset of Ω, and of sets O i of one of the following types: O i is a ball centered at one of the corners of Ω, which does not intersect the two other sides of Ω; or O i is a ball centered on one of the sides of Ω which does not intersect any of the three other sides of Ω.
If θ ∈ X m 1 (Ω), then θψ i ∈ X m 1 (Ω) by (2.3), so that we only need to approximate θψ i by smooth functions. Here the support of ψ i is contained in the set O i , and we start with considering the set O 0 , relatively compact in Ω, then we consider the balls O i centered on the boundary ∂Ω.
For any function v defined on Ω, here and again in the following we denote byṽ the function equal to v in Ω and to 0 in R 2 \Ω. We first consider the case ψ i = ψ 0 and O i = O 0 which is relatively compact in Ω. Let ρ be a mollifier such that ρ ≥ 0, ρ = 1, and ρ has compact support.
i) The function v = θψ 0 ∈ X m 1 (Ω) has compact support in O 0 . Since O 0 is relatively compact in Ω, then for ǫ small enough, ρ ǫ * v is supported in Ω. Using the characterization (2.18) for v, the standard mollifier theory (see e.g. Appendix C in [Eva98]) shows that for ǫ → 0:
(2.7) ρ ǫ * v → v, in H m (Ω), ρ ǫ * T ∂ α v → T ∂ α v, in L 2 (Ω), ∀ |α| = m.
Since the convolution and the operator T do not commute in the non-constant coefficient case, we need the following Friedrichs' lemma (see [Fri44] or [Hor61, Theorem 3.1]).
Lemma 2.1. Let U be an open set of R d . If ∇a ∈ L ∞ (U ) and u ∈ L 2 loc (U ), then for all 1 ≤ j ≤ d, a∂ x j (u * ρ ǫ ) − (a∂ x j u) * ρ ǫ → 0, when ǫ → 0,
in the sense of L 2 convergence on all compact subsets of U .
We then continue the proof of Proposition 2.2. Noting that v = θψ 0 has compact support in Ω, we apply Lemma 2.1 with U = Ω, a = λ and u = ∂ α v; we obtain
(2.8) T (ρ ǫ * ∂ α v) − ρ ǫ * T ∂ α v → 0, ∀ |α| = m,
in L 2 (Ω) as ǫ → 0. Combining (2.7) 2 and (2.8), we obtain that as ǫ → 0,
(2.9) T (ρ ǫ * ∂ α v) → T ∂ α v, in L 2 (Ω), ∀ |α| = m.
Therefore, ρ ǫ * v converges to v in X m 1 (Ω) by (2.4), (2.7) 1 and (2.9). ii) We then consider the case where ψ i = ψ 1 , and O i = O 1 which is a ball centered at the origin (0, 0); the other cases are similar or simpler. Set v = θψ 1 , and note that v does not vanish in general on the boundary ∂Ω of Ω. In order to extend v to the whole space R 2 , we use a well known extension result (see e.g. [Gri85, Theorem 1.4.3.1]).
Lemma 2.2 (Extension Theorem). Since the boundary ∂Ω of the domain Ω is Lipschitz continuous, there exists a continuous linear operator P = P m from H m (Ω) into H m (R 2 ) such that for all u ∈ H m (Ω), the restriction of P u to Ω is u itself, i.e. P u| Ω = u.
We denote byv the extension P v given in Lemma 2.2, and then observe that, for all |α| = m,
(2.10) ∂ αv = ∂ α v + ν α 1 , T ∂ α v = T ∂ α v + ν α 2 ,
where ν α 1 1 is a measure supported by O 1 \ Ω, and ν α 2 is a measure supported by O 1 ∩ ∂Ω. The two identities in (2.10) together show that
(2.11) T ∂ αv = T ∂ α v + µ α , ∀ |α| = m,
where the µ α are measures supported by O 1 \ Ω. Let ρ be the same mollifier as before, but now ρ is compactly supported in {x < 0, y < 0}; then mollifying (2.11) with this ρ gives
(2.12) ρ ǫ * (T ∂ αv ) = ρ ǫ * T ∂ α v + ρ ǫ * µ α , ∀ |α| = m.
By the choice of ρ, ρ ǫ * µ α is supported outside of Ω. Hence, restricting (2.12) to Ω implies that:
(2.13) (ρ ǫ * (T ∂ αv )) Ω = ρ ǫ * (T ∂ α v) → T ∂ α v, as ǫ → 0,
in L 2 (Ω). Applying Lemma 2.1 with U = R 2 , a = λ and u = ∂ αv , we obtain that as ǫ → 0,
(2.14)
T (ρ ǫ * ∂ αv ) − ρ ǫ * T ∂ αv → 0, ∀ |α| = m,
in L 2 (Ω), which, combined with (2.13), implies that
(2.15) T (ρ ǫ * ∂ αv ) Ω → T ∂ α v, as ǫ → 0, in L 2 (Ω). If we setv ǫ = ρ ǫ * v, then as ǫ → 0,v ǫ →v in H m (R 2 ), and (2.16) v ǫ Ω → v, in H m (Ω); T ∂ α (v ǫ Ω ) → T ∂ α v, in L 2 (Ω), ∀ |α| = m,
which shows thatv ǫ Ω converges to v in X m 1 (Ω). Since we have to prove a density theorem involving the boundary values of the functions on ∂Ω, we first need to show that the desired traces at the boundary make sense. We thus prove the following trace result. Proposition 2.3 (A trace theorem). We assume that λ = λ(x, y) satisfies the positive 0-condition. If θ ∈ X 0 1 (Ω), then the traces of θ are defined on all of ∂Ω, i.e. the traces of θ are defined at x = 0, L 1 , and y = 0, L 2 , and they belong to the respective spaces H −1 y (0, L 2 ) and H −1
x (0, L 1 ). Furthermore the trace operators are linear continuous in the corresponding spaces, e.g., θ ∈ X 0 1 (Ω) → θ| x=0 is continuous from X 0 1 (Ω) into H −1 y (0, L 2 ). Proof. Since θ ∈ L 2 (Ω), we see that θ y belongs to L 2 x (0, L 1 ; H −1 y (0, L 2 )), which implies that λθ x belongs to L 2 x (0, L 1 ; H −1 y (0, L 2 )) by observing that T θ = λθ x + θ y ∈ L 2 (Ω). Using assumptions (2.1) and (2.2) when m = 0 for λ, we obtain θ x ∈ L 2
x (0, L 1 ; H −1 y (0, L 2 )), which, in combination with θ ∈ L 2
x (0, L 1 ; L 2 y (0, L 2 )), shows that θ ∈ C x ([0, L 1 ]; H −1 y (0, L 2 )). Hence, the traces of θ are well-defined at x = 0 and L 1 , and belong to H −1 y (0, L 2 ). The continuity of the corresponding mappings is easy. The proof for the traces at y = 0 and L 2 is similar.
We are now going to introduce density theorems involving the boundary values of the functions on ∂Ω. Here and throughout this article we denote by Γ 1 , Γ 2 , Γ 3 , Γ 4 the boundaries x = 0, x = L 1 , y = 0, y = L 2 respectively, and define Γ to be Γ 1 ∪ Γ 3 = {x = 0} ∪ {y = 0}. We also write θ| Γ = 0 as a short notation for θ| x=0 = θ| y=0 = 0, and we introduce the function spaces: Note that when m = 0, the space H 0 Γ (Ω) is the space L 2 (Ω). We first have the following characterizations for the space X m Γ (Ω). Proposition 2.4. For all integer m ≥ 0, we have
X m Γ (Ω) = {θ ∈ H m Γ (Ω) : T θ ∈ H m Γ (Ω), ∂ α θ Γ = 0, ∀ |α| = m}; for all integer m ≥ 1, we have (2.18) X m Γ (Ω) = {θ ∈ H m Γ (Ω) : T θ ∈ H m Γ (Ω)}. Proof.
It is clear that the first statement holds for m = 0, we thus only need to show the second statement. By definition of the spaces H m Γ (Ω) and X m Γ (Ω), we see that
X m Γ (Ω) ⊂ {θ ∈ H m Γ (Ω) : T θ ∈ H m Γ (Ω)}.
In order to prove the converse inclusion, let θ belongs to the right-hand side of (2.18), then it is clear that we only need to show that θ satisfies the boundary conditions ∂ α θ Γ = 0 for all |α| = m. Furthermore, we only need to show that
(2.19)
∂ m x θ| Γ 1 = 0, and ∂ m y θ| Γ 2 = 0, since the other boundary conditions involve the derivatives with respect to tangential di-
rections. Since T θ ∈ H m Γ (Ω), on Γ 1 = {x = 0}, we have (2.20) 0 = ∂ m−1 x (λθ x + θ y ) = m−1 k=0 ∂ m−k−1 x (λ(x, y))∂ k x θ x + ∂ m−1 x θ y + λ∂ m x θ,
which, together with θ ∈ H m Γ (Ω), implies that λ∂ m x θ = 0. We thus have ∂ m x θ = 0 on Γ 1 . Similarly, we can also show that ∂ m y θ = 0 on Γ 2 . We thus completed the proof.
As an immediate consequence of Proposition 2.1, we also find the following equivalent characterizations of the space X m Γ (Ω). Proposition 2.5.
X m Γ (Ω) = {θ ∈ H m (Ω) : T ∂ α θ ∈ L 2 (Ω), ∀ |α| = m; ∂ α θ Γ = 0, ∀ |α| ≤ m} = {θ ∈ H m Γ (Ω) : T ∂ α θ ∈ L 2 (Ω), ∂ α θ Γ = 0, ∀ |α| = m}. Recall that Γ = Γ 1 ∪ Γ 3 = {x = 0} ∪ {y = 0}
, and then we state the density theorems:
Theorem 2.1. V Γ (Ω) ∩ H m Γ (Ω) is dense in H m Γ (Ω)
. Theorem 2.2. Suppose that λ = λ(x, y) satisfies the positive m-condition. Then we have Theorem 2.2 considers the density of the function space X m Γ (Ω) involving functions like T θ, hence this result is not included in Theorem 2.1.
V Γ (Ω) ∩ X m Γ (Ω) is dense in X m Γ (Ω).
The proof of Theorem 2.1 is similar to or simpler than the proof of Theorem 2.2, we thus only prove Theorem 2.2. To prove Theorem 2.2, we proceed similarly as in the proof of [HT14a, Theorem 1]
Proof of Theorem 2.2. Let ρ(x, y) be a mollifier such that ρ(x, y) ≥ 0, ρdxdy = 1 and ρ has compact support in {0 < 1 2 x < y < 2x}. For θ ∈ X m Γ (Ω) and all α satisfying |α| ≤ m, we observe that,
(2.21) ∂ αθ = ∂ α θ + ν α 1 , T ∂ α θ = T ∂ α θ + ν α 2 , where ν α 1 and ν α 2 are measures supported by {x = L 1 } ∪ {y = L 2 }. Therefore, we have (2.22) T ∂ αθ = T ∂ α θ + µ α , ∀ |α| ≤ m,
where µ α = T ν α 1 + ν α 2 is a measure also supported by {x = L 1 } ∪ {y = L 2 }. We now setθ ǫ = ρ ǫ * θ, and mollifying (2.22) with ρ (see [Hor65]) gives
(2.23) ρ ǫ * T ∂ αθ = ρ ǫ * T ∂ α θ + ρ ǫ * µ α , ∀ |α| ≤ m.
By the choice of ρ, we have that ρ ǫ * µ α is supported in Ω c , and hence restricting (2.23) to Ω implies that
(2.24) (ρ ǫ * T ∂ αθ ) Ω = (ρ ǫ * T ∂ α θ) Ω → T ∂ α θ, in L 2 (Ω) as ǫ → 0.
Direct computation shows that
(2.25) T ∂ α (ρ ǫ * θ) − ρ ǫ * T ∂ αθ = λ ∂(ρ ǫ * ∂ αθ ) ∂x − ρ ǫ * λ ∂(∂ αθ ) ∂x → 0, ǫ → 0,
where the convergence is in L 2 (Ω) and achieved by applying Lemma 2.1 with U = R 2 , a = λ, and u = ∂ αθ . Combining (2.24) and (2.25) yields
T ∂ α (ρ ǫ * θ) Ω → T ∂ α θ, in L 2 (Ω), as ǫ → 0,
that is for all |α| ≤ m,
(2.26) T ∂ α (θ ǫ | Ω ) → T ∂ α θ, in L 2 (Ω), as ǫ → 0.
Similarly, by (2.21), we have for all |α| ≤ m,
(2.27) ∂ α (θ ǫ | Ω ) = (ρ ǫ * ∂ αθ )| Ω = (ρ ǫ * ∂ α θ)| Ω → ∂ α θ, in L 2 (Ω) as ǫ → 0,
where we used the fact that the support of ρ ǫ * ν α 1 is in Ω c . In conclusion, there holds
(2.28) θ ǫ | Ω → θ, in H m (Ω), as ǫ → 0, T ∂ α (θ ǫ | Ω ) → T ∂ α θ, in L 2 (Ω), as ǫ → 0, ∀ |α| = m.
Finally,θ ǫ Ω vanishes in a neighborhood of Γ since the support ofθ ǫ Ω is away from Γ by the choice of ρ. We thus completed the proof of Theorem 2.2.
Remark 2.2. Looking back carefully at the proof of Theorem 2.2, we see that Theorem 2.2 is also valid if Γ = Γ 2 ∪ Γ 4 . Moreover, we say that λ(x, y) satisfies the negative m-condition if λ(x, y) satisfies (2.2) and the following condition:
(2.1 ′ ) − c 1 ≤ λ(x, y) ≤ −c 0 ,
where c 0 , c 1 are positive constants. Theorem 2.2 is also true if Γ is Γ 1 ∪ Γ 4 or Γ 2 ∪ Γ 3 , and λ(x, y) satisfies the negative m-condition provided we choose properly the support of the mollifier.
The time dependent shallow water equations operator
In this section, we aim to study the semigroup property of the (modified) SWE operator (see below) with variable coefficients in the supercritical case on the Hilbert space H m Γ (Ω) (see (2.17a)) with Γ = Γ 1 ∪ Γ 3 . We will successively consider the time-independent and the time-dependent cases. The linearized SWE operator that we consider reads
(3.1) A( U )U = û u x +vu y + gφ x uv x +vv y + gφ ŷ uφ x +vφ y +φ(u x + v y ) , where U = (û,v,φ) t , U = (u, v, φ) t ; we set E 1 ( U ) = û 0 g 0û 0 φ 0û , E 2 ( U ) = v 0 0 0v g 0φv . Note that E 1 2 , E 2 admit a symmetrizer S 0 = diag(1, 1, g/φ), i.e. S 0 E 1 , S 0 E 2 are both symmet- ric.
In order to take advantage of that, we consider the following modified SWE operator:
(3.2) A 0 ( U )U = E 0 1 ( U )U x + E 0 2 ( U )U y , where E 0 1 ( U ) = S 1/2 0 E 1 ( U )S −1/2 0 = û 0 gφ 0û 0 gφ 0û , E 0 2 ( U ) = S 1/2 0 E 2 ( U )S −1/2 0 = v 0 0 0v gφ 0 gφv .
The reason why we choose the form (3.2) will become clear in the next section.
In the following, we assume that m ≥ 3, the cases when m = 0, 1, 2 are similar or simpler. Here, we only consider the generic case when U does not vanish, and we first consider the time-independent case. We thus assume that U only depends on the space variables x, y and that U satisfies the positive (m + 1)-condition (see (2.1)-(2.2)) introduced in Section 2, i.e.
(3.3)û,v,φ satisfy the positive (m + 1)-condition; the reason why we assume one more level of regularity on U will be explained below. As we indicated before, we only study the supercritical case, and we thus assume that U also satisfies the enhanced supercritical condition:
(3.4)û 2 − gφ ≥ c 2 2 ,v 2 − gφ ≥ c 2 2 , where c 2 is a positive constant.
3.1. Boundary conditions. We aim to determine the boundary conditions which are suitable for the system
(3.5) A 0 ( U )U = E 0 1 ( U )U x + E 0 2 ( U )U y = F, where F ∈ H m Γ (Ω) 3 .
With assumption (3.4), we see that E 0 1 and E 0 2 are both positive definite. Thus, it is natural to treat either the x-or y-direction as the time-like direction. Let us choose the y-direction, which means that we first need to specify the boundary conditions at y = 0 (time-like initial conditions); choosing the x-direction would lead to the same result. Multiplying both sides of (3.5) by
(E 0 2 ) −1 gives (3.6) U y + E 0 2 ( U ) −1 E 0 1 ( U )U x = E 0 2 ( U ) −1 F.
We set κ 0 ( U ) = g(û 2 +v 2 − gφ)/φ, and we explicitly compute the eigenvalues of (E 0 2 ) −1 E 0 1 :
(3.7) λ 1 =ûv +φκ 0 v 2 − gφ , λ 2 =ûv −φκ 0 v 2 − gφ , λ 3 =û v .
Note that all the eigenvalues λ 1 , λ 2 , λ 3 of (E 0 2 ) −1 E 0 1 are positive under assumption (3.4). Therefore, from the general hyperbolic theory (see Chapter 4 in [BS07]), it is necessary and sufficient to specify the boundary conditions for U at x = 0 in order to solve (3.6) in U .
In conclusion, in order to solve (3.5) in U , we need to specify the boundary conditions for U at x = 0 and y = 0. We then consider the homogeneous case and thus choose to specify the boundary conditions for U :
(3.8) U = 0, on Γ = Γ 1 ∪ Γ 3 = {x = 0} ∪ {y = 0}.
As we will see in Lemma 3.2 and Section 5, any sufficiently regular solution U for (3.5) and for the nonlinear equations (5.1) will satisfy the following compatibility boundary conditions:
(3.9) ∂ k x U = 0, on Γ 1 = {x = 0}, ∀ 0 ≤ k ≤ m, ∂ k y U = 0, on Γ 3 = {y = 0}, ∀ 0 ≤ k ≤ m,
which, by differentiating with respect to the tangential direction, is equivalent to
(3.9 ′ ) ∂ α U = 0, on Γ = Γ 1 ∪ Γ 3 , ∀ |α| ≤ m.
Hence in the following, we use the compatibility boundary conditions (3.9) rather than the boundary conditions (3.8) for the domain of the unbounded operator A defined below.
We write H k Γ = H k Γ (Ω) 3 for k ≥ 0, in which the functions vanish on Γ (the part of the boundary ∂Ω), and we endow the space H m Γ with the Hilbert scalar product and norm of H m (Ω) 3 :
U, U H m Γ = |α|≤m ∂ α U, ∂ α U L 2 (Ω) , U H m Γ = { U, U H m Γ } 1/2 ; we then define the unbounded operator A on H m Γ , by setting AU = A 0 ( U )U, ∀U ∈ D(A) and D(A) = {U ∈ H m Γ : A 0 ( U )U = E 0 1 ( U )U x + E 0 2 ( U )U y ∈ H m Γ }.
Note that the compatibility boundary conditions (3.9) are already taken into account in the domain D(A) (see also Propositions 2.4-2.5) since m ≥ 3. We also introduce the corresponding smooth function space V(Ω)
:= V Γ (Ω) 3 . Note that V(Ω) is dense in H m
Γ , which is a direct consequence of Theorem 2.1. We also have the following results.
Lemma 3.1. We assume that U satisfies the assumptions (3.3) and (3.4). Then:
i) V(Ω) ∩ D(A) is dense in D(A). ii) D(A) is dense in H m Γ .
Lemma 3.1 is proven below. In order to prove it, we need an equivalent characterization of the domain D(A), which will allow us to use the density results established in Section 2. We introduce the notations κ, Ξ, P such that
κ( U ) = û 2 +v 2 − gφ, P( U ) −1 = v −û κ v −û −κ uv gφ , Ξ = ξ 1 ξ 2 ξ 3 = P −1 U ; (3.10) then direct computations give P t E 0 1 P = diag(û κ + gφv 2(û 2 +v 2 )κ ,û κ − gφv 2(û 2 +v 2 )κ ,û u 2 +v 2 ) =: diag(a 1 , a 2 , a 3 ), P t E 0 2 P = diag(v κ + gφû 2(û 2 +v 2 )κ ,v κ − gφû 2(û 2 +v 2 )κ ,v u 2 +v 2 ) =: diag(b 1 , b 2 , b 3 ).
(3.11)
We then rewrite the modified SWE operator as
P t A 0 ( U )U = P t E 0 1 (PΞ) x + P t E 0 2 (PΞ) y = P t E 0 1 PΞ x + P t E 0 2 PΞ y + P t E 0 1 P x Ξ + P t E 0 2 P y Ξ = diag(a 1 , a 2 , a 3 )Ξ x + diag(b 1 , b 2 , b 3 )Ξ y + P t E 0 1 P x Ξ + P t E 0 2 P y Ξ.
(3.12)
Direct computations also show that a i , b i , i ∈ {1, 2, 3} are all positive away from 0, and thus both E 0 1 and E 0 2 are symmetric and positive definite. Using repeatedly Lemma B.1 and noting that U belongs to H m+1 (Ω), we see that
E 0 1 , E 0 2 , κ, P, P −1 , a i , b i , i ∈ {1
, 2, 3} belong to H m+1 (Ω). Furthermore, the last two terms from the right-hand side of (3.12) both belong to H m (Ω) 3 , and also to H m Γ since P = P( U ) belongs to H m+1 (Ω), and that is the reason why we impose one more regularity on U . Therefore, saying that
A 0 ( U )U belongs to H m Γ is equivalent to saying that a i ξ i,x + b i ξ i,y belongs to H m Γ (Ω) for all i ∈ {1, 2, 3}. Hence, the equivalent characterization of the domain D(A) is D(A) = {U = PΞ : Ξ ∈ H m Γ , a i ξ i,x + b i ξ i,y ∈ H m Γ (Ω), ∀ i ∈ {1, 2, 3}}. Proof of Lemma 3.1. We remark that the statement ii) directly follows from i) since V(Ω) is dense in H m Γ .
We thus only need to prove i).
Using the new characterization of D(A) and applying Theorem 2.2 with λ = a i /b i for all i ∈ {1, 2, 3}, we see that each component of Ξ can be approximated by smooth functions which vanish in a neighborhood of Γ = Γ 1 ∪ Γ 3 . Then transforming back to the variable U , we obtain that U can also be approximated by smooth functions in V(Ω). The proof is complete.
3.2. Energy estimate for the operator A. In the following, we denote by ·, · the L 2 (Ω)-scalar product, and observe that if
U H m (Ω) ≤ M with m ≥ 3 (see−AU, U H m Γ = |α|≤m − ∂ α (E 0 1 U x + E 0 2 U y ), ∂ α U = |α|≤m − E 0 1 (∂ α U ) x + E 0 2 (∂ α U ) y , ∂ α U + |α|≤m − [∂ α , E 0 1 ]U x , ∂ α U + [∂ α , E 0 2 ]U y , ∂ α U .
(3.13)
Integrating by parts on the first summation at the right-hand side of (3.13) gives
|α|≤m 1 2 − L 2 0 E 0 1 ∂ α U, ∂ α U x=L 1 x=0 dy − L 1 0 E 0 2 ∂ α U, ∂ α U y=L 2 y=0 dx + E 0 1,x + E 0 2,y ∂ α U, ∂ α U .
(3.14)
Using the compatibility boundary conditions (3.9 ′ ) and that both E 0 1 and E 0 2 are positive definite to dispense with the boundary terms in (3.14), we find that (3.14) is less than
|α|≤m 1 2 E 0 1,x + E 0 2,y ∂ α U, ∂ α U , which is dominated by 1 2 E 0 1,x L ∞ + E 0 2,y L ∞ U 2 H m Γ , which is finally bounded by C(M ) U 2 H m Γ from the Sobolev embedding H 2 (Ω) ⊂ L ∞ (Ω)
, and m ≥ 3. Applying Lemma B.1 iii) with k = m on the commutators from the right-hand side of (3.13), we obtain that the second summation at the right-hand side of (3.13) is bounded by
(3.15) |α|≤m C E 0 1 H m U x H |α|−1 ∂ α U L 2 + E 0 2 H m U y H |α|−1 ∂ α U L 2 , which in turn is bounded by C(M ) U 2 H m Γ .
Gathering the estimates for (3.14) and (3.15), (3.13) implies that
(3.16) −AU, U H m Γ ≤ C 1 (M ) U 2 H m Γ , which is (3.17) AU, U H m Γ ≥ −C 1 (M ) U 2 H m Γ .
Thanks to Lemma 3.1 i), we conclude that (3.17) holds for all U in D(A).
Remark 3.1. In the cases when m = 0, 1, 2, we can easily check that the energy estimate (3.17) for the operator A is also valid. Indeed, the estimate for the boundary terms is the same, and the estimate for the commutators are simpler by direct calculation with the assumption that U satisfies the positive m-condition (m = 0, 1, 2 see (2.1)-(2.2)).
3.3. The surjectivity of ω + A. We set ω 0 = C 1 (M ), where C 1 (M ) is the constant appearing in (3.17), and we prove the following lemma.
Lemma 3.2. Let ω be a real number which is greater than ω 0 . Then if F belongs to H m Γ with m ≥ 3, the equation
(3.18) E 0 1 ( U )U x + E 0 2 ( U )U y + ωU = F, associated with the following boundary conditions (3.19) U = 0, on Γ = Γ 1 ∪ Γ 3 = {y = 0} ∪ {x = 0}, admits a unique solution U in D(A).
Proof. Since E 0 2 is nonsingular, multiplying by (E 0 2 ) −1 on both sides of (3.18) gives (3.20) 19) at y = 0, we can conclude by induction that ∂ k y U | y=0 = 0 for all 0 ≤ k ≤ m. Similar results also holds for the x-direction. Therefore, the solution U also satisfies the compatibility boundary conditions:
U y + E 0 2 ( U ) −1 E 0 1 ( U )U x + E 0 2 ( U ) −1 ωU = E 0 2 ( U ) −1 F.(3.21) ∂ k x U = 0, on Γ 1 = {x = 0}, ∀ 0 ≤ k ≤ m, ∂ k y U = 0, on Γ 3 = {y = 0}, ∀ 0 ≤ k ≤ m, since F belongs to H m Γ (i.e.(3.22) E 0 1 (∂ α U ) x + E 0 2 (∂ α U ) y + ω∂ α U = ∂ α F − [∂ α , E 0 1 ]U x − [∂ α , E 0 2 ]U y .
Taking the L 2 (Ω) scalar product of each side of (3.22) with ∂ α U and integrating by parts, we arrive at
ω ∂ α U 2 L 2 (Ω) + 1 2 L 2 0 E 0 1 ∂ α U, ∂ α U x=L 1 x=0 dy + L 1 0 E 0 2 ∂ α U, ∂ α U y=L 2 y=0 dx = ∂ α F, ∂ α U + 1 2 E 0 1,x + E 0 2,y ∂ α U, ∂ α U − [∂ α , E 0 1 ]U x , ∂ α U − [∂ α , E 0 2 ]U y , ∂ α U .
(3.23)
The compatibility boundary conditions (3.21) and the fact that E 0 1 and E 0 1 are both positive definite imply that the boundary terms in the left-hand side of (3.23) are nonnegative, and thus the left-hand side of (3.23) is larger than ω ∂ α U 2 L 2 (Ω) . For the right-hand side of (3.23), we use the Cauchy-Schwarz inequality to estimate the first term and the same arguments as for (3.14)-(3.17) to estimate the last three terms; then summing (3.23) for all |α| ≤ m yields:
ω U 2 H m Γ ≤ F H m Γ U H m Γ + C 1 (M ) U 2 H m Γ ≤ F H m Γ U H m Γ + ω 0 U 2 H m Γ , (3.24)
with ω 0 being a constant depending only on M . This implies that U belongs to H m Γ by the assumption ω > ω 0 . Finally, AU also belongs to H m Γ since ∂ α (AU )| Γ = 0, ∀|α| ≤ m − 1 and Under these new assumptions, we see that the unbounded operator −A defined in Subsection 3.1 generates a strongly continuous semigroup with the same arguments as above, once we treat the time variable t as a parameter. To be more precise, we define a family of unbounded operators A(t)U on the Hilbert space H with A(t)U = A 0 ( U (t))U, ∀U ∈ D(A(t)| H ) and
AU = E 0 1 ( U )U x + E 0 2 ( U )U y = F − ωU. ThereforeD(A(t)| H ) = {U ∈ H : A 0 ( U (t))U = E 0 1 ( U (t))U x + E 0 2 ( U (t))U y ∈ H}, where H = H k Γ (= H k Γ((resp. H m+1 Γ ) is −A(t) admissible for all t ∈ [0, T ] with respect to H m−1 Γ (resp. H m Γ ). That H m Γ ⊂ D(A(t)| H m−1 Γ ) and H m+1 Γ ⊂ D(A(t)| H m Γ ) holds for all t ∈ [0, T ]
is clear from the definition, and finally, that
the mapping t → −A(t) is continous in the L(H m Γ , H m−1 Γ )-norm or L(H m+1 Γ
, H m Γ )-norm follows from the second assumption on U (see (3.25b) above). In conclusion, we find the following result.
The linear shallow water system
In this section, we aim to study the well-posedness of the linear shallow water system in certain Sobolev spaces using the evolution semigroups technique. Keeping the notations introduced in Section 3, the linear shallow water system reads in compact form
(4.1) U t + E 1 ( U )U x + E 2 ( U )U y + ℓ(U ) = F,
where ℓ(U ) = (−f v, f u, 0) t , and f is the Coriolis parameter. Note that F which does not appear in the linearized shallow water system (1.1) is added here for mathematical generality and also for the study of the non-homogeneous boundary conditions or for the nonlinear case. Observe that the system (4.1) is Friedrichs symmetrizable (see Chapter 1 in [BS07]) with symmetrizer S 0 = diag(1, 1, g/φ), and in order to take advantage of that, we make as before a change of variables by setting U = S 1 2 0 U and substitute into (4.1); we obtain a new system for U which reads . Using Theorem A.6, we obtain that the following system
(4.2) U t + A 0 ( U ) U + B( U ) U = S 1 2 0 F, where A 0 ( U ) U = E 0 1 ( U ) U x + E 0 2 ( U ) U y , B( U ) U = S 1 2 0 (S − 1 2 0 ) t + E 1 ( U )(S − 1 2 0 ) x + E 2 ( U )(S − 1 2 0 ) y U + ℓ( U ).(4.4) d U (t) dt = −(A(t) + B(t)) U (t) + S 1 2 0 F (t), U (0) = U 0 admits a unique solution U = U (t) ∈ C([0, T ]; H m Γ ) if U 0 ∈ H m Γ and F = F (t) ∈ C([0, T ]; H m Γ )
. Transforming back to the original variables, we obtain the following: Remark 4.1. Using the system (4.1), the solution U in Theorem 4.1 satisfies the compatibility boundary conditions (3.9) by the same argument as in Lemma 3.2.
Remark 4.2. Notice that if U 0 ∈ C ∞ (Ω) 3 , and U , F ∈ C ∞ ([0, T ]; C ∞ (Ω)), then the solution U provided by Theorem 4.1 belongs to C([0, T ]; H k (Ω) 3 ) for all k ≥ 0, which implies that U belongs to C([0, T ]; C ∞ (Ω)), and then by using the system (4.1), we conclude by induction that U actually belongs to C ∞ ([0, T ]; C ∞ (Ω)).
We lost two space derivatives from U to the solution U for the linear system (4.1) in Theorem 4.1, which is not sufficient for us to study the nonlinear case. In order to gain these two derivatives back, we need some additional a priori estimates. 4.1. A priori estimates. With Remark 4.2, we assume that U , U 0 , F, U are smooth functions satisfying the following system (4.5) (4.6)
U t + E 1 ( U )U x + E 2 ( U )U y + ℓ(U ) = F, U (0) = U 0 , U | Γ = 0, with U 0 , U (t), F (t)c 0 ≤ U ≤ c 1 , u 2 − gφ ≥ c 2 2 ,v 2 − gφ ≥ c 2 2 ,
where c 0 , c 1 , c 2 are positive constants. We will first derive L 2 a priori estimates for the linear system (4.5) and then extend the L 2 -estimates to H m -estimates with m ≥ 3. For the sake of simplicity, we write Ω T = Ω × [0, T ], and L ∞ (H k ) = L ∞ (0, T ; H k (Ω)) for all k = 1, · · · , m and L ∞ (L 2 ) = L ∞ (0, T ; L 2 (Ω)). We assume that
(4.7) U L ∞ (H m ) ≤ M, U t L ∞ (H m−1 ) ≤ M.
Multiplying (4.5) 1 by S 0 and taking the scalar product in L 2 (Ω) with U gives
(4.8) S 0 U t , U + S 0 E 1 U x + S 0 E 2 U y , U + S 0 ℓ(U ), U = S 0 F, U .
We now calculate
S 0 U t , U = 1 2 d dt S 0 U, U − 1 2 S 0,t U, U , S 0 ℓ(U ), U = 0;
(4.9) and, using integration by parts, we find that
S 0 E 1 U x + S 0 E 2 U y , U = 1 2 L 2 0 S 0 E 1 U, U x=L 1 x=0 dy + 1 2 L 1 0 S 0 E 2 U, U y=L 2 y=0 dx − 1 2 (S 0 E 1 ) x + (S 0 E 2 ) y U, U ≥ − 1 2 (S 0 E 1 ) x + (S 0 E 2 ) y U, U , (4.10)
where the last inequality results from the boundary conditions (4.5) 3 and the fact that S 0 E 1 , S 0 E 2 are both positive definite. Finally, we obtain the following inequality by gathering the calculations (4.9)-(4.10):
(4.11) d dt S 0 U, U ≤ S 0,t U, U + (S 0 E 1 ) x + (S 0 E 2 ) y U, U + 2 S 0 F, U .
We set I 0 (t) = S 0 U, U = S 1 2 0 U 2 L 2 (Ω) , and then the first two terms in the right-hand side of (4.11) are bounded by
S 0,t + (S 0 E 1 ) x + (S 0 E 2 ) y S −1 0 L ∞ (Ω T ) S 0 U, U ≤ C( U t L ∞ (Ω T ) , U x L ∞ (Ω T ) , U y L ∞ (Ω T ) , U L ∞ (Ω T ) )I 0 (t), (4.12)
which is dominated by C( U t L ∞ (H 2 ) , U L ∞ (H 3 ) )I 0 (t) by using the Sobolev embedding H 2 (Ω) ⊂ L ∞ (Ω). Using the Cauchy-Schwarz inequality, we estimate the last term in the right-hand side of (4.11):
2 S 0 F, U ≤ S 1 2 0 F 2 L 2 (Ω) + S 1 2 0 U 2 L 2 (Ω) ≤ C( U L ∞ (H 3 ) ) F (t) 2 L 2 (Ω) + I 0 (t).
(4.13)
Combining with (4.12) and (4.13), (4.11) implies that
d dt I 0 (t) ≤ C U t L ∞ (H 2 ) , U L ∞ (H 3 ) + 1 I 0 (t) + C( U L ∞ (H 3 ) ) F (t) 2 L 2 (Ω) ≤ r 1 ( U ) I 0 (t) + F (t) 2 L 2 (Ω) ,
(4.14)
where the constant
r 1 ( U ) = C U t L ∞ (H 2 ) , U L ∞ (H 3 ) + 1 only depends increasingly on U t L ∞ (H 2 ) , U L ∞ (H 3 )
. We observe that
r 1 ( U ) = r 1 ( U t L ∞ (H m−1 ) , U L ∞ (H m ) ) ≤ r 1 (M, M ) = r 1 (M ),
with m ≥ 3 by the assumption (4.7), and we write r 1 = r 1 (M ) ≥ 1 for the sake of simplicity. Using Gronwall's lemma for (4.14), we obtain
I 0 (t) ≤ e r 1 t (I 0 (0) + r 1 t 0 F (s) 2 L 2 (Ω) ds) ≤ e r 1 t r 1 · U 0 2 L 2 (Ω) + t 0 F (s) 2 L 2 (Ω) ds .
(4.15)
Noticing that
I 0 (t) = Ω (u 2 + v 2 + ĝ φ φ 2 )dxdy ≥ min(1, g/c 1 ) U (t) 2 L 2 (Ω) ,
and setting r 2 = 1/min(1, g/c 1 ), (4.15) implies that
(4.16) U (t) 2 L 2 (Ω) ≤ e r 1 t r 1 r 2 ( U 0 2 L 2 (Ω) + t 0 F (s) 2 L 2 (Ω) ds).
Taking the L ∞ -norm of (4.16) over [0, T ] immediately gives We now turn to extending the L 2 -estimate (4.17) to H m -estimate. Applying ∂ α = ∂ α 1 x ∂ α 2 y with |α| = α 1 + α 2 ≤ m to (4.5) and recalling that U satisfies the compatibility boundary conditions (3.9 ′ ), we obtain that ∂ α U satisfies the following equations (4.18)
(∂ α U ) t + E 1 ( U )(∂ α U ) x + E 2 ( U )(∂ α U ) y + ℓ(∂ α U ) = F α , ∂ α U (0) = ∂ α U 0 , ∂ α U | Γ = 0, where F α = ∂ α F − [∂ α , E 1 ]U x − [∂ α , E 2 ]U y .
Observing that (4.18) has the same form as (4.5), therefore proceeding exactly as for (4.17), we find
(4.19) ∂ α U 2 L ∞ (L 2 ) ≤ C 0 (M, T ) ∂ α U 0 2 L 2 (Ω) + T F α 2 L ∞ (L 2 ) ,
where C 0 (M, T ) is the same as in (4.17).
We now need to estimate F α . Lemma B.1 iii) with k = m on the commutators in F α gives Summing (4.19) for all |α| ≤ m and using the estimates (4.20) for F α , we finally arrive at
F α (t) 2 L 2 (Ω) ≤ ∂ α F (t) 2 L 2 (Ω) + C E 1 ( U (t)) 2 H m (Ω) U x (t) 2 H |α|−1 (Ω) + E 2 ( U (t)) 2 H m (Ω) U y (t) 2 H |α|−1 (Ω) ≤ ∂ α F (t) 2 L 2 (Ω) + C(M ) U (t) 2 H |α| (Ω) ,(4.21) U 2 L ∞ (H m ) ≤ C 0 (M, T ) U 0 2 H m (Ω) + T F 2 L ∞ (H m ) + T C(M ) U 2 L ∞ (H m ) ,
where the constants C(M ) may be different at different places, but they enjoy the same property, i.e. they only depend on the bound of the L ∞ (H m )-norm of U in an increasing way.
We choose T small enough so that C 0 (M, T )T C(M ) ≤ 1/2; with this choice of T , we are able to absorb the term U L ∞ (H m ) in the right-hand side of (4.21) and we find that
(4.22) U 2 L ∞ (H m ) ≤ 2C 0 (M, T ) U 0 2 H m (Ω) + T F 2 L ∞ (H m ) ,
where C 0 (M, T ) is the same as in (4.17). We emphasize the fact that the choice of T only depends on the bound M of the L ∞ (H m )-norm of U and the L ∞ (H m−1 )-norm of U t . Finally, we estimate the L ∞ (H m−1 )-norm of U t . We write (4.5) 1 as (4.23)
U t = F − ℓ(U ) − E 1 ( U )U x − E 2 ( U )U y ,
We first take H m−1 (Ω)-norm of (4.23) and use Lemma B.1 i) with s = m − 1 and d = 2 to estimate the last two terms in the right-hand side of (4.23); then we take L ∞ -norm over [0, T ], and we find
U t L ∞ (H m−1 ) ≤ F L ∞ (H m−1 ) + C E 1 ( U ) L ∞ (H m−1 ) U x L ∞ (H m−1 ) + f U L ∞ (H m−1 ) + C E 2 ( U ) L ∞ (H m−1 ) U y L ∞ (H m−1 ) ≤ F L ∞ (H m−1 ) + C( U L ∞ (H m ) , f ) U L ∞ (H m ) , (4.24)
where f is the Coriolis parameter. The inequality (4.24) shows that
(4.25) U t 2 L ∞ (H m−1 ) ≤ 2 F 2 L ∞ (H m−1 ) + 2C 1 (M ) U 2 L ∞ (H m ) ,
where C 1 (M ) only depends on M -the bound of U in L ∞ (H m ). We also obtain the following
L ∞ (L 2 )-estimate (4.26) U t 2 L ∞ (L 2 ) ≤ 2 F 2 L ∞ (L 2 ) + 2C 1 (M ) U 2 L ∞ (H 1 ) .
Improved regularity.
With the H m -estimates (4.22) and (4.25) at hand, we are now able to gain back the derivatives lost in Theorem 4.1 by shrinking down the time T , and we prove the following theorem.
Theorem 4.2. Let there be given
U 0 ∈ H m (Ω), F, U ∈ L ∞ (0, T ; H m (Ω)), U t ∈ L ∞ (0, T ; H m−1 (Ω)),
and furthermore we also assume that U 0 , F (t) satisfy the compatibility boundary conditions Proof. Let ρ(x, y), σ(t) be mollifiers such that ρ(x, y), σ(t) ≥ 0, ρdxdy = σdt = 1 and ρ has compact support in {0 < 1 2 x < y < 2x}. For a function w defined on Ω, (ρ ǫ * w)| Ω stands for the restriction to Ω of ρ ǫ * w, wherew is the extension of w by 0 outside Ω, and similar notations are also used for the functions defined in Ω T , or the vector functions (with the notation applied to each component of the vector functions). We then set
U ǫ 0 = (ρ ǫ * U 0 )| Ω , F ǫ = ((ρσ) ǫ * F )| Ω , U ǫ = ((ρσ) ǫ * U )| Ω .
Standard mollifier theory shows that U ǫ 0 , F ǫ , U ǫ converge to U 0 , F, U respectively as ǫ → 0 in the corresponding spaces 4 . Hence for ǫ small enough, we can assume that
U ǫ 0 2 H m (Ω) ≤ 2 U 0 2 H m (Ω) , F ǫ 2 L ∞ (H m ) ≤ 2 F 2 L 2 (H m ) ; U ǫ L ∞ (H m ) ≤ 2M, U ǫ t L ∞ (H m−1 ) ≤ 2M.
In addition, with the choice of ρ, we have that the support of U ǫ 0 is away from Γ = {x = 0}∪{y = 0}, and so is the support of F ǫ (t) for all t ∈ [0, T ]. Therefore, U ǫ 0 , F ǫ (t) also satisfy (3.9) for all t ∈ [0, T ]. Then using Theorem 4.1 and Remarks 4.1-4.2, there exists a smooth solution U ǫ for system (4.5) with U 0 , F, U replaced by U ǫ 0 , F ǫ , U ǫ , and U ǫ also satisfies the compatibility boundary conditions (3.9). For T > 0 small enough only depending on the bound of the L ∞ (H m )-norm of U and the L ∞ (H m−1 )-norm of U t , then the a priori estimates (4.22) gives that
U ǫ 2 L ∞ (H m ) ≤ 2C 0 (2M, T ) U ǫ 0 2 H m (Ω) + T F ǫ 2 L ∞ (H m ) ≤ 4C 0 (2M, T ) U 0 2 H m (Ω) + T F 2 L ∞ (H m ) .
(4.27)
where 2M is the bound of the L ∞ (H m )-norm of U ǫ and the L ∞ (H m−1 )-norm of U ǫ t . The inequality (4.27) gives a uniform bound on the sequence {U ǫ }, which implies that there exists a subsequence of {U ǫ } converging weak-star in L ∞ (0, T ; H m (Ω)). The next point is to prove that the sequence {U ǫ } is Cauchy in L ∞ (0, T ; L 2 (Ω)). For that purpose, we write (4.28)
W ǫ = U ǫ − U ǫ ′ ,
and subtracting the corresponding equations of form (4.5) satisfied by U ǫ and U ǫ ′ , we obtain (4.29)
W ǫ t + E 1 ( U ǫ )W ǫ x + E 2 ( U ǫ )W ǫ y + ℓ(W ǫ ) = F , W ǫ (0) = U ǫ 0 − U ǫ ′ 0 , W ǫ | Γ = 0, where F = F ǫ − F ǫ ′ + E 1 ( U ǫ ′ ) − E 1 ( U ǫ ) U ǫ ′ x + E 2 ( U ǫ ′ ) − E 2 ( U ǫ ) U ǫ ′ y .
Noticing that (4.29) has the same form as (4.5), therefore proceeding exactly as for (4.17), we obtain
(4.30) W ǫ 2 L ∞ (L 2 ) ≤ C 0 (2M, T ) U ǫ 0 − U ǫ ′ 0 2 L 2 (Ω) + T F 2 L ∞ (L 2 )
. Using the explicit expressions for E 1 and E 2 , direct computation shows that (4.31)
E 1 ( U ǫ ′ ) − E 1 ( U ǫ ) 2 L ∞ (L 2 ) , E 2 ( U ǫ ′ ) − E 2 ( U ǫ ) 2 L ∞ (L 2 ) ≤ 3 U ǫ − U ǫ ′ 2 L ∞ (L 2 )
. Therefore, combining the estimates in (4.30) and (4.31), we obtain
U ǫ − U ǫ ′ 2 L ∞ (L 2 ) ≤ C 0 (2M, T ) U ǫ 0 − U ǫ ′ 0 2 L 2 (Ω) + T F ǫ − F ǫ ′ 2 L ∞ (L 2 ) + 3T U ǫ − U ǫ ′ 2 L ∞ (L 2 ) ( U ǫ ′ x 2 L ∞ (Ω T ) + U ǫ ′ x 2 L ∞ (Ω T ) ) ,
(4.32) which, with the use of the Sobolev embedding H 2 (Ω) ⊂ L ∞ (Ω) and noting that m ≥ 3, is furthermore bounded by (4.33)
C 0 (2M, T ) U ǫ 0 − U ǫ ′ 0 2 L 2 (Ω) + T F ǫ − F ǫ ′ 2 L ∞ (L 2 ) + 6T U ǫ ′ 2 L ∞ (H m ) U ǫ − U ǫ ′ 2 L ∞ (L 2 )
. Since {U ǫ 0 } is a Cauchy sequence in L 2 (Ω), and {F ǫ }, { U ǫ } are Cauchy sequences in L ∞ (L 2 ), and {U ǫ } is uniformly bounded in the L ∞ (H m )-norm by (4.27), we obtain from (4.32)-(4.33) that {U ǫ } is also a Cauchy sequence in L ∞ (0, T ; L 2 (Ω)). Hence by L 2 − H m interpolation, the sequence {U ǫ } converges strongly in L ∞ (0, T ; H m−1 (Ω)) to a function U which belongs to L ∞ (0, T ; H m (Ω)). Using Proposition 2.3, we obtain that U satisfies the compatibility boundary conditions (3.9) since U ǫ satisfies (3.9).
The a priori estimates (4.25) give a uniform bound on the sequence Proceeding exactly as for (4.26), we obtain that
{U ǫ t }, i.e. U ǫ t 2 L ∞ (H m−1 ) ≤ 2 F ǫ 2 L ∞ (H m−1 ) + 2C 1 (2M ) U ǫ 2 L ∞ (H m ) ≤ 4 F 2 L ∞ (H m−1 ) + 2C 1 (2M ) U ǫ 2 L ∞ (H m ) ,(4.35) U ǫ t − U ǫ ′ t 2 L ∞ (L 2 ) ≤ 2 F 2 L ∞ (L 2 ) + 2C 1 (2M ) U ǫ − U ǫ ′ 2 L ∞ (H 1 ) , which implies that {U ǫ t }
is also a Cauchy sequence in L ∞ (L 2 ) by using the above estimates for F and noting that {U ǫ } is Cauchy in L ∞ (H m−1 ) with m ≥ 3. Therefore, by L 2 − H m−1 interpolation, we obtain that {U ǫ t } converges strongly in L ∞ (H m−2 ) to a function V which belongs to L ∞ (H m−1 ). Now passing to the limit, we obtain that U solves (4.5), and U t = V at least in the sense of distributions. Finally, proceeding exactly as in Subsection 4.1, we see that the solution U satisfies the estimates (4.22) and (4.25); the uniqueness directly follows from the estimate (4.22). We thus completed the proof.
Remark 4.3 (Non-homogeneous boundary conditions). Using Remark 9.1 in [HT14a], the existence of a solution for the linear system (4.1) associated with non-homogeneous boundary conditions can be obtained, we omit the details here; see [RTT08b] for a similar situation.
The fully nonlinear shallow water system
In this section, we aim to investigate the well-posedness for Eqs. (1.1) associated with suitable initial conditions and homogeneous boundary conditions, and we will make a remark about the case of non-homogeneous boundary conditions. Keeping the notations introduced in Section 3 and 4, the fully nonlinear shallow water system reads in compact form
(5.1) U t + E 1 (U )U x + E 2 (U )U y + ℓ(U ) = 0.
5.1. Stationary solution. We want to study system (5.1) near a stationary solution, and we start by constructing such a stationary solution (u, v, φ) = (u s , v s , φ s ). These functions are independent of time and satisfy
(5.2) E 1 (U )U x + E 2 (U )U y + ℓ(U ) = 0.
The existence of the general stationary solution U s to (5.2), which satisfies the supercritical condition, is a 1-dimensional hyperbolic problem if we multiply by E −1 2 on both sides of (5.2) and treat the y-direction as the time-like direction as we already did in Lemma 3.2. The general results in [BS07,Chapter 11] guarantee the existence of the stationary solution U s if we specify suitable initial (y-direction) and boundary (x-direction) conditions. Actually, Subsection 2.1 in [HPT11] provides an y-independent stationary solution to (5.2) satisfying the supercritical condition. But in what follows, we think of U s in a general form (i.e. U s depends both x and y).
We thus choose our stationary solution U s = (u s , v s , φ s ) t of (5.2) satisfying a strong form of the supercritical condition, i.e.
(5.3) 2c 0 ≤ u s , v s , φ s ≤ 1 2 c 1 , u 2 s − gφ s ≥ 2c 2 2 , v 2 s − gφ s ≥ 2c 2 2 ,
where c 0 , c 1 , c 2 are given, positive constants which will play the same role as those in (4.6).
We set U = U s + U . Note that if we choose δ sufficiently small so that if | U | ≤ δ (i.e. |ũ|, |ṽ|, |φ| < δ), then U satisfies relations similar to (5.3), that is:
(5.4) c 0 ≤ũ + u s ,ṽ + v s ,φ + φ s ≤ c 1 , (ũ + u s ) 2 − g(φ + φ s ) ≥ c 2 2 , (ṽ + v s ) 2 − g(φ + φ s ) ≥ c 2 2 .
The relations (5.4) will guarantee that we remain in the supercritical case.
We then substitute U = U s + U into (5.1); we obtain a new system for U , and dropping the tildes, our new system reads:
(5.5) U t + E 1 (U + U s )U x + E 2 (U + U s )U y + ℓ(U ) = F U , where F U = −E 1 (U + U s )U s,x − E 2 (U + U s )U s,y − ℓ(U s ) = uu s,x + vu s,y uv s,x + vv s,y uφ s,x + vφ s,y + φ(u s,x + v s,y ) . (5.6)
In order to have the last equality, we use the fact that U s is a stationary solution satisfying (5.2). We supplement (5.5) with the following initial and homogeneous boundary conditions:
(5.7) I.C. U (0) = U 0 , B.C. U | x=0,y=0 = 0.
Observe that we can rewrite (5.5) as
(5.8) U x = E 1 (U + U s ) −1 F U − U t − E 2 (U + U s )U y − ℓ(U ) ;
at x = 0, we immediately see that U x | x=0 = 0. Applying ∂ k x with 1 ≤ k ≤ m − 1 to (5.8), we can conclude by induction that ∂ k x U | x=0 = 0, for all k = 0, · · · , m. Similarly, we also obtain that ∂ k y U | y=0 = 0 for all k = 0, · · · , m at y = 0. Therefore, if U satisfies (5.5) and (5.7), then U also satisfies the compatibility boundary conditions (3.9), i.e.
(5.9) ∂ k x U = 0, on Γ 1 = {x = 0}, ∀ 0 ≤ k ≤ m, ∂ k y U = 0, on Γ 3 = {y = 0}, ∀ 0 ≤ k ≤ m. 5.
2. Nonlinear shallow water system. In order to be able to solve the nonlinear system (5.5)-(5.7), we require the initial and boundary conditions to be compatible. We thus assume that U 0 satisfies the compatibility boundary conditions (5.9) (i.e. (3.9)). We are now on the stage to prove the following result. where ν m denotes the norm of the Sobolev embedding H m (Ω) ⊂ L ∞ (Ω). We are also given the initial condition U 0 ∈ H m (Ω) which satisfies (5.9) and
(5.10)
U 0 2 H m (Ω) ≤ min M 0 , M 0 2C(U s ) + 4C 1 (M )C 0 (M, 1) , M 0 4C 0 (M, 1) ,
where C 0 (M, 1) = e r 1 (M ) r 1 (M )r 2 (resp. C 1 (M )) is the constant appearing in (4.17) (resp. (4.25)), and C(U s ) only depends on the bound of the H m+1 (Ω)-norm of U s (see (5.18) below). Then there exists T > 0 only depending on the initial data U 0 and the stationary solution U s such that the system (5.5)-(5.7) admits a unique solution U satisfying U ∈ L ∞ (0, T ; H m (Ω)), U t ∈ L ∞ (0, T ; H m−1 (Ω)).
Proof. As a preliminary, we choose T such that T ≤ 1.
Considering the compatibility boundary conditions (5.9), the resolution of the nonlinear system (5.5)-(5.7) will be done using the following iterative scheme (5.11)
U k+1 t + E 1 (U k + U s )U k+1 x + E 2 (U k + U s )U k+1 y + ℓ(U k+1 ) = F U k , U k+1 (0) = U 0 , U k+1 | Γ = 0.
We initiate our iteration scheme by setting U 0 = U 0 , and then construct the approximate solutions U k by induction.
If we assume that then we have that F U k also satisfies (5.9). Using the Sobolev embedding H m (Ω) ⊂ L ∞ (Ω), the L ∞ -norm of U k is controlled by δ, which shows that U k satisfies the supercritical condition (5.4), i.e.
c 0 ≤ u k + u s , v k + v s , φ k + φ s ≤ c 1 , (u k + u s ) 2 − g(φ k + φ s ) ≥ c 2 2 , (v k + v s ) 2 − g(φ k + φ s ) ≥ c 2 2 ; furthermore, we have U k + U s L ∞ (H m ) ≤ M 0 + U s H m (Ω) ≤ M, (U k + U s ) t L ∞ (H m−1 ) ≤ M 0 ≤ M.
Therefore, for T > 0 small enough only depending on M , applying Theorem 4.2 to (5.11) with U = U k + U s , F = F U k , gives a solution U k+1 which satisfies (5.9), and that
U k+1 2 L ∞ (H m ) ≤ 2C 0 (M, T ) U 0 2 H m (Ω) + T F U k 2 L ∞ (H m ) ≤ (using Lemma B.1 i) with s = m and d = 2 for F U k ) ≤ 2C 0 (M, 1) U 0 2 H m (Ω) + T C U s 2 H m+1 (Ω) U k 2 L ∞ (H m ) ,(5.13)
and
U k+1 t 2 L ∞ (H m−1 ) ≤ 2 F U k 2 L ∞ (H m−1 ) + C 1 (M ) U k+1 2 L ∞ (H m ) , (5.14)
where C 0 (M, T ) (resp. C 1 (M )) is the constant appearing in (4.17) (resp. (4.25)), and M is the bound of the L ∞ (H m )-norm of U (= U k + U s ) and the L ∞ (H m−1 )-norm of U t (= U k t ). It follows from the explicit form (5.6) of F U that
(5.15) (F U ) t = F Ut .
In particular, (F U k ) t = F U k t . Using also that U k | t=0 = U 0 , we obtain with the mean value theorem that
(5.16) |F U k (t)| ≤ |F U k (t) − F U 0 | + |F U 0 | ≤ t|F U k t (t ′ )| + |F U 0 |, for all t ∈ [0, T ]
and for some t ′ ∈ (0, t), which implies that
F U k 2 L ∞ (H m−1 ) ≤ 2T 2 F U k t 2 L ∞ (H m−1 ) + 2 F U 0 2 L ∞ (H m−1 ) ≤ (using Lemma B.1 i) with s = m − 1 and d = 2) ≤ 2T 2 C U s 2 H m (Ω) U k t 2 L ∞ (H m−1 ) + 2C U s 2 H m (Ω) U 0 2 H m−1 (Ω) .
(5.17)
Gathering the estimates (5.13)-(5.17), and using the assumption (5.12), we finally arrive at
(5.18) U k+1 2 L ∞ (H m ) ≤ 2C 0 (M, 1) U 0 2 H m (Ω) + T C 0 (M, 1)M 0 C(U s ), U k+1 t 2 L ∞ (H m−1 ) ≤ C(U s ) + 2C 1 (M )C 0 (M, 1) U 0 2 H m (Ω) +T M 0 C(U s ) T + 2C 1 (M )C 0 (M, 1) ,
where C(U s ) only depends on the bound of the H m+1 (Ω)-norm of U s . Note that the first two terms in the right-hand side of (5.18) are less than M 0 /2 by the assumption (5.10), and both the second terms in the right hand side of (5.18) approach 0 when T → 0; we thus can choose T small enough again such that
(5.19) U k+1 2 L ∞ (H m ) ≤ M 0 , U k+1 t 2
L ∞ (H m−1 ) ≤ M 0 , U k+1 satisfies (5.9). Now U k+1 also satisfies (5.12); hence we can continue our construction. Let us emphasize that the choice of T only depends on M 0 , M , U s and is independent of k, therefore our iteration scheme can be conducted for all k, and we can construct the sequence {U k } as long as the starting point U 0 satisfies (5.12), which holds true by the assumption (5.10).
We now have an uniformly bounded sequence {U k } at hand, and the next point is to show that the sequence {U k } is Cauchy in L ∞ (0, T ; L 2 (Ω)), which is almost achieved in the proof of Theorem 4.2. Let us write (4.29), proceeding exactly as for (4.32)-(4.33) and noticing that W k+1 (0) = 0, we obtain
W k+1 = U k+1 − U k ; with W ǫ , U ǫ , U ǫ ′ , U ǫ , U ǫ ′ , F ǫ , F ǫ ′ replaced by W k+1 , U k+1 , U k , U k + U s , U k−1 + U s , F U k , F U k−1 inU k+1 − U k 2 L ∞ (L 2 ) ≤ C 0 (M, T )T F U k − F U k−1 2 L ∞ (L 2 ) + 6 U k 2 L ∞ (H m ) U k − U k−1 2 L ∞ (L 2 ) .
(5.20)
Using the explicit expression (5.6) for F U and the Sobolev embedding H m (Ω) ⊂ L ∞ (Ω), we estimate
F U k − F U k−1 2 L ∞ (L 2 ) ≤ C( U s L ∞ (Ω) ) U k − U k−1 2 L ∞ (L 2 ) ≤ C( U s H m (Ω) ) U k − U k−1 2 L ∞ (L 2 ) .
(5.21)
Combining (5.20) and (5.21) and using the uniform boundedness of the L ∞ (0, T ; H m (Ω))norm of U k , we obtain
(5.22) U k+1 − U k 2 L ∞ (L 2 ) ≤ C 0 (M, T )T C( U s H m (Ω) ) + 6M 0 U k − U k−1 2 L ∞ (L 2 )
. Upon reducing T again, we can assume that
(5.23) C 0 (M, T )T C( U s H m (Ω) ) + 6M 0 ≤ 1 4 ;
then the inequality (5.22) implies that
(5.24) U k+1 − U k L ∞ (L 2 ) ≤ 1 2 U k − U k−1 L ∞ (L 2 ) ≤ · · · ≤ ( 1 2 ) k U 1 − U 0 L ∞ (L 2 )
. Therefore {U k } is Cauchy in L ∞ (L 2 ); let U be the limit of this sequence. Note also that {U k } is uniformly bounded in L ∞ (H m ), so by L 2 − H m interpolation, the sequence {U k } converges strongly in L ∞ (H m−1 ) to U ∈ L ∞ (H m ). Similarly as for (4.35), we can obtain that {U k t } converges strongly in L ∞ (H m−2 ) to a function V ∈ L ∞ (H m−1 ). Now passing to the limit in (5.11), we obtain that U solves (5.5) and U t = V , and that U belongs to L ∞ (0, T ; H m (Ω)) and U t belongs to L ∞ (0, T ; H m−1 (Ω)). The uniqueness directly follows from (5.22). This completes the proof.
Remark 5.1 (Non-homogeneous boundary conditions). With Remark 4.3, the existence of a solution for the iterative scheme (5.11) associated with non-homogeneous boundary conditions can be obtained, and by passing to the limit, the nonlinear system (5.5) associated with non-homogeneous boundary conditions admits a unique solution; we omit the details here.
Remark 5.2. After completing this article, we found that we can also use a finite difference method to prove the existence and uniqueness of the fully nonlinear SWE (i.e. Theorem 5.1) by observing that we have the energy estimates (3.24) for the corresponding boundary value problem (although slightly different), which is the one we only need for the finite difference method. We omit the details here.
The finite difference method has the advantages that we do not need the density theorems in Section 2 and the evolution semigroup technique. However, the evolution semigroup technique has its own advantage that it tells us how we lost two space derivatives for the well-posedness of the linear SWE (see Theorem 4.1), and that explains why we only have local well-posedness for the fully nonlinear SWE in some sense.
Appendix A. Preliminary results about semigroups and evolution families
This appendix collects some basic facts on the semigroups and evolution families and the characterization of their generators. The main references are the classical books by K. Yosida [Yos80], and by E. Hille and R.S. Phillips [HP74], and by A. Pazy [Paz83] and the book by K.-J. Engel and R. Nagel [EN00].
Definition A.1. A family (S(t)) t≥0 of bounded linear operators on a Banach space X is called a strongly continuous (one-parameter) semigroup (or C 0 -semigroup) if it satisfies i) S(0) = I, S(t + s) = S(t)S(s) for all t, s ≥ 0; ii) ξ x : t → ξ x (t) := S(t)x is continuous from R + into X for every x ∈ X. Note that if ξ x (t) is right differentiable in t at t = 0, it is also differentiable at t, for any t ≥ 0.
Theorem A.1 (Hille-Yosida theorem). Let (A, D(A)) be a positive operator on a Hilbert space H such that ω + A is surjective for some ω > 0. Then −A generates a contraction semigroup.
We recall that on a Hilbert space, the linear operator A is called positive if Ax, x ≥ 0 for all x ∈ D(A).
Theorem A.2 (Uniqueness Theorem). Let (A, D(A)) be a closed, densely defined operator on a Banach space X, and let Y be a subspace of X which is continuously embedded in X (in symbols: Y ֒→ X). The part of A in Y is the operator A | defined by A | y := Ay with domain D(A | ) = {y ∈ D(A) ∩ Y : Ay ∈ Y }. Suppose that (A, D(A)) generates a strongly continuous semigroup (S(t)) t≥0 on X, and (A | , D(A | )) also generates a strongly continuous semigroup (R(t)) t≥0 on Y . Then (A.2) S(t)y = R(t)y holds for all y ∈ Y and t ≥ 0. Furthermore, the subspace Y is A-admissible, i.e. Y is an invariant subspace of S(t), t ≥ 0, and the restriction of S(t) to Y , which is R(t), is a strongly continuous semigroup on Y .
Proof. The identity (A.2) immediately follows from the uniqueness of the following Cauchy problem u(t) = Au(t), ∀ t ≥ 0, u(0) = y ∈ Y ;
and that Y is A-admissible follows from (A.2) and the assumption. Let I be the interval [0, T ], and (A(t), D(A(t)) t∈I be the family of infinitesimal generators of strongly continuous semigroups S t (s), s ≥ 0, on a Banach space X, and let Y be a Banach space which is densely and continuously embedded into X. The following stability definition appeared in [Kat70,Kat73]:
Definition A.5 (Kato-stability and Kato-condition). We say that the family {A(t)} t∈I is Kato-stable, if there exist constants M ≥ 1 and ω ∈ R such that Π n j=1 e s j A(t j ) ≤ M e ω n j=1 s j , holds for every time-ordered sequence (t 1 , · · · , t n ) in I and s j ≥ 0; and we say that the family {A(t)} t∈I satisfies the Kato-condition on I if the following conditions are satisfied: i) {A(t)} t∈I is Kato-stable in X. ii) Y is A(t)-admissible for all t ∈ I, and the family {A(t)| Y } of the part of A(t) in Y is Kato-stable in Y . iii) Y ⊂ D(A(t)) holds for all t ∈ I, and for all t ∈ I, A(t) is a bounded operator from Y into X and the mapping t → A(t) is continuous in the L(Y, X) norm · Y →X .
Remark A.1. If for all t ∈ I, A(t) is the infinitesimal generator of a quasi-contraction (see Definition A.2) semigroup R t (s) satisfying R t (s) ≤ e ωs , then the family {A(t)} t∈I is clearly Kato-stable with constants M = 1 and ω.
(E 2 ) ∂ + t W (t, s)v t=s = A(s)v, ∀ v ∈ Y, 0 ≤ s ≤ T,
if we choose F to be a C ∞ function such that F(x) = 0 for |x| ≤ ǫ 0 /2 and F(x) = 1/x for |x| ≥ ǫ 0 . iii) If k is an integer greater than d/2+1 and α is a d-tuple of length |α| ∈ [1, k], there exists C > 0 depending only on k and U such that for all a in H k (U ) and all u ∈ H |α|−1 (U ), we have the following estimate:
[∂ α , a]u L 2 (U ) ≤ C a H k (U ) u H |α|−1 (U ) .
Date: March 3, 2015. Key words and phrases. Shallow water equations, inviscid flow, initial and boundary value problems.
H m Γ (Ω) = {θ ∈ H m (Ω) : ∂ α θ Γ = 0, ∀ |α| ≤ m − 1}, (2.17a) X m Γ (Ω) = {θ ∈ H m (Ω) : T θ = λ(x, y)θ x + θ y ∈ H m (Ω), ∂ α θ Γ = 0, ∀ |α| ≤ m},(2.17b) V Γ (Ω) = {θ ∈ C ∞ (Ω) : θ vanishes in a neighborhood of Γ}. (2.17c)
Remark 2 . 1 .
21Theorem 2.1 generalizes the classical density results, i.e. that C ∞ (Ω) is dense in H m (Ω), and that C ∞ c (Ω) is dense in H m 0 (Ω), to the functions which vanish on part of the boundary ∂Ω.
the cases when m = 0, 1, 2), then the H m -norm of the functions E 0 1 ( U ), E 0 2 ( U ) are bounded by some constant C(M ). Here and again in this section, the constant C(M ) may be different at different places, but it only depends on M . Then for U smooth in D(A), we compute
Let us treat again the y-direction as the time-like direction; then (3.20) becomes a 1dimensional hyperbolic system. We observe that the boundary (x-direction only) is a regular open subset in R, and that the boundary conditions satisfy the uniform Lopatinskii condition (see [BS07, Chapter 9] or [CP82, Chapter 7]). Hence, the general results in [BS07, Chapter 9] (see also [CP82, Chapter 7]) guarantee the existence and uniqueness of a solution U for (3.20) and (3.19). Using (3.20) and the boundary condition (3.
U belongs to D(A), and the proof is complete. 3.4. Semigroup. We now set B = ω 0 +A, with D(B) = D(A); then (B, D(B)) is a positive operator on H m Γ by virtue of (3.17), and ω + B (=ω + ω 0 + A) is surjective for all ω > 0 thanks to Lemma 3.2. Hence, Theorem A.1 (the Hille-Yosida theorem) implies that the operator −B generates a contraction semigroup on H m Γ , and we then obtain the following result as a consequence of Theorem A.3 (Bounded Perturbation Theorem I).
Theorem 3. 1 .
1The operator (−A, D(A)) generates a quasi-contraction semigroup (R(t)) t≥0 on H m Γ = H m Γ (Ω) 3 satisfying R(t) ≤ e ω 0 t , ∀t ≥ 0.Remark 3.2. The constant ω 0 in Theorem 3.1 only depends on the H m -norm of U .3.5. Time-dependent modified SWE operator. In this subsection, we consider the case where U also depends on the time variable t, and we impose the following assumptions on U :(3.25a) U (t) satisfies the positive (m + 2)-condition for all t ∈ [0, T ], i.e. U (t) belongs to H m+2 (Ω) (m ≥ 3) and it satisfies the condition (2.1) with c 0 , c 1 independent of t ∈ [0, T ]; (3.25b) U belongs to C([0, T ]; H m+1 (Ω)); (3.25c) U satisfies the supercritical condition (3.4) with c 2 independent of t ∈ [0, T ].
Ω) 3
3) and k can be either m − 1, m or m + 1. Since the positive (m + 2)-condition implies the positive (m + 1)-and m-conditions, we thus obtain the following result as an immediate consequence of Theorem 3.1.
Corollary 3. 1 .
1The operators {−A(t)} t generate quasi-contraction semigroups (R t,1 (s)) s≥0 on H m−1 Γ , (R t (s)) s≥0 on H m Γ , and (R t,2 (s)) s≥0 on H m+1 Γ , and they satisfyR t,1 (s) ≤ e C(M 1 )s , R t (s) ≤ e C(M )s , R t,2 (s) ≤ e C(M 2 )s , for all s ≥ 0, where M 1 is the norm of U in C([0, T ]; H m−1 (Ω)), M is the norm of U in C([0, T ]; H m (Ω)), and M 2 is the norm of U in C([0, T ]; H m+1 (Ω)
Lemma 3 . 3 .
33Assume that U satisfies the assumptions (3.25a)-(3.25c) and m ≥ 3. Then the family {−A(t)} t∈[0,T ] satisfies the Kato-condition (see Definition A.5) with X = H m−1 Γ , Y = H m Γ or X = H m Γ , Y = H m+1 Γ .
(4. 3 )
3If we now assume that U satisfies the conditions introduced in Subsection 3.5, then the family ofoperators {−A(t)} t∈[0,T ] satisfies the Kato-condition (see Lemma 3.3). If we further assume that U belongs to C([0, T ]; H m+2 (Ω)) ∩ C 1 ([0, T ]; H m+1 (Ω)), then the operators {B(t)} t∈[0,T ] defined by B(t) U = B( U ) U are bounded operators on all the three spaces H m−1 Γ , H m Γ , H m+1 Γ by using the estimates in Lemma B.1. Therefore, with Theorem A.4 (Bounded Perturbation Theorem II), the family of operators {−A(t) − B(t)} t∈[0,T ] is a Kato-stable family, and furthermore, we have
Lemma 4 . 1 .
41The family {−A(t)−B(t)} t∈[0,T ] satisfies the Kato-condition with X = H m−1 Γ , Y = H m Γ or X = H m Γ , Y = H m+1 Γ .Combining TheoremA.5 and Lemma 4.1, we obtain an evolution family on H m−1 Γ and another evolution family on H m Γ . From the uniqueness in Theorem A.5, we see that these two evolution families coincide on H m Γ . Then this unique evolution family satisfies (E 1 ) − (E 3 ) with X = H m−1 Γ and satisfies (E 4 ) − (E 5 ) with Y = H m Γ (see Theorem A.5 and A.6)
Theorem 4 . 1 .
41Let there be given U 0 ∈ H m Γ = H m Γ (Ω) 3 and F = F (t) ∈ C([0, T ]; H m Γ ). We also assume that the U (t) are given for all t ∈ [0, T ] such that
( 1 )
1U (t) satisfies the positive m+2-condition for all t ∈ [0, T ], i.e. U (t) belongs to H m+2 (Ω) (m ≥ 3) and it satisfies the condition (2.1) with c 0 , c 1 independent of t ∈ [0, T ], (2) U belongs to C([0, T ]; H m+2 (Ω)) ∩ C 1 ([0, T ]; H m+1 (Ω)), (3) U satisfies the supercritical condition (3.4) with c 2 independent of t ∈ [0, T ]. Then the system (4.1) associated with the initial condition U (0) = U 0 has a unique solution U = U (t) which belongs to C([0, T ]; H m Γ ) ∩ C 1 ([0, T ];
satisfying the compatibility boundary conditions (3.9) (i.e. (3.9 ′ )) for all t ∈ [0, T ], and Γ = {x = 0} ∪ {y = 0}. In addition, U is positive away from 0 and satisfies the supercritical condition (3.4), i.e.
L
∞ (L 2 ) ≤ C 0 (M, T ) U 0 2 L 2 (Ω) + T F 2 L ∞ (L 2 ) ,where C 0 (M, T ) = e r 1 (M )T r 1 (M )r 2 , only depends on the bound of the L ∞ (H m )-norm of U and the L ∞ (H m−1 )-norm of U t .
(M ) only depends on M -the bound of the L ∞ (H m )-norm of U .
( 3 . 9 )
39for t ∈ [0, T ], and that U satisfies (4.6) and (4.7). Then there exists T > 0 small enough depending only on the bound of the L ∞ (H m )-norm of U and the L ∞ (H m−1 )-norm of U t such that the system (4.1) associated with the initial condition U (0) = U 0 and the homogeneous boundary conditions (3.8) has a unique solution U = U (t) such that U ∈ L ∞ (0, T ; H m (Ω)), U t ∈ L ∞ (0, T ; H m−1 (Ω)), and the solution U satisfies the compatibility and boundary conditions (3.9) and the estimates (4.22) and (4.25).
sequence {U ǫ } are uniformly bounded in the L ∞ (H m )-norm.
Theorem 5 . 1 .
51Let there be given the stationary solution U s ∈ H m+1 (Ω) with m ≥ 3, and two positive constants M 0 , M such that M 0 , M > 0, M 0 ∈ (0, δ ν m ], M = M 0 + U s H m (Ω) ,
L
∞ (0,T ;H m−1 (Ω)) ≤ M 0 , U k satisfies (5.9),
Proposition A. 1 .
1For every strongly continuous semigroup (S(t)) t≥0 , there exist constants ω ∈ R and M ≥ 1 such that(A.1) S(t) ≤ M e ωtfor all t ≥ 0.Definition A.2. A strongly continuous semigroup is called quasi-contraction if we can take M = 1 in (A.1), and called bounded if ω = 0, and called contraction if ω = 0 and M = 1 is possible. Definition A.3. The generator A : D(A) ⊂ X → X of a strongly continuous semigroup (S(t)) t≥0 on a Banach space X is the operatorAx :=ξ x (0) = lim h↓0 1 h (S(h)x − x)defined for every x in its domainD(A) := {x ∈ X : t → ξ x (t) is right differentiable in t at t = 0}.
Theorem A. 3 (
3Bounded Perturbation Theorem I). Let (A, D(A)) be the infinitesimal generator of a strongly continuous semigroup (S(t)) t≥0 on a Banach space X satisfyingS(t) ≤ M 0 e ωt , ∀ t ≥ 0, where ω ∈ R, M 0 ≥ 1. If B ∈ L(X), then C := A + B, with D(C) := D(A) generates a strongly continuous semigroup (R(t)) t≥0 satisfying R(t) ≤ M 0 e (ω+M 0 B )t , ∀ t ≥ 0.The following results are taken from [Paz83, Chapter 5].Definition A.4. A two parameter family of bounded linear operators W (t, s), 0 ≤ s ≤ t ≤ T , on a Banach space X is called an evolution system if the two following conditions are satisfied:i) W (s, s) = I, W (t, r)W (r, s) = W (t, s) for 0 ≤ s ≤ r ≤ t ≤ T ; ii) (t, s) → W (t, s) is strongly continuous for 0 ≤ s ≤ t ≤ T .
Theorem A. 4 (
4Bounded Perturbation Theorem II). Let {A(t)} t∈I be a Kato-stable family of infinitesimal generators with constants M and ω. Let {B(t)} t∈I be bounded linear operators on X. If B(t) ≤ K for all t ∈ I, then {A(t) + B(t)} t∈I is a Kato-stable family of infinitesimal generators with constants M and ω + KM . Theorem A.5. If the family {A(t)} t∈I satisfies the Kato-condition (see Definition A.5) then there exists a unique evolution system W (t, s), 0 ≤ s ≤ t ≤ T , in X satisfying (E 1 ) W (t, s) ≤ M e ω(t−s) , ∀ 0 ≤ s ≤ t ≤ T,
it satisfies (3.21) with m replaced by m − 1). It remains to show that the solution U actually belongs to H m Γ if F belongs to H m Γ . For all 0 ≤ |α| ≤ m, we deduce from (3.18) that ∂ α U satisfies the following equations
In fact, ν α 1 is the sum of a function with support in O1 \ Ω and a measure supported by O1 ∩ ∂Ω, but this additional information is not useful to us.
We sometimes write E1 = E1( U ) for the sake of conciseness, etc.
For the domain Ω = (0, L1) × (0, L2), we have that C ∞ (Ω) = ∩ ∞ k=0 H k (Ω), see [Gri85, Chapter 1].
See the details in a related situation in[HT14b]
AcknowledgmentsThis work was partially supported by the National Science Foundation under the grants NSF DMS-0906440 and DMS-1206438, and by the Research Fund of Indiana University.Appendix B. Classical lemmasIn this appendix, we collect some essential ingredients for Sobolev spaces (see e.g. Chapter 13 in[Tay97]or Appendix C in[BS07]). i) Consider u and v which both belong to L ∞ (U ) ∩ H s (U ) with s > 0. Then their product also belongs to H s (U ) and there exists C > 0 depending only on s and U such thatIf s > d/2, then the L ∞ assumption automatically follows from the Sobolev embedding, and we have the following estimate:ii) Let F be a C ∞ function on R such that F(0) = 0. Then there exists a continuous function C : [0, +∞) → [0, +∞) such that for all u ∈ H s (U ) ∩ L ∞ (U ) with s ≥ 0:If s > d/2, then the L ∞ assumption automatically follows from the Sobolev embedding, and if furthermore we assume that U is bounded and that u is positive away from 0, i.e. |u| ≥ ǫ 0 for some positive ǫ 0 , then we have 1 u ∈ H s (U ),
S Benzoni-Gavage, D Serre, Multi-dimensional Hyperbolic Partial Differential Equations. Oxford University PressS. Benzoni-Gavage and D. Serre, Multi-dimensional Hyperbolic Partial Differential Equations, Oxford University Press, 2007.
J Chazarain, A Piriou, Introduction to the Theory of Linear Partial Differential Equations. AmsterdamNorth-Holland Publishing CoTranslated from FrenchJ. Chazarain and A. Piriou, Introduction to the Theory of Linear Partial Differential Equations, North-Holland Publishing Co., Amsterdam, 1982. Translated from French.
Numerical approximation of the inviscid 3d primitive equations in a limited domain. Qingshan Chen, Ming-Cheng Shiue, Roger Temam, Joseph Tribbia, ESAIM: Mathematical Modelling and Numerical Analysis. 46Qingshan Chen, Ming-Cheng Shiue, Roger Temam, and Joseph Tribbia, Numerical approximation of the inviscid 3d primitive equations in a limited domain, ESAIM: Mathematical Modelling and Numerical Analysis 46 (2012), 619-646.
One-Parameter Semigroups for Linear Evolution Equations. K.-J Engel, R Nagel, Graduate Texts in Math. 194Springer-VerlagK.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., vol. 194, Springer-Verlag, 2000.
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| []
|
[
"Measurement of the ν e and Total 8 B Solar Neutrino Fluxes with the Sudbury Neutrino Observatory Phase-III Data Set",
"Measurement of the ν e and Total 8 B Solar Neutrino Fluxes with the Sudbury Neutrino Observatory Phase-III Data Set"
]
| [
"R Hazama ; E ",
"K M Heeger ; F ",
"W J Heintzelman ",
"\nTRIUMF\n4004 Wesbrook MallV6T 2A3VancouverBCCanada\n"
]
| [
"TRIUMF\n4004 Wesbrook MallV6T 2A3VancouverBCCanada"
]
| []
| This paper details the solar neutrino analysis of the 385.17-day Phase-III data set acquired by the Sudbury Neutrino Observatory (SNO). An array of 3 He proportional counters was installed in the heavy-water target to measure precisely the rate of neutrino-deuteron neutral-current interactions.This technique to determine the total active 8 B solar neutrino flux was largely independent of the methods employed in previous phases. The total flux of active neutrinos was measured to be 5.54 +0.33 −0.31 (stat.) +0.36 −0.34 (syst.) × 10 6 cm −2 s −1 , consistent with previous measurements and standard solar models. A global analysis of solar and reactor neutrino mixing parameters yielded the best-fit values of ∆m 2 = 7.59 +0.19 −0.21 × 10 −5 eV 2 and θ = 34.4 +1.3 −1.2 degrees. PACS numbers: 26.65.+t, 14.60.Pq, 13.15.+g, 95.85.Ry | 10.1103/physrevc.87.015502 | [
"https://arxiv.org/pdf/1107.2901v1.pdf"
]
| 14,889,134 | 1107.2901 | 9c6f5830e898c2b443c20ae297961b05caa11b20 |
Measurement of the ν e and Total 8 B Solar Neutrino Fluxes with the Sudbury Neutrino Observatory Phase-III Data Set
14 Jul 2011
R Hazama ; E
K M Heeger ; F
W J Heintzelman
TRIUMF
4004 Wesbrook MallV6T 2A3VancouverBCCanada
Measurement of the ν e and Total 8 B Solar Neutrino Fluxes with the Sudbury Neutrino Observatory Phase-III Data Set
Heise, 2, 918914 Jul 2011T. J. Sonley, 11, p T. D. Steiger, 18 L. C. Stonehill, 9, 18 (SNO Collaboration)numbers: 2665+t1460Pq1315+g9585Ry 3
This paper details the solar neutrino analysis of the 385.17-day Phase-III data set acquired by the Sudbury Neutrino Observatory (SNO). An array of 3 He proportional counters was installed in the heavy-water target to measure precisely the rate of neutrino-deuteron neutral-current interactions.This technique to determine the total active 8 B solar neutrino flux was largely independent of the methods employed in previous phases. The total flux of active neutrinos was measured to be 5.54 +0.33 −0.31 (stat.) +0.36 −0.34 (syst.) × 10 6 cm −2 s −1 , consistent with previous measurements and standard solar models. A global analysis of solar and reactor neutrino mixing parameters yielded the best-fit values of ∆m 2 = 7.59 +0.19 −0.21 × 10 −5 eV 2 and θ = 34.4 +1.3 −1.2 degrees. PACS numbers: 26.65.+t, 14.60.Pq, 13.15.+g, 95.85.Ry
I. INTRODUCTION
The Sudbury Neutrino Observatory (SNO) experiment [1] has firmly established that electron-type neutrinos (ν e ) produced in the solar core transform into other active flavors while in transit to the Earth [2][3][4][5][6][7]. This direct observation of neutrino flavor transformation, through the simultaneous observation of the disappearance of ν e and the appearance of other active neutrino types, confirmed the total solar neutrino flux predicted by solar models [8,9], and explained the deficit of solar neutrinos that was seen by other pioneering experiments [10][11][12][13][14]. The SNO results, when combined with other solar neutrino experiments and reactor antineutrino results from the KamLAND experiment [15], demonstrated that neutrino oscillations [16][17][18] are the cause of this flavor change.
In the first two phases of the SNO experiment the determination of the total active 8 B solar neutrino flux and its ν e component required a statistical separation of the Cherenkov signals observed by the detector's photomultiplier tube (PMT) array. In the third phase of the experiment, an array of 3 He proportional counters [19] was deployed in the detector's heavy-water target. The neutron signal in the inclusive total active neutrino flux measurement was detected predominantly by this "Neutral-Current Detection" (NCD) array, and was separate from the Cherenkov light signals observed by the PMT array in the ν e flux measurement. This technique to measure the total active 8 B solar neutrino flux was largely independent of the methods employed by SNO in previous phases.
The results from the third phase of the SNO experiment were reported in a letter [7], and confirmed those from previous phases. We present in this article the details of this measurement and an analysis of the neutrino oscillation parameters. In Sec. II we present an overview of the SNO experiment and the solar neutrino measurement with the NCD array. The Phase-III data set that was used in this measurement is described in Sec. III.
Details of the optical response of the PMT array and the reconstruction of its data are provided in Sec. IV. The electronic and energy response of the NCD array will be discussed in Sec. V. The determination of the neutron detection efficiencies for both the PMT and the NCD arrays, which are crucial to the measurement of the total active solar neutrino flux, is presented in Sec. VI. The evaluation of backgrounds in the measurement is summarized in Sec. VII. Alpha decays in the construction materials of the NCD array were a nonnegligible background for the detection of signal neutrons. We developed an extensive pulse-shape simulation and applied it to understand the response of the NCD counters to these alpha decays. The pulse-shape simulation model is presented in Sec. VIII. In Sec. IX, we discuss the analysis that determined the total active solar neutrino flux and the electron-type neutrino flux. These measured fluxes, along with results from previous SNO measurements and other solar and reactor neutrino experiments, were then used in the determination of the neutrino mixing parameters as described in Sec. X. A description of the cuts we used to remove instrumental backgrounds in the NCD array data can be found in Appendix A.
A discussion of the parameterization of nuisance parameters in the neutrino flux analysis is provided in Appendix B. non-imaging light concentrator [20] was mounted on each PMT to increase the effective photocathode coverage to nearly 55% of 4π. The AV and the PSUP were suspended in an underground cavity filled with approximately 7 kilotonnes of ultra-pure light water (H 2 O), which shielded the D 2 O volume against radioactive backgrounds from the cavity rock. The inner 1.7 kilotonnes of H 2 O between the AV and the PSUP also shielded the target against radioactive backgrounds from the geodesic structure and PMTs. On the outer surface of the PSUP, 91 outward-facing PMTs were installed to tag cosmic-ray events. An array of 23 PMTs were mounted in a rectangular frame that was suspended facing inwards in the outer H 2 O region. These PMTs, along with the 8 PMTs installed in the neck region of the AV, were used to reject instrumental background light. A full description of the SNO detector can be found in Ref. [1]. In the third phase of the SNO experiment, an array of 3 He proportional counters was deployed in the D 2 O volume. Details of this array are presented in Sec. II.2 and in Ref. [19].
The SNO detector detected solar neutrinos through the following processes:
CC : ν e + d → p + p + e − − 1.442 MeV NC : ν x + d → p + n + ν x − 2.224 MeV ES : ν x + e − → ν x + e −
where ν x refers to any active neutrino flavor (x = e, µ, τ ). The charged-current (CC) reaction is sensitive exclusively to ν e , whereas the neutral-current (NC) reaction is equally sensitive to all active neutrino flavors. Chen [21] realized that the NC measurement of the total active solar neutrino flux tests the solar model predictions independently of the neutrino-oscillation hypothesis, and a comparison of this flux to the CC measurement of ν e flux tests neutrino flavor transformation independently of solar models. The neutrino-electron elastic scattering (ES) reaction is used to observe neutrinos of all active flavors in SNO and other real-time water Cherenkov and liquid scintillator detectors. Its cross section for ν e is approximately six times larger than ν µ and ν τ for 8 B solar neutrinos, but is smaller than the CC or NC cross sections in the energy region of interest.
In the first phase of the SNO experiment, which used an unadulterated D 2 O target, NC interactions were observed by detecting the 6.25-MeV gamma ray following the capture of the neutron by a deuteron. Under the assumption of an undistorted 8 B neutrino spectrum, the hypothesis of the observed CC, NC and ES rates due solely to ν e interactions was rejected at 5.3σ. Details of the solar neutrino analysis in this phase can be found in Ref. [5].
Approximately two tonnes of sodium chloride (NaCl) were added to the D 2 O in the second phase of the SNO experiment. This addition enhanced the neutron detection efficiencies and allowed the statistical separation of CC and NC signals without making any assumption about the energy dependence of neutrino flavor change. As a result the accuracy of the ν e and the total active neutrino flux measurements were significantly improved. A full description of the solar neutrino analysis in this phase can be found in Ref. [6]. Recently the results of a solar neutrino analysis that combined the Phase-I and Phase-II data sets were reported in Ref. [22].
II.2. The Neutral-Current Detection (NCD) array
The NCD array, consisting of 36 strings of 3 He and 4 strings of 4 He proportional counters, was deployed in the D 2 O target, after the removal of NaCl, in the third phase of the experiment. Figure 1 shows a side view of the SNO detector with the NCD array in place.
The NCD counter strings were arranged on a square grid with 1-m spacing as shown in Fig. 2. The acrylic anchors to which the NCD strings were attached were bonded to the AV during its construction, and their positions were surveyed precisely by laser theodolite and were taken as reference. Details of the deployment of the NCD string can be found in Ref. [19].
Each NCD string was 9 to 11 meters in length, and was made up of three or four individual 5-cm-diameter counters that were laser-welded together. Ultra-low radioactivity nickel, produced by a chemical vapor deposition (CVD) process, was used in the construction of the counter bodies and end-caps. This process suppressed all but trace amounts of impurities.
The nominal thickness of the counter wall was ∼370 µm. Each counter was strung with a 50-µm-diameter low-background copper wire that was pretensioned with a 30-g mass. The gas in the counters was a mixture of 85% 3 He (or 4 He) and 15% CF 4 (by pressure) at 2.5 atmospheres (1900 Torr).
The coordinates of the top of the NCD strings were not a priori known because they were pulled slightly out of the anchors' reference positions by the cables. There were altogether three campaigns to measure the positions. Before and after data-taking in Phase III, laser range-finder (LRF) surveys were made of the counter tops using a custom-built LRF that could be introduced through the AV neck and immersed in the D 2 O. During data-taking, optical reconstruction of the average positions was obtained from the shadowing of calibration sources (see Sec. IV.1). These three measurements were in good agreement, with the shadowing results being about two times more precise than the LRF results.
Neutrons from the NC reaction were detected in the NCD array via the reaction n + 3 He → p + t + 764 keV.
The 4 He strings were not sensitive to neutrons and were used to characterize non-neutron backgrounds in the array.
During normal operation the anode wires were maintained at 1950 V, resulting in a gas gain of ∼220. A primary ionization of the counter gas would trigger an avalanche of The NCD array had two independently triggered readout systems. The "Shaper-ADC" system used a pulse-shaping and peak-detection network to integrate the signal pulse and measure its energy. This fast system was triggered by the pulse integral crossing a threshold and could handle the kilohertz event rates expected from a galactic supernova. The "Multiplexer-Scope" (MUX-scope) system digitized and recorded the entire 15-µs pulse. It consisted of four independent sets of electronics, or MUX boxes, each of which could accept signals from up to twelve strings. Each channel was triggered by the pulse amplitude crossing a threshold. Pulses were amplified by the logarithmic amplifier ("log-amp") in a MUX box to increase the range of pulse sizes that could be digitized at a 1-GHz sampling rate by the 8-bit digitizer in one of the two digital oscilloscopes. The multiplexer controller triggered the oscilloscope that was not busy (or toggled between them when neither was busy), allowing for a maximum digitization rate of 1.8 Hz. If the oscilloscopes were busy and the MUX system triggered, a "partial MUX" event was recorded without the digitized pulse. The MUX-scope system adopted in the 1990s is not a solution to be recommended today, but it was sufficient to handle typical solar-neutrino signal and background event rates. Signals from the PMT array and from these two readout systems of the NCD array were integrated in a global trigger system that combined the data streams with event timing information. Further details of the design and construction of these counters and their associated electronic systems can be found in Ref. [19].
III. DATA SET
III.1. Data selection
The measurements reported here are based on analysis of 385.17 ± 0.14 live days of data recorded between November 27, 2004 andNovember 28, 2006. The selection of solar neutrino data runs for analysis was based on evaluation as outlined in previous papers [5,6].
In addition to the offline inspections of run data from the PMT array, data-quality checks of the NCD array data were implemented. These checks validated the running condition, such as the trigger thresholds, of the NCD array.
To accurately determine the total active solar neutrino flux using the NCD array it was essential to utilize only data from strings that were operating properly. Six 3 He strings were defective and their data were excluded in the analysis. One of the counters in the string K5 was slowly leaking 3 He into an inter-counter space. Two strings, K2 and M8, had mechanical problems with the resistive coupling to the top of the counter string, as confirmed in postmortem examination at the end of the experiment, resulting in unstable responses.
The string K7 showed similar behavior, but a physical examination of this string at the end of the experiment did not indicate a loose coupling. The strings J3 and N4 were observed to produce anomalous instrumental background events in the neutron signal window. A loose resistive coupling was found during a physical examination of J3, but not in N4.
A variety of other kinds of instrumental events were recorded in the shaper and digitizeddata paths. Data reduction cuts were developed to remove these instrumental backgrounds.
In Appendix A examples of these background events and a summary of the cuts to remove them are provided. Physics events in a counter would trigger both the shaper-ADC and the MUX-scope subsystems; thus, a large fraction of instrumental backgrounds was removed simply by accepting only events with both triggers present. NCD array events that passed this selection criterion were subsequently analyzed by algorithms that were designed to identify non-ionization pulses such as micro-discharges and oscillatory noise. Two independent sets of cuts were developed. One of these sets examined the logarithmically amplified digitized waveforms in the time domain, while the other set utilized the frequency domain. The two sets of cuts were shown to overlap substantially, with 99.46% of cut events removed by both sets of cuts, 0.02% removed only by the time-domain cuts, and 0.52% removed only by the frequency-domain cuts. Both sets of cuts were used in reducing the data set. The number of raw triggers from the NCD array data stream was 1,417,811, and the data set was reduced to 91,636 "NCD events" after application of data reduction cuts. Figure 3 shows the energy dependence of the fractional signal loss determined from 252 Cf and Am-Be neutron calibration sources. A suite of instrumental background cuts for the PMT array data was developed in previous phases of the experiment [5,6]. These cuts were re-evaluated and re-calibrated to ensure their robustness in the analysis of the third-phase data. The number of raw triggers from the PMT array data stream was 146,431,347, with 2,381 "PMT events"
passing data reduction and analysis selection requirements similar to those in Ref. [5]. These selected PMT events have reconstructed radial distance R fit ≤ 550 cm and reconstructed electron effective kinetic energies T eff ≥ 6.0 MeV.
III.2. Live time
The raw live time of selected runs was calculated from the time differences of the first and last triggered event using the main trigger system's 10-MHz clock, which was synchronized to a global positioning system. Due to the combination of the data streams from the PMT and NCD arrays, a run boundary cut was applied to ensure that both systems were taking data by defining the start of the run as 1.1 seconds after the first event of either array, whichever came later. A reverse order cut was applied to define the end of each run. These calculated times were verified by comparing the results against those measured by a 50-MHz detector-system clock and a 10-MHz clock used by the NCD array's trigger system.
Several data selection cuts removed small periods of time from the data set during normal data taking conditions. The largest of these removed time intervals following high-energy cosmic-ray events and intervals containing time-correlated instrumental events. These cuts removed events from both the PMT and the NCD arrays. The final live time for the neutrino analysis was calculated by subtracting the total time removed by these cuts from the raw live time. This resulted in a reduction of 1.96% of the raw live time.
The final live time was checked with analyses of data from two detector diagnostic triggers:
the pulsed global trigger (PGT) and the NCD system's random pulser (NRP). The PGT was a detector-wide trigger issued at a frequency of 5 Hz based on timing from the 50-MHz system clock. The NRP randomly pulsed a spare channel on the NCD system at an average rate of 7.75 mHz. Systematic uncertainties in live time were evaluated by comparing the PGT and NRP measurements to the 10-MHz clock measurement. The total live time uncertainty was calculated to be 0.036%.
The "day" data set, which was acquired when the solar zenith angle cos θ z > 0, has a live time of 176.59 days. The night data set, for which cos θ z ≤ 0, has a live time of 208.58 days.
IV. RESPONSE OF THE PMT ARRAY
In the solar neutrino measurement, the SNO PMT array observed Cherenkov radiation from high energy electrons resulting from direct neutrino interactions, β decays of radioactive backgrounds, and Compton scattering of gamma rays from nuclear de-excitations and radiative captures. A thorough understanding of the propagation and the detection of Cherenkov photons in the SNO detector was vital to reconstruct the observables, such as energy and vertex position, of each triggered event in the PMT array. Details on the extensive optical and energy calibration of the PMT array in previous phases of SNO can be found in
Refs. [5,6].
In the third phase of SNO, optical and energy calibration procedures were modified from those in previous phases in order to account for complexities that did not exist before. The nickel body of the NCD strings scattered and absorbed Cherenkov photons. The orientation of the NCD array and its signal cables also accentuated the vertical (z) asymmetry in the response of the PMT array. These effects had to be incorporated in the reconstruction of Cherenkov events in order to precisely determine the energy and spatial distributions of neutrino signals and radioactive backgrounds in the heavy-water detector. In this section, we will first present the response of the PMT array to optical photons. This will be followed by a study of event vertex reconstruction, which required the arrival time of the detected photons as input, and a study of the energy response of the PMT array, which depended on reconstructed event vertex position.
IV.1. Optical response
The measurement of the SNO optical model parameters (see Sec. IV.A of Ref. [6]) was done by a χ 2 fit that minimized the differences between the measured and predicted light intensities at each PMT, for a set of calibration runs ("scan") taken with the "laserball" source [23] at a number of positions inside the D 2 O volume. These calibration methods had to be significantly modified with respect to the previous phases, in order to take into account the optical effects of the NCD array. The construction of a high-isotropy laserball source and the optimization of the calibration plans enabled good sampling of the PMT array despite partial shadowing from the NCD array. The changes in the analysis of the calibration data, and its results, are described below.
IV.1.1. Source and NCD string positions
The source position in individual laserball runs was determined by minimizing the differences between the calculated and measured PMT prompt peak time, similarly to what was done previously, adding cuts to remove PMTs with very few counts.
In SNO Phase III, the laserball source positions and the shadowing patterns observed in the PMT array were used to obtain the NCD string positions, which were in turn compared to the installation reference, or nominal, positions. These reference positions were the locations of the NCD string anchors that were surveyed by laser theodolite during the construction of the acrylic vessel. The method selected an ensemble of about 30 source positions in order to triangulate and reconstruct the position of each NCD string [24]. For each NCD string, sets of PMTs were selected so that the light path between them and the laserball lay near to the string, in x and y coordinates. The PMT occupancy for that run was filled in a two-dimensional map corresponding to the x − y line including the point of closest approach of the source-PMT path to the NCD string, and all the other points along that source-PMT direction (projected in the x − y plane). Since the count rate depended on the conditions of a run (such as laser pulse rate, source stability, PMT thresholds, source position, etc.), the occupancy of the selected PMTs in each run was normalized by their mean occupancy.
A map of these relative occupancies was built and fitted to a two-dimensional Gaussian function. Figure 4 shows an example of a reconstructed string position. The extraction of all NCD string positions gave an average difference between the fitted and nominal coordinates, the average horizontal displacement, of about 2 cm in the x and y directions, which was consistent with the estimated average uncertainty of the string positions.
The NCD string position fit was repeated by dividing the various trajectories into three z-bins, such that the (x, y)-coordinates were obtained as a function of z. The slope of (x, y) vs z gave the deviation of the counters from their nominal vertical position. The best-fit angular deviation of all the counters was found to be less than one degree, and was consistent with the measured average displacements. Therefore the rest of the analysis assumed the counters were perfectly vertical but had an average horizontal displacement.
This choice simplified the numerical simulation model.
The various systematic uncertainties of source position reconstruction, that summed up to approximately 2 cm, were propagated to get an estimated resolution of the (x, y)-coordinates in the triangulation method. The method yielded an average uncertainty of 2.2 ± 0.3 cm on the individual NCD string positions, projected at z = 0. The spread of 0.3 cm arose partly because of the geometry of the NCD array with respect to the calibration planes that limited the laserball positioning to (x, 0, z) and (0, y, z) coordinates. Thus strings in the outer rings were sometimes shadowed by other strings in inner rings, resulting in larger uncertainties for those strings. In addition, the uncertainty contained a small scan-to-scan variation which was taken into account in the spread. In the optical analysis, the 2.2 cm uncertainty and its spread were input parameters to the shadow-removal code that treated all laserball and string pairs in the same way, independently of their position, to remove the shadowed PMTs with an estimated efficiency of 99%.
IV.1.2. Optical effects of the NCD array
The attenuation lengths of various optical media and the PMT angular response were determined by analyzing the PMT data that were not affected by unwanted optical effects, including those caused by the NCD array. Given the fitted position of the laserball and NCD strings, the shadowed PMTs were removed from the analysis on a run-by-run basis.
x (cm) In addition to the shadowing effects from the NCD strings and their signal cables, the anchors that held them down to the bottom of the AV were also taken into account since they were made of UV-absorbing acrylic. The implementation of the anchor cut showed improvements in the determination of the efficiency for PMTs located at the bottom of the detectors.
In order to handle PMT-to-PMT variations in efficiency in previous phases, the light intensity at a PMT for a given run was always normalized to the intensity of that PMT in a run where the laserball was deployed at the center of the detector. In the third phase, this technique would result in the systematic removal of the shadowed PMTs in that central run. Therefore, the optical model used the PMT relative efficiencies measured in the "preparatory" phase, a run period after the removal of salt but prior to the installation of the NCD array, taking into account changes in PMT gain and threshold. Even when the NCD string shadow and anchor cuts removed many PMTs in the analysis, the new fit method and optimally chosen laserball positions compensated for this loss, resulting in the overall statistics in a given scan at most 50% lower than in previous phases.
Light reflections off the surface of NCD strings were predominantly diffuse, and it was impossible to associate specific trajectories between the source and the PMTs. Therefore, the effect of such reflections on the PMT counts must be estimated and corrected for runs at various source positions on a run-by-run basis. An analytic correction was derived for each source-PMT trajectory by calculating the fraction of solid angle corresponding to the optical paths between the source and a given PMT that included a reflection in one NCD with respect to the direct paths. The correction for diffuse reflection off the NCD strings was found to be less than 5% of a PMT's occupancy on average. The same correction could be inferred using various MC scenarios, employing the ratio of PMT calibrations from reflection-on and reflection-off simulations. The analytic and MC-based corrections agreed to within 10% and the difference between the two was applied as a systematic uncertainty on the optical parameters.
IV.1.3. Determination of the optical parameters
In addition to an optical data set taken during the preparatory phase as a reference of the detector state, eight optical calibration scans were performed during the third phase, among which five were selected for use in the analysis. For each scan, data were taken at six different wavelengths (337, 365, 386, 420, 500 and 620 nm) at multiple source positions in the SNO detector. The time span of the calibration sets allowed us to monitor the stability of the optical model parameters, which included the heavy-and light-water attenuation lengths, the PMT angular response, and the laserball's light isotropy.
The modeling of the angular response of the PMTs was improved using optical scan data from the preparatory phase. An empirical collection efficiency function in the simulation modifies the response of the PMT as a function of the position at which a photon strikes the photocathode, thus altering the angular response. This function has five tunable parameters, which were previously optimized to reproduce laserball scans at 386 nm, the most probable wavelength for registering a hit in the detector. In this phase, a joint χ 2 fit was performed at all six wavelengths at which laserball calibration data were taken, fitting for both the shape of the response at each wavelength and the relative normalizations. The calibration data at each wavelength were first normalized to the amplitude at normal incidence, and then scaled by the quantum efficiency for a typical PMT at that wavelength. The χ 2 of the empirical collection efficiency function fit at each wavelength was weighted by the relative likelihood of a successful hit being caused by a photon at that wavelength in a Cherenkov light event, in order to optimize the fit at the most probable wavelengths. This resulted in a greater weighting for the more probable wavelengths and a very small weighting for the data at 620 nm, for example, where the probability of a photon triggering a PMT was very low. The angular response shape thus produced by the simulation showed a significant improvement in the agreement with calibration data at all the most probable wavelengths. Figure 5 illustrates this shape and scale modeling improvement at 386 nm.
Angle of incidence (degrees) Figure 6 shows the relative PMT angular response for the five scans in the third phase.
The measurement from the preparatory phase is also shown for reference. analyzed scans, the uncertainties on the D 2 O and H 2 O attenuation coefficients and PMT response were generally well below 10%, 15%, and 1.5%, respectively. The effect of these uncertainties on event vertex position reconstruction accuracy and energy estimation was estimated to be less than 0.1% and 0.25% respectively.
IV.2. Vertex reconstruction of Cherenkov events
Algorithms that maximize the likelihood of event vertex position and direction, given the distribution of PMT trigger times and positions, were used to reconstruct Cherenkov events in SNO. The following sections describe event reconstruction in Phase III and the methodology used to determine the associated systematic uncertainties.
IV.2.1. Event vertex
Vertex reconstruction in the third phase was performed by maximizing the likelihood
function L = N hits i=1 f (t res |hit; r PMT , r fit ),(1)
where t res is the time-of-flight corrected PMT trigger time
t res = t PMT − t fit − | r fit − r PMT | c avg ,(2)
r fit and t fit are the reconstructed event position and time respectively, N hits is the number of selected PMT hits, and c avg (=21.87 cm/ns) is the group velocity of the mean detected photon wavelength at 380 nm. The function f (t res |hit; r PMT , r fit ) is the probability density function (PDF) that a particular PMT fires at time t res , given its position r PMT and the reconstructed event position. In SNO's third phase, the t res PDF was dependent on r PMT and r fit due to partial or complete shadowing by the NCD array. This shadowing effect was incorporated by generating t res distributions for non-shadowed and completely shadowed PMTs using Monte Carlo (MC) simulations and by interpolating between the two for partially shadowed PMTs using an algorithm that computed shadowing analytically. To reduce the effects of reflected photons, the PDF was approximated as a constant for t res values greater than 15 ns, and a time cut of ±50 ns around the median PMT hit time was imposed.
This determination of the PDF with shadowing effects allowed an overall improvement of 5% in spatial resolution.
To evaluate the differences between the true and reconstructed event vertex positions, or the "vertex shift", the average reconstructed event position of 16 N [25] calibration data relative to the source position was compared to that computed from simulated data. Figure 8 shows the difference between the data and Monte Carlo vertex shift as a function of the source position for scans along the main axes of the detector. It shows a spread of 4 cm in the three directions, and this value was taken as the vertex shift uncertainty. This uncertainty was found to be correlated to the PMT timing calibration of the detector. An overall offset of 5 cm was also observed in the z direction. The systematic uncertainty on "vertex scaling", a position-dependent inward or outward shift of reconstructed position, can have a direct effect on the fiducial volume for events detected by the PMT array. It was measured by determining the range of the slope of a first order polynomial that allowed the inclusion of 68% of the data points in Figure 8. This uncertainty was believed to be caused by physical factors such as a mismatch of the speed of light in different media and was expected to be the same in all directions. It was estimated to be 0.9% of the Cartesian coordinates.
Vertex resolution was another systematic uncertainty that could affect the fiducial volume. It was assessed by taking the difference between the data and the MC fitted position resolution for all 16 N calibration data taken inside the acrylic vessel, and propagated by smearing the coordinates of simulated events with a Gaussian random variable such that the width increased by the measured discrepancy. Figure 9 shows that this uncertainty Since most signals and backgrounds were not correlated with the direction of the Sun's position, the angular resolution uncertainty did not generally have a significant effect. The direction of electron-scattering (ES) events was however strongly correlated with the incoming neutrino direction and was well modeled by the function
P (cos θ ⊙ ) = α M β M e β M (cos θ ⊙ −1) 1 − e −2β M + (1 − α M ) β S e β S (cos θ ⊙ −1) 1 − e −2β S ,(3)
where β S is the parameter for the exponential component associated with the main peak and β M is associated with the multiple scattering component. To determine the systematic uncertainty on these parameters, the function in Eqn. 3 was fitted using 16 N calibration data and simulations. In this analysis, cos θ ⊙ was replaced by the cosine of the angle between the reconstructed and the true electron direction, the latter being approximated by the fitted vertex position relative to the source, for events reconstructed 120 cm or more away from the source position. Due to the correlation between these resolution parameters and the complexity associated with the smearing of angular resolution for single events using Eqn. 3, the uncertainty was propagated using the expression
cos θ ′ ⊙ = 1 + (cos θ ⊙ − 1)(1 ± δ),(4)
where δ = 0.12 is the relative uncertainty on β M and β S parameters. This parameterization was shown to be a good approximation for ES events.
IV.3. Energy calibration of Cherenkov events
The fundamental measure of event energy in SNO was the number of Cherenkov photons produced by fast electrons, and the most basic energy observable for Cherenkov events was the number of triggered PMTs (N hit ). The energy reconstruction algorithm discussed below attempts to determine the number of photons (N γ ) that would have been produced by an electron, given the reconstructed position ( r fit ) and direction (û fit ) of the event in the detector, to yield the number of triggered PMTs observed. An estimate of event kinetic energy (T eff ) can then be derived from the one-to-one relationship (F T ) between electron kinetic energy (T e ) and the mean of the corresponding distribution of N γ [26].
N i = N γ λ R i (λ, r,û) 1 λ 2 λ 1 λ 2 ,(5)
where R i , discussed in detail in Sec. IV.3.2, is the response of the i th PMT to a photon of wavelength λ. The sum over λ was done in 10 nm steps from 220 to 710 nm, the wavelength range over which the detector was sensitive. The total number of direct PMT hits (i.e. not from reflected light) predicted by the energy reconstruction (N predicted ) is then
N predicted = N PMTs i N i M (N i ) ,(6)
where M is a correction function that accounts for the possibility of multiple photons counting in the hit PMT. The initial estimate of N γ was modified by
N γ → N eff N predicted N γ ,(7)
where N eff is the effective number of PMT hits, the number of PMT hits within the prompt light window after corrections for dark noise were made. This process was iterated until agreement was reached between N predicted and the noise-corrected number of prompt PMT hits.
IV.3.2. PMT optical response (R i )
The optical response of the i th PMT, R i , in Eqn. 5 was calculated as
R i = ǫ i (λ, r fit , p i ,n) Ω i ( r fit , p i ,n) D(T e | r fit , p i ,û fit ) × F ( r fit , p i ) exp − 3 m=1 d m ( r fit , p i )α m (λ) ,(8)
where λ is photon wavelength, r fit is the reconstructed position,û fit is the reconstructed direction, p i is the vector between the PMT position and r fit ,n describes the orientation of the PMT, and T e is the true electron kinetic energy. The terms in the sum over the media The PMT efficiency (ǫ i ) was broken down into the factors
ǫ i = ǫ PCE E(λ, cos θ n )E optical i E electronic i ǫ NCD ,(9)
where ǫ PCE is the aforementioned average PMT collection efficiency, E is the relative response as a function of incidence angle (θ n ) for a typical PMT, and E optical
IV.3.3. Energy calibration
Once the detector optical parameters have been determined, the PMT collection efficiency (ǫ PCE ) and the energy calibration function (F T ) were still to be set. High rate (∼ 200 Hz) central 16 N source calibration runs were compared to Monte Carlo simulations using an initial estimate of ǫ PCE = 0.645 (as was determined for Phase II). This value was adjusted to match the means of the N γ distributions obtained from the source data and from simulations.
The result was a value of ǫ PCE =0.653 for the third phase. Figure 10 shows the relative photon collection efficiency of the detector using the same central 16 N calibration runs as above. The slight time variation, not accounted for by energy reconstruction, was described by the function
δ drift = 1.197 − 1.751 × 10 −5 t,(10)
IV.3.4. Energy systematic uncertainties
Two of the most important systematic uncertainties on the measured neutrino fluxes were the energy scale and energy resolution of the PMT array. These uncertainties were determined by comparing data and MC simulations using the 16 N source. This source was deployed on nearly a monthly basis and was used to probe not only the center of the detector but also to scan along the x, y, and z axes. A total of 1053 16 N runs were used in the energy systematics analysis.
In order to correctly determine the energy of an event, the energy estimator must use the number of working PMTs. The number of non-working PMTs considered as working by the energy estimator could be approximated by counting the number of PMTs that fell outside a region of 5σ from the average PMT occupancy. The ratio of this number by the total number of working PMTs for a run was taken as the potential influence of the uncertainty of the detector state on the estimated energy. The uncertainty due to the detector state was determined to be 0.03%.
The temporal stability of the energy response of the data and Monte Carlo was evaluated using the 16 N runs taken at the center of the detector. A comparison of the mean kinetic energy and resolution distributions in data and simulation was used to determine the temporal stability systematic uncertainty. The energy drift/stability uncertainty on the energy scale was found to be 0.40% and that on the resolution was determined to be 1.19%.
One of the largest contributions to the energy scale and to the energy resolution uncertainties was due to spatial variations in the detector. Figure 12 shows a comparison of T eff between data and MC simulations as a function of the volume-weighted position ρ=R fit /R AV , where R AV = 600 cm is the radius of the acrylic vessel. The point-to-point variations and radial biases of the detector response were determined by dividing the detector into radial and polar angle bins assuming an azimuthal symmetry. The average differences between data and MC simulations and the variance of the mean kinetic energy and resolution were determined in each of these bins. The volume-weighted average of these differences among the bins was then taken as the spatial variation on the energy scale and resolution. The uncertainty on the spatial variation of the energy scale was determined to be 0.64% and the spatial variation of the energy resolution was determined to be 1.04%.
The 16 N source was typically run at a rate on the order of several hundred hertz whereas neutrino data were taken with the detector operating at an event rate of an order of magnitude lower. In order to evaluate the potential rate dependence, the 16 N source was periodically run at 'low rate' (several hertz). Comparing the mean kinetic energy of low-rate runs taken close in time to high-rate runs, the uncertainty on the energy scale related to the rate dependence was determined to be 0.20%. The extracted timing peak means and widths were quite stable and similar to what were seen during previous phases. Since there were no indications that the timing had changed from the last phase, the uncertainty from the second phase, 0.10%, was used and was considered conservative.
V. ELECTRONIC CALIBRATION OF NCD ARRAY
As described in Sec. II.2 the data stream of the NCD array consisted of events from the shaper-ADC and the MUX-scope subsystems. The shapers could provide the total charge in an event and the MUX-scope subsystem could digitize the log-amplified waveform of the signal. The primary goal of the electronic calibration was to measure the parameters of the electronic model, so that the transformations of the counter signals as they propagated through the front-end electronic and data acquisition systems were quantified. A calibration system was implemented to pulse the preamplifiers. The output of a programmable waveform generator was attenuated by 30 dB and injected into a pulse distribution system (PDS) board, whose amplified outputs could be sent to selected preamplifiers through computer control.
V.1. Linearity
The gain and linearity of the shaper-ADC and MUX-scope channels were calibrated by sending rectangular pulses of known amplitudes to the preamplifiers, one preamplifier at a time. These calibrations were performed once a week at five different pulse amplitudes.
Extended electronic calibrations with twenty different pulse amplitudes over an expanded range were performed monthly. Rectangular pulses were used since their start and stop times were easily determined, which facilitated the integration of digitized waveforms in the analysis. The measured charge of the signal as a function of the calculated input charge was fit to a linear function, which measured the gain and offset of each channel and tested the channels' linear response. Since the digitized waveforms were logarithmically amplified and recorded by the digital oscilloscopes, they were first de-logged (inverting Eqn. 11 below)
for the MUX-scope channel linearity analysis. This tested the linearity of the MUX-scope system as well as the measured log-amp parameters. The shaper-ADC channels were found to be linear to within 0.5% across all channels. The transfer function of the logarithmic amplifier was responsible for an observed non-linearity of up to ∼5%, and a model was developed to account for this behavior. Figure 13 shows the temporal variation of the relative gain in the shaper-ADC and MUX channels, measured from the monthly extended calibration runs in Phase III, for string N1. channels for string N1. In these plots the gain measured from each monthly extended electronic calibration was normalized to the mean gain. The data shown here extended from the commissioning (prior to run 50000) to the completion of Phase-III data taking (∼run 67000).
The shaper-ADC channel corresponding to the mean of neutron signal peak in Am-Be calibrations was used to determine the conversion gain for the measured shaper-ADC charges to event energies in the 3 He counters. For the 4 He counters, which were insensitive to neutrons, the energy peak from 210 Po alpha decays was used.
V.2. Threshold
The threshold calibration involved injecting offset, single-cycle, sine waves with a constant width and varying amplitudes to all the preamplifiers simultaneously. The range of these pulser output amplitudes extended above and below each channel's threshold level.
Sine waves were chosen for this measurement because the calibration pulse amplitude and calculation of the total injected charge were more stable with sine waves than with signals that are not smooth, such as a rectangular or triangular pulses. The high-frequency components of non-smooth waveforms could produce transient currents, which could be difficult to calculate accurately for each channel.
The threshold levels of the shaper-ADC and MUX channels were determined by finding the pulser amplitude at which half of the expected events were observed. The algorithm searched over the range of pulser amplitudes, estimating the charge and current thresholds of the shaper-ADC and MUX channels. The thresholds were stable over the course of the experiment except for when they were intentionally changed to account for sporadic electromagnetic pickup or a malfunctioning string. In the latter case, the thresholds for the channels associated with the malfunctioning string were set to their maximum values to ensure that they were offline. Figure 14 shows the temporal variation of the shaper-ADC channel threshold for string N1 during Phase III.
V.3. Log-amp
The log-amp calibration pulse was an offset, single-cycle 1-µs-wide sine wave preceded 6 µs by a narrow rectangular trigger pulse. These pulses were injected to each preamplifier channel at 3 Hz for a duration of 15 seconds. A sine wave was selected because its smoothly varying shape and frequency characteristics were similar to the expected counter signals. If the sine wave were used to trigger the channel, it could be possible that the beginning of The data shown here extended from the commissioning (prior to run 50000) to the completion of Phase-III data taking (∼run 67000).
the pulse would not be recorded as the time for the sine wave to go from zero to the MUX threshold level might be longer than the electronic delay time. The width of the trigger pulse was set to a small value in order to reduce the amount of baseline offset produced by its integrated charge before the sine wave arrived at the input.
The logarithmic amplification in the MUX electronic chain was modeled as
V log (t) = A · log 10 1 + V lin (t − ∆t) B + C chan + V PreTrig ,(11)
where V log and V lin are the logarithmic and linear voltages, ∆t represents the time delay for each channel in the MUX, and A, B, C chan , and V PreTrig are constants determined by calibrations. Details of this parameterization of the MUX-scope electronic chain can be found in Ref. [27].
The log-amp calibration analysis involved a χ 2 -minimization that estimated the set of five log-amp parameters that best fit a simulated signal to each measured calibration pulse.
A weighted average of the parameter values extracted for each event, along with the uncertainty of the weighted average, was calculated for each NCD electronic channel. Because some of these parameters were dependent on which of the two oscilloscopes recorded the event, there were two sets of log-amp parameters to be measured for each string. The electronic calibrations measured these parameters and the current threshold level of digitization separately for each string.
VI. NEUTRON DETECTION EFFICIENCY CALIBRATION
The NC interaction produced a uniform distribution of neutrons in the D 2 O volume.
The primary method for determining the neutron capture efficiency of the NCD array was to deploy an evenly distributed 24 Na source in this volume [28]. Two such calibrations were was applied to compensate for the differences in the neutron energy and neutron radial distribution between the solar neutrino and the spike calibration data. After this correction, the neutron capture efficiency from the 24 Na calibration was equal to the efficiency for neutrons produced by the NC interaction. However, in calculating the efficiency by this technique and its uncertainty, possible deviations from perfect mixing that might occur near boundaries, such as the walls of the acrylic vessel and the NCD strings, were also considered as described below.
The temporal behavior of the NCD array response was monitored by deploying 252 Cf and AmBe sources in different parts of the D 2 O volume using the source manipulator system [1].
The data from these point calibrations were also used to calibrate the Monte Carlo code.
The technique that was used to tune the Monte Carlo is discussed in Sec. VI.5.
In our previous paper [7], the reported neutron capture efficiency of the The input elements needed to determine the neutron capture efficiency for the NCD array and the neutron detection efficiency for the PMT array are made explicit in the following formula:
ǫ sol = f non-unif · f edge · ǫ spike ,(12)
where ǫ sol is the capture or detection efficiency for neutrons produced by solar neutrinos and ǫ spike is the efficiency determined from the 24 Na spike calibration. The two factors multiplying ǫ spike correct for differences in the distribution of neutrons in solar neutrino and 24 Na calibration data. The factor f edge accounts for the differences in the neutron energy and neutron radial distribution, while f non-unif is a factor that accounts for the effect of possible non-uniformity of the activated brine in the D 2 O.
The neutron capture efficiency of the NCD array and the neutron detection efficiency of the PMT array measured from 24 Na calibration are given by the ratio of the observed signal rate R spike at a reference time and the 24 Na source strength A24 Na at that time:
ǫ spike = R spike A24 Na .(13)
VI.1.1. A24 Na : The 24 Na source strength measurement
The strength of the 24 Na source was determined using three different detectors: a germanium detector (ex situ measurement), the SNO PMT array (in situ), and the NCD array (in situ). In the following discussion, the quantity to be calibrated is the rate of neutrons produced by an encapsulated sample of the 24 Na brine placed at the center of the SNO detector.
For the calibration with the germanium detector, the total rate of 2.75-MeV gamma rays produced by the 24 Na brine sample was measured and a Monte Carlo simulation program was used to calculate the expected rate of neutrons produced in the heavy water if the 24 Na were positioned at the center of the SNO detector [30]. For the calibration measurement using the SNO PMT and NCD arrays, a comparison was made between the neutron rates observed in these detector arrays when the 24 Na brine sample was placed at the center of SNO and the rate observed when a well calibrated 252 Cf neutron source was placed at the same location, with a small further correction determined from simulations to deal with the different neutron spatial and energy distributions of the two sources.
In the germanium detector measurement, a small sample of the activated brine, with a mass measured to better than 0.5%, was placed on the detector (the liquid was contained in a Marinelli beaker [28]) and the rate of the 1. In the in situ SNO PMT array measurement, the PMT array detected the Cherenkov radiation produced by the 6.25-MeV gamma ray following the capture of these neutrons on deuterons. It was necessary to apply event selection cuts on the reconstructed radius and the energy in order to isolate these events from those due to background noise from the beta and gamma rays produced directly by the 24 Na decay. T eff was required to be between 5 and 9.5 MeV. This event selection criterion discriminated against radioactive background events, which had an average energy of about 3.0 to 3.5 MeV. The reconstructed event vertex was required to be 200 to 450 cm away from the source. This selection effectively removed background gamma-ray events whose range was limited by the Compton scattering length.
The calibration of the neutron rate from the brine sample was obtained by comparing this rate to the rate from the 252 Cf neutron source, whose strength was known with an uncertainty of 0.7% [6]. The rate measurement from the 252 Cf source was obtained within a few days of the 24 Na measurement so that the detector condition would be as similar as possible. Neutrons produced by the 252 Cf source had a similar, but not identical, radial capture profile in the D 2 O compared to that from the brine. Whereas the neutrons from the 252 Cf source were produced inside the source container, those from the 24 Na source were produced by photodisintegration in a sphere of radius about 30 cm. After production, the neutrons typically diffused by about 100 cm, so these initial differences were, to a great extent, mitigated. Yet, the capture profiles were different enough to introduce significant uncertainty in the 24 Na source strength measurement. Detector simulation was used to determine the effect of this difference; the capture efficiency of neutrons passing the radial selection cut was found to be about 2% smaller for the 24 Na source than for the 252 Cf source.
The combined statistical and systematic uncertainties were about as large as this correction.
The in situ measurement with the NCD array was performed almost identically as that with the PMT array and data from the same runs were analyzed. The main difference was the detection of the neutrons with the NCD array instead of with neutron captures by deuterons. As with the PMT array measurement, the neutron detection rate in the NCD array was measured with the activated brine in the detector, and this rate was divided by the measured rate of neutrons from the 252 Cf source. The ratio multiplied by the known 252 Cf source strength gave a good estimate of the brine source strength. Again, as with the PMT array data, the difference in the radial neutron capture profile required a correction which was obtained using detector simulation. The neutron capture efficiency for the 24 Na source was about 2% greater than that for the 252 Cf source. The uncertainty on this figure was much smaller (about 0.4%) than for the corresponding one for the PMT measurement because neutrons were captured much closer to the production region so that there was less reliance on the accuracy of modeling neutron diffusion to large radii.
The results for the total neutron production rate A24 Na from the 2005 and 2006 measurements are shown in Fig. 15 and Table IV. These numbers were derived from the actual measured values of the source strength in the following manner:
A24 Na = f P · m main m samp · e −∆t/τ 24Na · A samp ,(14)
where The neutron capture rate in the NCD array as a function of time for the 24 Na spike in 2005 is shown in Fig. 16; the plot for the spike in 2006 is similar. The horizontal axis can be divided into three regions:
1. During the first few hours, the spike was highly non-uniform, and the rate varied in an erratic manner.
2. For the two 24 Na lifetimes preceding the reference time t = 0, the spike was not quite uniform, as assessed using the distribution of Cherenkov light produced directly by the beta and gamma rays from the decay of 24 Na. However, the NCD array appeared to be quite insensitive to this moderate non-uniformity of the spike, as the measured rate was indistinguishable from the equilibrium distribution. During this time period, the rate appeared to decay exponentially as can be seen in the figure. Time t = 0 was 4.53 24 Na mean lifetimes after the spike was added.
3. After t = 0, the spike was determined to be well-mixed according to the Cherenkov light distribution. The rate continued to decay exponentially.
The neutron rate at t = 0 could be derived from the measured rate R(t i ) at the average time t i of the i th run as follows:
R i (0) = R(t i ) e t i /τ 24Na .(15)
The set of measurements {R i (0)} was found to be equal to each other within the statistical uncertainty; the result of a fit of a constant value to this set of measurements is shown by the horizontal line through the data in Fig. 16. The quantity R spike was obtained from this fit. The results from the runs in 2005 and 2006 are summarized in Table V.
Na decay lifetime 24 Time in units of In addition to the statistical uncertainty, several sources of systematic uncertainty existed for the neutron capture rate in the NCD array. In general, the rate can be written as follows:
R = N · f inst t data · L · ǫ comb ,(16)
where N is the number of detected events, t data is the length of time data were taken, L is the fraction of the time the detector was live (referred to as the "live fraction"), ǫ comb is a product of cut and threshold efficiencies, and f inst is a factor used to remove the estimated contribution of instrumental noise. Studies showed that uncertainties from these input terms were negligible in comparison to a 1% long-term fluctuation in the neutron detection rate, as assessed using standard point 252 Cf and Am-Be sources. Since the spike calibration runs were essentially two snapshots of the detector performance, the long-term fluctuation implied that the rate could have been different by ±1% if the calibration were performed at Although the activated brine produced a neutron distribution in D 2 O that was similar to that produced by solar neutrinos, the neutron density near the acrylic vessel was different. the vessel wall were significantly less likely to be detected by the NCD array than those produced elsewhere. Thus the volume-averaged capture efficiency of the NCD array for neutrons from the spike was somewhat larger than that for neutrons from solar neutrinos.
This difference near the acrylic vessel could be determined accurately by simulations, and was accounted for with the correction factor f edge :
f edge = 0.9702 ± 0.0078.(17)
VI.2.4. f non−unif : Correction factor for source non-uniformity
Another potential source of difference in the neutron distribution between that from the activated brine and from solar neutrinos was imperfect mixing of the brine. Although it was not possible to directly measure the salinity as a function of position in the D 2 O, there were a number of indications that the brine was well mixed and that any residual non-uniformity would not have a large impact on the measurements of the neutron capture efficiency. As can be seen in Fig. 16, the NCD array's neutron detection rate stabilized well after about a day of mixing, whereas the signal from the gamma rays emitted by the 24 Na did not stabilize completely for about three more days [28]. This indicated that the neutron capture rate of the NCD array was not sensitive to the remaining inhomogeneities for several days prior to the time t = 0 used for the start of the analysis of the 24 Na data and therefore could certainly
ǫ sol = 0.211 ± 0.005.(19)
This is in good agreement with the result from detector simulation, 0.210 ± 0.003 (see Sec. VI.5), and with our previously published value of 0.211 ± 0.007 [7]. In this paper, we have improved on the determination of the neutron capture efficiency by the NCD array and have further examined the assumptions that were made in the first Phase-III results reported in Ref. [7]. This resulted in a small reduction in the systematic uncertainty in the present analysis. This small improvement would have negligible effect on the measured NC flux, and was not incorporated in the solar neutrino flux analysis reported in this paper. C is the product of these individual factors:
C = l MUX · l scope · ǫ MUX · ǫ shaper · ǫ cut(20)
Table VII provides a summary of these factors.
The MUX live fraction in a data run was determined by comparing the readings on two live time scalars: one that determined the total run time, and the other that was stopped when the MUX system was unable to take in events. A pulser was installed to inject pulses at random times to provide additional validation. The mean MUX live fraction l MUX was the run-time-weighted average of the solar neutrino run measurements. The scope live fraction as a function of event rate was determined using the number of observed partial MUX events. The mean scope live fraction, l scope , was then established by numerically integrating the neutrino run time, weighted by the rate-dependent scope live fraction. This calculation of l scope was verified using the random pulser system. The methodology for measuring the MUX threshold efficiency has been described in Sec. V.2, and the results from regular calibration runs were averaged to provide an estimate of ǫ MUX . NCD array events with shaper energy E NCD > 0.4 MeV were selected for the solar neutrino flux measurement factor is the neutron signal acceptance of the data reduction cuts, which are described in Sec. III.1 and Appendix A. The overall correction factor is
C = 0.862 ± 0.004.(21)
VI.4. Neutron detection efficiency of the PMT array
Neutrons can also be captured by deuterons with the emission of a 6.25-MeV gamma ray that could be detected by the PMT array. The efficiency for this detection channel was much smaller than that for the NCD array because of the large difference in the thermal neutron capture cross section between the deuteron and 3 He. In this section, we present the analysis of the PMT array's neutron detection efficiency. The basic analysis approach,
i.e. the evaluation of individual terms in Eqn. 12, was nearly identical to that for the NCD array, discussed in Sec. VI.2 above.
The source non-uniformity factor f non-unif was evaluated in the same manner as the case for the NCD array. We assumed that the 24 Na spike was well mixed, but allowed for possible inhomogeneity near the acrylic vessel and the NCD strings. Based on simulation studies, these effects could lead to a combined uncertainty of 0.8% on the neutron detection efficiency for the PMT array, that is,
f non-unif = 1.000 ± 0.0080.(22)
The difference in the detection efficiency for NC neutrons and the 24 Na photodisintegration neutrons, characterized by f edge , was also different for the PMT array and the NCD array. This is because the radial dependence of the neutron capture efficiency was different for the two detection mechanisms. This difference was determined to be 0.92% by simulation, or a f edge value of:
f edge = 0.9789 ± 0.0090.(23)
For the measurement of R spike in Eqn. 13, a maximum likelihood analysis was performed to statistically separate the neutron signal from the 24 Na beta and gamma-ray backgrounds.
This is in agreement within uncertainties with the MC calculations (ǫ sol = 0.0485±0.0006)
in Sec. VI.5, which was used in our previously published results [7]. This small difference has a negligible effect on the solar neutrino flux results.
VI.5. Neutron calibration with discrete sources
Calibration data were used to tune the Monte Carlo simulation code, which was then used to predict the neutron detection efficiencies for both the PMT and NCD arrays. The uncertainties were calculated by propagating those on the tuning parameter and on other
Monte Carlo input parameters.
The calibration data used to tune the Monte Carlo were acquired using AmBe sources, which were periodically deployed around the D 2 O target during the data taking period.
In each of the calibrations data were taken with the source positioned at a series of welldefined, repeatable locations. The tuning parameter was the hydrogen concentration in the heavy water; its measured value, from Fourier transform infrared spectroscopy, was (9.8 ± 0.5) × 10 −4 atom of hydrogen per atom of deuterium.
The method for tuning the hydrogen concentration was to take averaged source data for a given source location and calculate the relative detection efficiency of rings of counters around the position. The relative capture efficiency of a ring of counters close to the source to one further away was sensitive to the neutron diffusion length and therefore to the concentration of hydrogen, which has a large neutron capture cross section. These ratios were independent of source strength and, because they were of rings of counters, they were relatively insensitive to the exact source position. Simulations were run with a range of hydrogen concentrations and source locations, and a maximum likelihood fit was used to extract the most probable hydrogen concentration. The measured hydrogen concentration was used as a constraint in the fit.
The central values for the neutron detection efficiencies in the PMT and NCD arrays were calculated using a simulation run at the best-fit hydrogen concentration of (9.45 +0.50 −1.05 )× 10 −4 atom of hydrogen per atom of deuterium, and the uncertainty propagated by rerunning the simulation with the hydrogen concentration set to its upper and lower bounds.
Additional sources of uncertainty in the Monte Carlo, such as parameters relating to the modeling of the counter and AV geometry, were studied separately. The only significant uncertainty came from the modeling of the shape of the NCD counter live regions, which was estimated to impart a 1.0% uncertainty on the NCD neutron detection efficiency.
The final prediction for the neutron capture efficiency of the NCD array was 0.210±0.003.
The prediction for the neutron detection efficiency of the PMT array was 0.0485 ± 0.0006.
VII. BACKGROUNDS
Several sources of radioactive backgrounds were present in the PMT array and NCD array data. The majority were associated with naturally occurring 238 U and 232 Th, cosmogenic activity and atmospheric neutrino interactions in the detector. The impact of these backgrounds on the neutrino analysis was reduced by optimization of analysis cuts. Those that remained were included in the fits for the background and the solar neutrino signals.
A summary of background contributions is given in Table VIII.
In this section, we will discuss the identification and measurement of the neutron and Cherenkov light backgrounds in the solar neutrino measurement. Alpha decays from the construction materials were the largest source of backgrounds in the neutron signal region in the NCD array. This background source was difficult to calibrate due to variation in the spatial distribution and in the composition of trace radioactivity in different counters. The treatment of alpha decay backgrounds will be presented in the next section (Sec. VIII).
VII.1. Photodisintegration backgrounds
Gamma rays with energy greater than 2.225 MeV can break apart a deuterium nucleus releasing a free neutron, which was indistinguishable from one produced by a NC interaction.
VII.1.2. In situ determination of radioactivity in D 2 O and NCD housing
The in situ technique measured the 232 Th and 238 U content of the water and NCD array housings directly from the Cherenkov light data [35]. In the energy window 4.0 < T eff ray and a beta with an end point of up to 1.8 MeV. 208 Tl events produced a more isotropic Cherenkov light distribution when compared with 214 Bi events and it was this difference in light isotropy that was used, in addition to differences in the radial distributions, to separate background components.
The light isotropy parameter was β 14 ≡ β 1 + 4β 4 , where
β l = 2 N(N − 1) N −1 i=1 N j=i+1 P l (cos θ ij ).(25)
In this expression P l is the Legendre polynomial of order l, θ ij is the angle between triggered
PMTs i and j relative to the reconstructed event vertex, and N is the total number of triggered PMTs in the event. Details of β 14 can be found in Ref. [6].
Radioactive decays originating from the NCD array had an exponential radial profile while those from the D 2 O had an approximately flat radial profile. The radial profiles were statistically indistinguishable for different radioisotopes that originated from the same location. Therefore, to distinguish between 208 Tl and 214 Bi, differences in event isotropy were used. By analyzing events that reconstructed with R fit < 450 cm they can be classified
VII.1.3. Radioactive hotspots on NCD strings
Two areas of increased activity (hotspots) were identified in strings K5 and K2. The in situ method identified an excess of events close to each of these strings, but it could not prove conclusively that these events were caused by radioactivity. The isotropy distribution of the events associated with K5 was more isotropic than that of the NCD bulk, implying either radioactivity or scintillant on the surface of the NCD strings. If the hotspot was However, K5 had a gain drift problem (Sec. III) and the number of neutrons captured by this string could not be quantified. No excess alphas were observed in the data from K2 suggesting that the contamination was embedded in the dead region of the counter, a conclusion that was supported by the in situ analysis. An extensive experimental program was developed to measure the radioactive content of these hotspots, and more details can be found in Ref. [37].
The lower-chain hotspot activities expressed in terms of equivalent masses of 232 Th and 238 U are summarized in Table IX. For the analysis presented here, the neutron rates have been calculated using the weighted average of the in situ and ex situ data. The updated analysis presented in Ref. [37] yields changes that are negligible relative to the uncertainties in the final result.
VII.2. Other neutron backgrounds
VII.2.1. Internal-source neutrons
In addition to photodisintegration backgrounds, there were other neutron backgrounds that were generated in the D 2 O. These included contributions from (α,n) reactions on nuclei, spontaneous fission from 238 U, cosmic-ray spallation and anti-neutrinos from nuclear reactor and atmospheric neutrinos. The SNO detector was also sensitive to CNO neutrinos from the Sun. It was estimated that 1 neutron per year would be produced by this process. A signal of < 0.2 event was expected in the NCD array.
VII.2.2. External-source neutrons
Radioactive backgrounds in the acrylic vessel and the surrounding H 2 O could bring forth photodisintegration neutrons in the D 2 O target. Neutrons could also be produced via (α, n)
reactions in the AV. The total neutron backgrounds due to these external sources were found to be 20.6 +10.2 −7.3 events for the PMT array data and 40.9 +20.6 −17.9 events for the NCD array data. During its construction the acrylic vessel was exposed to Rn in the underground laboratory air. The subsequent Rn daughters became embedded in the acrylic and could initiate (α, n) reactions on 13 C, 17 O and 18 O. The activity on the surface of the AV was directly counted using silicon counters. Results from measurements performed at the end of the third phase were in agreement with those performed at the end of the second phase. Thus the rate of these external-source neutrons from the vessel was taken to be the same as for
Phase II [6]. Adding the photodisintegration neutron backgrounds due to intrinsic Th and U in the acrylic, which were determined from ex situ assays [5], the total external-source neutron backgrounds from the AV were found to be 18.3 +10.2 −7.3 counts for the PMT array data and 33.8 +19.9 −17.1 counts for the NCD array data.
VII.3. Other Cherenkov light backgrounds
A 20-second veto following a tagged muon event removed the majority of radioactivity that followed. The residual background from the decay of cosmogenic 16 N was estimated at 0.61±0.61 event in the PMT array data. Other Cherenkov light background events inside and outside the fiducial volume were estimated using calibration source data, measured activities, Monte Carlo calculations, and controlled injections of Rn [28] into the detector.
These backgrounds were found to be small above the analysis energy threshold and within the fiducial volume, and were included as an additional uncertainty on the flux measurements.
Isotropic acrylic vessel background (IAVB) events were identified in previous phases [6]. It was estimated that < 0.3 IAVB event (68% CL) remained in the PMT array data after data reduction cuts.
VIII. SIMULATION OF PULSES IN THE NCD ARRAY
The largest source of backgrounds in the neutron signal region in the NCD counters was alpha decays from the construction materials of the array. This was a very difficult background to calibrate, as any alpha particles from external calibration sources would not have sufficient energy to penetrate the counter wall. The spatial distribution and the composition of trace radioactivity in the counters also varied from counter to counter; therefore, background samples from the 4 He counters were not sufficient to fully characterize this allimportant background to the neutron signal. An extensive Monte Carlo, discussed in this section, was developed to simulate ionization pulses in the NCD counters, and was used in defining the alpha background spectral shape for the solar neutrino analysis. Further details can be found in Refs. [41][42][43].
VIII.1. Physics model
The NCD counter simulation created ionization tracks for protons, tritons, alphas, and betas in the NCD counter gas. Alpha energy loss in the nickel wall was also calculated if necessary. Track formation for betas was handled by EGS4 [29]. Proton, triton, and alpha tracks were all calculated using the same procedure as follows. The track was divided into N (typically 5,000-20,000) 1-µm long segments such that each segment could be approximated as a point charge. The total current resulting from the whole track at time t was the sum of the individual currents from each track segment, i. The current induced on the anode wire from each segment was mainly a result of a positive charge, q i = en i , drifting towards the cathode [44]:
I track (t) = N i=1 G i n i q i 2 ln(b/a) 1 t − t 0 + τ ,(26)
where G i is the gas gain, n i is the number of electron-ion pairs created in segment i, a=25 µm is the anode radius, b=2.54 cm is the NCD-counter inner radius, t 0 is the "start time" for the current from the i th segment, and τ is the ion-drift time constant. The number of ion pairs depended on the stopping power, dE dx ; the mean energy required to produce an electron-ion pair in the gas, W ; and the segment length l, such that n i = dE dx l W . The description of an ionization track involved knowing where each segment was located and how much energy had been deposited there. Multiple scattering of the ionizing particle was simulated with the Ziegler-Biersack-Littmarck method [45]. The results of that simulation were in excellent agreement with the full TRIM Monte Carlo calculation [46]. Values of dE dx were determined with stopping-power tables from TRIM for protons, tritons, and alphas.
The average energy required to produce an electron-ion pair in the NCD counter gas was measured using neutron sources and undeployed NCD counters. Integrating over many current pulses with average energy E, the ratio G/W is proportional to the total current in a proportional counter, I [47]:
G W = I ηeE ,(27)
where η is the rate of neutron captures and e is the electron charge. E was determined with the NCD counter Monte Carlo to be (701 ±7) keV. W is a characteristic of the NCD counter gas. It is approximately energy-independent, and is approximately equal for protons, tritons, and alphas [48]. We measured W by operating the counter in the "ion saturation" mode (200-800 V), and G/W by operating the counter at the standard voltage (1950 V). In the first case, we found W = 34.1 ± 12.4 eV, and in the latter case, W = 34 ± 5 eV, which was used in the Monte Carlo.
A low-energy electron transport simulation was developed to evaluate the mean drift times, t d , of electrons in the NCD counter gas mixture as a function of radial distance from the anode wire r. The results were in good agreement with GARFIELD [49] (28) with t d in ns and r in cm.
Electron diffusion resulted in a radially-dependent smearing effect on all pulses, and dominated the time resolution. A smearing factor σ D was tabulated as a function of r and applied in pulse calculations. σ D and t d were linearly related:
σ D (t d ) = 0.0124 t d + 0.559(29)
The mean NCD counter gas gainḠ, as a function of voltage, was well described by the Diethorn formula [47]. However, ion shielding from the charge multiplication can significantly change the gas gain. A two-parameter model was developed to account quantitatively for this space-charge effect. The change in gas gain, δG, resulting from a change in wire charge density, δλ(r), due to the ions formed near the anode at voltage V is
δG ∝Ḡ ln(Ḡ) ln(b/a) 2πǫ • V 1 + 1 ln(r av /a) δλ(r),(30)
where ǫ • is the permittivity of free space and r av = 58 ± 10 µm is the mean avalanche radius. δλ can be obtained by dividing the induced charge by a characteristic shower width in the spatial dimension parallel to the anode wire, W. The other parameter that needs to be optimized is the constant of proportionality in this equation. Electrons originating from some segment of a track are affected by the density changes δλ j due to ions formed in previous electron cascades. Each of these ion clusters moves slowly towards the cathode while the primary electrons are being collected. In the presence of many ion clusters, the total change in the anode charge density at time t, experienced by electrons from the i th track segment is therefore:
δλ i = e W i−1 j=1 ln(b/r j (t)) ln(b/a) G j n j + e W ln(b/r) ln(b/a) n i ,(31)
where n j is the number of ion pairs formed in the j th segment. j loops over all previous ion clusters, which have moved to different radiir j (t) at time t.r j (t) is solved by integrating the relation
dr j dt = µ i E:r j (t) 2 = 2µ i V t ln(b/a) + r 2 av ,(32)
where µ i is the ion mobility and E is the cylindrical electric field.
The mean gas gain of the i th track segment isḠ i =Ḡ − δG i . The actual G i applied to the i th segment is sampled from an exponential distribution with meanḠ i .
The smaller ion mobility, relative to that of the electrons, results in the long tail that is characteristic of pulses from ionization in the NCD counters. The evolution of a current pulse in a cylindrical proportional counter is described by Eqn. 26. The ion time constant, τ , is inversely proportional to the ion mobility, µ:
τ = a 2 p ln(b/a) 2µV ,(33)
where p is the gas pressure.
We measured τ using neutron calibration data. Ionization tracks that are parallel to the anode wire have a relatively simple underlying structure; the primary ionization electrons all reach the anode at approximately the same time, with some spread due to straggling.
The ion-tail time constant was extracted by selecting the narrowest neutron pulses from calibration data sets and fitting each pulse with a Gaussian convolved with the ion tail, a reflection and the electronic model. This model fitted the peaks well enough to allow for a characterization of the ion tail. We found τ = 5.50±0.14 ns. This time constant corresponds to an ion mobility of µ = (1.082 ± 0.027) × 10 −8 cm 2 ns −1 V −1 .
VIII.2. Simulation of NCD array electronics
Propagation of the pulse along the NCD string was simulated with a lossy transmission Propagation in the NCD counter cable was simulated with a low-pass filter (RC ≈ 3 ns).
There was a small reflection (reflection coefficient = 15%) at the preamp input due to the slight impedance mismatch between the preamp input and the cable. This portion of the pulse traveled to the bottom of the NCD string and reflected back upwards.
The preamplifier was simulated with a gain (27,500 V/A), a low-pass filter (RC ≈ 22 ns) and a high-pass filter (RC = 58000 ns). All RC constants in the electronic model were measured by fitting the model to ex situ injected pulses.
The frequency response of the multiplexer system before the logarithmic amplifier was simulated with a low-pass filter (RC ≈ 13.5 ns). The constants used to parameterize the logarithmic amplification were the same constants used to "de-log" real data pulses, which were determined by regular in situ calibrations during data taking. The circuit elements after the logarithmic amplification were simulated with the final low-pass filter (RC ≈ 16.7 ns).
The pulse array values were rounded off to the nearest integer to replicate the digitization. Certain parts of the overall simulation were relatively slow due to the loops over the large (N = 17, 000) arrays containing the simulated pulses. As a result we implemented a fast alternative to the full simulation. The ionization track was simulated to determine the timing of the event and the energy deposited in the gas. That energy was converted directly to an approximate shaper-ADC measurement and was smeared with a Gaussian to roughly account for the missing physics and electronic noise. This option allowed simulations for preliminary comparisons with the data because they did not require pulse-shape calculations.
VIII.3. Verification and systematic uncertainties
The ratio of the number of bulk uranium and thorium alpha events to that from cathodesurface polonium in the NCD strings could not be fully represented in the 4 He string data due to string-to-string variation in these backgrounds. Simulations were used to calculate the alpha energy spectrum PDFs instead. Therefore, it was important to accurately simulate these PDFs, and to assess their systematic uncertainties for use in the region of interest for the solar neutrino analysis.
We optimized and validated the NCD array signal simulation by comparing with several types of data, including neutron source calibrations, high-energy alpha events, and 4 Hestring alpha data. This procedure was designed to ensure that the simulations accurately reproduced the data, without in any way tuning on a data set that contained the neutron signal, which was determined by signal extraction (Sec. IX).
The comparison of the simulation with neutron calibration data tested nearly all aspects of the simulation physics model. We compared 24 Na neutron calibration data with simulations for a number of pulse characteristics, such as the mean, width, skewness, kurtosis, amplitude, and integral, and timing variables, including the rise time, integral rise time, and full-width at half-maximum. These comparisons were used to estimate parameter values and uncertainties for electron and ion motion in the NCD counter gas, as well as the space charge model. Figure 20 shows a comparison of some of these pulse-shape variables between real and simulated 24 Na neutron data.
We estimated the fraction of surface polonium and bulk alpha events on each string by fitting the energy distribution above the neutron energy region, shown in the bottom plot in Fig. 21. After the fit, we calculated an event weight, which was a function of string number and alpha type (polonium, uranium, or thorium) describing the best-fit fraction of alphas on each string due to each source. In general, polonium comprised ∼60% of the alpha signal;
however, there were ±20% variations between strings. The best-fit alpha fractions and the MC energy scale correction were applied on an event-by-event basis to produce the alpha energy spectrum PDF.
The systematic uncertainties included the depth of alpha-emitting contaminations ("al- Monte Carlo systematic uncertainties was estimated from ex situ data, off-line measurements of the NCD array signal processing electronic response, and in situ constraints from the NCD array data. The most significant sources of systematic uncertainty were due to variations of the simulated alpha depth within the NCD counter walls and the data reduction cuts.
We calculated fractional first derivatives to describe the changes in the Monte Carlo alpha energy spectrum allowed by the systematic uncertainties [41]. For each of the systematic uncertainties, the first derivatives for each bin of the energy distribution were calculated by taking the difference between histograms containing the standard Monte Carlo prediction for the shape of the energy distribution, and the variational Monte Carlo energy-distribution shapes. Then, the total systematic uncertainty in each energy bin was assessed by summing the eight contributions in quadrature. The functional dependence of those derivatives on energy is shown in the top plot of Fig. 21.
The final simulated alpha energy spectrum in the solar neutrino analysis window for the NCD array is shown in the center plot of Fig. 21, with systematic uncertainties. This alpha spectrum was used as the alpha background PDF, together with the neutron signal template from 24 Na neutron calibration data, to determine the total number of neutron events.
IX. NEUTRINO SIGNAL DECOMPOSITION
This section describes the techniques used in the SNO Phase-III neutrino flux measurement (referred to as signal extraction). Three different signal extraction methods were developed. These extended log-likelihood techniques were designed to perform a joint analysis of the data from the PMT and the NCD arrays. The nuisance parameters (systematic uncertainties), weighted by external constraints determined from calibrations and simulations, were allowed to vary in the fit of the neutrino signals. This "floating" technique enabled the determination of the correlations between the observed signals and the nuisance parameters. In the three methods, the energy spectrum of NCD array events was fit with the Monte Carlo alpha background distribution described earlier, the neutron spectrum determined from 24 Na calibration, expected neutron backgrounds, and two instrumental background event distributions. Events from the PMT array were fitted in the reconstructed effective kinetic energy T eff , the cosine of the event direction relative to the vector from the Sun cos θ ⊙ , and the volume-weighted radius ρ.
There were in general two classes of systematic uncertainties: the first had a direct impact and total predicted number of events (grey, with systematic uncertainties) in the "open" data set (see Sec. IX). Bottom: energy spectrum above the NCD array data analysis energy window, for data (black, with statistical uncertainties), alpha "cocktail" (grey, with systematic uncertainties), polonium (red), and bulk (blue dashed) simulation. The "cocktail" is a collection of simulated alpha on the shapes of the probability density functions (PDF), such as energy scale or angular resolution for the PMT array data; the second are uncertainties on parameters that did not affect the PDF shapes, such as detection efficiencies. The three signal extraction techniques differed in the implementation of how the nuisance parameters were floated.
The first method was an extension to the signal extraction techniques [30] used in previous analyses [5,6]. Systematic uncertainties of the second kind were easily floated in this method.
But floating the first kind was much more challenging as it would require PDFs to be rebuilt between evaluation of the likelihood function in the minimization process. As a result, only a few significant systematic uncertainties were allowed to float in this method due to computational limitations. Those systematic uncertainties that were not floated were estimated by repeating the fit with the parameters varied by their positive and negative 1σ deviations.
In the second method the nuisance parameters were not varied during the fit. Instead the signal extraction fit to the data was done many times with the necessary PDFs built from a set of nuisance parameters, whose values were drawn randomly according to their distributions. After this ensemble of fits, the flux result from each trial was re-weighted in proportion to its likelihood. This sampling method circumvented the need to rebuild PDFs during the numerically intensive likelihood maximization process.
Markov-Chain Monte Carlo (MCMC) was the third method used in signal extraction. The flux results from this method were published in the original letter on this phase of SNO [7], and the details in its implementation are presented in the following. These discussions include a description of how the MCMC parameter estimation works, the extended loglikelihood function used to obtain parameter estimates, and finally the results of fits to the full third-phase data set.
It is noted that a blind analysis procedure was used. The data set used during the development of the signal extraction procedures excluded a hidden fraction of the final data set and included an admixture of neutron events from cosmic-ray muon interactions. The blindness constraints were removed after all analysis procedures, parameters and backgrounds were finalized. The finalization of the three signal extraction procedures before removing these constraints required an agreement of their flux results for the "open" data set to within the expected statistical spread. A comparison of the results from the three analysis methods after the blindness conditions had been removed revealed two issues, which were understood prior to the publication of the letter [7]. These issues will be discussed later in this section.
IX.1. Markov-Chain Monte-Carlo parameter estimation
In SNO's previous phases, the negative log-likelihood (NLL) function was simply minimized with respect to all parameters to get the best-fit value. Minimizing the NLL was very challenging with so many fit parameters, and the likelihood space near the minimum could be uneven. This means that common function minimization packages, such as MINUIT [52], could run into numerical convergence problems. The use of MCMC circumvented this slow convergence problem by interpreting NLL as the negative logarithm of a joint probability distribution P for all of the free parameters, i.e. P = exp(log(L)). The nuisance parameters were integrated to determine the posterior distributions for the fluxes. The origins of this procedure go back to Bayesian probability theory, and in fact our approach could be considered to be a Bayesian analysis with uniform priors assumed for the fluxes. Both the speed of convergence and the insensitivity to unevenness in the NLL space mean that the MCMC method is better suited to handling large numbers of nuisance parameters. PDFs for the PMT array data were three dimensional in ρ, cos θ ⊙ , and T eff . The exceptions were the photodisintegration backgrounds due to K2, K5 and the bulk nickel, which were factorized (i.e. P (T eff )P (cos θ ⊙ )P (ρ)) in order to avoid any problems associated with low statistics in their simulations.
IX.2.2. NCD array data
The NCD array signals and backgrounds in the fits were the NC signal and various sources of neutron backgrounds. These backgrounds include photodisintegration due to radioactivity in the D 2 O target, nickel housing of the NCD array, and hotspots on strings K2 and K5; external source neutrons due to radioactivity in the acrylic vessel and the H 2 O shield; neutrons from atmospheric neutrino interactions and cosmogenic muons; and instrumental backgrounds with characteristics of those seen in strings J3 and N4.
The only observable used in signal extraction was the summed energy spectrum of the shaper-ADC in the NCD array readout ("shaper energy", E NCD ), restricted to the range of 0.4 MeV< E NCD < 1.4 MeV. The shaper energy spectrum from a uniformly distributed 24 Na source was used as the NC PDF. The alpha background PDFs were derived from the simulation as discussed in Sec. VIII.
IX.3. The extended log-likelihood function
In the extended log-likelihood function, two different approaches were used to handle systematic uncertainties. To include systematic uncertainties associated with the PMT array data and uncertainties associated with the neutron PDFs in the NCD array data, the respective PDFs were rebuilt from an array of event observables on each evaluation of the log-likelihood function. For the systematic uncertainties associated with the alpha background PDFs in the NCD array data, the shapes of those PDFs were modified by a multiplicative function.
Twenty six systematic uncertainties were included in the fit. The systematic uncertainties due to neutron detection efficiency were applied to the neutral-current PDFs separately for the NCD and PMT data streams. The twelve systematic uncertainties associated with the PDF shapes for the PMT array were related to its energy scale and resolution, and the event vertex reconstruction algorithm's spatial accuracy, spatial resolution and angular resolution.
This last entity was applied to the ES signal only. The twelve systematic uncertainties associated with the PDF shapes for the NCD array were related to the energy scale (a E NCD The NLL function to be minimized was the sum of a NLL for the PMT array data − log L PMT and for the NCD array data (− log L NCD ):
− log L = − log L PMT − log L NCD .(34)
The following subsections describe how these two NLLs were handled in this analysis.
IX.3.1. PMT array -NLL with systematic uncertainties
The systematic uncertainties arose from differences in the simulations and reconstructed data. The PDFs associated with the PMT array (P mc ) were rebuilt using the scaled and smeared value of T eff , ρ and cos θ ⊙ . The goal was to allow the data to tell us how the scales and resolutions differed between data and simulation within the constraints from calibration data. We therefore modeled the differences as a possible re-mapping of the observables for the simulated events. We then fitted for the re-mapped parameters to determine the allowable range of this re-mapping while still matching the data observables.
The negative log-likelihood for the PMT array data can therefore be written as:
− log L PMT = M PMT i f PMT i S PMT i φ i − N PMT j log M PMT i f PMT i S PMT i φ i P i mc (ρ j , cos θ ⊙,j , T eff,j ) − log L PMT constraints ,(35)
where there were M P M T event classes (such as neutrino signals from different interactions and Cherenkov light backgrounds) and N P M T PMT events. For the i th signal, the factor to convert from flux to the number of events is f i , the fitted flux is φ i , the fiducial volume correction factor is S i , and the remapped normalized probability density function is P i MC . The constraint terms are:
− log L PMT constraints = 1 2 i ( p i − p i σ p i ) 2 + 1 2 i j (b i − b i ) (b j − b j ) (V −1 b ) ij(36)
where p i are all of the PMT systematic parameters other than two sets of correlated position resolution parameters denoted as b i in the second term. V b is the covariance matrix for these latter parameters. The parameterization of the nuisance parameters can be found in Appendix B.
IX.3.2. NCD array -NLL with systematic uncertainties
For the NCD array data the probability distribution functions Q were one dimensional functions of the shaper energy (E NCD ). For each evaluation of the likelihood, where the neutron PDF systematics were changed, the PDFs were rebuilt using E remap :
E remap = a E NCD 1 E NCD × (1 + N(0, b E NCD 0 ))(37)
where N(µ • , σ) is a Gaussian distribution with mean µ • and width σ. As mentioned previously, the simulated alpha background PDF was rebuilt by multiplying the unmodified PDF (Q α ) by a re-weighting factor (α i ) and a multiplicative function in shaper energy s i (E NCD ):
Q α mc (E NCD ) = Q α (E NCD )(1 + 8 i=1 α i s i (E NCD )).(38)
The eight reweighting functions s i included systematic effects in the depth and intensity variations of 210 Po and other natural alpha emitters in the NCD string body, ion mobility, avalanche width and gradient offset, drift time variation and data reduction cut efficiency.
In addition two instrumental background PDFs, based on instrumental background events observed in strings J3 and N4 (which were excluded from the neutrino candidate data set),
were parameterized as skewed Gaussian distributions. A summary of the parameterizations of these systematic uncertainties can be found in Appendix B.
The negative log-likelihood for the NCD array data can therefore be written as:
−logL NCD = M NCD i f NCD i S NCD i φ i − N NCD j log M NCD i f NCD i S NCD i φ i Q i MC (E NCD,j ) − log L NCD constraints ,(39)
where there are M NCD event classes and N NCD events in the NCD array data. For the i th event class, the flux-to-event conversion factor is f NCD i , the fitted flux is φ i , the fiducial volume correction factor is S i , and the remapped normalized probability density function is
Q i MC .
The constraint terms are:
− log L NCD constraints = 1 2 i ( p i − p i σ p i ) 2 ,(40)
where p i are all of the systematic parameters for the NCD array data.
In Eqns. 35 and 39 the factors S i were used to propagate the fiducial volume uncertainty.
They could change as the systematic parameters were changed, and were calculated each time the PDFs were rebuilt. The other set of conversion factors f i included live time and efficiencies that were used to convert the number of events in a fiducial volume into a flux above threshold. A full description of these conversion factors can be found in Ref. [5].
IX.3.3. Neutron background and other input constraints
The neutron backgrounds in the PMT array data were determined from the neutronbackground fits in the NCD array data. For N NCD i events fitted for a certain neutron background in the NCD array data, the number of background events from the same source
is f PMT i N NCD i , where f PMT i
is the ratio of the number expected in the NCD array data to that in the PMT array data. These conversion factors are summarized in Table X.
Several small Cherenkov light backgrounds required adjustments of the CC and ES fluxes.
These backgrounds were Cherenkov events from beta-gamma decays in the D 2 O and in the external regions (AV, H 2 O, and PMT support geodesic structure), and isotropic light background from the AV. To account for these small backgrounds at each step in the chain, their contributions were randomly drawn, assuming a Gaussian distribution with means and widths given in Table VIII. These contributions were then split between the CC and ES channels, with the latter assuming 10% of the total. This fraction was applied because the ES peak occupies a corresponding fraction in the cos θ ⊙ distribution for neutrino signal.
IX.4. Results of fits to the full Phase-III data set
Fits without systematic uncertainty evaluation were obtained by running a single fit Markov-chain with 320,000 steps where the first 20,000 steps were rejected to ensure convergence. The fits that allow systematic uncertainty evaluation were obtained by running 92 independent Markov-chains with 6,500 steps each. Each fit was started with different starting parameters near the best-fit point varied by a Gaussian distribution with width given by the estimated uncertainty on the parameter. The first 3,500 steps in this fit were rejected in order to minimize the effect of initial values on the posterior inference, and the remaining 3,000 steps of each fit were put into a histogram for each parameter. A total of 276,000 steps were used to estimate the parameter uncertainties.
To ensure the robustness of the MCMC signal extraction, an ensemble test with all systematic parameters floated was performed. Each of the mock data sets were assembled with PDFs that were regenerated with different randomly sampled systematic parameter values. To run this ensemble test of 100 runs with all systematic parameters included, each with a MCMC chain length of 22,000, a substantial amount of computational power was required. As a result, only the final fit configuration of the full Phase-III data set was tested.
The fits to the real data were performed with far more steps to ensure that the uncertainty estimates were more robust than those in the ensemble test. This ensemble test showed an acceptable level of bias and pull in the fit parameters, and established the reliability of this signal extraction method.
In the signal extraction fit, the spectral distributions of the ES and CC events were not constrained to the 8 B shape, but were extracted from the data. In Table XI
φ CC φ NC = 0.301 ± 0.033 (total).(42)
In Table XIII, the CC and ES electron differential energy spectra from the energyunconstrained fit are tabulated.
Projections of the best-fit distribution with the data are shown in Fig. 22. The fitted systematic parameter values are provided in Table XIV, while the constraints and fit results for the amplitude of different neutron and instrumental backgrounds are given in Table XV.
The uncertainties on the fitted values of the systematic parameters were for the most part the same as the width of the constraint that was used in the fit. There are three systematic parameters that have fit uncertainties that are considerably narrower than the constraint.
The shaper energy scale is narrower, and this appears to be a real effect of the neutron energy peak setting the energy scale. The shaper energy resolution has fit out at +1.2%, and is narrower only because of the combination of not being allowed to go negative, and having a constraint of +1.0 −0.0 %. The alpha Po depth have noticeably narrower fit uncertainties, most likely due to constraints from the spectral information of alpha backgrounds above about 0.9 MeV in the data. To better understand the contributions of the systematic uncertainties to the total uncertainties, we took the quadratic difference between the fit uncertainty including systematic parameter variations and the fit uncertainty without such variations. The equivalent neu- The MCMC fit results were checked against those from the other two independent methods described earlier in this section. A comparison of the results from these three analysis methods revealed two issues. A 10% difference between the NC flux uncertainties was found, and subsequent investigation revealed incorrect input parameters in two methods. After the inputs were corrected, the errors agreed, and there was no change in the fitted central values.
However, the ES flux determined from the MCMC method was 0.5σ lower than those from the other two analyses. This difference was found to be from the use of an inappropriate algorithm to provide a point estimation of the ES posterior distributions. A better method of fitting the posterior distribution to a Gaussian with different widths on either side of the mode was implemented, and the ES flux results agreed with those from the other two analyses.
The ES flux presented here is 2.2σ lower than that found by Super-Kamiokande-I [59].
This is consistent with a downward statistical fluctuation in the ES signal, as evidenced in the shortfall of signals near cos θ ⊙ = 1 in two isolated energy bins. The 8 B spectral and with the fluxes from the second phase [6]. Table XVII summarizes the CC, ES, and NC fluxes determined from energy-unconstrained fits in SNO's three phases, and Figure 23 shows a comparison of these NC measurements.
X. NEUTRINO MIXING MODEL INTERPRETATION OF RESULTS
The SNO measurements of the NC and CC fluxes for neutrinos originated from 8 neutrino flavor transitions due to the mixing of the massive neutrino states via the MSW effect [17,18]. Neutrino mixing parameters can be extracted by comparing the experimental data from SNO and other experiments, with the model predictions from the neutrino mixing hypothesis. The neutrino-mixing analyses presented in the following do not include solar neutrino results from a recent low-threshold analysis that combined the Phase-I and Phase-II data sets [22].
Full three-neutrino analyses are being carried out for all phases of SNO taken together and will be submitted for publication. For the purposes of the present work, a two-neutrino analysis is convenient because the mixing of the third mass eigenstate into ν e is small [62] and ∆m 2 sol ≪ ∆m 2 atm [63]. The two neutrino mixing parameters in this model are: the squared-mass difference of the neutrino mass eigenstates, ∆m 2 ≡ ∆m 2 21 , and the mixing angle between the appropriate mass eigenstates, θ ≡ θ 12 . The mixing angle is also given as tan 2 θ in order to compare its extracted value with our previous results, as well as with the results reported by others. The mixing model was used to propagate neutrinos inside the Sun, vacuum and the Earth, for each value of ∆m 2 and tan 2 θ. The model predictions for each experiment used in the global solar analysis were computed with the neutrino fluxes from the BS05(OP) solar model [64], which is in good agreement with the helioseismological data, and the latest 8 B spectrum shape with its associated uncertainties from Winter et al. [55].
The default approach in our analyses was the covariance χ 2 method. The χ 2 function was minimized at each point in the tan 2 θ − ∆m 2 plane with respect to the 8 B neutrino flux.
The least-square fit and the projection in the tan 2 θ − ∆m 2 plane were then performed by allowing any values for the 8 B neutrino flux for a given value of tan 2 θ and ∆m 2 . At the minimum value, χ 2 min , the best-fit values for the mixing parameters tan 2 θ and ∆m 2 were extracted, together with the corresponding value for the 8 B flux. Then, the 68%, 95% and 99.78% confidence level (CL) regions in the two-dimensional parameter space tan 2 θ − ∆m 2 were drawn. The uncertainties on the mixing parameters were determined by projecting the χ 2 function passing through the best-fit point on the tan 2 θ and ∆m 2 axes, separately. The one-dimensional (1D) projections were not a simple slice of the two-dimensional contour passing through χ 2 min , but instead a projection in which ∆χ 2 = χ 2 − χ 2 min was computed for each 1D axis allowing the other parameter to take any values. From these 1D projections the uncertainties on each parameter, the 1σ spreads were determined from the values at
χ 2 min + 1.
The solar neutrino data used in this analyses were: SNO Phase-I (SNO-I ) summed kinetic energy spectra (CC+ES+NC+backgrounds) for day and night [4], SNO Phase-II (SNO-II) CC kinetic energy spectra, ES and NC fluxes for day and night [6], SNO Phase-III (SNO-III) CC, ES and NC fluxes [7], Super-Kamiokande zenith binned energy spectra [59]; and the rate measurements from the Homestake [10], Gallex/GNO [65], SAGE [66] and Borexino experiments [67]. The global solar χ 2 function obtained by comparing these data with the corresponding model predictions were then combined with the 2881-ton-year KamLAND reactor anti-neutrino results [68], assuming CPT invariance. While the results from SNO are highly sensitive to the mixing angle through the measured ratio φ CC /φ NC , the KamLAND measurement has a higher sensitivity to the allowed values of the ∆m 2 parameter.
First, the constraint on the neutrino mixing parameters was placed by interpreting the measurements from the results of the three phases of SNO only. Detailed descriptions on the use of data sets from SNO-I and SNO-II to interpret neutrino mixing can be found in
Refs. [6,69]. SNO-III data were obtained from signal extraction (Sec. IX) as integrated CC, ES and NC fluxes (averaged over day and night), which are tabulated in Table XI. The statistical correlation coefficients between the three fluxes, which were needed in building the global χ 2 , are tabulated in Table XII.
For the SNO-III data sample, the χ 2 function is defined as [70]:
χ 2 = 3 i,j=1 (Y exp i − Y th i ) T [σ 2 ij (tot)] −1 (Y exp j − Y th j ),(44)
where Y exp i is the CC, ES or NC averaged flux measurement, and Y th i is the theoretical expectation obtained from the two-neutrino mixing model. The model prediction Y th i was calculated under the assumption of the two-neutrino oscillation hypothesis, thus it depended on the number of free parameters n in the fit. In the physics interpretation presented here and in Ref. [7], the free parameters were the neutrino mixing parameters (∆m 2 and tan 2 θ) and the total flux of the 8 B neutrinos φ8 B . The shape of the 8 B spectrum was fully constrained by the mixing parameters.
The covariance error matrix σ 2 ij (tot) was built as a sum of the squares of the statistical σ 2 ij (exp) and systematic σ 2 ij (syst) uncertainties:
σ 2 ij (tot) = σ 2 ij (stat) + σ 2 ij (syst),(45)
where the statistical covariance matrix is given by:
σ 2 ij (stat) = ρ ij u i u j ,(46)
where u i and ρ ij are the statistical uncertainty for the measurement Y exp
γ ik = ∆Y ik Y i .(47)
The relative uncertainties γ ik were then used to construct the systematic error matrix σ 2 ij (syst), which is defined as:
σ 2 ij (syst) = Y th i Y th j K k=1 r k ij γ ik γ jk ,(48)
where K is the number of systematic uncertainties affecting the observables i and j. A coefficient r k ij describes a correlation between the observables i and j induced by the systematic uncertainty k, within a single phase of a given experiment. Values of the correlation coefficients r k ij are summarized in Table XVIII for the SNO-III data sample. In our analyses, the correlations among the systematic uncertainties between the three phases of SNO was also accounted for. However, these correlations had little impact on the allowed regions in the tan 2 θ − ∆m 2 plane. The relative errors for the most important energy related systematic uncertainties, such as PMT energy scale and resolution, and also for the 8 B spectrum shape uncertainty, were computed for each value of the mixing parameters as γ ik = ∆Y th ik /Y th i . With the inclusion of SNO Phase-III results, the following best-fit neutrino mixing parameters were found for the SNO-only analyses: ∆m 2 = 4.57 +2.30 −1.22 × 10 −5 eV 2 and tan 2 θ = 0.447 +0.045 −0.048 . The flux of the 8 B neutrinos was floated with respect to the BS05(OP) model prediction of 5.69 × 10 6 cm −2 s −1 , and the best-fit value was φ8 B = 5.12 × 10 6 cm −2 s −1 . The flux of hep neutrinos was fixed at the BP05(OP) model value of 7.93 × 10 3 cm −2 s −1 . The minimum χ 2 at the best-fit point was 73.77 for 72 degrees of freedom. The allowed regions at 68%, 95% and 99.73% confidence level (CL) in the ∆m 2 − tan 2 θ plane from this fit, shown in Figure 24, were significant improvements compared to the Phase-II result [6]. The vacuum ("VAC") oscillation region was ruled out at the 99.73% CL for the first time using SNO data only. The remaining regions in the oscillation plane are significantly smaller than those presented in Ref. [6], with reduced marginalized 1σ uncertainties. The two best-fit results are given in Table XIX, with a comparison of the effects of including the SNO-III data sample in the SNO-only oscillation analysis.
In our first report of Phase-III results [7], the following changes in the global solar neutrino analysis were made with respect to our Phase-II analysis in Ref. [6]: the model predictions for all solar neutrino experiments were computed using the BS05(OP) model and the 8 B neutrino spectrum shape from Ref. [55], the inclusion of the 192 live-day results from Borexino [67], update of data from SK-1 using results from Ref. [59], and, most importantly, the new measurements from the third phase of SNO were incorporated. The global fit to these solar neutrino data led to the following neutrino mixing parameters: ∆m 2 = 4.90 +1.64 −0.93 × 10 −5 eV 2 and θ = 33.5 +1.3 −1.3 , with 8 B flux of φ8 B = 5.21 × 10 6 cm −2 s −1 . The minimum χ 2 at the best-fit point was 130.29 for 120 degrees of freedom. The allowed regions from this analysis are shown on the left panel in Fig. 25. The constraint on both neutrino mixing parameters was much better than our results in Phase II [6]. When the 2881-ton-year KamLAND results were included in this analysis [68], the best-fit parameters became: ∆m 2 = 7.59 +0. 21 −0.19 × 10 −5 eV 2 and θ = 34.4 +1.3 −1.2 degrees, and the 8 B flux of φ8 B = 4.92 × 10 6 cm −2 s −1 . The improvement in comparison with the former analysis was observed in the allowed regions from the combined fit shown in the right panel in Fig. 25. A summary of these results is given in Table XX.
In comparison to Phase-II results, this combined fit of the solar neutrino data and the 2881-ton-year results from KamLAND improved the constraints on the neutrino mixing parameters: mixing angle θ and ∆m 2 by 45% and 60%, respectively, at the time of the publication of Ref. [7]. This improvement on the mixing angle was dominated by the SNO experiment and the Phase-III results. The fitted values for the 8 B neutrino flux were in agreement with recent predictions from solar models. KamLAND data from [68], the best-fit point was at: ∆m 2 = 7.59 × 10 −5 eV 2 , tan 2 θ = 0.468, φ8 B = 4.92 × 10 6 cm −2 s −1 .
XI. SUMMARY
We have presented a detailed description of the SNO Phase-III results that were published in Ref. [7]. Neutrons from the NC reaction were detected predominantly by the NCD array.
The use of this technique, which was independent of the neutron detection methods in previous phases, resulted in reduced correlations between the CC, ES and NC fluxes and an improvement in the mixing angle uncertainty.
Several techniques to reliably calibrate the PMT and NCD arrays were developed and are detailed in this paper. The presence of the NCD array changed the optical properties, and hence the energy response, of the PMT array. Extensive studies and evaluation of the techniques used in calibrating the PMT array in previous phases were performed and reported. Radioactive backgrounds associated with the NCD array, the D 2 O target and other detector components were precisely quantified by in situ and ex situ measurements.
4.92
These measurements provided the constraints for the respective nuisance parameters in the determination of the ν e and the total active neutrino fluxes.
The total flux of active neutrinos was measured to be 5.54 +0.33 −0.31 (stat.) +0.36 −0.34 (syst.) × 10 6 cm −2 s −1 , and was consistent with previous measurements and standard solar models.
A global analysis of neutrino mixing parameters using solar and reactor neutrino results yielded the best-fit values of the neutrino mixing parameters of ∆m 2 = 7.59 +0. 19 −0.21 × 10 −5 eV 2 and θ = 34.4 +1.3 −1.2 degrees. A detailed paper that describes an analysis of data combined from all three phases of SNO is in preparation. Two independent sets of cuts were developed to remove instrumental backgrounds in the NCD array data. These cuts exploited the differences in the characteristics between ionization and non-ionization events. One of these two sets inspected the characteristics of the digitized waveforms in the time domain, while the other in the frequency domain. Cuts in both sets were used in the selection of the candidate event data set in the solar neutrino analysis. A summary of these cuts is provided in the following.
Bursts and overflow cuts
Two burst cuts were developed to remove events that occurred within a very short time window. If there were four or more shaper events within 100 ms, all events within this sequence were removed in the shaper burst cut. Similarly, if there were four or more MUXscope events within 100 ms, such events were removed in the MUX burst cut. In the shaper-overflow cut, shaper events that arrived within a short time (15 µs to 5 ms) after a previous event had saturated the shaper were removed. in which a small third reflection is seen in the tail of the pulse. Middle: a flat trace. Bottom: an oscillatory pulse.
Fork event cuts
The end of each counter string was attached to an open delay line. Pulses were reflected at this open termination; thus, some physics pulses exhibited a double-peak structure. There were also instrumental background pulses that exhibited similar double-peak characteristics, but with much different pulse width, time separation and amplitude ratio between the peaks.
The time-domain fork cut removed these "fork" events by exploiting these differences. Some fork events also featured a third reflection at the tail of the waveform. These events were removed by a cut that was specifically designed to search for such reflection in the time domain. The frequency-domain fork cut removed events with a peak around 12 MHz, which was the characteristic frequency of the fork events, in the power spectrum. A fork event example is shown in the top panel of Fig. 26.
Flat, oscillatory and narrow pulse event cuts
The flat-trace cut removed events that did not have a well-defined pulse profile. These pulses were mostly noise that crossed the trigger threshold. The dominant type of noise events during normal data taking was oscillatory pulses. These events were identified and removed by a cut on the number of times the waveform crossed the baseline. In the narrow pulse cut, pulses with widths that were too narrow to be ionization events were removed.
These pulses were mostly discharges and some of them carried a large amount of charge.
Spike events were also removed by identifying those with an abnormal ratio of their area and maximum amplitude. In the frequency domain, waveforms with unusual symmetry had a non-characteristic zero-frequency intercept of the phase in their Fourier transform. Waveforms with little power at low frequency were mostly flat or oscillation events. Waveforms with a large peak in the power spectrum at a frequency above 8 MHz (3.7 MHz) were mostly fork (oscillation) events. Three cuts were implemented to reject events with such anomalous characteristics in the power spectrum. Examples of flat and oscillatory trace events are shown in Fig. 26.
Parameterization of systematic for the NCD array data
There were ten systematic uncertainties associated with the shaper energy PDFs. For the uncertainties associated with the shaper-ADC energy E NCD and resolution, the energy PDFs were rebuilt using Eqn. 37.
For the alpha-background-related PDFs, the simulated alpha background PDF was rebuilt by multiplying the unmodified PDF (Q α ) by a re-weighting factor (α i ) and a multiplicative function in shaper energy s i (E NCD ):
In both cases the number of instrumental background events was allowed to float freely in the fit.
The numerical values of the constraints α i and the instrumental background parameters (p J3 1 and p N4 1 ) are tabulated in Table XIV
II. OVERVIEW OF THE SNO EXPERIMENTII.1. The SNO detector The SNO detector was located in Vale's Creighton Mine (46 • 28 ′ 30 ′′ N latitude, 81 • 12 ′ 04 ′′W longitude) near Sudbury, Ontario, Canada. The center of this real-time heavy-water ( 2 H 2 O, D 2 O hereafter) Cherenkov detector was at a depth of 2092 m (5890±94 meters of water equivalent). At this depth, the rate of cosmic-ray muons entering the detector was approximately three per hour. The solar neutrino target was 1000 metric tons (tonnes) of 99.92% isotopically pure D 2 O contained inside a 12-m-diameter acrylic vessel (AV). An array of 9456 20-cm Hamamatsu R1408 PMTs, installed on an 18-m diameter stainless steel geodesic structure (PSUP), was used to detect Cherenkov radiation in the target. A
FIG. 1 .FIG. 2 .
12Side view of the SNO detector in Phase III. The center of the acrylic vessel is the origin of the Cartesian coordinate system used in this paper. The NCD counter strings were arranged on a square grid with 1-m spacing (shown in Fig. 2). In this figure only the first row of NCD strings from the y − z plane are displayed. The positions of the strings in the NCD array projected onto the plane of the AV equator (x − y plane). The array was anchored on a square lattice with a 1-m grid constant. The strings labeled with the same letter denote strings of the same length and distance from the center of the AV. Strings I2, I3, I6 and I7 contained 4 He instead of 3 He. The outer circle is the AV equator and the inner circle is the neck of the AV through which the NCD strings were deployed. The NCD string markers are not drawn to scale. secondary ionizations, which led to a current pulse on the anode wire. This pulse traveled in both directions, up and down a counter string. The delay line at the bottom of each string added approximately 90 ns to the travel time of the portion of the pulse that traveled down the string. The termination at the end of the delay line was open, so the pulse was reflected without inversion. The direct and reflected portions of the pulse were separated by approximately 90-350 ns, depending on the origin of the pulse along the length of the NCD string. At the top of each counter string was a 93-Ω impedance coaxial cable that led to a current preamplifier. This preamplifier linearly transformed the signal to a voltage amplitude with a gain of 27.5 mV/µA.
FIG. 4 .
4An example of NCD string position reconstruction from the data of one of the laserball scans at 500 nm. The contours represent the 68% and 99% CL of the fitted position of NCD string M1. Also shown in this figure is the nominal position of the string. The difference between the fitted position and the nominal position is ∼2 cm in this case.
Figure 7
7shows the heavy-and light-water attenuation coefficients obtained at 421 nm. The time dependence of the attenuation lengths was very small, showing better stability than in Phase II. Therefore an average value of the optical parameters was used in all Monte Carlo simulations in the third phase. The new systematic uncertainties evaluated for the optical parameters included the precision of the PMT efficiency estimations, the NCD string shadow cut efficiency, the NCD string reflection corrections, and the up-down asymmetry in the PMT array response. The new systematic uncertainties contributed to a 10-25% increase of the uncertainties in the optical parameters compared to previous phases, and since the total uncertainties were dominated by the systematic component, the loss in statistics induced a negligible increase in the total uncertainties of the optical parameters. After averaging the results from all the
FIG. 7 .
7(a) D 2 O and (b) H 2 O inverse attenuation lengths as a function of time at 421 nm. The lines show the linear fits covering the commissioning and data-taking periods of Phase III.
FIG. 8 .
8Difference between data and MC vertex shifts in the three axis directions as a function of source position for scans along the main axes of the detector. The top, middle and bottom panels represent the shifts in x, y, and z coordinates, respectively.
varies significantly as a function of the z position of the source. Some smaller fluctuations were also observed for 16 N calibration scans in the x − y plane and were incorporated in the analysis. The apparent similarity of the results in x and y directions and the cylindrical symmetry of the detector in the third phase suggested the use of the same parameterization of the systematic effect for these two directions. The uncertainty on the vertex resolution was expressed as a second order polynomial for x and y directions and as a first order polynomial for the z direction.Tables I and II present the fitted values for the parameters ofthese polynomials, along with their associated correlation matrices.
FIG. 9 .
9Difference between data and MC fitted vertex width in the three axis directions (x, y and z, from top to bottom) as a function of the position of the source (z src ) for 16 N runs taken along the z axis of the detector.
The direction of events in the detector was determined at a later stage, decoupled from position reconstruction. It relied on a likelihood function composed of a Cherenkov light angular distribution function and a PMT solid angle correction. To avoid biases caused by scattered light, only PMT hits occurring within a time window of ±10 ns of the prompt light peak were selected.
(m = D 2
2O, acrylic, and H 2 O) are the product of the average optical path length (d m ) and the inverse attenuation length (α m ) measured in Sec. IV.1.3 above. The probability of photon transmission through the acrylic vessel (F ) was calculated from the Fresnel coefficients, D isthe Cherenkov angular distribution, Ω i is the solid angle of the PMT as seen from r fit , and ǫ i is the efficiency of photons to trigger a PMT upon their entering the light collecting region of the PMT and reflector assembly. Each factor in Eqn. 8 required the average optical path of a photon to be calculated from r fit , p i , and the detector geometry. The calculation was done to the center of the PMT photocathode surface. Ω i was determined from the optical paths to multiple points around the PMT light concentrator assembly.
and electronic channel efficiencies. E was determined at multiple wavelengths by a combination of optical calibrations and Monte Carlo simulations (see Sec. IV.1.3). It was normalized to the quantum efficiency for a typical PMT, which was also a function of wavelength, at normal incidence. In the central region of the detector, the obstruction of photons by the NCD array reduced the average number of direct PMT hits by up to 20%. The efficiency factor (ǫ NCD ), calculated based on a Monte Carlo simulation of Cherenkov light from single electrons in the D 2 O, accounted for this effect.
FIG. 10 .
10Relative photon collection efficiency of selected 16 N calibration runs as a function of time. The dashed line represents δ drift which was scaled to unity on September 27, 2005. for each set of simulated electron events, corrected by δ drift , was then matched to the known electron kinetic energy T e . F T consisted of the interpolation between these values. The average electron energy response can be characterized by a Gaussian function with resolution σ T = −0.2955 + 0.5031 √ T e + 0.0228T e , where T e is in MeV.
Figure 11
11shows the resulting mean T eff of selected 16 N calibration runs (solid points) after application of the drift correction. The mean T eff of the full Monte Carlo simulation with ǫ PCE = 0.653 δ drift are also shown (open points).
A
series of dedicated PMT high-voltage scans were performed to quantify the dependence of the detector response on the PMT gain. Inspecting the value of the upper half-maximum height of the single photoelectron charge distribution allowed for an estimation of the gain effects on the energy scale uncertainty. The ratio of the value of the upper half-maximum height of the single photoelectron charge distribution at the nominal 16 N energy compared to this slope allowed for a conservative estimation of the energy scale systematic due to gain changes. This uncertainty was determined to be 0.13%. A series of dedicated threshold scans were performed to quantify the dependence of the detector response on the PMT channel threshold. Comparing the ratio of the energy scale to the value of the lower half maximum height of the single photoelectron charge distribution allowed for an estimation of uncertainty on the energy scale due to threshold changes. This uncertainty was conservatively estimated at 0.11%. The energy scale could be affected by the change in the timing position of the prompt light peak. This is because the energy estimator utilized only the prompt light within a limited time window of 20 ns. In order to determine an uncertainty due to changes in the time residual position compared to the energy estimator's time window, the mean and width of the prompt timing peak were determined by a Gaussian fit on all the central 16 N runs.
FIG. 12 .
12The top plot shows the mean effective kinetic energy T eff versus volume-weighted radius ρ for the 16 N source deployed throughout the detector. The bottom plot shows the fractional difference between data and Monte Carlo (the ratio (data-MC)/data) for the16 N source deployed throughout the detector. The dashed line at ρ = 0.77 represents the edge of the fiducial volume at 550 cm. Points at the same value of ρ could have different energy response in data or Monte Carlo because of local point-to-point variation in detector response. The AV neck and the NCD array accentuated the variations at large ρ.
FIG. 14 .
14Temporal variation of the shaper-ADC channel threshold for string N1. In the threshold calibration, pulses from the pulser were attenuated by 30 dB. The channel threshold was determined by finding the attenuated pulser amplitude at which half of the expected events were observed.
performed in October 2005 and in November 2006. In these "spike" calibrations, about one liter of neutron-activated brine containing 24 Na was injected into the D 2 O volume and mixed. The 24 Na isotope, with a 14.96-hour halflife, decays to 24 Mg, almost always producing a beta decay electron with an end-point of 1.39 MeV and two gamma rays with energies of 1.37 MeV and 2.75 MeV. The 2.75-MeV gamma ray is capable of photodisintegrating a deuteron, the binding energy of which is 2.225 MeV. A perfectly mixed spike would produce a uniform distribution of neutrons up to within about 30 cm (one Compton scattering length) of the acrylic vessel, where the neutron intensity would drop off because the 2.75-MeV gamma ray has a significant chance of escaping the D 2 O volume. A correction factor, based on simulations of neutrons from NC interactions and from photodisintegration due to an evenly distributed 24 Na brine,
37 -
37MeV and 2.75-MeV gamma rays was measured. The measured rate was converted to the gamma radiation rate of the sample by correcting for the detector dead time, gamma ray acceptance, and the effect of the detector's dead layer. Given this measurement and the photodisintegration cross section of the deuteron at 2.75 MeV (with a 2% uncertainty [30]), the effective source strength if it were placed at the center of the SNO detector at a reference time could be determined in units of neutrons per second.Unlike the above measurement, the SNO PMT and NCD arrays were used to measure directly the neutron production rate due to a small activated brine sample in the D 2 O. A 10 g sample (measured to better than 0.5%) was placed in a sealed container and deployed to the center of the D 2 O volume using the calibration source manipulator. The 2.75-MeV gamma rays from the activated brine mostly escaped the container without any interactions; about one in 385 of these gamma rays photodisintegrated a deuteron to produce a free neutron.Gamma rays that interacted with the container lost energy by Compton scattering; when the effect of this energy loss was included, the average photodisintegration probability decreased to about 1/390 or 1/395, depending on the type of source container used (different ones were used in 2005 and 2006). The decrease in probability was estimated with a 0.7% uncertainty using Monte Carlo simulation.
FIG. 15 .
15A samp is the actual measured source strength of the sample of mass m samp (10-30 g), while A24 Na is the derived source strength of the main body of brine (m main = 500 or 1000 g) that was injected into the SNO detector several days later. The exponential correction factor e −∆t/τ 24Na corrects for the exponential decay of the source strength between the time of measurement and the reference time. The time offset ∆t for the run in 2005 was about 4 days, while in 2006, it was 6 days. The initial source strength was much larger in 2006, so more time was required to allow the source to cool down to levels that the SNO data acquisition system could handle. The quantity τ 24Na is the 24 Na lifetime, 21.58 hours. The quantity f P applies only to the in situ measurements, and corrects for the effects (discussed several paragraphs above) due to the radial distribution of neutron production and to a small fraction of gamma rays that scattered off the source container: f P = 1.0122 ± 0.0044 (stat.)± 0.0053 (syst.) for the 2005 run, and f P = 1.0288 ± 0.0050 (stat.) ± 0.0053 (syst.) for the 2006 run, where the statistical uncertainty is due to Monte Carlo statistics. The value of f P differed between 2005 and 2006 because different source containers were used. The brine source strength measurements from the three techniques were combined by taking a weighted average. The weighting of each data point was inversely proportional to the quadratic sum of the statistical and systematic uncertainties. Thus the PMT array measurement, which had the largest uncertainty, made only a minor contribution to the final result. The germanium detector and NCD array measurements, having similar uncertainty magnitudes, contributed about equally. The large uncertainty in the PMT array measure-The results of the brine source strength measurements in 2005 (top) and 2006 (bottom).The vertical axis in each frame shows the source strength A24 Na (in neutrons per second) in the SNO detector, deduced from the three different sample source strength A samp measurements (horizontal axis), at a reference time (see text and Eqn. 14 for details). This reference time (t = 0) was defined as the beginning of the first run when the brine was judged to be well mixed. The error bars show combined statistical and systematic uncertainties. The line through the data points is the result of the best fit, while the band shows the uncertainty of the fit.
FIG. 16 .
16Neutron capture rate in the NCD array as a function of time, from the 24 Na spike in 2005. The vertical axis is in units of neutron events per second, while the horizontal axis is time in units of 24 Na mean lifetimes (one lifetime = 21.58 hours). The exponentially decaying pointsare the actual measured rate R(t i ), while the data with the flat trend (R i (0)) is the rate with the exponential decay factor corrected for. The line through the points at time > 0 is the best fit to the data. Time t = 0 was 4.5324 Na mean lifetimes after the spike was added.
For 24
24Na decays occurring within about 30 cm (one Compton scattering length) of the acrylic vessel, the probability of the 2.75-MeV gamma ray to escape from the D 2 O volume was significant, so the neutron density dropped quickly as a function of the distance to the acrylic vessel wall. The neutrons produced by the 2.75-MeV gamma rays in 24 Na decays started with an energy of 260 keV and then were moderated by scattering, whereas those from the NC interaction started with a range of energies. This difference in the initial neutron energy could affect the neutron density at the edge of the vessel. Neutrons produced near
FIG. 17 .
17be considered stable after t = 0. Whereas the stability in the rate only indicated that the brine distribution reached some equilibrium configuration, the only reasonable regions where it might conceivably be non-uniform were near the inside of the acrylic vessel wall or near the NCD strings where boundary layers might have different salinity. Therefore these regions were considered carefully and used to establish systematic uncertainties on the uniformity of the brine.A possible way of measuring the brine distribution was by detecting the Cherenkov radiation from the Compton scattering of gamma rays and betas from the decay of 24 Na with the SNO PMT array and comparing with the expectations for a uniform distribution. This method, however, lacked precision because the calculations of the detector response were not very accurate for the low energies deposited by these decay products. When the Cherenkov light data were compared with Monte Carlo simulation of perfectly uniform brine, variations at the level of about ±10% were seen, which were consistent with our ability to model the detector at such low energies. Similar variations were seen in the studies of solar neutrino data at such energies.Another piece of evidence suggesting that the brine was well-mixed in the central regions away from the AV or the NCD array came from the comparison of the distribution of Cherenkov events from 24 Na gamma rays observed in the 2005 and 2006 data. The injection and mixing methods for the two years were very different[28]. In 2005, the brine was injected at several positions along the central vertical axis and then the water circulation was turned on. In this configuration D 2 O was pulled out from the bottom of the detector and returned at the top. In 2006, a more sophisticated method involving flow reversal and temperature inversion was employed. Using this method, an extensive eddy current was set up by causing the entire bulk of the D 2 O to rotate in one direction, then reversing the direction of the flow. In both years, the data showed that the brine distribution reached a stable equilibrium. If one examines the Cherenkov light data from each year alone, one cannot say with much confidence whether or not the equilibrium configuration was uniform because of the limitations associated with modeling the low-energy response. However, because the brine was mixed so differently, it is implausible that a non-uniform equilibrium configuration in 2005 could be the same as that in 2006.Figure 17shows, moreover, that the spatial distributions of the Cherenkov events in the brine were very similar. This comparison does not depend on the Monte Carlo simulations but is simply a study of the Cherenkov light data observed in the two cases. A plausible explanation for the similarity is that the brine was well-mixed in the central regions away from the NCD array and AV for both calibration sessions.Based on the above argument, we assumed that the spike was well-mixed, so the correctionfactor f non-unif has a central value of 1.0. The uncertainty on this value was obtained by considering the areas where stable inhomogeneities could possibly be established in the detector, namely near the NCD strings and near the wall of the acrylic vessel. In the vicinity of the NCD strings, calculations based on the laminar flow rates measured from the velocities observed from the Cherenkov light events indicated that the boundary layer could be on the order of 5 cm. A 0.5% effect of such a boundary layer on neutron capture efficiency was estimated from redistributing the salinity from within the layer uniformly in the D 2 O target. Careful studies of the fall-off of reconstructed gamma-ray signals from the 24 Na brine in the vicinity of the acrylic vessel, coupled with uncertainties in the knowledge of the optical properties of the detector in this region, led to an upper limit for a boundary layer thickness of 3.6 cm. Estimates based on flow rates in this vicinity gave smaller values for this boundary layer thickness. The effect of such a boundary layer on neutron capture efficiency was considered by redistributing the salinity from within the layer uniformly in the D 2 O target. This 1.5% effect was then combined in quadrature with the uncertainty from the NCD string region to give a full uncertainty 1.6% on the efficiency. From this conservative analysis of the systematic uncertainties, we obtained: f non-unif = 1.000 +0.016 −0.016 . (18) Ratio of the brine concentration as observed by reconstructed Cherenkov events in 2006 to that in 2005, as a function of detector coordinates. Top left: R = reconstructed radial distance from the center of SNO; top right: z, bottom left: x; bottom right: y. VI.2.5. ǫ sol : Capture efficiency for neutrons produced by solar neutrinosThe NCD array's capture efficiency for neutrons produced by solar neutrinos was obtained from the product of ǫ spike , f edge , and f non-unif (Eqn. 12). Before performing this multiplication, however, the value and uncertainties for ǫ spike from the runs in 2005 and 2006 were combined, taking into account uncertainty components that were correlated between the two years. The final result is:
VI. 3 .
3Neutron detection efficiency of the NCD array Several corrections must be applied to the capture efficiency, described in the previous section, in order to determine the NCD array's detection efficiency of NC neutrons in the solar neutrino analysis. These corrections, averaged over the duration of data-taking in Phase III, included the mean live fraction of the MUX (l MUX ) and the digitizing oscilloscope (l scope ), the average MUX threshold efficiency (ǫ MUX ), signal acceptance in the shaper energy window ( ǫ shaper ), and the acceptance of data reduction cuts ( ǫ cut ). The overall correction
Cherenkov light events were selected using the energy and fiducial volume cuts for the solar neutrino analysis, i.e. T eff ≥ 6 MeV and R fit ≤ 550 cm. A fit of the T eff spectrum from the combined data of the 24 Na runs in 2005 and 2006 to a combination of neutron signal and 24 Na beta-gamma background was performed. The number of neutrons was found to be n n = 8205.3±121.9 and the number of background events was found to be n γ = 1261.7±88.8 with a correlation of −0.61. Energy and reconstruction related systematic uncertainties in the solar neutrino analysis were propagated in this analysis, and their combined effect was found to be +1.71 −1.38 %. After symmetrizing this systematic uncertainty and combining it with the statistical uncertainty in the T eff spectral fit in quadrature, n n was determined to be 8205.3±185.4. With the neutron rates given inTable IV and the time span of the calibrationruns, ǫ spike could be evaluated, and the result for ǫ sol is ǫ sol = 0.0502 ± 0.0014.
Such gamma rays are emitted by beta-gamma decays of 208 Tl and 214 Bi from the 232 Th and 238 U chains respectively. An accurate measurement of these radioisotopes was crucial for the determination of the total 8 B neutrino flux. Concentrations of 3.8 × 10 −15 gTh/gD 2 O and 30 × 10 −15 gU/gD 2 O are each equivalent to the production of one neutron per day via photodisintegration. Two independent approaches were developed to measure these backgrounds. These are broadly classified as ex situ and in situ techniques. VII.1.1. Ex situ determination of radioactivity in D 2 O Three ex situ methods were developed to assay parent isotopes of 208 Tl and 214 Bi in the D 2 O and H 2 O regions of the detector. Common to all three techniques was extraction and filtering of a known amount of water from the detector and external counting of the resultant sample. Two methods extracted 224 Ra and 226 Ra, one using beads coated with manganese oxide (MnO x ) [31] and the other using filters loaded with hydrous titanium oxide (HTiO)
,33]. For MnO x and HTiO assays, up to 500 tonnes of water passed through the loaded columns over a 4 to 5-day period. In the MnO x technique, Ra isotopes were identified by alpha spectroscopy of Rn daughters. In the HTiO method, Ra isotopes were stripped from the filters, concentrated and identified using beta-alpha coincidence counting of the daughter nuclides.The equilibrium between 238 U and 214 Bi was broken by the ingress of 222 Rn (halflife = 3.82 d), primarily from the laboratory air and emanation from construction materials. The amount of 222 Rn in the water was measured by degassing, cryogenically concentrating the dissolved gases and counting the sample using a ZnS(Ag) scintillator[34]. Each Rn assay processed approximately 5 tonnes of water in a 5 hour period.During the third phase of SNO, 20 MnO x and 16 HTiO assays were conducted at regular intervals in the heavy water region. The results from each independent assay method were in good agreement. The activity measured by each assay was a combination of activity from the D 2 O and water systems piping. The variation of Th (Ra) activity in the water and piping was modeled as a function of time, taking into account other sources of Th (Ra) in the flow path. The resultant concentration was the live-time weighted combined HTiO and MnO x activity, which was 0.58 ± 0.35 × 10 −15 gTh/gD 2 O. The quoted uncertainty was combined from the systematic and statistical uncertainties. A total of 66 Rn assays were performed in the D 2 O region at regular intervals throughout the third phase. To calculate the mean 222 Rn concentration, the individual assay results were time and volume weighted. The equivalent mean 238 U concentration was 5.10 ± 1.80 × 10 −15 gU/gD 2 O, where the total uncertainty was combined from the systematic and statistical uncertainties.
Figure 18
18is a summary of the D 2 O assay results since the beginning of the SNO experiment. During SNO's Phase-III operation, an aggressive program of system purification combined with minimum recirculation of the heavy water led to a factor of five reduction in thorium. The concentration of 224 Ra was routinely measured at 0.1 atom/tonne.
< 4 . 5 FIG. 18 .
4518MeV, the selected events were dominated by 214 Bi and 208 Tl, from the 238 U and 232 Th chains respectively. The observed Cherenkov light was dominated by the direct beta decay of 214 Bi to the ground state of 214 Po with an end point of 3.27 MeV. 208 Tl decays almost always emitted a 2.614-MeV gamma ray, accompanied by one or more lower energy gamma D 2 O radioactivity measurements by ex situ assays are shown for 222 Rn (top), 224 Ra (middle) and 226 Ra (bottom). The dashed lines are the bounds for different SNO detector configurations (from periods A to J as indicated): A -Phase-I (unadulterated D 2 O) commissioning; B -Phase-I operation; C -salination; D -Phase-II (salt) operation; E -desalination; F -preparation for NCD array installation; G -NCD array deployment; H -NCD array commissioning; I -Phase-III (NCD array) operation; and J -removal of the NCD array. Periods highlighted in yellow were times with reduced access to the underground laboratory. For the radon data, color represents different sampling points: red is at the top (near the chimney-sphere interface), purple 1/3 down and blue at the bottom of the AV. The radon level was well below target for essentially the whole duration of the experiment, except for the high level at the beginning of the experiment, a large calibration spike in Phase-II, and during the deployment of the NCD array (when the cover gas protection was temporarily turned off). For the radium data, color indicates the technique used: red is for HTiO and blue for MnO x . Radium assays sampled the heavy water either from the top or the bottom of the AV. Again the initial higher concentrations quickly went below target. It should be noted that the data shown in this plot were not the only input to determining the radiopurity of the D 2 O target.
FIG. 19 .
19as 208 Tl and 214 Bi in the D 2 O or NCD strings. The in situ method provided continuous monitoring of backgrounds in the neutrino data set and a direct measurement of 214 Bi and 208 Tl, both of which could cause photodisintegration, without making any assumptions about equilibrium in the decay chain. The β 14 and the volume-weighted radial position, ρ, distributions for the different background sources in this in situ measurement are shown in Fig. 19. There were four main radioactive signals in the D 2 O region: uranium-and thorium-chain activities in the D 2 O and in the NCD strings. Assuming these were the dominant contributions in the in situ analysis signal region, a two-dimensional (radial and β 14 ) maximum likelihood fit was made to the data. The normalized PDF associated with each background was constructed from Monte Carlo simulations of 214 Bi and 208 Tl from the D 2 O and NCD strings. The simulated events were selected using the same cuts used on the data. Cherenkov light produced by 208 Tl decays in the NCD string bodies was dominated by the 2.614-MeV gamma rays. There was very little contribution from betas as nearly all of them were stopped in the nickel of the NCD counter housing. Thus, 208 Tl decay events in the NCD string bodies were less isotropic and had a higher average value of β 14 , when compared with events from 208 Tl decays in the D 2 O. The mean value of β 14 for these events was very similar to that of Probability distribution functions of β 14 (top) and ρ (bottom) for different background sources in the in situ measurement of D 2 O and NCD array backgrounds. 214 Bi decays in D 2 O. Bismuth decays in the NCD array produced even less light than 208 Tl decays. In addition to betas being stopped in the nickel bodies, gamma rays from 214 Bi decays have lower energies than those in 208 Tl decays. These 214 Bi decays were very similar in isotropy to 214 Bi D 2 O events. Therefore, 214 Bi D 2 O, 214 Bi and 208 Tl "NCD bulk" isotropy distributions were very similar, but were significantly different from that of 208 Tl D 2 O.Combining radial and isotropy information, the 208 Tl D 2 O, 214 Bi D 2 O and NCD bulk events could be separated. It was not possible to distinguish between 208 Tl and 214 Bi events originating from the NCD bulk. The ratio of U to Th in the NCD bulk obtained by coincidence studies of NCD array's alpha data[36] was used to separate the NCD events obtained by the 2D maximum likelihood fit into 208 Tl and 214 Bi. In the 232 Th chain, 220 Rn alpha decays to 216 Po which decays by alpha emission. The signature for this coincidence was a 6.288-MeV alpha followed by a 6.778-MeV alpha with a half-life of 0.15 seconds. In the 238 U chain the coincidence was between 222 Rn and 218 Po, the signature being a 5.49-MeV alphafollowed 3.10 minutes later by a 6.02-MeV alpha. The results from the alpha coincidence measurements were 2.8 +0.6 −0.8 × 10 −12 gU/gNi and 5.7 +1.0 −0.9 × 10 −12 gTh/gNi. For comparison with ex situ assay and alpha coincidence measurements, the number of 214 Bi and 208 Tl events were converted into equivalent amounts of 238 U and 232 Th by assuming secular equilibrium and using Monte Carlo simulations. The equivalent concentrations, integrated over the solar neutrino data set, were found to be 6.63 +1.05 −1.22 × 10 −15 gU/gD 2 O, 0.88 +0.27 −0.27 × 10 −15 gTh/gD 2 O, 1.81 +0.80 −1.12 × 10 −12 gU/gNi and 3.43 +1.49 −2.11 × 10 −12 gTh/gNi. The in situ results of the NCD bulk were in good agreement with those obtained from the alpha coincidence analysis. Results from the in situ and ex situ analyses of the D 2 O were found to be consistent. As the two methods and their systematic uncertainties were independent, the best measurement of the equivalent concentrations of 232 Th in the D 2 O was obtained by taking the weighted mean of the in situ and combined HTiO and MnO x results. The weighted mean of the in situ and ex situ Rn results were used to obtain the best measurement of the 238 U content in the D 2 O. The weighted mean concentrations were 6.14 ± 1.01 × 10 −15 gU/gD 2 O and 0.77±0.21×10 −15 gTh/gD 2 O. Only the in situ results for the NCD array bulk concentrations were used because the alpha coincidence method could not provide data for all NCD counters and it could not sample the whole array.
Neutrons can be produced by alpha reactions on 2 H,17 O and18 O. The most significant contribution to this background arose from the 5.3-MeV alpha produced by 210 Po decay. In the third phase, Po isotopes on the external surface of the NCD array were of particular concern. Taking the average alpha activity of 18 samples, a total surface area of 2.40 m 2 that was counted using a multi-wire proportional counter, yielded a neutron production rate of 1.32 ± 0.28 × 10 −2 neutron/day generated from the entire array, 64.72 m 2 , given that 6.4 × 10 −8 neutron were produced per alpha. This rate resulted in 5.1 ± 1.1 neutrons produced during Phase III.The contribution from spontaneous fission of 238 U was determined from the results of ex situ HTiO assays which placed limits on the concentration of 238 U in the detector.The NUANCE[38] neutrino Monte Carlo simulation package was used in the calculation of neutron backgrounds produced by atmospheric neutrino interactions. These atmospheric neutrino interactions were often associated with a burst of events in the detector. After applying time-correlation cuts that removed event bursts and other data reductions cuts to these simulated events, the expected number of observed neutrons from atmospheric neutrino interactions was determined to be 13.6 ±2.7 for the NCD array and 24.1±4.6 for the PMT array. The dominant systematic uncertainties associated with these estimates were those in the neutrino interaction cross section and atmospheric neutrino flux.The muon flux incident on the SNO detector was measured to be 3.31 ± 0.09 × 10 −10 cm −2 s −1[39,40]. The possibilities that the muon tag could be missed and that neutrons could be produced in the surrounding rock, led to an estimate of their total rate of 0.18 ± 0.02 neutron per year, a negligible background.Anti-neutrinos produced by nuclear reactors afar could also create neutrons in the D 2 O.The magnitude of this background was calculated assuming an average reactor anti-neutrino spectrum and an average power output of all commercial reactors within 500 km of the SNO detector. Oscillations were taken into account in the calculation, and the estimate was 1.4±0.2 neutrons per year. An estimate was also made of the number of neutrons produced by anti-neutrinos from radioactive decays within the Earth and this was found to be 0.02 neutron per year.
The in situ and ex situ techniques were applied to the H 2 O region between the AV and PSUP. In total, 29 MnO x and 25 HTiO H 2 O assays were conducted during the third phase.The results from the HTiO and MnO x assays were in good agreement. The results were corrected for the neutrino live time and the weighted average was calculated to produce a single ex situ 232 Th-chain measurement for the H 2 O region. The activity was found to be 26.9 ± 12.3 × 10 −15 gTh/gH 2 O. Due to hardware issues that were discovered towards the end of the phase, Rn assays, although performed, were not used in this analysis.The in situ analysis window for the H 2 O region was 4.0 < T eff < 4.5 MeV and 650 < R fit < 680 cm. The equivalent 238 U and 232 Th concentrations were determined using an isotropy fit to the data. The background levels determined by the in situ analysis were 30.0 +9.2 −19.4 × 10 −15 gTh/gH 2 O and 35.0 +9.9 −5.4 × 10 −14 gU/gH 2 O. The photodisintegration neutron background from Th activity in the H 2 O was determined from a weighted mean of the in situ and ex situ assay results, while that from Rn was determined exclusively from the in situ results. The photodisintegration neutron backgrounds due to Th and U in the H 2 O region were found to be 2.2 +0.8 −0.7 events for the PMT array data and 7.1 +5.4 −5.2 events for the NCD array data.
line model. Half of the pulse was propagated down the string, through the delay line, and back to the point-of-origin of the pulse. The delay-line attenuation was also simulated as a lossy transmission line. Both halves of the pulse (reflected and direct) were then transmitted up to the top of the NCD string. The model parameters were based on SPICE simulations [51] and ex situ measurements.
Noise was added to the pulses as the final stage in the simulation. It was added to the multiplexer and shaper branches of the electronics independently. For the multiplexer branch, the frequency spectrum of the noise on the current pulses was measured for each channel using the baseline portions of injected calibration data.The shaper-ADC branch of the electronics was simulated by a sliding-window integral of the preamplified pulse. This number was then converted to units of ADC counts by doing an inverse linearity calibration. The calibration constants used in this "uncalibration" were the same constants that were used to calibrate the data. Noise was added to the shaper value with a Gaussian-distributed random number. The mean and standard deviation of the noise for each channel were determined uniquely for each of the 40 NCD strings; the typical RMS noise (in units of ADC values) was 2.0, with a variance of 0.7 across the array.The multiplexer and shaper systems included independent triggers that used the true threshold values. Event triggers were determined by checking each pulse amplitude or shaper value against the appropriate threshold. The dead times of the two systems were then taken into account within each Monte Carlo event. The NCD string signals were integrated with the PMT trigger simulation by inserting each signal into the time-ordered array of PMT signals. As the simulation scanned over the combined PMT and NCD array signals, any individual NCD string signal was sufficient to cause a global trigger of the detector.
FIG. 20 .
20Comparison of pulse shape variables in 24 Na neutron calibration data (data points) and the NCD array Monte Carlo (histogram) in the neutron energy window, 0.4 to 1.2 MeV, with statistical uncertainties only. From top left to bottom right: fraction of events as a function of pulse amplitude, time-axis mean of the pulse, 10%-90% rise time, and full width at half maximum.All distributions are normalized to unit area. pha depth") within the NCD counter walls, the efficiency of data reduction cuts, the spacecharge model parameters, the electron-drift curve, the ion mobility, and the surface polonium to NCD bulk activity fraction on each string. These effects reflected uncertainties in the parameters of the NCD array simulation physics model. Systematic uncertainties were assessed by generating a large set of variation Monte Carlo samples, each with one input parameter varied by 1σ with respect to its default value. The size of the variation of the NCD array
FIG. 21 .
21Top: fractional first derivatives vs. event energy (MeV) for two systematic variations (space charge and alpha depth) of the simulation. Center: number of events versus energy in the NCD array data analysis energy window, for data (black, with statistical uncertainties), neutron template from calibration data (purple, dashed), alpha cocktail template from simulation (cyan),
The basic idea of MCMC is to take a random walk through the parameter space, where each step is taken with a probability given by the likelihood of the new step (L prop ) compared to the previous step (L curr ). The probability of accepting the proposed new parameters is min(1,L prop /L curr ). If the step is accepted the parameter values are updated, else the MCMC repeats the current point in the chain and generates a new proposal for the parameters. By the Metropolis-Hastings theorem[53,54], the resulting distribution of parameters from the chain will have a frequency distribution given by L. The choice of an appropriate proposal kernel is critical to this method. If the width of the proposal distribution were too wide, steps are rarely taken; and if the width of the proposal distribution were too narrow, then the MCMC would not sample enough of the parameter space to find the best-fit region, and instead could fall into a local minimum. In the MCMC signal extraction method, the step proposal distribution was a Gaussian of mean zero and width that was ∼ 1/3 of the expected statistical uncertainty or the constraint uncertainty. This choice was checked by the convergence and distributions of different MCMC chains that were started at different random starting points.IX.2. Observables, signals and fit rangesIX.2.1. PMT array dataFor the PMT array data, 33 signals and backgrounds were included in the fit. An energyunconstrained fit was done, meaning that CC and the ES fluxes were fitted for each T eff bin. The energy binning used for this unconstrained fit was 0.5-MeV bins between T eff of 6 MeV and 12 MeV, and a single bin from 12 MeV to 20 MeV, totaling 13 bins per signal.The other signal and backgrounds in the fit included the NC signal; photodisintegration neutron backgrounds due to radioactivity in the D 2 O target, the bulk nickel in the NCD strings, and the hotspots on the strings K2 and K5; photodisintegration and (α, n) neutron backgrounds in the acrylic vessel (which also included the backgrounds from the cables of the NCD array); and neutrons from atmospheric neutrino interactions.The observables used in the fit to the PMT data were each event's reconstructed T eff , ρ, and cos θ ⊙ . Cherenkov light candidate events were selected with the criteria ρ 77025, −1 ≤ cos θ ⊙ ≤ 1, and 6.0 MeV< T eff < 20.0 MeV. The signal and background
the counters, data reduction cut efficiency on alpha backgrounds, spatial distributions and intensity of different alpha background activities, energy spectrum of instrumental backgrounds, and parameters in the Monte Carlo that could affect the observed signal, such as ion mobility. Details of the parameterization of these systematic parameters can be found in Appendix B.A constraint term for each of the 26 systematic uncertainties was added to the loglikelihood function. The means and standard deviations of these constraint terms came from calibration measurements or Monte-Carlo studies and are listed inTable XIVin Sec. IX.4.
, the number of events in different signal and background classes determined from this "energyunconstrained" fit are tabulated. The equivalent neutrino fluxes, derived from the fitted number of CC, ES and NC events under the assumption of the 8 B neutrino spectrum in Ref. [55], were determined to be (in units of 10 6 /cm 2 /s) [56, 57]: φ CC = 1.67 +0.08 −0.09 φ ES = 1.77 +0.26 −0.23 (41) φ NC = 5.54 +0.48 −0.46where the uncertainties are the total uncertainties obtained from the posterior distributions.
FIG. 22 .
22One-dimensional projections of PMT array and NCD array data overlaid with best-fit results to signals. The χ 2 values for the ρ, cos θ ⊙ , T eff and shaper energy distributions are 7.3, 11.9, 0.3 and 17.0, respectively. Because these are one-dimensional projections in a multi-dimensional space, these χ 2 values are quoted as a qualitative demonstration of goodness-of-fit and cannot be simply evaluated.
FIG
.23. A comparison of the measured energy-unconstrained NC flux results in SNO's three phases. The horizontal band is the 1σ region of the expected total 8 B solar neutrino flux in the BS05(OP) model[64].
i and the statistical correlation coefficient between the observables Y exp i and Y exp j , respectively. The effect of each systematic uncertainty S k on the model expectation for the neutrino yield Y th i was estimated by computing the change in the expectation ∆Y ik with respect to the source of the uncertainty:
FIG. 24 .
24After SNO-III: SNO-only neutrino oscillation confidence level contours published in Reference[7]. This analysis includes the summed kinetic energy spectra from Phase I (day and night); NC and ES fluxes, and CC kinetic energy spectra from Phase II (day and night); and CC, NC and ES fluxes from Phase III. The best-fit point is at: ∆m 2 = 4.57 × 10 −5 eV 2 , tan 2 θ = 0.447, φ8 B = 5.12 × 10 6 cm −2 s −1 . The hep neutrino flux was fixed at 7.93 × 10 3 cm −2 s −1 .
25. After SNO-III: Neutrino oscillation parameters confidence level contours. (a) Global solar analysis including the rate measurements from Homestake, Gallex/GNO, SAGE and Borexino; SK-I zenith-energy spectra from [59], summed kinetic energy spectra from SNO-I (day and night); NC and ES fluxes, and CC kinetic energy spectra from SNO-II (day and night); and CC, ES and NC fluxes from SNO-III. The best-fit point was at: ∆m 2 = 4.90 × 10 −5 eV 2 , tan 2 θ = 0.437, φ8 B = 5.21 × 10 6 cm −2 s −1 . The hep neutrino flux was fixed at 7.93 × 10 3 cm −2 s −1 . (b) Including
2 .FIG. 26 .
226Time correlation of shaper-ADC and MUX-scope events For each real ionization in the proportional counter, there was a time-correlated pair of shaper and MUX-scope events. Instrumental background events exhibited a shorter time difference between the two events. A time-correlation cut was developed to remove shaper-MUX-scope event pairs from the same string that showed such an anomalous timing characteristic. Examples of instrumental background pulses in the NCD array data. Top: a fork event,
Q α MC (E NCD ) = Q α (E NCD )(1 + 7 i=0 α i s i (E NCD )).(B9)The reweighting functions, s i , are: Po alpha depth variations 0 = −2.06 + 6.58 E NCD − 6.depth variations 1 = −0.0684 + 0.0892 E NCD (B11) Drift time variations 2 = −0.131 + 0.252 E NCD − 0.117 E 2 NCD (B12) Avalanche width offset variations 3 = −0.0541 + 0.0536 E NCD (B13) Avalanche gradient offset variations 4 = −0.0138 (B14) Ion mobility variations 5 = −0.00930 (B15) Po/bulk fraction variations 6 = −0.00405 + 0.0386 E NCD (B16) Data reduction cut systematics 7 = 0.861 − 2.77 E NCD + 2.erf −1.59 (E NCD − p N4 1 .
FIG. 3. Energy dependence of fractional signal loss of NCD array data due to instrumental background cuts, as measured with 252 Cf and Am-Be neutron calibration data. The abscissa is the event energy recorded by the shaper-ADC system.Energy (MeV)
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
Fractional signal loss
0
0.01
0.02
0.03
0.04
0.05
FIG. 5. Angular response curves at 386 nm for the PMT-reflector assembly generated by the MC simulation both before and after the optimizations discussed in the text, in comparison to the optical scan in the preparatory phase in October 2003. Note that the y axis zero is suppressed.0
5
10
15
20
25
30
35
40
45
Relative angular response
0.118
0.120
0.122
0.124
0.126
0.128
0.130
0.132
0.134
0.136
0.138
Preparatory phase -October 2003
Previous MC
Modified MC
FIG. 6. PMT relative angular response at 421 nm. change in the response from the preparatory phase to the third phase, after analyzing the data at all six wavelengths, was about 4% at higher incidence angles. The change from the first to the last scan in Phase III was around 2%. The decrease in the response was consistent with observations in previous phases, where it was attributed primarily to aging of the light concentrators.The average
0
5
10
15
20
25
30
35
40
45
1.00
1.02
1.04
1.06
1.08
1.10
1.12
1.14
1.16
Preparatory phase -October 2003
Phase III average
October 2004
February 2005
May 2005
February 2006
August 2006
Incident angle (degrees)
0
5
10
15
20
25
30
35
40
45
Angular response (arb. units) / 1 degree
1.00
1.02
1.04
1.06
1.08
1.10
1.12
1.14
1.16
TABLE I .
IFitted values for the parameters in the polynomial a 0 + a 1 z + a 2 z 2 used to evaluate the
systematic uncertainty on vertex resolution.
Direction
a 0
a 1
a 2
[cm]
[×10 −2 ]
[×10 −5 cm −1 ]
x, y
1.19 ± 0.52
-0.10 ± 0.11
0.71 ± 0.36
z
1.29 ± 0.51
0.21 ± 0.15
NA
[cm]
src
z
-600
-400
-200
0
200
400
600
[cm]
MC
σ
-
data
σ
0
1
2
3
4
5
[cm]
src
z
-600
-400
-200
0
200
400
600
[cm]
TABLE II .
IICorrelation matrices for the fitted parameters of the polynomials used to evaluate the systematic uncertainty on vertex resolution.x, y directions
ρ
a 0
a 1
a 2
a 0
1.00
-0.13
-0.74
a 1
-0.13
1.00
0.31
a 2
-0.74
0.31
1.00
z direction
ρ
a 0
a 1
a 0
1.00
0.15
a 1
0.15
1.00
IV.2.2. Event direction
The derivation of F T was performed via Monte Carlo simulation. The only free parameter in the simulation, the average PMT collection efficiency (ǫ PCE ) for the PMT array, was determined by comparing the energy scale of 16 N calibration data and its simulation (seeSec. IV.3.3).
IV.3.1. The energy reconstruction algorithm
For an initial estimate of an event's equivalent electron kinetic energy T eff , a corresponding
estimate of N γ could be calculated via F T
−1 . The number of photons expected to trigger
the i th PMT, N i , was calculated as
TABLE III .
IIISummary of PMT array's energy scale and resolution systematic uncertainties.Scale uncertainty
Source
Uncertainty
Detector state
0.03%
Drift/stability (data-MC)
0.40%
Spatial variation
0.64%
Gain
0.13%
Threshold
0.11%
16 N source modeling
0.65%
Rate dependence
0.20%
Time calibration
0.10%
Total
1.04%
Resolution uncertainty
Source
Uncertainty
Spatial variation
1.04%
Resolution shift
Source
Shift
Drift/stability (data-MC)
1.19%
Incomplete modeling of the 16 N source contributed to the energy scale uncertainty as
Monte Carlo simulations of the source were used to compare to real data. Effects such as
approximations to the source geometry, uncertainties in the 16 N decay branching ratios, and
the finite step size in EGS4 [29] simulation were studied. Their combined contribution to
the energy scale uncertainty was determined to be 0.65%.
Table III summarizes the contributions to the systematic uncertainty on the energy scale
and resolution. The energy scale uncertainty contributions were added in quadrature to yield
an overall uncertainty of 1.04%. The resolution uncertainty contributions were determined
to be a shift of 1.19% along with an uncertainty of 1.04%.
NCD array was measured with the 24 Na spikes, and the neutron detection efficiency of the PMT array was calculated from the calibrated Monte Carlo code. A subsequent analysis of the PMT array's neutron detection efficiency using the 24 Na data is reported in Sec. VI.4. The results from direct 24 Na calibration and calculations from the tuned Monte Carlo are consistent. VI.1. Inputs for determining the neutron capture efficiency with a uniformly distributed 24 Na source
TABLE IV .
IVThe result of fitting a constant value to the data points inFig. 15.Data Set
Fit A24 Na (n/s)
Percent uncertainty
Fit χ 2 /d.o.f.
2005
1.241 ± 0.016
1.3%
0.22
2006
0.842 ± 0.011
1.3%
0.59
ment was, in large part, due to the difficulty in determining the neutron diffusion profile beyond 200 cm from the source, which was necessary to obtain a pure sample of neutron events.VI.2. Neutron capture efficiency of the NCD array VI.2.1. R spike : Neutron capture rate measurement by the NCD array
TABLE V .
VThe neutron capture rate by the NCD array at the reference time (t = 0), obtained by fitting a constant value to the set of measurements {R i (0)}. The statistical uncertainty is from the fit, while the systematic uncertainty is from the instability of the rate (see text for discussion).any other time. For this reason, we assigned a systematic uncertainty of 1% to R spike .VI.2.2. ǫ spike : Combining the rate and source strength measurementsThe capture efficiency by the NCD array of neutrons produced by the activated brine that was injected and mixed in the SNO detector is defined as the ratio of the number of neutrons captured in the NCD string live volume to the total number of neutrons produced. This ratio can be obtained experimentally from the two measured quantities A24 Na (Sec. VI.1.1) and R spike (Sec. VI.2.1): ǫ spike = R spike /A24 Na . The central value of ǫ spike can be obtained by simply dividing the numbers inTable V by those in Table IV. The error propagation was performed with care because the numerator and denominator depend at least partly on measurements in the NCD array. The systematic uncertainty of the rate in the NCD array was dominated by the long-term fluctuation of 1%. If the time between the source strength measurement and the spike rate measurement could be considered short, then the instability systematic should cancel out and the 1% systematic uncertainty in the numerator and denominator could be ignored. On the other hand, if this time period was not sufficiently short, then these uncertainties should be combined in quadrature. Although there was strong evidence for stability within a day or two of running, no data exist to demonstrate stability over a 4 to 6-day period, as was the case for the present analysis. Thus we decided to combine the uncertainties in quadrature. The result for ǫ spike is shown inTable VI.Year
Best-fit rate (n/s)
χ 2 /d.o.f.
2005
0.2708 ± 0.0020(stat.) ± 0.0027(syst.)
45.6/48
2006
0.1811 ± 0.0016(stat.) ± 0.0018(syst.)
35.6/38
TABLE VI .
VIThe NCD array's capture efficiency for neutrons produced by24 Na brine injected and well-mixed in the SNO detector. The combined statistical and systematic uncertainty is shown here.Year
ǫ spike
Percent uncertainty
2005
0.2182 ± 0.0046
2.1%
2006
0.2151 ± 0.0043
2.0%
VI.2.3. f edge : Correction factor for neutron density near the acrylic vessel
TABLE VII .
VIISummary of correction factors in the determination of the neutron detection efficiency.The NCD array's detection efficiency of NC neutrons, with a shaper energy threshold E NCD > 400 keV, is the product of the neutron capture efficiency (Sec. VI.2) and the combined factor C. described in Sec. IX. The fraction of shaper events above this energy threshold was evaluated for each NCD counter using AmBe calibration data. The mean acceptance ǫ shaper for NC neutron events was then calculated by averaging these individual counter estimates, weighted by the expected fraction of NC neutrons that each counter would capture. The last correctionThe factors shown in this table are the average values for solar neutrino data presented in this
paper.
Correction factor
Value
MUX live fraction (l MUX )
0.9980 ± 0.0001
Scope live fraction (l scope )
0.957 ± 0.004
Average MUX threshold efficiency ( ǫ MUX )
0.99491 ± 0.00031
Average shaper energy window acceptance ( ǫ shaper )
0.91170 ± 0.00014
Data reduction cut acceptance ( ǫ cut )
0.99521 ± 0.00011
Combined (C)
0.862 ± 0.004
TABLE VIII .
VIIISummary of backgrounds in the PMT and NCD arrays.Source
PMT events
NCD events
Neutrons generated inside D 2 O:
2 H photodisintegration [U,Th in D 2 O]
7.6 ± 1.2
28.7 ± 4.7
2 H photodisintegration [U,Th in NCD bulk]
4.3 +1.6
−2.1
25.8 +9.6
−12.3
2 H photodisintegration [U,Th in Hotspots]
17.7 ± 1.8
64.4 ± 6.4
2 H photodisintegration [U,Th in NCD Cables]
1.1 ± 1.0
8.0 ± 5.2
n from spontaneous fission [U]
0.2 ± 0.1
0.3 ± 0.1
2 H(α, αn) 1 H [Th, 222 Rn]
0.2 ± 0.1
0.4 ± 0.1
17,18 O(α, n) 20,21 Ne [Th]
0.3 ± 0.1
1.8 ± 0.4
Atmospheric ν
24.1 ± 4.6
13.6 ± 2.7
Cosmogenic muons
0.009 ± 0.002
0.04 ± 0.004
Reactor and terrestrial neutrinos
0.3 ± 0.1
1.4 ± 0.2
CNO solar ν
0.05 ± 0.05
0.2 ± 0.2
Total internal neutrons
55.8 +5.6
−5.4
144.7 +13.4
−15.5
Neutrons generated from AV and H 2 O radioactivity:
2 H photodisintegration [U,Th in H 2 O]
2.2 +0.8
−0.7
7.1 +5.4
−5.2
(α, n) in AV
18.3 +10.2
−7.3
33.8 +19.9
−17.1
Total external-source neutrons
20.6 +10.2
−7.3
40.9 +20.6
−17.9
Cherenkov events from radioactivity inside the D 2 O:
beta-gamma decays (U,Th)
0.70 +0.37
−0.38
N/A
Decays of spallation products in D 2 O: 16 N following muons
0.61 ± 0.61
N/A
Cherenkov backgrounds produced outside D 2 O:
beta-gamma decays (U,Th) in AV, H 2 O, PMTs
5.1 +9.7
−2.9
N/A
Isotropic acrylic vessel events
TABLE IX .
IXEquivalent masses of uranium and thorium for lower-chain activities in the K5 and K2 hotspots. More detailed and updated results may be found in Ref.[37].String
232 Th (µg)
238 U (µg)
Total ex situ
K5
1.28 ± 0.14
0.10 +0.05
−0.05
Total in situ
K5
1.48 +0.24
−0.27
0.77 +0.19
−0.23
Total ex situ
K2
1.43 ± 0.17
< 0.40
Total in situ
K2
< 0.93
≡ 0
radioactivity, an excess of neutrons should have been captured by the contaminated string.
TABLE X .
XMultiplicative conversion factors for determining the number of neutron background events in the PMT array data from the number in the NCD array data. Photodisintegration is denoted as "PD" in this table.Factor
Description
Value
f PMT
ex
AV, PD
0.5037
f PMT
d2opd
D 2 O, PD
0.2677
f PMT
ncdpd
NCD, PD
0.1667
f PMT
k2pd
K2, PD
0.2854
f PMT
k5pd
K5 , PD
0.2650
f PMT
cab
NCD Cables, PD
0.1407
f PMT
atmos
Atmospheric ν
1.8134
TABLE XI .
XIResults of fit to the full Phase-III data set. The fitted number of signal and background counts, along with the integral neutrino flux values are shown. In the data set, the total number of events in the PMT array and NCD array data sets were 2381 and 7302 respectively. The 8 Bspectrum from Ref. [55] was used in deriving the equivalent neutrino fluxes from the fitted number
of CC, ES and NC events. All the fluxes are in units of ×10 6 /cm 2 /s.
Fitted counts -PMT array
CC
1867 +91
−101
ES
171 +24
−22
NC
267 +24
−22
Backgrounds
77 +12
−10
Total
2382 +98
−107
Fitted counts -NCD array
NC
983 +77
−76
Neutron backgrounds
185 +25
−22
Alpha backgrounds
5555 +196
−167
Instrumental backgrounds
571 +162
−175
Total
7295 +82
−83
Integral flux
φ CC
1.67 +0.08
−0.09
φ ES
1.77 +0.26
−0.23
φ NC
5.54 +0.48
−0.46
TABLE XII .
XIIStatistical correlation coefficients for the CC, ES and NC fluxes in the full SNO TABLE XIII. The CC and ES electron differential energy spectrum. The fluxes in each of the 13 T eff bins are in units of 10 4 /cm 2 /s. The uncertainties shown are total uncertainties with correlations between all systematic uncertainties described in the text included. Their correlations are tabulated in Table XII. The ratio of the 8 B neutrino flux measured with the CC and NC reactions isPhase-III data set.
Signal
CC
ES
NC
CC
1.000
0.2376
-0.1923
ES
0.2376
1.000
0.0171
NC
-0.1923
0.0171
1.000
T eff (MeV)
CC
ES
TABLE XIV .
XIVSystematic parameters' constraints and fit results for the full Phase-III data signal extraction. Details of the parameterization of these nuisance parameters in the fit are described in Appendix B. Those constraints marked with an asterisk were handled by the second term in Eqn. 36, with the covariance matrices given in the same appendix. The 1σ width of the constraint for b θ 0 was input incorrectly in the analysis in Ref.[7]; the correct value should be 0.12. The fit value indicates that this error should not have any impact on the ES results.Gaussian constraint
Description
Mean
σ
Fit value
Systematic
Nuisance parameters -PMT array
f PMT
NC
NC flux to PMT NC events factor
0.46725
0.00603
0.46735 ± 0.00574
a x
0
x coordinate shift
0.0
4.0
1.0 ± 4.0
a y
0
y coordinate shift
0.0
4.0
-1.0 ± 3.9
a z
0
z coordinate shift
5.0
4.0
6.1 ± 3.6
a x
1
coordinate scale
0.000
0.006
-0.002 ± 0.008
b xy
0
xy resolution constant term
0.06546
0.02860 *
0.069 ± 0.029
b xy
1
xy resolution linear term
-0.00005501
0.00006051 *
-0.000053 ± 0.000058
b xy
2
xy resolution quadratic term
3.9×10 −7
0.2×10 −7 *
0.00000038 ± 0.00000020
b z
0
z resolution constant term
0.07096
0.02805 *
0.072 ± 0.027
b z
1
z resolution linear term
0.0001155
0.00008251 *
0.00012 ± 0.000082
b θ
0
a
PMT angular resolution
0.0
0.056
0.011 ± 0.059
a E
1
PMT energy scale
1.000
0.0109
1.0047 ± 0.0087
b E
0
PMT energy resolution (neutrons)
0.0119
0.0114
0.0121 ± 0.0104
Systematic
Nuisance parameters -NCD array
f NCD
NC
NC flux to NCD NC events factor
1.7669
0.0590
1.7713 ± 0.0586
a NCDE
1
NCD shaper energy scale
1.00
0.01
1.0047 ± 0.0035
b NCDE
0
NCD shaper energy resolution
0.00
+0.01 -0.00
0.0124 ± 0.0065
α 0
alpha PDF -alpha Po depth
0
1
1.21 ± 0.62
α 1
alpha PDF -alpha bulk depth
0
1
0.25 ± 0.86
α 2
alpha PDF -drift time
0
1
-0.11 ± 0.97
α 3
alpha PDF -avalanche width
0
1
0.27 ± 0.94
α 4
alpha PDF -avalanche gradient
0
1
-0.06 ± 1.00
α 5
alpha PDF -Po/bulk fraction
0
1
0.13 ± 0.97
α 6
alpha PDF -ion mobility
0
1
-0.06 ± 0.96
α 7
alpha PDF -data reduction cuts
0
1
-0.49 ± 0.93
p J3
1
J3-type background skew Gaussian mean
0.4584
0.0262
0.4724 ± 0.0231
p N4
1
N4-type background skew Gaussian mean
0.0257
0.0138
0.0333 ± 0.0112
a
TABLE XV .
XVConstraints and fit results for the amplitude of different neutron and instrumental background classes in the full Phase-III data signal extraction. trino fluxes with their statistical and systematic uncertainties are (in units of 10 6 cm −2 s −1 ): φ CC = 1.67 +0.05 −0.04 (stat.) +0.07 −0.08 (syst.) φ ES = 1.77 +0.24 −0.21 (stat.) +0.09Table XVIis a summary of these system uncertainties, categorized by different sources.Gaussian constraint
Description
Mean
σ
Fit value
Background
N NCD
ex
external n (AV, H 2 O backgrounds)
40.9
20.6
42.2 ± 19.3
N NCD
ncdpd
NCD bulk, cable
35.6
12.2
35.2 ± 12.1
N NCD
k2pd
K2
32.8
5.2
32.7 ± 5.1
N NCD
k5pd
K5
31.6
3.7
31.7 ± 3.7
N NCD
d2opd
D 2 O photodisintegration
31.0
4.8
30.9 ± 4.8
N NCD
atmos
atmospheric ν and cosmogenic muons
13.6
2.7
13.6 ± 2.7
N NCD
J3
J3-type instrumental background
unconstrained
355.6 ± 192.3
N NCD
N4
N4-type instrumental background
unconstrained
215.6 ± 170.5
−0.10 (syst.)
(43)
φ NC = 5.54 +0.33
−0.31 (stat.) +0.36
−0.34 (syst.).
TABLE XVI .
XVISources of systematic uncertainties on CC, ES and NC flux measurements. The total uncertainties differs from the individual uncertainties added in quadrature due to correlations.shape[55] used here differs from that[60] used in previous SNO results. The CC, ES and NC flux results are in agreement (p = 32.8%[61]) with the NC flux result of the first phase[2] Source
CC uncert.
ES uncert.
NC uncert.
(%)
(%)
(%)
PMT energy scale
±2.7
±3.6
±0.6
PMT energy resolution
±0.1
±0.3
±0.1
PMT radial energy dependence
±0.9
±0.9
±0.0
PMT radial scaling
±2.7
±2.7
±0.1
PMT angular resolution
±0.2
±2.2
±0.0
Background neutrons
±0.6
±0.7
±2.3
Neutron capture
±0.4
±0.5
±3.3
Cherenkov/AV backgrounds
±0.3
±0.3
±0.0
NCD instrumentals
±0.2
±0.2
±1.6
NCD energy scale
±0.1
±0.1
±0.5
NCD energy resolution
±0.3
±0.3
±2.7
NCD alpha systematics
±0.3
±0.4
±2.7
PMT data reduction cuts
±0.3
±0.3
±0.0
Total experimental uncertainty
±4.0
±4.9
±6.5
Cross section [58]
±1.2
±0.5
±1.1
B decays inside the Sun unambiguously proved that neutrinos change their flavor while traveling to the Earth. These results can be interpreted, as in previous SNO analyses [3, 4, 6], as
TABLE XVII .
XVIIEnergy-unconstrained CC, ES and NC flux results ( in units of 10 6 cm −2 s −1 ) from three phases of SNO. The T eff thresholds for the PMT array data in Phases I, II and III were 5.0, 5.5 and 6.0 MeV, respectively. "Energy-constrained" flux results, in which ES and CC events were constrained to an undistorted 8 B spectrum in the fit, can be found in Refs.[5] and[6] for Phases I and II respectively.Data set
φ CC
φ ES
φ NC
Phase I (306 live days)
-
-
6.42 +1.57
−1.57
+0.55
−0.58
Phase II (391 live days)
1.68 +0.06
−0.06
+0.08
−0.09
2.35 +0.22
−0.22
+0.15
−0.15
4.94 +0.21
−0.21
+0.38
−0.34
Phase III (385 live days)
1.67 +0.05
−0.04
+0.07
−0.08
1.77 +0.24
−0.21
+0.09
−0.10
5.54 +0.33
−0.31
+0.36
−0.34
TABLE XVIII .
XVIIICorrelation coefficients for the same sources of systematic uncertainties affecting different types of signals for the SNO-III data set.TABLE XIX. SNO-only neutrino oscillation best-fit parameters.Source of systematic
NC-CC
NC-ES
CC-ES
PMT energy scale
+1
+1
+1
PMT energy resolution
+1
+1
+1
PMT radial energy dependence
+1
+1
+1
PMT vertex resolution
+1
+1
+1
PMT vertex accuracy
+1
+1
+1
PMT angular resolution
+1
-1
-1
Background neutrons
+1
+1
+1
Neutron capture
+1
+1
+1
Cherenkov/AV backgrounds
+1
+1
+1
NCD instrumentals
+1
+1
+1
NCD energy scale
+1
+1
+1
NCD energy resolution
+1
+1
+1
NCD alpha systematics
+1
+1
+1
PMT data reduction cuts
0
0
+1
Analysis
∆m 2 (10 −5 eV 2 )
tan 2 θ
8 B flux (10 6 cm −2 s −1 )
Before SNO-III
5.0 +6.2
−1.8
0.45 +0.11
−0.10
5.11
After SNO-III
4.57 +2.30
−1.22
0.45 +0.05
−0.05
5.12
TABLE XX .
XXGlobal solar only and global solar+KamLAND best-fit parameters. The global solar results before the SNO-III phase do not include data from Borexino. The global solar+KamLAND results after SNO-III include the latest data from Borexino[67] and KamLAND[68].Analysis
∆m 2 (10 −5 eV 2 )
tan 2 θ
8 B flux (10 6 cm −2 s −1 )
Before SNO-III phase
Global solar
6.5 +4.4
−2.3
0.45 +0.09
−0.08
5.06
with KamLAND
8.0 +0.6
−0.4
0.45 +0.09
−0.07
4.93
After SNO-III phase
Global solar
4.90 +1.64
−0.93
0.44 +0.05
−0.04
5.21
with KamLAND
7.59 +0.21
−0.19
0.47 +0.05
−0.04
Sloan Foundation; UK: Science and Technology Facilities Council (formerly Particle Physics and Astronomy Research Council); Portugal: Fundação para a Ciência e a Tecnologia. We thank the SNO technical staff for their strong contributions and David Sinclair for careful review of the neutron efficiency analysis. We thank INCO (now Vale, Ltd.) for hosting this project in their Creighton mine. A: Instrumental background cuts for NCD array dataAppendix
. Present address: Center for Astrophysics and Space Astronomy, University of Colorado, Boulder, CO b Present address: Dept. of Physics, Case Western Reserve University, Cleveland, OH c Present address: CERN, Geneva, Switzerland d Present address: Lawrence Berkeley National Laboratory, Berkeley, CA e Present address: Dept. of Physics, Hiroshima University, Hiroshima, Japan f Present address: Dept. of Physics, University of Wisconsin, Madison, WI g Present address: Sanford Laboratory at Homestake, Lead, SD h Present address: Dept. of Physics, University of North Carolina, Chapel Hill, NC i Present address: Center of Cosmology and Particle Astrophysics, National Taiwan University, Taiwan j Present address: Black Hills State University, Spearfish, SD k Present address: Institute for Space Sciences, Freie Universität Berlin, Leibniz-Institute of Freshwater Ecology and Inland Fisheries, Germany l Present address: Dept. of Physics, University of Liverpool, Liverpool, UK m Present address: Center for Experimental Nuclear Physics and Astrophysics, and Dept. of Physics, University of Washington, Seattle, WA 98195 n Present address: Dept. of Physics, University of California, Santa Barbara, CA o Present address: Pacific Northwest National Laboratory, Richland, WA p Present address: Dept. of Physics, Queen's University, Kingston, Ontario, Canada q Present address: Dept. of Physics and Astronomy, University of Sussex, Brighton, UK r Present address: Dept. of Chemistry and Physics, Armstrong Atlantic State University, Savannah, GA s Present address: Dept. of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK t Present address: Dept. of Physics, Queen Mary University, London, UKa u Deceased
Appendix B: Parameterization of systematic uncertainties for the PMT and NCD array dataThe handling of systematic uncertainties in the negative log-likelihood (NLL) for the PMT and NCD array data are described in Secs. IX.3.1 and IX.3.2 respectively. In this Appendix, their parameterizations in the NLL are discussed. For the simulated values of the event kinetic energy (T g ), volume-weighted radius ρ g , and angle relative to the vector from the Sun θ g , the remapping functions are:ρ remap = ( x 2 remap + y 2 remap + z 2 remap /600cm)3, and (B5)where T 0 , (x, y, z), and θ are the nominal fitted effective kinetic energy, position and angle of the simulated event. The remapping of cos θ was applied to the ES channel only. In these expressions, the nuisance parameters were modeled as Gaussian distributions with means and widths, which are given inTable XIV
. J Boger, SNO CollaborationNucl. Instr. and Meth. 449172J. Boger et al. (SNO Collaboration), Nucl. Instr. and Meth. A449, 172 (2000).
. Q R Ahmad, SNO CollaborationPhys. Rev. Lett. 8771301Q. R. Ahmad et al. (SNO Collaboration), Phys. Rev. Lett. 87, 071301 (2001).
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| []
|
[
"ON CONVERGENCE OF OSCILLATORY ERGODIC HILBERT TRANSFORMS",
"ON CONVERGENCE OF OSCILLATORY ERGODIC HILBERT TRANSFORMS",
"ON CONVERGENCE OF OSCILLATORY ERGODIC HILBERT TRANSFORMS",
"ON CONVERGENCE OF OSCILLATORY ERGODIC HILBERT TRANSFORMS"
]
| [
"Ben Krause ",
"ANDMichael Lacey ",
"Máté Wierdl ",
"Ben Krause ",
"ANDMichael Lacey ",
"Máté Wierdl "
]
| []
| []
| We introduce sufficient conditions on discrete singular integral operators for their maximal truncations to satisfy a sparse bound. The latter imply a range of quantitative weighted inequalities, which are new. As an application, we prove the following ergodic theorem: let p(t) be a Hardy field function which grows "super-linearly" and stays "sufficiently far" from polynomials. We show that for each measure-preserving system, (X, Σ, µ, τ ), with τ a measure-preserving Zaction, the modulated one-sided ergodic Hilbert transform | 10.1512/iumj.2019.68.7615 | [
"https://arxiv.org/pdf/1610.04968v3.pdf"
]
| 53,644,891 | 1610.04968 | a3911fddf87797b9f99468450e3e705b175068f7 |
ON CONVERGENCE OF OSCILLATORY ERGODIC HILBERT TRANSFORMS
10 May 2017
Ben Krause
ANDMichael Lacey
Máté Wierdl
ON CONVERGENCE OF OSCILLATORY ERGODIC HILBERT TRANSFORMS
10 May 2017
We introduce sufficient conditions on discrete singular integral operators for their maximal truncations to satisfy a sparse bound. The latter imply a range of quantitative weighted inequalities, which are new. As an application, we prove the following ergodic theorem: let p(t) be a Hardy field function which grows "super-linearly" and stays "sufficiently far" from polynomials. We show that for each measure-preserving system, (X, Σ, µ, τ ), with τ a measure-preserving Zaction, the modulated one-sided ergodic Hilbert transform
∞ n=1 e 2πip(n) n τ n f (x)
converges µ-a.e. for each f ∈ L r (X), 1 ≤ r < ∞. This affirmatively answers a question of J. Rosenblatt [22].
In the second part of the paper, we establish almost sure sparse bounds for random one-sided ergodic Hilbert,
∞ n=1 X n n τ n f (x),
where {X n } are uniformly bounded, independent, and mean-zero random variables.
introduction
Our subject is discrete Harmonic Analysis. We give sufficient conditions for maximal truncations of discrete singular integral operators to have sparse bounds. Our argument has as its antecedents the Fefferman [13] and Christ [3] T T * approach to proving weak-type (1, 1) bounds for rough singular integral operators on Euclidean space; this approach has already appeared in the discrete context in the work of LaVictoire [19] where ℓ 1 → ℓ 1,∞ endpoint estimates for certain (random) maximal functions are proven, and in Urban and Zienkiewicz [24] and Mirek [20] where endpoint estimates for (deterministic) maximal functions taken over "thin" subsets of the integers are established.
We prove sparse bounds, which in turn easily imply quantitative weighted bounds, which are novel in this context. Moreover, we address maximal truncations, which is also new.
Sparse bounds are a relatively new topic, and we set some notation to describe sparse bounds. Say that I ⊂ Z is an interval if I = [a, b] ∩ Z for a, b ∈ R. Define f I,r = 1 |I| x∈I |f (x)| r 1/r .
If r = 1, we will frequently write f I,r = f I . We say that a collection of intervals S is sparse if for all S ∈ S there is a set E S ⊂ S so that the collection of sets {E S : S ∈ S} are disjoint, and |E S | ≥ 1 4 |S| for all S ∈ S. Define sparse bilinear forms by Λ S,r,s (f, g) = S∈S |S| f S,r g S,s , 1 ≤ r, s < ∞.
Given a (sub)linear operator T , we set T : (r, s) to be the smallest constant C so that for all finitely supported functions f, g there holds | T f, g | ≤ C sup Λ S,r,s (f, g),
where the supremum is over all sparse forms. Sparse operators are positive localized operators, hence their mapping properties are very easy to analyze. The following theorem is a remarkable refinement of the familiar fact that the Hilbert transform is weakly bounded. The theorem below implies all the standard weighted inequalities, as is explained in the references.
Theorem 1.1. [5,18] We have H Z, * : (1, 1) < ∞, where
H Z, * f (x) = sup N n : |n|>N f (x − n) n .
Our main theorem of this section concerns the maximal truncations
H Z, * a f := sup N ≥1 ∞ n=N a(n) n f (x − n) ,
where a : Z → {z ∈ C : |z| ≤ 1}. Hypotheses on a are of course required, and best expressed in terms of
µ j = µ a j := 2 j−1 <n≤2 j a(n) n δ n .
Letg(x) := g(−x) denote complex conjugation and reflection about the origin. Our main theorem (below) is proved in the following section.
Theorem 1.2. Suppose that we have the inequalities below, valid for some ǫ > 0, C ≥ 1, and all k > C.
|µ j * μ j (x)| 2 −(1+ǫ)j , x = 0, (1.3) µ j * μ k ∞ 2 −ǫk−j k + C ≤ j (1.4)
Then, we have the inequalities
H Z, * a : (1, r) 1 r − 1 , 1 < r < 2.
We stress that this result implies immediately not just the new weak (1,1) inequality, but a range of quantitative weighted inequalities, which are also new in this context. See [1,4,16] for these details, as well as more background and history of sparse bounds. Our theorem has antecedents in the works [4,16,17], which include the discrete setting, a sparse bound for maximal truncations, and the outline of a general theory of sparse bounds for "rough" singular integrals. Sparse quickly imply weighted inequalities, which are new in the ℓ p setting.
As an application of this maximal theory, we are able to prove pointwise convergence for a class of modulated -that is, oscillatory -(one-sided) ergodic Hilbert transforms, which we now proceed to describe:
Let (X, Σ, µ, τ ) be a measure-preserving system, i.e. a non-atomic σ-finite measure space, with τ a measure-preserving Z action. A celebrated result due to Cotlar [6] concerns the almost-everywhere convergence of the ergodic Hilbert transform,
Hf := n =0 τ n f n .
In particular, Cotlar [6] established the following theorem.
Theorem 1.5. For any measure-preserving system, and any f ∈ L r (X), 1 ≤ r < ∞, Hf converges µ-a.e.
Since Cotlar's result, Calderón [2] showed how to transfer analogous convergence results for the real-variable Hilbert transform to the ergodic setting, offering another proof; Petersen [21] has since offered an especially direct proof, using clever covering arguments. In all instances, the cancellation condition
lim N →∞ 0<|n|≤N 1 n = 0
played a crucial role in the arguments. Prior to Cotlar's result, a one-sided variant of the ergodic Hilbert transform was introduced by Izumi [15]:
H 1 f := ∞ n=1 1 n τ n f.
Izumi conjectured that for f ∈ L 2 (X) with f = 0, H 1 f would converge almost everywhere. Unfortunately, Halmos [14] proved that on any (non-atomic) probability space there always exists mean-zero f ∈ L 2 (X) for which H 1 f fails to converge even in the L 2 norm. In fact, Dowker and Erdös [11] exhibited mean-zero f ∈ L ∞ (X) so that
H * 1 f := sup N ≥1 N n=1
1 n τ n f = +∞ almost everywhere. Our Theorem 1.2 will allow us to prove pointwise convergence for "twisted" variants of the ergodic Hilbert transform,
H p f := ∞ n=1 e(p(n))
n τ n f, in the range 1 ≤ p < ∞. Here and throughout we let e(t) := e 2πit denote the complex exponential, and p(t) is taken to be a real-valued Hardy-field function, which grows super-linearly, in a quantifiable way, and stays sufficiently far from the class of polynomials. Good examples of such functions are fractional monomials, p(t) := t c , for non-integer c > 1. The presence of the phase n → p(n) introduces an element of "randomness" into the sequence { e(p(n)) n }. Indeed, although this sequence is not absolutely summable, a brief argument involving summation by parts and van der Corput's lemma (see below) shows that ∞ n=1 e(p(n)) n converges conditionally. Our main result in this direction is the following theorem.
Theorem 1.6. Suppose that p is an "admissible" Hardy field function. Then for any f ∈ L r (X), 1 ≤ r < ∞, H p f converges µ-a.e.
We defer the definition of "admissibile" to our section below on Hardy field functions.
We will prove Theorem 1.6 by establishing pointwise convergence for simple functions f : X → C and by noting the following proposition, which is an immediate consequence of Calderón's transference principle and our Theorem 1.2.
Theorem 1.7. For p as above, the maximally truncated modulated Hilbert transform,
H * p f := sup N ≥1
∞ n=N e(p(n)) n τ n f is bounded on L r (X) for 1 < r < ∞, and maps L 1 (X) → L 1,∞ (X).
In particular, we are able to give an elegant answer to the following question of Rosenblatt [22]. Problem 1.8. Does there exist a sequence {c n } with ∞ n=1 |c n | = ∞ such that for τ : X → X a measure-preserving transformation, and f ∈ L 1 (X), the series ∞ n=1 c n τ n f (x) converges for almost every x? See however the earlier solutions by Demeter [9] and Cuny [7,8], which are carefully constructed perturbations of the Hilbert transform kernel.
Finally, we consider random one-sided Hilbert transforms.
Theorem 1.9. Let {X n } be collection of uniformly bounded, independent, mean-zero random variables. Define
H Z, * X f := sup N ≥1 ∞ n=N X n n f (x − n) .
Almost surely, there holds
H Z, * X : (1, r) ω 1 r − 1 , 1 < r < 2.
By the work of Rosenblatt, and Calderón's transference principle, this immediately implies the following corollary. Corollary 1.10. Almost surely, for any f ∈ L 1 (X), for any (X, Σ, µ, τ ) measure preserving system, ∞ n=1 Xn n τ n f converges µ-a.e. Remark 1.11. Specializing {X n } to be i.i.d. ±1 random signs, we see that almost every choice of {c n = ± 1 n } provides an affirmative answer to Rosenblatt's Problem 1.8.
H Z, * X is too singular to fall under the purview of Theorem 1.2; in particular, (1.4) in general fails. Nevertheless, the approach used to establish Theorem 1.2 can be suitably modified. For further work in this direction, we refer the reader to the upcoming paper of the first two authors on oscillatory singular integrals.
The structure of our paper is as follows. In §2, below we will establish our Theorem 1.2 by presenting a general set of conditions for maximally truncated discrete singular integral operators to be bounded on ℓ r (Z), 1 < r < ∞, and to be weakly bounded on ℓ 1 (Z). The ℓ r (Z) theory is familar; the endpoint theory, which is the main novelty of this paper, is motivated by recent work of the first two authors [16,17].
In §3, we will introduce our class of "admissible" Hardy field functions, and prove pointwise convergence for the associated maximally truncated ergodic singular integral operators on simple functions.
Finally, in §4, we will prove almost sure sparse bounds for the random one-sided Hilbert transforms.
1.1. Acknowledgments. The first author would like to thank Ciprian Demeter for introducing him to random one-sided Hilbert transforms, and to both Ciprian Demeter and Terence Tao for helpful conversations and support.
1.2. Notation. As previously mentioned, we use e(t) := e 2πit . We will let M HL denote the Hardy-Littlewood maximal function acting on the integers.
We will make use of the modified Vinogradov notation. We use X Y , or Y X to denote the estimate X ≤ CY for an absolute constant C. If we need C to depend on a parameter, we shall indicate this by subscripts, thus for instance X r Y denotes the estimate X ≤ C r Y for some C p depending on r. We use X ≈ Y as shorthand for Y X Y .
The Maximal Theory
We present Theorem 1.2: Sufficient conditions for a discrete oscillatory operator to satisfy a sparse bound.
The Principle Recursive
Step. By a dyadic interval, we mean an interval Z ∩ I, where I is a dyadic interval in R of length at least 16. Set
T I f (x) = µ i * (f 1 1 3 I )(x), |I| = 2 i+3 .
Then, T I f is supported on I. We will show the sparse bound for
(2.1) T * f := sup ǫ I : |I|<ǫ T I f .
This is sufficient, for this reason. There are a choice of three dyadic grids D s , for s = 1, 2, 3 so that
µ i * f = 3 s=1 I∈Ds : |I|=2 i+3 T I f.
We fix one such dyadic grid D = D s in what follows.
The definition (2.1) is further adapted to different choices of dyadic intervals I ⊂ D. Set
(2.2) T * I f = sup ǫ I∈I : |I|<ǫ T I f .
On occasion, T I f denotes the full sum above, without a maximal truncation. Then, the main Lemma is Lemma 2.3. Suppose that I 0 is an interval, and I is a collection of intervals I ⊂ I 0 so that for function f supported on on I 0 we have
(2.4) sup I∈I f I ≤ K f I 0 , sup I∈I g I ≤ K g I 0 ,
Then,
(2.5) T * I f, g 1 r−1 |I 0 | f I 0 g I 0 ,r , 1 < r < 2.
Let us see how this Lemma proves the sparse bound for T * defined in (2.1), which in turn immediately implies Theorem 1.2.
Theorem 2.6. Assuming (1.3) and (1.4), we have T * : (1, r) 1 r−1 , 1 < r < 2.
Proof assuming Lemma 2.3. We can assume that f, g are supported on a dyadic in-
terval I 0 . Note that if x ∈ Z \ I 0 , we have T * f (x) M HL f (x)
. But it is a well known fact that the Hardy-Littlewood maximal function satisfes a (1, 1) sparse bound. We can therefore take I 0 to be in the sparse collection S that defines our sparse operator, and assume that the parent of I 0 in S is at least four times a big. It therefore remains to verify the sparse bound for T * I 0 f, g , where I 0 is the collection of all dyadic intervals contained in I 0 .
Let E be the maximal dyadic subintervals J of I 0 for which f J ≥ 10 f I 0 and/or g J ≥ 10 g I 0 . Setting E = J∈E J, we see that |E| ≤ 1 5 |I 0 |. And, letting I = {I ∈ I 0 : I ⊂ E}, we then have
T * I 0 f, g ≤ | T * I f, g | + J∈E | T * I 0 (J) f, g | where I 0 (J) = {I ∈ I 0 : I ⊂ J}.
But the first term is controlled by Lemma 2.3.
In particular, the right side of (2.5) is incorporated into the sparse form. And the collection E is added to the sparse collection. The proof follows by recursion.
Proof of Lemma 2.3.
We can assume
f I 0 = 1. Let B be the maximal dyadic subintervals J ⊂ I 0 for which f J ≥ K f I 0 . Make the Calderón-Zygmund decomposition, writing f = γ + b, where b = J∈B f 1 J .
(Note, no cancellative properties of b are needed. We can and do take f ≥ 0, so that b is as well.) The "good" function γ is bounded, so by Proposition 2.24, we have
T * I γ, g T * I γ r ′ g r 1 r−1 |I 0 | f I 0 g I 0 ,
r . It remains to consider the bad function.
For integers s let B(s) be the intervals J ∈ B with |J| = 2 s , and set
b = ∞ s=0 b s , where (2.7) b s = J∈B(s) f 1 J .
The principle points we have
(2.8) b s ∞ 2 s , ∞ s=0 b s 1 ≤ |I 0 |.
It is important to note that if I ∈ I, and J ∈ B, if I ∩ J = ∅ we necessarily have J I, just by construction. So, in particular,
T I b = s : 2 s <|I| T I b s
And, therefore
T I b = ∞ k=2 0≤s<k T I(k) b k−s .
Above, we are setting I(k) = {I ∈ I : |I| = 2 k+3 }. We will hold s fixed, obtaining geometric decay in that parameter. Thus, set
T I,s b(x) = T I b k−s (x) = µ k * (b k−s 1 1 3 I )(x), |I| = 2 k+3
We use the notation T I,s b and T * I,s b in a manner consistent with (2.2). Lemma 2.9. Under the assumptions of Lemma 2.3, and the notation (2.7) we have
(2.10) T * I,s b r ′ 2 − s 3(r−1) |I 0 | r−1 r , 1 < r < 2.
Summing over s ≥ 1 will give us the leading constant 1 r−1 .
We take up the proof of Lemma 2.9. Note that we have Recalling (1.3), one of our chief assumptions about µ j , the estimate of the right hand side naturally splits into two cases: The convolution is dominated byμ i * µ i (x)1 x =0 , the 'standard' case, or not, the 'non-standard' case.
T I,s b 2 2 = x∈I b k−s 1 1 3 I (x) ·μ i * µ i * (b k−s 1 1 3 I )(x). (2.11) Above, |I| = 2 i+3 .Write I = S s ∪ N s = k S s (k) ∪ k N s (k),
where I ∈ S s (k) if I ∈ I(k) and
(2.12)
T I b k−s 2 2 ≤ 64C 0 |I| −1−ǫ b k−s 1 I 2 1
; and, we collect I ∈ N s (k) if the above inequality (2.12) fails. Here, C 0 is the implied constant in (1.3), and K is as in (2.4).
Proposition 2.13. With the above notation, we have the bound
T * Ss,s b r ′ 2 − s 2(r−1) |I 0 | r−1 r , 1 < r < 2.
Proof. For fixed k, we have by (2.8)
T Ss(k),s b ∞ 2 −s .
And, by (2.12), and again (2.8),
T Ss(k),s b 2 2 −s/2 2 −kǫ/2 |I 0 | 1/2 .
Interpolating, and summing over k ≥ 1 completes the proof.
Thus, the core is the control of the non-standard collections. In the case that I ∈ N s (k), we have (2.14)
T I,s b 2 2 |I| −1 b k−s 1 I 2 2
. since the the convolution withμ i * µ i in (2.11) is dominated byμ i * µ i (0) ≃ |I| −1 . It is worth remarking that the purely ℓ 2 bound below T Ns,s b 2 2 −s/2 |I 0 | 1/2 is known [3,4], but a method to obtain a bound for maximal truncations is new, and adapted from [16].
The endpoint bound is easy.
Lemma 2.15. Assume that g satisfies (2.4) We have the bound uniformly in s.
(2.16) T * Ns,s b, g |I 0 | f I 0 g I 0 .
Proof. For any choice of measurable functions ε k : I → {z : |z| ≤ 1}, we have
k 0 k=s ε k T Ns(k) b k−s , g k 0 k=s I∈Ns b k−s 1 I 1 g I g I 0 s b k−s 1 |I 0 | f I 0 g I 0 .
This proves (2.16).
We need a good estimate for maximal truncations at ℓ 2 . Proof. One more definition is required to address maximal truncations. For integers t, say that I ∈ N s,t if I ∈ N s (k) for some integer k and
2 −t+c ≤ b k−s 1 I 1 |I| < 2 −t+c+1 .
Notce that if b k−s 1 I 1 = 0, we must have
b k−s 1 I 1 |I| 2 −s for I ∈ N s (k). Thus, t ≤ s + c. We show that (2.19) T * Ns,t,s b 2 s2 −2s/5 |I 0 | 1/2 . A summation over t ≤ s + c proves (2.18).
Crucially, this definition supplies us with a Carleson measure estimate. For all K ∈ N s,t we have For a sufficiently large constant C, we have |F t | < 1 4 |I 0 |. We claim that
(2.21) T * N ♯ s,t b 2 s2 −2s/5 |I 0 | 1/2 .
where N ♯ s,t = {I ∈ N s,t : I ⊂ F t }. To conclude (2.19), we recurse inside the set F t , but this step is easy, and we omit the details. It remains to prove (2.21).
To prove (2.21), we are in a position to apply the Rademacher-Menshov Lemma 2.25 below. It controls the maximal truncations, and its key assumption is an an orthogonality condition on the summands. To set up the application of this Lemma, we set M 1 to be the minimal elements of N ♯ s,t , and inductively set M u+1 to be the minimal elements of N ♯ s,t \ u v=1 M v . This collection will be empty for u > u 0 = C2 t . Then set
(2.22) β u = k I∈Mu(k) T I b k−s , where M u (k) := {I ∈ M u : |I| = 2 k+3 }. We will show that u 0 u=1 σ u β u 2 2 −2s/5 |I 0 | 1/2 , σ t ∈ {−1, 0, 1}.
In view of (2.26), we then conclude (2.21), after factoring a log u 0 ≃ t s into 2 −2s/5 . Now, we have from (2.14), (2.22), (2.8) and the definition of N s,t ,
u 0 u=1 σ u β u 2 2 u 0 u=1 k I∈Mu(k) |I| −1 b k−s 1 I 2 2 ≤ u 0 u=1 k I∈Mu(k) |I| −1 b k−s 1 I ∞ b k−s 1 I 1 (2.23) 2 −s u 0 u=1 k I∈Mu(k) b k−s 1 I 1 2 −s |I 0 |.
For u < v, we have by the off-diagonal assumption (1.4),
| σ u β u , σ v β v | = kv ku : ku<kv−c J∈Mv(kv ) I∈Mu(ku) I⊂J b ku−s , T * I T J b kv −s kv ku : ku<kv−c J∈Mv(kv) I∈Mu(ku) I⊂J |I| −ǫ |J| −1 b ku−s 1 I 1 b kv−s 1 J 1 2 −t kv ku : ku<kv−c J∈Mv(kv) I∈Mu(ku) I⊂J |I| −ǫ |I| |J| b kv−s 1 J 1 2 −t−ǫu kv J∈Mv(kv) b kv −s 1 J 1 ≤ 2 −t−ǫu b 1 2 −t−ǫu |I 0 |.
The last inequalities follow from the construction, and |I| ≥ 2 u , for I ∈ M u . By Cauchy-Schwarz, we have
4s/ǫ u=1 σ u β u 2 s 1/2 2 −s/2 |I 0 | 1/2 , u 0 4s/ǫ σ u β u 2 2 2 −s |I 0 | + 4s/ǫ≤u<v≤u 0 2 −t−ǫu |I 0 | 2 −s |I 0 |, since t ≤ s + c.H * a : ℓ r → ℓ r max{r, 1 r−1 } 1 < r < ∞.
Proof. We first observe that µ j * μ j (0) = µ j 2 2 2 −j ; together with the assumed bound on |µ j * μ j | away from zero, this implies that
µ j * f 2 2 2 −j f 2 2 + 2 −ǫj Z |f |(n)M HL f (n) 2 −ǫj f 2 2 .
Here, we used that µ j * μ j is supported in {|x| 2 j }. But we clearly have the ℓ 1 and ℓ ∞ estimate below without decay:
µ j : ℓ s → ℓ s 1, s = 1, ∞.
Interpolating, we see that µ j : ℓ r → ℓ r 2 −2ǫj r−1 r for 1 < r < 2. Majorizing
Hf ≤ ∞ j=0
|µ j * f | + M HL f and taking ℓ r norms yields the result. In fact, one may establish a sparse (r, r) bound for r(ǫ) < r < 2 due to the power gain in scale.
Mentioned above, this is a variant of the Rademacher-Menshov inequality that we used to control maximal truncations. This has been observed many times. See [10,Theorem 10.6].
Lemma 2.25. Let (X, µ) be a measure space, and {φ j : 1 ≤ j ≤ N} a sequence of functions which satisfy the Bessel type inequality below, for all sequences of coefficents
c j ∈ {0, ±1}, N j=1 c j φ j L 2 (X) ≤ A.
Then, there holds
(2.26) sup 1<n≤N n j=1 φ j L 2 (X)
A log(2 + N).
Specializing to Hardy Fields
We now introduce Hardy fields and some of their properties. We refer the reader to e.g. [12] and the references contained therein for further discussion of Hardy field functions and their applications to ergodic theory; in particular, the following introduction to Hardy field functions is taken from [12, §2].
We call two real valued functions of one real variable that are continuous for large values of s ∈ R equivalent if they coincide for large s ∈ R. We say that a property holds for large s (or eventually) if it holds for every s in an interval of the form [s 0 , ∞). The equivalence classes under this relation are called germs. The set of all germs we denote by B which is a ring. One can show that U contains the class L of logarithmico-exponential functions of Hardy, i.e., the class of functions which can be obtained by finitely many combinations of real constants, the variable s, log, exp, summation and multiplication. Thus, for example, it contains functions of the form s α = exp(α log s), α ∈ R.
Another property of Hardy fields is that each Hardy field is totally ordered with respect to the order < ∞ defined by f < ∞ g ⇐⇒ f (s) < g(s) for all large s.
Since the class L belongs to every maximal Hardy field, we conclude that every element of U is comparable to every logarithmico-exponential function. In particular, we can define the type of a function p ∈ U to be t(p) := inf{α ∈ R : |p(s)| < s α for large s}.
We say that p is subpolynomial if t(p) < +∞, i.e., if |p| is dominated by some polynomial. In particular, for eventually positive subpolynomial p with finite type there is α ∈ R such that for every η there is an s 0 so that s α−η < p(s) < s α+η holds for every s > s 0 . Note that considering eventually positive p is not a restriction since every nonzero p ∈ U is either eventually positive or eventually negative.
We now consider subpolynomial elements of U with positive non-integer type, such as for example p(s) = 5s π + s log s. More precisely, we introduce the following classes. We say that p is "admissible" if it is in some class N δ,M,m .
Remark 3.3. By l'Hôpital's rule, the condition s m+α−η p(s) s m+α+η is enough to guarantee that s m+α−j−η j p (j) (s) j s m+α−j+η for each j. We simply choose to make the implicit constant uniform over the first m + 2 derivatives.
We have the following lemma. Remark 3.5. As we will see from the proof, the ǫ > 0 of gain will depend only on δ, M, m.
For pointwise convergence reasons, we will need the following technical complement to Lemma 3.4: Lemma 3.6. For any κ > 0,
µ p j,κ := (1+κ) j−1 <n≤(1+κ) j e(p(n)) n δ n satisfies (3.7) |µ p j,κ * µ p j,κ | (1 + κ) −j δ 0 + (1 + κ) −(1+ǫ)j .
We defer the proofs of our technical lemmas to the following subsection, and complete the proof of pointwise convergence now.
Proof of Theorem 1.6, assuming Lemmas 3.4 and 3.6. Let p be an admissible Hardy field function, thus p ∈ N δ,M,m for some δ, M, m. By Lemma 3.4, Theorem 1.2, and the Calderón transference principle [2], we know that the maximal function
H * p f := sup N ∞ n=N e(p(n)) n τ n f
is weakly bounded on L r (X), 1 ≤ r < ∞. By a standard density argument, it therefore suffices to prove pointwise convergence for simple (bounded) functions, g.
In fact, by [23,Lemma 1.5], it suffices to prove only that for each simple g, and each κ > 1, the limit lim j→∞ (1+κ) j <n e(p(n)) n τ n g = 0 µ − a.e.
But this is straightforward. The technical estimate (3.7) implies
σ(g, j) := (1+κ) j <n≤(1+κ) j+1 e(p(n)) n τ n g (1 + κ) −ǫj g 2 ,
and thus
(1+κ) j <n e(p(n)) n τ n g 2 ≤ ∞ l=j σ(g, l) (1 + κ) −ǫj/2 g 2 .
Consequently, ∞ j=0 (1+κ) j−1 <n e(p(n)) n τ n g 2 is an integrable function, which proves our claim.
3.1. The Proof of Lemmas 3.4 and 3. 6. In what follows, we will need the following result of van der Corput which appears in [25,Satz 4] Lemma 3.8. Let k ≥ 2 be an integer and put K = 2 k . Suppose that a ≤ b ≤ a + N and that f : [a, b] → R has continuous kth derivative that satisfies the inequality
0 < λ ≤ |f (k) (x)| ≤ hλ for all x ∈ [a, b].
Then a≤n≤b e(f (n)) hN λ 1/(K−2) + N −2/K + (N k λ) −2/K .
With this tool in hand, we are prepared for the proof of our technical lemmas.
Proof of Lemma 3.4. We begin with the case where k < j, and seek to prove that
2 k−1 <n≤2 k ,2 j−1 <n+x≤2 j e(p(n + x) − p(n)) (n + x)n 2 −ǫk−j . Set A := max{2 k−1 , 2 j−1 − x}, B := min{2 k , 2 j − x}.
Note that B − A 2 k . By summation by parts, it suffices to show that max A<K≤B A<n≤K
e(p(n + x) − p(n)) 2 (1−ǫ)k .
We begin with the case m = 1. Note that we may assume that K − A 2 (1−ǫ)k . For large enough k, the phase f (n) = f x (n) := p(n+x)−p(n) has second derivative
(2 k ) α−1−η M |f ′′ (n)| M (2 k ) α−1+η for A < n ≤ B. By Lemma 3.8, max A<K≤B A<n≤K e(f (n)) 2 (1+2η−ǫ)k ,
for some ǫ = ǫ(α) bounded away from zero. We next turn to the second part of the lemma, where we consider diagonal interactions, j = k, evaluated at x = 0.
In this case, by the mean-value theorem, the phase f (n) has second derivative
|x|(2 j ) α−2−η M |f ′′ (n)| M |x|(2 j ) α−2+η . By Lemma 3.8, A<n≤K e(f (n)) 2 (1+2η−ǫ)j ,
for some ǫ = ǫ(α) bounded away from zero. The m ≥ 2 cases follow similarly from the (m + 1)th Van der Corput lemma.
The proof of Lemma 3.6 is similar; the details are left to the reader.
The Random One-sided Hilbert Transform
The goal of this section is to prove Theorem 1.9, reproduced below for the reader's convenience. First, we recall the {X n }: bounded, independent, and mean-zero random variables on a probability space Ω. Then: Theorem 4.1. Let {X n } be collection of uniformly bounded, independent, mean-zero random variables. Define
H Z, * X f := sup N ≥1 ∞ n=N X n n f (x − n) .
Almost surely, there holds
H Z, * X : (1, r) ω 1 r − 1 , 1 < r < 2.
Analogous to the previous sections, we let
µ i = µ ω i := 2 i−1 <n≤2 i X n n δ n ;
we will generally suppress the ω in our notation. Our first order of business is to establish good estimates on the convolutionsμ i * µ j , i ≤ j. We do so in the following subsection.
4.1. Random Preliminaries. We need the following well known large deviation inequality. where V N = N n=1 E|Z n | 2 . Using Lemma 4.2, we have the following control overμ i * µ j , i ≤ j. Lemma 4.3. Almost surely, the following hold.
• |μ i * µ i (x)| ω 2 −5i/4 for x = 0; • For i < j, |μ i * µ j | ω √ j 2 i/2+j 1 [0,2 j )
, and thus for all j 2 i/2 ,
|μ i * µ j | ω 2 −i/4−j 1 [0,2 j ) ; • For 2 i/2 ≪ j, (4.4) Z i,j := {|μ i * µ j | ≫ ω 2 −i/4−j } ⊂ m I m
where each |I m | = 2 i is dyadic, and the (disjoint) union is over at most a constant multiple of e −c 0 2 i/2 ×2 j−i intervals, for some (small) absolute constant c 0 .
Remark 4.5. We will use without comment the trivial upper bound (useful on the exceptional sets Z i,j ): |μ i * µ j | 2 −j .
Proof. The convolution in question is explicitly,
µ i * µ j (x) = 2 i−1 <n≤2 i , 2 j−1 <x+n≤2 j X x+n X n (x + n)n , i ≤ j.
Observe that the sum above is over uniformly bounded, mean zero random variables. And that
Eμ i * µ j (x) 2 2 −i−2j , x = 0
where the implied constant depends upon the uniform bound on the random variables {X n }. Since the sum in the definition ofμ i * µ j (x) can be separated into two sums, each over independent mean-zero random variables, the first point follows from Chernoff's inequality, Lemma 4.2, and a Borel-Cantelli argument. The second point is similar. For the third, for dyadic |I| = 2 i , I ⊂ [0, 2 j ), consider the random variables V i,j I := sup x∈I |μ i * µ j (x)|.
By subdividing into (say) ten subfamilies, we may assume that the {V i,j I : I} are independent. By Chernoff's inequality, we know that for each I, P(V i,j I ≥ 2 −i/4−j ) e −c2 i/2 for some absolute c > 0. By a binomial distribution argument, provided that c 0 is sufficiently small, there exists some c > c ′ > 0 so that P(|{I : V i,j I ≥ 2 −i/4−j }| ≥ e −c 0 2 i/2 2 j−i ) e −c ′ 2 j−i/2 e −c 0 2 i/2 .
Summing over j ≫ 2 i/2 and applying Borel-Cantelli yields the result.
4.2. The Proof. We follow the argument of §2, with the obvious notational changes; the key lemma needed is analogous to (2.21) in the proof of Lemma 2.17. We refer to §2 for relevant definitions. In treating Exceptional(u, v), the key estimates that we used were that b k ∞ 2 k , see (2.8), and this consequence of (4.4): For any y, |{I : |I| = 2 ku dyadic, I ∩ Z ku,kv + y = ∅}| 2 kv e −c 0 2 ku/2 .
The upshot is that we have majorized | σ u β u , σ v β v | by 2 −t−u/4 |I 0 |. Summing this over Cs ≤ u < v ≤ u 0 2 t yields the desired bound, completing the proof.
Interpolating between this bound and (2.16) completes the proof of (2.10), and hence the proof of Lemma 2.3.
J∈Ns,t : J⊂K |J| 2 t |K|. Indeed, if this is not so we have for some large constant M M · 2 t |K| ≤ J∈Ns,t : J⊂K |J| ≤ 2 t+c k J∈Ns,t(k) : J⊂K b k−s 1 J 1 2 t |K|, by construction, namely (2.4), and (2.8). This is a contradiction for large, but absolute choice of M. Therefore (2.20) holds.The Carleson measure condition(2.20) says that there are about 2 t overlapping intervals I ∈ N s,t . Set
Definition 3.1. A Hardy field is a subfield of B which is closed under differentiation. The union of all Hardy fields is denoted by U.
Definition 3. 2 .
2For δ ∈ (0, 1/2), M ≥ 1 and m ≥ 1 denote by N δ,M,m the set of all p ∈ U so that there exist α ∈ [δ, 1 − δ] and η ≪ δ,m 1 with 1 M s m+α−η−j ≤ p (j) (s) ≤ Ms m+α+η−j for all s ≥ 1 and j = 0, . . . , m + 2.
hypotheses of Theorem 1.2, namely (1.3) and (1.4).
Lemma 4. 2 (
2Chernoff's Inequality). Let {Z n } be mean-zero, independent random variables, all of which are almost surely bounded in magnitude by 1. Then there exists an absolute constant c >
Lemma 4. 6 .22
6There exists some δ > 0 so that we have the bound uniformly in −δ ′ s |I 0 | 1/2 , σ t ∈ {−1, 0, 1}, where u 0 2 t 2 s . This will allow us to conclude Lemma 4.6, by an application of the Rademacher-Menshov Lemma 2.25. The diagonal terms are estimated exactly as in(2.23). By Cauchy-Schwarz, we need only show that As for Exceptional(u, v), the key estimate is (−s 2 u e −c 0 2 u/2 b 1 2 −s−u |I 0 |.
−t−u/4 b 1 2 −t−u/4 |I 0 |.
Divide the summands on the left intowhere the inner products on the right are treated according to the estimates of Lemma 4.3. The two terms areWe begin by estimating Main(u, v) as follows:
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[
"Supernova rates from the Southern inTermediate Redshift ESO Supernova Search (STRESS) ⋆",
"Supernova rates from the Southern inTermediate Redshift ESO Supernova Search (STRESS) ⋆"
]
| [
"M T Botticella \nINAF -Osservatorio Astronomico di Collurania-Teramo\nI-64100TeramoV.M. MagginiItaly\n\nDipartimento di Scienze della Comunicazione\nUniversitá di Teramo\nviale Crucioli 122I-64100TeramoItaly\n\nAstrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT 71NNBelfastUnited Kingdom\n",
"M Riello \nInstitute of Astronomy\nMadingley RoadCambridgeUK\n",
"E Cappellaro \nINAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 5I-35122PadovaItaly\n",
"S Benetti \nINAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 5I-35122PadovaItaly\n",
"G Altavilla \nINAF -Osservatorio Astronomico di Bologna\nV. Ranzani 1, I-40127BolognaItaly\n",
"A Pastorello \nAstrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT 71NNBelfastUnited Kingdom\n",
"M Turatto \nINAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 5I-35122PadovaItaly\n",
"L Greggio \nINAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 5I-35122PadovaItaly\n",
"F Patat \nEuropean Southern Observatory\nK. Schwarzschild Str. 285748GarchingGermany\n",
"S Valenti \nEuropean Southern Observatory\nK. Schwarzschild Str. 285748GarchingGermany\n\nDipartimento di Fisica\nUniversitá di Ferrara\nvia del Paradiso 12I-44100FerraraItaly\n",
"L Zampieri \nINAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 5I-35122PadovaItaly\n",
"A Harutyunyan \nINAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 5I-35122PadovaItaly\n",
"G Pignata \nDepartamento de Astronomia\nUniversidad de Chile\nCasilla 36-DSantiagoChile\n",
"S Taubenberger \nMax-Planck-Institut für Astrophysik\nKarl-Schwarzschild-Str. 7D-85741Garching bei MünchenGermany\n"
]
| [
"INAF -Osservatorio Astronomico di Collurania-Teramo\nI-64100TeramoV.M. MagginiItaly",
"Dipartimento di Scienze della Comunicazione\nUniversitá di Teramo\nviale Crucioli 122I-64100TeramoItaly",
"Astrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT 71NNBelfastUnited Kingdom",
"Institute of Astronomy\nMadingley RoadCambridgeUK",
"INAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 5I-35122PadovaItaly",
"INAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 5I-35122PadovaItaly",
"INAF -Osservatorio Astronomico di Bologna\nV. Ranzani 1, I-40127BolognaItaly",
"Astrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT 71NNBelfastUnited Kingdom",
"INAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 5I-35122PadovaItaly",
"INAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 5I-35122PadovaItaly",
"European Southern Observatory\nK. Schwarzschild Str. 285748GarchingGermany",
"European Southern Observatory\nK. Schwarzschild Str. 285748GarchingGermany",
"Dipartimento di Fisica\nUniversitá di Ferrara\nvia del Paradiso 12I-44100FerraraItaly",
"INAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 5I-35122PadovaItaly",
"INAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 5I-35122PadovaItaly",
"Departamento de Astronomia\nUniversidad de Chile\nCasilla 36-DSantiagoChile",
"Max-Planck-Institut für Astrophysik\nKarl-Schwarzschild-Str. 7D-85741Garching bei MünchenGermany"
]
| []
| Aims. To measure the supernova (SN) rates at intermediate redshift we performed the Southern inTermediate Redshift ESO SupernovaSearch (STRESS). Unlike most of the current high redshift SN searches, this survey was specifically designed to estimate the rate for both type Ia and core collapse (CC) SNe. Methods. We counted the SNe discovered in a selected galaxy sample measuring SN rate per unit blue band luminosity. Our analysis is based on a sample of ∼ 43000 galaxies and on 25 spectroscopically confirmed SNe plus 64 selected SN candidates. Our approach is aimed at obtaining a direct comparison of the high redshift and local rates and at investigating the dependence of the rates on specific galaxy properties, most notably their colour. Results. The type Ia SN rate, at mean redshift z = 0.3, amounts to 0.22 +0.10+0.16 −0.08−0.14 h 2 70 SNu, while the CC SN rate, at z = 0.21, is 0.82 +0.31+0.30 −0.24−0.26 h 2 70 SNu. The quoted errors are the statistical and systematic uncertainties. Conclusions. With respect to local value, the CC SN rate at z = 0.2 is higher by a factor of ∼ 2 already at redshift , whereas the type Ia SN rate remains almost constant. This implies that a significant fraction of SN Ia progenitors has a lifetime longer than 2 − 3 Gyr. We also measured the SN rates in the red and blue galaxies and found that the SN Ia rate seems to be constant in galaxies of different colour, whereas the CC SN rate seems to peak in blue galaxies, as in the local Universe. SN rates per unit volume were found to be consistent with other measurements showing a steeper evolution with redshift for CC SNe with respect to SNe Ia. Finally we have exploited the link between SFH and SN rates to predict the evolutionary behaviour of the SN rates and compare it with the path indicated by observations. We conclude that in order to constrain the mass range of CC SN progenitors and SN Ia progenitor models it is necessary to reduce the uncertainties in the cosmic SFH. In addition it is important to apply a consistent dust extinction correction both to SF and to CC SN rate and to measure SN Ia rate in star forming and in passive evolving galaxies in a wide redshift range. | 10.1051/0004-6361:20078011 | [
"https://arxiv.org/pdf/0710.3763v1.pdf"
]
| 16,736,116 | 0710.3763 | d0bee2bc35bde01f18dd68f1bc2f5a91ed704650 |
Supernova rates from the Southern inTermediate Redshift ESO Supernova Search (STRESS) ⋆
19 Oct 2007 February 2, 2008
M T Botticella
INAF -Osservatorio Astronomico di Collurania-Teramo
I-64100TeramoV.M. MagginiItaly
Dipartimento di Scienze della Comunicazione
Universitá di Teramo
viale Crucioli 122I-64100TeramoItaly
Astrophysics Research Centre
School of Mathematics and Physics
Queen's University Belfast
BT 71NNBelfastUnited Kingdom
M Riello
Institute of Astronomy
Madingley RoadCambridgeUK
E Cappellaro
INAF -Osservatorio Astronomico di Padova
Vicolo dell'Osservatorio, 5I-35122PadovaItaly
S Benetti
INAF -Osservatorio Astronomico di Padova
Vicolo dell'Osservatorio, 5I-35122PadovaItaly
G Altavilla
INAF -Osservatorio Astronomico di Bologna
V. Ranzani 1, I-40127BolognaItaly
A Pastorello
Astrophysics Research Centre
School of Mathematics and Physics
Queen's University Belfast
BT 71NNBelfastUnited Kingdom
M Turatto
INAF -Osservatorio Astronomico di Padova
Vicolo dell'Osservatorio, 5I-35122PadovaItaly
L Greggio
INAF -Osservatorio Astronomico di Padova
Vicolo dell'Osservatorio, 5I-35122PadovaItaly
F Patat
European Southern Observatory
K. Schwarzschild Str. 285748GarchingGermany
S Valenti
European Southern Observatory
K. Schwarzschild Str. 285748GarchingGermany
Dipartimento di Fisica
Universitá di Ferrara
via del Paradiso 12I-44100FerraraItaly
L Zampieri
INAF -Osservatorio Astronomico di Padova
Vicolo dell'Osservatorio, 5I-35122PadovaItaly
A Harutyunyan
INAF -Osservatorio Astronomico di Padova
Vicolo dell'Osservatorio, 5I-35122PadovaItaly
G Pignata
Departamento de Astronomia
Universidad de Chile
Casilla 36-DSantiagoChile
S Taubenberger
Max-Planck-Institut für Astrophysik
Karl-Schwarzschild-Str. 7D-85741Garching bei MünchenGermany
Supernova rates from the Southern inTermediate Redshift ESO Supernova Search (STRESS) ⋆
19 Oct 2007 February 2, 2008Received .../ Accepted ...Astronomy & Astrophysics manuscript no. STRESS˙finsupernovae:general -star:formation -galaxy:evolution -galaxy:stellar content
Aims. To measure the supernova (SN) rates at intermediate redshift we performed the Southern inTermediate Redshift ESO SupernovaSearch (STRESS). Unlike most of the current high redshift SN searches, this survey was specifically designed to estimate the rate for both type Ia and core collapse (CC) SNe. Methods. We counted the SNe discovered in a selected galaxy sample measuring SN rate per unit blue band luminosity. Our analysis is based on a sample of ∼ 43000 galaxies and on 25 spectroscopically confirmed SNe plus 64 selected SN candidates. Our approach is aimed at obtaining a direct comparison of the high redshift and local rates and at investigating the dependence of the rates on specific galaxy properties, most notably their colour. Results. The type Ia SN rate, at mean redshift z = 0.3, amounts to 0.22 +0.10+0.16 −0.08−0.14 h 2 70 SNu, while the CC SN rate, at z = 0.21, is 0.82 +0.31+0.30 −0.24−0.26 h 2 70 SNu. The quoted errors are the statistical and systematic uncertainties. Conclusions. With respect to local value, the CC SN rate at z = 0.2 is higher by a factor of ∼ 2 already at redshift , whereas the type Ia SN rate remains almost constant. This implies that a significant fraction of SN Ia progenitors has a lifetime longer than 2 − 3 Gyr. We also measured the SN rates in the red and blue galaxies and found that the SN Ia rate seems to be constant in galaxies of different colour, whereas the CC SN rate seems to peak in blue galaxies, as in the local Universe. SN rates per unit volume were found to be consistent with other measurements showing a steeper evolution with redshift for CC SNe with respect to SNe Ia. Finally we have exploited the link between SFH and SN rates to predict the evolutionary behaviour of the SN rates and compare it with the path indicated by observations. We conclude that in order to constrain the mass range of CC SN progenitors and SN Ia progenitor models it is necessary to reduce the uncertainties in the cosmic SFH. In addition it is important to apply a consistent dust extinction correction both to SF and to CC SN rate and to measure SN Ia rate in star forming and in passive evolving galaxies in a wide redshift range.
Introduction
SNe, the catastrophic explosions that terminate the life of some stars, play a pivotal role in several astrophysical topics. They provide a crucial test for stellar evolution theory and, as the main contributors of heavy elements in the Universe, they are key actors of the chemical enrichment of galaxies. Their rapidly ex-panding ejecta sweep, compress and heat the interstellar medium causing gas outflows from galaxies and triggering the star formation process. Furthermore, in the last few years type Ia SNe have consolidated their prominence as cosmological probes providing the first direct evidence for the acceleration of the Universe (Perlmutter et al. 1999;Riess et al. 1998).
SN statistics is another important cosmological probe even though less exploited. In particular the SN rate as a function of the cosmic time is linked to some of the basic ingredients of the galaxy evolution such as mass, SFH, metallicity and environment.
The rate of CC SNe, including type II and type Ib/c, gives, for an assumed initial mass function (IMF), a direct measurement of the on-going SFR because of the short evolutionary life-times of their progenitors. On the other hand the rate of SNe Ia echoes the long term SFH because these SNe, originating from low mass stars in binary systems, are characterized by a wide range of delay times between progenitor formation and explosion (Yungelson & Livio 2000;Greggio 2005).
Conversely, if a SFH is assumed, the progenitor scenarios for SN Ia can be constrained measuring their rate as a function of galaxy type and redshift (Madau et al. 1998; Dahlén & Fransson 1999;Pain et al. 2002;Strolger et al. 2004;Tonry et al. 2003;Scannapieco & Bildsten 2005;Mannucci et al. 2005;Mannucci et al. 2006;Neill et al. 2006;Sullivan et al. 2006;Forster et al. 2006). Despite promising prospects, measurements of SN rates were very scanty so far.
A new opportunity for SN searches emerged in recent years thanks to the availability of panoramic CCD mosaic cameras mounted at medium/large size telescopes. These instruments allow the deep monitoring of wide areas in the sky and thus the collection of statistically significant samples of SNe at intermediate and high redshifts.
In this paper we report the results of the "Southern inTermediate Redshift ESO Supernova Search" (STRESS) which was carried out with ESO telescopes and designed to search for SNe in the range 0.05 < z < 0.6.
The paper is organized as follows: in Sect. 2 we illustrate the aims and the strategy of STRESS, in Sect. 3 we present the data set collected during our observing programme. The following three sections describe the data analysis, namely the selection of the galaxy sample (Sect. 4), the detection and classification of the SN candidates (Sect. 5) and the estimate of the control time of the search (Sect. 6). The measurement of the SN rates and the estimate of the statistical and systematic uncertainties are presented in Sect. 7. Our results are compared with published measurements and models predictions in Sect. 8. Finally, a brief summary is given in Sect. 9.
We remark that the measurements reported in this paper supersede the preliminary results published in Cappellaro et al. (2005) because of i) better SN statistics , ii) better filter coverage for characterization of the galaxy sample, iii) improved correction for the host galaxy extinction.
Thorough the paper we adopt the cosmological parameters H 0 1 = 70 km s −1 Mpc −1 , Ω M = 0.3, Ω Λ = 0.7. Magnitudes are in the Vega System.
Search goals and strategy
Most past and current high-redshift SN searches were/are aimed at detecting SNe Ia for measuring cosmological parameters, while the measurement of SN rates is a secondary goal. As such, the strategy of these searches are affected by several shortcomings (eg. Schmidt et al. 1998;Pain et al. 2002;Tonry et al. 2003;Dahlén et al. 2004;Neill et al. 2006). In particular, highredshift SN searches are optimized for the detection of unreddened SNe Ia near maximum light, so that all variable sources unlikely to fall into this category are assigned a lower priority for follow-up, when not completely neglected. This strategy introduces a severe bias in the SN sample; besides, in general, it prevents a reliable estimate of the rate of CC SNe.
STRESS was specifically designed to measure the rate of both SNe Ia and CC SNe at intermediate redshift, to be compared with the local values and estimate their evolution. To this aim we 1 h 70 = H 0 /70 km s −1 Mpc −1 tried to reduce as much as possible the biases which affect the different SN types.
To measure the SN rate we counted the events observed in a selected galaxy sample. This is the same approach used to derive the local rates, and offers two advantages. By preserving the link between the observed SNe and the monitored galaxies we can investigate the dependence of the rates on specific galaxy properties, most notably their colours, which depend primarily on the galaxy SFH, and on metallicity and dust extinction. In addition, the SN rates measured per unit of galaxy B luminosity, that is in SN unit (SNu 2 ), can be directly compared with the local ones.
For the selection and the characterization of the galaxy sample, we used multi-band observations and the photometric redshift technique (photo-z), which allow us to derive the redshift, the absolute luminosity, and the rest frame colours for all the galaxies in our sample.
In summary our approach involves the following steps:
1. the selection of the galaxy sample and its characterization, 2. the detection and classification of SN candidates, 3. the estimate of the control time of the galaxy sample, 4. the measurement of SN rates, the analysis of their dependence on the galaxy colours, and their evolution with redshift.
The observing programme
Observations were carried out using the Wide Field Imager (WFI) at the 2.2m MPG/ESO telescope at La Silla, Chile. WFI is a mosaic camera consisting of 2 × 4 CCDs, each of 2048 × 4096 pixels, with a pixel scale of 0.238 arcsec and a field of view of 34 × 33 arcmin 2 . The individual chips are separated by gaps of 23.8 arcsec and 14.3 arcsec along right ascension and declination respectively, for a resulting filling factor of 95.9%. We performed observations in the B, V, R, I bands using the following ESO/WFI broad-band filters: B/99, B/123,V/89,Rc/162,Ic/lwp. Our initial list included 21 sky-fields evenly distributed in right ascension to reduce the observing scheduling requirements. The fields were chosen at high galactic latitude to reduce stellar crowding and galactic extinction and with few bright sources to minimize CCD saturation and ghost effects.
The observing programme, distributed over a period of 6 years, from 1999 to 2005, can be divided into three phases. During the first year we carried out a pilot programme aimed at tuning the observing strategy and testing our software (at the beginning of our programme not all filters were available). In a second phase (2001)(2002)(2003)12 observing runs) we performed the SN search in the V band targeting events at redshift z ∼ 0.2−0.3. In the last phase (2004)(2005)4 observing runs), we performed the search in R band aiming to detect SNe at higher redshifts. A total of 34 nights were allocated to our programme out of which 9 were lost due to bad weather. Seeing was in the range 0.7 − 1.5 arcsec, with an average around 1 arcsec.
The temporal sampling, on average one observation every 3-4 months, was tuned to maximize event statistics. The observing log is reported in Tab 2.
Typically, in each run we observed the same fields for two consecutive nights, in the first one with the search filter, V or R, and in the second one with a different filter to obtain colour information both for SN candidates and galaxies. Each observation, typically of 45 min integration time, was split in a sequence of Schlegel et al. (1998) three exposures jittered by 5-10 arcsec to allow a better removal of cosmetic defects, cosmic rays, satellite tracks and fast moving objects. Due to technical and weather limitations and, occasionally, to scheduling constraints, in many cases we could not maintain our observational strategy. As a consequence, for many SN candidates we are missing colour information and for a few fields we have insufficient filter coverage to apply the photo-z technique to galaxy sample.
In order to secure the classification of the SN candidates spectroscopic observations were scheduled about one week after the search run. In the first year, spectra were taken at the ESO3.6m telescope equipped with ESO Faint Object Spectrograph and Camera (EFOSC); in the following years, we used the FOcal Reducer and low dispersion Spectrograph (FORS1/2) at the VLT. In total 2.5 nights were allocated at the ESO3.6m telescope and 12 nights at the VLT, 4 of which were lost due to bad weather conditions. For a better subtraction of the night sky emission lines we selected grisms of moderate resolution (FORS1/2 grism 300V and/or 300I) which allow the sampling of a wide wavelength range (445-865 nm and 600-1100 nm respectively) with a resolution of ∼ 10Å. The width of the slit was chosen to match the seeing but, to reduce the contamination of the host galaxy background, never exceeded 1 arcsec. Depending on the magnitude of the SN candidate, the exposure time of spectroscopic observations ranges from 900 seconds to 3 hours.
As we will detail in Sect. 5 we could obtain direct spectroscopic observation only for a fraction (∼ 40%) of the SN candidates. In order to obtain the redshifts of the host galaxies of the remaining SN candidates, and to check for signs of the presence of Active Galactic Nuclei (AGN), we carried out a follow-up program using FORS2. The observing strategy, tuned to make the best use of non-optimal seeing and sky transparency conditions, proved to be very successful: we obtained 44 spectra of SN candidate host galaxies in a single ESO period.
In addition, a dedicated service program using WFI (12 hours) was designed to complete the photometric band coverage of the fields monitored during SN search.
Eventually we were able to secure B, V, R, I imaging for 11 fields and B, V, R for 5 fields, whose coordinates, galactic extinction and band coverage are shown in Table 1. Further data analysis has been restricted to these 16 fields.
The galaxy sample
In this section we describe the steps required to select our galaxy sample, namely the detection of galaxies in the monitored fields (Sect. 4.1), the selection criteria and the application of the photoz technique (Sect. 4.3). We also present a number of crosschecks with other observations and galaxy catalogues that were performed to validate our catalogues (Sect. 4.2 and Sect. 4.3).
Source catalogues
For optimal source detection in each field we obtained deep images in each band. First we selected all the images with the best seeing and sky transparency for each field and band. Then we selected, for each band, a seeing range of < 0.15 arcsec with the requirement that it contains the maximum number of images which hereafter are co-added.
All images were preliminarily processed using the IRAF 3 package MSCRED, specifically designed to handle mosaic CCD images (Valdes 1998). We followed the standard recipes for CCD data reduction applying bias subtraction, flat field correction, trimming and, for I band images, de-fringing. Astrometric calibration for each image was performed using as reference the USNO-A2 catalog (Monet et al. 1997) with the task msccmatch. The r.m.s. dispersion of the absolute astrometry is ∼ 0.3 arcsec.
The photometric calibration was obtained with reference to Landolt standard fields (Landolt 1992) observed on photometric nights. Images obtained under non-photometric conditions were scaled to the flux of calibrated images by matching the photometry of the common stars.
After a proper match of the flux scale, the selected images of a given field and band were stacked together using the SWARP 4 software (Bertin et al. 2002) with a third-order Lanczos kernel for the resampling, and a median algorithm for the co-adding procedure.
The limiting magnitudes of the co-added images reflect the very different total exposure times for the search and complementary bands: the 3-σ limit for point sources was reached at B lim ∼ 23 mag, V lim ∼ 24 mag, R lim ∼ 23 mag and I lim ∼ 22 mag.
In order to consistently measure source colours, we have to take into account that the co-added images of a given field show different point spread functions (PSF) in each band. To avoid colour offsets, the co-added images in each band were convolved with an appropriate Gaussian kernel, using the IRAF task gauss, to match the PSF of the co-added image with the worst seeing.
For source detection and photometry we used SExtractor (Bertin & Arnouts 1996) in dual image mode: source detection and classification was obtained from the original V band images, since this is the band with the best S /N ratio, while the magnitudes of the sources were measured on the convolved images using the same adaptive aperture as in the original V images. This procedure assures that the fluxes are measured in the same physical region of each source in all bands. We adopted a source detection threshold of 3σ above the background noise and a minimum of 12 connected pixels above the detection threshold. To measure magnitude we chose a Kron-like elliptical aperture (with 2.5 pixel Kron radius) adaptively scaled to the object dimensions (mag auto). The instrumental magnitudes were normalized to 1 sec exposure time and corrected for the instrument zero point, atmospheric and Galactic extinction. For our purposes the color term correction in the photometric calibration is not required and, therefore, has not been applied.
For each field a general catalogue was created including all detected sources, their magnitude in all bands and the relevant SExtractor parameters. Close to very bright stars the efficiency and reliability of source detection and photometry are significantly lower than the average. These regions were masked automatically using SExtractor parameters that describe the shape and the size of the bright stars and sources detected in these areas were deleted from the catalogue. Finally, visual inspection of the detected sources was performed on the images in order to remove spurious detections and to verify the detection uniformity across the effective area.
To separate stars from galaxies we used the SExtractor neural-network classifier fed with isophotal areas and the peak intensity of the source. The classifier returns a parameter named class star with a value between 1 (for a point-like object) and 0 (for an extended profile). Fig. 1 shows the class star parameter as function of magnitude for the sources detected in one of our co-added image. For bright objects stars are clearly divided by galaxies while for fainter magnitudes the sequences spread out and merge. An intrinsically wide scatter in class star is typical of faint galaxies.
We defined as galaxies all sources having class star < 0.9 (Arnouts et al. 2001). In principle this choice may result in an overestimate of the number of galaxies at faint magnitudes but ensures that our sample is not biased against compact galaxies. Notice however that because of the relatively bright cut-off limit adopted for the galaxy sample selection (see Fig. 1 and Sect. 4.3) our analysis is confined to a regime where the star/galaxy separation is reliable.
Catalog crosschecks
To check the accuracy of the astrometric and photometric calibration of our catalogs we performed a number of comparisons with the results of galaxy surveys and with model predictions.
In particular one of our fields, dubbed AXAF, partially overlaps with the Chandra Deep Field South which has bee observed also with WFI at the ESO2.2m telescope during the ESO Imaging Survey (EIS, Arnouts et al. 2001) and the COMBO-17 survey (Wolf et al. 2003). In the overlapping area we identified ∼ 1200 galaxies in common with the COMBO-17 R band source list. The comparison of the astrometric calibration between our catalog and COMBO-17 ( Fig. 2) shows a small scatter (r.m.s. ∼ 0.15 arcsec) both in right ascension and in declination. Also the comparison of the R band photometric calibration (Fig. 3) shows a small zero point offset (∼ 0.01mag) and a small dispersion (rms ∼ 0.03mag) for magnitudes brighter than the cut-off limit (R = 21.8 mag).
To verify the photometric calibration for all the other fields, we selected all bright (non saturated) stars, i.e. sources with class star > 0.9 and brighter than V = 19 mag, and compared them to the Landolt's stars on two colour-colour (B−V, V −R,V − R, R − I) diagrams. The magnitudes of the Landolt's stars were transformed onto the WFI filter system using the corresponding color terms as reported in the instrument web page 5 . We also checked the observed number counts as a function of colour with those predicted by the galactic star count model of Girardi et al. (2005) TRILEGAL 6 for the WFI filter's set. Performing these tests for all fields we were able to confirm the photometric calibration to better that ∼ 0.1mag in all bands.
Fig. 3.
Difference between the R magnitudes of galaxies in our source list for the AXAF field and in the COMBO-17 catalog as a function of the R Combo magnitude for magnitudes brighter than R Combo = 21.8.
Galaxy sample selection and characterization
Defining a photometric redshift catalogue involves fulfilling two requirements: i) to photometrically define a sample of galaxies for which reliable photometric redshifts can be obtained and ii) to characterize the redshift error distribution.
In order to estimate the galaxy redshifts we used the spectral energy distribution (SED) template fitting technique. In practice, a grid of predicted galaxy colours as a function of redshift is constructed performing synthetic photometry on different galaxy spectral templates, either empirical or produced by spectral synthesis modeling. Then, a χ 2 test of the observed vs. the predicted colors is used to select the best fit redshift and spectral template in the grid for each galaxy of the catalogue.
The SED fitting procedure is based on the fit of the overall shape of spectra and on the detection of strong spectral features, in particular, the 4000Å, the Balmer or Lyman continuum breaks. Obviously the accuracy of this technique improves with the spectral range coverage: with B to R band photometry reliable redshifts can be obtained only in the range 0.2 < z < 0.8, which may be extended to z ∼ 1 by adding the I band photometry. Redshifts for both very nearby galaxies, which need the U band photometry, and more distant galaxies, which require infrared (IR) filter coverage, are definitely more uncertain.
In order to reduce the contamination from distant galaxies, which may be erroneously placed at low or intermediate redshift, we set a cut-off limit for the galaxy catalogue at an apparent magnitude of R = 21.8 mag. This corresponds to M * B at redshift z = 0.8, where M * B is a parameter of the Schechter function which fits the galaxy luminosity function (Wolf et al. 2003). We stress that for computating SN rates, we do not need the galaxy catalogue to be complete within a given volume. Nevertheless we note that at redshift z ∼ 0.2 − 0.3, where the observed SN distribution peaks (see Sect.5.2), the contribution to the total luminosity from galaxies fainter than R = 21.8 mag is only 25%.
Photometric redshifts for the selected galaxies were computed using the code hyper-z 7 (Bolzonella et al. 2000) running in the predefined redshift range 0 < z < 0.8. As spectral templates we tested both the observed spectra of Coleman, Wu & Weedman (CWW) (Coleman, Wu & Weedman 1980) as well as the synthetic models based on Bruzual & Charlot (BC) library (Bruzual & Charlot 1993). Along with the redshift, the code provides the rest frame B-band absolute magnitude for the best fitting template. To further reduce the contamination from outliers we set a prior on the galaxy absolute luminosity. When M B is out of the range (−23, −16) mag we examine the second solution provided by hyper-z, and if this is also inconsistent with the luminosity prior, we remove the galaxy from the catalog, This happened to 16% of the galaxies in the sample.
Our final galaxy sample counts 43283 objects. The accuracy of the photometric redshifts is estimated by comparing them to spectroscopic redshifts, when available, and to the photometric redshifts obtained by the COMBO-17 survey, which has a better SED sampling and used a different photo-z code (Wolf et al. 2003).
The spectroscopic redshifts for 470 galaxies of our sample were retrieved from the Nasa/IPAC Extragalactic Database 8 , and complemented with 92 unpublished redshift measurements from the EDISC survey (Poggianti, private communication). We quantify the reliability of the photometric redshifts by measuring the fractional error, ∆z = (z ph − z sp )/(1 + z sp ) where z ph and z sp are the photometric and spectroscopic redshifts, respectively. The distribution of the fractional error, both in the case of CWW templates (Fig. 4) and in that of BC templates, does not show systematic off-sets: for both cases a ∆z = 0.01 and σ(∆) ∼ 0.12 appear consistent with our limited SED sampling. This accuracy is sufficient for our statistical analysis. In addition, the fraction of "catastrophic" outliers, galaxies with ∆z ≥ 3σ, is relatively low, ∼ 3%.
We also tested wether there are systematic trends between the photometric redshift reliability and redshift range and the galaxy spectral type and magnitude. While there is no evidence of systematic trends with the magnitude (because we have selected bright galaxies), the dispersion and the fraction of outliers increase significantly at z < 0.2, as expected, since we lack the U band coverage. The dispersion of ∆z also increases going from early to late type galaxies, due to the fact that the photo-z technique strongly relies on the strength of the Balmer break, which is weaker in later type galaxies.
Similar conclusions are derived from the comparison between photometric redshifts of galaxies in the catalogue of AXAF and the redshifts for the same galaxies measured in the COMBO-17 survey (Wolf et al. 2003) as shown in
The SN sample
In this section we describe how we searched for SNe in our galaxy sample and obtained their redshift distribution. We first illustrate the detection of variable sources and the selection of SN candidates (Sect. 5.1) then their confirmation and classification (Sect. 5.2).
SN candidate detection and selection
In order to achieve a rapid, automatic detection of SN candidates we developed the S T RES S package, a collection of IRAF tasks and freely distributed software. After the standard data reduction (see Sect. 4.1), the three jittered exposures of a given field are mapped to a common geometrical grid, properly scaled in intensity and stacked together. From this search, a suitable archive image (template) obtained at a different epoch is subtracted, and the difference image is searched for variable sources.
After the images have been geometrically registered and photometrically scaled, an accurate match of the PSF of the two images is obtained using the mrj phot task from the ISIS package (Alard 2000) which derives a convolution kernel comparing selected sources from the two images. The best results in the image subtraction are obtained when the two images have similar seeing. For this reason we maintained an on−line archive of template images with different seeing. We note however that all the selected SN candidates were still detected even when using a different template image.
Variable sources are detected as positive residuals in the difference image using SExtractor with a source detection threshold of 3σ above the background noise and a minimum of 10 connected pixels above the threshold. The choice of the detection threshold is a trade-off between detecting sources as faint as possible and limiting the number of spurious detections. Indeed, the list produced by SExtractor is heavily contaminated by spurious sources due to imperfect removal of bright stars, cosmic rays, hot or dead pixels. In the catalogue obtained from the difference image, false detections outnumber real variable sources by a factor of ∼ 100. In addition, besides the SN candidates the detection list contains other variable sources such as fast moving objects, variable stars, variable AGNs.
To clean up the detection list we used a custom−made ranking program which assigns a score to each source based on several parameters measured by SExtractor in the search, template and difference images: the class star, the full width at half maximum (FWHM), the distance of the residual with respect to the center of the associate galaxy (if any), and apparent magnitudes of the residual measured with different prescriptions. The scoring algorithm, tuned through extensive artificial star experiments, produces a sorted detection list. The final selection of trusted variable sources, about ten per image, was performed through visual inspection by an experienced observer.
In particular, the residuals of fast moving objects, which were not completely masked by our jittering strategy, have irregular shape in the stacked image and are easily recognized by comparison of the individual dithered images. Conservatively, we classified as variable stars those sources showing a stellar profiles both in the search and in the template images. However, it is impossible to distinguish AGNs from SNe that exploded near galaxy nuclei by examining images taken at one epoch only.
We labeled all variable sources closer than 0.5 arcsec to the host galaxy nucleus as SNAGN candidates and maintained them in the follow-up target list to reduce biases in the SN candidate selection as much as possible. The role of AGN contamination for those SNAGN candidates without spectroscopic observations will be discussed in Sect. 5.2. Finally we classified as SN candidates those variables with a stellar profile in the difference image, no matter if they appeared projected on a (host) galaxy in the search and in the template images, or not. However, in the latter case we excluded the candidate from our analysis, since we concentrate on SNe occurring in the galaxies within our specific sample.
To deal effectively with the large number of epochs and candidates, we developed a MySQL 9 database with a web interface (Riello et al. 2003) to easily access the information remotely (e.g. during observing runs). In particular the database stores: the monitored fields, pointing coordinates, finding charts, log of observations (with seeing, exposure time, photometric zero point), all discovered variable sources, their identification chart and relevant parameters (sky coordinates, epoch, magnitude, stellarity index, etc.).
At the end of our SN search programme we reviewed all variable sources recorded in our database to obtain a final classification based on all information gathered during our observing programme. A search engine allows us to identify multiple detections of the same source in different epochs and filters. This was used for the identification of AGNs which in general show long term, irregular variability. Indeed, the relatively long duration of our search makes this approach fairly efficient for AGN removal.
SN candidate confirmation and classification
Spectroscopic observations were planned for all SN and SNAGN candidates but could not be completed due to limited telescope time and, in some cases, poor weather conditions. In general a higher priority was given to candidates flagged as SN. Spectrum extraction and calibration were performed using standard IRAF packages. Special care was devoted to removal of background contamination through a detailed analysis of the intensity profile along the slit. For the nuclear candidates it is impossible to separate the variable source from the background using this information only. In such cases, first the sky emissions was subtracted, then the spectrum was extracted by integrating all the light along the slit. A best fit galaxy spectral template extracted from the STSci Database of UV-Optical Spectra of Nearby Quiescent and Active Galaxies 10 was then subtracted from the candidate spectrum.
The classification of SN candidate spectra was facilitated by the use of Passpartou, a software package that automatically classifies the candidate spectrum after comparison with all the SN spectra in the archive of the Padova SN group (Harutyunyan et al. 2005).
We could obtain direct spectroscopic observations only for 38 candidates (∼ 40% of the total): 31 were confirmed to be SNe whereas 7 turned out to be AGN. In this paper we use only the information on SN type and redshift of the host galaxy while in a future paper we will give more details on the spectroscopic data analysis.
The properties of the 31 confirmed SNe are summarized in Table 3: 25 have been observed in our galaxy sample (9 SNe Ia and 16 SNe CC) whereas 6 SNe discovered in galaxies with R > 21.8 mag were excluded from the following analysis. The SN redshifts range from z = 0.056 to z = 0.61.
As mentioned in Sect.3, we carried out a complementary observing programme to estimate redshift and spectral type of SN host galaxies and to better constrain contamination of AGNs in the SNAGN candidate sample. We remark that the higher priority given to SN candidates for immediate follow-up introduces a bias against SNAGN candidates. Such bias is minimized by this late follow-up which mostly targets the host galaxies of the SNAGN candidates. We found that, out of 42, 22 host galaxies showed an active nucleus. All the candidates occurring in these galaxies were removed from the candidate list. Actually we have 10 http://www.stsci.edu/ftp/catalogs/nearby gal/sed.html to notice that with our approach a SN occuring in a galaxy with an active nucleus will be discharged if only late spectroscopy is available. In Table 4 we report coordinates, redshifts, search filter and classifications of the remaining 20 SN candidates with host galaxy spectroscopy.
Even after this complementary analysis, 13 SN and 28 SNAGN candidates still remained without direct or host galaxy spectroscopy. The list of these remaining candidates is reported in Tab. 5.
For all these candidates the redshift of the host galaxy was estimated using the photo-z technique. To avoid a possible contamination by the SN candidate in measuring the host galaxy colours we produced deep images stacking only those frames obtained 6 -12 months earlier than the epoch of discovery of the candidate.
In summary the analysis presented here is based on 86 objects: 25 spectroscopically confirmed SNe, 33 SN candidates and 28 SNAGN candidates. The redshift distribution of candidates peaks at z ∼ 0.2 − 0.3 and extends up to z ∼ 0.6 as shown in Fig 5. Combining information from the long term variability history, direct and host galaxy spectroscopy we found that ∼ 50% of the variable sources originally classified SNAGN were actually AGN. Hereafter, in the analysis of SN statistics, we assigned a weight of 0.5 to all SNAGN candidates with no spectroscopic observations. The unclassified SN and SNAGN candidates were distributed among type Ia and CC based on the observed fractions in each redshift bin for the SN subsample with spectroscopic confirmation. It turns out that, with respect to the total number, SNe Ia are 27 ± 18%, 47 ± 21% and 63 ± 36% at redshift z = 0.1, 0.3, 0.5 respectively. We quantify the effect of misclassi- fication for the candidates without spectroscopic observations by performing Monte Carlo simulations as illustrated in Sect. 7.2.
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The control time of the search
The time interval during which a SN occurring in a given galaxy can be detected (the control time) depends mainly on the SN detection efficiency (Sect. 6.1), the SN light curve (Sect. 6.2) and the amount of dust extinction (Sect. 6.4). In this section we describe the different ingredients needed to calculate the control time of each single observation and the recipe to obtain the total control time of our galaxy sample (Sect. 6.5).
The search detection efficiency
The SN detection efficiency depends mainly on the survey strategy, instrumental set up and observing conditions. For a given In each simulation artificial point-like sources of different magnitudes were added to an image which was then searched for variable sources using the same software as in the actual search. The detection efficiency at a given magnitude is computed as the ratio between the number of recovered and injected artificial sources.
Artificial point-like sources were generated using the IRAF Daophot package with the PSF determined from a number of isolated stars on each chip of each image. The artificial stars were then placed on the galaxies (one source per galaxy) in a random position with respect to the center, but following the the luminosity profile. Detailed simulations for a few selected fields allowed us to probe the most relevant parameters affecting the detection efficiency: the search image characteristics (observing conditions and effective search area), the search process (choice of template image and selection criteria) and the SN position in the host galaxy. 6. The magnitude at which the detection efficiency is 50%, m 50% , in R band as function of seeing.
In particular we found that, for a given seeing, differences in the sky transparency do not affect the shape of the efficiency curve. The magnitude at which the efficiency is 50% (m 50% ) becomes fainter when the sky transparency improves, and a variation of photometric zero point translates directly into a change in m 50% .
After accounting for the differences in sky transparency, a strong dependence of the detection efficiency on seeing is still present (Fig. 6). We found that m 50% is ∼ 1 mag brighter when the seeing degrades from 0.7 arcsec to 1.2 arcsec, as shown in Fig. 6. If the seeing is worse than 1.3-1.4 arcsec then m 50% decreases more rapidly.
The detection efficiency for different fields is very similar except for minor variations due to different effective search areas caused by the presence different numbers of bright stars (which were masked). This is also the reason why the detection efficiency is always lower than 100%, even for bright sources.
In order to reduce the overhead of determining the detection efficiency for each individual observations of a given field we performed an extensive set of Monte Carlo simulations for only one field, covering a wide range of source magnitudes and seeing, thereby generating a set of template efficiency curves. Then, for each individual observation of the others fields, we performed only few simulations to determine the shift to apply to the appropriate template curve.
Possible spurious effects caused by the choice of the template image were tested by performing the same analysis based on three other images with different seeing. We found that, in the range 21 −23 mag, 95% of the sources are recovered in all cases, while, at fainter magnitudes, this fraction quickly decreases.
As we mentioned in Sect. 5.1, the final selection of SN candidates was based on visual inspection. We verified that the detection efficiency is independent of the observer that carried out the visual inspection by comparing the list of candidates produced by different observers in several simulations. Finally, we found that the detection efficiency depends very little on the position of Nugent et al. (2006) the source with respect to the galaxy and with galaxy brightness except for nuclear candidates in very bright galaxies.
This analysis of the detection efficiency was carried out both on R-and V-band images with very similar results after considering the zero-point differences.
SN light curves
The observed light curve in the band F of a given SN type at redshift z is given by:
m SN F (t, z) = M SN B (0) + ∆M SN B (t ′ ) + K SN BF (z, t ′ ) +A G F + A h B + µ(z)(1)
where:
-M SN B (0) is the SN absolute magnitude at maximum in the B band, -∆M SN B (t ′ ) describes the light curve relative to the maximum, -K SN BF (z, t ′ ) is the K-correction from the B to the F band, (V or R), -A G F and A h B
are the Galactic and host galaxy absorption respectively, µ(z) is the distance modulus for the redshift z in the adopted cosmology.
The light curve is translated into the observer frame by applying the time dilution effect to the rest frame SN phase t ′ = t/(1 + z). Four basic SN types were considered: Ia, IIL, IIP, Ib/c. We adopted the same templates for the SN B-band light curves as in Cappellaro et al. (1997) and account for the observed dispersion of SN absolute magnitudes at maximum light by assuming a normal distribution with the mean M B,0 and σ listed in Tab. 6. In this table we also report the average absorption < A B > measured in a sample of nearby SNe.
Finally, Galactic extinction is taken into account using the Schlegel et al. (1998) extinction maps, whereas the correction of the host galaxy extinction is estimated through a statistical approach, described in Sect. 6.4.
K correction
For a given filter, K-correction is the difference between the magnitude measured in the observer and in the source rest frame which depends on the redshift and SED of the source. In principle, if the source redshift is known it is possible to design a filter properly tuned to minimise the need for a K-correction. In most practical cases, however, sources are distributed in a wide range of redshifts and this approach is inapplicable.
A detailed discussion of the formalism and application of K-correction to the case of type Ia SNe can be found in Nugent et al. (2002). SN Ia have received a special care as their use as cosmological tools requires high precision photometry.
Actually, for computing the control time, we can tolerate much larger uncertainties. However, we need to obtain the K correction also for type Ib/c and type II SNe, which have not been published so far.
In general, there are three main problems for an accurate estimate of the K-correction of SNe: i) the SED rapidly evolves with time ii) SNe, in particular type Ib/c and type II, show large SED diversities, iii) the UV spectral coverage, which is needed to compute optical K-correction at high redshift, is available only for few SNe, and typically only near maximum light.
We addressed these issues trying to use all available informations on SN SED, in particular:
1. we selected a sample of spectral sequences of different SN types with good S/N retrieved from the Asiago-Padova SN archive. The sample includes the type Ia SNe 1990N, 1991T, 1991bg, 1992A, 1994D, 1999ee, 2002bo, the type II SNe 1979C, 1987A, 1992H, 1995G, 1995AD, 1996W, 1999el, 2002GD, 2003G and the type Ib/c SNe 1990I, 1997B, 1997X, 1998bw. 2. UV spectral coverage was secured retrieving SN spectra from the ULDA SN spectra archive (SN Ia 1990N phase -9, 1981B phase 0) of the International Ultraviolet Explorer (Cappellaro et al. 1995) and from the HST archive (SN Ia 1992A at phase +5, +45, SN Ib/c 1994I phase +7d, SN 1999em phase 5d) . These were combined with optical and IR spectra to obtain the SED in a wide range of wavelengths. 3. we included models for standard SNIa at phase -7, 0, +15 d provided by Mazzali (2000) and models of the SNII 1999em at phase 0,+9,+11,+25 d, courtesy of Baron et al. (2004).
The observed and model spectra were shifted at different redshifts and synthetic photometry in the required bands was obtained using the IRAF package synphot. Eventually, we derived the K SN BV and K SN BR corrections for each SN type as a function of light curve phase with a step of 0.05 in redshift. Examples of the measured K correction as a function of the SN phase are shown in on line Figs. 1-18.
Uncertainties due to variances of the SN spectra, uncertain extinction correction and/or incomplete temporal/spectral coverage can be minimized by choosing a proper filter combination. Indeed as it can be seen in on-line Fig. 14, for the average redshift of our search, z ∼ 0.3, K SN BV is almost independent on the SN type and phase.
Host galaxy extinction
The host galaxy extinction correction is the most uncertain ingredient in the estimate of the SN rate. In the local Universe Cappellaro et al. (1999) relied on an empirical correction as a function of the host galaxy morphological type and inclination. This approach cannot be applied to galaxies of our sample because the relevant data are not available. Hence we resorted to a statistical approach, based on the modelling of SN and extinction distribution in galaxies.
In short, following the method described in Riello & Patat (2005), we performed a number of Monte Carlo simulations where artificial SNe were generated with a predefined spatial distribution function and were seen from uniformly distributed lines of sight. Integrating the dust column density along the line of sight for each SN we derived the total optical depth and the appropriate extinction was applied to the SN template spectrum. Repeating a number of simulations we obtained the expected distribution of SN absorptions. With respect to Riello & Patat (2005), we included in the modelling also CC SNe. Such extension was straightforward: we added spectral templates for CC SNe, and adopted a reasonable spatial distribution for the simulated CC SNe. Since CC SNe are thought to originate from massive progenitors, the spatial distribution is assumed to be concentrated in dust-rich regions along the spiral arms.
The parameters for SN and dust spatial distributions used in the simulations are summarized in Tab. 7. The top rows (1-3) of the Table show the value of the parameters that describe the bulge component, that is the bulge to total luminosity ratio (B/T ), the scale and the truncation radius of SN distribution (r b and n respectively). The bottom half of the table (rows 4-10) lists the parameters for the disk component of both SN and dust, namely the scale-length and scale-height of the disc (r d ,z d ), the respective truncation radii (n,m) and the spiral arm perturbation parameters. The reader should refer to Riello & Patat (2005) for a more detailed description of N a ,w,p parameters.
The two other parameters governing the dust properties are the total to selective extinction ratio R V and the total optical depth along the galaxy rotation axis τ(0), which provides a convenient parameterization for the total amount of dust in the galaxy. For the former we adopted the canonical value R V = 3.1. For the latter we considered two scenarios: a standard extinction scenario with τ(0) = 1.0 for both SN types and a high extinction scenario with τ(0) = 5.0 only for CC SNe.
For each SN type we ran a set of simulations with 10 5 artificial SNe covering the redshift range z = 0.05 − 0.80 with a step in redshift of 0.05. In order to speed-up the computation, rather than considering a library of SN spectra at different phases as in Riello & Patat (2005), we used only a single spectrum close to maximum light for each SN type. This is a reasonable approximation since the evolution of the SN SED with time has a negligible effect on the absorption (at least for the SN phases that are relevant for our search).
Once a simulation for a given parameter set was completed, we derived the observed distribution of SN absorptions, and used it to compute the control time (cf. Sect. 6.5). For τ(0) = 1.0, the average absorption from the modeling is < A h B >= 0.5 mag for SNe Ia and < A h B >= 0.7 mag for CC SNe, after excluding SNe with absorption larger than A h B = 3 mag (about 8% of SNe Ia and 12% of CC SNe in this model, which are severely biased against in typical optical searches). We note that these average absorptions are consistent with the observed ones, reported in Tab. 6. For τ(0) = 5.0 the average model absorption is < A h B >= 0.8 mag for CC SNe with A h B < 3 mag (about 50% of the CC SNe) which is still consistent with the observed value in the local Universe.
Our correction for dust extinctions is meant to provide a reasonable estimate of this effect. A more realistic model should consider the different amount and distribution of the dust in each galaxy type. This is far beyond the scope of the current analysis.
Control time calculation
We now detail how all the ingredients described in the previous sections have been combined to compute the control time of our galaxy sample. First, for a given SN type and filter F, the control time of the i-observation of the j-galaxy was computed as:
CT SN,F j,i = τ SN,F j (m) ǫ F i (m) dm(2)
where τ SN,F j (m) is the time spent by the SN in the magnitude range m and m + dm and ǫ F i (m) is the detection efficiency. More specifically, we convolved the distribution of the absolute magnitude at maximum (M SN B (0)) and of the absorption due to host galaxy extinction (A h B ), so as to determine the distribution of the quantity M SN B (0) + A h B appearing in Eq. 1, and computed τ SN,F j (m) as a weighted average of the individual times over this combined distribution.
Then, the total control time CT SN,F j of the j-galaxy was computed by summing the contribution of individual observations. If the temporal interval elapsed since the previous observation is longer than the control time, that contribution is equal to the control time of the observation, otherwise it is equal to the interval of time between the two observations (Cappellaro et al. 1999).
The total control time of the galaxy sample is obtained as the B band luminosity weighted (CT SN,F j ) average of the individual galaxies.
Since we merged all CC subtypes, including type Ib/c IIP and IIL, the control time for CC SNe is computed as follows:
CT CC j = f Ib/c CT Ib/c j + f IIL CT IIL j + f IIP CT IIP j (3)
where the relative fractions of the different CC subtypes is assumed to be constant with redshift and equal to that observed in the local Universe, namely 20% of Ib/c and 80% of II (Cappellaro et al. 1999), out of which 35% are IIL and 65% are IIP events (Richardson et al. 2002).
In order to illustrate the role of the host galaxy extinction, we calculated the CC SN rate by adopting the control time both for a standard (τ(0) = 1.0), and for a high (τ(0) = 5.0) extinction scenario.
SN rates
In this section we describe our approach to estimate SN rate, present our results (Sect. 7.1) and discuss their uncertainties (Sect. 7.2). We also compare our measurements to those obtained in the local Universe (Sect. 7.3).
SN rate estimate
The SN rate at redshift z is given by the ratio between the number of observed SNe and the control time of the monitored galaxies at that redshift. Since our SN sample spans a wide redshift range (0.056 − 0.61), we can obtain an observational constraint on evolution of the rate by analysing the redshift distribution of the events.
In analogy with Pain et al. (2002), and following analysis , we adopt the following power law for the reshift dependence of the rate:
r SN,F (z) = r SN,F (z) 1 + z 1 + z α SN,F(4)
where r(z) is the rate at the mean redshift of the search, z, and α is the evolution index. Writing the rate evolution in terms of its value at the average redshift of the search reduces the correlation between the two free parameters in Eq. (4).
The reference redshift z of the search is computed as the weighted average of the galaxy redshifts with weights given by the respective control times:
z SN,F = n j=1 z j CT SN,F j n j=1 CT SN,F j(5)
We obtained:
z Ia = 0.30 +0.14 −0.14 -z CC = 0.21 +0.08 −0.09 Clearly, the lower z for CC SNe is due to their being, on average, intrinsically fainter than SNe Ia; hence their control time at a given redshift is shorter.
For a given SN type and filter F, the number of expected SNe is given by the expression:
N SN,F (z) = r SN,F (z) (1 + z) n j=1 CT SN,F j (z)(6)
where the sum is extended over the n galaxies at redshift z and the factor 1 + z corrects the rate to the rest frame.
We compared the observed SN distribution, binned in redshift, to the expected distribution performing a Maximum Likelihood Estimate to derive the best fit values of the parameters r(z) and α, independently for SNe Ia and CC SNe.
Our candidates come from V and R search programs, with no overlap. The rates at the reference redshift based on V and R candidates differ by less than 20% both for type Ia and CC SNe, which is well below the statistical error. Therefore, V and R SN candidates were combined togheter to improve the statistics.
For a standard extinction, the best fit values and statistical uncertainties for r(z) and α are: Fig. 19 shows the 1,2,3 σ confidence levels for the two parameters describing the evolution of SN rate, whereas Fig. 20 shows the best fit of the observed redshift distribution separately for SNe Ia, CC SNe and then all together.
The assumption of a particular host galaxy extinction scenario is one of the most important sources of uncertainty for SN rate measurements. We tested the dependence of the results on this effect by performing the fit under extreme assumptions: no extinction and a high extinction scenario, the latter only for CC Fig. 19. Confidence levels for the parameters describing the redshift evolution of the SN rates, from the maximum-likelihood test.
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SNe.The results are reported in Tab. 8. With respect to the standard case, the type Ia and CC SN rates decrese respectively by a factor of 1.7 and 2 when no extinction is adopted. The effect is smaller for the SN Ia rate because SNe Ia occur, on average, in environments with a smaller amount of dust. If a high extinction correction is adopted, the CC rate increases by a factor of 2: this can be regarded as a solid upper limit.
Systematic Uncertainties
When the size of the SN sample is large enough, the statistical uncertainty is relatively small, and it becomes important to obtain an accurate estimate of systematic uncertainties. These were evaluated via Monte Carlo simulations. We identified the following sources of systematic uncertainties, for which we specify the distribution adopted for the simulations:
-re-distribution of unclassified SNe in SN types (Gaussian distribution with σ from the statistical error, as discussed in -AGN contamination factor (Gaussian distribution with an average value of 0.5 and σ = 0.25), -photometric redshifts (the number counts in each bin of the galaxy redshift distribution are left to vary by ±50% with an uniform distribution), -detection efficiency (Gaussian distribution with σ = 0.1 mag), -SNe absolute magnitude (Gaussian distribution with σ = 0.1 mag), -standard extinction correction (uniform distribution with ±50% variance with respect to the standard case).
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The fit described in the previous section was performed for each simulation, to yield a distribution of solutions for the r(z) and α. The results are shown in Tab. 9 for each source of systematic uncertainty, as derived by varying only one parameter at the time. The total systematic error has instead been derived by varying all parameters simultaneously.
Not surprinsingly, the major uncertainty is due to the lack of the spectroscopic classification for a large fraction of the SN candidates. For the CC SN rate also the estimate of the detec- tion efficiency and the dust extinction correction are important sources of uncertainty. Even for our relatively small SN sample, the statistical and systematic uncertainties are comparable.
Since, due to a growing number of detected SNe, the statistical uncertainty will decrease in the future, the systematic errors will soon dominate the overall uncertainty. A special care should then be devoted to reduce the systematic effects, in particular securing spectroscopic follow up of all the candidates, and obtaining more precise information on the extinction of the galaxy sample.
Comparison with the local SN rates
In order to measure SN rate in a galaxy sample one needs to specify how the rate scales with a physical parameter proportional to the stellar content of each galaxy. In the early '70 Tamman (1970;1974) and afterwards Cappellaro et al. (1993) showed that the SN rates scale with the galaxy B band luminosity and from then on the SN rates, in the local Universe, have been measured in SNu. For a direct comparison of the rate at intermediate redshift to the local value , we also chose to normalize to the galaxy B band luminosity. The local SN rates measured by collecting data of five photographic SN searches (Cappellaro et al. 1999; and reference therein) are: r Ia = 0.17 ± 0.04 and r CC = 0.41 ± 0.17 h 2 70 SNu at z = 0.01. As it can be seen the SN Ia rate in SNu appears constant, within the uncertainties, up to z = 0.3, whereas the SN CC rate increases by a factor 2 already at z = 0.21. The different evolutionary behaviour of CC and Ia SN rates implies that their ratio increases by a factor of ∼ 2 from the local Universe to a look-back time of "only" 3 Gyr (z = 0.25):
-(r CC /r Ia ) (0.01) = 2.4 ± 1.1 -(r CC /r Ia ) (0.25) = 5.6 ± 3.5
Considering that, in this same redshift range, the cosmic SFR nearly doubles, the evolution with redshift of the ratio r CC /r Ia requires that a significant fraction of SN Ia progenitors has a delay time longer that 2 − 3 Gyr (cf. Sect. 8).
The interpretation on the evolution of the rate in SNu is not straightforward, as it reflects both the redshift dependence of both the SFH and the B-band luminosity. Indeed, the B-band luminosity, with contribution from both old and young stars, evolves with a different slope in comparison with the on-going SFR.
We acknowledge that the estimate of the SN rate evolution with redshift depends on the adopted extinction correction. For instance, the ratio between the rate at z=0.21 and the local rates for CC SNe varies from 1.6 to 2.8, when no or high extinction correction is applied. Anyway, the fact that the CC SN rate increases faster than the SN Ia rate appears a robust result.
SN rates and galaxy colors
Integrated colours are valuable indicators of the stellar population and SFR in galaxies and, for galaxies at high redshift, the colour information is easier to derive than the morphological type. Therefore it is interesting to investigate the dependence of the SN rates on the galaxy colours.
In the local Universe, SN rates as a function of galaxy colours were derived by Cappellaro et al. (1999) for optical bands and by Mannucci et al. (2005) for optical-infrared bands. The SN Ia rate per unit B luminosity appears almost constant in galaxies with different U − V color, whereas the CC SN rate strongly increases from red to blue galaxies, a trend very similar to that of the SFR. This indicates that a fair fraction of SN Ia progenitors has long delay times, whereas CC SNe are connected to young massive stars.
On the other hand, when the local SN rates are normalized to the K band luminosity, or galaxy mass, they show a rapid increase with decreasing B − K galaxy color for all the SN types (Mannucci et al. 2005). This result indicates that a sizeable fraction of SNe Ia has short delay times. A population of SN Ia progenitors with short delay time has been proposed also to explain the high SN Ia rate in radio-loud galaxies (Della , and Mannucci et al. (2006) suggest that this fraction ("prompt" component) amounts to ∼ 50%. In this respect we notice that evolutionary scenarios for SN Ia progenitors do indeed predict a distribution of the delay times ranging from a few tens of Myr to the Hubble time, or more, and that such distributions are typically skewed at the early delay times (Greggio 1983;. Given the diagnostic power of the trend of the SN rates with the colour of the parent galaxy we investigate here this trend at intermediate redshift.
In the last decade the extensive study of large samples of galaxies at different redshifts has shown the existence of a bimodal distribution of galaxy colors (Strateva et al. 2001;Weiner et al. 2005;Bell et al. 2004). In both apparent and rest frame color magnitude diagrams, galaxies tend to separate into a broad blue sequence and a narrow red sequence. While the red sequence is adequately reproduced by passive evolution models, the blue sequence hosts galaxies with different rate of ongoing star formation . Actually, a minor fraction of the red galaxies are also star forming (e.g. edge-on disks or starbusts), their colours being red colours because of high dust extinction. The fraction of star forming galaxies in the red population is 15−20% (Strateva et al. 2001) at low redshift, and increases to 30% at z ∼ 0.7 (Bell et al. 2004;Weiner et al. 2005). The bimodality is not restricted to colour alone, but extends to many other galaxy properties as luminosity, mass and environment, which are all well correlated with colours.
We split up our galaxy sample into blue and red sub-samples, according to the observed B − V color. We took the rest frame color of Sa CWW template as reference. The local galaxy sample of Cappellaro et al. (1999) was divided in the same way. U − V or B − R colours could be more sensitive tracers of the stellar populations, but U-band photometry is not available for our galaxy sample and R-band is not available for the local sample.
The SN rates in the blue and red galaxy samples were computed by distributing the unclassified SN and SNAGN candidates among type Ia and CC SNe, based on the observed fractions of the spectroscopically confirmed SNe in each of the two galaxy samples (in the red sample SNe Ia are 67%, whereas they are only 25% in the blue sample).
The SN rates (in SNu) in the local Universe and at redshift z = 0.25 for the red and blue galaxy subsamples are listed in Tab. 10.
We found very similar trends in the local Universe and at redshift z = 0.25 both for type Ia and CC SNe. It appears that while the SN Ia rate is almost costant in galaxies of different colors, the CC SN rate always peaks in blue galaxies. This result is consistent with that of Sullivan et al. (2006) who, after comparison with the measurements of Mannucci et al. (2005), found that the SN Ia rate as a function of galaxy colours does not evolve significantly with redshift.
The rapid increase with redshift of the CC SN rate both in blue and red galaxies can be attributed to an increasing proportion of star forming galaxies in the red sample going from low to intermediate redshift.
Discussion
In this section we discuss the evolution of the SN rates and investigate on the link between SF and SN rates. First we verify the consistency of our estimate of the redshift evolution of SN rates with other measurements from the literature (Sec. 8.1). Then we compare the observed evolution of the CC SN rate with those expected for different SFHs (Sec. 8.2). Finally, we convolve the SFH of Hopkins & Beacom (2006) with the delay time distribution (DTD) for different progenitor scenarios, and compare the results to the measurements of the SN Ia rate evolution (Sec. 8.3).
Comparison with other estimates
Published measurements of the SN rates at intermediate and high redshift are expressed in units of co-moving volume. To convert our rates from SNu to volumetric rates we assumed that the SN rates are proportional to the galaxy B band luminosity. If this assumption is true, by multiplyng the rates in SNu by the total B band luminosity density in a given volume we derived the total rates in that volume, even if we did not sample the faint end of the galaxy luminosity function. We accounted for the evolution of the B band luminosity density with redshift. A compilation of recent measurements of the B band luminosity density is plotted as function of redshift in Fig 21, where we also show a linear where both statistical and systematic errors are indicated. Also local rates are converted into volumetric rates: r Ia (z = 0.01) = 0.18 ± 0.05 10 −4 h 3 70 yr −1 Mpc −3 and r CC (z = 0.01) = 0.43 ± 0.17 10 −4 h 3 70 yr −1 Mpc −3 . With respect to the rates in SNu's, the volumetric rates evolve more rapidly with redshift, due to the increase of the B band luminosity density. We find an increase of a factor of ∼ 2 at z = 0.3 for SNe Ia, and a factor of ∼ 3 at z = 0.21 for CC SNe.
Measurements of Ia and CC SN rate as function of redshift are shown in Fig. 22, where those originally given in SNu (Hardin et al. 2000;Blanc et al. 2004;Cappellaro et al. 2005) were converted into measurements per unit volume as above.
As it can be seen, the few measurements of the CC SN rate appear to be fully consistent, while those of the SN Ia rate show a significant dispersion which increases with redshift, in particular in the range 0.5 < z < 0.7 where the values of Barris & Tonry (2006) and Dahlén et al. (2004) are 2-3 times higher that those of Pain et al. (2002) and Neill et al. (2007). Our estimate of the SN Ia rate is consistent with all other measurements in the redshift range we explored; our result does not help to discriminate between the steep trend suggested by the Barris & Tonry (2006) and Dahlén et al. (2004) measurements and the slow evolution indicated by the Neill et al. (2007) measurement. The robust indication from the current data appears that the SN Ia rate per unit volume at redshift 0.3 is a factor of ∼ 2 higher that in the local Universe, while in the same redshift range the CC SN rate increases by a factor of ∼ 5.
Fig. 22.
Observed SN rates as function of redshift from different authors as indicated in the legend. The black (gray) symbols indicate SN Ia (SN CC) rate measurements. The shaded area represents the 1 σ confidence level of our rate evolution estimate as deduced from the MLE fit.
Comparison with the predicted evolution of the CC SN rate
The stellar evolution theory predicts that all stars more massive than 8-10 M ⊙ complete the eso-energetic nuclear burnings, up to the development of an iron core that cannot be supported by any further nuclear fusion reactions or by electron degenerate pressure. The subsequent collapse of the iron core results into the formation of a compact object, a neutron star or a black hole, accompanied by the high-velocity ejection of a large fraction of the progenitor mass. TypeII SNe originate from the core collapse of stars that, at the time of explosion, still retain their H envelopes, whereas the progenitors of type Ib/Ic SNe are thought to be massive stars which have lost their H (and He) envelope (Heger et al. 2003). Given the short lifetime of their progenitors (< 30 Myr), there is a simple, direct relation between the CC SN and the current SF rate:
r CC (z) = K CC × ψ(z)(7)
where ψ(z) is the SFR and K CC is the number of CC SN progenitors from a 1 M ⊙ stellar population:
K CC = m CC u m CC l φ(m)dm m U m L mφ(m)dm(8)
where φ(m) is the IMF, m L − m U is the total stellar mass range, and m CC l − m CC u is the mass range of CC SN progenitors. In principle, if accurate measurements of the CC SN and SF rates are available, it is possible to probe the possible evolution with redshift of either the CC SN progenitor scenarios or the IMF by determining the value of K CC . Here however we assume that K CC does not evolve significantly in the redshift range of interest and compare the observed evolution of the CC SN rate with that predicted by the SFH.
The estimate of the cosmic SFH is based on different SF indicators, depending on redshift range, and requires a suitable parameterization and accurate normalization. Since there is a large scatter between the measurements obtained by different SFR indicators (a factor 2-3 at z = 0.3 − 0.5 and increasing with redshift) it is hard to obtain a consistent picture of the SFH (Hopkins & Beacom 2006). Indeed observations made at different wavelengths, from X-rays to radio, sample different facets Fig. 23. Comparison between the SN CC and SF rate evolution. Symbols are as in Fig. 22 with additional open symbols (measurements not corrected for extinction) and filled black symbols (estimates for the high extinction correction). Lines are selected SFR evolutions from the literature. All SFHs have been scaled to the SalpeterA IMF.
of the SF activity and are sensitive to different time scales over which the SFR is averaged. Thus, different assumptions are required to convert the observed luminosities at the various wavelengths to the SFR, and different systematic uncertainties affect the SFR estimates. In particular the significant difference between the SFR inferred from the UV and Hα luminosity and that inferred from the far-IR (FIR) luminosity may be related to the effect of dust extinction (expecially for the UV) and/or to the contribution to the light from old stars and AGNs (for the FIR).
To illustrate this point, we select three representative prescriptions for the SFH, namely:
-the piecewise linear fit of selected SFR measurements in the range 0 < z < 6 by Hopkins & Beacom (2006), -the fit to the SFR measurements from the Hα emission line, by Hippelein et al. (2003), with an exponential increase from z = 0 to z = 1.2, -the prescriptions by Hernquist & Springel (2003), i.e. a double exponential function that peaks at redshift z ∼ 5.5, obtained from an analytical model and hydro-dynamic simulations.
All these SFHs were converted to the same IMF, a modified Salpeter IMF (SalA) with a turnover below 0.5 M ⊙ and defined in the mass range m L = 0.1M ⊙ to m U = 120 M ⊙ (Baldry & Glazebrook 2003). We assumed a mass range of 8 − 50M ⊙ for CC SN progenitors, which gives a scale factor K CC = 0.009.
The measurements of the CC SN rate per unit volume and the predicted evolutionary behaviours are shown in Fig 23. The observations confirm the steep increase with redshift expected by the SFH from Hopkins & Beacom (2006) and Hippelein et al. (2003). For a look-back time of 3 Gyr (z = 0.25) both the SFR and the CC SN rate increase by a factor of ∼ 3 compared with the local values. A flat evolution, as that proposed by Hernquist & Springel (2003), appears inconsistent with the observed CC SN rates in the overall range of redshift.
With the adopted K CC , the level of the CC SN rate predicted by the SFH of Hippelein et al. (2003) and, in general, by the UV and Hα based SFHs fits well the data. Instead, for the SFH of Hopkins & Beacom (2006) and in general the SFHs inferred through FIR luminosity, the predicted CC SN rate is higher than observed over the entire redshift range (see also Dahlén et al. 2004;Hopkins & Beacom 2006;Mannucci et al. 2007).
If we correct the observed CC SN rate according to the high extinction scenario, we obtain an acceptable agreement between the data and the predictions with Hopkins & Beacom (2006) SFH, as shown in Fig. 23. However, this correction would require an extremely high dust content in galaxies which is not favored by present measurements. Indeed, Mannucci et al. (2007) derived an estimate of the fraction of SNe which are likely to be missed in optical SN searches because they occur in the nucleus of starburst galaxies or, in general, in regions of very high extinction and found that, at the average redshift of our search, the fraction of missing CC SNe is only ∼ 10%, far too small to fill the gap between observed and predicted rates.
Alternatively we may consider the possibility of a narrower range for the CC SN progenitor masses: in particular, a lower limit of 10 − 12 M ⊙ would bring the observed CC SN rates in agreement with those predicted from FIR based SFHs. In this respect we notice that, from a theoretical point of view, there is the possibility that a fraction of stars between 7-8 M ⊙ and 10-12 M ⊙ avoids the collapse of the core and ends up as ONeMg White Dwarfs (Ritossa et al. 1998;Poelarends et al. 2007). On the other hand, estimates of the progenitor mass from the detection in pre-explosion (HST) images has been possible for a few SNe IIP (e.g. SN 2003gd (Hendry et al. 2005), SN 2005cs (Pastorello et al. 2006)): their absolute magnitudes and colours seem indicate a moderate mass (8−12 M ⊙ ) (Van Dyk et al. 2003;Smartt et al. 2004;Maund et al. 2005;Li et al. 2006).
Given these controversies, we conclude that in order to constrain the mass range of CC SN progenitors it is necessary to reduce the uncertainties in the cosmic SFH. In addition it is important to apply a consistent dust extinction correction both to SFH and to CC SN rate.
Comparison with the predicted evolution of SN Ia rate
According to the standard scenario SNe Ia originate from the thermonuclear explosion of a Carbon and Oxygen White Dwarf (C-O WD) in a binary system. In the first phase of the evolution the primary component, a star less massive than 8M ⊙ , evolves into a C-O WD. When the secondary expands and fills its Roche Lobe, two different paths are possible, depending on whether a common envelope forms around the two stars (double degenerate scenario, DD) or not (single degenerate scenario, SD). In the SD scenario, the WD remains confined within its Roche Lobe, grows in mass until it reaches the Chandrasekhar limit and explodes, while in the DD scenario the binary system evolves into a close double WD system, that merges after orbital shrinking due to the emission of gravitational wave radiation.
In both scenarios two basic ingredients are required to model the evolution of the SN Ia rate: the fraction of the binary systems that end up in a SN Ia, and the distribution of the time elapsed from star formation to explosion (delay-time). In the SD scenario, the delay time is the evolutionary lifetime of the secondary; in the DD, the gravitational radiation timescale has to be added. In both cases, the distribution of the delay times depends on the distribution of the binary parameters (Greggio 2005). In principle, the SD and DD scenarios correspond to different realization probabilities and different shape of the delay time distribution (DTD) functions, hence rather different evolutionary behavior of the SN Ia rate. As mentioned previously, the observations indicate that the distribution of the delay times of SN Ia progenitors is rather wide.
The cosmic SFH is the other critical ingredient that modulates the evolution of the SN Ia rate. Following Greggio (2005) the SN Ia rate is given by:
r Ia (t) = k α A Ia min(t,τ x ) τ i f Ia (τ)ψ(t − τ)dτ(9)
where k α is the number of stars per unit mass of the stellar generation, A Ia is the realization probability of the SN Ia scenario (the number fraction of stars from each stellar generation that end up as SN Ia), f Ia (τ) is the distribution function of the delay times and ψ(t −τ) is the star formation rate at the epoch t −τ. The integration is extended over all values of the delay time τ in the range τ i and min(t, τ x ), with τ i and τ x being the minimum and maximum possible delay times for a given progenitor scenario.
Here we assumed that both k α and A Ia do not vary with cosmic time.
A detailed analysis of the predicted evolution of the SN Ia rate for different SFHs and DTDs is presented elsewhere (Forster et al. 2006;Blanc & Greggio 2007). Here we consider only one SFH, the piecewise interpolation of Hopkins & Beacom (2006), since this is conveniently defined also at high redshift, and limit our analysis to few DTDs representative of different approaches to model the SN Ia rate evolution: three models from Greggio (2005), and two different parametrizations (Mannucci et al. 2006;Strolger et al. 2004) designed to address some specific observational constraints, regardless the correspondence to a specific progenitor scenario.
Specifically, among the Greggio (2005) models we select one SD and two DD-models, one of the "close" and the other of the "wide" variety, the latter being an example of a relatively flat DTD. This choice is meant to represent the full range of plausible DTDs. The minimum delay time for the SD model is the nuclear lifetime of the most massive secondary stars in the SN Ia progenitor's system, i.e. 8M ⊙ . In principle, for the DD model the minimum delay time could be appreciably larger than this because of the additional gravitational waves radiation delay. In practice, also for the DDs the minimum delay is of a few 10 7 yrs. The maximum delay time is quite sensitive to the model for the SN Ia progenitors. The reader should refer to Greggio (2005) for a more detailed description of these models.
The DTD parametrization proposed by Mannucci et al. (2006) is the sum of two distinct functions: a Gaussian centered at 5×10 7 yr, representative of a "prompt" progenitor population which traces the more recent SFR, and an exponentially declining function with characteristic time of 3 Gyr, a "tardy" progenitor population proportional to the total stellar mass. This DTD was introduced to explain, at the same time, the dependence of the SN Ia rate per unit mass on the galaxy morphological type, the cosmic evolution of the SN Ia rate and, in particular, the high SN Ia rate observed in radio-loud galaxies.
Finally, Strolger et al. (2004) showed that the best fit of the apparent decline of SN Ia rate at z > 1 is achieved for a DTD with a Gaussian distribution centered at about 3 Gyr. Nevertheless this DTD fails to reproduce the dependence of Greggio (2005) for SD and DD models, Mannucci et al. (2006) and Strolger et al. (2004). Fig. 25. SN Ia rate measurements fitted with different DTD functions and the SFH by Hopkins & Beacom (2006). Symbols for measurements are as in Fig. 22. the SN rate on galaxy colours which is observed in the local Universe (Mannucci et al. 2006).
The selected DTD functions are plotted in Fig. 24 while the predicted evolutionary behaviours of the SN Ia rate are compared with all published measurements in Fig. 25. In all cases, the value of k α A Ia was fixed to match the value of the local rate; depending on the model it ranges between 3.4-7.6×10 −4 . This normalization implies that, for the adopted SalA IMF, and assuming a mass range for the progenitors of 3 − 8 M ⊙ , the probability that a star with suitable mass becomes a SN Ia, is ∼ 0.01-0.03.
The models obtained with the different DTDs are all consistent with the observations with the exception of the "wide" DD model, whose redshift evolution is definitely too flat. On the other hand none of the DTD functions, with the adopted SFH, is able to reproduce at the same time the very rapid increase from redshift 0 to 0.5 suggested by some measurements (Barris & Tonry 2006;Dahlén et al. 2004) and the decline at redshift > 1 ( Dahlén et al. 2004). We note that a new measurement of Poznanski et al. (2007) suggests that the SN Ia rate decline at high redshift may be not as steep as estimated by Dahlén et al. (2004).
Given the current uncertainties of both SN Ia rate and SFH it is difficult to discriminate between the different DTD functions and hence between the different SN Ia progenitor models. To improve on this point, more measurements of SN Ia rate at high redshifts are required to better trace the rate evolution. At the same time measurements in star forming and in passive evolving galaxies in a wide redshift range can provide important evidence about the SN Ia progenitor models. In addition, it is essential to estimate the cosmic SFH more accurately because the position of the peak of the SFH was found to be the crucial parameter for the recovered delay time (Forster et al. 2006).
Summary
In this paper we have presented our SN search (STRESS) carried out with ESO telescopes and aimed at measuring the rate for different SN types at intermediate redshift.
Our approach, that consists in counting the SNe observed in a well defined galaxy sample, allows us to investigate the dependence of the rates on galaxy colours, and to perform a direct comparison with measurements in the local Universe, which are obtained with the same approach.
For the selection and the characterization of the galaxy sample we used multi-band observations and the photometric redshift technique. After 16 observing runs, 12 in V, 4 in R band, we discovered a few hundred variable sources from which we selected 86 SN candidates: 25 spectroscopically confirmed SNe (9 SNe Ia and 16 CC SNe), 33 SN candidates and 28 SNAGN candidates.
The control time of our galaxy sample has been evaluated through a detailed analysis of the SN detection efficiency of our search, with a number of MonteCarlo simulations. In addition, we have improved the handling of the correction for dust extinction by elaborating realistic modelling. The rates are presented for two scenarios, one stardard and one which, maximizing the effect, yield reasonable upper limits.
Since our SN sample spans a wide redshift range (0.05 < z < 0.6) we have obtained an observational constraint on the redshift evolution of the rates, described as a power law with two parameters: r(z), the rate at the mean redshift of the search z, and α, an evolution index. For SNIa and CC SNe we get: where both statistical and systematic uncertainties are reported.
Our results indicate that, compared to the local value, the CC SN rate per unit B band luminosity increases by a factor of ∼ 2 already at z ∼ 0.2, whereas the SN Ia rate is almost constant up to redshift z ∼ 0.3. The dependence of the SN rates per unit B band luminosity on the galaxy colour is the same as observed in the local Universe: the SN Ia rate seems to be almost constant from red to blue galaxies, whereas the CC SN rate seems to peak in the blue galaxies. Therefore, on the one hand, the observed evolution with redshift of the ratio r CC /r Ia requires a significant fraction of SN Ia progenitors with long delay times; on the other hand, the observed trend with galaxy colour requires a short delay time for a fraction of SN Ia progenitors.
Systematic uncertainties in our SN rate measurements have been investigated by considering the influence of the different sources. In particular we have analysed the important role played by the AGN contamination of SN candidate sample and the host galaxy extinction.
After accounting for the evolution with redshift of the blue luminosity density, we have compared our estimates with all other measurements at intermediate redshift available in the literature and found that they are consistent, within the relatively large errors.
Finally we have exploited the link between SFH and SN rates to predict the evolutionary behaviour of the SN rates and compare it with the path indicated by observations.
The predicted evolution of the CC SN rate with redshift has been computed assuming three representative SFHs and the mass range 8-50 M ⊙ for CC SN progenitors. The comparison with the observed evolution confirms the steep increase with redshift indicated by most recent SFH estimates. Specifically, we found a good agreement with the predictions from the SFH inferred through Hα luminosity (Hippelein et al. 2003), while the SFHs from FIR luminosity (Hopkins & Beacom 2006) overestimate the CC SN rate, unless a higher extinction correction and/or a narrower range for the progenitor masses is adopted. This result illustrates how interesting clues can be obtained by comparing the SN CC rate to other SFR tracers, in the same galaxy sample, to verify the reliability of the techniques used to derive SFR estimates and the adequacy of the dust extinction corrections.
The cosmic evolution of the SN Ia rate has been estimated by convolving the SFH of Hopkins & Beacom (2006) with various formulations of DTD related to different SN Ia progenitor models. All DTDs appear to predict a SN Ia rate evolution consistent with the observations, with the exception of the "wide" DD model, which appears too flat. At the same time none of the explored DTD functions, at least with the adopted SFH and k α A Ia factor are able to reproduce simultaneously both the rapid increase from redshift 0 to 0.5 and the decline at redshift > 1 suggested by some measurements.
With the current data of the rate evolution, it is difficult to discriminate between different DTDs and then between different SN Ia progenitor models. Measurements of the SN Ia rate in star forming and in passive evolving galaxies in a wide range of redshifts can provide more significant evidence about SN Ia progenitors.
An extensive survey to search all SN types in a well characterized sample of galaxies is of the highest priority to probe the link between the SN rates and the SFR in different contexts. This will yield important constraints on SN progenitor scenarios from a detailed analysis of SN rates as a function of the various properties of galaxies.
Online Material
Botticella et al.: STRESS, Online Material p 2
Fig. 1 .
1The stellarity index of the sources detected in one of our co-added image as function of magnitude. The vertical line indicates the cut-off magnitude for our galaxy sample.
Fig. 2 .
2Differences in the equatorial coordinates between our source list for the AXAF field and the COMBO-17 catalog for the objects in common.
Fig 4.
Fig. 4 .
4Distribution of the fractional error ∆z = (z ph −z)/(1 +z) in the comparison of our estimates z ph with spectroscopic redshift (thin line) and with photometric redshit of COMBO-17 (thick line).
Fig. 5 .
5The redshift distribution of SN candidates with (Ia and CC) and without spectroscopic confirmation (SN and SNAGN).
Fig.
Fig. 6. The magnitude at which the detection efficiency is 50%, m 50% , in R band as function of seeing.
Fig. 20 .
20Observed (points with statistical error bars) and expected (solid line) redshift distribution of SN events.
Fig. 21 .
21Measurements of the galaxy luminosity density at different redshifts. The DEEP2 and COMBO-17 data are taken fromTable 2inFaber et al. (2005). The line represents the linear least-square fit in the redshift interval z = 0 − 1.least-square fit to the data in the range z = 0 − 1: j B (z) = (1.03 + 1.76 × z) 10 8 L B ⊙ Mpc −3 . Multiplying our measurements by the value of j B at the average redshifts of the Ia and CC SN samples, the rates per unit of co-moving volume result: -r Ia (z = 0.30) = 0.34 +0.16+0.21 −0.15−0.22 10 −4 h 3 70 yr −1 Mpc −3 -r CC (z = 0.21) = 1.15 +0.43+0.42 −0.33−0.36 10 −4 h 3 70 yr −1 Mpc −3
Fig. 24 .
24Delay time distributions as derived by
Table 1 .
1List of the fields with multi-band coverage.Name
R.A. (J2000.0) Dec. (J2000.0) E(B-V) # Band
03Z3
03:39:31.1
−00:07:13
0.081
B, V, R, I
10Z2
10:46:45.8
−00:10:03
0.040
B, V, R
13Z3
13:44:28.3
−00:07:47
0.026
B, V, R
22Z3
22:05:05.8
−18:34:33
0.026
B, V, R
AXAF
03:32:23.7
−27:55:52
0.008
B, V, R, I
EF0427
04:29:10.7
−36:18:11
0.022
B, V, R, I
EisA1
22:43:25.6
−40:09:11
0.014
B, V, R
EisB
00:45:25.5
−29:36:47
0.018
B, V, R
EisC
05:38:19.8
−23:50:11
0.030
B, V, R, I
EisD
09:51:31.4
−20:59:25
0.040
B, V, R, I
Field2
19:12:51.9
−64:16:31
0.037
B, V, R
J1888
00:57:35.4
−27:39:16
0.021
B, V, R
white23
13:52:56.4
−11:37:03
0.067
B, V, R, I
white27
14:10:59.7
−11:47:35
0.057
B, V, R, I
white31
14:20:15.3
−12:35:43
0.096
B, V, R, I
whitehz2
13:54:06.1
−12:30:03
0.073
B, V, R, I
# Galactic extinction from
Table 3 .
3Spectroscopically confirmed SNe Not included in the rate computation because the host galaxy has R > 21.8.Design.
R.A.(J2000.0) Dec.(J2000.0) type
z
field
band
reference
1999ey #
00:58:03.43
−27:40:31.1
IIn
0.093
J1888
V
IAUC 7310
1999gt
03:32:11.57
−28:06:16.2
Ia
0.274
AXAF
V
IAUC 7346
1999gu
03:33:00.20
−27:51:42.7
II
0.147
AXAF
V
IAUC 7346
2000fc
00:58:33.55
−27:46:40.1
Ia
0.420
J1888
V
IAUC 7537
2000fp
05:38:01.27
−23:46:34.1
II
0.300
EisC
V
IAUC 7549
2001bd
19:13:10.94
−64:17:07.8
II
0.096
Field2
V
IAUC 7615
2001gf
22:04:21.27
−18:21:56.8
Ia
0.200
22Z3
V
IAUC 7762
2001gg
00:45:20.83
−29:45:12.2
II
0.610
EisB
V
IAUC 7762
2001gh
00:57:03.63
−27:42:32.9
II
0.160
J1888
V
IAUC 7762
2001gi #
04:28:07.06
−36:21:45.2
Ia
0.200 EF0427
V
IAUC 7762
2001gj
05:38:07.90
−23:42:34.4
II
0.270
EisC
V
IAUC 7762
2001io
00:45:53.64
−29:26:11.7
Ia
0.190
EisB
V
IAUC 7780
2001ip
03:31:13.03
−27:50:55.5
Ia
0.540
AXAF
V
IAUC 7780
2002cl
13:44:09.94
−00:12:57.8
Ic
0.072
13Z3
V
IAUC 7885
2002cm
13:52:03.67
−11:43:08.5
II
0.087
WH23
V
IAUC 7885
2002cn
13:53:21.24
−12:15:29.5
Ia
0.302 WHhz2
V
IAUC 7885
2002co #
14:10.53.04
−11:45:25.0
II
0.318
WH27
V
IAUC 7885
2002du
13:53:18.28
−11:37:28.8
II
0.210
WH23
V
IAUC 7929
2004ae
04:28:17.89
−36:18:55.0
II
0.480 EF0427
R
IAUC 8296
2004af
05:38:03.91
−23:59:00.2
Ic
0.056
EisC
R
IAUC 8296
2004ag
09:51:01.49
−20:50:37.5
II
0.362
EisD
R
IAUC 8296
2004ah
10:45:47.41
−00:06:58.1
Ia
0.480
10Z2
R
IAUC 8296
2004ai
13:54:26.09
−12:41:15.9
Ic
0.590 WHhz2
R
IAUC 8296
2004aj #
14:20:37.58
−12:24:14.4
Ia
0.247
WH31
R
IAUC 8296
2004cd #
13:44:50.70
−00:03:48.3
Ia
0.500
13Z3
R
IAUC 8352
2004ce
13:54:45.93
−12:14:36.3
Ia
0.465 WHhz2
R
IAUC 8352
2004cf
14:11:05.77
−11:44:09.4
Ib/c 0.247
WH27
R
IAUC 8352
2004cg
14:11:23.71
−12:01:08.4
II
0.264
WH27
R
IAUC 8352
2004dl
14:11:11.50
−12:02:36.9
Ia
0.250
WH27
R
IAUC 8377
2004dm
22:43:21.36
−40:19:46.7
Ib
0.225
EisA1
R
IAUC 8377
2005cq #
09:52:00.48
−20:43:26.5
Ia
0.310
EisD
R
IAUC 8551
# -
Table 4 .
4SN candidates with host galaxy spectraDesign.
R.A.(J2000.0) Dec.(J2000.0)
z
band
03Z3 B
03:39:27.76
−00:01:08.8
0.240
V
03Z3 F
03:40:36.80
−00:06:17.8
0.247
V
03Z3 H
03:40:09.43
−00:10:20.6
0.282
V
03Z3 J
03:40:37.11
−00:05:22.5
0.180
V
10Z2 B
10:45:42.76
00:00:27.9
0.295
V
10Z2 D
10:47:40.94
−00:13:52.6
0.350
V
EF0427 G
04:30:21.93
−36:06:49.9
0.359
V
EF0427 I
04:30:10.73
−36:25:47.4
0.420
V
EF0427 J
04:29:20.47
−36:32:16.7
0.185
V
EF0427 O
04:28:10.56
−36:13:54.4
0.450
V
EF0427 Q
04:28:17.09
−36:09:54.9
0.394
V
EisA1 B
22:44:01.94
−40:03:21.7
0.219
V
EisA1 C
22:44:25.26
−40:20:01.9
0.215
V
EisA1 J
22:44:02.82
−40:02:05.2
0.180
V
EisA1 Q
22:42:37.87
−39:57:54.4
0.380
V
EisA1 R
22:42:13.27
−40:03:19.9
0.530
V
EisC A
05:38:45.67
−23:44:41.0
0.643
V
EisC C
05:38:06.55
−23:36:57.1
0.133
V
EisC G
05:37:52.11
−23:38:14.4
0.156
V
WH27 B
14:11:02.77
−11:49:06.2
0.485
V
WHhz2 A
13:53:58.73
−12:43:04.8
0.444
V
WHhz2 B
13:54:25.27
−12:35:42.9
0.322
V
Table 5 .
5Candidates without direct or host galaxy spectroscopy instrumental set up and exposure time, the detection efficiency curve as a function of the SN candidate apparent magnitude was estimated via Monte Carlo simulations.Design.
R.A.(J2000.0) Dec.(J2000.0)
z
band class.
03Z3 Q
03:38:26.39
−00:04:53.1
0.30
V
snagn
10Z2 A
00:47:06.51
00:00:39.6
0.52
R
snagn
13Z3 R
13:44:01.94
00:00:03.9
0.35
R
snagn
13Z3 X
13:45:22.55
00:03:11.2
0.18
R
snagn
13Z3 AB
13:44:31.26
00:00:57.5
41
R
sn
13Z3 AC
13:45:11.26
−00:12:27.4
0.43
R
snagn
22Z3 I
22:06:03.26
−18:28:10.8
0.42
R
sn
22Z3 J
22:04:42.12
−18:36:38.3
0.19
R
sn
22Z3 K
22:04:35.87
−18:45:45.0
0.42
R
snagn
22Z3 N
22:04:49.66
−18:45:27.3
0.14
R
snagn
22Z3 R
22:04:03.87
−18:35:55.2
0.22
R
sn
AXAF B
03:32:31.16
−28:04:43.8
0.13
V
sn
AXAF D
03:31:28.98
−28:10:26.1
0.42
V
snagn
AXAF F
03:33:05.29
−27:54:09.2
0.17
V
snagn
AXAF G
03:32:39.23
−27:42:57.5
0.47
R
sn
AXAF N
03:32:47.68
−28:07:41.1
0.18
R
sn
EF0427 B
04:29:07.28
−36:28:01.8
0.50
V
sn
EF0427 F
04:29:49.31
−36:33:55.2
0.24
V
snagn
EF0427 M
04:27:57.15
−36:30:52.9
0.74
V
snagn
EisA1 E
22:43:44.86
−40:05:50.6
0.17
V
snagn
EisA1 I
22:44:48.37
−40:03:42.3
0.17
V
snagn
EisA1 O
22:42:01.73
−40:21:17.8
0.25
V
snagn
EisA1 AC
22:44:21.82
−40:08:21.4
0.52
R
sn
EisB B
00:45:09.35
−29:28:33.2
0.13
V
snagn
EisB P
00:44:14.57
−29:39:08.1
0.17
V
snagn
EisB S
00:45:51.29
−29:52:10.4
0.43
V
snagn
EisC H
05:39:28.83
−23:38:05.7
0.36
V
sn
EisC J
05:38:01.65
−23:51:40.7
0.20
V
snagn
EisC K
05:38:28.78
−23:46:29.4
0.80
V
snagn
EisC O
05:39:28.51
−24:00:05.4
0.80
V
snagn
EisC Q
05:37:50.49
−23:36:55.2
0.14
R
sn
EisD F
09:51:12.26
−20:56:14.0
0.16
R
snagn
Field2 C
19:11:16.62
−64:17:32.3
0.16
V
snagn
Field2 E
19:10:35.43
−64:21:36.0
0.28
R
sn
J1888 J
00:58:41.35
−27:50:38.1
0.30
V
sn
J1888 M
00:57:05.34
−27:45:57.7
0.63
V
snagn
WH23 A
13:53:18.79
−11:40:29.7
0.16
V
snagn
WH23 E
13:52:41.23
−11:51:24.7
0.21
R
snagn
WH27 N
14:11:47.81
−11:35:25.6
0.44
R
snagn
WHhz2 Q
13:54:31.46
−12:18:22.5
0.67
R
snagn
WHhz2 T
13:55:08.90
−12:18:38.3
0.23
R
snagn
Table 6 .
6Adopted B band, absolute magnitude at maximum for different SN types.SN type
M B,0
σ B,0 < A B >
Ref.
Ia
−19.37 0.47
0.4
Wang et al. (2006)
Ib/c
−18.07 0.95
0.7
Richardson et al. (2006)
IIP
−16.98 1.00
0.7 #
Richardson et al. (2002)
IIL
−18.17 0.53
0.7 #
"
"
# -
Table 7 .
7Parameters of our modelling of the host extinction for the four SN types considered in the simulations. The last column of the table provides the reference fromRiello & Patat (2005) where the meaning of each parameter is explained in detail.Parameter
Ia
Ib/c
IIP
Dust Ref.
B/T
0.5
0
0
-
§ 2.1
r b
1.8
-
-
-
Eq. (1)
n
8
-
-
-
§ 2.1.1
r d
4.0
4.0
4.0
4.0
Eq. (2)
z d
0.35 0.35 0.35 0.14 Eq. (2)
n
6
6
6
6
§ 2.1.2
m
6
6
6
6
§ 2.1.2
N a
0
2
2
2
Eq. (3)
w
-
0.2
0.2
0.4
Eq. (3)
p
-
20
20
20
Eq. (3)
Table 8 .
8The parameters of the rate evolution estimated for different extinction scenarios.SN type
z
Extinction r(z) [h 2
70 SNu]
α
Ia
0.30
none
0.13 +0.06
−0.04
2.2 +3.6
−3.8
standard
0.22 +0.10
−0.08
4.4 +3.6
−4.0
CC
0.21
none
0.42 +0.14
−0.12
5.4 +2.8
−3.4
standard
0.82 +0.31
−0.24
7.5 +2.8
−3.3
high
1.66 +0.32
−0.05
7.8 +2.8
−3.2
Table 9 .
9Summary of systematic uncertaintiesSNIa
SNCC
Source item
σ r
σ α
σ r
σ α
SN types distribution +0.08 +3.4
+0.18 +3.4
−0.08 −3.4
−0.20 −3.5
AGN fraction
+0.02 +0.2
+0.05 +0.2
−0.01 −0.2
−0.05 −0.1
Photometric redshifts +0.04 +0.7
+0.13 +0.6
−0.03 −0.7
−0.11 −0.8
Detection efficiency
+0.02 +0.6
+0.21 +0.6
−0.03 −0.6
−0.11 −0.8
Standard extinction
+0.07 +1.9
+0.26 +0.7
−0.05 −1.4
−0.22 −0.9
SN absolute mag
+0.02 +0.7
+0.06 +0.4
−0.02 −0.6
−0.06 −0.4
All systematic errors
+0.16 +3.3
+0.30 +3.6
−0.14 −3.5
−0.26 −3.7
Statistical errors
+0.10 +3.6
+0.31 +2.8
−0.08 −4.0
−0.24 −3.2
Table 10 .
10SN rates as a function of galaxy color [SNu h 2 ]SN type
r(z = 0.01)
r(z = 0.25)
red
blue
red
blue
Ia
0.20 ± 0.04 0.19 ± 0.04 0.16 +0.13
−0.18
0.23 +0.13
−0.09
CC
0.10 ± 0.04 0.99 ± 0.15 0.57 +0.48
−0.30
1.50 +0.66
−0.48
All
0.30 ± 0.06 1.18 ± 0.16 0.73 +0.50
−0.31
1.73 +0.67
−0.45
SN Ia: r(z = 0.3) = 0.22 +0.10+0.16−0.08−0.14 h 2
70 SNu, α = 4.4 +3.6+3.3
−4.0−3.5
CC SN: r(z = 0.2) = 0.82 +0.31+0.30
−0.24−0.26 h 2
70 SNu, α = 7.5 +2.8+3.6
−3.2−3.7
Table 2 .
2Observing log of the SN search fields. Botticella et al.: STRESS, Online Material p 3run
Field
Filter Nexp Texp Seeing
1999-02-23
10Z2
B
3
1200
1.12
13Z3
B
3
1200
1.10
1999-03-10
10Z2
B
3
1200
1.14
13Z3
B
3
1200
1.15
10Z2
V
3
1200
1.25
13Z3
V
3
1200
1.15
1999-03-19
10Z2
B
3
1200
1.04
13Z3
B
3
1200
0.96
10Z2
V
3
1200
0.86
13Z3
V
3
1200
0.84
1999-05-08
10Z2
B
3
1200
1.34
13Z3
B
3
1200
1.4
Field2
B
3
1200
1.91
Field2
R
3
1200
1.02
10Z2
V
3
1200
1.37
Field2
V
3
1200
1.62
1999-05-17
10Z2
B
3
1200
0.81
13Z3
B
3
1200
1.74
Field2
B
3
1200
0.74
10Z2
R
3
1200
0.96
13Z3
R
3
1200
0.91
10Z2
V
3
1200
0.90
13Z3
V
3
1200
0.98
Field2
V
3
1200
0.81
1999-08-03
13Z3
B
3
1200
1.43
22Z3
B
3
1200
1.17
Field2
B
3
1200
1.12
J1888
B
3
720
1.22
J1888
R
3
720
1.31
Field2
V
3
1200
1.8
J1888
V
3
720
1.41
1999-08-14
22Z3
B
2
1200
1.58
EisA1
B
3
1200
1.67
Field2
B
3
1200
2.15
1999-08-31
22Z3
B
3
1200
1.43
Field2
B
3
1200
2.44
1999-09-13
J1888
B
5
720
1.62
J1888
R
5
720
1.24
22Z3
V
3
1200
1.46
EisA1
V
3
1200
1.32
EisB
V
3
1200
1.27
Field2
V
3
1200
1.67
J1888
V
5
720
1.41
1999-10-30
AXAF
B
3
900
3.00
EisA1
B
3
900
1.32
EisB
B
3
900
2.02
EisA1
V
3
900
1.50
EisB
V
3
900
1.52
1999-11-09
03Z3
B
3
900
0.98
AXAF
B
3
900
1.13
EisA1
B
3
900
1.12
J1888
B
3
900
1.05
03Z3
V
3
900
1.07
AXAF
V
3
900
1.08
EisA1
V
3
900
0.84
EisB
V
3
900
0.76
J1888
V
3
900
0.81
1999-12-02
AXAF
B
3
900
1.03
AXAF
V
3
900
1.12
EisB
V
3
900
1.27
Table 2 .
2continued. Botticella et al.: STRESS, Online Material p 4run
Field
Filter Nexp Texp Seeing
EisC
V
3
900
1.15
EisD
V
3
900
1.2
J1888
V
3
900
1.06
1999-12-10
03Z3
B
3
900
1.6
03Z3
V
3
900
1.62
10Z2
V
3
900
1.28
EisA1
V
3
900
1.75
EisC
V
3
900
1.31
J1888
V
3
900
1.65
1999-12-28
EisC
B
3
900
1.18
03Z3
V
3
1200
2.18
AXAF
V
3
900
1.15
EisC
V
3
900
0.77
EisD
V
3
900
0.96
J1888
V
3
900
1.10
2000-01-08
EisC
B
3
900
1.06
EisD
B
3
900
0.91
EisC
V
3
900
0.85
EisD
V
3
900
0.90
10Z2
V
3
900
0.92
2000-11-16
03Z3
V
3
900
1.14
AXAF
V
3
600
0.92
EF0427
V
3
900
0.74
EisA1
V
3
900
0.92
EisB
V
3
900
0.86
EisC
V
3
900
0.64
EisD
V
3
900
0.83
J1888
V
3
900
1.05
2000-12-17
03Z3
V
3
600
1.14
AXAF
V
3
600
0.93
EF0427
V
3
900
0.93
EisA1
V
3
900
1.12
EisB
V
3
900
1.26
EisC
V
3
900
0.85
J1888
V
3
660
1.02
2001-04-18
10Z2
V
3
900
0.98
13Z3
V
3
900
0.85
22Z3
V
3
900
1.41
EisC
V
3
900
0.85
EisD
V
3
900
0.86
Field2
V
3
900
0.84
white23
V
3
900
0.82
2001-04-19 white27
R
3
1200
0.71
whitehz2
R
3
1200
0.67
white27
V
3
900
0.73
white31
V
3
900
1.01
whitehz2
V
3
900
0.79
2001-11-11
03Z3
V
3
900
0.90
22Z3
V
3
900
0.84
EF0427
V
3
900
0.75
EisA1
V
3
900
0.82
EisB
V
3
900
0.78
EisC
V
3
900
0.91
J1888
V
3
900
0.75
2001-11-12
22Z3
R
3
900
1.31
AXAF
R
3
900
0.92
EisA1
R
3
900
1.55
EisB
R
3
900
1.07
J1888
R
3
900
1.24
AXAF
V
3
900
1.03
2001-11-18
03Z3
V
3
900
0.90
Table 2 .
2continued. Botticella et al.: STRESS, Online Material p 5run
Field
Filter Nexp Texp Seeing
22Z3
V
3
900
1.10
AXAF
V
3
900
0.84
EF0427
V
3
900
0.93
EisA1
V
3
900
1.20
EisB
V
3
900
0.83
EisC
V
3
900
0.65
J1888
V
3
900
1.02
2001-12-08
03Z3
V
3
900
0.94
AXAF
V
3
900
0.95
EF0427
V
3
900
0.95
EisB
V
3
900
0.88
EisC
V
3
900
0.95
J1888
V
3
900
0.94
2001-12-09
22Z3
R
1
900
2.00
AXAF
R
3
900
0.82
EF0427
R
3
900
1.1
EisB
R
3
900
0.89
EisC
R
3
600
1.05
J1888
R
3
900
0.84
22Z3
V
1
900
1.90
EisD
V
3
900
0.92
2002-04-07
EisC
V
3
900
1.32
EisD
V
3
900
0.92
10Z2
V
3
900
0.91
13Z3
V
3
900
0.78
white27
V
3
900
0.80
white31
V
3
900
0.85
Field2
V
3
900
1.30
2002-04-08
EisD
R
3
900
0.85
10Z2
R
3
900
0.88
13Z3
R
3
900
1.1
white31
R
3
900
0.87
Field2
R
3
900
1.31
white23
V
3
900
0.96
whitehz2
V
3
900
0.91
2004-02-18 EF0427
R
3
900
0.85
EisC
R
3
900
0.91
EisD
R
3
900
0.93
white27
R
3
900
0.82
white31
R
3
900
0.94
white31
R
3
900
0.91
whitehz2
R
3
900
0.83
10Z2
R
3
900
0.87
2004-02-19 EF0427
I
3
900
1.05
EisC
I
3
900
0.80
EisD
I
3
900
1.03
10Z2
I
3
900
0.73
whitehz2
I
3
900
0.76
white27
I
3
900
0.67
white31
I
3
900
0.75
2004-05-14
13Z3
R
3
900
0.78
EisD
R
3
900
1.11
22Z3
R
3
900
0.86
white31
R
3
900
1.18
whitehz2
R
3
900
0.78
white27
R
3
900
0.81
2004-05-15
EisD
B
3
900
1.57
white31
B
3
900
1.38
whitehz2
B
3
900
1.38
white27
B
3
900
1.55
2004-07-14 white27
R
3
900
1.01
Table 2 .
2continued.run
Field
Filter Nexp Texp Seeing
white31
R
3
900
1.12
22Z3
R
3
900
0.94
whitehz2
R
3
900
0.86
EisA1
R
3
900
0.90
EisB
R
3
900
1.02
2005-05-13 white23
R
3
900
1.27
white27
R
3
900
1.48
white31
R
3
900
1.55
whitehz2
R
3
900
1.18
13Z3
R
3
900
1.29
EisD
R
3
900
1.56
10Z2
R
3
900
1.39
Field2
R
3
900
1.32
S Nu = S N (100yr) −1 (10 10 L B ⊙ ) −1
IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. 4 www.terapix.iap.fr
http://www.ls.eso.org/lasilla/sciops/2p2/E2p2M/WFI/zeropoints 6 http://trilegal.ster.kuleuven.be/cgi-bin/trilegal
www.web.ast.obs-mip.fr 8 http://nedwww.ipac.caltech.edu/
MySQL is an open source database released under the GNU General Public License (GPL), http://www.mysql.com/.
Acknowledgements. MTB acknowledges Amedeo Tornambé for useful discussions and the European Southern Observatory for hospitality during the development of this work. This work has been supported by the Ministero dell'Istruzione, dell'Universitá e della Ricerca (MIUR-PRIN 2004-029938,2006 and by the National Science Foundation under Grant No. PHY05-51164. G.P acknowledges support by the Proyecto FONDECYT 3070034. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.Fig. 16. Same asFig.1for the K correction from the observed R band to the rest frame B band for redshift 0.1.Fig. 18. K(BR) correction for redshift 0.6.
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| []
|
[
"Improved Method for Individualization of Head-Related Transfer Functions on Horizontal Plane Using Reduced Number of Anthropometric Measurements",
"Improved Method for Individualization of Head-Related Transfer Functions on Horizontal Plane Using Reduced Number of Anthropometric Measurements"
]
| [
"Wahidin Hugeng ",
"Dadang Wahab ",
"Gunawan "
]
| []
| [
"JOURNAL OF TELECOMMUNICATIONS"
]
| An important problem to be solved in modeling head-related impulse responses (HRIRs) is how to individualize HRIRs so that they are suitable for a listener. We modeled the entire magnitude head-related transfer functions (HRTFs), in frequency domain, for sound sources on horizontal plane of 37 subjects using principal components analysis (PCA). The individual magnitude HRTFs could be modeled adequately well by a linear combination of only ten orthonormal basis functions. The goal of this research was to establish multiple linear regression (MLR) between weights of basis functions obtained from PCA and fewer anthropometric measurements in order to individualize a given listener's HRTFs with his or her own anthropomety. We proposed here an improved individualization method based on MLR of weights of basis functions by utilizing 8 chosen out of 27 anthropometric measurements. Our objective experiments' results show a superior performance than that of our previous work on individualizing minimum phase HRIRs and also better than similar research. The proposed individualization method shows that the individualized magnitude HRTFs could approximated well the the original ones with small error. Moving sound employing the reconstructed HRIRs could be perceived as if it was moving around the horizontal plane. | null | [
"https://arxiv.org/pdf/1005.5137v1.pdf"
]
| 11,169,112 | 1005.5137 | 261dccf726f61238ffeac2f5f16da8d56e837195 |
Improved Method for Individualization of Head-Related Transfer Functions on Horizontal Plane Using Reduced Number of Anthropometric Measurements
MAY 2010
Wahidin Hugeng
Dadang Wahab
Gunawan
Improved Method for Individualization of Head-Related Transfer Functions on Horizontal Plane Using Reduced Number of Anthropometric Measurements
JOURNAL OF TELECOMMUNICATIONS
231MAY 2010Index Terms-HRIRHRTF IndividualizationPrincipal Components AnalysisMultiple Linear Regression
An important problem to be solved in modeling head-related impulse responses (HRIRs) is how to individualize HRIRs so that they are suitable for a listener. We modeled the entire magnitude head-related transfer functions (HRTFs), in frequency domain, for sound sources on horizontal plane of 37 subjects using principal components analysis (PCA). The individual magnitude HRTFs could be modeled adequately well by a linear combination of only ten orthonormal basis functions. The goal of this research was to establish multiple linear regression (MLR) between weights of basis functions obtained from PCA and fewer anthropometric measurements in order to individualize a given listener's HRTFs with his or her own anthropomety. We proposed here an improved individualization method based on MLR of weights of basis functions by utilizing 8 chosen out of 27 anthropometric measurements. Our objective experiments' results show a superior performance than that of our previous work on individualizing minimum phase HRIRs and also better than similar research. The proposed individualization method shows that the individualized magnitude HRTFs could approximated well the the original ones with small error. Moving sound employing the reconstructed HRIRs could be perceived as if it was moving around the horizontal plane.
INTRODUCTION
ithout two eyes, direction of sound source can be recognized by a person by utilizing his or her two ears. The primary cues in localizing the direction of a sound are interaural time difference (ITD), interaural level difference (ILD), and spectral modification caused by pinna, head, and torso. These primary sound cues are encrypted in HRTF. On the horizontal plane, ITD and ILD are two main cues due to the perception of sound direction [1]. HRTF is defined as the acoustic filter of human auditory system, in frequency domain, from a sound source to the entrance of ear canal. The counterpart of HRTF in time domain is known as head-related impulse response (HRIR). One key implementation of binaural HRTFs is in the creation of Virtual Auditory Display (VAD) in virtual reality to filter monaural sound. This fact is based on the human psychoacoustic characteristic, i.e. a convincing spatial sound can be obtained sufficiently using two channels. As suggested by [3] and [4], HRTF changes with directions of sound sources and varies from subject to subject due to inter-individual difference in anthropometric measurements.
Synthesis of ideal VAD systems needs a series of empirical measurements of individual HRTFs for every listener. These measurements are not practical because of the requirements of heavy and expensive equipments as well as a long measurement time. Most commercial virtual auditory systems are recently synthesized using generic/nonindividualized HRTFs that ignore inter-subject difference. However, non-individualized HRTFs suffer from distortions such as in-head localization when using headphones, inaccurate lateralization, poor vertical effects, and weak front-back distinction caused by unsuitable HRTFs applied to a listener [1], [4]. Thus, it is needed and a priority to develop an individualization method to estimate proper HRIRs for a listener, that present adequate sound cues without measurement of the individual HRIRs.
The individualization of HRTF in frequency domain or HRIR in time domain is nowadays a challenging subject of much research. Several HRTF individualization methods have been developed, such as HRTF clustering and selection of a few most representative ones [5], HRTF scaling in frequency [6], a structural model of composition and decomposition of HRTFs [7], HRTF database matching [8], the boundary element method [9], HRIR subjective customization of pinna responses [10] and of pinna, head, and torso responses [11] in the median plane, and HRTF personalization based on multiple regression analysis (MRA) in the horizontal plane [12]. Shin and Park [10] suggested HRIR customization method based on subjective tuning of W only pinna responses (0.2 ms out of entire HRIR) in the median plane using PCA of the CIPIC HRTF Database [2]. They achieved the customized pinna responses by letting a subject tune the weight on each basis function. Hwang and Park [11] follow the similar method as [10], but they fed PCA with the entire median HRIRs; each HRIR is 1.5 ms long (67 samples) since the arrival of direct pulse. This HRIR includes the pinna, head, and torso responses. They tuned subjectively the weights of three dominant basis functions due to the three largest standard deviations at each elevation. Hu et al. [12] personalized the estimated log-magnitude responses of HRTFs by MRA. At the beginning, the log-magnitude responses are estimated using PCA as linear combination of weighted basis functions. The weights of the basis functions are then estimated using anthropometric measurements based on MRA.
Our individualization method is similar to the method in [12], but we employed in the PCA modeling, the magnitude responses of HRTFs, instead of the log-magnitude responses of HRTFs utilized by Hu et al., however, our selection procedure of anthropometric measurements is also different. Entire horizontal magnitude HRTFs calculated from the original HRIRs in the CIPIC HRTF Database are included in a single analysis. Thus, all horizontal magnitude HRTFs for both ears share the same set of basis functions, which cover not only the inter-individual variation but also the inter-azimuth variation. This paper presents an individualization method by developing the statistical PCA model of magnitude HRTFs and MLR between weights of basis functions and selected few anthropometric measurements, that was different and showed improved performance from [12]. Section 2 describes the proposed algorithm of individualization method, database used, minimum phase analysis, PCA of magnitude HRTFs, minimum phase reconstruction and synthesis of HRIR models, individualization of magnitude HRTFs using MLR, and correlation analyses for the selection process of independent variables and dependent variables of MLR models. Section 3 discusses experiments' results, which consist of discussions of resulted basis functions and weights of basis functions from PCA, and the performance of the proposed individualization method.
PROPOSED INDIVIDUALIZATION METHOD
The goal of our research is to develop an improved individualization method of HRTFs on the horizontal plane, by using multiple regression models between magnitude HRTFs and a few anthropometric measurements. This method individualizes magnitude HRTF models into suitable HRIRs for a given listener, by using a few of his or her own anthropometric measurements. The suitable individualized HRIRs are necessary when the listener uses a spatial audio application.
The schematic diagram of the proposed HRTFs individualization method is shown in Fig. 1. The database of HRIRs used in the research was provided by CIPIC Interface Laboratory of California University at Davis [2], [3]. This database is reviewed briefly in the subsection below. Firstly, as seen in Fig. 1, we obtained from the database the entire original HRIRs on horizontal plane of 37 subjects, which consists of 50 HRIRs of each ear and each subject. We used a total number of 3700 HRIRs in modeling and individualizing HRIRs of a listener. Each HRIR was processed by 256-points fast Fourier transform (FFT) to transform it into its corresponding complex HRTF.
As the object of HRTF modeling using PCA, we took only 128 frequency components of magnitude of the complex HRTF. At this step, the phase of the complex HRTF was discarded. Then, we computed the mean of the entire magnitude HRTFs. This mean was substracted from each magnitude HRTF to obtain its corresponding direct transfer function (DTF). This substraction was performed in order to have centered data of magnitude HRTF, called DTF, which was necessary for PCA to get a good result.
For HRTFs modeling purpose, all DTFs were thus fed into PCA. The PCA delivered 128 ordered basis functions or principal components (PCs) and their weights (PCWs). The PCs were ordered from the PC with largest eigen value to the PC with smallest eigen value. It must be kept in mind that each eigen value determined the percentage variance of all DTFs explained by its corresponding PC. The first PC that corresponds to the largest eigen value explained largest percentage variance of the entire DTFs. To attain later individualized HRTFs of a new listener, we performed multiple linear regression (MLR) between the PCWs resulted from PCA and a few anthropometric measurements of 37 subjects in the database. Detailed selection process of anthropometric measurements from a total of 27 measurements is explained in the separated subsection below. The MLR method provided regression coefficients that correlated the PCWs and selected anthropometric measurements. These regression coefficients were thus applied to a set of anthropometric measurements of a new listener to obtain estimated PCWs for that listener. A linear combination of weighted PCs using these estimated PCWs resulted in an individualized DTF.
The desired individualized HRIRs of a listener were attained using the reconstruction process shown by the dashed lines in Fig. 1. Each individualized DTF that was achieved from the MLR method and PCA, was added to the mean of DTFs calculated before to yield its individualized magnitude HRTF. Minimum-phase was inserted to the individualized magnitude HRTF to result in an individualized complex HRTF. Here we followed the assumption that the phase of the HRTF can be approximated by minimum-phase [13]. The inverse Fourier transform finally was applied to obtain individualized HRIRs from the corresponding complex HRTFs. The initial left-and right-ear time delay due to the distance from the sound source in a particular direction to each ear drum were inserted respectively to the left-ear HRIR and to the right-ear HRIR.
The database used, the minimum phase analysis, reconstruction, PCA of the magnitude HRTFs in the frequency domain, minimum phase reconstruction and synthesis of HRIRs, MLR method, and selection of anthropometric measurements are explained in the following subsections.
The Database Used
Most commercial VAD systems convolve input signals with a pair of standard HRIRs, which ordinarily come from a serial of studies that used public HRIR data of acoustic manikin called Knowles Electronics Manikin for Auditory Research (KEMAR). HRIRs vary significantly among individuals, hence a database which results from sufficiently large number of HRIRs measurements is needed in order to perform HRIRs modeling. CIPIC Interface Laboratory at California University, Davis -USA, had measured HRIRs with a high spatial resolution from more than 90 subjects [2], [3]. They has released CIPIC HRTF Database Release 1.2, which is a subset of database for only 45 subjects. This database is downloadable from their website and can be used freely for academic research purpose. CIPIC HRTF Database not only consists of impulse responses from 1250 spatial directions for each ear and each subject, but also includes a set of anthropometric measurements of all subjects. We used the CIPIC HRTF Database in our research because of its extent features.
The number of subjects involved in the HRIRs measurements is 43 people, consists of 27 males and 16 females. Two other subjects are KEMAR with small pinnae and KEMAR with large pinnae. All impulse responses were measured with condition that the subject sat in the center of a circle with radius 1 meter. The position of head was not fixed, but the subject could monitor his or her head position.
As sound sources, Bose Acoustimass TM loudspeakers, with cone diameter of 5.8 cm, were mounted at different positions on the half-circled hoop. Golay-code signals were generated by a modified Snapshot TM system from Crystal River Engineering. Each ear canal was blocked and ‚Etymotic Research ER-7C' probe microphones were used to pick up the Golay-code signals. Output of a microphone was sampled with frequency 44,100 Hz, 16 bit resolution, and processed by Snapshot's oneshot function to produce a raw HRIR. A modified Hanning window was applied on the raw HRIR to eliminate room reflections and then the result was free-field compensated to improve the spectral charateristics of the transducers used. The length of each HRIR is 200 samples with duration of about 4.5 ms.
Direction of a sound source was determined by azimuth angle, θ, and elevation angle, ø, in interaural- Although these measurements are not very accurate, but they allow investigation about possible correspondence or correlation among physical dimensions and HRTF characteristics. Following the approach suggested by Genuit [5], there are 27 anthropometric measurements in the database, which consists of 17 measurements of head and torso, and 10 measurements of pinna as shown in Fig. 2 [2], [3]. Generally, histogram of subjects' measurements indicates a normal distribution of values. Discarding the offset measurements x4, x5, x13, where the percentages of deviation can be ignored, mean of percentages of deviation is ±26%. Thus, there are a sufficient number of variations in the measurements and sizes of subjects in the database used.
Minimum Phase Analysis
Each HRIR in the dababase used was measured with a distance of one meter from the sound source to the center of subject's head. From the graph of HRIR versus time, it is observed a time delay due to the distance mentioned before, which is needed by sound wave to propagate from its source to the ear drum, before a maximum amplitude of HRIR occurs. To eliminate this time delay, HRIR can be reconstructed into a minimum-phase HRIR using Hilbert transform. In the minimum-phase HRIR, the phase is allowed to be arbitrary or else it is set in such a way that the magnitude response of HRIR is made easier to achieve. A linear time invariant filter, H(z) = B(z)/A(z), is said to have minimum phase if all of its poles and zeros are inside the unit circle, |z|=1, in the zplane. Equivalently, a filter H(z) has minimum phase if not only itself but also its inverse, 1/H(z), are stable. A minimum phase filter is also causal since noncausal terms in the transfer function correspond to poles at infinity. The simplest example of minimum phase filter would be the unit-sample advance, H(z) = z, which consists of a zero at z = 0 and a pole at z = oo. A filter is called to have minimum phase if both the numerator and denominator of its transfer function are minimum phase polynomials in z -1 , i.e. a polynomial of the form,
B(z) = b0 + b1 z -1 + b2 z -2 + . . . + bM z -M = b0(1-θ1z -1 )(1-θ2z -1 )...(1-θMz -1 )(1)
is said to have minimum phase if all of its roots, θi, i=1,2,...,M, lie inside the unit circle, i.e. |θi |<1. A general property of minimum phase impulse responses is that among all impulse responses, hi(n), having identical magnitude spectra, impulse responses with minimum phases experience the fastest decay in the sense that,
∑ ∑ = = ≥ K n K n i mp n h n h 0 0 2 2 ) ( ) ( , n=0, 1, 2, ..., K,(2)
where hmp(n) is a minimum phase impulse response.
The equation above represents that the energy in the first K + 1 samples of the minimum-phase case is at least as large as any other causal impulse response having the same magnitude spectrum. Thus, minimum-phase impulse responses are maximally concentrated toward time t=0 among the space of causal impulse responses for a given magnitude spectrum. Because of this property, minimum-phase impulse responses are sometimes called minimum-delay impulse responses. It is known that in a minimum phase filter, H(z) = e a(z) e i b(z) , the relations, b(z) = -H {a(z)} and a(z) = -H {b(z)}, are also valid, where H {} is the Hilbert transform. The logarithmic change of these relations was obtained mainly through the calcultion of real cepstrum.
It is proposed by Kulkarni et al. [13], that the phase of HRIR can be approximated by minimum phase. A minimum phase system function, H(z), of an HRIR, h(n), has all poles and all zeros that are placed inside the unit circle |z| =1 in the z-plane. The calculation of real cepstrum of an original HRIR, which has arbitrary phase, results in a minimum phase HRIR, hmp(n). We can say that the minimum phase HRIR is the removed initial time delay version of the correspond original HRIR. But both kinds of HRIR have the same magnitude spectrum in the frequency domain. The real cepstrum, v(n), of HRIR, h(n), is calculated as follow,
v(n) = Re{F 1 − D {ln|FD{h(n) }|}},(3)
where ln and Re{} denote respectively natural logarithm and the real part of a complex variable, FD{} and F 1 − D {} are the discrete Fourier transform and its inverse respectively. This real cepstrum is then weighted by the following window function,
0 if n < 0, w(n) = 1 if n = 0,(4)
2 if n > 0.
In case of a rational H(z), the window function can be seen as a complex conjugate inversion of the zeros outside the unit circle, so that a minimum phase HRIR is provided. Hence the desired minimum phase HRIR, hmp(n), is resulted from:
hmp(n) = Re{exp(FD{w(n).v(n)})}.(5)
PCA of Magnitude HRTFs in Frequency Domain
Complex HRTFs were attained by implementing fast Fourier transform (FFT) to HRIRs of the database used. The entire complex HRTFs were computed from left-ear and right-ear HRIRs of 37 subjects on horizontal plane. There are 50 HRIRs from different directions (50 azimuths) on horizontal plane for each ear of a subject, so that a total of 3700 complex HRTFs were produced by 256-points FFT. We took only magnitudes of all complex HRTFs as the input of PCA modeling. Only 128 first frequency components of a magnitude HRTF were taken into analysis because of the symmetry property of a magnitude spectrum. A matrix composed of DTFs is needed by PCA. The original data matrix, H (NxM), is composed of magnitudes of HRTFs on horizontal plane, in which, each column vector, hi (i=1,2,…,M), represents a magnitude HRTF of an ear of a subject in a direction on horizontal plane. The number of magnitude HRTFs of each subject on horizontal plane is 100 (2 ears x 50 azimuths). Hence, the size of H is 128 x 3700 (N=128, M=3700). The empirical mean vector (µ: Nx1) of all magnitude HRTFs is given by,
µ = (1/M) ∑ = M i 1 hi.(6)
The DTFs matrix, D, is the mean-subtracted matrix and is given by,
D = H -µ.y,(7)
where y is a 1xM row vector of all 1's. The next step is to compute a covariance matrix, S, that is given by
S = D.D*/ (M-1)(8)
where * indicates the conjugate transpose operator. The basis functions or PCs, vi (i=1,2,…,q), are the q eigenvectors of the covariance matrix, S, corresponding to q largest eigenvalues. If q = N, then the DTFs can be fully reconstructed by a linear combination of the N PCs. However, q is set smaller than N because the goal of PCA is to reduce the dimension of dataset. An estimate of the original dataset is obtained here by only 10 PCs, which account for 93.93% variance in the original data D. By using only 10 PCs to model magnitude HRTFs, we expected to obtain satisfactory good results. The PCs matrix, V = [v1 v2 … vN], that consisted of complete set of PCs can be obtained by solving the following eigen equation,
S V = Λ V(9)
where Λ = diag{ λ 1 ,…,λ 128 }, is a diagonal matrix formed by 128 eigen values, where each eigen value, λ i , represents sample variance of DTFs that was projected onto i-th eigen vektor or PC, vi.
Then, the weights of PCs (PCWs), W(10x3700), that correspond to all DTFs, D, can be obtained as,
W = V*.D,(10)
where PCs matrix now was reduced to V = [v1 v2 … v10].
PCWs represent the contribution of each PC to a DTF. They contain both the spatial features and the interindividual difference of DTF. Thus, the matrix consisted of models of magnitude HRTFs, Ĥ, is given by,
Ĥ = V.W + µ.y.(11)
Tabel 1 shows the percentage variance and the cummulative percentage variance of DTFs in the database explained by PC-1 to PC-20 (v1, v2, … , v20) respectively. The application of more PCs would reduce the modeling error between the magnitude HRTF of database and the model of magnitude HRTF, but on the other hand, it costed more computing time and larger memory space. The PCsmatrix, V, that at first has 128x128 elements was reduced into a matrix of only 128x10 elements. We used only the first 10 PCs out of all 128 PCs. In this way automatically we needed only 10 PCWs to perform the model. Hence, one can see obviously the advantage of PCA in reducing significantly the memory space needed. Fig. 3 shows a left magnitude HRTF of Subject 003 and its PCA model due to direction with azimuth -80 o and elevation 0 o (top panel). On the bottom panel, it is shown the right magnitude HRTF and its PCA model due to the same direction. We can see that the models approximate well the corresponding magnitude HRTFs.
Minimum Phase Reconstruction and Synthesis of HRIR Models
As explained in the previous subsection, we obtained PCs matrix, V, and PCWs matrix, W, from the PCA method. Both matrices together with the empirical mean vector, µ, were applied to yield the matrix of models of magnitude HRTFs, Ĥ, as suggested by (11). By now, we could calculate the models of magnitude HRTFs of both ears. In order to synthesize the models of complex HRTFs, the phase information of left-and right-ear model of magnitude HRTF should be inserted into those models. We reconstructed the models of complex HRTFs based on the approach made by Kulkarni et al. [13]. They assumed that the phase of a HRTF was minimum phase. The phase function for a given model of magnitude HRTF was calculated by using Hilbert transform of natural logarithm of the model of magnitude HRTF. The minimum phase, ϕmp, of a model of magnitude HRTF, ĥi ((i=1,2,…,M)), is given by,
ϕmp = Imag{ H {-ln(ĥi)}},(12)
where Imag{} denotes the imaginary part of a complex number and ln is the natural logarithm. Thus, the model of minimum phase complex HRTF, ĥc, can be calculated using,
ĥc = ĥi . exp(j. φmp),(13)
where exp() denotes the exponential function. And the corresponding model of minimum phase HRIR, ĥmp(n), is given by the inverse fast Fourier transform (IFFT) of its complex HRTF, ĥc, from (13). Furthermore, in reconstructing the model of left-ear minimum phase HRIR and the model of right-ear minimum phase HRIR for a parti-cular direction of sound source into related model of left-ear HRIR and model of right-ear HRIR respectively, we needed to insert respective time delay related to the distance travelled by sound wave from the sound source to each ear drum of a subject, into each model of minimum phase HRIR. The time delays to be inserted were obtained from the means of time delays of respective directions on the horizontal plane from all subjects in the database used. The difference between left-ear time delay and right-ear time delay is called interaural time difference (ITD), which is needed by human to determine sound source direction. Fig. 4 shows, on the left panel, the original HRIRs of subject 003 due to direction with azimuth -80 o and elevation 0 o . On the right panel, we can see related models of left and right HRIR. These models re-sulted from the reconstructions of the PCA models of magnitude HRTFs into their corresponding HRIRs, as explained before. However, the models of magnitude HRTFs attained had not been individualized.
Individualization of Magnitude HRTFs Using MLR
As shown in Fig. 1, the individualization of the models of magnitude HRTFs, which were resulted from PCA, were done through MLR of PCWs matrix, W, using anthropometric measurements of a listener. From the matrix W of (10), we can extract a weights vector, wi,θ (37x1), which is a vector consisted of the i-th weights of the i-th PC, vi, of an ear of all subjects with azimuth θ on the horizontal plane, where i=1,2,...,10. In this research, we employed only 8 anthropometric measurements of a subject in the individualization process. The selection of these 8 measurements will be discussed in detail in the separate subsection below. These selected measurements of all subjects being analyzed were then gathered together in the columns of an anthropometric matrix, X (37x9), where the first column of X consists of all 1's. Suppose that the relation between the weights vector, wi,θ, and the anthropometric matrix, X, is given by,
wi,θ = X . βi,θ + Ei,θ ,(14)
where βi,θ (9x1) is the regression coefficients vector and Ei,θ (9x1) is the estimation errors vector. The regression coefficients were found by implementing least-square estimation. This estimation is performed by solving the optimization problem min{Ei,θ(n)}, where Ei,θ(n) is the n-th dependent variable's estimation error. PCWs and anthropometric measurements are respectively the model's dependent and independent variables. From (14), the regression coefficients due to i-th PCWs in azimuth θ, Bi,θ, can be estimated as, Bi,θ = (X T .X) -1 .X T . wi,θ.
As suggested by (15), enhancing the performance of the MLR method, it is needed to select both dependent and independent variables carefully. By applying PCA on magnitude HRTFs, the dimensions of independent variables were reduced significantly, so was the complexity of the models. Many correlation analyses were employed to select the independent variables in obtaining more accurate and simpler MLR method, as explained further in the subsection 2.6.
Correlation Analyses for Selection of Anthropometric Measurements
We employed the CIPIC HRTF Database, which are composed of both the measured HRIRs and some anthropometric measurements for 45 subjects, including the KE-MAR mannequin with both small and large pinna. The detail definitions of the all 27 anthropometric measurements are given in [2], [3] and can be seen in Fig. 2. Modeling of the listener's own HRIRs via his or her own anthropometric measurements will directly affect the feasibility and complexity of the system. It is obviously not advisable to implement all measurements into the model. Some useful information will be concealed by the unnecessary measurements, which results in a worse regression model. Besides, many measurements are very difficult to be measured correctly.
There are three parameters that psychoacoustically important in the perception of natural sound, i.e. interaural time difference (ITD), interaural level difference (ILD) and pinna notch frequency, fpn. ITD is the time difference between the arrival of first pulse of sound source from a particular direction on the left ear drum and that of the right ear drum. At the directions of sound source on median plane, ITD is near zero, where for a perfect symmetric head, there is no ITD on that plane. Thus, one can say that ITD is a function of azimuth on planes with fixed elevation. ITD can be calculated from the time delay of maximum cross correlation of the left HRIR and right HRIR at a particular direction. Then, ILD is defined as level or magnitude difference (in dB) in frequency domain between the left magnitude HRTF and the right magnitude HRTF at a particular direction of sound source. For a particular direction, we obtained ILD from each frequency component in the range of 0 -22050 Hz. ILDs generally are analyzed for a determined frequency component on the horizontal plane and on the median plane. Another significant psychoacoustic parameter is pinna notch frequency, fpn. Pinna notch frequency is the notch frequency in the magnitude spectrum of HRTF caused by diffraction and reflection of sound wave on a pinna. ITD and ILD are significant for the perception of azimuth of sound source. They affect much the variation of HRTF on the horizontal plane. But ILD and fpn play important role in the perception of elevation of sound source and affect the variation of HRTF on the median plane. It is difficult to characterize the range of HRTF variation among subjects. However, maximum ITD, ITDmax, maximum ILD, ILDmax, and fpn are simple and perceptually relevant parameters that characterize existing HRTF variation. Correlation analyses were applied to determine which anthropometric measurements have strong correlations with ITDmax, ILDmax, and fpn. From a few strongest correlated anthorpometric measurements, four measurements were chosen from head and torso sizes, i.e. x1 with ρ = 0.736, x3 with with ρ = 0.706, x6 with ρ = 0.726, and x12 with ρ = 0,768, where ρ denotes the correlation coefficient between the measurement and ITDmax. These 4 measurements were employed in the individualization of magnitude HRTFs using MLR method. Correlation analyses between ILDmax and head and torso sizes provided weaker correlations but confirmed the chosen of x1, x6, and x12. The selection of x3, x6, and x12 was also confirmed with the correlation analyses on the horizontal plane between the first PCWs, w1,θ, from the PCA of magnitude HRTFs and anthropometric measurements. We focused on first PCWs because they have largest variation through the azimuths.
The effects of pinna sizes are stronger with HRTFs on the median plane than HRTFs on the horizontal plane [1]. But overall the pinna sizes affect HRTFs in all directions. The correlation analyses between fpn and anthropometric measurements provided in general weaker correlations than those of ITDmax. Four pinna sizes had strongest correlations with fpn and that's why to be chosen; i.e. d1 with ρ = 0.435, d3 with ρ = 0.360, d5 with ρ = 0.204, and d6 with ρ = 0.280. These selected sizes of pinna are easy to be measured and represent measures of height and width. Hence, eight anthropometric measurements, x1, x3, x6, x12, d1, d3, d5, and d6 were chosen and fed in the MLR method in order to calculate regression coefficients. These eight anthropometric measurements are the same as the measurements that we used in our previous work [14]. Then the regression coefficients were applied in estimating the PCWs of a DTF at each direction on the horizontal plane.
EXPERIMENTS' RESULTS AND DISCUSSION
In this section, we discussed the performance of the proposed individualization method from the objective simulation experiments between the original magnitude HRTFs of the database and the individualized models of magnitude HRTFs. The experiments were done by employing only the data on the horizontal plane of 37 subjects out of all 45 subjects in the database. This occurred because the database had not included the complete set of anthropometric measurements of all subjects and the selected 8 anthropometric measurements were included only for 37 subjects.
Basis Functions Resulted from PCA
The inputs of the PCA were 3700 DTFs processed from HRIRs on the horizontal plane of 37 subjects. By solving eigen equation, we attained 10 basis functions or PCs to model the given DTFs. Fig. 5 shows the first five basis functions, v1,...,v5. As shown in this figure, that all five basis functions can be said roughly constant and approx-imate zero at frequencies below 1-2 kHz. This reflects the fact that there is almost no direction-dependent variability in the DTFs in this frequency range. Regardless of the weights employed to the basis functions, the resulting weighted sum will be close to zero in this range. Above about 2 kHz, all five basis functions have nonzero values. It is obvious that with the exception of the first PC, the high-frequency variability in these basis functions represents the direction-dependent high-frequency peaks and notches in the DTFs. The higher order basis function has more ripples and more details especially for the frequencies above about 2 kHz. The trends explained above are similar for the sixth to tenth basis functions. Taken together, all basis functions seem to capture the high frequency spectral variability. They also reflect spectral differences between sources in front and sources behind the subject.
Weights of Basis Functions
Based on PCA, assumed that DTFs can be represented by a relatively small number of basic spectral shapes of PCs, it seems reasonable to expect that the amount each basic shape contributes to the DTF at a given source position would related, in a simple way, to source azimuth and elevation. In the case of source position on horizontal plane, this amount or weight is related to azimuth only. On the other side, contralateral sources have negative weights. The magnitudes of weights for ipsilateral sources are much larger than those for contralateral sources. This distribution of PC-1 weights is similar across the 37 subjects, which has low intersubject variability. As seen in Fig. 5, that the first basis function has almost flat magnitude through all frequencies, so it can be said that PC-1 weights are functioning as the amplification in HRTF modeling. The remaining nine PC weights have larger variability in the ipsilateral side and, in the contralateral side, beginning at about azimuth 0 o , the weights have almost constant values near zero. Higher order of PC has corresponding flatter weights pattern for sources on the horizontal plane. It is observed, that the patterns of PC weights are roughly similar across subjects and ears.
Performance of Proposed Individualization
Method The performances of the estimated magnitude HRTFs on the horizontal plane, obtained either from PCA or individualization, were evaluated by the comparison of mean-square error of the differences between the estimated magnitude HRTFs, and the original magnitude HRTFs, calculated from database, to the mean-square error of the original magnitude HRTFs in percentage, which is defined by,
e j (θ) = 100 % x || h j (θ) -ĥ j (θ) || 2 / || h j (θ)|| 2 (16)
where hj(θ) is the j-th original magnitude HRTF with azimuth θ on horizontal plane, ĥj(θ) is the corresponding estimated magnitude HRTF of hj(θ). If the error is larger, the performance of the estimated magnitude HRTF is worse, where better localization results will be achieved with small error, ej(θ).
Before individualizing magnitude HRTFs using MLR, mean error from PCA modeling of magnitude HRTFs was calculated across all data in the database. At first, PCA modeling was performed for all data from all source directions of 45 subjects. This experiment resulted in mean error of 3.31% across all directions and subjects, but mean error across directions on horizontal plane was 3.65%. Second, modeling was performed using data at all directions of only 37 subjects, which resulted in mean error of 3.32% and mean error on horizontal plane was 3.68%. From these two experiments, it can be said that the corresponding mean errors were the same. Third, data of both ears of 45 subjects at directions only on horizontal plane were used and mean error of 3.67% was obtained. At last, PCA modeling was performed using data of both ears of only 37 subjects at directions only on horizontal plane. This experiment resulted in mean error of 3.68%. Again we obtained the same mean errors from the last two experiments. It is summarized that using data of 45 subjects or 37 subjects, yielded the same mean errors across related directions. Mean errors on horizontal plane were the same either data from all directions used or only from directions on horizontal plane. These mean errors are less than half of the related mean errors obtained from our previous work on PCA modeling of minimum phase HRIRs [14].
In individualizing magnitude HRTFs, we used only the data of both ears of 37 subjects at directions on horizontal plane, which meant that we used the results of fourth experiment mentioned above for individualization. We obtained here significantly small mean error of PCA models of magnitude HRTFs, i.e. 3.68% compared to 8.32% as in [14].
In turn, we individualized the PCA model of magnitude HRTFs using MLR with eight chosen measurements. The mean error of a subject was different from that of another subject in the database and also noted that a good performance of the individualized left-ear magnitude HRTFs of a subject was not always followed by the same performance of the right-ear ones. The overall mean error was only 12.17%, which was much better than 22.50% as in [14]. Fig. 7 shows the left-and right-ear errors as a where positive azimuth is due to source direction in the right side. However, the right-ear errors of subject 003 are mostly very good about 5% across azimuths. The left-ear errors of subject 163 seem to be much better than its rightear errors. Under the assumption stated below, if the spectral distortion (SD) score defined by Hu et al. [12], was applied to determine the performance of the individualized magnitude HRTFs, our SD scores of subject 003 and subject 163 on the front horizontal plane are no larger than 1 dB, which is much better than that in [12]. Using logarithm properties, i.e. 20.log(|a|/|b|) = (20.log|a| -20.log|b|), we assumed that the difference of logmagnitudes (20.log|a| -20.log|b|) in SD score might be replaced by the difference of magnitudes (log|a| -log|b|) in our case because we resulted in individualized magnitudes of HRTFs and Hu et el. resulted in individualized log-magnitudes of HRTF from PCA.
There were overall additional errors introduced by the proposed individualization method. These additional errors were introduced by the MLR. The unsystematic behavior of weights of PCs across subjects and across directions had caused MLR quite difficult to estimate adequately accurate regression coefficients. Besides, we performed here linear regression of anthropometric measurements to estimate the weights of PCs. Higher order regression might provide better estimates of these weights.
The individualized magnitude HRTFs of subject 003 could well approximate the corresponding original magnitude HRTFs particularly at frequencies below about 8 kHz. Fig. 8 shows the individualized and original magni- tude HRTFs for both the left and right ear in the extreme directions on the front horizontal plane. The top, middle, and bottom panel corresponds to azimuth -80°, 0°, and 80° respectively. Informal listening tests done by five subjects had shown a good and natural perceived moving sound around the horizontal plane by all subjects when the subjects' individualized reconstructed HRIRs, due to the sound source directions, were implemented in the headphone simulation.
CONCLUSION
In this paper, a simple and efficient individualization method of magnitude HRTFs for sources on horizontal plane, based on principal components analysis and multiple linear regression, was proposed. The proposed method shows better performance in the objective simulation experiments than that of similar research and was superior compared to our previous work. The additional errors introduced by MLR to PCA model might be lowered by applying higher order regression or other algorithm for MLR than the least square.
Dadang Gunawan received the B.Sc. degree in electrical engineering from University of Indonesia in 1983, and M.Eng and Ph.D degrees from Keio University, Japan, and Tasmania University, Australia in 1989 and 1995, respectively. He is the Head of Telecommunication Laboratory and of the Wireless and Signal Processing Research Group of the Electrical Engineering Department, University of Indonesia. His research interests are wireless communication and signal processing. Prof. Dr. D. Gunawan is a senior member of IEEE and IEEE Signal Processing Society.
•
Hugeng is with the Department of Electrical Engineering, University of Indonesia, Depok 16424 -Indonesia. • W. Wahab is with the Department of Electrical Engineering, University of Indonesia, Depok 16424 -Indonesia. • D. Gunawan is with the Department of Electrical Engineering, University of Indonesia, Depok 16424 -Indonesia.
Fig. 1 .
1Proposed HRTFs Individualization Method.
polar coordinate system. Elevation was sampled at 360/64 = 5,625 o step from -45 o to +230.625 o , while azimuth was sampled at -80 o , -65 o , -55 o , from -45 o to +45 o in 5 o step, at 55 o , 65 o , and 80 o . Hence, 1250-points spatial samples were obtained from the measurements of each ear of a subject. The published CIPIC HRTF Database contains anthropometric measurements of each subject.
Fig. 2 .
2Subject
Fig. 4 .
4Comparison of Original HRIRs and Models of HRIRs Obtained by Reconstruction of Models of Magnitude HRTFs.
Fig. 5 .
5The first five basis functions or PCs extracted from PCA of 3700 DTFs from both ears of 37 subjects on horizontal plane.
Fig. 6 .
6Left ear PC-1 weights for DTFs of Subject 003 on horizontal plane.
Fig. 6
6shows left ear PC-1 weights for DTFs of Subject 003, which were plotted as function of source azimuth on the front horizontal plane. It is shown, that there is a tendency for the weights to decrease in magnitude as the source moves from the median plane (azimuth 0 o ). Ipsilateral sources, sources with negative azimuths, have positive weights with exception the sources with azimuth -180 o to -150 o have negative weights.
azimuths on the front horizontal plane of subject 003, in the top panel, and of subject 163, in the bottom panel, after individualizing magnitude HRTFs. The mean errors for both ears of subject 163 are worse than those for both ears of subject 003, i.e. 12.92% and 21.20% repectively for left ear and right ear of subject 163, but only 8.27% and 5.18% repectively for left ear and right ear of subject 003. The left-ear errors of subject 003 on the front horizontal plane are about 10%, except for azimuth 65 o ,
Fig. 8 .
8Magnitude Responses of the Original and Individualized HRTFs of Subject 003 on the Front Horizontal Plane.
TABLE 1 The
1Percentage of Variance Explained by
Basis Functions
© 2010 JOT http://sites.google.com/site/journaloftelecommunications/
ACKNOWLEDGMENTThe authors wish to thank all staffs of CIPIC Interface Laboratory of California University at Davis, USA for providing the CIPIC HRTF Database. This work was supported in part by a grant from Electrical Engineering Department, Industrial Technology Faculty, Trisakti University, Jakarta, Indonesia.
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| []
|
[
"Towards Bin Packing (preliminary problem survey, models with multiset estimates) *",
"Towards Bin Packing (preliminary problem survey, models with multiset estimates) *"
]
| [
"Mark Sh Levin [email protected] \nInst. for Information Transmission Problems\nRussian Academy of Sciences\n19 Bolshoj Karetny Lane127994MoscowRussia\n"
]
| [
"Inst. for Information Transmission Problems\nRussian Academy of Sciences\n19 Bolshoj Karetny Lane127994MoscowRussia"
]
| []
| The paper described a generalized integrated glance to bin packing problems including a brief literature survey and some new problem formulations for the cases of multiset estimates of items. A new systemic viewpoint to bin packing problems is suggested: (a) basic element sets (item set, bin set, item subset assigned to bin), (b) binary relation over the sets: relation over item set as compatibility, precedence, dominance; relation over items and bins (i.e., correspondence of items to bins). A special attention is targeted to the following versions of bin packing problems: (a) problem with multiset estimates of items, (b) problem with colored items (and some close problems). Applied examples of bin packing problems are considered: (i) planning in paper industry (framework of combinatorial problems), (ii) selection of information messages, (iii) packing of messages/information packages in WiMAX communication system (brief description). | null | [
"https://arxiv.org/pdf/1605.07574v1.pdf"
]
| 18,171,133 | 1605.07574 | 7862f7667e267981eed3db1f6ba16cf6d07956a6 |
Towards Bin Packing (preliminary problem survey, models with multiset estimates) *
24 May 2016
Mark Sh Levin [email protected]
Inst. for Information Transmission Problems
Russian Academy of Sciences
19 Bolshoj Karetny Lane127994MoscowRussia
Towards Bin Packing (preliminary problem survey, models with multiset estimates) *
24 May 2016combinatorial optimizationbin-packing problemssolving frameworksheuristicsmultiset estimatesapplication
The paper described a generalized integrated glance to bin packing problems including a brief literature survey and some new problem formulations for the cases of multiset estimates of items. A new systemic viewpoint to bin packing problems is suggested: (a) basic element sets (item set, bin set, item subset assigned to bin), (b) binary relation over the sets: relation over item set as compatibility, precedence, dominance; relation over items and bins (i.e., correspondence of items to bins). A special attention is targeted to the following versions of bin packing problems: (a) problem with multiset estimates of items, (b) problem with colored items (and some close problems). Applied examples of bin packing problems are considered: (i) planning in paper industry (framework of combinatorial problems), (ii) selection of information messages, (iii) packing of messages/information packages in WiMAX communication system (brief description).
Introduction
Bin-packing problem is one of the well-known basic combinatorial optimization problems (e.g., [40,70,120,203]). The problem is a special case of one-dimensional "cutting-stock" problem [123,261] and the "assembly-line balancing" problem [69]. Fig. 1 illustrates the relationship of one-dimensional bin-packing problems and some other combinatorial optimization problems.
Fig. 1. Bin-packing problems and their relationship
Cutting-stock problems (e.g., [123,261]) ❄ One-dimensional cutting--stock problem (e.g., [123]) ❄ Assembly-line balancing problem (e.g., [69])
✠ ✛ ✚ ✘ ✙ ✛ ✚ ✘ ✙
Bin packing problem(s) (e.g., [66,120,150,194,195,203])
"Neighbor" problems: (a) multiple knapsack problem (e.g., [49,120,144,159,203,226]), (b) generalized assignment problem (e.g., [46,120,159,235,238,256]), (c) multi-processor scheduling (e.g., [59,120]) ✠ ❄ ❅ ❅ | Multicontainer packing problems (e.g., [67,114]): (a) bin packing problem, (b) multiple knapsack problem, (c) bin covering problem (basic dual bin packing), (d) min-cost covering problem (multiprocessor or makespan scheduling)
The bin packing problem can be described as follows (Fig. 2). Initial information involves the following: (i) a set of items A = {a 1 , ..., a i , ..., a n }, each item a i has a weight w i ∈ (0, 1]; (ii) a set of bins (or onedimensional containers, blocks) B = {B 1 , ..., B κ , ..., B m }, capacity of each bin B κ equals 1. The basic (classical) bin packing problem is (e.g., [150,151,152,259]):
Find a partition of the items such that: (a) each part of the item set is packed into the same bin while taking into account the bin capacity constraint (i.e., the sum of packed items in each bin ≤ 1), (b) the total number of used bins is minimized.
This problem is one of basic NP-hard combinatorial optimization problems (e.g., [120,157]). Bins (blocks, containers, knapsacks)
Bin 1 1 2 4 ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ Bin 2 3 ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ 5 ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ Bin 3 6 ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ Bin 4 . . .
Note the following basic types of items are examined (e.g., [19,24,25,30,67,89,90,115,117,120,193,196,203,204,227,243,265]): rectangular items, 2D items, irregular shape items, variable sizes items, composite 2D items (including items with common components), 3D items, multidimensional items, items as cylinders, items as circles, etc. A generalized illustration for bin packing problem is depicted in Fig. 3. II. Binary relation over items: 2.1. conflicts as a binary relations for item pairs that can not be assigned into the same bin: R conf l A×A (this can be considered as a part of the next relation), 2.2. compatibility (e.g., by type/color) as binary relation for items which are compatible (e.g., for assignment to the same bin, to be neighbor in the same bin): R compt L×L , here a weighted binary relation can be useful (e.g., for colors, including non-symmetric binary relation for neighborhood), 2.3. compatibility (e.g., by common components, for multi-component items), close to previous case (this may be crucial for "intersection" of items): R compt−com A×A , 2.4. precedence over items (this is important in the case of ordering of items which are assigned into the same bin): R prec A×A , 2.5. importance (dominance, preference) of items from the viewpoint of the first assignment to bins, as a linear ordering or poset-like structure over items: G(A, E dom ) (the poset-like structure may be based on multicriteria estimates or multiset estimates of items).
III. Binary relations over bins: 3.1. importance of bins from the viewpoint of the first usage, as a linear ordering or poset-like structure) over bins: G(B, E imp ) (the poset-like structure may be based on multicriteria estimates or multiset estimates of bins).
Numerical examples of the above-mentioned relations are presented as follows (on the basis of example from Fig. 2: six items and four bins): (i) correspondence of items to bins R A×B (Table 1), (ii) relation of item conflict R conf l A×A (Table 2), (iii) relation of item compatibility R comp A×A (e.g., by type/color) ( Table 3), (iv) precedence relation over items R prec A×A (Fig. 4 ), (v) (relation of dominance over items G(A, E dom ) (Fig. 5), and (vi) relation of importance over bins G(B, E imp ) (Fig. 6).
Further, the solution of the bin packing problem can be examined as the following (i.e., assignment of items into bins, a Boolean matrix):
S = {A 1 , ..., A κ , ..., A k , ..., A m } where |A κ1 A κ2 | = 0 ∀κ 1 , κ 2 = 1, m (i.e., the intersection is empty), A = m κ=1 A κ .
For classic bin packing problem (i.e., minimization of used bins), |A κ | = 0 ∀κ = k + 1, m (the first k bins are used) and A = m κ=1 A κ . In inverse bin packing problem (maximization of assigned items into the limited number of bins), a part of the most important items are assigned into m bins:
m κ=1 A κ ⊆ A.
Additional requirements to packing solutions are the following (i.e., fulfilment of the constraints): 1. Correspondence of item to bin. The following has to be satisfied: a i ∈ A κ If (a i , B κ ) ∈ R A×B . 2. Importance/dominance of items. This corresponds to inverse problem: If (a i1 , a i2 ) ∈ R dom A×A (i.e., a i1 a i2 Then three cases are correct: (a) both a i1 and a i2 are assigned to bin(s), (b) both a i1 and a i2 are not assigned to bin(s), (c) a i1 is assigned to bin and a i2 is not assigned to bin.
3. Item precedence. In the case of precedence constraint(s) according the above-mentioned precedence relations over items R prec A×A , the items have to be linear ordered in each bin (for each bin, i.e., ∀κ):
If (a i1 , a i2 ) ∈ R prec A×A and a i1 , a i2 ∈ A κ Then a i1 → a i2 . 4. Item conflicts. In the case of conflict constraints, the following has to be satisfied: a i1 , a i2 ∈ A κ If (a i1 , a i2 )∈R conf l A×A . 5. Item compatibility. In the case of compatibility constraints, the following has to be satisfied:
a i1 , a i2 ∈ A κ If (a i1 , a i2 ) ∈ R comp
A×A . In general, it is possible to use some penalty functions in the cases when the constraints are not satisfied. Table 2. Relation on item conflict R conf l A×A Item a i \ item a j a 1 a 2 a 3 a 4 a 5 a 6 a 1 ⋆ 1 Table 3. Relation on item compatibility R comp A×A Item a i \ item a j a 1 a 2 a 3 a 4 a 5 a 6 a 1 ⋆ 1
Item a i \ bin B κ B 1 B 2 B 3 B 4 a 1 3 2 1 0 a 2 3 1 0 0 a 3 1 3 2 0 a 4 3 2 2 0 a 5 1 3 1 1 a 6 2 3 3 11 1 0 0 a 2 1 ⋆ 1 1 1 0 a 3 1 3 ⋆ 4 1 0 a 4 1 1 1 ⋆ 1 0 a 5 1 1 1 1 ⋆ 0 a 6 1 0 0 0 0 ⋆1 1 0 0 a 2 1 ⋆ 1 1 1 0 a 3 1 3 ⋆ 4 1 0 a 4 1 1 1 ⋆ 1 0 a 5 1 1 1 1 ⋆ 0 a 6 1 0 0 0 0 ⋆ Fig. 4. Precedence over items R prec A×A r ❞ Item 1 ✲ ❍ ❍ ❍ ❍ ❍ ❍ • r ❞ Item 2 ❳ ❳ ❳ ❳ ❳ ❳ ③ r ❞ Item 3 ✲ r ❞ Item 5 ✘ ✘ ✘ ✘ ✘ ✘ ✿ r ❞ Item 4 ✲ r ❞ Item 6 Fig. 5. Importance of items G(A, E dom ) r ❞ Item 1 ✲ ❍ ❍ ❍ ❍ ❍ ❍ • ❍ ❍ ❍ ❍ ❍ ❍ • r ❞ Item 3 ✲ ✟ ✟ ✟ ✟ ✟ ✟ ✯ r ❞ Item 2 ✲ r ❞ Item 2 ✲ r ❞ Item 5 r ❞ Item 6 Fig. 6. Importance of bins G(B, E imp ) r ❞ Bin 1 ✲ r ❞ Bin 2 ✘ ✘ ✘ ✘ ✘ ✘ ✿ ❳ ❳ ❳ ❳ ❳ ❳ ③ r ❞ Bin 3 r ❞ Bin 4
Numerous publications have already addressed and analyzed various versions of static and dynamic bin packing problems. Many surveys on BPPs have been published (e.g., [23,64,65,66,67,68,81,116,120,195,203,252]). The basic taxonomies/typologies of bin packing problems have been examined in [67,89,90,193,265].
Some basic versions of bin packing problems (BPPs) are listed in Table 4 and special classes of bin packing problems (e.g., with relations among items) are pointed out in Table 5. Table 6 contains a list of main applications of bin packing problems.
Many surveys on algorithms for bin packing problems have been published (e.g., [23,63,66,68,81,101,114,138,141,150,203,231,255]). Basic algorithmic approaches are listed in Table 7, Table 8, and Table 9. Multidimensional bin packing problems: 2.1. 2D bin-packing [5,47,77,53,78,120,139,140] [ 141,193,194,195,199,203,255] 2.2. Oriented 2D bin packing [192,193] [80,239,222] A general classification scheme for bin packing problems has been suggested in [67]:
arena | objective function | algorithm class | results | constraints
where the scheme components are as follows: (a) arena describes types of bins (e.g., sizes, etc), (b) objective function describes types of problem (i.e., minimum of bin, minimum of "makespan", etc.), (c) algorithm class describes types of algorithm (e.g., offline, online, complexity estimate, greedy-type, etc.), (d) constraints describes quality of solution, e.g., asymptotic worst case ratios, absolute worst case, average case, etc., (e) constraints describes bounds on item sizes, bound on the number of items which can be packed in a bin, binary relation over item set (e.g., items a ι1 and a ι2 can not be put into the same bin), etc. Fig. 7 illustrates the basic trends in modifications of bin packing problems: (1) multicriteria (multiobjective) bin packing, (2) bin packing problems under uncertainty (e.g., fuzzy set usage of estimates), (3) examination of additional relations over items and over bins, (4) dynamic bin packing. This paper addresses the bin packing problem survey and some new formulations of bin packing problems: (a) with relations over item set, (b) with multiset estimates of items.
✻ ❍ ❍ ( ✟ ✟ ❅ ❅ | ✻ ❍ ❍ ( ✟ ✟ ✯ ✻ ❍ ❍ ( ✟ ✟ ✯ ✻ ❍ ❍ ( ✟ ✟ ✯ ✻ ✻ ✻ ✻
Basic multicontainer packing problems: (a) bin packing problem, (b) multiple knapsack problem, (c) bin covering problem (basic dual bin packing), (d) min-cost covering problem (multiprocessor or makespan scheduling) Table 9. Main algorithmic approaches, part III: online and evolutionary methods No. Solving approach Some source(s) VI.
Online and dynamic algorithms for bin packing:
Preliminary information
Basic problem formulations
The classical formal statement of BPP is the following (e.g., [150,151,152,259]). Given a bin S of size V and a list of n items with sizes a 1 , ..., a n to pack.
Find an integer number of bins B and a B-partition S 1
... S B of set {1, ..., n} such that i∈S k a i ≤ V for all k = 1, ..., B.
A solution is optimal if it has minimal B. The B-value for an optimal solution is denoted OPT below. A possible integer linear formulation of the problem is [203]:
min B = n i=1 y i s.t. B ≥ 1, n j=1 a j x ij ≤ V y i , ∀i ∈ {1, ..., n} n i=1 x ij = 1, ∀j ∈ {1, ..., n} y i ∈ {0, 1}, ∀i ∈ {1, ..., n} x ij ∈ {0, 1}, ∀i ∈ {1, ..., n}, ∀j ∈ {1, ..., n}
where y i = 1 if bin i is used and x ij = 1 if item j is put into bin i.
Maximizing the number of packed items (inverse problems)
The inverse bin packing problem is targeted to maximization of the number of packed items. Here, two basic kinds of the problems have been considered:
(i) maximization of the number of packed items (the number of bins is fixed) (e.g., [58]); (ii) "maximization" of the total preference estimate for packed items (the number of bins is fixed, (preference relation over item set) (e.g., [115]).
The description of inverse bin packing problem will be examined in further section.
Interval multiset estimates
Interval multiset estimates have been suggested by M.Sh. Levin in [183]. A brief description of interval multiset estimates is the following [183,185]. The approach consists in assignment of elements (1, 2, 3, ...) into an ordinal scale [1, 2, ..., l]. As a result, a multi-set based estimate is obtained, where a basis set involves all levels of the ordinal scale: Ω = {1, 2, ..., l} (the levels are linear ordered: 1 ≻ 2 ≻ 3 ≻ ...) and the assessment problem (for each alternative) consists in selection of a multiset over set Ω while taking into account two conditions:
1. cardinality of the selected multiset equals a specified number of elements η = 1, 2, 3, ... (i.e., multisets of cardinality η are considered); 2. "configuration" of the multiset is the following: the selected elements of Ω cover an interval over scale [1, l] .., l}. The number of multisets of cardinality η, with elements taken from a finite set of cardinality l, is called the "multiset coefficient" or "multiset number" ( [164,272]
): µ l,η = l(l+1)(l+2)...(l+η−1) η! .
This number corresponds to possible estimates (without taking into account interval condition 2). In the case of condition 2, the number of estimates is decreased. Generally, assessment problems based on interval multiset estimates can be denoted as follows: P l,η .
A poset-like scale of interval multiset estimates for assessment problem P 3,3 is presented in Fig. 8. 8a illustrates the scale-poset and estimates for problem P 3,3 (assessment over scale [1,3] with three elements, estimates (2, 0, 2 and (1, 0, 2) are not used) [183,185]. For evaluation of multi-component system, multi-component poset-like scale composed from several poset-like scale may be used [183,185]. The following operations over multiset estimates are used [183,185] as well: integration, vector-like proximity, aggregation, and alignment.
e 3,3 1 ☛ ✡ ✟ ✠ ☛ ✡ ✟ ✠ {1, 1, 1} or (3, 0, 0) 1 2 3 e 3,3 2 ☛ ✡ ✟ ✠ {1, 1, 2} or (2, 1, 0) 1 2 3 e 3,3 3 ☛ ✡ ✟ ✠ {1, 2, 2} or (1, 2, 0) 1 2 3 e 3,3 4 ☛ ✡ ✟ ✠ {2, 2, 2} or (0, 3, 0) 1 2 3 e 3,3 6 ☛ ✡ ✟ ✠ {2, 2, 3} or (0, 2, 1) 1 2 3 e 3,3 7 ☛ ✡ ✟ ✠ {2, 3, 3} or (0, 1, 2) 1 2 3 e 3,3 8 ☛ ✡ ✟ ✠ {3, 3, 3} or (0, 0, 3) 1 2 3 ❅ ❅ ❅ ✏ ✏ ✏ e 3,3 5 ☛ ✡ ✟ ✠ {1, 2, 3} or (1, 1, 1) 1 2 3 (b) integrated poset-like scale by elements & compatibility r ✓ ✓ ✓ ✓ ✓ ✓ ❙ ❙ ❙ ❙ ❙ ❙ ✓ ✓ ✓ ✓ ✓ ✓ ❙ ❙ ❙ ❙ ❙ ❙ ✓ ✓ ✓ ✓ ✓ ✓ ❙ ❙ ❙ ❙ ❙ ❙ r ❢ Ideal point w = 1 w = 2 w = 3 Fig.
Integration of estimates (mainly, for composite systems) is based on summarization of the estimates by components (i.e., positions). Let us consider n estimates (position form): estimate e 1 = (η 1 1 , ..., η 1 ι , ..., η 1 l ), . . ., estimate e κ = (η κ 1 , ..., η κ ι , ..., η κ l ), . . ., estimate e n = (η n 1 , ..., η n ι , ..., η n l ). Then, the integrated estimate is: estimate e I = (η I 1 , ..., η I ι , ..., η I l ), where η I ι = n κ=1 η κ ι ∀ι = 1, l. In fact, the operation is used for multiset estimates: e I = e 1 ... e κ ... e n .
Further, vector-like proximity is described. Let A 1 and A 2 be two alternatives with corresponding interval multiset estimates e(A 1 ), e(A 2 ). Vector-like proximity for the alternatives above is:
δ(e(A 1 ), e(A 2 )) = (δ − (A 1 , A 2 ), δ + (A 1 , A 2 ))
, where vector components are: (i) δ − is the number of one-step changes: element of quality ι + 1 into element of quality ι (ι = 1, l − 1) (this corresponds to "improvement"); (ii) δ + is the number of one-step changes: element of quality ι into element of quality ι + 1 (ι = 1, l − 1) (this corresponds to "degradation"). It is assumed:
|δ(e(A 1 ), e(A 2 ))| = |δ − (A 1 , A 2 )| + |δ + (A 1 , A 2 )|.
Now let us consider median estimates (aggregation) for the specified set of initial estimates (traditional approach). Let E = {e 1 , ..., e κ , ..., e n } be the set of specified estimates (or a corresponding set of specified alternatives), let D be the set of all possible estimates (or a corresponding set of possible alternatives) (E ⊆ D). Thus, the median estimates ("generalized median" M g and "set median" M s ) are: M g = arg min M∈D n κ=1 |δ(M, e κ )|; M s = arg min M∈E n κ=1 |δ(M, e κ )|. In recent decade, the significance of multiset studies and applications has been increased. Some recent studies in multisets and their applications are pointed out in Table 9. [179,180,183,185]). An examined composite (modular, decomposable) system consists of components and their interconnection or compatibility (IC). Basic assumptions of HMMD are the following: (a) a tree-like structure of the system; (b) a composite estimate for system quality that integrates components (subsystems, parts) qualities and qualities of IC (compatibility) across subsystems; (c) monotonic criteria for the system and its components; (d) quality of system components and IC are evaluated on the basis of coordinated ordinal scales. The designations are: (1) design alternatives (DAs) for leaf nodes of the model; (2) priorities of DAs (ι = 1, l; 1 corresponds to the best one); (3) ordinal compatibility for each pair of DAs (w = 1, ν; ν corresponds to the best one). Let S be a system consisting of m parts (components): R(1), ..., R(i), ..., R(m). A set of design alternatives is generated for each system part above. The problem is:
Problems with multiset estimates
3.1. Some combinatorial optimization problems with multiset estimates 3.1.1. Knapsack problem with multiset estimates The basic knapsack problem (i.e., "0 − 1 knapsack problem") is (e.g., [120,159,203]): (i) given item set A = {1, ..., i, ..., m} with parameters ∀i ∈ A: profit (or utility) γ i , resource requirement (e.g., weight) a i ; (ii) given a resource (capacity) of knapsack b. Thus, the model is as follows:
max m i=1 γ i x i s.t. m i=1 a i x i ≤ b, x i ∈ {0, 1}, i = 1, m where x i = 1 if item i is selected
, and x i = 0 otherwise. Often nonnegative coefficients are assumed.
In the case of multiset estimates of item "utility" e i , i ∈ {1, ..., i, ..., n} (instead of γ i ), the following aggregated multiset estimate can be used for the objective function ("maximization") (e.g., [183,185]): (a) an aggregated multiset estimate as the "generalized median", (b) an aggregated multiset estimate as the "set median", and (c) an integrated multiset estimate. Knapsack problem with multiset estimates and the integrated estimate for the solution is (solution
S = {i|x i = 1}): max e(S) = i∈S={i|xi=1} e i , s.t. m i=1 a i x i ≤ b; x i ∈ {0, 1}.
In the case of objective function based on median estimate for solution, the problem is:
max e(S) = max M = arg min M∈D | i∈S={i|xi=1} δ(M, e i )| s.t. m i=1 a i x i ≤ b, x i ∈ {0, 1}.
In addition, it is reasonable to consider a new problem formulation while taking into account the number of the selected items (i.e. a special two-objective knapsack problem with multiset estimates)
(solution S = {i|x i = 1}): max e(S) = max M = arg min M∈D | i∈S={i|xi=1} δ(M, e i )| max n i=1 x i s.t. m i=1 a i x i ≤ b, x i ∈ {0, 1}.s ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ Lattice for element quality (poset-like scale) ✟ ✟ ✟
The ideal point by elements (e.g., "median")
Multiple choice problem with interval multiset estimates
In multiple choice problem, items are divided into groups (without intersection) and items are selected in each group under total resource constraint (e.g., [120,159,203]). Here, one item is selected in each group. This version of multiple choice problem is (Boolean variable
x i,j equals 1 if item (i, j) is selected): max m i=1 qi j=1 γ ij x ij s.t. m i=1 qi j=1 a ij x ij ≤ b, qi j=1 x ij = 1, i = 1, m, x ij ∈ {0, 1}.
A special case of multiple choice problem is considered [183,185]: (1) multiset estimates of item "utility" e ij (i = 1, m, j = 1, q i ∀i) (instead of c ij ); (2) an aggregated multiset estimate as the "generalized median" (or "set median") is used for the objective function ("maximization"). The item set is: 1), (i, 2), ..., (i, q i )}. The solution is a subset of the initial item set:
A = m i=1 A i , A i = {(i,S = {(i, j)|x i,j = 1}. Formally, max e(S) = max M = arg min M∈D (i,j)∈S={(i,j)|xi,j =1} |δ(M, e i,j )| s.t. m i=1 qi j=1 a ij x i,j ≤ b, qi j=1 x ij = 1, x ij ∈ {0, 1}.
Evidently, this problem is similar to the above-mentioned combinatorial synthesis problem without compatibility of the selected items (objects, alternatives) [183,185].
Multiple knapsack problem with multiset estimates
The basic multiple knapsack problem is the following (e.g., [49,120,144,159,203,226]): (i) item set A = {1, ..., i, ..., m}; (ii) knapsack set B = {B 1 , ..., B j , ..., B k } (k ≤ m); (iii) parameters ∀i ∈ A: profit c i , resource requirement (e.g., weight) a i ; and (iv) resource (capacity) of knapsack B j ∈ B: b j . This problem is a special case of generalized assignment problem (multiple knapsack problem contains bin packing problem as special case). The model (i.e., "0 − 1 multiple knapsack problem") is:
max k j=1 m i=1 γ i x ij s.t. m i=1 a i x ij ≤ b j , ∀j = 1, k, k j=1 x ij ≤ 1, ∀i = 1, m, x ij ∈ {0, 1}, i = 1, m, j = 1, k,(i,j)∈S={(i,j)|xi,j =1} e i , s.t. m i=1 a i x ij ≤ b j , ∀j = 1, k, k j=1 x ij ≤ 1, ∀i = 1, m, x ij ∈ {0, 1}, i = 1, m, j = 1, k.
In the case of objective function based on median estimate for solution, the problem is:
max e(S) = max M = arg min M∈D | (i,j)∈S={(i,j)|xij =1} δ(M, e i )| s.t. m i=1 a i x ij ≤ b j , ∀j = 1, k, k j=1 x ij ≤ 1, ∀i = 1, m, x ij ∈ {0, 1}, i = 1, m, j = 1, k.
In addition, it is reasonable to consider a new problem formulation while taking into account the number of the selected items (i.e., a special two-objective knapsack problem with multiset estimates)
(solution S = {(i, j)|x ij = 1}): max e(S) = max M = arg min M∈D | (i,j)∈S={(i,j)|xi,j =1} δ(M, e i )| max n i=1 x i,j s.t. m i=1 a i x ij ≤ b j , ∀j = 1, k. k j=1 x ij ≤ 1, ∀i = 1, m, x ij ∈ {0, 1}, i = 1, m, j = 1, k.
Here, "two"-dimensional space of solution quality (Fig. 9) can be considered as well.
Assignment and generalized assignment problems with multiset estimates
The basic assignment problem is the following (e.g., [120,246]). Simple assignment problem involves nonnegative correspondence matrix Υ = ||γ ij || (i = 1, m, j = 1, m) where c ij is a profit ('utility') to assign element i to position j. The problem is (e.g., [120]):
Find the assignment π = (π(1), ..., π(m)) of elements i (i = 1, m) to positions π(i) which corresponds to a total effectiveness:
m i=1 γ iπ(i) → max.
The simplest algebraic problem formulation is:
max m i=1 m j=1 γ i,j x i,j s.t. m i=1 x i,j ≤ 1, j = 1, m; m j=1 x i,j = 1, i = 1, m; x i,j ∈ {0, 1}, i = 1, m, j = 1, m.
Here x i,j = 1 if element i is assigned into position j, c ij is a profit ("utility") of this assignment. The problem can be solved efficiently, for example, on the basis of Hungarian method (e.g., [169]). Note this problem is the matching problem for a bipartite graph (e.g., [120]).
In the generalized assignment problem, each item i (i = 1, m) can be assigned to k (k ≤ m) positions (knapsacks, bins) and a capacity is considered for each position j (j = 1, k) (with corresponding capacity constraint ≤ b j ) (Fig. 10).
Items ✤ ✣ ✜ ✢ t t t t t t ❳ ❳ ❳ ❳ ❳ ❳ ③ ✲ ✲ ✲ ✲ ✘ ✘ ✘ ✘ ✘ ✘ ✿ Positions (e.g., bins, knapsacks) ✬ ✫ ✩ ✪ ❡ r ❡ r ❡ r ❡ r Formally, max m i=1 k j=1 γ i,j x i,j s.t. m i=1 x i,j ≤ 1, j = 1, k; k j=1 x i,j ≥ 1, i = 1, m; x i,j ∈ {0, 1}, i = 1, m, j = 1, k.
In the case of multiset estimates, item "utility" e ij (i = 1, m j = 1, k) instead of c ij is considered. The generalized assignment problem with multiset estimates and the integrated estimate for the solution
is (solution S = {(i, j)|x ij = 1}): max e(S) = (i,j)∈S={(i,j)|xi,j =1} e i , s.t. m i=1 a i x ij ≤ b j , ∀j = 1, k, k j=1 x ij = 1, ∀i = 1, m, x ij ∈ {0, 1}, i = 1, m, j = 1, k.
In the case of objective function based on median estimate for solution, the problem is:
max e(S) = max M = arg min M∈D | (i,j)∈S={(i,j)|xij =1} δ(M, e i )| s.t. m i=1 a i x ij ≤ b j , ∀j = 1, k, k j=1 x ij = 1, ∀i = 1, m, x ij ∈ {0, 1}, i = 1, m, j = 1, k.
In addition, it is reasonable to consider a new problem formulation while taking into account the number of the selected items (i.e. a special two-objective generalized assignment problem with multiset estimates) (solution S = {(i, j)|x ij = 1}):
max e(S) = max M = arg min M∈D | (i,j)∈S={(i,j)|xi,j =1} δ(M, e i )| max n i=1 x i,j s.t. m i=1 a i x ij ≤ b j , ∀j = 1, k. k j=1 x ij = 1, ∀i = 1, m, x ij ∈ {0, 1}, i = 1, m, j = 1, k.
Here, "two"-dimensional space of solution quality (Fig. 9) can be considered as well.
Inverse bin packing problem with multiset estimates
Generally, the inverse bin packing problem can be formulated as multiple knapsack problem with equal knapsack (i.e., bins).
First, the basic inverse bin packing problem (with maximization of packed items), i.e., maximum cardinality bin packing problem, is considered as follows (e.g., [7,41,58,52,94,171,172,221]). Problem components are: (i) item set A = {1, ..., i, ..., m}; (ii) set of equal bins B = {B 1 , ..., B j , ..., B k } (usually, k ≤ m); (iii) parameters ∀i ∈ A: profit γ i , resource requirement (e.g., weight) a i ; and (iv) equal resource (capacity) of each bin B j ∈ B: b. The model is:
max k j=1 m i=1 γ i x ij s.t. m i=1 a i x ij ≤ b, ∀j = 1, k, k j=1 x ij ≤ 1, ∀i = 1, m, x ij ∈ {0, 1}, i = 1, m, j = 1, k,
where x ij = 1 if item i is selected for knapsack B j , and x ij = 0 otherwise.
In the case of multiset estimates, item "utility" e i , i = 1, m (instead of c i ) is considered. The inverse bin packing problem with multiset estimates and the integrated estimate for the solution is (solution
S = {(i, j)|x ij = 1}): max e(S) = (i,j)∈S={(i,j)|xi,j =1} e i , s.t. m i=1 a i x ij ≤ b, ∀j = 1, k, k j=1 x ij ≤ 1, ∀i = 1, m, x ij ∈ {0, 1}, i = 1, m, j = 1, k.
In the case of objective function based on median estimate for solution, the problem is:
max e(S) = max M = arg min M∈D | (i,j)∈S={(i,j)|xij =1} δ(M, e i )| s.t. m i=1 a i x ij ≤ b, ∀j = 1, k, k j=1 x ij ≤ 1, ∀i = 1, m, x ij ∈ {0, 1}, i = 1, m, j = 1, k.
The problem formulation while taking into account the number of the selected items (i.e., a special two-objective inverse bin packing problem with multiset estimates) (solution S = {(i, j)|x ij = 1}) is:
max e(S) = max M = arg min M∈D | (i,j)∈S={(i,j)|xi,j =1} δ(M, e i )| max n i=1 x i,j s.t. m i=1 a i x ij ≤ b, ∀j = 1, k. k j=1 x ij ≤ 1, ∀i = 1, m, x ij ∈ {0, 1}, i = 1, m, j = 1, k.
Here, "two"-dimensional space of solution quality (Fig. 9) can be considered as well.
Bin packing with conflicts
The bin packing problem with conflict consists in packing items into the minimum number of bins subject to incompatibility constraints. (e.g., [98,107,121,149,237]). The description of the problem is the following. Given a set of n items A, corresponding their weights w 1 , w 2 , ... w n , and a set of identical bins (k = 1, 2, ...) with capacity b. It can be assumed: w 1 ≥ w 2 ≥ ... ≥ w n . Given conflict relation over items as conflict graph G = (A, E), where an edge (ι 1 , ι 2 ) ∈ E exists if and only if items ι 1 , ι 2 ∈ A conflict or w ι1 + w ι2 > b. Let y k be a binary variable: y k = 1 if bin k is used, and x ιk be a binary variable: x ιk = 1 if item i is assigned to bin k. Formally,
min z = n ι=1 y k s.t. n ι=1
w ι x ιk ≤ by k ∀k = 1, n; n ι=1
x ιk = 1 ∀ι = 1, n; x ι2k + x ι2k ≤ 1 ∀(ι 1 , ι 2 ) ∈ E, ∀k = 1, n; y k ∈ {0, 1} ∀k = 1, n; x ιk ∈ {0, 1} ∀ι = 1, n, ∀k = 1, n.
Evidently, the problem generalizes the classic bin packing problem and is HP-hard (e.g., [203]).
In inverse bin packing problem (maximization of the number of packed items subject to fixed set of bins), The problem is as follows. Let γ ι be an importance (utility, profit) of packing item ι ∈ A. Formally, x ιk ≤ 1 ∀ι = 1, n; x ι2k + x ι2k ≤ 1 (ι 1 , ι 2 ) ∈ E, ∀k = 1, n;
x ιk ∈ {0, 1} ∀ι = 1, n, ∀k = 1, n.
Let e ι be an importance multiset estimate (utility, profit) of packing item ι ∈ A. The inverse bin packing problem with multiset estimates and the integrated estimate for the solution is (solution S = {(ι, k)|x ιk = 1}):
max e(S) = (ι,k)∈S={(ι,k)|x ι,k =1} e ι , s.t. n ι=1 q k=1 w ι,k x ιk ≤ b, ∀k = 1, n; n ι=1
x ι,k ≤ 1 ∀ι = 1, n; x ι2k + x ι2k ≤ 1 (ι 1 , ι 2 ) ∈ E, ∀k = 1, n;
x ιk ∈ {0, 1} ∀ι = 1, n, ∀k = 1, n.
In addition, objective function can be examine: max n ι=1 q k=1 x ι,k ∀ι = 1, n, ∀k = 1, n.
Colored bin packing
Basic colored bin packing
Now consider the basic colored bin packing problem (e.g., [51,257]). A set of items A = {a 1 , ..., a i , ..., a n } of different sizes (e.g., w i ∈ (0, 1] ∀i = 1, n) is given. It is necessary to pack the items above into bins of equal size so that a few bins is used in total (at most α times optimal), and that the items of each color span few bins (at most β times optimal). The obtained allocations are called α, β-approximate. The colored bin packing problem corresponds to many significant applications, for example (e.g., [257]):
(1) allocating files in P2P networks, (2) allocating related jobs (i.e., related jobs are of the same color) to processors, (3) allocating related items in a distributed cache, and (4) allocating jobs in a grid computing system. Fig. 11 illustrates the colored bin packing problem: eleven items, three colors (λ, µ, θ). The illustrative solution is: (i) color λ for bin 1, bin 2; (ii) color µ for bin 3; and (iii) color θ for bin 4, bin 5. . . .
Recently, some versions of colored bin packing problem have been examined: (1) basic colored bin packing [51,257], (2) offline colored bin packing [257], (3) online colored bin packing [257], and (4) online bin coloring (packing with minimum colors) [167].
Two auxiliary graph coloring problems 4.2.1. Auxiliary vertex graph coloring problem with ordinal color proximity
First, the vertex coloring problem is considered as a basic one. The problem can be described as the following (e.g., [44,93,120,134,135,168,218,266]). Given undirected graph G = (A, E) (a node/vertices set A and an edge set E, |A| = n). There is a set of colors (labels, numbers) X = {x 1 , ..., x l , ..., x k }. Let C(G) = {C(a 1 ), ..., C(a i ), ..., C(a i )} (C ai ∈ X) (or < C(a 1 ) ⋆ ... ⋆ C(a i ) ⋆ ... ⋆ C(a i ) > ) be a color configuration (i.e., assignment of a color for each vertex). The problem is:
Assign for each vertex ∀a i ∈ A label or color (i.e., C(a i )) such that no edge connects two identical colored vertices, i.e., ∀a i , a j ∈ A if (a i , a j ) ∈ E (i.e., adjacent vertices) then C(a i ) = C(a j ).
Thus, color configuration (e.g., C(G) = {C(a 1 ), ..., C(a i ), ..., C(a n )}) for a given graph G = (A, E) is searched for. Clearly, |C(G)| equals the number of used colors (labels). (The minimal number of required colors for a graph G is called chromatic number of the graph χ(G)). Note, other coloring problems can be transformed into the vertex version. Fig. 12 illustrates the vertex coloring problem:
G = (A, E), A = {p, q, u, v, w}, E = {(p, q), (p, u), (q, v), (u.v), (w, p)(w, q)(w, u)(w, v)} and three colors {x 1 , x 2 , x 3 } (i.e.,❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ p P 1 P 2 P 3 ✞ ✝ ☎ ✆ V 1 V 2 V 3 v ✞ ✝ ☎ ✆ U 1 U 2 U 3 u ✞ ✝ ☎ ✆ Q 1 Q 2 Q 3 q ✞ ✝ ☎ ✆ W 1 W 2 W 3 w ✞ ✝ ☎ ✆
The resultant color configuration (solution) is:
C(G) = {P 2 , W 1 , V 2 , Q 3 , U 3 }.
The number of possible resultant color configurations (three colors) equals 6:
(
1) C 1 (G) = {P 1 , W 2 , V 1 , Q 3 , U 3 }, (2) C 2 (G) = {P 1 , W 3 , V 1 , Q 2 , U 2 }, (3) C 3 (G) = {P 3 , W 2 , V 3 , Q 1 , U 1 }, (4) C 4 (G) = {P 3 , W 1 , V 3 , Q 2 , U 2 }, (5) C 5 (G) = {P 2 , W 1 , V 2 , Q 3 , U 3 }, (6) C 6 (G) = {P 2 , W 3 , V 2 , Q 1 , U 1 }.
In addition, an aggregated weight (e.g., additive function) of used colors (each color has its nonnegative weight w(x l ) ∀x l ∈ X, l = 1, k) can be considered as well. As a result, the following minimization problem formulation can be examined: This problem is NP-hard (e.g., [93,120,153,266]). Let C * (G) = {c * θ } be the set of used colors (i.e., C * (G) ⊆ C * (G)). In the case of weighted colors (and additive aggregation function), the following model can be considered:
min ∀c * θ ∈C * (G) w(c * θ ) s.t. C * (a i ) = C * (a j ) ∀(a i , a j ) ∈ E, i = j.
Clearly, if w(x l ) = 1 ∀x l ∈ X this problem formulation is equivalent to the previous one. In the case of vector-like color weight ( w 1 (c θ ), ..., w µ (c θ ), ..., w λ (c θ ) ) ∀c θ ∈ C and additive aggregation functions, the objective vector function is:
( ∀c * θ ∈C * (G) w 1 (c * θ ) , ..., ∀c * θ ∈C * (G) w µ (c * θ ) , ..., ∀c * θ ∈C * (G) w λ (c * θ ) )
and Pareto-efficient solutions by the vector function are searched for. Generally, it may be prospective to consider a set of objective functions (criteria) as follows (e.g., [179,185]): (i) number of used colors, (ii) an aggregated weight of used colors, (iii) correspondence of colors to vertices (e.g., the worst correspondence, average correspondence) (e.g., [179]); (iv) quality of compatibility of colors, which were assigned to the neighbor (i.e., adjacent) vertices (e.g., the worst case, average case) (e.g., [179]); and (v) conditions at a distance that equals three, four, etc.
The author's version of graph (vertex) coloring problem (while taking into account color compatibility and correspondence of colors to vertices) is described in [179] (numerical example, Fig. 13). Here, the solving approach is based on morphological clique problem (i.e., HMMD). Six colors are used: x 1 , x 2 , x 3 , x 4 , x 5 , and x 6 . Estimates of correspondence of colors to vertices are shown in parentheses in Fig. 13 (1 corresponds to the best level). Table 10 contains compatibility estimates for colors (4 corresponds to the best level).
✚ ✚ ✚ ✚ ✚ ✚ ) ) ) ) ) ) p P 1 (3) P 2 (1) P 3 (1) P 4 (3) P 5 (3) ✞ ✝ ☎ ✆ V 1 (3) V 2 (1) V 3 (2) V 4 (3) V 5 (3) v ✞ ✝ ☎ ✆ Q 1 (3) Q 2 (3) Q 3 (2) Q 4 (2) Q 5 (1) q ✞ ✝ ☎ ✆ W 1 (3) W 2 (3) W 3 (3) W 4 (2) W 5 (2) w ✞ ✝ ☎ ✆
If the edge between vertices is absent the corresponding compatibility estimates of colors equal to the best level (i.e., 4 for vertex pair (p, v)). Two examples of color combinations (color compositions) and their quality vectors are the following (Fig. 14): Table 10. Compatibility estimates of colors
(a) C * 1 (G) =< P 2 ⋆ Q 3 ⋆ V 3 ⋆ W 5 >, N (C * 1 (G)) = (4; 1, 3, 0); (b) C * 2 (G) =< P 3 ⋆ Q 5 ⋆ V 2 ⋆ W 4 >, N (C * 2 (G)) = (2; 3, 1, 0); (c) C * 3 (G) =< P 2 ⋆ Q 5 ⋆ V 2 ⋆ W 5 >, N (C * 2 (G)) = (2; 3, 1, 0).Q 1 Q 2 Q 3 Q 4 Q 5 V 1 V 2 V 3 V 4 V 5 W 1 W 2 W 3 W 4 WQ 1 0 1 2 3 4 0 1 2 3 3 Q 2 1 0 4 2 3 1 0 1 2 4 Q 3 4 1 4 1 2 2 1 0 1 4 Q 4 3 2 1 0 3 3 2 1 0 2 Q 5 4 3 2 3 0 4 3 2 3 3 V 1 0 1 2 3 3 V 2 1 0 1 2 4 V 3 2 1 0 1 4 V 4 3 2 1 0 2 V 5 4 3 2 3 2 Fig. 14. Poset-like scale for color configuration r ✓ ✓ ✓ ✓ ✓ ✓ ❙ ❙ ❙ ❙ ❙ ❙ ✓ ✓ ✓ ✓ ✓ ✓ ❙ ❙ ❙ ❙ ❙ ❙ ✓ ✓ ✓ ✓ ✓ ✓ ❙ ❙ ❙ ❙ ❙ ❙ ✓ ✓ ✓ ✓ ✓ ✓ ❙ ❙ ❙ ❙ ❙ ❙ q ❞ N (C * 2 (G)), N (C * 3 (G)) q ❞ N (C * 1 (G)) r ❢Ideal point w = 1 w = 2 w = 3 w = 4
Partition coloring problem
Here the partition coloring problem (i.e., selective graph clustering over clustered graph) is considered as a close auxiliary problem [113,142,188,219]. The problem formulation is as follows. Given a non-directed graph G = (V, E), where V is the set of vertices (nodes) and E is the set of edges. Let {V 1 , V 2 , ..., V q } be a partition of V into q subsets with V = q ι=1 V ι and |V ι1 V ι2 | = 0 ∀ι 1 , ι 2 = 1, 2, ..., q with ι 1 = ι 2 . Clearly, V ι (∀ι = 1, q) is a graph part or a graph component. The partition coloring problem is:
Find a subset V ′ ⊆ V such that |V ′ V ι | = 1 ∀ι = 1, q (i.e., V ′ contains one vertex from each component V ι ), and the chromatic number of the graph induced in G by V ′ is minimum.
Evidently, the problem is a generalization of the graph coloring problem and belongs to class of NP-hard problems (e.g., [188]). Several formal models for this problem have been proposed: (a) binary integer programming problem (e.g., [112,113,142]), (b) model based on the independent set problem [142], and (c) two integer programming formulations using representatives [10]. Fig. 15 depicts an instance of partition coloring problem (graph with ten vertices and four graph parts). Here, the resultant colorings are (two colors: c 1 , c 1 ):
Q 1 =< 2(c 1 ), 6(c 2 ), 9(c 1 ), 5(c 2 ) >, Q 2 =< 2(c 2 ), 6(c 1 ), 9(c 2 ), 5(c 1 ) >.
✤ ✣ ✜ ✢ ✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ s s 9 10 ✓ ✒ ✏ ✑ ❍ ❍ ❍ ) ) ) ) ) ) ✚ ✚ ✚ ✚ ✚ ✚ s s 4 5 ✛ ✚ ✘ ✙ ✟ ✟ ✟ ✚ ✚ ✚ ✚ ✚ ✚ ❍ ❍ ❍ ) ) ) ) ) ) ❍ ❍ ❍ ❳ ❳ ❳ ❳ ❳ ❳ s s s 6 7 8 ✬ ✫ ✩ ✪ (b) solution (two colors) q ❝ 2 ✤ ✣ ✜ ✢ q ❝ 9 ✓ ✒ ✏ ✑ ❍ ❍ ❍ s ❢ 5 ✛ ✚ ✘ ✙ ✚ ✚ ✚ ✚ ✚ ✚ ❍ ❍ ❍ ❳ ❳ ❳ ❳ ❳ ❳ s ❢6 ✬ ✫ ✩ ✪
Some solving approaches proposed for the partition coloring problem are listed in Table 11. Engineering heuristics [188,191] In real world, this problem corresponds to routing and wavelength assignment in all-optical networks (i.e., computation of alternative routes for the lightpaths, followed by the solution of a partition colorings problem in a conflict graph) (e.g., [188,191,219].
In fact, the partition coloring problem is very close to representative problems (e.g., [10]). Generally, this kind of problems is based on selection of elements from graph parts (components) (e.g., vertices) while taking into account compatibility of the selected elements (i.e., construction of a clique or quasiclique). In addition, it is possible to examine some preference relation(s) over elements for graph part. Thus, the problem can be considered as a morphological clique problem (i.e., hierarchical morphological design or combinatorial synthesis) [179,180,185].
In the future, it may be very interesting to examine a new multistage partition coloring problem with costs of changes of vertex colors as restructuring of partition coloring problem. (i.e., a version of dynamical partition coloring problem).
Some applications
Composite planning framework in paper production system
Here a composite planning framework is described that was prepared by the author for a seminar of Institute for Industrial Mathematics in May 1992 (Beer Sheva, Israel). Fig. 16 depicts an illustrative solution of the composite planning problem for three machines. In the problem, there are a set of paper horizontal bar for each machine. It is necessary to cut it (by special knifes) to obtain a set of 2D items of the required sizes and colors (by coloring). Seven colors are considered: white (col 1 ), blue (col 2 ), red (col 3 ), green (col 4 ), magenta (col 5 ), brown (col 6 ), and yellow (col 7 ). Table 12 contains ordinal estimates of color change: col i ⇒ col j (i = 1, 7, j = 1, 7). Item parameters are presented in Table 13: twenty five 2D items (required items of required sizes and colors) (the width of the paper horizontal bar equals 20).
Evidently, two objective functions are considered: (i) minimizing the volume of non-used domain in bins, (ii) minimizing the total cost of color changes (e.g., as a total sum of color change estimates in the solution) (this function can be transformed to non-used bin domains as well). The following heuristic solving scheme is considered:
Stage 1.
Grouping of initial items by colors.
Stage 2. For each color: forming the general items (combinations of items of the same color) as packed bins (bin size equals 20). For the items in the same bin, their heights/lenghts are about close. Some initial items can be integrated (as items 3 and 4 in the example, Fig. 16). Here, bin packing problem can be used. As e result, a set of general items (the same color for each item) are obtained. In Fig. 16, the following 8 general items are depicted: (i) items 1, 2, and 3&4 (color col 1 ); (ii) items 5, 6, and 7 (color col 4 ); (iii) items 8, 9, 10, and 11 (color Stage 3. Forming the bins for each machine and for one period (from the general items): bin size corresponds to time period). Here bin packing problem can be used.
Stage 4. For each obtained bin: linear ordering of the generalized items while taking into account color changes. Here the traveling salesman problem can be used (while taking into account the ordinal estimates of color change as element distance, Table 12). Note the considered composite planning framework can be extended/modifued to use in communication systems (e.g., multiple channel systems).
Planning in communication system
The basic multi-processor scheduling problems based on bin packing have been described in [59,60,69,143]. Here, some combinatorial planning problems as 2D bin packing for communications (one-channel communications, telecommunication WiMAX systems). Note, close problems are used in resource allocation in multispot satellite networks (e.g., [4]).
Selection of messages/information packages
First, the basic simplified planning problem can be considered as the well-known secretary problem. Given a set of items n (e.g., messages) A = {a 1 , ..., a i , ..., a n }, each item a i has a weight w i (e.g., time for processing). The problem is (Fig. 17): Evidently, the algorithm to obtain the optimal solution is based on ordering of the items by nondecreasing of w i (i.e., the item with minimal weight has to be processed as the 1st, and so on) (complexity estimate of the algorithm is O(n log n)). This is the algorithm: 'smallest weight first'.
Note, the solution can be defined by Usually the described secretary problem is used for planning in one-channel communication system. In this case, there is a time interval (i.e., planning period T ) and the initial set of items A is ordered to send via the channel. If all messages can be send during period T (i.e., n i=1 w i ≤ T ) the considered algorithm can be successfully used. Unfortunately, if period T is not sufficient to send all message (i.e., n i=1 w i ≥ T ), a subset of items with highest weights have to wait the next period (i.e., a wait set). Here, the problem can be formulated as a knapsack model: Here, the algorithm above leads to the optimal solution. Note, the first objective function requires linear ordering of items in the solution by non-decreasing of w i (as in previous problem). After using the algorithm the items which do not belong to the solution can be considered as a wait set. Thus, it is reasonable to examine an extension of the problem above. Let each item a i ∈ A has two parameters: (i) the weight (i.e., processing time) w i and (ii) the number of wait periods γ i = 0, 1, ... . The problem statement can be considered as two-criteria knapsack model: This problem is NP-hard. The selection of items for sending (i.e., solution) can be based on detection of Pareto-efficient items by two parameters: (a) minimum weight w i (rule: smallest weight first) and (b) maximum number γ i (rule: longest wait first). The following heuristic algorithm can be considered:
Stage 1. Definition A = A. Stage 2.
Deletion of Pareto-efficient items in A by two parameters weight w i (minimum) and importance γ i (maximum) to obtain the subset A P ⊆ A (the current items layer by Pareto rule). Complexity estimates for the above-mentioned version hierarchical clustering algorithm (by stages) is presented in Table 14. Afterhere, the first objective function min t(S) = 1 n n ι=1 τ s[ι] x ai,s[ι] will not be considered because items of the solution can be ordered to take into account n the objective function.
Evidently, each item (message) can have other parameters, for example, importance (it will leads to an additional objective function in the model above). In this case, the model is: In addition, precedence binary relation over items can be examined as well. This leads to an additional logical constraint in the model above and corresponding algorithm scheme is based on linear ordering of the selected items while taking account the precedence constraint.
max n i=1 n ι=1 β ai x ai,
Two-dimensional packing in WiMAX system
In recent decade, two-dimensional packing problems have been used in contemporary telecommunication systems (IEEE 802.16/WiMAX standard) [55,56,57,197,206].
An illustrative structure of WiMAX system is depicted in Fig. 18. A general description of the above-mentioned approach is presented in [206] as follows. Information transmission process is based on rectangular frames "down link zones": time (width) × frequency (height). Thus, data packages correspond to 2D items (i.e., rectangular) which are stored in "down link zones" (i.e., bins). In [206], a general three phase solving scheme is examined:
Phase 1. Selection of information packages (messages) for the current transmission period. Phase 2. Arranging the selected packets into rectangular regions (as general items). Phase 3. Allocation of the resultant regions to the rectangular frame.
Note, the above-mentioned phase 1 can be based on model and solving approach from the previous section as selection of Pareto-efficient messages (information packages) for the current transmission period. The allocation problem above (i.e., phase 3) is studied in [55,56,57] (including problem statement, complexity, heuristic algorithms, computing experiments). In mobile broadband wireless access systems like IEEE 802.16/WiMAX, Orthogonal Frequency Division Multiple Access (OFDMA) is used in order to exploit frequency and multi-user diversity (i.e., improving the spectral efficiency). MAC (medium access control) frame extends in two dimensions, i.e., time and frequency. At the beginning of each frame, i.e., every 5 ms, the base station is responsible both for scheduling packets, based on the negotiated quality of service requirements, and for allocating them into the frame, according to the restrictions imposed by 802.16 OFDMA.
Here, a two-stage solving scheme for resource allocation is applied (e.g., [55,56,57]): (a) scheduling of packets in a given time frame, (b) allocation of packets across different subcarriers and time slots. The second stage above can be examined as a special 2D bin packing problem [55,56,57].
Evidently, integrated solving scheme for the two above-mentioned stages is a prospective research direction. In general, it is necessary to study the integrated approach for three-phase for planning in WiMAX system from [206]. In addition, it may be prospective to examine ordinal and/or multiset estimates for problem elements including lattice-based quality domain(s) for problem solutions.
Conclusion
In this paper, a generalized integrated glance to bin packing problems is suggested. The approach is based on a system structural problem description: (a) element sets (i.e., item set, bin set, item subset assigned to bin), (b) binary relation over the sets above: relation over item set(s) as compatibility, precedence, dominance; relation over items and bins (i.e., correspondence of items to bins). Here, the following objective functions can be examined: (1) traditional functions (i.e., minimizing the number of used bins, maximizing the number of assigned items), (2) weighted and vector versions of the functions above, and (3) Generally, it is necessary to point out the following. In recent decades there exists a trend in applied combinatorial optimization (e.g., [182,184,185]):
"FROM basic combinatorial problem TO composite framework consisting of several interconnected combinatorial problems". A well-known example of the composite frame is the following: timetabling problem that is usually based a combination of basic combinatorial optimization problems (e.g., assignment, clustering, graph coloring, scheduling). From this viewpoint, bin packing problems and their extensions/modifications can be examined as a basis of various applied composite frameworks. Thus, our material may be useful to build the applied composite frameworks above.
In the future, it may be reasonable to investigate the following research directions: 1. further examination of bin packing problems with multiset estimates; 2. study of various versions of colored bin packing problems (e.g., various color proximities, various objective functions);
3. examination of multi-stage bin packing problems (i.e., models, methods, applications); 4. examination of new applied composite frameworks based on bin packing problems; 5. execution of computing experiments to compare many solving schemes for various bin packing problems with ordinal/multiset estimates;
6. examination of multi-period (or cyclic) multi-channel scheduling problems based on various bin packing models;
7. study of resource allocation in multispot satellite networks on the basis of various bin packing problems;
8. analysis of applied multicriteria bin packing problems and bin packing problems with multiset estimates; and 9. usage of our material in educational courses (e.g., applied mathematics, computer science, engineering, management).
problem formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Maximizing the number of packed items (inverse problems) . . . . . . . . . . . . . . . . . 11 2.3 Interval multiset estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Support model: morphological design with ordinal and interval multiset estimates . . . . 13 3 Problems with multiset estimates 15 3.1 Some combinatorial optimization problems with multiset estimates . . . . . . . . . . . . . 15 3.1.1 Knapsack problem with multiset estimates . . . . . . . . . . . . . . . . . . . . . . . 15 3.1.2 Multiple choice problem with interval multiset estimates . . . . . . . . . . . . . . . 15 3.1.3 Multiple knapsack problem with multiset estimates . . . . . . . . . . . . . . . . . . 16 3.1.4 Assignment and generalized assignment problems with multiset estimates . . . . . 17 3.2 Inverse bin packing problem with multiset estimates . . . . . . . . . . . . . . . . . . . . . 18 3.3 Bin packing with conflicts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4 Colored bin packing 20 4.1 Basic colored bin packing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 Two auxiliary graph coloring problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2.1 Auxiliary vertex graph coloring problem with ordinal color proximity . . . . . . . . 20 4.2.2 Partition coloring problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Fig. 2
2illustrates the classic bin packing (i.e., packing the items into the minimal number of bins): 6 items are packed into 3 bins.
Fig. 2 .
2Illustration
Fig. 3 .
3Generalized items A = {a 1 , a 2 , ...Further, it is necessary to point out binary relations: I. Binary relations over initial items and bins (items A = {a 1 , a 2 , ..., a n }, bins B = {B 1 , ..., B κ , ..., B m }): 1.1. correspondence of items to bins or preference (for each item) as binary relation (or weighted binary relation): R A×B .
(i.e., "interval multiset estimate"). Thus, an estimate e for an alternative A is (scale [1, l], position-based form or position form): e(A) = (η 1 , ..., η ι , ..., η l ), where η ι corresponds to the number of elements at the level ι (ι = 1, l), or e(A) = { η1 1, ..., 1, η2 2, ...2, η3 3, ..., 3, ..., η l l, .
Fig. 8 .
8Multiset based scale, estimates (based on [183,185]) (a) interval multset based poset-like scale by elements (P 3,3 )
Fig. 8b depicts the integrated poset-like scale for tree-component system (ordinal scale for system component compatibility is [0, 1, 2, 3]).
a composite design alternative S = S(1) ⋆ ... ⋆ S(i) ⋆ ... ⋆ S(m) of DAs (one representative design alternative S(i) for each system component/part R(i), i = 1, m ) with non-zero compatibility between design alternatives.A discrete "space" of the system excellence (a poset) on the basis of the following vector is used: N (S) = (w(S); e(S)), where w(S) is the minimum of pairwise compatibility between DAs which correspond to different system components (i.e., ∀ R j1 and R j2 ,1 ≤ j 1 = j 2 ≤ m) in S, e(S) = (η 1 , ..., η ι , ..., η l ),where η ι is the number of DAs of the ιth quality in S. Further, the problem is described as follows: max e(S), max w(S), s.t. w(S) ≥ 1. As a result, we search for composite solutions which are nondominated by N (S) (i.e., Pareto-efficient). "Maximization" of e(S) is based on the corresponding poset. The considered combinatorial problem is NP-hard and an enumerative solving scheme is used. Here, combinatorial synthesis is based on usage of multiset estimates of design alternatives for system parts. For the resultant system S = S(1) ⋆ ... ⋆ S(i) ⋆ ... ⋆ S(m) the same type of the multiset estimate is examined: an aggregated estimate ("generalized median") of corresponding multiset estimates of its components (i.e., selected DAs). Thus, N (S) = (w(S); e(S)), where e(S) is the "generalized median" of estimates of the solution components. Finally, the modified problem is: max e(S) = M g = arg min M∈D m i=1 |δ(M, e(S i ))|, max w(S), s.t. w(S) ≥ 1.
Fig. 9
9depicts the corresponding "two"-dimensional space of solution quality.
Fig. 9 .
9"
where x ij = 1 if item i is selected for knapsack B j , and x ij = 0 otherwise.In the case of multiset estimates, item "utility" e i , i = 1, m (instead of c i ) is considered. Multiple knapsack problem with multiset estimates and the integrated estimate for the solution is (solution S = {(i, j)|x ij = 1}): max e(S) =
Fig. 10 .
10Generalized assignment problem
Fig. 11 .
11Illustration
corresponding indices for colors of vertices).
Fig. 12 .
12Example
Find
coloring of vertices C * (G) for a given graph G = (A, E) with the minimum number of used colors (labels): min {C(G)} |C * (G = (A, E))| s.t. C * (a i ) = C * (a j ) ∀(a i , a j ) ∈ E, i = j.
Fig. 13 .
13Coloring
Fig. 15 .
15Instance
Fig. 16 .
16Composite
col 1 col 2 col 3 col 4 col 5 col 6 col 7 col 1
col 5 ); (iv) items 12, 13, and 14 (color col 2 ); (v) items 15, 16, and 17 (color col 6 ); (vi) items 18 and 19 (color col 3 ); (vii) items 20, 21, and 22 (color col 3 ); and (viii) items 23, 24, and 25 (color col 7 ).
Find
the schedule (i.e., ordering of items as permutation) of the items from A: S =< s[1], ..., s[ι], ..., s[n] > (s[ι]) corresponds to an item a i that is processed at the ι-th place in schedule S) such that average completion time for each item a i ∈ A (i.e., sum of waiting time and processing time) is minimal: t(S) = 1 n n ι=1 τ s[ι] , where the waiting time is as follows (τ s[1] = w s[1] , ι = 2, n): τ s[ι] = w s[ι] + ι−1 κ=1 w s[κ] ∀κ = 1, n.
,s[ι] ≤ 1 ∀i = 1, n, x ai,s[ι] ∈ {0, 1}.
ai,s[ι] w i ≤ T, x ai,s[ι] ∈ {0, 1}.
Stage 3 .
3Assignment of items from A P to bins. Stage 4. Definition subset A = A\A P . If | A| = 0 that GO TO Stage 5 Otherwise GO TO Stage 2. Stage 5. Stop.
Fig
Fig. 18. Structure of WiMAX system
the objective functions based on lattices. Some new problem statements with multiset estimates of items are presented. Two applied examples are considered: (i) planning in paper industry, (ii) planning in communication systems (selection of messages, packing of massages in WiMAX).
Table 1 .
1Correspondence of items to bins R L×B
Table 4 .
4Main bin packing problem formulationsNo. Problem
Table 5 .
5Binpacking problems with multi-component items, with binary relations
No. Problem
Some source(s)
I.
Problems with multi-component items/items fragmentation:
1.1. Bin packing with multi-component items
[115]
1.2. Packing with item fragmentation
[245]
1.3. Packet scheduling with fragmentation
[216]
II. Colored bin packing:
2.1. Offline black and white bin packing
[14]
2.2. Online black and white bin packing
[15]
2.3. Colored bin packing
[51,257]
2.4. Offline colored bin packing
[257]
2.5. Online colored bin packing
[34,257]
2.6. Online bin coloring (packing with minimum colors)
[167]
2.7. Composite planning framework in paper production system
This paper
III. Multcriteria/multobjective bin packing, relations over items:
3.1. Bin packing with conflicts
[121,98,107,149,237]
3.2. Bin packing with multicriteria items
[115]
3.3. Multi-objective bin packing
[189,217]
3.4. Multi-objective bin packing with rotations
[108]
3.5. Problems with preference over items
[115]
3.6. Problems with precedence among items
Table 6 .
6Some applications of bin packing problemsNo. Domain(s)/Problem(s)
Table 7 .
7Mainalgorithmic approaches, part I: basic methods
No. Solving approach
Some source(s)
I.
Fitting algorithms (i.e., classical ones) and their combinations:
[63,67,68,120]
1.1. Next Fit (NF) algorithm
[277]
1.2. Next-fit (NFD) decreasing algorithm
[277]
1.3. First-Fit (FF) (on-line)
[12,84]
1.4. First-Fit decreasing (FFD) (off-line)
[12,83,248]
1.5. Best-Fit (BF) (on-line)
[26,84,277]
1.6. Best-Fit decreasing (BFD) algorithm (off-line)
[248,277]
1.7. Worst Fit (WF) algorithm (makespan context)
[67]
1.8. Worst Fit decreasing (makespan context)
[67]
1.9. Shelf algorithms (for 2D bin packing problems)
[13,75]
II.
Exact enumerative methods:
2.1. Surveys
[81]
2.2. Branch-and-bound algorithms
[91,171,214,222,240,260]
2.3. Branch-and-price algorithms
[221,262,263]
2.4. Exact column generation and branch-and-bound method
[260,263]
2.5. Bin completion algorithm
[114]
(bin-oriented branch-and-bound strategy)
III.
Basic approximation algorithms:
3.1. Surveys
[63,68,81,234]
3.2. Near-optimal algorithms for bin packing
[150,234]
3.3. Fast algorithms for bin packing
[151]
3.4. Linear-time approximation algorithms for bin packing
[276]
3.5. Efficient approximation scheme
[156]
3.6. Efficient approximation scheme for variable sized bin packing
[215]
3.7. Asymptotic Polynomial Time Approximation Scheme (APTAS) [14,21,106,257]
3.8. Asymptotic Fully Polynomial Time Approximation Scheme
[14,156]
(AFPTAS)
3.9. Augmented asymptotic PTAS
[71]
3.10. Robust APTAS (for classical bin packing)
[99]
3.11. Approximation schemes for multidimensional problems
[18,19]
Table 8 .
8Main Combinations of evolutionary algorithms and hyper-heuristics[42,198,236] 5.5. Combination of Lagrangian relaxation and column generation[92] algorithmic approaches, part II: heuristics
No. Solving approach
Some source(s)
IV.
Heuristics:
4.1. Surveys and heuristics comparison
[65,86,133]
4.2. Basic heuristics
[79,121,122,140,193,199]
4.3. Local search algorithms
[187,220]
4.4. Greedy procedures
[54]
4.5. Variable neighborhood search procedures
[54,109]
4.6. Dynamic programming based heuristics
[222]
4.7. Simulated annealing based algorithms
[22,154,254]
4.8. Tabu search algorithms
[6,196,240]
4.9. GRASP algorithms
[175]
4.10. Ant colony algorithms
[187]
4.11. Quantum inspired cuckoo search algorithms
[176]
4.12. Set-covering-based heuristics
[19,213]
4.13. Average-weight-controlled bin-oriented heuristics
[110]
4.14. Bottom-left bin packing heuristic (for 2D problem)
[48]
4.15. Heuristic for 2D and 3D large bin packing
[201]
V.
Hybrid approaches, metaheuristics and hyper-heuristics:
5.1. Hybrid approach, metaheuristrics for 2D bin packing
[77,137,140,141,193]
5.2. Hyper-heuristics, generalized hyper-heuristics
[247,255]
5.3. Unified hyper-heuristic framework
[200]
5.4.
Table 9 .
9Studies in multisets and their applications No. Research direction(s) Interval multiset estimates, operations over multisets [183,185] (e.g., proximity, summarization, aggregation) 1.10. Perturbation of multisets (measure of remoteness between multisets) [165] 1.11. Multiset processing (general) Classification (e.g., classification of credit cardholders) [224] 2.9. Applications in decision making (e.g., multicriteria ranking/sorting) [20,223,225] 2.10. Processing of data streams [9,126,127] 2.11. Evaluation of composite system(s)/alternative(s) [179,180,183,185] 2.12. Knapsack problem [183,185] 2.13. Multiple choice knapsack problem [183,185] 2.14. Combinatorial synthesis (morphological system design) [179,180,183,185] 2.4. Support model: morphological design with ordinal and interval multiset estimates A brief description of combinatorial synthesis (Hierarchical Morphological Multicriteria Design -HMMD) with ordinal estimates of design alternatives is the following (Source(s)
Table 11 .
11Algorithmsfor partition coloring problem
No. Approach
Source(s)
1.
Branch-and-price approach [10,112,113,142]
2.
Tabu search heuristic
[219]
3.
Two-phase heuristic
[219]
4.
Table 12 .
12Ordinal estimates of color change (col i ⇒ col j ) col i \col j
Table 13 .
13Items and their parameters
Item Width Height/lenght Color General item Machine Time interval
1
8
43
col 1
I
1
1
2
5
30
col 1
I
1
1
3
6
21
col 1
I
1
1
4
5
21
col 1
I
1
1
5
5
36
col 4
II
1
2
6
7
33
col 4
II
1
2
7
7
28
col 4
II
1
2
8
4
25
col 5
III
2
1
9
5
24
col 5
III
2
1
10
6
23
col 5
III
2
1
11
5
22
col 5
III
2
1
12
5
26
col 2
IV
2
2
13
8
25
col 2
IV
2
2
14
5
23
col 2
IV
2
2
15
8
26
col 6
V
2
3
16
6
25
col 6
V
2
3
17
5
23
col 6
V
2
3
18
10
24
col 3
V I
3
1
19
9
23
col 3
V I
3
1
20
6
24
col 3
V II
3
2
21
5
23
col 3
V II
3
2
22
7
22
col 3
V II
3
2
23
6
30
col 7
V III
3
3
24
8
27
col 7
V III
3
3
25
6
25
col 7
V III
3
3
Boolean variables: x ai,s[ι] ∈ {0, 1}, where x ai,s[ι] = 1 if item a i is assigned into place s[ι] in the solution. Thus, the solution is defined by Boolean matrix: X = ||x ai,s[ι] ||, i = 1, n, ι = 1, n.Fig. 17. Illustration for secretary problem✲
0
t
s[1]
w s[1]
...
...
s[ι − 1]
w s[ι−1]
s[ι]
w s[ι]
...
...
s[n]
w s[n]
✛
✲
τ s[ι]
Table 14 .
14Complexity Deletion of current Pareto-efficient items layer A P ⊆ A in A O(n 2 ) (by parameters w i and γ i ) Stage 3. Assignment of items from A P to bins. O(n) Stage 4 A = A\A P . If all items are processed GO TO Stage 2.estimates
s[ι] , max ai,s[ι] w i ≤ T, x ai,s[ι] ∈ {0, 1}, where β ai is importance parameter of the corresponding item i. Note the importance parameter may by dependent on scheduling place s[ι]: β ai,s[ι] . In the case of multiset estimate of the importance parameter e ai,s[ι] , the model is: max M = arg min ai,s[ι] w i ≤ T, x ai,s[ι] ∈ {0, 1},n
i=1
n
ι=1
x ai,s[ι] ,
max
n
i=1
n
ι=1
x ai,s[ι] γ i
s.t.
n
i=1
n
ι=1
x M∈D
|
i∈{i|x a i ,s[ι] =1}
δ(M, e i )|,
max
n
i=1
n
ι=1
x ai,s[ι] ,
max
n
i=1
n
ι=1
x ai,s[ι] γ i
s.t.
n
i=1
n
ι=1
x
. 18. Structure of WiMAX systemWiMAX
base
station
t ❢ ✐
✟
✟ ✟ ✯
To Internet
✁
✁
✁
✁
✁
✁
✁ ✁ ❆
❆
WiMAX
base
station
t ❢ ✐
✁ ✁ ❆
❆
✏ ✏ ✏ ✏ ✏
❳
❳
❳
❳
❳
❳
❍
❍
❍
❍
❍
❍
✘
✘
✘
✘
✘
✘
❍ ❍
❍ ❍
❍ ❍
Office
network
(WiFi)
s ❣ ✟ ✟
❍ ❍
r
r
...
Cafe network
(WiFi)
s ❣
❅ ❅
r
r
...
Network
of
mobile
users
r
r
...
Private (home)
network (WiFi)
s ❣
❅ ❅
r
r
...
Enumeration methods or heuristics can be used ([179,180,183,185]).
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| []
|
[
"Improving the Simulation of Quark and Gluon Jets with Herwig 7",
"Improving the Simulation of Quark and Gluon Jets with Herwig 7"
]
| [
"Daniel Reichelt \nInstitut für Kern-und Teilchenphysik\nTechnische Universität Dresden\n\n",
"Peter Richardson \nTheory Department\nCERN\nGeneva\n\nDepartment of Physics\nIPPP\nDurham University\n\n",
"Andrzej Siodmok "
]
| [
"Institut für Kern-und Teilchenphysik\nTechnische Universität Dresden\n",
"Theory Department\nCERN\nGeneva",
"Department of Physics\nIPPP\nDurham University\n"
]
| []
| The properties of quark and gluon jets, and the differences between them, are increasingly important at the LHC. However, Monte Carlo event generators are normally tuned to data from e + e − collisions which are primarily sensitive to quark-initiated jets. In order to improve the description of gluon jets we make improvements to the perturbative and the non-perturbative modelling of gluon jets and include data with gluon-initiated jets in the tuning for the first time. The resultant tunes significantly improve the description of gluon jets and are now the default in Herwig 7.1. | 10.1140/epjc/s10052-017-5374-8 | [
"https://arxiv.org/pdf/1708.01491v1.pdf"
]
| 59,038,969 | 1708.01491 | 8f97363e6074149a9d5eb9116e84fea95d322cf7 |
Improving the Simulation of Quark and Gluon Jets with Herwig 7
Daniel Reichelt
Institut für Kern-und Teilchenphysik
Technische Universität Dresden
Peter Richardson
Theory Department
CERN
Geneva
Department of Physics
IPPP
Durham University
Andrzej Siodmok
Improving the Simulation of Quark and Gluon Jets with Herwig 7
Received: date / Accepted: dateEur. Phys. J. C manuscript No. (will be inserted by the editor)
The properties of quark and gluon jets, and the differences between them, are increasingly important at the LHC. However, Monte Carlo event generators are normally tuned to data from e + e − collisions which are primarily sensitive to quark-initiated jets. In order to improve the description of gluon jets we make improvements to the perturbative and the non-perturbative modelling of gluon jets and include data with gluon-initiated jets in the tuning for the first time. The resultant tunes significantly improve the description of gluon jets and are now the default in Herwig 7.1.
Introduction
Monte Carlo generators are essential tools, both for the design of future experiments and the analysis of data from the LHC, and previous collider experiments. Modern event generators [1][2][3] provide a simulation of exclusive events based on the combination of fixed-order perturbative results, resummation of large logarithms of scales using the partonshower approach and non-perturbative models of hadronization and multiple-parton scattering. 1 These simulations rely on universality and factorization in order to construct a simulation of the complex final states observed in hadronic collisions. This allows the simulation of final-state radiation in the parton shower and the nonperturbative hadronization models to be first developed, and the parameters of the the model tuned, using the simpler and cleaner environment of e + e − collisions, and then applied to more complicated hadronic collisions. These models are then combined with the parton-shower simulation of initialstate radiation, a multiple scattering model of the underlying a e-mail: [email protected] b e-mail: [email protected] c e-mail: [email protected] 1 For a recent review of modern Monte Carlo event generators see [4]. event and a non-perturbative colour reconnection model in order to describe hadronic collisions. In principle universality requires that the colour reconnection model is also used to describe leptonic collisions. In practice however colour reconnection has little effect on the distributions which so far have been used to develop and tune the models. These models are therefore usually either not included at all for the simulation of leptonic collisions, or if they are, the parameters are determined by tuning to hadronic data sensitive to multiple partonic scattering.
As the LHC accumulates data at an unprecedented rate there are a number of observables which are not well described by current Monte Carlo event generators, and where the limitations of this approach have started to become obvious, for example:
the difference in the properties of jets initiated by quarks and gluons is not well described with generators predicting either a larger or smaller difference between the jets than is observed by the LHC experiments [5]; the transverse momentum spectra of identified baryons and strange hadrons which are not well described by current generators. [6]; long-range correlations in high multiplicity events [7,8].
In this paper we will focus on improvements to the perturbative and non-perturbative modelling to give a better description of both quark-and gluon-initiated jets, as well as the differences between them in Herwig 7. Beyond leading order there is no clear distinction between quark and gluon jets and the definition will depend on the analysis. 2 As e + e − annihilation to hadrons starts with an initial partonic quark-antiquark configuration the data used to develop the final-state parton-shower algorithm, tune its parameters and those of the hadronization model, are dominated by quarkinitiated jets. However at the LHC jets initiated by gluons can often dominate, depending on the production process, rapidity and transverse momentum of the jets. Regrettably while there is great interest in the differences between quark and gluon jets at the LHC most of the experimental studies have concentrated on differentiating between quark and gluon jets using neural network, or similar, techniques which makes a direct comparison with simulated hadron-level events impossible. We will therefore use some recent data from the ATLAS experiment [10] which is sensitive to both quark and gluon jet properties, together with data on gluon jets in e + e − collisions from the OPAL experiment [11,12] which has not previously been used in the development and tuning of the current generation of Monte Carlo event generators to study the properties of gluon jets.
In the next section we will first recap the default partonshower algorithm used in Herwig 7 focusing on recent changes we have made to improve the simulation of both quark and gluon jets. In Section 3 we will briefly review the important parameters in the cluster hadronization model used in Herwig 7 and identify the issues which may lead to different treatments of quark and gluon jets. We will then discuss the tuning strategy used to produce the tunes presented in this paper. We present our results in Section 5 3 followed by our conclusions.
Herwig 7 Parton-Shower Algorithm
The default Herwig 7 parton-shower algorithm [13] is an improved angular-ordered parton shower. In this approach the momenta of the partons produced in the parton shower are decomposed in terms of the 4-momentum of the parton initiating the jet, p (p 2 = m 2 , the on-shell parton mass-squared), a light-like reference vector, n, in the direction of the colour partner of the parton initiating the jet and the momentum transverse to the direction of p and n. The four momentum of any parton produced in the evolution of the jet can be decomposed as
q i = α i p + β i n + q ⊥i ,(1)
where α i and β i are coefficients and q ⊥i is the transverse four momentum of the parton (q ⊥i · p = q ⊥i · n = 0). If we consider the branching of a final-state parton i to two partons j and k, i.e. i → jk as shown in Fig. 1, the branching is described by the evolution variablẽ
q 2 i = q 2 i − m 2 i z i (1 − z i ) ,(2)
where q 2 i is the square of the virtual mass developed by the parton i in the branching, m i is the physical mass of parton i, 3 Additional results on quark and gluon jet discrimination power are included in the Appendix.q iq k q j z i 1 − z i Fig. 1 Branching of the parton i to produce the partons j, k which then undergo subsequent branching. and z i is the momentum fraction of the parton j defined such that
α j = z i α i , α k = (1 − z i )α i .(3)
The transverse momenta of the partons produced in the branching are
q ⊥ j = z i q ⊥i + k ⊥i q ⊥k = (1 − z i )q ⊥i − k ⊥i ,(4)
where k ⊥i is the transverse momentum generated in the branching. In this case the virtuality of the parton i is
q 2 i = p 2 Ti z(1 − z) + m 2 j z + m 2 k 1 − z ,(5)
where p T is the magnitude of the transverse momentum produced in the branching defined such that k 2 ⊥i = −p 2 Ti . In this case the probability for a single branching to happen is
dP = dq 2 ĩ q 2 i α S 2π dφ i 2π dz i P i→ jk (z,q),(6)
where P i→ jk (z,q) is the quasi-collinear splitting function, and φ i is the azimuthal angle of the transverse momentum k ⊥i generated in the splitting. As the branching probability is singular for massless partons an infrared cut-off is required to regularise the singularity. In HERWIG 6 [14] and early versions of Herwig++ [15] the cut-off was implemented by giving the partons an infrared mass. However while this remains an option in later versions of Herwig++ and Herwig 7 [1] the default cut-off is now on the minimum transverse momentum of the branching [16].
In order to resum the dominant subleading logarithms [17] the transverse momentum of the branching is used as the scale for the strong coupling constant. This also means that the strong coupling used in the parton shower is that defined in the Catani-Marchesini-Webber (CMW) scheme which includes the subleading terms via a redefinition of QCD scale, Λ QCD .
While this specifies both the branching probability and kinematics of the partons for a single emission in the case of subsequent emission from the daughter partons j and/or k we must decide which properties of the originally generated kinematics to preserve once the masses of j and/or k in Eqn. 5 are no longer the infrared cut-off masses but the virtualities generated by any subsequent emissions. While this choice is formally subleading it can have a large effect on physical observables.
In Herwig++ the transverse momentum of the branching was calculated using Eqn. 5 and the infrared cut-off masses when the emission was generated and then preserved during the subsequent evolution of the daughter partons. In Herwig 7.0 the default option was to instead preserve the virtuality of the branching and calculate the transverse momentum of the branching using the virtual masses the daughter partons develop due to subsequent emissions. This means that if the daughter partons develop large virtual masses the transverse momentum of the branching is reduced, and in some cases the branching has to be vetoed if there is no solution of Eqn. 5. However, this choice inhibits further soft emission and significantly changes the evolution by vetoing emissions and leads to incorrect evolution of observables. We therefore consider a further choice in which if it is possible to preserve the virtuality and still have a solution for p 2 T > 0 we do so, however if this is not possible instead of vetoing the emission we set p T = 0 and allow the virtuality to increase.
The most important parameters which affect the behaviour of the parton shower and which we will tune in this paper are:
the choice of whether to preserve p T or q 2 during the subsequent evolution; the value of the strong coupling constant AlphaMZ, taken to be α CMW S (M Z ), value of the coupling constant in the CMW scheme at the mass of the Z boson, M Z ; the cut-off in the parton shower 4 . For a cut-off in p T this is the minimum transverse momentum allowed for the branchings in the shower, p min T . For a virtuality cut-off we parameterize the threshold for different flavours as
Q g = max δ − am q b , c ,(7)
where a and b are parameters chosen to give a threshold which is slightly reduced for heavier quarks. The parameter c = 0.3 GeV is chosen to prevent the cutoff becoming too small, we also keep the default value of b = 2.3. Only the parameters δ (cutoffKinScale) and a (aParameter) are tuned to the data. There is one other major feature of the angular-ordered parton shower which we need to consider. The angular ordering of the parton shower, which is used to implement the phenomenon of colour coherence, leads to regions of phase space in which there is no gluon emission. Consider for example the process e + e − → qqg. In this case there is a deadzone which is not filled by one emission from the parton shower, as shown in Fig. 2. Given this deficit of hard, wideangle emission it is necessary to combine the parton-shower with the fixed-order calculation of e + e − → qqg. There are now a range of techniques which can achieve this including both the next-to-leading order normalization of the total cross section, or including the fixed-order results for multiple emissions. However, for our purposes it is sufficient to consider the simplest matrix-element correction approach where the dead-zone is filled using the leading-order matrix element for e + e − → qqg, as shown in Fig. 3, together with the reweighting of emission probability, Eqn. 6, to the exact leading-order result, for any emission which could have the highest transverse momentum in the parton shower. 5 The choice of whether to preserve the transverse momentum or virtuality of the branching affects the phase-space region which is filled by the shower in the case of multiple emission. In this case we cluster the partons using the Durham jet algorithm [19], using the p-scheme as implemented in FastJet [20], keeping track of the partons emitted by the quark and antiquark and then take the hardest additional jet to be the gluon. The resulting Dalitz plots of e + e − → qq show that while the choice to preserve the transverse momentum of the branching leads to a significant number of events in the dead-zone, Fig. 4, if the virtuality of the branching is preserved, Fig. 5, there is little emission outside the original angular-ordered region.
x i = 2E i /Q
where E i is the energy of parton i and Q is the centre-of-mass energy of the collision.
Hadronization and Colour Reconnection
All the Herwig family of event generator generators use the cluster hadronization model [21]. This model is based on the phenomena of colour pre-confinement, i.e. if we nonperturbatively split the gluons left at the end of the parton shower into quark-antiquark pairs and cluster quarks and antiquarks into colour-singlet clusters the mass spectrum of these clusters is peaked at masses close to the cut-off in the parton shower, falls rapidly as the cluster mass increases, and is universal, i.e. the mass distribution of these clusters is independent of the hard scattering process and its centre-ofmass energy. The cluster model assumes that these clusters are a superposition of heavy hadronic states and uses a simple phase-space model for their decay into two hadrons. The main parameters of the model are therefore:
the non-perturbative gluon mass, which is not very sensitive and we do not tune; the parameters which control the probability of producing baryons and strange quarks during cluster decay; the parameter which controls the Gaussian smearing of the direction of the hadrons produced which contain a parton from the perturbative evolution about the direction of that parton, with separate values for light, charm and bottom quarks.
There are however a small fraction of large mass clusters for which the two hadron decay ansatz is not reasonable and these must first be fissioned into lighter clusters. While only a small fraction of clusters undergo fission due to the larger masses of these clusters they produce a significant fraction of the hadrons.
A cluster is split into two clusters if the mass, M, is such that
M Cl pow ≥ Cl max Cl pow + (m 1 + m 2 ) Cl pow ,(8)
where Cl max and Cl pow are parameters of the model, and m 1,2 are the masses of the constituent partons of the cluster. For clusters that need to be split, a qq pair is selected to be popped from the vacuum. The mass distribution of the new clusters is given by
M 1 = m 1 + (M − m 1 − m q )R 1/P split 1 ,(9a)M 2 = m 2 + (M − m 2 − m q )R 1/P split 2 ,(9b)
where m q is the mass of the parton popped from the vacuum, M 1,2 are the masses of the clusters formed by the splitting and R 1,2 are pseudo-random numbers uniformly distributed between 0 and 1. The distribution of the masses of the clusters is controlled by the parameter P split .
In order to improve the description of charm and bottom hadron production these parameters for cluster fission all depend on the flavour of the partons in the cluster so that there are separate parameters for light, charm and bottom quarks.
In practice there is always a small fraction of clusters that are too light to decay into two hadrons. Before Herwig 7.1 these clusters were decayed to the lightest hadron, with the appropriate flavours. However in some cases, for example for clusters containing a charm or bottom quarkantiquark pair, or a bottom quark and a light antiquark, there can be a number of hadrons of the appropriate flavour below the threshold. In these cases the lightest meson with the appropriate flavours is the pseudoscalar 1 S 0 state and the vector 3 S 1 state is also below the threshold 6 which leads to a lower production rate for the vector state with respect to the pseudoscalar state than expected. For the mesons composed of a bottom quark and a light quark the rate is significantly less than that expected from the counting of spin states, or indeed observed experimentally [22][23][24][25]. For charmonium and bottomonium states as this mechanism is the only way the vector states can be produced via hadronization it leads to a complete absence of direct J/ψ and ϒ production. In Herwig 7.1 we therefore include the possibility that instead of just producing the lightest state all states below the threshold are produced with a probability proportional to 2S + 1, where S is the spin of the particle.
In order to improve the behaviour at the threshold for charm and bottom clusters the option exists of allowing clusters above the threshold mass, M threshold , for the production of two hadrons to decay into a single hadron such that a single hadron can be formed for masses
M < M limit = (1 + SingleHadronLimit)M threshold ,(10)
where SingleHadronLimit is a free parameter of the model. The probability of such a single-meson cluster decay is assumed to decrease linearly for M threshold < M < M limit and there are separate parameters for charm and bottom clusters. In order to explain the rising trend of p t vs N ch (average transverse momentum as a function of the number of charged particles in the event) observed already by UA1 [26] and describe Underlying Event [27][28][29][30] and the Minimum Bias data [31][32][33][34], the hadronization model is supplemented with a model of colour reconnections (CR) [35]. The default version of the model implemented in Herwig 7.0 is not very sophisticated. The colour reconnection model defines the distance between two partons based on their invariant mass, i.e. the distance is small when their invariant mass (cluster mass) is small. The aim of the CR model is to reduce the colour length λ ≡ ∑ N cl i=1 m 2 i , where N cl is the number of clusters in an event and m i is the invariant mass of cluster i. The colour reconnection of the clusters leading to a reduction of λ is accepted with a given probability which is a parameter of the model. Although the default model is quite simple it should be stressed that its results resemble the more sophisticated statistical colour reconection model [35] which implements the minimization of λ as Metropolis-like algorithm and requires a quick "cooling" of the random walk.
In this model the only possible reconnections which are not allowed are connecting the quark and antiquark produced in the non-perturbative splitting of the gluon. It is therefore possible that the colour lines of a gluon produced at any other stage of the shower can be reconnected leading to the production of a colour-singlet object. While this is physically possible we would expect that it occurs at a rate which is suppressed in the number of colours, N C , as ∼ 1
N 2 C = 1 9
, not the much higher reconnection rate ∼ 2/3 7 which is necessary to describe the underlying event data. This can lead to the production of a colour-singlet gluon jet at a much higher rate than expected. This is particularly problematic in the theoretically clean, but experimentally inaccessible, colour-singlet gluon pair production processes often used to study gluon jets [9].
Consider, for example, the simple process of coloursinglet gluon pair production followed by the branching of all the gluons via g → gg, shown in Fig. 6a. After the nonperturbative splitting of the gluons into quark-antiquark pairs, as shown in Fig. 6b, without colour reconnection the quarks and antiquarks will be formed into colour-singlet clusters as (q 1 ,q 3 ), (q 3 ,q 4 ), (q 4 ,q 2 ) and (q 2 ,q 1 ). Given the configuration it is likely that the clusters containing partons from the 7 The value from the tune of Herwig 7.1 with a new soft and diffractive model [36]. and (q 4 ,q 2 ), will have large masses and the rearrangement to give the clusters (q 1 ,q 2 ) and (q 4 ,q 3 ) will be kinematically favoured, although it means the original gluons will effectively become colour singlets rather than octets. In Herwig 7.1 we have therefore included the possibility to forbid the colour reconnection model making any reconnection which would lead to a gluon produced in any stage of the parton-shower evolution becoming a colour-singlet after hadronization. We will investigate the effect of this change on the simulation of quark and gluon jets.
Tuning
The Rivet [37] program was used to analyse the simulated events and compare the results with the experimental measurements. The Professor program [38] was then used to interpolate the shower response and tune the parameters by minimising the chi-squared. 8 In general we use a heuristic chi-squared function
χ 2 (p) = ∑ O w O ∑ b∃O ( f b (p) − R b ) 2 ∆ 2 b(11)
where p is the set of parameters being tuned, O are the observables used each with weight w O , b are the different bins in each observable distribution with associated experimental measurement R b , error ∆ b and Monte Carlo prediction f b (p). Weighting of those observables for which a good description of the experimental result is important is used in most cases. The parameterisation of the event generator response, f (p), is used to minimize χ 2 and find the optimum parameter values. We take w O = 1 in most cases except for the particle multiplicities where we use w O = 10 and total charged particle multiplicities where we use w O = 50. This ensures that particle multiplicities influence the result of the fit and are required due to the much higher quantity of event 8 While tuning the parameters sensitive to bottom quarks it proved impossible to get a reliable interpolation of the generator response with Professor and therefore a random scan of the bottom parameters was performed and the values adjusted by hand about the minimum to minimise the χ 2 .
shape and spectrum data used in the tuning. Given the aim of this paper is to improve the description of gluon jets this data was also included with w O = 10 in order to avoid the fit being dominated by the large quantity of data sensitive to quark jets. In addition as we do not except a Monte Carlo event generator to give a perfect description of all the data and in order to avoid the fit being dominated by a few observables with very small experimental errors we use
∆ eff b = max(0.05 × R b , ∆ b ),(12)
rather than the true experimental error, ∆ b , in the fit. The standard procedure which was adopted to tune the shower and hadronization parameters of the Herwig++ and Herwig 7 event generators to data is first the shower and those hadronization parameters which are primarily sensitive to light quark-initiated processes are tuned to LEP1 and SLD measurements of event shapes, the average charged multiplicity and charged multiplicity distribution, and identified particle spectra and rates which only involve light quark mesons and baryons; the hadronization parameters for bottom quarks are tuned to the bottom quark fragmentation function measured by LEP1 and SLD together with LEP1 and SLD measurements of event shapes and identified particle spectra from bottom events; the hadronization parameters involving charm quarks are then tuned to identified particle spectra, from both the Bfactories and LEP1, and LEP1 and SLD measurements of event shapes and identified particle spectra from charm events; the light quark parameters are then retuned using the new values of the bottom and charm parameters together with different weights for the charged multiplicity distributions in e + e − collisions at energies between 12 GeV and 209 GeV due to the difficulty in fitting the charged multiplicity.
Only e + e − annihilation data from the continuum region near the ϒ (4s) meson, for charm meson spectra, and at the Z-pole from LEP1 and SLD were used in the tune.
In this paper we have extended this approach in order to better constrain the energy evolution to include data from a wider range of centre-of-mass energies both below the Zpole, from the JADE and TASSO experiments, and above the Z-pole, from LEP2.
In order to tune the shower and light quark hadronization parameters we used data on jet rates and event shapes for centre-of-mass energies between 14 and 44 GeV [39][40][41], at LEP1 and SLD [41][42][43][44][45] and LEP2 [41,44,45], particle multiplicities [42,43] and spectra [42,43,[46][47][48][49][50][51][52][53][54][55][56] at LEP 1, identified particle spectra below the ϒ (4S) from Babar [57], the charged particle multiplicity [58,59] and particle spectra [58,60,61] in light quark events at LEP1 and SLD, the charged particle multiplicity in light quark events at LEP2 [62,63], the charged particle multiplicity distribution at LEP 1 [64], and hadron multiplicities at the Z-pole [65]. We also implemented in Rivet and made use of the data on the properties of gluon jets [11,12] for the first time.
The hadronization parameters for charm quarks were tuned using the charged multiplicity in charm events at SLD [59] and LEP2 [62,63], the light hadron spectra in charm events at LEP1 and SLD [58,60,61], the multiplicities of charm hadrons at the Z-pole [42,65], and charm hadron spectra below the ϒ (4S) [66,67] and at LEP1 [68].
The hadronization parameters for bottom quarks were tuned using the charged multiplicity in bottom events at SLD [59] and LEP2 [62,63], the light hadron spectra in bottom events at LEP1 and SLD [58,60,61], the multiplicities of charm and bottom hadrons at the Z-pole [42,65], charm hadron spectra at LEP1 [68] and the bottom fragmentation function measured at LEP1 and SLD [69][70][71].
In order to tune the evolution of the total charged particle multiplicity in e + e − collisions as a function of energy the results of Refs. [42,45,59,62,63,[72][73][74][75][76][77][78] spanning energies from 12 to 209 GeV were used.
In order to study the various effects we have discussed we have produced tunes for the shower and hadronization parameters in the case that either the transverse momentum or virtuality in the shower is preserved. In each case we first tuned the shower and light quark parameters without the data on charged particle multiplicities as centre-ofmass energies below the mass of the Z 0 boson. In the final stage of the process where we retune these parameters three tunes were produced for each choice of cut-off and preserved quantity, one (labelled A) without the low-energy charged multiplicity data, one (labelled B) where all the charged multiplicity data was included with in the tune with weight w O = 100 and a final tune (labelled C) where this data had weight w O = 1000.
Unfortunately due to the CPU time required it is impossible to include the ATLAS data [10] directly in the tune, therefore we compare the results of the different tunes to this data.
Results
We have produced 12 tunes for different choices of the cutoff variable in the shower, the choice of which quantity to preserve in the parton shower, and different weightings of the charged particle multiplicities. The parameters obtained in the fits are given in Table. 1 while the χ 2 values are given in Table. 2.
The effects of changing the colour reconnection model can be seen in Fig. 7. In the results of Herwig++ 2.7.1 or Herwig 7.0 there is an unphysical tendency of the gluon jets to contain an even number of charged particles due to the production of colour-singlet gluons by the reconnection model, this feature is not present in any of the new tunes which provide a much better description of the distribution of charged particles in the gluon jets, see also the Appendix.
The choice of which tune and choice of cut-off variable and preserved quantity has to be a balance between how well we wish to describe the various different data sets, as unfortunately no choice provides a good description of all the data sets.
If we first consider the choice of cut-off it is clear that using a virtual mass provides a larger χ 2 for all sets of observables used in the tuning apart from those sensitive to bottom quarks. In addition it displays an unphysical energy dependence in the difference in charged particle multiplicities between bottom (or charm) quark and light quark events, as shown in Fig. 8 where the results which use a cutoff on the virtual mass, Herwig++ 2.7.1 and the new tune q 2 -q 2 -B, show a strong dependence on the centre-of-mass energy while those which use a p ⊥ cut-off, Herwig 7.0 and the new tune p ⊥ -q 2 -B, are relatively independent of energy. We therefore prefer a cut-off on the minimum transverse momentum of the branching.
In order to obtain a reasonable evolution of the number of charged particles with centre-of-mass energy in e + e − collisions, see Fig. 9, without ruining the description of particle spectra and event shape observables we choose to use the B tune as our default.
The choice of whether to preserve the p ⊥ or q 2 of the branching is more complicated. While the data on light quark jets, in particular event shapes measured at LEP (for example the thrust Fig. 10), favour preserving q 2 the data on the charged particle multiplicity in gluon jets at LEP Fig. 11, and in jets at the LHC Figs. 12,13 favours preserving the p ⊥ of the branching.
Our preferred choice, in particular in the presence of higher-order matching, is to preserve the q 2 of the branching in order to ensure that the parton shower does not overpopulate the dead-zone. This also ensures a more reasonable value of strong coupling, α CMW S (M Z ) = 0.126 which gives α MS S (M Z ) = 0.118. However given the better description of gluon jets it is reasonable to also consider the alternative of Table 1 The Monte Carlo parameters obtained for different choices of the cut-off option, the preserved quantity in the shower and weight of the charged particle multiplicity data. Table 2 The values of χ 2 per degree of freedom obtained in the fit for different choices of the cut-off option, the preserved quantity in the shower and weight of the charged particle multiplicity data. The values are χ 2 as described in the text for the tuning observables, normalised to the sum of the weights for the different bins, and the true χ 2 using the experimental error for the charged particle multiplicities. The number of degrees of freedom for each set of observables is given together with the sum including weights in brackets, where this is different.
Cut-Off p ⊥ Virtual Mass Preserved p ⊥ q 2 p ⊥ q 2 Tune A B C A B C A B C A B C
preserving the p ⊥ , see for example Fig. 16 from the Appendix.
Conclusions
We have performed a tuning the the Herwig 7 event generator using data on gluon jets from LEP for the first time. To-gether with changes to the non-perturbative modelling this gives a significantly better description of gluon jets, in particular their charge particle multiplicity. It is however impossible to get a good description of the LEP particle spectra and the charged particle multiplicities, particularly in gluon jets, at the same time. We therefore choose the tune p ⊥ -q 2 -B as the default for Herwig 7.1. However for jets at the LHC the tune p ⊥ -p ⊥ -B gives a better description of jet properties. The data is from [59, 61-63, 73, 79-88] as compiled in [89] While the tunes presented in this paper are an improvement on their predecessors there is a tension between the data on charged particle multiplicities, for both quark and gluon initiated jets, and the data on event shapes and particle spectra from LEP. The cluster hadronization model also continues to have problems describing final states in events with bottom quarks. Any further improvement in the description of this data will require improvements to the nonperturbative modelling.
Acknowledgements This work was supported in part by the European Union as part of the FP7 and H2020 Marie Skłodowska-Curie Initial Training Networks MCnetITN and MCnetITN3 (PITN-GA-2012-315877 and 722104). Daniel Reichelt thanks CERN for the award of a summer studentship during which this work was initiated and acknowl- Appendix A: Generalized angularities and quark and gluon jet discrimination power
In this appendix we investigate how the improvements of the simulation of quark and gluon proposed in the manuscript affect the quark and gluon jet discrimination power recently studied in [9] 9 . For this purpose, we present results for five generalized angularities λ κ β [90]:
(κ, β ) (0, 0) (2, 0) (1, 0.5) (1, 1) (1, 2) λ κ β : multiplicity p D T
LHA width mass
where λ κ β = ∑ i∈jet z κ i θ β i , i runs over the jet constituents, z i ∈ [0, 1] is a momentum fraction, and θ i ∈ [0, 1] is an angle to the jet axis. To quantify discrimination performance, we use 9 The results and the analysis code used for this study is available as a RIVET routine [37], which can be downloaded from https://github. com/gsoyez/lh2015-qg. Fig. 13 The difference between the average number of particles in central and forward jets compared to data from the ATLAS experiment [10]. classifier separation:
∆ = 1 2 dλ p q (λ ) − p g (λ ) 2 p q (λ ) + p g (λ ) ,
where p q (p g ) is the probability distribution for λ in a generated quark jet (gluon jet) sample. ∆ = 0 corresponds to no discrimination power and ∆ = 1 corresponds to perfect discrimination power.
We start with an idealized case of e + e − collisions (see Section 5 of [9] for details). In Fig. 14 we show the discrimination power as a function of an angularity predicted by PYTHIA 8.215 [2], HERWIG++ 2.7.1 [16], SHERPA 2.2.1 [3] , the NNL analytical calculation from [9] and the both p ⊥ -q 2 -B and p ⊥ -p ⊥ -B tunes of Herwig 7.1. Firstly, we see that the both Herwig 7.1 tunes give significantly different results compared to HERWIG++ 2.7.1. In order to understand the source of the difference, in Fig 15 we investigate, for p ⊥ -q 2 -B tune, the following settings variations:
-HERWIG: NO g → qq. Turning off g → qq splittings in the parton shower. -HERWIG: NO CR. The variation turns off color reconnections.
We can see that the results are not very sensitive to the change of the settings. This was not the case for HERWIG++ 2.7.1 where the colour reconnection had a huge effect on the discrimination power, see [9]. Therefore, we can conclude that the difference is due to the improvements of the CR model described in Section 3, which as expected reduce effects of CR in the case of e + e − collisions. Secondly, the results of the both Herwig 7.1 tunes are quite similar and closer to the other predictions giving more constrained prediction on the quark/gluon jet discrimination power in e + e − collisions.
In fact just before finishing this paper the new tune was used in [91] confirming that indeed that improvements introduced in the manuscript reduced the tension between Pythia and Herwig and bring Herwig results closer to NNLL' results from [91]. Next, in Fig. 16 we show the results for ∆ in the case of quark/gluon tagging at the LHC (see Section 6 of [9] for details). Here we can see that the differences between HER-WIG++ 2.7.1 and the both Herwig 7.1 tunes are more modest when compared to the previous case of e + e − collisions. However, as expected the largest differences between generators appear for IRC-unsafe observables like multiplicity (0,0) and p D T (2,0), where nonperturbative hadronization plays an important role. It is also worth to notice that the p ⊥ -p ⊥ -B tune which is preferred by the data on the charged particle multiplicity in gluon jets at LEP Fig. 11, and in jets at the LHC Figs. 12,13 gives slightly better discrimination power reducing the gap between predictions of Pythia and the other generators. Finally, it would be interesting to estimate the parton-shower uncertainties [92][93][94][95] in the context of the quark and gluon jet discrimination observables to see whether the remaining discrepancy in the predictions is covered by the uncertainty band. Fig. 16 Classifier separation ∆ for the five angularities, determined from the various generators at hadron level in the case of quark/gluon tagging at the LHC (see Section 6 of [9] for details). The first two columns correspond to IRC-unsafe distributions (multiplicity and p D T ), while the last three columns are the IRC-safe angularities.
Fig. 2 Fig. 3
23Dalitz plot for e + e − → qqg showing the region of phase space filled by one emission from the quark and antiquark in the angularordered parton shower. The line shows the limits for the parton-shower emission. x i = 2E i /Q where E i is the energy of parton i and Q is the centre-of-mass energy of the collision. Dalitz plot for e + e − → qqg showing the emission from the hard matrix element correction into the dead-zone which is not populated by parton-shower emission. The line shows the limits for the partonshower emission. x i = 2E i /Q where E i is the energy of parton i and Q is the centre-of-mass energy of the collision.
Fig. 4
4Dalitz plot for e + e − → qq showing the region of phase space filled after multiple emission from the quark and antiquark in the angular-ordered parton shower. The transverse momentum of the branchings was preserved in the case of multiple emission. The line shows the limits for the parton-shower emission for a single emission.x i = 2E i /Qwhere E i is the energy of parton i and Q is the centre-ofmass energy of the collision.
Fig. 5
5Dalitz plot for e + e − → qq showing the region of phase space filled after multiple emission from the quark and antiquark in the angular-ordered parton shower. The virtuality of the branchings was preserved in the case of multiple emission. The line shows the limits for the parton-shower emission for a single emission.
Fig. 6
6Example of colour-singlet gluon pair production followed by the branching of all the colours via g → gg. The Feynman diagram is shown in (a) whereas the colour flows, including the non-perturbative splitting of the gluons into quark-antiquark pairs is shown in (b). parton shower of each of the original gluons, i.e. (q 1 ,q 3 )
Fig. 7
7Multiplicity distribution of charged particles in gluons jets for two different gluon energies compared to data from OPAL[11].
Fig. 8
8Difference between the charged multiplicity in bottom and light quark events in e + e − collisions as a function of centre-of-mass energy.
Fig. 9
9The evolution of the number of charged particles in e + e − → hadrons as a function of the centre-of-mass energy. edges support from the German Research Foundation (DFG) under grant No. SI 2009/1-1. Andrzej Siodmok acknowledges support from the National Science Centre, Poland Grant No. 2016/23/D/ST2/02605. We thank our collaborators on Herwig for many useful discussions. The tuning of Herwig to experimental data would not have been possible without the use of GRIDPP computer resources.
Fig. 10
10The thrust at the Z-pole compared to data from the DEL-PHI[42] experiment.
Fig. 11
11The evolution of the number of charged particles in gluon jets as a function of twice the energy of the gluon jet.
Fig. 12
12The average number of charged particles in jets as a function of the jet transverse momentum compared to data from the ATLAS experiment[10].
Fig. 14 Fig. 15
1415Classifier separation ∆ for the five angularities, determined from the various generators at hadron level for an idealized case of e + e − collisions. The first two columns correspond to IRC-unsafe distributions (multiplicity and p D T ), while the last three columns are the IRC-safe angularities. Settings variations for Herwig 7.1 p ⊥ -q 2 -B tune. Hadron-level results for the classifier separation ∆ derived from the five benchmark angularities.
Hadron-level, plain jet Pythia 8.215 Hw++ 2.7.1 Hw 7.1 p ⊥ -q 2 -B Hw 7.1 p ⊥ -p ⊥ -B Sherpa 2.2.1
See Ref.[9] for a more detailed discussion.
There is an option to extend the parton-shower radiation to the nonperturbative region and effectively remove the cut-off, see[18].
Due to the choice of ordering variable the hardest emission may not be the one that has the highest value of the ordering variable, i.e. the hardest emission may be not the first emission.
For charmonium and bottomonium states there are a number of other states below the threshold.
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| []
|
[
"Colloid-polymer mixtures in the protein limit",
"Colloid-polymer mixtures in the protein limit"
]
| [
"Peter G Bolhuis ",
"EvertJan Meijer ",
"Ard A Louis \nDept. of Chemistry\nUniversity of Cambridge\nLensfield RoadCB2 1EWCambridgeUK\n",
"\nDept. of Chemical Engineering\nUniversity of Amsterdam\nNieuwe Achtergracht 1661018 WVAmsterdamNetherlands\n"
]
| [
"Dept. of Chemistry\nUniversity of Cambridge\nLensfield RoadCB2 1EWCambridgeUK",
"Dept. of Chemical Engineering\nUniversity of Amsterdam\nNieuwe Achtergracht 1661018 WVAmsterdamNetherlands"
]
| []
| We computed the phase-separation behavior and effective interactions of colloid-polymer mixtures in the "protein limit", where the polymer radius of gyration is much larger than the colloid radius. For ideal polymers, the critical colloidal packing fraction tends to zero, whereas for interacting polymers in a good solvent the behavior is governed by a universal binodal, implying a constant critical colloid packing fraction. In both systems the depletion interaction is not well described by effective pair potentials but requires the incorporation of many-body contributions. | 10.1103/physrevlett.90.068304 | [
"https://arxiv.org/pdf/cond-mat/0210528v1.pdf"
]
| 17,527,988 | cond-mat/0210528 | bb205d88d274f72e136cd566f861eeda76849d39 |
Colloid-polymer mixtures in the protein limit
23 Oct 2002 (Dated: February 1, 2008)
Peter G Bolhuis
EvertJan Meijer
Ard A Louis
Dept. of Chemistry
University of Cambridge
Lensfield RoadCB2 1EWCambridgeUK
Dept. of Chemical Engineering
University of Amsterdam
Nieuwe Achtergracht 1661018 WVAmsterdamNetherlands
Colloid-polymer mixtures in the protein limit
23 Oct 2002 (Dated: February 1, 2008)numbers: 8235Np6125Hq8270Dd
We computed the phase-separation behavior and effective interactions of colloid-polymer mixtures in the "protein limit", where the polymer radius of gyration is much larger than the colloid radius. For ideal polymers, the critical colloidal packing fraction tends to zero, whereas for interacting polymers in a good solvent the behavior is governed by a universal binodal, implying a constant critical colloid packing fraction. In both systems the depletion interaction is not well described by effective pair potentials but requires the incorporation of many-body contributions.
We computed the phase-separation behavior and effective interactions of colloid-polymer mixtures in the "protein limit", where the polymer radius of gyration is much larger than the colloid radius. For ideal polymers, the critical colloidal packing fraction tends to zero, whereas for interacting polymers in a good solvent the behavior is governed by a universal binodal, implying a constant critical colloid packing fraction. In both systems the depletion interaction is not well described by effective pair potentials but requires the incorporation of many-body contributions. Adding polymers to suspensions of micro-and nanoparticles induces depletion interactions that profoundly affect their physical properties. This phenomenon has important scientific and (bio-)technological applications. Polymers such as polyethylene glycol are routinely added to protein solutions to enable protein crystallization [1,2], a poorly understood process and of great importance in structural biology [3]. In cell biology depletion interactions are key in the process of macro-molecular crowding [4]. Food and paint production are among the industrial sectors where depletion phenomena play a role.
In this Letter we focus on mixtures of hard sphere (HS) colloids with a radius R c and non-adsorbing polymers with a radius of gyration R g , in the regime where q = R g /R c > 1. This is often called the nano-particle or "protein limit", because in practice small particles such as proteins or micelles are needed to achieve large sizeratios q. Whereas the opposite "colloid limit" (q < ∼ 1) has been well studied, the physics in the protein limit is less established. This imbalance is partially due to the lack of well characterized experimental model systems for the protein limit and partially to a poor theoretical understanding. The colloid limit can be well described within the framework of effective depletion pair potentials [5,6], in contrast to the protein limit, where the interactions cannot be reduced to a pairwise form [7,8]. Nevertheless, for biological and industrial applications, this regime is at least as important as the colloid limit.
One of the first theoretical treatments of colloidpolymer mixtures in the protein limit was by de Gennes [9], who showed that the insertion free-energy F (1) c of a single hard, non-adsorbing sphere into an athermal polymer solution scales as
βF (1) c ∼ (R c /ξ) 3−1/ν(1)
when R c < ξ, with the polymer correlation length [10]. Here, β = 1/k B T is the reciprocal temperature, ν ≈ 0.59 is the Flory exponent and φ p = ρ p 4 3 πR 3 g is the polymer volume fraction for a polymer number density ρ p , so that φ p ≈ 1 at the crossover from a dilute to a semi-dilute solution [10]. The prefactors can be calculated by the renormalization group (RG) theory [11], yielding: βF (1) c ≈ 4.39φ p q −1.3 which has been verified by computer simulations for small q [12]. Based on this description of F (1) c , de Gennes [10] and Odijk [8] predicted extensive miscibility for colloid-polymer mixtures in the large q limit if R c < ξ. However, it is well known that proteinpolymer mixtures do phase-separate [13,14]. Recently, Odijk et al. [15] suggested that a poor solvent could facilitate phase-separation. Sear [16] altered the form of F (1) c to include effects when R c ≫ ξ, and also predicted phase-separation with a truncated virial theory. The same author recently proposed an alternative theory [17] where the colloids induce depletion attractions between the polymers, leading to a poorer effective solvent and eventually phase-separation. Mean-field cell model calculations also predict demixing [18]. Another promising approach uses integral equation techniques [19] to predict spinodal curves and critical points. However, all these theories suffer from several uncontrolled approximations leading to different predictions for the causes and properties of the phase-separation. To clarify this situation, we performed computer simulations with as few simplifying assumptions as possible, on which we report in this Letter.
ξ(φ p ) ∼ R g φ −ν/(3ν−1) p ≈ R g φ −0.77 p
We have recently used a coarse-graining technique [20] to study the colloid limit, and found quantitative agreement with experimental fluid-fluid binodals [21], and significant qualitative differences between interacting (IP) and non-interacting (NIP) polymers. Here, we study the same athermal model of HS colloids and non-adsorbing polymers in the protein limit, and calculate, for the first time, the full fluid-fluid binodals by direct simulation. The results for the IP and NIP show even larger qualitative differences, and many-body depletion interactions must be invoked to understand the phase-behavior.
The simulation model consists of polymers on a simple cubic lattice mixed with HS colloids. The interacting polymers in a good solvent are modeled as self avoid- ing walks (SAW) of length L, which have a radius of gyration R g ∼ L ν . The non-interacting polymers are modeled as random-walks, for which R g ∼ L 0.5 . In both models there is an excluded-volume interaction between the colloidal HS and the polymer segments. The simulations were performed on a D 3 lattice with periodic boundary conditions, where D = 48 and D = 100 for the NIP and IP system, respectively. Throughout this Letter, we use the lattice spacing as the unit of length. For the NIP the colloidal HS diameter was σ c = 5.5 and the polymer length was L = 50, 100, 200 and 500, corresponding to q = 1.03, 1.45, 2.05 and 3.2, respectively. For the IP L = 2000, and σ c = 10, 14 and 20, yielding q = 3.86, 5.58 and 7.78, respectively. Colloidal positions have continuous values, but when we calculated the interaction between colloid and polymer the colloids were shifted such that they occupied a constant number of lattice sites to prevent spurious attractive positions for single colloids (other lattice effects, although unavoidable, are expected to be small.) Thermodynamic state points were calculated in the grand-canonical ensemble, i.e. at fixed volume V , colloid chemical potential µ c and polymer chemical potential µ p using Monte Carlo (MC) techniques. The NIP were sampled using an (exact) lattice propagation method [22,23], while the IP configurations were generated using translation, pivot moves and configurational bias MC [24] in an expanded ensemble to facilitate insertion of long chains [25]. The open symbols are the colloid limit (q < ∼ 1) results from Ref. [21]. Solid lines are a guide to the eye. Inset: The same binodals plotted in a reduced polymer density representation. The dotted curve corresponds to a simple theory for the universal binodal when polymers are in a good solvent while the dashed line is for polymers in a poorer solvent. Crosses and stars as in Fig. 1. for several values of µ p and locate the µ c for which a sudden density change occurred. Subsequently 8-10 (µ p , µ c ) coexistence state points were simulated simultaneously using parallel tempering [25]. When the estimated coexistence points are sufficiently close to the true binodal and to each other, and near the critical point, this scheme results in proper ergodic sampling of both phases. If necessary, the chemical potentials were adjusted towards coexistence. We used the multiple histogram reweighting [25,26] technique to determine the precise location of the (µ c , µ p ) coexistence line, and the phase boundaries in the (φ c , φ p ) plane, where φ c = ρ c 4 3 πR 3 c is the colloid volume fraction, with ρ c the colloid number density.
Figs. 1 and 2 contrast the calculated phase diagrams for NIP and IP for several size-ratios q. Firstly, we note that both models show extensive immiscibility, in agreement with experiment [14]. Secondly, the two systems exhibit striking differences: for the NIP, the critical colloid volume fraction φ crit c tends to zero with increasing size ratio q, while the IP exhibit a nearly constant value of φ crit c . For both systems the critical polymer concentration φ crit p increases with increasing q. The phase-separation occurs well into the semi-dilute regime for the IP, again in qualitative agreement with experiment [13]. Properties of semi-dilute polymer solutions are independent of polymer length, being instead determined by the correlation length ξ, which is a function of the monomer density c = Lρ p . The phase behavior of the polymer-HS mixture should therefore only be a function of the ratio R c /ξ [16]. Indeed, when the phase lines in Fig. 2 are rescaled with an accurate prescription for ξ(ρ p ) [12], the binodals nearly collapse onto a "universal binodal", as shown in the inset of Fig. 2. This explains why the critical colloid packing fraction is nearly constant in the simulations. Similarly, φ crit p scales as φ crit p ∼ q (3ν−1)/ν ≈ q 1.3 . For comparison, we have also included results for q < ∼ 1 from Ref. [21] in Fig. 2. These results do not exhibit the same scaling behavior, since they are not in the semi-dilute regime.
The differences between NIP and IP phase behavior can be rationalized with some simple theories. Consider a Helmholtz free-energy F of the form βF/V = f = f HS c + f p + f cp . Here, the HS free energy f HS c is given by the accurate Carnahan-Starling expression [27], and the polymer free energy f p for either IP or NIP solutions is well understood [10]. The contribution due to the HS-polymer interactions f cp is non-trivial. A first approximation truncates after the second cross-virial coefficient, yielding f cp ∼ ρ c F (1) c . For NIP the insertion free-energy F (1) c is exactly known [11], so that f cp takes the form f id cp = ρ p φ c 1 + 6q √ π + 3q 2 ≡ ρ p φ cbcp , which defines the reduced cross-virial coefficientb cp . Since f id cp grows with increasing q, immiscibility sets in at lower colloid packing fraction φ c . The theory can be improved by realizing that the polymers only exist in the free volume left by the colloids [28]. Simply taking this free volume to be 1 − φ c is an adequate first approximation for the protein limit. The trends for the binodal lines calculated from this simple theory, shown in Fig. 1, agree qualitatively with the simulations. For example, the critical point shifts to smaller φ c and larger φ p for increasing q, and the binodal lines cross at a low φ c . For computational reasons the simulations only go up to q = 3.2 and we expect better quantitative agreement for larger q since φ crit c decreases so that the second-virial theory should become more accurate. In the q → ∞ limit, this theory yields φ crit c → 1/b cp ∼ 1/(3q 2 ), and φ crit p = q 3 /b cp ∼ q/3. Note that in the same limit, the penetrable sphere or Asakura-Oosawa model [28] scales somewhat differently: φ crit c → 1/q 3 and φ crit p → 1. Sear [7] already pointed out the φ crit c → 0 behavior using a slightly different prescription for the free volume than we employ here. Here we claim that the limiting results are a general feature of free-volume theories.
In the IP case, f cp is more difficult to estimate, even for a second cross-virial theory. The R c ≪ ξ limit is given by Eq. (1) with the prefactors from RG theory. For R c ≥ ξ we have previously shown that F (1) c is given by F
(1) c = 4
3 πR 3 c Π + 6πR 2 c γ s [12], where the polymer osmotic pressure Π ∼ ξ −3 is well known [10]. However, since Eq. (1) is essentially a surface (depletion layer) contribution, we use a simple approximate second crossvirial term f cp = ρ c βΠ(ρ) 4 3 πR 3 c + 4.39φ p q −1.3 , which reduces to the correct form in both the R c ≪ ξ and the R c ≫ ξ limit [33]. As with our treatment of NIP, we take the effect of the colloid excluded volume into account by computing the polymer densities in the free volume fraction 1 − φ c (see Ref. [29] for a complimentary approach). The theoretical binodals were calculated using accurate expressions for ξ and Π [12] and are compared with the simulation results in Fig. 2, in the R c /ξ versus φ c plane.
The qualitative agreement suggests that we can also use this theory to estimate the effect of a poorer solvent on the binodals. Following Ref. [15], we alter the scaling of
ξ to ξ ∼ φ −δ/3 p so that Π ∼ ξ −3 ∼ ρ p φ δ−1 p .
Interestingly, Fig. 2 shows that using δ ∼ 1.5 instead of the appropriate exponent for polymers in a good solvent (δ ∼ 2.3), does not result in important differences in the binodals. Of course, the differences will appear larger in the (φ c , φ p ) plane due to the different scaling of ξ. One must keep in mind, however, that these predictions follow from a simple scaling theory and qualitatively different behavior may emerge when one approaches the θ-point (where δ = 1).
To illustrate the many-body nature of the depletion interaction we estimated the phase behavior by approximating the system by colloids interacting via pairwise effective potentials. We computed the effective pair interaction v(r) between two colloids in a bath of IP's, by βv(r) = − ln g(r) for ρ c → 0. The colloid radial distribution function g(r) was estimated by measuring the insertion probability of a HS at a distance r from a second fixed HS in a SAW polymer solution using the above MC techniques. Results for a single size-ratio q = 7.78 as a function of φ p are shown in Fig. 3 [30]. Inset: Reduced second osmotic virial coefficient B * 2 = B2/( 2 3 πR 3 c ) as a function polymer densities for several size ratio's. similar to the colloid limit [30]: the range shortens and the well-depth increases with increasing φ p . Interestingly, our simple depletion potential [30], derived for the colloidal limit, also works semi-quantitatively in this regime. A good measure for the attractive strength of an effective pair potentials is given by the second osmotic virial coefficient [27], shown in Fig. 3. The saturation of B * 2 for larger q is an interesting qualitative feature: apparently the shortening of the range compensates the deepening of the attraction, so that the total cohesion does not increase with increasing φ p , something also found in RG [11] and integral equation calculations [19]. For pairwise interacting systems, phase-separation typically sets in when B * 2 ≤ −1.5 [31]. Here, the saturation of B * 2 suggests that for large q the pair interactions do not provide enough cohesion to explain the phase-separation. We arrive at the same conclusions with simple mean-field theories [27], which should be relatively trustworthy given the long range of the pair potentials. Obviously, for q ≫ 1 a pair-level description is not sufficient, and many-body interactions must be invoked.
For the NIP, one might also expect many-body interactions to be important for large q. A good approximation to the pair-potentials exists [23,30], from which the second virial coefficients at the calculated critical points follow: B * 2 (q = 1.03) ≈ −13.1; B * 2 (q = 1.45) ≈ −16.4; B * 2 (q = 2.05) ≈ −22.7. Even though the actual critical φ c 's are very low, so that a second-virial description might be thought to be sufficient, the analysis above shows that for NIP the pair-interactions provide too much cohesion, opposite to the IP case. Clearly, many-body interactions must also be invoked to describe the phase-behavior correctly, as suggested by other authors [7,8,19,23,32].
In conclusion, we have shown by computer simulations that a mixture of polymers and non-adsorbing HS colloids shows extensive immiscibility in the protein limit, where the polymer-colloid size-ratio q ≫ 1. For IP the phase-behavior is dictated by a universal binodal in the semi-dilute regime. For NIP, the colloid packing fraction tends to zero for increasing polymer length. In contrast to the better studied colloid limit, pair interactions are not sufficient to rationalize the phase behavior. We hope that future experiments on HS colloids with nonadsorbing polymer will test these predictions. Future work might include extensions to non-spherical particles, poor solvents, and adsorbing systems.
A.A.L. acknowledges the Isaac Newton Trust for financial support. E.J.M. acknowledges the Royal Netherlands Academy of Art and Sciences for financial support. We acknowledge support from the Stichting Nationale Computerfaciliteiten (NCF) and the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) for the use of supercomputer facilities. We thank M. Fuchs, R. Sear and R. Tuinier for helpful discussions.
PACS numbers: 82.35.Np,61.25.Hq,82.70.Dd
FIG. 1 :
1Fluid-fluid binodals for a mixture of non-interacting polymers and HS colloids with different size-ratios q. Crosses indicate the estimated critical point, obtained by extrapolating the calculated phase boundaries. The full lines are a guide to the eye. Dashed lines denote the simple theory described in the text, with stars marking the critical points.
FIG. 2 :
2Typical simulations lengths are 10 9 Monte Carlo moves per state point. In order to determine the liquid-liquid binodals we first estimated the coexistence line by scanning a series of µ c Fluid-fluid binodals for a mixture of interacting polymers and HS colloids at different size-ratios q. Filled symbols are direct simulation data.
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"Quarkonium Spectral Functionś",
"Quarkonium Spectral Functionś"
]
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"Agnes Mócsy \nDepartment of Mathematics and Science\nPratt Institute\n11205BrooklynNYUSA\n"
]
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"Department of Mathematics and Science\nPratt Institute\n11205BrooklynNYUSA"
]
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| In this talk I summarize the progress achieved in recent years on the understanding of quarkonium properties at finite temperature. Theoretical studies from potential models, lattice QCD, and effective field theories are discussed. I also highlight a bridge from spectral functions to experiment. | 10.1016/j.nuclphysa.2009.10.035 | [
"https://arxiv.org/pdf/0908.0746v2.pdf"
]
| 15,471,432 | 0908.0746 | e7721d7a9bf14eea77f8c53352684927cc7bdf67 |
Quarkonium Spectral Functionś
27 Sep 2009
Agnes Mócsy
Department of Mathematics and Science
Pratt Institute
11205BrooklynNYUSA
Quarkonium Spectral Functionś
27 Sep 2009
In this talk I summarize the progress achieved in recent years on the understanding of quarkonium properties at finite temperature. Theoretical studies from potential models, lattice QCD, and effective field theories are discussed. I also highlight a bridge from spectral functions to experiment.
Introduction
It has been 23 years since what is today the most cited paper in heavy ion physics [1] was publised. In this work Matsui and Satz conclude that "J/ψ suppression in nuclear collisions should provide an unambiguous signature of quark-gluon plasma formation". This is how the story of quarkonium at finite temperature began. One of the important and necessary consequences of this story is the quest for determining properties of quarkonium states at finite temperature. Quarkonium properties can be conveniently studied through spectral functions. In this talk I discuss properties of quarkonium inside an equilibrated plasma, as obtained from the three main theoretical approaches for determining the spectral functions. These are, in historic order, potential models, lattice QCD, and effective field theories. I then discuss what possible implications for the experiments the results might have.
Quarkonium states, made of a heavy quark and its antiquark (m Q ≫ Λ QCD ), are tightly bound; their size is much smaller than the typical hadronic scale, r J/ψ ≃ 0.4 fm and r Υ ≃ 0.2 fm ≪ 1 fm. Due to the heavy quark mass the quark velocity is very small, v ≪ 1, which allows for a non-relativistic treatment of the bound states. Properties of quarkonium have been customarily obtained by solving the Schrödinger equation, where the interaction between the quark Q and antiquark Q at distance r away from each other is implemented as a potential, V(r). This prescription works very well at zero temperature and it has been somewhat naively used also at finite temperature. There, the idea is that in a high temperature deconfined medium there is a rearrangement of color around the heavy quarks, due to which the effective charge of the Q and Q is reduced. So light quarks and gluons screen the potential between the Q and the Q, analogous to the known Debye-screening in QED plasmas. The original Matsui-Satz idea is that when the range of screeing becomes smaller than the size of the state, r D < r QQ , the quark and antiquark cannot see each other and the two are no longer bound. Customarily, these effects were described with a temperature-dependent potential V(r, T ).
This simple and attractive idea leaves a few questions: How good is it to describe medium effects with a temperature dependent potential? Is there Debye-screening in the plasma? Is screening the only relevant effect? And what do we mean by a bound state at finite temperature?
Spectral Functions
Quarkonium properties are conveniently described using spectral functions. These contain all the relevant information about a given channel, providing a unified treatment of bound-states, continuum, and threshold. Bound states show up as peaks in the spectral function. At zero temperature states are well defined, the widths of the peaks can be narrow, the lifetimes of the states can be large. The energy gap between the continuum threshold and a peak position defines the binding energy, E bin for that state. Peaks are expected to broaden with increasing temperature. The disappearance of a peak from the spectral function means the bound state is dissociated (often referred to as the state is melted). Here I want to emphasize, that bound states may disappear well before the binding energy vanishes. Thus a melting (or dissociation) temperature should be defined by a less severe condition than zero binding energy, contrary to what has generally been considered in the literature.
There are three main theoretical approaches to determine spectral functions. 1) Finite temperature potential models date back to the Matsui and Satz paper. Besides they are easy to handle, a great advantage of the newer versions of potential models is that they easily accomodate non-perturbative information from lattice QCD calculations of the free energy of static quarks, in form of different "lattice-based potentials". The drawback is, however, their ad-hoc nature.
2) The first lattice QCD calculations for quarkonium spectral functions at finite temperature appeared in 2003. Although the computation of quarkonium correlators is exact, it is really difficult to extract spectral functions on quantitative and sometimes even qualitative level from these. 3) Effective field theories have only appeared in the last few years. These are obtained directly from QCD and they can clarify the domain of applicability for the simpler potential model approach. However, currently derivations of the potentials from effective theories must assume some scale separation (between the bound state-and temperature-scales), and utilize weak-coupling techniques, which makes their application near the critical temperature problematic. With the advantages and draw-backs of each of these approaches, a combination of all their results provides a newly emerging complete picture of quarkonium at finite temperature. The chart on figure 1 illustrates the interconnectedness of these approaches.
Color Screening from Lattice QCD
A rigorous way to study the modification of inter-quark forces with temperature T and quark separation r is provided from lattice QCD by computing the heavy quark singlet free-energy, F 1 (r, T ) [2]. This is shown in the left panel Figure 2. At short distances, r < r med (T ), the singlet free energy is independent of temperature and coincides with the zero temperature potential. The range r med (T ) where F 1 (r, T ) = F 1 (r) decreases with increasing temperature. At large distances r > r scr (T ) and temperatures above the critical T c the singlet free-energy is screened, F 1 (r, T ) = F 1 (T ). With increasing T the strong modification of the static Q − Q interaction sets in at shoter and shorter distances r scr (T ). As illustrated on the right panel of Figure 2 the range of Q − Q interaction is comparable to the size of QQ charmonium states. This generic observation together with the original Matsui-Satz argument suggests that the charmonium states, such as the J/ψ, and excited bottomonium states, do not exist as bound states above T c , while the ground state bottomonium, Υ, is small enough to survive deconfinement up to higher temperatures. One problem with this line of reasoning is that the singlet free energy, F 1 (r, T ), in general, does not coincide with the potential V(r, T ). It has thus been argued to consider the internal energy, obtained from the free energy, as the heavy quark potential. However, the internal energy is also 2 Figure 1: Flow chart of in-mdeium quarkonium calculations.
not the potential. The right panel of Figure 2 further illustrates that at a given temperature the internal energy is more binding than the corresponding free energy, leading possibly to survival of quarkonium states when identified as the heavy quark potential. In what follows, I will discuss spectral functions obtained from a variety of lattice-based potentials, and argue that survival of the J/ψ well above T c is unlikely.
Spectral Functions from Potential Models
In potential models the basic assumption is that all medium effects are accounted for as a temperature-dependent potential. The potential that describes heavy quark-antiquark interaction in a hot medium is not yet known. Furthermore, even the applicability domain of the traditional potential model approach is questionable. The newest advancements towards deriving the potential from QCD are from effective field theories, discussed in a following section. Lacking an exact form, potentials are either phenomenological [3] or lattice-based [4,5,6,7,8,9,10].
The main advantage of lattice-based potentials is that non-perturbative effects can be conveniently accommodated using lattice data on the heavy quark-antiquark singlet free energy discussed above and shown in the left panel of Fig. 2. The true potential at a given temperature and large distances (if it can be addressed at all in this approach) lies somewhere between the freeand the internal-energy (see right panel of Fig. 2). As said in the previous section, at short distances it corresponds to the zero temperature potential, but we do not have information about its behavior at intermediate distances. Utilizing these constraints one can interpolate at intermediate distances and construct lattice-based potentials. Clearly, the shortcoming of this approach is its somewhat ad hoc nature. We can however, look at the most binding potential still consistent with lattice and learn about the quarkonium spectral functions in this upper limit. Figure 3 shows quarkonium spectral functions obtained from a non-relativistic Greens function calculation using the most confining potential in full QCD [10] (left panel) and from a Tmatrix calculation using the internal energy as potential in quenched QCD [8] (right panel). Both results are for the S-wave charmonium channel. The spectral function in the left panel shows no peak structures at high temperature, indicating that all charmonium states are dissolved in the deconfined phase [9,10]. The result shown in the right panel finds the ground state peak at somewhat higher temperatures [8].
Even though there are some differences in the details, there are some essential features common to all models: For one, there is a large threshold (rescattering) enhancement beyond what corresponds to free quark propagation. This enhancement is present even at high temperatures and it indicates correlations persisting between the quark and antiquark even in the absence of bound states. Threshold enhancement has been identified in all of the channels (charmonium and bottomonium S-and P-states) [9,10]. Second, there is a strong decrease with increasing temperature of the binding energies (the distance between peak position and continuum threshold in the spectral function) determined from potential models [11]. The decreased binding energy implies a large increase in the thermal width as well. As mentioned earlier in this talk, bound states disappear before the binding energy goes to zero [11]. One can think that this happens when the width becomes larger than the binding energy. In other words the time it takes for a quark and antiquark to bind is larger than the time it takes for the bound state to decay. G/G rec Figure 4: S-wave bottomonium (left) and charmonium (right) spectral functions calculated in potential model [9]. Insets: correlators compared to lattice data. Charmonium is compared to lattice data from [20] shows a ground state bottomonium peak up to higher temperatures into the deconfined medium, but with a largely reduced binding energy. I note that binding energies are somewhat increased in a viscous medium [12]. For further details on anisotropy effects on quarkonium I refer the reader to [13]. I further note that potential model calculations of spectral functions do not include a realistic temperature-dependent width for the states. Widths naturally arise from effective theories [14,15], as discussed in a later section. We calculated the thermal widths from the binding energies [10] following [16]. Thermal broadening of quarkonium has been addressed also in a NLO perturbative QCD [17] and a QCD sum rule [18] analysis. All of these calculations show that the J/ψ for example, is significantly broadened at temperatures right above that of deconfinement. From the analysis of the spectral functions one can provide an upper bound for the dissociation temperatures, i.e. the temperatures above which no bound states peaks can be seen in the spectral function. Above these temperatures bound state formation is suppressed. Conservative upper limit dissociation temperatures for the different quarkonium states have been obtained from a full QCD calculation [10]. Accordingly, charmonium states are dissolved in the deconfined phase, while the bottomonium ground state may persist up to temperature of about 2T c .
Spectral Functions from Lattice QCD
In lattice QCD, current-current correlation functions of mesonic currents in Euclidean-time are calculated [19,20,21]. These correlators are related to the spectral function through the integral:
G(τ, T ) = dωσ(ω, T ) cosh (ω(τ − 1/2T )) / sinh (ω/2T ) .(1)
It is customary to present the temperature-dependence of the correlators as the ratio G/G recon , where G recon (τ, T = 0) is the correlator reconstructed from spectral function at zero (or some low) temperature. Since correlators are computed with high accuracy, and they can be determined in potential models by integrating the spectral function using equation (1), it has been suggested to compare correlators from potential models to the ones from lattice QCD [5]. Somewhat surprisingly, Euclidean correlation functions show very little temperature dependence, irrespective of whether a state is there (such as the Υ) or not (such as the J/ψ). Note also, that correlators from potential models are in accordance with the lattice calculations (see insets in Fig. 3) (for a review see [11]). Originally, the small temperature dependence of the correlators was considered as evidence for survival of different quarkonium states [19,22]. It is now clear that this conclusion was premature. So flat correlators do not tell about the survival or melting of states, since these kind of changes do not show up in it. We now understand that the threshold enhancement compensates for the absence of bound states and leads to Euclidean correlation functions which show only very small temperature dependence [9].
From the current-current correlation functions for different quarkonium channels the quarkonium spectral functions are extracted [19,22,20]. The extraction of spectral function using the Maximum Entropy Method is still difficult due to discretization effects, statistical errors, default model dependence (for a review see [23]). So the uncertainties in the spectral function are significant and details of this cannot be resolved. Moreover, a seemingly surviving ground state peak is in perfect agreement with a mere threshold enhancement obtained from potential models, as shown juxtaposed on the right panel of figure 4. Again, a 1 GeV wide peak obtained with huge uncertainties from lattice QCD, and peak, whose details cannot be resolved, is most likely not a true bound state peak. Improved calculations are on the way [24].
Spectral Functions from Effective Field Theories
In effective field theories the quarkonium potential is derived from the QCD Lagrangian. The basis of these theories is the existence of scales related to the bound state and scales related to the temperature. At T=0 for heavy quarks the existence of different energy scales related to the heavy quark mass m, the inverse size mv ∼ 1/r , and the binding energy mv 2 ∼ E bin makes it possible to construct a sequence of effective theories. The effective theory which emerges after integrating out the scale m and mv 2 is pNRQCD, which delivers the potential model at T = 0 [25]. Recently this approach has been extended to finite temperature, where the presence of the scales T , the Debye mass m D ∼ gT , and the magnetic scale g 2 T , makes computations more difficult. With the assumption of m ≫ T and weak coupling g ≪ 1 these scales are well separated. Depending on how these thermal scales compare with the bound state scales, the different hierarchies allow for the derivation of different effective theories for quarkonium bound states at finite temperature [15]. A finding common to all of the theories is that the quarkonium potential has not just a real part but also an imaginary part. The temperature dependent imaginary part determines the thermal width, and thus the smearing out, of the quarkonium states in the spectral function. Several physical processes contribute to this thermal width. For instance the scattering of particles in the medium with gluons (Landau damping) [14], or the thermal breakup of a color singlet Q − Q into a color octet state and gluons (octet transition) [15]. Smearing of states, leading to their dissolution, even before the onset of the exponential Debye-screening of the real part of the potential (see e.g. discussion in [14]).
Another discovery from the new effective field theory calculations is that finite temperature effects can be other than the originally thought exponential screening of the real part of the 6 potential; in the weak coupling approach thermal corrections to the potential are obtained only when the temperature is larger than the binding energy [15]. The current shortcomings of effective field theories are that near T c the applicability of weak coupling techniques is problematic and theories must incorporate also non-perturbative effects.
Relevance for experiments
Knowing the quarkonium spectral functions in equilibrium QCD is necessary, but alone is not sufficient to predict their production in heavy ion collisions. In principle, there is a simple relation between quarkonium spectral functions and quarkonium production rates measurable in experiments. This relation will hold only in thermal equilibrium, which is not likely the achieved for heavy quarks in heavy ion collisions. The bridge between theoretical spectral functions and experimental yield measurements requires additional dynamical modeling (see figure 1). There are several such models with different underlying assumptions [26,27,28,29].
Let us look for instance at RHIC energies, where we learned from potential models that no J/ψ peak in the spectral function is seen, only threshold enhancement [9,10]. This means that J/ψ are not formed, only spatially correlated c and c [9,10]. This correlation can be modeled classically, using Langevin dynamics which includes a drag force and a random force on the heavy quark (antiquark) from the medium, as well as the forces acting between the quark and anti-quark (described by the potential). The distribution of the separation between c − c pairs with and without threshold enhancement in the spectral function is shown in the left panel of Figure 5. Some of these c − c pairs will stay correlated throughout the evolution of the system. Those pairs that are not diffused away they can bind together into a J/ψ at hadronization.
The right panel of Figure 5 shows the result of a Langevin simulation of evolving c − c pairs on top of a hydrodynamically expanding quark-gluon plasma, which describes the RHIC data on charmonium suppression quite well [29]. In particular, this model can explain why, despite the fact that a deconfined medium is created at RHIC, we see only 40 − 50% suppression in the charmonium yield.
Summary
Potential model calculations based on lattice QCD, as well as resummed perturbative QCD calculations indicate that all charmonium states and excited bottomonium states dissolve in the deconfined medium. Lattice data is consistent with J/ψ screened just above T c . Dissolved states lead to the reduction of the quarkonium production yield in heavy ion collisions compared to the binary-scaled proton-proton collisions. Due to possible recombination effects, however, the yield will not be zero. Potential models indicate the decrease in the binding energy of quarkonium states and a sizable threshold enhancement leading to residual quark-antiquark correlations persisting at high temperatures. Implication of this for the understanding of experimental data can be investigated in dynamical models, which suggest that recombination of residually correlated pairs is possible when the temperature cools down sufficiently. Effective field theory calculations indicate temperature-dependent widths and temperature effects beyond what potential models have accounted for.
Figure 2 :
2Heavy quark singlet free energy versus quark separation calculated in 2+1 flavor QCD for different temperatures[2](left) and constraints from lattice on the potential at a given temperature (right).
Figure 3 :
3S-wave charmonium spectral functions calculated in potential model from[10] (left) and from[8] (right).
Figure 4
4
Figure 5 :
5Distribution of c − c pairs versus the inter-quark distance (left) and predicted suppression (right) from[29].
AcknowledgmentsI thank the Organizers of Quark Matter 2009 for inviting me to give this talk.
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| []
|
[
"Quantum harmonic oscillator in option pricing",
"Quantum harmonic oscillator in option pricing"
]
| [
"Liviu-Adrian Cotfas \nFaculty of Economic Cybernetics, Statistics and Informatics\nAcademy of Economic Studies\n6 Piata Romana010374BucharestRomania\n",
"Nicolae Cotfas \nPhysics Department\nUniversity of Bucharest\nP.O. Box MG-11077125BucharestRomania\n"
]
| [
"Faculty of Economic Cybernetics, Statistics and Informatics\nAcademy of Economic Studies\n6 Piata Romana010374BucharestRomania",
"Physics Department\nUniversity of Bucharest\nP.O. Box MG-11077125BucharestRomania"
]
| []
| The Black-Scholes model anticipates rather well the observed prices for options in the case of a strike price that is not too far from the current price of the underlying asset. Some useful extensions can be obtained by an adequate modification of the coefficients in the Black-Scholes equation. We investigate from a mathematical point of view an extension directly related to the quantum harmonic oscillator. In the considered case, the solution is the sum of a series involving the Hermite-Gauss functions. A finite-dimensional version is obtained by using a finite oscillator and the Harper functions. This simplified model keeps the essential characteristics of the continuous one and uses finite sums instead of series and integrals. | null | [
"https://arxiv.org/pdf/1310.4142v2.pdf"
]
| 117,572,340 | 1310.4142 | 4d7a598a0bc8166db01f53d05b753fceb9222012 |
Quantum harmonic oscillator in option pricing
23 Oct 2013
Liviu-Adrian Cotfas
Faculty of Economic Cybernetics, Statistics and Informatics
Academy of Economic Studies
6 Piata Romana010374BucharestRomania
Nicolae Cotfas
Physics Department
University of Bucharest
P.O. Box MG-11077125BucharestRomania
Quantum harmonic oscillator in option pricing
23 Oct 2013econophysicsBlack-Scholes equationquantum financefinite quantum systemsquantum harmonic oscillator 2010 MSC: 91B8091G80
The Black-Scholes model anticipates rather well the observed prices for options in the case of a strike price that is not too far from the current price of the underlying asset. Some useful extensions can be obtained by an adequate modification of the coefficients in the Black-Scholes equation. We investigate from a mathematical point of view an extension directly related to the quantum harmonic oscillator. In the considered case, the solution is the sum of a series involving the Hermite-Gauss functions. A finite-dimensional version is obtained by using a finite oscillator and the Harper functions. This simplified model keeps the essential characteristics of the continuous one and uses finite sums instead of series and integrals.
Introduction
Black-Scholes (BS) equation, one of the most important equations in econophysics, is usually solved by using its direct connection with the heat equation. The option price V (S, t) at the moment t is a function of the stock price S at the considered moment of time. The function V (S, t) can be expressed through a change of independent and dependent variable in terms of a function u(x, t) satisfying the heat equation. The solution V (S, t) of BS equation which coincides with the payoff function at the maturity time T is directly obtained from a solution of heat equation satisfying a certain initial condition.
BS equation is very useful in finance, but it describes only an idealized case (constant volatility and risk-free interest rate). Some additional effects can be included by using an adequate modification of coefficients [1]. In this paper, we consider only the case of a generalized Black-Scholes (GBS) equation obtained by adding a certain potential to the last coefficient. In the case of a GBS equation a direct reduction to the heat equation seems to be impossible. The method of separation of variables can be used as an alternative tool. By looking for solutions having a particular form (separated variables), the GBS equation which is a differential equation with partial derivatives is reduced to a family of ordinary differential equations in one variable x and depending on a parameter ε. More than that, for the considered GBS equations the corresponding equations in one variable are Schrödinger equations.
Two particular cases of GBS equation have been investigated by Jana and Roy in [12]. Their purpose was to obtain a supersymmetric partner for each of them and to compute the pricing kernel in all the cases. We consider the case related to the quantum harmonic oscillator and we present: -a solution in terms of Hermite-Gauss functions, -a class of supersymmetric partners depending on a continuous parameter α, and the corresponding solutions, -a finite-dimensional approach based on a finite-difference operator, the finite Fourier transform and Harper functions.
Black-Scholes equation
We present, by following [15], a short review concerning the use of the heat equation in order to offer the possibility to compare it with the approach based on the separation of variables. It is well-known that BS equation
∂V ∂t + σ 2 2 S 2 ∂ 2 V ∂S 2 + rS ∂V ∂S − rV = 0(1)
for the European option price V (S, t) is equivalent to the heat equation
∂u ∂τ = ∂ 2 u ∂x 2(2)
whose solution for the initial condition u(x, 0) = u 0 (x) is
u(x, τ ) = 1 √ 4πτ ∞ −∞ e − (x−ξ) 2 4τ u 0 (ξ) dξ.(3)
The solution of the Black-Scholes equation can be obtained in closed-form by using the change of independent and dependent variables
S = Ke x , t = T − 2τ σ 2 , V (S, t) = K e −γx−(γ+1) 2 τ u(x, τ )(4)
where
γ = r σ 2 − 1 2 (5)
T is the maturity time, S the stock price, K the strike price, σ the volatility, and r is the risk-free interest rate. In terms of the new variables, the payoff functions
V C (S, T ) = max{S −K, 0}, V P (S, T ) = max{K −S, 0}(6)
become u C (x, 0) = max{e (γ+1)x −e γx , 0}, u P (x, 0) = max{e γx −e (γ+1)x , 0} (7) and by denoting
Φ(ζ) = 1 √ 2π ζ −∞ e −η/2 dη(8)
the Black-Scholes formulas for the values of a European call and put option can be written in the closed form
V C (S, t) = S Φ(d 1 ) − K e −r(T −t) Φ(d 2 ) V P (S, t) = K e −r(T −t) Φ(d 1 ) − S Φ(d 2 )(9)
with
d 1 = log(S/K)+(r+ 1 2 σ 2 )(T−t) σ √ T −t , d 2 = log(S/K)+(r− 1 2 σ 2 )(T−t) σ √ T −t .(10)
A generalized version of Black-Scholes equation
Let us consider the more general version of Black-Scholes equation
∂V ∂t + σ 2 2 S 2 ∂ 2 V ∂S 2 + rS ∂V ∂S + (σ 2 U(ln S) − r)V = 0(11)
defined by using a function U : R −→ R, called a potential [1]. The function with separated variables
V (S, t) = e εt S −γ φ(ln S)(12)
where γ is the constant used in the previous section, is a solution of equation (11) if and only if φ(x) is a solution of the Schrödinger equation
− 1 2 d 2 φ dx 2 + U(x) φ = λ φ (13) with λ = ε σ 2 − 1 2 (γ +1) 2 .(14)
The explicitly solvable cases
U(x) = 0 and U(x) = ∞ for x ≤ a 0 for a < x < b ∞ for x > b(15)
have been analyzed in [12] by using the factorization method. A supersymmetric partner has been obtained for each of them. Our purpose is to investigate the case U(x) = x 2 2 lying, in a certain sense, between the cases (15). We present a solution in terms of Hermite-Gauss functions, a family of supersymmetric partners depending on a continuous parameter α, and the corresponding solutions.
A version related to the quantum harmonic oscillator
The modified Black-Scholes equation
∂V ∂t + σ 2 2 S 2 ∂ 2 V ∂S 2 + rS ∂V ∂S + σ 2 2 (ln S) 2 − r V = 0(16)
is directly related to the Schrödinger equation of the quantum oscillator
− 1 2 d 2 φ dx 2 + x 2 2 φ = λ φ.(17)
The function
Ψ n : R −→ R, Ψ n (x) = 1 n! 2 n √ π H n (x) e − 1 2 x 2 (18)
where H n is the Hermite polynomial
H n (x) = (−1) n e x 2 d n dx n e −x 2(19)
satisfies the equation
− 1 2 d 2 Ψ n dx 2 + x 2 2 Ψ n = n+ 1 2 Ψ n for any n ∈ {0, 1, 2, ...}.(20)
The system of Hermite-Gaussian functions
{Ψ n } n∈{0,1,2,...} is orthonormal ∞ −∞ Ψ n (x) Ψ k (x) dx = δ nk = 1 if n = k 0 if n = k(21)
and complete in the Hilbert space of square integrable functions
L 2 (R) = ψ : R −→ C ∞ −∞ |ψ(x)| 2 dx < ∞ .(22)
In view of the result obtained in the previous section, the function
V n (S, t) = e εnt S −γ Ψ n (ln S)(23)
where
ε n = nσ 2 + σ 2 2 + σ 2 2 (γ +1) 2(24)
is a solution of equation (16). If the coefficients c n are such that the function
V (S, t) = S −γ ∞ n=0 c n e εnt Ψ n (ln S)(25)
exists and can be derived term-by-term then it is also a solution of (16). We choose 0 < a < b such that the interval (a, b) is large enough to contain all the possible values of the stock price S, and consider the square integrable payoff functions
v C (S) = 0 for S < K S −K for K ≤ S ≤ b 0 for S > b (26) v P (S) = 0 for S < a K −S for a ≤ S ≤ K 0 for S > K(27)
Since
δ nk = ∞ −∞ Ψ n (x) Ψ k (x) dx = ∞ 0 1 S Ψ n (ln S) Ψ k (ln S) dS ,(28)from the relation v C (S) = S −γ ∞ n=0 c n e εnT Ψ n (ln S)(29)
we get
c n = e −εnT ∞ 0 v C (s) s γ−1 Ψ n (ln s) ds = e −εnT b K (s−K) s γ−1 Ψ n (ln s) ds.(30)
The solution of the equation (16) is
V (S, t) = S −γ ∞ n=0 e εn(t−T ) Ψ n (ln S) b K (s−K) s γ−1 Ψ n (ln s) ds.(31)
In the case of the put option the solution of (16) is
V (S, t) = S −γ ∞ n=0 e εn(t−T ) Ψ n (ln S) K a (K −s) s γ−1 Ψ n (ln s) ds.(32)
If the real parameter α is such that |α| > 1 2 √ π then the potential [13]
U α (x) = x 2 2 − dg α dx (x)(33)
where,
g α : R −→ R, g α (x) = e −x 2 α + x 0 e −u 2 du ,(34)
is a supersymmetric partner of U(x) = x 2 2 . The functions
ϕ 0 , ϕ 1 = AΨ 0 , ϕ 2 = AΨ 1 , ϕ 3 = AΨ 2 , ... where ϕ 0 (x) = e − x 2 2 exp x 0 g α (u) du(35)
and A is the first order differential operator
A = 1 √ 2 − d dx +x+g α (x) ,(36)
belong to L 2 (R) and are orthogonal. The corresponding orthonormal system
Φ 0 = ϕ 0 ||ϕ 0 || , Φ 1 = ϕ 1 ||ϕ 1 || , Φ 2 = ϕ 2 ||ϕ 2 || , ...
is complete in L 2 (R) and [8,13] − 1 2
d 2 Φ n dx 2 + U α (x) Φ n = n+ 1 2 Φ n for any n ∈ {0, 1, 2, ...}. (37)
The solution of the modified Black-Scholes equation
∂V ∂t + σ 2 2 S 2 ∂ 2 V ∂S 2 + rS ∂V ∂S + (σ 2 U α (ln S) − r)V = 0 (38) satisfying the condition V (S, T ) = v C (S) is V (S, t) = S −γ ∞ n=0 e εn(t−T ) Φ n (ln S) b K (s−K) s γ−1 Φ n (ln s) ds(39)
and the solution satisfying the condition V (S,
T ) = v P (S) is V (S, t) = S −γ ∞ n=0 e εn(t−T ) Φ n (ln S) K a (K −s) s γ−1 Φ n (ln s) ds.(40)
An approach based on a finite quantum oscillator
The Hamiltonian of the quantum oscillator can be written as
H = − 1 2 D 2 + 1 2 x 2 , where D = d dx(41)
The inverse of the Fourier transform
ψ → F [ψ], F [ψ](x) = 1 √ 2π ∞ −∞ e −ixξ ψ(ξ) dx.(42)
is the adjoint transformation
ψ → F + [ψ], F + [ψ](x) = 1 √ 2π ∞ −∞ e ixξ ψ(ξ) dx(43)
and
F [Dψ] (x) = ix F [ψ](x), F D 2 ψ (x) = −x 2 F [ψ](x).(44)
If in the last formula we put F + [ψ] instead of ψ then we get
F D 2 F + [ψ](x) = −x 2 ψ(x)(45)
and we have
H = − 1 2 (D 2 + F D 2 F + ).(46)
In order to obtain a finite counterpart of H, we consider a positive odd integer d = 2ℓ+1 and the set [7]
R d = {−ℓ √ κ, (−ℓ+1) √ κ, . . . , (ℓ−1) √ κ, ℓ √ κ} with κ = 2π d .
The space l 2 (R d ) of all the functions ψ : R d −→ C considered with the inner product
ψ 1 , ψ 2 = ℓ n=−ℓ ψ 1 (n √ κ) ψ 2 (n √ κ)(47)
is a Hilbert space isomorphic to the d-dimensional Hilbert space C d . Since
lim d→∞ √ κ = 0 and lim d→∞ (±ℓ) √ κ = ±∞(48)
we can consider that, in a certain sense,
R d d→∞ − −−→ R and l 2 (R d ) d→∞ − −−→ L 2 (R).(49)
Each function ψ : R d −→ C can be regarded as the restriction to R d of a periodic function ψ :
Z √ κ −→ C with period d √ κ.
The inverse of the finite Fourier transform
l 2 (R d ) −→ l 2 (R d ) : ψ → F [ψ], where F [ψ](n √ κ) = 1 √ d ℓ k=−ℓ e − 2πi d nk ψ(k √ κ).(50)
is the adjoint transformation
l 2 (R d ) −→ l 2 (R d ) : ψ → F + [ψ], defined by F + [ψ](n √ κ) = 1 √ d ℓ k=−ℓ e 2πi d nk ψ(k √ κ).(51)
The finite-difference operator D 2 , where
D 2 ψ(n √ κ) = ψ((n+1) √ κ) − 2ψ(n √ κ) + ψ((n−1) √ κ) κ (52)
is an approximation of D 2 , and we have
F D 2 F + ψ(n √ κ) = d π cos 2πn d − 1 ψ(n √ κ).(53)
The finite-difference Hamiltonian
H d = − 1 2 (D 2 + F D 2 F + )(54)
with the matrix − d 4π 2(cos 2πn d −2)δ nm +δ n,m+1 +δ n,m−1 +δ n,m−2ℓ +δ n,m+2ℓ −ℓ≤n,m≤ℓ (55) that is,
− d 4π 2 cos 2π(−ℓ) d −4 1 0 · · · 1 1 2 cos 2π(−ℓ+1) d −4 1 · · · 0 0 1 2 cos 2π(−ℓ+2) d −4 · · · 0 . . . . . . . . . . . . . . . 1 0 0 · · · 2 cos 2πℓ d −4
is a finite counterpart of the Hamiltonian H and, in a certain sense [4],
H d d→∞ − −−→ H.F Ψ m = (−i) m Ψ m and F h m = (−i) m h m .(57)
The functions h m , called Harper functions [4], are eigenfunctions of H d corresponding to certain eigenvalues λ n , that is,
H d h m = λ m h m , for any m ∈ {0, 1, 2, ..., d−1}.(58)
The eigenvalues λ n and the functions h n are available only numerically by diagonalizing the matrix (55). Nevertheless, they play an important role in the theory of fractional Fourier transform, optics and signal processing [14].
In practice, the option price is a discrete variable, not a continuous one. It is an integer multiple of a certain minimal quantity, a sort of quantum of cash (usually, 1/100 or 1/1000 of the currency unit). In our simplified approach, we distinguish only a finite number of possible values, namely,
e −ℓ √ κ , e (−ℓ+1) √ κ , . . . e (ℓ−1) √ κ , e ℓ √ κ .(59)
An acceptable description is obtained for d = 2ℓ+1 large enough. In the case of the strike price K = e k √ κ we consider the payoff functions
v C (e n √ κ ) = 0 for −ℓ ≤ n < k e n √ κ −e k √ κ for k ≤ n ≤ ℓ (60) v P (e n √ κ ) = e k √ κ −e n √ κ for −ℓ ≤ n ≤ k 0 for k < n ≤ ℓ(61)
and assume that the solutions of (16) can be approximated by functions of the form (see (14) and (25)) with ε n = σ 2 λ n + σ 2 2 (γ +1) 2 .
V (e m √ κ , t) = e −mγ √ κ d−1 n=0 c n e εnt h n (m √ κ)(62)
Since h n , h m = δ nm , from V (S, T ) = v P (S) we get
c n = e −εnT ℓ q=k e q √ κ −e k √ κ e qγ √ κ h n (q √ κ)(64)
and the corresponding solution of (16) can be approximated by
V C (e m √ κ , t) = e −mγ √ κ d−1 n=0 e εn(t−T ) h n (m √ κ) ℓ q=k e q √ κ −e k √ κ e qγ √ κ h n (q √ κ).
The solution of (16) satisfying V (S, T ) = v P (S) can be approximated by
V P (e m √ κ , t) = e −mγ √ κ d−1 n=0 e εn(t−T ) h n (m √ κ) k q=−ℓ e k √ κ −e q √ κ e qγ √ κ h n (q √ κ)
In Fig. 1 we present the values of V C (e m √ κ , t) (left hand side) and V P (e m √ κ , t) (right hand side) for t = 3 (squares), t = 4 (rhombus) and maturity time T = 5 (bullets) in the case d = 21, σ = 0.25, r = 0.03 and a strike price K = e 8 √ κ . In a neighbourhood of the strike price, our results agree with those obtained by using the standard BS equation.
Concluding remarks
BS equation represents a 'gold vein' for finance, and it is worth to dig around it because some other very interesting things may exist. We investigate from a mathematical point of view a modified version of the BS equation, without to know any possible financial interpretation. Our main contribution is a finite-dimensional approach to a GBS equation which keeps the essential characteristics of the continuous case.
The mathematical modeling of price dynamics of the financial market is a very complex problem. We could never take into account all economic and non-economic conditions that have influences to the market [9,10]. Therefore, we usually consider some very simplified and idealized models, a kind of toy models which mimic certain features of a real stock market [2,3]. We think that the finite-dimensional models [5,6,7,16] may offer enough accuracy and are more accessible numerically . They use linear operators with finite spectrum, finite sums instead of series and integrals, finite-difference operators instead of differential operators, etc.
The harmonic oscillator is among the most studied physical systems. Our results open a way to use in quantum finance the rich mathematical formalism developed around the quantum oscillator. We can use, for example, coherent states and the coherent state quantization [11], finite frames and the finite frame quantization [7] in order to define mathematical objects with a financial meaning.
Figure 1 :
1Time evolution of the option price (see the text).
The operators H and H d are both Fourier invariantF H = HF
and
F H d = H d F .
(56)
The eigenvalues of H d are distinct, and the normalized eigenfunctions h m of
H d , considered in the increasing order of the number of sign alternations, can
be regarded as a finite version of Hermite-Gaussian functions Ψ 0 , Ψ 1 , ..., Ψ d−1 .
For example, we have [4]
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| []
|
[
"A second order differential equation for a point charged particle",
"A second order differential equation for a point charged particle"
]
| [
"Ricardo Gallego Torromé \nDepartamento de Matemática\nUniversidade de São Paulo\n\n"
]
| [
"Departamento de Matemática\nUniversidade de São Paulo\n"
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| A model for the dynamics of a classical point charged particle interacting with higher order jet electromagnetic fields is described by an implicit ordinary second order differential equation. We show that such equation is free of run-away and pre-accelerated solutions of Dirac's type. The theory is Lorentz invariant and is compatible with Newton's first law and Larmor's power radiation law. | 10.1142/s0219887817500499 | [
"https://arxiv.org/pdf/1207.3627v8.pdf"
]
| 117,085,356 | 1207.3627 | 4e2942ab950be3e7821d973297bcc242b62734fd |
A second order differential equation for a point charged particle
6 Feb 2014 May 2, 2014
Ricardo Gallego Torromé
Departamento de Matemática
Universidade de São Paulo
A second order differential equation for a point charged particle
6 Feb 2014 May 2, 2014
A model for the dynamics of a classical point charged particle interacting with higher order jet electromagnetic fields is described by an implicit ordinary second order differential equation. We show that such equation is free of run-away and pre-accelerated solutions of Dirac's type. The theory is Lorentz invariant and is compatible with Newton's first law and Larmor's power radiation law.
Introduction
The existence of some fundamental problems in the electrodynamics theory of point charged particles interacting with its own radiation reaction field is one of the oldest open problems in field theory (see for instance [14,20,30,34] and references therein). The investigation of the radiation reaction of point charged particles led to the Abraham-Lorentz-Dirac equation [12] (in short, the ALD equation). However, a main conceptual difficulty is that, although the derivation of the ALD is based on general principles, the equation has un-physical solutions. Such problems on the standard ALD equation suggest that classical electrodynamics of point charged particles should not be based on the ALD equation together with Maxwell's theory. Ideas to overcome the difficulties range from reduction of order schemes for the ALD equation [22], re-normalization group schemes [33,34], extended models of the charge particle [7,36], higher order derivative field theories [4,24,11], the Feynman-Wheeler's absorber theory of electrodynamics [15], models where the observable mass is variable on time [3,19,23], non-linear electrodynamic theories [5], dissipative force models [27,19] among other proposals (for an account of some of the above approaches, see [14] and [31]). Some of the above mentioned proposals contain serious deficiencies that prevent them to be considered as consistent solutions of the radiation reaction problem. The imposition of the Dirac's asymptotic condition seems an ad hoc procedure to eliminate the undesired solutions. Such point of view puts in disadvantage those approaches based on the Lorentz-Dirac equation as fundamental equation, in particular the models of reduction of order. On the other hand, simplicity in the assumptions makes natural to consider point charged particle as a model for the probe particle (this excludes extended models of particles). Otherwise, if we adopt more sophisticated models for the probe charged particle, one will need to explain the stability of the extended structure of the probe charged particle and prove the mathematical consistency of the theory. Therefore, in order to avoid some of such complications we will concentrate on the case that probe particles are described by point charged particles. For a probe point charged particles, the simplest dynamical equation that is Lorentz invariant, compatible with conservation of relativistic four momentum and with the hypothesis that the electromagnetic field are solutions of the Maxwell equations is the ALD equation [12]. Thus, the deep paradox is that the ALD equation is properly derived from first principles but is physically untenable. One wonders whether the problems of the ALD is inherent to the classical description of point charged particles, since at a more fundamental level, the description of a charged particle should be in the framework of non-relativistic quantum electrodynamics. However, we are not pursuing here a quantum description of the problem. This is because our question is a classical issue, posed at the classical level and therefore it should be solvable (in positive or negative way) at the same level. Also, it is not clear that a quantum formulation could provide a consistent description of the radiation reaction phenomena. Even if there are some arguments favoring a semi-classical quantum treatment of the problem (see for instance [28] and references therein), there is no a complete proof of the resolution in the framework of the quantum theory. Thus, in the absence of a rigorous result showing the impossibility of solving the above problems in a classical framework or showing that they can be completely solved in the quantum mechanical framework, we find justified to formulate the problem and try to solve it in the simpler classical framework. The fact that the ALD equation is properly derived from fundamental principles of classical field theory suggests that the way to obtain a consistent classical description of radiation reaction of point charged particles requires to relax some of the fundamental assumptions of classical electrodynamics. One of such new ideas was depicted by R. Feynman and Wheeler. In Wheeler-Feynman's theory, the electromagnetic field in vacuum is identically zero, there is a complete time reversal symmetry not only in the equations for the potential. In their theory, the electromagnetic field appears only as a secondary, mathematically convenient object. That is, in Wheeler-Feynman proposal there are only particles and not free external fields.
In the companion paper [18], a generalization of the classical Maxwell-Lorentz electrodynamics in the form of a theory of abelian generalized higher order jets elec-tromagnetic fields was developed 2 . It was shown that such notion of generalized higher order field is more flexible than the usual notion of classical field. Thus, in our proposal apart from probe particles, there is a smooth generalized electromagnetic field, whose dynamics and value is not independent of the dynamics of the probe particle used to measure it. It is in this sense that our theory resembles the Wheeler-Feynman's theory, since the probe particle affects directly the field.
In the case of electromagnetic phenomena, such weaker notion of field can be partially justified as follows. The standard notion of classical fields as sections of a vector bundle over manifold M presupposes the independence of the field from the dynamical state of the probe particles used to measure the fields. However, we have learned from the quantum theory that the action of a measurement on a physical can disturb in a fundamental way the system being measured, such that after the measurement, the system has changed drastically the state. In a similar way, one can imagine that the notion of local classical electromagnetic field (that is, a closed, differential 2-form living on the spacetime manifold M and independent of the motion of the probe particle) fails to accommodate the effects of the dynamics of the probe charged particle, due to radiation reaction effects, that distort the original electromagnetic field. The relevant point is how this is accomplished. For instance, in the Wheeler-Feynman theory, the fields are still living on the manifold M and the particle-field interaction is formulated in a time-symmetric way. As a result, the radiation damping term is the same than in the standard Abraham-Lorentz theory. Such damping term it is the problematic one. Thus, Wheeler-Feynman's appears more as a justification for the theory of Abraham-Lorentz-Dirac than a solution to the problem. One needs to modify the way the probe particle changes the field. A notion of classical field that can be adapted to the dynamics of the probe charged point particle is of fundamental importance for a better understanding of feedback phenomena and in particular the radiation reaction in classical electrodynamics. We find that in this case, a field that depends on the details of the trajectory of the probe particle is the adequate notion. This lead us to a convenient notion of physical fields as a generalized differential form with values in higher order jet bundles along curves [18]. Such fields are non-local in the sense that they do not live on the spacetime manifold M (the generalized fields does not depend only on where they are measured, but also on how they are measured). As a consequence, the description of electrodynamics using generalized electromagnetic fields will not be local, since the fields depend not only on the position with time x(t) but also on the derivatives of the world-line particle probe.
In this work we adopt the mathematical framework introduced in [18] to obtain in detail an implicit second order differential equation for point charged particles that is free of the run away and pre-accelerated problems of the ALD equation. We found that the new implicit differential equation is compatible with Larmor's power radiation law, a fundamental constraint in searching for models of radiation reaction. The derivation of the new equation of motion for probe point charged particles is based on two new assumptions. First, the notion of generalized higher order fields is applied to the case of electromagnetic fields. Second, the hypothesis that proper acceleration (measured with the Lorentzian metric η) of a point charged particle is bounded, an idea that convey us to the use of a maximal acceleration geometry [18]. This is a natural requirement to avoid run-away solutions. It can also be seen that such metrics appears in a natural way if one requires to preserve the first and second laws of dynamics (see [18], section 3). With the notion of generalized higher order fields and maximal acceleration on hand and demanding that the equation of motion of a probe point charged particle is of second order, it is possible to eliminate the Schott term in the ALD equation consistently. The strategy of eliminating the Schott term in the equation of motion of a point particle was first brought to light by J. Larmor and later by W. Bonnor. In their theory, the observable rest mass of the point charged particle could vary on time, being this the source for the radiation. On the other hand, the bare mass was constant. In contrast, in our proposal the bare mass can vary during the evolution but the observable rest mass is constant. However, the variable bare mass should be thought as a consistent requirement, and not as a main principle (as it was in the variable mass hypothesis in Bonnor's theory). Thus, in our theory the higher order electromagnetic fields contain additional degrees of freedom corresponding to the higher order corrections, that originate the change in the bare mass (see [18], section 7) and allow also for the radiation of energy-momentum. In addition, we keep the minimal higher order corrections necessary to obtain an implicit second order differential equation. The two new assumptions (generalized fields and maximal acceleration geometry) are necessary to solve the problem of the motion of a point charge in a consistent way. We will show that the notion of generalized higher order electromagnetic field is not sufficient to provide an appropriate dynamical description of a point charged particle. Similarly, the hypothesis of maximal acceleration geometry is also not enough (see Section 4). However, in combination with the hypothesis of maximal proper acceleration one can obtain a model compatible with the power radiation law of a point charged particle, obtaining equation (5.10). Thus, maximal acceleration is useful for controlling the grow of the acceleration and the order of the differential equation.
The new differential equation (5.10) is compatible with the first Newton's law, it does not have un-physical solutions as the ALD equation and it is compatible with energy-momentum conservation. We show that the equation (5.10) can be approximated by an ordinary differential equation where the second time derivative is explicitly isolated. We argue that in the theoretical domain of applicability of the theory, the approximate equation is equivalent to the original new equation of motion. This approximated differential equation opens the possibility to use standard ODE theory to prove existence and uniqueness, bounds and regular properties of the solutions.
The derivation of equation (5.10) breaks down in two dynamical regimes. The first corresponds to curves of maximal acceleration. Thus the domain of applicability of the theory corresponds to world-lines far from the maximal acceleration regime. The second corresponds to covariant uniform acceleration. In the case of covariant uniform acceleration, a similar analysis as in the non covariant uniform acceleration case leads us to a differential equation which is a Lorentz force equation with a constant total electromagnetic field [16]. However, covariant uniform motion of point charged particles is unstable: in order to compensate the radiation reaction field and provide the total constant force of the uniform motion, an infinite precision in the calibration of the external fields is required, which seems to be an unstable regime. The structure of this paper is the following. In section 2, the notion of generalized higher order electromagnetic field is motivated and briefly discussed. We have used local coordinates, since it is enough for our objective of finding an ODE for point charged particles. The necessary notation on jet bundle theory is put in a form of Appendix A in a way that the reader un-familiar with that theory can follow the local description. In section 2 we also provided the fundamental equations of the generalized Maxwell's theory. In section 3, the basic notions of the geometry of metrics of maximal acceleration is presented. An heuristic argument in faubour of a principle of maximal acceleration is presented. The perturbation scheme of our theory is also introduced. Note that we will not develop in full the perturbation scheme, since we do not provide any definite value for the maximal proper acceleration. However, to consider the proper acceleration very large compared with the value of the maximal acceleration it will be enough to provide an effective model. In addition, to show the consistency of the notion of maximal acceleration geometry, we provide Appendix B, where it is proved that the proper-time of a metric of maximal acceleration is invariant under reparameterizations. Section 4 is devoted to explain a geometric method to obtain the ALD equation. This is based on the use of adapted frames and the imposition of consistency with Larmor's radiation law of a point particle. In section 5, the method is used in the framework of generalized fields and maximal acceleration, obtaining the equation (5.10) (or the covariant version (5.11)). A short discussion of the theory is presented in section 6. Notation. M will be a four dimensional spacetime manifold. T M is the tangent bundle of M and η is the Minkowski metric with signature (−1, 1, 1, 1). The null cone bundle of η isπ : N C → M . Local coordinates on M are denoted by (x 0 , x 1 , x 2 , x 3 ) or simply by x µ . We will find useful to identify points by its coordinate, since we are working on single homeomorphic open domains to R 4 . Natural coordinates on T M and N C are denoted as (x, y). Further notation will be introduced in later sections, Appendix A and Appendix B.
Notion of generalized higher order fields
The notion of generalized field adopted in this paper is conveyed by the idea of linking the mathematical model of probe particle with an economical in postulates description of fields capable to support a consistent dynamics with probe particles. For classical fields, the simplest model of probe particle is the point charged particle, described by a one dimensional world-line curve x : I → M . By considering the departure of the world-line from being a geodesic, one can obtain information on the value of the electromagnetic field. However, this inverse process is not enough to completely determine the electromagnetic field and in general, it is necessary to consider more than one probe particle to completely fix the field variables. Therefore, a classical electromagnetic field could be thought as a functional (with co-domain in a convenient space) defined over a quotient space of the path space of world-lines of physical probe particles. Examples of such quotient spaces are the spacetime manifold M itself and the corresponding jet bundles over the spacetime manifold. From the mathematical side, the main problem for obtaining a consistent theory is to determine functionals and quotient spaces compatible with experiment and that admit a consistent mathematical formulation; from the physical side, the main concern is that the logical system should be predictive and falsifiable. Such functionals could be difficult to analyze. In this contest, jet bundle theory is quite adequate, since it deals systematically with Taylor expansions of functions and sections with arbitrary order. Moreover, jet bundle formalism is useful when dealing with geometric objects that are defined along maps, independently of their geometric character. For instance, jet theory is extensively used in problems of calculus of variations (see Appendix A and references therein), in the formulation of generalized field theories like Bopp-Podolsky theory [11] and more generally, in the formulation of problems involving partial differential equations.
Jet fields have a definite transformation rule under local coordinate transformations and in some circumstances, a section of a vector bundle can be approximated locally by jet sections. This is as a consequence of Peetre's type theorems (see [21], pg. 176). Thus, for non-increasing operators acting on smooth sections, it is equivalent to work with operators acting on convenient higher order jet fields. An additional motivation for considering jet fields comes from the fact that, at the quantum level description, the relevant dynamical variables in gauge theories are holonomy variables or loop variables (see for instance the discussion in Chapter 1 of [9]). Thus, one could expect some degree of non-locality in an intermediate description of physical systems between quantum and classical systems. Jet fields are a natural mathematical tool accomplishing such properties, since they are classical (they live on the world-line curve x : I → M , which corresponds to a location of the particle in spacetime) and at the same time they depend not only on the spacetime point manifold x(I) ֒→ M but also on the higher derivatives associated with the time-like curve x : I → M . Note that although the comparison between the nonlocality of the holonomy variables in one hand and the corresponding for jet fields is merely formal, it suggests a description of the holonomy variables in terms of infinite higher order jets fields.
Notions of generalized electromagnetic field and current
Although we will use mainly the coordinate formulation of the generalized electromagnetic field, there is a coordinate-free formulation for such objects these objects. In particular, the Hodge star operator ⋆ and the nilpotent exterior derivative d 4 are well defined geometric objects acting on generalized forms. For the notation and basic definitions of jet bundle theory we refer to Appendix A. Here we will adopt the following generalization of the Faraday and excitation tensor and electromagnetic density current,
Definition 2.1
The electromagnetic fieldF along the lift k x : I → J k 0 (M ) is a 2-form that in local natural coordinates can be written as
F ( k x) =F (x,ẋ,ẍ, ... x , ..., x (k) ) = F µν (x) + Υ µν (x,ẋ,ẍ, ... x , ..., x (k) ) d 4 x µ ∧ d 4 x ν , (2.1) with F (x) ∈ ΓΛ 2 M . The excitation tensor along the lift k x : I → J k 0 (M ) is the 2-form G( k x) =Ḡ(x,ẋ,ẍ, ... x , ..., x (k) ) = G µν (x) + Ξ µν (x,ẋ,ẍ, ... x , ..., x (k) ) d 4 x µ ∧ d 4 x ν . (2.
2) The density current in electrodynamics is represented by a 3-form
J(x,ẋ,ẍ, ... x , ..., x (k) ) = J µνρ (x) +Φ µνρ (x,ẋ,ẍ, ... x , ..., x (k) ) d 4 x µ ∧d 4 x ν ∧d 4 x ρ . (2.3)
These generalized fields are localization of sections of Λ p (M, F(J k 0 (M ))) (see [18] or the Appendix A). A generalized field associates to each pair of tangent vectors e µ , e ν and curve x : I → M an element of the k-jet along x, e µ (x(τ )), e ν (x(τ )) →F ( k x(τ )). That F is defined on the Grasmannian G(2, M ) allows to define fluxes and to associate with them the outcome of macroscopic measurements. The fact that the fields take values on higher jet bundles over the spacetime manifold is the mathematical implementation of our idea that fields depend on the state of motion of the probe particle in a fundamental way. At his stage we did not fix the value of the order k. We will keep it free until later sections, where it will be fixed to be k = 3 by physical constraints. With the aid of the nilpotent exterior derivative d 4 and the Hodge star operator ⋆ defined by η, one can write in complete analogy with Maxwell's equations the following homogeneous and non-homogeneous equations [18]:
• The generalized homogeneous equations are
d 4F = 0. (2.4)
• Let us assume the simplest constitutive relationḠ = ⋆F . Then the generalized inhomogeneous Maxwell's equations are
d 4 ⋆F = J + d 4 ⋆ Υ. (2.5)
Using some constrains on Υ coming from compatibility with the equation of motion of point charged particles, one obtains an effective theory which is equivalent to the standard Maxwell's theory. Thus the homogeneous equations are
dF = 0. (2.6)
Similarly, the non-homogeneous equations are
d ⋆ F = J, (2.7)
and the conservation of the current density are
dJ = 0. (2.8)
Since the fields are originally in higher order spaces, there are additional degrees of freedom to define a consistent particle dynamics 3 . This also shows that the effective theory of higher order fields, in the sector of Maxwell's fields, is reduced to the standard Maxwell's theory. This will be different for the particle law of motion, as we will see in this work.
3 Maximal acceleration geometry
Why maximal acceleration?
The notion of maximal acceleration is not new. It was developed first in the work of E. Caianiello and co-workers, that provided heuristic motivation for such notion in their framework of quantum geometry (see [6] and references there in). We give here an heuristic argument for the existence of a maximal acceleration based on the assumption that there is a minimal length L min and a maximal speed. The minimal length is the assumed scale of the spacetime region that can produce an effect on the system in the shortest period of time. This idea is not necessarily related with a quantification of spacetime, but requires a notion of extended local domain where cause-effect relations are originated. Therefore, the maximal acceleration could be relational, depending on the physical system. This is in contrast with universal maximal acceleration. However, we will require that whatever the maximal acceleration is, they will be very large compare with the acceleration of the probe particle. In this way, our perturbative scheme will be perfectly applicable. By adopting the above hypotheses, the action on a particle done by its surrounding (assuming a certain form of locality) and one obtains for a maximal work as a result of such action to be
L min m a ∼ δm v 2 max ,
where a is the value of the acceleration in the direction of the total exterior effort is done. Then one associates this value to the work over any fundamental degree of freedom evolving in M , caused by rest of the system. Since the speed must be bounded, v max ≤ c. Also, the maximal work produced by the system on a point particle is δm = −m. Thus, there is a bound for the value of the acceleration,
a max ≃ c 2 L min . (3.1)
Maximal acceleration in electrodynamics has been studied before, in particular in the theory of extended charged particles. The first instance on the limitation of acceleration is found in the Lorentz's theory of the electron, where it appears as a causal condition for the evolution of the charge elements defining the electron (for a modern discussion of the theory see [34]). Another instance where maximal acceleration appears is in Caldirola's extended model of charge particles [8], where the existence of a maximal speed and minimal elapsed time (called chronon) implies the maximal acceleration. In contrast, our probe particle is a point particle and the maximal acceleration is a postulate to allow us to define a perturbative model, with phenomenological consequences.
Universal maximal acceleration has appeared recently as a direct consequence of covariant loop gravity, in a way compatible with local Lorentzian geometry [32]. Before, it was argued that some models of string theory contains maximal acceleration too [29]. Thus, the notion of proper bound for maximal acceleration must naturally contained in the physical geometry of the spacetime.
Elements of covariant maximal acceleration geometry
In the same way that the electromagnetic field is described by a generalized higher order field, it is natural to describe the spacetime structures by generalized higher order metrics. The motivation for this is two-fold. First, when coupling gravity with generalized higher order fields, it will be natural to consider the gravitational field described by a generalized higher order field. Second, when considering maximal acceleration kinematics it turns out that the corresponding path structure is a generalized metric. In this paper we will consider the second point, leaving the relation between generalized fields and gravity for future work.
Let Dẋ be the covariant derivative in the directionẋ associated with the Levi-Civita connection of the Minkowski metric η. It induces a Levi-Civita connection γ µ νρ on M and a non-linear connection on the bundle π : T M \ N C → M whose coefficients are given by
N µ ν (x, y) = γ µ νρ y ρ , µ, ν = 0, 1, 2, 3 (3.2)
with γ µ νρ the Christoffel symbols. The induced vertical forms are
{δy µ = dy µ + N µ ν dx ν , µ, ν = 0, 1, 2, 3}. The set of local sections { δ δx 1 | u , ..., δ δx n | u , u ∈ π −1 (x), x ∈ U } generates the local horizontal distribution H U over T U , with U an open set of M .
The covariant acceleration vector field of the curve x : I → M is the vector Dẋẋ ∈ T x M that satisfies the differential equation in local coordinates
Dẋẋ µ :=ẍ µ + γ µ νρẋ νẋρ . (3.3)
The covariant formalism for geometries of maximal acceleration that we use was developed in [17] as a geometric formulation of Caianiello's quantum geometry. In the way related with generalized tensors was developed in [18]. The Sasaki-type metric on the bundle T M \ N C is the pseudo-Riemannian metric There is a bilinear, non-degenerate, symmetric form g along the lifted timelike curve 1 x : I → T M , such for the corresponding horizontal lifted vector hẋ conformaly equivalent to η such that the following relation holds,
g S = η µν dx µ ⊗ dx ν + 1 A 2 max η µν δy µ ⊗ δy ν .g µν (x(τ )) = 1 − η σλ Dẋẋ σ (τ )Dẋẋ λ (τ ) A 2 max η µν + higher order terms. (3.5)
Proof. The tangent vector of the lift (x(τ ),ẋ(τ )) is (ẋ,ẍ). Using the non-linear connection, one considers the vector field T =ẋ µ δ δx µ +ẍ µ ∂ ∂y µ ∈ T J 1 M . Then the metric g S acting on the vector field T at the point (x(τ ),ẋ(τ )) has the value
g S (T, T ) = η µν dx µ ⊗ dx ν + 1 A 2 max η µν δy µ ⊗ δy ν T, T = η(ẋ,ẋ) 1 + 1 A 2 max η µν Dẋẋ µ Dẋẋ ν η(ẋ,ẋ) ≃ − 1 − 1 A 2 max η µν Dẋẋ µ Dẋẋ ν ,
which coincides with the value (g µν dx µ ⊗ dx ν )(ẋ,ẋ), with g µν given by (3.5) at leading order.
✷ The generalized metric η is called metric of maximal acceleration, because of the following property,
Proposition 3.2 If g(ẋ,ẋ) < 0, then g(Dẋẋ, Dẋẋ) < A 2 max .
Proof. From the expression (3.5) and since g is non-degenerate, the factor
1 − η σλ Dẋẋ σ (τ )Dẋẋ λ (τ ) A 2 max
cannot be zero. Thus, if g(ẋ,ẋ) < 0, then it follows that η(Dẋẋ, Dẋẋ) < A 2 max , from which follows directly the thesis.
✷ Since the metric η is flat, there is a local coordinate system where the connection coefficients γ µ νρ are zero (normal coordinate system). In such coordinate system ne also has that N µ ν (x, y) = 0 and the bilinear form (3.5) reduces to
g(τ ) = 1 − η σλẍ σ (τ )ẍ λ (τ ) A 2 max η µν dx µ ⊗ dx ν . (3.6)
This non-covariant version was first discussed by Caianiello's and co-workers [6].
For timelike curves, the domain D of the metric of maximal acceleration is the intersection of the open domains
D 1 = { 2 x ∈ J 2 0 (M ) s.t.η(ẋ,ẋ) < 0} and D 2 = { 2 z ∈ J 2 0 (M ) s.t. η(ẍ,ẍ) < A 2 max }. Therefore, the domain of definition of g is an open domain D = D 1 ∩ D 2 ⊂ J 2 0 (M )
. This is in concordance with the definition of g as a generalized metric (see Appendix B and for more details [18]). For lightlike vectors η(ż,ż) = 0, one also has g(ż,ż) = 0. Indeed, using the expression (3.4), we have for a lightlike vector of η,
g s (ẋ,ẋ) = η(ẋ,ẋ) + 1 A 2 max η(ẍ,ẍ) = 1 A 2 max η(ẍ,ẍ).
On the other hand
g(ẋ,ẋ) = g s (ẋ,ẋ) + O(ǫ 2 0 ) = 1 − η σλẍ σ (τ )ẍ λ (τ ) A 2 max η(ẋ,ẋ) + O(ǫ 2 0 ) = O(ǫ 2 0 ).
Thus, for lightlike curves respect to η, the two expressions are compatible iff η(ẍ,ẍ) = 0. In the regime η(ẋ,ẋ) << A 2 max and g(ẍ,ẍ) = 0, this condition is equivalent tö x = 0, or in covariant form, Dẋẋ = 0. Therefore, for lightlike trajectories, the domain of the maximal acceleration g are the lightlike geodesics.
On the validity of the clock hypothesis and geometry of maximal acceleration
It is well known that the existence of ideal clocks and rods associated with accelerated particles is an assumption, which is commonly named the clock hypothesis. It is logically independent of the principle of relativity and the principle of constance of the speed of light in vacuum. Indeed, Einstein's words, a standard clock and rod is such assumed that its behavior depends only upon velocities, and not upon accelerations, or at least, that the influence of acceleration does not counteract that of velocity (see [13], footnote in page 64).
Let us consider a normal coordinate system associated with the Minkowski metric η and let us denote by τ the proper-time parameter along a given curve respect to g,
τ [x] = t t 0 −g(x ′ (t), x ′ (t)) dt. (3.7)
Clearly, the clocks whose proper-time are calculated by using a generalized metric (3.5). Therefore, the justification of the use of proper-time (3.7) requires first to justify why the clock hypothesis is not universally valid. Such step was recognized by B. Mashhoon long time ago in [25]. After analyzing the equivalent hypothesis of locality, it was shown that such hypothesis is not valid when the so called acceleration time and length (a short of measure of the tidal effects on measurement devices) are not much more large compared with the intrinsic length and time of the physical system acting as a measurement device. However, this is exactly the situation for a radiating electron , that is the case of investigation in this paper: for a radiating electron the acceleration length and time are of the same order than the intrinsic length and time [25]. Therefore, for the situations that we are interesting in, the hypothesis of locality (or equivalently the ideal clock hypothesis) will not work.
Another relevant instance where the hypothesis will not work is on the singularity points. That is, when the accelerations become extremely high or un-bounded. In this situation, the proper-time given by equation (3.7) is compatible with Lorentz invariance, diffeomorphism reparameterization invariance and with maximal proper acceleration. Therefore, we adopt as the physical proper-time parameter. Note that doing this, we need to disregard the Lorentzian proper-time as the physical time. However, because the above properties of τ , this is indeed a moderate change. Moreover, due to the high value of the maximal acceleration, in usual situations the difference τ [γ] − s[γ] is small, being increasing as the proper acceleration η(ẍ,ẍ) increases. Moreover, since the existence of maximal acceleration, it is prevented the existence of singularities, allowing to define τ in the full domain of the curves γ.
Perturbation scheme
We can parameterize each world-line by the corresponding proper time τ associated to g and this should be understood below in this work if anything else is not stated. The acceleration square function is defined by the expression
a 2 (τ ) := η µρẍ µẍρ . (3.8)
Then the function ǫ is defined in a normal coordinate system of η by the relation
ǫ(τ ) = η σλẍ σ (τ )ẍ λ (τ ) A 2 max . (3.9)
In an arbitrary coordinate system the parameter ǫ(τ ) is given by the relation
ǫ(τ ) := η(Dẋẋ, Dẋẋ) A 2 max .
(3.10)
The relation between g and η along x : I → M is a conformal factor, g(τ ) = (1 − ǫ(τ ))η. (3.11) ǫ(τ ) determines a bookkeeping parameter ǫ 0 by the relation ǫ(τ ) = ǫ 0 h(τ ), where
ǫ 0 = max{ ǫ(τ ), τ ∈ I}. (3.12)
Also, the relation between the proper parameter of τ and the proper parameter of g is determined by the relation
ds = (1 −ǫ) −1 dτ. (3.13)
For compact curves this definition always makes sense. However, we will need to bound higher order derivatives in order to keep such parameter finite for non compact curves. Then one can speak of asymptotic expansions on powers of O(ǫ l 0 ), with the basis for asymptotic expansions being {ǫ l 0 , l = −∞, ..., −1, 0, 1, ..., +∞}. In order to define the perturbative scheme, let us consider a normal coordinate system for η. We assume that the dynamics happens in a regime such that
a 2 (τ ) ≪ A 2 max .
(3.14)
The curves X : I → M with a 2 (τ ) = A 2 max are curves of maximal acceleration. Since we assume that all the derivatives (ǫ,ǫ,ǫ, ...) are small, the dynamics of point charged particles will be away from the sector of maximal acceleration. The metric g determines different kinematical relations than η. Let us assume that the parametrization of the world-line x : I → M is such that g(ẋ,ẋ) = −1 and the monomials in powers of the derivatives of ǫ define a complete generator set for asymptotic expansions. In a geometry of maximal acceleration, the kinematical constrains are
g(ẋ,ẋ) = −1, (3.15) g(ẋ,ẍ) ≃ǫ 2 η(ẋ,ẋ), (3.16) g( ... x ,ẋ) + g(ẍ,ẍ) ≃ d dτ ǫη(ẋ,ẋ) 2 +ǫη(ẍ,ẋ) (3.17)
and similar conditions hold for higher derivatives obtained by derivation of the previous ones. Therefore, in a geometry of maximal acceleration (M, g), given the normalization g(ẋ,ẋ) = −1, the following approximate expressions hold:
x ρẋ ρ := g µρẋ µẋρ = −1, (3.18) x ρẋ ρ := g µρẍ µẋρ =ǫ 2 + O(ǫ 2 0 ),(3.19)
...
x ρẋ ρ +ẍ ρẍ ρ = g µρ ...
x µẋρ + g µρẍ µẍρ = d dτ ǫη(ẋ,ẋ) 2 +ǫ + O(ǫ 2 0 ). (3.20)
4 A simple derivation of the ALD force equation
Rohrlich's derivation of the ALD equation
Let us assume that the spacetime is the Minkowski spacetime (M, η). In a normal coordinate system of η, the ALD equation is the third order differential equation
mẍ µ = eF µ νẋ ν + 2 3 e 2 ... x µ − (ẍ ρẍ ρ )ẋ µ ,ẍ µẍ µ :=ẍ µẍσ η µσ . (4.1)
with e being the charge of the electron, m is the experimental inertial mass and the time parameter t is the proper time respect to the metric η. The ALD has run-away and pre-accelerated solutions [12,20], both against what is observed in everyday experience and in contradiction with the first Newton's law of classical dynamics. We present an elementary derivation of the ALD equation (4.1). This derivation is what we have called Rohrlich's argument [31]. It illustrates an application of Cartan's adapted frame method [35] that we will use later in the contest of generalized higher order fields in combination with maximal acceleration geometry. In addition, it does not require the formal introduction of the energy-momentum tensor for the generalized fields. Indeed, one deals formally with the geometry of point particles and the covariant Larmor's law.
One starts with the Lorentz force equation for a point charged particle interacting with an electromagnetic field F µν ,
m bẍ µ = eF µνẋ ν ,(4.2)
where m b is the bare mass and e the electric charge of the particle. Note that both sides are orthogonal toẋ by using the Minkowski spacetime metric η. Also, in this sub-section, the parameter t is the proper parameter associated with η. In order to generalize the equation (4.2) to take into account the radiation reaction, one can add to the right hand side of (4.2) a vector field Z along the curve x : I → M ,
m bẍ µ = eF µνẋ ν + Z( 3 x(t))
The orthogonality condition η(Z(t),ż(t)) = 0 implies the following general expression for Z,
Z µ (t) = P µ ν (t) a 1ẋ ν (t) + a 2ẍ ν (t) + a 3 ... x ν ), P µν = η µν +ẋ µ (t)ẋ ν (t),ẋ µ = η µνẋ ν . (4.3)
We can prescribe a 1 = 0. Then using the kinematical relation ...
x ρẋσ η ρσ = −ẍ ρẍσ η ρσ , one obtains the relation
Z µ (t) = a 2ẍ µ (t) + a 3 ( ... x µ − (ẍ ρẍσ η ρσ )ẋ µ )(t).
The term a 2ẍ combines with the left hand side to renormalize the bare mass,
(m b − a 2 )ẍ µ = mẍ µ . (4.4)
This procedure will be assumed to be valid independently of the values of m b and a 2 (that both could have infinite values but the different be finite). The argument from Rohrlich follows by requiring that the right hand side is compatible with the relativistic power radiation formula [20,31],
P µ rad (t) = − 2 3 e 2 (ẍ ρẍσ η ρσ )(t)ẋ µ (t). (4.5)
This condition is satisfied if
a 3 = 2/3 e 2 .
In order to recover this relation, the minimal piece required in the equation of motion of a charged particle is −2/3e 2 (ẍ ρẍρ η ρσ )ẋ µ . The Schott term 2 3 e 2 ...
x is a total derivative. It does not contribute to the averaged power emission of energymomentum. However, in the above argument, the radiation reaction term and the Schott term are necessary, due to the kinematical constraints of the metric η.
Rohrlich's derivation of the Abraham-Lorentz-Dirac equation with maximal acceleration
Using the kinematical constraints for metrics of maximal acceleration, we can repeat Rohrlich's argument. However, there is a slightly modification caused by the bound in the acceleration and because now the kinematic relations are the corresponding to a geometry of maximal acceleration. If one writes know
Z µ (τ ) = P µ ν (τ ) λ 1ẋ ν (τ ) + λ 2ẍ ν (τ ) + λ 3 ... x ν ),
and by an analogous procedure as before, λ 1 is arbitrary and we can prescribe λ 1 = 0. The equation is consistent with Larmor's law iḟ
ǫ 2 λ 2 + λ 3 d dτ (g(ẋ,ẍ)) = 0, λ 3 = 2 3 e 2 , λ k = 0, ∀k ≥ 4. (4.6)
The corresponding modified ALD equation is
mẍ µ = eF µ νẋ ν + 2 3 e 2 ... x µ − (ẍ ρẍσ η ρσ )ẋ µ + O(ǫ 2 0 ),(4.7)
with F µ ν := η µρ F ρσ . This equation is formally identical to the ALD equation. This fact implies that only maximal acceleration hypothesis is not enough to solve the problem of the Schott term in the ALD equation. Similar considerations holds if we repeat the calculation in the framework of generalized higher order fields. In that case one can see that without the requirement of bounded acceleration one does not obtain a second order differential equation for the motion of the point charged probe. In that case, one obtains again a differential equation that is formally the same than the ALD equation. However, we will see in next section that if one combines maximal acceleration geometry with generalized higher order fields, it is possible to obtain an implicit second order differential equation for point charged particles.
A second order differential equation for point charged particles
Let us assume that the physical trajectory of a point charged particle is a smooth curve of class C k such that g(ẋ,ẋ) = −1, withẋ 0 < 0 and such that the square of the acceleration vector field a 2 is bounded from above. The motivation for this assertion is the hypothesis that the physical measurable metric structure is the corresponding to the maximal acceleration metric. We will follow closely an analogous argument to Rohrlich's argument for generalized higher order fields in a maximal geometry back-ground. Thus one has the following general expression,
Υ µν (x,ẋ,ẍ,ẍ, ...) = B µẋν − B νẋµ + C µẍν − C νẍµ + D µ ... x ν − D ν ... x µ + ..., withẋ µ = g µνẋ ν .
This implies that we will have the expression
m bẍ µ = e F µ ν + B µẋ ν −ẋ µ B ν ẋ ν + C µẍ ν −ẍ µ C ν ẋ ν + D µ ... x ν − ... x µ D ν ẋ ν + ...,
and with F µ ν = g µρ F ρσ = η µρ F ρσ . On the right hand side of the above expression all the contractions that appear in expressions like B µẋ ν −ẋ µ B ν ẋ ν , etc, are performed with the metric g instead of the Minkowski metric η. The other terms come from the higher order terms of the expression (2.1) of the electromagnetic field. The general form of the k-jet field along a smooth curve x :
R → M B(τ ), C(τ ), D(τ ) are B µ (τ ) = β 1ẋ µ (τ ) + β 2ẍ µ (τ ) + β 3 ... x µ (τ ) + β 4 .... x µ (τ ) + · · ·, C µ (τ ) = γ 1ẋ µ (τ ) + γ 2ẍ µ (τ ) + γ 3 ... x µ (τ ) + γ 4 .... x µ (τ ) + · · ·, D µ (τ ) = δ 1ẋ µ (τ ) + δ 2ẍ µ (τ ) + δ 3 ... x µ (τ ) + δ 4 .... x µ (τ ) + · · ·.
Let us assume that there are not derivatives higher than 2 in the differential equation of a point charged particle. One way to achieve this is to impose that all the coefficients for higher derivations are equal to zero,
γ k = δ k = 0, k ≥ 0, β k = 0, k > 3. (5.1)
With this choice and using the kinetic relations for g, one obtains the expression
m bẍ µ = e F µ ν − β 2ẍ µ − β 3 ... x µ − 1 2 β 2ǫẋ µ − β 3 (−a 2 (τ ) + ... ǫ )ẋ µ . (5.2)
Under the assumption that the equation of motion of a point charged particle must be of second order and compatible with the power radiation formula (4.5) and foṙ ǫ(τ ) = 0, one obtains the relations
β 2 = 4 3 e 2 a 2 (τ ) 1 ǫ , (5.3) β k = 0, ∀k ≥ 3. (5.4)
At leading order in ǫ 0 , we obtain the following differential constraint:
m b (τ )ẍ µ = eF µ νẋ ν − 2 3 e 2 a 2 (τ )ẋ µ − 2 3 e 2 a 2 (τ ) 1 ǫẍ µ . (5.5)
There is a re-normalization of the bare mass,
m b (τ ) + 2 3 e 2 a 2 (τ ) 1 ǫ = m,ǫ = 0. (5.6)
In this expression, the term m b contains already the infinite term from the Coulomb field, that is the infinite electrostatic mass has been already renormalized (for instance the termẍ µ C νẋ ν is a term that can renormalize the Coulomb self-energy). The constraint (5.6) implies differential constraints on the derivatives, obtaining by taking the derivative respect to τ in both sides. Thus, in order to keep consistency, the bare mass m b will be defined such that the derivativesṁ b ,m b , ..., following the corresponding constraint. Using thatṁ = 0 and the relation (3.9). For the first derivative of the bare massṁ b , one has the constrain on the first derivative,
d k dτ k m b + 2 3 e 2 A 2 max d k dτ k ǫǫ = 0, k = 1, 2, 3, ... (5.7)
All such initial conditions are satisfied if one imposes the condition
ǫ = 0 ⇒ m b (τ ) = m = constant. (5.8)
With this assumption, the renormalization of mass relations (5.6) coincides with solution of the constraint (5.7). The first derivative of the bare mass must be of the form,
dm b dτ τ =0 = − 2 3 e 2 A 2 max d dτ ǫǫ τ =0 . (5.9)
In a similar way, one can compute the defining properties for higher order derivatives of the bare mass m b , that by definition will have the formal expression
m b (τ ) = ∞ k=0 m (k) | τ =0 k! τ k .
Since m b is not observable, each coefficient m (k) | τ =0 can be adjusted to be compatible with the derivative constraints like (5.7). Thus the bare mass m b is an element of the infinite jet bundle m b (τ ) ∈ J ∞ 0 (M ). Therefore, forǫ = 0 the following implicit differential equation should hold,
mẍ µ = e F µ νẋ ν − 2 3 e 2 η ρσẍ ρẍσẋµ , F µ ν = g µρ F ρν . (5.10)
The derivation of this implicit second order differential equation breaks down foṙ ǫ = 0 and we will consider this case separately.
The covariant form of equation (5.10) in any coordinate system is m Dẋẋ = e ιẋF (x(τ )) − 2 3 e 2 η(Dẋẋ, Dẋẋ), (5.11) with ιẋF (x(τ )) the dual of the contraction ιẋF ∈ ΓΛ 1 M . Note that since we are considering metrics of maximal acceleration avoids the orthogonality problem of taking away the Schott term in the ALD equation. The derivatives in equations (5.10) and (5.11) are taken with respect to the propertime determined by the metric of maximal acceleration. Thus, we need to prove that such parameter exists. This is done in Appendix B. In particular the proof applies to the reparameterization invariance between proper time of g and time coordinate. Moreover, it is important to remark that in our framework the physical metric, the time elapsed by an observer at rest with the particle, is measured by hypothesis by a metric of maximal acceleration, that depends explicitly of the acceleration of the particle measured by an inertial system. Thus, the clock hypothesis is not maintain in our theory. This corresponds with profound departure from Special and General Relativity, where the metric of the spacetime is Lorentzian. In contrast, in our framework, the Lorentzian metric η plays only a secondary role, and it is not associated with the physical proper-time ( [18], section 3).
It is in this context that one can ask for the transformation rule of equation (5.10) under the parameter change from the proper parameter s determined by η and the proper-time τ determined by g. Such transformation rule must be determined by the fundamental relation (3.13). In the transformation rule of the equation (5.10) under such parameter change, since the dependence on higher derivative, they will appear higher derivatives in the transformed equation of motion. However, the transformation ϕ(τ ) = s is not a diffeomorphism of the form ψ : I →Ĩ, I,Ĩ ⊂ R. That is, s = ϕ(τ ) is not a reparameterization of the parameter τ as usually is understood, in accordance with the identification of g as the physical measurable metric and η as only a convenient geometric device 4 .
Properties of the equation (5.10)
Let us consider a normal coordinate system for η. Let us multiply equation (5.10) by itself and contract with the metric g. Using the kinetic relations of proposition
one obtains
m 2 a 2 (1 − ǫ) = e 2 F µ ρẋ ρ F ν λẋ λ (1 − ǫ)η µν + ( 2 3 e 2 ) 2 (a 2 ) 2ẋµẋν g µν − 2e 2 3 e 2 F µ ρẋ ρẋν (1 − ǫ)η µν = (1 − ǫ) e 2 F µ ρẋ ρ F ν λẋ λ η µν − ( 2 3 e 2 ) 2 (a 2 ) 2 − 2e 2 3 e 2 F µ ρẋ ρẋν (1 − ǫ)η µν = (1 − ǫ) F 2 L − 1 1 − ǫ ( 2 3 e 2 ) 2 (a 2 ) 2 .
with the magnitude of the Lorentz force F L given by
F 2 L = e 2 F ν µ F µ ρẋνẋλ η λρ .
Proposition 5.1 For any curve solution of equation (5.10) one has the following consequences,
1. The Lorentz force is always spacelike or zero,
F 2 L ≥ 0. (5.12)
2. In the case the Lorentz force is zero, the magnitude of the acceleration is zero,
F 2 L = 0 ⇔ a 2 = 0. (5.13)
3. If there is an external electromagnetic field, the acceleration is bounded by the strength of the corresponding Lorentz force.
Proof. For ǫ = 0, one can re-write the expression
m 2 a 2 (1 − ǫ) = (1 − ǫ) F 2 L − 1 1 − ǫ ( 2 3 e 2 ) 2 (a 2 ) 2
as the following
F 2 L = 1 1 − ǫ ( 2 3 e 2 ) 2 (a 2 ) 2 + m 2 a 2 ,(5.14)
from which follows the three consequences. ✷ Contracting both sides of the differential equation withẋ
mẍ µẋν g µν = F µ ρẋ ρẋν g µν − 2 3 e 2 a 2 g(ẋ,ẋ) = 2 3 e 2 a 2 . (5.15)
Using the kinetic relations from g, the relation (5.15) reduces to
m g(ẍ,ẋ) = − 2 3 e 2 a 2 (τ ) + O(ǫ 2 ).
Defining the characteristic time
τ 0 := 2 3m e 2 ,
the relation can be re-written as
g(ẍ,ẋ) ≃ −τ 0 a 2 (τ ). (5.16)
The constraint (5.10) is an implicit differential equation. Locally, one can solveẍ in the left side under some approximations in the following way. First, note that equation (5.10) can be read as
x ρ (mδ µ ρ + 2 e 2 3ẍ ρẋ µ ) = eF µ νẋ ν .
If the radiation reaction term 2 3 e 2 a 2 x µ is small compared with the Lorentz force term one can treat it as a perturbation. As a first approximation one can consider that the Lorentz equation holds,ẍ µ = e m F µ ρẋ ρ and substitute in the above expression. The inversion of the operator
M µ ρ = (mδ µ ρ + 2e 3 3mẋ µ F ρλẋ λ ) (5.17)
is given by an expression in local coordinates as
O µ κ = 1 m δ µ κ − 2 e 3 3m 2ẋ µ F κρẋ ρ . (5.18)
Thus, the second derivativeẍ can be isolated as
x µ = eẋ ν F ρ ν 1 m δ µ ρ − 2 e 2 3m 2ẋ µ F ρσẋ σ . (5.19)
This equation can be seen as an approximation for (5.10) at leading order in ǫ 0 . It is useful, since for this approximated equation (5.19) one can use existence and uniqueness theorem of ODE theory [10] to state the following result, Proposition 5.2 Let z 1 and z 2 be two solutions of the equation of (5.10) with the same initial conditions. Then they differ by a smooth function on the radiation reaction term 2 3 e 2 a 2 .
Proof. Each of the solutions of equation (5.10) can be approximated by a solution of equation of the equation (5.19) and the difference is given locally as a power on ǫ = 2 3 e 2 a 2 , starting at least at power 1 in ǫ. By uniqueness of solutions of ODE [10], given a fixed initial conditions (x µ (τ 0 ),ẋ µ (τ 0 )) the solution exits and is unique (in a finite interval of time τ ). In such interval the two solutions differ by a polynomial on 2 3 e 2 a 2 in a short time interval. ✷ Equation (5.19) can be written as a geodesic equation for a suitable connection. Let us consider a connection 2 ∇ on T J 2 0 (M ) that in the local holonomic frame
{∂ µ , ∂ ∂ẋ µ , ∂ ∂ẍ µ } 3 µ=0 of T J 2 0 (M ) has connection coefficients 2 Γ µ νσ := ( 2 ∇ ∂ν ∂ σ ) µ = γ µ νσ + K µ νσ + L µ νσ ,(5.20)
with the tensors K and L defined by the expressions
K µ νσ = − 1 2m F µ ν η σλẋ λ + F µ ν η σλẋ λ 1 1 − η κξẍ κẍξ A 2 max , (5.21) L µ νσ = − 2e 3 3m 2 F ρσ F ρ νẋ µ . (5.22)
All the rest of connection coefficients of 2 ∇ are put equal to zero. Therefore, The geometric interpretation of equation (5.19) opens the possibility to investigate approximations to (5.10) using geometric and kinetic methods.
Covariant uniform acceleration
The case of world-lines withǫ = 0 requires special care. Let us consider the following definition of covariant uniform acceleration, Definition 5.4 A covariant uniform acceleration curve is a map x : I → M such that along it, the following constrain holds:
ǫ = d dτ η(Dẋẋ, Dẋẋ) A 2 max = 0. (5.24)
This notion of covariant uniform motion is general covariant. Thus, in order to make some computations we can adopt normal coordinates associated to η. Then the kinetic relation (3.19) holds. Although the four-acceleration is in general not orthogonal by g to the four velocity vector field, in the case of covariant uniform accelerated motion one has that g(ẍ,ẋ) = 0. By the relation (3.20),
g(ẍ,ẍ) = −g( ... x ,ẋ).
Since the right hand side is necessarily zero,
g( ... x ,ẋ) = (1 − ǫ) η( ... x ,ẋ) = (1 − ǫ)ǫ = 0,
x is also lightlike vector with the metric g, The implication of this fact is that the four-acceleration cannot be constant in a normal coordinate system, except for a 2 = 0.
For maximal accelerations such that A 2 max ≤ ∞ our definition coincides with the standard definition of covariant uniform acceleration appearing in standard references (see for instance [22]). It also implies the same notion of covariant uniform motion as a solution of an homogeneous differential equation [16].
Equation (5.10) fails to describe covariant uniform motion. The reason for this is that in order to apply the mass renormalization procedure (5.6) and the perturbation scheme the requirementǫ = 0 is need. Therefore, one needs to consider separately the caseǫ = 0. Let us consider the following consequence of equations (5.6) and (5.5),
m bẋ µ = − eF µ νẋ ν +β 2ẍ µ .
Thus one has the following differential equation for covariant uniform motion,
mẍ µ = e F µ νẋ ν , m = m b (τ ) −β 2 (τ ). (5.25)
For covariant uniform acceleration one can see easily that d dτ g µρ = 0 (5.26) and this relation implies
0 = d dτ a 2 = d dτ F 2 L m . (5.27)
We will discuss below that this notion of covariant uniform motion is highly unstable under small perturbations. Note that if the point τ 0 withǫ(τ 0 ) = 0, τ 0 is an isolated critical point of a 2 , one can extend by continuity the solutions of (5.10) to the full world-line by continuity arguments.
Absence of pre-acceleration of Dirac's type for the equation (5.10)
Let us adopt a normal coordinate system associated to η. We discuss the absence of non-physical solutions on (5.10). Run away solutions are solutions that have the following peculiar behavior: even if the external forces have a compact domain in the spacetime, the charged particle follows accelerating indefinitely. This is a pathological behavior of the ALD equation. However, since the condition (5.13) holds for equation (5.10), we show that our model is free of such problems for asymptotic conditions where the exterior field is bounded. In order to investigate the existence of pre-acceleration in some of the solutions for the equation (5.10), let us consider the example of a pulsed electric field [12]. For an external electric pulse E = (κ δ(τ ), 0, 0), the equation (5.10) in the non-relativistic limit reduces to
aẍ 0 = κ δ(τ ), a = 3m 2e 2 . (5.28)
The solution of this equation is the Heaviside's function,
aẋ =κ, τ ≥ 0, 0, τ < 0,
that do not exhibit pre-acceleration behavior. In order to prove uniqueness on the space of smooth functions, let us consider two solutions to the equation (5.28). Clearly, the two solutions must differ by an affine function and the requirement of having the same initial conditions, implies that affine function must be trivial. This shows that in the non-relativistic limit the equation (
Discussion
Combining the notion of generalized higher order electromagnetic field with maximal acceleration geometry, we have obtained the implicit differential equation (5.10) as a description of the dynamics of a point charged particle that takes into account the radiation reaction. Equation (5.10) is free of run-away solutions and pre-accelerate solutions of Dirac's type. The hypothesis of maximal acceleration is necessary in order to keep under control the value of the acceleration. It also provides a bookkeeping parameter that allows to construct our perturbation method. The hypothesis of generalized higher order fields is fundamental too, since it provides us with the degree of freedom that we need to eliminate the Schott term in the ALD equation. Moreover, the physical interpretation of such fields is clear: fields should not be defined independently of the way they are measured or detected.
The hypotheses of generalized higher order fields for k = 3 and maximal acceleration do not fix completely the dynamics of a point charged particle. However, the requirements that
• The dynamics is compatible with the covariant power radiation law (4.5),
• The differential equation is second order and
• The notion of field is extended minimally to keep the lowest extension on the derivatives possible are enough to fix the dynamics of point charged particles, except for covariant uniform motion (which is a degenerate case in our framework). This different behavior of the uniform motion is because the method that we follow breaks down whenǫ = 0. For the case of covariant uniform acceleration one needs to have a special treatment. Indeed, for a covariant uniform motion of a point charged particle, one needs to provide not only the initial conditions (x µ (τ 0 ),ẋ µ (τ 0 )), but also a mechanism to fix in anticipation the external field, such that along the world-line of the charged particle the total field is constant. This requires of infinite precision on the determination of the external electromagnetic field F L in such a way that it compensates the radiation reaction field. This fine tuning is highly unstable, and the point charged particle will not behave uniformly accelerated under small fluctuations in the external field. Thus, a model of motion is not structurally stable. On the other hand, a stable description for the dynamics of a point charged particle is given by equation (5.10) only. This argument and the assumption of requiring an stable model 5 force us to abandon the notion of covariant uniform acceleration as a possible physical dynamics for point charged particles.
It is natural to consider the differential equation (5.19) as an approximation to the motion of a point charge when the radiation reaction is negligible compared with the external field. The difference on the solutions of (5.19) and (5.10) is given by the tensor L µ νσ , which is by hypothesis, a small perturbation. Thus, for short time evolution and for initial conditions such that the radiation reaction is small compared with the total field Lorentz force, the difference on the solutions of the two equations is small. Also, the difference between the solutions of (5.10) and (5.25) is small. The generalized fields are assumed to be well defined on the trajectory of the probe particle. This is contradictory with the fact that electromagnetic fields are infinite at the localization in spacetime of the point charged particle world-line, due to the Coulomb singularity. In order to treat this problem we have adopted the method of mass renormalization, with a time variable bare mass m b (τ ). Although we do not think that is the last word on this, the method allows us to consider well defined fields over the world-line of charged particle probes, without the singularity of Coulomb type, which is renormalized to give the observable constant mass m. Therefore, the considerations in this model concern finite fields in the whole spacetime manifold M . A theory which in some sense resembles the one proposed in this lines, is Bonnor's theory [3]. In such theory, the observable mass is time variable, and the constraint on the variable mass m 0 is given by such constraint is not true for metrics of maximal acceleration. Finally, let us remark that the notion of generalized higher order electromagnetic field can be extended to non-abelian Yang-Mill fields, following a similar scheme. Also, the type of metric structure g is a generalized tensor. This is consistent with a general picture of generalized metrics and fields, coupled by a generalized version of Einstein's field equations, still to be developed.
d ds m 0 = − 2 3 e 2 A 2 max ǫ.
A Jet bundles and generalized tensors
In this appendix we collect some of the notions and definitions that we need to define generalized higher order fields. An extended version of the theory was developed in [18]. Given a smooth curve x : I → M , the set of derivatives (p = x(0), dx dσ , d 2 x dσ 2 , ..., d k x dσ k ) determines a point (jet) in the space of jets J k 0 (p) over the point p ∈ M ,
J k 0 (p) := (x(0), dx dσ 0 , ..., d k x dσ k 0 ), ∀ C k -curve x : I → M, x(0) = p ∈ M, 0 ∈ I .
Thus the set of higher derivatives (p = x(0), dx dσ , d 2 x dσ 2 , ..., d k x dσ k ) determines a jet at the point p = x(0). The jet bundle J k 0 (M ) over M is the disjoint union
J k 0 (M ) := x∈M J k 0 (x).
The projection map is
k π : J k 0 (M ) → M, (x(0), dx dσ 0 , d 2 x dσ 2 0 , ..., d k x dσ k 0 ) → x(0). Example A.1
The simplest example where jets appear is when we use Taylor's expansions of smooth functions. This kind of approximation is an example of the type of approximation.
Example A.2 Some of the applications of jet bundle formalism requires the previous introduction of more sophisticated notation and notion. However, there is an example that being natural and of fundamental relevance, we can mention in this Appendix as an example of use of jets in the investigation of ordinary differential equations. It is the case of the perturbative study of geodesic deviation equations of a connection (not necessarily affine) on the manifold M . In that case, the condition for the difference on the coordinates ξ µ = x µ − X µ is of the geodesic is written as
ξ µ + Γ µ νσ (X + ξ,Ẋ +ξ) Ẋ ν +ξ ν Ẋ σ +ξ σ − Γ µ σν (X)Ẋ σẊ ν = 0.
Considering a bookeeping parameter ǫ on the functions {ξ µ ,ξ µ ,ξ µ , ...}. Then developing by Taylor's expansion all the functions in the above expression, one obtains a series in terms of ǫ.
∞ k=1 ǫ k G k (Ξ µ ,Ξ µ ,Ξ µ ) = 0.
Equating to zero each term, a hierarchy of ordinary differential equations is obtained,
G k (Ξ µ ,Ξ µ ,Ξ µ ) = 0, k = 1, 2, 3, ... (A.1)
The equation obtained from the first order G 1 (Ξ µ ,Ξ µ ,Ξ µ ) = 0 is the Jacobi equation of the connection. Higher order deviation equations are obtained by equating to zero the expressions G k (Ξ µ ,Ξ µ ,Ξ µ ) = 0 for k = 2, 3, .... This method was pioneered by S.L. Bażański [2]. We observe clearly that this is a recursive system of ODE's, and that to each rorder in ǫ corresponds to an equation for an r-jet order fields approximations of ξ µ .
Given a curve x : I → M , the k-lift is the curve k x : I → J k 0 (M ) such that the following diagram commutes,
J k 0 (M ) k π I k x < < ② ② ② ② ② ② ② ② ② x / / M.
There are also the notions of lift of tangent vectors and smooth functions,
• Let X ∈ T x M be a tangent vector at x ∈ M . A lift k l u (X) at u ∈ k π −1 (x) is a tangent vector at u such that k π * ( k l u (X)) = X.
• Let us denote by F(J k 0 (M )) the algebra of real smooth functions over J k 0 (M ). Then there is defined the lift of a function f ∈ F(M ) to F(J k 0 (M )) by k π * (f )(u) = f ( x), ∀u ∈ k π −1 (x).
Each point on the lifted curve k x : I → J k 0 (M ) has local coordinates given by (x(σ), dx(σ) dσ , ..., d k x(σ) dσ k ), where σ is the parameter of the curves x : I → M and k x : I → J k 0 (M ). Also, the linear map k π : J k 0 (M ) → M is differentiable and the differential of the projection k π u at u ∈ k π −1 (x) is the linear map
( k π * ) u : T u J k 0 (M ) → T x M.
We will denote by k π * the projection ( k π * ) : T J k 0 (M ) → T M such that at each u it is the linear map ( k π * ) u . The kernel of ( k π * ) x at x ∈ M is the vector space
k V x := ( k π * ) −1 (0 x ).
where 0 x is the zero vector in T x M . Then vertical bundle over M is
k V := x∈ M k V x and the vertical bundle over J k 0 (M ) is determined by the surjection k π V : k V → J k 0 (M ), ( k π * ) −1 (0 x ) ∋ ξ v → u ∈ (π k ) −1 (x).
k V is a real vector bundle over J k 0 (M ), since it is the kernel of k π * . The composition of k π V with k π determines also a real vector bundle over M ,
k π • k π V : k V → M.
One can introduce the notation kπ V = k π • k π V . Note that we are interested in the case when the probe particles are described by world-lines. In the case that the probe particles have spacetime extension, the maps describing the particles must be other type of sub-manifolds Θ : R d → M and the jets bundles that must be considered are J k 0 (R d , M ). Thus, we have considered in this paper the paper the case d = 1 only. A relevant property of jet bundles is Peetre's theorem. Recall that the support supp of a section S : M → E of a vector bundle is the closure of the sets {x ∈ M ; S(x) = 0}. An operator D between the bundles π 1 : E 1 → M and π 2 :
E 2 → M D : C ∞ (E 1 ) → C ∞ (E 2 )
is said to be support non-increasing if supp(DS) ⊂ supp(S) for every section S ∈ ΓS. Theorem A.3 (Peetre, 1960) Consider two vector bundles π 1 : E 1 → M and π 2 : E 2 → M and a non-increasing operator . Then D : C ∞ (E 1 ) → C ∞ (E 2 ). For every compact K ∈ suppD, there is an integer r(K) such that if the jets j r (S 1 ) = j r (S 2 ), then DS 1 = DS 2 .
A tensor field of (p, q)-type T is a special type of section of the bundle T (p,q) J k 0 (M ). To define it properly, it is necessary to introduce a connection kĤ on J k 0 (M ) (see for instance [21] and [26]). Given a connection kĤ on J k 0 (M ), one can define the notions of horizontal and vertical tensors of any order. For instance, a tensor field T of type (2, 0) is a section of the bundle T (2,0) J k 0 (M ) and ia defined independently of the connection: in a local neighborhood, the sections of T (2,0) J k 0 (M ) are spanned (with coefficients on F(J k 0 (M )) by the tensor product of elements of the frame
{e 1 ( k x), ..., e n(k+1) ( k x)} T (2,0) J k 0 (M ) = span e i ( k x) ⊗ e j ( k x), i, j = 1, ..., n(k + 1) .
On the other hand, a (0, 2)-horizontal tensor fields are locally spanned as
T (2,0) h J k 0 (M ) = span δ δx µ k x ⊗ δ δx ν k x , µ, ν = 1, ..., n ,
where the horizontal local sections δ δx 1 , ..., δ δx n are defined by the expressions
δ δx ν (x,y) = ∂ ∂x ν (x,y) − N A ν (x, y) ∂ ∂y A (x,y) . (A.2)
The coefficients N A ν (x, y) are the non-linear connection coefficients [26]. Similarly, the tensor bundle of (2, 0)-vertical tensors is locally spanned by the local frame
T (2,0) h J k 0 (M ) = span ∂ ∂y A k x ⊗ ∂ ∂y B k x , A, B = 1, ..., nk .
The bundle over J k 0 (M ) of hv tensors of type (1, 1) is locally spanned by the local frame
T (1,1) h J k 0 (M ) = span δ δx µ k x ⊗ ∂ ∂y A k x
, µ = 1, ..., n, A = 1, ..., nk .
One can consider an horizontal (1, 1) tensor, that generically will have the following expression in local coordinates
T ( k x) = T i j ( k x) δ δx i k x ⊗ δx j k x , T i j ( k x) ∈ F(J k 0 (M )).
Horizontal p-forms can be spanned locally in a similar way. For instance, a 2-form horizontal can be expressed in local coordinates as
ω( k x) = ω ij ( k x)dx i | k x ∧ dx j | k x .
Note that horizontal forms does not depend on the specific connection kĤ that we can choose. The tensor product of horizontal tensors is an horizontal tensor. In a similar way, the exterior product of horizontal forms is an horizontal form. There is a notion of horizontal exterior derivative, for which definition we need a connection on J k 0 (M ) [18]. The following definition provides the fundamental notion of generalized higher order field, . It can be shown that this notion is equivalent to the notion of generalized higher order field as horizontal fields.
Example A.5 A generalized electromagnetic field is a section of Λ 2 (M, F(J 3 0 (M ))). There is a well defines Cartan's calculus for generalized forms in a similar way as the standard Cartan calculus of smooth differential forms. The usual formulation of the inner derivation, exterior product and exterior derivative can be introduced in a coordinate free form and are independent of the connection used (for the construction and general properties of Cartan's calculus of generalized forms, the reader can see [18]). For instance, there is a coordinate free definition of the operator d 4 . The realization in local coordinates of the operator d 4 is straightforward: if φ is a generalized k-form, its exterior derivative is d 4 φ = d 4 φ i (x(s),ẋ,ẍ, ..., x (k) )d 4 x i = d φ i (x(s),ẋ,ẍ, ..., x (k) ) ∧ d 4 x i = ∂ j φ i (x(s),ẋ,ẍ, ...,
x (k) ) d 4 x j ∧ d 4 x i .
This operator is nil-potent, (d 4 ) 2 = 0. There is a definition of integration of differential forms which is diffeomorphic invariant and a corresponding Stoke's theorem.
B Re-parameterization invariance of the proper-time associated with the metric of maximal acceleration
The general theory of generalized metrics is considered in [18]. In this paper, we consider the particular case of a metric of maximal acceleration (3.5), showing that the associated proper-time is re-parameterization invariant. The proper time of the form (3.5) is
τ [ (2) x] = t t 0 −g(x ′ (t), x ′ (t)) dt.
The parameter t ∈ I ⊂ R is arbitrary. Thus the natural question is if the propertime τ [ (2) x] is invariant under an arbitrary re-parameterization φ : I →Ĩ, t → φ(t) = s. (B.1)
The natural way to check this is through the definition of generalized metric and some of its fundamental properties. A convenient definition of generalized metric is the following (see [18]), in particular subsection (2.6)):
Definition B.1 A generalized metric is a sectionḡ ∈ Γ T (0,2) (M, F(J k 0 (M ))) such that the following condition holds:
1.ḡ is smooth in the sense that for all X 1 , X 2 smooth vector fields along the curve x : I → M , the functionḡ(X 1 , X 2 ) is smooth except when it takes the zero value. for all X ∈ T x M and λ i > 0, i = 1, ..., k.
3.ḡ is symmetric, in the sense that g(X 1 , X 2 ) =ḡ(X 2 , X 1 )
for all X 1 , , X 2 smooth vector fields along the curve x : I → M .
It is bilinear,
g(X 1 + f ( k x)X 2 , X 3 ) =ḡ(X 1 , X 3 ) + f ( k x)ḡ(X 2 , X 3 ),
for all X 1 , X 2 , X 3 arbitrary smooth vector fields along x : I → M and f ∈ F(M ).
5.
It is non-degenerate, in the sense that ifḡ(X, Z) = 0 for all Z smooth along x : I → M , then X = 0.
The metric of maximal acceleration g determines a section of Γ T (0,2) (M, F(J 2 0 (M ))), with connection components g µν ( (2) x) = (1 − η(ẍ,ẍ) A 2 max ) η µν (x), µ, ν = 1, ..., n.
The homogeneity property of the Definition B.1 when applied to the metric of the maximal acceleration reads,
g(x µ (s),ẋ µ (s),ẍ µ (s))(X, X) = g(x µ (s), λ 1ẋ µ (s), λ 21ẍ µ (s)+λ 2ẋ µ (s))(X, X).
where the (2)x is the second jet of the mapx :Ĩ → M, s →x(s), X ∈ Tx (s) M and λ 1 , λ 21 , λ 2 are arbitrary real functions alongx :Ĩ → M . In particular we have that by the bilinearity property on X, g(x µ (s),ẋ µ (s),ẍ µ (s))(λ 1 X, λ 1 X) = (λ 1 ) 2 g(x µ (s), λ 1ẋ µ (s), λ 21ẍ µ (s) + λ 2ẋ µ (s))(X, X).
In short, g( 2x )(λ 1 X, λ 1 X) = (λ 1 ) 2 g(x µ (s), λ 1ẋ µ (s), λ 21ẍ µ (s)+λ 2ẋ µ (s))(X, X). where (2)x = (x µ (s), λ(x) ′µ (s), λ 2 (x ′′ ) µ (s)+λλ ′ (x ′ ) µ (s)),ṽ ′′ = λ 2 (x ′′ ) µ (s)+λλ ′ (x ′ ) µ (s) and λ = dφ dt . Using the homogeneity condition for the generalized metric g, * =
consider a timelike curve x : I → M , parameterized such that for the lifted vectors g S ( 1ẋ , 1ẋ ) = −1. Then the metric (3.4) induces a bilinear, non-degenerate, symmetric form g along the lift 1 x : I → T M [17], Theorem 3.1
Proposition 5. 3
3The coefficients 2 ∇ define a covariant derivative on J the solutions of the equation(5.19).
5.10) does not have pre-accelerated solutions of Dirac's type, but since the theory is Lorentz covariant, the equation (5.10) does not have pre-accelerate solutions of Dirac's type in any coordinate system. It is open the question if (5.10) is free of any other type of pre-accelerated solutions.
this differential equation is completely different form the equation for the bare mass m b , equation(5.7), that is trivially satisfied if the condition (5.8) holds. Apart from the structural differences between the two models, there are at least two important formal properties in Bonnor's theory that differ from our model. The first one is that in our model is the bare mass changes with time, while the observable mass is constant in time. This is in contrast with Bonnor's model, where de observable mass changes with time. The second relevant formal difference is on the spacetime metric used and on the orthogonal relations used: in Bonnor's model, the metric is the usual Minkowski metric and the orthogonality condition are η(ẋ,ẍ)
Definition A.4 A generalized tensor T of type (p, q) with values on F(J k 0 (M )) is a smooth section of the bundle of F(M )-linear homomorphisms T (p,q) (M, F(J k 0 (M ))) := Hom(T * M × ... p ... × T * M × T M × ... q ... × T M, F(J k 0 (M ))). A p-form ω with values on F(J k 0 (M )) is a smooth section of the bundle of F(M )linear completely alternate homomorphisms Λ p (M, F(J k 0 (M ))) := Alt(T M × ... p ... × T M, F(J k 0 (M ))). The space of 0-forms is Γ Λ 0 (M, F(J k 0 (M ))) := F(J k 0 (M ))
2 .
2It is homogeneous of degree zero: if k x : I → M has local coordinates (x µ (s),ẋ µ (s),ẍ µ (s)...), thenḡ (x µ (s), λ 1ẋ µ (s), λ 2 1ẍ µ (s)+λ 2ẋ µ (s), λ 3ẋ + 3λ 2 λ 1ẍ + λ 3 1 ... x , ...)(X, X) = g(x µ (s),ẋ µ (s),ẍ µ (s), ... x (k) (s))(X, X)
property of re-parameterization invariance of the proper-time of the metric of maximal acceleration g ∈ Γ T (0,2) (M, F(J 2 0 (M ))) follows directly, Theorem B.2 The proper-time (3.7) is invariant under re-parameterizations. Proof. Let us consider two different curves x : I → M andx :Ĩ → M , related by re-parameterization φ : I →Ĩ. Then we have that τ [ (2) x(ṽ ′′ ,ṽ ′′ ) A 2 max η(λx ′ (s), λx ′ (s)) |λ −1 |ds = * .
x(x ′′ ,x ′′ ) A 2 max η(λx ′ (s), λx ′ (s)) |λ x ′ (s), x ′ (s)) ds = τ [ (2)x ].
✷
email: [email protected]; Currently at the Departamento de Matemática, Universidade Federal de São Carlos, Brazil. Work financially supported by FAPESP, process 2010/11934-6 and Riemann Center for Geometry and Physics, Leibniz University Hanover.
The mathematical foundations for generalized higher order tensors and differential forms, the fundamental elements of the geometry of maximal acceleration, as well as the fundamental mathematical aspects of generalized fields and generalized higher order electrodynamics are developed in more detail in the companion paper[18].
A detailed derivation of the equations (2.4) to (2.8) from the generalized Maxwell's equations is contained in ref.[18],Section 7.
Note that any second differential equation respect to a parameter r can be changed to an arbitrary order differential equation where the derivatives are taken respect a second parameter l, if the relation l(s) contains higher order than zero derivatives. Thus it is not a surprise that this happens in the case of equation (5.10) when passing from τ to s.
By a stable model we mean a weakly structurally stable[1].
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| []
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[
"Ternary codes, biplanes, and the nonexistence of some quasi-symmetric and quasi-3 designs",
"Ternary codes, biplanes, and the nonexistence of some quasi-symmetric and quasi-3 designs"
]
| [
"Akihiro Munemasa ",
"Vladimir D Tonchev "
]
| []
| []
| The dual codes of the ternary linear codes of the residual designs of biplanes on 56 points are used to prove the nonexistence of quasi-symmetric 2-(56, 12, 9) and 2-(57, 12, 11) designs with intersection numbers 0 and 3, and the nonexistence of a 2-(267, 57, 12) quasi-3 design. The nonexistence of a 2- (149, 37,9)quasi-3 design is also proved. | 10.1002/jcd.21740 | [
"https://arxiv.org/pdf/2003.04453v1.pdf"
]
| 212,647,399 | 2003.04453 | d7c3fe8ce7427197ee9196231bae3648a994f833 |
Ternary codes, biplanes, and the nonexistence of some quasi-symmetric and quasi-3 designs
9 Mar 2020 March 11, 2020
Akihiro Munemasa
Vladimir D Tonchev
Ternary codes, biplanes, and the nonexistence of some quasi-symmetric and quasi-3 designs
9 Mar 2020 March 11, 2020linear codequasi-symmetric designsymmetric designresidual de- signbiplanequasi-3 design
The dual codes of the ternary linear codes of the residual designs of biplanes on 56 points are used to prove the nonexistence of quasi-symmetric 2-(56, 12, 9) and 2-(57, 12, 11) designs with intersection numbers 0 and 3, and the nonexistence of a 2-(267, 57, 12) quasi-3 design. The nonexistence of a 2- (149, 37,9)quasi-3 design is also proved.
Introduction
We assume familiarity with basic facts and notions from combinatorial design theory and coding theory ( [1], [2], [4], [10], [21]). A combinatorial design (or an incidence structure) is a pair D = (X, B) of a finite set X = {x i } v i=1 of points, and a collection B = {B j } b j=1 of subsets B j ⊆ X, called blocks. The (points by blocks) incidence matrix A = (a i,j ) of a design D with v points and b blocks is a (0, 1)-matrix with v rows indexed by the points and b columns indexed by the blocks, where a i,j = 1 if the ith point belongs to the jth block, and a i,j = 0 otherwise. The transposed matrix A T is called the blocks by points incidence matrix of D. If p is a prime number, the p-rank of A (or rank p A), is defined as the rank of A over a finite field of characteristic p.
Given a design D with v points and b blocks, and a finite field F = GF (q), one can define two linear codes over F associated with D: the code of length b spanned by the rows of the v by b incidence matrix A is called the code of D spanned by the points, while the code of length v spanned by the columns of A is called the code of D spanned by the blocks.
Let D = (X, B) be a design, and let B ∈ B be a block of D. The incidence structure D B = (X ′ , B ′ ), where is called the derived design of D with respect to block B.
The incidence structure D B = (X ′′ , B ′′ ), where
X ′′ = X \ B, B ′′ = {B j \ (B j ∩ B) | B j ∈ B, B j = B},
is called the residual design of D with respect to block B.
Definition 1.1. Let A be the v by b incidence matrix of a design D with v points and b blocks, and let A ′′ be the incidence matrix of a residual design D B with respect to a block B. The residual design D B is said to be linearly embeddable [19] Given a design D = (X, B), its dual design D * is the incidence structure having as points the blocks of D, and having as blocks the points of D, where a point and a block of D * are incident if and only if the corresponding block and point of D are incident. If A is the incidence matrix of D, then A T is the incidence matrix of the dual design D * .
over F = GF (p), (p prime), if rank p A = rank p A ′′ + 1.(1)
Given integers v ≥ k ≥ t ≥ 0, λ ≥ 0, a t-(v, k, λ) design (or briefly, a t-design) D is an incidence structure with v points and blocks of size k such that every t-subset of points is contained in exactly λ blocks. A t-(v, k, λ) design is also an s-(v, k, λ s ) design for every integer s in the range 0 ≤ s ≤ t, with λ s = λ v−s t−s / k−s t−s . Let D = (X, B) be a t-(v, k, λ) design, and let x ∈ X be a point. The derived design
D x with respect x is the (t − 1)-(v − 1, k − 1, λ) design with point set X \ {x}, and block set {B \ {x} | B ∈ B, x ∈ B}. The residual design D x with respect to x is the (t − 1)-(v − 1, k, λ t−1 − λ) design with point set X \ {x}, and block set {B | B ∈ B, x / ∈ B}. If D is a 2-(v, k, λ) design with v > k > 0, the number of blocks b = v(v − 1)λ/(k(k − 1)) satisfies the Fisher inequality: b ≥ v,(2)
and the equality b = v holds if and only if every two blocks of D share exactly λ points.
A 2-(v, k, λ) design D with b = v is called symmetric.
The dual design D * of a symmetric 2-(v, k, λ) design D is a symmetric design having the same parameters as D. A symmetric design is self-dual if it is isomorphic to its dual design. A biplane is a symmetric design with λ = 2.
Note 2. Assume that D is a 2-(v, k, λ) design. If x is a point of D, the derived design D x is a 1-(v − 1, k − 1, λ) design, while the residual design D x is a 1-(v − 1, k, r − λ) design, where r = λ 1 = λ(v − 1)/(k − 1) is the number of blocks of D that contain x.
If D is a symmetric 2-(v, k, λ) design, then r = k and D x is a 1-
(v − 1, k, k − λ) design. In addition, if B is a block, the derived design D B is a 2-(k, λ, λ − 1) design, while the residual design D B is a 2-(v − k, k − λ, λ) design.
A 2-(v, k, λ) design is quasi-symmetric with intersection numbers x, y, (x < y), if every two blocks intersect in either x or y points. A brief survey on quasi-symmetric designs is given in [17]. Links between quasi-symmetric designs and error-correcting codes are discussed in [20].
Examples of quasi-symmetric designs are: (1) unions of identical copies of symmetric 2-designs; (2) non-symmetric 2-(v, k, 1) designs; (3) strongly resolvable designs; (4) residual designs of biplanes. A quasi-symmetric 2-(v, k, λ) design with 2k < v which does not belong to any of these four classes is referred to as exceptional.
The classification of exceptional quasi-symmetric designs is a difficult open problem. A table of admissible parameters for exceptional quasi-symmetric designs with number of points v ≤ 70 is given in [17, Two admissible parameters sets for exceptional quasi-symmetric designs whose existence has been unknown since 1982, are 2-(56, 12,9), (x = 0, y = 3), and 2-(57, 12, 11), (x = 0, y = 3). It is the goal of this paper to show that quasi-symmetric designs with these parameters do not exist.
The nonexistence of a quasi-symmetric 2-(57, 12, 11) design implies also the nonexistence of a quasi-3 design with parameters 2-(267, 57, 12). Similarly, using the nonexistence of quasi-symmetric 2-(37, 9, 8) designs with intersection numbers 1 and 3 [9], we show that a quasi-3 design with parameters 2-(149, 37, 9) does not exist. The existence of quasi-3 designs with these parameters was a long standing open question [14].
2 Residual 2-(45, 9, 2) designs and their ternary codes Our proof of the nonexistence of a quasi-symmetric 2-(56, 12,9) design is based on the following lemmas.
Lemma 3. Suppose that D = (X, B) is a quasi-symmetric 2-(56, 12, 9) design with intersection numbers 0 and 3, and let z ∈ X be a point of D.
(i) The derived design D z is a 1-(55, 11,9) design with 45 blocks whose dual design (D z ) * is a 2-(45, 9, 2) design.
(ii) The residual design D z is a 1-(55, 12, 36) design with 165 blocks. The columns of the 55 × 165 incidence matrix of D z belong to the dual code C ⊥ of the linear code C over GF (3) spanned by the columns of the 55 × 45 incidence matrix of D z .
Proof. (i) Since every two non-disjoint blocks of D share exactly three points, every two blocks of D z share exactly two points. This implies that (D z ) * is a 2-(45, 9, 2) design.
(ii) The inner product of the incidence vector of every block B of D z with the incidence vector of every block of D z is either 0 or 3.
By a theorem of Hall and Connor [7], every 2-(45, 9, 2) design is a residual design with respect to a block of a biplane with parameters 2-(56, 11, 2). There are five nonisomorphic biplanes Bi, (1 ≤ i ≤ 5) with these parameters [6, 15.8], all five being self-dual. The first biplane, B1, was found by Hall, Lane and Wales [8], B2 was found by Mezzaroba and Salwach [16], B3 and B4 were found by Denniston [5], and B5 was found by Janko and Trung [11]. The residual 2-(45, 9, 2) designs of the five biplanes fall into 16 isomorphism classes (see [13,Table 2]).
It was shown by an exhaustive computer search (Kaski andÖstergård [12]), that up to isomorphism, there are exactly five biplanes with 56 points, and consequently, exactly 16 nonisomorphic 2-(45, 9, 2) designs.
Using Lemma 3, the existence question for a 2-(56, 12, 9) design can be resolved by computing the sets of all (0, 1)-vectors of weight 12 in the dual codes of the ternary linear codes spanned by the rows of the 45 × 55 incidence matrices of the 16 nonisomorphic 2-(45, 9, 2) designs, and checking if any of these sets contains a subset of 165 vectors that form the incidence matrix of a 1-(55,12,36) design with 165 blocks of size 12, such that every two blocks are either disjoint or share exactly three points. (3), where A ′′ is the 45 × 55 incidence matrix of D B , and A ′ is the 11 × 55 incidence matrix of the derived 2-(11, 2, 1) design D B .
A = 0 A ′′ . . . 0 1 A ′ . . . 1 .(3)
If c = (c 1 , . . . , c 55 ) is a (0, 1)-codeword of weight 12 in the dual code (L ′′ ) ⊥ of the ternary linear code L ′′ spanned by the rows of A ′′ , then c * = (c 1 , . . . , c 55 , 0) belongs to the dual code L ⊥ of the ternary linear code L spanned by the rows of A.
Proof. The ternary codes of the five biplanes with parameters 2-(56, 11, 2) were computed in [13], and all five codes have minimum distance 11. Thus, by Definition 1.1 and Note 1, every residual 2-(45, 9, 2) design is linearly embeddable over GF (3), and rank 3 A = rank 3 A ′′ + 1. vectors of the matrix in the right-hand side of (4), to which c * is orthogonal. Therefore, c * ∈ L ⊥ . Lemma 3 and Lemma 4 imply the following (see Fig. 1 for an illustration). (ii) A quasi-symmetric 2-(57, 12, 11) design with block intersection numbers 0, 3 does not exist.
This implies
dim L = rank 3 0 A ′′ . . .
Proof. (i) By a theorem of Hall and Connor [7], every 2-(45, 9, 2) design is a residual design with respect to a block of a biplane with parameters 2-(56, 11, 2). Thus, by Lemma 5, it is sufficient to inspect the sets of all (0, 1)-codewords of weight 12 in the dual codes of the ternary codes of the five biplanes with 56 points. The number of such codewords can be found by computing with Magma [3] the complete weight enumerator of the dual codes. This was done in [13] for three of the biplanes, B1, B2 and B4, while upper bounds 91 and 22 were found for the codes of the biplanes B3 and B5 [13, Table 1], and these results were used to prove that none of the five biplanes can be extended to a 3-(57, 12, 2) design. Using Magma, we were able to compute the exact numbers of such codewords for the dual codes of B3 and B5 (84 and 20, respectively). We reproduce some of the properties of these codes in Table 1, where the first column lists the corresponding biplane, column two gives the 3-rank of its incidence matrix, or the code dimension, the third column gives the order of the automorphism group of the biplane, the fourth column gives the minimum distance of the code, and the last column gives the total number of (0, 1)-codewords of weight 12 in the dual code.
Since the dual codes of the biplanes B3, B4 and B5 each contains less than 165 (0, 1)codewords of weight 12, it follows from Lemma 5 that none of the dual designs of their residual 2-(45, 9, 2) designs can be a derived 1-(55, 11,9) design of a quasi-symmetric 2-(56, 12,9) design with intersection numbers 0, 3.
Since the automorphism group of B1 acts transitively on the set of blocks, all residual 2-(45, 9, 2) designs of B1 are isomorphic. We define a graph Γ 1 having as vertices the 2100 (0, 1)-codewords of weight 12 in the dual code of the ternary code spanned by the points of B1, where two codewords are adjacent in Γ if their supports are either disjoint or share exactly three points. It follows from Lemma 5 that if a quasi-symmetric 2-(56, 12,9) design exists and has a derived design with respect to a point which is the dual design of a residual 2-(45, 9, 2) design of B1, then Γ 1 will contain a clique of size 165. A quick computation with Cliquer [15] shows that the maximum clique size of Γ 1 is 22, thus none of the residual designs of B1 is embeddable in a quasi-symmetric 2-(56, 12,9) design.
The graph Γ 2 having as vertices the 516 (0, 1)-codewords of weight 12 in the dual code of the ternary code spanned by the points of of B2, where two codewords are adjacent in Γ 2 if their supports are either disjoint or share exactly three points, has maximum clique size 18. It follows by Lemma 5 that a quasi-symmetric 2-(56, 12, 9) design having a derived design which is the dual design of some residual 2-(45, 9, 2) design of B2, does not exist. This completes the proof of part (i).
(ii) Suppose there exists a quasi-symmetric 2-(57, 12, 11) design D with block intersection numbers 0, 3. Then the residual design of D with respect to a point p is a quasisymmetric 2-(56, 12,9) design with block intersection numbers 0, 3. Such a design does not exist by (i). 4 The nonexistence of some quasi-3 designs Definition 4.1. A symmetric 2-(v, k, λ) design D is a quasi-3 design [14] with triple intersection numbers x and y (x < y) if every three blocks of D intersect in either x or y points.
Clearly, D is a quasi-3 design if and only if every of its derived (with respect to a block) dim C | Aut Bi| min # wt 12 in C ⊥ B1 20 80640 11 2100 B2 22 288 11 516 B3 26 144 11 84 * B4 24 64 11 148 B5 26 24 11 20 * Table 1: The ternary codes of the five biplanes 2-(k, λ, λ − 1) designs is a quasi-symmetric design with block intersection numbers x and y.
According to [14,Table 47.14], there are 12 parameter sets of quasi-3 designs with number of points v ≤ 400, for which the existence of a quasi-3 design is unknown. Two of these twelve open cases are the parameters 2-(149, 37, 9), (x = 1, y = 3), and 2-(267, 57, 12), (x = 0, y = 3).
Theorem 7. A quasi-3 design with parameters 2-(267, 57, 12), (x = 0, y = 3), does not exist.
Proof. Any derived design with respect to a block of a quasi-3 2-(267, 57, 12) design with triple intersection numbers x = 0, y = 3 is a quasi-symmetric 2-(57, 12, 11) design with block intersection numbers x = 0, y = 3. By Theorem 6, part (ii), a quasi-symmetric design with the latter parameters does not exist.
Theorem 8. A quasi-3 design with parameters 2-(149, 37, 9), (x = 1, y = 3), does not exist.
Proof. Any derived design with respect to a block of a quasi-3 2-(149, 37, 9) design with triple intersection numbers x = 1, y = 3 is a quasi-symmetric 2-(37, 9, 8) design with block intersection numbers x = 1, y = 3. However, it was proved in [9] that quasi-symmetric designs with the latter parameters do not exist.
Lemma 4 .
4Let D B be a 2-(45, 9, 2) design with point set {1, 2, . . . , 45}, being the residual design of a 2-(56, 11, 2) biplane D with respect to a block B = {46, . . . , 56}. Let A be an incidence matrix of D given by
Figure 1 :
1sum of all rows of A over GF (3) is the constant vector with all entries equal to 2, the all-one vector1 56 =(1 55 , 1) of length 56 belongs to L. Thus, L is spanned by the rows ofD z ⊆ S B ⊆ S ⊆ L ⊥ Lemma 5
Lemma 5 .
5Let D be a 2-(56, 11, 2) biplane, and let S be the set of all (0, 1)-vectors of weight 12 in the dual code of the ternary linear code spanned by the points of D. Let D B be a residual 2-(45, 9, 2) design of D with respect to a block B.A necessary condition for the existence of a quasi-symmetric 2-(56, 12, 9) designD having D B * as a derived design is that the subset S B ⊂ S consisting of all vectors having 0 in the position labeled by B, contains a set of 165 vectors that are the incidence vectors of the blocks of a 1-(55, 12, 36) designD z , such that every two blocks ofD z are either disjoint or share exactly three points.3 The nonexistence of quasi-symmetric 2-(56,12,9) and 2-(57, 12, 11) designs Theorem 6. (i) A quasi-symmetric 2-(56,12,9) design with block intersection numbers 0, 3 does not exist.
Note 1. A sufficient condition for a residual design D B to be linearly embeddable is that the minimum distance of the linear code over F spanned by the blocks of D is equal to |B| [19, Theorem 2.2].
Table 48 .
4825]. This table is an updated version of Neumaier's table [18] published in 1982.
E F AssmusJr, J D Key, Designs and their codes. CambridgeE. F. Assmus, Jr., J. D. Key, Designs and their codes, Cambridge, 1992.
T Beth, D Jungnickel, H Lenz, Design theory. Cambridge University Press2nd editionT. Beth, D. Jungnickel, H. Lenz, Design theory (2nd edition). Cambridge University Press, 1999.
Handbook of Magma Functions. W Bosma, J Cannon, School of Mathematics and Statistics. University of SidneyW. Bosma, J. Cannon, Handbook of Magma Functions, School of Mathematics and Statistics, University of Sidney, Sidney, July 22 (1999).
C J Colbourn, J H Dinitz, Handbook of Combinatorial Designs. Chapman & Hall/CRCC.J. Colbourn and J.H. Dinitz, Handbook of Combinatorial Designs, 2nd ed., Chap- man & Hall/CRC, 2007.
R H F Denniston, On biplanes with 56 points. 9R.H.F. Denniston, On biplanes with 56 points, Ars Combin. 9 (1980), 167-179.
. M Hall, Jr Theory, WileyNew YorkSecond EditionM. Hall, Jr., Combinatorial Theory, Second Edition, Wiley, New York 1986.
An embedding theorem for balanced incomplete block designs. M Hall, Jr , W S Connor, Canad. J. Math. 6M. Hall, Jr., and W.S. Connor, An embedding theorem for balanced incomplete block designs, Canad. J. Math. 6 (1953), 35-41.
Designs derived from permutation groups. M HallJr, R Lane, D Wales, J. Combin. Theory. 8M. Hall, Jr., R. Lane, and D. Wales, Designs derived from permutation groups, J. Combin. Theory 8 (1970), 12-22.
Self-dual codes and the nonexistence of a quasi-symmetric 2-(37, 9, 8) design with intersection numbers 1 and 3. M Harada, A Munemasa, V D Tonchev, J. Combin. Designs. 25M. Harada, A. Munemasa, and V.D. Tonchev, Self-dual codes and the nonexistence of a quasi-symmetric 2-(37, 9, 8) design with intersection numbers 1 and 3, J. Combin. Designs 25 (2017), 469-476.
W C Huffman, V Pless, Fundamentals of Error-Correcting Codes. Cambridge University PressW. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, 2003.
A new biplane of order 9 with a small automorphism group. Z Janko, T Van Trung, J. Combin. Theory Ser. A. 42Z. Janko and T. Van Trung, A new biplane of order 9 with a small automorphism group, J. Combin. Theory Ser. A 42 (1986), 305-309.
There are exactly five biplanes with k = 11. P Kaski, P R J Östergård, J. Combin. Des. 162P. Kaski and P.R. J.Östergård. There are exactly five biplanes with k = 11. J. Combin. Des., 16(2):117-127, 2008.
Computational results for the known biplanes of order 9. J Key, V D Tonchev, London Math. Soc. Lecture Note Ser. 245Cambridge Univ. PressJ. Key and V.D. Tonchev, Computational results for the known biplanes of order 9, London Math. Soc. Lecture Note Ser. 245, Cambridge Univ. Press, Cambridge, 1997, pp. 113-122.
Quasi-3 designs. G Mcguire, Handbook of Combinatorial Designs. C.J. Colbourn and J.H. DinitzChapman & Hall/CRC2nd EditionG. McGuire, Quasi-3 designs, Chapter VI.47 in: Handbook of Combinatorial De- signs, 2nd Edition, C.J. Colbourn and J.H. Dinitz, eds., Chapman & Hall/CRC, 2007, pp. 576-578.
Cliquer User's Guide, Version 1.0. S Niskanen, P R J Östergård, T48Communications Laboratory, Helsinki University of Technology, Espoo, FinlandTech. Rep.Niskanen, S.,Östergård, P. R. J.: Cliquer User's Guide, Version 1.0. Tech. Rep. T48, Communications Laboratory, Helsinki University of Technology, Espoo, Fin- land, 2003.
The four known biplanes with k = 11. C J Salwach, J A Mezzaroba, Intern. J. Math. & Math. Sci. 2C.J. Salwach and J.A. Mezzaroba, The four known biplanes with k = 11, Intern. J. Math. & Math. Sci. 2 (1979), 251-260.
Quasi-Symmetric Designs. M S Shrikhande, Handbook of Combinatorial Designs. C.J. Colbourn and J.H. DinitzChapman & Hall/CRC2nd EditionM.S. Shrikhande, Quasi-Symmetric Designs, Chapter VI.48 in: Handbook of Com- binatorial Designs, 2nd Edition, C.J. Colbourn and J.H. Dinitz, eds., Chapman & Hall/CRC, 2007, pp. 578-582.
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. V D Tonchev, Combinatorial Configurations. WileyV. D. Tonchev, Combinatorial Configurations, Wiley, New York, 1988.
| []
|
[
"Arbitrary degree regular graphs of large girth",
"Arbitrary degree regular graphs of large girth",
"Arbitrary degree regular graphs of large girth",
"Arbitrary degree regular graphs of large girth"
]
| [
"Xavier Dahan [email protected] \nFaculty of mathematics\nKyûshû university\nJapan\n",
"Xavier Dahan [email protected] \nFaculty of mathematics\nKyûshû university\nJapan\n"
]
| [
"Faculty of mathematics\nKyûshû university\nJapan",
"Faculty of mathematics\nKyûshû university\nJapan"
]
| []
| For every integer d ≥ 10, we construct infinite families {G n } n∈N of d + 1-regular graphs which have a large girth ≥ log d |G n |, and for d large enough ≥ 1, 33 · log d |G n |. These are Cayley graphs on P GL 2 (F q ) for a special set of d + 1 generators whose choice is related to the arithmetic of integral quaternions. These graphs are inspired by the Ramanujan graphs of Lubotzky-Philips-Sarnak and Margulis, with which they coincide when d is prime. When d is not equal to the power of an odd prime, this improves the previous construction of Imrich in 1984 where he obtained infinite families {I n } n∈N of d + 1-regular graphs, realized as Cayley graphs on SL 2 (F q ), and which are displaying a girth ≥ 0, 48 · log d |I n |. And when d is equal to a power of 2, this improves a construction by Morgenstern in 1994 where certain families {M n } n∈N of 2 k + 1-regular graphs were shown to have a girth ≥ 2 3 log 2 k |M n |. * Supported by the GCOE project "Maths-for-Industry" of Kyûshû university | 10.1007/s00493-014-2897-6 | [
"https://arxiv.org/pdf/1110.5259v4.pdf"
]
| 5,826,460 | 1110.5259 | 4c9ad7fec2d9c54ca5abd670cc7e5dfa0c2aeedd |
Arbitrary degree regular graphs of large girth
25 Nov 2011
Xavier Dahan [email protected]
Faculty of mathematics
Kyûshû university
Japan
Arbitrary degree regular graphs of large girth
25 Nov 2011
For every integer d ≥ 10, we construct infinite families {G n } n∈N of d + 1-regular graphs which have a large girth ≥ log d |G n |, and for d large enough ≥ 1, 33 · log d |G n |. These are Cayley graphs on P GL 2 (F q ) for a special set of d + 1 generators whose choice is related to the arithmetic of integral quaternions. These graphs are inspired by the Ramanujan graphs of Lubotzky-Philips-Sarnak and Margulis, with which they coincide when d is prime. When d is not equal to the power of an odd prime, this improves the previous construction of Imrich in 1984 where he obtained infinite families {I n } n∈N of d + 1-regular graphs, realized as Cayley graphs on SL 2 (F q ), and which are displaying a girth ≥ 0, 48 · log d |I n |. And when d is equal to a power of 2, this improves a construction by Morgenstern in 1994 where certain families {M n } n∈N of 2 k + 1-regular graphs were shown to have a girth ≥ 2 3 log 2 k |M n |. * Supported by the GCOE project "Maths-for-Industry" of Kyûshû university
Introduction
The "Moore bound" follows from a simple counting argument, and permits to show that a d-regular graph G of size |G|, admits the following upper bound on its girth (see [
It is not known if this bound is tight. A convenient way to formulate what should be understood by "tight", is to consider large graphs, and even better, an infinite family of constant degree regular graphs. Let us recall the following definition: a family of d-regular graphs {G n } n∈N is said to have large girth if there exists a constant c > 0 independent of n (eventually dependent on d), such that: girth(G n ) ≥ (c + o(1)) log d−1 |G n |.
Given an infinite family {G n } n of d-regular graph, let us define: What the bound (2) says is that γ d ≤ 2, for any d ≥ 3. As for lower bounds, it was proved that γ d ≥ 1 by Erdös and Sachs [7] for any d ≥ 3 (see also [1,Ch. III] Theorem 1.4 and the discussion above it). Their proof, of probabilistic nature, did not provide explicit families of graphs {G n } n . Currently, the best lower bounds for γ d that are deduced from explicit examples of family of graphs, are:
1. γ d ≥ 12 7 for d = p 3 + 1, p an odd prime (this is found in [4]).
2. γ d ≥ 4 3 for d = p k + 1, p an odd prime and k ∈ N ⋆ , (for d = p + 1 where p is an odd prime; This was first achieved by Lubotzky-Philips-Sarnak [11] and independently by Margulis [13], then later also by Lazebnik-Ustimenko [10] with a different construction. Finally, Morgenstern [14] treated the case d − 1 equal to any power of an odd prime).
3. γ d ≥ 2 3 for d = 2 k + 1 with k ∈ N ⋆ . This is also due to Morgenstern [14,. 4. γ d ≥ 0, 48 for other values of d (this is due to Imrich [9], extending the method of Margulis [12] where it was proved that γ d ≥ 4 9 for odd d). These are the best results we are aware of. Is proposed in this paper an improvement on the lower bounds on γ d+1 in the cases 3-4, that is when d is not equal to the power of an odd prime. For other values of d + 1, the lower bounds that would be obtained do not improve the best ones shown in the cases 1 and 2. That is why we focus only on the cases not equal to the power of an odd prime, and henceforth consider only d ≥ 10 (lower values are either odd prime powers or non manageable by our method). Theorem 1.1 For any integer d ≥ 10, which is not a prime power, there is an explicit infinite family {G n } n of d + 1-regular graphs, bipartite and connected, as well as having large girth. Precisely:
girth(G n ) ≥ c(d) log d |G n | − log d 4,(3)
where c(d) is a constant independent of n, such that c(d) ≤ 4 3 and:
case d odd if d ≥ 1335, c(d) ≥ 1, 33 if 35 ≤ d ≤ 1331 c(d) ≥ 1, 3 if 15 ≤ d ≤ 31, c(d) ≥ 1, 27 case d even if d ≥ 4826, c(d) ≥ 1, 33 if 184 ≤ d ≤ 4824 c(d) ≥ 1, 3 if 44 ≤ d ≤ 182, c(d) ≥ 1, 25 if 22 ≤ d ≤ 42, c(d) ≥ 1, 1 c(10) ≥ 1, 28 c(12) ≥ 1, 12 c(14) ≥ 1, 19 c(18) ≥ 1, 3 c(20) ≥ 1, 061.
Related to the families {G n } n , there are also explicit families of d + 1-regular graphs {H n } n , connected and non-bipartite, for which the girth verifies:
girth(H n ) ≥ c(d) 2 log d |H n |.
The family {G n } n will be X d and {H n } n will be Y d that are both introduced in Definition 1.3. The values in the theorem are indicative, having been chosen for their readability. More precise values of c(d) for each d can be obtained, but they are of limited interest. More interesting is to mention that c(d) → 4 3 when d becomes large. These values on c(d) provide significantly better lower bounds for γ d+1 than was previously known: γ d+1 ≥ c(d), improving upon γ d+1 ≥ 0, 48 in the case 4, and improving upon γ d+1 ≥ 2 3 in the case 3. The fact that c(d) ≤ 4 3 shows that no further improvement can be expected from the trick introduced in the present paper (but this trick applied to the graphs of [14] gives slightly better estimates sometimes, see the discussion "Further improvement 1" in Conclusion).
Furthermore, these explicit families of graphs do even better than what the probabilistic method [7] is able to achieve, namely a γ d ≥ 1. When dealing with Cayley graphs on P GL 2 (F q ), it was proved in Theorem 9 of [8] that random Cayley graphs 1 have a girth ≥ ( 1 3 − o(1)) log d |P GL 2 (F q )| for q sufficiently large. What is the exact value is not known but the new graphs of the present paper have likely much larger girth than the one for the corresponding random Cayley graph.
The main inequality. This paragraph presents the main intermediate result (4), and the next paragraph will show how to deduce from it the bounds of Theorem 1. Given q > Q d (p) another prime, there is a symmetric 2 subset D p,q of P GL 2 (F q ) of cardinal d + 1, such that if we define G d,p,q := Cay(P GL 2 (F q ), D p,q ) for values of q modulo which p is not a quadratic residue, and G d,p,q := Cay(P SL 2 (F q ), D p,q ) for values of q modulo which p is a quadratic residue (see Definition 1.3 for more details on G d,p,q ), then:
• G d,p,q is a d + 1-regular graph of size |P GL 2 (F q )| = q 3 − q or |P SL 2 (F q )| = 1 2 (q 3 − q)
according to the sign of the Legendre symbol p q .
• G d,p,q is connected, bipartite if p q = −1, and not bipartite if p q = 1.
• the girth of G d,p,q verifies the main inequality:
girth(G d,p,q ) ≥ 2 3κ log d |G d,p,q | if p q = 1 4 3κ log d |G d,p,q | − log p 4 if p q = −1(4)
Let us point out here that girth(G d,p,q ) ≤ 4 3 log d |G d,p,q | + 1 or girth(G d,p,q ) ≤ 2 3 log d |G d,p,q | + 1, for any d. Indeed, these lower bounds already occur for the Ramanujan graphs [13,Last proposition], from which are deduced the graphs G d,p,q . This is why c(d) ≤ 4 3 in the theorem 1.1. Fixing d and p, we can consider the following two kinds of infinite families of d + 1-regular graphs, indexed by q:
X d,p := {G d,p,q } q prime, q>Q d (p), p q =−1 ,(5)
and
Y d,p := {G d,p,q } q prime, q>Q d (p), p q =1 .(6)
From Main Inequality (4) above, comes: γ(X d,p ) ≥ 4 3κ and γ(Y d,p ) ≥ 2 3κ , where κ = log d p.
Main Inequality implies Theorem 1.1. It is quite easy to recover the bounds on c(d) of Theorem 1.1 from Main Inequality (4). The lower bound on the girth in (4) is indeed the largest when κ is the smallest. To minimize κ, let us first introduce some notations: Then, for each d ≥ 10, we consider 2 families of d + 1-regular graphs X d and Y d among the families introduced in (5) and (6):
if d is even:
X d := X d,p(d) , Y d := Y d,p(d) , and if d is odd: X d := X d,p 3 (d) , Y d := Y d,p 3 (d)
The real number κ of Definition 1.
2 verifies then κ = log d p(d) if d is odd, and κ = log d p 3 (d) if d is even.
Then, minimizing κ brings in the question: Given u odd, how big is the smallest prime p(u) larger than u ? Similarly , if u is even, how big can be p 3 (u) ?
Bertrand's postulate affirms that p(u) < 2u, but for u ≥ 3275, the better estimate p(u) < u(1 + 1 2(log u) 2 ) holds (see [6, p. 14]). It implies that: κ ≤ log u (u(1 + Setting ǫ = 0, 002811, it comes for x ≥ 10 10 and any y:
y 4 − ǫ x 4 ≤ θ(y; 8, 3) ≤ ǫ x 4 + y 4 .
It follows that for all b > a ≥ 10 10 ,
θ(b; 8, 3) − θ(a; 8, 3) ≥ b 4 (1 − 2ǫ) − a 4 .
This insures that for a ≥ 10 10 there is a prime equal to 3
max 1≤y≤x |θ(y; 8, 3) − y 4 | ≤ 1, 82 √ x, for 1 ≤ x ≤ 10 10 . It follows that θ(b; 8, 3) − θ(a; 8, 3) ≥ b−a 4 − 2 · 1, 82 √ b for b > a.
Proof of the main inequality
It remains to show that Main Equality (4) holds. All the necessary material is contained in the monograph [5]. To make this section a minimum self-contained, many results appearing therein are recalled.
Unique factorization of quaternions and regular trees
The construction of Ramanujan graphs by Lubotzky-Philips-Sarnak is achieved by taking finite quotients of a "mother graph", which is a regular tree. They used simply the factorization of quaternions to build these regular trees.
We recall briefly this here, referencing to Ch. 2.6 of the aforementioned monograph [5] for the details.
Quaternions. For R a commutative ring, let H(R) denotes the Hamilton quaternion algebra over R:
H(R) := R + Ri + Rj + Rk, i 2 = j 2 = k 2 = −1, k = ij = −ji. The conjugate of an element α = a 0 +a 1 i+a 2 j+a 3 k is α := 2a 0 −α = a 0 −a 1 i−a 2 j−a 3 k, and the norm of α is N (α) = αα = a 2 0 + a 2 1 + a 2 2 + a 2 3 .
The rules of the multiplication of quaternions make the norm multiplicative: N (αβ) = N (α)N (β). Given a quaternion α = a 0 + a 1 i + a 2 j + a 3 k the non-negative integer gcd(a 0 , a 1 , a 2 , a 3 ) is called the content of α and is denoted c(α). If c(α) = 1, then α is primitive.
Let us set R = Z. We introduce a property of unique factorization for integral quaternions H(Z), yet in a special easy case that is sufficient for the purpose of this article. This restriction is to consider only quaternions whose norm is a power of an odd prime p (instead of considering any quaternion in H(Z) 3 ).
Given an odd prime p, and a primitive quaternion α ∈ H(Z), of norm p k , there exist prime quaternions π 1 , . . . , π k (prime means that if π = αβ, then either α or β is a unit in H(Z)) such that: α = π 1 · · · π k . In a word, this follows from the possibility to perform a Euclidean division in H(Z) of 2 such quaternions whose norm is a power of p; A non-commutative Euclidean algorithm (one "one the right", one "on the left" ) is deduced, in order to compute left and right gcds. This permits to show that prime quaternions are precisely those whose norm is a prime number. Then the existence of a factorization follows easily by induction on the exponent k of the norm p k = N (α).
The default of uniqueness is completely related to the units of H(Z) (which are ±1, ±i, ±j, ±k). What this means is that two distinct factorizations π 1 · · · π k and µ 1 · · · µ k of α verify: π i = ǫ i µ i , for some ǫ i ∈ H(Z) ⋆ and for 1 ≤ i ≤ k. The group of 8 units H(Z) ⋆ acts on the set of quaternions of norm p. By isolating one quaternion per orbit, uniqueness can be recovered. Since the number of quaternions of norm p is 8(p + 1) by a famous theorem of Jacobi (indeed, such
quaternions x 0 + x 1 i + x 2 j + x 3 k give a solution in Z 4 of f (x) = p, where x = (x 0 , x 1 , x 2 , x 3 ) and f (x) = x 2 0 + x 2 1 + x 2 2 + x 2 3 )
. As perfectly explained in p. 67-68 of [5], a quite natural way to isolate one quaternion per orbit is to introduce:
P(p) = {π ∈ H(Z) primitive : N (π) = p, π 0 > 0, π − 1 ∈ 2H(Z)} if p ≡ 1 mod 4, (7) P(p) = {π ∈ H(Z) primitive : N (π) = p, π 0 > 0 if π 0 = 0, or π 1 > 0 if π 0 = 0, and π − i − j − k ∈ 2H(Z)} if p ≡ 3 mod 4 (8)
The fact that π 0 = 0, or π 1 = 0 if π 0 = 0 is made clear by the explanations coming hereafter.
Remark 2.1 Some general remarks about this set:
(a) if α ∈ P(p), then ǫα and αǫ are not in P(p), for all ǫ ∈ H(Z) ⋆ different from 1.
(b) Similarly, given β ∈ H(Z), N (β) = p, there are exactly two ε, ε ′ ∈ H(Z) ⋆ that yields εβ ∈ P(p) and βε ′ ∈ P(p).
(c) this implies that |P(p)| = p + 1.
(d) given π ∈ P(p), if π 0 = 0 then π ∈ P(p) (easy to check). If π is such that π 0 = 0, as it may happen when π ≡ 3 mod 4 (actually when p ≡ 3 mod 8 after Proposition 2.3), then π = −π ∈ P(p), in conformity with the two points (a) and (b) above.
Remark that the first point (a) allows a form of uniqueness of the factorization of quaternions [5, 2.6.13 Theorem].
Theorem 2.2 Given an odd prime p and a quaternion α of norm p k , of content c(α) = p ℓ , there exists unique π 1 , . . . , π k−2ℓ ∈ P(p) and a unique unit ǫ ∈ H(Z) ⋆ such that: α = c(α) ǫ π 1 · · · π k−2ℓ , with π i = π i−1 if π i ∈ P(p), and with π i = π i−1 else.
Let us stress that under these conditions, the quaternion π 1 · · · π k−2ℓ is primitive (motivating later the definition of irreducible product in Definition 2.6).
We focus now on the case π ∈ P(p) and π ∈ P(p), which may happen when p ≡ 3 mod 4 as mentioned in (d) of Remark 2.1.
Proposition 2.3
There is an element π = π 0 + π 1 i + π 2 j + π 3 k ∈ P(p) for which π 0 = 0 (equivalently π = π, or π ∈ P(p)) if and only if p ≡ 3 mod 8.
Proof: By definition of P(p) this can only happens if p ≡ 3 mod 4, since else π 0 ≡ 1 mod 2. For such a π, N (π) = π 2 1 + π 2 2 + π 2 3 and consequently p is a sum of 3 squares. Reciprocally, a sum of 3 squares x 2 1 + x 2 2 + x 2 3 equal to p gives a quaternion x = x 1 i + x 2 j + x 3 k ∈ H(Z) of norm p, which is also necessarily primitive (because p is prime). Since p ≡ 3 mod 4, p is not the sum of 2 squares. Hence necessarily x 1 ≡ x 2 ≡ x 3 ≡ 1 mod 4, implying x ∈ P(p).
We have proved that such a π exists in P(p) if and only if p ≡ 3 mod 4 and p is the sum of 3 squares. This is true if and only if p ≡ 3 mod 8, as the Legendre's theorem on sum of 3 squares shows:
Theorem 2.4 (Legendre) An integer n is the sum of 3 squares if and only if n is not equal to 4 k (8ℓ + 7) for any k, ℓ ∈ N.
Hence, p ≡ 3 mod 4 is the sum of 3 squares if and only if p = 4 k (8ℓ + 7). Suppose p = 4 k (8ℓ + 7), then p ≡ 3 mod 4 gives k = 0, and p = 8ℓ + 7, implying p ≡ 3 mod 8. This proves that p ≡ 3 mod 4 is sum of 3 squares if and only if p ≡ 3 mod 8, achieving the proof of Proposition 2.3
In the case p ≡ 3 mod 8, we denote a quaternion in P(p) of the form shown in Proposition 2.3 with a letter ν, and the others by the letter µ. One has: if p ≡ 3 mod 8, P(p) = {µ 1 , . . . , µ s , ν 1 , . . . , ν t }, with s + t = p + 1 and t > 0.
Note that s is even because each µ i comes along with its conjugate, that is there is an i ′ = i such that µ i = µ i ′ . Hence t = p + 1 − s is also even.
Trees built on quaternions. The unique factorization theorem 2.2 permits to build infinite regular trees of arbitrary degree d. Proof: If d is odd, then the definitions (8)-(7) of P(p) when p ≡ 3 mod 8 makes it clear: it suffices to choose d+1 2 elements pairwise not conjugate, as well as their d+1 2 conjugates (that are also in P(p) in this case). For the case p ≡ 3 mod 8, let us use the two even integers s and t defined in (9). We first choose k 1 := max{ d+1 2 , s 2 } couple of conjugates in P(p), and, if necessary, d + 1 − 2k 1 elements π such that π ∈ P(p).
If d is even, then p ≡ 3 mod 8 by Definition 1.2. A way to choose the set D(d) is as follows. First choose k 1 := max{ d 2 , s 2 } couples of conjugates, completed with d + 1 − 2k 1 elements π such that π ∈ P(p). -are not conjugate,
α i = α i+1 , if α i ∈ P(p) -are not equal, α i = α i+1 , if α i ∈ P(p). The set of all irreducible products over D(d) is denoted Λ D .
The motivation of this terminology comes from the following fact, resulting of the factorization theorem 2.2: the product of a sequence of elements in D(d) that does not verify the conditions mentioned in the definition can be reduced, yielding a non primitive quaternion.
Furthermore, the unique factorization theorem 2.2 also tells that two different irreducible products yields two different quaternions. This allows to define a d + 1-regular tree T d in the following way:
• the vertex set V (T d ) is identified with the irreducible products Λ D over D(d) ⊂ P(p)
• given a vertex identified with the irreducible product α 1 · · · α ℓ , we define d adjacent vertices whose irreducible products are:
α 1 · · · α s α s+1 , α s+1 ∈ D(d) where α s+1 = α s if α s ∈ P(p) α s+1 = α s if α s ∈ P(p)
• and the last adjacent vertex is the irreducible product α 1 · · · α s−1
Algebraic construction of the tree and definition of the graphs G d,p,q
It is necessary to give an interpretation of the tree T d constructed above more algebraically. Indeed, the graphs G d,p,q of Main Inequality (4) are naturally defined algebraically.
Algebraic construction of the trees T d . It consists in seeing the tree T d as Cayley graphs on free groups. These free groups are:
Proposition 2.7 Given an integer d ≥ 10, the set Λ D of all irreducible products over D(d) can be endowed of a structure of free groups on the generators D(d).
Proof: Given two irreducible products α := α 1 · · · α n , and β := β 1 · · · β m , we associate a quaternion denoted α × β which is an irreducible product, defined as follows: -there is no integer i ≥ 0 such that α n−i = β i+1 if β i+1 ∈ P(p), or α n−i = β i+1 if β i+1 ∈ P(p). Then we define α × β = 1.
-else, let ℓ ≥ 0 be the largest such integer i. The content of αβ is then c(αβ) = p ℓ , and αβ p ℓ is primitive. Its unique factorization is given by: αβ p ℓ = ±α 1 · · · α n−ℓ β ℓ+1 · · · β m . This allows to define, α × β := α 1 · · · α n−ℓ β ℓ+1 · · · β m .
Note that this is an irreducible product of length m + n − 2ℓ, over D(d).
It is easy to check that × defines an associative law on Λ D with unit element 1 (the void irreducible product). The inverse of an irreducible product α := α 1 · · · α n is β :=α n · · ·α 1
whereα i = α i ∈ D(d) if α i ∈ P(p), andα i = α i if α i = −α i ∈ P(p). The content of αβ is then p n , hence α × β = 1.
It remains to show that the group (Λ D , ×) is free. This follows by the definition 2.6 of irreducible products on D(d), that yields different quaternions by the unique factorization theorem 2.2.
Remark 2.8 Using the notations in (9), D(d) consists of elements µ 1 , . . . , µ u , ν 1 , . . . , ν v with u ≤ s and v ≤ t, such that ν i ∈ D(d) and µ i = µ i ′ ∈ D(d). Let (K, ×) be the subgroup of (Λ D , ×) generated by µ 1 , . . . , µ u . This is a free group for ×, and we have:
(Λ D , ×) ≃ (K, ×) * ν 1 * · · · * ν v ,
where ν i is the subgroup of (Λ D , ×) generated by ν i , and * is the free product on subgroups of (Λ D , ×).
The combinatorial definition of the tree T d given at the end of Section 2.2, and the above, shows that T d is the Cayley graph of the group (Λ D , ×) with "Cayley set" D(d). D(d)).
T d = Cay(Λ D ,
Graphs G d,p,q as finite quotients of the tree T d . As above, we let d be an integer greater than 10, and p a prime greater than d, ordinary if d is odd, and equal to 3 modulo 8 if d is even. Now we let q > Q d (p) where Q d (p) is the constant introduced in Definition 1.2.
The next step consists in taking finite quotients of the tree T d . Let
τ q : H(Z) → H(F q )(10)
the reduction map modulo q. When restricted to Λ D , we observe the following:
• τ q (Λ D ) ⊂ H(F q ) ⋆
• τ q (αβ) and τ q (α × β) differs by ±τ q (p ℓ ), where p ℓ is the content of αβ, which is in the center Z of the group H(F q ) ⋆ . Indeed,
Z = {α ∈ H(F q ) ⋆ | α = α}.
Hence, by taking the quotient group H(F q ) ⋆ /Z the following map:
µ q : Λ D → H(F q ) ⋆ /Z,
is a group homomorphism. Next, we identify the image of this group homomorphism. Recall that since p = 2, the quaternion algebra over F q as was defined in Section 2.1 is isomorphic to the algebra of 2-by-2 matrices over F q . Indeed, in F q there are two elements x and y such that x 2 + y 2 + 1 = 0 (see Prop. 2.5.2 and 2.5.3 in [5]). The following map is an isomorphism of F q -algebra:
φ : H(F q ) → M 2 (F q ), α 0 + α 1 i + α 2 j + α 3 k → α 0 + α 1 x + α 3 y −α 1 y + α 2 + α 3 x −α 1 y − α 2 + α 3 x α 0 − α 1 x − α 3 y .
Moreover N (α) = det φ(α). We deduce the following group isomorphism ψ from φ:
ψ : H(F q ) ⋆ /Z → P GL 2 (F q ),
and we let:
µ q := ψµ q , and ker µ q := Λ D (q), so that Λ D /Λ D (q) ֒→ P GL 2 (F q ).
Lemma 2.9
If q is such that p is a quadratic residue modulo q, then µ q (D(d)) ⊂ P SL 2 (F q ). Else, µ q (D(d)) ⊂ P GL 2 (F q ) − P SL 2 (F q ).
Proof: The group homomorphism ǫ :
H(F q ) ⋆ → {−1, 1}, x → N (α) q
takes the same value on each class modulo the center Z. The factor map ǫ : H(F q ) ⋆ /Z → {−1, 1}, xZ → ǫ(x), is well-defined. The set of quaternions in H(F q ) ⋆ of norm 1, denoted H 1 , are sent to 1 by ǫ, and hence ker ǫ ⊃ H 1 /H 1 ∩ Z. Now, given π ∈ D(d), p q and ǫ(µ q (π)) are equal. This shows that if p q = 1, then µ q (D(d)) ⊂ ker ǫ, and if p q = −1, then µ q (D(d)) ⊂ H(F q ) ⋆ /Z − ker ǫ. Using the isomorphism ψ, we obtain µ q (D(d)) ⊂ P SL 2 (F q ) if p q = 1, and µ q (D(d)) ⊂ P GL 2 (F q ) − P SL 2 (F q ) else.
By the above discussion, comes:
Λ D /Λ D (q) ֒→ P SL 2 (F q ) if p q = 1 P GL 2 (F q ) if p q = −1(11)
Lemma 2.10 Let D p,q := µ q (D(d)). One has |D p,q | = |D(d)| = d + 1
Proof: The map ψ being an isomorphism it suffices to show that |D(d)| = |µ q (D(d))|. Since D(d) ⊂ P(p), this will certainly follows from |P(p)| = |µ q (P(p))|. The later is (easily) proved in [5,4.2.1 Lemma], under the assumption that q > 2 √ p, verified because q > Q d (p) ≥ p 8 .
Already mentioned in Introduction, we now give a precise definition of the graph G d,p,q : Definition 2.11 Given the three integers d, p and q as defined above, the graph G d,p,q is :
G d,p,q := Cay(P GL 2 (F q ) , D p,q ) if p q = −1 Cay(P SL 2 (F q ) , D p,q ) if p q = 1
By Lemma 2.10, the graphs G d,p,q are d + 1-regular.
Lemma 2.12
The graphs G d,p,q are bipartite when p q = −1. Moreover, assuming that G d,p,q is connected when p q = 1, G d,p,q is non-bipartite.
Proof: In the first case, a bipartition A∪B of the vertices V (G d,p,q ) is given by A := P SL 2 (F q ), and B := P GL 2 (F q ) − P SL 2 (F q ). If x ∈ A, written as x = µ q (α) for an α ∈ Λ D , then a neighbor y is written y = µ q (α × π) for a π ∈ D(d). Using the notations of Lemma 2.9, one sees that H 1 /Z ∩ H 1 = ψ −1 (P SL 2 (F q )) ⊂ ker ǫ, and thus N (α) q = 1, and therefore
N (α×π) q = p q = −1,
showing that µ q (α × π) / ∈ A and y ∈ B.
As for th case p q = 1, saying that G d,p,q is connected is equivalent to saying that D p,q generates P SL 2 (F q ). Then a bipartition would imply a non-trivial group homomorphism P SL 2 (F q ) → {−1, 1}, whose kernel would be a proper normal subgroup of P SL 2 (F q ), excluded since P SL 2 (F q ) is simple [5, 3.2.2 Theorem].
To end this subsection, all these Cayley graphs are actually connected (this is Proposition 2.15, in particular, G d,p,q is non-bipartite when p q = 1, by the lemma above). This point is important for estimating the girth, and is not trivial. In [11], they make use of a deep and technical result of Malyshev on the number of integer solutions of quadratic definite positive forms; the construction of Margulis [13] differs slightly from the one of [11], where rather a density argument (strong approximation theorem) was used. In our modified construction of graphs, the connectedness is also crucial, but none of these 2 theorems would work. Fortunately, later appeared in [5] (see discussion p. 6 therein) a simple argument to prove the connectedness, based on the properties of the subgroups of P SL 2 (F q ), whose observation goes back to Dickson. This will be instrumental in the present work.
Connectedness and final proof
Following the method of Ch. 4.3 in [5], this is achieved by showing logarithmic girth.
Let X denotes the connected component of G d,p,q containing the identity.
Lemma 2.13 Let D ′ (d) denotes the image of D(d) ⊂ Λ D through the group homomorphism for × Λ D → Λ D /Λ D (q).
One has the isomorphism of graphs:
X ≃ Cay(Λ D /Λ D (q) , D ′ (d)),
Proof: By definition of Cayley graphs G d,p,q , we see that X = Cay( D p,q , D p,q ), where D p,q denotes the subgroup of P GL 2 (F q ) generated by D p,q . On the other hand, since D(d) generates Λ D , D ′ (d) generates Λ D /Λ D (q). The embedding (11) shows that Λ D /Λ D (q) is isomorphic to a subgroup of P GL 2 (F q ), which is precisely D p,q . This induces the graph isomorphism
Cay(Λ D /Λ D (q) , D ′ (d)) ≃ Cay( D p,q , D p,q )
concluding the proof.
In a Cayley graph on a group, the closed paths of length ℓ (starting and ending) at a vertex x and the ones (starting and ending) at a vertex y are in one-one correspondence. In particular a closed path of minimal length in the graph is found at each vertex, including the vertex 1. Thanks to Lemma 2.13, a closed path starting at the identity of Λ D /Λ D (q) corresponds to a product α = α 1 × · · · × α t ∈ Λ D , with α i ∈ D(d), such that α ∈ Λ D (q). Thus: girth(X) := inf{t ∈ N ⋆ : α 1 × · · · × α t ∈ Λ D (q), α i ∈ D(d)}.
The computations that follow are classical. They already appeared in [11]. Note that x = x 0 + x 1 i + x 2 j + x 3 k ∈ Λ D (q) implies that q|x i for i = 1, 2, 3. If we write x i = qy i , appears that N (x) = x 2 0 + q 2 (y 2 1 + y 2 2 + y 2 3 ) = p t . At least one y i = 0 among the values of i = 1, 2, 3, else x ∈ Λ D . Hence, t ≥ 2 log p q = 2 3 log q q 3 . In the case where p q = −1, the graphs G d,p,q are bipartite by Lemma 2.12 and the girth, as is the length of any cycle path, is an even number. Hereafter, the girth is equal to 2t. A basic refinement is possible in this case: as before, we get p 2t = x 2 0 + q 2 (y 2 1 + y 2 2 + y 2 3 ), with at least one y i = 0 among y 1 , y 2 , y 3 . Hence, p 2t ≡ x 2 0 mod q 2 . This is equivalent to p t ≡ ±x 0 mod q 2 , the group (Z/q 2 Z) ⋆ being cyclic. Therefore, p t = ±x 0 + mq 2 for a positive integer m. A simple calculation yields 2p t − mq 2 > 0, from which t ≥ 2 log p q − log p 2 follows. The girth in this case verifies girth(X) ≥ 4 3 log p q 3 − 2 log p 2. Recall that X is the connected component of G d,p,q containing 1. Its cardinality verifies |X| ≤ |P GL 2 (F q )| = q 3 − q, and even |X| ≤ |P SL 2 (F q )| = 1 2 (q 3 − q) when p q = 1. The definition 1.2 of κ along with the above show that 2 3 log p |X| = 2 3κ log d |X| ≤ 2 3κ log d q 3 ≤ girth(X), when p q = 1. And similarly, 4 3κ log d |X| − log p 4 ≤ girth(X) when p q = −1. The graph X has logarithmic girth. A trick that first appeared in [5, 3.3.4 Theorem] proves that it implies connectedness. We recall this theorem resulting from the properties of subgroups of SL 2 (F q ) due to Dickson; a group is said to be metabelian if it admits a normal subgroup N such that both N and H/N are abelian. Hence, to prove that H = P SL 2 (F q ), it suffices to prove that |H| > 60 and that H is not metabelian.
Proposition 2.15
Since d ≥ 10 and q > max{d 8κ , (120d) κ } = max{p 8 , 120 κ p}, one has that the graph G d,p,q is connected.
Proof: It amounts to show that X = G d,p,q . Thanks to Lemma 2.13, it suffices to show that the embedding (11) is onto, that is:
Λ D /Λ D (q) ≃ P SL 2 (F q ) if p q = 1 P GL 2 (F q ) if p q = −1
This is equivalent to show that µ q (Λ D ) = P SL 2 (F q ) or P GL 2 (F q ). Since P SL 2 (F q ) is an index 2 normal subgroup of P GL 2 (F q ) and that µ q (Λ D ) ⊂ P SL 2 (F q ) if p q = −1, it suffices to show that µ q (Λ D ) ∩ P SL 2 (F q ) = P SL 2 (F q ).
Let L := µ q (Λ D )) ∩ P SL 2 (F q ). First, we have |L| > 60. Indeed, by Equation (1) and the bound on the girth of X obtained above, 2 log p q ≤ girth(X) < 2 log d |X| + 2, from which follows log p q − 1 < log d |X|, then |X| > d log p q−1 = p 1 κ (log p q−1) and finally |X| > ( q p ) 1 κ . Next, holds |X| ≤ 2|L|. The equality may occur if G d,p,q is connected, i.e. X = G d,p,q , and when p q = −1. Follows |L| > 1 2 ( q p ) 1 κ . Since, q ≥ 120 κ p, this implies |L| > 60. The second step is to show that L is not metabelian, that is there exists four elements ℓ 1 , ℓ 2 , ℓ 3 , ℓ 4 in L such that: Suppose α 1 , α 2 , α 3 , α 4 verifies the latter. Let ℓ i := µ q (α i ) ∈ D p,q ⊂ P GL 2 (F q ). Then by construction of Cayley graphs, the commutator [[ℓ 1 , ℓ 2 ], [ℓ 3 , ℓ 4 ]] (performed in P GL 2 (F q )) yields a backtrackless path of length 16 in Cay( D p,q , D p,q ) ≃ X. Beforehand, we have proved that girth(X) ≥ 2 log p q which is strictly greater than 16 considering that q > p 8 . Hence, we have [[ℓ 1 , ℓ 2 ], [ℓ 3 , ℓ 4 ]] = 1 concluding the proof of (12), under the existence of the α i s in D(d).
[[ℓ 1 , ℓ 2 ], [ℓ 3 , ℓ 4 ]] = 1.(12)
It is actually always possible to find such α i s as soon as |D(d)| > 6, as perfectly explained in the proof of [5] p. 120, paragraphs (a) and (b). This is the case since d ≥ 10 by assumption.
Since X = G d,p,q , it follows that girth(G p,d,q ) ≥ 2 log p q = 2 3κ log d q 3 > 2 3κ log d |G p,d,q | if p q = 1, and girth(G d,p,q ) ≥ 4 log p q − log p 4 > 4 3κ log q |G d,p,q | − log q 4 if p q = −1, achieving the proof of Main Inequality (4).
As for the non-bipartite graphs d + 1-regular graphs H n mentioned in Theorem 1.1, they correspond to the families Y d of Definition 1.3. It has not be proved yet that they are not bipartite. Going back to the second point above Main Inequality (4), we must show that G d,p,q is non-bipartite when p q = 1. It was not possible to prove it at the time of the proof of Lemma 2.12, because of the lack of knowledge of the connectedness. Granted by Proposition 2.15, this concludes the proof of Theorem 1.1.
Concluding remarks
On the previous work. By a simple modification made on the classical construction of Ramanujan graphs of [11], the lower bounds on the girth of regular graphs of degree d ≥ 10 not a prime power were largely increased. Indeed, is obtained γ d ≥ 1, 06 and even γ d ≥ 1, 33 for larger values of d. This improves upon the 30 years old γ d ≥ 0, 48 proved in [9], for d = 2 k + 1. For d = 2 k +1, this improves upon the γ d ≥ 2 3 of [14]. It even outperforms what the probabilistic method [7] is able to give, namely γ d ≥ 1.
The construction of Imrich [9] is inspired by the previous work of Margulis [12]. The families that are built therein are derived from a "mother" graph, seen as a Cayley graph on a suitable free subgroup of SL 2 (Z). This prevents to use quaternions as done here and in [11,14,13], because the Hamilton quaternion algebra H(Q) is not split (no isomorphism with the 2-by-2 matrices). Thanks to quaternions, it is comparatively possible to do better. The lower bound obtained on the girth of the non-bipartite Cayley graphs on P SL 2 (F q ) H n in Theorem 1.1, is Further improvement 2. In the recent work [4] the record on the lower bound for γ d was beaten, from γ d ≥ 4 3 to γ d ≥ 12 7 for d = p 3 + 1, p an odd prime. This new construction is based on octonions and follows the main steps of the construction made in [11]. Since the improvement proposed in the present paper is based on a simple modification of [11], it is reasonable to hope that a similar modification of [4] would provide a comparable improvement on the lower bounds for γ d , d = p 3 + 1, than the one given in the present paper.
There are 2 obstacles to do so:
1. Similarly to quaternions, there is a special subset of prime octonions P(p) that is used for unique factorization. For an odd d, we can also isolate a subset of octonions D(d) ⊂ P(p) of size d+1, stable by conjugation, and it would allow to define a d+1-regular infinite tree. But this does not define a suitable "free algebraic structure" (not exactly a free group, because of lack of associativity of octonions) that would allow to take finite quotients on which the same analysis done here would work. Very roughly, this is because if π 1 , π 2 , π 3 , π 4 would be in D(d), the octonion (π 1 π 2 )(π 3 π 4 ) admits a unique factorization ((µ 1 µ 2 )µ 3 )µ 4 with µ i ∈ P(p), but nothing says if the µ i 's lie also in D(d).
2. To prove the connectedness, some knowledge on the proper subgroups of P SL 2 (F p ) was crucial. If we consider octonions, there is no such similar result available yet.
If these 2 obstacles came to be overcome or circumvented, let us mention roughly what one could expect:
Conjecture: For any integer d > C, C a fixed constant, there is an explicit family of d + 1-regular graphs X = {X n } n based on octonions such that:
girth(X n ) ≥ c(d) log d |X n |,
with c(d) ≥ 1, 7 and c(d) → 12 7 . This would prove that γ d ≥ 1, 7 for any d > C.
d−1 |G| + 1 if girth(G) is odd, 2 log d−1 |G| + 2 − 2 log d−1 2 if girth(G) is even.(1)This implies that for d ≥ 5,girth(G) ≤ (2 + 2 log d−1 |G| ) log d−1 |G| = (2 + o(1)) log d−1 |G|
γ({G n }) = lim inf n→∞ girth(G n ) log d−1 |G n | ,and γ d := sup {Gn}n family of d-regular graphs γ({G n }).
Definition 1. 3
3Given an integer u > 5, let p(u) := min{p ≥ u : p prime}, and p 3 (u) := min{p ≥ u : p prime ≡ 3 mod 8}.
15 ,
15u) 2 )) for u ≥ 3275. And proves that c(d) = 4 3κ ≥ 1, 33 for d ≥ 3275. For smaller values of d, I used a computer and found the following. The smallest integer d 1 for which d ≥ d 1 ⇒ 4 3κ ≥ 1, 33 with κ = log d p(d) is 1335, and then p(1335) = 1361. The smallest integer d 2 for which d ≥ d 2 ⇒ 4 3κ ≥ 1, 3 with κ = log d p(d) is 35, and then p(35) = 37. Between 15 and 31, it is easy to check that 4 3κ ≥ 1, 27. There is no integer smaller than 15 and greater than 10 which is not a prime power. This achieves the proof of the bound on c(d) in Theorem 1.1, when d is odd. As for p 3 (u), I used results of [15]. This requires to introduce the classical arithmetic function θ(x; k, ℓ) := p≡ℓ mod k p≤x ln(p), where p denotes a prime number. Indeed, there is a prime number equal to 3 modulo 8 in the interval [a; b] if and only if θ(b; 8, 3)− θ(a; 8, 3) > 0. The estimate of [Theorem for x ≥ 10 10 .
Remark that in general, there are several other possible ways of choosing D(d) inside P(p). Definition 2.6 An irreducible product of length ℓ over D(d) is the product of ℓ elements α 1 , . . . , α ℓ in D(d) where two consecutive elements:
Given an integer d, let a prime p ≥ d equal to 3 modulo 8 if d is even, and which is ordinary if d is odd.Let the real number κ ≥ 1 equal to log p d, so that p = d κ , and let Q d (p) := max{p 8 , 120 κ p}.1. A few more
notations are necessary:
Definition 1.2
For values d ≤ 10 10 , a laptop computer may not be powerful enough to check what the maximal values of log d p 3 (d) are. Again from [15, Theorem 2], in this case:modulo 8 in each interval [a; a
1−2ǫ ].
For d ≥ 10 10 , this clearly proves that 4
3κ ≥ 1, 33, since then κ = log d p 3 (d) ≤ log d
d
1−2ǫ .
This shows that in the interval [a; a(1 + 8·1,82 √ a−8·1,82 )] there is a prime equal to 3 modulo 8. Hence, κ = log d p 3 (d) ≤ 1 + log d (1 + 8·1,82 The other values of c(d) of Theorem 1.1 in the case d even, for d ≤ 228050 are easily obtained with the help of a computer. This concludes the proof of Theorem 1.1 on top of the main inequality (4).√
d−8·1,82
), showing that 4
3κ ≥ 1, 33 if d ≥ 228050.
It is easy to see that H is metabelian if and only if for any four elements h 1 , h 2 , h 3 , h 4 ∈ H one has such that |H| > 60. Then H is metabelian.[[h 1 , h 2 ], [h 3 , h 4 ]] = 1,
(where [a, b] = aba −1 b −1 ).
Theorem 2.14 ([5], 3.3.4 Theorem) Let q be a prime. Let H be a proper subgroups of
P SL 2 (F q ),
Let 4 elements α 1 , α 2 , α 3 , α 4 in D(d). The commutator [[α 1 , α 2 ], [α 3 , α 4 ]] performed in the group (Λ D , ×), yields an irreducible product of length smaller than 16. And it is equal to 16 if and only if [[α 1 , α 2 ], [α 3 , α 4 ]] performed this time in H(Z) is primitive (that is no reduction occurred).
the model of random Cayley graphs is described p. 2 of[8] 2 that is if x ∈ Dp,q, then x −1 ∈ Dp,q as well
For the full story about factorization of quaternions, see[3, Ch. 5]
AcknowledgmentI am indebted to J.-P. Tillich who initiated me to the subject of Ramanujan graphs.
As already mentioned, this is better than for the Cayley graphs on SL 2 (F q ) in [9], where the lower bound on the girth is worked out directly on matrices of SL 2 (Z) (see Proposition 4 of [2] for more details). Log D |h N | For D Large, Enough, and not on integral quaternions as done here2 · log d |H n | for d large enough. As already mentioned, this is better than for the Cayley graphs on SL 2 (F q ) in [9], where the lower bound on the girth is worked out directly on matrices of SL 2 (Z) (see Proposition 4 of [2] for more details) and not on integral quaternions as done here.
It should be mentioned that the families of non-bipartite d + 1-regular graphs Y d,p defined in (6) are expander families, at least when d is odd. This is due to their large girth property, for which the theorem of Bourgain & Gamburd [2, Theorem 3] holds. In particular, the non-bipartite graphs G d,p,q do not have a small chromatic number. Expander graphs. but have a small diameter in the order of O(log |G d,p,q |Expander graphs. It should be mentioned that the families of non-bipartite d + 1-regular graphs Y d,p defined in (6) are expander families, at least when d is odd. This is due to their large girth property, for which the theorem of Bourgain & Gamburd [2, Theorem 3] holds. In particular, the non-bipartite graphs G d,p,q do not have a small chromatic number, but have a small diameter in the order of O(log |G d,p,q |).
a set of special prime quaternions of cardinal p k + 1 used to prove unique factorization. This allows to take a subset D(d) of cardinal d + 1 as done in the present paper and to define similarly Cayley graphs on P GL 2 (F p k ) on d + 1 elements. The connectedness of these graphs could be proved also by using properties of subgroups of P SL 2 (F p k ). Ramanujan graphs by Lubotzky-Philips-Sarnak [11] and Margulis [13] a unique factorization property similar to Theorem 2.2), and to find a suitable reduction map similar to the map τ q in (10), which yields a split quaternion algebra (isomorphic to 2-by-2 matrices). indeed Dickson's result hold for subgroups of P SL 2 over any finite field, not only for prime finite fieldsFurther improvement 1. In 1994, Morgenstern in [14] has extended the construction of families of p + 1-regular Ramanujan graphs by Lubotzky-Philips-Sarnak [11] and Margulis [13] a unique factorization property similar to Theorem 2.2), and to find a suitable reduction map similar to the map τ q in (10), which yields a split quaternion algebra (isomorphic to 2- by-2 matrices). Similarly to the set of special prime quaternions P(p) of (8)-(7), there is also in [14, Equation (9)] a set of special prime quaternions of cardinal p k + 1 used to prove unique factorization. This allows to take a subset D(d) of cardinal d + 1 as done in the present paper and to define similarly Cayley graphs on P GL 2 (F p k ) on d + 1 elements. The connectedness of these graphs could be proved also by using properties of subgroups of P SL 2 (F p k ) (indeed Dickson's result hold for subgroups of P SL 2 over any finite field, not only for prime finite fields).
. Moreover, for primes p = 2 , the girth of the graphs of Morgenstern is comparable to the girth of the graphs of [11, 13. with a similar constant 4 3 (see Theorem 4.13, point b.3 in [14])Moreover, for primes p = 2 , the girth of the graphs of Morgenstern is comparable to the girth of the graphs of [11, 13], with a similar constant 4 3 (see Theorem 4.13, point b.3 in [14]).
the constant κ would be equal to log d (p k ) smaller than log d (p ′ ) as it is here, yielding a better lower bound on the girth. Despite this appealing fact, we found out that the use of Morgenstern's construction may not be worth, considering the tradeoff between simplicity and sharpness of the bounds, as explained below: • for an even number d, to build a d + 1-regular tree was required some "idempotents" in P(p), as explained in Remark 2.8. They were proved to exist only if p ≡ 3 mod 8. There is no such idempotent in the similar special set of prime quaternions of Equality. graphs of Morgenstern would yield graphs displaying a better lower bound on the girth than the one shown in Theorem 1.1. Indeed, given an integer d the next prime power p k is always smaller than the next prime p ′ : p k ≤ p ′ . Looking back to the Main Inequality. But in this case, roughly because P SL 2 (F 2 k ) = P GL 2 (F 2 k ), the Cayley graphs Γ g obtained are non-bipartite and only of girth ≥ 2 3 log q |Γ g |. see Theorem 5.13). This does not compete with the girth of the graphs described in the present paper, even in the non-bipartite caseThis indicates that the same modification brought in here to the construction of [11], but applied to the graphs of Morgenstern would yield graphs displaying a better lower bound on the girth than the one shown in Theorem 1.1. Indeed, given an integer d the next prime power p k is always smaller than the next prime p ′ : p k ≤ p ′ . Looking back to the Main Inequality (4), the constant κ would be equal to log d (p k ) smaller than log d (p ′ ) as it is here, yielding a better lower bound on the girth. Despite this appealing fact, we found out that the use of Morgenstern's construction may not be worth, considering the tradeoff between simplicity and sharpness of the bounds, as explained below: • for an even number d, to build a d + 1-regular tree was required some "idempotents" in P(p), as explained in Remark 2.8. They were proved to exist only if p ≡ 3 mod 8. There is no such idempotent in the similar special set of prime quaternions of Equality (9) of [14] (see Definitions 4.3 and 4.6 therein). Hence, to build a d + 1-regular tree we are led to consider the prime p = 2, and to choose d + 1 elements in the set defined in Equality (18) and Definition 5.3 of [14] (indeed, by Corollary 5.7 they yield such idempotents). But in this case, roughly because P SL 2 (F 2 k ) = P GL 2 (F 2 k ), the Cayley graphs Γ g obtained are non-bipartite and only of girth ≥ 2 3 log q |Γ g | (see Theorem 5.13). This does not compete with the girth of the graphs described in the present paper, even in the non-bipartite case.
• for an odd number d, the use of Morgenstern graphs could make sense, however the values of c(d) for d odd shown in Theorem 1.1 are not too bad. becoming close to the upper limit• for an odd number d, the use of Morgenstern graphs could make sense, however the values of c(d) for d odd shown in Theorem 1.1 are not too bad, becoming close to the upper limit
• the use of the construction of Morgenstern would induce a jump in technicality. without a significant strengthening of the results, as shown by the two previous points• the use of the construction of Morgenstern would induce a jump in technicality, without a significant strengthening of the results, as shown by the two previous points.
Extremal graph theory. B Bollobás, London Mathematical Society Monographs. Academic Press Inc11LondonHarcourt Brace Jovanovich PublishersB. Bollobás. Extremal graph theory, volume 11 of London Mathematical Society Mono- graphs. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1978.
Uniform expansion bounds for Cayley graphs of SL 2 (F p ). J Bourgain, A Gamburd, J. Bourgain and A. Gamburd. Uniform expansion bounds for Cayley graphs of SL 2 (F p ).
. Ann. of Maths. 1672Ann. of Maths, 167(2):625-642, 2008.
On quaternions and octonions. J Conway, D Smith, A.K. PetersJ. Conway and D. Smith. On quaternions and octonions. A.K. Peters, 2003.
Ramanujan graphs of very large girth based on octonions. X Dahan, J.-P Tillich, arXiv:1011.2642PreprintX. Dahan and J.-P. Tillich. Ramanujan graphs of very large girth based on octonions. Preprint, arXiv:1011.2642, November 2010.
Elementary number theory, group theory, and Ramanujan graphs. G Davidoff, P Sarnak, A Valette, London Math. Soc. Student Texts. Cambridge U. Press55G. Davidoff, P. Sarnak, and A. Valette. Elementary number theory, group theory, and Ramanujan graphs, volume 55 of London Math. Soc. Student Texts. Cambridge U. Press, 2003.
Autour de la fonction qui compte le nombre de nombre premiers. P Dusart, Université de LimogesPhD thesisP. Dusart. Autour de la fonction qui compte le nombre de nombre premiers. PhD thesis, Université de Limoges, 1998.
Reguläre Graphen gegebener Tailenweite mit minimaler Knollenzahh. P Erdös, H Sachs, Wiss. Z. Univ. Halle-Willenberg Math. Nat. 12P. Erdös and H. Sachs. Reguläre Graphen gegebener Tailenweite mit minimaler Knollen- zahh. Wiss. Z. Univ. Halle-Willenberg Math. Nat., 12:251-258, 1963.
On the girth of random Cayley graphs. A Gamburd, S Hoory, M Shahshahani, A Shalev, B Virg, Random Structures and Algorithms. 351A. Gamburd, S. Hoory, M. Shahshahani, A. Shalev, and B. Virg. On the girth of random Cayley graphs. Random Structures and Algorithms, 35(1):100 -117, 2009.
Explicit construction of regular graphs without small cycles. W Imrich, Combinatorica. 41W. Imrich. Explicit construction of regular graphs without small cycles. Combinatorica, 4(1):53-59, 1984.
Explicit construction of graphs with an arbitrary large girth and of large size. F Lazebnik, V A Ustimenko, ARIDAM VI and VII. New Brunswick, NJ60F. Lazebnik and V. A. Ustimenko. Explicit construction of graphs with an arbitrary large girth and of large size. Discrete Appl. Math., 60(1-3):275-284, 1995. ARIDAM VI and VII (New Brunswick, NJ, 1991/1992).
Ramanujan graphs. A Lubotzky, R Phillips, P Sarnak, Combinatorica. 83A. Lubotzky, R. Phillips, and P. Sarnak. Ramanujan graphs. Combinatorica, 8(3):261-277, 1988.
Explicit constructions of graphs without short cycles and low density codes. G A Margulis, Combinatorica. 21G. A. Margulis. Explicit constructions of graphs without short cycles and low density codes. Combinatorica, 2(1):71-78, 1982.
Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators. G A Margulis, Problemy Peredachi Informatsii. 241G. A. Margulis. Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators. Problemy Peredachi In- formatsii, 24(1):51-60, 1988.
Existence and explicit constructions of q + 1-regular Ramanujan graphs for every prime power q. M Morgenstern, J. Combin. Theory Ser. B. 621M. Morgenstern. Existence and explicit constructions of q + 1-regular Ramanujan graphs for every prime power q. J. Combin. Theory Ser. B, 62(1):44-62, 1994.
O Ramaré, R Rumely, Primes in arithmetic progressions. Mathematics of Computation. 65O. Ramaré and R. Rumely. Primes in arithmetic progressions. Mathematics of Computa- tion, 65(213):397-425, 1996.
| []
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"Calibration of the MaGIXS experiment I: Calibration of the X-ray source at the X-ray and Cryogenic Facility (XRCF)"
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"Ken Kobayashi \nNASA Marshall Space Flight Center\n35812HuntsvilleAL\n",
"Genevieve D Vigil \nNASA Postdoctoral Program\nNASA Marshall Space Flight Center\n35812HuntsvilleAL\n",
"Harlan Haight \nNASA Marshall Space Flight Center\n35812HuntsvilleAL\n",
"Steven Johnson \nNASA Marshall Space Flight Center\n35812HuntsvilleAL\n",
"Christian Bethge \nUniversities Space Research Association\nNASA Marshall Space Flight Center\n35812HuntsvilleAL\n",
"Laurel A Rachmeler \nNASA Marshall Space Flight Center\n35812HuntsvilleAL\n\nNational Centers for Environmental Information (NOAA)\n80305BoulderCO\n",
"Sabrina Savage \nNASA Marshall Space Flight Center\n35812HuntsvilleAL\n",
"Brent Beabout \nNASA Marshall Space Flight Center\n35812HuntsvilleAL\n",
"Dyana Beabout \nNASA Marshall Space Flight Center\n35812HuntsvilleAL\n",
"William Hogue \nNASA Marshall Space Flight Center\n35812HuntsvilleAL\n",
"Anthony Guillory \nNASA Marshall Space Flight Center\n35812HuntsvilleAL\n",
"Richard Siler \nNASA Marshall Space Flight Center\n35812HuntsvilleAL\n",
"Ernest Wright \nNASA Marshall Space Flight Center\n35812HuntsvilleAL\n",
"Jeffrey Kegley \nNASA Marshall Space Flight Center\n35812HuntsvilleAL\n"
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| The Marshall Grazing Incidence Spectrometer (MaGIXS) is a sounding rocket experiment that will observe the soft X-ray spectrum of the Sun from 24 -6.0Å (0.5 -2.0 keV) and is scheduled for launch in 2021. Component and instrument level calibrations for the MaGIXS instrument are carried out using the X-ray and Cryogenic Facility (XRCF) at NASA Marshall Space Flight Center. In this paper, we present the calibration of the incident X-ray flux from the electron impact source with different targets at the XRCF using a CCD camera; the photon flux at the CCD was low enough to enable its use as a "photon counter" i.e. the ability to identify individual photon hits and calculate their energy. The goal of this paper is two-fold: 1) to confirm that the flux measured by the XRCF beam normalization detectors is consistent with the values reported in the literature and therefore reliable for MaGIXS calibration and 2) to develop a method of counting photons in CCD images that best captures their number and energy. | 10.3847/1538-4357/abc268 | [
"https://arxiv.org/pdf/2010.09823v1.pdf"
]
| 224,803,124 | 2010.09823 | 36319f6b57efeba5c3737ab1fb3e4a46578623c5 |
Calibration of the MaGIXS experiment I: Calibration of the X-ray source at the X-ray and Cryogenic Facility (XRCF)
P S Athiray
NASA Postdoctoral Program
NASA Marshall Space Flight Center
35812HuntsvilleAL
Amy R Winebarger
NASA Marshall Space Flight Center
35812HuntsvilleAL
Patrick Champey
NASA Marshall Space Flight Center
35812HuntsvilleAL
Ken Kobayashi
NASA Marshall Space Flight Center
35812HuntsvilleAL
Genevieve D Vigil
NASA Postdoctoral Program
NASA Marshall Space Flight Center
35812HuntsvilleAL
Harlan Haight
NASA Marshall Space Flight Center
35812HuntsvilleAL
Steven Johnson
NASA Marshall Space Flight Center
35812HuntsvilleAL
Christian Bethge
Universities Space Research Association
NASA Marshall Space Flight Center
35812HuntsvilleAL
Laurel A Rachmeler
NASA Marshall Space Flight Center
35812HuntsvilleAL
National Centers for Environmental Information (NOAA)
80305BoulderCO
Sabrina Savage
NASA Marshall Space Flight Center
35812HuntsvilleAL
Brent Beabout
NASA Marshall Space Flight Center
35812HuntsvilleAL
Dyana Beabout
NASA Marshall Space Flight Center
35812HuntsvilleAL
William Hogue
NASA Marshall Space Flight Center
35812HuntsvilleAL
Anthony Guillory
NASA Marshall Space Flight Center
35812HuntsvilleAL
Richard Siler
NASA Marshall Space Flight Center
35812HuntsvilleAL
Ernest Wright
NASA Marshall Space Flight Center
35812HuntsvilleAL
Jeffrey Kegley
NASA Marshall Space Flight Center
35812HuntsvilleAL
Calibration of the MaGIXS experiment I: Calibration of the X-ray source at the X-ray and Cryogenic Facility (XRCF)
Submitted to ApJDRAFT VERSION OCTOBER 21, 2020 Typeset using L A T E X default style in AASTeX63Sun:coronaX-raysmethods: data analysis
The Marshall Grazing Incidence Spectrometer (MaGIXS) is a sounding rocket experiment that will observe the soft X-ray spectrum of the Sun from 24 -6.0Å (0.5 -2.0 keV) and is scheduled for launch in 2021. Component and instrument level calibrations for the MaGIXS instrument are carried out using the X-ray and Cryogenic Facility (XRCF) at NASA Marshall Space Flight Center. In this paper, we present the calibration of the incident X-ray flux from the electron impact source with different targets at the XRCF using a CCD camera; the photon flux at the CCD was low enough to enable its use as a "photon counter" i.e. the ability to identify individual photon hits and calculate their energy. The goal of this paper is two-fold: 1) to confirm that the flux measured by the XRCF beam normalization detectors is consistent with the values reported in the literature and therefore reliable for MaGIXS calibration and 2) to develop a method of counting photons in CCD images that best captures their number and energy.
INTRODUCTION
For over four decades, X-ray, EUV, and UV spectral observations have been used to measure physical properties of the solar atmosphere. During this time, there has been substantial improvement in the spectral, spatial, and temporal resolution of the observations in the EUV and UV wavelength ranges. At wavelengths below 100Å, however, observations of the solar corona with simultaneous spatial and spectral resolution are limited, and not since the late 1970's have spatially resolved solar X-ray spectra been measured. Because the soft X-ray regime is dominated by emission lines formed at high temperatures, X-ray spectroscopic techniques yield insights to fundamental physical processes that are not accessible by any other means.
The Marshall Grazing Incidence X-ray Spectrometer (MaGIXS) will measure, for the first time, the solar spectrum from 6-24Å with a spatially resolved component along an > 8 slit (Kobayashi et al. 2010(Kobayashi et al. , 2018Champey et al. 2016). MaGIXS will provide a good diagnostic capability for high-temperature, low-emission measure plasma, which will help to determine the frequency of heating events in coronal structures . The instrument, comprised of a Wolter-I telescope mirror, a grazing incidence spectrograph, and a CCD detector, was aligned using a UV centroiding instrument designed for the Chandra X-ray Observatory (Glenn 1995) and a visible-light auto-collimating theodolite. Several small reference mirrors and alignment reticles were co-aligned to the optical axis of the grazing incidence mirrors during the mounting process. These references were used later to co-align the telescope mirror with the spectrometer assembly (Champey et al., in preparation). A series of X-ray tests were performed iteratively during the alignment process to confirm alignment, focus the telescope on the slit, and measure throughput. X-ray testing of the mounted telescope mirror was performed in the Stray Light Facility (SLF) (Champey et al. 2019), while the integrated instrument was tested in the X-ray and Cryogenic Facility (XRCF) at NASA Marshall Space Flight Center. The remaining set of end-to-end X-ray tests are designed to calibrate the spectrograph dispersion (wavelength calibration) and measure throughput of MaGIXS. These tests will use the XRCF's electron impact point source (EIPS) and the four targets listed in Table 1.
Though absolute radiometric calibration is not required for this sounding rocket instrument, the data gathered at the XRCF make radiometric calibration possible if the beam normalization detectors (BND), available at the XRCF, can be used to provide an accurate measure of the photon flux of the X-ray beam. Because these detectors have not been calibrated since 1998 (Weisskopf & O'Dell 1997), we performed an experiment to verify the BND performance, as well as develop and verify our method to detect photons and calculate their energies in a CCD detector identical to the MaGIXS X-ray detector.
We present our experiments, methods of analysis, and results. In Section 2 we review the experiment details and data collection. In Section 3 we discuss the Monte Carlo simulation used to develop the event selection algorithm. In Section 4, we present the results of applying this method to the test data. We find good agreement (to within 20%) between the photon flux measured by the BND and the CCD. In a follow-on paper, we will apply these techniques to the MaGIXS instrument, tested in different configurations, determine the component level radiometry, and predict the solar signal for typical active regions.
EXPERIMENTS
Setup
A schematic representation of the experimental setup is shown in Figure 1. The test setup includes an X-ray source (EIPS), the BND, and a CCD with back-end readout electronics and a liquid nitrogen (LN2) cooling system. The EIPS has selectable targets and a filter wheel populated with continuum reducing filters of nominally 2 × mean free path thickness at the desired spectral line. The X-ray beam from the source is transmitted through an evacuated, 518 m long guide tube to an evacuated 7.3 m diameter × 22.9 m long instrument chamber. The source is operated with an anode voltage that is 4 × the L-shell or 5 × the K-shell binding potential of the target at a permissible current. A list of the calibration targets, along with corresponding line energies useful for MaGIXS calibration, is given in Table 1. The targets employed in the EIPS are the same ones used in the calibration of Chandra X-ray mirrors, except for target Mg, which has been replaced with a new Mg target. The BND is mounted on a translation stage within the guide tube at a distance of 38 meters from the source, and outside of the EIPS-CCD beam path. The BND is preceded by a positionable masking plate with apertures of calibrated diameters (see Table 1), used to avoid saturation of the detector. The CCD, along with readout avionics and cooling hardware, are mounted on a translation stage and are placed within the instrument chamber, 538 m from the source.
Data collection and pre-processing
The BND is a flow proportional counter (FPC) using flowing P10 gas (90% Argon, 10% Methane) at a pressure of 400 Torr. It has a 15 × 5 Aluminum-coated polyimide window supported by a gold-coated tungsten wire grid. The FPC anode voltage controls its avalanche gain, and can be set to a voltage appropriate for the energy of the line being monitored. A signal processing chain follows the FPC, consisting of a preamplifier, shaping amplifier, and multi-channel buffer analog to digital converter. The data collected are transferred to a computer for archival and analysis. For a complete description of the FPC, see Wargelin et al. (1997).
The CCD employed for this test is a back-illuminated, ultra-thinned, astro-processed sensor with an active area that contains 2K × 1K pixels with 15 µm pixel size, similar to the MaGIXS flight camera. The CCD is mounted in a copper carrier that is connected via a copper strap to a cold block. The cold block is actively cooled to roughly -100 • C using liquid nitrogen, which results in a CCD carrier average temperature of roughly -70 • C. The carrier temperature is controlled such that it is maintained constant within ± 5 • C. The low-noise camera has been developed by NASA Marshall Spaceflight Center as one of a series of cameras developed for suborbital missions (Rachmeler et al. 2019;Champey et al. 2014Champey et al. , 2015. The CCD is operated in frame transfer mode, with 1k × 2k pixels exposed and two 500 × 2k readout regions that are mechanically masked. There are four read-out taps on the detector; it requires roughly 1.2 s to read out 1k×2k pixels. All exposures are 2 s long, meaning the CCD is operating with ∼100% duty cycle. More than 200 frames are collected for every target in Table 1 for adequate statistics. Additionally, during each data run, dark images are acquired by closing a gate valve between the source and CCD detector. The raw CCD images are processed using standard reduction procedures to remove bias level, dark current, and fixed pattern noise in the images. The gain of the camera was determined prior to XRCF testing using the Mn K-α and K-β lines from a sealed radioactive source Fe 55 and found to be ∼ 2.58 electrons per data number (DN). The variability of gain between the quadrants are less than 0.03 electron DN −1 , which means the systematic uncertainty introduced from gain is less than ∼ 2%. The images are converted from Data Number (DN) to electrons using the known gain of the camera. A histogram of the residual distribution indicates the RMS read noise is approximately 9 e − as shown in Figure 2.
If the photon flux is low and the energy of the photons are much larger than the noise in the detector, we can detect individual photons and measure their associated energy. These conditions are satisfied by controlling the source strength with the adjustment of current/voltage settings. Appropriate filters are used at the source end to reduce the low energy bremsstrahlung continuum and lines arising from any anode contaminants. The optimal settings used for our experiment are listed in Table 1 (Column 3, 4 and 5). We mention that during the experiment, no filter was placed when we operated target Zn, which was unintentional.
EVENT RECOGNITION
In order to successfully perform photon counting using CCD, events registered from photon hits are to be selected properly and eliminate all other sources of noise that are present in CCD. One of the challenges with the CCD is the relatively small pixel size (15 µm). The charge clouds produced from the absorbed X-ray photons drift under the electric field and undergo lateral diffusion. As a result, charge induced by the X-ray events may be shared over many pixels, known as charge shared events. To calculate the energy of a photon, it is important that the charge collected in multiple pixels are measured accurately and added with appropriate selection criteria to precisely determine the incident photon flux. Event reconstruction demands a good understanding of detector noise and uniformity in the pixel response. Typically thresholds are applied to exclude noisy outliers and maximize event recovery from multipixel events. Our goal is to develop a method and demonstrate how well we can determine the incident photon flux from CCD images by maximizing the event recovery from multipixel events.
Simulation
To understand the effect of pixel size and detector noise, and to optimize event reconstruction from charge shared events, we performed Monte-Carlo simulations of photon interaction and charge propagation in the CCD, which is assumed to have properties similar to the MaGIXS detector. The model assumes a simple 1D electric field obtained by solving Poisson's equation. Some of the basic assumptions and calculations for the electric field are obtained from Athiray et al. (2015). We mention that our model is simplified with many assumptions and does not simulate charge loss mechanisms. We consider this 'toy' model as an approximation to simulate multipixel events by predicting the response to incident X-ray photons in a CCD. This tool provides a method with which to test and validate the event selection method, which can be further fine-tuned for 'real' experiment data. The physical device parameters of the CCD used in the simulation are listed in Table 2. The variance in the production of number of electron-hole per X-ray photon will always be less than the Poisson statistical variance, which is known as the Fano noise. The Fano noise determines the inherent line width of an X-ray detector. To incorporate this effect, we simulated the incident spectrum with a mean energy E i along with the Fano noise using:
E f = E i + R n (0) F ωE i (1)
where E f is the energy distribution with Fano noise added, R n (0) is the normally distributed random number with mean 0 and variance 1, F is the Fano factor = 0.115 and ω is the average energy required to produce an electron-hole pair in Si= 3.65 eV. A mono-energetic spectrum with a mean energy at 1.25 keV, incorporated with Fano noise, is simulated at normal incidence to the CCD. Figure 3 (left) shows the histogram of the incident photon spectrum. Each photon is simulated to hit a random pixel location on the CCD. The depth at which each photon interaction occur inside the CCD is determined from:
z 0 = − 1 µ(E) ln(R u )(2)
where µ(E) is the linear mass absorption coefficient of Si at photon energy E, R u is a random number with uniform distribution. Photons interacting at depths (z 0 ) beyond the substrate thickness are considered to be lost in the simulation. The absorbed photon produces a charge cloud, which is assumed to be spherical with a Gaussian distributed charge density. The 1-σ radius of the initial spherical charge cloud is calculated using (Townsley et al. 2002):
r i 0.0062 E 1.75 f (in µm)(3)
where E f is the photon energy (in keV). The charge cloud drifts under the influence of electric field and undergo random thermal motions resulting in the expansion of the charge cloud radius determined by (Hopkinson 1987). The typical final mean-square charge cloud radius for a photon in the MaGIXS energy range lies around ∼ 2 to 5 µm. The amount of charge collected in each pixel for a given event is computed by integrating the Gaussian distributed charge cloud by using Equation 9 from (Athiray et al. 2015). Further, we add random noise to the charge collected in each pixel, which is the residual of measured average CCD darks.
The synthetic X-ray image simulated with source photons and noise events is shown in Figure 3 (right). We use this as our test case to reconstruct the incident photon energy and determine the number of X-ray hits on the CCD. We emphasize that photons travelling beyond the depletion depth and photons that hit channel stops are considered to be lost and hence are not included in the event selection. A more detailed charge transport simulation with interactions at different layers on the CCD would help to understand the spectral response (e.g. Townsley et al. 2002;Athiray et al. 2015;Godet et al. 2009;Haberl et al. 2004), which is beyond the scope of our current investigation.
Event Characterization
For a typical charge cloud radius of ∼5 µm, photons interacting near to the pixel center register as a single event. Depending on the location of interaction away from the pixel center, events can be registered in more than one pixel. Based on the pixel size and typical charge cloud size we deduce a search grid ± 1 pixel around the center hit pixel would be suffice for event characterization. Reconstructing the total energy deposited from a charge shared event by summing the signal over many pixels is further complicated by the detector readout noise. Typically, most of the X-ray imaging instruments using CCDs implement event grading methods to classify pixel patterns. In this approach, a 'baseline' threshold is first applied to identify pixels above certain intensity as a valid X-ray event. We have adopted 3 × σ noise as our baseline threshold (∼ 0.12 keV) and marked pixels with intensity above this value to be an X-ray event; σ noise is the RMS readout noise. Starting with the brightest pixel, we search for additional neighbor hits or 'split' events registered within ± 1 pixel from the center bright pixel. This scanning width implies a search over a grid of 3 × 3 pixels. We then classify events based on the number of pixels that are above the baseline threshold within each grid as '1 pix','2 pix' '3 pix', '4 pix', etc. To recover the total energy deposited and ensure charge is not lost from the shared events, all the pixels with charge deposit greater than zero within each grid are summed up. A schematic representation of the event classification is shown in Figure 4. Pixels that satisfy baseline threshold are marked as green boxes, the search grid (3 × 3 pixels) is shown in peach boxes, blue boxes represent pixels within the search grid that have a charge deposit greater than zero and less than baseline threshold, and labels ('S', 'D', 'T' and 'Q') denote event classification in Figure 4. Thresholds are carefully chosen so that noise events do not influence the counting of 'real' X-ray photon events. Complications arising from charge transfer inefficiency are excluded in the model for simplicity. Figure 5 (left) shows the histogram of energy measured in individual pixels that form single or multipixel events. Though there are a few single pixel events detected (black line), a significant fraction of events undergo charge sharing and are close to detector noise levels. This measurement implies that proper energy reconstruction of the incident photon is critical to precisely determine the incident photon flux. It also conveys that the charge induced by 1.25 keV photons can be distributed over as many as 4-pixels. The energy reconstructed spectrum is shown in Figure 5 (right) along with the simulated incident spectrum overplotted. This . Representation of single and multipixel events in the CCD and event classification. The labeled Green boxes represent pixels with values greater than the baseline threshold (3-σnoise). Peach color boxes represent the search grid around the bright pixels. Blue color boxes represent pixels within the search grid that have a charge deposit greater than zero and less than baseline threshold, which are summed up with pixels that satisfy the criteria of baseline threshold to reconstruct the photon energy. Figure 5. The histogram of simulated events registered on the CCD satisfying the event selection criteria and classification. (Left) Histogram of the observed events classified based on the number of pixels above baseline threshold in each search grid, without recovering energy from the charge shared events. It is evident that significant fraction of multipixel events are close to detector noise levels. (Right) Histogram of events processed through energy recovery by summing up pixels with charge deposits greater than zero in each search grid. The energy reconstructed from the multipixel events using the event selection algorithm matches well with the incident energy. demonstrates the energy reconstructed from multipixel events using the event selection algorithm described above matches well with the incident photon energy.
Further, we also tried to understand the sensitivity of Si substrate thickness to result in single and multipixel events, within MaGIXS's wavelength range (6 -24Å). For this, we performed a simulation with different substrate thicknesses ranging from 10 µm to 30 µm, retaining all other parameters such as the pixel geometry etc fixed. The simulation assumes the substrate thicknesses to be a field-region, where there is a strong electric field to drift the charge cloud. The fraction of single pixel and multipixel events varying as a function of substrate thickness for photon energy E = 1.25 keV is shown in Figure 6. This clearly indicates the fraction of single pixel events increases with substrate thickness, which means more collection efficiency as seen from the detected event fraction. It also suggests the fraction of multipixel events is much less sensitive to substrate thickness. This result could be explained as the substrate thickness increases, the active volume of the detector i.e. region with Figure 6. Si substrate thickness Vs fraction of single and multipixel events. The fraction of single pixel events systematically increase with Si thickness. This implies increase in collection efficiency, which is shown as increase in the total detected event fraction. However, the fraction of multipixel events are less sensitive to substrate thickness. electric field increases, which improves the probability of photon detection from a relatively greater depth, thereby increasing the overall quantum efficiency of the detector. The strong electric field created while increasing the active volume of the detector yields a relatively small cloud radius at these photon energies, which improves the fraction of single-pixel events, but does not significantly affect the fraction of multipixel events arising from the weak electric field regions. We interpret the observed charge shared events as mainly due to photon interactions near the pixel boundaries, and therefore describe this as a 3D geometrical effect.
RESULTS FROM EXPERIMENTAL DATA
Applying Event Detection to CCD Data
We processed the experimental data through the same event selection logic used in the simulation and determined the incident photon flux for different targets listed in Table 1. The best comparison of simulations and real data is the distribution of multipixel events. A sample of distributions of 1 pix, 2 pix, 3 pix, 4 pix, and 5 pix events recovered from the event processing for different targets is shown in Figure 7. The results indicate a significant fraction of photon hits lead to multipixel events, which is consistent with our simulation result. However, we observe that multipixel events are more pronounced in real data as compared to simulations. Energy reconstruction from single pixel events matches well with the incident photon energy, which is marked with a vertical arrow in Figure 7. However, summing up of pixels from all of the charge shared events does not lead to proper reconstruction of the incident photon energy. The reconstructed energy from multipixel events show consistently less than the incident photon energy, which is not the case in our simulation. This discrepancy could be due to incorrect modeling of charge diffusion and/or lack of charge loss mechanism in our 'ideal' simulation. Also additional effects, such as pixel non-uniformity, charge transfer inefficiency, and distortion in electric field distributions, could account for loss in the total energy collected from multipixel events. It has already been shown that energy reconstruction of X-rays below 2 keV is highly complicated for detectors with large depletion and small pixel sizes (Miller et al. 2018), which is consistent with our observations.
Comparison of spectra with BND
Finally, we compare the spectra from the processed CCD data described above with the spectra obtained from the BND. Note that these two detectors are completely different in terms of noise characteristics, spectral response function, and quantum efficiency, which are energy dependent parameters. These distinctions are acceptable since the spectral comparisons are semiqualitative considering our objectives 1) to determine and compare the X-ray intensity at target's line energy and 2) to look for similarities in the spectral profile, including the line to continuum.
As the detectors are placed at different distances from the source, a scale factor is applied to the BND data to project the spectra at the appropriate CCD distance. The comparison of spectra from the CCD and BND (after taking into account respective integration times, and collection area) is shown in Figure 8. Spectra from targets Ni, Cu, and Mg are taken with respective filters, which preferentially transmit the target's characteristic X-rays and suppress the continuum. In contrast, spectra from the Zn target without a filter (Figure 8 lower left) clearly shows the line riding over a dominant bremsstrahlung continuum. It is evident the Figure 7. Histograms of X-ray events reconstructed for targets Mg-K, Ni-L, Cu-L, and Zn-L. The photon energy derived from single pixel events is clearly conspicuous at the respective line energies of the target. For multipixel events the reconstructed energy is systematically less than the line energy and exhibits a broad spectral response, consistent with (Miller et al. 2018). overall spectral profile for different targets from both detectors looks similar. However, there are discrepancies that are clearly visible between the spectra, especially at the high energy tail of the line and the continuum. We ascribe the discrepancy in the continuum between the detectors chiefly arising from the inherent spectral response of the detector.
It has been already shown by Auerhammer et al. (1998) that the spectral response of the BND exhibits a skewed peak and an energy dependent low energy shelf, as measured at several discrete monochromatic energies. Further studies indicate that the skewness of the peak can be best modeled by a Prescott function (Prescott 1963), which has a heritage of application for proportional counters (Budtz-Jørgensen et al. 1995). Other factors, such as a small fraction of pile-up events or energy-dependent spectral response function of the detectors, can introduce additional second order effects that cause discrepancies between the observed spectra from the two detectors. We modeled the line peak in the CCD spectra with a Gaussian function along with a Prescott function for the BND spectra to determine the intensity at target's line energy and then compare the X-ray flux as given in Table 3. We used the quantum efficiency of the BND (Weisskopf & O'Dell 1997) and the quantum efficiency of the CCD determined from the simulation to convert the line fluxes from counts to respective photon units. The quantum efficiency of the CCD obtained from the simulation closely agrees with the published quantum efficiency for the back-illuminated, astro-processed CCDs from e2V Technologies Ltd. (Moody et al. 2017), which are demonstrated to be reliable and consistent in the MaGIXS energy range. The error in the CCD flux is derived from the quadrature sum of statistical uncertainty and systematic uncertainty from the CCD gain variations. The error in BND flux is determined from the quadrature sum of statistical uncertainty and from Section 3, we demonstrated the confidence on the event selection method for CCD in the recovery of photon energy accurately and hence the incident flux. Therefore, the idea for comparing the BND flux against CCD is to determine the level of uncertainty in the BND flux. From Table 3, we observe the incident X-ray flux determined from the CCD agrees within 20% of the incident flux measured from the BND. NOTE-The BND fluxes listed here are determined after correcting for distance from the CCD. Figure 8. Comparison of spectra from different targets with the histograms of X-ray events reconstructed for the corresponding targets. The line peaks are modeled with a Gaussian function for the CCD spectra along with a Prescott function for the BND spectra to determine the intensities at respective target line energies, which are compared in Table 3.
SUMMARY
In this paper, we presented the experiments and data processing techniques implemented to verify the absolute number of photons entering the instrument from the X-ray source at the XRCF. This characterization of the source throughput is critical for its utilization in the alignment and calibration activities of the MaGIXS experiment, as measured by a CCD detector. Simultaneous measurement of the incident spectra was obtained using a BND that was calibrated in 1998, which is used to cross-calibrate the incident X-ray flux. Using Monte-Carlo simulations of a CCD detector with simple approximations, we first created synthetic multipixel events with realistic detector noise and optimized our event selection algorithm. We find that knowledge of pixel-based noise sources is critical for soft X-ray photons to achieve proper energy reconstruction and absolute photon counting. Applying the algorithm to the simulated data, we verified that proper energy reconstruction could be achieved and demonstrated photon counting. Furthermore, we studied the dependence of event distribution with Si substrate thickness by performing simulations with different Si substrate thicknesses modeled as field-region. Our findings indicate that the fraction of single pixel events increase with substrate thickness, while the fraction of multipixel events appears to be less sensitive to substrate thickness and mainly depends on the interactions near to pixel boundaries.
We then applied the event selection algorithm on the real experiment data from the CCD and classified multipixel events. The observed multipixel events are more pronounced in real data than our simulations. We find that the energy reconstructed from multipixel events systematically appear at lower energy than the incident photon energy. We then compared spectra from different targets obtained from both the BND and the CCD after taking into account their respective distances, integration time, and quantum efficiency. Though the overall spectral profile from both detectors showed similarities, discrepancies are noticeable in the spectral redistribution function between the BND and the CCD. This disparity warrants advanced spectral modeling including a detailed charge transport simulation, which are beyond the purview of our current investigation. With the confidence gained from the CCD event selection method for precise photon energy and flux estimation, we compared the intensity of different targets, at the respective line energies, observed by both detectors. The measured incident photon flux from both the BND and the CCD show agreement to within 20%. This result of validating or cross-calibrating the incident photon flux measured simultaneously by the BND will enable radiometric calibration for the MaGIXS instrument and for any future space instrument characterization.
Figure 1 .
1Schematic representation of experiment setup at the X-ray and Cryogenic Facility at NASA Marshall.
Figure 2 .
2Histogram of the residuals from the processed data (bias, dark current, and fixed pattern noise subtracted) demonstrates that the measured RMS read noise is ∼ 9 e − .
Figure 3 .
3(Left) The incident photon spectrum with a mean energy at 1.25 keV incorporated with Fano noise using the Fano factor F = 0.115 simulated to enter the CCD. (Right) Synthetic X-ray image showing simulated photon hits (Cyan) with mean energy of 1.25 keV combined with a noise spectrum shown inFigure 2.
Figure 4
4Figure 4. Representation of single and multipixel events in the CCD and event classification. The labeled Green boxes represent pixels with values greater than the baseline threshold (3-σnoise). Peach color boxes represent the search grid around the bright pixels. Blue color boxes represent pixels within the search grid that have a charge deposit greater than zero and less than baseline threshold, which are summed up with pixels that satisfy the criteria of baseline threshold to reconstruct the photon energy.
Table 1 .
1List of calibration targets and respective line energies.S.No Target Line energy Voltage Current
Filter
BND aperture diameter
keV
kV
mA
cm
1
Ni-L
0.85
3.40
5.8
Ni-L2
0.4
2
Cu-L
0.93
3.70
7.7
Cu-L2
0.4
3
Zn-L
1.01
4.00
3.0
No filter
0.1
4
Mg-K
1.25
6.50
1.5
Mg-K2
0.4
Table 2 .
2Physical parameters for the simulated photon interaction and charge propagation in the MaGIXS CCD.Parameters
Values
Mean photon energy
1.25 keV
Dopant concentration (Na) 4 × 10 12 cm −3
Temperature
203 K
CCD dimensions
2 K × 1K pixels
Pixel size
15 µm
Table 3 .
3Incident X-ray flux determined from the BND and the CCD for different calibration targets at respective target's line energy .Target Line energy Voltage Current
Filter
BND flux
CCD flux
Percentage
keV
kV
mA
Ph/s/cm 2 /mA Ph/s/cm 2 /mA difference
Ni-L
0.85
3.40
5.8
Ni-L2
4.6 ± 0.16
3.72 ± 0.04
-19.7
Cu-L
0.93
3.70
7.7
Cu-L2
3.4 ± 0.12
3.08 ± 0.04
-8.4
Zn-L
1.01
4.00
3.0
No filter 145.0 ± 5.1 124.78 ±1.26
-14.0
Mg-K
1.25
6.50
1.5
Mg-K2
14.4 ± 0.56
12.67 ± 0.15
-11.4
ACKNOWLEDGMENTS P. S. Athiray's research is supported by an appointment to the NASA Postdoctoral Program at the Marshall Space Flight Center, administrated by Universities Space Research Association under contract with NASA. The MaGIXS instrument team is supported by the NASA Low Cost Access to Space program. The authors would like to thank Dr. Jeffery Kolodziejczak at MSFC for helpful discussions and insights in preparing this manuscript. The authors gratefully acknowledge the many people at XRCF who have contributed to the testing. The authors appreciate helpful and insightful comments and suggestions from an anonymous referee.
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| []
|
[
"Optimization over Structured Subsets of Positive Semidefinite Matrices via Column Generation",
"Optimization over Structured Subsets of Positive Semidefinite Matrices via Column Generation"
]
| [
"Ali Amir \nIBM Research\nPrinceton, PrincetonORFE, ORFE\n",
"Ahmadi \nIBM Research\nPrinceton, PrincetonORFE, ORFE\n",
"Sanjeeb Dash [email protected] \nIBM Research\nPrinceton, PrincetonORFE, ORFE\n",
"Georgina Hall \nIBM Research\nPrinceton, PrincetonORFE, ORFE\n"
]
| [
"IBM Research\nPrinceton, PrincetonORFE, ORFE",
"IBM Research\nPrinceton, PrincetonORFE, ORFE",
"IBM Research\nPrinceton, PrincetonORFE, ORFE",
"IBM Research\nPrinceton, PrincetonORFE, ORFE"
]
| []
| We develop algorithms to construct inner approximations of the cone of positive semidefinite matrices via linear programming and second order cone programming. Starting with an initial linear algebraic approximation suggested recently by Ahmadi and Majumdar, we describe an iterative process through which our approximation is improved at every step. This is done using ideas from column generation in large-scale linear programming. We then apply these techniques to approximate the sum of squares cone in a nonconvex polynomial optimization setting, and the copositive cone for a discrete optimization problem. | 10.1016/j.disopt.2016.04.004 | [
"https://arxiv.org/pdf/1512.05402v2.pdf"
]
| 6,438,784 | 1512.05402 | 19c5c38c94fb5b8a029a8d9c309c75ebdbfef810 |
Optimization over Structured Subsets of Positive Semidefinite Matrices via Column Generation
March 14, 2016
Ali Amir
IBM Research
Princeton, PrincetonORFE, ORFE
Ahmadi
IBM Research
Princeton, PrincetonORFE, ORFE
Sanjeeb Dash [email protected]
IBM Research
Princeton, PrincetonORFE, ORFE
Georgina Hall
IBM Research
Princeton, PrincetonORFE, ORFE
Optimization over Structured Subsets of Positive Semidefinite Matrices via Column Generation
March 14, 2016
We develop algorithms to construct inner approximations of the cone of positive semidefinite matrices via linear programming and second order cone programming. Starting with an initial linear algebraic approximation suggested recently by Ahmadi and Majumdar, we describe an iterative process through which our approximation is improved at every step. This is done using ideas from column generation in large-scale linear programming. We then apply these techniques to approximate the sum of squares cone in a nonconvex polynomial optimization setting, and the copositive cone for a discrete optimization problem.
Introduction
Semidefinite programming is a powerful tool in optimization that is used in many different contexts, perhaps most notably to obtain strong bounds on discrete optimization problems or nonconvex polynomial programs. One difficulty in applying semidefinite programming is that state-of-the-art general-purpose solvers often cannot solve very large instances reliably and in a reasonable amount of time. As a result, at relatively large scales, one has to resort either to specialized solution techniques and algorithms that employ problem structure, or to easier optimization problems that lead to weaker bounds. We will focus on the latter approach in this paper.
At a high level, our goal is to not solve semidefinite programs (SDPs) to optimality, but rather replace them with cheaper conic relaxations-linear and second order cone relaxations to be precise-that return useful bounds quickly. Throughout the paper, we will aim to find lower bounds (for minimization problems); i.e., bounds that certify the distance of a candidate solution to optimality. Fast, good-quality lower bounds are especially important in the context of branch-and-bound schemes, where one needs to strike a delicate balance between the time spent on bounding and the time spent on branching, in order to keep the overall solution time low. Currently, in commercial integer programming solvers, almost all lower bounding approaches using branch-and-bound schemes exclusively produce linear inequalities. Even though semidefinite cuts are known to be stronger, they are often too expensive to be used even at the root node of a branch-and-bound tree. Because of this, many high-performance solvers, e.g., IBM ILOG CPLEX [16] and Gurobi [1], do not even provide an SDP solver and instead solely work with LP and SOCP relaxations. Our goal in this paper is to offer some tools that exploit the power of SDP-based cuts, while staying entirely in the realm of LP and SOCP. We apply these tools to classical problems in both nonconvex polynomial optimization and discrete optimization.
Techniques that provide lower bounds on minimization problems are precisely those that certify nonnegativity of objective functions on feasible sets. To see this, note that a scalar γ is a lower bound on the minimum value of a function f : R n → R on a set K ⊆ R n , if and only if f (x) − γ ≥ 0 for all x ∈ K. As most discrete optimization problems (including those in the complexity class NP) can be written as polynomial optimization problems, the problem of certifying nonnegativity of polynomial functions, either globally or on basic semialgebraic sets, is a fundamental one. A polynomial p(x) := p(x 1 , . . . , x n ) is said to be nonnegative, if p(x) ≥ 0 for all x ∈ R n . Unfortunately, even in this unconstrained setting, the problem of testing nonnegativity of a polynomial p is NP-hard even when its degree equals four. This is an immediate corollary of the fact that checking if a symmetric matrix M is copositive-i.e., if x T M x ≥ 0, ∀x ≥ 0-is NP-hard. 1 Indeed, M is copositive if and only if the homogeneous quartic polynomial p(x) = i,j M ij x 2 i x 2 j is nonnegative.
Despite this computational complexity barrier, there has been great success in using sum of squares (SOS) programming [35], [25], [33] to obtain certificates of nonnegativity of polynomials in practical settings. It is known from Artin's solution [7] to Hilbert's 17th problem that a polynomial p(x) is nonnegative if and only if
p(x) = t i=1 q 2 i (x) r i=1 g 2 i (x) ⇔ ( r i=1 g 2 i (x))p(x) = t i=1 q 2 i (x)(1)
for some polynomials q 1 , . . . , q t , g 1 , . . . , g r . When p is a quadratic polynomial, then the polynomials g i are not needed and the polynomials q i can be assumed to be linear functions. In this case, by writing p(x) as
p(x) = 1 x T Q 1 x ,
where Q is an (n + 1) × (n + 1) symmetric matrix, checking nonnegativity of p(x) reduces to checking the nonnegativity of the eigenvalues of Q; i.e., checking if Q is positive semidefinite. More generally, if the degrees of q i and g i are fixed in (1), then checking for a representation of p of the form in (1) reduces to solving an SDP, whose size depends on the dimension of x, and the degrees of p, q i and g i [35]. This insight has led to significant progress in certifying nonnegativity of polynomials arising in many areas. In practice, the "first level" of the SOS hierarchy is often the one used, where the polynomials g i are left out and one simply checks if p is a sum of squares of other polynomials. In this case already, because of the numerical difficulty of solving large SDPs, the polynomials that can be certified to be nonnegative usually do not have very high degrees or very many variables. For example, finding a sum of squares certificate that a given quartic polynomial over n variables is nonnegative requires solving an SDP involving roughly O(n 4 ) constraints and a positive semidefinite matrix variable of size O(n 2 ) × O(n 2 ). Even for a handful of or a dozen variables, the underlying semidefinite constraints prove to be expensive. Indeed, in the absence of additional structure, most examples in the literature have less than 10 variables.
Recently other systematic approaches to certifying nonnegativity of polynomials have been proposed which lead to less expensive optimization problems than semidefinite programming problems. In particular, Ahmadi and Majumdar [4], [3] introduce "DSOS and SDSOS" optimization techniques, which replace semidefinite programs arising in the nonnegativity certification problem by linear programs and secondorder cone programs. Instead of optimizing over the cone of sum of squares polynomials, the authors optimize over two subsets which they call "diagonally dominant sum of squares" and "scaled diagonally dominant sum of squares" polynomials (see Section 2.1 for formal definitions). In the language of semidefinite programming, this translates to solving optimization problems over the cone of diagonally dominant matrices and scaled diagonally dominant matrices. These can be done by LP and SOCP respectively. The authors have had notable success with these techniques in different applications. For instance, they are able to run these relaxations for polynomial optimization problems of degree 4 in 70 variables in the order of a few minutes. They have also used their techniques to push the size limits of some SOS problems in controls; examples include stabilizing a model of a humanoid robot with 30 state variables and 14 control inputs [29], or exploring the real-time applications of SOS techniques in problems such as collision-free autonomous motion planning [5].
Motivated by these results, our goal in this paper is to start with DSOS and SDSOS techniques and improve on them. By exploiting ideas from column generation in large-scale linear programming, and by appropriately interpreting the DSOS and SDSOS constraints, we produce several iterative LP and SOCPbased algorithms that improve the quality of the bounds obtained from the DSOS and SDSOS relaxations. Geometrically, this amounts to optimizing over structured subsets of sum of squares polynomials that are larger than the sets of diagonally dominant or scaled diagonally dominant sum of squares polynomials. For semidefinite programming, this is equivalent to optimizing over structured subsets of the cone of positive semidefinite matrices. An important distinction to make between the DSOS/SDSOS/SOS approaches and our approach, is that our approximations iteratively get larger in the direction of the given objective function, unlike the DSOS, SDSOS, and SOS approaches which all try to inner approximate the set of nonnegative polynomials irrespective of any particular direction.
In related literature, Krishnan and Mitchell use linear programming techniques to approximately solve SDPs by taking a semi-infinite LP representation of the SDP and applying column generation [24]. In addition, Kim and Kojima solve second order cone relaxations of SDPs which are closely related to the dual of an SDSOS program in the case of quadratic programming [23]; see Section 3 for further discussion of these two papers.
The organization of the rest of the paper is as follows. In the next section, we review relevant notation, and discuss the prior literature on DSOS and SDSOS programming. In Section 3, we give a high-level overview of our column generation approaches in the context of a general SDP. In Section 4, we describe an application of our ideas to nonconvex polynomial optimization and present computational experiments with certain column generation implementations. In Section 5, we apply our column generation approach to approximate a copositive program arising from a specific discrete optimization application (namely the stable set problem). All the work in these sections can be viewed as providing techniques to optimize over subsets of positive semidefinite matrices. We then conclude in Section 6 with some future directions, and discuss ideas for column generation which allow one to go beyond subsets of positive semidefinite matrices in the case of polynomial and copositive optimization.
Preliminaries
Let us first introduce some notation on matrices. We denote the set of real symmetric n × n matrices by S n . Given two matrices A and B in S n , we denote their matrix inner product by A · B := i,j A ij B ij = Trace(AB). The set of symmetric matrices with nonnegative entries is denoted by N n . A symmetric matrix A is positive semidefinite (psd) if x T Ax ≥ 0 for all x ∈ R n ; this will be denoted by the standard notation A 0, and our notation for the set of n × n psd matrices is P n . A matrix A is copositive if x T Ax ≥ 0 for all x ≥ 0. The set of copositive matrices is denoted by C n . All three sets N n , P n , C n are convex cones and we have the obvious inclusion N n + P n ⊆ C n . This inclusion is strict if n ≥ 5 [14], [13]. For a cone K of matrices in S n , we define its dual cone K * as {Y ∈ S n : Y · X ≥ 0, ∀X ∈ K}.
For a vector variable x ∈ R n and a vector q ∈ Z n + , let a monomial in x be denoted as x q := Π n i=1 x q i i , and let its degree be n i=1 q i . A polynomial is said to be homogeneous or a form if all of its monomials have the same degree. A form p(x) in n variables is nonnegative if p(x) ≥ 0 for all x ∈ R n , or equivalently for all x on the unit sphere in R n . The set of nonnegative (or positive semidefinite) forms in n variables and degree d is denoted by P SD n,d . A form p(x) is a sum of squares (sos) if it can be written as p(x) = r i=1 q 2 i (x) for some forms q 1 , . . . , q r . The set of sos forms in n variables and degree d is a cone denoted by SOS n,d . We have the obvious inclusion SOS n,d ⊆ P SD n,d , which is strict unless d = 2, or n = 2, or (n, d) = (3, 4) [21]. Let z(x, d) be the vector of all monomials of degree exactly d; it is well known that a form p of degree 2d is sos if and only if it can be written as p(x) = z T (x, d)Qz(x, d), for some psd matrix Q [35], [34]. The size of the matrix Q, which is often called the Gram matrix, is n+d−1 d × n+d−1 d . At the price of imposing a semidefinite constraint of this size, one obtains the very useful ability to search and optimize over the convex cone of sos forms via semidefinite programming.
DSOS and SDSOS optimization
In order to alleviate the problem of scalability posed by the SDPs arising from sum of squares programs, Ahmadi and Majumdar [4], [3] 2 recently introduced similar-purpose LP and SOCP-based optimization problems that they refer to as DSOS and SDSOS programs. Since we will be building on these concepts, we briefly review their relevant aspects to make our paper self-contained.
The idea in [4], [3] is to replace the condition that the Gram matrix Q be positive semidefinite with stronger but cheaper conditions in the hope of obtaining more efficient inner approximations to the cone SOS n,d . Two such conditions come from the concepts of diagonally dominant and scaled diagonally dominant matrices in linear algebra. We recall these definitions below.
Definition 2.1. A symmetric matrix A = (a ij ) is diagonally dominant (dd) if a ii ≥ j =i |a ij | for all i.
We say that A is scaled diagonally dominant (sdd) if there exists a diagonal matrix D, with positive diagonal entries, such that DAD is diagonally dominant.
We refer to the set of n × n dd (resp. sdd) matrices as DD n (resp. SDD n ). The following inclusions are a consequence of Gershgorin's circle theorem:
DD n ⊆ SDD n ⊆ P n .
We now use these matrices to introduce the cones of "dsos" and "sdsos" forms and some of their generalizations, which all constitute special subsets of the cone of nonnegative forms. We remark that in the interest of brevity, we do not give the original definitions of dsos and sdsos polynomials as they appear in [4] (as sos polynomials of a particular structure), but rather an equivalent characterization of them that is more useful for our purposes. The equivalence is proven in [4]. 3,4]). Recall that z(x, d) denotes the vector of all monomials of degree exactly d. A form p(x) of degree 2d is said to be (i) diagonally-dominant-sum-of-squares (dsos) if it admits a representation as
Definition 2.2 ([p(x) = z T (x, d)Qz(x, d), where Q is a dd matrix, (ii) scaled-diagonally-dominant-sum-of-squares (sdsos) if it admits a representation as p(x) = z T (x, d)Qz(x, d), where Q is an sdd matrix, (iii) r-diagonally-dominant-sum-of-squares (r-dsos) if there exists a positive integer r such that p(x)( n i=1 x 2 i ) r is dsos, (iv) r-scaled diagonally-dominant-sum-of-squares (r-sdsos) if there exists a positive integer r such that p(x)( n i=1 x 2 i ) r is sdsos.
We denote the cone of forms in n variables and degree d that are dsos, sdsos, r-dsos, and r-sdsos by DSOS n,d , SDSOS n,d , rDSOS n,d , and rSDSOS n,d respectively. The following inclusion relations are straightforward:
DSOS n,d ⊆ SDSOS n,d ⊆ SOS n,d ⊆ P SD n,d , rDSOS n,d ⊆ rSDSOS n,d ⊆ P SD n,d , ∀r.
The multiplier ( n i=1 x 2 i ) r should be thought of as a special denominator in the Artin-type representation in (1). By appealing to some theorems of real algebraic geometry, it is shown in [4] that under some conditions, as the power r increases, the sets rDSOS n,d (and hence rSDSOS n,d ) fill up the entire cone P SD n,d . We will mostly be concerned with the cones DSOS n,d and SDSOS n,d , which correspond to the case where r = 0. From the point of view of optimization, our interest in all of these algebraic notions stems from the following theorem. 3,4]). For any integer r ≥ 0, the cone rDSOS n,d is polyhedral and the cone rSDSOS n,d has a second order cone representation. Moreover, for any fixed d and r, one can optimize a linear function over rDSOS n,d (resp. rSDSOS n,d ) by solving a linear program (resp. second order cone program) of size polynomial in n.
Theorem 2.3 ([
The "LP part" of this theorem is not hard to see. The equality
p(x)( n i=1 x 2 i ) r = z T (x, d)Qz(x, d)
gives rise to linear equality constraints between the coefficients of p and the entries of the matrix Q (whose size is polynomial in n for fixed d and r). The requirement of diagonal dominance on the matrix Q can also be described by linear inequality constraints on Q. The "SOCP part" of the statement comes from the fact, shown in [4], that a matrix A is sdd if and only if it can be expressed as
A = i<j M ij 2×2 ,
where each M ij 2×2 is an n×n symmetric matrix with zeros everywhere except for four entries M ii , M ij , M ji , M jj , which must make the 2 × 2 matrix M ii M ij M ji M jj symmetric and positive semidefinite. These constraints are rotated quadratic cone constraints and can be imposed using SOCP [6], [27]:
M ii ≥ 0, 2M ij M ii − M jj ≤ M ii + M jj .
We refer to optimization problems with a linear objective posed over the convex cones DSOS n,d , SDSOS n,d , and SOS n,d as DSOS programs, SDSOS programs, and SOS programs respectively. In general, quality of approximation decreases, while scalability increases, as we go from SOS to SDSOS to DSOS programs. Depending on the size of the application at hand, one may choose one approach over the other. In related work, Ben-Tal and Nemirovski [10] and Vielma, Ahmed and Nemhauser [42] approximate SOCPs by LPs and produce approximation guarantees.
Column generation for inner approximation of positive semidefinite cones
In this section, we describe a natural approach to apply techniques from the theory of column generation [9], [19] in large-scale optimization to the problem of optimizing over nonnegative polynomials. Here is the rough idea: We can think of all SOS/SDSOS/DSOS approaches as ways of proving that a polynomial is nonnegative by writing it as a nonnegative linear combination of certain "atom" polynomials that are already known to be nonnegative. For SOS, these atoms are all the squares (there are infinitely many). For DSOS, there is actually a finite number of atoms corresponding to the extreme rays of the cone of diagonally dominant matrices (see Theorem 3.1 below). For SDSOS, once again we have infinitely many atoms, but with a specific structure which is amenable to an SOCP representation. Now the column generation idea is to start with a certain "cheap" subset of atoms (columns) and only add new ones-one or a limited number in each iteration-if they improve our desired objective function. This results in a sequence of monotonically improving bounds; we stop the column generation procedure when we are happy with the quality of the bound, or when we have consumed a predetermined budget on time.
In the LP case, after the addition of one or a few new atoms, one can obtain the new optimal solution from the previous solution in much less time than required to solve the new problem from scratch. However, as we show with some examples in this paper, even if one were to resolve the problems from scratch after each iteration (as we do for all of our SOCPs and some of our LPs), the overall procedure is still relatively fast. This is because in each iteration, with the introduction of a constant number k of new atoms, the problem size essentially increases only by k new variables and/or k new constraints. This is in contrast to other types of hierarchies-such as the rDSOS and rSDSOS hierarchies of Definition 2.2-that blow up in size by a factor that depends on the dimension in each iteration.
In the next two subsections we make this general idea more precise. While our focus in this section is on column generation for general SDPs, the next two sections show how the techniques are useful for approximation of SOS programs for polynomial optimization (Section 4), and copositive programs for discrete optimization (Section 5).
LP-based column generation
Consider a general SDP max y∈R m b T y s.t. C − m i=1 y i A i 0,(2)
with b ∈ R m , C, A i ∈ S n as input, and its dual
min X∈Sn C · X s.t. A i · X = b i , i = 1, . . . , m, X 0.(3)
Our goal is to inner approximate the feasible set of (2) by increasingly larger polyhedral sets. We consider LPs of the form
max y,α b T y s.t. C − m i=1 y i A i = t i=1 α i B i , α i ≥ 0, i = 1, . . . , t.(4)
Here, the matrices B 1 , . . . , B t ∈ P n are some fixed set of positive semidefinite matrices (our psd "atoms"). To expand our inner approximation, we will continually add to this list of matrices. This is done by considering the dual LP
min X∈Sn C · X s.t. A i · X = b i , i = 1, . . . , m, X · B i ≥ 0, i = 1, . . . , t,(5)
which in fact gives a polyhedral outer approximation (i.e., relaxation) of the spectrahedral feasible set of the SDP in (3). If the optimal solution X * of the LP in (5) is already psd, then we are done and have found the optimal value of our SDP. If not, we can use the violation of positive semidefiniteness to extract one (or more) new psd atoms B j . Adding such atoms to (4) is called column generation, and the problem of finding such atoms is called the pricing subproblem. (On the other hand, if one starts off with an LP of the form (5) as an approximation of (3), then the approach of adding inequalities to the LP iteratively that are violated by the current solution is called a cutting plane approach, and the associated problem of finding violated constraints is called the separation subproblem.) The simplest idea for pricing is to look at the eigenvectors v j of X * that correspond to negative eigenvalues. From each of them, one can generate a rank-one psd atom B j = v j v T j , which can be added with a new variable ("column") α j to the primal LP in (4), and as a new constraint ("cut") to the dual LP in (5). The subproblem can then be defined as getting the most negative eigenvector, which is equivalent to minimizing the quadratic form x T X * x over the unit sphere {x| ||x|| = 1}. Other possible strategies are discussed later in the paper.
This LP-based column generation idea is rather straightforward, but what does it have to do with DSOS optimization? The connection comes from the extreme-ray description of the cone of diagonally dominant matrices, which allows us to interpret a DSOS program as a particular and effective way of obtaining n 2 initial psd atoms.
Let U n,k denote the set of vectors in R n which have at most k nonzero components, each equal to ±1, and define U n,k ⊂ S n to be the set of matrices This theorem tells us that DD n has exactly n 2 extreme rays. It also leads to a convenient representation of the dual cone:
U n,k := {uu T : u ∈ U n,k }.
For a finite set of matrices
T = {T 1 , . . . , T t }, let cone(T ) := { t i=1 α i T i : α 1 , . . . , α t ≥ 0}.DD * n = {X ∈ S n : v T i Xv i ≥ 0,
for all vectors v i with at most 2 nonzero components, each equal to ±1}.
Throughout the paper, we will be initializing our LPs with the DSOS bound; i.e., our initial set of psd atoms B i will be the n 2 rank-one matrices u i u T i in U n,2 . This is because this bound is often cheap and effective. Moreover, it guarantees feasibility of our initial LPs (see Theorems 4.1 and 5.1), which is important for starting column generation. One also readily sees that the DSOS bound can be improved if we were to instead optimize over the cone U n,3 , which has n 3 atoms. However, in settings that we are interested in, we cannot afford to include all these atoms; instead, we will have pricing subproblems that try to pick a useful subset (see Section 4).
We remark that an LP-based column generation idea similar to the one in this section is described in [24], where it is used as a subroutine for solving the maxcut problem. The method is comparable to ours inasmuch as some columns are generated using the eigenvalue pricing subproblem. However, contrary to us, additional columns specific to max cut are also added to the primal. The initialization step is also differently done, as the matrices B i in (4) are initially taken to be in U n,1 and not in U n,2 . (This is equivalent to requiring the matrix C − m i=1 y i A i to be diagonal instead of diagonally dominant in (4).) Another related work is [40]. In this paper, the initial LP relaxation is obtained via RLT (Reformulation-Linearization Techniques) as opposed to our diagonally dominant relaxation. The cuts are then generated by taking vectors which violate positive semidefiniteness of the optimal solution as in (5). The separation subproblem that is solved though is different than the ones discussed here and relies on an LU decomposition of the solution matrix.
SOCP-based column generation
In a similar vein, we present an SOCP-based column generation algorithm that in our experience often does much better than the LP-based approach. The idea is once again to optimize over structured subsets of the positive semidefinite cone that are SOCP representable and that are larger than the set SDD n of scaled diagonally dominant matrices. This will be achieved by working with the following SOCP
max y∈R m ,a j i b T y s.t. C − m i=1 y i A i = t i=1 V i a 1 i a 2 i a 2 i a 3 i V T i , a 1 i a 2 i a 2 i a 3 i 0, i = 1, . . . , t.(6)
Here, the positive semidefiniteness constraints on the 2×2 matrices can be imposed via rotated quadratic cone constraints as explained in Section 2.1. The n × 2 matrices V i are fixed for all i = 1, . . . , t. Note that this is a direct generalization of the LP in (4), in the case where the atoms B i are rank-one. To generate a new SOCP atom, we work with the dual of (6):
min X∈Sn C · X s.t. A i · X = b i , i = 1, . . . , m, V T i XV i 0, i = 1, . . . , t.(7)
Once again, if the optimal solution X * is psd, we have solved our SDP exactly; if not, we can use X * to produce new SOCP-based cuts. For example, by placing the two eigenvectors of X * corresponding to its two most negative eigenvalues as the columns of an n × 2 matrix V t+1 , we have produced a new useful atom. (Of course, we can also choose to add more pairs of eigenvectors and add multiple atoms.) As in the LP case, by construction, our bound can only improve in every iteration.
We will always be initializing our SOCP iterations with the SDSOS bound. It is not hard to see that this corresponds to the case where we have n 2 initial n × 2 atoms V i , which have zeros everywhere, except for a 1 in the first column in position j and a 1 in the second column in position k > j. We denote the set of all such n × 2 matrices by V n,2 .
The first step of our procedure is carried out already in [23] for approximating solutions to QCQPs. Furthermore, the work in [23] shows that for a particular class of QCQPs, its SDP relaxation and its SOCP relaxation (written respectively in the form of (3) and (7)) are exact. Figure 1 shows an example of both the LP and SOCP column generation procedures. We produced two 10 × 10 random symmetric matrices E and F . The outer most set is the feasible set of an SDP with the constraint I + xE + yF 0. (Here, I is the 10 × 10 identity matrix.) The SDP wishes to maximize x + y over this set. The innermost set in Figure 1(a) is the polyhedral set where I + xE + yF is dd. The innermost set in Figure 1(b) is the SOCP-representable set where I + xE + yF is sdd. In both cases, we do 5 iterations of column generation that expand these sets by introducing one new atom at a time. These atoms come from the most negative eigenvector (resp. the two most negative eigenvectors) of the dual optimal solution as explained above. Note that in both cases, we are growing our approximation of the positive semidefinite cone in the direction that we care about (the northeast). This is in contrast to algebraic hierarchies based on "positive multipliers" (see the rDSOS and rSDSOS hierarchies in Definition 2.2 for example), which completely ignore the objective function.
Nonconvex polynomial optimization
In this section, we apply the ideas described in the previous section to sum of squares algorithms for nonconvex polynomial optimization. In particular, we consider the NP-hard problem of minimizing a form (of degree ≥ 4) on the sphere. Recall that z(x, d) is the vector of all monomials in n variables with degree d. Let p(x) be a form with n variables and even degree 2d, and let coef(p) be the vector of its coefficients with the monomial ordering given by z(x, 2d). Thus p(x) can be viewed as coef(p) T z(x, 2d). Let s(x) := ( n i=1 x 2 i ) d . With this notation, the problem of minimizing a form p on the unit sphere can be written as
max λ λ s.t. p(x) − λs(x) ≥ 0, ∀x ∈ R n .(8)
With the SOS programming approach, the following SDP is solved to get the largest scalar λ and an SOS certificate proving that p(x) − λs(x) is nonnegative:
max λ,Y λ s.t. p(x) − λs(x) = z T (x, d)Y z(x, d),(9)
Y 0.
The sum of squares certificate is directly read from an eigenvalue decomposition of the solution Y to the SDP above and has the form
p(x) − λs(x) ≥ i (z T (x, d)u i ) 2 ,
where Y = i u i u T i . Since all sos polynomials are nonnegative, the optimal value of the SDP in (9) is a lower bound to the optimal value of the optimization problem in (8). Unfortunately, before solving the SDP, we do not have access to the vectors u i in the decomposition of the optimal matrix Y . However, the fact that such vectors exist hints at how we should go about replacing P n by a polyhedral restriction in (9)
: If the constraint Y 0 is changed to Y = u∈U α u uu T , α u ≥ 0,(10)
where U is a finite set, then (9) becomes an LP. This is one interpretation of Ahmadi and Majumdar's work in [3,4] where they replace P n by DD n . Indeed, this is equivalent to taking U = U n,2 in (10), as shown in Theorem 3.1. We are interested in extending their results by replacing P n by larger restrictions than DD n . A natural candidate for example would be obtained by changing U n,2 to U n,3 . However, although U n,3 is finite, it contains a very large set of vectors even for small values of n and d. For instance, when n = 30 and d = 4, U n,3 has over 66 million elements. Therefore we use column generation ideas to iteratively expand U in a manageable fashion. To initialize our procedure, we would like to start with good enough atoms to have a feasible LP. The following result guarantees that replacing Y 0 with Y ∈ DD n always yields an initial feasible LP in the setting that we are interested in.
λ ∈ R such that p(x) − λ( n i=1 x 2 i ) d is dsos. Proof. As before, let s(x) = ( n i=1 x 2 i ) d .
We observe that the form s(x) is strictly in the interior of DSOS n,2d . Indeed, by expanding out the expression we see that we can write s(x) as z T (x, d)Qz(x, d), where Q is a diagonal matrix with all diagonal entries positive. So Q is in the interior of DD ( n+d−1 d ) , and hence s(x) is in the interior of DSOS n,2d . This implies that for α > 0 small enough, the form
(1 − α)s(x) + αp(x)
will be dsos. Since DSOS n,2d is a cone, the form
(1 − α) α s(x) + p(x)
will also be dsos. By taking λ to be smaller than or equal to − 1−α α , the claim is established.
As DD n ⊆ SDD n , the theorem above implies that replacing Y 0 with Y ∈ SDD n also yields an initial feasible SOCP. Motivated in part by this theorem, we will always start our LP-based iterative process with the restriction that Y ∈ DD n . Let us now explain how we improve on this approximation via column generation.
Suppose we have a set U of vectors in R n , whose outerproducts form all of the rank-one psd atoms that we want to consider. This set could be finite but very large, or even infinite. For our purposes U always includes U n,2 , as we initialize our algorithm with the dsos relaxation. Let us consider first the case where U is finite: U = {u 1 , . . . , u t }. Then the problem that we are interested in solving is
max λ,α j λ s.t. p(x) − λs(x) = z T (x, d)Y z(x, d), Y = t j=1 α j u j u T j , α j ≥ 0 for j = 1, . . . , t.
Suppose z(x, 2d) has m monomials and let the ith monomial in p(x) have coefficient b i , i.e., coef(p) = (b 1 , . . . , b m ) T . Also let s i be the ith entry in coef(s(x)). We rewrite the previous problem as
max λ,α j λ s.t. A i · Y + λs i = b i for i = 1, . . . , m, Y = t j=1 α j u j u T j , α j ≥ 0 for j = 1, . . . , t.
where A i is a matrix that collects entries of Y that contribute to the i th monomial in z(x, 2d), when z T (x, d)Y z(x, d) is expanded out. The above is equivalent to
max λ,α j λ s.t. j α j (A i · u j u T j ) + λs i = b i for i = 1, . . . , m,(11)
α j ≥ 0 for j = 1, . . . , t.
The dual problem is
min µ m i=1 µ i b i s.t. ( m i=1 µ i A i ) · u j u T j ≥ 0, j = 1, . . . , t, m i=1 µ i s i = 1.
In the column generation framework, suppose we consider only a subset of the primal LP variables corresponding to the matrices u 1 u T 1 , . . . , u k u T k for some k < t (call this the reduced primal problem). Let (ᾱ 1 , . . . ,ᾱ k ) stand for an optimal solution of the reduced primal problem and letμ = (μ 1 , . . . ,μ m ) stand for an optimal dual solution. If we have
( m i=1μ i A i ) · u j u T j ≥ 0 for j = k + 1, . . . , t,(12)
thenμ is an optimal dual solution for the original larger primal problem with columns 1, . . . , t. In other words, if we simply set α k+1 = · · · = α t = 0, then the solution of the reduced primal problem becomes a solution of the original primal problem. On the other hand, if (12) is not true, then suppose the condition is violated for some u l u T l . We can augment the reduced primal problem by adding the variable α l , and repeat this process.
Let B = m i=1μ i A i . We can test if (12) is false by solving the pricing subproblem:
min u∈U u T Bu.(13)
If u T Bu < 0, then there is an element u in U such that the matrix uu T violates the dual constraint written in (12). Problem (13) may or may not be easy to solve depending on the set U. For example, an ambitious column generation strategy to improve on dsos (i.e., U = U n,2 ), would be to take U = U n,n ; i.e., the set all vectors in R n consisting of zeros, ones, and minus ones. In this case, the pricing problem (13) becomes min u∈{0,±1} n u T Bu.
Unfortunately, the above problem generalizes the quadratic unconstrained boolean optimization problem (QUBO) and is NP-hard. Nevertheless, there are good heuristics for this problem (see e.g., [12], [17]) that can be used to find near optimal solutions very fast. While we did not pursue this pricing subproblem, we did consider optimizing over U n, 3 . We refer to the vectors in U n,3 as "triples" for obvious reasons and generally refer to the process of adding atoms drawn from U n,3 as optimizing over "triples". Even though one can theoretically solve (13) with U = U n,3 in polynomial time by simple enumeration of n 3 elements, this is very impractical. Our simple implementation is a partial enumeration and is implemented as follows. We iterate through the triples (in a fixed order), and test to see whether the condition u T Bu ≥ 0 is violated by a given triple u, and collect such violating triples in a list. We terminate the iteration when we collect a fixed number of violating triples (say t 1 ). We then sort the violating triples by increasing values of u T Bu (remember, these values are all negative for the violating triples) and select the t 2 most violated triples (or fewer if less than t 2 are violated overall) and add them to our current set of atoms. In a subsequent iteration we start off enumerating triples right after the last triple enumerated in the current iteration so that we do not repeatedly scan only the same subset of triples. Although our implementation is somewhat straightforward and can be obviously improved, we are able to demonstrate that optimizing over triples improves over the best bounds obtained by Ahmadi and Majumdar in a similar amount of time (see Section 4.2).
We can also have pricing subproblems where the set U is infinite. Consider e.g. the case U = R n in (13). In this case, if there is a feasible solution with a negative objective value, then the problem is clearly unbounded below. Hence, we look for a solution with the smallest value of "violation" of the dual constraint divided by the norm of the violating matrix. In other words, we want the expression u T Bu/norm(uu T ) to be as small as possible, where norm is the Euclidean norm of the vector consisting of all entries of uu T . This is the same as minimizing u T Bu/||u|| 2 . The eigenvector corresponding to the smallest eigenvalue yields such a minimizing solution. This is the motivation behind the strategy described in the previous section for our LP column generation scheme. In this case, we can use a similar strategy for our SOCP column generation scheme. We replace Y 0 by Y ∈ SDD n in (9) and iteratively expand SDD n by using the "two most negative eigenvector technique" described in Section 3.2.
Experiments with a 10-variable quartic
We illustrate the behaviour of these different strategies on an example. Let p(x) be a degree-four form defined on 10 variables, where the components of coef(p) are drawn independently at random from the normal distribution N (0, 1). Thus d = 2 and n = 10, and the form p(x) is 'fully dense' in the sense that coef(p) has essentially all nonzero components. In Figure 2, we show how the lower bound on the optimal value of p(x) over the unit sphere changes per iteration for different methods. The x-axis shows the number of iterations of the column generation algorithm, i.e., the number of times columns are added and the LP (or SOCP) is resolved. The y-axis shows the lower bound obtained from each LP or SOCP. Each curve represents one way of adding columns. The three horizontal lines (from top to bottom) represent, respectively, the SDP bound, the 1SDSOS bound and the 1DSOS bound. The curve DSOS k gives the bound obtained by solving LPs, where the first LP has Y ∈ DD n and subsequent columns are generated from a single eigenvector corresponding to the most negative eigenvalue of the dual optimal solution as described in Section 3.1. The LP triples curve also corresponds to an LP sequence, but this time the columns that are added are taken from U n,3 and are more than one in each iteration (see the next subsection). This bound saturates when constraints coming from all elements of U n,3 are satisfied. Finally, the curve SDSOS k gives the bound obtained by SOCP-based column generation as explained just above.
Larger computational experiments
In this section, we consider larger problem instances ranging from 15 variables to 40 variables: these instances are again fully dense and generated in exactly the same way as the n = 10 example of the previous subsection. However, contrary to the previous subsection, we only apply our "triples" column generation strategy here. This is because the eigenvector-based column generation strategy is too computationally expensive for these problems as we discuss below.
To solve the triples pricing subproblem with our partial enumeration strategy, we set t 1 to 300,000 and t 2 to 5000. Thus in each iteration, we find up to 300,000 violated triples, and add up to 5000 of them. In other words, we augment our LP by up to 5000 columns in each iteration. This is somewhat unusual as in practice at most a few dozen columns are added in each iteration. The logic for this is that primal simplex is very fast in reoptimizing an LP when a small number of additional columns are added to an LP whose optimal basis is known. However, in our context, we observed that the associated LPs are very hard for the simplex routines inside our LP solver (CPLEX 12.4) and take much more time than CPLEX's interior point solver. We therefore use CPLEX's interior point ("barrier") solver not only for the initial LP but for subsequent LPs after adding columns. Because interior point solvers do not benefit significantly from warm starts, each LP takes a similar amount of time to solve as the initial LP, and therefore it makes sense to add a large number of columns in each iteration to amortize the time for each expensive solve over many columns. Table 1 is taken from the work of Ahmadi and Majumdar [4], where they report lower bounds on the minimum value of fourth-degree forms on the unit sphere obtained using different methods, and the respective computing times (in seconds). In Table 2, we give our bounds for the same problem instances. We report two bounds, obtained at two different times (if applicable). In the first case (rows labeled R1), the time taken by 1SDSOS in Table 1 is taken as a limit, and we report the bound from the last column generation iteration occuring before this time limit; the 1SDSOS bound is the best non-SDP bound reported in the experiments of Ahmadi and Majumdar. In the rows labeled as R2, we take 600 seconds as a limit and report the last bound obtained before this limit. In a couple of instances (n = 15 and n = 20), our column generation algorithm terminates before the 600 second limit, and we report the termination time in this case. Table 2: Lower bounds on the optimal value of a form on the sphere for varying degrees of polynomials using Triples on a 2.33 GHz Linux machine with 32 GB of memory.
We observe that in the same amount of time (and even on a slightly slower machine), we are able to consistently beat the 1SDSOS bound, which is the strongest non-SDP bound produced in [4]. We also experimented with the eigenvalue pricing subproblem in the LP case, with a time limit of 600 seconds. For n = 25, we obtain a bound of −23.46 after adding only 33 columns in 600 seconds. For n = 40, we are only able to add 6 columns and the lower bound obtained is −61.49. Note that this bound is worse than the triples bound given in Table 2. The main reason for being able to add so few columns in the time limit is that each column is almost fully dense (the LPs for n=25 have 20,475 rows, and 123,410 rows for n = 40). Thus, the LPs obtained are very hard to solve after a few iterations and become harder with increasing n. As a consequence, we did not experiment with the eigenvalue pricing subproblem in the SOCP case as it is likely to be even more computationally intensive.
Inner approximations of copositive programs and the maximum stable set problem
Semidefinite programming has been used extensively for approximation of NP-hard combinatorial optimization problems. One such example is finding the stability number of a graph. A stable set (or independent set) of a graph G = (V, E) is a set of nodes of G, no two of which are adjacent. The size of the largest stable set of a graph G is called the stability number (or independent set number) of G and is denoted by α(G). Throughout, G is taken to be an undirected, unweighted graph on n nodes. It is known that the problem of testing if α(G) is greater than a given integer k is NP-hard [22]. Furthermore, the stability number cannot be approximated to a factor of n 1− for any > 0 unless P=NP [20]. The natural integer programming formulation of this problem is given by
α(G) =max x i n i=1 x i s.t. x i + x j ≤ 1, ∀(i, j) ∈ E, x i ∈ {0, 1}, ∀i = 1, . . . , n.(14)
Although this optimization problem is intractable, there are several computationally-tractable relaxations that provide upper bounds on the stability number of a graph. For example, the obvious LP relaxation of (14) can be obtained by relaxing the constraint x i ∈ {0, 1} to x i ∈ [0, 1]:
LP (G) =max x i i x i s.t. x i + x j ≤ 1, ∀(i, j) ∈ E, x i ∈ [0, 1], ∀i = 1, . . . , n.(15)
This bound can be improved upon by adding the so-called clique inequalities to the LP, which are of the form x i 1 + x i 2 + . . . + x i k ≤ 1 when nodes (i 1 , i 2 , . . . , i k ) form a clique in G. Let C k be the set of all k-clique inequalities in G. This leads to a hierarchy of LP relaxations:
LP k (G) = max i x i ,
x i ∈ [0, 1], ∀i = 1, . . . , n, C 2 , . . . , C k are satisfied.
Notice that for k = 2, this simply corresponds to (15), in other words, LP 2 (G) = LP (G).
In addition to LPs, there are also semidefinite programming (SDP) relaxations that provide upper bounds to the stability number. The most well-known is perhaps the Lovász theta number ϑ(G) [28], which is defined as the optimal value of the following SDP:
ϑ(G) :=max X J · X s.t. I · X = 1, X i,j = 0, ∀(i, j) ∈ E X ∈ P n .(17)
Here J is the all-ones matrix and I is the identity matrix of size n. The Lovász theta number is known to always give at least as good of an upper bound as the LP in (15), even with the addition of clique inequalities of all sizes (there are exponentially many); see, e.g., [26, Section 6.5.2] for a proof. In other words,
ϑ(G) ≤ LP k (G), ∀k.
An alternative SDP relaxation for stable set is due to de Klerk and Pasechnik. In [18], they show that the stability number can be obtained through a conic linear program over the set of copositive matrices. Namely,
α(G) = min λ λ s.t. λ(I + A) − J ∈ C n ,(18)
where A is the adjacency matrix of G. Replacing C n by the restriction P n + N n , one obtains the aforementioned relaxation through the following SDP
SDP (G) := min λ,X λ s.t. λ(I + A) − J ≥ X, X ∈ P n .(19)
This latter SDP is more expensive to solve than the Lovász SDP (17), but the bound that it obtains is always at least as good (and sometimes strictly better). A proof of this statement is given in [18,Lemma 5.2], where it is shown that (19) is an equivalent formulation of an SDP of Schrijver [39], which produces stronger upper bounds than (17).
Another reason for the interest in the copositive approach is that it allows for well-known SDP and LP hierarchies-developed respectively by Parrilo [34, Section 5] and de Klerk and Pasechnik [18]-that produce a sequence of improving bounds on the stability number. In fact, by appealing to Positivstellensatz results of Pólya [36], and Powers and Reznick [37], de Klerk and Pasechnik show that their LP hierarchy produces the exact stability number in α 2 (G) number of steps [18,Theorem 4.1]. This immediately implies the same result for stronger hierarchies, such as the SDP hierarchy of Parrilo [34], or the rDSOS and rSDSOS hierarchies of Ahmadi and Majumdar [4].
One notable difficulty with the use of copositivity-based SDP relaxations such as (19) in applications is scalibility. For example, it takes more than 5 hours to solve (19) when the input is a randomly generated Erdós-Renyi graph with 300 nodes and edge probability p = 0.8. 3 Hence, instead of using (19), we will solve a sequence of LPs/SOCPs generated in an iterative fashion. These easier optimization problems will provide upper bounds on the stability number in a more reasonable amount of time, though they will be weaker than the ones obtained via (19).
We will derive both our LP and SOCP sequences from formulation (18) of the stability number. To obtain the first LP in the sequence, we replace C n by DD n + N n (instead of replacing C n by P n + N n as was done in (19)) and get
DSOS 1 (G) := min λ,X λ s.t. λ(I + A) − J ≥ X, X ∈ DD n .(20)
This is an LP whose optimal value is a valid upper bound on the stability number as DD n ⊆ P n . Proof. We need to show that for any n × n adjacency matrix A, there exists a diagonally dominant matrix D, a nonnegative matrix N , and a scalar λ such that
λ(I + A) − J = D + N.(21)
Notice first that λ(I + A) − J is a matrix with λ − 1 on the diagonal and at entry (i, j), if (i, j) is an edge in the graph, and with −1 at entry (i, j) if (i, j) is not an edge in the graph. If we denote by d i the degree of node i, then let us take λ = n − min i d i + 1 and D a matrix with diagonal entries λ − 1 and off-diagonal entries equal to 0 if there is an edge, and −1 if not. This matrix is diagonally dominant as there are at most n − min i d i minus ones on each row. Furthermore, if we take N to be a matrix with λ − 1 at the entries (i, j) where (i, j) is an edge in the graph, then (21) is satisfied and N ≥ 0.
Feasibility of this LP is important for us as it allows us to initiate column generation. By contrast, if we were to replace the diagonal dominance constraint by a diagonal constraint for example, the LP could fail to be feasible. This fact has been observed by de Klerk and Pasechnik in [18] and Bomze and de Klerk in [11].
To generate the next LP in the sequence via column generation, we think of the extreme-ray description of the set of diagonally dominant matrices as explained in Section 3. Theorem 3.1 tells us that these are given by the matrices in U n,2 and so we can rewrite (20) as
DSOS 1 (G) := min λ,α i λ s.t. λ(I + A) − J ≥ X, X = u i u T i ∈U n,2 α i u i u T i , α i ≥ 0, i = 1, . . . , n 2 .(22)
The column generation procedure aims to add new matrix atoms to the existing set U n,2 in such a way that the current bound DSOS 1 improves. There are numerous ways of choosing these atoms. We focus first on the cutting plane approach based on eigenvectors. The dual of (22) is the LP
DSOS 1 (G) := max X J · X, s.t. (A + I) · X = 1, X ≥ 0, (u i u T i ) · X ≥ 0, ∀u i u T i ∈ U n,2 .(23)
If our optimal solution X * to (23) is positive semidefinite, then we are obtaining the best bound we can possibly produce, which is the SDP bound of (19). If this is not the case however, we pick our atom matrix to be the outer product uu T of the eigenvector u corresponding to the most negative eigenvalue of X * . The optimal value of the LP DSOS 2 (G) := max X J · X, s.t. (A + I) · X = 1,
X ≥ 0, (u i u T i ) · X ≥ 0, ∀u i u T i ∈ U n,2 , (uu T ) · X ≥ 0(24)
that we derive is guaranteed to be no worse than DSOS 1 as the feasible set of (24) is smaller than the feasible set of (23). Under mild nondegeneracy assumptions (satisfied, e.g., by uniqueness of the optimal solution to (23)), the new bound will be strictly better. By reiterating the same process, we create a sequence of LPs whose optimal values DSOS 1 , DSOS 2 , . . . are a nonincreasing sequence of upper bounds on the stability number.
Generating the sequence of SOCPs is done in an analogous way. Instead of replacing the constraint X ∈ P n in (19) by X ∈ DD n , we replace it by X ∈ SDD n and get
SDSOS 1 (G) := min λ,X λ s.t. λ(I + A) − J ≥ X, X ∈ SDD n .(25)
Once again, we need to reformulate the problem in such a way that the set of scaled diagonally dominant matrices is described as some combination of psd "atom" matrices. In this case, we can write any matrix X ∈ SDD n as
X = V i ∈V n,2 V i a 1 i a 2 i a 2 i a 3 i V T i ,
where a 1 i , a 2 i , a 3 i are variables making the 2 × 2 matrix psd, and the V i 's are our atoms. Recall from Section 3 that the set V n,2 consists of all n×2 matrices which have zeros everywhere, except for a 1 in the first column in position j and a 1 in the second column in position k = j. This gives rise to an equivalent formulation of (25):
SDSOS 1 (G) := min λ,a j i λ s.t. λ(I + A) − J ≥ X X = V i ∈V n,2 V i a 1 i a 2 i a 2 i a 3 i V T i a 1 i a 2 i a 2 i a 3 i 0, i = 1, . . . , n 2 .(26)
Just like the LP case, we now want to generate one (or more) n × 2 matrix V to add to the set {V i } i so that the bound SDSOS 1 improves. We do this again by using a cutting plane approach originating from the dual of (26):
SDSOS 1 (G) := max X J · X s.t. (A + I) · X = 1, X ≥ 0, V T i · XV i 0, i = 1, . . . , n 2 .(27)
Note that strong duality holds between this primal-dual pair as it is easy to check that both problems are strictly feasible. We then take our new atom to be
V = (w 1 w 2 ),
where w 1 and w 2 are two eigenvectors corresponding to the two most negative eigenvalues of X * , the optimal solution of (27). If X * only has one negative eigenvalue, we add a linear constraint to our problem; if X * 0, then the bound obtained is identical to the one obtained through SDP (19) and we cannot hope to improve. Our next iterate is therefore
SDSOS 2 (G) := max X J · X s.t. (A + I) · X = 1, X ≥ 0, V T i · XV i 0, i = 1, . . . , n 2 , V T · XV 0.(28)
Note that the optimization problems generated iteratively in this fashion always remain SOCPs and their optimal values form a nonincreasing sequence of upper bounds on the stability number.
To illustrate the column generation method for both LPs and SOCPs, we consider the complement of the Petersen graph as shown in Figure 3(a) as an example. The stability number of this graph is 2 and one of its maximum stable sets is designated by the two white nodes. In Figure 3(b), we compare the upper bound obtained via (19) and the bounds obtained using the iterative LPs and SOCPs as described in (24) and (28). Note that it takes 3 iterations for the SOCP sequence to produce an upper bound strictly within one unit of the actual stable set number (which would immediately tell us the value of α), whereas it takes 13 iterations for the LP sequence to do the same. It is also interesting to compare the sequence of LPs/SOCPs obtained through column generation to the sequence that one could obtain using the concept of r-dsos/r-sdsos polynomials. Indeed, LP (20) (resp. SOCP (25)) can be written in polynomial form as
DSOS 1 (G) (resp. SDSOS 1 (G)) = min λ λ s.t. x 2 1 . . . x 2 n T (λ(I + A) − J) x 2 1 . . . x 2 n
is dsos (resp. sdsos). (29) Iteration k in the sequence of LPs/SOCPs would then correspond to requiring that this polynomial be k-dsos or k-sdsos. For this particular example, we give the 1-dsos, 2-dsos, 1-sdsos and 2-sdsos bounds in Table 3.
Iteration r-dsos bounds r-sdsos bounds r = 0 4.00 4.00 r = 1 2.71 2.52 r = 2 2.50 2.50 Table 3: Bounds obtained through rDSOS and rSDSOS hierarchies.
Though this sequence of LPs/SOCPs gives strong upper bounds, each iteration is more expensive than the iterations done in the column generation approach. Indeed, in each of the column generation iterations, only one constraint is added to our problem, whereas in the rDSOS/rSDSOS hierarchies, the number of constraints is roughly multiplied by n 2 at each iteration.
Finally, we investigate how these techniques perform on graphs with a large number of nodes, where the SDP bound cannot be found in a reasonable amount of time. The graphs we test these techniques on are Erdös-Rényi graphs ER(n, p); i.e. graphs on n nodes where an edge is added between each pair of nodes independently and with probability p. In our case, we take n to be between 150 and 300, and p to be either 0.3 or 0.8 so as to experiment with both medium and high density graphs. 4 In Table 4, we present the results of the iterative SOCP procedure and contrast them with the SDP bounds. The third column of the table contains the SOCP upper bound obtained through (27); the solver time needed to obtain this bound is given in the fourth column. The fifth and sixth columns correspond respectively to the SOCP iterative bounds obtained after 5 mins solving time and 10 mins solving time. Finally, the last two columns chart the SDP bound obtained from (19) and the time in seconds needed to solve the SDP. All SOCP and SDP experiments were done using Matlab, the solver MOSEK [2], the SPOTLESS toolbox [30], and a computer with 3.4 GHz speed and 16 GB RAM. From the table, we note that it is better to run the SDP rather than the SOCPs for small n, as the bounds obtained are better and the times taken to do so are comparable. However, as n gets bigger, the SOCPs become valuable as they provide good upper bounds in reasonable amounts of time. For example, for n = 300 and p = 0.8, the SOCP obtains a bound that is only twice as big as the SDP bound, but it does so 30 times faster. The sparser graphs don't do as well, a trend that we will also observe in Table 5. Finally, notice that the improvement in the first 5 mins is significantly better than the improvement in the last 5 mins. This is partly due to the fact that the SOCPs generated at the beginning are sparser, and hence faster to solve.
In Table 5, we present the results of the iterative LP procedure used on the same instances. All LP results were obtained using a computer with 2.3 GHz speed and 32GB RAM and the solver CPLEX 12.4 [16]. The third and fourth columns in the table contain the LP bound obtained with (23) and the solver time taken to do so. Columns 5 and 6 correspond to the LP iterative bounds obtained after 5 mins solving time and 10 mins solving time using the eigenvector-based column generation technique (see discussion around (24)). The seventh and eighth columns are the standard LP bounds obtained using (16) and the time taken to obtain the bound. Finally, the last column gives bounds obtained by column generation using "triples", as described in Section 4.2. In this case, we take t 1 = 300, 000 and t 2 = 500. Table 5: LP bounds obtained on the same ER(n, p) graphs.
We note that in this case the upper bound with triples via column generation does better for this range of n than eigenvector-based column generation in the same amount of time. Furthermore, the iterative LP scheme seems to perform better in the dense regime. In particular, the first iteration does significantly better than the standard LP for p = 0.8, even though both LPs are of similar size. This would remain true even if the 3-clique inequalities were added as in (16), since the optimal value of LP 3 is always at least n/3. This is because the vector ( 1 3 , . . . , 1 3 ) is feasible to the LP in (16) with k = 3. Note that this LP would have order n 3 constraints, which is more expensive than our LP. On the contrary, for sparse regimes, the standard LP, which hardly takes any time to solve, gives better bounds than ours. Overall, the high-level conclusion is that running the SDP is worthwhile for small sizes of the graph. As the number of nodes increases, column generation becomes valuable, providing upper bounds in a reasonable amount of time. Contrasting Tables 4 and 5, our initial experiments seem to show that the iterative SOCP bounds are better than the ones obtained using the iterative LPs. It may be valuable to experiment with different approaches to column generation however, as the technique used to generate the new atoms seems to impact the bounds obtained.
Conclusions and future research
For many problems of discrete and polynomial optimization, there are hierarchies of SDP-based sum of squares algorithms that produce provably optimal bounds in the limit [35], [25]. However, these hierarchies can often be expensive computationally. In this paper, we were interested in problem sizes where even the first level of the hierarchy is too expensive, and hence we resorted to algorithms that replace the underlying SDPs with LPs or SOCPs. We built on the recent work of Ahmadi and Majumdar on DSOS and SDSOS optimization [4], [3], which serves exactly this purpose. We showed that by using ideas from linear programming column generation, the performance of their algorithms is improvable. We did this by iteratively optimizing over increasingly larger structured subsets of the cone of positive semidefinite matrices, without resorting to the more expensive rDSOS and rSDSOS hierarchies.
There is certainly a lot of room to improve our column generation algorithms. In particular, we only experimented with a few types of pricing subproblems and particular strategies for solving them. The success of column generation often comes from good "engineering", which fine-tunes the algorithms to the problem at hand. Developing warm-start strategies for our iterative SOCPs for example, would be a very useful problem to work on in the future.
Here is another interesting research direction, which for illustrative purposes we outline for the problem studied in Section 4; i.e., minimizing a form on the sphere. Recall that given a form p of degree 2d, we are trying to find the largest λ such that p(x) − λ( n i=1 x 2 i ) d is a sum of squares. Instead of solving this sum of squares program, we looked for the largest λ for which we could write p(x) − λ as a conic combination of a certain set of nonnegative polynomials. These polynomials for us were always either a single square or a sum of squares of polynomials. There are polynomials, however, that are nonnegative but not representable as a sum of squares. Two classic examples [31], [15] are the Motzkin polynomial M (x, y, z) = x 6 + y 4 z 2 + y 2 z 4 − 3x 2 y 2 z 2 , and the Choi-Lam polynomial CL(w, x, y, z) = w 4 + x 2 y 2 + y 2 z 2 + x 2 z 2 − 4wxyz.
Either of these polynomials can be shown to be nonnegative using the arithmetic mean-geometric mean (am-gm) inequality, which states that if x 1 , . . . , x k ∈ R, then
x 1 , . . . , x k ≥ 0 ⇒ ( k i=1 x i )/k ≥ (Π k i=1 x i )
For example, in the case of the Motzkin polynomial, it is clear that the monomials x 6 , y 4 z 2 and y 2 z 4 are nonnegative for all x, y, z ∈ R, and letting x 1 , x 2 , x 3 stand for these monomials respectively, the am-gm inequality implies that x 6 + y 4 z 2 + y 2 z 4 ≥ 3x 2 y 2 z 2 for all x, y, z ∈ R.
These polynomials are known to be extreme in the cone of nonnegative polynomials and they cannot be written as a sum of squares (sos) [38]. It would be interesting to study the separation problems associated with using such non-sos polynomials in column generation. We briefly present one separation algorithm for a family of polynomials whose nonnegativity is provable through the am-gm inequality and includes the Motzkin and Choi-Lam polynomials. This will be a relatively easy-to-solve integer program in itself, whose goal is to find a polynomial q amongst this family which is to be added as our new "nonnegative atom".
The family of n-variate polynomials under consideration consists of polynomials with only k+1 nonzero coefficients, with k of them equal to one, and one equal to −k. (Notice that the Motzkin and the Choi-Lam polynomials are of this form with k equal to three and four respectively.) Let m be the number of monomials in p. Given a dual vector µ of (11) of dimension m, one can check if there exists a nonnegative degree 2d polynomial q(x) in our family such that µ · coef(q(x)) < 0. This can be done by solving the following integer program (we assume that p(
x) = m i=1 x α i ): min c,y m i=1 µ i c i − m i=1 kµ i y i(30)
s.t.
i:α i is even Here, we have α i ∈ N n and the variables c i , y i form the coefficients of the polynomial q(x) = m i=1 c i x α i − k m i=1 y i x α i . The above integer program has 2m variables, but only n + 2 constraints (not counting the integer constraints). If a polynomial q(x) with a negative objective value is found, then one can add it as a new atom for column generation. In our specific randomly generated polynomial optimization examples, such polynomials did not seem to help in our preliminary experiments. Nevertheless, it would be interesting to consider other instances and problem structures.
α i c i = k m i=1 α i y i , m i=1 c i = k,
Similarly, in the column generation approach to obtaining inner approximations of the copositive cone, one need not stick to positive semidefinite matrices. It is known that the 5 × 5 "Horn matrix" [14] for example is extreme in the copositive cone but cannot be written as the sum of a nonnegative and a positive semidefinite matrix. One could define a separation problem for a family of Horn-like matrices and add them in a column generation approach. Exploring such strategies is left for future research.
Acknowledgments
We are grateful to Anirudha Majumdar for insightful discussions and for his help with some of the numerical experiments in this paper.
Theorem 3 . 1 (
31Barker and Carlson [8]). DD n = cone(U n,2 ).
(a) LP starting with DSOS and adding 5 atoms.(b) SOCP starting with SDSOS and adding 5 atoms.
Figure 1 :
1LP and SOCP-based column generation for inner approximation of a spectrahedron.
Theorem 4. 1 .
1For any form p of degree 2d, there exists
Figure 2 :
2Lower bounds for a polynomial of degree 4 in 10 variables obtained via LP and SOCP based column generation
Theorem 5. 1 .
1The LP in(20) is always feasible.
Figure 3 :
3bounds on the stable set number α(G) Bounds obtained through SDP(19) and iterative SOCPs and LPs for the complement of the Petersen graph.
∈ {0, 1}, y i ∈ {0, 1}, i = 1, . . . , m, c i = 0 if α i is not even.
n p
nSDSOS 1 time (s) SDSOS k (5 mins) SDSOS k (10 mins) SDP (G)Table 4: SDP bounds and iterative SOCP bounds obtained on ER(n,p) graphs.time (s)
150 0.3
105.70
1.05
39.93
37.00
20.43
221.13
150 0.8
31.78
1.07
9.96
9.43
6.02
206.28
200 0.3
140.47
1.84
70.15
56.37
23.73
1,086.42
200 0.8
40.92
2.07
12.29
11.60
6.45
896.84
250 0.3
176.25
3.51
111.63
92.93
26.78
4,284.01
250 0.8
51.87
3.90
17.25
15.39
7.18
3,503.79
300 0.3
210.32
5.69
151.71
134.14
29.13
32,300.60
300 0.8
60.97
5.73
19.53
17.24
7.65
20,586.02
n p
nDSOS 1 time (s) DSOS k (5m) DSOS k (10m) LP 2 time (s) LP triples (10m)150 0.3
117
< 1
110.64
110.26
75
< 1
89.00
150 0.8
46
< 1
24.65
19.13
75
< 1
23.64
200 0.3
157
< 1
147.12
146.71
100
< 1
129.82
200 0.8
54
< 1
39.27
36.01
100
< 1
30.43
250 0.3
194
< 1
184.89
184.31
125
< 1
168.00
250 0.8
68
< 1
55.01
53.18
125
< 1
40.19
300 0.3
230
< 1
222.43
221.56
150
< 1
205.00
300 0.8
78
< 1
65.77
64.84
150
< 1
60.00
Weak NP-hardness of testing matrix copositivity is originally proven by Murty and Kabadi[32]; its strong NP-hardness is apparent from the work of de Klerk and Pasechnik[18].
The work in[4] is currently in preparation for submission; the one in[3] is a shorter conference version of[4] which has already appeared. The presentation of the current paper is meant to be self-contained.
The solver in this case is MOSEK[2] and the machine used has 3.4GHz speed and 16GB RAM; seeTable 4for more results. The solution time with the popular SDP solver SeDuMi[41] e.g. would be several times larger.
All instances used for these tests are available online at http://aaa.princeton.edu/software.
k .
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| []
|
[
"Estimating Open Access Mandate Effectiveness: I. The MELIBEA Score",
"Estimating Open Access Mandate Effectiveness: I. The MELIBEA Score"
]
| [
"Philippe Vincent-Lamarre \nUniversité du Québec à Montréal\n\n\nUniversité d'Ottawa\n\n",
"Jade Boivin \nUniversité du Québec à Montréal\n\n",
"Yassine Gargouri \nUniversité du Québec à Montréal\n\n",
"Vincent Larivière \nUniversité de Montréal\n\n",
"Stevan Harnad \nUniversité du Québec à Montréal\n\n\nUniversity of Southampton\n\n"
]
| [
"Université du Québec à Montréal\n",
"Université d'Ottawa\n",
"Université du Québec à Montréal\n",
"Université du Québec à Montréal\n",
"Université de Montréal\n",
"Université du Québec à Montréal\n",
"University of Southampton\n"
]
| []
| MELIBEA is a Spanish database that uses a composite formula with eight weighted conditions to estimate the effectiveness of Open Access mandates (registered in ROARMAP). We analyzed 68 mandated institutions for publication years 2011-2013 to determine how well the MELIBEA score and its individual conditions predict what percentage of published articles indexed by Web of Knowledge is deposited in each institution's OA repository, and when. We found a small but significant positive correlation (0.18) between MELIBEA score and deposit percentage. We also found that for three of the eight MELIBEA conditions (deposit timing, internal use, and opt-outs), one value of each was strongly associated with deposit percentage or deposit latency (immediate deposit required, deposit required for performance evaluation, unconditional opt-out allowed for the OA requirement but no opt-out for deposit requirement). When we updated the initial values and weights of the MELIBEA formula for mandate effectiveness to reflect the empirical association we had found, the score's predictive power doubled (.36). There are not yet enough OA mandates to test further mandate conditions that might contribute to mandate effectiveness, but these findings already suggest that it would be useful for future mandates to adopt these three conditions so as to maximize their effectiveness, and thereby the growth of OA. | 10.1002/asi.23601 | [
"https://arxiv.org/pdf/1410.2926v1.pdf"
]
| 8,144,721 | 1410.2926 | fbec02b40b76235e2d0722df9ad310afa158ef38 |
Estimating Open Access Mandate Effectiveness: I. The MELIBEA Score
Philippe Vincent-Lamarre
Université du Québec à Montréal
Université d'Ottawa
Jade Boivin
Université du Québec à Montréal
Yassine Gargouri
Université du Québec à Montréal
Vincent Larivière
Université de Montréal
Stevan Harnad
Université du Québec à Montréal
University of Southampton
Estimating Open Access Mandate Effectiveness: I. The MELIBEA Score
MELIBEA is a Spanish database that uses a composite formula with eight weighted conditions to estimate the effectiveness of Open Access mandates (registered in ROARMAP). We analyzed 68 mandated institutions for publication years 2011-2013 to determine how well the MELIBEA score and its individual conditions predict what percentage of published articles indexed by Web of Knowledge is deposited in each institution's OA repository, and when. We found a small but significant positive correlation (0.18) between MELIBEA score and deposit percentage. We also found that for three of the eight MELIBEA conditions (deposit timing, internal use, and opt-outs), one value of each was strongly associated with deposit percentage or deposit latency (immediate deposit required, deposit required for performance evaluation, unconditional opt-out allowed for the OA requirement but no opt-out for deposit requirement). When we updated the initial values and weights of the MELIBEA formula for mandate effectiveness to reflect the empirical association we had found, the score's predictive power doubled (.36). There are not yet enough OA mandates to test further mandate conditions that might contribute to mandate effectiveness, but these findings already suggest that it would be useful for future mandates to adopt these three conditions so as to maximize their effectiveness, and thereby the growth of OA.
Introduction
The Open Access (OA) movement (BOAI 2002) arose as a result of two concurrent developments: (1) the "serials crisis," which made research journals increasingly unaffordable, hence inaccessible, to researchers' institutions (Okerson et al 1995;Miller et al 2010), even the richest ones (Harvard University Library 2012) and (2) the advent of the online medium, which made it possible in principle to make all research journal articles freely accessible to all users online.
The primary target content of OA is refereed research journal articles, and that is the only kind of item analyzed in this study. Researchers can provide OA to their journal articles in two different ways-by publishing in an OA journal ("Gold OA," often for a publication fee) or by publishing in a subscription journal and, in addition, self-archiving the final, peer-reviewed draft online ("Green OA") (Harnad et al 2004). OA also comes in two degrees: "Gratis" OA is free online access. "Libre" OA is free online access plus certain re-use rights (Suber 2008). We will only be considering Gratis OA in this study, because Gratis OA is (i) a prerequisite for Libre OA, (ii) faces fewer publisher restrictions, and (iii) is the most urgently needed by researchers. We will also only be considering Green OA (self-archiving), because (i) the majority of OA to date is Green, (ii) it does not entail any payment of fees, (iii) it can be provided by researchers themselves, and, as will be explained below, (iv) providing it can be mandated by researchers' institutions and funders (without having to pay any extra fees).
One might have expected that because of the many advantages provided by OA (Hitchcock 2013) -free access for all users, enhanced research uptake and impact, relief from the serials crisis, and the speed and power of the online mediumresearchers would all have hastened to make their papers (Green) OA ever since it became possible. But the growth of OA -possible since even before the birth of the Web in 1989 (Berners-Lee 1989) --has actually been surprisingly slow: For example, in 2009, already two decades after the Web began, Björk et al estimated the percentage of OA to be only 20.4% (the majority of it Green). In 2013 Archambault announced close to 50% OA, but Chen (2014) found only 37.8% and Khabsa & Giles (2014) even less (24%). All these studies reported great variation across fields and none took timing adequately into account (publication date vs. OA date): Björk et al (2014) pointed out that 62% of journals endorse immediate Green OA self-archiving by their authors, 4% impose a 6-month embargo, and 13% impose a 12-month embargo; so at least 79% of articles published in any recent year could already have been OA within 12 months of their date of publication via Green OA alone, 62% of them immediately, if authors were actually providing it.
There are many reasons why researchers have been so slow to provide Green OA even though they themselves would be its biggest potential beneficiaries (Harnad 2006). The three principal reasons are that (i) researchers are unsure whether they have the legal right to self-archive, (ii) they fear that it might put their paper's acceptance for publication at risk and (iii) they believe that self-archiving may be a lot of work. All these concerns are groundless but it has become increasingly clear with time that merely pointing out how and why they are groundless is not enough to induce authors to go ahead and self-archive of their own accord. Researchers' funders and institutions worldwide are beginning to realize that they need to extend their existing "publish or perish" mandates so as to make it mandatory to provide OA to researchers' publications, not only for the benefit of (i) the researchers themselves and of (ii) research progress, but also to maximize the (iii) return to the tax-paying public on its investment in funding the research.
Funders and institutions are accordingly beginning to adopt OA policies: Starting with NIH in the US and the Wellcome Trust in the UK, soon followed by the Research Funding Councils UK, the European Commission, and now President Obama's Directive to all the major US federal funding agencies, research funders the world over are beginning to mandate OA (ROARMAP 2014). In addition, research institutions, the providers of the research, are doing likewise, with Harvard, MIT, University College London and ETH Zurich among the vanguard, adopting OA mandates of their own that require all their journal article output, across all disciplines, funded and unfunded, to be deposited in their institutional OA repositories.
These first OA mandates, however, differ widely, both in their specific requirements and in their resultant success in generating OA. Some mandates generate deposit rates of over 80%, whereas others are doing no better than the global baseline for spontaneous (un-mandated) self-archiving (Gargouri et al 2013). As mandate adoption grows worldwide, it is therefore important to analyze the existing mandates to determine which conditions are essential to making a mandate effective.
We have accordingly analyzed the institutional mandates indexed by MELIBEA, a Spanish database that classifies OA mandates in terms of their specific conditions as well as providing a score for overall mandate strength. This score is based on a composite formula initialized with a-priori values and weights for eight conditions that MELIBEA hypothesizes to be predictive of mandate effectiveness (see Figure 3 and Table 1). The purpose of our study was (i) to test the predictive power of the overall MELIBEA score for mandate strength in terms of deposit rate and deposit timing, (ii) to test individually the association of each of the conditions with deposit rate and timing, and (iii) to update the initialized values and weights of the MELIBEA formula to reflect the empirical findings of (ii).
What we found was a small but significant positive correlation between MELIBEA score and deposit rate. We also found that for three of the eight MELIBEA conditions (deposit timing, internal use, and opt-outs), one of their values was most associated with deposit rate and latency (immediate deposit required, deposit required for performance evaluation, unconditional opt-out allowed for the OA requirement but no opt-out for deposit requirement). When we updated the initial values and weights of the MELIBEA formula for mandate strength to reflect this association, the score's predictive power was doubled. This suggests that it would be useful to adopt OA mandates with these conditions in order to maximize their effectiveness, and thereby the growth of OA.
Databases Used
The data for our analysis were drawn from several databases. The ROAR Registry of OA Repositories provided a database of all open access repositories (ROAR, http://roar.eprints.org). The ROARMAP Registry of OA Repository Mandatory Archiving Policies provided the subset of the ROAR repositories that had an OA mandate (ROARMAP, http://roarmap.eprints.org). An in-house version of Thomson Reuters Web of Science (WoS) database hosted at the Observatoire des sciences et des technologies (OST-UQAM) provided the bibliographic metadata for all (WoS-indexed) articles published in years 2011-2013 by any author affiliated with the ROARMAP subset of institutions with OA mandates that had been adopted by 2011. The MELIBEA Directory and Estimator of OA Policies provided a classification of the OA mandates in terms of their specific conditions (assigning a numerical value to each option for each condition) plus an overall score based on a weighted combination of eight of those conditions as an estimate of mandate strength (MELIBEA (http://accesoabierto.net/politicas) (Table 1).
WoS indexes the 12,000 most cited peer-reviewed academic journals in each research area. (This corresponds to slightly more than 11% of the 105,000 peer-reviewed journals listed in Ulrich's Global Serials Directory, http://ulrichsweb.com.) We also used the Webometrics Ranking Web of Universities, which estimates institutions' "excellence" based on how many of their published articles are among the top 10% most cited articles (the lower the score the higher the rank). (http://webometrics.info). We excluded the bottom 19 institutions from our sample as outliers (i.e., those with Webometrics ranks beyond 5000th; see Figure 1). To balance representativeness and sample size, we also excluded institutions that had fewer than 30 publications during the 2011-13 time window under study. This yielded 68 institutional mandates for analysis. Figure 1: Distribution of mandated institutions according to Webometrics "excellence" ranking based on citedness (lower score means higher rank).
Deposit Rate and Deposit Latency
Each of the 68 institutional repositories was crawled to determine what percentage of its WoS-indexed articles had been deposited (deposit rate), and when they were deposited (deposit latency, i.e., the delay between publication date, estimated by WoS indexation date, and deposit date) for publication years 2011years , 2012years and 2013years (crawled in April 2014. Deposits could be of two kinds: Open Access (OA) or Restricted Access (RA), to comply with publisher OA embargoes. We subtracted publication date from deposit date, so negative values mean an article was deposited before publication and positive values mean it was deposited after publication ( Figure 2). Figure 2: Distribution of institutions' deposit rates (percentage of WoS articles deposited) and deposit latencies (in days, relative to publication date) for Open Access (OA) and Restricted Access (RA) Deposits. The peak for RA latency is negative (i.e., before publication) and the peak for OA latency is positive (i.e., after publication).
MELIBEA Score for Mandate Strength
We used each mandating institution's overall MELIBEA score for mandate strength as of April 6 2014. This score is a weighted combination of eight specific individual conditions and their respective options ( Figure 3) for each OA policy for each mandating institution (Table 1). We tested how well the conditions jointly predicted deposit rate and deposit latency. Then we analyzed the conditions separately with Analyses of Variance (ANOVA) to determine which individual options contributed to deposit rate or latency.
40%
( 1) 10%( 2) 5%( 3) 10%( 4) 5% ( 5) 10%( 6) 10%( 7) 10% ( 8) Figure 3: The eight mandate conditions (renamed C1-C8 here) and their initial weights and values in MELIBEA's formula for mandate strength (See Table 1 below to identify the eight conditions). Table 1: The eight conditions (re-numbered) together with the initial values assigned by MELIBEA for each option for each condition in the formula for the MELIBEA score for mandate strength Figure 4). Separate ANOVAS were done (see below) for the three boldface conditions and options (2,4,7). The other conditions either did not have enough mandate instances (3, 5, 6) or were irrelevant to this study (1, 8).
We excluded from this study all OA policies that were not mandates (i.e., compliance was not required, nor requested) because our specific interest here is in which of the conditions of mandatory OA policies make a contribution to increasing deposit rate and/or decreasing deposit latency. For the individual t-tests and ANOVAs testing the eight MELIBEA conditions and their respective options separately, three of the conditions (Version (C3), Embargo Length (C5), Copyright (C6)) did not have enough institutional mandate instances to be tested with an ANOVA; one condition (Theses (C8)) was irrelevant to our target content, which was only refereed journal articles; and for one condition (Required or Request (C1)), our study concerned only the mandatory (required) policies, not recommended ones. We accordingly analyzed only conditions C2 (Opt-Out), C4 (Deposit Timing) and C7 (Internal Use) with their respective options:
Value
--2 12 months after publication 1 Request 0.5 6 months after publication 2 Requirement --2 More than 12 months after publication --2 Unspecified ('after period stipulated by the publisher')
2
No deposit opt--out but unconditional OA opt--out 2
No deposit opt--out but conditional OA opt--out 1 Allow opt--out from copyright reservation case by case --1 Deposit opt--out and unconditional OA opt--out 2 1 Any publishing or copyright agreements concerning articles have to comply with the OA policy 2 Authors retain the non--exclusive rights of explotation to allow self--archiving 0. Condition C2 (Opt-Out): Deposit and OA are both required and (2a) opt-out is not allowed from the deposit requirement but conditional opt-out is allowed from the OA requirement, on a case-by-case basis; (2b) opt-out is not allowed from the deposit requirement but unconditional opt-out is allowed from the OA requirement; (2c) unconditional opt-out is allowed from both the deposit and the OA requirements Condition C4 (Deposit Timing): Deposit is required (4a) at time of acceptance, (4b) at time of publication, or (4c) time unspecified.
Condition C7 (Internal Use): Deposit is (7a) required for internal use or (7b) not required for internal use.
Power of Initialized MELIBEA Score to Predict Deposit Rate and Latency
To test the correlation between the MELIBEA score for mandate strength and deposit (1) rate and (2) latency, we analyzed OA and RA deposits separately as well as jointly (OA + RA). For deposit latency, which was normally distributed, we used the Pearson product moment correlation coefficient. For deposit rate, which was not normally distributed ( Figure 2), we used permutation tests (Edgington & Onghena, 2007).
The individual mandate conditions were then tested using a combination of t-tests and one-way ANOVAs, with and without permutation testing. Permutation testing consists of permuting the values in a sample randomly a large number of times (e.g. 10,000) and computing the test statistic for each new sample. The p value is the probability that the test statistic of the original sample will occur within the distribution generated. Permutation testing does not presuppose homogeneity of variance or normality.
The overall MELIBEA score is a weighted combination of the eight specific conditions of OA policies (Table 1), assigning a value to each of their options, and then combining them into the weighted formula in Figure 1 (40%(C 1 ) x 10%(C 2 ) x 5%(C 3 ) x 10%(C 4 ) x 5% (C 5 ) x 10%(C 6 ) x 10%(C 7 ) x 10% (C 8 )).
MELIBEA initialized the overall score with these weightings on the 8 conditions as well as with their respective option values (on the basis of theory and what prior evidence was available). Each of these initialized values and weightings is now awaiting evidence-based validation of their power to predict the effectiveness of OA mandates (i.e., how many OA and RA deposits they generate, and how soon) so that either the values or the weightings can be updated to maximize their predictive power as the evidence base grows. At the end of this section we will illustrate how these initial values and weightings can be updated based on the present findings so as to make the MELIBEA score more predictive of deposit rate and latency.
Our analysis has found that this initial overall MELIBEA score has a small significant positive correlation with deposit rate for OA deposits ( Figure 5) but not for RA deposits (nor for OA + RA jointly). For deposit latency there is no correlation at all with the overall MELIBEA score. Figure 5: Correlation between MELIBEA mandate strength score and deposit rate, by publication year, for RA deposits, OA deposits and RA + OA deposits jointly
Effect of Deposit-Timing Condition (C4) on Deposit Rate and Latency
To update and optimize the weightings on the specific OA policy conditions, we need to look at their predictive power individually. As noted, two of the eight conditions are not relevant for this study, and for three of the conditions the number of OA policies that have adopted them is so far still too small to test their effectiveness. We report here the results for only those three conditions (Opt-Out (C2), Deposit Timing (C4) and Internal Use (C7)) that have reached testable sample sizes.
For OA and RA deposits combined, mandates that required deposit "At time of acceptance" had a significantly higher deposit rate than mandates that required deposit "At time of publication" or "Unspecified." The same pattern is present for each of the three years, significant for year 2011 and almost significant for all three years combined ( Figure 6). The effect for RA deposits alone and OA deposits likewise shows exactly the same pattern, for each of the three years, but without reaching statistical significance. Figure 6: Average deposit rate when deposit is required at time of acceptance, at time of publication, or unspecified (Condition C4). For each of the three years, requiring deposit at time of acceptance consistently generates a higher deposit rate for RA + OA considered jointly (upper histogram) as well as for OA (left) and RA (right) considered separately (lower histogram).
When the deposit must be done is an especially important parameter in determining mandate effectiveness. Our interpretation is that requiring deposit at time of acceptance gives authors a much clearer and more specific time-marker than requiring deposit at time of publication. After submission, refereeing, revision and resubmission, successful authors receive a dated letter from the journal notifying them that their final draft has now been accepted for publication. That is the natural point in authors' workflow to deposit their final draft: They know the acceptance date then, and they have the final draft in hand. Publication date, in contrast, is uncertain: Authors don't know when the published version will appear, the delay ("publication lag") following acceptance can be quite long (sometimes months or even years), the calendar date on the published issue may not correspond to the actual date it appeared, and by the time the article is published the author may no longer have the final draft available for deposit. Requiring deposit at an unspecified date is even vaguer, and probably closer to not requiring deposit at all.
Figure 7: Average deposit latencies (delays relative to publication date = 0). There is no interpretable effect of deposit timing requirements (Condition C4) on average deposit latencies, either for RA + OA (shown), or for RA and OA separately.
One would expect the deposit timing requirement (Condition C4) to have an effect not only on deposit rate but also on deposit latency, i.e., the time at which the deposit is done; yet no interpretable pattern or statistical significance was observed across the three years for RA, OA or RA + OA (Figure 7). RA deposits are eventually converted to OA (as the OA embargo elapses) so it may be that the latency effect takes the form of more early RA deposits, which later become OA, rather than making either RA or OA deposits themselves occur earlier. Below, however, we will find that it is the internal-use condition (C7) that has a significant effect on deposit latency.
Which version must be deposited (the author's final draft or the publisher's version of record --MELIBEA Condition C3) might also be expected to have an effect on deposit rate and/or latency, but it was not possible to make the comparison, because of our exclusion criteria: Before excluding the institutions that had fewer than 30 publications or Webometric rank beyond 5000, mandates requiring deposit of the author's version did have a significantly higher deposit rate than those requiring deposit of the publisher's version. But once the weaker institutions were excluded, almost all the remaining policies required the author's version, so no comparison was possible. This pattern is consistent, however, with the interpretation that requiring deposit immediately upon acceptance generates more deposits, and hence more deposits of the author's final draft. As there are more publisher restrictions on the later publisher's version than on the earlier author's version, and as many authors are reluctant to put their chances of getting published at risk by challenging their publishers' restrictions, it is to be expected that the compliance rate for the author's version will be higher, and also that deposit will be done sooner, with papers initially deposited as RA and then made OA after the publisher OA embargo has elapsed.
Effect of Internal-Use Condition (C7) on Deposit Rate and Latency
When it is stipulated that deposit is required "For internal use" (Condition C7), deposit rates are significantly higher for OA and RA deposits combined (Figure 8). The effect is larger and significant for RA deposits alone and smaller but in the same direction for OA deposits alone. With an internal-use requirement, deposit latency is also significantly shorter for OA and RA combined, but this time the effect is larger and significant for OA deposits alone, smaller and not significant for RA deposits alone ( Figure 9). (Note that especially for 2013 this measurement in April 2014 was probably too early for a reliable estimate because the average delay of about 175 days for mandates without an internal-use requirement had not yet been reached in April 2014. Similarly, the deposit rate for 2011 may be inflated because papers published in that year had the longest time to get deposited by April 2014.) Figure 8: Average deposit rates, with and without internal use. For each of the three years, average OA + RA deposit rates are higher when the deposit is required for internal institutional use (e.g., research evaluation; Condition C7). The pattern is the same for OA alone (left) and RA alone (right), for each of the three years, but the deposit rate difference is bigger for RA. Figure 9: For each of the three years the average OA + RA deposit latencies are shorter when there is a requirement to deposit for internal use (Condition C7). (Higher is later, lower is earlier; 0 is the date of publication; latency is in days, relative to publication date). The pattern is the same for OA alone (left) and RA alone (right) for each of the three years, but the effect is bigger for OA.
The internal-use requirement pertains mostly to research performance evaluation, on which a researcher's rank and salary often depend. Hence it is predictable that researchers will be eager to make their papers available for this purpose by depositing them (Rentier & Thirion 2011). And since internal use only requires deposit, not OA, it is also predictable that this requirement will have a stronger effect on RA deposit than on OA deposit. What is a little surprising is that an internal-use requirement accelerates OA deposits more than RA deposits. This may be because our latency data were noisier and had more gaps (missing data) than our deposit data. It could also be because RA deposits occur earlier than OA deposits in any case (because of publisher OA embargoes), leaving less room for shortening their latency even further. Or it may be because the internal-use requirement (C7) has the effect of reinforcing the depositupon-acceptance requirement (C4) for OA deposits (those that have no publisher OA embargo with which the author wishes to comply): "You have to deposit immediately on acceptance anyway, for performance evaluation, so you may as well make it OA immediately too, rather than RA."
Effect of Opt-Out Condition (C2) on Deposit Rate and Latency
The last policy condition that proved significant concerned the right to waive or opt out of a requirement (Condition C2): We looked at the subset of mandates where the author was required both to deposit and to make the deposit OA, but was allowed to opt out of the OA requirement (only) (i.e., the deposit could be RA instead of OA). In particular, the effect concerned whether (i) unconditional opt-out from the OA requirement was allowed or (ii) the author had to negotiate each opt-out on a conditional, case-by-case basis. We found that an OA requirement allowing unconditional opt-out generates a higher deposit rate than an OA requirement allowing only a conditional opt-out ( Figure 10). This result might seem paradoxical at first, because in a sense a "requirement" that allows an unconditional opt-out is not a requirement at all! So why would it generate a higher deposit rate than a requirement allowing only a conditional opt-out?
We think the answer is in the component of the requirement from which no opt-out at all is allowed, namely, the deposit requirement itself: The internal-use requirement (C7) reinforces compliance with the immediate-deposit requirement (C4). In contrast, having to negotiate opt-out from the OA requirement case-by-case (conditional OA opt-out) inhibits compliance with the immediate-deposit requirement, whereas the possibility of unconditional opt-out from the OA requirement (C2) reinforces compliance with the immediate-deposit requirement. Figure 10: When deposit is required (no opt out) but authors can opt out of OA unconditionally (Condition C2), deposit rates are higher than when each individual OA opt-out has to be negotiated on a case-by-case basis. The effect is especially strong for RA (lower right); for each of the three years OA (lower left) and OA+RA (above) show the same pattern as RA (above). All the data on deposit rates and deposit latencies can be thought of as measures of mandate compliance rates. In the case of the effect of allowing unconditional opt-out from the OA requirement, the biggest observed increase is in RA deposits, and our interpretation is that authors are more likely to comply with a deposit requirement if they can choose to deposit as RA rather than OA whenever they feel it is necessary, without the prospect of having to do case-by-case negotiation for a waiver of the OA requirement. Knowing that making the deposit OA is optional makes it more likely that an author will comply with the requirement to deposit-on-acceptance -or even to deposit at all.
Power of Updated MELIBEA Score to Predict Deposit Rate
The original option-values and condition-weightings in the formula for the MELIBEA score were initialized largely on the basis of guesswork. Those initial values and weightings can and should be updated as empirical evidence accrues, so as to increase their power to predict mandate strength. Using the present findings on the association of Conditions C2 (Opt-Out), C4 (Deposit Timing) and C7 (Internal Use), we have updated option-values and condition-weightings in order to reflect the findings of this study: Higher values have been assigned to those options that our t-tests and ANOVAs showed to be predictive of deposit rate and latency (Table 2). We have also re-weighted the MELIBEA formula for mandate strength, assigning zero weight to the two irrelevant conditions (C1, Mandate/Request and C8, Theses) and equal weights to the rest ( Figure 12). 0( 1) 1( 2) 1( 3) 1( 4) 1( 5) 1( 6) 1( 7) 0( 8) Figure 12: Updates of the initial weights on the eight conditions (C1-C8) in MELIBEA's formula for mandate strength based on the findings of the present study. Condition C1 (Mandate or Request) dropped because this study only considers mandates. Condition C8 (Theses) dropped because this study considers only journal articles. All other weights are set as one. We make no claim that the new values and weightings are as yet optimal. A larger sample may well bring other conditions and their options into play as more mandates are adopted. In addition, latency effects will be clearer with a more complete and accurate sample than the present one. Nor are deposit rates and latencies the only criterion against which one might wish to validate mandate strength estimators. However, it is clear that our evidence-based updates of the initial values and weights already do increase the power of the MELIBEA Score to predict deposit rate (though not latency). The correlations in Figure 13 are all higher than those in Figure 5, and especially dramatically so for RA deposits.
Limitations
This is an exploratory study in which we made predictions in advance as to which variables we expected would have an effect on deposit rates and latency, in which direction, and why. As only the three predicted variables had significant effects, we did not apply a Bonferroni correction for multiple tests (though we tested many other variables). We take the replication of the direction of the effects (whether significant or insignificant) --across the three independent time periods, and for RA + OA jointly as well as for RA and OA separately --to further decrease the likelihood that the observed pattern of effects was due to chance.
(1) The data from multiple databases have limitations because they are incomplete, noisy and approximate. We have publication data from the top 12,000 journals, but we only have access to the full deposit and publication date of a portion of them. The latency data are especially noisy and incomplete.
(2) The short ( (3) MELIBEA's classification and weighting of OA mandates was provisional: the initialized option-values and condition-weights were not yet validated and somewhat subjective. As the number and age of the mandates increases and the ranking and classification system is updated and optimized, the data should become less noisy and variable. Moreover, if new mandates adopt the conditions that have already emerged as most important from the present analysis, both the rate and speed of deposit are likely be enhanced.
Summary and Conclusions
Our first finding was that the higher an institution's MELIBEA score for OA mandate strength, the higher was the rate of deposit in its institutional repository for each year, but the correlation was small, and was significant only for OA + RA deposits jointly and for OA deposits alone, but not for RA deposits alone. There was also no correlation with latency.
In an effort to increase the predictive power of the MELIBEA score for mandate strength and to determine the effects of its conditions individually, we analyzed three of the eight conditions separately to test their effects on deposit rate and latency. (Two of the other MELIBEA conditions were irrelevant because they pertained to non-mandates or to theses, respectively, and for the remaining three the sample was too small to test them separately.) The three conditions we examined were (C4) when the author had to do the deposit (upon acceptance, upon publication, or unspecified; (C7) whether the deposit was required for internal use (such as performance evaluation) and (C2) whether the author could opt out of the requirement to make the deposit OA unconditionally or only conditionally, on a case-by-case basis.
For the deposit-timing condition (C4), the requirement to deposit immediately upon acceptance generated significantly higher deposit rates for OA + RA deposits combined. The pattern was also the same for RA and OA separately, and for each individual year, but there was no consistent or significant effect on deposit latency.
For the internal-use condition (C7), requiring deposit for internal use generated significantly higher deposit rates for OA + RA deposits combined. The pattern was also the same for RA and OA separately, and for each individual year, but the rate-increase was larger for RA than for OA. The internal-use requirement had an effect on latency as well: Deposit is done earlier when it is required for internal use (same pattern for RA + OA, RA, OA, and for each year), but the latency-decrease effect is larger for OA than for RA.
Our interpretation for C7 is that when the deposit is required for internal use, more authors who may feel inhibited by a publisher OA embargo from depositing OA go ahead and deposit RA rather than not depositing at all or waiting for the end of the publisher OA embargo to deposit. This is what increases the rate of RA deposit. For the OA deposits, which tend to be done later than RA deposits (because of publisher OA embargoes), the effect of their being needed for internal use is to reduce their latency (i.e., speed them up). RA deposits tend to be done much earlier than OA deposits.
For the opt-out condition (C2), allowing unconditional opt-out from the OA requirement (but not the deposit requirement) generated significantly higher deposit rates for OA + RA deposits combined. The pattern was the same for RA and OA separately, and for each individual year, but the effect was especially marked for RA. (There was no effect on latency.) Our interpretation for C2 is that authors are more likely to comply with a deposit requirement if they know they can choose to deposit as RA rather than OA unconditionally.
Using our findings on the effects of these three conditions, we updated the initial optionvalues and condition-weightings of the MELIBEA formula for mandate strength with the result that its correlation with deposit rate increased overall, and especially for RA deposits. There is still no correlation with deposit latency, but as RA deposits eventually become OA deposits, increases in RA may reflect earlier deposit of papers that might otherwise only have been deposited later, as OA. (Note that RA deposits are indirectly accessible even during the OA embargo through individual user requests via the repository's request-a-copy Button; Sale et al 2014.)
Our updating of a subset of the MELIBEA values and weights shows that in combination, three at least of the MELIBEA parameters can already predict deposit rate -and hence reflect mandate strength --better than the original MELIBEA overall score. This is just an illustration of how further research can be used to keep optimizing the predictive power of estimators of mandate strength.
Practically speaking, it also emerges from this analysis that the deposit-on-acceptance requirement, internal-use requirement, and the possibility of opting out of making the deposit OA unconditionally are each very important factors in the effectiveness of an OA mandate in generating greater author compliance and hence more deposits and more OA, sooner. As the rate of adoption of OA policies is now growing too, these findings make it possible for policy-makers to make them both more evidence-based and more effective by taking these findings into account in designing their mandates.
This study was based on institutional mandates rather than funder mandates, because it is easier to identify total journal output for an institution's researchers than for a funder's grantees, but there is no reason to think the conclusions do not apply to both kinds of mandates.
Figure 4 :
4Distribution of MELIBEA scores for OA mandate strength
Figure 11 :
11There is no consistent, interpretable effect of conditional vs. unconditional OA opt-out (Condition C2) on deposit latencies across the years. (Only OA + RA shown here)
Figure 13 :
13Correlation between updated MELIBEA mandate strength score and deposit rate, by publication year, for RA deposits, OA deposits and RA + OA deposits jointly, with values and weights updated to reflect the findings of this study. (Cf.Figure 5). (The correlations with latency remained inconsistent and uninterpretable.) opt--out but unconditional OA opt--out 5No deposit opt--out but conditional OA opt--out 0.5 Allow opt--out from copyright reservation case by case 0 Deposit opt--out and unconditional OA opt--out 0 Any publishing or copyright agreements concerning articles have to comply with the OA policy 0.5Authors retain the non--exclusive rights of explotation to allow self-
3-year) time window for which we have deposit rate and latency data provides a narrowing picture of changes across time, especially for the most recent year, 2013. For the articles published earliest in the 3-year time window (2011) the deposit rates have stabilized and the latency averages are based on a long enough time-base (2.5 years), whereas the deposit rates for articles published in the second half of 2013 are based on a time window of less than a year.
Table 2 :
2Updated values for the MELIBEA mandate strength score (Table
1), based on the findings of the present study. Higher values have been
assigned to the options that t-tests and ANOVAs showed to be predictive
of deposit rate and latency.
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[
"New sizes of complete arcs in P G(2, q)",
"New sizes of complete arcs in P G(2, q)"
]
| [
"Alexander A Davydov \nInstitute for Information Transmission Problems\nDipartimento di Matematica e Informatica, Università degli Studi di Perugia\nRussian Academy of Sciences\nBol'shoi Karetnyi per. 19, GSP-4, Via Vanvitelli 1127994, 06123Moscow, PerugiaRussia, Italy\n",
"Adav@iitp Ru \nInstitute for Information Transmission Problems\nDipartimento di Matematica e Informatica, Università degli Studi di Perugia\nRussian Academy of Sciences\nBol'shoi Karetnyi per. 19, GSP-4, Via Vanvitelli 1127994, 06123Moscow, PerugiaRussia, Italy\n",
"Giorgio Faina [email protected] \nInstitute for Information Transmission Problems\nDipartimento di Matematica e Informatica, Università degli Studi di Perugia\nRussian Academy of Sciences\nBol'shoi Karetnyi per. 19, GSP-4, Via Vanvitelli 1127994, 06123Moscow, PerugiaRussia, Italy\n",
"Stefano Marcugini \nInstitute for Information Transmission Problems\nDipartimento di Matematica e Informatica, Università degli Studi di Perugia\nRussian Academy of Sciences\nBol'shoi Karetnyi per. 19, GSP-4, Via Vanvitelli 1127994, 06123Moscow, PerugiaRussia, Italy\n",
"Fernanda Pambianco [email protected] \nInstitute for Information Transmission Problems\nDipartimento di Matematica e Informatica, Università degli Studi di Perugia\nRussian Academy of Sciences\nBol'shoi Karetnyi per. 19, GSP-4, Via Vanvitelli 1127994, 06123Moscow, PerugiaRussia, Italy\n"
]
| [
"Institute for Information Transmission Problems\nDipartimento di Matematica e Informatica, Università degli Studi di Perugia\nRussian Academy of Sciences\nBol'shoi Karetnyi per. 19, GSP-4, Via Vanvitelli 1127994, 06123Moscow, PerugiaRussia, Italy",
"Institute for Information Transmission Problems\nDipartimento di Matematica e Informatica, Università degli Studi di Perugia\nRussian Academy of Sciences\nBol'shoi Karetnyi per. 19, GSP-4, Via Vanvitelli 1127994, 06123Moscow, PerugiaRussia, Italy",
"Institute for Information Transmission Problems\nDipartimento di Matematica e Informatica, Università degli Studi di Perugia\nRussian Academy of Sciences\nBol'shoi Karetnyi per. 19, GSP-4, Via Vanvitelli 1127994, 06123Moscow, PerugiaRussia, Italy",
"Institute for Information Transmission Problems\nDipartimento di Matematica e Informatica, Università degli Studi di Perugia\nRussian Academy of Sciences\nBol'shoi Karetnyi per. 19, GSP-4, Via Vanvitelli 1127994, 06123Moscow, PerugiaRussia, Italy",
"Institute for Information Transmission Problems\nDipartimento di Matematica e Informatica, Università degli Studi di Perugia\nRussian Academy of Sciences\nBol'shoi Karetnyi per. 19, GSP-4, Via Vanvitelli 1127994, 06123Moscow, PerugiaRussia, Italy"
]
| []
| New upper bounds on the smallest size t2(2, q) of a complete arc in the projective plane P G(2, q) are obtained for 853 ≤ q ≤ 2879 and q = 3511, 4096, 4523, 5003, 5347, 5641, 5843, 6011. For q ≤ 2377 and q = 2401, 2417, 2437, the relation t2(2, q) < 4.5 √ q holds. The bounds are obtained by finding of new small complete arcs with the help of computer search using randomized greedy algorithms. Also new sizes of complete arcs are presented. | null | [
"https://arxiv.org/pdf/1004.2817v5.pdf"
]
| 115,180,834 | 1004.2817 | 98ea404b731de7bf4ece3c9fd88f41bdef61bb7d |
New sizes of complete arcs in P G(2, q)
28 Aug 2010
Alexander A Davydov
Institute for Information Transmission Problems
Dipartimento di Matematica e Informatica, Università degli Studi di Perugia
Russian Academy of Sciences
Bol'shoi Karetnyi per. 19, GSP-4, Via Vanvitelli 1127994, 06123Moscow, PerugiaRussia, Italy
Adav@iitp Ru
Institute for Information Transmission Problems
Dipartimento di Matematica e Informatica, Università degli Studi di Perugia
Russian Academy of Sciences
Bol'shoi Karetnyi per. 19, GSP-4, Via Vanvitelli 1127994, 06123Moscow, PerugiaRussia, Italy
Giorgio Faina [email protected]
Institute for Information Transmission Problems
Dipartimento di Matematica e Informatica, Università degli Studi di Perugia
Russian Academy of Sciences
Bol'shoi Karetnyi per. 19, GSP-4, Via Vanvitelli 1127994, 06123Moscow, PerugiaRussia, Italy
Stefano Marcugini
Institute for Information Transmission Problems
Dipartimento di Matematica e Informatica, Università degli Studi di Perugia
Russian Academy of Sciences
Bol'shoi Karetnyi per. 19, GSP-4, Via Vanvitelli 1127994, 06123Moscow, PerugiaRussia, Italy
Fernanda Pambianco [email protected]
Institute for Information Transmission Problems
Dipartimento di Matematica e Informatica, Università degli Studi di Perugia
Russian Academy of Sciences
Bol'shoi Karetnyi per. 19, GSP-4, Via Vanvitelli 1127994, 06123Moscow, PerugiaRussia, Italy
New sizes of complete arcs in P G(2, q)
28 Aug 20101
New upper bounds on the smallest size t2(2, q) of a complete arc in the projective plane P G(2, q) are obtained for 853 ≤ q ≤ 2879 and q = 3511, 4096, 4523, 5003, 5347, 5641, 5843, 6011. For q ≤ 2377 and q = 2401, 2417, 2437, the relation t2(2, q) < 4.5 √ q holds. The bounds are obtained by finding of new small complete arcs with the help of computer search using randomized greedy algorithms. Also new sizes of complete arcs are presented.
Introduction
Let P G(2, q) be the projective plane over the Galois field F q . An n-arc is a set of n points no 3 of which are collinear. An n-arc is called complete if it is not contained in an (n + 1)-arc of P G(2, q). Surveys of results on arcs can be found in [9,10]. In [10] the close relationship between the theory of complete n-arcs, coding theory and mathematical statistics is presented. In particular a complete arc in a plane P G(2, q), points of which are treated as 3-dimensional q-ary columns, defines a parity check matrix of a q-ary linear code with codimension 3, Hamming distance 4 and covering radius 2. Arcs can be interpreted as linear maximum distance separable (MDS) codes and they are related to optimal coverings arrays [8] and to superregular matrices [11].
One of the main problems in the study of projective planes, which is also of interest in Coding Theory, is the determination of the spectrum of possible sizes of complete arcs. Especially the problem of determining t 2 (2, q), the size of the smallest complete arc in P G (2, q), is interesting.
In Section 2 we give upper bounds on t 2 (2, q) for 853 ≤ q ≤ 2879 and q = 3511, 4096, 4523, 5003, 5347, 5641, 5843, 6011. These bounds are new for almost all q. For q ≤ 2377 and q = 2401, 2417, 2437, the relation t 2 (2, q) < 4.5 √ q holds. For smaller q slightly smaller bounds hold. The upper bounds have been obtained by finding of new small complete arcs with the help of the randomized greedy algorithms described in [1,Sect. 2], [5,Sect. 2]. In Section 3 we present new sizes of complete arcs in P G(2, q) with 169 ≤ q ≤ 349 and q = 1013, 2003. 2 Small complete k-arcs in P G(2, q), 853 ≤ q ≤ 2879
In the plane P G(2, q), we denote t 2 (2, q) the smallest known size of complete arcs. For q ≤ 841, the values of t 2 (2, q) < 4 √ q are collected in [2, Tab. 1].
In Tables 1 and 2, the values of t 2 (2, q) for 853 ≤ q ≤ 2879 and q = 3511, 4096, 4523, 5003, 5347, 5641, 5843, 6011 are given. We denote A q = 4.5 √ q − t 2 (2, q) , B q a superior approximation of t 2 (2, q)/ √ q. Also,
C q = 5 √ q − t 2 (2, q)
. For all q in Table 1 and q = 2401, 2417, 2437 in Table 2, it holds that t 2 (2, q) < 4.5 √ q.
In [7], complete k-arcs are obtained with k = 4( √ q−1), q = p 2 odd, q ≤ 1681 or q = 2401. For even q = 2 h , 10 ≤ h ≤ 15, the smallest known sizes of complete k -arcs in P G(2, q) are obtained in [3], see also [2, p. 35]. They are as follows: k = 124, 201, 307, 461, 665, 993, for q = 2 10 , 2 11 , 2 12 , 2 13 , 2 14 , 2 15 , respectively. Also, 6( √ q − 1)-arcs in PG(2, q), q = 4 2h+1 , are constructed in [4]; for h ≤ 4 it is proved that they are complete. It gives a complete 3066-arc in PG (2, 2 18 ). In Tables 1 and 2, we use the results of [3,7] for q = 31 2 , 37 2 , 41 2 , 7 4 , 2 10 , 2 11 . The rest of sizes k for small complete arcs in Tables 1 and 2 is obtained in this work by computer search with the help of the randomized greedy algorithms.
From Tables 1 and 2, we obtain Theorems 1 and 2. Our methods allow us to obtain small arcs in P G(2, q) for q ≤ 6011, using our present computers. We plan to write on these arcs sizes in a journal paper.
Let c be a constant independent of q. Let t(P q ) be the size of the smallest complete arc in any projective plane P q of order q. In [12], for sufficiently large 3 Table 1. The smallest known sizes t 2 = t 2 (2, q) < 4.5
√ q of complete arcs in Table 2. The smallest known sizes t 2 = t 2 (2, q) < 5 √ q of complete arcs in planes q, it is proved that t(P q ) ≤ √ q log c q, c = 300. The logarithm basis is not noted as the estimate is asymptotic. For definiteness, we use the binary logarithms. We introduce D q (c) and D q (c) as follows:
planes P G(2, q). A q = 4.5 √ q − t 2 (2, q) , B q > t 2 (2, q)/ √ q q t 2 A q B q q t 2 A q B q q t 2 A q B= t 2 (2, q) < 4.5 √ q of complete arcs in planes P G(2, q). A q = 4.5 √ q − t 2 (2, q) , B q > t 2 (2, q)/ √ q q t 2 A q B q q t 2 A q B q q t 2 A q BP G(2, q). A q = 4.5 √ q − t 2 (2, q) , B q > t 2 (2, q)/ √ q, C q = 5 √ q − t 2 (2, q) q t 2 A q C q B q q t 2 C q B q q t 2 C q Bt 2 (2, q) = D q (c) √ q log c 2 q, t 2 (2, q) = D q (c) √ q log c 2 q.
From Tables 1, 2 (1)
Moreover, let
t 2 (2, q) = 0.73331 √ q log 0.75 2 q, ∆ q = t 2 (2, q) − t 2 (2, q), P q = 100∆ q t 2 (2, q) %.
It holds that
− 1.86 ≤ ∆ q ≤ 1.23. (2) −1.73% < P q < 1.31% if 173 ≤ q ≤ 997, −0.80% < P q < 0.93% if 1009 ≤ q ≤ 1999, −0.53% < P q < 0.54% if 2003 ≤ q ≤ 2879.(3)
In other words, t 2 (2, q) = 0.73331 √ q log 0.75 2 can be treated as a predicted value of t 2 (2, q). Then ∆ q is the difference between the smallest known size t 2 (2, q) of complete arcs and the predicted value. Finally, P q is this difference in percentage terms of the smallest known size.
By (2),(3), the magnitude of the difference ∆ q is smaller than two. The magnitude of the percentage value P q is smaller than two for q < 1000 and smaller than one for q > 1000. The region of P q is decreasing with growth of q. Also, by (1), the region of D q (0.75) is decreasing with growth of q.
The graphs of values of D q (0.75), ∆ q , and P q are shown on Figures 1-3. Examples for great q are given in Table 3. Table 3. The smallest known sizes t 2 = t 2 (2, q) < 5 √ q of complete arcs in The examples confirm Observation 1. So, along with B q , the values D q (c), in particular with c = 0.75, can be useful for estimates of complete arcs sizes.
planes P G(2, q) with great q. B q > t 2 (2, q)/ √ q, C q = 5 √ q − t 2 (2, q) q t 2 C q B q D q (1) D q ( 1 2 ) D q ( 3 4 ) q t 2 C q B q D q (1) D q ( 1 2 ) D q (
Note that a complete 302-arc of Table 3 improves the result of [3] for q = 2 12 . From Tables 1-3 Taking into account (1) and Table 3, we assume that the following upper bound on the smallest size t 2 (2, q) of complete arc in the plane P G(2, q) holds. Let m 2 (2, q) be the greatest size of complete arcs in P G(2, q). For odd q, m 2 (2, q) = q + 1. For even q, m 2 (2, q) = q + 2. For q = p 2 there is the complete (q − √ q + 1)-arc [10]. For q odd there is a complete 1 2 (q + 5)-arc [13]. For q ≡ 2 (mod 3) odd, 11 ≤ q ≤ 3701 [2], and for q ≡ 1 (mod 4), q ≤ 337 [6], there is a complete 1 2 (q + 7)-arc. For even q ≥ 8 there is a complete 1 2 (q + 4)-arc [9]. For even q, let M q = 1 2 (q + 4). For odd q, let M q = 1 2 (q + 7) if either q ≡ 2 (mod 3), 11 ≤ q ≤ 3701, or q ≡ 1 (mod 4), q ≤ 337. Else, M q = 1 2 (q + 5). Below we suppose that t 2 (2, q) is given in [2, Tab. 1] for q ≤ 841, q = 343, and in Tables 1 and 2 Conjecture 2. Let 353 ≤ q ≤ 2879 be an odd prime. Then in P G(2, q) there are complete k-arcs of all the sizes in the region t 2 (2, q) ≤ k ≤ M q . Moreover, complete k-arcs with t 2 (2, q) ≤ k ≤ 1 2 (q + 5) can be obtained by the randomized greedy algorithms of [1,5] with a new approach to creation of starting data.
Our methods are applicable using our present computers for q ≤ 5171. For reason of space we plan to write more complete results and to describe this new approach to creation of starting conditions and data in a journal paper.
K-ARCS IN P G(2, Q), 853 ≤ Q ≤ 2879
and [ 2 ,
2Tab. 1], we obtain Observation 1. Observation 1. Let 173 ≤ q ≤ 2879, q = 5 4 , 3 6 , 29 2 , 31 2 , 2 10 , 37 2 , 41 2 , 7 4 .Then (i) 0.45 > D q (1) > 0.397. Also, 0.428 > D q (1) if 467 ≤ q; 0.415 > D q (1) if 1013 ≤ q; 0.41 > D q (1) if 1399 ≤ q; 0.405 > D q (1) if 1889 ≤ q. So, D q (1) has a tendency to decreasing. (ii) 1.202 < D q ( 1 2 ) < 1.355. Also, D q ( 1 2 ) < 1.27 if q ≤ 443; D q ( 1 2 ) < 1.32 if q ≤ 1291; D q( 1 2 ) < 1.325 if q ≤ 1327; D q ( 1 2 ) < 1.335 if q ≤ 1801. So, D q ( 1 2 ) has a tendency to increasing. (iii) 0.720 < D q (0.75) < 0.743. The values of D q (0.75) oscillate about the average value 0.73331. It holds that 0.720 < D q (0.75) < 0.743 if 173 ≤ q ≤ 997, 0.727 < D q (0.75) < 0.741 if 1009 ≤ q ≤ 1999, 0.729 < D q (0.75) < 0.738 if 2003 ≤ q ≤ 2879.
Figure 1 :
112 4.83 0.392 1.375 0.7345 6011 377 10 4.87 0.387 1.372 0The values of D q (0.75) = t 2 (2,q) √ q log 0.75 2 q , 173 ≤ q ≤ 2879, q = 5 4 , 3 6 , 29 2 , 31 2 , 2 10 , 37 2 , 41 2 , 7 4
Figure 2 :
2The values of ∆ q = t 2 (2, q) − 0.73331 √ q log 0.75 2 q, 173 ≤ q ≤ 2879, q = 5 4 , 3 6 , 29 2 , 31 2 , 2 10 , 37 2 , 41 2 , 7 4 3 On the spectrum of possible sizes of complete arcs in P G(2, q)
Figure 3 :
3of this paper for 853 ≤ q ≤ 2879. Also, in this work we The values of P q = 100∆q t 2 (2,q) %, 173 ≤ q ≤ 2879, q = 5 4 , 3 6 , 29 2 , 31 2 , 2 10 , 37 2 , 41 2 , 7 4 have obtained the value t 2 (2, 343) = 66 that improves the result of[2].
Theorem 4 .
4In P G(2, q) with 25 ≤ q ≤ 251, 257 ≤ q ≤ 349, and q = 1013, 2003, there are complete k-arcs of all the sizes in the region t 2 (2, q) ≤ k ≤ M q . In P G(2, 256) there are complete k-arcs of sizes k = 55 − 123, 130, 241, 258. Proof. For 25 ≤ q ≤ 167 the assertion follows from [1, Tab. 2] and [2, Tab. 2]. For 169 ≤ q ≤ 349 and q = 1013, 2003, all the results are obtained in this work by the randomized greedy algorithms of [1, 5] with a new approach to creation of starting conditions and data.
Table 1(continue). The smallest known sizes t 2q
853 118 13 4.05 1087 137 11 4.16 1327 155
8 4.26
857 119 12 4.07 1091 138 10 4.18 1331 155
9 4.25
859 119 12 4.07 1093 138 10 4.18 1361 157
9 4.26
863 119 13 4.06 1097 138 11 4.17 1367 158
8 4.28
877 120 13 4.06 1103 138 11 4.16 1369 144 22 3.90
881 121 12 4.08 1109 138 11 4.15 1373 158
8 4.27
883 121 12 4.08 1117 140 10 4.19 1381 159
8 4.28
887 121 13 4.07 1123 139 11 4.15 1399 160
8 4.28
907 123 12 4.09 1129 140 11 4.17 1409 160
8 4.27
911 123 12 4.08 1151 142 10 4.19 1423 161
8 4.27
919 124 12 4.10 1153 142 10 4.19 1427 162
7 4.29
929 125 12 4.11 1163 143 10 4.20 1429 161
9 4.26
937 126 11 4.12 1171 144
9 4.21 1433 161
9 4.26
941 126 12 4.11 1181 144 10 4.20 1439 161
9 4.25
947 127 11 4.13 1187 145 10 4.21 1447 162
9 4.26
953 127 11 4.12 1193 145 10 4.20 1451 163
8 4.28
961 120 19 3.88 1201 146
9 4.22 1453 164
7 4.31
967 128 11 4.12 1213 147
9 4.23 1459 164
7 4.30
971 128 12 4.11 1217 147
9 4.22 1471 164
8 4.28
977 129 11 4.13 1223 147 10 4.21 1481 164
9 4.27
983 129 12 4.12 1229 148
9 4.23 1483 165
8 4.29
991 130 11 4.13 1231 148
9 4.22 1487 166
7 4.31
997 130 12 4.12 1237 148 10 4.21 1489 166
7 4.31
1009 132 10 4.16 1249 149 10 4.22 1493 166
7 4.30
1013 131 12 4.12 1259 150
9 4.23 1499 166
8 4.29
1019 132 11 4.14 1277 151
9 4.23 1511 166
8 4.28
1021 132 11 4.14 1279 151
9 4.23 1523 168
7 4.31
1024 124 20 3.88 1283 152
9 4.25 1531 169
7 4.32
1031 132 12 4.12 1289 152
9 4.24 1543 169
7 4.31
1033 133 11 4.14 1291 152
9 4.24 1549 170
7 4.32
1039 134 11 4.16 1297 153
9 4.25 1553 170
7 4.32
1049 134 11 4.14 1301 153
9 4.25 1559 170
7 4.31
1051 135 10 4.17 1303 153
9 4.24 1567 171
7 4.32
1061 135 11 4.15 1307 153
9 4.24 1571 171
7 4.32
1063 136 10 4.18 1319 154
9 4.25 1579 172
6 4.33
1069 136 11 4.16 1321 154
9 4.24 1583 172
7 4.33
2 SMALL COMPLETE K-ARCS IN P G(2, Q), 853 ≤ Q ≤ 2879
2 SMALL COMPLETE K-ARCS IN P G(2, Q), 853 ≤ Q ≤ 2879
Computer search in projective planes for the sizes of complete arcs. A A Davydov, G Faina, S Marcugini, F Pambianco, J. Geom. 82A. A. Davydov, G. Faina, S. Marcugini, and F. Pambianco, Computer search in projective planes for the sizes of complete arcs, J. Geom., 82, 50-62, 2005.
On sizes of complete caps in projective spaces P G(n, q) and arcs in planes P G(2, q). A A Davydov, G Faina, S Marcugini, F Pambianco, J. Geom. 94A. A. Davydov, G. Faina, S. Marcugini, and F. Pambianco, On sizes of complete caps in projective spaces P G(n, q) and arcs in planes P G(2, q), J. Geom., 94, 31-58, 2009.
New inductive constructions of complete caps in P G(N, q), q even. A A Davydov, M Giulietti, S Marcugini, F Pambianco, J. Comb. Des. 183A. A. Davydov, M. Giulietti, S. Marcugini, and F. Pambianco, New induc- tive constructions of complete caps in P G(N, q), q even, J. Comb. Des., 18, no. 3, 176-201, 2010.
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Small complete caps in P G(2, q) for q an odd square. M Giulietti, J. Geom. 69M. Giulietti, Small complete caps in P G(2, q) for q an odd square, J. Geom., 69, 110-116, 2000.
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The packing problem in statistics, coding theory and finite geometry: update 2001. J W P Hirschfeld, L Storme, Proc. of the Fourth Isle of Thorns Conf. A. Blokhuis, J. W. P. Hirschfeld, D. Jungnickel and J. A. Thas, Kluwerof the Fourth Isle of Thorns ConfChelwood GateFinite GeometriesJ. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite geometry: update 2001, in Finite Geometries, Developments of Mathematics, 3, (Proc. of the Fourth Isle of Thorns Conf., Chelwood Gate, July 16-21, 2000), 201-246, Eds. A. Blokhuis, J. W. P. Hirschfeld, D. Jungnickel and J. A. Thas, Kluwer, 2001.
Types of superregular matrices and the number of n-arcs and complete n-arcs in P G(r, q). G Keri, J. Comb. Des. 14G. Keri, Types of superregular matrices and the number of n-arcs and complete n-arcs in P G(r, q), J. Comb. Des., 14, 363-390, 2006.
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| []
|
[
"Nuclear Physics A",
"Nuclear Physics A"
]
| [
"Pasi Huovinen \nInstitut für Theoretische Physik\nJohann Wolfgang Goethe-Universität\n60438Frankfurt am MainGermany\n\nFrankfurt Institute for Advanced Studies\n60438Frankfurt am MainGermany\n",
"Péter Petreczky \nPhysics Department\nBrookhaven National Laboratory\nUpton11973NYUSA\n",
"Christian Schmidt \nFakultät für Physik\nUniversität Bielefeld\n33615BielefeldGermany\n"
]
| [
"Institut für Theoretische Physik\nJohann Wolfgang Goethe-Universität\n60438Frankfurt am MainGermany",
"Frankfurt Institute for Advanced Studies\n60438Frankfurt am MainGermany",
"Physics Department\nBrookhaven National Laboratory\nUpton11973NYUSA",
"Fakultät für Physik\nUniversität Bielefeld\n33615BielefeldGermany"
]
| [
"Nuclear Physics A"
]
| We employ the lattice QCD data on Taylor expansion coefficients up to sixth order to construct an equation of state at finite netbaryon density. When we take into account how hadron masses depend on lattice spacing and quark mass, the coefficients evaluated using the p4 action are equal to those of hadron resonance gas at low temperature. Thus the parametrised equation of state can be smoothly connected to the hadron resonance gas equation of state. We see that the equation of state using Taylor coefficients up to second order is realistic only at low densities, and that at densities corresponding to s/n B 40, the expansion converges by the sixth order term. | 10.1016/j.nuclphysa.2014.08.069 | [
"https://arxiv.org/pdf/1407.8532v1.pdf"
]
| 118,503,452 | 1407.8532 | 5d7baba8a572d2d7558ac948df6e2c92553c8f7a |
Nuclear Physics A
2014
Pasi Huovinen
Institut für Theoretische Physik
Johann Wolfgang Goethe-Universität
60438Frankfurt am MainGermany
Frankfurt Institute for Advanced Studies
60438Frankfurt am MainGermany
Péter Petreczky
Physics Department
Brookhaven National Laboratory
Upton11973NYUSA
Christian Schmidt
Fakultät für Physik
Universität Bielefeld
33615BielefeldGermany
Nuclear Physics A
Nuclear Physics A
002014Equation of state at finite net-baryon density using Taylor coefficients up to sixth orderlattice QCDequation of statehadron resonance gas
We employ the lattice QCD data on Taylor expansion coefficients up to sixth order to construct an equation of state at finite netbaryon density. When we take into account how hadron masses depend on lattice spacing and quark mass, the coefficients evaluated using the p4 action are equal to those of hadron resonance gas at low temperature. Thus the parametrised equation of state can be smoothly connected to the hadron resonance gas equation of state. We see that the equation of state using Taylor coefficients up to second order is realistic only at low densities, and that at densities corresponding to s/n B 40, the expansion converges by the sixth order term.
One of the methods to extend the lattice QCD calculations to non-zero chemical potential is Taylor expansion of pressure in chemical potentials:
P T 4 = i j c i j (T ) µ B T i µ S T j .(1)
The coefficients of this expansion are derivatives of pressure, P, with respect to baryon and strangeness chemical potentials, µ B and µ S , respectively:
c i j (T ) = 1 i! j! T i+ j T 4 ∂ i ∂µ i B ∂ j ∂µ j S P(T, µ B = 0, µ S = 0),(2)
where T is temperature 1 . The lattice QCD calculations of these coefficients have matured to a level where both Budapest-Wuppertal [1] and hotQCD [2] collaborations have published the final continuum extrapolated results for the second order Taylor coefficients, see Fig. 1. As seen in Fig. 1 stout HRG Figure 3. The fitted trace anomaly (solid curve) compared to the HRG trace anomaly (dotted) and the continuum extrapolated lattice QCD result using stout action [11].
sixth order Taylor coefficients, and compare to the actual HRG pressure. The result is shown in Fig. 2, where the pressure at constant T = 150 MeV temperature is shown as a function of inverse of entropy per baryon, n B /s. We use n B /s as variable to facilitate easy comparison to heavy-ion collisions at various energies since, unlike net-baryon density n B , or baryon chemical potential µ B , s/n B is (approximately) constant during the entire expansion stage of the collision. We remind that at midrapidity the relevant entropy per baryon is s/n B = 400, 100, 65, and 40 at collision energies √ s NN ≈ 200, 64, 39, and 17 GeV/fm 3 , respectively. Thus an equation of state based on second order Taylor coefficients only [3] can be expected to be realistic only at relatively low net-baryon densities, s/n B 100, i.e., at collisions with collision energy √ s NN 64 GeV.
The Taylor coefficients have been evaluated on lattice up to sixth order [4,5], but unfortunately the fourth and sixth order coefficients suffer from large discretisation errors. As we have argued previously [6,7,8,9], these errors are mostly due to the lattice discretisation effects on hadron masses: When the hadron mass spectrum is modified accordingly (for details see [10]), the HRG model reproduces the lattice data, see Fig. 1 of Ref. [8]. Interestingly this change can be accounted for by shifting the modified HRG result of purely baryonic coefficients towards lower temperature by 30 MeV [8,9]. Based on this finding, and because the lattice data agree so well with the modified HRG, we suggest that cutoff effects can be accounted for by shifting the p4 lattice data by 30 MeV. The fourth and We parametrise the shifted p4 data, and the unshifted, continuum extrapolated stout and HISQ data, using an inverse polynomial of five (second order) or six (fourth and sixth order coefficients) terms:
c i j (T ) = m k=1 a ki ĵ T n ki j + c SB i j ,(3)
where c SB i j is the Stefan-Boltzmann value of the particular coefficient, a ki j are the parameters, and the powers n ki j are required to be integers between 1 and 23.T = (T − T s )/R with scaling factors T s = 0 and R = 0.15 GeV for the second order coefficients and T s = 0.1 GeV and R = 0.05 GeV for all other coefficients. We match the parametrisation of second order coefficients to the HRG value at temperature T SW = 160 MeV, and the fourth and sixth order coefficients at T SW = 155 MeV by requiring that the Taylor coefficient and its first, second, and third derivatives are continuous. The switching temperatures have been chosen to optimise the fit and lead to smooth behaviour of the speed of sound (see Fig. 5). These constraints fix four (or five) of the parameters a ki j . The remaining parameters are fixed by a χ 2 fit to the lattice data. The resulting parametrizations are shown as solid curves in Figs. 1 and 4.
We obtain the pressure at µ B = 0, i.e. the coefficient c 00 , from the continuum extrapolated stout data for the trace anomaly ( − 3P)/T 4 [11], which agree with the very recent HISQ data [12] within errors. As in our earlier parametrisation of trace anomaly [7], we fit the lattice result using an inverse polynomial with four terms, and connect it to the HRG trace anomaly at T SW = 167 MeV temperature, see Fig. 3.
We characterise the equation of state in Fig. 5a similar speed of sound, but below T ≈ 150 MeV, the second order result deviates strongly from the fourth and sixth order results. This shows that at low temperatures equation of state based on the second order coefficients only is not sufficient. Furthermore, the small difference between the fourth and sixth order equation of state indicates that if s/n B 40, the expansion has basically converged by the sixth order term.
To summarise we have argued that an equation of state based on the Taylor expansion up to second order, is realistic only in collisions with larger collision energy than at the RHIC beam energy scan. We argue that the temperature shift of 30 MeV is a good approximation of the discretisation effects in the lattice QCD data obtained using p4 action. We have constructed an equation of state for finite baryon densities based on hadron resonance gas and lattice QCD data. In such an equation of state, the Taylor expansion essentially converges by the sixth order term if s/n B 40.
Figure 2 .
2Pressure at constant temperature T = 150 MeV in hadron resonance gas (HRG) and using Taylor coefficients of HRG up to second, fourth and sixth order in Eq.
Figure 4 .
4The parametrised (solid line) fourth and sixth order Fourier coefficients compared to HRG values (dashed line) and the shifted p4 data[4,5] (see the text).sixth order coefficients are shown inFig. 4, where the data below 206 MeV has been shifted by 30 MeV, the data point at 209 MeV by 15 MeV (open symbol), and the points above 209 MeV have not been shifted. At low temperatures the shifted data now agrees with the unmodified HRG.
by showing the square of the speed of sound, c 2 s = ∂P/∂ | s/n B , on various isentropic curves with constant entropy per baryon. The curves at s/n B = 400, 65, and 40 are relevant at collision energies √ s NN = 200, 39 and 17 GeV, respectively. At s/n B = 400 (dotted line), the equation of state is basically identical to the equation of state at µ B = 0 (thin solid line). At larger baryon densities the effect of finite baryon density is no longer negligible. The larger the density, the stiffer the equation of state above, and softer below the transition temperature. We see some ripples forming in the transition region with decreasing s/n B . These ripples grow fast with increasing density when one goes beyond s/n B = 40, and therefore we consider s/n B = 40 to give a practical maximum density for our parametrisation. OnFig. 5bwe have evaluated the square of the speed of sound along the isentropic s/n B = 40 curve using Taylor coefficients up to second, fourth and sixth order. At temperatures above the transition region they all lead to very
Figure 5 .
5The square of the speed of sound, c 2 s , as a function of temperature on various isentropic curves with constant entropy per baryon (a), and on s/n B = 40 curve evaluated using Fourier coefficients up to second, fourth or sixth order (b).
, at low temperatures the coefficients evaluated using the hadron resonance gas (HRG) model agree with the lattice QCD results. Thus we may expect that HRG is an acceptable approximation of the physical equation of state also at finite net-baryon densities. To check how soon one may truncate the expansion in Eq. (1), we calculate the Taylor coefficients in HRG up to sixth order, evaluate the pressure using up to second, fourth or0.0
0.2
0.4
0.6
0.8
150 200 250 300
T [MeV]
c 20 × 5
HISQ
150 200 250 300
T [MeV]
-c 11 × 3
stout
150 200 250 300
T [MeV]
c 02 × 2
HRG
Figure 1. The parametrised (solid line) second order Fourier coefficients compared to HRG values (dashed) and the continuum extrapolated
HISQ [2] and stout [1] data.
We use natural units where c = = k B = 1 throughout the text.
AcknowledgementsThis work was supported by BMBF under contract no. 06FY9092, and by the U.S. Department of Energy under contract DE-AC02-98CH1086.
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| []
|
[
"Effects of baryonic and dark matter substructure on the Pal 5 stream",
"Effects of baryonic and dark matter substructure on the Pal 5 stream"
]
| [
"Nilanjan Banik \nInstitute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics\nGRAPPA Institute\nUniversity of Amsterdam\nScience Park 9041098 XHAmsterdamThe Netherlands\n\nLorentz Institute\nLeiden University\nNiels Bohrweg 2NL-2333Leiden, CAThe Netherlands\n",
"Jo Bovy \nDepartment of Astronomy and Astrophysics\nUniversity of Toronto\n50 St. George StreetM5S 3H4TorontoONCanada\n",
"Alfred P Sloan Fellow "
]
| [
"Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics\nGRAPPA Institute\nUniversity of Amsterdam\nScience Park 9041098 XHAmsterdamThe Netherlands",
"Lorentz Institute\nLeiden University\nNiels Bohrweg 2NL-2333Leiden, CAThe Netherlands",
"Department of Astronomy and Astrophysics\nUniversity of Toronto\n50 St. George StreetM5S 3H4TorontoONCanada"
]
| [
"MNRAS"
]
| Gravitational encounters between small-scale dark matter substructure and cold stellar streams in the Milky Way halo lead to density perturbations in the latter, making streams an effective probe for detecting dark matter substructure. The Pal 5 stream is one such system for which we have some of the best data. However, Pal 5 orbits close to the center of the Milky Way and has passed through the Galactic disk many times, where its structure can be perturbed by baryonic structures such as the Galactic bar and giant molecular clouds (GMCs). In order to understand how these baryonic structures affect Pal 5's density, we present a detailed study of the effects of the Galactic bar, spiral structure, GMCs, and globular clusters on the Pal 5 stream. We estimate the effect of each perturber on the stream density by computing its power spectrum and comparing it to the power induced by a CDM-like population of dark matter subhalos. We find that the bar, spiral structure, and GMCs can each individually create power that is comparable to the observed power on large scales, leaving little room for dark matter substructure. On degree scales, the power induced by the bar and spirals is small, but GMCs create small-scale density variations that are much larger in amplitude than the dark-matter induced variations but otherwise indistinguishable from it. These results demonstrate that Pal 5 is a poor system for constraining the dark matter substructure fraction and that observing streams further out in the halo will be necessary to confidently detect dark matter subhalos. | 10.1093/mnras/stz142 | [
"https://arxiv.org/pdf/1809.09640v2.pdf"
]
| 119,426,331 | 1809.09640 | 4cdff0ef515c79abb4e9b58654ad297a27339967 |
Effects of baryonic and dark matter substructure on the Pal 5 stream
2018
Nilanjan Banik
Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics
GRAPPA Institute
University of Amsterdam
Science Park 9041098 XHAmsterdamThe Netherlands
Lorentz Institute
Leiden University
Niels Bohrweg 2NL-2333Leiden, CAThe Netherlands
Jo Bovy
Department of Astronomy and Astrophysics
University of Toronto
50 St. George StreetM5S 3H4TorontoONCanada
Alfred P Sloan Fellow
Effects of baryonic and dark matter substructure on the Pal 5 stream
MNRAS
0002018Accepted XXX. Received YYY; in original form ZZZPreprint 27 September 2018 Compiled using MNRAS L A T E X style file v3.0Cosmology: dark matter -Galaxy: evolution -Galaxy: halo -Galaxy: kinematics and dynamics -Galaxy: structure
Gravitational encounters between small-scale dark matter substructure and cold stellar streams in the Milky Way halo lead to density perturbations in the latter, making streams an effective probe for detecting dark matter substructure. The Pal 5 stream is one such system for which we have some of the best data. However, Pal 5 orbits close to the center of the Milky Way and has passed through the Galactic disk many times, where its structure can be perturbed by baryonic structures such as the Galactic bar and giant molecular clouds (GMCs). In order to understand how these baryonic structures affect Pal 5's density, we present a detailed study of the effects of the Galactic bar, spiral structure, GMCs, and globular clusters on the Pal 5 stream. We estimate the effect of each perturber on the stream density by computing its power spectrum and comparing it to the power induced by a CDM-like population of dark matter subhalos. We find that the bar, spiral structure, and GMCs can each individually create power that is comparable to the observed power on large scales, leaving little room for dark matter substructure. On degree scales, the power induced by the bar and spirals is small, but GMCs create small-scale density variations that are much larger in amplitude than the dark-matter induced variations but otherwise indistinguishable from it. These results demonstrate that Pal 5 is a poor system for constraining the dark matter substructure fraction and that observing streams further out in the halo will be necessary to confidently detect dark matter subhalos.
INTRODUCTION
A crucial prediction of the ΛCDM framework is the presence of a large number of subhalos orbiting within a Milky Way sized host halo (Klypin et al. 1999;Moore et al. 1999;Diemand et al. 2008;Springel et al. 2008). Detecting these subhalos would not only prove that dark matter is a form of matter capable of clustering on sub-galactic scales, but would also give crucial insight into its particle nature and interactions. One purely gravitational method for detecting dark matter substructures is gravitational lensing (e.g., Mao & Schneider 1998;Dalal & Kochanek 2002;Vegetti et al. 2012). Gravitational lensing can allow us to measure the abundance of low-mass substructures around external galaxies (Hezaveh et al. 2016;Daylan et al. 2018). E-mail: [email protected]
An alternate but equally promising purely gravitational method for detecting these subhalos is to use cold stellar streams that originate as a globular cluster falls into our Galaxy's gravitational potential and gets tidally disrupted. The density of such a stream is largely uniform along its length in the absence of perturbations. A gravitational encounter with a dark matter subhalo perturbs the stream density resulting in gaps in the density (Siegal-Gaskins & Valluri 2008;Carlberg 2009). Much work has been done in the last few years towards modeling and analyzing these gaps and inferring the properties of the subhalos that the stream encountered (e.g., Yoon et al. 2011;Carlberg 2012Carlberg , 2013Erkal & Belokurov 2015a,b;Sanders et al. 2016).
Recently, a statistical approach for inferring the properties of the dark matter subhalos using the stream density power spectrum and bispectrum was proposed by Bovy et al. (2017). Applying this approach to data on the density of the Pal 5 stream from Ibata et al. (2016), the authors computed the observed power spectrum of the stream density and by matching this to simulations used this to constrain the number of cold dark matter subhalos orbiting within the Galactic volume of the Pal 5 orbit. In doing so however, the authors neglected any effects from the baryonic perturbers in the Galaxy such as the central bar, the spiral structure, giant molecular clouds (GMCs), and the globular cluster (GC) system. Because of this neglect, they pointed out that their measurement of the number of dark matter subhalos was in fact a robust upper limit to the amount of dark matter substructure. The effect of the bar on stellar streams orbiting near the center of the Galaxy has been shown to be potentially large (e.g., Hattori et al. 2016;Erkal et al. 2017;Pearson et al. 2017), especially for the Pal 5 stream because it is in a prograde orbit with respect to the disk and everything orbiting within it. Therefore the density of the Pal 5 stream can be affected by the Galactic bar (Pearson et al. 2017), GMCs (Amorisco et al. 2016), and likely spiral structure as well. All these findings then beg the question: is the Pal 5 stream a good probe for detecting dark matter substructures in our Galaxy? To answer this question, in this paper we perform a detailed investigation of the possible baryonic perturbers individually, using up-to-date constraints on their properties, and we compute the effect each one has on the Pal 5 stream.
The paper is structured as follows: In Section 2, we introduce the Pal 5 stream, observations of its density, and a brief description of the CDM subhalo model used for the stream-subhalo encounters. In Section 3, we discuss the bar model setup and decide on the intervals over which the bar model parameters will be varied. In the subsection 3.2 we discuss how we model the effects of the bar on the Pal 5 stream density, followed by subsection 3.3, where we present the results of the mock Pal 5 stream's power spectrum as a result of varying the bar models. Next, in Section 4, we describe the model of the spiral potential and present the results of varying the spiral arm potential's model parameters in subsection 4.2. In Section 5, we discuss how the GMCs are included in our simulations and in subsection 5.2, we explore how their effect on Pal 5 stream's density changes on varying Pal 5's pericenter within a range that is allowed by observations. We present the results of the GMC impacts on Pal 5 stream in subsection 5.3. Then in Section 6, we describe how we incorporated the Galactic population of GCs in the stream simulations and discuss the results. Finally, in Section 7, we discuss all the results and present our conclusions.
All of our modeling is done using tools available as part of the galpy galactic dynamics Python package 1 (Bovy 2015).
THE PAL 5 STREAM
The Pal 5 stream is a cold stellar stream emanating from its namesake progenitor, the Pal 5 globular cluster. It was discovered by Odenkirchen et al. (2001) using data from the 1 Available at https://github.com/jobovy/galpy . Sloan Digital Sky Survey (SDSS) (York et al. 2000). Its trailing arm spans over ∼ 14 • while its leading arm is only around ∼ 8 • (Bernard et al. 2016). Since its discovery, there has been a number of follow up studies to measure its stellar density (e.g., Odenkirchen et al. 2003;Carlberg et al. 2012;Ibata et al. 2016). In what follows, we briefly describe how we model the smooth Pal 5 stream in this paper.
Following Bovy (2014), we generate a mock Pal 5 stream using a frequency-angle (Ω, θ) framework in the MWPotential2014 (Bovy 2015). This method requires the phase space coordinates of the progenitor, the velocity dispersion σ v of the stars and the time t d since disruption commenced. Following Fritz & Kallivayalil (2015), we set the phase space coordinates of the Pal 5 globular cluster to (RA, Dec, D, µ α cos δ, µ δ , V los ) = (229 • .018, −0 • .124, 23.2 kpc, −2.296 mas yr −1 , −2.257 mas yr −1 , − 58.7 km s −1 ). Following Bovy et al. (2017), we set σ v = 0.5 km/s and t d = 5 Gyr, because they demonstrated that this gives a close match to all of the data on the Pal 5 stream. The stream generated in the (Ω, θ) space is then transformed to rectangular Galactocentric coordinates using the approach of Bovy (2014) and from there to the custom (ξ, η) stream coordinates defined by Ibata et al. (2016). For the rest of this paper we will focus only on the trailing arm of the stream in the range 0.65 • < ξ < 14.35 • , because this is the part of the stream for which the best density data exists and it is the part studied in detail by Bovy et al. (2017).
Throughout this paper, we compare the effect of baryonic perturbers on the Pal 5 stream to that expected from dark matter subhalos. The modeling of the subhalo population and how it affects the Pal 5 stream is discussed in detail by Bovy et al. (2017). Here we briefly describe important aspects of this modeling that are relevant for the discussion in the subsequent sections. Following the approach in Bovy et al. (2017), we use the CDM subhalo mass function, dn/dM ∝ M −2 and model subhalos as Hernquist spheres whose scale radius depends on the subhalo mass according to the fitting relation r s (M) = 1.05 kpc (M/10 8 M ) 0.5 ; this relation was obtained by fitting Hernquist profiles to the circular velocity-M relation from Via Lactea II simulations (Diemand et al. 2008). The amplitude of the subhalo mass function in the range 10 5 to 10 9 M is determined from the number of dark matter subhalos within 25 kpc in the Via Lactea II simulations. Because the number of subhalos is based on a dark-matter-only simulation, this number does not take into account the possible disruption of some fraction of the subhalo population in the inner Galaxy due to tidal shocking by the disk and bulge, which in simulations leads to a factor of two to four lower subhalo abundance (e.g., D'Onghia et al. 2010;Sawala et al. 2017).
THE EFFECT OF THE GALACTIC BAR
Bar models
We model the Galactic bar with the triaxial, exponential density profile from Wang et al. (2012):
ρ bar = ρ 0 exp(−r 2 1 /2) + r −1.85 2 exp(−r 2 ) .
(1) The dashed, dotted and dash-dotted curves represent the density for the disk, the halo and the full Milky Way potential respectively, in the MWPotential2014 model for the potential that we use. The bottom panel displays the relative difference between each reconstructed density and the analytic density of the bar. The expansion with n = 9 and l = 19 gives an excellent match to the input density of the bar model.
Where the functions r 1 and r 2 are defined as
r 1 = (x/x 0 ) 2 + (y/y 0 ) 2 + (z/z 0 ) 4 1/4
(2)
r 2 = q 2 (x 2 + y 2 ) + z 2 z 2 0 1/2(3)
with x 0 = 1.49 kpc, y 0 = 0.58 kpc and z 0 = 0.4 kpc, q = 0.6 and ρ 0 is the normalization for a given mass of the bar. To compute the potential from the density, one needs to solve the Poisson equation. We do this by following the basis-function expansion method from Hernquist & Ostriker (1992), in which we expand the potential and density into a set of orthogonal basis functions of potential-density pairs consisting of spherical harmonics indexed by (l, m) and a radial set of basis functions indexed by n. The method requires a single distance scale parameter r s to be set as well. This method is implemented in galpy and we compute the expansion coefficients by setting the scale length r s = 1 kpc. To find the minimum order of expansion coefficients required to get a close reconstruction of the density from the potential, we reconstructed the density for a range of expansion orders and compared the resulting density to the analytic form of the density in Equation (1). The colored curves in Figure 1 show the reconstructed density for some of the expansion orders. The bottom panel displays the percentage difference between each case of expansion order and the analytic density (Eq.
[1]). The departure of the reconstructed density at Galactocentric r > 5 kpc does not affect the analysis since at that radial distance the disk's contribution to the density and hence the potential is much more important than that of the bar, as shown in the same figure. From visual inspection, we found that for n = 9 and l = 19 we get an excellent reconstruction of the density and therefore we used these values for the rest of the analysis. Pearson et al. (2017) also used the basis function expansion technique to model the bar, however they used expansion order up to n = 2 and l = 6. As shown in Figure 1, this gives a poorer reconstruction of the density. We consider five bar models with masses between 6 × 10 9 M and 1.4 × 10 10 ] M in increments of 2 × 10 9 M . To incorporate the bar while keeping the same total baryonic mass in our Milky Way mass model, we remove the bulge of mass 5 × 10 9 M from the MWPotential2014 model and any additional mass of the bar above this value is removed from the disk component. Figure 2 shows the resulting circular velocity curve for each model of the bar. It is clear that the circular velocity of all of the models is very similar outside of the bar region and only slightly changes within it. The measurement shown represents the circular velocity constraint of 218 ± 6 km s −1 in the solar neighborhood as obtained from APOGEE data by Bovy et al. (2012). In the same plot, we list the value of the vertical force in each bar model at 1.1 kpc above the plane at this position, which was constrained by Zhang et al. (2013) from the kinematics of K-type dwarfs to be |F z | = 67 ± 6 (2πG M pc −2 ). Therefore, our bar models do not significantly change MWPotential2014 outside of the bar region, and our barred models are therefore approximately as good mass models for the Milky Way as MWPo-tential2014. In addition to varying the mass of the bar, we vary its pattern speed over the grid between the values of 35 and 61 km s −1 kpc −1 in increments of 2 km s −1 kpc −1 and we consider ages for the bar between 1 and 5 Gyr in 1 Gyr increments. In each case, the amplitude of the bar is smoothly grown from 0 to its full amplitude following the prescription of Dehnen (2000) over two rotation periods of the bar. As a fiducial model, We consider the model of the bar to be 5 Gyr old, with a mass of 10 10 M , and rotating at a pattern speed of 39 km s −1 kpc −1 (Portail et al. 2016). The angle of the bar's major axis with respect to the Sun-Galactic-center line at the present day is in all bar models set to 27 • (Wegg & Gerhard 2013).
As the non-axisymmetric bar rotates, it imparts kicks to orbiting stars, thereby altering their kinematics. To give an indication of how a barred potential affects an orbiting star, we compare the orbital evolution of the Pal 5 globular cluster in the barred potential to that in an axisymmetric version of the same potential. We construct the latter by setting the expansion coefficients with m 0 to zero in the basis function expansion. To evolve the orbit of Pal 5 globular cluster, we first integrated it back for 5 Gyr in the past in both the axisymmetric and non-axisymmetric potentials and then integrated it forward to the present in the same potentials. The resulting orbits are plotted in Figure 3. In the presence of the bar, the orbit is only slightly altered.
Effect of the Galactic bar on the Pal 5 stream
To compare the density structure of the Pal 5 stream induced by the bar to that due to dark matter subhalos and to the observed density, we evolve a mock Pal 5 stream in the barred Milky Way potentials described above and compute the power spectrum of the stream density. We make use of the same technique as discussed in Bovy et al. (2017) to compute the power spectrum. The frequency-angle framework for modeling tidal streams does not support nonaxisymmetric potentials. We therefore generate the mock stream in the axisymmetric version of each bar potential and sample from the phase-space coordinates at the present time and the time each star was stripped from the progenitor cluster for many stars. We then integrate each star backwards in time in the same axisymmstric potential to the time of stripping (which is different for each star). Finally, we integrate each star forward in time, now in the barred Milky Way potential until today. This gives the phase-space coordinates of stream stars today due to their evolution in the barred potential. We then transform the coordinates of these stars to (ξ, η) coordinates. After selecting all the stars that lie in the observed part of the trailing arm, we bin the sample of stars in 0.1 • bins in the ξ coordinate, to mimic the analysis of Bovy et al. (2017). To minimize the shot noise in the density resulting from sampling only a finite number of stars, we sample ∼ 500,000 stars. Following the arguments presented in Bovy et al. (2017), we normalize the binned density by fitting a third order polynomial to the density and divide the density by this fit. This is done to remove large scale variations in the stream such as could be expected from variations in the stripping rate, which in our analysis is assumed to be constant. We use this normalized density to compute the power spectrum. Figure 4 shows the star counts of a smooth stream and a stream perturbed by the fiducial bar model. As expected, the unperturbed stream has a largely uniform star count along its length. Evolution in the barred potential results in large density perturbations along the stream.
For each case of pattern speed and mass of the bar that we consider, we compute the stream density and its power. For certain pattern speeds, the Pal 5 stream is so heavily perturbed as to appear far shorter than what is observed, because many stars get large perturbations due to repeated interactions with the bar that remove them far from the observed portion of the stream. Therefore, we first compute the length of the stream and only consider bar models that do not lead to a significantly shorter stream. We define the length of the stream as the ξ at which the stellar density drops below 20% of the mean stellar density within 0.65 • < ξ < 3 • of the stream generated in the axisymmetric potential. The length of the Pal 5 stream for different pattern speeds and different bar masses is shown in Figure 5; the bar is assumed to be 5 Gyr old. We find that for a few combinations of pattern speed and bar mass the stream length is much shorter than what is observed. This effect of stream shortening could potentially be used to constrain the pattern speed of the bar, as pattern speeds that severely shorten the stream are disfavored. However, this may be degenerate with other parameters such as the dynamical age of the stream and this therefore requires a deeper investigation to become a useful constraint. For the remaining analysis, we remove the few pattern speeds for which the stream is severely shortened and only consider the cases which lead to a stream length that is comparable to the observed angular extent of the Pal 5 stream which is ∼ 14 • in ξ (Ibata et al. 2016). The approach for computing the density and its power spectrum described above does not take into account the fact that the progenitor's orbit is slightly different in the barred potential compared to that in the axisymmetric potential as shown in Figure 3. Therefore, the phase space coordinates of the stripped stars at the time of stripping will be different in the barred potential. This may lead to a different stellar density today along the stream. To investigate the effects of the progenitor's different orbit on the stream density, we generate mock streams using an implementation of the particle spray technique described in Fardal et al. (2015). In this approach, stripped stars are generated along the progenitor's orbit by offsetting them at the time of stripping from the progenitor in the instantaneous orbital plane (perpen- dicular to the angular momentum), with offsets in position and velocity some fraction of the tidal radius and circular velocity. For consistency with the method with which the effect from dark matter subhalos and GMCs on the Pal 5 stream is computed, which uses the frequency-angle framework for mock stream generation (see below), we use the particle-spray mock streams only for determining the size of the effect of including the progenitor's perturbation on the resulting power spectrum-nevertheless, the induced power in the mock perturbed streams is similar with both methods.
Thus, to determine the impact of the perturbation to the progenitor's orbit, we generate mock Pal 5 streams using the particle spray technique based on the progentor's orbit in both the barred and axisymmetric potential, but with orbits of stream stars computed in the barred potential in each case. To quantify the effect of Pal 5's different orbit on the stream density we compute the difference in the power spectrum between these two cases ∆ P δδ (k ξ ) = √ P δδ | with progenitor − √ P δδ | without progenitor .
Results
In Figure 6, we show the power spectrum for a 5 Gyr old bar of mass 10 10 M , for different values of the pattern speed: 39, 43, 47, 51, 57, 61 km s −1 kpc −1 . The power at the higher end of the angular scale is very sensitive to the pattern speed of the bar. There is no clear trend of increase or decrease of power with pattern speed. This suggests that a resonancelike condition is responsible for the structure we see. This is similar to what is seen in simulations of the bar's effect on the evolution of the Ophiuchus stream (Hattori et al. 2016) where the stream members in resonance with the bar suffer maximum torque from it, which results in more den- sity perturbations. For the faster rotating bars with pattern speeds 50 km s −1 kpc −1 , the power is smaller compared to the other cases. This is interesting, because until recently such fast pattern speeds were the preferred value, because they explain the presence of the Hercules stream in the solar neighborhood (e.g., Dehnen 2000;Bovy 2010;Hunt et al. 2018).
For the fiducial pattern speed of 39 km s −1 kpc −1 (green curve) the power is comparable to the power induced by dark matter subhalos, which is of the same order as the observed power of Pal 5. The predicted power from dark matter subhalos is shown by the thick dashed cyan line, which represents the median power spectrum of the stream density as a result of impacts with CDM subhalos in mass range 10 5 − 10 9 M . The shot noise power for our bar simulations in all cases is at the level of 10 −2 and is shown by the gray dashed horizontal line. The shot noise is a limitation stemming from only using 500,000 stream particles; the true power induced by the bar on small scales is below this noise floor and therefore smaller than that from dark matter subhalos. The bottom panel in Figure 6 displays the difference in power when the effect of the progenitor's different orbit is considered. For a pattern speed of 57 km s −1 kpc −1 this effect is most prominent indicating considerable departure from the orbit in the axisymmetric potential. In most cases the power difference is below zero, indicating that including the effect of Pal 5's orbit should result in lowering the power, and so the power presented in the top panel are an overestimate, although on the logarithmic scale of the upper panel this difference is small. Next, we explore the effect of varying the mass of the bar on the power spectrum of Pal 5. In Figure 7, we show the power spectrum for different bar masses with pattern speed set to 39 km s −1 kpc −1 . The sub-panel in each figure again displays the difference in the power spectrum ∆ P δδ (k ξ ) due to the effect of the bar on the progenitor's orbit. There is a clear trend of increasing power with the mass of the bar. This is expected, because a bar with more mass imparts stronger perturbations to the stream. In this case, including the progenitor's motion lowers the power by almost the same amount for all the different mass bars. The effect of varying the age of the bar on Pal 5 stream's power spectrum is shown in Figure 8. It was shown in Cole & Weinberg (2002), that the Galactic bar is less than 6 Gyr old and likely less than 3 Gyr. We vary the age of the bar between 1 and 5 Gyr and compute the power spectrum of Pal 5 in each case. The power of Pal 5 is virtually unaffected by the age of the bar as long as it is at least 2 Gyr old.
EFFECT OF THE SPIRAL ARMS ON PAL 5 4.1 Spiral structure models
In this section we investigate the possible effect of spiral structure on the density of Pal 5 stream. We model the gravitational potential due to spiral structure using galpy's Spi-ralArmsPotential which is based on the analytic model of Cox & Gómez (2002). The potential has the following form : . Same as Figure 6 but for a Milky Way potential that includes spiral structure rather than a bar. The plot shows the effect of varying the age and pattern speed of a two-armed spiral that contributes 1 % of the radial force at the Sun.
Φ(R, φ, z) = −4πGH ρ 0 exp r 0 − R R s × n C n K n D n cos(nγ) sech K n z β n β n(4)
where
K n = nN R sin(α) (5) β n = K n H(1 + 0.4K n H)(6)D n = 1 + K n H + 0.3(K n H) 2 1 + 0.3K n H (7) γ = N φ − φ ref − ln(R/r 0 ) tan(α)(8)
N denotes the number of spiral arms, ρ 0 sets the amplitude, and r 0 is a reference radius, which we took to be 8 kpc. The pitch angle α is set to 9.9 which we set to 7 kpc (Cox & Gómez 2002) and H is the vertical scale height set to 0.3 kpc. The C n determine the profile of the spiral arms: if C n = 1, then we get a sinusoidal potential profile, whereas if C n = [8/3π, 1/2, 8/15π] then the density takes approximately a cosine squared profile in the arms with flat interarm separations. Following Monari et al. (2016b), the amplitude ρ 0 is set such that the radial force at the location of the Sun due to the spiral arms is around one percent of the radial force due to the axisymmetric Milky Way potential (MWPotential2014). We explore the effects of varying the following parameters: (a) number of arms, either N = 2 or 4 , (b) amplitude such that the local radial force is 0.5% or 1% of the total local radial force (Monari et al. 2016b), and (c) pattern speed of 19.5 and 24.5 km s −1 kpc −1 on Pal 5's density power spectrum. As for the bar above, we grow the amplitude of the spiral potential from zero to full over two rotation periods of the spiral arms following the prescription of Dehnen (2000). The spiral potential is in all cases added to the axisymmetric MWPotential2014. We then follow the same set of steps as for the Galactic bar above. The results are shown in Figures 9 and 10 for 2 arms and 4 arms spiral potential respectively. In each case the amplitude is set such the local radial force from the spirals is 1% of the radial force of the background axisymmetric potential. For the lower value of the radial force, we found the power to be consistently lower than all the 1% cases, as expected, and hence we do not show them.
Results
From Figure 9, it is clear that for a two-armed spiral arm potential contributing 1% of the radial force at the Sun, the power induced on the Pal 5 stream is around an order of magnitude lower than the power induced due to CDM subhalo impacts. Varying the age and the pattern speed does not show any strict trend in the power. However, varying the age of the spiral arms above 3 Gyr has almost negligible effect on the power. The power difference in the subplot implies that the motion of the Pal 5 progenitor has very little effect on the progenitor's orbit and leads to lowering of the power. Increasing the number of spiral arms to four significantly increases the power induced in the Pal 5 stream as shown in Figure 10. A 3 or 5 Gyr old spiral arms results in identical power at the large scales regardless of the pattern speed. In this case, the motion of the progenitor again has almost no effect on the power.
EFFECT OF THE GIANT MOLECULAR CLOUDS
Modeling the Milky Way's population of GMCs
In this section, we explore how the Galactic population of giant molecular clouds (GMCs) affects the Pal 5 stream. Amorisco et al. (2016) demonstrated that GMCs confined to the razor thin disk can impact globular cluster streams such as Pal 5 and give rise to gaps in their density. Because the size and mass of the largest GMCs is similar to that of low-mass dark matter subhalos, GMC-induced gaps are similar to the ones that result from dark matter subhalo impacts and therefore will introduce large uncertainties when using stellar streams as probes for dark matter subhalos. We investigate the cumulative effect of gravitational encounters of GMCs with the Pal 5 stream over its dynamical age by computing the power spectrum of density perturbations induced by GMCs rather than dark matter. Rather than using a simple model of the GMC population in the Milky Way, we directly use a recent catalog of 8,107 GMCs from Miville-Deschênes et al. (2016), which is close to complete for the largest GMCs that are of highest interest here (as we discuss below, GMCs with masses 10 5 M have very little effect). We setup their orbits by positioning them at their present-day location in the Galaxy and placing them on a circular orbit in the MWPotential2014 potential. We then evolve them back in time for the dynamical age of the Pal 5 stream. Next, we evolve both the GMCs and the Pal 5 stream forward in the same potential and compute the impacts of the GMCs on the stream during this time.
The mass M and the physical radius R of each GMC in the Miville-Deschênes et al. (2016) catalog corresponds to its entire angular extent on the sky. We fit M versus R from the catalog, we find
R(M) ∼ 295 pc M 10 7 M 0.45 .(9)
This is very close to the CDM subhalo mass versus scale radius relation obtained in Erkal et al. (2016). However, the radius R is not the scale radius, but the full radius, and therefore we model the GMCs as Plummer spheres with scale radius
because for a Plummer sphere, 90% of mass is contained within 3 times the scale radius. Thus, Equation (10) ensures that most (∼ 90%) of the mass of the GMCs will be contained within their physical radius R.
To compute the effect of the GMCs on Pal 5 stream, we only consider GMCs with M > 10 5 M , because we found that including the lower mass GMCs resulted in neg-ligible change in the power. Following Bovy et al. (2017), we consider impacts up to a maximum impact parameter b max = 5 × r s (M). This takes into account the effect that smaller (low mass) GMCs need to pass more closely by the stream compared to bigger (more massive) ones to have an observable effect. The GMC impacts are modeled by the impulse approximation and the resulting stream density is computed using the line-of-parallel-angles approach as described in Bovy et al. (2017). Following the same reference, to save computational time, we re-sample impacts on a discrete grid of time over the dynamical age of the stream. To properly resolve the interactions between the GMCs and the stream, it is necessary to compute the impact parameters-time of impact, closest approach-with a time resolution at least equal to the typical time scale over which a GMC interacts with a stream, which is of order r s /v (few 100pc)/(200 km/s) ∼ 1 Myr). We have checked that the density power has converged using this time resolution for computing impact parameters.
The lifetime of a typical GMC is between 10 to 50 Myr (Jeffreson & Kruijssen 2018). Therefore, the present day population of GMCs did not exist during the entire dynamical lifetime of the Pal 5 stream and is thus at best a proxy for the population of GMCs that may have interacted with the stream. To compute the effect of an evolving population of GMCs impacting the stream, we create new realizations of the GMC population by adding random rotations to the Galactocentric cylindrical φ coordinates of the present GMCs and then follow the same steps as above to find the overall density perturbations imparted on the stream. This φ randomization maintains the spatial and mass distribution of GMCs, but allows us to study the range of possible histories of GMC interactions. We generate 42 different random φ realizations. Figure 12 displays the resulting density contrast in two cases. Bovy et al. (2016) performed a detailed investigation of the orbits and Milky Way potential models that are consistent with the Pal 5 stream and other dynamical data in the Milky Way. The full range of possible Pal 5 progenitor's phasespace coordinates were sampled using MCMC. We use all of the generated MCMC chains to explore differences in Pal 5's orbit from our fiducial orbit model. We find that Pal 5's pericenter varies between 4.68 and 8.01 kpc; for the fiducial orbit that we have been considering so far, the pericenter radius is 7.34 kpc. Because GMCs are distributed non-uniformly in radius, with especially a much larger number of highmass GMCs at radii 7 kpc, variations in Pal 5's pericenter radius can have a big impact on the predicted effect from GMCs. Therefore, we consider 5 different pericenter values in the allowed range: 4.68, 5.52, 6.44, 7.18, and 7.92 kpc.
Effect of varying Pal 5's pericenter
To study these, we randomly pick 5 chains that correspond to these values out of all the MCMC chains from all the potentials. For each chain and its corresponding potential model, we follow the same procedure as for the fiducial orbit and potential, and compute GMC impacts as described above. Finally, for each chain, we impact the stream in each case with 21 different realizations of the GMC population by adding random φ rotations to their current coordinates as described above. The resulting power spectrum for all the cases are shown in Figure 13 by the colored curves.
Results
In Figure 13 we show the median power spectrum of the 42 GMC realizations and their 2 − σ scatter (gray-shaded region) for the fiducial Pal 5 orbit. The upper bound power at the largest scale is comparable to that of the CDM case, which itself matches the observed Pal 5 stream's power. The lower bound is an order of magnitude less, indicating the wide range of power that the GMCs can impart to the stream. Compared to the CDM case, there is a lot more power due to the GMCs at lower angular scale. This seems counterintuitive in the light of Figure 14, which indicates the stream has a similar number of hits by low mass (< 10 6 M ) subhalos (dashed black line) as by GMCs (dashed blue line). The difference arises because the GMCs are much more compact (∼ 5 times) than the subhalos, so they are capable of inflicting small scale perturbations to the stream, as also seen in their density in Figure 12. This difference in power at small scales can be used to statistically set the GMC impacts from CDM subhalo impacts. At large scales there is more power for the CDM case, because a stream has a signif- icant chance of encountering a massive (> 10 8 M ) subhalo during its evolution as seen in Figure 14, while such massive GMCs do not exist.
Varying the pericenter of the stream, we find that for orbits with pericenters that are less than the fiducial 7.34 kpc, the stream encounters many more GMC impacts. As a result the power in all these cases is much higher. In general, the number of GMCs increases as one goes closer to the Galactic center. However, the number of impacts depends not only on the number of GMCs, but also on their orbit relative to the stream, because that decides whether a GMC will fly by the stream with an impact parameter less than b max .
The upper limit of the 2 − σ dispersion of power of all the different pericenter cases is at the level of ∼ 1.2, which is higher than the observed power of the Pal 5 stream. From all these results, we can conclude that our ignorance of the evolution of the GMCs over the dynamical age of the stream makes them the biggest source of uncertainty in using the Pal 5 stream as a probe for dark matter subhalos.
IMPACTS DUE TO THE GLOBULAR CLUSTERS
The final baryonic component whose effect on the Pal 5 stream we consider is the population of globular clusters (GCs). The Milky Way hosts 157 GCs (Harris 1996(Harris , 2016, which as dense, massive concentrations of stars may affect stellar streams. To determine their effect on the Pal 5 stream, we follow the same procedure as for the GMCs above. We obtain approximate orbits for the GCs as follows. For 75 of the GCs, we use the proper motions and other kinematic information from Helmi et al. (2018), who determined proper motions of these GCs using data from Gaia Data Release 2 (DR2). For 72 of the remaining GCs, we obtain the same information from the recent catalog by Vasiliev (2018), who similarly used Gaia DR2 data. For the remaining 10 GCs we were unable to find complete kinematic information and we do not consider them further. Aside from the proper motions, most of the phase-space coordinate information in both the Helmi et al. (2018) and Vasiliev (2018) comes from the Harris (2016) globular cluster catalog, aside from some minor modifications from Baumgardt & Hilker (2018).
We obtained masses for 112 of the GCs in the sample from the catalog by Baumgardt & Hilker (2018). For GCs without a mass measurement in this catalog, we conservatively assign them the highest of any GC in the catalog: 3.5 × 10 6 M . Just like their kinematic information, the angular size of the GCs are taken from their respective catalogs and their physical radius then follows from multiplying the angular size by the distance. We then model the GCs as Plummer spheres with scale radius set equal to their physical radius. As in the case of GMCs, by modeling the GCs as Plummer spheres, we make the assumption that most (∼ 90%) of the mass is within their angular size and so the scale radius is set by dividing the angular radius by 3. Using the kinematic information, we compute the past orbit of each GC in MWPotential2014 and use the same steps as for the GMCs to compute their impacts with the Pal 5 stream. Given that the maximum size of the GCs is ∼ 100 pc, we consider impacts out to 0.5 kpc from the stream in all cases.
The density perturbation arising from GC impacts on the Pal 5 stream are very small. The power of the relative density fluctuation is 10 −3 on all scales. This is below the contribution from all of the other baryonic perturbers. This conclusion is unsurprising, since most GCs have low masses and that they are sparsely distributed throughout the halo. Because the population of GCs is not expected to change much over the last 5 Gyr, therefore their effect on the Pal 5 stream's density is not expected to be any different.
DISCUSSION AND CONCLUSION
In this paper, we have presented an in-depth analysis of the effects of the baryonic structures in our Galaxy on the density of the Pal 5 stream. We considered the effect from the Galactic bar, the spiral arms, and the Galactic population of GMCs and GCs. We examined the effect of each perturber separately by varying their model parameters within limits set by observations and quantified the perturbation imparted to the stream by computing the power spectrum of the stream's density relative to a smooth fit. Figure 15 Galactic Bar Spiral arms GMC CDM Figure 15. Summary of the results of this work. Each curve shows the power spectrum of the mock Pal 5 stream's density as a result of perturbations from the different baryonic structures considered in this paper. The black curve shows the power induced by a 5 Gyr old bar of mass 10 10 M rotating with a pattern speed 39 km s −1 kpc −1 . The green curve is the power due to fourarmed, 3 Gyr old spiral structure whose amplitude corresponds to 1% of the radial force at the location of the Sun due to the axisymmetric background, and rotating with a pattern speed of 19.5 km s −1 kpc −1 . Both the bar and spiral curves are constructed using 5×10 6 points along the stream to minimize the shot noise (the noise is therefore 1/ √ 10 times lower than in the results in Section 3.2 and 4.2). The red curve is the median power imparted by the Galactic population of GMCs on the Pal 5 stream and the blue curve indicates the median power due to CDM subhalo impacts. The black dots are the power and the gray horizontal line is the noise power of the observed Pal 5 stream as computed in Bovy et al. (2017).
presents a summary of our findings. In this figure, the density power spectrum of the Pal 5 stream for the four different types of perturbers is shown and compared to the observed power spectrum from Bovy et al. (2017). For the bar and spiral structure models we choose a representative example, while for GMCs and dark-matter subhalos we present the median expectation from different realizations of the population (we do not show the GCs, because their power is negligible).
On large scales, where the current observations are dominated by signal rather than noise, all different types of perturbers-baryonic and dark-matter alike-can produce power in the density similar to the observed power. This implies that constraining the CDM subhalo population using the large-scale power, as was done by Bovy et al. (2017), is complicated. Because the exact parameters of the bar, spi-ral structure, and the GMC population are still uncertain, it is difficult to predict exactly how much power they induce. But for the bar models in particular, the generic prediction from our modeling in Section 3 is that much power is induced on large scales and the bar must therefore contribute much of the power. Thus, little room is left for dark-matter subhalos to contribute to the power on large scales.
On small scales, the effect of the bar and spiral structure diminishes strongly and on ≈ 1 • scales, they drop below the predicted power from dark-matter substructure. But the power due to impacts with the GMC population remains about an order of magnitude larger than that due to dark matter subhalos on these scales. As demonstrated by Figure 14, this is due to the combination of the fact that the stream interacts with a much larger number of M ≈ 10 6 M GMCs than dark matter subhalos and that these GMCs are more concentrated than subhalos of the same mass. GMC impacts are difficult to distinguish from those from dark matter subhalos. The only difference is that (a) they occur when the stream passes through the disk, while dark matter subhalo impacts occur while the stream is in the halo, and (b) GMCs are about five times more compact. While these differences may in principle be used to distinguish the GMCs and dark matter subhalos, in practice Bovy et al. (2017) showed that the exact time of impact matters little after a few orbits and that the effect of the concentration on the power spectrum is degenerate with the number of perturbers. Thus, the large effect of GMCs on the Pal 5 stream's density is a largely insurmountable issue when using the stream to constrain the dark matter subhalo population.
Based on the analyses in this paper, it is clear that the Pal 5 stream is heavily affected by the baryonic structures in our Galaxy. While better constraints on the properties of the bar and spiral structure in the Milky Way might allow us to account for their deterministic effect, the stochastic nature of the Pal 5 stream's interaction with the Galactic population of GMCs severely limits its usefulness for constraining the dark matter subhalo population in the inner Milky Way. Figure 15 demonstrates that better observations of Pal 5's density should uncover large small-scale density fluctuations like those in Figure 12 that are due to GMC impacts. This could provide a useful constraint on the total population of high-mass (M 10 5 M ) GMCs and on the evolution of the GMC mass function. That we can measure the properties of the high-mass GMC population using the Pal 5 stream will also be useful in determining the, hopefully lesser, effect of GMCs on other cold streams.
For constraining dark matter substructure, it is necessary to consider other cold streams. A prime example is the GD-1 stream (Grillmair & Dionatos 2006), which may be a more suitable candidate, because it is situated farther away from the Galactic center with a perigalacticon of ≈ 14 kpc. In addition, GD-1 is on a retrograde orbit with respect to the disk and therefore the effect of the bar, spiral structure, and GMC population is expected to be minimal (e.g., Amorisco et al. 2016;Erkal et al. 2017). However, baryonic effects may still play a minor role and the methodology in this paper could be applied with few changes to determine their effect on GD-1 or any other stream.
Figure 1 .
1Reconstructed density of the bar model for different expansion orders in the basis-function approach to obtaining the potential. The black curve show the analytic density of the bar.
2 Figure 2 .
22Mbar =6 × 10 9 M , Fz = −71.38 × 2πG M pc −2 Mbar =8 × 10 9 M , Fz = −70.38 × 2πG M pc −2 Mbar =10 × 10 10 M , Fz = −69.38 × 2πG M pc −2 Mbar =1.2 × 10 10 M , Fz = −68.38 × 2πG M pc −2 Mbar =1.4 × 10 10 M , Fz = −67.39 × 2πG M pc −Circular velocity curve for models with different bar masses. The black dot with the error bar shows the constraint on the circular velocity at the location of the Sun. The legend of the plot lists the vertical force at (R, z) = (8, 1.1) kpc.
Figure 3 .
3Orbit of the Pal 5 globular cluster evolved in the fiducial barred potential (red) versus in the axisymmetric potential (black). The top and the middle panel shows the orbits in Galactocentric (R, z) plane and (x, y) plane for the last 5 Gyr of evolution. The bottom panel compares the orbital evolution in custom (ξ, η) coordinates from 50 Myr in the past to 50 Myr in the future, the range which approximately spans the observed Pal 5 stream; there is almost no difference in the present orbit between the barred and axisymmetric potential.
Figure 4 .
4Star counts of mock Pal 5 streams. The top panel shows the star count of the stream evolved in the axisymmetric potential. The middle panel shows the star count of the stream evolved in the Milky Way potential with a 5 Gyr old bar of mass 10 10 M and of pattern speed 39 km s −1 kpc −1 . The blue curve is the 3 rd order polynomial fit to the star counts with which we normalize the density. The bottom panel shows the perturbed stream star count divided by the polynomial, of which we compute the power spectrum shown in subsequent figures. In each case the sample size is 500,000 and the ξ bin width is 0.1 • . The red error bars are the shot noise in each bin.
Figure 5 .
5Length of the Pal 5 stream as a result of varying the mass and pattern speed of the bar. For certain pattern speeds such as 35, 45 and 55 km s −1 kpc −1 , the stream is cut short if the mass of the bar is greater than 6×10 9 M . For the case M bar = 1.4×10 10 M and with a pattern speed of 55 km s −1 kpc −1 , the stream has a length of ∼ 14 • which is out of trend. This happened because a large number of stars from the leading arm are perturbed past the progenitor and they end up at the location of the trailing arm.
Figure 6 .
6Power spectrum of the density of the Pal 5 stream evolved in a barred Milky Way potential with different pattern speeds. We only consider pattern speeds for which the stream length is close to the observed length of the Pal 5 stream. In each case the bar is 5 Gyr old and has a mass of 10 10 M . The top panel shows the power spectrum of the stream density. The gray dashed horizontal line shows the noise power as a result of the shot noise. The cyan dashed curve is the median power spectrum of 1,000 simulations of the stream density as result of impacts with CDM subhalos of mass in the range 10 5 − 10 9 M fromBovy et al. (2017) for comparison. The bottom panel displays the difference between the power in the case where the effect of the perturbation on the progenitor orbit is considered and the case in which it is not considered, as described in the text. The bar induces power on large scales that is similar or larger than that induced by dark matter subhalos, but drops significantly on small scales.
Figure 7 .
7Same asFigure 6, but showing the effect of varying the mass of the bar on the density power spectrum. The pattern speed of the bar is 39 km s −1 kpc −1 .
Figure 8 .
8Same asFigure 6, but showing the effect of varying the age of the bar on the density power spectrum. The mass of the bar is 10 10 M and its pattern speed is 39 km s −1 kpc −1 . The age of the bar has only a minor effect on the induced power, especially on small scales.
Figure 9
9Figure 9. Same as Figure 6 but for a Milky Way potential that includes spiral structure rather than a bar. The plot shows the effect of varying the age and pattern speed of a two-armed spiral that contributes 1 % of the radial force at the Sun.
Figure 10 .
10• and the reference angle φ ref is set to 26 •(Siebert et al. 2012;Faure et al. 2014;Monari et al. 2016a). R s is the radial scale length of the spiral density, Same asFigure 9but for a four-armed spiral potential.
Figure 11 .
11Size R vs. mass M relation for GMCs in the Miville-Deschênes et al. (2016) catalog. The red line is the fit described by Equqation (9) and the black line is 3 times the scale radius of the Plummer spheres used to model the GMCs.
Figure 12 .
12Normalized stream density of two realizations from the 42 different realizations of the Milky Way GMC population as explained in the text. There are a number of small scale perturbations which results in high power at small angular scales as shown inFigure 13.
rFigure 13 .
13peri = 4.68 kpc r peri = 5.52 kpc r peri = 6.44 kpc r peri = 7.18 kpc r peri = 7.34 kpc r peri = 7.92 kpc CDM Median power spectrum of density perturbations from GMCs for the fiducial Pal 5 orbit (pericenter = 7.34 kpc) from 42 different realizations of the Galactic population of GMCs with mass greater than 10 5 M (black curve); the shaded region displays the 2 σ range spanned by the 42 realizations. Each colored curve represents the median power (over 21 realizations) of the Pal 5 stream with orbits with different pericenter radii. The cyan dashed curve shows the power of the stream density as a result of CDM subhalo encounters.
Figure 14 .
14Histogram showing the average number of impacts for different mass GMCs/subhalos that the Pal 5 stream encounters in different setups. The red solid line denotes the mean number of impacts over the 5 non-fiducial Pal 5 orbits (each orbit had 21 realizations) whose median powers are shown inFigure 13. The blue dashed line denotes the mean number of GMC impacts of the fiducial Pal 5 orbit over the 42 realizations. The black dashed line denotes the mean number of impacts over the 1000 CDM-like simulations whose median power is shown by the dashed cyan curve inFigure 13.
MNRAS 000, 1-13(2018)
ACKNOWLEDGEMENTSWe thank Gianfranco Bertone for several useful discussions and comments. NB acknowledges the support of the D-ITP consortium, a programme of the Netherlands Organization for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW). JB acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number RGPIN-2015-05235, and from an Alfred P. Sloan Fellowship.This paper has been typeset from a T E X/L A T E X file prepared by the author.
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"SELLING A SINGLE ITEM WITH NEGATIVE EXTERNALITIES TO REGULATE PRODUCTION OR PAYMENTS?",
"SELLING A SINGLE ITEM WITH NEGATIVE EXTERNALITIES TO REGULATE PRODUCTION OR PAYMENTS?"
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"Tithi Chattopadhyay [email protected] \nCenter for Information Technology Policy\nDepartment of Computer Science\nPrinceton University\n08540PrincetonNJ\n",
"Nick Feamster [email protected] \nDepartment of Computer Science\nPrinceton University\n08540PrincetonNJ\n",
"Matheus V X Ferreira \nDepartment of Computer Science\nPrinceton University\n08540PrincetonNJ\n",
"Danny Yuxing Huang \nDepartment of Computer Science\nPrinceton University\n08540PrincetonNJ\n",
"S Matthew Weinberg [email protected] \nPrinceton University\n08540PrincetonNJ\n"
]
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"Department of Computer Science\nPrinceton University\n08540PrincetonNJ",
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| We consider the problem of regulating products with negative externalities to a third party that is neither the buyer nor the seller, but where both the buyer and seller can take steps to mitigate the externality. The motivating example to have in mind is the sale of Internet-of-Things (IoT) devices, many of which have historically been compromised for DDoS attacks that disrupted Internet-wide services such as Twitter Brian Krebs (2017); Nicky Woolf (2016). Neither the buyer (i.e., consumers) nor seller (i.e., IoT manufacturers) was known to suffer from the attack, but both have the power to expend effort to secure their devices. We consider a regulator who regulates payments (via fines if the device is compromised, or market prices directly), or the product directly via mandatory security requirements. Both regulations come at a cost-implementing security requirements increases production costs, and the existence of fines decreases consumers' values-thereby reducing the seller's profits. The focus of this paper is to understand the efficiency of various regulatory policies. That is, policy A is more efficient than policy B if A more successfully minimizes negatives externalities, while both A and B reduce seller's profits equally. We develop a simple model to capture the impact of regulatory policies on a buyer's behavior. In this model, we show that for homogeneous markets-where the buyer's ability to follow security practices is always high or always low-the optimal (externality-minimizing for a given profit constraint) regulatory policy need regulate only payments or production. In arbitrary markets, by contrast, we show that while the optimal policy may require regulating both aspects, there is always an approximately optimal policy which regulates just one. Sand-Zantman. 2016. A mechanism design approach to climate-change agreements. | 10.1145/3308558.3313692 | [
"https://arxiv.org/pdf/1902.10008v1.pdf"
]
| 67,856,244 | 1902.10008 | a75126378755d669d6b07c1f4b28a46b0d288de9 |
SELLING A SINGLE ITEM WITH NEGATIVE EXTERNALITIES TO REGULATE PRODUCTION OR PAYMENTS?
February 27, 2019
Tithi Chattopadhyay [email protected]
Center for Information Technology Policy
Department of Computer Science
Princeton University
08540PrincetonNJ
Nick Feamster [email protected]
Department of Computer Science
Princeton University
08540PrincetonNJ
Matheus V X Ferreira
Department of Computer Science
Princeton University
08540PrincetonNJ
Danny Yuxing Huang
Department of Computer Science
Princeton University
08540PrincetonNJ
S Matthew Weinberg [email protected]
Princeton University
08540PrincetonNJ
SELLING A SINGLE ITEM WITH NEGATIVE EXTERNALITIES TO REGULATE PRODUCTION OR PAYMENTS?
February 27, 2019Mechanism Design and ApproximationAuction DesignNegative ExternalitiesTragedy of the CommonsPolicy and Regulation
We consider the problem of regulating products with negative externalities to a third party that is neither the buyer nor the seller, but where both the buyer and seller can take steps to mitigate the externality. The motivating example to have in mind is the sale of Internet-of-Things (IoT) devices, many of which have historically been compromised for DDoS attacks that disrupted Internet-wide services such as Twitter Brian Krebs (2017); Nicky Woolf (2016). Neither the buyer (i.e., consumers) nor seller (i.e., IoT manufacturers) was known to suffer from the attack, but both have the power to expend effort to secure their devices. We consider a regulator who regulates payments (via fines if the device is compromised, or market prices directly), or the product directly via mandatory security requirements. Both regulations come at a cost-implementing security requirements increases production costs, and the existence of fines decreases consumers' values-thereby reducing the seller's profits. The focus of this paper is to understand the efficiency of various regulatory policies. That is, policy A is more efficient than policy B if A more successfully minimizes negatives externalities, while both A and B reduce seller's profits equally. We develop a simple model to capture the impact of regulatory policies on a buyer's behavior. In this model, we show that for homogeneous markets-where the buyer's ability to follow security practices is always high or always low-the optimal (externality-minimizing for a given profit constraint) regulatory policy need regulate only payments or production. In arbitrary markets, by contrast, we show that while the optimal policy may require regulating both aspects, there is always an approximately optimal policy which regulates just one. Sand-Zantman. 2016. A mechanism design approach to climate-change agreements.
Introduction
The Tragedy of the Commons is a well-documented phenomenon where agents act in their own personal interests, but their collective action brings detriments to the common good Hardin (1968). One motivating example that we will keep referencing in the paper is the sale of Internet-of-Things (IoT) devices, such as Internet-connected cameras, light bulbs, and refrigerators. Recent years have seen a proliferation of these "smart-home" devices, many of which are known to contain security vulnerabilities that have been exploited to launch high-profile attacks and disrupt Internet-wide services such as Twitter and Reddit Brian Krebs (2017); Nicky Woolf (2016). Both the owners and manufacturers of IoT devices have the ability to protect the common good (i.e., Internet-wide service for all users) from being attacked by securing their devices, but have little incentive to do so. For the manufacturers, implementing security features, such as using encryption or having no default passwords, introduces extra engineering cost August et al. (2016). Similarly, security practices, such as regularly updating the firmware or using complex and difficult-to-remember passwords, can be a costly endeavor for the consumers Choi et al. (2010); Redmiles et al. (2018). The results of their actions cause a negative externality, where Internet service is disrupted for other users.
One way to reduce the negative externality is regulation. In the context of IoT sales, a regulator can, for instance, set minimum security standards for the manufacturers or impose fines on owners of hacked IoT devices that engage in attacks. Fines could come in a few forms: direct levies on the consumer, or indirect monetary incentives. For instance, ISPs could offer discounts to users whose networks have not displayed any signs of malicious activities. One might argue that such penalty-based policies could be too futuristic, but it is worth noting that similar practices are being adopted in other industries to mitigate negative externalities Baumol (1972). One example is the levying of fines on users (such as cars and factories) that cause pollution Fullerton (1997). While there are a lot of practicalities that have to be kept in mind and the decision of when to implement consumer/user fines depends on various factors, this is certainly one of the various policy alternatives that is worthwhile to study. Such regulations, however, can potentially increase the cost of production, discourage consumers from purchasing IoT devices, and reduce the manufacturer's profit. Our focus is to compare the efficiency of various regulatory policies: for two policies which equally hurt the seller's profits, which one better mitigates externalities? We will also be interested in understanding the optimal policy: the minimum security standards for the manufactures and fines on owners that together best mitigate externalities subject to a minimum seller's profit.
We first develop a model that consists of a buyer (e.g., consumers interested in purchasing IoT devices) and a single product for sale (e.g., IoT device). The product may come with some (costly to increase) level of security, c, and a consumer purchasing the device may choose to spend additional effort h to further secure the device. We consider a mechanism for regulating the market through incentives, for example, by requiring that the product being sold implement security features that cost the seller c dollars, imposing a fine of y dollars on the buyer if the product is later compromised and used in attacks, or both. Which intervention is more appropriate depends on how efficient buyers are in securing the product. The goal of the regulator is to minimize the negative externalities subject to a cap on the negative impact on the seller's profits-the idea being that any policy which too negatively impacts the seller could be unimplementable due to industry backlash.
Understanding the effects of such regulations on the behavior of a single consumer is relatively straightforward. For example: as fines go up, consumers adjust (upwards) the optimal level of effort to expend, lowering their total value for the item. Yet, reasoning about how an entire market of consumers will respond to changes, and how these responses impact seller profits becomes more complex.
Our contributions are as follows: (i) We model the sale of a single item with negative externalities, using the sale of IoT devices as the motivating example (Section 3). (ii) We show in Sections 5 and 6 that when the population of consumers is homogeneous (i.e., all consumers are comparably effective at translating effort into security) that optimal policies need only to regulate either the product (via minimum security standards) or the payments (via fines). (iii) We provide an example of non-homogeneous markets where the optimal policy regulates both product and payments, but prove that in all markets, it is always approximately optimal to regulate only one (Sections 7.1 and 7.2). The technical sections additionally contain numerous examples witnessing the subtleties in reasoning about these problems, and that any assumptions made in our theorem statements are necessary.
Related Work
Auction Design with Externalities There is ample prior work studying auction design with network externalities in the following sense: if the item for sale is a phone, then one consumer's value for the phone increases when another consumer purchases a phone as well (which is a positive externality, because they can talk to more people). Similarly, the item could be advertising space, in which case one consumer's value for advertising space could decrease as other consumers receive space (which is a negative externality, as now each unit of space is less likely to grab attention) Haghpanah et al. (2013); Jehiel et al. (1996); Mirrokni et al. (2012); Candogan et al. (2012); Bhattacharya et al. (2011); Hartline et al. (2008). Our work differs in that it is a third party, who is neither selling nor purchasing an item, who suffers the externalities.
Improving the Commons There is also a large body of work studying the regulation of common goods (e.g., clean air, security, spectrum access) in the form of taxes or licenses. For example, a government agency can regulate the emission of pollution by auctioning licenses (perhaps towards minimizing the total social cost-regulation cost plus negative externalities) Montero (2008); Martimort and Sand-Zantman (2016); Seabright (1993); Lehr and Crowcroft (2005); Feldman et al. (2013); Weitzman (1974). Our work differs in that our regulations are constrained to guarantee minimum profit to the seller, rather than focusing exclusively on the social good.
Approximation in Auction Design
Owing to the inherent complexity of optimal auctions for most settings of interest, it is now commonplace in the Economics and Computation community to design simple but approximately auctions. Our work too follows this paradigm. We refer the reader to previous work Hartline (2013) for an overview of this literature.
Mitigating Security Problems Computer security is a particular example of the Tragedy of the Commons, where a software or hardware provider sells an insecure product, and where consumers may purchase the product without considering or taking actions to reduce the security risks. In addition, users might be unable to distinguish insecure products from insecure ones Akerlof (1978). One mitigation strategy is to have the vendor release updates with security features, although this could be a costly process, as August observes August et al. (2016). However, identifying the existence of security vulnerabilities in the first place may take time for the vendors; for instance, a common software vulnerability known as buffer overflow remained in more than 800 open-source products for a median period of two years before the vendors fixed the problems, according to a study by Li Li and Paxson (2017).
An alternative to relying on a vendor to implement security features or releasing updates is to incentivize the users to follow security practices. Redmiles has found that users who adopt security practices, like using two-factor authentication, have a lower overall utility for themselves than if they adopt no security practices at all, as security practices may introduce inconvenience Redmiles et al. (2018). Even if users were notified of security problems that they were presumably unaware of, it took as long as two weeks for fewer than 40% of the users to take remedial actions, according to a study Li et al. (2016). To introduce incentives, vendors could, for instance, offer discounts to users who adopt security behaviors August et al. (2016); regulators, on the other hand, could incur fines to users whose software or devices were hacked Kunreuther and Heal (2003), which is a part of our model in this paper.
Model
In this section, we introduce our model, which consists of a population of rational buyers and a single item for sale. After introducing each of the concepts one-by-one, we include a table (Table 1) at the end of this section to remind the reader of each of the components.
Buyer properties Buyers in our model have two parameters: (v, k) ∈ R 2 + . v denotes the buyer's value for the item (i.e., how much value does the buyer derive from the IoT device in isolation, independent of fines, etc.). k denotes the buyer's effectiveness in translating effort into improved security. That is, a buyer with high k can spend little effort and greatly reduce the risk of being hacked (e.g. because they are well-versed in security measures). A buyer with low k requires significant effort for minimal security gains. We will often use t := (v, k) to denote a buyer's type.
Security A buyer who chooses to purchase an item will spend some level of effort h ≥ 0 securing it, which causes disutility h to the buyer. The seller may also include some default security level c. If the buyer has effectiveness k, we then denote the combined effort by EFFORT(k, c, h) := c + kh. The idea is that buyers with higher effectiveness are more effective at securing the device for the same disutility. Note that buyers with effectiveness k > 1 are more effective than the producer, and buyers with effectiveness k < 1 are less effective. Highly effective buyers should not necessarily be interpreted as "more skilled" than producers, but some security measures (e.g., password management) are simply more effective for consumers than producers to implement.
We model the probability that a device is compromised as a function g(·) of EFFORT, with g(x) := e −x . This modeling decision is clearly stylized, and meant as an approximation to practice which captures the following two important features: (a) as effort x approaches ∞, g(x) → 0 (that is, it is possible to shrink the probability of being compromised arbitrarily small with sufficient effort), and (b) g (x) ≥ 0. That is, the initial units of effort are more effective (i.e. g (x) is larger in absolute value) than latter ones (when g (x) is smaller in absolute value). The idea is that consumers/producers will take the highest "bang-for-buck" steps first (e.g., setting a password). Note that our results do not qualitively change if, for instance, g(x) := λ 1 e λ2x for some constants λ 1 ∈ (0, 1], λ 2 > 0, but since the model is stylized anyway we set λ 1 = λ 2 = 1 for simplicity of notation.
Regulatory Policy
The regulator selects a policy/strategy s = (y, c, p) ∈ ∈R 3 + . Here, c denotes the security standards the producer must include which is equivalent to the production cost. y denotes the fine the consumer pays should their device be compromised. p denotes the price of the item. Conceptually, one should think of the regulator inducing the producer to set security standard c and price p via particular regulatory policies (e.g., requiring a minimum security level c , or mandating purchase of insurance). Mathematically, we will not belabor exactly how the regulator arrives at (y, c, p). We will also be interested in "simple" policies, which regulate either y or c.
Definition 1 (Simple Policy). For a policy s = (y, c, p), we say s is a fine policy if c = 0, a cost policy if y = 0 and a simple policy if s is either a fine policy or a cost policy.
Utilities Recall that so far our buyer has value v and efficiency k, and chooses to put in effort h. The regulator mandates security c (which is equivalent to the production cost) on the item (which has price p) and imposes fine y for compromised items. The probability that an item is compromised is g(EFFORT(k, c, h)) = e −c−kh . The buyer's utility is therefore: v − p − h − y · e −c−kh . Observe that the buyer is in control of h (but not v, p, y, c, k). So the buyer will optimize over h ≥ 0 to minimize h + y · e −c−kh . By taking the derivative with respect to h, we get a closed form for the choice of effort h * (t, s) (recalling that we denote the buyer's type t = (v, k) and the regulator's strategy s = (y, c, p)):
h * (t, s) = max 0, ln(yk) − c k(1)
We can now see that the probability that the buyer's item is compromised, conditioned on expending the optimally chosen effort is:
RISK(t, s) := min e −c , 1 yk .(2)
We will additionally refer to the buyer's (security) loss as the expected fines they suffer plus the effort they spend. That is:
(t, s) := y · RISK(t, s) + h * (t, s) := ln(yk)−c+1 k , yk ≥ e c ye −c , yk < e c(3)
It then follows that the buyer's utility (value minus price minus expected fines) is:
u(t, s) := v − p − (t, s) = v − p − 1/k − ln(yk)−c k , yk ≥ e c v − p − ye −c , yk < e c(4)
Population of Buyers We model the population of buyers as a distribution D over types t. Additionally, we make the now-typical assumption in the multi-dimensional mechanism design literature (e.g. (Chawla et al., 2007;Hart and Nisan, 2017, and follow-up work)) that the parameters v and k are drawn independently, so that D := D v × D k . 1 The seller's profits are then:
PROF D (s) := (p − c) · Pr t←D [u(t, s) ≥ 0](5)
Externalities Finally, we define the externalities caused and the regulator's objective function. Each device sold has some probability of being compromised, and the regulator wishes to minimize the total fraction of compromised devices. 2 That is, we measure the externalities caused as: 3
EXT D (s) := E t←D [RISK(t, s) · I(u(t, s) ≥ 0)] Pr t←D [u(t, s) ≥ 0](6)
Optimization The regulator's objective is to propose an s = (y, c, p) that minimizes EXT D (s). Observe that, if left unconstrained, the regulator can simply propose c → ∞, resulting in 0 externalities. Such a policy is completely unrealistic, as it would cause costs to approach ∞ and destroy the industry. Similarly, taking y → ∞ would cause consumers to have negative utility even to get the item for free (again destroying the industry). We therefore impose a Figure 1: Seller's optimal profits under different distributions for efficiency, k. We plot the seller's profits on the vertical axis and the default security c on the horizontal axis. Each curve corresponds to different fines, y. Importantly, observe that when the fine is zero, the seller achieves greatest profits with lower default security. However, when the fine is non-zero, the seller may actually increase their profits with default security, but the benefits (to the seller) of default security decrease as the buyer population becomes more efficient.
minimum profit constraint for a policy to be considered feasible. Indeed, this forces the regulator to trade off profits for externalities as effectively as possible. Therefore, our regulator is given some profit constraint R, and aims to find:
arg min s,PROF D (s)≥R {EXT D (s)}
We will only consider cases where there is some feasible s (that is, we will only consider R such that there exists a p with R ≤ PROF D (0, 0, p). If no such p exists, then the profit constraint exceeds the optimal achievable profit without regulation, and the problem is unsolvable). Table 1 recaps the parameters of our model for future reference. Note also that many parameters (e.g.
Recap of model
(·, ·) are formally defined as a function of t = (v, k) and s = (y, c, p), but only depend on (e.g.) k, y, c. As such, it will often be clearer to overload notation and write (k, y, c), rather than defining a new t = (v, k) with a meaningless parameter. Sometimes, though, it will be clearer to use the defined notation for a type t that was just defined. In the interest of clarity, we will overload notation for these variables, but it will be clear from context what they refer to.
t = (v, k) (value, effectiveness) N/A D := D v × D k buyer population N/A s = (y, c, p) (fine, security, price) N/A h * (t, s) buyer optimal effort max{0, ln(yk)−c k },(1) RISK(t, s) compromise prob. min{e −c , 1 yk }, (2) (t, s) buyer security loss Equation (3) u(t, s) buyer utility v − p − (t, s), (4) PROF D (s) seller profits Equation (5) EXT D (s)
frac. compromised Equation (6) Final Thoughts on Model We propose a stylized model to capture the following salient aspects of this market: (a) neither buyer nor seller suffer externalities when the item is compromised, (b) the regulator can regulate both the product (via c) and payments (via y, p), (c) there is a population of buyers, each with different value v and effectiveness k at translating effort into security, and (d) the regulator must effectively trade off externalities with profits by minimizing negative externalities, subject to a minimum profit constraint R. The goal of this model is not to capture every potentially relevant parameter, but to isolate the salient features above.
An Intuitive Example
In this section, we provide one example to help give intuition for the interaction between the fines y, default security c, and seller's profits PROF D (s). In particular, Figure 1 plots the maximum achievable PROF D (s) over all s with a fixed c (the x-axis) and y (the color of the plot). In all three examples, D v is the uniform distribution on [0, 20], and k is drawn from either the uniform distribution on [0, 1], [0, 3], or [2, 3], respectively. Note that k ≥ 1 is the threshold when a buyer is more efficient than the seller in mitigating externalities, so these examples cover two homogeneous populations, where all consumers are more (respectively, less) efficient than the producer, and one heterogeneous population, where some consumers are more efficient, and others are not.
For each possible (partial) regulation (y, c), the profit-maximizing choice of p is essentially a classic single-item problem (e.g. Myerson (1981)), as the buyer's "modified value" v is simply v − (t, s) − c, and the seller's profit for setting price p is just p · Pr t←D [v ≥ p]. Therefore, for each partial regulation (y, c), we can construct the modified distribution and simply maximize p · Pr t←D [v ≥ p] as above.
Observe in Figure 1, when y = 0, the seller gets greater profits with lower c. This should be intuitive, as neither the buyer nor seller suffer when the device is compromised. When y > 0, and D k = U ([0, 1]), the seller's profits can increase with c. This should also be intuitive: now that the buyer suffers when the device is compromised, they prefer to buy a secure device.
On the other hand, when the market contains only efficient buyers (k > 1 always), the buyer prefers to provide her own security; any increased cost will always decrease the buyer's utility. Indeed, observe that ∂ (t,s) ∂c is either 0 (if yk < e c ) or 1 − 1/k (otherwise). If k > 1, then this is always positive, so higher c results in (weakly) higher loss for the consumer, and lower utility.
Example of Efficiency Distribution
In our model, we will not make any assumptions on the efficiency distribution D k , but we provide an example of how one could model such distributions. To construct D k , we can isolate the different features (e.g. encryption and security practices) that affects security of a population and how they combined affects the buyer's effectiveness in providing security.
Consider the case where some IoT devices, such as security cameras, allow the use of two-factor authentication, where the first factor is password-based authentication, and the second factor is based on, for instance, SMS. A user has the choice of using passwords alone for authentication or using the two factors. For users that use passwords alone, the efficiency may depend on the strength of their passwords or how likely the passwords are re-used. Figure 2(a) illustrates an example where buyers are generally more efficient than sellers if the buyers pick strong passwords.
For two-factor authentication, the efficiency may depend on the robustness of the second factor; in particular, an SMS-based second factor might be more prone to compromise than hardware-token-based solutions, as SMS messages could be intercepted. 4 Depending on which second factor is implemented by the seller, k's density varies, an example of which is shown in Figure 2(b). In general, however, because the seller has the control over which second factor to use, the seller is more efficient than the buyer. In Figure 2 (c), we use mixture distribution to model D k when password authentication is combined with SMS authentication. In general, systems with two factor authentication allow users to reset passwords (1 st -factor) through SMS (2 nd -factor) which suggests the second factor carries higher weight than passwords in the buyer's efficiency. We define P r[D k = x] = 2 3 P r[D 1 = x] + 1 3 P r[D 2 = x] where the weights model the fact the second 2 nd factor can override the 1 st factor even though SMS authentication can be vulnerable.
Preliminary Observations
We conclude with two observations which allow an easy comparison between the profits of certain policies. Intuitively, Observation 1 claims that any policy which makes every single consumer in the population have lower loss generates greater profits for the seller. We will make use of Observation 1 repeatedly throughout the technical sections to modify existing policies into ones which improve profits (ideally while also improving externalities, although that is not covered by Observation 1).
Observation 1. Let s = (y, c, p), s = (y , c , p ) be such that p − c = p − c ≥ 0 and for all k ∈ support(D k ), (k, s) + c ≤ (k, s ) + c . Then for all D v , PROF Dv×D k (s) ≥ PROF Dv×D k (s ). The efficiency distribution of a population. We plot the effectiveness of a buyer when restricted only to password authentication (left) and SMS authentication (center). By combining the features, and taking into account their contributions in minimizing externalities, we construct the efficiency distribution in the right.
Proof. Observe that for both s and s , the seller's profit per sale is identical (as p − c = p − c). So we just wish to show that the probability of sale for s is larger than that for s. Indeed, observe that for all t:
u(t, s) = v − (k, s) − p = v − ( (k, s) + c) + c − p ≥ v − ( (k, s ) + c ) + c − p = v − (k, s ) − p = u(t, s ).
Therefore, any consumer (v, k) who chooses to purchase the item under policy s will also choose to purchase under policy s, and therefore the probability of sale is at least as large for s as s .
Observation 2 below claims that the profit of any policy s is larger in populations D where every consumer is more effective than in D .
Observation 2. Let D k stochastically dominate D k . 5 Then for all policies s = (y, c, p) with p ≥ c, and all D v , PROF Dv×D k (s) ≥ PROF Dv×D k (s).
Proof. As D k stochastically dominates D k , it is possible to couple draws (t, t ) from (D v × D k , D v × D k ) such that v = v and k ≥ k . Observe simply that u(t, s) ≥ u(t , s) always. Therefore, Pr[u(t, s) ≥ 0] ≥ Pr[u(t , s) ≥ 0], and PROF Dv×D k (s) ≥ PROF D×D k (s).
Observe, however, that Observation 2, perhaps counterintuitively, does not hold if we replace profits with externalities. That is, for a fixed policy s, we might increase all consumers' effectiveness yet also increase the externalities caused. Intuitively, this might happen (for instance) in a fine policy which successfully only sells the item to extremely effective consumers who effectively secure their purchase. Ineffective consumers choose not to purchase the product to avoid fines. However, if these ineffective consumers are instead somewhat effective, they may now choose to purchase the item, thereby increasing externalities. Below is a concrete instantiation:
Example 1. Consider the population where D v is a point-mass at e, and D k takes on effectiveness 0 with probability 1/2 and x > 1 with probability 1/2. Consider the policy s = (e, 0, e − 2.5). Then the (e, 0) consumer chooses not to purchase: (e, 0, s) = e, so their utility would be e − e − (e − 2.5) < 0. The (e, x) consumer chooses to purchase, as their loss is 2+ln(x)
x < 2 (as x > 1). So EXT Dv×D k (s) = 1 ex . Consider now improving the effectiveness of the k = 0 consumers to k = 1 (so D k now takes on 1 with probability 1/2 and x with probability 1/2). The (e, 1) consumer now chooses to purchase, as their loss is 2 (so their utility is e − 2 − (e − 2.5) = 1/2). So now EXT Dv×D k (s) = ( 1 e + 1 ex )/2. As x > 1, the externalities have gone up. If x ≥ 1, the externalities may have gone up quite significantly.
In Example 1, of course "the right" thing to do is to also change the policy. Indeed, it is still the case that, for a fixed consumer who purchases the item, increasing effectiveness can only decrease externalities. But without fixing whether the consumer has purchased the item or not, the claim is false. Observation 3 captures what we can claim about risk, loss, etc. on a per-consumer basis. Proofs for the claims in Observation 3 all follow immediately from the definitions in Section 3.
Observation 3. Let k > k , then for all s:
• RISK(k, s) ≤ RISK(k , s). • (k, s) ≤ (k , s). • h * (k, s) ≥ h * (k , s). • u(k, s) ≥ u(k , s).
Roadmap of Technical Sections
Now that we have the appropriate technical language, we provide a brief roadmap of the results to come.
• In Section 5, we provide a technical warmup to get the reader familiar with how to reason about our problem.
The main result of this section is Theorem 1, which claims that the optimal policy when D k is a point-mass is simple. The proof of this theorem helps illustrate one key aspect of our later arguments, and will also be used as a building block for later proofs.
• In Section 6, we prove our first main result (Theorem 2): as a function of R and D v , there exists a cutoff T .
If D k is supported on [0, T ], then a cost policy is optimal. If D k is supported on [T, ∞)
, then a fine policy outperforms all profits-maximizing policies (we define this term in the relevant section -intuitively a policy is profits-maximizing if the price is the seller's best response to (y, c)). Section 6 also contains a surprising example witnessing that the additional profits-maximizing qualification is necessary.
• In Section 7, we consider general distributions. Unsurprisingly, simple policies are no longer optimal. Perhaps surprisingly, if one insists on exceeding the profits benchmark exactly, no simple policy can guarantee any bounded approximation to the optimal externalities (Corollary 13). However, we also show (Theorem 3) that it is possible to get a bicriterion approximation: if one is willing to approximately satisfy the profits constraint, it is possible to approximately minimize externalities with a simple policy. That is, for any s, D, there is a simple policy s with PROF D (s ) = Ω(1) · PROF D (s) and EXT D (s ) = O(1) · EXT D (s).
• We include complete proofs for our results on point-mass and homogeneous distributions, as these convey many of the key ideas. By Theorem 3, the proofs get quite technical so we defer them to the appendix.
Warm-up: Point-Mass Effectiveness
As a warm-up, we first study the case where D k is a point mass (that is, all buyers in the population have the same effectiveness k). In this case, we show that a simple policy is optimal. The proof is fairly intuitive, with one catch. The intuitive part is that every consumer will put in the same effort, conditioned on buying the item. It therefore seems intuitive that if k < 1, it is better for all parties involved if any effort spent by the consumer is transferred to the producer instead (and this is true). It also seems intuitive that if k > 1, it is again better for all parties involved if any effort spent by the producer is "transferred" to the consumer instead (e.g. by raising fines so that the consumer chooses to spend the desired level of effort). This is not quite true: the catch is that the fine required to induce the desired buyer behavior may be too high to satisfy the profit constraint. But, the above argument does work for sufficiently large k. Importantly, there is some cutoff T such that for all k ≤ T , the optimal policy is a cost policy (y = 0), while for all k ≥ T , the optimal policy is a fine policy (c = 0). Below, when we write D v × {k}, we mean the distribution which draws v from D v and outputs (v, k).
Theorem 1. For all D v , R, and k, the externality-minimizing policy for D v × {k} is a simple policy. Moreover, for all R, D v , there is a cutoff T such that if k ≤ T , then the optimal policy is a cost policy. If k ≥ T , then the optimal policy is a fine policy.
Proof. Consider any policy s = (y, c, p). Because all consumers have the same effectiveness k, s induces the same loss for all consumers. We first claim the following:
Lemma 1. Let k ≤ 1.
Then for all D v and any policy s = (y, c, p), there is an alternative policy s = (0, c , p) with PROF Dv×{k} (s ) ≥ PROF Dv×{k} (s) and EXT Dv×{k} (s ) ≤ EXT Dv×{k} (s).
Proof. In policy s, all consumers have the same loss (k, s). This therefore is a good opportunity to try and make use of Observation 1. First, consider the possibility that yk < e c . In this case, h * (k, s) = 0, (k, s) = ye −c , and RISK(k, s) = e −c . This implies that EXT Dv×{k} (s) = e −c . Consider instead the policy s = (0, c, p). Then (k, s ) = 0, but RISK(k, s) = e −c and EXT Dv×{k} (s) = e −c like before. So the externalities are the same. An application of Observation 1 concludes that the profits have improved (indeed, (c, p) are the same in both policies, and the loss decreases as we switch from policy s to s ).
Consider now the possibility that yk ≥ e c . In this case, h * (k, s) = ln(yk)−c k , (k, s) = ln(yk)−c+1 k , and RISK(k, s) = 1 yk . Consider instead the policy s = (0, ln(yk), p − c + ln(yk)). In this new policy, (k, s ) = 0 and RISK(k, s ) = 1 yk . So indeed, the new policy has the same externalities. We just need to ensure that we can apply Observation 1. To this end, observe that:
(k, s) + c − ( (k, s ) + c ) = ln(yk) − c + 1 k + c − ln(yk) = (1/k − 1) · (ln(yk) − c) + 1/k ≥ 0.
The last line follows because k ≤ 1 and ln(yk) ≥ c (because yk ≥ e c ). So the hypotheses of Observation 1 hold, and we can apply Observation 1 to conclude that the profits improve from s to s as well.
Lemma 1 covers the cases when k ≤ 1: there is always an optimal cost policy. We now move to the case when k > 1.
There are two cases to consider: one where the optimal policy will be a cost policy, and one where the optimal policy will be a fine policy. The distinguishing feature between these cases will be for a given c, how big of a fine is necessary to incentivize the consumer to put in effort c/k, and what the consumer's loss looks like for this choice of y. Below, c * is defined to be the maximum c such that there exists a p such that PROF Dv×{0} (0, c, p) ≥ R. Observe that c * is also equal to the maximum such that there exists a p such that PROF (Dv− )×{0} (0, 0, p) ≥ R (here, D v − denotes the distribution which samples v from D v and then subtracts , taking a maximum with 0 if desired). That is, c * is the maximum loss that can be uniformly applied to all consumers (drawn from D v ) while still resulting in a distribution for which profit ≥ R is achievable.
Lemma 2. Let c * denote the maximum c such that there exists a p such that PROF Dv×{0} (0, c, p) ≥ R. Then a cost policy is optimal for
D v × {k} if k ∈ [1, 1 + 1/c * ].
Proof. First, observe that the lemma hypothesis implies that any feasible policy must have (k, s) + c ≤ c * (if not, then an application of Observation 1 lets us contradict the lemma's hypothesis with a feasible c = (k, s) + c > c * ).
Consider now k ∈ [1, 1 + 1/c * ], and start from some policy s = (y, c, p). If this policy has h * (k, s) = 0, then certainly we can just update s = (0, c, p) and get better profits with the same externalities (by Observation 1). If instead h * (k, s) > 0, then (k, s) = ln(yk)−c+1 k , and RISK(k, s) = 1 yk . Consider instead s * = (0, c * , p * ), for whichever p * witnesses PROF D (s * ) ≥ R (we know that such a p * exists by the lemma's hypothesis). So now we just need to compare externalities. Assume for contradiction that RISK(k, s * ) > RISK(k, s). Then we get:
RISK(k, s * ) > RISK(k, s) ⇒ e −c * > 1 yk ⇒ c * < ln(yk) ⇒ ln(yk) − c + 1 k > c * − c + 1 k ⇒ (k, s) + c > c * − c + 1 k + c ⇒ (k, s) + c > c * + 1 k ⇒ (k, s) + c > c * ⇒⇐ .
The last implication uses the fact that k ≤ 1 + 1/c * . The line before this uses that k ≥ 1. The contradiction arises because this would imply a scheme (s) with profit ≥ R with loss > c * , contradicting the definition of c * by the reasoning in the first paragraph of this proof.
Lemma 3. Let c * denote the maximum c such that there exists a p such that PROF Dv×{0} (0, c, p) ≥ R. Then a fine policy is optimal for D v × {k} if k ≥ 1 + 1/c * .
Proof. Again start from some policy s = (y, c, p), inducing some loss (k, s). First, maybe h * (k, s) > 0. In this case, the risk is 1 yk and the loss plus cost is ln(yk)−c+1 k + c. In particular, observe that the partial derivative of the loss plus cost with respect to c is 1 − 1/k > 0. So the policy s = (y, 0, p − c) has RISK(k, s ) = RISK(k, s) but also (k, s ) + c < (k, s) + c. So Observation 1 claims that this policy gets at least as much profits (and the risk is the same).
If instead, h * (k, s) = 0, then the risk is e −c and the loss is y · e −c . In this case, consider instead y * such that ln(y * k)+1 k = c * and using s * = (y * , 0, p * ), for the p * satisfying PROF D (s * ) ≥ R (again, such a p * must exist by definition of c * , and the fact that (k, s * ) = c * , plus Observation 1). We just need to analyze the risk. Similar to the previous proof, assume for contradiction that RISK(k, s * ) > RISK(k, s). Then:
RISK(k, s * ) > RISK(k, s) ⇒ e −c < 1 y * k → c > ln(y * k) ⇒ ln(y * k) + 1 k < c + 1 k ⇒ c * < c + 1 k ⇒ c * > c * · c + 1 1 + c * ⇒⇐ .
The last inequality uses the fact that k ≥ 1 + 1/c * , and derives a contradiction as c ≤ c * (if c > c * , then certainly (k, s) + c > c * , contradicting the definition of c * ).
All three cases together prove Theorem 1. The T prescribed in the theorem statement is exactly 1 + 1/c * , where c * is the maximum c such that there exists a p for which PROF Dv×{0} (0, c, p) ≥ R.
We conclude with one last proposition regarding the behavior of the threshold with respect to the profits constraints R.
Proposition 4 below states that as R increases, the threshold beyond which a fine policy is optimal increases as well. Proposition 4. Let T (D v , R) denote the threshold such that both a fine policy and cost policy are optimal for
D v × {T (D v , R)} subject to profits constraints R. Then T (D v , R) is monotone increasing in R.
Proof. To see this, let c * (D v , R) denote the maximum c such that there exists a p such that PROF Dv×{0} (0, c, p) ≥ R. Then c * (D v , R) is decreasing in R (as the profits constraint goes up, we can't afford as much security). So
1 + 1/c * (D v , R) is increasing in R.
This means that the threshold T (D v , R) beyond which a fine policy is optimal for D v × {T } is increasing as a function of the profits constraint R (because T = 1 + 1/c * (D v , R)).
This concludes our treatment of the case where k is a point-mass. Theorem 1 should both be viewed as a warm-up to introduce some of our core techniques, and also as a building block towards our stronger theorems (in the following sections). The main technique we introduced is the ability to reduce risk and loss simultaneously to improve both profits and externalities. The idea was that if the buyer is less effective than the seller, everyone prefers that the seller put in effort (y = 0, c > 0). If the buyer is more effective than the seller, everyone prefers that the buyer put in effort. However, the regulator can not directly mandate that the buyer put in effort, and unfortunately the fines required to extract the desired buyer behavior may too negatively affect the profit. This is why the transition from cost to fine policies is 1 + 1/c * instead of 1.
Homogeneous Distributions
In this section, we show that for populations that are sufficiently homogeneous in effectiveness, the optimal policy remains simple. The second half of Theorem 2 requires a technical assumption. Specifically, we say that a policy (y, c, p) is profits-maximizing if, conditioned on y, c, p is set to maximize the seller's profits (that is, PROF D (y, c, p) ≥ PROF D (y, c, p ) for all p ). Theorem 2. For all D v , R, there exists a cutoff T such that • For all D k supported on [0, T ], the externality-minimizing policy for D v × D k subject to profits R is a cost policy.
• For all D k supported on [T, ∞), the externality-minimizing policy for D v × D k subject to profits R is either a fine policy, or it is not profits-maximizing.
The proof of Theorem 2 will follow from Lemmas 4 and 6, which handle the two claims in the theorem separately. Finally, we show in Section 6.3 that the profits-maximizing qualification in part two of Theorem 2 is necessary: Proposition 5. There exist distributions D v , D k , and profits constraint R such that:
• T is such that for all k ≥ T , the externality-minimizing policy for D v × {k} subject to profits constraints R is a fine policy.
• D k is supported on [T, ∞).
• No fine policy is externality-minimizing policy for D v × D k subject to profits constraints R.
• The externality-minimizing policy for D v × D k subject to profits constraints R is not simple, and not profitsmaximizing (the latter is implied by the second bullet of Theorem 2).
Proposition 5 is perhaps surprising: a fine policy is externality-minimizing for D v × {T }, and D k stochastically dominates T , so the same fine policy has even lower externalities, and potentially greater profit for D v × D k . Indeed, the optimal fine policy for D v × D k achieves lower externalities than that of D v × {T }. The catch is that an even better non-simple policy becomes viable, and achieves still lower externalities. Theorem 2 claims, however, that the optimal non-simple policy must not be profits-maximizing. Even more surprising, we show in Appendix B that if we constrain the optimization problem to only profits-maximizing prices then Proposition 5 is still true which implies the negation of the second bullet of Theorem 2.
Extension Lemma for small k
The small k case follows roughly from the following intuition. For cost policies, neither the buyer's loss nor her risk depend on k. So whichever cost policy is optimal for D v × {T } achieves the same profits and externalities as D v × D k . Intuitively, going from {T } to D k supported on [0, T ] cannot possibly increase the profits of any scheme (formally:
Observation 2), so the initial cost policy should remain optimal. Lemma 4 (Extension of Cost Policy). Let s be a cost policy that is optimal for D v × {T } subject to profits R. Then for all D k supported on [0, T ], s is optimal for D v × D k subject to profits R.
Proof. First, we observe that PROF Dv×{T } (s) = PROF Dv×D k (s). This is simply because the loss of consumers is independent of k (as y = 0). Similarly, EXT Dv×{T } (s) = EXT Dv×D k (s). This is again because the risk of consumers is independent of k.
Now, assume for contradiction that there is some policy s with profits PROF Dv×D k (s ) ≥ R and also EXT Dv×D k (s ) < EXT Dv×{T } (s). Then we have the following inequality from Observation 2:
R ≤ PROF Dv×D k (s ) ≤ PROF Dv×{T } (s ).
Therefore, as s is optimal for D v × {T } subject to profits R, we must have:
EXT Dv×{T } (s ) ≥ EXT Dv×{T } (s).
This now lets us conclude the following chain of inequalities, where the first line is a corollary of Observation 3: the consumer in a population with D k supported on [0, T ] whose device is least likely to be compromised is a consumer with k = T . The third line follows from the reasoning above (that s achieves profits at least R on D v × {T }, and is therefore feasible). The final line follows because the externalities of a cost policy are independent of k.
EXT Dv×D k (s ) ≥ RISK(T , s ) = EXT Dv×{T } (s ) ≥ EXT Dv×{T } (s) = EXT Dv×D k (s).
Lemma 4 proves the first bullet of Theorem 2.
Extension Lemma for large k
In this section, we proof for the large k case of Theorem 2. The proof will be a little more involved this time, since we can no longer claim that the externalities of a fine policy are independent of k (whereas this does hold for cost policies). The intuition for this case is the same though: if a fine policy is optimal for D v × {k} for all k ≥ T , and D k is supported on [T, ∞), fine policies should remain optimal for D v × D k . Most of the proof does not make use of the technical assumption that the s we are competing with is a profits-maximizing policy: this assumption only arises at the very end.
The first step in our proof is the following concept, which captures the change in loss for a consumer (v, k) for regulation s versus s : Definition 2 (Policy Comparison Function). For two policies s and s , we define the policy comparison function g s,s (·) so that g s,s (k) = (k, s) − (k, s ).
The policy comparison function takes as input an effectiveness k, and outputs the change in loss for a consumer under one policy versus another. Our first lemma argues that for certain pairs (s, s ), the policy comparison function is monotone in k. That is, consumers with more effectives have greater preference for one policy over another.
Lemma 5. Let s = (y, c, p) and s = (y , c , p ) be such that ye −c ≤ y e −c . Then g s,s (·) is monotone non-decreasing.
Observe that the hypothesis holds if y ≤ y and c ≥ c .
Proof of Lemma 5. There are three regions of k to consider: k ∈ [0, e c /y ), k ∈ [e c /y , e c /y), and k ≥ e c /y. In the first region the consumer has effort 0 for both policies. In the middle region, the consumer has effort 0 for one policy.
In the last region, the consumer has non-zero effort for both policies.
First consider k ≤ e c /y ≤ e c /y. Then also y k ≤ e c and yk ≤ e c , so the consumer's effort is 0. In this case,
∂ (k,s) ∂k = 0 = ∂ (k,s ) ∂k
, because the loss is independent of k. So g s,s is monotone non-decreasing in this range (in fact it is constant).
Next, there are k such that yk < e c , y k ≥ e c . Then ∂ (k,s) ∂k = 0, and ∂ (k,s ) ∂k = − ln(y k)−c k 2 ≤ 0. So g s,s is also monotone non-decreasing in this range (because it is equal to 0 minus a non-increasing function). . And we get:
∂ (k, s) ∂k − ∂ (k, s) ∂k = ln(y k) − c − ln(yk) + c k 2 = ln(y /y) + c − c k 2 ≥ 0.
The final inequality comes because by hypothesis: y e −c ≥ ye −c ⇒ ln(y ) − c ≥ ln(y) − c. So in all regions, g s,s is monotone non-decreasing.
We use Lemma 5 to claim the following corollary, which essentially states that if a policy change universally lowers loss and risk, then it is possible to adjust the price so that the profits go up and externalities go down.
Corollary 6. Let (y, c), (y , c ) be such that (a) ye −c ≤ y e −c and (b) for all k in the support of D k , (k, y, c) + c ≥ (k, y , c ) + c and RISK(k, y, c) ≥ RISK(k, y , c ). Then for all p and all D v , there exists a p such that:
PROF Dv×D k (y , c , p ) ≥ PROF Dv×D k (y, c, p), EXT Dv×D k (y , c , p ) ≤ EXT Dv×D k (y, c, p).
Proof of Corollary 6. First, consider setting p := p − c + c . Then the profits generated per sale are equal under s = (y, c, p) and s = (y , c , p ). Observe also that the probability of sale is at least as large under s as s, as we have u(t, s)
= v − p + c − c − (k, s) ≤ v − p + c − c − (k, s ) = u(t,
s ) for all t. Therefore we have that PROF Dv×D k (y , c , p − c + c ) ≥ PROF Dv×D k (y, c, p). But unfortunately we can't yet say anything about the externalities. Indeed, the problem might be that there are many additional consumers with poor effectiveness who previously did not purchase the item under s but who now purchase it under s (recall Example 1). So the plan from here is to raise the price until the probability of sale is back to its original level (clearly the profits must still be larger, as now the probabilities of sale match, but the profit-per-sale of our new scheme is better). We'll use Lemma 5 to claim that the set of consumers who remain are only more secure than what we started with.
So formally, raise the price p until the probability of sale for s = (y , c , p ) is the same as s. 6 Now we have two schemes: s = (y, c, p) and s = (y , c , p ). Both sell the item with the same probability, q. If both schemes sold to exactly the same q fraction of consumers, then the lemma hypothesis that RISK(k, s ) ≤ RISK(k, s) for all k would suffice to let us claim that EXT Dv×D k (s ) ≤ EXT Dv×D k (s). However, it could be a completely different q fraction of consumers. Still, it turns out that because y e −c ≥ ye −c , the fraction of consumers that purchase only have larger k.
Indeed, observe that if some consumer t = (v, k) purchases under s but not s , and some other consumer t = (v , k ) purchases under s but not s, then we have:
g s,s (k) < 0 < g s,s (k )
But by Lemma 5, we know that g s,s (·) is monotone increasing, so k > k. In particular, this means that every consumer in the mass which purchased under s but not s has lower k than any consumer which purchased under s but not s. As RISK(·, s) is clearly monotone decreasing in k, and the fraction of buyers purchasing under s and s is the same, we conclude that we must have EXT Dv×D k (s ) ≤ EXT Dv×D k (s) as desired.
Now we are ready to prove the extension lemma for large k.
Lemma 6 (Extension of Fine Policy). Let D k be supported on [T, ∞), where T is such that a fine policy is optimal for D v × {T } subject to profits R. Then there is a fine policy s with PROF D (s ) ≥ R such that for all profits-maximizing
s with PROF D (s) ≥ R, EXT D (s ) ≤ EXT D (s).
Proof of Lemma 6. First, observe that we necessarily have T > 1 if the hypothesis is to hold, by Theorem 2.
Consider any proposed optimal policy s = (y, c, p). Let s * = (y * , 0, p * ) denote the optimal fine policy for D v × {T } subject to profits R. Then maybe (T, s) + c ≥ (T, s * ). If so, let s = (y , 0, p − c) be such that (T, s ) = (T, s) + c.
As T > 1, observe that decreasing c decreases (T, s) + c. Therefore, decreasing c to 0 results in y ≥ y for the equality to hold. As such, s and s satisfy the hypotheses of Lemma 5, and we can conclude that (k, s ) ≤ (k, s) + c for all k ≥ T . We now just need to show that RISK(T , s ) ≤ RISK(T , s). This is surprisingly tricky, and carried out in the subsequent paragraph.
Indeed, observe that (y, c) is some partial policy, and setting the pricep which maximizes profits yields R(y, c) := PROF Dv×{T } (y, c,p) on population D v ×{T }. Observe that as (T, s)+c ≥ (T, s * ), we have R(y, c) ≤ R (otherwise s * would not be optimal for D v × {T } subject to constraint R, as we could increase y * ). We can now ask what is the optimal policy for D v × {T } subject to constraint R(y, c)? By Lemma 3, we claim it must be a fine policy, and that this fine policy is exactly (y , 0,p). To see this, we use Proposition 4, which asserts that T (D v , R(y, c)) ≤ T (D v , R) = T . As T ≥ T (D v , R(y, c)) now, we conclude that a fine policy must be optimal for D v × {T } subject to profit constraint R(y, c). This policy s must have (T, s ) = (T, s) + c (if it is bigger, then the profit will be < R(y, c). If it is smaller, then the loss should be increased to get more profit). (y , 0,p) is exactly this policy. As it is optimal for D v × {T }. We may now conclude that RISK(T , s ) ≤ RISK(T , s). In particular, this necessarily implies that the risk is 1 y T (because c = 0, the only other alternative would be to have risk = 1, which is clearly not ≤ e −c for c > 0). To conclude that RISK(k, s ) ≤ RISK(k, s) for all k ≥ T , simply observe that we must now have RISK(k, s ) = 1 y k ≤ 1 y T ≤ e −c , and also 1 y k ≤ 1 yk as y ≥ y. So whether or not a consumer with effectiveness k has h * (k, s) > 0, the risk 1 y k is better. Now we can apply Lemma 5: we have come up with a new policy where everyone's risk is (weakly) lower, and everyone's loss is (weakly) lower, so we can increase the price until the probability of sale is the same, and this will (weakly) increase the profit and (weakly) decrease the risk.
So now we've covered the case that (T, s) + c ≥ (T, s * ). We just now need to consider the case that (T, s) + c < (T, s * ). This is the only case where we'll assume that the s we started with was profits-maximizing. Observe that if (T, s) + c < (T, s * ), then there exists a price p such that (y, c, p ) generates profits strictly exceeding R. Indeed, if (T, s) + c < (T, s * ), then (k, s) + c < (T, s * ) for all k ≥ T , so the distribution of v − (k, s) − c strongly stochastically dominates the distribution of v − (T, s * ) in the following sense: for any probability q, the value v q such that Pr[v − (k, s) − c ≥ v q ] = q exceeds w q such that Pr[v − (T, s * ) ≥ w q ] = q. As there exists a price p such that p · Pr[v − (T, s * ) ≥ p] ≥ R, that same probabilility of sale with a strictly increased price p guarantees that
Pr[v − (k, s) − c ≥ p ] · p > R.
Finally, Lemma 7 (stated below) implies that every optimal policy s for D v × D k subject to profits constraint R has PROF Dv×D k (s) = R. This is because for all policy s where the profit > R, there is a ε > 0 such that we can construct a policy s with profit ≥ R but strictly less externalities. This should perhaps not be surprising, as intuitively one should be able to decrease externalities at the cost of a little ε (although one should be careful to do this properly this for arbitrary ε). Therefore, the optimal policy we started with achieved profits R, while the previous paragraph observes that there necessarily exists a price which achieves profits > R. So our original policy must not be profit-maximizing.
Definition 3 (Invariant Transformation). Given a policy s = (y, c, p), define
INV(s, α) := ye (p−c)(1−α) , αc + (1 − α)p, p where α ∈ [0, p p−c ]. Lemma 7 (Invariant Property). Let s = INV(s, α), then for all k ∈ R + • h * (k, s ) = h * (k, s).
• (k, s ) = (k, s).
• u(t, s) = u(t, s ).
In addition,
PROF D (s ) = αPROF D (s) EXT D (s ) = e −(1−α)(p−c) EXT D (s, p)
Proof of Lemma 7. First note that for all types t, their optimal effort under policy s is the same as under policy s: (k, s ) = ye (1−α)(p−c) RISK(k, s ) + h * (k, s ) = yRISK(k, s) + h * (k, s) = (k, s) Which implies that:
h * (k, s ) = ln(yk) + (1 − α)(p − c) − αc − (1 − α)p k + = ln(yk) − c k + = h * (k,u(t, s ) = v − (t, s ) − p = v − (t, s) − p = u(t, s).
We can conclude that a type t purchases in policy s iff type t purchases in policy s . Therefore:
PROF D (s ) = (p − αc − (1 − α)p)P r t←D [u(t, s ) ≥ 0] = α(p − c)P r t←D [u(t, s) ≥ 0] = αPROF D (s)
and finally:
EXT D (s ) = E t←D [RISK(t, s )|u(t, s ) ≥ 0] = e −(1−α)(p−c) E t←D [RISK(t, s)|u(t, s) ≥ 0] = e −(1−α)(p−c) EXT D (s)
which concludes the proof.
This concludes the proof of bullet two of Theorem 2.
Example: The Profits-Maximizing Qualification is Necessary
In this section we provide the example promised in Proposition 5. Consider the following distribution, and profits constraint R := 0.5:
D v = v 1 = 1 w. p. 1 2 v 2 = 16/15 w. p. 1 2 D k = k 1 = 3 w. p. 1 2 k 2 = x → ∞ w. p. 1 2
Above, x will be finite, but approaching ∞, and ε will be finite but approaching 0). The proposition will follow from the following sequence of claims. First, we will establish bullet one for T := 3. Claim 7. A fine policy is optimal for D v × {3}.
Proof of Claim 7. To establish the claim, we will directly find the c * which is maximal among those such that PROF(0, c, p) ≥ 0.5, and observe that T = 1 + 1/c * .
Indeed, if c = 0.5, then D v − c takes value ≥ 1/2 with probability 1, so profits 0.5 is indeed achievable. However, for any c > 0.5, D v − c takes value w 1 < 1/2 with probability 1/2, and w 2 < 1 with probability 1/2. So setting either price w 1 or w 2 yields profits < 1/2. So c * = 1/2 for this example, and 3 = 1 + 1/c * as desired.
Bullet two now immediately follows, as D k is indeed supported on [3, ∞). We now just need to find the optimal fine policy for D v × D k , and establish a better policy that is not simple. We now search for the optimal fine policy. Such a policy might sell only to (16/15, x), but then the profits is at most 4/5, which is too little. Such a policy might sell only to (16/15, x) and (1, x). But since x is finite, such a policy certainly charges price < 1 (unless y = 0, in which case the policy sells to all four types), and sells with probability ≤ 1/2, so the profits are also too small. Such a policy might sell to all four types, which we analyze below. Or it might sell to all types except (1, 3), which we analyze after. Claim 8. The optimal fine policy s which sells to all four types has EXT Dv×D k (s) ≥ 1 2 √ e .
Proof of Claim 8. Such a policy necessarily has (3, y) ≤ 1/2, which means that we must have ln(3y)+1 3 ≤ 1/2, or y ≤ √ e/3. Such a policy has externalities at least 1 2 · 3 3· √ e = 1 2 √ e .
Claim 9. The optimal fine policy s which sells to all types except (1, 3) has EXT Dv×D k (s) ≥ e −1/5 /3.
Proof of Claim 9. Such a policy certainly has 16/15 − (3, y) ≥ 2/3, as we are now selling with probability 3/4, so we must charge a price at least 2/3 in order to get profits ≥ 1/2. Observe that 16/15 = 2/3 + 2/5, so we must now have (3, y) ≤ 2/5. That yields ln(3) + ln(y) + 1 ≤ 6/5, or y ≤ e 1/5 /3. So the externalities are at least 1 3 · 3 e 1/5 3 = e −1/5 /3.
Corollary 10. The optimal fine policy s has EXT Dv×D k (s) ≥ e −1/5 /3.
Here's now some intuition for how we're going to design a better non-simple policy: given that we wish to sell to all types except (1, 3), we can set y very close to 0 and have RISK(x, s) ≈ 0, because x is so large. The remaining question is then whether we wish to use y or c to make the risk of (16/15, 3) as small as possible. Note that we must keep their loss under 2/5 < 1/2 (as above). But for k = 3, a loss of 1/2 is exactly the cutoff when it becomes more efficient to use a fine policy instead of a cost policy. So if we use c instead, we can get the risk lower for the same loss.
Claim 11. Let ε be such that ln(x)+1
x ≤ ε. Then set c = 1/3−ε, and y = (2/5−c)e c . Then EXT Dv×D k (y, c, 2/3+c) = 2 3yx + e −1/3+ε /3 and PROF Dv×D k (y, c, 2/3 + c) = 1/2. x . Because ln(x)+1+ln(2/5−c)
x ≤ ln(x)+1
x ≤ ε, the loss is ≤ ε = 1/3 − c, so (1, x) is willing to pay c + 2/3. Finally, we just need to compute the externalities and profits. The profits are exactly 1/2, as it sells with probability 3/4 and achieves profit 2/3 when selling. The externalities are exactly 2 3 · 1 yx + 1 3 · e −c . Now, we just need to compare e −1/5 /3 and e −1/3+ε /3 + 2 3yx . Observe that as x → ∞, ε → 0 and e −1/3+ε /3 approaches e −1/3 /3. So 2 3yx + e −1/3+ε /3 → 0 + e −1/3 /3 < e −1/5 /3, and the externalities are indeed lower.
As a sanity check, we'll show that ((2/5 − c)e 1/3−ε , 1/3 − ε, 2/3 + c) is not profits-maximizing (technically, Theorem 2 doesn't imply this, since we didn't prove that the scheme is optimal. But as this scheme is better than all fine policies, certainly the optimal policy is not simple, and therefore not profits-maximizing by Theorem 2. So the fourth bullet is already proven).
Claim 12. ((2/5 − c)e 1/3−ε , 1/3 − ε, 2/3 + c) is not profits-maximizing.
Proof of Claim 12. The four quantities of value minus loss are equal to: {16/15 − ε, 1 − ε, 1 − ε, 14/15 − ε}. The seller generates profits 1/2 by setting price 2/3 + c. If instead they set price 3/5 + c, the item would sell with probability 1 and yield profits 3/5.
General Distributions: An Approximation
In this section, we consider general distributions. Clearly, one should not expect a simple policy to be optimal in general. Given that simple policies are optimal for homogeneous populations, one might reasonably expect that simple policies are approximately optimal for general distributions by simply ignoring half of the population and targeting the half that is responsible for most of the externalities. This idea works in one direction: if the "low k" region is responsible for most of the externalities in the optimum solution, then using a cost policy for the entire distribution is a good idea: the high k consumers may have significantly higher risk than previously, but this doesn't outweigh the original risk from the low k region.
This idea fails horribly, however, if the "high k" region is responsible for most of the externalities in the optimum solution. The problem is that while we can choose a policy to exclusively target this subpopulation, any low k (think: k = 0) consumers who choose to purchase anyway may have enormous risk in comparison to before (i.e. it could now be 1 when it was previously e −c for large c). We first show that this intuition can indeed manifest in a concrete example by presenting a lower bound in Section 7.1. This rules out a single-criterion approximation that satisfies the profits constraint exactly, and approximates the externalities. In Section 7, we present a bicriterion approximation which approximately satisfies the profits constraint and also approximately minimizes externalities. This approximation is our most technical result. As such, we provide mainly proof sketches to overview the key steps.
Lower Bound on Heterogeneous Distributions
The key insight for our example is to make the profits constraint so binding that the only way to match it exactly is for the entire population to purchase the item. Part of the population will have k = 0, and part will have k → ∞. With both c and y, it will be feasible to get the k → ∞ consumers to have risk essentially 0, while the k = 0 consumers will have reasonably small risk. But with either c = 0 or y = 0, one of these will be lost, which causes significant risk increase.
Example 2. Let D v be a point mass at v 0 = 2e x/2 · (x + e −x ). Let D k be a distribution with two point masses, one at k = 0 with probability e −x/2 , one at e xe x/2 with probability 1 − e −x/2 . Let R :
= v 0 − e −x − x.
Lemma 8. The policy (1, x, R + x) achieves profit R in Example 2, and has externalities ≤ e −x/2 · e −x + 1 · e −xe x/2 .
Proof. The utility of (v 0 , 0) is exactly v 0 − e −x − R − x = 0, so they will choose to purchase. (v 0 , e xe x/2 ) has only larger utility, so they will purchase as well. Therefore, the profit is indeed R.
The externalities are computed simply as the probability of having consumer (v 0 , 0) times their risk (e −x ) plus (upper bound on the) probability of consumer (v 0 , e xe x/2 ) times their risk (e −xe x/2 ).
Lemma 9. Any cost policy that achieves profit R has externalities at least e −x+1
Proof. The maximum security we can set and still have profit R is x + e −x . If we set this, then the risk of all consumers (which is now independent of k) is e −x+e −x ≥ e −x+1 .
Lemma 10. Any fine policy that achieves profit R has externalities at least e −x/2 .
Proof. To achieve profit R, the policy must sell to the entire population. The consumer with k = 0 will not put in any effort, and therefore their risk will be one, and the externalities will be at least e −x/2 .
Corollary 13. For all x, there exists a distribution D v × D k and profits constraint R such that the optimal policy is not simple, and any simple policy that satisfies profits constraints R has externalities at least a factor of x larger than the optimum.
Corollary 13 is the main result of this section. Clearly the distribution witnessing Corollary 13 is highly contrived and unrealistic. And clearly, the way to get around this is to allow for a slight relaxation in the profits constraint so that we don't have to sell to the entire market (indeed, even allowing to relax the constraint by a (1 − e −x/2 ) fraction in this case would suffice). So the subsequent section shows that by relaxing the profits constraint, an approximation guarantee is possible.
A Bicriterion approximation
Given the lower bound in Section 7.1, we show that simple policies guarantee a bicriterion approximation. As is traditional with worst-case approximation guarantees, our constants are not particularly close to 1, but are still relatively small. This is not meant to imply that the seller should be happy with (e.g.) a 1/8-fraction of the original profits, but more qualitatively to conclude that simple policies can reap many of the benefits of optimal ones (see Hartline (2013) for further discussion about the role of approximation in mechanism design). As referenced previously, the proof of Theorem 3 is quite technical, so we sketch the key steps and left the proof to Appendix A. Proof Sketch. Given an arbitrary policy s = (y, c, p), consider the conditional distribution of buyers that purchase under s. If with constant probability a buyer has efficiency k ≤ 1, then we output the cost policy s 1 := (0, c + (σ, s), p + (σ, s)) where σ is chosen such that a buyer continues to purchase with constant probability. We can show that c + (σ, s) is sufficiently large such that RISK(D k , s ) ≤ RISK(D k , s) with constant probability.
For the case where with constant probability a buyer has efficiency k > 1, we define a blowup of the fines such that with constant probability a buyer continues to purchase but with the hope that inefficient buyers stop to purchase. The blowup can fail in two conditions: (1) D k is not heavy tail, (2) D v is heavy tail. For (1), we cannot derive a significant blowup if D k is concentrated close to 1. For (2), we cannot drive inefficient buyers out of the market if they have high value. Either condition allow us to construct cost policies that give good externality guarantees.
Summary
We propose a stylized model to study regulation of single item sales with negative externalities, from which neither the buyer nor seller suffer. We first show that a simple policy is optimal in homogenous markets: That is, for all D v , R, there exists a cutoff T such that when the effectiveness of consumers ranges in [0, T ], the optimal policy regulates only the product (and does not impose fines). Similarly, if all consumers have effectiveness in [T, ∞), a policy which regulates only payments (via fines, and does not impose default security features) outperforms all profits-maximizing policies. Importantly, T is not necessarily the cutoff at which the consumers are more effective than the producer (which would be T = 1), but actually depends on the value distribution D v and profit constraint R.
We then show in general markets that while a simple policy may not be optimal, one is always approximately optimal. In particular, we show that while no simple scheme can guarantee any finite approximation while satisfying the profit constraint exactly, a bicriterion approximation exist, which approximately satisfies the profit constraint and also approximately minimizes externalities. Going forward, we must better understand the effectiveness of consumers to decide which regulation strategy is more appropriate to approximately minimizes externalities.
While stylized, our model captures the key salient features of this problem. We chose to study the single seller/single item setting in order to isolate these features without bringing in additional complexities (and the numerous examples throughout our paper demonstrate that even the single seller/single item setting is quite rich). Now that our results develop this understanding, a good direction for future work is to consider competing sellers or multiple items.
A Proof of Theorem 3
We define the necessary tools for the case where a constant fraction of the population that purchase has efficiency k ≤ 1 in Appendix A.1 and proof approximation guarantees in Appendix A.1.1. For the case where a constant fraction of the population that purchase has efficiency k > 1, we define the necessary tools in Appendix A.2 and proof approximation guarantees in Appendix A.2.4. In Appendix A.3, we combine the approximation guarantees to complete the proof of Theorem 3.
Notation Policy s induces a threshold k 0 (s) := inf{k|h * (s, t) > 0} of types with zero effort. Let k h (s) := max{1, k 0 (s)}, and define the events:
A(s) := {t ← D purchase under s} B(s) := A(s) ∩ {t ← D has efficiency k > k h (s)}
We define the buyer's value after regulation:
VALUE(t, s) := v − (t, s)
then event A(s) is equivalent to VALUE(t, s) ≥ p or u(t, s) ≥ 0.
Space Partition Given (D, R), and some arbitrary policy s = (y, c, p). Let's look over the distribution of buyers that purchase under s. More formally, we will consider the following partition of the probability space:
ε 1 = P r t←D [k ≤ k 0 (s)|u(t, s) ≥ 0] ε 2 = P r t←D [k 0 (s) < k ≤ 1|u(t, s) ≥ 0] ε 3 = P r t←D [k > k h (s)|u(t, s) ≥ 0]
The key idea behind our approximation mechanism, Algorithm 1, consists of defining three transformations of s. The cost policy COST 1 , Equation (7) in Appendix A.1, guarantees a constant approximation proportional to ε 1 and ε 2 . Algorithm 2, Appendix A.2, denoted by FINE outputs a simple policy that guarantees a constant approximation proportional to ε 3 .
Input: s = (y, c, p), D 1: Let β = 1 2 2: if ε 1 ≥ 1 8 then 3:
Output COST 1 (s, 1) 4: end if 5: if ε 2 ≥ 1 8 then 6:
Output COST 1 (s, ε 1 + ε 2 ) 7: else 8:
Output FINE(s) 9: end if Algorithm 1: APPROX
A.1 Approximation with Low Efficiency Buyers
Next, we define the tools to proof Theorem 3 for the case where ε 1 + ε 2 = O(1) and postpone the proof to Appendix A.1.1. The following policy clearly meets the profit guarantees when ε = O(1).
COST 1 (s, ε) := (0, c + (σ, s), p + (σ, s)) (7) where ε ∈ [0, 1], and we choose σ such that P r t←D [v ≥ (σ, s)
+ p] = εP r t←D [v ≥ (k, s) + p].
Zero efficiency case For COST 1 (s, 1), we get a good approximation to the externalities whenever a constant fraction ε 1 of the consumers who purchase have h * (t, s) = 0. This is because the externalities are at least ε 1 · e −c under s, and our new externalities are just e −c . So if ε 1 is big enough, we get our desired approximation, Corollary 14.
Non-Zero efficiency case If ε 2 is big, then ε 2 1 y is a good lower bound for EXT D (s). This implies we must target a cost proportional to ln y. For COST 1 (s, ε 2 ), we can argue (σ, s) is at least 1 + ln y which implies good externality bounds, Corollary 15.
A.1.1 Proof of Approximation with Low Efficiency Buyers
Lemma 11. Let ε ∈ [0, 1], then PROF D (COST 1 (s, ε)) = εPROF D (s, ε)
EXT D (COST 1 (s, ε)) = e −c− (σ,s) E t←D [RISK(t, s)|u(t, s) ≥ 0] EXT D (s)
Proof. By our choice of σ,
P r t←D [v ≥ p + (σ, s)] = εP r t←D [v ≥ p + (k, s)]
This implies the following bounds in the profit,
PROF D (COST 1 (s, ε)) = (p + (σ, s) − c − (σ, s))P r t←D [v ≥ p + (σ, s)] = (p − c)εP r t←D [u(t, s) ≥ 0] = εPROF D (s)
For the externality,
EXT D (COST 1 (s, ε)) EXT D (s) = e −c− (σ,s) E t←D [RISK(t, s)|u(t, s) ≥ 0]
which concludes the proof.
Corollary 14.
PROF D (COST 1 (s, 1)) = PROF D (s)
PROF D (COST 1 (s, 1)) ≤ 1 ε 1 EXT D (s)
Proof. The profit bound follows directly from Lemma 11. For the externality,
EXT D (s) ≥ ε 1 E t←D [RISK(t, s)|u(t, s) ≥ 0, k ≤ k 0 (s)] = ε 1 e −c then, EXT D (COST 1 (s, 1)) EXT D (s) = e −c− (σ,s) EXT D (s) ≤ e −c ε 1 e −c
which concludes the proof.
Corollary 15. Assume ε 2 > 0, then
PROF D (COST 1 (s, ε 2 )) ≥ (ε 1 + ε 2 )PROF D (s) EXT D (COST 1 (s, ε 2 )) ≤ 1 ε 2 e EXT D (s)
Proof. We will first claim (σ, s) ≥ (1, s). Assume for contradiction (σ, s) < (1, s), then it must be
(ε 1 + ε 2 )P r t←D [v ≥ p + (k, s)] = P r t←D [v ≥ p + (σ, s)] > P r t←D [v ≥ p + (1, s)] ≥ P r t←D [v ≥ p + (k, s)|k ≤ 1] = P r t←D [k ≤ 1|u(t, s) ≥ 0]P r t←D [u(t, s) ≥ 0] P r t←D [k ≤ 1] ≥ P r t←D [k ≤ 1|u(t, s) ≥ 0]P r t←D [u(t, s) ≥ 0] = (ε 1 + ε 2 )P r t←D [v ≥ p + (k, s)] ⇒⇐
where the first equality follows by the choice of σ in Equation (7) and the second equality follows by Bayes' theorem.
Next, we bound the externality with Lemma 11. We use the fact σ ≤ 1 and since ε 2 > 0, it must be h * (1, s) > 0.
EXT D (COST 1 (s, ε 2 )) EXT D (s) = e −c− (σ,s) E t←D [RISK(t, s)|u(t, s) ≥ 0] ≤ e −c− (1,s) ε 2 E t←D [1/yk|u(t, s) ≥ 0, k 0 (s) < k ≤ 1] = ye −c−1−ln y+c ε 2 E t←D [1/k|u(t, s) ≥ 0, k 0 (s) < k ≤ 1] ≤ 1 eε 2
The profit bound follow directly from Lemma 11 which concludes the proof.
A.2 Approximation with High Efficiency Buyers
In this section, we define the tools to proof Theorem 3 for the case where ε 3 = O(1). We will construct FINE(s), Algorithm 2, which targets the population that purchases under s and have high efficiency k > k h (s). To construct FINE(s), we will further define three additional transformations described in this section.
As motivation consider Example 2. Ideally, we would like to make the inefficient buyer to stop to purchase. FINE(s) will first consider a blowup of the fines, BLOWUP(s) in Definition 4, dependent on the population that purchase under s. The blowup cannot be too high; otherwise, the utility of the efficient buyer decreases too much, hurting profit.
If the inefficient buyer is still willing to purchase after the blowup, it must be because the buyer value distribution has a heavy tail. In that case, we can derive the transformation HEAVY(s), Equation (11), that leverages the tail of the distribution to impose high security regulation directly on the product.
A.2.1 Preliminaries
Let G(x|E) := P r[k ≤ x|E] denote the cumulative distribution function of efficiency conditioned on event E. We will assume ε = P r t←D [k > k h (s)|VALUE(t, s) ≥ p(s)]. The parameter β ∈ [0, 1] parameterize the fraction of the profit we are willing to compromise.
A.2.2 Blowup
Definition 4 (Blowup transformation). We define, if ye −c < 2 then 13:
σ := max 1, G −1 (1 − β|B(s))(8)
Output COST 1 (s, 1)
where q := inf{x ≥ 1|P r t←D [A(xye σ−1 , 0, p − c)] ≤ P r t←D [VALUE(t, s) ≥ p]} We define y sk (s) as the fine of BLOWUP(s), y sk (s) := qye c(σ−1)
We definek(s) as the efficiency k such that RISK(k(s), BLOWUP(s)) = e −c , k(s) := e c y sk (s) By construction, if β is big, then BLOWUP(s) will always provide good profit guarantees, Claim 16. For the externalities, we can also ensure BLOWUP(s) will provide good externality guarantees for the population that used to purchase under s and had efficiency k ≥k(s), Claim 17. This is because y sk (s) ≥ y; therefore, if h * (k, s) > 0, we must have RISK(k, BLOWUP(s)) ≤ RISK(k, s). If h * (k, s) = 0, then we can have RISK(k, BLOWUP(s)) > RISK(k, s) if k is small.k(s) precisely captures this phase change such that if k ≥k(s), then RISK(k, BLOWUP(s)) ≤ RISK(k, s) and if k <k(s), RISK(k, BLOWUP(s)) > RISK(k, s).
To proof approximation bounds for BLOWUP(s) for the population where k <k(s), we first discuss the case where σ ≥ 2. We discuss the case where σ < 2 on Appendix A.2.3.
Assuming σ is sufficiently large, BLOWUP(s) would still fail to provide good externality guarantees if the probability that a buyer with efficiency k <k(s) purchase under BLOWUP(s) is high.
Unfortunately, for arbitrary distributions, we should not expect buyers with efficiency k <k(s) will have a small contribution to externalities under BLOWUP(s). However, we can define an upper bound on their externalities that would be sufficient to proof externality guarantees for BLOWUP(s), Claim 18. Definition 5 (Good Blowup). BLOWUP(s) is good if buyers with efficiency at mostk(s) give a small contribution to externalities:
E t←D [RISK(t, BLOWUP(s)) · I(k ≤k(s))|A(BLOWUP(s))] ≤ 4(p − c) (1 − β)(1 + c)ε EXT D (s)(10)
Composition Observe the dependence on p − c on the externality bound which can be arbitrarily large. This can be easily be solved by applying a composition of BLOWUP with INV. This is because Lemma 7 states the probability space of s and INV(s, ·) is the same and by sacrificing a constant fraction of the profit we reduce externalities by a factor of O 1 p−c .
FEBRUARY 27, 2019
Heavy Tail If BLOWUP(s) fails to push inefficient buyers out of the market (buyers with efficiency k <k(s) cause high externalities), it must be because the value distribution D v has a heavy tail. In that scenario, we define the policy HEAVY(s) that impose high regulation in the product and ensure constant approximation ratio, Claim 19.
HEAVY(s) := 0, 1 2 p HEAVY , p HEAVY (11)
where p HEAVY := (k(s), BLOWUP(s)) = (1 + c)qye cσ e 2c (12)
A.2.3 When the Blowup is Small
For the case where σ < 2, we might hope cost policies can still provide a good approximation since it suggests D k has a short tail. Next, we construct a family of cost policies H(s) such that if no policy s ∈ H(s) gives good profit guarantees, we can proof externality guarantees for BLOWUP(s), Claim 24.
Under BLOWUP(s), in order for a buyer to have risk x ∈ [e −c , 1], she must draw efficiency k = 1 y sk (s)x and observe k ≤k(s).
We define an alternative expression for the loss of k under fine y and cost c = 0 as function of its risk x:
RISK (x, y) := (ln 1/x + 1)xy
Let H(x) be the probability a value is greater or equal to the loss RISK (x, y sk (s)).
H(x) := P r t←D v ≥ ln 1 x + 1 xy sk (s)(14)
We define the cost policy COST 3 x (s) where the probability of sale under COST 3 because (k, COST 3 x (s)) = 0 for all k. Since D k has a short tail (σ is small), all policies in s ∈ H(s) would give good externality guarantees when compared to s, Claim 22; however, the probability of sale might be too small to get good profit guarantees. Bellow, we define a sufficient condition to proof a policy in H would give good profit guarantees and the formal statement is given in Claim 22. Definition 6 (Good COST 3
x (s) is equivalent to H(x),
x (s)). We define COST 3 x (s) as good if its probability of sale is bigger than
H(x) ≥ 2βεPROF D (s) (ln 1/x + 1)xy sk (s)
If Definition 6 is not satisfied for any policy in H, it implies the probability of sale to buyers with efficiency k <k(s) under BLOWUP(s) is bounded. This implies, we can directly bound the externalities contributed by k < k(s) and proof externality guarantees for BLOWUP(s), Claim 23.
A.2.4 Proof of Approximation for High Efficiency Buyers
In this section, we bound the profit and the externality for the output of Algorithm 2. Claim 16. P r t←D [A(BLOWUP(s))] ≥ βεP r t←D [u(t, s) ≥ 0] PROF D (BLOWUP(s)) ≥ βεPROF D (s)
Proof. We will first bound the probability that a type t that purchase under policy s and has efficiency at least k h (s), also purchase under policy s = BLOWUP(s). Given that t purchase under policy s, we must have p ≤ VALUE(t, s), then
P r t←D [VALUE(t, s ) ≥ p − c|B(s)] ≥ P r t←D [VALUE(t, s ) ≥ VALUE(t, s) − c|B(s)] = P r t←D [v − 1 + ln y + ln k − c k − cσ k ≥ v − 1 + ln y + ln k − c k − c|B(s)] = P r t←D [k ≥ σ|B(s)] = β
In the last step, we use the fact σ = G −1 (1 − β|B(s)).
Next, we lower bound the probability of sale and the profit under policy s .
P r t←D [VALUE(t, s ) ≥ p − c] ≥ P r t←D [u(t, s) ≥ 0] · P r t←D [k > k h (s)|u(t, s) ≥ 0] · P r t←D [VALUE(t, s ) ≥ p − c|B(s)] ≥ βεP r t←D [u(t, s) ≥ 0] PROF D (s ) = (p − c)P r t←D [VALUE(t, s ) ≥ p − c] ≥ βε(p − c)P r t←D [u(t, s) ≥ 0] = βεPROF D (s)
which concludes the proof.
Claim 17.
E t←D [RISK(t, BLOWUP(s)) · I(k >k(s))|A(BLOWUP(s))] ≤ 1 εβ E t←D [RISK(t, s) · I(k >k(s))|u(t, s) ≥ 0]
Proof. Let s = BLOWUP(s). If k >k(s), we can have either h * (k, s) > 0 or h * (k, s) = 0. In the first case, by definition of BLOWUP(s), y ≥ y, and together with h * (k, s) > 0 implies RISK(k, s ) ≤ RISK(k, s). If instead, h * (k, s) = 0, r(k, s) = e −c , but since k >k(s), r(k, BLOWUP(s)) ≤ e −c . So in all cases, r(k, s ) ≤ r(k, s).
Still by construction of BLOWUP(s), the probability VALUE(t, BLOWUP(s)) ≥ p − c is at most the probability u(t, s) ≥ 0. By a similar argument in Corollary 6, g s,s (k) is monotone decreasing which implies the set of types that start to purchase under s can only be more efficient than the set of types that stops to purchase. We can conclude
E t←D [RISK(t, s ) · I(k >k(s), u(t, s ) ≥ 0)] ≤ E t←D [RISK(t, s) · I(k >k(s), u(t, s) ≥ 0)]
We can re-write the expectations as
E t←D [RISK(t, s ) · I(k >k(s))|u(t, s ) ≥ 0] ≤ E t←D [RISK(t, s) · I(k >k(s))|u(t, s) ≥ 0] P r t←D [u(t, s) ≥ 0] βεP r t←D [u(t, s) ≥ 0]
which concludes the proof.
Claim 18. If BLOWUP(s) is good, then
EXT D (BLOWUP(s)) ≤ 4(p − c) (1 − β)(1 + c)ε + 1 βε EXT D (s)(15)
Proof. By combining Definition 5, Claim 17 that bounds the expected risk of buyers with efficiency in the intervals [0,k(s)], (k(s), ∞) respectively, the result follows.
Claim 19. If BLOWUP(s) is bad and σ ≥ 2, c ≥ 1, then PROF D (HEAVY(s)) ≥ βεPROF D (s)
EXT D (HEAVY(s)) ≤ 4 (1 − β)(c + 1)ε EXT D (s)
Proof. Let n = 4(p−c) (1−β)(1+c)ε , s = BLOWUP(s). We first bound the probability of sale of HEAVY(s). By Equation (12), p HEAVY = (k(s), s ) which implies
P r t←D [v ≥ (k(s), s ) + p(s )] ≤ P r t←D [v ≥ (k(s), s )] ≤ P r t←D [u(t, HEAVY(s)) ≥ 0]
where the last inequality follows from VALUE(t, HEAVY(s)) = v since HEAVY(s) is a cost policy.
For a fixed price p, the probability of sale must decrease as we decrease efficiency, then P r t←D [A(HEAVY(s))] ≥ P r t←D [v ≥ (k(s), s ) + p(s )]
≥ P r t←D [u(t, s ) ≥ 0|k ≤k(s)] = P r t←D [k ≤k(s)|u(t, s ) ≥ 0]P r t←D [u(t, s ) ≥ 0] P r t←D [k ≤k(s)]
where the equality comes from Bayes' theorem.
Next, we bound P r t←D [k ≤k(s)|u(t, s ) ≥ 0]. The inequality comes from the fact RISK(t, s ) ≤ 1,
E t←D [RISK(t, s ) · I(k ≤k(s))|u(t, s ) ≥ 0] ≤ E t←D [I(k ≤k(s))|u(t, s ) ≥ 0] = P r t←D [k ≤k(s)|u(t, s ) ≥ 0] This implies, P r t←D [A(HEAVY(s))] ≥ P r t←D [k ≤k(s)|u(t, s ) ≥ 0]P r t←D [u(t, s ) ≥ 0] ≥ E t←D [RISK(t, s ) · I(k ≤k(s))|u(t, s ) ≥ 0]P r t←D [u(t, s ) ≥ 0]
Using the assumption that s is bad, and Claim 16 to lower bound P r t←D [u(t, s ) ≥ 0], we get Equation (16)
P r t←D [u(t, HEAVY(s)) ≥ 0] ≥ βεnP r t←D [u(t, s) ≥ 0]EXT D (s)(16)
Next, we lower bound the profit of HEAVY(s). In the first equality, we use the definition of p HEAVY , Equation (12). In the first inequality, we lower bound the probabily of sale of HEAVY(s), Equation (16) (1 − β)(1 + c)e 2c+(c−1)(σ−2)+(σ−2) σ(p − c)e 2c PROF D (s)
E t←D [1/k|u(t, s) ≥ 0, k > k h (s)] ≥ 1 − β σ EXT D (s) ≥ ε(1 − β) yσ Proof. E t←D [1/k|B(s)] ≥ P r t←D [k ≤ σ|B(s)]E t←D [1/k|u(t, s) ≥ 0, k ∈ (k h (s), σ]] ≥ (1 − β) σ
Using the fact c ≥ 1, σ ≥ 2, e σ /σ ≥ e 2 /2 and substituting for n,
PROF(HEAVY(s)) ≥ (1 − β)βε 2 n(1 + c)e σ σ2e 2 (p − c) PROF D (s) ≥ βε 2 n(1 + c) 2e 2 (p − c) (1 − β)e σ σ PROF D (s) ≥ βεPROF D (s)
Similarly, we bound the externalities,
EXT D (HEAVY(s)) = e −pHEAVY/2 ≤ e − (1+c)ye cσ 2e 2c = e − (1+c)ye 2c+c(σ−2) 2e 2c ≤ e − (1+c)ye c(σ−2) 2 < 2 (1 + c)ye c(σ−2) ≤ 2 (1 + c)ye (c−1)(σ−2)+(σ−2)
Using the fact c ≥ 1, σ ≥ 2, e σ /σ ≥ e 2 /2 and dividing and multiplying by E t←D [1/k|B(s)], EXT D (HEAVY(s)) ≤ 2e 2 εE t←D [1/k|B(s)] (c + 1)yE t←D [e σ /k|B(s)]
≤ 4e 2 (1 − β)(c + 1)εe 2 EXT D (s)
where the second inequality follows from Proposition 20. Claim 21. If ye −c < 2, σ < 2 then
PROF D (COST 1 (s, 1 − δ)) = (1 − δ)PROF D (s) EXT D (COST 1 (s, 1 − δ)) ≤ 4 ε(1 − β) EXT D (s) Proof. We will proof ye −c < 2 implies E t←D [1/yk · I(k > k h )|u(t, s) ≥ 0] ≥ e −c ε(1−β) 4
, and the statement follows directly by Lemma 11.
E t←D [1/yk · I(k > k h )|u(t, s) ≥ 0] > εe −c 2 E t←D [1/k|u(t, s) ≥ 0, k > k h ] ≥ εe −c (1 − β) 2σ > εe −c (1 − β) 4
where the second inequality follows by Proposition 20.
Claim 22. Assume ye −c ≥ 2, and σ ≤ 2. If for some x ∈ [e −c , 1], COST 3
x (s) is good, then
PROF D (COST 3 x (s)) ≥ βεPROF D (s) EXT D (COST 3 x (s)) ≤ 2 ε(1 − β) EXT D (s)
Proof. To bound ln 1 x + 1 x when x ∈ [e −c , 1], note it achieves its minimum value when x = e −c which implies ln 1 x + 1 x ≥ (c + 1)e −c . We can then bound the net profit of policy COST 3 x (s). More precisely,
we claim p COST 3 x (s) − ln y ≥ ln 1 x +1 xy sk (s) 2
. By definition p COST 3 x (s) = ln 1 x + 1 xy sk (s) and we will show p COST 3
x (s) ≥ 2 ln y.
ln 1 x + 1 xy sk (s) ≥ (c + 1)e −c ye −c e cσ ≥ (c + 1)ye −c ≥ 2c + ye −c by the fact ye −c ≥ 2
Write ln y = ln ye −c + c and observe ye −c ≥ 2 ln ye −c by the fact ye −c ≥ 2. We can conclude 2c + ye −c ≥ 2(c + ln ye −c ) = 2 ln y which implies p COST 3
x (s) − ln y ≥ ln 1 x +1 xy sk (s) 2 . By the fact COST 3 x (s) is good, H(x) ≥ 2βεPROF D (s) (ln 1/x+1)xy sk (s) , then PROF D (COST 3 x (s)) = (p COST 3 x (s) − ln y)H(x) ≥ ln 1 x + 1 xy sk (s) 2 2βεPROF D (s) ln 1 x + 1 xy sk (s) ≥ βεPROF D (s)
Next, we bound the externalities,
EXT D (COST 3 x (s)) EXT D (s) = 1 yEXT D (s) ≤ yσ yε(1 − β) < 2 ε(1 − β)
where the first inequality follows from Proposition 20 and the second inequality follows from σ < 2.
Claim 23. If σ ≤ 2, and ∀x ∈ [e −c , 1], COST 3
x (s) is bad, then
E t←D [RISK(t, BLOWUP(s)) · I(k ≤k(s))|A(BLOWUP(s))] ≤ 4(p − c) ε(1 − β) EXT D (s)
Proof. Define µ s RISK (x) = P r k←D k [RISK(k, s) = x] and observe we can sample k by sampling a risk x from the distribution µ s RISK and computing k. Let's compute E t←D [RISK(t, s ) · I(k ≤k(s), u(t, s ) ≥ 0)] where below, in the first equality, we apply the tower rule by sampling a risk x from µ BLOWUP(s) RISK . In the second equality, if type k has risk x < e −c , by definition ofk(s), we must have k >k(s) which implies I(k ≤k(s)) = 0. In the first inequality, we use the fact H(x) upper bounds the probability of sale for a buyer with risk x.
Proof. By combining Claim 23 and 17, that bound the expected risk of buyers with efficiency in the intervals [0,k(s)], (k(s), ∞) respectively, the result follows.
A.3 Proof of Theorem 3
Proof of Theorem 3. If APPROX(s) outputs COST 1 (s, 1), then by Lemma 11, the profit is PROF D (APPROX(s)) = PROF D (s) Still by Lemma 11, the externalities are at most EXT D (APPROX(s)) EXT D (s) ≤ e −c EXT D (s) ≤ e −1/2(p−c) max 4(p − 1 2 c − 1 2 p) (1 − β)(1 + 1 2 c + 1 2 p)ε 3
+ 1 βε 3 , 4(p − 1 2 c − 1 2 p) (1 − β)ε 3 + 1 βε 3 ≤ 4(p − c) (p − c)(1 − β)ε 3 + 1 βε 3 = 4 (1 − β)ε 3 + 1 βε 3 ≤ 40/3
In the worst case, the profit is 3/16 of PROF D (s) and the externality is 40/3 times higher than EXT D (s) which concludes the proofs.
To conclude, for the profit, in the worst case, APPROX(s) outputs COST 2 (s) at a compromise of at most 1/8 of the profit. For the externalities, FINE(s) will have 40/3 times more externalities than s which completes the proof.
B Profits-Maximizing Seller
In this section, we consider the a variant of our main model. Here, the seller always selects the profits-maximizing price given y and c, and we also compute externalities differently. Specifically, we consider the three stage game where the regulator commits on a fine y and cost c and in sequence the seller is free to select the profit optimal price p. In the last step, the buyer decides to purchase or not. Finally, we will assume externalities are measured as the expected risk conditioned on a buyer to purchase times the probability of purchase (i.e. the total probability of compromise, versus compromise conditioned on purchase). One purpose of this section is to explore a variant of our model. The other purpose is to highlight that results do not significantly change in related models.
Definition 7 (Externality). Given a policy s = (y, c, p), we define Observe that in Example 2, the impossibility result is shown by deriving a distribution D and profit constraint R that can only be satisfied if everyone purchase; therefore, the impossibility follows to the profits-maximizing case.
Next, we will proof Proposition 5 for the profits-maximizing seller.
Proposition 26. For a profits-maximizing seller, there exist distributions D v , D k , and profits constraint R such that:
• T is such that for all k ≥ T , the externality-minimizing policy for D v × {k} subject to profits constraints R is a fine policy.
• D k is supported on [T, ∞).
• No fine policy is externality-minimizing policy for D v × D k subject to profits constraints R.
Under profit constraint R, we can show that the optimal policy gives profits R, Corollary 27. Before we proof this result, we will extend the invariant property, Lemma 7, to the profits-maximizing setting. In the last step, if the profit is non-zero then p > c which implies c(s ) > c because c(s ) is a convex combination of p and c. We conclude p(s) ≤ p(s ). Since p(s ) gives more profit than p(s), we must have PROF D (s ) ≥ αPROF D (s).
In addition, the fact the distribution of values is identical and we only sell with lower probability implies EXT D (s ) ≤ e −(p−c)(1−α) EXT D (s) which completes the proof.
Corollary 27. If policy s is optimal, then PROF D (s) = R.
Proof. Suppose PROF D (s) > R, then there is a sufficiently small ε > 0 such that s = INV(s, (1 − ε)) and PROF D (s ) ≥ R by Lemma 12. Yet by Lemma 12 EXT D (s ) < EXT D (s) contradicting s was optimal.
Figure 2 :
2Figure 2: The efficiency distribution of a population. We plot the effectiveness of a buyer when restricted only to password authentication (left) and SMS authentication (center). By combining the features, and taking into account their contributions in minimizing externalities, we construct the efficiency distribution in the right.
Finally
, there are k such that yk ≥ e c . Then ∂ (k
k, s ) = e −αc−(1−α)p−kh * (k,s ) = e −(1−α)(p−c) · e −kh * (k,s )−c = e −(1−α)(p−c) RISK(k, s) And:
Proof of Claim 11. (3, (y, c)) ≤ ye −c = (2/5 − c)e c · e −c = 2/5 − c. So (16/15, 3) is willing to pay c + 2/3. (x, (y, c)) = ln(xy)−c+1 x = ln(x(2/5−c))+ln(e c )−c+1 x = ln(x)+ln(2/5−c)+1
Theorem 3 .
3For all distributions D, and all policies s, there exists a simple policy s such that PROF D (s ) ≥ PROF D (s)/8, EXT D (s ) ≤ 40/3 · EXT D (s).
BLOWUP(s) := (qye c(σ−1) , 0, p − c)(9)Input: β ∈ [0, 1], s = (y, c, p), D 1: ε = P r t←D [k > k h (s)|VALUE(t, s) ∃x ∈ [e −c , 1], COST 3x (s) is good then 12:
xy sk (s) We define the family of cost policies H(s), H(s) := {COST 3 x (s) : x ∈ [e −c , 1]} We can verify P r t←D [A(COST 3 x (s))] = H(x), P r t←D VALUE(t, COST 3 x (s)) ≥ ln 1 x + 1 xy sk (s) = H(x)
E
t←D [RISK(t, s ) · I(k >k(s), u(t, s ) ≥ 0)] = E t←D [RISK(t, s ) · I(k >k(s))|u(t, s ) ≥ 0]P r t←D [u(t, s ) ≥ 0]By Claim 16, P r t←D [u(t, s ) ≥ 0] ≥ βεP r t←D [u(t, s) ≥ 0], then
By definition of EXT D (s), and the previous bound EXT D (s) ≥ εE t←D [1/yk|B(s)]
E t←D [RISK(t, BLOWUP(s)) · I(k ≤k(s), A(BLOWUP(s)))]= E x←µ BLOWUP(s) RISK [E t←D [RISK(t, BLOWUP(s)) · I(k ≤k(s), A(BLOWUP(s)))|RISK(k, BLOWUP(s)) = x]] = E x←µ BLOWUP(s) RISK [x · I(x ∈ [e −c , 1]) · E t←D [I(A(BLOWUP(s)))|RISK(k, BLOWUP(s)) = x]] = E x←µ BLOWUP(s) RISK [x · I(x ∈ [e −c , 1]) · P r t←D [v ≥ xy sk (s)(1 + ln 1/x) + p(BLOWUP(s))]] ≤ E x←µ BLOWUP(s) RISK [x · I(x ∈ [e −c , 1]) · H(x)]For all x ∈ [e −c , 1], because COST 3x (s) is bad, we must haveH(x) < 2βεPROF D (s) (ln 1/x + 1)xy sk (s) ≤ 2βε(p − c)P r t←D [u(t, s) ≥ 0] xy sk (s) This implies, E t←D [RISK(t, BLOWUP(s)) · I(k ≤k(s), A(BLOWUP(s)))] < E x←µ BLOWUP(s) RISK [x · I(x ∈ [e −c , 1]) · 2βε(p − c)P r t←D [u(t, s) ≥ 0] xy sk (s) ] = 2βε(p − c)P r t←D [u(t, s) ≥ 0]y sk (s) Bellow, we use the fact P r t←D [A(BLOWUP(s))] ≥ βεP r t←D [u(t, s) ≥ 0], Claim 16.
E t←D [RISK(t, BLOWUP(s)) · I(k ≤k(s), A(BLOWUP(s)))]= E t←D [RISK(t, BLOWUP(s)) · I(k ≤k(s))|A(BLOWUP(s))]P r t←D [A(BLOWUP(s))] ≥ βεP r t←D [u(t, s) ≥ 0]E t←D [RISK(t, BLOWUP(s)) · I(k ≤k(s))|A(BLOWUP(s))]Combining the previous bounds, we have E t←D [RISK(t, BLOWUP(s)) · I(k ≤k(s))|A(BLOWUP(s))]EXT D (s) ≤ 2(p − c) y sk (s)EXT D (s) ≤ 2yσ(p − c) y sk (s)(1 − β)ε ≤ 4(p − c) (1 − β)εwhere the last inequality follows from Proposition 20 which completes the proof. Claim 24. If σ ≤ 2, and ∀x ∈ [e −c , 1], COST 3 x (s) is bad thenEXT D (BLOWUP(s)) ≤ 4(p − c) ε(1 − β) + 1 βε EXT D (s)
E t←D [RISK(t, s)|VALUE(t, s) ≥ p(s)] ≤ e −c E t←D [RISK(t, s) · I(k < k 0 (s))|VALUE(t, s) ≥ p(s)] = e −c ε 1 e −c 25 proves approximation guarantees when APPROX(s) outpus FINE(s).Claim 25.PROF D (FINE(s)) ≥ 3 16 PROF D (s) EXT D (FINE(s)) ≤ 24EXT D (s)Proof. If APPROX(s) outputs FINE(s) then ε 3 ≥ 3/4. Let's first bound the profit of FINE(s). If FINE(s) outputs INV(s, p, p/(p − c)), then it must be c = 1. By Lemma 7,PROF D (FINE(s)) = p p − c PROF D (s) ≥ PROF D (s)If FINE(s) outputs BLOWUP • INV(s, p, 1/2), by Claim 16, and Lemma 7,PROF D (FINE(s)) ≥ 1 2 βε 3 PROF D (s) ≥ 3 16 PROF D (s)If FINE(s) outputs HEAVY(s), COST 1 (s, 1) or COST x (s), then by Claim 21, 22, and Lemma 11,PROF D (FINE(s)) ≥ βε 3 PROF D (s) ≥ 3 8 PROF D (s)Let's now bound the externality of FINE(s). If FINE(s) outputs COST 1 (s, 1),then E t←D [1/yk · I(k > k h (s))|A(s)] ≥ e −c ε3(1−β) 4and by Lemma 11EXT D (FINE(s)) EXT D (s) ≤ e −c E t←D [RISK(t, s)|VALUE(t, s) ≥ p(s)] ≤ e −c E t←D [1/yk · I(k > k h (s))|A(s)] If FINE(s) outputs INV(s, p, p/(p − c)), then it must be c ≤ 1 and by Lemma 7, EXT D (FINE(s)) EXT D (s) = e c ≤ e If FINE(s) outputs HEAVY(s), by Claim 19, EXT D (FINE(s)) EXT D (s) If FINE(s) outputs BLOWUP • INV(s, p, 1/2), by Claim 18, 24 and Lemma 7, EXT D (FINE(s))
EXT D (s) := E t←D [RISK(t, s) · I(VALUE(t, s) ≥ p)]
Lemma 12 (
12Augmented Invariant Property). Given policy s, let s = INV(s, α), α ∈ [0, 1], then PROF D (s ) ≥ αPROF D (s), and the optimal price of s is p ≥ p which implies EXT D (s ) ≤ e −(p−c)(1−α) EXT D (s).Proof. Let p(s) denote the optimal price under policy s. Let c(s ) = αc + (1 − α)p. It follows,(p(s) − c)P r t←D [VALUE(t, s) ≥ p(s)] ≥ (p(s ) − c)P r t←D [VALUE(t, s) ≥ p(s )] (p(s ) − c(s ))P r t←D [VALUE(t, s ) ≥ p(s )] ≥ (p(s) − c(s ))P r t←D [VALUE(t, s ) ≥ p(s)]By adding the inequalities and observing that by Lemma 7, for all price p, P r t←D [VALUE(t, s)≥ p] = P r t←D [VALUE(t, s ) ≥ p], we have (p(s) − p(s) + c(s ) − c)P r t←D [VALUE(t, s) ≥ p(s)] ≥ (p(s ) − p(s ) + c(s ) − c)P r t←D [VALUE(t, s) ≥ p(s )] ⇐⇒ (c(s ) − c)P r t←D [VALUE(t, s) ≥ p(s)] ≥ (c(s ) − c)P r t←D [VALUE(t, s) ≥ p(s )]⇐⇒ P r t←D [VALUE(t, s) ≥ p(s)] ≥ P r t←D [VALUE(t, s) ≥ p(s )]
Table 1 :
1Model VariablesVariable
Text Definition
Formal Definition
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. In the second inequality, we usethe fact PROF D (s) = (p − c)P r t←D [u(t, s) ≥ 0]. − c)e 2c EXT D (s)PROF D (s)Next, we bound E t←D [1/k|B(s)] and EXT D (s),PROF(HEAVY(s)) =
p HEAVY
2
P r t←D [v ≥ p HEAVY ]
=
(1 + c)qye cσ
2e 2c
P r t←D [v ≥ p HEAVY ]
≥
(1 + c)ye cσ
2e 2c
βεnP r t←D [u(t, s) ≥ 0]EXT D (s)
≥ βεn
(1 + c)ye cσ
2(p Proposition 20.
This assumption is even more justified in our setting than usual, as it is hard to imagine correlation between the value a consumer derives from using a smart refridgerator and their ability to secure IoT devices.2 It would be equally natural for the regulator to aim to minimize the total mass of compromised devices. Most of our results do not rely on optimizing one objective versus the other, but we stick with one in order to unify the presentation.3 Below, I(·) denotes the indicator function, which takes value I(X) = 1 whenever event X occurs, and 0 otherwise.
See https://www.theverge.com/2017/9/18/16328172/sms-two-factor-authentication-hack-password-bitcoin
That is, it is possible to couple draws (k, k ) from (D k , D k ) so that k ≥ k with probability 1. Equivalently: for all x,Pr[k ≥ x, k ← D k ] ≥ Pr[k ≥ x, k ← D k ].
Note that we are assuming that for any desired probability q, we can set a price that sells with probability exactly q. When either Dv or D k has no point masses, this is clearly true. When both have point masses, observe that if we set a price so that a positive mass of consumers are indifferent between purchasing the item and not, we will assume that we can have some buyers purchase the item and some not (as they are indifferent, either is a best response).
SUPP(D) := {x 1 , ..., x n } then for i ∈ [|SUPP(D v )|], j ∈ [|SUPP(D k |], let t ij := (v i , k j ), r ij (s) := VALUE(t ij , s)P r t←D [VALUE(t, s) ≥ VALUE(t ij , s)] In the discrete case, we have PROF D (s) := max i,j r ij (s). To compute r ij (s), we must compute if for some i ∈ [|SUPP(D v )|], j ∈ [|SUPP(D k |], VALUE(t ij , s) < VALUE(t i j , s) or VALUE(t ij , s) ≥ VALUE(t i j , s). Definition 8 (Type Total Order). We define a type total order < s where t ij < s t i j denote VALUE(t ij , s) ≤ VALUE(t i j , s). Observe that for all regulation s, fix indexes i and j, then for allBefore we proof Proposition 26, let's get intuition why one might expect it to be true. Assume D v and D k have each support of cardinality 2. When there is no regulation s 0 = (0, 0), we have the total order of types t 11 < s0 t 12 < s0 t 21 < s0 t 22 As we increase fines, for some y > 0 and policy s = (y, 0), we will have VALUE(t 12 , s) = VALUE(t 21 , s) since t 12 is more efficient than t 21 . However, assume at the highest possible fine, the seller prefers to sell to {t 12 , t 21 , t 22 }. When fines are slightly smaller, the seller might prefer to sell only to t 21 , and t 22 if v 2 is sufficiently larger than v 1 . If that is the case, selling at lower probability might be better to decrease externalities than increasing fines. If that is the case, the optimal fine policy will give more profit than the profit constraint R, and by Corollary 27 such policy cannot be optimal.Proof of Proposition 26. Define the value distribution D v , and efficiency distribution D k ,ek . Let the profit constraint be R = r 12 (1.2 · y(3), 0). We will have R ∈ (0.51, 0.52), but to compute R, we must first compute the typs total order when fines are 1.2 · y(3), Claim 28. We will then show that setting externalities to y(3) yields strictly lower externalities then setting fines to 1.2 · y(3), Claim 31. Further we argue that when fines are 1.2 · y(x), D v × {3} has optimal fine policy, Claim 32.Claim 28. If s = (y, 0), y ≤ 1.2 · y(3) then t 11 < s t 12 < s t 21 < s t 22 .Proof. Clearly t 11 < s t 12 and t 21 < s t 22 . It is sufficient to show for y = 1.2 · y(3), t 12 < s t 21 and the claim follows for all fines smaller than 1.2 · y(3) because ∂ 2 VALUE(t,s) ∂y∂k ≥ 0. Observe, VALUE(t 21 , s) − VALUE(t 12 , s) = 1.58 − 1 + (3, s) − (9, s) > 0.83 > 0 which completes the proof.Let's now proof when the fines are in the range [y(3), 1.2 · y(3)], the seller never sells with probability 1.Claim 29. For y ∈ [y(3), 1.2 · y(3)], then r 21 (y, 0) > r 11 (y, 0)Proof. By Claim 28, if y ≤ 1.2 · y(3), then t 11 < s t 12 < s t 21 < s t 22 . We can then compute the profit, Next, we claim if the fine is 1.2 · y(3), the seller prefers to sell to t 12 , t 21 , t 22 with probability 3 4 and when the fine is y(3), the seller sells only to t 21 and t 22 .Claim 30. r 21 (y(3), 0) > r 12 (y(3), 0) and r 12 (1.2 · y(3), 0) > r 21 (1.2 · y(3), 0)Proof. For arbitrary y, r 21 (y, 0) − r 12 (y, 0) = 1 2 (1.58 − (3, y)) − 3 4(1 − (9, y))By computing the left hand side for y = y(3) and y = 1.2 · y(3), we have r 21 (y(3), 0) − r 12 (y(3), 0) > 0 and r 21 (1.2 · y(3), 0) − r 12 (1.2 · y(3), 0) < 0.Next, we show that having slightly lower fines yields lower externalities.Claim 31.Proof. For all fines y, if the seller sells at price VALUE(t 22 , y, 0), the profit is at most 1/4 which is smaller than the profit constraint R > 1/2 . We have shown that for y ∈ [y(3), 1.3 · y(3)] the seller will not prefers to sell to everyone; therefore, the seller either sells to {t 12 , t 21 , t 22 } or to {t 21 , t 22 }. In particular, when y = y(3), the seller sells to t 21 and t 22 only. The externalities areWhen y = 1.2 · y(3), the seller sells to t 12 , t 21 , t 22 and the externalities areObserve 1.2 · y(3) is the highest fine of any feasible fine policy by our profit constraint R = r 12 (1.2 · y(3), 0). This is because r 12 (1.2 · y(3), 0) is the highest profit we can get under policy (1.2 · y(3), 0). For fines y strictly larger than 1.2 · y(3), VALUE(t, 1.2 · y(3), 0) strictly stochastic dominate VALUE(t, y, 0) implying the profit is strictly smaller than R. By the same argument, policy (y(3), 0) gets profit > R, and by Corollary 27, there is a non-simple policy that yields lower externalities than (y(3), 0) which implies a non-fine policy is optimal. We conclude by claiming the distribution D v × {3} with profit constraint R has optimal fine policy.Claim 32. A fine policy is optimal on D v × {3}, R.Proof. Another way to state Lemma 3 is to observe that given a profit constraint R, if 1 T −1 is the highest loss we can have, then T is the threshold where for all D v × {k}, k ≥ T , the optimal policy is a fine policy. Observe that (k, y(k), 0) = 1 k−1 . This implies, if y(3) is feasible on (D v × {3}, R) then a fine policy is optimal. Let s = (y(3), 0) and suppose the seller sells at price 1.58 − (3, y(3), 0) having a sale with probability 1 2 and getting profit 0.54. We have R < 0.52; therefore, the policy s is feasible. By definition of y(k), since (y(3), 0) is feasible, the threshold T where fine policies are optimal is at least 3. We can conclude, with distribution D v × {3} and profit constraint R, we have an optimal fine policy with fine at least y(3).this completes the proof of Proposition 26.
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| []
|
[
"When renormalizability is not sufficient: Coulomb problem for vector bosons",
"When renormalizability is not sufficient: Coulomb problem for vector bosons"
]
| [
"V V Flambaum [email protected]†email:[email protected] ",
"M Yu Kuchiev ",
"\nSchool of Physics\nPhysics Division\nUniversity of New South Wales\n2052SydneyAustralia\n",
"\nArgonne National Laboratory\n60439-4843ArgonneIllinoisUSA\n"
]
| [
"School of Physics\nPhysics Division\nUniversity of New South Wales\n2052SydneyAustralia",
"Argonne National Laboratory\n60439-4843ArgonneIllinoisUSA"
]
| []
| The Coulomb problem for vector bosons W ± incorporates a known difficulty; the boson falls on the center. In QED the fermion vacuum polarization produces a barrier at small distances which solves the problem. In a renormalizable SU (2) theory containing vector triplet (W + , W − , γ) and a heavy fermion doublet F with mass M the W − falls on F + , to distances r ∼ 1/M , where M can be made arbitrary large. To prevent the collapse the theory needs additional light fermions, which switch the ultraviolet behavior of the theory from the asymptotic freedom to the Landau pole. Similar situation can take place in the Standard Model. Thus, the renormalizability of a theory is not sufficient to guarantee a reasonable behavior at small distances for non-perturbative problems, such as a bound state problem. PACS numbers: 12.15.Ji, 12.15.Lk, 12.20.Ds It is usually believed that a renormalizable theory automatically exhibits good physical behavior at large momenta and small distances. This claim is definitely correct in low orders of the perturbation theory. However, in higher orders, which are necessary, for example, in a bound state problem, the situation is not so obvious. We present here an example, in which the renormalizability by itself fails to define a proper behavior of the theory at small distances.Consider a negatively charged vector boson, which propagates in the Coulomb field created by a heavy pointlike charge Z|e| assuming that the boson is massive. A bound state problem for this boson needs summation in all orders in Zα. Since the electrodynamics for massive vector particles is non-renormalizable, one should expect problems here. One of them, found long time ago, is particularly interesting for our discussion. Soon after Proca formulated theory for vector particles [1] it became clear that it produces inadequate results for the Coulomb problem: the W wave function is so singular that the integral over the charge density of W is divergent near the origin[2,3,4]. Corben and Schwinger [5] modified the Proca theory, tuning the Lagrangian and equations of motion for vector bosons in such a way as to force the gyromagnetic ratio of the vector boson to acquire a favorable value g = 2. It is well known now that g = 2 is the gyromagnetic ratio of the W -boson in the Standard model. This modification allowed Corben and Schwinger to obtain a physically acceptable spectrum for the Coulomb problem, which is described by the Sommerfeld formula similar to the spectrum of Dirac particles, but with integer values of the total angular momentum j = 0, 1, ....Corben and Schwinger found also that their modifi- * cation did not resolve the main problem, the W -boson still falls to the center for two series of quantum states; one with j = 0, and the other one with "l"= 0 (if "l" is defined appropriately). The wave function of W is so singular at the origin for these states that the integral over W charge density is divergent near the origin. In our works [6] we found a cure for this problem. The QED fermion vacuum polarization was shown to produce an effective potential barrier for the W boson at small distances. The charge density of the W boson in the j = 0 state decreases as exp (−const/r) under this barrier and vanishes at the origin; similar improvement exhibits the l = 0 state. As a result the Coulomb problem for vector particles becomes well defined. The corresponding correction to the Sommerfeld spectrum proves to be small. The effective potential of Ref.[6] is repulsive only when the running coupling constant exhibits the Landau-pole behavior; in contrast, for asymptotic freedom, the collapse is inevitable.In[6]we derived the Corben-Schwinger Lagrangian from the Lagrangian of the Standard Model, where the mass of W is produced by the Higgs mechanism, which preserves the renormalizability of theory. However, the applicability of the Corben-Schwinger wave equation to W requires that the Coulomb center does not interact with the Z-boson and Higgs particle. (It may be taken, for example, as a small charged black hole.) However, such Coulomb center is not described by the Standard Model, preventing the theory from being a complete renormalizable one.In the present work we consider an example of a fully renormalizable model, which exhibits a similar phenomenon. Take an SU (2) gauge theory and a triplet of real Higgs scalars Φ L Boson = − 1 4 G a µν G a µν + 1 2 D µ Φ a * D µ Φ a + ... .(1)Here G a µν and D µ are the gauge field and the covariant derivative, which includes the gauge potential A a µ ; | 10.1142/s0217732307025984 | [
"https://arxiv.org/pdf/hep-ph/0609194v1.pdf"
]
| 119,489,116 | hep-ph/0609194 | d5ab0a3379e3e115996767be19b302eb72aba6fe |
When renormalizability is not sufficient: Coulomb problem for vector bosons
V V Flambaum [email protected]†email:[email protected]
M Yu Kuchiev
School of Physics
Physics Division
University of New South Wales
2052SydneyAustralia
Argonne National Laboratory
60439-4843ArgonneIllinoisUSA
When renormalizability is not sufficient: Coulomb problem for vector bosons
(Dated: October 1, 2018)arXiv:hep-ph/0609194v1 19 Sep 2006
The Coulomb problem for vector bosons W ± incorporates a known difficulty; the boson falls on the center. In QED the fermion vacuum polarization produces a barrier at small distances which solves the problem. In a renormalizable SU (2) theory containing vector triplet (W + , W − , γ) and a heavy fermion doublet F with mass M the W − falls on F + , to distances r ∼ 1/M , where M can be made arbitrary large. To prevent the collapse the theory needs additional light fermions, which switch the ultraviolet behavior of the theory from the asymptotic freedom to the Landau pole. Similar situation can take place in the Standard Model. Thus, the renormalizability of a theory is not sufficient to guarantee a reasonable behavior at small distances for non-perturbative problems, such as a bound state problem. PACS numbers: 12.15.Ji, 12.15.Lk, 12.20.Ds It is usually believed that a renormalizable theory automatically exhibits good physical behavior at large momenta and small distances. This claim is definitely correct in low orders of the perturbation theory. However, in higher orders, which are necessary, for example, in a bound state problem, the situation is not so obvious. We present here an example, in which the renormalizability by itself fails to define a proper behavior of the theory at small distances.Consider a negatively charged vector boson, which propagates in the Coulomb field created by a heavy pointlike charge Z|e| assuming that the boson is massive. A bound state problem for this boson needs summation in all orders in Zα. Since the electrodynamics for massive vector particles is non-renormalizable, one should expect problems here. One of them, found long time ago, is particularly interesting for our discussion. Soon after Proca formulated theory for vector particles [1] it became clear that it produces inadequate results for the Coulomb problem: the W wave function is so singular that the integral over the charge density of W is divergent near the origin[2,3,4]. Corben and Schwinger [5] modified the Proca theory, tuning the Lagrangian and equations of motion for vector bosons in such a way as to force the gyromagnetic ratio of the vector boson to acquire a favorable value g = 2. It is well known now that g = 2 is the gyromagnetic ratio of the W -boson in the Standard model. This modification allowed Corben and Schwinger to obtain a physically acceptable spectrum for the Coulomb problem, which is described by the Sommerfeld formula similar to the spectrum of Dirac particles, but with integer values of the total angular momentum j = 0, 1, ....Corben and Schwinger found also that their modifi- * cation did not resolve the main problem, the W -boson still falls to the center for two series of quantum states; one with j = 0, and the other one with "l"= 0 (if "l" is defined appropriately). The wave function of W is so singular at the origin for these states that the integral over W charge density is divergent near the origin. In our works [6] we found a cure for this problem. The QED fermion vacuum polarization was shown to produce an effective potential barrier for the W boson at small distances. The charge density of the W boson in the j = 0 state decreases as exp (−const/r) under this barrier and vanishes at the origin; similar improvement exhibits the l = 0 state. As a result the Coulomb problem for vector particles becomes well defined. The corresponding correction to the Sommerfeld spectrum proves to be small. The effective potential of Ref.[6] is repulsive only when the running coupling constant exhibits the Landau-pole behavior; in contrast, for asymptotic freedom, the collapse is inevitable.In[6]we derived the Corben-Schwinger Lagrangian from the Lagrangian of the Standard Model, where the mass of W is produced by the Higgs mechanism, which preserves the renormalizability of theory. However, the applicability of the Corben-Schwinger wave equation to W requires that the Coulomb center does not interact with the Z-boson and Higgs particle. (It may be taken, for example, as a small charged black hole.) However, such Coulomb center is not described by the Standard Model, preventing the theory from being a complete renormalizable one.In the present work we consider an example of a fully renormalizable model, which exhibits a similar phenomenon. Take an SU (2) gauge theory and a triplet of real Higgs scalars Φ L Boson = − 1 4 G a µν G a µν + 1 2 D µ Φ a * D µ Φ a + ... .(1)Here G a µν and D µ are the gauge field and the covariant derivative, which includes the gauge potential A a µ ;
The Coulomb problem for vector bosons W ± incorporates a known difficulty; the boson falls on the center. In QED the fermion vacuum polarization produces a barrier at small distances which solves the problem. In a renormalizable SU (2) theory containing vector triplet (W + , W − , γ) and a heavy fermion doublet F with mass M the W − falls on F + , to distances r ∼ 1/M , where M can be made arbitrary large. To prevent the collapse the theory needs additional light fermions, which switch the ultraviolet behavior of the theory from the asymptotic freedom to the Landau pole. Similar situation can take place in the Standard Model. Thus, the renormalizability of a theory is not sufficient to guarantee a reasonable behavior at small distances for non-perturbative problems, such as a bound state problem. It is usually believed that a renormalizable theory automatically exhibits good physical behavior at large momenta and small distances. This claim is definitely correct in low orders of the perturbation theory. However, in higher orders, which are necessary, for example, in a bound state problem, the situation is not so obvious. We present here an example, in which the renormalizability by itself fails to define a proper behavior of the theory at small distances.
Consider a negatively charged vector boson, which propagates in the Coulomb field created by a heavy pointlike charge Z|e| assuming that the boson is massive. A bound state problem for this boson needs summation in all orders in Zα. Since the electrodynamics for massive vector particles is non-renormalizable, one should expect problems here. One of them, found long time ago, is particularly interesting for our discussion. Soon after Proca formulated theory for vector particles [1] it became clear that it produces inadequate results for the Coulomb problem: the W wave function is so singular that the integral over the charge density of W is divergent near the origin [2,3,4]. Corben and Schwinger [5] modified the Proca theory, tuning the Lagrangian and equations of motion for vector bosons in such a way as to force the gyromagnetic ratio of the vector boson to acquire a favorable value g = 2. It is well known now that g = 2 is the gyromagnetic ratio of the W -boson in the Standard model. This modification allowed Corben and Schwinger to obtain a physically acceptable spectrum for the Coulomb problem, which is described by the Sommerfeld formula similar to the spectrum of Dirac particles, but with integer values of the total angular momentum j = 0, 1, ....
Corben and Schwinger found also that their modifi- * Email:[email protected] † Email:[email protected] cation did not resolve the main problem, the W -boson still falls to the center for two series of quantum states; one with j = 0, and the other one with "l"= 0 (if "l" is defined appropriately). The wave function of W is so singular at the origin for these states that the integral over W charge density is divergent near the origin. In our works [6] we found a cure for this problem. The QED fermion vacuum polarization was shown to produce an effective potential barrier for the W boson at small distances. The charge density of the W boson in the j = 0 state decreases as exp (−const/r) under this barrier and vanishes at the origin; similar improvement exhibits the l = 0 state. As a result the Coulomb problem for vector particles becomes well defined. The corresponding correction to the Sommerfeld spectrum proves to be small. The effective potential of Ref. [6] is repulsive only when the running coupling constant exhibits the Landau-pole behavior; in contrast, for asymptotic freedom, the collapse is inevitable.
In [6] we derived the Corben-Schwinger Lagrangian from the Lagrangian of the Standard Model, where the mass of W is produced by the Higgs mechanism, which preserves the renormalizability of theory. However, the applicability of the Corben-Schwinger wave equation to W requires that the Coulomb center does not interact with the Z-boson and Higgs particle. (It may be taken, for example, as a small charged black hole.) However, such Coulomb center is not described by the Standard Model, preventing the theory from being a complete renormalizable one.
In the present work we consider an example of a fully renormalizable model, which exhibits a similar phenomenon. Take an SU (2) gauge theory and a triplet of real Higgs scalars Φ
L Boson = − 1 4 G a µν G a µν + 1 2 D µ Φ a * D µ Φ a + ... . (1)
Here G a µν and D µ are the gauge field and the covariant derivative, which includes the gauge potential A a µ ;
the dots refer to the nonlinear self-interaction of scalars, which produces the vacuum expectation value for the Higgs field Φ vac = (0, 0, v). Then one gauge boson A 3 µ remains massless, call it the photon, A µ = A 3 µ . The two other bosons, which we call W ± µ = (A 1 µ ∓ iA 2 µ )/ √ 2 acquire the Higgs mass m = g 2 v 2 /2, see e.g. [7]. To allow the Coulomb center to appear in the model, consider a heavy fermion doublet F = (F + , F − ) with charges e/2 and −e/2 for its two components. Presume for simplicity that the parity is conserved and the fermion doublet does not interact with the Higgs field; its large mass, M ≫ m, is a free parameter in the Lagrangian
L Fermi =F ( iγ µ D µ − M ) F .(2)
Our goal is to demonstrate that the interaction between W − and the heavy fermion F + results in the collapse of the boson onto the fermion. But firstly, let us consider the high-energy dependence of the lowest-order scattering amplitudes, which emphasizes a difference between renormalizable and non-renormalizable models. Consider the diagrams (a) and (b) in Fig. 1, which describe scattering of W on F . Conventional calculations show that at high collision energy their amplitudes satisfy
M (a) ≃ −M (b) ≃ − e 2 4m 2 ( p µ + p ′ µ )F γ µ F .(3)
Separately, each one of them grows with energy, violating the unitarity limit; note that the diagram (b) contains only one partial wave. The increase is due to the longitudinal polarization of the W-boson, ǫ W µ = k µ /m + O(m/p 0 ), p 0 is the W energy -compare e.g. [7]. The energy increase of the photon exchange diagram (a) signals the non-renormalizability of the pure vector electrodynamics. However, the sum of the two diagrams M (a) + M (b) does not possess this problem, a compensation of the two diagrams results in a reasonable behavior of the scattering amplitude at high energy, in accord with the renormalizability of the SU(2) model introduced in Eq. 1. It is important that the cancellation of the diagrams (a) and (b) manifests itself only when the collision energy is taken as a large parameter, p 0 ≫ m.
However, in the bound-state problem the energy is fixed, p 0 ≃ m. Moreover, the wave function of the W -boson at distances 1/M ≪ r ≪ 1/m is represented by the off shell diagrams, in which legs of the W -boson carry large momenta m ≪ |p|,
|p ′ | ∼ 1/r ≪ M . The invariant t of the scattering problem is large in this case, t = −|p− p ′ | 2 ≫ m 2 , while the s-invariant remains fixed.
In this kinematic region the diagram (a) is not compensated by (b). Therefore the verified above renormalizability of the theory cannot shed light on the behavior of the wave function at 1/M ≪ r ≪ 1/m. To establish this behavior one needs to consider a set of all ladder diagrams of the type shown in Fig. 1 (c), which are known to produce dominant contribution to the off-shell amplitude when t is large. Clearly summation of this ladder is equivalent to the solution of the wave equation for the W -boson in an attractive Coulomb field created by the heavy fermion. The necessary wave equation can be derived from the Lagrangian Eq.(1). Keeping there only those terms, which describe W -bosons and their interaction with photons we find an effective Lagrangian, which proves to be identical to the Lagrangian introduces by Corben-Schwinger
( ) a W W ( ) b F W 0 | | p m m M p F F F F F ' 0 | ' | p m m M p ( ) c p ' p 2 2 2 ' p p m p ' p 2 2 2 ' p p m p ' p F W W FIGL W = − 1 2 (∇ µ W ν − ∇ ν W µ ) + (∇ µ W ν − ∇ ν W µ ) +i e F µν W + µ W ν + m 2 W + µ W µ ,(4)
Here F µν = ∂ µ A ν − ∂ ν A µ is the electromagnetic field, and ∇ µ = ∂ µ + ieA µ is the covariant derivative in this field (e < 0). From Eq.(4) one derives the Corben-Schwinger wave equation, which can be written in the following form
∇ 2 + m 2 W µ + 2ieF µν W ν + ie m 2 ∇ µ (j ν W ν ) = 0. (5)
The last term here includes explicitly an external current, j ν = ∂ µ F µν , which creates the electromagnetic field, see details in Ref. [6]. To derive conclusions from Eq.(5) several points need to be specified. Since the boson is massive, it is necessary to rewrite the equation in terms of a 3-vector, for example via spatial components W of the four-vector W µ = (W 0 , W). It is necessary also to specify the equation for a static, spherically symmetrical external potential U (r), in which the W -boson propagates with angular momentum j = 0, 1, · · · , these details can also be found in Ref. [6]. Consider the most interesting for us partial wave j = 0. In this state the three-vector W satisfies W(r) = n v(r), n = r/r. It can be verified that Eq.(5) imposes the following equation
on v = v(r) v ′′ + G v ′ + H v = 0 ,(6)
where the coefficients G = G(r) and H = H(r) are
G = 2 r − U ′ U − U ′ + Υ ′ U + Υ ,(7a)H = − 2 r 2 − 2 r U ′ U + U ′ + Υ ′ U + Υ + ( U + Υ ) U . (7b)
In the region r ≪ 1/m ≃ 1/p 0 the energy and mass do not manifest themselves in the wave equation. The functions G, H can depend only on the potential U = U (r), in which the boson propagates, and on an additional term Υ = Υ(r), which originates from the zero-th component of the external current j µ in Eq.(5)
Υ = e j 0 /m 2 = −∆U/m 2 .(8)
Let us apply firstly Eq.(6) to the pure Coulomb potential, when U (r) = −Zα/r, where Z = 1/2 is the charge of F + , and G = 4/r and H = (2 + Z 2 α 2 )/r 2 . Consequently we find the solution in the region 1/
M ≪ r ≪ 1/m, v(r) ≃ r γ−3/2 ,(9)
where γ = (1/4 − Z 2 α 2 ) 1/2 . Straightforward calculations show that this solution results in a major problem, forcing the charge density of the W -boson ρ W = ρ W (r) to diverge at small distances, ρ W ∝ r 2γ−4 . Since 2γ < 1 a divergence of the integral of this charge density signals the collapse of the W -boson to the Coulomb center, or at least into the region r ∼ 1/M . This makes the pure Coulomb problem poorly defined, in accord with conclusions of Ref. [5]. At this point it is instructive to return to the Comptontype diagram (b) in Fig.1, which was so important in the high energy limit for on shell processes. However, it is unable to remedy the problem of the collapse of the W -boson on the Coulomb center. The reason is clear. We saw that the collapse takes place in the region 1/M ≪ r ≪ 1/m, which is well separated from the heavy fermion, while the diagram (b) operates only when the distance r between the W -boson and the heavy fermion F + is small, r ∼ 1/M [9]. Clearly the short-range interaction described by this diagram cannot prevent the fast increase of the wave function of the W -boson at larger distances r > 1/M .
Consider now the radiative corrections. The most important phenomenon, which takes place at small distances (large momenta) is related to the renormalization of the coupling constant, which in the case considered results in the renormalization of the Coulomb charge. It suffices to consider the vacuum polarization in the lowestorder approximation, when it is described by the known Uehling potential, which at small distances is represented via a conventional logarithmic function, see e. g. [8]. A combined potential energy of the Coulomb and Uehling potentials read
U (r) = U C + U U = −[ 1 − αβ ln (mr) ]Zα/r , (10)
where β is the lowest order coefficient of the Gell-Mann -Low beta-function. The polarization produces small variation of the potential in Eq.(10), but makes the Υterm Eq.(8) large at small distances
Υ = Zα 2 β/(m 2 r 3 ) ≫ |U | .(11)
The functions G, H in Eq.(7) calculated with account of this Υ-term read,
G ≃ 6/r , H ≃ −Z 2 α 3 β/(m 2 r 4 ) ,(12)
which results in the following asymptotic solution of Eq.
(6) v ∝ 1 r 2 × exp(−φ), β > 0, cos( |φ| + δ ), β < 0 ,(13)
where φ = Zα(αβ) 1/2 /(m r), and δ is a constant phase defined by the behavior of the solution at r → 0, which we do not discuss here. We see that the sign of β plays a crucial role. In pure QED it is positive, β = 2/(3π) > 0 for one generation of the Dirac fermions in the normalization adopted in Eq.(10). Eq.(13) indicates in this case that v(r) is exponentially suppressed at small distances, which makes the Coulomb problem stable, well defined in accord with conclusions of Ref. [6]. In contrast, the considered SU(2) model is asymptotically free, β = −22/(3π) < 0, which makes v(r) a growing, strongly oscillating function at small distances. This clearly indicates the collapse of the W -boson. Therefore the Coulomb problem cannot be formulated in that case. We observe an unexpected result. For an attractive Uehling potential U U < 0 (that characterizes the pure QED, β > 0) the Coulomb problem turns out to be stable. In contrast, the repulsive Uehling potential U U > 0 (SU(2) model, β < 0) results in the collapse of the Wboson, which makes the Coulomb problem unstable. In other words, the situation looks as if there is an effective potential, which sign is opposite to the sign of the Uehling potential. This surprising behavior finds its origin in the properties of the Υ-potential, which describes the zero-th component of the external current as shows Eq. (8). A presence of this current in the wave equation, see Eq.(5), distinguishes the case of vector particles from scalars and spinors [10].
The collapse of the W -boson on the Coulomb center is not related to particular properties of the model discussed. It manifests itself similarly within, for example, the Standard Model SU (2) × U (1), if it includes heavy fermions [11]. At small distances r < 1/m the mass of the Z-boson may be neglected. In this situation one may use any linear combinations of degenerate eigenstates. In our case it is convenient to use the original bosons B 3 µ from SU(2) and W (0) from U(1), instead of the eigenstates Z and γ. W − interacts with W 0 only, which reduces the problem to the SU(2) sector, where the W -boson collapses on the heavy fermion. We verified this claim by direct calculations, which show that at small distances the Weinberg mixing angle θ W is canceled out. The Coulomb problem can be remedied only if a sufficient number of light fermions, which change the sign of the vacuum polarization, is added. The point is that this condition does not follow from the renormalizability of the theory.
Finally, let us consider another aspect of the problem. Eqs. (6) -(8) contain ∆U C ∝ δ(r), which was neglected in previous works. Working with this term it is convenient to introduce a finite nuclear size R and then take the limit R → 0 (for simplicity we assume infinite M ). Inside the "nucleus" an effective potential U eff = −H given in Eq.(12) dominates in the wave equation. This attractive potential produces large number of states (infinite for the zero nuclear size), which are localized inside the nucleus. Their energies are well below the groundstate energy given by the Sommerfeld formula. These levels would be populated via creation of W + W − pairs, similar to the vacuum breakdown for the Dirac particles. The difference is that for vector bosons the vacuum breakdown happens for any, however small charge of the nucleus. Note that this contact potential was neglected when the Sommerfeld spectrum was derived for the point-like nucleus. However, we see that this potential drastically modifies the spectrum.
In pure QED the problem is saved by the fermion vacuum polarization, which produces the impenetrable potential barrier (for R = 0). This eliminates any contact interaction with the nucleus, and the Sommerfeld spectrum survives. In the case of the SU (2) the situation may seem different since the Compton diagram Fig.1 (b) produces the repulsive interaction inside the nucleus. One may hope that this interaction eliminates the negativeenergy states located inside the nucleus and brings the spectrum to the Sommerfeld form. However, the contact interaction does not influence the wave function of W at the distances r ≫ R. Therefore, the collapse of W to a vicinity of the nucleus is inevitable. As was pointed above, the collapse can be prevented by addition of light fermions, which switch the ultraviolet behavior of the theory from the asymptotic freedom to the Landau pole, thus preventing the collapse. equation for scalars, S = 0, includes only the potential Aµ. The wave equation for spinors, S = 1/2, if written as the second-order differential equation, includes also the field Fµν , which is represented by first derivatives of the potential. In the wave equation for vector bosons, S = 1, there appears also the external current jµ, which can be expressed via second derivatives of the potential.
[11] The inclusion of new heavy particles is motivated by the dark matter observations and possible extensions of the Standard model, e.g. supersymmetry.
[12] Zero-mass particles have no bound states with a Coulomb center since the binding energy is proportional to the mass. The spontaneous symmetry breaking, which creates the mass via the Higgs mechanism generates the bound states. The gluon acquires a dynamical mass inside a hadron and, in principle, may form a bound state with a heavy quark. This problem may also be relevant to the quark-gluon plasma.
PACS numbers: 12.15.Ji, 12.15.Lk, 12.20.Ds
. 1 :
1Scattering of the W − -boson on the heavy fermion F + , (a) the photon exchange, (b) -Compton-type scattering; (a) and (b) compensate each other for high energy of W on the mass shell p 2 = m 2 ; (c) one of the ladder-type diagrams, which are important for a short-distance behavior of the Wboson wave function at m 2 ≪ −p 2 ≪ M 2 .
This work was supported by the Australian Research Council. One of us (VF) appreciates support from the Department of Energy, Office of Nuclear Physics, Contract No. W-31-109-ENG-38. VF is also grateful to C. Roberts for a discussion of a possible link between this work and QCD [12].
. A Proca, Compt.Rend. 2021490A. Proca, Compt.Rend. 202, 1490 (1936).
. H F W Massey, H C Corben, Proc.Camb.Phi.Soc. 35463H. F. W. Massey and H. C. Corben, Proc.Camb.Phi.Soc. 35, 463 (1939).
. J R Oppenheimer, H Snyder, R Serber, Phys.Rev. 5775J. R. Oppenheimer, H. Snyder and R. Serber, Phys.Rev. 57, 75 (1940).
. I E Tamm, Phys. Rev. 58952I. E. Tamm. Phys. Rev. 58, 952 (1940);
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. M Yu, V V Kuchiev, Flambaum, Mod.Phys.Lett.A. 21781M. Yu. Kuchiev and V. V. Flambaum. Mod.Phys.Lett.A 21, 781 (2006);
. Phys. Rev. D. 7393009Phys. Rev. D 73, 093009 (2006).
Ta-Pei Cheng, Ling-Fong Li Gauge theory of elementary particles. OxfordClarendon PressTa-Pei Cheng, Ling-Fong Li Gauge theory of elementary particles, Clarendon Press, Oxford, 1984.
V B Berestetsky, E M Lifshits, L P Pitaevsky, Quantum electrodynamics. PergamonPressV. B. Berestetsky, E. M. Lifshits, and L. P. Pitaevsky, Quantum electrodynamics, PergamonPress, 1982.
In the non-relativistic limit the Compton diagram gives a repulsive contact interaction ∼ (|e|/p0)(W † 0 W0)ρ(r) energy of the W -boson p0 ≈ m. In the non-relativistic limit the Compton diagram gives a repulsive contact interaction ∼ (|e|/p0)(W † 0 W0)ρ(r) energy of the W -boson p0 ≈ m.
| []
|
[
"Electromagnetic moments of quasi-stable particle",
"Electromagnetic moments of quasi-stable particle"
]
| [
"Tim Ledwig \nInstitut für Kernphysik\nJohannes Gutenberg Universität Mainz\nD-55099MainzGermany\n",
"Vladimir Pascalutsa \nInstitut für Kernphysik\nJohannes Gutenberg Universität Mainz\nD-55099MainzGermany\n",
"Marc Vanderhaeghen \nInstitut für Kernphysik\nJohannes Gutenberg Universität Mainz\nD-55099MainzGermany\n"
]
| [
"Institut für Kernphysik\nJohannes Gutenberg Universität Mainz\nD-55099MainzGermany",
"Institut für Kernphysik\nJohannes Gutenberg Universität Mainz\nD-55099MainzGermany",
"Institut für Kernphysik\nJohannes Gutenberg Universität Mainz\nD-55099MainzGermany"
]
| []
| We deal with the problem of assigning electromagnetic moments to a quasi-stable particle (i.e., a particle with mass located at particle's decay threshold). In this case, an application of a small external electromagnetic field changes the energy in a non-analytic way, which makes it difficult to assign definitive moments. On the example of a spin-1/2 field with mass M * interacting with two fields of masses M and m, we show how a conventionally defined magnetic dipole moment diverges at M * = M + m. We then show that the conventional definition makes sense only when the values of the applied magnetic field B satisfy |eB|/2M * ≪ |M * − M − m|. We discuss implications of these results to existing studies in electroweak theory, chiral effective-field theory, and lattice QCD. | 10.1103/physrevd.82.091301 | [
"https://arxiv.org/pdf/1004.5055v1.pdf"
]
| 118,480,228 | 1004.5055 | 72b4790102070ef75c1c9cdb497b848d51b5febb |
Electromagnetic moments of quasi-stable particle
28 Apr 2010 (Dated: April 29, 2010)
Tim Ledwig
Institut für Kernphysik
Johannes Gutenberg Universität Mainz
D-55099MainzGermany
Vladimir Pascalutsa
Institut für Kernphysik
Johannes Gutenberg Universität Mainz
D-55099MainzGermany
Marc Vanderhaeghen
Institut für Kernphysik
Johannes Gutenberg Universität Mainz
D-55099MainzGermany
Electromagnetic moments of quasi-stable particle
28 Apr 2010 (Dated: April 29, 2010)
We deal with the problem of assigning electromagnetic moments to a quasi-stable particle (i.e., a particle with mass located at particle's decay threshold). In this case, an application of a small external electromagnetic field changes the energy in a non-analytic way, which makes it difficult to assign definitive moments. On the example of a spin-1/2 field with mass M * interacting with two fields of masses M and m, we show how a conventionally defined magnetic dipole moment diverges at M * = M + m. We then show that the conventional definition makes sense only when the values of the applied magnetic field B satisfy |eB|/2M * ≪ |M * − M − m|. We discuss implications of these results to existing studies in electroweak theory, chiral effective-field theory, and lattice QCD.
Electromagnetic (e.m.) moments of a particle are determined through observations of the particle's behavior in an applied electromagnetic field. For example, the magnetic moment is measured by observing the spin precession in a magnetic field. In doing so, one assumes that the uniform magnetic field B induces a linear response in the energy:
∆E = − µ · B,(1)
with µ being the magnetic moment. This method works perfectly well for stable particles (electron, proton), as well as for many unstable particles (muon, neutron, etc.), which live long enough for their spin precession to be observed. In this letter we examine the case of a "quasistable" particle, i.e., a particle with mass M * that could decay into two (for simplicity) particles with masses M and m, such that
M * = M + m .(2)
It turns out that applying the magnetic field in this situation does not lead to a polynomial energy shift but to a response which is non-analytic in B, typically ∆E ∼ | B| 1/2 . The square-root behavior is characteristic for the particle-production cut. In a more general situation, when M * ≈ M + m, a polynomial expansion in B can be made as long as
| B| × [magneton] ≪ |M * − M − m|,(3)
which thus becomes a condition for the magnetic moment to be observable. We do not yet know of examples in nature where the masses of particles would be tuned to such an extent that the condition Eq. (3) would be violated. For example, the neutron mass is less than 1 MeV above the threshold (M n −M p −m e ≈ 0.8 MeV), but this number is huge when compared to any reasonable value of the magnetic field measured in units of nuclear magneton: µ N ≃ 3 × 10 −14 MeV/Tesla. Nevertheless, situations where the condition Eq. (3) is violated are sometimes encountered in theoretical studies. In the studies of the W -boson's magnetic and quadrupole moments as a function of bottom-and topquark masses m b and m t , a singularity at m b +m t = M W arises from the bt (or tb) loop contributions. This singularity was reported firtstly in [1,2] at a time when the value of m t was not known yet. In lattice Quantum Chromodynamics (QCD), the e.m. moments of hadron are computed for various values of light quark masses and, as calculations based on chiral perturbation theory show, cups and singularities arise too [3,4]. In this work we find that the singularities arise in the region where the electromagnetic moments are ill-defined, because the condition Eq. (3) is not satisfied.
Our findings are best demonstrated on a simple toy model of three fields: a scalar ϕ and two Dirac spinors ψ and Ψ , interacting via the Yukawa type of coupling:
L int = g Ψ ψ ϕ + ψ Ψ ϕ * ,(4)
with g ≪ 1, a small coupling constant. We denote the masses of ϕ, ψ and Ψ respectively as: m, M , and M * , and will later on focus on the region specified by Eq. (2). Suppose the field Ψ , as well as one of the other two fields, has an electric charge e, and couples minimally to electromagnetism. We look for its anomalous magnetic moment (a.m.m.) κ * at leading order in the coupling g. Depending on whether ϕ or ψ is charged we ought to consider the electromagnetic vertex corrections shown in Fig. 1, and obtain (unprimed: ϕ charged, ψ neutral, or primed: ψ charged, ϕ neutral):
κ * = 2g 2 (4π) 2 1 0 dx −(r + x) x(1 − x) xµ 2 − x(1 − x) + (1 − x)r 2 , (5) κ ′ * = 2g 2 (4π) 2 1 0 dx (r + x)(1 − x) 2 xµ 2 − x(1 − x) + (1 − x)r 2 ,(6)
where r = M/M * , µ = m/M * . We have checked that for M * = −M = M N and m = m π being respectively the mass of the nucleon and the pion, these expressions reproduce results of the meson theory (the same result also arises in chiral perturbation theory at next-to-leading order [5]). The minus sign in front of M appears due to the pseudo-scalar nature of pion.
At M * = m + M (or, 1 = µ + r), the denominator in the integrands takes the form [xµ − (1 − x)r] 2 , which leads to an essential singularity in these expressions for any positive µ and r. This can explicitly be seen, for instance, in Fig. 2 where κ * is plotted as a function of µ. If ϕ is pseudo-scalar, M flips the sign in these expressions, and the singularity is replaced by a cusp.
The singularity is clearly unphysical, since an infinite value of the magnetic moment would correspond to a infinite-energy response to an external magnetic field. To find the correct answer we consider the selfenergy of the Ψ -field in a constant electromagnetic field,
F µν ≡ ∂ µ A ν − ∂ ν A µ = const.
Calculations of this sort have been done before, most notably by Sommerfield and Schwinger [6,7] as a technique to obtain the correction term of order To cast this technique into a modern field-theoretic language, we introduce the sources Θ , and j for the fields Ψ , ψ and ϕ, respectively, and write down the generating functional of the theory,
Z[Θ , , j; A] = exp − g d 4 z δ 3 δj * (z) δ¯(z) δΘ (z) + δ 3 δj(z) δΘ (z) δ(z) exp − d 4 x d 4 y ×Θ (x) S(x − y; A) Θ (y) + . . . ,(7)
where
S(x − y; A) = iγ µ ∂ ∂x µ − eA µ (x)γ µ − M * −1 δ (4) (x − y) .(8)
is the propagator of a charged Dirac particle in the presence of an e.m. field. We then calculate the energy shift induced by the Ψ -field self-energy correction in the presence of a constant e.m. field. The dependence on the e.m. field comes in the form of the γ µ γ ν F µν structure sandwiched between the free Ψ -field states. When the electric contribution is zero, this structure simply yields the projection of the magnetic field onto the spin direction.
The resulting energy-shift, to leading order in g, is for the two cases given by:
∆Ẽ = g 2 (4π) 2 1 0 dx (r + x) (9) × ln 1 + x(1 − x)B xµ 2 − x(1 − x) + (1 − x)r 2 , ∆Ẽ ′ = g 2 (4π) 2 1 0 dx (r + x)(10)× ln 1 − (1 − x) 2B xµ 2 − x(1 − x) + (1 − x)r 2 ,
where the following dimensionless variables are used:
B = eB z M 2 * , ∆Ẽ = ∆E M * + 1 2B ,(11)
with B z the projection of the magnetic field on the spin direction. The quantity ∆Ẽ is the energy shift (in units of M * ) due to the a.m.m. effect. In the following we will discuss the unprimed contribution, the primed one can be obtained analogously. The e.m. field is assumed to be small in comparison with the mass-scale of particles, and therefore some terms which are higher-order inB 2 can be neglected. Nevertheless, one can still see that a naive perturbative expansion inB does not always work. In the naive expansion, one finds
∆Ẽ = − κ * 2B + . . . ,(12)
with κ * given by Eq. In Fig. 4 we again compare the perturbative and non-perturbative results, but now as a function of the magnetic-field strength. The masses are fixed such that the Ψ particle is stable for solid and long-dashed curves and unstable for medium-and short-dashed curves. In either situation there is a kink appearing at some value of the magnetic field, which indicates the crossing over the decay threshold. When Ψ is quasi-stable, µ + r = 1, the kink appears at B = 0, which makes it impossible to define the moments as derivatives of the energy response with respect to the e.m. field.
Integration over the Feynman-parameter x in Eq. (9) yields more insight into the non-analytic dependence on the e.m. field. The result can be written as
∆Ẽ = g 2 (4π) 2 (r + α) (Ω + A) − [(r + α) (Ω + A)]B =0 ,(13)
where Ω is non-analytic inB : with
Ω = λ ln (α + λ)(β + λ) (α − λ)(β − λ) ,(14)α = 1 2(1 −B) 1 + r 2 − µ 2 −B , β = 1 2(1 −B) 1 − r 2 + µ 2 −B ,(15)λ = α 2 − r 2 /(1 −B) 1/2 .
while the analytic terms are contained in
A = −2 + β ln µ 2 + α ln r 2 − µ 2 (1 − ln µ 2 ) − r 2 (1 − ln r 2 ) 2(α + r)(1 −B) .(16)
From the expression for Ω we can readily see that a Taylor expansion in B only make sense when the condition of Eq. (3) is satisfied. The masses of particles are rarely tuned to the extent that the condition Eq. (3) is in danger. One field of applications where one does need to pay attention is lattice QCD. In modern lattice studies the e.m. moments of hadrons can directly be accessed using the background e.m. field method [8]. However, the field strength cannot be arbitrarily small, the periodicity condition poses a lower bound. In the case of magnetic field the bound is: eB ≥ 2π/(a 2 L), or in best case [9]: eB ≥ 2π/(aL) 2 , with length a and integer L being respectively the lattice spacing and size. For typical modern lattices the lowest possible value of the magnetic field can be as large as 10 14 Tesla. Certainly in such strong e.m. fields the problem raised here becomes relevant and should be studied on a case-by-case basis. One typical example would be the case of the ∆(1232) isobar, which magnetic moment has recently been computed using the background field method for various pion masses [10,11]. Figure 5 shows how the condition
eB 2M ∆ (M ∆ − M N − m π ) ≪ 1(17)
can be violated in this type of studies, but of course for very specific values of pion mass and the background magnetic field. We emphasize that the actual parameters in [10,11], do not violate the above condition, mainly thanks to the large values of pion mass used in these works. However, current lattice calculations begin to approach the pion-mass range where this condition would be violated. It would be interesting to see how the nonanalytic B 1/2 behavior emerges in these calculations. Of course one can expect this behavior to be shielded by the finite volume effects, the question is to which extent. To conclude, the singularities found in calculations of the e.m. moments of particles, such as W-boson in the Standard Model (prior to the top-quark discovery) or some of the hadrons in chiral effective-field theory, reflect only the limitation of the calculational technique. When the mass of the particle is near a decay threshold (quasi-stable state), a small external e.m. field may induce the decay instead of interacting with the particle's e.m. moments. We have formulated an exact condition for this effect to occur. In this situation an extra care should be taken in defining and determining the moments, as has been described in this work. The present and future lattice QCD calculations of hadron e.m. moment using the background e.m. field technique are a very likely subject to this problem.
We acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG). The work of T.L. was partially supported by the Research Centre "Elementarkraefte und Mathematische Grundlagen" at the Johannes Gutenberg University Mainz.
Figure 1 :
1One-loop electromagnetic vertex corrections. Double lines, single and dotted lines denote the propagators of Ψ , ψ, and ϕ, respectively. Dots denote the Yukawa coupling and rectangles the minimal electromagnetic coupling.
.
α 2 em ≃ (1/137) 2 to the electron's a.m.m.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .
Figure 2 :
2The anomalous magnetic moment κ * of Ψ -field as function of ϕ-field mass µ, at fixed value r = 0.9. The red (solid) curve shows the real part and the blue (dashed) curve the imaginary part of κ. (The sign of the imaginary part is determined by the iε prescription.)
( 5 )
5, which recovers the conventional result. However, around the (in)stability threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 3 :
3The real part of the energy shift ∆Ẽ as function of µ for a fixed magnetic field strengthB. The red (solid) curve is obtained from Eq.(9) while the blue (dotted) from Eq. (12) and Eq. (5). The parameters are chosen r = 0.9 and B = 0.05. M * = M + m, the naive expansion breaks down, as can be seen from Fig. 3 where we plot the energy shift Eq. (9) compared to the result of the naive perturbative expansion: Eq. (12) with Eq. (5). It is clear that the two results are very different around the threshold which here is at µ = 0.1. The size of the region where the two results are different is proportional to the strength of the magnetic field.
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Figure 4 :
4The real part of the energy shift ∆Ẽ as function of the magnetic fieldB for a fixed µ. The parameters are chosen as r = 0.9. The red (solid) curve is obtained from Eq. (9) and the blue (long-dashed) from Eq. (12) with Eq. (5) for µ = 0.09. The brown (medium-dashed) curve is obtained from Eq. (9) while the purple (short-dashed) curve from Eq. (12) for µ = 0.11.
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Figure 5 :
5The condition Eq. (3) for the ∆-nucleon-pion system, | eB 2M ∆ (M ∆ −M N −mπ ) | ≪ 1, plotted for the range of fields used in [10]: |eB| = 0.00108/a 2 . . . 0.00864/a 2 , with 1/a = 2 GeV as function of the pion mass. Red (solid) curve corresponds to the stronger field and the blue (dashed) to the weaker field. The Delta-nucleon mass difference is taken to be constant: M∆ − MN = 0.293 GeV.
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.0
∆Ẽ [g 2
/(4π) 2
]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .
. G Couture, J N Ng, Z. Phys. C. 3565G. Couture and J.N. Ng, Z. Phys. C 35, 65 (1987).
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| []
|
[
"A Unified Finite Element Method for Fluid-Structure Interaction",
"A Unified Finite Element Method for Fluid-Structure Interaction"
]
| [
"Yongxing Wang \nSchool of Computing\nUniversity of Leeds\nLS2 9JTLeedsUK\n",
"Peter Jimack \nSchool of Computing\nUniversity of Leeds\nLS2 9JTLeedsUK\n",
"Mark Walkley \nSchool of Computing\nUniversity of Leeds\nLS2 9JTLeedsUK\n"
]
| [
"School of Computing\nUniversity of Leeds\nLS2 9JTLeedsUK",
"School of Computing\nUniversity of Leeds\nLS2 9JTLeedsUK",
"School of Computing\nUniversity of Leeds\nLS2 9JTLeedsUK"
]
| []
| In this article, we present a new unified finite element method (UFEM) for simulation of general Fluid-Structure interaction (FSI) which has the same generality and robustness as monolithic methods but is significantly more computationally efficient and easier to implement. Our proposed approach has similarities with classical immersed finite element methods (IFEMs), by approximating a single velocity and pressure field in the entire domain (i.e. occupied by fluid and solid) on a single mesh, but differs by treating the corrections due to the solid deformation on the left-hand side of the modified fluid flow equations (i.e. implicitly). The method is described in detail, followed by the presentation of multiple computational examples in order to validate it across a wide range of fluid and solid parameters and interactions. the two velocities on each interface node should be consistent. There are typically two methods to handle this: partitioned/segregated methods [2, 3] and 15 monolithic/fully-coupled methods[4,5,6]. The former solve the fluid and solid equations sequentially and iterate until the velocities become consistent at the interface. These are more straightforward to implement but can lack robustness and may fail to converge when there is a significant energy exchange between the fluid and solid [3]. The latter solve the fluid and solid equations simultane-20 ously and often use a Lagrange Multiplier to weakly enforce the continuity of velocity on the interface [6]. This has the advantage of achieving accurate and stable solutions, however the key computational challenge is to efficiently solve the large systems of nonlinear algebraic equations arising from the fully-coupled implicit discretization of the fluid and solid equations. Fitted mesh methods can 25 accurately model wide classes of FSI problems, however maintaining the quality of the mesh for large solid deformations usually requires a combination of arbitrary Lagrangian-Eulerian (ALE) mesh movement and partial or full remeshing [7]. These add to the computational expense and, when remeshing occurs, can lead to loss of conservation properties of the underlying discretization [8].30Unfitted mesh methods use two meshes to represent the fluid and solid separately and these do not generally conform to each other on the interface. In this case, the definition of the fluid problem may be extended to an augmented domain which includes the solid domain. Similarly to the fitted case, there are also two broad approaches to treat the solid domain: partitioned methods and 35 monolithic methods. On an unfitted mesh, there is no clear boundary for the solid problem, so it is not easy to enforce the boundary condition and solve the solid equation. A wide variety of schemes have been proposed to address this issue, including the Immersed Finite Element Method (IFEM)[9,10,11,12,13], the Fictitious Domain (FD) method [14,15,16,17], and the mortar approach 40 [16, 18]. The IFEM developed from the Immersed Boundary method first introduced by Peskin [19], and has had great success with applications in bioscience and biomedical fields. The classical IFEM does not solve solid equations at all. Instead, the solid equations are arranged on the right-hand side of the fluid equations as an FSI force, and these modified fluid equations are solved on the 45 augmented domain (occupied by fluid and solid). There is also the Modified IFEM [13], which solves the solid equations explicitly and iterates until convergence. Reference [14] presents a fractional scheme for a rigid body interacting with the fluid, whilst [15] introduces a fractional step scheme using Distributed Lagrange Multiplier (DLM)/FD for fluid/flexible-body interactions. In the case 50 of monolithic methods, [16] uses a FD/mortar approach to couple the fluid and structure, but the coupling is limited to a line (2D) representing the structure. Reference [18] uses a mortar approach to solve fluid interactions with deformable and rigid bodies, and [17] also solves a fully-coupled FSI system with hierarchical B-Spline grids. There are also other monolithic methods based on unfitted 65 solid are unified in one equation in which only one velocity variable is solved;(2) a range of solid materials, from the very soft to the very hard, may be considered in this one scheme.The term "semi-explicit" also has two components: (1) we linearize the solid constitutive model (an incompressible neo-Hookean model) explicitly using the 70 value from the last time step; (2) we couple the FSI interaction implicitly by arranging the solid information on the left-hand side of control equations.The main idea of UFEM is as follows. We first discretize the control equations in time, re-write the solid equation in the form of a fluid equation (using the velocity as a variable rather than the displacement) and re-write the solid 75 constitutive equation in the updated coordinate system. We then combine the fluid and solid equations and discretize them in an augmented domain. Finally the multi-physics problem is solved as a single field.The UFEM differs from the classical IFEM approach which puts all the solid model information from the last time step explicitly on the right-hand side of 80 the fluid equations. This typically requires the use of a very small time step to simulate the whole FSI system. This IFEM approach works satisfactorily when the solid behaves like a fluid, such as a very soft solid, but can lead to significant errors when the solid behaves quite differently from the fluid, such as a hard solid. The UFEM scheme includes the solid information on the left-hand side 85 and, as we will demonstrate, can simulate a wide range from very soft to very hard solids both accurately and efficiently.As noted above, monolithic methods strongly couple the fluid and solid models, and discretize them into one implicit nonlinear equation system at each time step. The unknowns include velocity, displacement and a Lagrangian multiplier 90 to enforce consistency of velocity on an interface (fitted mesh)[4,5,6]or in a solid domain (unfitted mesh)[16,17,18]. One may gain both a stable and an accurate solution from such fully-coupled schemes. However, it is clear that this strategy is very costly, especially for the unfitted mesh case, in which the so called mortar integrals are involved [18]. The UFEM only solves for velocity 95 as unknowns, which is cheaper, but does not lose stability or accuracy as shown by the numerical experiments reported in this paper.The following sections are organized as follows. In section 2, the control equations and boundary conditions for fluid-structure interactions are introduced; In section 3, the weak form of the FSI system is presented based on the 100 augmented fluid domain. In section 4, details of the linearization of the FSI equations are discussed and the numerical scheme is presented. In section 5, numerical examples are described to validate the proposed UFEM. | null | [
"https://arxiv.org/pdf/1608.04998v1.pdf"
]
| 13,417,394 | 1608.04998 | 784c0ccc94bb08e8aa066db3b86ece15180afc52 |
A Unified Finite Element Method for Fluid-Structure Interaction
Yongxing Wang
School of Computing
University of Leeds
LS2 9JTLeedsUK
Peter Jimack
School of Computing
University of Leeds
LS2 9JTLeedsUK
Mark Walkley
School of Computing
University of Leeds
LS2 9JTLeedsUK
A Unified Finite Element Method for Fluid-Structure Interaction
Fluid-Structure interactionFinite element methodImmersed finite element methodMonolithic methodUnified finite element method
In this article, we present a new unified finite element method (UFEM) for simulation of general Fluid-Structure interaction (FSI) which has the same generality and robustness as monolithic methods but is significantly more computationally efficient and easier to implement. Our proposed approach has similarities with classical immersed finite element methods (IFEMs), by approximating a single velocity and pressure field in the entire domain (i.e. occupied by fluid and solid) on a single mesh, but differs by treating the corrections due to the solid deformation on the left-hand side of the modified fluid flow equations (i.e. implicitly). The method is described in detail, followed by the presentation of multiple computational examples in order to validate it across a wide range of fluid and solid parameters and interactions. the two velocities on each interface node should be consistent. There are typically two methods to handle this: partitioned/segregated methods [2, 3] and 15 monolithic/fully-coupled methods[4,5,6]. The former solve the fluid and solid equations sequentially and iterate until the velocities become consistent at the interface. These are more straightforward to implement but can lack robustness and may fail to converge when there is a significant energy exchange between the fluid and solid [3]. The latter solve the fluid and solid equations simultane-20 ously and often use a Lagrange Multiplier to weakly enforce the continuity of velocity on the interface [6]. This has the advantage of achieving accurate and stable solutions, however the key computational challenge is to efficiently solve the large systems of nonlinear algebraic equations arising from the fully-coupled implicit discretization of the fluid and solid equations. Fitted mesh methods can 25 accurately model wide classes of FSI problems, however maintaining the quality of the mesh for large solid deformations usually requires a combination of arbitrary Lagrangian-Eulerian (ALE) mesh movement and partial or full remeshing [7]. These add to the computational expense and, when remeshing occurs, can lead to loss of conservation properties of the underlying discretization [8].30Unfitted mesh methods use two meshes to represent the fluid and solid separately and these do not generally conform to each other on the interface. In this case, the definition of the fluid problem may be extended to an augmented domain which includes the solid domain. Similarly to the fitted case, there are also two broad approaches to treat the solid domain: partitioned methods and 35 monolithic methods. On an unfitted mesh, there is no clear boundary for the solid problem, so it is not easy to enforce the boundary condition and solve the solid equation. A wide variety of schemes have been proposed to address this issue, including the Immersed Finite Element Method (IFEM)[9,10,11,12,13], the Fictitious Domain (FD) method [14,15,16,17], and the mortar approach 40 [16, 18]. The IFEM developed from the Immersed Boundary method first introduced by Peskin [19], and has had great success with applications in bioscience and biomedical fields. The classical IFEM does not solve solid equations at all. Instead, the solid equations are arranged on the right-hand side of the fluid equations as an FSI force, and these modified fluid equations are solved on the 45 augmented domain (occupied by fluid and solid). There is also the Modified IFEM [13], which solves the solid equations explicitly and iterates until convergence. Reference [14] presents a fractional scheme for a rigid body interacting with the fluid, whilst [15] introduces a fractional step scheme using Distributed Lagrange Multiplier (DLM)/FD for fluid/flexible-body interactions. In the case 50 of monolithic methods, [16] uses a FD/mortar approach to couple the fluid and structure, but the coupling is limited to a line (2D) representing the structure. Reference [18] uses a mortar approach to solve fluid interactions with deformable and rigid bodies, and [17] also solves a fully-coupled FSI system with hierarchical B-Spline grids. There are also other monolithic methods based on unfitted 65 solid are unified in one equation in which only one velocity variable is solved;(2) a range of solid materials, from the very soft to the very hard, may be considered in this one scheme.The term "semi-explicit" also has two components: (1) we linearize the solid constitutive model (an incompressible neo-Hookean model) explicitly using the 70 value from the last time step; (2) we couple the FSI interaction implicitly by arranging the solid information on the left-hand side of control equations.The main idea of UFEM is as follows. We first discretize the control equations in time, re-write the solid equation in the form of a fluid equation (using the velocity as a variable rather than the displacement) and re-write the solid 75 constitutive equation in the updated coordinate system. We then combine the fluid and solid equations and discretize them in an augmented domain. Finally the multi-physics problem is solved as a single field.The UFEM differs from the classical IFEM approach which puts all the solid model information from the last time step explicitly on the right-hand side of 80 the fluid equations. This typically requires the use of a very small time step to simulate the whole FSI system. This IFEM approach works satisfactorily when the solid behaves like a fluid, such as a very soft solid, but can lead to significant errors when the solid behaves quite differently from the fluid, such as a hard solid. The UFEM scheme includes the solid information on the left-hand side 85 and, as we will demonstrate, can simulate a wide range from very soft to very hard solids both accurately and efficiently.As noted above, monolithic methods strongly couple the fluid and solid models, and discretize them into one implicit nonlinear equation system at each time step. The unknowns include velocity, displacement and a Lagrangian multiplier 90 to enforce consistency of velocity on an interface (fitted mesh)[4,5,6]or in a solid domain (unfitted mesh)[16,17,18]. One may gain both a stable and an accurate solution from such fully-coupled schemes. However, it is clear that this strategy is very costly, especially for the unfitted mesh case, in which the so called mortar integrals are involved [18]. The UFEM only solves for velocity 95 as unknowns, which is cheaper, but does not lose stability or accuracy as shown by the numerical experiments reported in this paper.The following sections are organized as follows. In section 2, the control equations and boundary conditions for fluid-structure interactions are introduced; In section 3, the weak form of the FSI system is presented based on the 100 augmented fluid domain. In section 4, details of the linearization of the FSI equations are discussed and the numerical scheme is presented. In section 5, numerical examples are described to validate the proposed UFEM.
Introduction
Numerical simulation of fluid-structure interaction is a computational challenge because of its strong nonlinearity, especially when large deformation is considered. Based on how to couple the interaction between fluid and solid, existing numerical methods can be broadly categorized into two approaches: 5 partitioned/segregated methods and monolithic/fully-coupled methods. Similarly, based on how to handle the mesh, they can also be broadly categorized into two further approaches: fitted mesh/conforming methods and unfitted/nonconforming mesh methods [1].
A fitted mesh means that the fluid and solid meshes match each other at the interface, and the nodes on the interface are shared by both the fluid and the solid, which leads to the fact that each interface node has both a fluid velocity and a solid velocity (or displacement) defined on it. It is apparent that meshes [20,21]. It can be seen that the major methods based on unfitted meshes either avoid solving the solid equations (IFEM) or solve them with additional variables (two velocity fields and Lagrange multiplier) in the solid domain. However, physically, there is only one velocity field in the solid domain. In this article, we develop a semi-explicit Unified FEM (UFEM) approach which only solves one velocity variable in the whole/augmented domain. We shall use unfitted meshes to introduce our UFEM, although the methodology can also be applied to fitted meshes.
The word "unified" here has two meanings: (1) the equations for fluid and
Governing equations for FSI
In the following context, let
(u, v) Ω = Ω uvdΩ,(1)
where u and v are functions defined in domain Ω. 105 All subscripts, such as i, j, and k, represent spatial dimension. If they are repeated in one term (including the bracket defined in (1)), it implies summation over the spatial dimension; if they are not repeated, they take the value 1 and 2 for 2D, and 1 to 3 for 3D. All superscripts are used to distinguish fluid and solid (f and s respectively), distinguish different boundaries (Γ D and Γ N ) or In our model we assume an incompressible fluid governed by the following equations in Ω f as shown in Figure 1:
ρ f Du f i Dt − ∂σ f ij ∂x j = ρ f g i ,(2)∂u f j ∂x j = 0,(3)σ f ij = µ f ∂u f i ∂x j + ∂u f j ∂x i − p f δ ij = τ f ij − p f δ ij .(4)
We also assume an incompressible solid that is governed by the following equations in Ω s as shown in Figure 1:
ρ s Du s i Dt − ∂σ s ij ∂x j = ρ s g i ,(5)det (F) = 1,(6)σ s ij = µ s ∂x s i ∂X k ∂x s j ∂X k − δ ij − p s δ ij = τ s ij − p s δ ij .(7)
In the above τ f ij and τ s ij are the deviatoric stress of the fluid and solid re- 115 spectively, ρ f and ρ s are the density of the fluid and solid respectively, µ f is the fluid viscosity, and g i is the acceleration due to gravity. Note that (5)-(7) describe an incompressible neo-Hookean model that is based on [16] and is suitable for large displacements. In this model, µ s is the shear modulus and p s is the pressure of the solid (p f being the fluid pressure in (4)). We denote by x i 120 the current coordinates of the solid or fluid, and by X i the reference coordinates of the solid, whilst F = ∂xi ∂Xj is the deformation tensor of the solid and D Dt represents the total derivative of time.
On the interface boundary Γ s :
u f i = u s i ,(8)σ f ij n s j = σ s ij n s j ,(9)
where n s j denotes the component of outward pointing unit normal, see Figure 1. Dirichlet and Neumann boundary conditions may be imposed for the fluid:
125 u f i =ū i on Γ D ,(10)
σ f ij n j =h i on Γ N .
Finally, initial conditions are typically set as:
u f i t=0 = u s i | t=0 = 0,(12)
though they may differ from (12). Remark 1 Using Jacobi's formula [22]:
d dt det (F) = det (F) tr F −1 dF dt ,(13)we have d dt det (F) = det (F) ∂u s j ∂x j ,(14)
which, using (6), gives ∂u s j ∂x j = 0.
We choose that the reference configuration is the same as the initial configuration, so (6 ) also implies (6). In our UFEM model, the incompressibility constraint (6 ) will be used instead of (6).
Weak form of FSI equations
where R (u i ) = u i | Ω s is the restriction map.
Notice that p f and p s are not uniquely determined in (15). In fact, taking p f + c and p s + c instead of p f and p s respectively, the left-hand side of (15) does not change. This situation can be avoided by fixing the pressure at a selected point (P 0 ) or by imposing the following constraint [23]:
Ω f p f dΩ + Ω s p s dΩ = Ω pdΩ = 0.(16)
We shall use the former approach therefore define the trial space for pressure in Ω as:
L 2 0 (Ω) = p : p ∈ L 2 (Ω), p | P0 = 0 .
The weak form of the FSI system in the augmented domain Ω can now be 145 reformulated by rearranging equation (15) to yield the following formulation.
Find u i ∈ W and p ∈ L 2 0 (Ω) such that
ρ f Du i Dt , v i Ω + τ f ij , ∂v i ∂x j Ω − p, ∂v j ∂x j Ω − ∂u j ∂x j , q Ω + ρ s − ρ f Du i Dt , v i Ω s + τ s ij − τ f ij , ∂v i ∂x j Ω s = h i , v i Γ N + ρ f (g i , v i ) Ω + ρ s − ρ f (g i , v i ) Ω s ,(17)
∀v i ∈ W 0 and ∀q ∈ L 2 (Ω).
Remark 2
The fluid deviatoric stress τ f ij is generally far smaller than the solid deviatoric stress τ s ij , so we choose to neglect the fluid deviatoric stress τ f ij in Ω s in what follows. Note that the classical IFEM neglects the whole fluid 150 stress σ f ij when computing the FSI force [9]. An equivalent way of interpreting neglecting τ f ij in Ω s is to view the solid as being slightly visco-elastic, having the same viscosity as the fluid.
Remark 3 We treat the solid as a freely moving object in a fluid, so u s i , v s i ∈ H 1 (Ω s ) without any boundary constraints in the definition of W and 155 W 0 respectively. Physically, however, if part of solid boundary is fixed, this fixed boundary can also be regarded as a fixed fluid boundary and implemented as a zero velocity condition in the fluid domain, hence the solid still can be treated as if it were freely moving. Furthermore, the interface boundary condition (8) is automatically built into the solution because we use an augmented solution
160 space W which requires u i | Ω s = u s i .
Computational scheme
The integrals in equation (17) are carried out in two different domains as illustrated in Figure 1. We use an Eulerian mesh to represent Ω and an updated Lagrangian mesh to represent Ω s , therefore the total time derivatives in these two different domains have different expressions, i.e:
Du i Dt = ∂u i ∂t + u j ∂u i ∂x j in Ω,(18)
and
Du s i Dt = ∂u s i ∂t in Ω s .(19)
Standard FEM isoparametric interpolation may be used to transfer data between the two meshes. Firstly, based on the above two equations (18) and (19), we discretize (17) in time using a backward finite difference. Then omiting the superscript n + 1, showing the solution is at the end of the time step, for convenience, we obtain:
ρ f u i − u n i ∆t + u j ∂u i ∂x j , v i Ω + τ f ij , ∂v i ∂x j Ω − p, ∂v j ∂x j Ω − ∂u j ∂x j , q Ω + ρ s − ρ f u i − u n i ∆t , v i Ω s + τ s ij , ∂v i ∂x j Ω s = h i , v i Γ N + ρ f (g i , v i ) Ω + ρ s − ρ f (g i , v i ) Ω s .(20)
Using the splitting method of [24,Chapter 3], equation (20) can be expressed in the following two steps.
(1) Convection step:
ρ f u * i − u n i ∆t + u * j ∂u * i ∂x j , v i Ω = 0;(21)
(2) Diffusion step:
ρ f u i − u * i ∆t , v i Ω + τ f ij , ∂v i ∂x j Ω − p, ∂v j ∂x j Ω − ∂u j ∂x j , q Ω + ρ s − ρ f u i − u n i ∆t , v i Ω s + τ s ij , ∂v i ∂x j Ω s = h i , v i Γ N + ρ f (g i , v i ) Ω + ρ s − ρ f (g i , v i ) Ω s .(22)
The treatment of the above two steps is described separately in the following 165 subsections.
Linearization of the convection step
In this section, two methods are introduced to treat the convection equation: the implicit Least-squares method and the explict Taylor-Galerkin method, both of which can be used in the framework of our UFEM scheme. Some numerical 170 results for comparison between these two methods are discussed subsequently in section 5.
Implicit Least-squares method
It is possible to linearize (21) using the value of u i from the last time step:
u * j ∂u * i ∂x j ≈ u * j ∂u n i ∂x j + u n j ∂u * i ∂x j − u n j ∂u n i ∂x j .(23)
Substituting (23) into equation (21) gives,
u * i + ∆t u * j ∂u n i ∂x j + u n j ∂u * i ∂x j , v i Ω = u n i + ∆tu n j ∂u n i ∂x j , v i Ω .(24)
For the Least-squares method [25], we may choose the test function in the following form:
v i = L (w i ) = w i + ∆t w j ∂u n i ∂x j + u n j ∂w i ∂x j ,(25)
where w i ∈ W 0 . In such a case, the weak form of (21) is:
(L (u * i ) , L (w i )) Ω = u n i + ∆tu n j ∂u n i ∂x j , L (w i ) Ω .(26)
In our UFEM a standard biquadratic finite element space is used to discretize equation (26) directly, although other spaces could be used.
Explicit Taylor-Galerkin method
It is also possible to linearize equation (21) as:
u * i − u n i ∆t + 1 2 u n j ∂ ∂x j (u * i + u n i ) , v i Ω = 0,(27)
or
u * i − u n i ∆t + u n j ∂u n i ∂x j , v i Ω = 0.(28)
Re-write (28) as:
u * i = u n i − ∆tu n j ∂u n i ∂x j ,(29)
and substitute (29) into equation (27), we have
u * i − u n i ∆t + u n j ∂u n i ∂x j − ∆t 2 u n j ∂ ∂x j u n k ∂u n i ∂x k , v i Ω = 0.(30)
Notice that a second order derivative exists in the last equation. In practice, one does not need to calculate the second order derivative, instead, Integration by parts may be used to reduce the order:
∂ ∂x j u k ∂u i ∂x k , v i Ω = u k ∂u i ∂x k , v i Γ N − u k ∂u i ∂x k , ∂v i ∂x j Ω .(31)
The boundary integral in the last equation can be neglected if u i is the solution of the previous diffusion step, which means no convection exists on the boundary after the diffusion step. Using (31), equation (30) may be approximated as:
u * i − u n i ∆t + u n j ∂u n i ∂x j , v i Ω = − ∆t 2 u n k ∂u n i ∂x k , u n j ∂v i ∂x j Ω .(32)
At last the weak form of the Taylor-Galerkin method [24,Chapter 2] can be expressed, by rearranging the last equation, as:
(u * i , v i ) Ω = u n i − ∆tu n j ∂u n i ∂x j , v i Ω − ∆t 2 2 u n k ∂u n i ∂x k , u n j ∂v i ∂x j Ω .(33)
This Taylor-Galerkin method is explicit, however a small time step is usually needed to keep the scheme stable.
Linearization of the diffusion step
In both the above and the following context, the derivative ∂ ∂xi on the updated solid mesh is computed at the current known coordinates x n i , that is to say ∂ ∂xi = ∂ ∂x n i . Furthermore, τ s ij in equations (22), has a nonlinear relationship with x i , i.e.:
τ s ij = τ s ij n+1 = µ s ∂x n+1 i ∂X k ∂x n+1 j ∂X k − δ ij .(34)
Using a chain rule, the last equation can also be expressed as:
+ µ s ∂x n+1 i ∂x n k ∂x n k ∂X m ∂x n l ∂X m − δ kl ∂x n+1 j ∂x n l ,(36)
and then τ s ij n+1 can be expressed by the current coordinate x n i as follows:
τ s ij n+1 = µ s ∂x n+1 i ∂x n k ∂x n+1 j ∂x n k − δ ij + ∂x n+1 i ∂x n k (τ s kl ) n ∂x n+1 j ∂x n l .(37)Using x n+1 i − x n i = u n+1
i ∆t which is the displacement at the current step, the last equation can also be expressed as:
τ s ij n+1 = µ s ∆t ∂u n+1 i ∂x n j + ∂u n+1 j ∂x n i + ∆t ∂u n+1 i ∂x n k ∂u n+1 j ∂x n k + τ s ij n + ∆t 2 ∂u n+1 i ∂x n k (τ s kl ) n ∂u n+1 j ∂x n l + ∆t ∂u n+1 i ∂x n k τ s kj n + ∆t (τ s il ) n ∂u n+1 j ∂x n l .(38)
There are two nonlinear terms in the last equation. Using a Newton method, they can be linearized as follows.
∂u n+1 i ∂x n k ∂u n+1 j ∂x n k = ∂u n+1 i ∂x n k ∂u n j ∂x n k + ∂u n i ∂x n k ∂u n+1 j ∂x n k − ∂u n i ∂x n k ∂u n j ∂x n k(39)
and
∂u n+1 i ∂x n k (τ s kl ) n ∂u n+1 j ∂x n l = ∂u n+1 i ∂x n k (τ s kl ) n ∂u n j ∂x n l + ∂u n i ∂x n k (τ s kl ) n ∂u n+1 j ∂x n l − ∂u n i ∂x n k (τ s kl ) n ∂u n j ∂x n l .(40)
Substituting (38)-(40) into (22) and dropping off the superscripts n + 1 of u n+1 i for notation convenience, this may be expressed as:
ρ f u i − u * i ∆t , v i Ω + ρ s − ρ f u s i − (u s i ) n ∆t , v i Ω s + µ f ∂u i ∂x j + ∂u j ∂x i , ∂v i ∂x j Ω − p, ∂v j ∂x j Ω − ∂u j ∂x j , q Ω + µ s ∆t ∂u i ∂x j + ∂u j ∂x i + ∆t ∂u i ∂x k ∂u n j ∂x k + ∆t ∂u n i ∂x k ∂u j ∂x k , ∂v i ∂x j Ω s + ∆t 2 ∂u i ∂x k (τ s kl ) n ∂u n j ∂x l + ∂u n i ∂x k (τ s kl ) n ∂u j ∂x l , ∂v i ∂x j Ω s + ∆t ∂u i ∂x k τ s kj n + (τ s il ) n ∂u j ∂x l , ∂v i ∂x j Ω s = h i , v i Γ N + ρ f (g i , v i ) Ω + ρ s − ρ f (g i , v i ) Ω s + µ s ∆t 2 ∂u n i ∂x k ∂u n j ∂x k + ∆t 2 ∂u n i ∂x k (τ s kl ) n ∂u n j ∂x l − τ s ij n , ∂v i ∂x j Ω s .(41)
The spatial discretization of the above linearized weak form will be discussed 180 in the following section.
Discretization in space
In the 2D case, which is considered in the remainder of this paper, a standard Taylor-Hood element Q2Q1 (9-node biquadratic quadrilateral for velocity and 4-node bilinear quadrilateral for pressure) is used to discretize in space. We first discretize the domain Ω to get Ω h , then define finite dimensional subspaces of W and W 0 as follows.
The solution space for each component of velocity:
W h = u h i : u h i ∈ H 1h Ω h , R h u h i = u sh i , u h i Γ D =ū h i , whilst test space for velocity is W h 0 = v h i : v h i ∈ H 1h Ω h , R h v h i = v sh i , v h i Γ D = 0 .
We also discretize the domain Ω s to get Ω sh , and both the discretized trial space and test space on the solid domain are H 1h Ω sh based on the discussion of Remark 3.
190
The solution and test spaces for pressure are L 2h 0 Ω h and L 2h Ω h respectively, which represent the finite dimensional subspaces of L 2 0 (Ω) and L 2 (Ω), respectively, based on continuous piecewise bilinear functions. H 1h Ω h H 1h Ω sh represents the finite dimensional subspace of H 1 (Ω) H 1 (Ω s ) based upon continuous piecewise biquadratic functions. Then equation (41) can be discretized as:
ρ f u h i − u * h i ∆t , v h i Ω h + ρ s − ρ f u sh i − u sh i n ∆t , v sh i Ω sh + µ f ∂u h i ∂x j + ∂u h j ∂x i , ∂v h i ∂x j Ω h − p h , ∂v h j ∂x j Ω h − ∂u h j ∂x j , q h Ω h + µ s ∆t ∂u sh i ∂x j + ∂u sh j ∂x i + ∆t ∂u sh i ∂x k ∂u n j ∂x k + ∆t ∂u n i ∂x k ∂u sh j ∂x k , ∂v sh i ∂x j Ω sh + ∆t 2 ∂u sh i ∂x k (τ s kl ) n ∂u n j ∂x l + ∂u n i ∂x k (τ s kl ) n ∂u sh j ∂x l , ∂v sh i ∂x j Ω sh + ∆t ∂u sh i ∂x k τ s kj n + (τ s il ) n ∂u sh j ∂x l , ∂v sh i ∂x j Ω sh = h i , v h i Γ N h + ρ f g i , v h i Ω h + ρ s − ρ f g i , v sh i Ω sh + µ s ∆t 2 ∂u n i ∂x k ∂u n j ∂x k + ∆t 2 ∂u n i ∂x k (τ s kl ) n ∂u n j ∂x l − τ s ij n , ∂v sh i ∂x j Ω sh .(42)
Notice that in the continuous space W , we have the restriction map R (
u i ) = u i | Ω s = u s i , while in the discretized space W h , we use the standard FEM isopara- metric transformation R h to represent the map, i.e. u sh i = R h u h i ,(43)
where subscript i denotes the velocity components in each space dimension.
Let u i = ( u i1 , u i2 · · · u iN f ) T and u s i = ( u s i1 , u s i2 · · · u s iN s )
T denote the i th components of the nodal velocity vectors on the fluid and solid meshes respectively, and ϕ = (ϕ 1 , ϕ 2 · · · ϕ N f ) T and ϕ s = (ϕ s 1 , ϕ s 2 · · · ϕ s N s ) T denote the vector of velocity basis functions on the fluid and solid meshes respectively, where N f and N s are the number of nodes of fluid and solid mesh respectively. Then equation (43) can be expressed as:
u s ik ϕ s k = R h ( u ik ϕ k ) .(44)
The FEM isoparametric transformation defines R h from u i to u s i as follows:
u s ik = R h ( u il ) = u il R lk ,(45)
where R lk = ϕ l (x k ), x k (k = 1, 2 · · · N s ) is the current coordinate of the k th node on the solid mesh. Therefore,
u sh i = u s ik ϕ s k = u il R lk ϕ s k .(46)
For velocity test functions, we similarly have
v sh i = v s ik ϕ s k = v il R lk ϕ s k ,(47)
where
v i = ( v i1 , v i2 · · · v iN f )
T is an arbitrary nodal velocity (virtual velocity) vector on the fluid mesh, which satisfies the homogeneous Dirichlet boundary condition.
On the fluid mesh, velocity and pressure can also be expressed as follows:
u h i = u ik ϕ k ,(48)v h i = v ik ϕ k ,(49)p h = p k ψ k ,(50)q h = q k ψ k ,(51)where ψ = (ψ 1 , ψ 2 · · · ψ N p ) T is the vector of pressure basis functions, p = ( p 1 , p 2 , · · · p N p )
T is the nodal pressure vector, and q = ( q 1 , q 2 , · · · q N p ) T is an arbitrary nodal pressure vector. N p denotes the number of nodes on the fluid mesh at which only pressure is defined.
Substituting (46)-(51) into (42), we have ρ f u ik − u * ik ∆t ϕ k , v im ϕ m Ω h + ρ s − ρ f u il − u n il ∆t R lk ϕ s k , v ir R rm ϕ s m Ω sh + µ f u ik ∂ϕ k ∂x j + u jk ∂ϕ k ∂x i , v im ∂ϕ m ∂x j Ω h − p k ψ k , v jm ∂ϕ m ∂x j Ω h − u jk ∂ϕ k ∂x j , q m ψ m Ω h + µ s ∆t u il R lk ∂ϕ s k ∂x j + u jl R lk ∂ϕ s k ∂x i , v ir R rm ∂ϕ s m ∂x j Ω sh + µ s ∆t 2 u ia R ab ∂ϕ s b ∂x k ∂u n j ∂x k + u ja R ab ∂u n i ∂x k ∂ϕ s b ∂x k , v ir R rm ∂ϕ s m ∂x j Ω sh + ∆t 2 u ia R ab ∂ϕ s b ∂x k (τ s kl ) n ∂u n j ∂x l , v ir R rm ∂ϕ s m ∂x j Ω sh + ∆t 2 u ja R ab ∂u n i ∂x k (τ s kl ) n ∂ϕ s b ∂x l , v ir R rm ∂ϕ s m ∂x j Ω sh + ∆t u ia R ab ∂ϕ s b ∂x k τ s kj n + (τ s il ) n u ja R ab ∂ϕ s b ∂x l , v ir R rm ∂ϕ s m ∂x j Ω sh = h i , v im ϕ m Γ N h + ρ f (g i , v im ϕ m ) Ω h + ρ s − ρ f (g i , v ir R rm ϕ s m ) Ω sh + µ s ∆t 2 ∂u n i ∂x k ∂u n j ∂x k + ∆t 2 ∂u n i ∂x k (τ s kl ) n ∂u n j ∂x l − τ s ij n , v ir R rm ∂ϕ s m ∂x j Ω sh .(52)Let u = u T 1 , u T 2 T and v = v T 1 , v T 2
T , we then express (52) in the following
matrix form: v T M u − u * ∆t + v T D T M s D u − u n ∆t + v T K u + v T B p + q T B T u + v T D T K s D u = v T f + v T D T f s ,(53)or v T , q T A B B T 0 u p = v T , q T b 0 ,(54)
where
A = M/∆t + K + D T (M s /∆t + K s ) D and b = f + D T f s + M u * /∆t + D T M s D u n /∆t. The matrix M = ρ f M 11 M 22(55)
is the velocity mass matrix of the fluid, where
(M 11 ) km = (M 22 ) km = (ϕ k , ϕ m ) Ω h , k, m = 1, 2, · · · N f .
The matrix
M s = ρ s − ρ f M s 11 M s 22(56)
is the velocity mass matrix of the solid, where
(M s 11 ) km = (M s 22 ) km = (ϕ s k , ϕ s m ) Ω sh , (k, m = 1, 2, · · · N s ) .
K is the stiffness matrix of the fluid:
K = µ f K 11 K 12 K 21 K 22 ,(57)
where
(K 11 ) km = 2 ∂ϕ k ∂x 1 , ∂ϕ m ∂x 1 Ω h + ∂ϕ k ∂x 2 , ∂ϕ m ∂x 2 Ω h , (K 22 ) km = 2 ∂ϕ k ∂x 2 , ∂ϕ m ∂x 2 Ω h + ∂ϕ k ∂x 1 , ∂ϕ m ∂x 1 Ω h , (K 12 ) km = ∂ϕ k ∂x 1 , ∂ϕ m ∂x 2 Ω h , (K 21 ) km = (K 12 ) mk = ∂ϕ k ∂x 2 , ∂ϕ m ∂x 1 Ω h ,
and k, m = 1, 2, · · · N f . K s is the stiffness matrix of the solid:
K s = K s 11 K s 12 K s 21 K s 22 ,(58)
where
(K s 11 ) bm = µ s ∆t2 ∂ϕ s b ∂x 1 , ∂ϕ s m ∂x 1 Ω sh + µ s ∆t ∂ϕ s b ∂x 2 , ∂ϕ s m ∂x 2 Ω sh + 2µ s ∆t 2 ∂ϕ s b ∂x k ∂u n 1 ∂x k , ∂ϕ s m ∂x 1 Ω sh + µ s ∆t 2 ∂ϕ s b ∂x k ∂u n 2 ∂x k , ∂ϕ s m ∂x 2 Ω sh + 2∆t 2 ∂ϕ s b ∂x k (τ s kl ) n ∂u n 1 ∂x l , ∂ϕ s m ∂x 1 Ω sh + ∆t 2 ∂ϕ s b ∂x k (τ s kl ) n ∂u n 2 ∂x l , ∂ϕ s m ∂x 2 Ω sh + 2∆t ∂ϕ s b ∂x k (τ s k1 ) n , ∂ϕ s m ∂x 1 Ω sh + ∆t ∂ϕ s b ∂x k (τ s k2 ) n , ∂ϕ s m ∂x 2 Ω sh .
It can be seen from the pattern of the above matrices that one can get K s 22 by changing the subscript 1 to 2, and changing 2 to 1 in the formula of K s 11 .
Similarly, the elements of K s 12 can be expressed as:
(K s 12 ) bm = µ s ∆t ∂ϕ s b ∂x 1 , ∂ϕ s m ∂x 2 Ω sh + µ s ∆t 2 ∂u n 1 ∂x k ∂ϕ s b ∂x k , ∂ϕ s m ∂x 2 Ω sh + ∆t 2 ∂u n 1 ∂x k (τ s kl ) n ∂ϕ s b ∂x l , ∂ϕ s m ∂x 2 Ω sh + ∆t (τ s 1k ) n ∂ϕ s b ∂x k , ∂ϕ s m ∂x 2 Ω sh ,
and (K s 21 ) bm = (K s 12 ) mb , (b, m = 1, 2, · · · N s ). The matrix B has the following expression.
B = B 1 B 2 ,(59)
where
(B 1 ) mk = ψ k , ∂ϕ m ∂x 1 Ω h , (B 2 ) mk = ψ k , ∂ϕ m ∂x 2 Ω h (k = 1, 2, · · · N p and m = 1, 2, · · · N f . The vector f = f 1 f 2 (60)
is the fluid force vector, where
(f 1 ) m = ρ f (g 1 , ϕ m ) Ω h + h 1 , ϕ m Γ N h , and (f 2 ) m = ρ f (g 2 , ϕ m ) Ω h + h 2 , ϕ m Γ N h m = 1, 2, · · · N f . The vector f s = f s 1 f s 2(61)
is the solid force vector, where
(f s 1 ) m = ρ s − ρ f (g 1 , ϕ s m ) Ω sh + µ s ∆t 2 ∂u n 1 ∂x k ∂u n j ∂x k + ∆t 2 ∂u n 1 ∂x k (τ s kl ) n ∂u n j ∂x l − τ s 1j n , ∂ϕ s m ∂x j Ω sh and (f s 2 ) m = ρ s − ρ f (g 2 , ϕ s m ) Ω sh + µ s ∆t 2 ∂u n 2 ∂x k ∂u n j ∂x k + ∆t 2 ∂u n 2 ∂x k (τ s kl ) n ∂u n j ∂x l − τ s 2j n , ∂ϕ s m ∂x j Ω sh
(m = 1, 2, · · · N s ). Finally, matrix D is the FEM interpolation matrix which can be expressed as:
D = R T R T .(62)
Using the arbitrariness of our test vectors v and q, one can obtain the following linear algebraic equation for the whole FSI system from equation (54):
A B B T 0 u p = b 0 .(63)
The UFEM algorithm
Having derived a discrete system of equations we now describe the solution 200 algorithm at each time step.
(1) Given the solid configuration (x s ) n and velocity field u n =
u f n in Ω f ( u s ) n
in Ω s at time step n.
(2) Discretize the convection equation (26) or (33) and solve it to get an intermediate velocity u * .
205
(3) Compute the interpolation matrix and solve equation (63) using u * and ( u s ) n as initial values to get velocity field u n+1 .
(4) Compute solid velocity ( u s ) n+1 = D u n+1 and update the solid mesh by
(x s ) n+1 = (x s ) n + ∆t ( u s ) n+1
, then go to step (1) for the next time step.
Remark 4 When implementing the UFEM algorithm, it is unnecessary to 210 perform the matrix multiplication D T K s D in (53) globally, because the FEM interpolation is locally based. All the matrix operations can be computed based on the local element matrices only. Alternatively, if an iterative solver is used, it is actually unnecessary to compute D T K s D. What an iterative step needs is to compute D T K s D u for a given vector u, therefore one can compute Du 215 first, then K s (Du), and last D T (K s Du).
Numerical experiments
In this section, we present some numerical examples that have been selected to allow us to assess our proposed UFEM. We shall demonstrate the convergence of UFEM in time and space, and compare results obtained by the UFEM 220 with those obtained using monolithic approaches and IFEM, as well as compare against results from laboratory experiment.
In order to improve the computational efficiency, an adaptive spatial mesh with hanging nodes is used in all the following numerical experiments. Readers can reference Appendix A for details of the treatment of hanging nodes. 225
Oscillation of a flexible leaflet oriented across the flow direction
This numerical example is used by [15,16,17] to validate their methods. We first use the same parameters as used in the above three publications in order to compare results and test convergence in time and space, then use a range of parameters to show the robustness of our UFEM. The implicit Least-squares 230 method is used to treat the convection step in all these tests unless otherwise stated. The computational domain and boundary conditions are illustrated in Figure 2.
The inlet flow is in the x-direction and given by u x = 15.0y (2 − y) sin (2πt). Gravity is not considered in the first test (i.e. g = 0), and other fluid and solid 235 properties are presented in Table 1. The leaflet is approximated with 1200 linear triangles with 794 nodes (medium mesh size), and the corresponding fluid mesh is adaptive in the vicinity of the leaflet so that it has a similar size. A stable time step ∆t = 5.0 × 10 −4 s is used in these initial simulations. The configuration of the leaflet is illustrated 240 at different times in Figure 3.
Previously published numerical results are qualitatively similar to those in Figure 3 but show some quantitative variations. For example, [16] solved a fully-coupled system but the coupling is limited to a line, and the solid in their results (Figure 7 (l)) behaves as if it is slightly harder. Alternatively, [15] used 245 a fractional step scheme to solve the FSI equations combined with a penalty method to enforce the incompressibility condition. In their results (Fig. 3 (h)) the leaflet behaves as if it is slightly softer than [16] and harder than [17]. In [17] a beam formulation is used to describe the solid. The fluid mesh is locally refined using hierarchical B-Splines, and the FSI equation is solved monolithically. The 250 leaflet in their results (Fig. 34) behaves as softer than the other two considered here. Our results in Figure 3 are most similar to those of [17]. This may be seen more precisely by inspection of the graphs of the oscillatory motion of the leaflet tip in Figure 4 corresponding to Fig. 32 in [17]. We point out here that the explicit Taylor-Galerkin method is also used to solve the convection step 255 for this test, and we gain almost the same accuracy using the same time step ∆t = 5.0 × 10 −4 s.
Having validated our results for this example against the work of others, we shall use this test case to further explore more details of our method.
We commence by testing the influence of the ratio of fluid and solid mesh We next consider convergence tests undertaken for refinement of both the fluid and solid meshes with the fixed ratio of mesh sizes r m ≈ 3.0. Four different levels of meshes are used, the solid meshes are: coarse (584 linear triangles with 386 nodes), medium (1200 linear triangles with 794 nodes), fine (2560 275 linear triangles with 1445 nodes), and very fine (3780 linear triangles with 2085 nodes). The fluid meshes have the corresponding sizes with the solid at their maximum refinement level. As can be seen in Figure 6 and Table 2, the velocity is converging as the mesh becomes finer. Between different mesh sizes Difference of maximum horizontal velocity at t = 0.5s coarse and medium 0.01497 medium and fine 0.00214 fine and very fine 0.00190 Table 2: Comparison of maximum velocity for different meshes.
In addition, we consider tests of convergence in time using a fixed ratio of 280 fluid and solid mesh sizes r m ≈ 3.0. Using the medium solid mesh size and the same fluid mesh size as above, results are shown in Figure 7 and Table 3. From these it can be seen that the velocities are converging as the time step decreases.
Steps sizes compared Difference of maximum horizontal velocity at t = 0. Finally, in order to assess the robustness of our approach, we vary each of the physical parameters using three different cases as shown in Figure 8. A The dimensionless parameters shown in Figure 8 are defined as:
ρ r = ρ s ρ f ,μ s = µ s ρ f U 2 , Re = ρ f U H µ f
and F r = gH U 2 where the average velocity U = 10 in this example. T = 1 is the period of inlet flow.
It can be seen from the results of group (a) that the larger the value of 290 shear modulusμ s the harder the solid behaves, however a smaller time step is required. For the case ofμ s = 10 9 , the solid behaves almost like a rigid body, as we would expect. From results of group (b), it is clear that the Reynolds Number (Re) has a large influence on the behavior of the solid. The density and gravity have relatively less influence on the behavior of solid in this problem 295 which can be seen from the results of group (c) and group (d).
Oscillation of a flexible leaflet oriented along the flow direction
The following test problem that we consider is taken from [26], which describes an implementation on a ALE fitted mesh. It has since been used as a benchmark to validate different numerical schemes [17,18]. The geometry and 300 boundary conditions are shown in Figure 9. For the fluid, the viscosity and density are µ f = 1.82 × 10 −4 and ρ f = 1.18×10 −3 respectively. For the solid, we use shear modulus µ s = 9.2593×10 −5 and density ρ s = 0.1. The leaflet is divided by 1063 3-node linear triangles with 666 nodes, and the corresponding fluid mesh locally has a similar node density 305 to the leaflet (r m ≈ 3.0). First the Least-squares method is tested and a stable time step ∆t = 1.0 × 10 −3 s is used. A snapshot of the leaflet deformation Figure 11: Distribution of pressure across the leaflet on the three lines in Figure 10 and fluid pressure at t = 5.44s are illustrated in Figure 10. In Figure 11, the distributions of pressure across the leaflet corresponding to the three lines (AB, CD and EF) in Figure 10 (b) are plotted, from which we can observe that the 310 sharp jumps of pressure across the leaflet are captured.
The evolution of the vertical displacement of the leaflet tip with respect to time is plotted in Figure 12(a). Both the magnitude (1.34) and the frequency (2.94) have a good agreement with the result of [26], using a fitted ALE mesh and of [17], using a monolithic unfitted mesh approach. The Taylor-Galerkin 315 method is also tested which uses ∆t = 2.0 × 10 −4 s as a stable time step, and a corresponding result is shown in 12(b) which has a similar magnitude (1.24) and frequency (2.86). These results are all within the range of values in [17, Table 4]. Note that since the initial condition before oscillation for these simulations is an unstable equilibrium, the first perturbation from this regime is due to 320 numerical disturbances. Consequently, the initial transient regimes observed for the two methods (implicit Least-squares and explicit Taylor-Galerkin methods) are quite different. It is possible that an explicit method causes these numerical perturbations more easily, therefore makes the leaflet start to oscillate at an earlier stage than when using the implicit Least-squares approach. 325
Solid disc in a cavity flow
This numerical example is used to compare our UFEM with the IFEM, which is cited in [11,27]. In order to compare some details, we also implement the IFEM, but we implemented it on an adaptive mesh with hanging nodes, and we use the isoparametric FEM interpolation function rather than the discretized 330 delta function or RKPM function of [9,10].
The fluid's density and viscosity are 1 and 0.01 respectively, and the following solid properties are chosen to undertake the tests: ρ s =1 and µ s =0.1 or 1. The horizontal velocity on the top boundary of the cavity is prescribed as 1 and the vertical velocity is fixed to be 0 as shown in Figure 13. The velocities on the other three boundaries are all fixed to be 0, and pressure at the bottom-left point is fixed to be 0 as a reference point. In order to compare the UFEM and IFEM, we use the same meshes for fluid and solid: the solid mesh has 2381 nodes and the fluid mesh locally has a similar number of nodes (adaptive, see Figure 14). First the implicit Least-squares 340 method is used to solve the convection step, and the time step is ∆t = 1.0×10 −3 . Figure 15 and Figure 16 show the configuration of the disc at different stages, from which we do not observe significant differences of the velocity norm even for a long run as shown in Figure 16 (b). Then the explicit Taylor-Galerkin method is tested, and we achieve almost the same accuracy by using the same 345 time step. The magnitudes of velocity at the same stages of Figure 16 are presented in Figure 17.
We should mention that for the case µ s = 0.1, as the disc arrives at the top of the cavity (time > 3.0) the quality of the solid mesh does begin to deteriorate using our UFEM. We do not currently seek to improve the mesh 350 quality (using an arbitrary Lagrangian-Eulerian (ALE) update [7], for example) however this would be necessary in order to reduce the shear modulus further without compromising the quality of the solid mesh.
Conversely, a large µ s makes the solid behave like a rigid body. For the proposed UFEM, we can use µ s = 100 or larger without changing the time step, 355 whereas for the IFEM the simulation always breaks down for µ s = 100, however small the time step, due to the huge FSI force on the right-hand side of the FSI system.
Solids in a channel with gravity
We first simulate a falling disc due to gravity in order to further validate the 360 accuracy of the UFEM. We then show a simulation of the evolution of different shapes of solids falling and rising in a channel in order to show the flexibility and robustness of the proposed UFEM.
The test of a falling disc in a channel is cited by [10,18] in order to validate the IFEM and a monolithic method respectively. The computational domain 365 and parameters are illustrated in Figure 18 and Table 4 respectively. The fluid velocity is fixed to be 0 on all boundaries except the top one. There is also an empirical solution of a rigid ball falling in a viscous fluid [18], for which the terminal velocity, u t , under gravity is given by
u t = ρ s − ρ f gr 2 4µ f ln L r − 0.9157 + 1.7244 r L 2 − 1.7302 r L 4 ,(64)
where ρ s and ρ f are the density of solid and fluid respectively, µ f is viscosity of the fluid, g = 980 cm/ s 2 is acceleration due to gravity, L = W / 2 and r is the radius of the falling ball. We choose µ s = 10 8 dyne/ cm 2 to simulate a Table 4: Fluid and material properties of a falling disc.
rigid body here, and µ s = 10 12 dyne/ cm 2 is also applied, which gives virtually identical result. Three different meshes are used: the disc boundary is represented with 28 nodes (coarse), 48 nodes (medium), or 80 nodes (fine). The fluid mesh near the solid boundary has the same mesh size, and a stable time step t = 0.005s 375 is used for all the three cases. The Least-squares method is used to treat the convection step in all these tests. A local snapshot of the vertical velocity with the adaptive mesh is shown in Figure 20. From the fluid velocity pattern around the disc, we can observe that the disc behaves like a rigid body as expected. In addition, the evolution of the velocity of the mid-point of the disc is shown in 380 Figure 21, from which it can be seen that the numerical solution converges from below to the empirical solution.
Reference [18] uses a monolithic method to simulate multiple rigid and deformable discs in a gravity channel. We have implemented this example and obtain very similar results. Rather than replicate these here however, we in-385 stead show a more complex example, as illustrated in Figure 19. The computational domain, boundary conditions and the fluid properties are the same as the above one-disc test. All the solids are numbered at their initial positions as Table 5.
A high resolution of each solid boundary is used in this simulation as shown in Figure 22 (a), which can guarantee the mesh quality during the whole process 395 of evolution, and a stable time step t = 0.002s is used. Snapshots of the solids at different times are shown in Figure 22 and 23.
Conclusion and future works
In this article we introduce a new unified finite element method (UFEM) for fluid-structure interaction, which can be applied to a wide range of problems, methods solve solid equations, however UFEM solves one velocity field in the solid domain using FEM interpolation, while monolithic methods solve one velocity field and one displacement field in the solid domain using Lagrangian multipliers. In summary therefore we believe that UFEM has the potential to offer the robustness and range of operation of monolithic methods, but at a 415 computational cost that is much closer to that of the immersed finite element methods.
The following generalizations of our proposed UFEM approach will be considered in the future: (1) Implementation in 3D using adaptive mesh with hanging nodes; (2) implementation for non-Newtonian flow; (3) an efficient precon-420 ditioned iterative solver for the UFEM algebraic system; (4) a second order splitting scheme in time.
Appendix A. A method to treat hanging nodes
An adaptive mesh with hanging nodes reduces the number of degrees of freedom compared to uniform refinement, hence, decreases the cost of computation.
425
However, the nature of hanging nodes has the potential to cause discontinuity and breaks the framework of the finite element shape functions, which, therefore, needs special treatment in finite element codes.
In order to treat the hanging nodes, one can construct a conforming shape function [28,29] or constrain and cancel the degree of freedom at the hanging 430 nodes [29,30]. The former is very appealing and leads to optimal convergence, but it is difficult to extend to high-order shape functions [31]. In this article we will adopt the latter method and only use 2-level hanging nodes, which means at most 2 hanging nodes are allowed in one element (this can be guaranteed by imposing safety layers to ensure that neighbouring element nodes differ by more 435 than one level of refinement). The implementation of arbitrary-level hanging nodes can be found in [31,32,33]. For a quadrilateral element, when the velocity is interpolated by biquadratic shape functions and the pressure is interpolated by bilinear shape functions, the implementation of hanging nodes must be different for each, as shown in Figure ( A.1. For example, when velocity is interpolated, point D is a hanging node for element II, and point E is a hanging node for element III. When pressure is interpolated, point C is a hanging node for both the element II and III. Take element II for example, if we use the constraint method to cancel the hanging
p C = 1 2 p A + 1 2 p B (A.2)
where u i and p are velocity components and pressure respectively defined at the corresponding nodes. The interpolation coefficients can be calculated by putting edge AB in a one dimensional finite element reference coordinate system. Notice that when computing the element matrix II, point B is outside of the
u 1 i u 2 i u 3 i u 4 i u 5 i u 6 i u 7 i u 8 i u 9 i = D v u 1 i u 2 i u 3 i u 4 i u 5 i u B i u 7 i u 8 i u 9 i , D v = 1 1 1 1 1 3 4 3 8 − 1 8 1 1 1 . (A.3) p 1 p 2 p 3 p 4 = D p p 1 p B p 3 p 4 , D p = 1 1 2 1 2 1 1 . (A.4)
One should use matrices D v and D p to modify the element matrix II. Suppose K e is the stiffness matrix of element II without consideration of hanging nodes, and the unknowns are arranged in the following column vector. u 1 1 , u 2 1 , · · · u 9 1 , v 1 1 , v 2 1 · · · v 9 1 , p 1 , p 2 · · · p 4 T . (A.5)
It is clear that K e = [k ij ] is a n×n (n=22) matrix, and it could be modified by 460 the following pseudocode, which distribute the contribution of hanging nodes to the corresponding nodes according to formula (A.1).
for j=1 to n for j=1 to n for j=1 to n k i1j = k i1j + k i0j · 3/8 k ji1 = k ji1 + k ji0 · 3/8 k i0j = −k i0j /8 k i2j = k i2j + k i0j · 3/4 k ji2 = k ji2 + k ji0 · 3/4 k ji0 = −k ji0 /8 end end end
Let i 0 = 6, i 1 = 3, and i 2 = 2 (based on (A.5)), sequentially executing the above three pieces of codes would modify the matrix K e corresponding to the first component of velocity, and let i 0 = 15, i 1 = 12, and i 2 = 11 (based on 465 (A.5)), executing the above codes would modify the matrix K e corresponding to the second component of velocity. Similarly, in order to modify the matrix corresponding to pressure, one can execute the following codes which are based on formula (A.2): for j=1 to n for j=1 to n for j=1 to n k i1j = k i1j + k i2j /2 k ji1 = k ji1 + k ji2 /2 k i2j = −k i2j /2; k ji2 = −k ji2 /2 end end end where i 1 = 21 and i 2 = 20 based on (A.5). Executing all the above pieces of 470 codes is equivalent to performing the following matrix multiplication.
D v D v D p T K e D v D v D p . (A.6)
The modification of the mass matrix is similar but easier if a lumped mass is adopted, though it is unnecessary to present details here. Once the element matrix is modified, it can then be assembled directly to the global matrix and therefore implement the constraint of the hanging nodes, because the hanging 475 node shares the same equation number with its related node in the neigbouring element.
110 represent time step (n). For example, u f i and u s i denote the velocity components of fluid and solid respectively, σ f ij and σ s ij denote the stress tensor components of fluid and solid respectively, and (u s i ) n is a solid velocity component at time t n .
Figure 1 :
1Schematic diagram of FSI, Ω = Ω f ∪ Ω s , Γ = Γ D ∪ Γ N .
Figure 2 :
2Computational domain and boundary conditions, taken from[16].
= 100 kg/ m 3 ρ s = 100 kg/ m 3 µ f = 10 N · s/ m 2 µ s = 10 7 N / m 2
Figure 3 :
3Configuration of leaflet and magnitude of velocity on the adaptive fluid mesh.
Figure 4 :
4sizes r m =(local fluid element area)/(solid element area). Fixing the fluid mesh size, three different solid mesh sizes are chosen: coarse (640 linear triangles with Evolution of horizontal and vertical displacement at top right corner of the leaflet. 403 nodes r m ≈ 1.5), medium (1200 linear triangles with 794 nodes r m ≈ 3.0) and fine (2560 linear triangles with 1445 nodes r m ≈ 5.0), and a stable time step ∆t = 5.0 × 10 −4 s is used. From these tests we observe that there is a slight 265 difference in the solid configuration for different meshes, as illustrated at t = 0.6s inFigure 5, however the difference in displacement decreases as the solid mesh becomes finer. Further, we found that 1.5 ≤ r m ≤ 5.0 ensures the stability of the proposed UFEM approach. Note that we use a 9-node quadrilateral for the fluid velocity and 3-node triangle for solid velocity, so r m ≈ 3.0 means the fluid 270 and solid mesh locally have a similar number of nodes for velocity.
Figure 5 :
5Configuration of leaflet for different mesh ratio rm, and contour plots of displacement magnitude at t = 0.6s.
Figure 6 :
6Contour plots of horizontal velocity at t = 0.5s.
5s ∆t = 2.0 × 10 −3 and ∆t = 1.0 × 10 −3 0.00854 ∆t = 1.0 × 10 −3 and ∆t = 5.0 × 10 −4 0.00517 ∆t = 5.0 × 10 −4 and ∆t = 2.5 × 10 −4 0.00263
(c) ∆t = 5.0 × 10 −4 s.(d) ∆t = 2.5 × 10 −4 s.
Figure 7 :
7Contour plots of horizontal velocity at t = 0.5s. (a) ρ r = 1, Re = 100 and F r = 0. (b) ρ r = 1,μ s = 10 3 and F r = 0. (c) Re = 100,μ s = 10 3 and F r = 0. (d) Re = 100 andμ s = 10 3 .
Figure 8 :
8Parameters sets and results, ∆t = 5.0 × 10 −4 s for Group (b)∼(d).
Figure 9 :
9Computational domain and boundary condition for oscillation of flexible leaflet. (a) Leaflet displacement and fluid pressure. (b) Mesh refinement near the structure.
Figure 10 :
10Contour plots of leaflet displacement and fluid pressure.
Figure 11: Distribution of pressure across the leaflet on the three lines in Figure 10 (b).
Figure 12 :
12Displacement of leaflet tip as a function of time.
Figure 13 :
13Computational domain for cavity flow, taken from[27].
Figure 14 :
14Adaptive mesh for cavity flow.
Figure 15 :
15Velocity norm for a soft solid (µ s = 0.1) in a driven cavity flow using UFEM (left) and IFEM (right).
Figure 16 :
16Velocity norm for a soft solid (µ s = 1.0) in a driven cavity flow using UFEM (left) and IFEM (right), Least-squares method for convection step. (a) t = 5.0s (b) t = 25.0s
Figure 17 :
17Velocity norm for µ s = 1.0, Taylor-Galerkin method for convection step.
Figure 18 :
18Computational domain for a falling disc.
Figure 19 :
19Computational domain for different shapes of solids with different properties. Fluid Disc W = 2.0 cm d = 0.0125 cm H = 4.0 cm h = 0.5 cm ρ f = 1.0 g/ cm 3 ρ s = 1.2 g/ cm 3 µ f = 1.0 dyne · s/ cm 2 µ s = 10 8 dyne/ cm 2 g = 980 cm/ s 2 g = 980 cm/ s 2
Figure 20 :
20Contour of vertical velocity at t = 1s (fine mesh).
Figure 21 :
21Evolution of velocity at the center of a falling disc. (The blue solid line represents the empirical solution from formula (64).) shown in Figure 19 with A(0, −1), B(0.2, −1.2), C(−0.5, −1.1), D(−0.5, −1.5), E(−0.2, −1.3), F (−0.7, −2.9) and G(0, −3). The center and radius (r 1 ) of the 390 3 rd solid (circle) are (0, −2) and 0.2 respectively, and the center and radius (r 2 ) the 4 th solid (octagon) are (0.3, −2.7) and 0.2 respectively. The solid properties are illustrated in
Figure 22 :
22Contours of vertical velocity at different times.
Figure 23 :
23Contours of vertical velocity at different times (continued).
Figure A.1: Elements with hanging nodes.
Figure A. 2 :
2Element II in Figure A.1 in the reference coordinate system.
element, but the element matrix II still contributes to node B because of the hanging node D. So we can treat the two points, B and D, as a master-slave couple, which means letting them share the same equation number in the final global linear equation system. However one should modify the element matrix II according to (A.1) and (A.2) in the following way before assembling it to the 455 global matrix. Suppose the element II is enumerated in the reference coordinate system as shown in Figure A.2. Then, formulae (A.1) and (A.2) imply the following equations:
Table 1 :
1Properties and domain size for test problem with a leaflet oriented across the flow direction.
Table 3 :
3Comparison of maximum velocity for different time step size.
No. of solid Density g/ cm 3Shear modulus dyne/ cm 2
1
1.3
10 4
2
1.2
10 3
3
1.0
10
4
0.8
10 6
5
0.7
10 2
Table 5 :
5Properties for multi-solids falling in a channel as shown inFigure 19.from small deformation to very large deformation and from very soft solids through to very rigid solids. Several numerical examples, which are widely used in the literature of Immersed FEM and monolithic methods, are implemented to validate the proposed UFEM. The UFEM combines features from the IFEM and from monolithic methods. Nevertheless, it differs from each of them in the following aspects. Firstly, UFEM is a semi-explicit (explicitly linearizing the constitutive equation of solid and implicitly coupling FSI interaction) scheme, similar to IFEM, however UFEM solves the solid equations and fluid equations together while the classical IFEM does not solve the solid equations; secondly, both UFEM and monolithic410
medium mesh size with fixed r m ≈ 3.0 is used to undertake all of these tests.
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| []
|
[
"On a canonical construction of tesselated surfaces via finite group theory, Part I",
"On a canonical construction of tesselated surfaces via finite group theory, Part I"
]
| [
"Mark Herman ",
"Jonathan Pakianathan ",
"Ergün Yalçın "
]
| []
| []
| This paper is the first part in a 2 part study of an elementary functorial construction from the category of finite non-abelian groups to a category of singular compact, oriented 2-manifolds. After a desingularization process this construction results in a collection of compact, connected, oriented tesselated smooth surfaces equipped with a closed-cell structure which is face and edge transitive and which has at most 2 orbits of vertices. These tesselated surfaces can also be viewed as abstract 3-polytopes (or as graph embeddings in the corresponding surface) which are either equivar or dual to abstract quasiregular polytopes. This construction generally results in a large collection of tesselated surfaces per group, for example when the construction is applied to Σ6 it yields 4477 tesselated surfaces of 27 distinct genus and even more varieties of tesselation cell structure.We study the distribution of these surfaces in various groups and some interesting resulting tesselations. In a second paper, we show that extensions of groups result in branched coverings between the component surfaces in their decompositions. We also exploit functoriality to obtain interesting faithful, orientation preserving actions of subquotients of these groups and their automorphism groups on these surfaces and in the corresponding mapping class groups. | null | [
"https://arxiv.org/pdf/1310.3848v1.pdf"
]
| 119,148,224 | 1310.3848 | 5cbbeed862593ee84efbbe56163f6adfdb2f71b7 |
On a canonical construction of tesselated surfaces via finite group theory, Part I
14 Oct 2013 October 16, 2013
Mark Herman
Jonathan Pakianathan
Ergün Yalçın
On a canonical construction of tesselated surfaces via finite group theory, Part I
14 Oct 2013 October 16, 2013Riemann surface tesselationsregular graph mapsstrong sym- metric genus 2010 Mathematics Subject Classification Primary: 20D99; Secondary: 05B4555M9957M20
This paper is the first part in a 2 part study of an elementary functorial construction from the category of finite non-abelian groups to a category of singular compact, oriented 2-manifolds. After a desingularization process this construction results in a collection of compact, connected, oriented tesselated smooth surfaces equipped with a closed-cell structure which is face and edge transitive and which has at most 2 orbits of vertices. These tesselated surfaces can also be viewed as abstract 3-polytopes (or as graph embeddings in the corresponding surface) which are either equivar or dual to abstract quasiregular polytopes. This construction generally results in a large collection of tesselated surfaces per group, for example when the construction is applied to Σ6 it yields 4477 tesselated surfaces of 27 distinct genus and even more varieties of tesselation cell structure.We study the distribution of these surfaces in various groups and some interesting resulting tesselations. In a second paper, we show that extensions of groups result in branched coverings between the component surfaces in their decompositions. We also exploit functoriality to obtain interesting faithful, orientation preserving actions of subquotients of these groups and their automorphism groups on these surfaces and in the corresponding mapping class groups.
Introduction
There is a long tradition in the area of algebraic combinatorics of associating geometries or simplicial complexes to algebraic objects such as groups as a means of studying them. In group cohomology, various combinatorial complexes associated to groups such as the Quillen complex and the Brown complex have proven very useful tools. (See [B], [Be])
In this paper we explore such a construction which takes a finite nonabelian group G and constructs a 2-dimensional simplicial complex X(G) in a canonical manner from it. This complex turns out to have a nice geometric structure as a union of finitely many 2-dimensional pseudomanifolds which pairwise intersect in finitely many points. The pseudomanifolds are compact, connected and oriented and hence surfaces with at most singularities due to self-pointintersections, for example a sphere with 3 points identified or a torus with three meridian circles squashed to points. Here the construction X(G) is canonical in the sense it is part of a covariant functor from the category of nonabelian finite groups to the category of compact, connected, oriented, 2-dimensional triangulated manifolds with "singularities" of a type described precisely in Section 2 in detail.
These singularities can be resolved in a functorial way to yield an associated complex Y (G) which is a disjoint union of finitely many compact, connected, oriented, triangulated 2-manifolds (we'll call these Riemann surfaces in this paper for brevity at the expense of slight abuse of notation as we will not be using complex structure at all.) The manifolds in Y (G) hence form a natural set of invariants for the group. More specifically the number of times the surface of genus g occurs in Y (G) is a natural invariant of the group which we study. Furthermore the original complex X(G) can be shown to be homotopy equivalent to a bouquet of these manifolds with some circles due to the connections between them.
Additionally, the Riemann surfaces arising via this functorial construction are equipped with interesting closed-cell structures naturally associated to their triangulations and indeed we show that a slight functorial modification of the triangulations of the components yield closed-cell structures where all faces consist of a given type of polygonal face. We show these satisfy the conditions of what is called an abstract 3-polytope in the combinatorics literature. Any of these component polytopes is shown to have a face and edge transitive automorphism group and at most 2 orbits of vertices (and hence two types of vertex valency). We identify group theoretic conditions for a component polytope to be equivar (have a single valency) and identify the Schläfli symbol in this case. We also identify conditions for when the component forms a regular abstract 3-polytope such as a Platonic solid. In this case the component can also be considered as a regular map, i.e., a regular embedding of a graph in the underlying surface. For example this is the case when G is the extraspecial p-group of order p 3 where p is an odd prime which results in a regular tesselation of the surface of genus g = p(p−3) 2 + 1 as the union of p 2p-gons.
Constructions that yield such tesselations on Riemann surfaces are not new. There are classical constructions using triangle or Fuchsian groups, hyperbolic geometric tesselations of the Poincare disk, Cayley graphs or fundamental group arguments to construct similar objects (see [Cox], [Cox2] for example). Often a set of generators or specific presentation is chosen -our construction seems like an elementary variant, constructed in an algorithmic and functorial way from any nonabelian group. The initial construction uses the group itself in a very direct straightforward manner with no reference to specific presentation or external geometry -it clearly displays these tesselations as intrinsic to the group itself.
On the other hand, since X(G) and Y (G) are determined functorially from G, one can show that any automorphism of G determines an orientation preserving, simplicial automorphism of X(G). These simplicial automorphisms can permute the surface "components" of Y (G) but must preserve the genus and indeed the tessellation type. Thus, one can use this construction to find faithful actions of subquotients of Aut(G) on various surfaces and hence obtain embeddings of these subquotients into the group of orientation preserving diffeomorphisms of X g , the surface of genus g. In the second paper [HP], we explore some examples of orientation preserving group actions afforded by this construction and some corresponding embeddings into mapping class groups. We also discuss some examples pertaining to the strong symmetric genus (see [TT], [MZ1]) of the group, i.e. the smallest genus orientable closed surface for which the group acts on faithfully via orientation preserving diffeomorphisms. See [HP] for details.
We attempt to quantify the number and type of surface components that occur in the decomposition Y (G) for a given group G. For example when G is the alternating group on 7 letters, the construction Y (G) discussed in this paper results in 16813 surface components of 58 distinct genus, with even more distinct cell-structures. Tables summarizing the various polytopes that occur in Y (G) for various groups G can be found in the appendix of [HP].
Finally in [HP], we show that extensions of groups yield branched coverings between their constituent surfaces in their decompositions.
The complex X(G)
Let G be a finite nonabelian group and let Z(G) be its center. We will now describe the construction of X(G) in detail -the reader is encouraged to draw pictures to help with the definitions in this section as at first they seem elaborate though with some study, the nice structure of the complex will emerge. We will use some basic notation from the world of simplicial complexes -for a good reference on the terminology, see [Mu].
Before we construct the simplicial complex X(G), we construct an associated subgraph G 1 = (V 1 , E 1 ). The vertices of the graph G 1 are given by the noncentral elements of G, i.e., V 1 = G − Z(G) and are labelled as type 1 vertices, thus a typical vertex is labelled (x, 1) where x ∈ G − Z(G). We declare [(x, 1), (y, 1)] to be an edge in the simplicial graph G 1 if and only if x and y do not commute in G, i.e., xy = yx.
This graph is the 1-skeleton of the simply connected complex BN C(G) considered in [PY] and hence is path connected but we provide a simple proof of this here:
Lemma 2.1. G 1 = (V 1 , E 1 ) is a path connected graph.
Proof. Let (x, 1), (y, 1) be two vertices in G 1 . Then x, y are not central in G so their centralizer subgroups C(x) and C(y) are proper subgroups of G and hence have at most half the elements of G in them. Thus |C(x) ∪ C(y)| < |G| as C(x), C(y) each have at most half the elements of G in them and they have the identity element in common. Thus there must exist z / ∈ C(x) ∪ C(y) which means there is a path of length two from (x, 1) to (y, 1) in G 1 through (z, 1) as z does not commute with either x or y. Since (x, 1), (y, 1) were arbitrary vertices of the graph G 1 , we conclude that G 1 is path connected.
We now extend the graph G 1 to get the simplicial complex X(G). First we include a second set of vertices V 2 which also consists of the noncentral elements of G but with label 2, written as (x, 2), for x ∈ G − Z(G). Thus the vertex set of X(G) consists of the disjoint union of V 1 and V 2 i.e. two copies of the non central elements of G. To describe a simplicial complex, it is enough to describe its maximal faces.
The maximal faces of X(G) are of the form [(x, 1), (y, 1), (xy, 2)] where x and y do not commute -we orient these 2-simplices in the order indicated also. Note that the three vertices x, y, xy of this face pairwise do not commute in G.
Notice every edge in the graph G 1 lies in two distinct faces of X(G) i.e., [(x, 1), (y, 1), (xy, 2)] and [(y, 1), (x, 1), (yx, 2)]. Also note that the orientations on these two faces cancel along this edge. Furthermore if α is a noncentral element, there must be an element x which does not commute with it by definition. Then y = x −1 α does not commute with x and [(x, 1), (y, 1), (α, 2)] is a face of X(G). Thus every type 2 vertex does lie in a face also. Notice from this argument that for any α, x which do not commute, the edge [(x, 1), (α, 2)] is part of the complex. Furthermore such an edge lies again in exactly two faces [(x, 1), (x −1 α, 1), (α, 2)] and [(αx −1 , 1), (x, 1), (α, 2)] whose orientations cancel along that edge. To summarize, we have defined a 2-dimensional simplicial complex X(G) whose vertex set V is the disjoint union of V 1 and V 2 where
V 1 = {(x, 1)|x ∈ G − Z(G)} and V 2 = {(x, 2)|x ∈ G − Z(G)}
and whose edges are of the form [(x, 1), (y, 1)] and [(x, 1), (y, 2)] where x and y do not commute. Finally the (oriented) faces in the complex are of the form [(x, 1), (y, 1), (xy, 2)] and every edge lies in exactly two faces whose orientations cancel along that edge. Note in particular every face of X(G) shares an edge with the graph G 1 considered earlier.
Finally note that X(G) is path-connected as any point in X(G) lies in a face and any point in a face can be connected by a straight line path to a point on the graph G 1 . As we have seen that the graph G 1 is path-connected, it follows that any two points in X(G) are connected by a path.
We summarize the properties of X(G) here that will be relevant to the rest of the analysis of this complex in this section. We postpone the proof of the functoriality of this construction till later.
Proposition 2.2. Let G be a finite nonabelian group and Z(G) be its center. The complex X(G) is a compact, 2-dimensional simplicial complex such that:
(1) Every edge lies in exactly two faces.
(2) The faces can be oriented so that along any edge, the two orientations of the adjacent faces cancel.
(3) It is path-connected, thus in particular every vertex lies in an edge.
We will now show that any simplicial complex that satisfies the criteria of Proposition 2.2 like X(G) does is a union of finitely many compact, connected, oriented, pseudomanifolds who pairwise intersect in a finite collection of vertices.
Pseudomanifolds
Recall (see [Mu]) that an n-dimensional pseudomanifold is a n-dimensional simplicial complex X which satisfies:
(1) X is the union of its n-simplices.
(2) Every n − 1 simplex in X lies in exactly two n-simplices.
(3) For any two n-simplices σ 0 , σ k in X there is a sequence of n-simplices σ 1 , . . . , σ k−1 such that σ i and σ i+1 share a common (n−1)-face for 0 ≤ i ≤ k−1. Furthermore a pseudomanifold is orientable if one can orient all the n-simplices in such a way that for any n − 1 simplex τ in X, the two orientations from the two adjacent n-simplices cancel along τ . Note that conditions (1) and (3) imply that all pseudomanifolds are pathconnected.
It is well known that every path connected, triangulated topological nmanifold is a n-dimensional pseudomanifold. However pseudomanifolds also allow certain types of singularites which we will talk about below.
Proposition 2.3. Let X be a simplicial complex satisfying the conditions of Proposition 2.2 such as X(G). Define an equivalence relation on the 2-faces of X by saying σ 0 and σ k are equivalent if there is a sequence of 2-faces σ 1 , . . . , σ k−1 such that σ i and σ i+1 share a common edge for 0 ≤ i ≤ k − 1.
The union of faces in an equivalence class then gives an oriented compact 2-dimensional pseudomanifold. Furthermore X has finitely many equivalence classes and so is a union of finitely many oriented, compact, pseudomanifolds any two of which can meet only in a finite set of vertices.
Proof. As every vertex of X lies in an edge and every edge lies in a face (2simplex), it is clear that X is the union of its 2-simplices. The relation described on these faces is easily checked to be an equivalence relation and the equivalence classes partition the 2-simplices of X. As every edge of X lies in exactly two faces and faces can be oriented so that orientations cancel along the edges in common, it is clear that the union of the 2-simplices in one of the equivalence classes forms a 2-dimensional oriented pseudomanifold. It consists of finitely many simplices and is compact as X is. Thus X is a union of finitely many compact, oriented, 2-dimensional pseudomanifolds as claimed. If Y and Z are two of these, the intersection Y ∩ Z is a subcomplex. If this complex contained any edge, it would bound two faces, one of which has to be in Y and one which has to be in Z, and these faces would be equivalent contradicting the construction of the pseudomanifolds Y and Z as unions from distinct equivalence classes of faces. Thus the intersection contains no edges and hence no faces and so consists of at most a finite collection of vertices as claimed.
Note from the proof above, it is clear that the pseudomanifolds whose union is X(G) can be algorithmically determined and hence are unique. We will call them the pseudomanifold components of X(G).
The following three examples are examples of simplicial complexes satisfying the hypothesis of Proposition 2.2 and give a good indication of the type of spaces that can be X(G):
(1) A compact, connected, oriented, triangulated 2-manifold. Here there is a single pseudomanifold component which happens to be a manifold. We will call these examples, with slight abuse of notation, "Riemann surfaces" -in [HP], we show that our constructions inherit unique complex structures compatible with functoriality so this abuse is justifiable.
(2) Any compact, oriented 2-dimensional pseudomanifold. For example a sphere with 3 points identified to a single point. This can be triangulated so that it is a compact, oriented pseudomanifold and is a typical example of such. Every compact, oriented 2-dimensional pseudomanifold can be obtained from a triangulated Riemann surface by a finite number of vertex identifications. We will see this soon as a byproduct of other analysis. Thus basically in dimension 2 an oriented compact pseudomanifold is nearly an oriented compact connected manifold aside from some point self-intersections.
(3) Take a torus and squash three meridian circles to three distinct points. This space can be triangulated so that it satisfies the hypothesis of Proposition 2.2. It has three pseudomanifold components which are spheres and any two of these pseudomanifold components intersect in a single vertex but no point lies in all three pseudomanifold components.
Let X be any simplicial complex satisfying the conditions of Proposition 2.2. Notice that if x is a point in the interior of a face (2-simplex) then it is clear it has an open neighborhood homeomorphic to the unit open ball of R 2 . If x is a point on an edge other than its two end points, then as every edge lies in exactly two faces, it is again clear that x has an open neighborhood homeomorphic to the unit open ball of R 2 .
Thus the only points of X that might not have open neighborhoods homeomorphic to open subsets of R 2 are the vertices. In particular X−{Vertex set of X} is a 2-dimensional oriented topological manifold whose connected components, we will see later, are punctured Riemann surfaces. The next proposition shows that the connected components of X −{ vertex set of X} are in natural bijective correspondence with the pseudomanifold components of X.
Proposition 2.4. Let X be a simplicial complex satisfying the conditions of Proposition 2.2 and let Y = X − {Vertex set of X} be given the subspace topology. Then Y is a 2-dimensional oriented topological manifold and the connected components of Y are in natural bijective correspondence with the pseudomanifold components of X. More precisely y 1 , y 2 ∈ Y lie in the same component of Y if and only if they lie in the same pseudomanifold component of X.
Proof. It is clear from the preceeding paragraph that Y is an oriented 2-dimensional topological manifold. Fix y 1 , y 2 in Y . As X was the union of its pseudomanifold components X 1 , . . . X m which pairwise intersected in at most finitely many vertex points, we see that Y is the disjoint union of X 1 − V, . . . , X m − V where V is the vertex set of X. As each X i is closed in X, each X i − V is closed in Y = X − V and hence also open as there are a finite number of pseudomanifold components. Thus if y 1 , y 2 lie in distinct pseudomanifold components of X they must lie in distinct connected components of Y . Conversely, suppose that y 1 , y 2 lie in the same pseudomanifold component X j of X. Then there must be a face σ 0 and a face σ k both in X j joined by a sequence of faces σ 1 , . . . , σ k−1 in X j such that y 1 ∈ σ 0 and y 2 ∈ σ k and such that σ i , σ i+1 share a common edge for 0 ≤ i ≤ k − 1. As y 1 , y 2 are not vertices, it is clear we can construct a path along this sequence of faces which avoids vertices and joins y 1 , y 2 . Thus y 1 , y 2 lie in the same path component and hence component of Y . This concludes the proof.
Closed stars of vertices
Throughout this section, we will be concerned with the structure of the closed star of vertices in a simplicial complex X which satisfies the conditions of Proposition 2.2. Throughout this section X is such a complex.
Definition 2.5 (m-stars). Let m ≥ 3. Let V = {0} ∪ {e 2πik/m |0 ≤ k < m, k an integer } be the vertex set containing zero and the mth roots of unity viewed within the complex plane. Consider the collection of edges E obtained by joining 0 to the m, mth roots of unity. This simplicial graph G m = (V, E) will be called the m-star graph. Note it is star-convex so as a space, it is contractible to its middle point 0.
Definition 2.6 (m-disks). Let m ≥ 3. Let V be the vertex set containing 0 and the mth roots of unity viewed within the complex plane. Let D m be the convex hull of V (i.e, a m-gon) triangulated using its m boundary edges and the m edges joining 0 to the mth roots of unity as edge set. D m has m faces (2-simplices) consisting of the m triangles cut out by the edges mentioned above. D m is homeomorphic to the standard closed unit disk in the complex plane but triangulated by this specific triangulation. We will call such a triangulated disk, an m-disk or a disk of type m in this paper. 0 is called the center of this m-disk. The boundary of a m-disk is a triangulated circle which we will call an m-circle.
Definition 2.7 ((m 1 , m 2 , . . . , m k )-disk bouquet). Let m 1 , . . . , m k be integers ≥ 3. A simplicial complex T is called a (m 1 , m 2 , . . . , m k )-disk bouquet if it is simplicially isomorphic to the simplicial complex obtained by taking k disjoint disks of types m 1 , . . . , m k respectively and identifying their centers to a common center vertex. The individual disks in a disk bouquet are called the sheets of the bouquet.
We have seen neighborhoods of nonvertex points of X look like open disks in R 2 , we are now ready to describe neighborhoods of vertices in X, including their simplicial structure.
Proposition 2.8 (Closed stars of vertices). Let X be a simplicial complex satisfying the conditions of Proposition 2.2 and let v be a vertex of X. Then there exist integers k ≥ 1, m 1 , . . . , m k ≥ 3 such that the closed star of v,St(v) is a (m 1 , m 2 , . . . , m k )-disk bouquet. The link of v, Lk(v) is a disjoint union of k circles, of simplicial types m i , 1 ≤ i ≤ k.
Proof. The vertex v is contained in a face σ 1 . Let E be the edge opposite v in this face. Note that by the structure of simplicial complexes, no other face can contain both v and E or it would have to be equal to σ 1 . Now take an edge of σ 1 that contains v. As every edge lies in exactly two faces, there is a unique face σ 2 that shares this edge with σ 1 and is not σ 1 . Note v is contained in this face also. Repeating the argument we can find a sequence of faces, each containing v of the form σ 1 , . . . , σ k where σ i and σ i+1 share an edge for 1 ≤ i ≤ k − 1. Eventually by finiteness and as every edge lies in exactly two faces, we must have σ k share an edge with the first face σ 1 for some k = m 1 . The union of faces σ 1 , . . . , σ m1 hence forms a m 1 -disk centered at v. If this exhausts all faces in X we are done. If not pick another unused face containing v and proceed to find a m 2 disk centered at v using a similar process. Proceed in this way till one has found k disks all centered at v and there are no unused faces containing v. (This must occur eventually as there are finite number of faces in X).
Note that as no two faces which have v as a vertex, can intersect in an edge opposite v, and all the edges adjacent to v can only be in two faces which are part of the same disk, it is clear that the disks obtained are disjoint except for the common vertex v. Thus the union of all faces containing v, i.e. the closed star of v, forms a (m 1 , m 2 , . . . , m k )-disk bouquet. m i ≥ 3 for all i by general structure conditions of simplicial complexes.
The statement on links follows immediately from this so we are done.
Corollary 2.9. Let X be a simplicial complex satisfying the conditions of Proposition 2.2 and let Y = X − {Vertex set of X}. Then the components of Y are punctured Riemann surfaces. We may fill in each distinct puncture of Y with a distinct point to obtain a disjoint union of finitely many Riemann surfaces which are in bijective correspondence with the pseudomanifold components of X.
Proof. We already know that Y is an oriented 2-dimensional manifold with finitely many connected components in bijective correspondence with the pseudomanifold components of X. Note from Proposition 2.8, we see that there is open neighborhood of each puncture homeomorphic to a punctured disk. From this it is easy to see that if the punctures are filled in with distinct points, we will obtain components which are connected, oriented 2-manifolds with finite triangulations (and hence compact). Adding these distinct points do not change connected components so we result in a finite disjoint union of Riemann surfaces, which are in bijective correspondence with the pseudomanifold components of X.
Definition 2.10. Let G be a finite nonabelian group. Then X(G) satisfies the conditions of Proposition 2.2 and so we may remove its vertices and fill in the resulting punctures with distinct points. The resulting space will be denoted by Y (G) and is a disjoint union of Riemann surfaces which are in bijective correspondence with the pseudomanifold components of X(G). We will see later that Y (G) can be triangulated in a way that preserves the basic functoriality of the construction. As a Riemann surface is determined up to homeomorphism by its genus g which is a nonnegative integer, the quantity m g (G), the number or Riemann surfaces of genus g that occur in the complex Y (G) is a natural invariant of the group G which we will study later.
Y (G) will be called the desingularization of X(G).
Main Structure Theorem
In this subsection, we will prove the main structure theorem for the complex X(G).
Theorem 2.11. Let G be a finite nonabelian group. Let m g (G) denote the number of surfaces of genus g that occur in Y (G), the desingularization of X(G). Then X(G) is homotopy equivalent to a wedge product (bouquet) of Riemann surfaces and a finite number of circles where the surface of genus g occurs m g (G) times in the wedge product. In other words,
X(G) ≃ ( g≥0 X [mg] g ) (S 1 ∨ · · · ∨ S 1 ) where X [mg] g
is the m g = m g (G)-fold bouquet of the Riemann surface X g of genus g with itself.
Proof. Let Z(G) be the simplicial complex obtained as follows: First take out all the vertices from X(G) resulting in a punctured oriented topological 2-manifold. Label each puncture point by the vertex in X(G) it resulted from (note if the closed star of v in X(G) is a bouquet of k disks, there will be k punctures with label v). Now for each vertex v in X(G) which lies in k sheets, i.e. whose closed star is a bouquet of k disks, take a disjoint k-star graph and identify its ends with the k points in the desingularization Y (G) filling in the punctures labelled by v. The result is a path connected space consisting of the Riemann surfaces in Y (G) connected by a finite number of star graphs (any two of which are disjoint from each other), one for each vertex in X(G). This is the space Z(G).
Now note that if one collapses all the star graphs in Z(G) to their central points, one obtains a homotopy equivalence Z(G) ≃ X(G). Now Z(G) consists of the finite set of Riemann surfaces X 1 , . . . , X m of Y (G) connected together with various star graphs. Note that any edge in the star graph can be thought of as obtained by adjunction of the unit interval via a gluing map on its two boundary points. By basic facts on adjunction spaces, we can move the point of attachment of the interval along a continuous path without changing the homotopy type of the whole space. Thus if one has a k-star graph in Z(G) with k ≥ 3, note that two of the edges form an interval whose end points lie on the Riemann surfaces. Thus using adjunction deformations, we can move the central point attachment of the other k − 2 edges so that they attach to Riemann surfaces in Y (G) on both ends. Thus up to homotopy equivalence, Z(G) and hence X(G) is homotopy equivalent to Y (G) with a finite number of edges attached where both end points lie in Y (G) and whose interiors are disjoint. Now fix a basepoint x i in each Riemann surface X i of Y (G). Using adjunction space deformations, we can up to homotopy equivalence assume all the edges have endpoints in the set {x 1 , . . . , x m } as the Riemann surfaces X i are path connected.
At this stage as X(G) is connected, the union of these edges forms a connected graph with vertices {x 1 , . . . , x m }. Collapsing a spanning tree of this graph (which is contractible) one obtains a final homotopy equivalence. Notice that under this collapse, the points x 1 , . . . , x m become a common bouquet point to which the Riemann surfaces X 1 , . . . X m from the desingularization Y (G) are attached. Any edges of the graph not in the spanning tree of the graph become circles in this bouquet. The theorem is hence proven.
Corollary 2.12. The integral homology groups of X(G) are free abelian groups of finite rank. If β i denotes the ith Betti number, i.e., the rank of H i (X(G); Z) then β 0 = 1, β 2 = g≥0 m g (G) and
β 1 = g≥0 2gm g (G) + L
where m g (G) denotes the number of times the surface of genus g occurs in the desingularization Y (G) of X(G) and L denotes the number of circles occurring in the homotopy decomposition of Theorem 2.11.
Proof. Follows immediately from Theorem 2.11 and the known homology of Riemann surfaces.
The Euler characteristic of X(G)
In this section we find some useful formulas for the Euler characteristic of X(G). One of these follows immediately from the homology computation of the last section and the other from a simplex count of the simplicial complex which we go over now.
Vertices of X(G): There are |G| − |Z(G)| many vertices of type 1 and the same number of vertices of type 2 for a total of V = 2(|G| − |Z(G)|) vertices in X(G).
Let E 1 denote the number of edges joining two type 1 vertices and let E 2 denote the number of edges joining a type 1 and type 2 vertex. Finally let F denote the number of faces (2-simplices). As each face consists of 3 edges, (two in E 2 and one in E 1 ) and every edge lies in exactly two faces, we conclude 2F = 2E 2 and F = 2E 1 and E = E 1 + E 2 is the total number of edges. Thus the Euler
characteristic of X(G), χ(X(G)) = V − E + F = V − (E 1 + E 2 ) + E 2 = V − E 1 .
Note that there is an edge in E 1 for every (unordered) set of two noncommuting elements of G.
Thus E 1 = 1 2 x∈G (|G| − |C(x)|) as for each x ∈ G, there are |G| − |C(x)| elements y which don't commute with x. Here C(x) is the centralizer subgroup of x. So E 1 = 1 2 (|G| 2 − x∈G |C(x)|).
The following lemma evaluates this sum: Lemma 2.13. For G a finite group, x∈G |C(x)| = |G|c where c is the number of conjugacy classes of G.
Proof. Ler x 1 , . . . , x c denote a complete list of conjugacy class representatives of G. Note that |C(x)| = |C(x ′ )| when x, x ′ lie in the same conjugacy class. Thus
x∈G |C(x)| = c i=1 |C(x i )|c i where c i is the size of the conjugacy class of x i . Note that c i = |G| |C(xi)| and so x∈G |C(x)| = c i=1 |G| = |G|c
as we set out to show.
We record a side corollary of Lemma 2.13:
Corollary 2.14. Let G denote a finite group and P denote the probability that two elements picked independently and uniformly from G commute. Then P = c |G| where c is the number of conjugacy classes of G.
Proof. As any ordered pair of elements in G × G are equally likely to be selected we have
P = Number of (x, y) where x, y commute |G| 2 = sum x∈G |C(x)| |G| 2 = c |G| .
where we used Lemma 2.13 in the last step.
We now record two formulas regarding the Euler characteristic of X(G).
Theorem 2.15. Let G be a finite nonabelian group. Then
V − E + F = χ(X(G)) = β 0 − β 1 + β 2 becomes 2(|G| − |Z(G)|) − 1 2 (|G| 2 − |G|c) = χ(X(G)) = 1 − L + g≥0 (1 − 2g)m g (G)
where c is the number of conjugacy classes of G and m g (G) denotes the number of times the surface of genus g occurs in Y (G) the desingularization of X(G). L denotes the number of circles that occur in the homotopy decomposition of X(G) given by Theorem 2.11.
Proof. Follows as the Euler characteristic can either be computed by the alternating sum V − E + F or the alternating sum of Betti numbers. The Betti sum is evaluated using corollary 2.12 while V − E + F = V − E 1 is evaluated using the calculations made earlier in this subsection.
Group Theoretic Analysis of Closed Stars of X(G)
Definition 2.16. Let G be a finite group and α ∈ G. The cyclic subgroup C α =< α > generated by α acts on G by conjugation. The orbits are called α-conjugacy classes and have size dividing the order of α. Two elements of the group are said to be α-conjugate if they lie in the same α-conjugacy class. An element {x} forms an α-conjugacy class of size 1 if and only if x commutes with α if and only if x ∈ C(α), the centralizer group of α. We will denote the conjugate of x by α −1 i.e., α −1 xα by x α .
Let us study the closed star of a type 2 vertex
v = (α, 2) in X(G). A face in St(v) is of the form σ 1 = [(x, 1), (y, 1), (α, 2)
] where x, y do not commute and α = xy. Note α does not commute with either x or y. Now let us consider the adjacent face σ 2 = [(y, 1), (z, 1), (α, 2)]. Then yz = α and so yz = xy i.e. z = y −1 xy = y −1 x −1 xxy = α −1 xα = x α . Thus the adjacent face is σ 2 = [(y, 1), (x α , 1), (α, 2)]. Now taking the equation α = xy corresponding to the first face σ 1 and conjugating it by α −1 we see that α = x α y α . Thus the face σ 3 other than σ 1 which is adjacent to σ 2 which contains the vertex (α, 2) is σ 3 = [(x α , 1), (y α , 1), (α, 2)]. Note that σ 3 was obtained from σ 1 by conjugating its 3 vertices by α −1 .
We now have made an important observation. In a given sheet of the closed star of a type 2 vertex, given one triangle in the sheet, the triangle two away in the same sheet is obtained by conjugating all vertices by α −1 . Thus the entire sheet is made by α-conjugacy applied to the two initial triangles [(x, 1), (y, 1), (α, 2)] and [(y, 1), (x α , 1), (α, 2)].
Figure 4: A representative sheet inSt(v) for v = (α, 2). Α xy, 2 x, 1 y, 1 x Α , 1 y Α , 1
There are two possibilities: Either the α-conjugacy orbits of these two triangles are distinct and the sheet is made from two α-conjugacy orbits of triangles or these two triangles are α-conjugate and the sheet consists of a single α-conjugacy orbit of triangles.
The latter happens if and only if x and y = x −1 α are α-conjugate. This happens if and only if there is a positive integer k such that x α k = y i.e. xx α k = α. The next lemma shows that this is equivalent to saying α is the product of two distinct elements in the α-conjugacy class of x.
Lemma 2.17. Let G be a finite group and α, x ∈ G be two elements which don't commute. Then the following are equivalent:
(1) xx α k = α for some positive integer k.
(2) α is the product of two distinct elements in the α-conjugacy class of x.
Proof. Clearly (1) implies (2). Thus we only have to show (2) implies (1). Let m be the size of the α-conjugacy class of x, then x α s x α t = α for some 0 ≤ s, t < m with s = t. Conjugating this equation by α s we find xx α t−s = α. t − s is congruent modulo m to a unique positive integer k with 1 ≤ k < m and xx α k = α. Thus we are done.
Definition 2.18. Let G be a finite group and α ∈ G. There are three mutually exclusive, exhaustive possibilities for an α-conjugacy class C:
(1) C has size 1 and consists of a single element in C(α).
(2) α is the product of two distinct elements in C. We call such an α-conjugacy class productive.
(3) α is not the product of two distinct elements in C and C contains at least 2 elements. We call such an α-conjugacy class nonproductive. Let p α denote the number of productive α-conjugacy classes and n α denote the number of nonproductive α-conjugacy classes. Thus p α + n α + |C(α)| is the total number of α-conjugacy classes.
We now summarize the earlier analysis of the closed star of a type-2 vertex.
Proposition 2.19 (Closed star of type 2 vertices). Let α be a noncentral element of a finite group G and let v = (α, 2) be the corresponding type 2 vertex in X(G). Then for each sheet inSt(v), the vertices of Lk(v) in that sheet consist of either:
(1) Two non-productive α-conjugacy classes of type 1 vertices. In this case there are exactly two α-conjugacy classes of triangles in that sheet and the two α-conjugacy classes have the same size. Thus the number of triangles in the sheet is 2ℓ, an even number, where ℓ ≥ 2 is a divisor of the order of α.
(2) A single productive α-conjugacy class of type 1 vertices. In this case there is exactly one α-conjugacy class of triangles in the sheet. Thus the number of triangles in the sheet is ℓ where ℓ ≥ 3 is an odd divisor of the order of α. The number of triangles on the sheet in this case is odd.
Proof. Most of this proposition was proven in the preceding paragraphs. Only a few final comments are in order. In case (1), the fact that the two orbits of triangles are interlaced shows that they have the same number of triangles which shows the two non-productive α-conjugacy classes in the link must have the same size. In case (2), the fact that a sheet must have at least 3 triangles shows that productive α-conjugacy classes must have size ≥ 3. (A direct algebraic argument can also show that the class cannot have size 2 but we appeal to the geometry directly!) The divisibility comments follow as the size of any α-conjugacy orbit divides the order of α. The fact that there are an odd number of triangles in the case of one α-conjugacy class follows because the action of α-conjugacy always moves one 2 steps along the rim of a n-triangle sheet, thus there are two or one α-conjugacy classes in a sheet, depending if the subgroup generated by 2 has index two or one in the cyclic group of order n which depends exactly if n is even or odd respectively.
Note from proposition 2.19, whether a sheet about a type 2 vertex consists of two or one α-conjugacy orbits, depends completely on whether it consists of an even or odd number of triangles.
Corollary 2.20. If α is a non-central element of a finite group G then: (1) n α is even.
(2) The closed star of (α, 2) in X(G) consists of nα 2 + p α sheets. (3) If α has prime order p, each sheet in the closed star of (α, 2) has either p or 2p triangles depending on if it corresponds to a productive orbit or two nonproductive orbits respectively. (4) If α has order 2 then each sheet in the closed star of (α, 2) has 4 triangles and consists of two non-productive orbits. In particular α has no productive orbits and |G| ≡ |C(α)| mod 4.
Proof. Given x that does not commute with α there is a unique y such that xy = α. This implies that (x, 1) is on a unique sheet ofSt((α, 2). The vertices in the link of this sheet then consist of the α −1 -orbit of (x, 1) and (y, 1) and hence consist of exactly the union of the two (not necessarily distinct) α-conjugacy classes of x and y.
By Proposition 2.19, the non-productive α-conjugacy classes must pair up to make sheets and so there are an even number of them. Thus (1) is proven.
(2) also follows immediately from the same proposition. (3) follows similarly once one notes that the only ℓ ≥ 2 dividing a prime p is ℓ = p. (4) follows as we have seen that productive α-conjugacy classes must have size ≥ 3 but also size dividing the order of α which is 2 and is hence impossible in this case. G − C(α) then decomposes as an even number of non-productive orbits each of size 2 which implies 4 divides |G| − |C(α)|. Now we consider the closed star of vertices of type 1. Let w = (α, 1) be a type 1 vertex of X(G). Let [(x, 1), (α, 1), (xα, 2)] be a face contaning w.
[(α, 1), (x α , 1), (xα, 2)] and [(α, 1), (x, 1), (αx, 2)] are the adjacent faces containing w also. As (αx) α = xα, it is easy to check that the 2-labelled vertices in the part of Lk(w) in this sheet consists of the α-conjugacy orbit of αx. It also follows that the 1-labelled vertices in the part of Lk(w) in this sheet consists of the α-conjugacy orbit of x. The α-conjugacy orbit of the 1-vertices in the link for this sheet has the same size as the α-conjugacy orbit of the 2-vertices in the link as the edges along the link of w alternate between type 1 and type 2 vertices. Furthermore the α-conjugacy class of the 2-vertices is uniquely determined from the one for the 1-vertices by the fact that there exist u in the type 2 vertex orbit and v in the type 1 vertex orbit whose "difference" v −1 u is α.
We have thus proven:
Proposition 2.21 (Closed star of type 1 vertices). Let G be a nonabelian group and let w = (α, 1) be a type 1 vertex in X(G). Then each sheet ofSt(w) has 2ℓ triangles where ℓ ≥ 2 divides the order of α. The vertices along Lk(w) in any sheet alternate between type 1 and type 2 vertices and there is a single α-conjugacy orbit of type 1 vertices and a single α-conjugacy orbit of type 2 vertices in the link of any given sheet. These orbits have the same size. One orbit determines the other by the fact the type 2 orbit contains u and the type 1 orbit contains v such that v −1 u = α. As each sheet contains one α-conjugacy class of size > 1 each of type 1 and type 2 vertices, the total number of sheets inSt(w) is the total number of α-conjugacy classes of size > 1 i.e., n α + p α .
Definition 2.22. Let G be a nonabelian group and v a vertex of X(G). Let s(v) be the number of sheets the closed star of v has and let V denote the vertex set of X(G).
(
1) If v = (α, 2) then s(v) = nα 2 + p α . (2) If v = (α, 1) then s(v) = n α + p α . (3) v∈V s(v) = α∈G ( 3nα 2 + 2p α ).
We are now ready to present another formula for the Euler characteristic of X(G).
Theorem 2.23. Let G be a nonabelian group then
χ(X(G)) = 2(|G| − |Z(G)|) − α∈G ( 3n α 2 + 2p α ) + g≥0 (2 − 2g)m g (G).
Combining this with previous formulas for χ(X(G)) yields the formula:
α∈G (3n α + 4p α ) + g≥0 (4g − 4)m g (G) = |G|(|G| − c)
where c is the number of conjugacy classes of G.
Proof. Recall that Y (G), the desingularization of X(G) is the disjoint union of m g (G) Riemann surfaces of genus g over all g ≥ 0. Thus χ(Y (G)) = g≥0 (2 − 2g)m g (G). Recall that Z(G) was a complex obtained by attaching disjoint closed star graphs to Y (G) where there was one such graph for each vertex in X(G) and the type of star graph used for any given vertex depended on the number of sheets in the closed star of that vertex. This means Z(G) is obtained from Y (G) by adding |V | vertices where V is the vertex set of X(G) and by adding v∈V s(v) edges. This means χ(Z(G)) = χ(Y (G))− v∈V s(v)+|V |. As X(G) is homotopy equivalent to Z(G) we have χ(X(G)) = χ(Z(G)). Combining this with the fact that |V | = 2(|G| − |Z(G)|) and putting everything together we get the formula claimed for χ(X(G)). Comparing this to previous formulas for χ(X(G)) yields the second formula immediately.
Functoriality
A group homomorphism f : G → H is said to be injective on commutators if the restriction f : G ′ → H is injective where G ′ is the commutator subgroup of G. It is easy to check that the composition of two homomorphisms which are injective on commutators is also injective on commutators. If f : G → H is such a homomorphism then g 1 commutes with g 2 if and only if f (g 1 ) commutes with f (g 2 ) for all g 1 , g 2 ∈ G.
Let C denote the category of finite groups and homomorphisms which are injective on commutators. Let D be category of finite oriented 2-dimensional simplicial complexes and orientation preserving simplicial maps. Here by an oriented 2-dimensional simplicial complex we mean one whose faces (2-simplices) have all been given an orientation and when we say a simplicial map preserves orientation we mean it takes 2-simplices to 2-simplices in a manner that preserves orientation of the individual faces.
Proposition 2.24. The construction X(G) is part of a covariant functor from the category C of finite groups and homomorphisms injective on commutators to the category D of finite oriented 2-dimensional simplicial complexes and orientation preserving simplicial maps. In particular if G 1 is isomorphic to G 2 then X(G 1 ) is simplicially isomorphic to X(G 2 ) and if H ≤ G then X(H) is a subcomplex of X(G).
Proof. We have already described X on the level of objects so let f : G → H be a homomorphism injective on commutators. For vertices (either of type 1 or type 2) define X(f )((v, i)) = (f (v), i) for i = 1, 2. As f takes noncommuting elements to noncommuting elements, it takes edges of X(G) to those of X(H). If [(x, 1), (y, 1), (xy, 2)] is an oriented face of X(G), then [(f (x), 1), (f (y), 1), (f (xy), 2)] is an oriented face of X(H) as f (xy) = f (x)f (y). Thus X(f ) defines an orientation preserving simplicial map between X(G) and X(H). It is now easy to check that X respects compositions and identity maps and defines a covariant functor from C to D as desired. The rest follows readily.
Corollary 2.25. For G a finite nonabelian group, Aut(G) acts on X(G) through orientation preserving simplicial automorphisms. In particular G acts on X(G) simplicially by conjugation. Furthermore an anti-automorphism of G like θ(g) = g −1 induces an orientation reversing simplicial automorphism of X(G).
The covariant functor X induces another functor Y , the desingularization of X. We have already described the desingularization Y (G) on the level of objects.
For a homomorphism f : G → H which is injective on commutators, we have already defined an orientation preserving simplicial map X(f ) : X(G) → X(H). As such a map takes faces to faces, it is easy to see that it takes pseudomanifold components to pseudomanifold components and induces a continuous map X(G)−{vertices} → X(H)−{vertices} which takes punctures to punctures. As each puncture arises from a unique sheet of the closed star of a unique vertex, or equivalently from a unique circle of the link of a unique vertex, to see if there is a well-defined continuous extension of X(f ) to a simplicial map Y (G) → Y (H), we need only note that each circle in a link of the vertex (v, i) must map to a unique circle in the link of the vertex (f (v), i) under X(f ). We then define Y (f ) in such a way as to map the puncture associated to a particular circle in the link of (v, i) in X(G) to the puncture associated to the circle in the link (f (v), i) in X(H) which X(f ) takes the first circle to.
After doing this, it is easy to check Y (f ) is a well-defined orientation preserving simplicial map from Y (G) to Y (H) and that the construction preserves compositions and identity maps. Thus we have proven:
Proposition 2.26. The construction Y (G) is part of a covariant functor Y from C to D.
From this it follows that if H is a subgroup of G, then the manifold components of Y (H) are a subset of those of Y (G). Thus m g (H) ≤ m g (G) for every genus g. Thus we have proven:
Corollary 2.27 (Monotonicity). Let H ≤ G be finite non-abelian groups, then m g (H) ≤ m g (G) for every genus g ≥ 0.
The next corollary follows from monotonicity as every finite group is a subgroup of a symmetric group Σ n . It shows that a particular genus g surface can occur as a component of Y (G) for finite groups if and only if it can occur for symmetric groups.
Corollary 2.28. If m g (G) > 0 for some genus g ≥ 0 and finite non-abelian group G then m g (Σ n ) > 0 for some n ≥ 3.
We also record the important fact that every component of X(G) originates from a component of X (H) where H ≤ G is a non-abelian subgroup generated by 2 elements. In this regard much like Coxeter's hyperbolic Cayley graph construction, the Riemann surfaces that make up X(G) originate from various 2generated subgroups. However the triangulations on these surfaces and method of generation seems to differ greatly so it is unclear what the exact relationship is between the two constructions.
Corollary 2.29. Let G be a finite non-abelian group, then X(G) = ∪ H∈A X(H) where A denotes the collection of subgroups of G which are non-abelian and generated by 2-elements.
Proof. Starting from a triangle in X(G), say [(x, 1), (y, 1), (xy, 2)] it is easy to verify by induction that all triangles in the pseudomanifold component which contains that triangle have vertices which lie in the subgroup H =< x, y > which is generated by x and y. As x and y don't commute, H is non-abelian and generated by 2-elements. As X(G) is the union of its pseudomanifold components, the corollary is proved.
Abstract 3-polytope structure of components
In this section, we discuss a modification of the cell structure of the triangulated Riemann surfaces that occur as components of Y (G) and show this cell structure satisfies the conditions of what is called an abstract 3-polytope in the combinatorics literature. Fix a nonabelian finite group G in this section.
Recall, we have seen that the components of the desingularization Y (G) are triangulated Riemann surfaces. The closed star of a type 2 vertex in this surface is a single sheet which is simplicially isomorphic to some simplicial m-disk.
LetŶ (G) be the same collection of Riemann surfaces in Y (G) except that the cell-structure is modified as follows. For each component, all type 2-vertices and edges joining type 1 to type 2 vertices are erased resulting in a new cell structure for the component where the vertices consist of only the type 1 vertices of the original construction, the edges only those joining type 1 vertices in the original construction and where the faces are polygonal consisting of the sheet about a type 2 vertex in the original construction with its nonboundary vertices and edges erased. Topologically we are not changing the component at all, we are just changing the cell structure. The final cell structure will be called the 2-cell structure of the component. It is no longer a triangulation but does give the component the structure of a CW -complex. This cell structure forms a "closed cell structure", i.e., the closed cells are homeomorphic to 2-disks and no boundary identifications occur between points on the boundary of the same face.
We first show that all 2-sheets in a given component of Y (G) are simplicially isomorphic and hence all the 2-faces in the corresponding cell structure of a given component ofŶ (G) consist of the same type of polygonal face.
Theorem 2.30. Let G be a nonabelian finite group and let T be the unique component of Y (G) which contains the triangle [(x, 1), (y, 1), (xy, 2)]. Then all type 1-vertices in T are conjugate to either x or y and all type 2-vertices in T are conjugates of xy. Furthermore all sheets about type 2-vertices in T are simplicially isomorphic via a conjugation and hence are n-gons for the same n ≥ 3. These conjugations are by elements in the subgroup H generated by x and y.
In the corresponding 2-cell structure, all vertices are conjugate to (and hence have the same valency as) either (x, 1) or (y, 1). All faces are n-gons for the same n. All oriented edges are conjugate to the edge [(x, 1), (y, 1)] or the edge [(y, 1), (y −1 xy, 1)] and all unoriented edges are conjugate. Thus the cell automorphism group of the component is both face and edge transitive.
Proof. Let α = xy. A quick calculation shows that the three triangles adjacent to [(x, 1), (y, 1), (α, 2)] in T are [(y, 1), (x α , 1), (α, 2)], [(y, 1), (x, 1), (α x , 2)] and [(xyx −1 , 1), (x, 1), (α, 2)]. The vertices in these 3 triangles are conjugate to those in the original triangle using conjugation by elements in the group H generated by x and y. As every triangle in the component can be joined to the original one by a sequence of adjacent triangles, and the union of the triangles is the component, the conjugacy result for vertices follows.
Now consider the sheet about (α, 2) which contains the triangle [(x, 1), (y, 1), (α, 2)] and the adjacent sheet that shares the edge [(x, 1), (y, 1)] i.e. the sheet about (yx, 2) which contains the triangle [(y, 1), (x, 1), (yx, 2)]. Conjugating the triangle [(x, 1), (y, 1), (α, 2)] by x −1 and using functoriality yields a simplicial isomorphism of Y (G) which takes the triangle to the one [(x, 1), (x −1 yx, 1), (yx, 2)] which is adjacent to [(y, 1), (x, 1), (yx, 2)]. From this it follows that this simplicial automorphism must take the component T back to itself and the sheet about (α = xy, 2) to the one about (yx, 2) in the same component. In particular two adjacent sheets in a given component are conjugate and hence simplicially isomorphic. As any sheets about type 2-vertices in a given component can be joined by a sequence of adjacent sheets, the result on sheets follows.
To get the result on edges, first note every edge is conjugate to one that lies in the original sheet. Then previous results show that under conjugation by the center element of the sheet there are at most two orbits of oriented edges depending if the conjugacy classes x and y are a pair of nonproductive α-conjugacy classes or a single productive one. These (unoriented) edges are represented by orbit representatives [(x, 1), (y, 1)] and [(y, 1), (y −1 xy, 1)] in the case of two α-orbits. Conjugation by y −1 takes the first edge to the second (with orientation flipped) and is easily seen to induce an automorphism of the component. Thus the cell automorphism group of the component is (unoriented) edge transitive and the theorem is proven.
Corollary 2.31. Let G be a finite nonabelian group then the number of components of Y (G) is greater or equal to the number of conjugacy classes of noncentral elements in G.
Proof. By Theorem 2.30, there is at most one conjugacy class represented by the type 2 vertices in a given component. As there has to be at least one type 2 vertex for each noncentral element of G, the corollary follows. (Note the desingularization process can cause there to be more than one type 2-vertex corresponding to a given noncentral element of G. Also examples show a conjugacy class can be spread out over more than one component. Both these factors cause the inequality just proved to often not be equality.)
Recall the modified cell structure of components denotedŶ (G) where the vertices are only the type 1 vertices, the edges only those joining two type 1 vertices and 2-faces being n-gons which were sheets about type 2 vertices in Y (G). These n-gons will carry an implicit label given by the original middle element of the corresponding sheet. Theorem 2.30 shows that a given component with this cell structure has face and edge-transitive cell automorphism group (bijections of vertices which carry edges to edges and boundaries of 2-faces to boundaries of 2-faces) and either one or two vertex orbits. The next theorem collects important formulas for the quantities in this component.
Theorem 2.32. Let G be a finite nonabelian group and let T be the unique component ofŶ (G) which corresponds to the triangle [(x, 1), (y, 1), (α = xy, 2)] where x, y ∈ G do not commute. T is a compact, connected, oriented 2-manifold of genus g with cell structure consisting of 2-faces which are all n-gons for some fixed n ≥ 3. The cell automorphism group always acts face and (unoriented) edge transitively on T .
(1) If x and y lie in different α-conjugacy classes (n even case), then n = 2(size of α − conjugacy class of x) and the automorphism group acts with at most 2-orbits of vertices represented by (x, 1) and (y, 1) respectively. The valency of (x, 1) in T with the 2-cell structure is given by λ 1 ≥ 2, the size of the xconjugacy class of α = xy. The valency of (y, 1) in T in this modified 2-cell structure is λ 2 ≥ 2, the size of the y-conjugacy class of α = xy. In the case λ 1 = λ 2 we define λ = λ1V1+λ2V2 V1+V2 ≥ 2 to be the average valency of the component where V i denotes the number of vertices of valency λ i in the component. In this case each edge in the component joins a vertex of valency λ 1 with a vertex of valency λ 2 and we also have E = λ i V i for i = 1, 2. Finally the average valency can be computed as either λ = 2E V or as the harmonic average of the valencies λ 1 , λ 2 , i.e., 2 λ = 1 λ 1 + 1 λ 2 .
(2) If x and y lie in the same α-conjugacy class (n odd case), then n = size of α − conjugacy class of x and the automorphism group acts transitively on edges and vertices also. The common valency λ ≥ 2 of all vertices is given by the size of the x-conjugacy class of α.
(3) If V, E, F denote the number of vertices, edges and 2-faces of the 2-cell structure of the component T and the face type of the component is n-gons, with average vertex valency λ then the following equations hold:
nF = 2E nF = λV 2 − 2g = V − E + F = 2( 1 λ + 1 n − 1 2 )E
Proof. The formulas for n follow from results in previous sections concerning the number of triangles in a sheet about a type 2-vertex. The formulas for valency follow once one notes that the sheet centered about (xy, 2) and the one about (yx, 2) containing (x, 1) are adjacent (share an edge) and are conjugate to each other by conjugation by x −1 . Repeating this observation, one finds that the 2-faces containing (x, 1) in T consist of those centered at the x-conjugacy orbit of (α = xy, 2). Thus the number of faces containing (x, 1) which is the same as the number of edges incident to (x, 1) in the 2-cell structure is given by the size of this x-conjugacy class. Similar arguments work for the vertex (y, 1). As the cell automorphism group of the component is edge transitive, every edge joins a vertex conjugate to (x, 1) to a vertex conjugate to (y, 1) where the edge [(x, 1), (y, 1)] is the original one determining the component. Thus each edge joins a vertex of valency λ 1 to one of valency λ 2 . When λ 1 = λ 2 note that each edge contributes a total of one to the valency count of vertices of valence λ 1 . As λ 1 of these edges are incident on a given such vertex we find E = λ 1 V 1 . Similarly E = λ 2 V 2 . Note also that V = V 1 + V 2 . Now it follows that 2E = λ 1 V 1 + λ 2 V 2 = λV by definition so λ = 2E V . Now note:
1 λ 1 + 1 λ 2 = V 1 E + V 2 E = V E = 2 λ
and so λ is the harmonic average of λ 1 and λ 2 . Finally the first formula in (3) follows from the observation that each polygonal face contributes n edges to the component and each edge lies in exactly 2 such faces. We will prove the second formula in the harder case of two types of vertex valencies -the other case follows similarly. First recall that each edge will contribute two vertices, one of which is in the orbit of (x, 1) and the other in the orbit of (y, 1). Now each face contributes n vertices with an equal number of valency λ 1 as with valency λ 2 . Again taking account that a vertex of valency λ i lies in λ i such faces we get nF 2 = λ i V i . Adding these equations for i = 1, 2 yields nF = λ 1 V 1 + λ 2 V 2 = λV and so the second formula is proven. Finally the Euler characteristic of a Riemann surface of genus g is 2 − 2g and equals the alternating sum V − E + F in any cell decomposition. Thus the final formula follows upon plugging in the previous formulas.
It follows from face and edge transitivity that many of the quantities in the last theorem have stringent divisibility conditions. The next proposition records these:
Proposition 2.33. Let G be a finite nonabelian group and let T be a component ofŶ (G) with the 2-cell structure discussed in Theorem 2.32 consisting of n-gon faces and with vertex, edge and face counts V, E, F respectively. Then:
(1) E ≥ 3 and F ≥ 3 are divisors of |G|.
(2) In the case of two distinct vertex orbits V 1 , V 2 , λ 1 , λ 2 ≥ 2 are divisors of E and hence of |G|. Furthermore n ≥ 4 is even and divides 2E and hence 2|G|. Proof. G acts onŶ (G) cellularly by conjugation. Thus it shuffles components around taking components to isomorphic components. The G-conjugacy orbit of the component T is thus a union of ℓ ≥ 1 components all isomorphic as cell complexes with T . Taking an edge in e in the component T , Theorem 2.30 shows that the G-conjugation orbit of e includes all the edges in T . It is then easy to see that the G-conjugation orbit of e is exactly the set of edges in the ℓ components conjugate to T . Since each of these components has the same edge count E we have ℓE = |G| |S| where S is the stabilizer subgroup of the edge e. Thus |G| = ℓE|S| and so E divides |G|. Furthermore E = |G| if and only if |S| = 1 and ℓ = 1 i.e. T is invariant under G-conjugation and G acts freely and transitively on the set of edges of T . As any element of C(x) ∩ C(y) fixes the edge [(x, 1), (y, 1)] under conjugation, we must have C(x) ∩ C(y) = {1} in this case and in particular G must have trivial center. An analogous argument works for faces and so (1) and (4) are proved. In case (2), we have E = λ i V i and 2E = nF and so the result follows. Also n is even in this case as the proof of Theorem 2.32 shows. In case (3), we have 2E = λV = nF and that case follows immediately also. Finally (5) follows as there are only finitely many divisors of a positive integer |G| and 2 − 2g = V 1 + V 2 − E + F = V − E + F is an "alternating" sum of three or four of these divisors.
The reader is warned that in general V = V 1 + V 2 does not divide |G| as examples in later sections show.
The 2-cell structures on the components ofŶ (G) can be modified trivially to form what is called an abstract 3-polytope in the combinatorics literature. The definition of these objects is a bit long so we will not repeat it here but the reader may find it in [Cox2] and [MS]. We will verify these conditions briefly here.
First note the cell structure on a given component T ofŶ (G) can be used to form a poset consisting of the set F −1 ∪ F 0 ∪ F 1 ∪ F 2 ∪ F 3 where F k will be the set of k-faces in the poset ordered by inclusion. Here F 0 is the set of vertices of the component, F 1 is the set of edges of the component and F 2 is the set of polygonal 2-faces of the component. F −1 consists of a single empty face, which is the smallest element of the poset and F 3 consist of a formal "biggest" face containing all other faces.
A flag in this poset consists of ∅ ⊂ v ⊂ e ⊂ f ⊂ D where v is a vertex, e is an edge, f is a 2-face and D is the formal greatest 3-face. All such flags have length 5.
This poset satisfies the "diamond condition" of a polytope as every edge contains exactly two vertices, every edge is contained in exactly two 2-faces and for any v ∈ f there are exactly two edges e, e ′ such that v ∈ e ⊂ f, v ∈ e ′ ⊂ f .
Finally the poset is strongly connected as each component is a path-connected Riemann surface with path connected 2-faces as the reader can verify.
Thus the 2-cell structure of any component extended formally to a poset with smallest emptyset face and greatest 3-face satisfies the properties of an abstract 3-polytope. The automorphism group of this poset is easily seen to be the same as the group of bijections of the vertex set to itself which takes edges to edges and faces to faces i.e. is the same as the group of cell automorphisms of the 2-cell complex T . We have already seen this automorphism group is always face and edge transitive. This translates to the sections S(∅, f ) of this poset to be all n-polygons for fixed n ≥ 3. There are at most 2 orbits of vertices, the sections S(v, D) thus are polygons of at most two different types.
To be equivar in the language of polytopes, these valencies must be the same. In addition to be regular, the automorphism group must act transitively on flags. We identify conditions for these situations in the next "dictionary" theorem:
Theorem 2.34. Let G be a nonabelian group and T the unique component of Y (G) corresponding to the triangle [(x, 1), (y, 1), (xy, 2)] with the extended cell structure discussed in the previous paragraphs. Recall all 2-cells of T are n-gons. Then:
(1) This cell structure of T forms an abstract 3-polytope with 2-face and edge transitive automorphism group.
(2) The abstract 3-polytope is equivar if and only if all vertices have the same valency λ. This happens when the size of the x and y-conjugacy classes of α = xy are the same. The Schläfli index of this equivar 3-polytope is {n, λ}.
(3) The abstract 3-polytope has 2-face, edge and vertex transitive automorphism group if x and y are α-conjugate.
(4) The abstract 3-polytope has flag-transitive automorphism group (i.e. is regular) if x and y are α-conjugate and there is an automorphism of the group G taking x to y −1 and y to x −1 (5) The results of (3) and (4) still hold when x and y are not α-conjugate as long as there exists an automorphism of the group G taking x to y and y to y −1 xy and another taking x to y −1 and y to x −1 .
Proof.
(1) and (2) follow from the work in Theorem 2.30 and the previous paragraphs. To see (3), note that any edge or vertex of the component can be mapped by an automorphism to lie in the 2-cell centered at (α, 2). Then α-conjugacy will move this image edge or vertex to any chosen representative edge or vertex on that 2-cell as long as all vertices on the boundary of this 2-cell are α-conjugate. This happens if and only if x and y are α-conjugate. For (4), given a reference flag v ∈ e ⊂ f (we will surpress the empty face and greatest face in this proof as they do not come into play) and another flag v ′ ∈ e ′ ⊂ f ′ we can first apply an automorphism to take f ′ to f by face transitivity. Thus we many assume f = f ′ . Then as all vertices along the rim of the corresponding 2-cell are conjugate, we may conjugate fixing f so that e ′ moves to e. Thus it remains to show that there is a automorphism of the polytope taking the flag (x, 1) ∈ [(x, 1), (y, 1)] ⊂ [(x, 1), (y, 1), (xy, 2)] to the flag (y, 1) ∈ [(x, 1), (y, 1)] ⊂ [(x, 1), (y, 1), (xy, 2)]. As such a map has to be orientation reversing, to achieve it using functoriality we have to use an antiautomorphism of the group. If I is the inversion map I(x) = x −1 , then any anti-automorphism is the composition of I with an automorphism. A quick calculation shows that if φ is an automorphism of G taking x to y −1 and y to x −1 , the simplicial automorphism arising from I • φ does the job! Finally for (5), the stated automorphism maps the edge [(x, 1), (y, 1)] to its adjacent edge on the sheet about (xy, 2). This together with previous comments gives vertex transitivity. Then regularity follows as in the proof of (4).
The duality operation of reversing the poset ordering in the poset underlying an abstract 3-polytope has the effect of interchanging the roles of vertices and 2-faces while leaving the edges alone. It takes an equivar 3-polytope of Schläfli symbol {n, λ} to one of symbol {λ, n}. It corresponds to the classical dual cell structure construction behind Poincare duality of the component. The vertices in this new structure correspond to the centers of the 2-faces in the original. Edges are drawn between these vertices when the 2-faces they came from were adjacent. The new 2-faces hence come from arrangements of faces around vertices in the original and are λ i -gons if the vertex had valency λ i . Thus the dual of one of our complexes would be vertex and edge transitive and have either one or two orbits of faces. In the case of two orbits, the faces would consist of either λ 1 -gons or λ 2 -gons and around every vertex these types would alternate with total even vertex valency n. Thus all in all, the role of face type and valency interchange and the role of vertices and faces change under duality. Thus if the reader prefers, they can consider the dual to our construction which would be like a "soccerball", consisting of at most two types of polygonal faces with vertex and edge transitive automorphism group. In other words the complexes that arise as components in the constructionŶ (G) that have two valencies are in general dual to abstract quasiregular (i.e., vertex, edge transitive with two types of faces arranged alternatingly about a vertex) 3-polytopes.
Valence Two and Doubling
In this section we will discuss the case when one or both valencies in Theorem 2.32 are equal to two.
Lemma 2.35. Let X be a tesselated Riemann surface as those arising is Theorem 2.32 i.e., edge and face transitive and with at most two orbits of vertices. If the valencies λ 1 = λ 2 = 2 then F = 2, E = V = n, g = 0 and X is a sphere obtained by gluing two n-gon faces along their common rim.
Proof. λ 1 = λ 2 = 2 implies λ = 2. Using this in the equations in part (3) of Theorem 2.32, we find 2 − 2g = 2E n . As this quantity is positive, this forces g = 0 and then E = n. Then nF = 2E forces F = 2 and λV = 2E forces E = V . The lemma follows.
We now consider the case of a tesselated Riemann surface which is edge/face transitive and has two orbits of vertices of valency λ 1 = 2 and λ 2 = k > 2. Recall when we have two valencies we have an even face type 2s. We will denote the Schläfli symbol of such a complex as {2s, 2-k} where 2s is the face type.
If we have such a complex, the two orbits of vertices are distinguishable due to their distinct valencies. Notice each vertex of valency 2 lies in two faces and is adjacent to two vertices of valency k > 2 which also lie in both of these faces. Thus we can remove all valency 2 vertices and combine the two incident edges to any of them into a single edge. This construction doesn't change the underlying surface so the genus is unchanged. The face type changes from 2s-gon to sgon and the new edge count is half the original one. The new vertex count is equal to the count of valency k vertices in the original complex. The resulting complex is equivar with a single vertex valency k. Thus we have changed a {2s, 2-k} complex into an equivar {s, k} complex. Conversely given an equivar complex of the form {s, k} one can add a midpoint vertex to each edge to obtain a {2s, 2-k} one and it is easy to see these processes are inverse processes.
We will refer to the {2s, 2-k} complex obtained from the {s, k} one as the "double" of the {s, k}-complex. We will sometimes write {2s, 2-k} = D{s, k}. Although the complexes are so similar, it is important to note that if both occur in the decomposition Y (G) for a given group G, they are distinct functorially, i.e., no group automorphism can interchange the two types. Also note that {4, 2-k} = D{2, k}; in this case only, after removing valency two vertices, one obtains an equivar complex whose faces are 2-gons. While the {2, k} complexes do not arise inŶ (G) since n < 3, the {2, k} complexes are duals of the {k, 2} complexes described in Lemma 2.35. Since the doubling and duality operations are genus preserving, the {4, 2-k} complexes have genus zero (g = 0).
A Finiteness Theorem
In this section we show that except the genus one (g = 1), and the {n, 2} and {4, 2-k} families described above, for a given genus there are only finitely many distinct tesselations on the closed surface of genus g of the sort arising in Theorem 2.32. Recall that these tesselations are closed cell structures, i.e., the closed cells are homeomorphic to 2-disks, that is to say, no self-identifications occur along the boundary of the faces.
Theorem 2.36. Let g, V, E, F, n, λ 1 , λ 2 be as in Theorem 2.32. The distinct closed-cell tesselations on the closed surface of genus g which are edge and face transitive having n ≥ 3, can be categorized as follows: (i) For each fixed genus g ≥ 2, there are only finitely many possibilities for all the data (V i , E, F, n, λ i ).
(ii) For g = 1, there are only finitely many possibilities for the Schläfli symbol {n, λ} or {n, λ 1 -λ 2 }. There are infinitely many possibilities for V i , E, and F . (iii) For g = 0, there are infinite families when λ 1 = λ 2 = 2 and {n, λ 1 -λ 2 } = {4, 2-k}, k ≥ 3. Otherwise, there are only finitely many possibilities for all the data (V i , E, F, n, λ i ).
Proof. The {n, 2} (i.e. λ 1 = λ 2 = 2) and {4, 2-k} cases are described by Lemma 2.35 and the discussion that followed. Furthermore, the doubling operation shows that the remaining valence two cases where λ 1 = 2 and λ 2 = k > 2, are in one-to-one correspondence with the equivar complexes {s, k}, where s ≥ 3. Hence, we need only prove the finiteness assertions for λ 1 , λ 2 ≥ 3.
We utilize the equations presented in Theorem 2.32:
nF = 2E (1) nF = λV (2) 2(1 − g) = V − E + F = 2( 1 λ + 1 n − 1 2 )E,(3)
where λ = λ 1 = λ 2 is the common valency in the equivar case, and 1 λ = 1 2λ1 + 1 2λ2 in the case of two valencies. In general n ≥ 3, and in the two valency case n must be even (hence n ≥ 4). Let g ≥ 1. Using n ≥ 3 in Equation 3 we have
E( 1 6 − 1 λ ) ≤ g − 1 .(4)
Since V ≥ 3 we have E = λV 2 ≥ 3λ 2 . Applying this to Equation 4, we obtain λ ≤ 2(2g + 1). Note that F ≥ 3 since we assume λ ≥ 3. Then n ≤ 2(2g + 1) by the exact same argument with the roles of λ and n interchanged. In particular, this proves that there are only finitely many possibilities for λ and n in the equivar case, for each g ≥ 1. It remains to show there are only finitely many possibilities for 3 ≤ λ 1 < λ 2 in the two valency case. Using λ 1 < λ and n ≥ 4, Equation 3 becomes
E( 1 4 − 1 λ 1 ) < g − 1
Since V 1 ≥ 2 in this case, E = λ 1 V 1 ≥ 2λ 1 . Applying this we obtain 3 ≤ λ 1 < 2g + 2. Proceeding a similar manner using n ≥ 4, we have
E( 1 4 − 1 2λ 1 − 1 2λ 2 ) ≤ g − 1
and with λ 1 ≥ 3 and E ≥ 2λ 2 we obtain 3 < λ 2 ≤ 6g. So for g ≥ 1 there are finitely many possibilities for λ 1 , λ 2 , n. For g ≥ 2, all of the data (V i , E, F, n, λ i ) is determined by g, n, λ 1 , λ 2 through equations 1-3. This proves (i). For g = 1, E is not determined by g, n, λ 1 , λ 2 , and there are infinitely many possibilities for E, V, and F . This proves (ii).
For g = 0, equation 3 implies 1 λ + 1 n > 1 2 . Using n ≥ 3 we have λ < 6, and from λ ≥ 3 we obtain n < 6. So there are only finitely many possibilities for n and λ in the equivar case. In the case of two valencies 3 ≤ λ 1 < λ 2 , applying n ≥ 4 and λ 1 < λ, 1 λ + 1 n > 1 2 implies 1 λ1 + 1 4 > 1 2 or 3 ≤ λ 1 < 4. So λ 1 = 3. Using this with n ≥ 4, 1 λ + 1 n > 1 2 implies 1 6 + 1 2λ2 + 1 4 > 1 2 or λ 2 < 6. So for g = 0 there are finitely many possibilities for λ 1 , λ 2 , n. This proves (iii) since all of the data (V i , E, F, n, λ i ) is determined by g, n, λ 1 , λ 2 through Equations 1-3.
Corollary 2.37. A closed cell tesselation on the closed surface of genus 0 which is edge and face transitive having n ≥ 3, must have data (V i , E, F, n, λ i ) given by one of the rows of the following: [HP].
Proof. The proof of Theorem 2.36 provides an algorithm for finding all of the cases. We have the infinite families {n, 2} and {4, 2-k}. Considering λ 1 , λ 2 ≥ 3, in the equivar cases we found the bounds 3 ≤ n, λ ≤ 5, or n, λ = 3, 4, or 5. Then on each pair n, λ we use equation 3 to solve for E, or rule out the case if no integer solution is found. If an integer E is found, we then use equations 1 and 2 to solve for V and F . This leads to the well-known five Platonic Solids as shown in the chart. We also have the doubles of these five. In the two valency case 3 ≤ λ 1 < λ 2 we found n = 4, λ 1 = 3, λ 2 = 4 or 5. Again we use equation 3 to solve for E, then Equations 1 and 2 to solve for V and F . This provides all cases.
Corollary 2.38. If G is an odd order group then no surface of genus 0 (sphere) occurs in Y (G) and hence X(G) is a K(π, 1)-space. On the other hand, using the odd order theorem, it follows that for any nonabelian simple group G, there exists a surface of genus 0 (sphere) in the decomposition Y (G) and π 2 (X(G)) = 0.
Proof. The possible cell-structures of the surface of genus 0 that can arise in Y (G) are captured in Corollary 2.37. All of these have an even number of faces or an even number of edges which implies |G| is even if one of these occurs in Y (G) by Proposition 2.33. Thus if G is an odd order group, no spheres occur in Y (G) and hence X(G) is homotopy equivalent to a bouquet of closed surfaces of genus g ≥ 1 and circles and hence is a K(π, 1)-space, i.e., all higher homotopy groups vanish. On the other hand, if G is a nonabelian simple group, then by the odd order theorem, |G| is even and G possesses an element of order 2. If all elements of order two commuted with each other in G, they would form a normal elementary abelian subgroup which is impossible as G is simple and so there exist two noncommuting elements of order two which generate a dihedral subgroup. As spheres occur in Y (H) when H is dihedral (see the example section under dihedral groups for a proof of this), spheres occur in Y (G) also by monotonicity. Thus X(G) is homotopy equivalent to a bouquet of a positive number of spheres with a K(π, 1)-space and so has π 2 (X(G)) = 0.
Note the odd order theorem was used in the 2nd part of the argument of the last corollary. In fact, if an independent argument could be made to show that π 2 (X(G)) = 0 or equivalently that Y (G) contained a sphere when G is a nonabelian simple group then it would provide a proof of the odd order theorem.
One can similarly solve for all possible Schläfli Symbols in higher genus cases. We present the genus g = 1 case in Corollaries 2.39. Full lists of the possibilities in the regular case (which are limited to single valency) are available for genus 2 through 15 and are contained in work of Conder and Dobcsányi (see [CD]). The reader is warned that in [CD], cell structures do not have to be closed, that is to say that the interior of faces are open disks but their closure need not be disks in the space: self-identifications along the boundary are allowed. As noted in the proof of Theorem 2.36, Equation 3 does not constrain E when g = 1, and there are infinitely many possibilities for V, E, and F . For example, it is well-known that a torus can be given a closed-cell tesselation with pq squares, p, q ≥ 2, (a composite number) simply by subdividing a rectangle into a p × q grid, then gluing the top and bottom edges, and left and right edges. It is interesting to note that Σ 5 contains a g = 1 component with (V, E, F ) = (5, 10, 5) and {n, λ} = {4, 4}. That is, one can give a closed-cell tesselation on the torus using five 4-gons (a prime number), see Figure 5. In fact, this is possible with p 4-gons, for any prime p ≥ 5. For example Figure 6 shows p = 7. Note since these are closed-cell tesselations, the faces are genuinely 4-gons. (If self-identifications were allowed on the boundary of faces, then such tesselations would exist trivially.) Corollary 2.39. A closed-cell tesselation on the closed surface of genus 1 which is edge and face transitive having n ≥ 3, must have one of the following Schläfli Symbols:
{3, 6}, D{3, 6} = {6, 2-6}, {4, 4}, D{4, 4} = {8, 2-4}, {6, 3}, D{6, 3} = {12, 2-3}, {4, 3-6}
Note: All cases above exist as tesselations of the torus, however we are not sure if they all occur forŶ (G), for some group G.
Examples
Dihedral groups
Let D 2n , n ≥ 3 denote the dihedral group of order 2n. It consists of n rotations and n reflections which are elements of order 2. It is generated by two reflections σ 1 , σ 2 whose product is a rotation of order n. Since a product of an even number of reflections is a rotation and the product of an odd number of reflections is a reflection, each triangle in the triangulation of X(D 2n ) has vertices which consist of two reflections and a rotation. Let τ denote a generator of the cyclic subgroup of rotations of order n and σ a fixed reflection, then στ σ = τ −1 . The set of reflections is then {στ k |0 ≤ k < n}. The center of D 2n is trivial if n is odd and has order two generated by τ n 2 if n is even. Let τ k be a noncentral rotation. A simple computation shows that the sheets centered at the vertex (τ k , 2) have rim vertices of the form (στ s , 1), (στ s+k , 1), (στ s+2k , 1), . . . . Thus these sheets form d-gons where d is the order of τ k and there are n d of them corresponding to the cosets of the cyclic group of order d generated by τ k in the cyclic group of n rotations. Each of these sheets fits together with a corresponding sheet of its inverse (τ −k , 2) to make a sphere which is the suspension of a d-gon. In the cell structure ofŶ (D 2n ), the corresponding Schläfli symbol is {d, 2}.
Let φ(d) be Euler's Phi function, denoting the number of primitive dth roots of unity or equivalently the number of generators of a cyclic group of order d. Then when n is odd, for every divisor 1 < d|n we have φ(d) 2 pairs of elements of order d, each pair leading to n d spheres of Schläfli symbol {d, 2} as mentioned above. When n is even we only need to exclude the case d = 2 which corresponds to the nontrivial central rotation of order 2. The analysis so far accounts for all 2-sheets centered at rotations.
Now a completely similar analysis shows that a sheet about (τ k , 1) is of the form (στ s , 1), (στ s+k , 2), (στ s+2k , 1), (στ s+3k , 2), . . . and hence consists of d ′ type 1 vertices and d ′ type 2 vertices where d ′ is the order of τ 2k .The number of such distinct sheets as s varies is n d ′ and each of these fits together with a paired sheet about (τ −k , 1) to form a sphere which is the suspension of a 2d ′gon. In the corresponding cell structure inŶ (D 2n ), the 2-cells are 4-gons, the d ′ equatorial vertices have valency 2 and the north and south poles (τ ±k , 1) have valency d ′ . Thus the Schläfli symbol is {4, 2-d ′ } where d ′ is the order of τ 2k . Now when τ k has order d then τ 2k has order d ′ = d when d is odd and d ′ = d 2 when d is even.
These observations can then be put together to get the following theorem:
Theorem 3.1. Let D 2n , n ≥ 3 denote the dihedral group of order 2n. Then:
(1) All components in Y (D 2n ) are spheres (genus g = 0).
(2) If n is odd, then for any 1 < d|n there are φ(d)n
Quaternions
Let Q 8 = {±1, ±i, ±j, ±k} denote the quaternionic group of order 8. One component of Y (Q 8 ) is an octahedron with (k, 2), (−k, 2) as north and south pole and with the 4 vertices (i, 1), (j, 1), (−i, 1), (−j, 1) along the equator. There are two more similar components obtained by cyclically permuting the roles of i, j and k. In the corresponding cell structure these three spheres have Schläfli symbol {4, 2}. ThusŶ (D 8 ) andŶ (Q 8 ) are isomorphic as cell complexes. It is not hard to check that Y (D 8 ) and Y (Q 8 ) are isomorphic as simplicial complexes and so are X(D 8 ) and X(Q 8 ) which consist of three octahedra which pairwise meet in a pair of antipodal vertices. Up to homotopy equivalence we have X(Q 8 ) ≃
X(D 8 ) ≃ ∨ [3] S 2 ∨ ∨ [4] S 1 .
Extraspecial p-groups
Let p be an odd prime and F p be the field of p elements. Consider U 3 (p) the group of 3 × 3 upper triangular matrices with entries in F p and 1's on the diagonal. Thus
U 3 (p) = { 1 a b 0 1 c 0 0 1 | a, b, c ∈ F p }
It is easy to see that U 3 (p) has order p 3 and exponent p (any nonidentity element has order p). To see this just note that any matrix in U 3 (p) can be written as I + A where A is a strictly upper triangular 3 × 3 matrix and so A is nilpotent with A 3 = 0. Then (I + A) p = I follows from the binomial theorem and the fact that p 1 and p 2 are congruent to zero modulo p. G = U 3 (p) is sometimes called the extra special p-group of exponent p and order p 3 . , y] = 1 >. The abelianization of U 3 (p) is an elementary abelian p-group of rank 2 (i.e. vector space of dimension 2 over F p ) generated by the imagesx,ȳ of x and y. Thus U 3 (p) fits into a central short exact sequence:
1 → C → U 3 (p) → E = F p × F p → 1
where C is the center of U 3 (p) and is cyclic of order p. It is clear from this short exact sequence that C = F rat(G), the Frattini subgroup of G. Corresponding to this extension is a commutator map [·, ·] : E × E → C which takes two elements of E, lifts them to U 3 (p) and looks at their commutator which lies in C. This map is readily checked to be well-defined and bilinear, alternating, (see for example [BrP] for details). From this map it follows that any two elements u, v that do not commute in U 3 (p) must map to a basis in E under the abelianization and hence generate U 3 (p) by properties of Frattini quotients. They clearly satisfy the same presentation that x and y did and hence there must be an automorphism of U 3 (p) taking any noncommuting pair of elements to any other noncommuting pair of elements. Finally let us note that if α and β are conjugate in U 3 (p) they must map to the same element in the abelianization E. Thus β = αc where c is some element in the center C. In particular conjugate elements commute in U 3 (p) and every noncentral element has exactly p elements in its conjugacy class. In fact if x is a non central element, xC is its conjugacy class.
We summarize these observations in the next lemma as we will use them in determining the 2-cell structure of the components ofŶ (U 3 (p)).
Lemma 3.3. Let p be an odd prime and G = U 3 (p) be the extraspecial group of order p 3 and exponent p.
(1) If (u, v), (u ′ , v ′ ) are pairs of noncommuting elements in G, then there exists an automorphism φ of G such that φ(u) = u ′ and φ(v) = v ′ .
(2) If α, β are conjugate in G, then α and β commute.
(3) Every noncentral element α has exactly p elements in its conjugacy class.
(3) If α does not commute with x then the α-conjugacy class of x consists of exactly p elements.
Proof. Parts (1), (2) and (3) were proved in the paragraph before the lemma. Part (4) follows as the size of an α-conjugacy class must divide the order of α. If α does not commute with x then α is not the identity element and hence has order p and the α-conjugacy class of x must have size > 1 and dividing the order of α. As |α| = p is prime, this size must be p.
The next theorem determines the structure ofŶ (U 3 (p)) completely.
Theorem 3.4. Let p be an odd prime and G = U 3 (p), the extra special p-group of order p 3 and exponent p. Then inŶ (G) we have:
(1) All components are isomorphic as cell-complexes and there are (p 2 −1)(p 2 −p) 2 of them.
(2) The cell structure of each component ofŶ (G) is a regular abstract 3-polytope which Schläfli symbol {2p, p}.
(3) These cell structures hence tessellate the Riemann surface of genus g = p(p−3) 2 + 1 with 2p-gons. The face, edge and vertex count of this regular tesselation is given by F = p, E = p 2 , V = 2p and vertex valency p.
Proof. If T is the component of Y (G) determined by the triangle [(x, 1), (y, 1), (xy, 2)] and T ′ is the component of Y (G) determined by the triangle [(u, 1), (v, 1), (uv, 2)], then Lemma 3.3 guarantees the existence of a group automorphism and hence simplicial automorphism of Y (G) (and also X(G)) which takes one triangle to the other and hence induces a simplicial isomorphism of the component T with the component T ′ . This simplicial isomorphism induces a cell-isomorphism between the cell structures of these two components inŶ (G) also.
In the component determined by the triangle [(x, 1), (y, 1), (xy, 2)], all type 2vertices are conjugate to xy. Since the x-conjugacy class of xy must all occur as type 2-vertices by Proposition 2.21 there are at least p of these in the component. However xy only has p-conjugates so all conjugates of xy occur. As x and y do not commute, they are not conjugate by Lemma 3.3. Thus by Proposition 2.19, the sheets in the closed star of (xy, 2) in that component consist of two distinct xy-conjugacy classes of triangles each of order p and hence forms a 2p-gon in the corresponding cell-structure. Now there is a 2p-gon face in the component for each type-2 vertex in that component and these consist of the p conjugates of (xy, 2) with possible multiplicities due to the desingularization process (there can be more than one type-2 vertex labeled with the same group element in a given component because of the desingularization).
We wish to show there is no multiplicity of type 2-vertices in a fixed component and hence that there are exactly p of these faces. To do this, by symmetry (we have already shown there are automorphisms which will take any triangle to any other in X(G)), it is enough to show that there is exactly one sheet about a type-2 vertex labelled (xy, 2) in the pseudo-manifold component of X(G) that contains the triangle [(x, 1), (y, 1), (xy, 2)] (before desingularization). In other words there is only one disk in the closed star of (xy, 2) which is a bouquet of disks that lies in that pseudomanifold component. Note that all the type-1 vertices in the given pseudomanifold-component are conjugate to either x or y by Theorem 2.30 and hence of the form xz a , yz b where z is a generator of the center and 0 ≤ a, b < p. Thus any triangle in a sheet about a type-2 vertex labeled (xy, 2) in the given pseudo-manifold component of X(G) (before desingularization) has to be of the form [(xz a , 1), (yz b , 1), (xy, 2)] or of the form [(yz b , 1), (y −1 xyz a , 1), (xy, 2)] which forces b ≡ −a mod p. This means there there are at most 2p such triangles and hence exactly one such sheet in that pseudo manifold component. Thus there are no multiply labelled type 2-vertices in a given component in Y (G) and we can conclude that each component ofŶ (G) has cell-structure given by exactly p many 2p-gon faces.
By part (5) of Theorem 2.34 and Lemma 3.3 we see that the cell structure of each component forms a regular abstract 3-polytope. By Theorem 2.32 the vertex valency is the size of the x-conjugacy class of xy which is p. Thus this regular abstract 3-polytope has Schläfli symbol {2p, p}. As F = p, the formulas in part (3) of the same theorem can then be used to yield the stated values of E, V and g.
As V = 2p in the cell structure, we see that the p conjugates of (x, 1) and (y, 1) occur without multiplicity in the component (alternatively one can mimic the proof used for type 2-vertices earlier). Thus each component consists exactly of two noncommuting conjugacy classes in U 3 (p) and is determined by the unordered pair of these. These pairs in turn are determined uniquely by the two linearly independent vectors in the abelianization E of U 3 (p) that they project to. Thus the number of components of Y (U 3 (p)) is the same as the number of unordered pairs of noncommuting conjugacy classes which is the same as the number of unordered basis of E, a F p -vector space of dimension 2. This number is easily computed as (p 2 −1)(p 2 −p) 2 . Note when p = 3 in the last theorem, we see that U 3 (3) provides a regular tesselation of the torus by 3 hexagons with vertex valancy 3. When p = 5, U 3 (5) provides a regular tesselation of the surface of genus 6 by five 10-gons with vertex valancy 5. Corollary 3.5. As one varies the construction X(G) over all finite nonabelian groups, the set of genuses of components that occur is infinite.
Proof. Theorem 3.4 shows that the set of genuses obtained when looking at extraspecial groups over all odd primes p is infinite and so the corollary follows.
For G a group, let Aut(G) denote the extended automorphism group which consists of automorphisms and anti-automorphims of the group G under composition. Aut(G) is a subgroup of Aut(G) of index 2 (hence normal) as any anti-automorphism of G is the composition of the inversion map I(x) = x −1 with an automorphism. Furthermore it is easy to verify that if φ is an automorphism then φ(x −1 ) −1 = φ(x) and so φ commutes with I. Thus I generates a central subgroup of order 2 complementary to Aut(G) in Aut(G) and so Aut(G) ∼ = Aut(G) × Z/2Z.
The anti-automorphism group of U 3 (p) acts transitively on the set of components ofŶ (G). If H is the stabilizer of a component in this action, then H acts transitively on the flags in the cell-structure of this component by Theorem 2.34 and its proof. As the group elements represented by the vertices of this component generate the group U 3 (p), this action is also faithful. Thus H is the abstract flag symmetry group of the regular abstract 3-polytope represented by this component. Just as the isometry group of classical regular polyhedra are reflection groups, it has been shown that the automorphism groups of abstract regular polyhedra are generated by involutions. It follows from this theory (see [MS]) that H is a C-string group generated by three involutions ψ 0 , ψ 1 , ψ 2 such that (ψ 0 ψ 1 ) 2p = 1 and (ψ 1 ψ 2 ) p = 1 and ψ 0 , ψ 2 commute.
Thus we have obtained the following corollary also:
Corollary 3.6. Let p be an odd prime and G = U 3 (p) be the extraspecial group of exponent p and order p 3 . Then there exists a subgroup H of Aut(G)×Z/2Z of index (p 2 −1)(p 2 −p) 2 such that H is a C-string group generated by three involutions ψ 0 , ψ 1 , ψ 2 which satisfy (ψ 0 ψ 1 ) 2p = 1 = (ψ 1 ψ 2 ) p and ψ 0 , ψ 2 commute.
. . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Closed stars of vertices . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Main Structure Theorem . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 The Euler characteristic of X(G) . . . . . . . . . . . . . . . . . . 12
Figure 1 :
1The
Figure 2 :
2A representative simplicial complex X(G).
Figure 3 :
3The resulting desingularization Y (G) of the complex X(G) in figure 2.
( 3 )
3In the case of one vertex orbit λ ≥ 2, V, n ≥ 3 are divisors of 2E and hence of 2|G|.(4) If either E or F is equal to |G| then the component T must be invariant under conjugation by G and G must have trivial center. In fact if the edge [(x, 1), (y, 1)] lies in the component then C(x) ∩ C(y) = {1}.(5) There are finitely many possibilities for all the data of the component (g, V i , E, F, λ i , n) given |G| determined by these simple divisibility conditions.
Figure 5 :
5A tesselation of the Torus using five 4-gons.
Figure 6 :
6A tesselation of the Torus using seven 4-gons.
symbol {d, 2} and another φ(d)n 2d spherical components with Schläfli symbol {4, 2-d} = D{2, d}. (3) If n is even, then for any 3 ≤ d|n there are φ(d)n 2d spherical components with Schläfli symbol {d, 2}. For every odd such d there is another φ(d)n 2d spherical components with symbol {4, 2-d} = D{2, d} and for every even such d there is another φ(d)n d spherical components with symbol {4, 2-d 2 }. Corollary 3.2.Ŷ (D 6 ) =Ŷ (Σ 3 ) consists of two spheres with cell structure of type {3, 2} and {4, 2-3} = D{2, 3}. Y (D 8 ) consists of three spheres with cell structure of type {4, 2}.
x and y generate U 3 (p) and have commutator [x, y] = xyx −1 y is central and in fact generates the center of U 3 (p) which is a cyclic group of order p. From this one can easily verify that U 3 (p) has presentation < x, y|x p = y p = [x, y] p = [[x, y], x] =[[x, y]
Table 1 :
1Possible Tesselations on the Riemann Surface of Genus 0# Faces
Schläfli Symbol
# Vertices # Edges
Solid Type
2
{n, 2}
n
n
dual Hosahedron
k
{4, 2-k}
k + 2
2k
double Hosahedron
4
{3, 3}
4
6
Tetrahedron
4
{6, 2-3} = D{3, 3}
10
12
double Tetrahedron
6
{4, 3}
8
12
Cube
6
{8, 2-3} = D{4, 3}
20
24
double Cube
8
{3, 4}
6
12
Octahedron
8
{6, 2-4} = D{3, 4}
18
24
double Octahedron
12
{5, 3}
20
30
Dodecahedron
12
{10, 2-3} = D{5, 3}
50
60
double Dodecahedron
12
{4, 3-4}
14
24
Rhombic Dodecahedron
20
{3, 5}
12
30
Icosahedron
20
{6, 2-5} = D{3, 5}
42
60
double Icosahedron
30
{4, 3-5}
32
60
Rhombictriacontahedron
Note: All cases in table 1 are realized in our construction, in particular these
cases exist. For example, these are generated by Σ 5 , see table 9 of
Cambridge Studies in advanced mathematics. D J Benson, Representations and Cohomology II. Cambridge University Press31D.J. Benson, Representations and Cohomology II, Cambridge Studies in advanced mathematics, 31, Cambridge University Press 1991.
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Cohomology of Groups. K S Brown, Springer Verlag GTM87New York-Heidelberg-BerlinK. S. Brown, Cohomology of Groups, Springer Verlag GTM 87, New York-Heidelberg-Berlin, 1994.
Determination of all regular maps of small genus. M Conder, P Dobcsányi, J. Combinatorial Theory, Series B. 81M. Conder, P. Dobcsányi, Determination of all regular maps of small genus, J. Combinatorial Theory, Series B, 81 (2001), 224-242.
H S M Coxeter, Regular Polytopes. Dover Publications IncH. S. M. Coxeter, Regular Polytopes, Dover Publications Inc., (1973).
H S M Coxeter, Regular Complex Polytopes. Cambridge University PressH. S. M. Coxeter, Regular Complex Polytopes, Cambridge University Press, (1991).
On a canonical construction of tesselated surfaces via finite group theory, Part II. M Herman, J Pakianathan, PreprintM. Herman, J. Pakianathan, On a canonical construction of tesselated surfaces via finite group theory, Part II, Preprint.
Groups of small strong symmetric genus. C May, J Zimmerman, J. Group Theory. 33C. May, J. Zimmerman, Groups of small strong symmetric genus, J. Group Theory, 3(3) (2000), 233-245.
Abstract Regular Polytopes. P Mcmullen, E Schulte, of Encyclopedia of Mathematics and It's Applications. CambridgeCambridge University Press92P. McMullen, E. Schulte, Abstract Regular Polytopes, Volume 92 of Encyclopedia of Mathematics and It's Applications, Cambridge Uni- versity Press, Cambridge, 2002.
. Munkres, Elements of Algebraic Topology. Addison-Wesley Publishing Co., IncMunkres, Elements of Algebraic Topology, Addison-Wesley Publishing Co., Inc., 1984.
On Commuting and Non-Commuting Complexes. J Pakianathan, E Yalcın, Journal of Algebra. 236J. Pakianathan, E. Yalcın, On Commuting and Non-Commuting Com- plexes, Journal of Algebra, 236 (2001), 396-418.
Finite Groups Acting on Surfaces and the Genus of a Group. T W Tucker, Journal of Combinatorial Theory, Series B. 34T. W. Tucker, Finite Groups Acting on Surfaces and the Genus of a Group, Journal of Combinatorial Theory, Series B, 34, (1983) 82-98.
E-mail address: yalcine@fen. Ankara, TurkeyDept. of Mathematics Bilkent Universitybilkent.edu.trDept. of Mathematics Bilkent University, Ankara, Turkey. E-mail address: [email protected]
| []
|
[
"DYNAMICAL SYSTEMS AND UNIFORM DISTRIBUTION OF SEQUENCES",
"DYNAMICAL SYSTEMS AND UNIFORM DISTRIBUTION OF SEQUENCES"
]
| [
"Manfred G Madritsch ",
"Robert F Tichy "
]
| []
| []
| We give a survey on classical and recent applications of dynamical systems to number theoretic problems. In particular, we focus on normal numbers, also including computational aspects. The main result is a sufficient condition for establishing multidimensional van der Corput sets. This condition is applied to various examples. | 10.1007/978-3-319-28203-9_17 | [
"https://arxiv.org/pdf/1501.07411v1.pdf"
]
| 119,160,376 | 1501.07411 | 6b6404e582197afa3ce6f93e5837346d8a3a9f66 |
DYNAMICAL SYSTEMS AND UNIFORM DISTRIBUTION OF SEQUENCES
29 Jan 2015
Manfred G Madritsch
Robert F Tichy
DYNAMICAL SYSTEMS AND UNIFORM DISTRIBUTION OF SEQUENCES
29 Jan 2015Dedicated to the memory of Professor Wolfgang Schwarz
We give a survey on classical and recent applications of dynamical systems to number theoretic problems. In particular, we focus on normal numbers, also including computational aspects. The main result is a sufficient condition for establishing multidimensional van der Corput sets. This condition is applied to various examples.
Dynamical systems in number theory
In the last decades dynamical systems became very important for the development of modern number theory. The present paper focuses on Furstenberg's refinements of Poincaré's recurrence theorem and applications of these ideas to Diophantine problems.
A (measure-theoretic) dynamical system is formally given as a quadruple (X, B, µ, T ), where (X, B, µ) is a probability space with σ-algebra B of measurable sets and µ a probability measure; T : X → X is a measure-preserving transformation on this space, i.e. µ(T −1 A) = µ(A) for all measurable sets A ∈ B. In the theory of dynamical systems, properties of the iterations of the transformation T are of particular interest. For this purpose we only consider invertible transformations and call such dynamical systems invertible.
The first property, we consider, originates from Poncaré's famous recurrence theorem (see Theorem 1.4 of [32] or Theorem 2.11 of [13]) saying that starting from a set A of positive measure µ(A) > 0 and iterating T yields infinitely many returns to A. More generally, we call a subset R ⊂ N of the positive integers a set of recurrence if for all invertible dynamical systems and all measurable sets A of positive measure µ(A) > 0 there exists n ∈ R such that µ(A ∩ T −n A) > 0. Then Poincaré's recurrence theorem means that N is a set of recurrence.
A second important theorem for dynamical systems is Birkhoff's ergodic theorem (see Theorem 1.14 of [32] or Theorem 2.30 of [13]). We call T ergodic if the only invariant sets under T are sets of measure 0 or of measure 1, i.e. T −1 A = A implies µ(A) = 0 or µ(A) = 1. Then Birkhoff's ergodic theorem connects average in time with average in space, i.e.
lim N →∞ 1 N N −1 n=0 f • T n (x) = X f (x)dµ(x)
for all f ∈ L 1 (X, µ) and µ-almost all x ∈ X.
Let us explain an important application of this theorem to number theory. For q ≥ 2 a positive integer, consider T :
[0, 1) → [0, 1) defined by T (x) = {qx}, where {t} = t − ⌊t⌋ denotes the fractional part of t. If x ∈ R is given by its q-ary digit expansion x = ⌊x⌋ + ∞ j=1 a j (x)q −j , then the digits a j (x) can be computed by iterating this transformation T : a j (x) = i if T j−1 x ∈ i q , i+1 q with i ∈ {0, 1, . . . , q − 1}.
Moreover, since a j (T x) = a j+1 (x) for j ≥ 1 the transformation T can be seen as a left shift of the expansion. Now we call a real number x simply normal in base q if lim N→∞ 1 N #{j ≤ N : a j = d} = 1 q for all d = 0, . . . , q −1, i.e. all digits d appear asymptotically with equal frequencies 1/q. A number x is called q-normal if it is simply normal with respect to all bases q, q 2 , q 3 , . . .. This is equivalent Date: January 30, 2015. to the fact that the sequence ({q n x}) n∈N is uniformly distributed modulo 1 (for short: u.d. mod 1), which also means that all blocks d 1 , d 2 , . . . , d L of subsequent digits appear in the expansion of x asymptotically with the same frequency q −L (cf. [8,12,16]). For completeness, let us give here one possible definition of u.d. sequences (x n ): a sequence of real numbers x n is called u.d. mod 1 if for all continuous functions f :
[0, 1] → R (1.1) lim N →∞ 1 N N n=1 f (x n ) = 1 0 f (x)dx.
Note, that by Weyl's criterion the class of continuous functions can be replaced by trigonometric functions e(hx) = e 2πihx , h ∈ N or by characteristic functions 1 I (x) of intervals I = [a, b). Applying Birkhoff's ergodic theorem, shows that Lebesgue almost all real numbers are q-normal in any base q ≥ 2. Defining a real number to be absolutely normal if it is q-normal for all bases q ≥ 2, this immediately yields that almost all real numbers are absolutely normal.
In particular, this shows the existence of absolutely normal numbers. However, it is a different story to find constructions of (absolutely) normal numbers. It is a well-known difficult open problem to show that important numbers like √ 2, ln 2, e, π etc. are simply normal with respect to some given base q ≥ 2. A much easier task is to give constructions of q-normal numbers for fixed base q. Champernowne [9] proved that 0.1 2 3 4 5 6 7 8 9 10 11 12 . . . is normal to base 10 and later this type of constructions was analysed in detail. So, for instance, for arbitrary base q ≥ 2
0. ⌊g(1)⌋ q ⌊g(2)⌋ q . . . is q-normal, where g(x)
is a non-constant polynomial with real coefficients and the q−normal number is constructed by concatenating the q−ary digit expansions ⌊g(n)⌋ q of the integer parts of the values g(n) for n = 1, 2, . . .. These constructions were extended to more general classes of functions g (replacing the polynomials) (see [11,17,18,22,23,29]) and the concatenation of [g(p)] q along prime numbers instead of the positive integers (see [10,18,19,24]).
All such constructions depend on the choice of the base number q ≥ 2, and thus they are not suitable for constructing absolutely normal numbers. A first attempt to construct absolutely normal numbers is due to Sierpinski [30]. However, Turing [31] observed that Sierpinski's "construction" does not yield a computable number, thus it is not based on a recursive algorithm. Furthermore, Turing gave an algorithm for a construction of an absolutely normal number. This algorithm is very slow and, in particular, not polynomially in time. It is very remarkable that Becher et al. [2] established a polynomial time algorithm for the construction of absolutely normal numbers. However, there remain various questions concerning the analysis of these algorithms. The discrepancy of the corresponding sequences is not studied and the order of convergence of the expansion is very slow and should be investigated in detail. Furthermore, digital expansions with respect to linear recurring base sequences seam appropriate to be included in the study of absolute normality from a computational point of view.
Let us now return to Poincaré's recurrence theorem which shortly states that the set N of positive integers is a recurrence set. In the 1960s various stronger concepts were introduced:
(i) R ⊆ N is called a nice recurrence set if for all invertible dynamical systems and all measurable sets A of positive measure µ(A) > 0 and all ε > 0, there exist infinitely many n ∈ R such that
µ(A ∪ T −n A) > µ(A) 2 − ε.
(ii) H ⊆ N is called a van der Corput set (for short: vdC set) if for all h ∈ H the following implications holds:
(x n+h − x n ) n∈N is u.d. mod 1 =⇒ (x n ) n∈N is u.d. mod 1.
Clearly, any nice recurrence set is a recurrence set. By van der Corput's difference theorem (see [12,16]) the set H = N of positive integers is a vdC set. Kamae and Mendès-France [15] proved that any vdC set is a nice recurrence set. Ruzsa [25] conjectured that any recurrence set is also vdC. An important tool in the analysis of recurrence sets is their equivalence with intersective (or difference) sets established by Bertrand-Mathis [5]. We call a set I intersective if for each subset E ⊆ N of positive (upper) density, there exists n ∈ I such that n = x − y for some x, y ∈ E. Here the upper density of E is defined as usual by
d(E) = lim sup N →∞ #(E ∩ [1, N ]) N .
Bourgain [7] gave an example of an intersective set which is not a vdC set, hence contradicting the above mentioned conjecture of Ruzsa. Furstenberg [14] proved that the values g(n) of a polynomial g ∈ Z[x] with g(0) = 0 form an intersective set and later it was shown by Kamae and Mendès-France [15] that this is a vdC set, too. It is also known, that for fixed h ∈ Z the set of shifted primes {p ± h : p prime} is a vdC set if and only if h = ±1. ( [20,Corollary 10]). This leads to interesting applications to additive number theory, for instance to new proofs and variants of theorems of Sárkőzy [26][27][28]. A general result concerning intersective sets related to polynomials along primes is due to Nair [21].
In the present paper we want to extend the concept of recurrence sets, nice recurrence sets and vdC sets to subsets of Z k , following the program of Bergelson and Lesigne [4] and our earlier paper [3]. In section 2 we summarize basic facts concerning these concepts, including general relations between them and counter examples. Section 3 is devoted to a sufficient condition for establishing the vdC property. In the final section 4 we collect various examples and give some new applications.
Van der Corput sets
In this section we provide various equivalent definitions of van der Corput sets in Z k . In particular, we give four different definitions, which are k-dimensional variants of the one dimensional definitions, whose equivalence is due to Ruzsa [25]. These generalizations were established by Bergelson and Lesigne [4]. Then we present a set, which is not a vdC set in order to give some insight into the structure of vdC sets. Finally, we define the higher-dimensional variant of nice recurrence sets.
2.1.
Characterization via uniform distribution. Similarly to above we first define a van der Corput set (vdC set for short) in Z k via uniform distribution.
Definition 2.1. A subset H ⊂ Z k \ {0}
is a vdC set if any family (x n ) n∈N k of real numbers is u.d. mod 1 provided that it has the property that for all h ∈ H the family (x n+h − x n ) n∈N k is u.d. mod 1.
Here the property of u.d. mod 1 for the multi-indexed family (x n ) n∈N k is defined via a natural extension of 1.1:
(2.1) lim N1,N2,...,N k →+∞ 1 N 1 N 2 · · · N k 0≤n<(N1,N2,...,N k ) f (x n ) = 1 0 f (x)dx
for all continuous functions f : [0, 1] → R. Here in the limit N 1 , N 2 , . . . , N k are tending to infinity independently and < is defined componentwise.
Using the k-dimensional variant of van der Corput's inequality we could equivalently define a vdC set as follows:
Definition 2.2. A subset H ⊂ Z k \ {0}
is a van der Corput set if for any family (u n ) n∈Z k of complex numbers of modulus 1 such that ∀h ∈ H, lim N1,N2,...,N k →+∞ 1 N 1 N 2 · · · N k 0≤n<(N1,N2,...,N k ) u n+h u n = 0 the relation lim N1,N2,...,N k →+∞ 1 N 1 N 2 · · · N k 0≤n<(N1,N2,...,N k ) u n = 0 holds.
2.2.
Trigonometric polynomials and spectral characterization. The first two definitions are not very useful for proving or disproving that a set H is a vdC set. Similar to the one dimensional case the following spectral characterization involving trigonometric polynomials is a better tool. if for all ε > 0, there exists a real trigonometric polynomial P on the k-torus T k whose spectrum is contained in H and which satisfies P (0) = 1, P ≥ −ε.
The set of polynomials fulfilling the last theorem for a given ε forms a convex set. Moreover the conditions may be interpreted as some infimum. Therefore we might expect some dual problem, which is actually provided by the following theorem. For details see Bergelson and Lesigne [4] or Montgomery [20]. If H is a vdC set, then H 1 or H 2 also has to be a vdC set. Suppose there exists a q ∈ N such that H ∩ qN is finite. Then we may split H into the sets H ∩ qN and H \ qN. The first one is finite and the second one contains no multiples of q. Therefore both are not vdC sets and hence H is not a vdC set.
The first counter example deals with arithmetic progressions. The sufficiency (and also the necessity) of the requirement a | b follows from the following result of Kamae and Mendès-France [15] (cf. Corollary 9 of [20]). Lemma 2.5. Let P (z) ∈ Z[z] and suppose that P (z) → +∞ as z → +∞. Then H = {P (n) > 0 : n ∈ N} is a vdC set if and only if for every positive integer q the congruence P (z) ≡ 0 (mod q) has a root. Now we want to establish a similar result for sets of the form {ap + b : p prime}. In this case the following result is due to Bergelson and Lesigne [4] which is a generalization of the case f (x) = x due to Kamae and Mendès-France [15]. Proof. It is clear from Lemma 2.6 that {ap + b : p ∈ P} is a vdC set if |a| = |b|.
On the contrary a combination of Lemma 2.3 and Lemma 2.4 yields that a | b. Now we consider the sequence modulo b. Then by Lemma 2.3 we get that ap + b ≡ ap ≡ 0 mod b infinitely often.
Since (p, b) > 1 only holds for finitely many primes p we must have b | a. Combining these two requirements yields |a| = |b|.
A sufficient condition
In this section we want to formulate a general sufficient condition which provides us with a tool to show for plenty of different examples that they generate a vdC set. This is a generalization of the conditions of Kamae and Mendès-France [15] and Bergelson and Lesigne [4]. Before stating the condition we need an auxiliary lemma. Our main tool is the following general result. Applications are given in the next section.
Proposition 3.2. Let g 1 , . . . , g k : N → Z be arithmetic functions. Suppose that g i1 , . . . , g im is a basis of the Q-vector space span(g 1 , . . . , g k ). For each q ∈ N, we introduce D q := (g i1 (n), . . . , g im (n) : n ∈ N and q! | g ij (n) for all j = 1, . . . , m .
Suppose further that, for every q, there exists a sequence (h (q) n ) n∈N in D q such that, for all x = (x 1 , . . . , x m ) ∈ R m \ Q m , the sequence (h (q) n · x) n∈N is uniformly distributed mod 1. Then D := {(g 1 (n), . . . , g k (n)) : n ∈ N} ∈ Z k is a vdC set.
Proof. We first show that the set D := {(g i1 (n), . . . , g im (n)) : n ∈ N} is a vdC set in Z m . For q, N ∈ N we define a family of trigonometric polynomials
P q,N := 1 N N n=1 e h (q) n · x .
By hypothesis, lim N →∞ P q,N (x) = 0 for x ∈ Q m . For fixed q there exists a subsequence (P q,N ′ ) which converges pointwise to a function g q . Since g q (x) = 1 (for x ∈ Q m and q sufficiently large) and g q (x) = 0 (for x ∈ Q m ), the sequence (g q ) is pointwise convergent to the indicator function of Q m . For a positive measure σ on the m-dimensional torus with vanishing Fourier transform σ on D, we have P q,N dσ = 0 for all q, N . Thus σ(Q m ) = 0 follows from the dominating convergence theorem, obviously σ({0, 0, . . . , 0}) = 0, and thus D is a vdC set.
In order to prove that D is a vdC set we apply Lemma 3.1 twice. Since g i1 , . . . , g im is a base of span(g 1 , . . . , g k ), we can write each g j as a linear combination (with rational coefficients) of g i1 , . . . , g im . Multiplying with the common denominator of the coefficients yields a j g j = b j,1 g i1 + · · · + b j,m g im for j = 1, . . . , k and certain a j , b j,ℓ ∈ Z. Considering the transformation L : Z m → Z k given by the matrix (b j,ℓ ) and applying part (1) of Lemma 3.1 shows that {(a 1 g 1 (n), . . . , a k g k (n)) : n ∈ N} is a vdC set for certain integers a 1 , . . . , a k . Now consider the transformation L : Z k → Z k given by the k × k diagonal matrix with entries a 1 , . . . , a k in the diagonal. Then by part (2) of Lemma 3.1 also D is a vdC set and the proposition is proved.
Various examples and applications to additive problems
In this section we consider multidimensional variants of prime powers, entire functions and x α log β x sequences. 4.1. Prime powers. In a recent paper the authors together with Bergelson, Kolesnik and Son [3] consider sets of the form
{(α 1 (p n ± 1) θ1 , . . . , α k (p n ± 1) θ k ) : n ∈ N},
where α i , β i ∈ R and p n ∈ P runs over all prime numbers. These sets are vdC, however, we missed the treatment of a special case in the proof. In particular, if for some i = j the exponents satisfy θ i = θ j =: θ, then the vector (p θ n , p θ n ) is not uniformly distributed mod 1. Here we close this gap. The central tool is the following result of Baker [1]. Theorem 2]). Let f be a transcendental entire function of logarithmic order 1 < λ < 4 3 . Then the sequence (f (p n )) n≥1
D 1 = { (p − 1) α1 , · · · , (p − 1) α k , [(p − 1) β1 ], · · · , [(p − 1) β ℓ ] | p ∈ P},Theorem 4.2 ( [1,
is uniformly distributed mod 1.
Our second example of a class of vdC sets is the following. First we show, that for every q ∈ N the set
D (q) := {(d 1 , . . . , d k ) ∈ D : q | d i }
has positive relative density in D. We note that if 0 ≤ fi(pn) q < 1 q for 1 ≤ i ≤ k, then d n ∈ D (q) . By Theorem 4.2 the sequence
f 1 (p n ) q , . . . , f k (p n ) q n≥1
is uniformly distributed and thus D (q) has positive density in D.
For each q ∈ N we enumerate the elements of D (q!) = (d
(q!) n ) n≥1 , such that d (q!) n
is increasing. Since the logarithmic orders are distinct we immediately get that the functions f i are Q-linearly independent. Thus by Proposition 3.2 it is sufficient to show that for all q ∈ N and all x = (x 1 , . . . ,
x k ) ∈ R k \ Q k the sequence (d (q!) n · x) n≥1 is u.d. mod 1.
Using the orthogonality relations for additive characters we get for any non-zero integer h, that
1 {n ≤ N : d n ∈ D (q!) } n≤N e h d (q!) n · x = 1 {n ≤ N : d n ∈ D (q!) } 1 (q!) k q! j1=1 · · · q! j k =1 1 N n≤N e d n · hx + j 1 q! , . . . , j k q! .
The innermost sum is of the form n≤N e(g(p n )), with g(x) = k i=1 α i ⌊f i (x)⌋ for a certain (α 1 , . . . , α k ) ∈ R k \ Q k . By relabeling the terms we may suppose that there exists an ℓ such that α 1 , . . . , α ℓ ∈ Q and α ℓ+1 , . . . , α k ∈ Q. Furthermore we may write α j = aj q for ℓ + 1 ≤ j ≤ m. Then
e(g(p n )) = e k i=1 α k ⌊f i (p n )⌋ = ℓ j=1 s j (α j f j (p n ), f j (p n )) k j=ℓ+1 t j (⌊f j (p n )⌋),
where s j (x, y) = e(x − {y}α j ) (1 ≤ j ≤ ℓ) and t j (z) = e a j z q (ℓ + 1 ≤ j ≤ k).
Since s j (x, y) is Riemann-integrable on T 2 for j = 1, . . . , ℓ and t j (z) is continuous on Z q = Z/qZ, the function
ℓ j=1 s j k j=ℓ+1 t j is Riemann-integrable on T 2ℓ × Z k−ℓ q .
Now an application of Theorem 4.2 yields that for any u ∈ N the sequence α 1 f 1 (p n ), f 1 (p n ), . . . , α ℓ f ℓ (p n ), f ℓ (p n ), f ℓ+1 (p n ) u , . . . , f k (p n ) u n≥1 is u.d. in T 2ℓ × T k−ℓ . Since ⌊x⌋ ≡ a (mod q) is equivalent to x q ∈ [ a q , a+1 q ], we deduce that (α 1 f 1 (p n ), f 1 (p n ), . . . , α ℓ f ℓ (p n ), f ℓ (p n ), ⌊f ℓ+1 (p n )⌋, . . . , ⌊f k (p n )⌋) n≥1 is u.d. in T 2ℓ × Z k−ℓ q , and Weyl's criterion implies that
4.3.
Functions of the form x α log β x. In the one-dimensional case Boshernitzan et al. [6] showed, among other things, that these sets are vdC sets. Our aim is to show an extended result for the k-dimensional case. Therefore we use the following general criterion, which is a combination of Fejer's theorem and van der Corput's difference theorem. Theorem 4.4 ( [16,Theorem 3.5]). Let f (x) be a function defined for x > 1 that is k-times differentiable for x > x 0 . If f (k) (x) tends monotonically to 0 as x → ∞ and if lim x→∞ x f (k) (x) = ∞, then the sequence (f (n)) n≥1 is u.d. mod 1.
Applying this theorem we get the following Corollary 4.5. Let α = 0 and • either σ > 0 not an integer and τ ∈ R arbitrary • or σ > 0 an integer and τ ∈ R \ [0, 1]. Then the sequence (αn σ log τ n) n≥2 is u.d. mod 1.
Our third example is the following class of vdC sets. Theorem 4.6. Let α 1 , . . . , α k > 0 and β 1 , . . . , β k ∈ R, such that β i ∈ [0, 1] whenever α i ∈ Z for i = 1, . . . , k. Then the set D := {(⌊n α1 log β1 n⌋, . . . , ⌊n α k log β k n⌋) : n ∈ N} is a vdC set.
Proof. Following the same arguments as is the proof of Theorem 4.3 and replacing the uniform distribution result for entire functions (Theorem 4.3) by the corresponding result for n α log β n sequences (Corollary 4.5) yields the proof.
Theorem 2. 2 ( [ 4 ,
24Theorem 1.8]). Let H ⊂ Z k \ {0}. Then H is a van der Corput set if and only if for any positive measure σ on the k-torus T k such that, for all h ∈ H, σ(h) = 0, this implies σ({(0, 0, . . . , 0)}) = 0. 2.3. Examples. The structure of vdC sets is better understood by first giving a counter example. The following lemma shows to be very useful in the construction of counter examples. Lemma 2.3. Let H ⊂ N. If there exists q ∈ N such that the set H ∩ qN is finite, then the set H is not a vdC set. Proof. The proof is a combination of the following two observations of Ruzsa [25] (see Theorem 2 and Corollary 3 of [20]): (1) Let m ∈ N. The sets {1, . . . , m} and {n ∈ N : m ∤ n} are both not vdC sets. (2) Let H = H 1 ∪ H 2 ⊂ N.
Lemma 2 . 4 .
24Let a, b ∈ N. If the set {an + b : n ∈ N} is a vdC set, then a | b.Proof. Let b ∈ N and H = {an + b : n ∈ N} be a vdC set. Then by Lemma 2.3 we must havean + b ≡ b ≡ 0 mod a infinitely often.This implies that a | b.
1.22]). Let f be a (non zero) polynomial with integer coefficients and zero constant term. Then the sets {f (p − 1) : p ∈ P} and {f (p + 1) : p ∈ P} are vdC sets in Z.We show the converse direction.
Lemma 2 . 7 .
27Let a and b be non-zero integers. Then the set {ap + b : p ∈ P} is a vdC set if and only if |a| = |b|, i.e. ap + b = a(p ± 1).
1.15]). Let d and e be positive integers, and let L be a linear transformation from Z d into Z e (represented by an e × d matrix with integer entries). Then the following assertions hold: (1) If D is a vdC set in Z d and if 0 ∈ L(D), then L(D) is a vdC set in Z e . (2) Let D ∈ Z d . If the linear map L is one-to-one, and if L(D) is a vdC set in Z e , then D is a vdC set in Z d .
Theorem 4 . 1 .
41If α i are positive integers and β i are positive and non-integers, then
and D 2 = { (p + 1) α1 , · · · , (p + 1) α k , [(p + 1) β1 ], · · · , [(p + 1) β ℓ ] | p ∈ P} are vdC sets in Z k+ℓ .Proof. Since x θ1 and x θ2 are Q-linear dependent for all x ∈ Z if and only if θ 1 = θ 2 , an application of Proposition 3.2 yields that it suffices to consider the case where all exponents are different.However, this follows by the same arguments as in the proof of Theorem 4.1 in [3]. 4.2. Entire functions. In this section we consider entire functions of bounded logarithmic order. We fix a transcendental entire function f and denote by S(r) := max |z|≤r |f (z)|. Then we call λ the logarithmic order of f if lim sup r→∞ log S(r) log r = λ.
Theorem 4. 3 .
3Let f 1 , .. . , f k be entire functions with distinct logarithmic orders 1 < λ 1 , λ 2 , . . . , λ k <4 3 , respectively. Then the set D := {(⌊f 1 (p n )⌋, . . . , ⌊f k (p n )⌋) : n ∈ N} is a vdC set.Proof. We enumerate D = (d n ) n≥1 , where d n := (⌊f 1 (p n )⌋, . . . , ⌊f k (p n )⌋) .
⌊f i (p n )⌋ = 0,proving the theorem.
AcknowledgmentThis research work was done when the first author was a visiting lecturer at the Department of Analysis and Computational Number Theory at Graz University of Technology. The author thanks the institution for its hospitality.The second author acknowledges support of the project F 5510-N26 within the special research area "Quasi Monte-Carlo Methods and applications" founded by the Austrian Science Fund.
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On difference sets of sequences of integers. III. Acta Math. Acad. Sci. Hungar. 313-4, On difference sets of sequences of integers. III, Acta Math. Acad. Sci. Hungar. 31 (1978), no. 3-4, 355-386.
Discrepancy of normal numbers. J Schiffer, Acta Arith. 472J. Schiffer, Discrepancy of normal numbers, Acta Arith. 47 (1986), no. 2, 175-186.
W Sierpinski, Démonstrationélémentaire du théorème de M. Borel sur les nombres absolument normaux et détermination effective d'une tel nombre. 45W. Sierpinski, Démonstrationélémentaire du théorème de M. Borel sur les nombres absolument normaux et détermination effective d'une tel nombre, Bull. Soc. Math. France 45 (1917), 125-132.
A note on normal numbers. A M Turing, A.M. Turing (J. BrittonNorth HollandAmsterdamA. M. Turing, A note on normal numbers, Collected Works of A.M. Turing (J. Britton, ed.), North Holland, Amsterdam, 1992, pp. 117-119.
An introduction to ergodic theory. P Walters, Graduate Texts in Mathematics. 79Springer-VerlagP. Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982.
. UMR. 7502G. Madritsch) 1. Université de Lorraine ; Institut Elie Cartan de LorraineG. Madritsch) 1. Université de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, Vandoeuvre- lès-Nancy, F-54506, France;
Vandoeuvre-lès-Nancy, F-54506, France E-mail address: [email protected] (R. F. Tichy) Department for Analysis and Computational Number Theory, Graz University of Technology, A-8010 Graz. Institut Cnrs, Lorraine Elie Cartan De, 7502Austria E-mail address: [email protected], Institut Elie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France E-mail address: [email protected] (R. F. Tichy) Department for Analysis and Computational Number Theory, Graz University of Tech- nology, A-8010 Graz, Austria E-mail address: [email protected]
| []
|
[
"Ego-Motion Sensor for Unmanned Aerial Vehicles Based on a Single-Board Computer Ego-Motion Sensor for Unmanned Aerial Vehicles Based on a Single-Board Computer",
"Ego-Motion Sensor for Unmanned Aerial Vehicles Based on a Single-Board Computer Ego-Motion Sensor for Unmanned Aerial Vehicles Based on a Single-Board Computer"
]
| [
"Gaëlécorchard [email protected] ",
"Adam Heinrich ",
"Libor Přeučil ",
"Gaëlécorchard ‡ ",
"Adam Heinrich ",
"Libor Přeučil ",
"\nCzech Institute for Informatics, Robotics, and Cybernetics\nCzech Institute for Informatics, Robotics, and Cybernetics Czech\nCzech Technical University\nPragueCzech Republic\n",
"\nTechnical University\nPragueCzech Republic\n"
]
| [
"Czech Institute for Informatics, Robotics, and Cybernetics\nCzech Institute for Informatics, Robotics, and Cybernetics Czech\nCzech Technical University\nPragueCzech Republic",
"Technical University\nPragueCzech Republic"
]
| [
"Proceedings of CLAWAR 2017: 20th International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines"
]
| This paper describes the design and implementation of a groundrelated odometry sensor suitable for micro aerial vehicles. The sensor is based on a ground-facing camera and a single-board Linux-based embedded computer with a multimedia System on a Chip (SoC). The SoC features a hardware video encoder which is used to estimate the optical flow online. The optical flow is then used in combination with a distance sensor to estimate the vehicle's velocity. The proposed sensor is compared to a similar existing solution and evaluated in both indoor and outdoor environments. | 10.1142/9789813231047_0025 | [
"https://arxiv.org/pdf/1901.07278v1.pdf"
]
| 58,981,545 | 1901.07278 | 9e968fec6e4a043d2fe281db915cc760d2a385fd |
Ego-Motion Sensor for Unmanned Aerial Vehicles Based on a Single-Board Computer Ego-Motion Sensor for Unmanned Aerial Vehicles Based on a Single-Board Computer
22 Jan 2019
Gaëlécorchard [email protected]
Adam Heinrich
Libor Přeučil
Gaëlécorchard ‡
Adam Heinrich
Libor Přeučil
Czech Institute for Informatics, Robotics, and Cybernetics
Czech Institute for Informatics, Robotics, and Cybernetics Czech
Czech Technical University
PragueCzech Republic
Technical University
PragueCzech Republic
Ego-Motion Sensor for Unmanned Aerial Vehicles Based on a Single-Board Computer Ego-Motion Sensor for Unmanned Aerial Vehicles Based on a Single-Board Computer
Proceedings of CLAWAR 2017: 20th International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines
CLAWAR 2017: 20th International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines22 Jan 201910.1142/9789813231047The article has been published in This is an author submitted version of the paper. For the revised version, please go to https://doi.org/10.1142/9789813231047 0025 or contact the authors at 1 https://www.ciirc.cvut.czvisual odometryoptical flowego-motion
This paper describes the design and implementation of a groundrelated odometry sensor suitable for micro aerial vehicles. The sensor is based on a ground-facing camera and a single-board Linux-based embedded computer with a multimedia System on a Chip (SoC). The SoC features a hardware video encoder which is used to estimate the optical flow online. The optical flow is then used in combination with a distance sensor to estimate the vehicle's velocity. The proposed sensor is compared to a similar existing solution and evaluated in both indoor and outdoor environments.
Introduction
The ability to estimate the velocity is a fundamental task for the control of micro aerial vehicles (MAVs). Such information can be provided, e.g. by the PX4Flow sensor [1], which is an optical flow sensor based on a microcontroller. The sensor provides the linear velocities at a rate of 400 Hz but does not provide changes in heading (yaw). A similar solution is the already-discontinued ArduEye [2]. The visual odometry method described by Kazik and Goktoganin [3] is based on a groundfacing camera and an approach based on a Fourier-Mellin transform which recovers both rotation and translation instead of using the optical flow.
Optical flow sensors based on chips used in optical computer mice used to be quite popular especially in the hobby community, probably ‡ GaëlÉcorchard's work is supported by the SafeLog project funded by the European Union's Horizon 2020 research and innovation programme under grant agreement No. 688117. § Libor Přeučil's work is supported by the Technology Agency of the Czech Republic under Project TE01020197 Center for Applied Cybernetics.
due to the good availability and low cost of these sensors, such as the ADNS-3080 chip [4]. One of the available solutions using such technology is a part of the ArduPilot system [5], however, a rotation around the center of the sensor (yaw) can not be recovered and is said to confuse the sensor. Some authors, e.g. Briod et al. [6] or Kim and Brambley [7], combine several such sensors and an inertial unit, thus removing the need for a separate distance sensor.
A common approach which enables the usage of otherwise CPUintensive algorithms in real-time systems is their implementation in FPGA, cf. Krajnik et al. [8]. The displacement of SURF features between consecutive frames is used to estimate the MAV's displacement.
Next section introduces some theoretical background related to opticalflow sensors. Then, we present our implementation before presenting the real-world results and concluding.
Theoretical Background
Optical Flow
An optical flow is the displacement of pixel values in the image sequence induced by a movement of a camera or a scene observed by it. Let I(u, v, t) be an image function of the pixel position (u, v) and time t. The optical flow between two frames captured at times t and t + ∆t can then be represented by the displacement (∆u, ∆v) and time difference ∆t.
Most approaches to optical flow estimation are based on a brightness constancy constraint. This constraint assumes that moving pixels keep the same brightness between consecutive frames.
The brightness constancy can be linearized using the Taylor approximation [9], which yields in:
0 = I u V u + I v V v + I t ,(1)
where I u , I v and I t are partial derivatives of the image function with respect to u, v and t, respectively and V u and V v are the velocities of the optical flow, V u = ∆u ∆t V v = ∆v ∆t . As Equation 1 has two variables, it has an infinite number of solutions, this is known as the aperture problem. This ambiguity means that another constraints have to be enforced, such as the spatial smoothness constraint. The spatial smoothness constraint assumes that neighboring pixels belong to the same objects and therefore represent the same motion.
The block matching algorithm is one of the simplest methods to compute the optical flow. For every pixel (u, v) in the original image, the closest match (u + ∆u, v + ∆v) in the subsequent image is found by minimizing the Sum of Absolute Differences (SAD). The SAD value is computed by comparing a small (usually square) window (M × N ) around the pixel:
SAD = M 2 i=− M 2 N 2 j=− N 2 |I(u + ∆u + i, v + ∆v + j, t + ∆t) − I(u + i, v + j, t)| (2)
This method can be made faster by computing the flow only for a subset of image pixels instead of the full image matrix, producing only a sparse optical flow. It can be well parallelized as the flow can be computed independently for each pixel.
Camera Model
The camera model is simplified by the pinhole camera model. By supposing that the vertical position of the camera is approximately constant between two image frames, the displacement of a point in space [∆X, ∆Y, 0] T between two image frames is associated to the displacement of the point projected by a pinhole camera onto the image plane [∆u, ∆v, 0] T in pixels, which can be computed as
∆X = − s f ∆uZ, ∆Y = − s f ∆vZ,(3)
where f is the focal length, s is the pixel size, Z is the distance from the camera to the ground. The ground distance Z must be obtained from an external sensor, such as ultrasonic, laser, or barometric pressure sensors.
Compensation of Angular Velocities
It is necessary to compensate small rotations between consecutive frames which manifest as an optical flow in the image plane. Assuming that the camera has been rotated around its x and y axes between two consecutive frames, the displacements in the image plane induced by the rotations are
∆x = f tan(ω y ∆t), ∆y = f tan(ω x ∆t),(4)
where ω x and ω y are the angular velocities which can be obtained from a gyroscope. The displacements ∆x and ∆y have to be subtracted from the resulting optical flow in order to compensate for the angular motion. The rotation around the optical axis (z-axis) does not have to be corrected as the induced optical flow is useful for the estimation of the vehicle's heading. By approximating tan(x) ≈ x for the small involved angles, Equation 3 becomes then
∆X = −( s f u − f ω y ∆t)Z, ∆Y = −( s f v − f ω x ∆t)Z.(5)
Implementation
The algorithm presented in the previous section was implemented on the Raspberry Pi 3 Single-Board Computer as a mixed CPU-GPU solution [10]. The Raspberry Pi was chosen for its low cost and the ability to obtain the optical flow from its integrated hardware H.264 encoder, through its undocumented Coarse Motion Estimator (CME). The other required hardware used in the current setup is the Raspberry Pi camera module v2, with a Sony IMX219 image sensor, used at a resolution of 1640 × 1232 at 40 fps or 1280 × 720 at 90 fps, the MaxBotix HRLV-EZ4 ultrasonic distance sensor, and the L3GD20H 3-axis MEMS digital gyroscope by STMicroelectonics. The complete setup is presented in Fig. 1.
Figure 1: Raspberry Pi with camera and sensors
The different steps of the algorithm are described in Fig. 2 The CME divides the image into 16×16-pixel macroblocks and provides the motion vectors as two 8-bit values and the SAD as a 16-bit value per macroblock. Only the motion vectors are used by our algorithm.
Although the displacements can theoretically be in the range ±127 pixels, a closer analysis with a rapidly moving video sequence showed that it is in fact in the range ±64 pixels from the macroblock's center with a two-pixel resolution.
With the camera parameters obtained by calibration and the range of the motion vector, one can compute the minimum and maximum theoretical detectable velocities. These results are presented in Fig. 3 and compared to those of the PX4Flow. Although our system's frame rate is low when compared to the PX4Flow, it operates with a larger resolution of 480 × 480, so its maximum theoretical detectable velocity is even higher.
Indoor Testing
In order to compare the output of the developed sensoric system with an absolute and precise measurement, we tested it within an arena, referred as the WhyCon system, developed within our group that uses a downwards-looking camera and a pattern recognition algorithm to compute the position of a ring in a plane [11]. For this purpose, we mounted the Raspberry Pi and the pattern on a wheeled carrier and used a circular trajectory and a square one. Figure 4 shows the velocities measured by the Raspberry Pi compared to derived and filtered positions measured by the WhyCon system. Average errors and standard deviations are shown in Table 1.
The processing time is approximately constant and its average value is 5.36 ms. The number of iterations was set to 210 so that the probability of selecting an uncontaminated sample of two motion vectors from the set with up to 85% outliers is 0.99. This could be further lowered by using the adaptive method described by Hartley and Zisserman [12].
Outdoor Testing
Outdoor tests have been performed in a park environment using a commercially available hexacopter DJI F550 equipped with a GPS unit and the PX4Flow sensor, remotely controlled by an operator. The altitude data were obtained from the PX4Flow and integrated in post-processing because the ultrasonic sensor on our system would otherwise interfere with the one of the PX4Flow. The orientation was ignored during this experiment in order to compare measurements with the PX4Flow sensor which does not recover orientation from the optical flow. 6 shows a different trajectory recorded during a flight above pavement to demonstrate the ability to recover changes in orientation. Fig. 7 shows the position integrated by the Raspberry Pi compared to the position recorded from the GPS receiver. The geographic coordinates have been approximately converted to meters using the WGS 84 spheroid [13], both trajectories have been aligned to have the same ori-gin and, arguably, they have been rotated to have the best fit over the complete trajectory. The comparison shows that our sensoric system provides coherent data also on the long term.
Conclusion and Perspectives
We presented an ego-motion sensor based on the Raspberry Pi 3 and other off-the-shelf components. The accuracy of the system is comparable with the one of the PX4Flow and coherent when compared to some GPS data. Its average power consumption is 390 mA at 5 V, compared to 115 mA for the PX4Flow. The advantages of our system over the PX4Flow, however, is that our system is able to provide the changes in orientation around the vertical axis and is cheaper. Moreover, as the average CPU usage is approximately 23 % at 30 fps and the memory footprint is under 30 MB, the computing resources of the Raspberry Pi are not saturated and let room for other algorithms. To foster further developments of the system we plan to publish the source code with an open-source license.
Our further work will be the integration in the control-loop of the MAV and the integration of a ego-motion computation algorithm based on feature detection to improve the low-speed behavior, particularly its inherent drift. Tests over different surfaces also belong to the plans for future work.
Figure 2 :
2Pipeline of the designed solution
Figure 3 :
3Theoretical velocity limits with respect to the ground distance 4 Results
Figure 4 :
4Measured velocity for trajectory "Circles" compared to WhyCon
Fig. 5 Figure 5 :
55shows the comparison of velocities between the Raspberry Pi and the PX4Flow. The average difference is 0.076 m·s −1 . Computed velocity compared to PX4Flow during outdoor experimentFig.
Figure 6 :Figure 7 :
67Integrated Integrated trajectory compared to GPS
. The RANSAC algorithm computes the rigid transform while eliminating outliers.Camera
Angular
correction
Encoder CME
RANSAC
Scaling
(Δx, Δy)
Δθ
Image
Gyroscope
Sonar
(ω x , ω y )
Ground distance
(Δu,Δv)
Table 1 :
1Measured velocity errors
Trajectory µ [m·s −1 ] σ [m·s −1 ]
"Squares"
0.050
0.044
"Circles"
0.042
0.036
An open source and open hardware embedded metric optical flow CMOS camera for indoor and outdoor applications. D Honegger, L Meier, P Tanskanen, M Pollefeys, Intl. Conf. on Robotics and Automation. D. Honegger, L. Meier, P. Tanskanen and M. Pollefeys, An open source and open hardware embedded metric optical flow CMOS camera for indoor and outdoor applications, in Intl. Conf. on Robotics and Automation, 2013.
Computing Optic Flow with ArduEye Vision Sensor. K Schneider, J Conroy, W Nothwang, tech. rep., Army Research LaboratoryK. Schneider, J. Conroy and W. Nothwang, Computing Optic Flow with ArduEye Vision Sensor, tech. rep., Army Research Laboratory (2013).
Visual odometry based on the Fourier-Mellin transform for a rover using a monocular ground-facing camera. T Kazik, A H Goktogan, IEEE Intl. Conf. on Mechatronics. T. Kazik and A. H. Goktogan, Visual odometry based on the Fourier- Mellin transform for a rover using a monocular ground-facing camera, in IEEE Intl. Conf. on Mechatronics, 2011.
Avago Technologies, ADNS-3080. High-Performance Optical Mouse Sensor. Avago Technologies, ADNS-3080. High-Performance Optical Mouse Sensor [Datasheet], (2008).
Mouse-based optical flow sensor (adns3080) (2016). Ardupilot, OnlineArduPilot, Mouse-based optical flow sensor (adns3080) (2016), [Online].
Optic-flow based control of a 46g quadrotor. A Briod, J.-C Zufferey, D Floreano, Workshop on Vision-based Closed-Loop Control and Navigation of Micro Helicopters in GPS-denied Environments, IROS. A. Briod, J.-C. Zufferey and D. Floreano, Optic-flow based control of a 46g quadrotor, in Workshop on Vision-based Closed-Loop Control and Naviga- tion of Micro Helicopters in GPS-denied Environments, IROS , 2013.
Dual optic-flow integrated inertial navigation for small-scale flying robots. J Kim, G Brambley, Australasian Conf. on Rob. and Autom. J. Kim and G. Brambley, Dual optic-flow integrated inertial navigation for small-scale flying robots, in Australasian Conf. on Rob. and Autom., 2017.
T Krajnik, M Nitsche, S Pedre, L Preucil, M E Mejail, A simple visual navigation system for an UAV, in Intl. Multi-Conf. on Systems, Signals & Devices. T. Krajnik, M. Nitsche, S. Pedre, L. Preucil and M. E. Mejail, A sim- ple visual navigation system for an UAV, in Intl. Multi-Conf. on Systems, Signals & Devices, 2012.
Stereo Scene Flow for 3D Motion Analysis. A Wedel, D Cremers, Springer-Verlag GmbHA. Wedel and D. Cremers, Stereo Scene Flow for 3D Motion Analysis (Springer-Verlag GmbH, 2011).
The Raspberry Pi Foundation. Online; accessed 1The Raspberry Pi Foundation, The raspberry pi https://www. raspberrypi.org, (2017), [Online; accessed 1 February 2017].
Vizuální lokalizace pro experimentaci v mobilní robotice (in Czech), bachelor's thesis. T Pivoňka, Czech Technical University in PragueT. Pivoňka, Vizuální lokalizace pro experimentaci v mobilní robotice (in Czech), bachelor's thesis, Czech Technical University in Prague (2016).
Multiple View Geometry in Computer Vision. R Hartley, A Zisserman, Cambridge University Pr.R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge University Pr., 2003).
User's Handbook on Datum Transformations Involving WGS 84. International Hydrographic, Bureau, 3rd edn.International Hydrographic Bureau, User's Handbook on Datum Transfor- mations Involving WGS 84, 3rd edn.(July, 2003).
| []
|
[
"End extending models of set theory via power admissible covers",
"End extending models of set theory via power admissible covers"
]
| [
"Zachiri Mckenzie [email protected] ",
"Ali Enayat [email protected] "
]
| []
| []
| Motivated by problems involving end extensions of models of set theory, we develop the rudiments of the power admissible cover construction (over ill-founded models of set theory), an extension of the machinery of admissible covers invented by Barwise as a versatile tool for generalising model-theoretic results about countable well-founded models of set theory to countable ill-founded ones. Our development of the power admissible machinery allows us to obtain new results concerning powersetpreserving end extensions and rank extensions of countable models of subsystems of ZFC. The canonical extension KP P of Kripke-Platek set theory KP plays a key role in our work; one of our results refines a theorem of Rathjen by showing that Σ P 1 -Foundation is provable in KP P (without invoking the axiom of choice). | 10.1016/j.apal.2022.103132 | [
"https://arxiv.org/pdf/2108.02677v3.pdf"
]
| 236,924,291 | 2108.02677 | afa89c6f698b1e41e0a6465e36bd1f037e2f99b3 |
End extending models of set theory via power admissible covers
24 Mar 2022 March 28, 2022
Zachiri Mckenzie [email protected]
Ali Enayat [email protected]
End extending models of set theory via power admissible covers
24 Mar 2022 March 28, 2022arXiv:2108.02677v3 [math.LO]
Motivated by problems involving end extensions of models of set theory, we develop the rudiments of the power admissible cover construction (over ill-founded models of set theory), an extension of the machinery of admissible covers invented by Barwise as a versatile tool for generalising model-theoretic results about countable well-founded models of set theory to countable ill-founded ones. Our development of the power admissible machinery allows us to obtain new results concerning powersetpreserving end extensions and rank extensions of countable models of subsystems of ZFC. The canonical extension KP P of Kripke-Platek set theory KP plays a key role in our work; one of our results refines a theorem of Rathjen by showing that Σ P 1 -Foundation is provable in KP P (without invoking the axiom of choice).
Introduction
The admissible cover machinery was introduced by Barwise in the Appendix of his venerable book [Bar75] on admissible set theory. Admissible covers allow one to extend the range of infinitary compactness arguments from the domain of countable well-founded models of KP (Kripke-Platek set theory) to countable ill-founded models of KP. For example, Barwise uses admissible covers in his book to prove a striking result: Every countable model of ZF has an extension to a model of ZF + V = L. 1 Admissible covers also appear in the work of Ressayre [Res], who showed that the results presented in the Appendix of [Bar75] pertaining to KP do not depend on the availability of the full scheme of foundation among the axioms of KP; more specifically, they only require the scheme of foundation for Σ 1 ∪ Π 1 -formulae. 2 Admissible covers were used more recently by Williams [Wil18], to show that certain class theories (including Kelley-Morse class theory) fail to have minimum transitive models (this result of Williams also appears in 1 This end extension result, together with certain elaborations of it, first appeared in an earlier paper of Barwise [Bar71]. It is also noteworthy that, as shown recently by Hamkins [Ham18], Barwise's end extension theorem can also be proved using more classical techniques (without appealing to methods of admissible set theory).
2 Note that the full scheme of foundation is included in the axioms of KP in Barwise's treatment [Bar75]. However, we follow the convention proposed by Mathias to only include Π1-Foundation in the axiomatisation of KP; this is informed by the fact, demonstrated by Mathias [Mat01], that many (but not all) results about Barwise's KP can be carried out within Mathias' KP. their paper [Wil19], but with a different proof). In this paper we explore the variant power admissible cover of the notion of admissible cover in order to obtain new results in the model theory of set theory. The main inspiration for our results on end extensions arose from our joint work with Kaufmann [EKM] on automorphisms of models of set theory (see Theorem 5.10).
The canonical extension KP P of Kripke-Platek set theory KP plays a key role in our work. KP P is intimately related to Friedman's so-called power admissble system PAdm s , whose well-founded models are the the so-called power admissible sets [Fri]. 3 These two systems can accommodate constructions by Σ 1 -recursions relative to the power set operation. The system KP P has been closely studied by Mathias [Mat01] and Rathjen [Rat14], [Rat20]. In the latter paper Rathjen proves that Σ P 1 -Foundation is provable in KP P + AC (where AC is the axiom of choice).
The highlights of the paper are as follows. In Corollary 3.3 we refine Rathjen's aforementioned result by showing that Σ P 1 -Foundation is provable outright in KP P . The rudiments of power admissible covers are developed in Section 4. In Section 5 the machinery of power admissible covers is put together with results of earlier sections to establish new results about powerset-preserving end extensions and rank extensions of models of set theory. For example in Theorem 5.7 we show that every countable model of M |= KP P has a topless rank extension, i.e., M has a proper rank extension N |= KP P such that Ord N \ Ord M has no least element. This result generalises a classical theorem of Friedman that shows that every countable well-founded model of KP P has a topless rank extension.
Background
We use L throughout the paper to denote the language {∈, =} of set theory. We will make reference to generalisations of the Lévy hierarchy of formulae in languages extending L that possibly contain constant and function symbols.
Let L ′ be a language extending L. We use ∆ 0 (L ′ ) to denote the smallest class of L ′ -formulae that is closed under the connectives of propositional logic and quantification in the form ∃x ∈ t and ∀x ∈ t, where t is a term of L ′ and x is a variable that does not appear in t. The classes Σ 1 (L ′ ), Π 1 (L ′ ), Σ 2 (L ′ ), . . . are defined inductively from ∆ 0 (L ′ ) in the usual way. We will write ∆ 0 , Σ 1 , Π 1 , . . . instead of ∆ 0 (L), Σ 1 (L), Π 1 (L), . . . , and we will use Π ∞ and Π ∞ (L ′ ) to denote the class of all L-formulae and L ′formulae respectively. An L ′ -formula is ∆ n (L ′ ), for n > 0, if it is equivalent to both a Σ n (L ′ )-formula and a Π n (L ′ )-formula.
The class ∆ P 0 is the smallest class of L-formulae that is closed under the connectives of propositional logic and quantification in the form Qx ⊆ y and Qx ∈ y where Q is ∃ or ∀, and x and y are distinct variables. The Takahashi classes ∆ P 1 , Σ P 1 , Π P n , . . . are defined from ∆ P 0 in the same way as the classes ∆ 1 , Σ 1 , Π 1 , . . . are defined from ∆ 0 . If Γ is a collection of L ′ -formulae and T is an L ′ -theory, then we write Γ T for the class of L ′ -formulae that are provably in T equivalent to a formula in Γ.
We will use capital calligraphic font letters (M, N , . . . ) to denote L-structures. If M is an L-structure, then, unless we explicitly state otherwise, M will be used to denote the underlying set of M and E M will be used to denote the interpretation of ∈ in M.
Let L ′ be a language extending L and let M be an L ′ -structure with underlying set M . If a ∈ M , then a * is defined as follows:
a * := {x ∈ M | M |= (x ∈ a)},
as long as the structure M is clear from the context. Let Γ be a class of formulae. We
say that A ⊆ M is Γ-definable over M if there exists a Γ-formula φ(x, z) and a ∈ M such that A = {x ∈ M | M |= φ(x, a)}.
Let M and N be L ′ -structures. We will partake in the common abuse of notation and write M ⊆ N if M is a substructure of N .
• We say that N is an end extension of M, and write M ⊆ e N , if M ⊆ N and for all x, y ∈ N , if y ∈ M and N |= (x ∈ y), then x ∈ M .
• We say that N is a powerset-preserving end extension of M, and write M ⊆ P e N , if M ⊆ e N and for all x, y ∈ N , if y ∈ M and N |= (x ⊆ y), then x ∈ M .
• We say that N is a topless powerset-preserving end extension of M, and write
M ⊆ P topless N , if M ⊆ P e N , M = N and for all c ∈ N , if c * ⊆ M , then c ∈ M .
• We say that N is a blunt powerset-preserving end extension of M, and write M ⊆ P blunt N , if M ⊆ P e N , M = N and N is not a topless powerset-preserving end extension of M.
Let Γ be a class of L-formulae. The following define the restriction of the ZF-provable schemes Separation, Collection, and Foundation to formulae in the class Γ:
(Γ-Separation) For all φ(x, z) ∈ Γ, ∀ z∀w∃y∀x(x ∈ y ⇐⇒ (x ∈ w) ∧ φ(x, z)).
(Γ-Collection) For all φ(x, y, z) ∈ Γ, ∀ z∀w((∀x ∈ w)∃yφ(x, y, z) ⇒ ∃C(∀x ∈ w)(∃y ∈ C)φ(x, y, z)).
(Γ-Foundation) For all φ(x, z) ∈ Γ, ∀ z(∃xφ(x, z) ⇒ ∃y(φ(y, z) ∧ (∀x ∈ y)¬φ(x, z))).
If Γ = {x ∈ z} then we will refer to Γ-Foundation as Set-foundation.
We will also make reference to the following fragments of Separation and Foundation for formulae that are ∆ n with parameters:
(∆ n -Separation) For all Σ n -formulae, φ(x, z), and for all Π n -formulae, ψ(x, z),
∀ z(∀x(φ(x, z) ⇐⇒ ψ(x, z)) ⇒ ∀w∃y∀x(x ∈ y ⇐⇒ (x ∈ w) ∧ φ(x, z))).
(∆ n -Foundation) For all Σ n -formulae, φ(x, z), and for all Π n -formulae, ψ(x, z), ∀ z(∀x(φ(x, z) ⇐⇒ ψ(x, z)) ⇒ (∃xφ(x, z) ⇒ ∃y(φ(y, z) ∧ (∀x ∈ y)¬φ(x, z))).
Similar definitions can also be used to express ∆ P n -Separation and ∆ P n -Foundation. We use TCo to denote the axiom that asserts that every set is contained in a transitive set.
We will consider extensions of the following subsystems of ZFC:
• S 1 is the L-theory with axioms: Extensionality, Emptyset, Pair, Union, Set difference, and Powerset.
• M is obtained from S 1 by adding TCo, Infinity, ∆ 0 -Separation, and Set-foundation.
• Mac is obtained from M by adding AC (the axiom of choice).
• M − is obtained from M by removing Powerset.
• KP is the L-theory with axioms: Extensionality, Pair, Union, ∆ 0 -Separation, ∆ 0 -Collection and Π 1 -Foundation.
• KP − is obtained from KP by removing Π 1 -Foundation.
• KPI is obtained KP by adding Infinity.
• KP P is obtained from M by adding ∆ P 0 -Collection and Π P 1 -Foundation.
• MOST is obtained from M by adding Σ 1 -Separation and AC.
In subsystems of ZFC that include Infinity we can also consider the following restriction of Γ-Foundation:
(Γ-Foundation on ω) For all φ(x, z) ∈ Γ, ∀ z((∃x ∈ ω)φ(x, z) ⇒ (∃y ∈ ω)(φ(y, z) ∧ (∀x ∈ y)¬φ(x, z))).
The second family of theories that we will be concerned with are extensions of the variant of Kripke-Platek Set Theory with urelements that is introduced in [Bar75,Appendix].
Let L * be obtained from L by adding a second binary relation E, a unary predicate U, and a unary function symbol F. The intended interpretation of U is to distinguish urelements from sets. The binary relation E is intended to be a membership relation that holds between urelements, and ∈ is intended to be a membership relation that can hold between sets or urelements and sets.
Let L * P be obtained from L * by adding a new unary function symbol P. An L * Pstructure is a structure
A M = M; A, ∈ A , F A , P A , where M = M, E A , M is the extension of U,
A is the extension of ¬U, ∈ A is the interpretation of ∈, E A is the interpretation of E, F A is the interpretation of F, and P A is the interpretation of P.
L * -structures will be presented in the same format, but without an interpretation of P. The L * -and L * P -theories presented below will ensure that E A ⊆ M × M , and
∈ A ⊆ (M ∪ A) × A.
Following [Bar75], we simplify the presentation of L * -and L * P -formulae by treating these languages as two-sorted rather than one-sorted.
When writing L * -and L * P -formulae we will use the convention below of Barwise [Bar75].
• The variables p, q, p 1 , . . . range over elements of the domain that satisfy U (urelements).
• the variables a, b, c, d, f, . . . range over elements of the domain that satisfy ¬U (sets); and
• the variables x, y, z, w . . . range over all elements of the domain.
Therefore, ∀p(· · · ) is an abbreviation of ∀x(U(x) ⇒ · · · ), ∃a(· · · ) is an abbreviation of ∃x(¬U(x) ∧ · · · ), etc.
In section 4, we will see that certain L-structures can interpret L * -and L * P -structures in which the urelements are isomorphic to the original L-structure. It is this interaction that motivates our unorthodox convention of using E M , E N , . . . to denote the interpretation of ∈ in the L-structures M, N , . . . It should be noted that this convention differs from Barwise [Bar75] where E is consistently used to denote the interpretation of ∈ in L-structures.
The following are analogues of axioms, fragments of axiom schemes and fragments of theorem schemes of ZFC in the languages L * and L * P : (Extensionality for sets) ∀a∀b(a = b ⇐⇒ ∀x(x ∈ a ⇐⇒ x ∈ b)).
(Pair) ∀x∀y∃a∀z(z ∈ a ⇐⇒ z = x ∨ z = y). (Union) ∀a∃b(∀y ∈ a)(∀x ∈ y)(x ∈ b). Let Γ be a class of L * P -formulae. (Γ-Separation) For all φ(x, z) ∈ Γ, ∀ z∀a∃b∀x(x ∈ b ⇐⇒ (x ∈ a) ∧ φ(x, z)).
(Γ-Collection) For all φ(x, y, z) ∈ Γ, ∀ z∀a((∀x ∈ a)∃yφ(x, y, z) ⇒ ∃b(∀x ∈ a)(∃y ∈ b)φ(x, y, z)).
(Γ-Foundation) For all φ(x, z) ∈ Γ, ∀ z(∃xφ(x, z) ⇒ ∃y(φ(y, z) ∧ (∀w ∈ y)¬φ(w, z))).
The following axiom in the language L * describes the desired behaviour of the function symbol F:
( †) ∀p∀x(xEp ⇐⇒ x ∈ F(p)) ∧ ∀a(F(a) = ∅).
The next axiom, in the language L * P , says that the function symbol P is the usual powerset function:
(Powerset) ∀a∀b(b ∈ P(a) ⇐⇒ b ⊆ a).
We will have cause to consider the following theories:
• KPU Cov is the L * -theory with axioms: ∃a(a = a), ∀p∀x(x / ∈ p), Extensionality for sets, Pair, Union, ∆ 0 (L * )-Separation, ∆ 0 (L * )-Collection, Π 1 (L * )-Foundation and ( †).
• KPU P
Cov is the L * P -theory obtained from KPU Cov by adding Powerset, ∆ 0 (L * P )-Separation, ∆ 0 (L * P )-Collection and Π 1 (L * P )-Foundation.
Definition 2.1 Let M = M, E M be an L-structure.
An admissible set covering M is an L * -structure
A M = M; A, ∈ A , F A |= KPU Cov .
such that ∈ A is well-founded. A power admissible set covering M is an L * P -structure
A M = M; A, ∈ A , F A , P A |= KPU P Cov such that ∈ A is well-founded.
We use Cov M = M; A M , ∈, F M to denote the smallest admissible set covering M whose membership relation ∈ coincides with the membership relation of the metatheory.
We use Cov P M = M; A M , ∈, F M , P M to denote the smallest power admissible set covering M whose membership relation coincides with the membership relation of the metatheory.
Note that if A M = M; A, ∈ A , F A , . . . is an admissible set covering M, then A M is isomorphic to a structure whose membership relation ∈ is the membership relation of the metatheory.
A M = M; A, ∈ A , F A , P A |= KPU P Cov .
We use WF(A) to denote the largest B ⊆ e A such that B, ∈ A is well-founded. The well-founded part of A M is the L * P -structure
WF(A M ) = M; WF(A), ∈ A , F A , P A .
Note that WF(A M ) is always isomorphic to an L * P -structure whose membership relation ∈ coincides with the membership relation of the metatheory.
As usual, in the theories M − , KP − and KPU Cov the ordered pair x, y is coded by the set {{x}, {x, y}}. This definition ensures that there is a ∆ 0 -formula OP(x) that says that x is an ordered pair, and functions fst( x, y ) = x and snd( x, y ) = y, whose graphs are defined by ∆ 0 -formulae. In KPU Cov the rank function, ρ, and support function, sp, are defined by recursion: ρ(p) = 0 for all urelements p, and ρ(a) = sup{ρ(x) + 1 | x ∈ a} for all sets a; sp(p) = {p} for all urelements p, and sp(a) = x∈a sp(x) for all sets a.
The theory KPU Cov proves that both of these are total and their graphs are ∆ 1 (L * ). In the theory KP, in which everything is a set, the rank function, ρ, is ∆ 1 and remains provably total. We say that x is a pure set if sp(x) = ∅. We say that x is an ordinal if x is a hereditarily transitive pure set; where:
Transitive(x) ⇐⇒ ¬U(x) ∧ (∀y ∈ x)(∀z ∈ y)(z ∈ x), and Ord(x) ⇐⇒ (Transitive(x) ∧ (∀y ∈ x)(Transitive(y)).
Therefore, both 'x is transitive' and 'x is an ordinal' can be expressed using ∆ 0 (L * )formulae. In the theories M − and KP, we can omit the reference to the predicate U in the definition of 'x is transitive', thus making both the property of being transitive and the property of being an ordinal into ∆ 0 properties.
The rank function allows us to strengthen the notion of powerset-preserving end extensions for models of KP. Let L ′ be a language extending L. Let M and N be L ′ -structures that satisfy KP.
• We say that N is a rank extension of M, and write M ⊆ rk e N , if M ⊆ P e N and for all x, y ∈ N , if y ∈ M and N |= (ρ(x) ≤ ρ(y)), then x ∈ M . • We say that N is a topless rank extension of M, and write M ⊆ rk topless N , if M ⊆ rk e N , M = N and for all c ∈ N , if c * ⊆ M , then c ∈ M . • We say that N is a blunt rank extension of M, and write M ⊆ rk blunt N , if M ⊆ rk e N , M = N and N is not a topless rank extension of M.
Note that KP − is a subtheory of M − + ∆ 0 -Collection. We will make use of the following results:
• A consequence of [Mat01, Theorem Scheme 6.9(i)] is that M proves ∆ P 0 -Separation.
• The availability of the collection scheme for the relevant class of formulae means that the class of formulae that are equivalent to a Σ 1 -formula and the class of formulae that are equivalent to a Π 1 -formula are closed under bounded quantification in the theory KP − ; the class of formulae equivalent to a Σ P 1 -formula and the class of formulae that are equivalent to a Π P 1 -formula are closed under bounded quantification in the theory M − + ∆ P 0 -Collection; the class of formulae equivalent to a Σ 1 (L * )-formula and the class of formulae that are equivalent to a Π 1 (L * )formula are closed under bounded quantification in the theory KPU Cov ; and the class of formulae equivalent to a Σ 1 (L * P )-formula and the class of formulae that are equivalent to a Π 1 (L * P )-formula are closed under bounded quantification in the theory KPU P Cov .
• The proof of [Bar75,I.4.4] shows:
1. KP − ⊢ Σ 1 -Collection; 2. M − + ∆ P 0 -Collection ⊢ Σ P 1 -Collection; 3. KPU Cov ⊢ Σ 1 (L * )-Collection; and 4. KPU P Cov ⊢ Σ 1 (L * P )-Collection.
• The argument used in [Bar75,I.4.5] shows:
1. KP − ⊢ ∆ 1 -Separation; 2. M − + ∆ P 0 -Collection ⊢ ∆ P 1 -Separation; 3. KPU Cov ⊢ ∆ 1 (L * )-Separation; and 4. KPU P Cov ⊢ ∆ 1 (L * P )-Separation.
The following is Mathias's calibration [Mat01, Proposition Scheme 6.12] of [Tak,Theorem 6].
Theorem 2.3 The following inclusions hold between the indicated classes of formulae (n ≥ 1):
1. Σ 1 ⊆ (∆ P 1 ) MOST and ∆ P 0 ⊆ ∆ S 1 2 . 2. Σ n+1 ⊆ (Σ P n ) MOST . 3. Π n+1 ⊆ (Π P n ) MOST . 4. ∆ n+1 ⊆ (∆ P n ) MOST . 5. Σ P n ⊆ Σ S 1 n+1 . 6. Π P n ⊆ Π S 1 n+1 . 7. ∆ P n ⊆ ∆ S 1 n+1 .
• As noted by Mathias in [Mat01, Corollary 6.15], MOST+Π 1 -Collection and MOST+ ∆ P 0 -Collection axiomatise the same theory. This fact follows from part 1 of Theorem 2.3 and the results mentioned above and will be repeatedly used throughout this paper.
The Σ P 1 -Recursion Theorem [Mat01,Theorem 6.26] shows that the theory KP P is capable of constructing the levels of the cumulative hierarchy: V 0 = ∅ and for all ordinals α,
V α+1 = P(V α ) and, if α is a limit ordinal, V α = β∈α V β .
More precisely, let RK(α, f ) be the L-formula:
(f is a function) ∧ (α is an ordinal) ∧ dom(f ) = α ∧ (∀β ∈ α) (β is a limit ordinal) ⇒ f (β) = γ∈β f (γ) ∧ (∃γ ∈ β)(β = γ + 1) ⇒ ((∀x ⊆ f (γ))(x ∈ f (β)) ∧ (∀x ∈ f (β))(x ⊆ f (γ))) . Note that RK(f, α) is a ∆ P 0 -formula.
Lemma 2.4 The theory KP P proves (I) for all ordinals α, there exists f such that RK(α, f );
(II) for all ordinals α and for all f , if RK(f, α + 1), then
f (α) = {x | ρ(x) < α}.
Therefore, the theory KP P proves that the function α → V α is total and that the graph of this function is ∆ P 1 -definable. In contrast, the theory MOST does not prove that the function α → V α is total (see Example 2.7 below). Note that the availability of AC in MOST allows us to identify cardinals with initial ordinals. Consider the ∆ P 0 -formula BFEXT(R, X) defined 4 by:
(R is an extensional relation on X with a top element) ∧ (∀S ⊆ X)(S = ∅ ⇒ (∃x ∈ S)(∀y ∈ S)( y, x / ∈ R)).
The following lemma captures two important features of the theory MOST that follow from [Mat01,Theorem 3.18].
Lemma 2.5 The theory MOST proves the following statements:
(I) for all X, R with BFEXT(X, R), there exists a transitive set T such that X, R ∼ = T, ∈ ;
(II) there exist arbitrarily large initial ordinals;
(III) for all cardinals κ, the set H ≤κ = {x | |TC(x)| ≤ κ} exists.
In the theory MOST, the formula "X = H ≤κ " is ∆ P 1 with parameters X and κ:
(κ is a cardinal)∧ (∀R ⊆ κ × κ)(BFEXT(R, κ) ⇒ (∃x, f, T ∈ X)(T = TC({x}) ∧ f : R ∼ =∈↾ T ))∧ (∀x ∈ X)(∃T, f ∈ X)(T = TC({x}) ∧ (f : T −→ κ is injective)) .
The next result is a special case of [Gor,Corollary 6.11]:
Lemma 2.6 Let M and N be models of KP P . If M ⊆ P e N , then M ⊆ rk e N . ✷
The following examples show that neither of assumptions that M in Lemma 2.6 satisfies ∆ P 0 -Collection and Π P 1 -Foundation can be removed. The structure M defined in Example 2.7 satisfies all of the axioms of KP P except ∆ P 0 -Collection. The structure M defined in Example 2.8 satisfies all of the axioms of KP P except Π P 1 -Foundation.
Example 2.7 Let N = N, E N |= ZF + V = L, and M = (H N ℵω ) * , E N . Then M |= MOST + Π ∞ -Separation, and M ⊆ P blunt N , but N is not a rank extension of M. Example 2.8 Let N = N, E N be an ω-nonstandard model of ZF + V = L. Let M = M, E N , where M = n∈ω (H N ℵn ) * .
Then M |= MOST + Π 1 -Collection, and M ⊆ P topless N , but N is not a rank extension of M.
The following recursive definition can be carried out within KPU Cov thanks to the ability of KPU Cov to carry out Σ 1 (L * )-recursions. The recursion defines an operation · for coding the infinitary formulae of L ee ∞ω , where L ee be the language obtained from L by adding new constant symbolsā for each urelement a and a new constant symbol c.
• for all ordinals α, v α = 0, α ,
• for all urelements a, ā = 1, a ,
• c = 2, 0 , • if φ is an L ee ∞ω -formula and x is a free variable of φ, then ∃xφ = 3, x , φ , • if φ is an L ee ∞ω -formula and x is a free variable of φ, then ∀xφ = 4, x , φ ,
• if Φ is a set of L ee ∞ω -formulae such that only finitely many variables appear as a free variable of some formula in Φ, then
φ∈Φ φ = 5, Φ * , where Φ * = { φ | φ ∈ Φ}, • if Φ is a set of L ee
∞ω -formulae such that only finitely many variables appear as a free variable of some formula in Φ, then
φ∈Φ φ = 6, Φ * , where Φ * = { φ | φ ∈ Φ}, • if φ is an L ee ∞ω -formula, then ¬φ = 7, φ ,
• if s and t are terms of L ee ∞ω , then s = t = 8, s , t ,
• if s and t are terms of L ee ∞ω , then s ∈ t = 9, s , t .
Let M = M, E M be an L-structure and let A M = M; A, ∈, F A , P A be a power admissible set covering M. We use L ee A M to denote the fragment of L ee ∞ω that is coded in A M . The L ee A M -formulae in the form s = t or s ∈ t, where s and t are L ee A M -terms, are the atomic formulae of L ee A M .
The formula that identifies the codes of the atomic formulae of L ee
A M is ∆ 1 (L * )-definable over A M .
Similarly, other important properties of codes of L ee A M constituents, such as being a variable, constant, well-formed formula, sentence, . . . , are all ∆ 1 (L * )-definable over A M . We will often equate an L ee A M -theory T with that subset of A M of codes of L ee A M -sentences in T . The following is the Barwise Compactness Theorem ([Bar75, III.5.6]) tailormade for countable admissible L * P -structures.
Theorem 2.9 (Barwise Compactness Theorem) Let A M = M; A, ∈, F A , P A be a power admissible set covering M. Let T be an L ee A M -theory that is Σ 1 (L * P )-definable over A M and such that for all T 0 ⊆ T , if T 0 ∈ A, then T 0 has a model. Then T has a model.
The scheme of Σ P 1 -Foundation
Motivated by the apparent reliance of the constructions presented in the next section on Σ P 1 -Foundation, this section investigates the status of this scheme in the theories MOST + Π 1 -Collection and KP P . We begin by showing that KP P proves Σ P 1 -Foundation. In contrast, Σ P 1 -Foundation is not provable in MOST + Π 1 -Collection but does hold in every ω-standard model of this theory.
In [Rat20,Lemma 4.4] it is shown that KP P + AC proves Σ P 1 -Foundation 5 . Here we use a modification of a choiceless scheme of dependant choices introduced in [Rat92] to show that Σ P 1 -Foundation can be proved in KP P . The following is [Rat92, Definition 3.1]:
Definition 3.1 Let φ(x, y, z) be an L-formula. Define δ φ (a, b, f, z) to be the formula:
a is an ordinal ⇒ f is a function ∧ dom(f ) = a + 1 ∧ f (0) = {b}∧ (∀u ∈ a) (∀x ∈ f (u))(∃y ∈ f (u + 1))φ(x, y, z)∧ (∀y ∈ f (u + 1))(∀x ∈ f (u))φ(x, y, z) .
By considering the variables z to be parameters, φ(x, y, z) defines a directed graph. The formula δ φ (a, b, f, z) says that for all 0 ≤ i ≤ a, f (i) is a collection of vertices lying at a stage i on a directed path of length a starting at b in this graph. In the next definition we introduce a formula that, given b, f and z, says that f is a function with domain ω and for all n ∈ ω, δ φ (n + 1, b, f ↾ (n + 1), z).
Definition 3.2 Let φ(x, y, z) be an L-formula. Define δ φ ω (b, f, z) to be the formula: f is a function ∧ dom(f ) = ω ∧ f (0) = {b}∧ (∀u ∈ ω)
(∀x ∈ f (u))(∃y ∈ f (u + 1))φ(x, y, z)∧ (∀y ∈ f (u + 1))(∀x ∈ f (u))φ(x, y, z)
.
Note that if φ(x, y, z) is a ∆ P 0 -formula (∆ 0 -formula) then both δ φ (a, b, f, z) and δ φ ω (b, f, z) are both ∆ P 0 -formulae (respectively ∆ 0 -formulae). The following is a modification of Rathjen's ∆ 0 -weak dependant choices scheme (∆ 0 -WDC) from [Rat92]:
(∆ P 0 -WDC ω ) For all ∆ P 0 -formulae, φ(x, y, z), ∀ z(∀x∃yφ(x, y, z) ⇒ ∀w∃f δ φ ω (w, f, z)).
The next result is based on the proof of [Rat92, Proposition 3.2]:
Theorem 3.3 The theory M + ∆ P 0 -WDC ω proves Σ P 1 -Foundation.
Proof Work in the theory M + ∆ P 0 -WDC ω . Suppose, for a contradiction, that there is an instance of Σ P 1 -Foundation that fails. Let φ(x, y, z) be a ∆ P 0 -formula and let a be a finite sequence of sets such that the class C = {x | ∃yφ(x, y, a)} is nonempty and has no ∈-least element. Let b and d be such that φ(b, d, a) holds. Now, since C has no ∈-least element, ∀x∀u∃y∃v(φ(x, u, a) ⇒ (y ∈ x) ∧ φ(y, v, a)).
Therefore, we have have ∀x∃yθ(x, y, a) where θ(x, y, a) is y 1 , a)).
x = x 0 , x 1 ∧ y = y 0 , y 1 ∧ (φ(x 0 , x 1 , a) ⇒ (y 0 ∈ x 0 ) ∧ φ(y 0 ,
Note that θ(x, y, a) is a ∆ P 0 -formula. Therefore, using ∆ P 0 -WDC ω , let f be such that δ θ ω ( b, d , f, a). Now, ∆ P 0 -Separation facilitates induction for ∆ P 0 -formulae and proves that for all n ∈ ω,
f (n) = ∅ ∧ (∀x ∈ f (n))(x = x 0 , x 1 ∧ φ(x 0 , x 1 , a))∧ (∀x ∈ f (n))(∃y ∈ f (n + 1))(x = x 0 , x 1 ∧ y = y 0 , y 1 ∧ y 0 ∈ x 0 )∧ (∀y ∈ f (n + 1))(∃x ∈ f (n))(y = y 0 , y 1 ∧ x = x 0 , x 1 ∧ y 0 ∈ x 0 )
. Let B = TC({b}). Induction for ∆ 0 -formulae suffices to prove that for all n ∈ ω,
(∀x ∈ f (n))(x = x 0 , x 1 ∧ x 0 ∈ B). Consider A = x ∈ B | (∃n ∈ ω)(∃z ∈ f (n)) ∃y ∈ z (z = x, y ) ,
which is a set by ∆ 0 -Separation. Now, let x ∈ A. Let y and n ∈ ω be such that x, y ∈ f (n). Therefore, there exists w ∈ f (n + 1) such that w = u, v and u ∈ x. So u ∈ A and u ∈ x, which shows that A has no ∈-least element. This contradicts Set-Foundation in M and proves the theorem. ✷
The fact that KP P proves Σ P 1 -Foundation follows from the fact that KP P proves ∆ P 0 -WDC ω . The proof of Theorem 3.5 is inspired by the argument used in the proof of [FLW,Theorem 4.15]. The stratification of the universe into ranks allows us to select sets of paths through a relation defined by a ∆ P 0 -formula φ(x, y, z) with parameters z.
Definition 3.4 Let φ(x, y, z) be an L-formula. Define η φ (a, b, f, z) by
δ φ (a, b, f, z)∧ (∀u ∈ a)∃α∃X (α is an ordinal) ∧ (X = V α ) ∧ (∀x ∈ f (u + 1))(x ∈ X) ∧ (∀y ∈ X)(∀x ∈ f (u))(φ(x, y, z) ⇒ y ∈ f (u + 1)) ∧ (∀β ∈ α)(∀Y ∈ X) Y = V β ⇒ (∃x ∈ f (u))(∀y ∈ Y )¬φ(x, y, z) .
The formula η φ (a, b, f, z) asserts that f is a function with domain a + 1 such that f (0) = {b} and for all u ∈ a, f (u + 1) is the set of y of rank α such that there exists x ∈ f (u) with φ(x, y, z) and α is the minimal ordinal such that for all x ∈ f (u), there exists y of rank α such that φ(x, y, z). Recall that, in the theory KP P , the formula 'X = V α ' is ∆ P 1 with parameters X and α. Therefore, if φ(x, y, z) is a ∆ P 0 -formula, then η φ (a, b, f, z) is equivalent to a Σ P 1 -formula in the theory KP P .
Theorem 3.5 The theory KP P proves ∆ P 0 -WDC ω .
Proof Work in the theory KP P . Let φ(x, y, z) be a ∆ P 0 -formula. Let a be sets such that ∀x∃yφ(x, y, a) holds. Let b be a set. We begin by claiming that for all n ∈ ω, ∃f η φ (n, b, f, a). Suppose, for a contradiction, that this does not hold. Using Π P 1 -Foundation, there exists a least m ∈ ω such that ¬∃f η φ (m, b, f, a). It is straightforward to see that m = 0. Therefore, there exists a function g with dom(g) = m such that η φ (m − 1, b, g, a) holds. Consider the class A = {α ∈ Ord | ∀X(X = V α ⇒ (∀x ∈ g(m − 1))(∃y ∈ X)φ(x, y, a))}.
Applying ∆ P 0 -Collection to the formula φ(x, y, a) shows that A is nonempty. Therefore, by Π P 1 -Foundation, there exists a least element β ∈ A. Let
C = {y ∈ V β | (∃x ∈ g(m − 1))φ(x, y, a)},
which is a set by ∆ P 0 -Separation. Now, let f = g ∪ { m, C }. So, η φ (m, b, f, a), which is a contradiction. Therefore, for all n ∈ ω, ∃f η φ (n, b, f, a). Note that for all n ∈ ω and for all f and g, if η φ (n, b, f, a) and η φ (n, b, g, a), then f = g. Now, using Σ P 1 -Collection, we can find a set D such that (∀n ∈ ω)(∃f ∈ D)η φ (n, b, f, a). Let
f = { n, X ∈ ω × TC(D) | (∃g ∈ D)(η φ (n, b, g, a) ∧ g(n) = X)} = { n, X ∈ ω × TC(D) | (∀g ∈ D)(η φ (n, b, g, a) ⇒ g(n) = X)}.
Now, f is a set by ∆ P
1 -Separation and f is the function required by ∆ P 0 -WDC ω . ✷
Combining Theorems 3.3 and 3.5 yields:
Corollary 3.6 KP P ⊢ Σ P 1 -Foundation. ✷
We now turn to investigating Σ P 1 -Foundation in the theory MOST + Π 1 -Collection. The following is an instance of [PK,Proposition 2] in the context of set theory:
Theorem 3.7 Let Σ denote Σ P 1 -Foundation on ω, and Π denote Π P 1 -Foundation on ω. Then we have:
M − + ∆ P 0 -Collection + Π ⊢ Σ, and M − + Σ ⊢ Π.
Proof To see that Π implies Σ, work in the theory M − + ∆ P 0 -Collection. We prove the contrapositive. Let φ(x, z) be a Π P 1 -formula and let a be sets such that the class {x ∈ ω | φ(x, a)} is nonempty and has no least element. Let p ∈ ω be such that φ(p, a). Let C = {x ∈ ω | ∃w(x + w = p ∧ (∀y ∈ w)¬φ(y, a))}.
Note that ∆ P 0 -Collection implies that C is a Σ P 1 -definable subclass of ω. Moreover, p ∈ C and 0 / ∈ C. Identical reasoning to that used above shows that C has no least element. Therefore Σ P 1 -Foundation on ω fails. To see that Σ implies Π, work in the theory M − . Again, we prove the contrapositive. Let φ(x, z) be a Σ P 1 -formula and let a be the sequence of set parameters such that the class {x ∈ ω | φ(x, a)} is nonempty and has no least element. Let p ∈ ω be such that φ(p, a). Let C = {x ∈ ω | ∀w(x + w = p ⇒ (∀y ∈ w)¬φ(y, a)}.
Note that C is a Π P 1 -definable subclass of ω, p ∈ C and 0 / ∈ C. Suppose that q ∈ C is a least element of C. Let u ∈ ω be such that q + u = p. Now, φ(u, a), since q is the least of C, and (∀y ∈ u)¬φ(y, a). But then u is a least element of {x ∈ ω | φ(x, a)}, which is a contradiction. Therefore Π P 1 -Foundation on ω fails. ✷
An examination of the proof of [Mat01, Proposition 9.22] yields:
Theorem 3.8 The consistency of Mac is provable in M + Π P 1 -Foundation on ω. ✷
The results of [Mat01] and [M] (see [M,Corollary 3.5]) show that Mac and MOST + Π 1 -Collection have the same consistency strength. Therefore, Theorem 3.7 yields:
Theorem 3.9 The consistency of MOST+Π 1 -Collection is provable in MOST+Π 1 -Collection+ Σ P 1 -Foundation. ✷ Therefore, Σ P 1 -Foundation is not provable in MOST+ Π 1 -Collection. However, we can show that Σ P 1 -Foundation does hold in every model of MOST + Π 1 -Collection in which the natural numbers are standard.
In the context of the theory MOST + Π 1 -Collection, we can use the stratification of the universe into the sets H ≤κ in the same way that we used the stratification of the universe into ranks in the proof of Theorem 3.5.
Definition 3.10 Let φ(x, y, z) be an L-formula. Define χ φ (a, b, f, z) by
δ φ (a, b, f, z)∧ (∀u ∈ a)∃κ∃X (X = H ≤κ ) ∧ (∀x ∈ f (u + 1))(x ∈ X)∧ (∀y ∈ X)(∀x ∈ f (u))(φ(x, y, z) ⇒ y ∈ f (u + 1))∧ (∀λ ∈ κ)(∃x ∈ f (u))(∀R ⊆ λ × λ) BFEXT(R, λ) ⇒ ∃T, f, y T = TC({y})∧ f : R ∼ =∈↾ T ∧ ¬φ(x, y, z) .
The formula χ φ (a, b, f, z) asserts that f is a function with domain a + 1 such that f (0) = {b} and for all u ∈ a, f (u + 1) is the set of all y in H ≤κ such that there exists an x ∈ f (u) with φ(x, y, z) and κ is the minimal cardinal such that for all x ∈ f (u), there exists y ∈ H ≤κ with φ(x, y, z). Recall that the formula expressing "X = H ≤κ " is ∆ P 1 with parameters X and κ in the theory MOST. Therefore, if φ(x, y, z) is a ∆ P 0 -formula, then χ φ (a, b, f, z) is equivalent to a Σ P 1 -formula in the theory MOST + Π 1 -Collection. In the proof of the next theorem the formula χ φ (a, b, f, z) plays the role of η φ (a, b, f, z) in the proof of Theorem 3.5. Proof Let φ(x, y, z) be a ∆ P 0 -formula. Let a be sets such that M |= ∀x∃yφ(x, y, a). Let b a set. We begin by showing that
M |= (∀n ∈ ω)∃f χ φ (n, b, f, a).
Work inside M. Suppose, for a contradiction, that there exists n ∈ ω such that ¬∃f χ φ (n, b, f, a) holds. Therefore, since M is ω-standard, there is a least m ∈ ω such that ¬∃f χ φ (m, b, f, a). It is straightforward to see that m = 0. Therefore, there exists a function g with dom(g) = m such that χ φ (m − 1, b, g, a). Consider A = {κ ∈ Ord M | M |= ∀X(X = H ≤κ ⇒ (∀x ∈ g(m − 1))(∃y ∈ X)φ(x, y, a))} = {κ ∈ Ord M | M |= ∃X(X = H ≤κ ∧ (∀x ∈ g(m − 1))(∃y ∈ X)φ(x, y, a))}.
Applying ∆ P 0 -Collection to φ(x, y, a) shows that A is nonempty. Therefore, ∆ P 1 -Separation ensures that A has an ∈-least element λ. Let
C = {y ∈ H ≤λ | (∃x ∈ g(m − 1))φ(x, y, a)}, which is a set by ∆ P 0 -Separation. Now, let f = g ∪ { m, C }. So, χ φ (m, b, f, a)
which is a contradiction. This shows that M |= (∀n ∈ ω)∃f χ φ (n, b, f, a).
Work inside M.
Note that for all n ∈ ω and for all f and g, if χ φ (n, b, f, a) and χ φ (n, b, g, a), then f = g. Now, using Σ P 1 -Collection, there exists D such that
(∀n ∈ ω)(∃f ∈ D)χ φ (n, b, f, a). Let f = { n, X ∈ ω × TC(D) | (∃g ∈ D)(χ φ (n, b, g, a) ∧ g(n) = X)} = { n, X ∈ ω × TC(D) | (∀g ∈ D)(χ φ (n, b, g, a) ⇒ g(n) = X)},
which is a set by ∆ P 1 -Separation. Therefore δ ω (b, f, a) holds, which completes the proof that ∆ P 0 -WDC ω holds in M. ✷ Combining Theorems 3.3 and 3.11: Bar75,Appendix] shows how the admissible cover, Cov M , can be built from an Lstructure M that satisfies KP+Σ 1 -Foundation. The construction proceeds in two stages. The first stage interprets a model of KPU Cov inside M. The second stage takes the wellfounded part of this interpreted model of KPU Cov to obtain an admissible set covering M that [Bar75,Appendix] shows is minimal. It should be noted that [Bar75,Appendix] starts with a structure M that satisfies full Π ∞ -Foundation. It is noted in [Res,Chapter 2] that all of the elements of Barwise's construction of Cov M can be carried out when M satisfies Π 1 ∪ Σ 1 -Foundation. The aim of this section is to review the construction of Cov M from M and investigate the influence of the theory of M on Cov M . In particular, we will show that if M is a model of KP + powerset + ∆ P 0 -Collection + Σ P 1 -Foundation, then P can be interpreted in Cov M to make it a power admissible set.
Corollary 3.12 If M is an ω-standard model of MOST + Π 1 -Collection, then M |= Σ P 1 -Foundation.✷ 4 Obtaining Cov M from M [
Throughout this section we will work with a fixed L-structure M = M, E M that satisfies KP + Powerset + ∆ P 0 -Collection + Σ P 1 -Foundation. We begin by expanding the interpretation of the theory KPU Cov inside M presented in [Bar75,Appendix Section 3] to obtain L * P -structure that satisfies Powerset. Working inside M, define the unary relations N and Set, the binary relations E and E ′ , and unary function symbolsF andP by:
N(x) iff ∃y(x = 0, y );
xE ′ y iff ∃w∃z(x = 0, w ∧ y = 0, z ∧ w ∈ z);
Set(x) iff ∃y(x = 1, y ∧ (∀z ∈ y)(N(z) ∨ Set(z)));
xEy iff ∃z(y = 1, z ∧ x ∈ z);
F(x) = 1, X where X = { 0, y | ∃w(x = 0, w ∧ y ∈ w)}; P(x) = 1, X where X = { 1, y | ∃w(x = 1, w ∧ y ⊆ w)}.
[ Bar75,Appendix Section 3] notes that N, E ′ , E andF are defined by ∆ 0 -formulae in M, and, using the Second Recursion Theorem ([ Bar75,V.2.3.]), Set can be expressed using a Σ 1 -formula. [Res,Chapter 2] notes that the Second Recursion Theorem can be proved in KP + Σ 1 -Foundation. The function y =P(x) is defined by a ∆ P 0 -formula:
y =P(x) iff OP(x) ∧ OP(y) ∧ fst(x) = 1 ∧ fst(y) = 1∧
(∀z ⊆ snd(x))( 1, z ∈ snd(y)) ∧ (∀w ∈ snd(y))(snd(w) ⊆ snd(x)).
These definitions yield an interpretation, I, of an L * P -structure that is summarised in
L * P -structure in M L * P Symbol L expression under I ∀x ∀x(N(x) ∨ Set(x) ⇒ · · · ) = = U(x) N(x) xEy xE ′ y x ∈ y xEy F(x)F(x) P(x)P(x)
In other words,
A N = N ; Set M , E M ,F M ,P M , where N = N M , (E ′ ) M , is an L * P -structure.
If φ is an L * P -formula, then we write φ I for the translation of φ into an L-formula of M described in Table 1. Note that the map x → 0, x is an isomorphism between M and N . The following is the refinement of [Bar75, Appendix Lemma 3.2] noted by [Res,Chapter 2]:
Theorem 4.1 A N |= KPU Cov . ✷
We now turn to showing that axioms and axiom schemes transfer from M to A N .
Lemma 4.2 A N |= Powerset.
Proof Let a be a set of A N . To see thatP(a) exists, note that a = 1, a 0 andP(a) = 1, X where X = {1} × P(a 0 ). Therefore, the powerset axiom in M ensures thatP is total in A N . Now, let b be a set of A N . Work inside M.
Now, b = 1, b 0 . And, bEP(a) iff b 0 ⊆ a 0 ,
iff for all x, if xEb, then xEa,
iff (b ⊆ a) I . Therefore, A N satisfies Powerset. ✷ Lemma 4.3 Let φ( x) be a ∆ 0 (L * P )-formula. Then φ I ( x) is equivalent to a ∆ P 1 -formula in M.
Proof We prove this lemma by induction on the complexity of φ I . Note that, by the above observations, N(x) and xE ′ y can be written as ∆ 0 -formulae. Moreover, y =F(x) is equivalent to a ∆ 0 -formula, and y =P(x) is equivalent to a ∆ P 0 -formula. Now, yEF(x) iff fst(y) = 0 ∧ snd(y) ∈ snd(x),
which is ∆ 0 . Similarly, yEP(x) iff fst(y) = 1 ∧ snd(y) ⊆ snd(x),
which is also ∆ 0 . Now, suppose that t(x) is an L * P -term and both y = t I (x) and yEt
I (x) are ∆ P 1 in M. Now, y =P(t I (x)) iff ∃w(w = t I (x) ∧ y =P(w)), iff ∀w(w = t I (x) ⇒ y =P(w)). Similarly, yEP(t I (x)) iff ∃w(w = t I (x) ∧ yEP(w)), iff ∀w(w = t I (x) ⇒ yEP(w)).
Therefore, both y =P(t I (x)) and yEP(t I (x)) are ∆ P 1 in M. Now, y =F(t I (x)) iff ∃w(w = t I (x) ∧ y =F(w)), iff ∀w(w = t I (x) ⇒ y =F(w)).
And, yEF(t I (x)) iff ∃w(w = t I (x) ∧ yEF(w)),
iff ∀w(w = t I (x) ⇒ yEF(w)).
SinceF andP are both unary functions, this shows that for every L * P -term t(x), both y = t I (x) and yEt I (x) are ∆ P 1 in M. Finally, we need an induction step that allows us to deal with bounded quantification. Let ψ(x 0 , . . . , x n−1 ) be an L * P -formula such that
ψ I (x 0 , . . . , x n−1 ) is ∆ P 1 in M. Now, (∃x 0 Ex n )ψ I (x 0 , . . . , x n−1 ) iff (∃x 0 ∈ snd(x n ))ψ I (x 0 , . . . , x n−1 ). Therefore, (∃x 0 Ex n )ψ I (x 0 , . . . , x n−1 ) = ((∃x 0 ∈ x n )ψ(x 0 , . . . , x n−1 )) I is ∆ P 1 in M. Let t(x) be an L * P -term. Now, (∃x 0 Et I (x n ))ψ I (x 0 , . . . , x n−1 )
iff ∃w(w = t I (x n ) ∧ (∃x 0 ∈ snd(w))ψ I (x 0 , . . . , x n−1 )), iff ∀w(w = t I (x n ) ⇒ (∃x 0 ∈ snd(w))ψ I (x 0 , . . . , x n−1 )).
Therefore (∃x 0 Et I (x n ))ψ I (x 0 , . . . , x n−1 ) = ((∃x 0 ∈ t(x n ))ψ(x 0 , . . . , x n−1 )) I is ∆ P 1 in M. The Lemma now follows by induction. ✷ Lemma 4.4 A N |= ∆ 0 (L * P )-Separation.
Proof Let φ(x, z) be a ∆ 0 (L * P )-formula, v be sets and/or urelements of A N and a a set of A N . Work inside M. Now, a = 1, a 0 . Let
b 0 = {x ∈ a 0 | φ I (x, v)}, which is a set by ∆ P 1 -Separation. Let b = 1, b 0 . Therefore, for all x such that Set(x), xEb iff xEa ∧ φ I (x, v). Therefore, A N satisfies ∆ 0 (L * P )-Separation. ✷ Lemma 4.5 A N |= ∆ 0 (L * P )-Collection.
Proof Let φ(x, y, z) be a ∆ 0 (L * P )-formula. Let v be a sequence of sets and/or urelements of A N and let a be a set of A N such that A N |= (∀x ∈ a)∃yφ(x, y, v).
Work inside M.
Since a is a set of A N , a = 1, a 0 . We have (∀xEa)∃y((N(y) ∨ Set(y)) ∧ φ I (x, y, v)).
And,
(∀x ∈ a 0 )∃y((N(y) ∨ Set(y)) ∧ φ I (x, y, v)).
So, since (N(y) ∨ Set(y)) ∧ φ I (x, y, v) is equivalent to a Σ P 1 -formula, we can apply ∆ P 0 -Collection to obtain b such that
(∀x ∈ a 0 )(∃y ∈ b)((N(y) ∨ Set(y)) ∧ φ I (x, y, v)) (b) .
Let b 0 = {y ∈ b | (N(y) ∨ Set(y)) (b) }, which is a set by ∆ 0 -Separation. Let b 1 = 1, b 0 . Therefore Set(b 1 ) and (∀xEa)(∃yEb 1 )φ I (x, y, v).
So,
A N |= (∀x ∈ a)(∃y ∈ b 1 )φ(x, y, v).
This shows that A N satisfies ∆ 0 (L * P )-Collection. ✷ Lemma 4.6 A N |= Σ 1 (L * P )-Foundation.
Proof Let φ(x, z) be a Σ 1 (L * P )-formula. Let v be a sequence of sets and/or urelements be such that
{x ∈ A N | A N |= φ(x, v)} is nonempty. Work inside M. Consider θ(α, z) defined by (α is an ordinal) ∧ ∃x((Set(x) ∨ N(x)) ∧ ρ(x) = α ∧ φ I (x, z)).
Note that θ(α, z) is equivalent to a Σ P 1 -formula. Therefore, using Σ P 1 -Foundation, let β be an ∈-least element of {α ∈ M | M |= θ(α, v)}.
Let y be such that (N(y) ∨ Set(y)), ρ(y) = β and φ I (y, v). Note that if xEy, then ρ(x) < ρ(y). Therefore y is an E-least element of
{x ∈ A N | A N |= φ(x, v)}. ✷
The following combines [Bar75,II.8.4] Consider the formula θ(β, z) defined by (β is an ordinal) ∧ (∀x ∈ a)(∃α ∈ β)∃y(ρ(y) = α ∧ φ(x, y, z).
Since WF(A N ) ⊆ P e A N , if β is a nonstandard ordinal of A N , then A N |= θ(β, v). Using ∆ 0 (L * P )-Collection, θ(β, z) is equivalent to a Σ 1 (L * P )-formula in A N . Therefore, by Σ 1 (L * P )-Foundation, {β | A N |= θ(β, v)
} has a least element γ. Note that γ is an ordinal of WF(A N ). Now, consider the formula ψ(x, y, z, γ) defined by φ(x, y, z) ∧ (ρ(y) < γ).
Note that
A N |= (∀x ∈ a)∃yψ(x, y, v, γ).
Using ∆ 0 (L * P )-Collection in A N , there exists a set b of A N such that A N |= (∀x ∈ a)(∃y ∈ b)ψ(x, y, v, γ).
Let c = {x ∈ b | ρ(x) < γ}, which is a set in A N by ∆ 1 (L * P )-Separation. Now, c is a set of WF(A N ) and WF(A N ) |= (∀x ∈ a)(∃y ∈ c)φ(x, y, v).
Therefore, WF(A N ) satisfies ∆ 0 (L * P )-Collection. And so, WF(A N ) is a power admissible set covering N . Finally, since the L * -reduct of WF
(A N ) is isomorphic to Cov M , WF(A N ) is isomorphic to Cov P M . ✷
The following theorem summarises the analysis undertaken in this section:
Theorem 4.9 If M |= KP + Powerset + ∆ P 0 -Collection + Σ P 1 -Foundation, then there is an interpretation of P in Cov M that yields the power admissible set Cov P M . ✷
This yields a version of [Bar75,Corollary 2.4.] that will be useful for the compactness arguments in the next section.
End extension results
In this section we use the Barwise Compactness Theorem for L ee Cov P M to show that every countable model of KP + powerset + ∆ P 0 -Collection + Σ P 1 -Foundation has a powersetpreserving end extension.
The following is an immediate consequence of Theorem 4.10:
Lemma 5.1 Let M = M, E M |= KP+ Powerset + ∆ P 0 -Collection + Σ P 1 -Foundation, and let T 0 be an L ee Cov P M -theory. If T 0 ∈ Cov P M , then there exists b ∈ M such that b * = {a ∈ M |ā is mentioned in T 0 }. ✷
The next result expands on comments made in [Bar75,p. 637] and connects definability in M to definability in Cov P M .
Lemma 5.2 Let M = M, E M |= KP + Powerset + ∆ P 0 -Collection + Σ P 1 -Foundation, and let φ( z) be a ∆ P 0 -formula. Then there exists a formulaφ( z) that is ∆ 1 (L * P ) in the theory KPU P Cov such that for all z ∈ M ,
M |= φ( z) iff Cov P M |=φ( z).
Proof Let φ( z) be a ∆ 0 -formula. We prove the lemma by structural induction on the complexity of φ. Without loss of generality we can assume that the only connectives of propositional logic appearing in φ are ¬ and ∨. If φ(z 1 , z 2 ) is z 1 ∈ z 2 , then letφ(z 1 , z 2 ) be the ∆ 0 (L * P )-formula z 1 Ez 2 . Therefore, for all z 1 , z 2 ∈ M , M |= φ(z 1 , z 2 ) iff Cov P M |=φ(z 1 , z 2 ).
If φ( z) is ¬ψ( z) and the lemma holds for ψ( z), then letφ( z) = ¬ψ( z). So,φ( z) is ∆ 1 (L * P ) in the theory KPU P Cov and for all z ∈ M ,
M |= φ( z) iff Cov P M |=φ( z).
Suppose that φ( z) is ψ 1 ( z) ∨ ψ 2 ( z) and the lemma holds for ψ 1 ( z) and ψ 2 ( z). Letφ( z) beψ 1 ( z) ∨ψ 2 ( z). Therefore,φ( z) is ∆ 1 (L * P ) in the theory KPU P Cov and for all z ∈ M ,
M |= φ( z) iff Cov P M |=φ( z).
Suppose φ(y, z) is (Qx ∈ y)ψ(x, y, z), where Q ∈ {∃, ∀}, and the lemma holds for ψ(x, y, z). Letφ(y, z) be (Qx ∈ F(y))ψ(x, y, z). So,φ( z) is ∆ 1 (L * P ) in the theory KPU P Cov . Since Cov P M satisfies ( †), for all y, z ∈ M ,
M |= φ(y, z) iff M |= (Qx ∈ y)ψ(x, y, z) iff Cov P M |= (Qx ∈ F(y))ψ(x, y, z) iff Cov P M |=φ(y, z).
Suppose that φ(y, z) is (Qx ⊆ y)ψ(x, y, z), where Q ∈ {∃, ∀}, and the lemma holds for ψ(x, y, z). Letφ(y, z) be (Qx ∈ P(F(y)))∃p(F(p) = x ∧ψ(p, y, z)). Note that, in the theory KPU P Cov , for all urelements z, (Qx ∈ P(F(y))) ∃p(F(p) = x ∧ψ(p, y, z)) ⇐⇒ (Qx ∈ P(F(y))) ∀p(F(p) = x ⇒ψ(p, y, z)).
Therefore, in the theory KPU P Cov ,φ(y, z) is ∆ 1 (L * P ). Moreover, by Theorem 4.10 and ( †), for all z ∈ M , M |=φ(y, z) iff M |= (Qx ⊆ y)ψ(x, y, z)
iff (Qx ∈ P(F(y))) ∃p(F(p) = x ∧ψ(p, y, z))
iff Cov P M |=φ(y, z). Therefore, the lemma follows by induction. ✷
We are now able to use the machinery we have developed to establish the following result.
Theorem 5.3 Let S be a recursively enumerable L-theory such that
S ⊢ KP + Powerset + ∆ P 0 -Collection + Σ P 1 -Foundation,
and let M be a countable model of S. Then there exists an L-structure N such that M ⊆ P e N |= S, and for some d ∈ N , and for all x ∈ M , N |= (x ∈ d).
Proof Let T be the L ee • for all a ∈ M ,ā ∈ c.
Lemma 5.2 shows that T ⊆ Cov P M is Σ 1 (L * P )-definable over Cov P M . Let T 0 ⊆ T be such that T 0 ∈ Cov P M . Using Lemma 5.1, there exists c ∈ M such that c * = {a ∈ M |ā is mentioned in T 0 }.
Therefore, by interpreting eachā that is mentioned in T 0 by a ∈ M and interpreting c by c, we can expand M to a model M ′ that satisfies T 0 . Therefore, by the Barwise Compactness Theorem, there exists N |= T . It is straightforward to see that the Lreduct of N is the desired extension of M. ✷
We first apply this result to show that countable models of KP P have topless rank extensions that satisfy KP P . This generalises [Fri,Theorem 2.3], which shows that every countable transitive model of KP P has a topless rank extension that satisfies KP P .
Theorem 5.4 (Friedman) Let S be a recursively enumerable L-theory such that S ⊢ KP P . If M is a countable transitive model of S, then there exists N |= S such that M ⊆ rk topless N . ✷ It follows from [Gor,Theorem 4.8] that every countable nonstandard model of KP P + Σ P 1 -Separation, N , is isomorphic to substructure M of N such that M ⊆ rk topless N . We will make use of this result in the following form:
Theorem 5.5 (Gorbow) Let M be a countable nonstandard model of KP P +Σ P 1 -Separation. Then there exists N ≡ M such that M ⊆ rk topless N . ✷ We next note that if a model of KP P has a blunt rank extension, then that model must satisfy the full scheme of separation. So, M = (V N α ) * and every instance of Π ∞ -Separation in M can be reduced to an instance of ∆ 0 -Separation in N and, since M ⊆ P e N , the resulting set will be in M. Therefore, M |= Π ∞ -Separation. ✷ Theorem 5.7 Let S be a recursively enumerable L-theory such S ⊢ KP P . If M is a countable model of S, then there exists N |= S such that M ⊆ rk topless N .
Proof Let M be a countable model of S. If M is well-founded, then M is isomorphic to a transitive model of KP P and we can use Theorem 5.4 to find an L-structure N |= S such that M ⊆ rk topless N . Therefore, assume that M is nonstandard. By Corollary 3.6, M satisfies KP + Powerset + ∆ P 0 -Collection + Σ P 1 -Foundation. Therefore, using Theorem 5.3, we can find an L-structure N |= S such that M = N and M ⊆ P e N . So, by Lemma 2.6, M ⊆ rk e N . If M ⊆ rk topless N , then we are done. Alternatively, if M ⊆ rk blunt N , then, by Lemma 5.6, M |= Π ∞ -Separation. Therefore, since M is nonstandard, we can apply Theorem 5.5 to obtain an L-structure N ′ ≡ M such that M ⊆ rk topless N ′ . ✷
We now turn to showing that every countable model of MOST + Π 1 -Collection + Σ P 1 -Foundation has a topless powerset-preserving end extension that satisfies MOST + Π 1 -Collection + Σ P 1 -Foundation. We first prove an analogue of Lemma 5.6 for models of MOST. which is a set by Σ 1 -Separation. Let µ = sup A and note that µ is an initial ordinal. Work in the metatheory again. If µ ∈ M , then so is (µ + ) N = (µ + ) M ∈ M . So, H N µ + ∈ M and N |= (c ⊆ H µ + ). And, c ∈ M , which is a contradiction. Therefore µ / ∈ M . Now, So, M = (H N µ ) * and every instance of Π ∞ -Separation in M can be reduced to an instance of ∆ 0 -Separation in N and, since M ⊆ P e N , the resulting set will be in M. Therefore, M |= Π ∞ -Separation. ✷ Theorem 5.9 Let S be an L-theory such that S ⊢ MOST+Π 1 -Collection+Σ P 1 -Foundation. If M is a countable model of S, then there exists a model N such that M ⊆ P topless N |= S.
Proof This can be proved using an identical argument to the proof of Theorem 5.7 after observing that every transitive model of MOST + Π 1 -Collection + Σ P 1 -Foundation is a model of KP P and KP P + Σ P 1 -Separation is a subtheory of MOST + Π 1 -Collection + Π ∞ -Separation. ✷
The work [EKM] studies the class C of structures I fix(j) where j : M −→ M is a nontrivial automorphism, M is an L-structure that satisfies MOST, j fixes every point in (ω M ) * and I fix(j) is the substructure of M that consists of elements x of M such that j fixes every point in (TC M ({x})) * . The results of [EKM,Section 3] show that every structure in C satisfies MOST + Π 1 -Collection. Conversely, [EKM,Section 4] shows that a sufficient condition for a countable structure M that satisfies MOST + Π 1 -Collection to be in C is that there exists M ⊆ P topless N such that N satisfies MOST + Π 1 -Collection. Theorem 5.9 allows us to extend [EKM, Theorem B] by showing that C contains all countable models of MOST + Π 1 -Collection + Σ P 1 -Foundation.
Theorem 5.10 Let M = M, E M be a countable model of MOST + Π 1 -Collection + Σ P 1 -Foundation. Then there exists a model N = N, E N that satisfies MOST and a nontrivial automorphism j : N −→ N such that M ∼ = I fix(j) , where I fix(j) is the substructure of N with underlying set I fix(j) = {x ∈ N | (∀y ∈ (TC N ({x})) * )(j(y) = y)}.
✷ Combined with Corollary 3.12 and [EKM,Theorem 5.6] this shows that the class C contains every countable recursively saturated model of MOST+Π 1 -Collection and every countable ω-standard model of MOST+Π 1 -Collection, providing a partial positive answer to Question 5.1 of [EKM]. A positive answer to the following question would positively answer Question 5.1 of [EKM]:
Question 5.11 Does every countable ω-nonstandard model of MOST + Π 1 -Collection have a topless powerset-preserving end extension that satisfies MOST + Π 1 -Collection?
Note that [EKM,Theorem 5.6] shows that this question has a positive answer when the countable model is recursively saturated.
Definition 2. 2
2Let M = M, E M be an L-structure, and let
Theorem 3.11 Let M = M, E M be an ω-standard model of MOST + Π 1 -Collection. Then M |= ∆ P 0 -WDC ω .
The L * -reduct of WF(A N ), WF − (A N ) = N ; WF(Set M ), E M ,F M is an admissible set covering N that is isomorphic to Cov M . ✷ We now turn to extending this result to show that WF(A N ) is a power admissible set covering N and therefore the least power admissible set covering N .Theorem 4.8 The structure WF(A N ) = N ; WF(Set M ), E M ,F M ,P M is a power admissible set covering N . Moreover, WF(A N ) is isomorphic to Cov P M . Proof Note that it follows immediately from Theorem 4.7 that WF(A N ) = N ; WF(Set M ), E M ,F M ,P M satisfies all of the axioms of KPU Cov plus full Foundation. The fact that WF(A N ) ⊆ P e A N implies that WF(A N ) satisfies Powerset and ∆ 0 (L * P )-Separation. To show that WF(A N ) satisfies ∆ 0 (L * P )-Collection, let φ(x, y, z) be a ∆ 0 (L * P )-formula. Let v be sets and/or urelements of WF(A N ) and let a be a set of WF(A N ) such that WF(A N ) |= (∀x ∈ a)∃yφ(x, y, v).
Theorem 4.10 Let M = M, E M |= KP + Powerset + ∆ P 0 -Collection + Σ P 1 -Foundation. For all A ⊆ M , there exists a ∈ M such that a * = A if and only if A ∈ Cov P M . ✷
•
for all a, b ∈ M with M |= (a ∈ b),ā ∈b; • for all a ∈ M , ∀x x ∈ā ⇐⇒ b∈a (x =b) ;• for all a ∈ M ,
Lemma 5. 6
6Let M = M, E M and N = N, E N be models of KP P . If M ⊆ rk blunt N , then M |= Π ∞ -Separation. Proof Assume that M ⊆ N , E M = E N ↾ M and M ⊆ rk blunt N . Let c ∈ N be such that c * ⊆ M and c / ∈ M . Working inside N , let α = ρ(c). Therefore, since M ⊆ rk blunt N , x ∈ (V N α ) * if and only if N |= (ρ(x) < α)if and only if N |= (∃y ∈ c)(ρ(x) ≤ ρ(y)) if and only if x ∈ M.
Let M = M, E M and N = N, E N be models of MOST. If M ⊆ P blunt N , then M |= Π ∞ -Separation. Proof Assume that M ⊆ N , E M = E N ↾ M and M ⊆ rk blunt N . Let c ∈ N be such that c * ⊆ M and c / ∈ M . Work inside N . Let κ = |TC(c)| and note that κ / ∈ M . Consider A = {λ ∈ κ | (∃y ∈ c)(λ = |TC(y)|)},
x
∈ (H N µ ) * if and only if N |= (|TC(x)| < µ) if and only if N |= (∃y ∈ c)(|TC(x)| < |TC(y)|) if and only if x ∈ M.
Table 1
1that extends the table in [Bar75, p. 373]: Table 1: The interpretation I of an
The precise relationship between Friedman's system and KP P is worked out in Section 6.19 of[Mat01].
The abbreviation BFEXT has long been used by NF-theorists for well-founded extensional relations with a top, it is an abbreviation of Bien Fondée Extensionnelle, extensively employed by the Frenchspeaking NF-ists in Belgium.
Rathjen proves the scheme that asserts that set induction holds for all Π P 1 -formulae, which, in the theory KP P , is equivalent to Σ P 1 -Foundation.
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Admissible Sets and Structures. Perspectives in Mathematical Logic. Jon Barwise, Springer-VerlagBerlin-Heidelberg-New YorkBarwise, Jon. Admissible Sets and Structures. Perspectives in Mathematical Logic. Springer-Verlag, Berlin-Heidelberg-New York. 1975.
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Countable models of set theories. Harvey M Friedman, Cambridge Summer School in Mathematical Logic. A. R. D. Mathias and H. Rogers Jr. Springer337SpringerLecture Notes in Mathematics.Friedman, Harvey M. "Countable models of set theories". Cambridge Summer School in Mathematical Logic, August 1-21, 1971. Edited by A. R. D. Mathias and H. Rogers Jr. Springer Lecture Notes in Mathematics. Vol. 337. Springer, Berlin. 1973. pp 539-573.
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| []
|
[
"IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 1 Context Based Emotion Recognition using EMOTIC Dataset",
"IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 1 Context Based Emotion Recognition using EMOTIC Dataset"
]
| [
"Ronak Kosti ",
"Jose M Alvarez ",
"Adria Recasens ",
"Agata Lapedriza "
]
| []
| []
| In our everyday lives and social interactions we often try to perceive the emotional states of people. There has been a lot of research in providing machines with a similar capacity of recognizing emotions. From a computer vision perspective, most of the previous efforts have been focusing in analyzing the facial expressions and, in some cases, also the body pose. Some of these methods work remarkably well in specific settings. However, their performance is limited in natural, unconstrained environments. Psychological studies show that the scene context, in addition to facial expression and body pose, provides important information to our perception of people's emotions. However, the processing of the context for automatic emotion recognition has not been explored in depth, partly due to the lack of proper data. In this paper we present EMOTIC, a dataset of images of people in a diverse set of natural situations, annotated with their apparent emotion. The EMOTIC dataset combines two different types of emotion representation: (1) a set of 26 discrete categories, and (2) the continuous dimensions Valence, Arousal, and Dominance. We also present a detailed statistical and algorithmic analysis of the dataset along with annotators' agreement analysis. Using the EMOTIC dataset we train different CNN models for emotion recognition, combining the information of the bounding box containing the person with the contextual information extracted from the scene. Our results show how scene context provides important information to automatically recognize emotional states and motivate further research in this direction. Dataset and Code is open-sourced and available on https://github.com/rkosti/emotic. Link for the published article https://ieeexplore.ieee.org/document/8713881. | 10.1109/tpami.2019.2916866 | [
"https://arxiv.org/pdf/2003.13401v1.pdf"
]
| 157,057,606 | 2003.13401 | a1223d53683bca97d7868dc216851461f9f34365 |
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 1 Context Based Emotion Recognition using EMOTIC Dataset
Ronak Kosti
Jose M Alvarez
Adria Recasens
Agata Lapedriza
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 1 Context Based Emotion Recognition using EMOTIC Dataset
Index Terms-Emotion recognitionAffective computingPattern recognition !
In our everyday lives and social interactions we often try to perceive the emotional states of people. There has been a lot of research in providing machines with a similar capacity of recognizing emotions. From a computer vision perspective, most of the previous efforts have been focusing in analyzing the facial expressions and, in some cases, also the body pose. Some of these methods work remarkably well in specific settings. However, their performance is limited in natural, unconstrained environments. Psychological studies show that the scene context, in addition to facial expression and body pose, provides important information to our perception of people's emotions. However, the processing of the context for automatic emotion recognition has not been explored in depth, partly due to the lack of proper data. In this paper we present EMOTIC, a dataset of images of people in a diverse set of natural situations, annotated with their apparent emotion. The EMOTIC dataset combines two different types of emotion representation: (1) a set of 26 discrete categories, and (2) the continuous dimensions Valence, Arousal, and Dominance. We also present a detailed statistical and algorithmic analysis of the dataset along with annotators' agreement analysis. Using the EMOTIC dataset we train different CNN models for emotion recognition, combining the information of the bounding box containing the person with the contextual information extracted from the scene. Our results show how scene context provides important information to automatically recognize emotional states and motivate further research in this direction. Dataset and Code is open-sourced and available on https://github.com/rkosti/emotic. Link for the published article https://ieeexplore.ieee.org/document/8713881.
INTRODUCTION
O Ver the past years, the interest in developing automatic systems for recognizing emotional states has grown rapidly. We can find several recent works showing how emotions can be inferred from cues like text [1], voice [2], or visual information [3], [4]. The automatic recognition of emotions has a lot of applications in environments where machines need to interact or monitor humans. For instance, automatic tutors in an online learning platform would provide better feedback to a student according to her level of motivation or frustration. Also, a car with the capacity of assisting a driver can intervene or give an alarm if it detects the driver is tired or nervous.
In this paper we focus on the problem of emotion recognition from visual information. Concretely, we want to recognize the apparent emotional state of a person in a given image. This problem has been broadly studied in computer vision mainly from two perspectives: (1) facial expression analysis, and (2) body posture and gesture analysis. Section 2 gives an overview of related work on these perspectives and also on some of the common public datasets for emotion recognition.
Although face and body pose give lot of information on the affective state of a person, our claim in this work is that scene context information is also a key component for understanding emotional states. Scene surroundings of the person, like the place category, the place attributes, the objects, or the actions occurring around the person. Fig. 1 illustrates the importance of scene context for emotion recognition. When we just see the kid it is difficult to recognize his emotion (from his facial expression it seems he is feeling Surprise). However, when we see the context ( Fig. 2.a) we see the kid is celebrating his birthday, blowing the candles, probably with his family or friends at home. With this additional information we can interpret much better his face and posture and recognize that he probably feels engaged, happy and excited.
The importance of context in emotion perception is well supported by different studies in psychology [5], [6]. In general situations, facial expression is not sufficient to determine the emotional state of a person, since the perception of the emotion is heavily influenced by different types of context, including the scene context [2], [3], [4].
In this work, we present two main contributions. Our first contribution is the creation and publication of the EMOTIC (from EMOTions In Context) Dataset. The EMOTIC database is a collection of images of people annotated according to their apparent emotional states. Images are spontaneous and unconstrained, showing people doing different things in different environments. Fig. 2 shows some examples of images in the EMOTIC database along with their corresponding annotations. As shown, annotations combine 2 different types of emotion representation: Discrete Emotion Categories and 3 Continuous Emotion Dimensions Valence, Arousal, and Dominance [7]. The EMOTIC dataset is now publicly available for download at the EMOTIC website 1 . Details of the dataset construction process and dataset statistics can be found in section 3.
Our second contribution is the creation of a baseline system for the task of emotion recognition in context. In particular, we present and test a Convolutional Neural Network (CNN) model that jointly processes the window of the person and the whole image to predict the apparent emotional state of the person. Section 4 describes the CNN model and the implementation details while section 5 presents our experiments and discussion on the results. All the trained models resulting from this work are also publicly available at the EMOTIC website 1 .
This paper is an extension of the conference paper "Emotion Recognition in Context", presented at the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR) 2017 [8]. We present here an extended version of the EMOTIC dataset, with further statistical dataset analysis, an analysis of scene-centric algorithms on the data, and a study on the annotation consistency among different annotators. This new release of the EMOTIC database contains 44.4% more annotated people as compared to its previous smaller version. With the new extended dataset we retrained all the proposed baseline CNN models with additional loss 1. http://sunai.uoc.edu/emotic/ functions. We also present comparative analysis of two different scene context features, showing how the context is contributing to recognize emotions in the wild.
RELATED WORK
Emotion recognition has been broadly studied by the Computer Vision community. Most of the existing work has focused on the analysis of facial expression to predict emotions [9], [10]. The base of these methods is the Facial Action Coding System [11], which encodes the facial expression using a set of specific localized movements of the face, called Action Units. These facial-based approaches [9], [10] usually use facial-geometry based features or appearance features to describe the face. Afterwards, the extracted features are used to recognize Action Units and the basic emotions proposed by Ekman and Friesen [12]: anger, disgust, fear, happiness, sadness, and surprise. Currently, state-of-the-art systems for emotion recognition from facial expression analysis use CNNs to recognize emotions or Action Units [13].
In terms of emotion representation, some recent works based on facial expression [14] use the continuous dimensions of the V AD Emotional State Model [7]. The VAD model describes emotions using 3 numerical dimensions: Valence (V), that measures how positive or pleasant an emotion is, ranging from negative to positive; Arousal (A), that measures the agitation level of the person, ranging from non-active / in calm to agitated / ready to act; and Dominance (D) that measures the level of control a person feels of the situation, ranging from submissive / non-control to dominant / in-control. On the other hand, Du et al. [15] proposed a set of 21 facial emotion categories, defined as different combinations of the basic emotions, like 'happily surprised' or 'happily disgusted'. With this categorization the authors can give a fine-grained detail about the expressed emotion.
Although the research in emotion recognition from a computer vision perspective is mainly focused in the analysis of the face, there are some works that also consider other additional visual cues or multimodal approaches. For instance, in [16] the location of shoulders is used as additional information to the face features to recognize basic emotions. More generally, Schindler et al. [17] used the body pose to recognize 6 basic emotions, performing experiments on a small dataset of non-spontaneous poses acquired under controlled conditions. Mou et al. [18] presented a system of affect analysis in still images of groups of people, recognizing group-level arousal and valence from combining face, body and contextual information.
Emotion Recognition in Scene Context and Image Sentiment Analysis are different problems that share some characteristics. Emotion Recognition aims to identify the emotions of a person depicted in an image. Image Sentiment Analysis consists of predicting what a person will feel when observing a picture. This picture does not necessarily contain a person. When an image contains a person, there can be a difference between the emotions experienced by the person in the image and the emotions felt by observers of the image. For example, in the image of Figure 2.b, we see a kid who seems to be annoyed for having an apple instead of chocolate and another who seems happy to have chocolate. However, as observers, we might not have any of those sentiments when looking at the photo. Instead, we might think the situation is not fair and feel disapproval. Also, if we see an image of an athlete that has lost a match, we can recognize the athlete feels sad. However, an observer of the image may feel happy if the observer is a fan of the team that won the match.
Emotion Recognition Datasets
Most of the existing datasets for emotion recognition using computer vision are centered in facial expression analysis. For example, the GENKI database [19] contains frontal face images of a single person with different illumination, geographic, personal and ethnic settings. Images in this dataset are labelled as smiling or non-smiling. Another common facial expression analysis dataset is the ICML Face-Expression Recognition dataset [20], that contains 28, 000 images annotated with 6 basic emotions and a neutral category. On the other hand, the UCDSEE dataset [21] has a set of 9 emotion expressions acted by 4 persons. The lab setting is strictly kept the same in order to focus mainly on the facial expression of the person.
The dynamic body movement is also an essential source for estimating emotion. Studies such as [22], [23] establish the relationship between affect and body posture using as ground truth the base-rate of human observers. The data consist of a spontaneous set of images acquired under a restrictive setting (people playing Wii games). The GEMEP database [24] is multi-modal (audio and video) and has 10 actors playing 18 affective states. The dataset has videos of actors showing emotions through acting. Body pose and facial expression are combined.
The Looking at People (LAP) challenges and competitions [25] involve specialized datasets containing images, sequences of images and multi-modal data. The main focus of these datasets is the complexity and variability of human body configuration which include data related to personality traits (spontaneous), gesture recognition (acted), apparent age recognition (spontaneous), cultural event recognition (spontaneous), action/interaction recognition and human pose recognition (spontaneous).
The Emotion Recognition in the Wild (EmotiW) challenges [26] host 3 databases: (1) The AFEW database [27] focuses on emotion recognition from video frames taken from movies and TV shows, where the actions are annotated with attributes like name, age of actor, age of character, pose, gender, expression of person, the overall clip expression and the basic 6 emotions and a neutral category; (2) The SFEW, which is a subset of AFEW database containing images of face-frames annotated specifically with the 6 basic emotions and a neutral category; and (3) the HAPPEI database [28], which addresses the problem of group level emotion estimation. Thus, [28] offers a first attempt to use context for the problem of predicting happiness in groups of people.
Finally, the COCO dataset has been recently annotated with object attributes [29], including some emotion categories for people, such as happy and curious. These attributes show some overlap with the categories that we define in this paper. However, COCO attributes are not intended to be exhaustive for emotion recognition, and not all the people in the dataset are annotated with affect attributes.
EMOTIC DATASET
The EMOTIC dataset is a collection of images of people in unconstrained environments annotated according to their apparent emotional states. The dataset contains 23, 571 images and 34, 320 annotated people. Some of the images were manually collected from the Internet by Google search engine. For that we used a combination of queries containing various places, social environments, different activities and a variety of keywords on emotional states. The rest of images belong to 2 public benchmark datasets: COCO [30] and Ade20k [31]. Overall, the images show a wide diversity of contexts, containing people in different places, social settings, and doing different activities. Fig. 2 shows three examples of annotated images in the EMOTIC dataset. Images were annotated using Amazon Mechanical Turk (AMT). Annotators were asked to label each image according to what they think people in the images are feeling. Notice that we have the capacity of making reasonable guesses about other people's emotional state due to our capacity of being empathetic, putting ourselves into another's situation, and also because of our common sense knowledge and our ability for reasoning about visual information. For example, in Fig. 2.b, the person is performing an activity that requires Anticipation to adapt to the trajectory. Since he is doing a thrilling activity, he seems excited about it and he is engaged or focused in this activity. In Fig. 2.c, the kid feels a strong desire (yearning) for eating the chocolate instead of the apple. Because of his situation we can interpret his facial expression as disquietness and annoyance. Notice that images are also annotated according to the continuous dimensions V alence, Arousal, and Dominance. We describe the emotion annotation modalities of EMOTIC dataset and the annotation process in sections 3.1 and 3.2, respectively.
After the first round of annotations (1 annotator per image), we divided the images into three sets: Training (70%), Validation (10%), and Testing (20%) maintaining a similar affective category distribution across the different sets. After that, Validation and Testing were annotated by 4 and 2 extra annotators respectively. As a consequence, images in the Validation set are annotated by a total of 5 annotators, while images in the Testing set are annotated by 3 annotators (these numbers can slightly vary for some images since we removed noisy annotations).
We used the annotations from the Validation to study the consistency of the annotations across different annotators. This study is shown in section 3.3. The data statistics and algorithmic analysis on the EMOTIC dataset are detailed in sections 3.4 and 3.5 respectively.
Emotion representation
The EMOTIC dataset combines two different types of emotion representation:
Continuous Dimensions: images are annotated according to the V AD model [7], which represents emotions by a combination of 3 continuous dimensions: Valence, Arousal and Dominance. In our representation each dimension takes an integer value that lies in the range [1 − 10]. Fig. 4 shows examples of people annotated by different values of the given dimension. Table 1). The person in the red bounding box is annotated by the corresponding category. The list of emotion categories has been created as follows. We manually collected an affective vocabulary from dictionaries and books on psychology [32], [33], [34], [35]. This vocabulary consists of a list of approximately 400 words representing a wide variety of emotional states. After a careful study of the definitions and the similarities amongst these definitions, we formed cluster of words with similar meanings. The clusters were formalized into 26 categories such that they were distinguishable in a single image of a person with her context. We created the final list of 26 emotion categories taking into account the Visual Separability criterion: words that have a close meaning could not be visually separable. For instance, Anger is defined by the words rage, furious and resentful. These affective states are different, but it is not always possible to separate them
Emotion Category
Continuous Dimension "Consider each emotion category separately and, if it is applicable to the person in the given context, select that emotion category" "Consider each emotion dimension separately, observe what level is applicable to the person in the given context, and select that level" Notice that the final list of affective categories also includes the 6 basic emotions (categories 2, 5, 16, 17, 21, 24), but we used the more general term Aversion for the category Disgust. Thus, the category Aversion also includes the subcategories dislike, repulsion, and hate apart from disgust.
Collecting Annotations
We used Amazon Mechanical Turk (AMT) crowd-sourcing platform to collect the annotations of the EMOTIC dataset. We designed two Human Intelligence Tasks (HITs), one for each of the 2 formats of emotion representation. The two annotation interfaces are shown in Fig. 5. Each annotator is shown a person-in-context enclosed in a red bounding-box along with the annotation format next to it. Fig. 5.a shows the interface for discrete category annotation while Fig. 5.b displays the interface for continuous dimension annotation. Notice that, in the last box of the continuous dimension interface, we also ask AMT workers to annotate the gender and estimate the age (range) of the person enclosed in red bounding-box. The designing of the annotation interface has two main focuses: i) the task is easy to understand and ii) the interface fits the HIT in one screen which avoids scrolling.
To make sure annotators understand the task, we showed them how to annotate the images step-wise, by explaining two examples in detail. Also, instructions and examples were attached at the bottom on each page as a quick reference to the annotator. Finally, a summary of the detailed instructions was shown at the top of each page ( Table 2). We adopted two strategies to avoid noisy annotations in the EMOTIC dataset. First, we conduct a qualification task to annotator candidates. This qualification task has two parts: (i) an Emotional Quotient HIT (based on standard EQ task [36]) and (ii) 2 sample image annotation tasks -one for each of our 2 emotion representations (discrete categories and continuous dimensions). For the sample annotations, we had a set of acceptable labels. The responses of the annotator candidates to this qualification task were evaluated and those who responded satisfactorily were allowed to annotate the images from the EMOTIC dataset. The second strategy to avoid noisy annotations was to insert, randomly, 2 control images in every annotation batch of 20 images; the correct assortment of labels for the control images was know beforehand. Annotators selecting incorrect labels on these control images were not allowed to annotate further and their annotations were discarded.
a) b) Back
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Peace
(fond feelings/tenderness/love/compassion) Expectation (state of anticipating/hoping on something or someone) Esteem (favorable opinion or judgment/gratefulness/admiration/respect) (feeling of being certain/proud/encouraged/optimistic) Engagement (occupied/absorbed/interested/paying attention to something) Pleasure (feeling of delight in the senses) Happiness (feeling delighted/enjoyment/amusement) Excitement (pleasant and excited state/stimulated/energetic/enthusiastic) Surprise (sudden discovery of something unexpected) (distressed/perturbed/anguished) Disapproval (think that something is wrong or reprehensible/contempt/hostile) Yearning (strong desire to have something/jealous/envious) Fatigue (weariness/tiredness/sleepy) Pain Doubt/Confusion Fear (feeling afraid of danger/evil/pain/horror) Vulnerability (feeling of being physically or emotionally wounded) Disquitement (unpleasant restlessness/tense/worried/upset/stressed) Annoyance (bothered/iritated/impatient/troubled/frustrated) Anger (intense displeasure or rage/furious/resentful) Disgust (feeling dislike or repulsion/feeling hateful) Sadness (feeling unhappy/grief/disappointed/discouraged) Disconnection Embarrassment (feeling ashamed or guilty)
Back
Go to Next Image (Image 1 of 20)
Agreement Level Among Different Annotators
Since emotion perception is a subjective task, different people can perceive different emotions after seeing the same image. For example in both Fig. 6.a and 6.b, the person in the red box seems to feel Affection, Happiness and Pleasure and the annotators have annotated with these categories with consistency. However, not everyone has selected all these emotions. Also, we see that annotators do not agree in the emotions Excitement and Engagement. We consider, however, that these categories are reasonable in this situation. Another example is that of Roger Federer hitting a tennis ball in Fig. 6.c. He is seen predicting the ball (or Anticipating) and clearly looks Engaged in the activity. He also seems Confident in getting the ball. After these observations we conducted different quantitative analysis on the annotation agreement. We focused first on analyzing the agreement level in the category annotation. Given a category assigned to a person in an image, we consider as an agreement measure the number of annotators agreeing for that particular category. Accordingly, we calculated, for each category and for each annotation in the validation set, the agreement amongst the annotators and sorted those values across categories. Fig. 7 shows the distribution on the percentage of annotators agreeing for an annotated category across the validation set.
We also computed the agreement between all the annotators for a given person using Fleiss' Kappa (κ). Fleiss' Kappa is a common measure to evaluate the agreement level among a fixed number of annotators when assigning categories to data. In our case, given a person to annotate, there is a subset of 26 categories. If we have N annotators per image, that means that each of the 26 categories can be selected by n annotators, where 0 ≤ n ≤ N . Given an image we compute the Fleiss' Kappa per each emotion category first, and then the general agreement level on this image is computed as the average of these Fleiss' Kappa values across the different emotion categories. We obtained that more than 50% of the images have κ > 0.30. Fig. 8.a shows the distribution of kappa values across the validation set for all the annotated people in the validation set, sorted in decreasing order. Random annotations or total disagreement produces κ ∼ 0, however for our case, κ ∼ 0.3 (on average) suggesting significant agreement level even though the task of emotion recognition is subjective. For continuous dimensions, the agreement is measured by the standard deviation (SD) of the different annotations. The average SD across the Validation set is 1.04, 1.57 and 1.84 for Valence, Arousal and Dominance respectively -indicating that Dominance has higher (±1.84) dispersion than the other dimensions. It reflects that annotators disagree more often for Dominance than for the other dimensions which is understandable since Dominance is more difficult to interpret than Valence or Arousal [7]. As a summary, Fig. 8.b shows the standard deviations of all the images in the
Dataset Statistics
EMOTIC dataset contains 34, 320 annotated people, where 66% of them are males and 34% of them are females. There are 10% children, 7% teenagers and 83% adults amongst them. Fig. 9.a shows the number of annotated people for each of the 26 emotion categories, sorted by decreasing order. Notice that the data is unbalanced, which makes the dataset particularly challenging. An interesting observation is that there are more examples for categories associated to positive emotions, like Happiness or Pleasure, than for categories associated with negative emotions, like Pain or Embarrassment. The category with most examples is Engagement. This is because in most of the images people are doing something or are involved in some activity, showing some degree of engagement. Figs. 9.b, 9.c and 9.d show the number of annotated people for each value of the 3 continuous dimensions. In this case we also observe unbalanced data but fairly distributed across the 3 dimensions which is good for modelling. Fig. 10: Co-variance between 26 emotion categories. Each row represents the occurrence probability of every other category given the category of that particular row. Fig. 10 shows the co-occurrence rates of any two categories. Every value in the matrix (r, c) (r represents the row category and c column category) is a co-occurrence probability (in %) of category r if the annotation also contains the category c, that is, P (r|c). We observe, for instance, that when a person is labelled with the category Annoyance, then there is 46.05% probability that this person is also annotated by the category Anger. This means that when a person seems to be feeling Annoyance it is likely (by 46.05%) that this person might also be feeling Anger. We also used a K-Means clustering on the category annotations to find groups of categories that occur frequently. We found, for example, that these category groups are common in the EMOTIC annotations: {Anticipation, Engagement, Confidence}, {Affection, Happiness, Pleasure}, {Doubt/Confusion, Disapproval, Annoyance}, {Yearning, Annoyance, Disquietment}. Fig. 11 shows the distribution of each continuous dimension across the different emotion categories. For every plot, categories are arranged in increasing order of their average values of the given dimension (calculated for all the instances containing that particular category). Thus, we observe from Fig. 11.a that emotion categories like Suffering, Annoyance, Pain correlate with low Valence values (feeling less positive) in average whereas emotion categories like Pleasure, Happiness, Affection correlate with higher Valence values (feeling more positive). Also interesting is to note that a category like Disconnection lies in the mid-range of Valence value which makes sense. When we observe Fig. 11.b, it is easy to understand that emotional categories like Disconnection, Fatigue, Sadness show low Arousal values and we see high activeness for emotion categories like Anticipation, Confidence, Excitement. Finally, Fig. 11.c shows that people are not in control when they show emotion categories like Suffering, Pain, Sadness whereas when the Dominance is high, emotion categories like Esteem, Excitement, Confidence occur more often.
Cooccurence of Categories
An important remark about the EMOTIC dataset is that there are people whose faces are not visible. More than 25% of the people in EMOTIC have their faces partially occluded or with very low resolution, so we can not rely on facial expression analysis for recognizing their emotional state.
Algorithmic Scene Context Analysis
This section illustrates how current scene-centric systems can be used to extract contextual information that can be potentially useful for emotion recognition. In particular, we illustrate this idea with a CNN trained on Places dataset [37] and with the Sentibanks Adjective-Noun Pair (ANP) detectors [38], [39], a Visual Sentiment Ontology for Image Sentiment Analysis. As a reference, Fig. 12 shows Places and ANP outputs for sample images of the EMOTIC dataset. We used AlexNet Places CNN [37] to predict the scene category and scene attributes for the images in EMOTIC. This information helps to divide the analysis into place category and place attribute. We observed that the distribution of emotions varies significantly among different place categories. For example, we found that people in the 'ski slope' frequently experience Anticipation or Excitement, which are associated to the activities that usually happen in this place category. Comparing sport-related and workingenvironment related images, we find that people in sport- Fig. 12: Illustration of 2 current scene-centric methods for extracting contextual features from the scene: AlexNet Places CNN outputs (place categories and attributes) and Sentibanks ANP outputs for three example images of the EMOTIC dataset. related images usually show Excitement, Anticipation and Confidence, however they show Sadness or Annoyance less frequently. Interestingly, Sadness and Annoyance appear with higher frequency in working environments. We also observe interesting patterns when correlating continuous dimensions with place attributes and categories. For instance, places where people usually show high Dominance are sport-related places and sport-related attributes. On the contrary, low Dominance is shown in 'jail cell' or attributes like 'enclosed area' or 'working', where the freedom of movement is restricted. In Fig. 12, the predictions by Places CNN describe the scene in general, like in the top image there is a girl sitting in a 'kindergarten classroom' (places category) which usually is situated in enclosed areas with 'no horizon' (attributes).
We also find interesting patterns when we compute the correlation between detected ANPs and emotions labelled in the image. For example, in images with people labelled with Affection, the most frequent ANP is 'young couple', while in images with people labelled with Excitement we found frequently the ANPs 'last game' and 'playing field'. Also, we observe a high correlation between images with Peace and ANP like 'old couple' and 'domestic scenes', and between Happiness and the ANPs 'outdoor wedding', 'outdoor activities', 'happy family' or 'happy couple'.
Overall, these observations suggest that some common sense knowledge patterns related with emotions and context could be potentially extracted, automatically, from the data.
CNN MODEL FOR EMOTION RECOGNITION IN SCENE CONTEXT
We propose a baseline CNN model for the problem of emotion recognition in context. The pipeline of the model is shown in Fig. 13 and it is divided in three modules: body feature extraction, image (context) feature extraction and fusion network. The first module takes the whole image as input and generates scene-related features. The second module takes the visible body of the person and generates bodyrelated features. Finally, the third module combines these features to do a fine-grained regression of the two types of emotion representations (section 3.1). Fig. 13: Proposed end-to-end model for emotion recognition in context. The model consists of two feature extraction modules and a fusion network for jointly estimating the discrete categories and the continuous dimensions.
The body feature extraction module takes the visible part of the body of the target person as input and generates body-related features. These features include important cues like face and head aspects and pose or body appearance. In order to capture these aspects, this module is pre-trained with ImageNet [40], which is an object centric dataset that includes the category person.
The image feature extraction module takes the whole image as input and generates scene-context features. These contextual features can be interpreted as an encoding of the scene category, its attributes and objects present in the scene, or the dynamics between other people present in the scene. To capture these aspects, we pre-train this module with the scene-centric Places dataset [37].
The fusion module combines features of the two feature extraction modules and estimates the discrete emotion categories and the continuous emotion dimensions.
Both feature extraction modules are based on the onedimensional filter CNN proposed in [41]. These CNN networks provide competitive performance while the number of parameters is low. Each network consists of 16 convolutional layers with 1-dimensional kernels alternating between horizontal and vertical orientations, effectively modeling 8 layers using 2-dimensional kernels. Then, to maintain the location of different parts of the image, we use a global average pooling layer to reduce the features of the last convolutional layer. To avoid internal-covariant-shift we add a batch normalizing layer [42] after each convolutional layer and rectifier linear units to speed up the training.
The fusion network module consists of two fully connected (FC) layers. The first FC layer is used to reduce the dimensionality of the features to 256 and then, a second fully connected layer is used to learn independent representations for each task [43]. The output of this second FC layer branches off into 2 separate representations, one with 26 units representing the discrete emotion categories, and second with 3 units representing the 3 continuous dimensions (section 3.1).
Loss Function and Training Setup
We define the loss function as a weighted combination of two separate losses. A predictionŷ is composed by the prediction of each of the 26 discrete categories and the 3 continuous dimensions,ŷ = (ŷ disc ,ŷ cont ). In particular, y disc = (ŷ disc . Given a predictionŷ, the loss in this prediction is defined by L = λ disc L disc + λ cont L cont , where L disc and L cont represent the loss corresponding to learning the discrete categories and the continuous dimensions respectively. The parameters λ (disc,cont) weight the contribution of each loss and are set empirically using the validation set.
Criterion for Discrete categories (L disc ): The discrete category estimation is a multilabel problem with an inherent class imbalance issue, as the number of training examples is not the same for each class (see Fig 9.a).
In our experiments, we use a weighted Euclidean loss for the discrete categories. Empirically, we found the Euclidean loss to be more effective than using Kullback−Leibler divergence or a multi-class multi-classification hinge loss. More precisely, given a predictionŷ disc , the weighted Euclidean loss is defined as follows
L 2disc (ŷ disc ) = 26 i=1 w i (ŷ disc i − y disc i ) 2(1)
whereŷ disc i is the prediction for the i-th category and y disc i is the ground-truth label. The parameter w i is the weight assigned to each category. Weight values are defined as w i = 1 ln(c+pi) , where p i is the probability of the i-th category and c is a parameter to control the range of valid values for w i . Using this weighting scheme the values of w i are bounded as the number of instances of a category approach to 0. This is particularly relevant in our case as we set the weights based on the occurrence of each category for each batch. Experimentally, we obtained better results using this approach compared to setting the global weights based on the entire dataset.
Criterion for Continuous dimensions (L cont ): We model the estimation of the continuous dimensions as a regression problem. Due to multiple annotators annotating the data based on subjective evaluation, we compare the performance when using two different robust losses: (1) a margin Euclidean loss L 2cont , and (2) the Smooth L 1 SL 1cont . The former defines a margin of error (v k ) when computing the loss for which the error is not considered. The margin Euclidean loss for continuous dimension is defined as:
L 2cont (ŷ cont ) = 3 k=1 v k (ŷ cont k − y cont k ) 2 ,(2)
whereŷ cont k and y cont k are the prediction and the groundtruth for the k-th dimension, respectively, and v k ∈ {0, 1} is a binary weight to represent the error margin. v k = 0 if |ŷ cont k − y cont k | < θ. Otherwise, v k = 1. If the predictions are within the error margin, i.e. error is smaller than θ, then these predictions do not contribute to update the weights of the network.
The Smooth L 1 loss refers to the absolute error using the squared error if the error is less than a threshold (set to 1 in our experiments). This loss has been widely used for object detection [44] and, in our experiments, has been shown to be less sensitive to outliers. Precisely, the Smooth L 1 loss is defined as follows
SL 1cont (ŷ cont ) = 3 k=1 v k 0.5x 2 , if |x k | < 1 |x k | − 0.5, otherwise(3)
where x k = (ŷ cont k − y cont k ), and v k is a weight assigned to each of the continuous dimensions and it is set to 1 in our experiments.
We train our recognition system end-to-end, learning the parameters jointly using stochastic gradient descent with momentum. The first two modules are initialized using pretrained models from Places [37] and Imagenet [45] while the fusion network is trained from scratch. The batch size is set to 52 -twice the size of the discrete emotion categories. We found empirically after testing multiple batch sizes (including multiples of 26 like 26, 52, 78, 108) that batchsize of 52 gives the best performance (on the validation set).
EXPERIMENTS
We trained four different instances of our CNN model, which are the combination of two different input types and the two different continuous loss functions described in section 4.1. The input types are body (i.e., upper branch in Fig. 13), denoted by B, and body plus image (i.e., both branches shown in Fig. 13 Results for discrete categories in the form of Average Precision per category (the higher, the better) are summarized in Table 3. Notice that the B+I model outperforms the B model in all categories except 1. The combination of body and image features (B+I(SL 1 ) model) is better than the B model.
Results for continuous dimensions in the form of Average Absolute Error per dimension, AAE (the lower, the better) are summarized in Table 4. In this case, all the models provide similar results where differences are not significant. Fig. 14 shows the summary of the results obtained per each instance in the testing set. Specifically, Fig. 14.a shows Jaccard coefficient (J C) for all the samples in the test set. The JC coefficient is computed as follows: per each category we use as threshold for the detection of the category the value where Precision = Recall. Then, the JC coefficient is computed as the number of categories detected that are also present in the ground truth (number of categories in the intersection of detections and ground truth) divided by the total number of categories that are in the ground truth or detected (union over detected categories and categories in the ground truth). The higher this JC is the better, with a maximum value of 1, where the detected categories and the ground truth categories are exactly the same. In the graphic, examples are sorted in decreasing order of the JC coefficient. Notice that these results also support that the B+I model outperforms the B model.
For the case of continuous dimensions, Fig. 14.b shows the Average Absolute Error (AAE) obtained per each sample in the testing set. Samples are sorted by increasing order (best performances on the left). Consistent with the results shown in Table 4, we do not observe a significant difference among the different models.
Finally, Fig. 15 shows qualitative predictions for the best B and B+I models. These examples were randomly selected among samples with high JC in B+I (a-b) and samples with Fig. 15.c.
Context Features Comparison
The goal of this section is to compare different context features for the problem of emotion recognition in context.
A key aspect for incorporating the context in an emotion recognition model is to be able to obtain information from the context that is actually relevant for emotion recognition.
Since the context information extraction is a scene-centric task, the information extracted from the context should be based in a scene-centric feature extraction system. That is why our baseline model uses a Places CNN for the context feature extraction module. However, recent works in sentiment analysis (detecting the emotion of a person when he/she observes an image) also provide a system for scene feature extraction that can be used for encoding the relevant contextual information for emotion recognition.
To compute body features, denoted by B f , we fine tune an AlexNet ImageNet CNN with EMOTIC database, and use the average pooling of the last convolutional layer as features. For the context (image), we compare two different feature types, which are denoted by I f and I S . I f are obtained by fine tunning an AlexNet Places CNN with EMOTIC database, and taking the average pooling of the last convolutional layer as features (similar to B f ), while I S is a feature vector composed of the sentiment scores for the ANP detectors from the implementation of [39].
To fairly compare the contribution of the different context features, we train Logistic Regressors for the following features and combination of features: (1) B f , (2) B f +I f , and (3) B f +I S . For the discrete categories we obtain mean APs AP = 23.00, AP = 27.70, and AP = 29.45, respectively. For the continuous dimensions, we obtain AAE 0.0704, 0.0643, and 0.0713 respectively. We observe that, for the discrete categories, both I f and I S contribute relevant information to the emotion recognition in context. Interestingly, I S performs better than I f , even though these features have not been trained using EMOTIC. However, these features are smartly designed for sentiment analysis, which is a problem closely related to extracting relevant contextual information for emotion recognition, and are trained with a large dataset of images.
CONCLUSIONS
In this paper we pointed out the importance of considering the person scene context in the problem of automatic emotion recognition in the wild. We presented the EMOTIC database, a dataset of 23, 571 natural unconstrained images with 34, 320 people labeled according to their apparent emotions. The images in the dataset are annotated using two different emotion representations: 26 discrete categories, and the 3 continuous dimensions V alence, Arousal and Dominance. We described in depth the annotation process and analyzed the annotation consistency of different annotators. We also provided different statistics and algorithmic analysis on the data, showing the characteristics of the EMOTIC database. In addition, we proposed a baseline B (L2) Ground Truth B+I (SL1) CNN model for emotion recognition in scene context that combines the information of the person (body bounding box) with the scene context information (whole image). We also compare two different feature types for encoding the contextual information. Our results show the relevance of using contextual information to recognize emotions and, in conjunction with the EMOTIC dataset, motivate further research in this direction. All the data and trained models are publicly available for the research community in the website of the project.
Fig. 1 :
1How is this kid feeling? Try to recognize his emotional states from the person bounding box, without scene context.
Fig. 2 :
2Sample images in the EMOTIC dataset along with their annotations.
Fig. 3 :
3Examples of annotated people in EMOTIC dataset for each of the 26 emotion categories (
Fig. 4 :
4Examples of annotated images in EMOTIC dataset for each of the 3 continuous dimensions Valence, Arousal & Dominance. The person in the red bounding box has the corresponding value of the given dimension. Emotion Categories: in addition to VAD we also established a list of 26 emotion categories that represent various state of emotions. The list of the 26 emotional categories and their corresponding definitions can be found inTable 1. Also,Fig. 3shows (per category) examples of people showing different emotional categories.
Fig. 5 :
5Valence: Negative vs. Positive Arousal (awakeness): Calm vs. Ready to act Dominance: Dominated vs. In control Gender and age of the person in the yellow box AMT interface designs (a) For Discrete Categories' annotations & (b) For Continuous Dimensions' annotations
Fig. 6 :
6Annotations of five different annotators for 3 images in EMOTIC.
Fig. 7 :
7Representation of agreement between multiple annotators. Categories sorted in decreasing order according to the average number of annotators who agreed for that category.
Fig. 8: (a) Kappa values (sorted) and (b) Standard deviation (sorted), for each annotated person in validation set
Fig. 9 :
9Dataset Statistics. (a) Number of people annotated for each emotion category; (b), (c) & (d) Number of people annotated for every value of the three continuous dimensions viz.Valence, Arousal & Dominance validation set for all the 3 dimensions, sorted in decreasing order.
Fig. 11 :
11Distribution of continuous dimension values across emotion categories. Average value of a dimension is calculated for every category and then plotted in increasing order for every distribution.
), denoted by B+I. The continuous loss types are denoted in the experiments by L 2 for Euclidean loss (equation 2) and SL 1 for the Smooth L 1 (equation 3).
Fig. 14 :
14Results per each sample (Test Set, sorted): (a) Jaccard Coefficient (J C) of the recognized discrete categories (b) Average Absolute Error (AAE) in the estimation of the three continuous dimensions.
Fig. 15 :
15Ground truth and results on images randomly selected with different JC scores.
context includes the • Ronak Kosti & Agata Lapedriza are with Universitat Oberta de Catalunya, Spain. Email: [email protected], [email protected]. • Adria Recasens is with the Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, USA. Email: [email protected]. • Jose M. Alvarez is with NVIDIA, USA. Email: [email protected] • Project Page: http://sunai.uoc.edu/emotic/
1 .
1Affection: fond feelings; love; tenderness 2. Anger: intense displeasure or rage; furious; resentful 3. Annoyance: bothered by something or someone; irritated; impatient; frustrated 4. Anticipation: state of looking forward; hoping on or getting prepared for possible future events 5. Aversion: feeling disgust, dislike, repulsion; feeling hate 6. Confidence: feeling of being certain; conviction that an outcome will be favorable; encouraged; proud 7. Disapproval: feeling that something is wrong or reprehensible; contempt; hostile 8. Disconnection: feeling not interested in the main event of the surrounding; indifferent; bored; distracted 9. Disquietment: nervous; worried; upset; anxious; tense; pressured; alarmed 10. Doubt/Confusion: difficulty to understand or decide; thinking about different options 11. Embarrassment: feeling ashamed or guilty 12. Engagement: paying attention to something; absorbed into something; curious; interested 13. Esteem: feelings of favourable opinion or judgement; respect; admiration; gratefulness 14. Excitement: feeling enthusiasm; stimulated; energetic 15. Fatigue: weariness; tiredness; sleepy 16. Fear: feeling suspicious or afraid of danger, threat, evil or pain; horror 17. Happiness: feeling delighted; feeling enjoyment or amusement 18. Pain: physical suffering 19. Peace: well being and relaxed; no worry; having positive thoughts or sensations; satisfied 20. Pleasure: feeling of delight in the senses 21. Sadness: feeling unhappy, sorrow, disappointed, or discouraged 22. Sensitivity: feeling of being physically or emotionally wounded; feeling delicate or vulnerable 23. Suffering: psychological or emotional pain; distressed; anguished 24. Surprise: sudden discovery of something unexpected 25. Sympathy: state of sharing others emotions, goals or troubles; supportive; compassionate 26. Yearning: strong desire to have something; jealous; envious; lust
TABLE 1 :
1Proposed emotion categories with definitions.
TABLE 2 :
2Instruction summary for each HIT visually in a single image. Thus, our list of affective categories can be seen as a first level of a hierarchy, where each category has associated subcategories.
TABLE 3 :
3Average Precision (AP) obtained on test set per category. Results for models where the input is just the body B, and models where the input are both the body and the whole image B+I. The type of L cont used is indicated in parenthesis (L 2 refers to equation 2 and SL 1 refers to equation 3).Continuous
Dimensions
CNN Inputs and Lcont type
B (L 2 )
B (SL 1 )
B+I (L 2 )
B+I (SL 1 )
Valence
0.0537
0.0545
0.0546
0.0528
Arousal
0.0600
0.0630
0.0648
0.0611
Dominance
0.0570
0.0567
0.0573
0.0579
Mean
0.0569
0.0581
0.0589
0.0573
TABLE 4 :
4Average Absolute Error (AAE) obtained on test set per each continuous dimension. Results for models where the input is just the body B, and models where the input are both the body and the whole image B+I. The type of L cont used is indicated in parenthesis (L 2 refers to equation 2 and SL 1 refers to equation 3).low JC in B+I (g-h). Incorrect category recognition is indicated in red. As shown, in general, B+I model outperforms B, although there are some exceptions, like
ACKNOWLEDGMENT This work has been partially supported by the Ministerio de Economia, Industria y Competitividad (Spain), under the Grants Ref. TIN2015-66951-C2-2-R and RTI2018-095232-B-C22, and by Innovation and Universities (FEDER funds). The authors also thank NVIDIA for their generous hardware donations.
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Ronak Kosti is pursuing his PhD at Universitat Oberta de Catalunya, Spain advised by Prof. Ronak Kosti is pursuing his PhD at Universitat Oberta de Catalunya, Spain advised by Prof.
| [
"https://github.com/rkosti/emotic."
]
|
[
"Anomalous Transport in a Superfluid Fluctuation Regime",
"Anomalous Transport in a Superfluid Fluctuation Regime"
]
| [
"Shun Uchino \nRIKEN Center for Emergent Matter Science\n351-0198WakoSaitamaJapan\n",
"Masahito Ueda \nRIKEN Center for Emergent Matter Science\n351-0198WakoSaitamaJapan\n\nDepartment of Physics\nUniversity of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan\n"
]
| [
"RIKEN Center for Emergent Matter Science\n351-0198WakoSaitamaJapan",
"RIKEN Center for Emergent Matter Science\n351-0198WakoSaitamaJapan",
"Department of Physics\nUniversity of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan"
]
| []
| Motivated by a recent experiment in ultracold atoms [ S. Krinner et al., Proc. Natl. Acad. Sci. U.S.A 113, 8144 (2016)], we analyze transport of attractively interacting fermions through a one-dimensional wire near the superfluid transition. We show that in a ballistic regime where the conductance is quantized in the absence of interaction, the conductance is renormalized by superfluid fluctuations in reservoirs. In particular, the particle conductance is strongly enhanced and the plateau is blurred by emergent bosonic pair transport. For spin transport, in addition to the contact resistance the wire itself is resistive, leading to a suppression of the measured spin conductance. Our results are qualitatively consistent with the experimental observations. | 10.1103/physrevlett.118.105303 | [
"https://arxiv.org/pdf/1608.01070v2.pdf"
]
| 40,754,352 | 1608.01070 | 8f6bd508cf24844250cb9b8118ba039e4d345f55 |
Anomalous Transport in a Superfluid Fluctuation Regime
Shun Uchino
RIKEN Center for Emergent Matter Science
351-0198WakoSaitamaJapan
Masahito Ueda
RIKEN Center for Emergent Matter Science
351-0198WakoSaitamaJapan
Department of Physics
University of Tokyo
7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan
Anomalous Transport in a Superfluid Fluctuation Regime
j=L,R [ p σ=↑,↓ ξ j,p,σ c † j,p,σ c j,p,σ + V j ] + H T , (1) V j = −g p,p ,q c † j,p+q,↑ c † j,−p,↓ c j,−p ,↓ c j,p +q,↑ , (2)
Motivated by a recent experiment in ultracold atoms [ S. Krinner et al., Proc. Natl. Acad. Sci. U.S.A 113, 8144 (2016)], we analyze transport of attractively interacting fermions through a one-dimensional wire near the superfluid transition. We show that in a ballistic regime where the conductance is quantized in the absence of interaction, the conductance is renormalized by superfluid fluctuations in reservoirs. In particular, the particle conductance is strongly enhanced and the plateau is blurred by emergent bosonic pair transport. For spin transport, in addition to the contact resistance the wire itself is resistive, leading to a suppression of the measured spin conductance. Our results are qualitatively consistent with the experimental observations.
Transport measurements often play crucial roles in revealing the fundamental nature of matter. In condensed matter physics, superconductivity, the Kondo effect, and the quantum Hall effect were all discovered with transport measurements. A two-terminal setup realized in ultracold atoms has opened up yet another avenue to explore strongly correlated systems through transport [1][2][3][4][5][6][7]. In ultracold atoms, quantum transport typically occurs in the clean limit. The bulk conductivity cannot distinguish between different quantum states, since the f -sum rule and the momentum conservation dictate that the conductivity involves the delta-function singularity at zero frequency whose weight does not depend on detailed states of matter [8]. However, transport through a constriction such as a quantum point contact allows one to distinguish between different states due to the breakdown of the momentum conservation at the constriction. Indeed, such a setup has unveiled different transport properties for non-interacting [5] and superfluid fermions [6].
Recently, particle conductance and spin conductance have been measured with a quantum point contact in ultracold fermions [7]. There, with increasing attractive interaction, both of them deviate significantly from quantized values [9] just above the superfluid critical temperature. More specifically, compared with the noninteracting limit, the particle conductance is enhanced, whereas the spin conductance is suppressed. These remarkable features stand in sharp contrast with the conventional wisdom that a conductance is not renormalized by an interaction in a one-dimensional wire [10][11][12][13][14][15]. Meanwhile, different from condensed matter situations, fluctuation and interaction effects in reservoirs may be significant in cold atom experiments.
On another front, to cope with effects of an interaction at reservoirs in ballistic transport presents a theoretical challenge, since Landauer's approach does not operate with an interaction, and phenomenological approaches [11][12][13][14][15] used to explain the interaction effect in a one-dimensional wire cannot directly answer the question. When the conductance of the wire is small, a tun-neling Hamiltonian approach is widely used to investigate the effect of interactions in reservoirs on transport [16][17][18][19]. However, to discuss the ballistic limit realized in Ref. [7], we must go beyond the linear response theory which has been widely used in tunneling experiments with correlated materials [20].
In this Letter, motivated by the ETH experiment [7] and the theoretical challenge mentioned above, we examine the effects of superfluid fluctuations in reservoirs on transport through a one-dimensional wire. To deal with ballistic transport, we apply the nonlinear response theory [21][22][23][24][25] to demonstrate that the breakdown of the quantization of conductance occurs by superfluid fluctuations. We show that transport of preformed pairs induced by superfluid fluctuations is essential to account for the breakdown. We also point out that in addition to the contact resistance, the resistance in the one-dimensional wire plays an important role in spin transport.
The Model.-We consider a system where two macroscopic reservoirs with superfluid fluctuations are connected by a quantum point contact (a one-dimensional channel). In the ETH experiment, the constriction has potential variations that take place over length scales larger than 1/k F with the Fermi momentum k F . This implies that when transport near the Fermi energy is concerned, the adiabatic approximation is justified in which the detailed shape in the constriction is irrelevant [9,26]. Thus, for the single channel case, we can start with the following Hamiltonian ( = k B = 1):
H = j=L,R [ p σ=↑,↓ ξ j,p,σ c † j,p,σ c j,p,σ + V j ] + H T , (1) V j = −g p,p ,q c † j,p+q,↑ c † j,−p,↓ c j,−p ,↓ c j,p +q,↑ ,(2)H T = dxdy σ t(x, y)ψ † L,σ (x)ψ R,σ (y) + h.c.,(3)
where c j,p,σ (c † j,p,σ ) is the fermionic annihilation (creation) operator with momentum p and spin σ in reservoir j, and the energy ξ j,p,σ = p 2 2m − µ j,σ is measured arXiv:1608.01070v2 [cond-mat.quant-gas] 11 May 2017 from the chemical potential µ j,σ . In addition, ψ j,σ is the operator in the real space with its argument x (y) representing a position in the left (right) reservoir. The first and second terms on the right-hand side of Eq. (1) are the single-particle Hamiltonian and the interaction with an attractive coupling −g (g > 0), respectively, and describe the system with a broad-Feshbach resonance used in the ETH experiment [7]. Below, we focus on the BCS regime (1/(k F a) < 0 with the s-wave scattering length a) [26][27][28][29] above the superfluid transition temperature T c . To discuss the case of a single conducting channel, we set t(x, y) = tδ(x − x 0 )δ(y − y 0 ) [26,30,31], where near the Fermi energy the tunneling amplitude t can be chosen to be a real constant without loss of generality, and x 0 (y 0 ) is the entrance (exit) point in the quantum point contact. In fact, this tunneling Hamiltonian can precisely reproduce the known universal conduction properties in the quantum point contact including ballistic transport with superfluid reservoirs [6,30]. In terms of Eq. (3), the mass and spin current operators are given by
= + = + !"# !$# ≈ · · · !%# !&#I mass = − σṄ L,σ = − σ i[H T , N L,σ ], and I spin = −i[H T , N L,↑ ] + i[H T , N L,↓ ], where N L,σ = dxψ † L,σ (x)ψ L,σ (x)
is the number operator with spin σ in the reservoir L. We note that the number operator in each reservoir commutes with the Hamiltonian except for H T . In the presence of a chemical-potential difference between the reservoirs,
V ≡ µ L,↑ − µ R,↑ = µ L,↓ − µ R,↓ = 0 (V ≡ µ L,↑ − µ R,↑ = −µ L,↓ + µ R,↓ )
, the mass (spin) current is induced. Then, the averages of the mass and spin currents at time τ are given by
I mass/spin (x 0 , y 0 , τ ) = 2Im{e −iV τ A ↑ (x 0 , y 0 , τ ) H ±e ±iV τ A ↓ (x 0 , y 0 , τ ) H }, (4) where A σ (x 0 , y 0 , τ ) = tψ † R,σ (y 0 , τ )ψ L,σ (x 0 , τ )
, and · · · H means the thermal average for the Hamiltonian (1).
In the presence of superfluid fluctuations, we should consider contributions arising from fermionic quasiparticles and fluctuation pairs [17]. Below, such fluctuations are considered up to the Gaussian level in each propagator, which is reasonable in a regime 10 −3 (T −T c )/T c 10 −1 for the case of three-dimensional reservoirs [17] relevant to the ETH experiment.
Fermionic quasiparticle current.-We now examine a steady current induced by fermionic quasiparticles. By the assumption of the steady state, we can put τ = 0 without loss of generality. Then, the mass and spin currents can be expressed as [25], and we use dωReG R σ (x 0 , y 0 , ω) = 0 for the retarded Green's function G R σ [24,25,31]. As shown in Fig. 1 (b), the single-particle Green's function is renormalized by the fluctuation pair propagator ( Fig. 1 (a)) up to the Gaussian level. As pointed out in Ref. [30], the effect of t must be incorporated to all orders in the ballistic limit. By using an analysis similar to the case of noninteracting fermions [22,[30][31][32] the fermionic quasiparticle contribution to the conductance per spin is obtained as [26]
I mass/spin = t 2π dωRe[G K ↑ (x 0 , y 0 , ω) ± G K ↓ (x 0 , y 0 , ω)], where G K σ (x 0 , y 0 , ω) = −i dτ e iωτ [ψ L,σ (x 0 , τ ), ψ † R,σ (y 0 , 0)] H is the Keldysh Green's functionG q ≈ 1 h ∞ −∞ dω 4π 2 t 2 ρ 2 (ω) |1 + π 2 t 2 ρ 2 (ω)| 2 − ∂n F (ω) ∂ω ,(5)
where n F (ω) = (e ω/T + 1) −1 is the Fermi distribution function at temperature T , g R (ω) = p g R (p, ω) with the retarded Green's function g R (p, ω) in the reservoirs, and ρ(ω) = −Im[g R (ω)]/π is the density of state (DOS). We note that the conductance depends neither on x 0 nor y 0 [26]. To obtain the above result, an expansion up to linear order in V is considered, since V /µ L(R) 0.1 and no significant deviation from the linear order is found in the ETH experiment. We note that the same expression holds for the mass and spin currents. In the case of small transmittance where |1 + π 2 t 2 ρ 2 (ω)| 2 ≈ 1 in the denominator, Eq. (5) essentially reduces to the Ambegaokar-Baratoff formula [17,33]. On the other hand, in the absence of the fluctuations, the conductance is reduced to G q = T0 h , and is equivalent to Landauer's formula with transmittance T 0 = 4π 2 t 2 ρ 2 0 (0)/(1 + π 2 t 2 ρ 2 0 (0)) 2 , where ρ 0 is the DOS for noninteracting fermions and we use the fact that the change of ρ 0 around the Fermi level is much smaller than that of ∂n F ∂ω [22,30,32]. In this limit, the quantized conductance, 1/h, is obtained for t = 1/(πρ 0 (0)).
The superfluid fluctuations renormalize the conductance of fermionic quasiparticles and generate that of preformed pairs. The fermionic quasiparticle conductance, in general, tends to be suppressed due to pseudogap effect [34,35]; however, in the case of three-dimensional reservoirs, such suppression is found to be negligible in the experimentally relevant regime (T − T c )/T c ∼ 10 −1 [26].
Fluctuation pair current.-We now consider a current carried by the fluctuating (preformed) pairs that makes a dominant contribution to the conductivity in dirty superconductors [17]. As shown on the left of Fig. 1 (c), the lowest-order diagram already contains a factor t 4 . In usual tunneling experiments where πtρ 0 (0) 1 [16][17][18][19][20], this contribution is negligible compared with the fermionic quasiparticle current, and has not been considered for realistic situations. However, in the ballistic regime in which πtρ 0 (0) ≈ 1, one needs to consider it seriously. To evaluate the fourth-order diagram, we calculate the third-order response function, which is related to an imaginary-time correlation function through analytic continuation [16,21]. Up to linear order in V , the fluctuation pair current at the fourth order in t behaves as I (4) p ∼ t 4 (2V )/(T − T c ). Here, the factor 2V originates from the pair exchange between the reservoirs, and the factor 1/(T − T c ) reflects the superfluid fluctuations. We note that as in the case of spin conductivity [8], the fluctuation pair contribution in the spin current vanishes. This reflects the fact that the pair exchange is not caused by a spin bias. Thus, the enhancement of the current by fluctuation pairs only occurs for mass transport.
We also note that the left-hand side of Fig. 1 (c) can be replaced by the right-hand side of Fig. 1 (c) up to linear order in V . Namely, the fluctuation pair contribution can be expressed in terms of an effective hopping amplitudet = (πtρ 0 (0)) 2 /(2T ) [36] and the retarded pairfluctuation propagator whose expression in the vicinity of T c is given by [17]
L R (q, ω) = 8T πρ 0 (0) 1 iω − (τ −1 GL + 8T ξ 2 π q 2 ) ,(6)
where τ −1 GL = 8(T − T c )/π and ξ 2 = 7ζ(3)v 2 F /(16dπ 2 T 2 ) with the Fermi velocity v F and the dimension of the system d. Thus, this contribution can be calculated as tunneling of the preformed pairs for a given effective hopping amplitude and bias 2V , and therefore the multiple tunneling processes of the preformed pairs represented by power series int can be systematically evaluated as depicted in Fig. 1 (d). By using the nonlinear response theory, we obtain the conductance contributed from the
!"# !$# !"# !"$ !"% !"& !"' $"# #"! #"$ #"( #"% #") #"& !!! ! ! !"#$ !!""# !"# !"$ !"% !"& !"' $"# !G p ≈ 1 h ∞ −∞ dω sinh 2 ( ω 2T ) 2t 2 T ( q Im[L R (q, ω)]) 2 {1 −t 2 ( q Re[L R (q, ω)]) 2 } 2 .(7)
We note that the above formula indeed reflects bosonic transport, since the term 1/ sinh 2 ( ω 2T ) in the integrand is the derivative of the Bose distribution function with respect to ω (note that such a term is absent in Eq. (5)). Thus, fluctuation pairs make a positive contribution to the mass conductance. Such a contribution is already visible in the regime (T − T c )/T c ∼ 10 −1 [26].
Conductances in the single-channel regime.-We now compare our theory with the ETH experiment in the ballistic single-channel regime. To this end, one may also consider an interaction effect inside the wire.
For mass transport, the mass current operator commutes with the bulk Hamiltonian containing an interaction in the wire [8,26] (except, of course, for the tunneling term), the wire resistance is expected to be negligible, and the conductance calculation obtained above is directly applicable to the ETH experiment. An essential input parameter in the theory is the ratio T /T F with the Fermi temperature T F , which is extracted from the experiment [7,26]. By assigning the ballistic limit πtρ 0 (0) = 1, we compare the theory with the experiment and find excellent agreement as shown in Fig. 2 (a). A crucial point here is that the conductance is enhanced due to the bosonic fluctuation-pair contribution. Since our theory is based on an expansion from T c , some deviation is expected at T /T c 2.
For spin transport, the spin current operator and the Hamiltonian do not commute even in the absence of the tunneling term, giving rise to the wire resistance [8,[37][38][39][40][41]. In the presence of an attractive interaction, a spin gap, ∆ s shows up. A typical estimation suggests 10nK ∆ s 500nK, where the lower bound is estimated with the Yang-Gaudin model at the density n ∼ 10 6 /m and the upper bound is determined from the binding energy of the confinement-induced resonance ∼ 0.6 ω ⊥ , where ω ⊥ is the transverse confinement frequency [42,43]. In Fig. 2 (b), we show the spin conductance G spin whose resistance is the sum of the contact resistance and wire resistance. The wire resistance is estimated so as to be compatible with the experiment by assuming R s /h ∼ e ∆s/T [44]. Our result shows that the wire resistance for spin transport is of the order of the contact resistance, implying that even in the ballistic limit a nonnegligible chemical potential drop occurs inside the wire due to the interaction between ↑ and ↓ spin components.
We also comment on an effect of the spin gap near the contacts. The spin gap in the wire originates from the strong nesting effect allowed in a one-dimension system [44]. On the other hand, near the contact, multiple channels that render the spin gap smeared out through the dimensional crossover are present. Thus, the effect of the spin gap near the contacts is expected to make negligible contributions to the contact resistance.
Effects of the gate potential, trapping, and interaction on particle conductance-Now, we discuss how the particle conductance is affected by the gate potential, trap potential, and interaction. Since the gate and trap potentials shift the energy levels of the conducting channels, these effects can be incorporated by introducing multiple tunneling amplitudes, each of which depends on the gate and trap potentials [7,26,45,46]. The tunneling amplitudes as the input parameters are determined so as to reproduce the weakest-interaction data in the experiment based on Landauer's formula for noninteracting fermions, since there, (T − T c )/T c > 1 and the superfluid fluctuations are expected to be minuscule [26]. On the other hand, the interaction strength 1/(k F a) is directly related to how close the system is to T c , since increasing 1/(k F a) towards the unitarity leads to an enhancement of T c [7,26]. Figure 3 compares the results of our theory with the ETH experiments for different interaction strengths (T /T F = 0.1 in Fig. 3 (a) and T /T F = 0.075 in Fig. 3 (b) [7]). For the weakest interaction 1/(k F a) = −2.1, the grey curves are obtained by fittings of the experimental data by assuming Landauer's formula. For and as a function of the gate potential (b). Solid curves are theoretical predictions, and circles with error bars represent experimental data whose color is identical to that of the corresponding theory curve. Grey curves are obtained with Landauer's formula for noninteracting fermions [26]. The parameter T /TF = 0.1 in Fig. 3 (a) and T /TF = 0.075 in Fig. 3 (b).
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stronger interaction strengths, where (T − T c )/T c < 1, we use our theory by incorporating the superfluid fluctuations to calculate the particle conductance by assigning tunneling amplitudes estimated from the data at 1/(k F a) = −2.1. As shown in Fig. 3, the particle conductance is enhanced and deviates from Landauer's formula, which is consistent with the experiment [7]. As in the case of the single-channel regime, enhancement is caused by the preformed pairs. The discrepancy between the theory and the experiment occurring at the larger gate potential in Fig. 3 (b) may be due to an effect of the higher transverse channels that are not treated in the theory. Summary.-We have shown that superfluid fluctuations cause two competing effects in two-terminal transport through a quantum point contact; suppression of the conductance of fermionic quasiparticles and enhancement due to bosonic preformed pairs. The former is negligible in the ETH experiment, since the depletion of the DOS near the Fermi level is negligible. The latter in the ballistic regime is shown to be significant due to the absence of the Pauli exclusion principle. Thus, the net conductance can exceed the upper bound of the quantized conduc-tance 1/h for noninteracting fermions through multiple tunneling processes that are captured with the nonlinear response theory. Such transport is ideally realized in an impurity-free system with perfect transmittance such as cold atoms and high-mobility semiconductors in which diffusive properties in a one-dimensional wire, which tends to suppress the bosonic transport, can be ignored. We have also shown that spin transport is affected by the wire resistance originating from the spin gap whose determination with no ambiguity requires the more precise knowledge of the particle density in the wire.
FIG. 1 .
1(a) Dyson's equation for a pair-fluctuation propagator L(q, ω). The dot and the solid line represent the interatomic coupling (−g) and the single-particle Green's function, respectively. (b) Diagrammatic representation of the singleparticle Green's function with the first-order correction of pair fluctuations represented by the double line. (c) Lowest-order diagram on the fluctuation pair tunneling. Each circle with a cross mark represents the tunneling amplitude t. This process can be replaced by the direct pair exchange diagram shown on the right with a renormalized tunneling amplitudet (double circle with a cross mark). (d) Higher-order diagram of the fluctuation pair tunneling.
FIG. 2 .
2Particle conductance Gmass (a) and the spin conductance Gspin (b) as a function of T /Tc in the single-mode regime. Circles with error bars and solid curves represent the ETH experimental data[7] and our theoretical calculations, respectively. The blue and red colors show the experimental results with T /TF = 0.075 and 0.1, respectively. In (b) the wire resistance is estimated so as to be compatible with the experiment by the relation Rs/h ∼ e ∆s/T . fluctuation pair per spin up to linear order in V as[26]
FIG. 3 .
3Comparison of the present theory with the ETH experiment in the particle conductance for different interaction strengths 1/(kF a). Conductances as a function of the transverse confinement frequency of the quantum point contact (a)
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| []
|
[
"Robust estimation of location and concentration parameters for the von Mises-Fisher distribution",
"Robust estimation of location and concentration parameters for the von Mises-Fisher distribution"
]
| [
"Shogo Kato \nThe Institute of Statistical Mathematics\nJapan\n",
"Shinto Eguchi \nThe Institute of Statistical Mathematics\nJapan\n"
]
| [
"The Institute of Statistical Mathematics\nJapan",
"The Institute of Statistical Mathematics\nJapan"
]
| []
| Robust estimation of location and concentration parameters for the von Mises-Fisher distribution is discussed. A key reparametrisation is achieved by expressing the two parameters as one vector on the Euclidean space. With this representation, we first show that maximum likelihood estimator for the von Mises-Fisher distribution is not robust in some situations. Then we propose two families of robust estimators which can be derived as minimisers of two density power divergences. The presented families enable us to estimate both location and concentration parameters simultaneously. Some properties of the estimators are explored. Simple iterative algorithms are suggested to find the estimates numerically. A comparison with the existing robust estimators is given as well as discussion on difference and similarity between the two proposed estimators. A simulation study is made to evaluate finite sample performance of the estimators. We consider a sea star dataset and discuss the selection of the tuning parameters and outlier detection. | 10.1007/s00362-014-0648-9 | [
"https://arxiv.org/pdf/1201.6476v1.pdf"
]
| 88,512,340 | 1201.6476 | 8710779965c413cd1ee4c33e156ae5fa2809ab9d |
Robust estimation of location and concentration parameters for the von Mises-Fisher distribution
31 Jan 2012 January 30, 2012
Shogo Kato
The Institute of Statistical Mathematics
Japan
Shinto Eguchi
The Institute of Statistical Mathematics
Japan
Robust estimation of location and concentration parameters for the von Mises-Fisher distribution
31 Jan 2012 January 30, 2012arXiv:1201.6476v1 [stat.ME]
Robust estimation of location and concentration parameters for the von Mises-Fisher distribution is discussed. A key reparametrisation is achieved by expressing the two parameters as one vector on the Euclidean space. With this representation, we first show that maximum likelihood estimator for the von Mises-Fisher distribution is not robust in some situations. Then we propose two families of robust estimators which can be derived as minimisers of two density power divergences. The presented families enable us to estimate both location and concentration parameters simultaneously. Some properties of the estimators are explored. Simple iterative algorithms are suggested to find the estimates numerically. A comparison with the existing robust estimators is given as well as discussion on difference and similarity between the two proposed estimators. A simulation study is made to evaluate finite sample performance of the estimators. We consider a sea star dataset and discuss the selection of the tuning parameters and outlier detection.
Introduction
Observations which take values on the p-dimensional unit sphere arise in various scientific fields. In meteorology, for example, wind directions measured at a weather station (Johnson and Wehrly, 1977) can be considered two-dimensional spherical or, simply, circular data. Other examples include directions of magnetic field in a rock sample (Stephens, 1979), which can be expressed as unit vectors on the three-dimensional sphere.
For the analysis of spherical data, some probability distributions have been proposed in the literature. Among them, a model which has played a central role is the von Mises-Fisher distribution which is also called the Langevin distribution. It has density f µ,κ (x) = κ (p−2)/2 (2π) p/2 I (p−2)/2 (κ) exp κµ ′ x , x ∈ S p , with respect to surface area on the sphere, where µ ∈ S p , κ ≥ 0, S p = {x ∈ R p ; x = 1}, y ′ is the transpose of y, and I q (·) denotes the modified Bessel function of the first kind and order q (Gradshteyn and Ryzhik, 2007, Equations (8.431) and (8.445)). The parameter µ controls the centre of rotational symmetry, or the mean direction, of the distribution, while the other parameter κ determines the concentration of the model. The distribution is unimodal and rotationally symmetric about x = µ. See Watson (1983), Fisher et al. (1987) and Mardia and Jupp (1999) for book treatments of the model. Although numerous works have been done on robust estimation for models for R pvalued data in the literature, considerably little attention have been paid to the robust estimation for models for data on a bounded space. A typical example is a p-dimensional sphere which shows some different features from the usual linear space. Since the unit sphere is a compact set, the gross error sensitivity of the maximum likelihood estimator is bounded. However, as pointed out, for example, in Watson (1983) and discussed later in this paper, there is strong need for the robust estimation for spherical data especially when observations are concentrated toward a certain direction.
There have been some discussion on robust estimation of the parameters for the von Mises-Fisher distribution in the literature. Robust estimators of the location parameter µ for the circular, or two-dimensional, case were proposed by Mardia (1972, p.28) and Lenth (1981). Fisher (1985), Ducharme and Milasevic (1987) and Chan and He (1993) discussed the estimation of µ for the general dimensional case. The estimation of the concentration parameter κ was considered by Fisher (1982), Ducharme and Milasevic (1990) and Ko (1992).
As described above, most of these existing works concern robust estimation of either location or concentration parameter for the von Mises-Fisher distribution. However, comparatively little work has been done to estimate both location and concentration parameters simultaneously. Lenth (1981) briefly discussed a numerical algorithm which estimates both parameters for the circular case. A nonparametric approach is taken in Agostinelli (2007) for estimation for the circular case. To our knowledge, robust estimation of both parameters for the general dimensional case have never been considered before.
In this paper we propose two families of robust estimators of both location and concentration parameters for the general dimensional von Mises-Fisher distribution. To achieve this, we first reparametrise the parameters so that they can be expressed as one R p -valued parameter and then derive the estimators as minimisers of density power divergences developed by Basu et al. (1998) or Jones et al. (2001). These approaches enable us to estimate both location and concentration parameters simultaneously. With this parametrisation, some measures of robustness of estimators, such as influence function, are discussed. To estimate the parameters numerically, we provide simple iterative algorithms. Some desirable properties such as consistency and asymptotic normality hold for the proposed estimators. Influence functions and asymptotic covariance matrices are available, and it is shown that they can be expressed in using only the modified Bessel functions of the first kind if a distribution underlying data is a mixture of the von Mises-Fisher distributions.
Subsequent sections are organised as follows. In Section 1 we discuss maximum likelihood estimation for the von Mises-Fisher distribution and show some problems about the robustness of the estimator. Also, we briefly consider what is an outlier for spherical data and provide the motivation for our study. In Sections 2 and 3, we propose two classes of robust estimators of location and concentration parameters and discuss their properties. A comparison among the two proposed estimators and an existing estimator of Lenth (1981) is made in Section 4. In Section 5 a simulation study is given to compare the finite sample performance of the proposed estimators. In Section 6 a sea star dataset is considered to illustrate how our estimators can be utilised to estimate the parameters and detect outliers. A prescription for choosing the tuning parameters is given. Finally, concluding remarks are made in Section 7.
The von Mises-Fisher distribution 2.1 Reparametrisation
Before we embark on discussion on robust estimators, we consider parametrisation of the von Mises-Fisher distribution. In the most literature on this model, the parameters are represented as a unit vector µ and a scalar κ. Each parameter has clear-cut interpretation; The parameter µ controls the mean direction of the model, while κ determines the concentration.
In this paper, however, for the sake of discussion on robustness of the estimator, we consider the following reparametrisation:
ξ = κµ.
Clearly, ξ takes a value in R p . It is easy to see that the Euclidean norm of ξ, ξ , represents the concentration of the model, while the standardised vector, ξ/ ξ , denotes the mean direction. Then the density of the von Mises-Fisher distribution can be written as
f ξ (x) = ξ (p−2)/2 (2π) p/2 I (p−2)/2 ( ξ ) exp ξ ′ x , x ∈ S p ; ξ ∈ R p .(1)
For brevity, write X ∼ vM p (ξ) if an S p -valued random variable X follows a distribution with density (1). With this convention, it is clearer to evaluate how an outlier influences the estimators of both location and concentration parameters. For example, the influence function, which is commonly used to discuss robustness, is more interpretable if the parameter is expressed in this manner. See Sections 2.3, 3.3 and 4.3 for details. Throughout the paper we denote the density (1) by f ξ .
Maximum likelihood estimation
In this subsection we discuss maximum likelihood estimation for the von Mises-Fisher distribution. Let X 1 , . . . , X n be random samples from vM p (ξ). Then the maximum likelihood estimator of ξ is known to bê
ξ = A −1 p 1 n n j=1 X j n j=1 X j n j=1 X j ,(2)
where A p (x) = I p/2 (x)/I (p−2)/2 (x), x ∈ [0, ∞). See, for example, Mardia and Jupp (1999, Section 10.3.1) for the derivation of the estimator. The following hold for A p :
(i) A p (0) = 0 and lim x→∞ A p (x) = 1, (ii) A p (x) is strictly increasing with respect to x.
See Watson (1983, Appendix A2) for proofs. From this result, it follows that there exists a unique solutionξ which satisfies (2). These properties are also attractive to solve the inverse function, i.e. x = A −1 p (y), numerically. Maximum likelihood estimation is associated with minimum divergence estimation based on the Kullback-Leibler divergence. Let f ξ be density (1) and G the distribution underlying the data having density g. Then the Kullback-Leibler divergence between f ξ and g is defined as
d KL (g, f ξ ) = log (g/f ξ ) dG(x).(3)
Here and in many expressions in this paper, we omit the variable of integration. If we assume that G is the empirical distribution function, i.e., G = G n (X 1 , . . . , X n ), then the minimiser of the divergence, argmin ξ∈R p d KL (g, f ξ ), is the same as the maximum likelihood estimator (2).
Influence function of the maximum likelihood estimator
The influence function of the maximum likelihood estimator (2) for the reparametrised von Mises-Fisher distribution (1) is given in the following theorem. See Appendix for proof.
Theorem 1. The influence function of the maximum likelihood estimator (2) at G is given by
IF(G, x) = {M (ξ)} −1 x − A p ( ξ ) ξ ξ ,(4)
where
M (ξ) = A p ( ξ ) ξ I + 1 − A 2 p ( ξ ) − p ξ A p ( ξ ) ξξ ′ ξ 2 .
Note that the above influence function is different from the ones seen in Wehrly and Shine (1981) and Ko and Guttorp (1988). Their papers discuss the influence functions of the estimators of location and concentration parameters separately, whereas we summarise these two m.l.e.'s as 'one estimator of one parameter' and discuss its influence function.
Given the influence function in Theorem 1, a natural question to address concerns the gross error sensitivity. Because the unit sphere is a compact set, it is clear that the gross error sensitivity of estimator (2) is bounded. Nevertheless the following results points out the need for the robust estimation for the model defined on this special manifold. The proof is straightforward from Theorem 1 and omitted.
Theorem 2. The following properties hold for maximum likelihood estimator (2):
(i) For any ξ ∈ R p \ {0}, it holds that argmax x∈Sp IF(G, x) = − ξ ξ and argmin x∈Sp IF(G, x) = ξ ξ .
(ii) Let ξ/ ξ be a fixed vector. Then
lim ξ →∞ sup x∈Sp IF(G, x) − inf x∈Sp IF(G, x) = ∞ and lim ξ →∞ sup x∈Sp IF(G, x) inf x∈Sp IF(G, x) = ∞.
This result implies that we need to develop a robust method to estimate ξ when the concentration of the distribution, ξ , is large. * * * Figure 1 about here * * * Figure 1 plots influence functions (4) of maximum likelihood estimators for some selected values of ξ. It seems that the direction of the influence function is close to that of x for small ξ . The direction is strongly attracted towards −ξ/ ξ when ξ is large. The figure also suggests that, for small ξ , the range of the norms of the influence functions is fairly narrow. The greater the value of ξ , the wider the range of the norm. As Theorem 2 shows, the norm of the influences functions is maximised if x = −ξ/ ξ . Also, it can confirmed that the norms of the influence functions tend to infinity as ξ approaches infinity. This provides strong motivation for robust estimation of the parameter for the von Mises-Fisher distribution.
Outliers in directional data
Since a unit sphere is a compact set, unlike linear data, it is not clear what is an outlier in directional data. For example, if a distribution underlying data is the uniform distribution on the sphere, it seems difficult to identify an outlier. However, if a distribution is highly concentrated, then an outlier can be defined in a similar manner as in linear data.
Here we consider an area where a sample from the von Mises-Fisher density (1) is not likely to be observed. Let α (∈ [0, 1]) be probability which determines the size of the area. The area, which we denote by Ar p , is defined by the interesection of the unit sphere S p and the sphere with centre at −ξ/ ξ , namely, Ar p = [x ∈ S p ; x+ξ/ ξ < {2(1−cos δ)} 1/2 ]. Here δ = δ(α) (∈ [0, π)) is the solution of the following equation
Arp f ξ (x)dx = α.
The left-hand side of the equation can be simplified to
Arp f ξ (x)dx = x∈Sp ξ ′ x/ ξ ≤− cos δ f ξ (x)dx = ξ (p−2)/2 (2π) p/2 I (p−2)/2 ( ξ ) π 0 exp(κ cos θ 1 ) sin p−2 θ 1 dθ 1 × 2π 0 π 0 · · · π 0 sin p−3 θ 2 · · · sin θ p−2 dθ 2 · · · dθ p−1 = ( ξ /2) (p−2)/2 Γ( 1 2 )Γ{ 1 2 (p − 1)}I (p−2)/2 ( ξ ) − cos δ −1 e ξ t (1 − t 2 ) (p−3)/2 dt.(5)
Hence, it follows that δ can be obtained as the solution of the following integral equation
− cos δ −1 e ξ t (1 − t 2 ) (p−3)/2 dt = π 1/2 α I (p−2)/2 ( ξ ) Γ 1 2 (p − 1) 1 2 ξ −(p−2)/2 .(6)
Since the integral in the left-hand side is bounded and strictly increasing with respect δ, it is possible to find the unique solution δ numerically. * * * Figure 2 about here * * * Figure 2(a) demonstrates how the area Ar 2 is determined from density (1) and probability α. From this frame, it can be easily understood that the area Ar 2 or, equivalently, (−δ, δ) increases as α increases. Figure 2(b) plots how ξ influences the size of area (5). As this frame clearly shows, the size of the area, which is monotonically increasing with respect to δ, increases as ξ increases. It can be confirmed that, for any p, as ξ tends to infinity, δ approaches π, meaning that a sample is likely to be observed only in a neighbour of x = ξ/ ξ . Therefore we conclude that robust estimation for the von Mises-Fisher is necessary especially when the parameter ξ is large. This statement is also supported from discussion given in Figure 1 in the previous subsection.
3 Minimum divergence estimator of Basu et al. (1998) In this section we propose a family of estimators of the parameter for the von Mises-Fisher distribution. Our estimator can be derived as a minimiser of the divergence proposed by Basu et al. (1998). An iterative algorithm is presented to estimate the parameter numerically. The influence function and asymptotic distribution of the estimator are considered. Basu et al. (1998) Let f θ be a parametric density and g a density underlying the data. Basu et al. (1998) define the density power divergence between g and f θ to be
The divergence of
d β (g, f θ ) = 1 β(1 + β) g 1+β − 1 β gf β θ + 1 1 + β f 1+β θ dx, β > 0,(7)d 0 (g, f θ ) = lim β→0 d β (g, f θ ) = g log(g/f θ )dx.
This divergence is called the β-divergence as seen in Minami and Eguchi (2002) and Fujisawa and Eguchi (2006). The divergence between the von Mises-Fisher density and a density underlying the data is given in the following theorem. The proof is given in Appendix.
Theorem 3. Let f θ in (7)
d β (g, f ξ ) = 1 β(1 + β) g 1+β dx − 1 β ξ (p−2)/2 (2π) p/2 I (p−2)/2 ( ξ ) β exp(βξ ′ x)g(x)dx + ξ (p−2)β/2 (2π) pβ/2 (1 + β) p/2 I (p−2)/2 {(1 + β) ξ } I 1+β (p−2)/2 ( ξ ) , β > 0.(8)
If β equals 0, then d 0 (g, f ξ ) = d KL (g, f ξ ), where d KL (g, f ξ ) is as in (3).
As a density underlying the data, consider the following mixture model
g(x) = (1 − ε)f ξ (x) + εf η (x),(9)
where 0 ≤ ε ≤ 1 and f ζ denotes the von Mises-Fisher vM p (ζ) density. In this case the divergence can be expressed as
d β (g, f ξ ) = {(1 − ε)f ξ + εf η } 1+β dx + 1 β ξ (p−2)/2 (2π) p/2 I (p−2)/2 ( ξ ) β × ε I (p−2)/2 {(1 + β) ξ } (1 + β) (p−2)/2 I (p−2)/2 ( ξ ) − η (p−2)/2 βξ + η (p−2)/2 I (p−2)/2 ( βξ + η ) I (p−2)/2 ( η ) − I (p−2)/2 {(1 + β) ξ } (1 + β) p/2 I (p−2)/2 ( ξ ) .
Note that the second and third terms of the divergence can be expressed by using only the modified Bessel functions of the first kind. In general, the first term should be evaluated numerically. If β is an integer, then the first term is of the form
{(1 − ε)f ξ + εf η } 1+β dx = 1+β k=0 1 + β k (1 − ε) k ε 1+β−k ( ξ k η 1+β−k ) (p−2)/2 (2π) βp/2 kξ + (1 + β − k)η (p−2)/2 × I (p−2)/2 { kξ + (1 + β − k)η } I k (p−2)/2 ( ξ )I 1+β−k (p−2)/2 ( η )
.
It is remarked here that, in this case, the first term also does not involve any special functions other than the modified Bessel functions of the first kind.
Estimating equation
The estimating equation derived from the Basu et al. (1998) divergence is known to be
ψ β (x, ξ)dG(x) = 0,(10)
whereψ
β (x, ξ) = f β ξ u ξ − f 1+β ξ u ξ dy and u ξ = ∂ ∂ξ log f ξ = x − A p ( ξ ) ξ ξ .
Following the convention in Jones et al. (2001), we call the solution of this equation function the type 1 estimator. From the general theory, it immediately follows that the estimator is consistent for ξ.
Theorem 4. The functionψ β (x, ξ) can be expressed as
ψ β (x, ξ) = C β x − A p ( ξ ) ξ ξ exp(βξ ′ x) − I (p−2)/2 {(1 + β) ξ } (1 + β) (p−2)/2 I (p−2)/2 ( ξ ) [A p {(1 + β) ξ } − A p ( ξ )] ξ ξ ,
where C is the normalising constant of the von Mises-Fisher vM p (ξ) density, i.e., C = ξ (p−2)/2 /{(2π) p/2 I (p−2)/2 ( ξ )}.
See Appendix for the proof. From this form, it immediately follows that Fisher consistency holds for the estimator. For simplicity, we redefine the ψ-function as
ψ β (x, ξ) = C −βψ β (x, ξ).(11)
Then Equation (10) holds for ψ β replaced byψ β . Since C does not depend on x, it is clear that M -estimation based upon ψ β is essentially the same as the one based onψ β . Substituting G for an empirical distribution G n (X 1 , . . . , X n ) in Equation (10) in which ψ β is replaced by ψ β , we have
n −1 n j=1 x j − A p ( ξ ) ξ ξ exp(βξ ′ x j ) − I (p−2)/2 {(1 + β) ξ } (1 + β) (p−2)/2 I (p−2)/2 ( ξ ) [A p {(1 + β) ξ } − A p ( ξ )] ξ ξ = 0.
Thus the estimator which minimises divergence (7) satisfies the following relationship:
ξ = A −1 p j w j,β (ξ)x j − nDξξ j w j,β (ξ) j w j,β (ξ)x j j w j,β (ξ)x j ,(12)
where
w j,β (ξ) = exp(βξ ′ x j ) andD ξ = I (p−2)/2 {(1+β) ξ }[A p {(1+β) ξ }−A p ( ξ )]/{(1+ β) (p−2)/2 I (p−2)/2 ( ξ ) ξ }. Note that, since A p (x)(≡ y) is strictly increasing with respect to x, there exists a unique solution x satisfying x = A −1 p (y)
. Then an algorithm induced from the above relationship is suggested as follows.
Algorithm for the type 1 estimate
Step 1. Take an initial value ξ 0 .
Step 2. Compute ξ 1 , . . . , ξ N as follows until the estimate ξ N remains virtually unchanged from the previous estimate ξ N −1 ,
ξ t+1 = A −1 p j w j,β (ξ t ) x j − nD ξt ξ t j w j,β (ξ t ) j w j,β (ξ t ) x j j w j,β (ξ t ) x j .
Step 3. Record ξ N as an estimate of ξ.
Our simulation study implies that the above algorithm converges if an initial value is set properly and β is not too large. As an initial value, the maximum likelihood estimator (2) may be one promising choice. The tuning parameter β can be estimated by using the cross-validation (Hastie et al., 2009, Section 7.10.1). We will discuss more details of the selection of β in Section 7.
Influence function
In this subsection we consider the influence function of the type 1 estimator and compare this estimator with m.l.e. in terms of the influence function.
Theorem 5. The influence function of the type 1 estimator at G is given by
IF(x, G) = {M β (ξ, G)} −1 ψ β (x, ξ),(13)
where
M β (ξ, G) = − exp(βξ ′ x) βxx ′ − βA p ( ξ ) ξx ′ ξ − A p ( ξ ) ξ I − 1 − p ξ A p ( ξ ) − A 2 p ( ξ ) ξξ ′ ξ 2 dG(x) − (1 + β) (p−2)/2 I (p−2)/2 ( ξ ) −1 × ξ −1 [A p {(1 + β) ξ } − A p ( ξ )] I (p−2)/2 {(1 + β) ξ }I + (1 + β)I p/2 {(1 + β) ξ } [A p {(1 + β) ξ } − A p ( ξ )] +I (p−2)/2 {(1 + β) ξ } β − (1 + β)A 2 p {(1 + β) ξ } − p ξ A p {(1 + β) ξ } − A p ( ξ )A p {(1 + β) ξ } + 2A 2 p ( ξ ) + p ξ A p ( ξ ) ξξ ′ ξ 2 ,(14)
and ψ β (x, ξ) is as in (11).
Proof. The influence function (13) can be obtained in a similar approach as in Theorem 1. Some calculations to obtain M β can be done by using Theorem 4, Equations (22) and (23), and Equation (8.431.1) of Gradshteyn and Ryzhik (2007).
Here we consider the mixture model (9) as a distribution of G. Then the integral part of function M β (ξ, G) in the influence function is given by
− exp(βξ ′ x) βxx ′ − βA p ( ξ ) ξx ′ ξ − A p ( ξ ) ξ I − 1 − p ξ A p ( ξ ) − A 2 p ( ξ ) ξξ ′ ξ 2 dG(x) = 1 − ε (1 + β) (p−2)/2 I (p−2)/2 ( ξ ) × A p ( ξ )I (p−2)/2 {(1 + β) ξ } − βI p/2 {(1 + β) ξ } (1 + β) ξ I − β − 1 + pA p ( ξ ) ξ + A 2 p ( ξ ) I (p−2)/2 {(1 + β) ξ } + p (1 + β) ξ + A p ( ξ ) I p/2 {(1 + β) ξ } ξξ ′ ξ 2 − η (p−2)/2 ζ 1,β (p−2)/2 ε I (p−2)/2 ( η ) βI p/2 ( ζ 1,β ) − I (p−2)/2 ( ζ 1,β ) A p ( ξ ) ξ I − I (p−2)/2 ( ζ 1,β ) − p ζ 1,β I p/2 ( ζ 1,β ) ζ 1,β ζ ′ 1,β ζ 1,β 2 +A p ( ξ )I p/2 ( ζ 1,β ) ξζ ′ 1,β ξ ζ 1,β +I (p−2)/2 ( ζ 1,β ) 1 − A 2 p ( ξ ) − pA p ( ξ ) ξ ξξ ′ ξ 2 ,
where ζ j,α = jαξ + η. Note that, in this case, the influence functions do not involve any special functions other than the modified Bessel functions of the first kind and orders (p − 2)/2 and p/2. * * * Figure 3 about here * * *
β = 0.25, IF(G V M , −ξ/ ξ ) − IF(G V M , ξ/ ξ ) and IF(G V M , −ξ/ ξ ) / IF(G V M , ξ/ ξ ) take smaller values than those for the maximum likelihood estimator, where G V M is the distribution function of vM 2 {(2.37, 0) ′ }.
This result implies that the type 1 estimator is more robust than the maximum likelihood estimator.
Asymptotic normality
The asymptotic normality of the estimator can be shown from the M -estimation theory. Let X 1 , . . . , X n be random samples from the von Mises-Fisher vM p (ξ) distribution. Suppose thatξ is the type 1 estimator of ξ. Then
n 1/2 (ξ − ξ) d −→ N (0, V β ) as n → ∞, where V β = M β (ψ β , G) −1 Q β (ψ β , G){M β (ψ β , G) ′ } −1 , Q β (ψ β , G) = ψ β (x, ξ){ψ β (x, ξ)} ′ dG(x),
and ψ β (x, ξ) and M β (ψ β , G) are defined as in (11) and (14), respectively. In particular, if G is the distribution function of the mixture model (9), then Q β (ψ β , G) is given by
Q β (ψ β , G) = 1 − ε I (p−2)/2 ( ξ ) I p/2 {(1 + 2β) ξ } (1 + 2β) p/2 ξ I + A p ( ξ ) (1 + 2β) (p−2)/2 A p ( ξ )I (p−2)/2 {(1 + 2β) ξ } − 2I p/2 {(1 + 2β) ξ } − 1 − A p ( ξ ) (1 + β) p−2 I 2 p/2 {(1 + β) ξ } I (p−2)/2 ( ξ ) [A p {(1 + β) ξ } − A p ( ξ )] ξξ ′ ξ 2 + ε I (p−2)/2 ( η ) η (p−2)/2 ζ 2,β (p−2)/2 I p/2 ( ζ 2,β ) ζ 2,β I + I (p−2)/2 ( ζ 2,β ) − p ζ 2,β I p/2 ( ζ 2,β ) ζ 2,β ζ ′ 2,β ζ 2,β 2 −A p ( ξ )I p/2 ( ζ 2,β ) ζ 2,β ξ ′ + ξζ ′ 2,β ζ 2,β ξ + A 2 p ( ξ )I (p−2)/2 ( ζ 2,β ) ξξ ′ ξ 2 − η (p−2)/2 ζ 1,β (p−2)/2 I (p−2)/2 {(1 + β) ξ } (1 + β) (p−2)/2 I (p−2)/2 ( ξ ) [A p {(1 + β) ξ } − A p ( ξ )] × I p/2 ( ζ 1,β ) ζ 1,β ξ ′ + ξζ ′ 1,β ζ 1,β ξ + 2A p ( ξ )I (p−2)/2 ( ζ 1,β ) ξξ ′ ξ 2 + I 2 (p−2)/2 {(1 + β) ξ } (1 + β) p−2 I 2 (p−2)/2 ( ξ ) [A p {(1 + β) ξ } − A p ( ξ )] 2 ξξ ′ ξ 2 .
Remark that the asymptotic covariance matrix can be expressed in a form of the modified Bessel functions of the first kind and orders (p − 2)/2 and p/2. Next we consider another divergence which is also based on density power. It is defined as
d γ (g, f θ ) = 1 γ(1 + γ) log g 1+γ dx − 1 γ log gf γ θ dx + 1 1 + γ log f 1+γ θ dx, γ > 0, (15) d 0 (g, f θ ) = lim γ→0 d γ (g, f θ ) = g log(g/f θ )dx.
This divergence was briefly considered by Jones et al. (2001, Equation (2.9)) as a special case of a general family of divergences. Fujisawa and Eguchi (2008) investigated detailed properties of the divergence with emphasis on the case in which the underlying distribution contains heavy contamination. The divergence between the von Mises-Fisher density and a density underlying the data can be calculated as follows. The procedure to derive this divergence is similar to that to obtain the Basu et al. (1998) divergence given in Theorem 3, and therefore the proof is omitted.
Theorem 6. Let f ξ be the von Mises-Fisher vM p (ξ) density. Then divergence (15) between f ξ and an arbitrary density g is given by
d γ (g, f ξ ) = 1 γ(1 + γ) log g 1+γ dx − 1 γ log exp(γξ ′ x)g(x)dx + 1 1 + γ log (2π) p/2 I (p−2)/2 { (1 + γ)ξ } (1 + γ)ξ (p−2)/2 .(16)
Note that the expression for this divergence is slightly simpler than that for the Basu et al. (1998)
divergence.
As a special case of the underlying density in the Jones et al. divergence, consider again the mixture of the two von Mises-Fisher densities (9). Then the divergence can be expressed as
d γ (g, f ξ ) = 1 γ(1 + γ) log {(1 − ε)f ξ + εf η } 1+γ dx − 1 γ log (1 − ε) I (p−2)/2 {(1 + γ) ξ } (1 + γ) (p−2)/2 I (p−2)/2 ( ξ ) + ε ξ (p−2)/2 I (p−2)/2 ( ξ ) I (p−2)/2 ( γξ + η ) γξ + η (p−2)/2 − 1 1 + γ log (2π) p/2 I (p−2)/2 { (1 + γ)ξ } (1 + γ)ξ (p−2)/2 .
From discussion in Fujisawa and Eguchi (2008), one can ignore the contamination f η if the second term in the second logarithm is sufficiently small. This condition is satisfied, for example, when ξ is large and γξ + η ≃ 0 holds.
Estimating equation
The estimating equation of the Jones et al. divergence is
f γ ξ u ξ g dx f γ ξ g dx − f 1+γ ξ u ξ dx f 1+γ ξ dx = 0,(17)
where u ξ is defined as in Section 2.3. Or equivalently,
ψ γ (x, ξ)dG(x) = 0, whereψ γ (x, ξ) = f γ ξ u ξ − f 1+γ ξ u ξ dy f 1+γ ξ dy .
It is remarked here that this equation has been discussed by Windham (1994) although the divergence which the estimating equation is based on was not considered there. As in Jones et al. (2001), the estimator derived from this estimation equation is called the type 0 estimator in the paper.
Theorem 7. The functionψ γ (x, ξ) is given bỹ
ψ γ (x, ξ) = C γ exp(γξ ′ x) x − A p {(1 + γ) ξ } ξ ξ ,
where C is as in Theorem 4.
The proof is similar to that for Theorem 4 and omitted. In a similar manner as in Section 3.2, we redefine the ψ-function as ψ γ (x, ξ) =ψ γ (x, ξ) and consider the Mestimation based on ψ γ .
On substituting G for an empirical distribution G n (X 1 , . . . , X n ) in (17), it follows that
n j=1 exp(γξ ′ x j )x j n j=1 exp(γξ ′ x j ) − A p {(1 + γ) ξ } ξ ξ = 0.
Therefore it can be seen that the minimum divergence estimator satisfies the following equationξ
= 1 1 + γ A −1 p j w j,γ (ξ)x j j w j,γ (ξ) j w j,β (ξ)x j j w j,β (ξ)x j .(18)
Given estimating Equation (18), the following algorithm is naturally induced.
Algorithm for the type 0 estimate
Step 1. Take an initial value ξ 0 .
Step 2. Compute ξ 1 , . . . , ξ N as follows until the estimate ξ N remains virtually unchanged from the previous estimate ξ N −1 ,
ξ t+1 = 1 1 + γ A −1 p j w j,γ (ξ t )x j j w j,γ (ξ t ) j w j,γ (ξ t )x j j w j,γ (ξ t )x j .
Step 3. Record ξ N as an estimate of ξ.
This algorithm can also be derived from an iterative algorithm of Fujisawa and Eguchi (2008, Section 4), and from their discussion, the monotonicity of the algorithm follows:
d γ (g, f ξ 0 ) ≥ d γ (g, f ξ 1 ) ≥ · · · ≥ d γ (g, f u ),
where g denotes the empirical density.
As for a prescription for choosing the tuning parameter γ, cross-validation is available. See Section 7 for details.
Influence function
The influence function of the type 0 estimator (17) is provided in the following theorem. The basic process to obtain the divergence is similar to that in Theorem 5 and omitted.
Theorem 8. The influence function of the type 0 estimator at G is given by
IF(x; ξ, G) = {M γ (ξ, G)} −1 ψ γ (x, ξ),(19)
where
M γ (ξ, G) = − exp(γξ ′ x) γ xx ′ − A p {(1 + γ) ξ } ξx ′ ξ − A p {(1 + γ) ξ } ξ I − (1 + γ) 1 − A 2 p {(1 + γ) ξ } + pA p {(1 + γ) ξ } ξ ξξ ′ ξ 2 dG(x).
If G is the distribution function of the mixture model with density (9), then the function M γ (ξ, G) in Theorem 8 can be expressed as Figure 4 about here * * *
M γ (ξ, G) = 1 − ε (1 + γ) (p−2)/2 I (p−2)/2 ( ξ ) 1 1 + γ I p/2 {(1 + γ) ξ } ξ I + p (1 + γ) ξ + A p {(1 + γ) ξ } γ I p/2 {(1 + γ) ξ } + I (p−2)/2 {(1 + γ) ξ } × (1 + γ)[1 − A 2 p {(1 + γ) ξ }] − pA p {(1 + γ) ξ } ξ − γ ξξ ′ ξ 2 − η (p−2)/2 ζ 1,γ (p−2)/2 ε I (p−2)/2 ( η ) γI p/2 ( ζ 1,γ ) ζ 1,γ − I (p−2)/2 ( ζ 1,γ ) A p {(1 + γ) ξ } ξ I +γ I (p−2)/2 ( ζ 1,γ ) − p ζ 1,γ I p/2 ( ζ 1,γ ) ζ 1,γ ζ ′ 1,γ ζ 1,γ 2 −γ A p {(1 + γ) ξ }I p/2 ( ζ 1,γ ) ξζ ′ 1,γ ξ ζ 1,γ −I (p−2)/2 ( ζ 1,γ ) (1 + γ)[1 − A 2 p {(1 + γ) ξ }] − pA p {(1 + γ) ξ } ξ ξξ ′ ξ 2 . * * *
Asymptotic normality
In a similar way as in Section 3.4, one can show the asymptotic normality of the estimator. Since ψ γ (x, ξ) and M γ (ψ γ , G) have already been given in Theorem 8, the function Q γ (ψ γ , G) and the asymptotic covariance matrix V γ can be calculated in a straightforward manner. In particular, if G is the distribution function of the mixture model with density (9), then Q γ (ψ γ , G) is given by
Q γ (ψ γ , G) = 1 − ε (1 + 2γ) (p−2)/2 I (p−2)/2 ( ξ ) I p/2 {(1 + 2γ) ξ } (1 + 2γ) ξ I + I (p−2)/2 {(1 + 2γ) ξ } − pI p/2 {(1 + 2γ) ξ } (1 + 2γ) ξ − 2A p {(1 + γ) ξ }I p/2 {(1 + 2γ) ξ } +A 2 p {(1 + γ) ξ }I (p−2)/2 {(1 + 2γ) ξ } ξξ ′ ξ 2 + η (p−2)/2 ζ 2,γ (p−2)/2 ε I (p−2)/2 ( η ) I p/2 ( ζ 2,γ ) ζ 2,γ I + I (p−2)/2 ( ζ 2,γ ) − pI p/2 ( ζ 2,γ ) ζ 2,γ ζ 2,γ ζ ′ 2,γ ζ 2,γ 2 − A p {(1 + γ) ξ }I p/2 ( ζ 2,γ ) ζ 2,γ ξ ′ + ξζ ′ 2,γ ξ ζ 2,γ +A 2 p {(1 + γ) ξ }I (p−2)/2 ( ζ 2,γ ) ξξ ′ ξ 2 .
Again, the asymptotic covariance matrix does not involve any special functions other than the modified Bessel functions of the first kind and orders (p − 2)/2 and p/2.
Comparison among the robust estimators
Comparison between types 0 and 1 estimators
In this subsection we compare the two proposed estimators of the parameters of the von Mises-Fisher distribution. A detailed comparison between the two estimators for the general family of distributions has been given in Jones et al. (2001). Here we consider some properties of the estimators which are special for the von Mises-Fisher distribution. The numerical algorithms for both types of estimates presented in Sections 3.2 and 4.2 have some similarities and differences. A similarity is that both algorithms are expressed in relatively simple forms and require to calculate the inverse of the function A p , i.e., the ratio of the modified Bessel functions. However, as discussed in Section 2.2, it seems fairly easy to calculate A −1 p numerically since it is bounded and strictly increasing. A slight difference is that the algorithm for the type 0 estimator appears to be slightly simpler than that for the type 1 estimator as it does not involve subtraction in the argument of A −1 p in Step 2. Another special feature of the type 0 estimator is the monotonicity of the algorithm as shown in Section 4.2.
Next we discuss the influence functions and asymptotic covariance matrices of both estimators, which can be expressed by using the modified Bessel functions of the first kind and orders (p − 2)/2 and p/2. Their expression for the type 0 estimator is simpler although both require the calculations of the aforementioned functions. A comparison between Figures 3 and 4 suggests that the behaviour of the influence functions of both estimators looks similar, at least, for the practical choices of the tuning parameters. When the tuning parameters of both estimators are large, e.g., β = γ = 0.75, a simulation study, which will be given in the next section, implies that the influence functions of each estimator behave in a different manner. However, in most of the realistic cases in which the tuning parameters are moderately small, the performance of the estimators and their influence functions seems quite similar. As for the asymptotic covariance matrices, as discussed in Jones et al. (2001, Section 3.3), the large-sample variances matrices of both estimators show relatively small loss of efficiency when the tuning parameters are equal and small. We will provide further comparison of these estimators through a simulation study and an example in the later sections. Lenth's (1981) estimator Lenth (1981) proposed a robust estimator of a location parameter of the two-dimensional von Mises-Fisher distribution and briefly considered an algorithm to estimate both location and concentration parameters. The estimator is defined as follows. Assume that θ 1 , . . . , θ n are random samples from the von Mises distribution vM 2 (κ cos µ, κ sin µ). Define
Similarities to and differences from
C w = j w j cos θ j j w j , S w = j w j sin θ j j w j , R w = C 2 w + S 2 w 1/2 , w j = ψ{t(θ j − µ; κ)} t(θ j − µ; κ) , t(φ; κ) = ±{2κ(1 − cos φ)}, ψ H (t) = t, |t| ≤ c c sign(t), |t| > c , ψ A (t) = c sin(t/c), |t| ≤ cπ 0, |t| > cπ .(20)
with " + " or " − " chosen according to φ (mod 2π) ∈ (or / ∈) [0, π). Then the estimator is defined as solutions of the following estimating equations: n j=1 w j sin(θ j −μ) = 0 andκ = A −1 2 (R w ).
This estimator is somewhat associated with the types 0 and 1 estimators discussed in the paper. All of these three estimators are related in the sense that the parameters are estimated by introducing some weight functions in the estimating equations. Also, all estimators can be obtained numerically through fairly simple algorithms.
However our two families of estimators are different from Lenth's one. One obvious distinction is that our estimators discuss the general dimensional case of the von Mises-Fisher distribution, while the Lenth estimator, as it stands, can be used only for the two-dimensional case. In addition there are some other differences between the Lenth estimator and ours even for the two-dimensional case. As seen in Equations (12) and (18), our estimators adopt the power of the densities as weight functions, whereas Lenth (1981) used the weight functions (20) proposed by Huber (1964) or Andrews (1974).
This distinction makes a difference in discussing Fisher consistency of the estimators. As shown in Sections 3.2 and 4.2 of the paper, our estimators are Fisher consistent. On the other hand, as shown below, Fisher consistency does not hold for the Lenth estimator. To prove this, we first show the following general result, which helps us evaluate theoretical first cosine moment for the Lenth estimator. The proof is given in Appendix.
Lemma 1. Let f be a probability density function on the circle [−π, π). Assume that w is a function on [−π, π) which satisfies the following properties:
1. w is symmetric about 0, i.e., w(θ) = w(−θ) for any θ ∈ [−π, π).
2. If cos θ 1 > cos θ 2 , then w(θ 1 ) ≥ w(θ 2 ).
3. 0 < π −π w(θ)f (θ)dθ < ∞. Then π −π w(θ) cos θf (θ)dθ −π −π w(θ)f (θ)dθ ≥ π −π cos θf (θ)dθ.
The equality holds if and only if w(θ) = c.
Using Lemma 1, we immediately obtain the following result. See Appendix for the proof.
Theorem 9. Assume ψ is not a constant function. Then Lenth's estimator is not Fisher consistent.
However, we should note that Lenth's estimator of the location µ, which is the main focus of the paper, is Fisher consistent if the concentration κ is known, and the estimator can be useful in that situation.
Simulation study
In this section a simulation study is carried out to compare the finite sample performance of the two proposed estimators.
We consider the performance of the estimators in the following two cases: (i) random samples of some selected sizes are gathered from the von Mises-Fisher distributions (without contamination), and (ii) 100 random samples are generated from the contaminated von Mises-Fisher distributions with some selected contamination ratios. The case (i) is to discuss how much efficiency of the estimators is lost when random samples of small sizes do not include any outliers. The more attention will be paid to the case (ii) as robustness is the main theme of the paper. We do not discuss Lenth's (1981) estimator here since, as shown in the previous section, the estimator for the concentration parameter can be biased due to the fact that Fisher consistency does not hold. Table 1 about here * * * Table 1 shows the estimates of the relative mean squared errors of the types 0 and 1 estimators for some selected values of the tuning parameters. A comparison of these two estimators suggests that, when the tuning parameters of these estimators are equal, the estimates of the relative mean squared errors generally take similar values. An exception is a case in which the sample size is small and the tuning parameters of the estimators are large. In this case the type 1 estimator generally outperforms the type 0 estimator although both estimators do not seem satisfactory. Except for this special case, however, it might be appreciated that only a little efficiency is lost for these robust estimators. The table suggests that the relative mean squared error diminishes as the tuning parameter decreases. Also it is noted that, for large sample sizes, the relative mean squared errors of both estimators are almost equal to one regardless of the values of tuning parameters, confirming consistency of the estimators.
Next we consider the case (ii) in order to discuss the robustness of these two families of estimators is discussed. Two families of distributions are chosen as contaminations, namely, the uniform and von Mises-Fisher distributions. The uniform distribution or the von Mises-Fisher distribution with small concentration ξ is often used as a contamination as seen in Ducharme and Milasevic (1987) and Chan and He (1993). It seems less common to assume the von Mises-Fisher distribution with fairly large concentration parameter as a contamination, but this model also appears to be a reasonable choice if we choose its parameter such that most observations from the model lie on an area where samples from the von Mises-Fisher of interest are not likely to be observed.
First consider the uniform contamination. Generate 100 random samples from a mixture of the von Mises-Fisher and uniform distributions having the form (1 − ε) vM p (ξ) + ε U p for some selected values of ε, p and ξ. Then we calculate the estimates of the relative mean squared error in a similar way as in the previous simulation. * * * Table 2 about here * * * Table 2 displays the estimates of the relative mean squared errors of types 0 and 1 estimators with respect to maximum likelihood estimator for some selected values of tuning parameters. It seems from the table that, when the concetration parameter or, equivalently, A p ( ξ ) is small, the relative mean squared errors are close to one. This can be mathematically validated from the fact that the von Mises-Fisher distribution approaches the uniform distribution as ξ tends to 0. On the other hand, when A p ( ξ ) is large, then the robust estimators outperform the maximum likelihood estimator. In particular, when A p ( ξ ) is close to 1, the relative mean squared errors of the proposed estimators take much smaller values than one. It is also noted that the tuning parameters which minimise the relative mean squared errors increase as the contamination ratio ε increases.
Second, we discuss the robustness of the estimators when the true distribution follows a mixture of two von Mises-Fisher distributions. This time, generate 100 samples from a mixture of the two von Mises-Fisher distributions having the form (1 − ε) vM p (ξ) + ε vM p (ζ) with some selected values of ε, p, ξ and ζ. * * * Table 3 about here * * * Estimates of the relative mean squared errors of the types 0 and 1 estimators with respect to maximum likelihood estimator for some selected contamination ratios and tuning parameters for a mixture of two von Mises-Fisher distributions are given in Table 3. Note that the contaminating distribution is assumed to follow the von Mises-Fisher, not the uniform distribution, so that almost all observations lie in the area where observations from the distribution of interest are not likely to be observed. This table implies that the two estimators outperform the maximum likelihood estimator if the tuning parameters are chosen correctly. In particular, if ε is large, both of the proposed estimators show much better results than the maximum likelihood estimator. It seems from the table that both estimators behave quite similarly, especially for small values of tuning parameters. Since the contaminating distribution is concentrated toward the opposite direction of ξ, the tuning parameters minimising the relative mean squared errors are greater than those given in Table 2 for the fixed values of ε.
Example
To illstrate how our methods can be utilised to real data, we consider a dataset of directions of sea stars (Fisher, 1993, Example 4.20). The dataset consists of the resulant directions of 22 Sardinian sea stars 11 days after being displaced from their natural habitat. * * * Figure 5 about here * * * Figure 5(a) plots measurements of resultant directions of sea stars. Since the dataset shows symmetry and unimodality, it seems reasonable to fit the von Mises distribution to this dataset. However, as this frame suggests, there are two observations which can be considered possible outliers. Of these two samples, one at 2.57 seems to be an apparent outlier on the assumption that the observations follow a von-Mises distribution, while the other one at 5.20 appears to be much more difficult to judge. We fit the von Mises distribution based on the maximised likelihood and types 1 and 0 divergences and discuss how these results can be utilised for detecting outliers. To select the tuning parameters of the types 1 and 0 estimators, we use the three-fold cross-validation (Hastie et al., 2009, Section 7.10.1) implemented as follows. First, divide the dataset D into three subsets D 1 , D 2 and D 3 . Define
CV(f ξ , α) = 1 N N j=1 L y j ,f −τ (j) ξ (x j , α) ,(21)
wheref −ι ξ (x, α) is the estimated density with a tuning parameter α based on a subset of the data D \ D ι , N is the sample size of the dataset, and τ (k) is an index function defined as τ (k) = l for x k ∈ D l . Here we define the loss functions L for the types 1 and 0 estimators as L y,f −ι ξ (x, α) = d 0.6 (g y ,f −ι ξ ) where d 0.6 are the Basu et al. divergence (8) with β = 0.6 and Jones et al. divergence (16) with γ = 0.6, respectively, in which g y is a probability function of a point distribution with singularity at Y = y. Then the estimate of the tuning parameter is given by a minimiser of CV(f ξ , α). Figure 5 Table 4 about here * * * Table 4 shows the estimates of the parameters and tuning parameters for the maximum likelihood and types 1 and 0 estimators. The maximum likelihood estimators are obtained for some subsets of the data which exclude no samples, one at 2.57 and ones at 2.57 and 5.20. A comparison among the maximum likelihood estimates suggests that these possible outliers do not influence the estimate of the location parameter µ significantly. On the other hand, the estimate of the concentration parameter κ seems to be influenced by the possible outliers. Both types 1 and 0 estimates are similar to the maximum likelihood estimate for the dataset excluding the one sample, implying that the dataset includes only one outlier at 2.57 actually. This conclusion coincides with the one given by Fisher (1993, Example 4.20) who derived the same consequence from his outlier test for discordancy for von Mises data. Figure 5(d) and (e) display Q-Q plots for the data excluding one outlier for types 1 and 0 estimators, respectively, where quantiles of the robust estimators (horizontal axis) and of the empirical distribution (vertical axis) are plotted. This figure shows that the estimated model provides a satisfactory fit to the dataset.
Discussion
As pointed out in Watson (1983) and some other references, it is known that maximum likelihood estimator of the parameter for the von Mises-Fisher distribution is not robust. In particular, as discussed in Section 2.3, a robust estimator is required especially when observations are concentrated toward a certain direction. Lenth (1981) briefly considered an algorithm to estimate both location and concentration parameters simultaneously. However, as discussed in Section 5.2, Lenth's estimator of the parameters can be used only for the circular case of the distribution and is not Fisher consistent. In this paper we provided two families of robust estimators which enable us to estimate both location and concentration parameters simultaneously for the general case of the von Mises-Fisher distribution. Both estimators can be derived as the minimisers of divergences proposed by Basu et al. (1998) and Jones et al. (2001). It follows from the general theory that some properties, including consistency and asymptotic normality, hold for the estimators. In addition it was shown our estimators have some special features. For example, the presented estimators can be obtained through fairly simple algorithms numerically. Also, the influence functions and asymptotic covariance matrices of both estimators can be expressed using the modified Bessel functions of the first kind. Some simulations suggest that the performance of both estimators is quite satisfactory and, in particular, the proposed estimators greatly outperform the maximum likelihood estimator if the distributions underlying data are concentrated toward a certain direction. Possible future works include robust estimation of parameters for the extended families of distributions such as the ones proposed by Kent (1982) and Jones and Pewsey (2005). In particular robust methods for distributions with weak symmetry properties would be desired.
where the third equality derives from the following formula (Abramowitz and Stegun, 1970, 9.6.26):
d ds I p (s) t = p t I p (t) + I p+1 (t).(22)
Using this result, M (ξ) can be calculated as
M (ξ) = − ∂ ∂ζ ′ ψ(x, ζ) ξ dF (x) = − ∂ ∂ζ ′ x − A p ( ζ ) ζ ζ ξ dF (x) = ∂ ∂t A p (t) ξ ξξ ′ ξ 2 + A p ( ξ ) 1 ξ I − ξξ ′ ξ 2 = 1 − A 2 p ( ξ ) − p − 1 ξ A p ( ξ ) ξξ ′ ξ 2 + A p ( ξ ) 1 ξ I − ξξ ′ ξ 2 = A p ( ξ ) ξ I + 1 − A 2 p ( ξ ) − p ξ A p ( ξ ) ξξ ′ ξ 2 ,
where the fourth equality holds due to the following formula (Mardia and Jupp (1999, Appendix 1, (A.14)); Jammalamadaka and SenGupta (2001, p.289
)) d ds A p (s) t = 1 − A 2 p (t) − p − 1 t A p (t).(23)
Thus we obtain Theorem 1. ✷
B Proof of Theorem 3
It is clear that d 0 (g, f ξ ) = d KL (g, f ξ ). We consider a case β > 0.
d β (g, f ξ ) = 1 β(1 + β) g 1+β − 1 β gf β ξ + 1 1 + β f 1+β ξ dx = 1 β(1 + β) g 1+β dx − 1 β ξ (p−2)/2 (2π) p/2 I (p−2)/2 ( ξ ) β exp(βξ ′ x)g(x)dx + 1 1 + β ξ (p−2)/2 (2π) p/2 I (p−2)/2 ( ξ ) 1+β exp{(1 + β)ξ ′ x}dx
Using the fact that f (1+β)ξ dx = 1, the integral in the third term of the equation can be expressed as
1 1 + β f 1+β ξ dx = 1 1 + β ξ (p−2)/2 (2π) p/2 I (p−2)/2 ( ξ ) 1+β 2π p/2 I (p−2)/2 {(1 + β) ξ } {(1 + β) ξ /2} (p−2)/2 = ξ (p−2)β/2 (2π) pβ/2 (1 + β) p/2 I (p−2)/2 {(1 + β) ξ } I 1+β (p−2)/2 ( ξ )
. ✷
C Proof of Theorem 4
It is easy to obtain the first term ofψ β . We consider the second term ofψ β , namely, − f 1+β ξ u ξ dy. To be more specific, it can be expressed as
f 1+β ξ u ξ dy = C 1+β exp{(1 + β)ξ ′ y} x − A p ( ξ ) ξ ξ dy = C 1+β y exp{(1 + β)ξ ′ y}dy − A p ( ξ ) ξ ξ · exp{(1 + β)ξ ′ y}dy .(24)
The first integration can be calculated by using the fact that, if X ∼ vM p (ζ), then E(X) = A p (ζ) ζ/ ζ . (See, for example, Mardia and Jupp (1999, p.169)). From this, it immediately follows that
y exp{(1 + β)ξ ′ y}dy = (2π) p/2 I p/2 {(1 + β) ξ } {(1 + β) ξ } (p−2)/2 ξ ξ .
The process to calculate the second integration of (24) is essentially the same as that to obtain the normalising constant of the von Mises-Fisher density vM p {(1 + β)ξ}. Thus we obtain Theorem 4. ✷
D Proof of Lemma 1
For convenience, write
T w = π −π w(θ) cos θf (θ)dθ −π −π w(θ)f (θ)dθ and T = π −π cos θf (θ)dθ.
Then T w can be expressed as T w = w ′ (θ) cos θf (θ)dθ, where w ′ (θ) = w(θ) / π −π w(u)f (u)du. With this convention, it holds that
T w = π −π w ′ (θ) cos θf (θ)dθ = T + π −π {w ′ (θ) − 1} cos θf (θ)dθ.(25)
The second term of (25) can be decomposed into two terms as
π −π {w ′ (θ) − 1} cos θf (θ)dθ = w ′ (θ)≥1 + w ′ (θ)<1 .(26)
Due to Properties 1-3 of w(θ), it is easy to see that there exists a constant d ∈ [−1, 1) such that {θ ∈ [−π, π) | w ′ (θ) ≥ 1} = {θ ∈ [π, π) | cos θ ≥ d}. Then the following inequality holds for (26):
w ′ (θ)≥1 + w ′ (θ)<1 = cos θ≥d {w ′ (θ) − 1} cos θf (θ)dθ + cos θ<d {w ′ (θ) − 1} cos θf (θ)dθ ≥ d cos θ≥d {w ′ (θ) − 1}f (θ)dθ + cos θ<d {w ′ (θ) − 1}f (θ)dθ = d π −π w ′ (θ)f (θ)dθ − π −π f (θ)dθ = 0.
Thus we obtain T w ≥ T . Since f (θ) > 0 for some subset A which is not a null set, it can be seen that the equality holds if and only if w ′ (θ) = 1. ✷
E Proof of Theorem 7
We show thatκ is not a Fisher consistent estimator. Without loss of generality, assume µ = 0. Then it is easy to see that w(θ) sin θf V M (θ)dθ = 0, where f V M is the von Mises vM 2 (κ, 0) density since the integrand is an odd function. Then, from Lemma 1, we immediately obtain (
w(θ) cos θf V M (θ)dθ w(θ)f V M (θ)dθ > cos θf V M (θ)dθ = A 2 (κ). Therefore R w = M w > A 2 (κ). ✷tributions (1 − ε) vM p (ξ) + ε U p with
Figure 3
3displays influence functions (13) of the type 1 estimator at the two-dimensional von Mises-Fisher model for some selected values of β. From four frames of the figure andFigure 1(d), it seems that the norms of influence functions are not large for moderately large β. In particular, it seems that, for
4
Minimum divergence estimator ofJones et al. (2001) 4.1 The divergence ofJones et al. (2001)
Figure 4
4plots the influence functions (13) of the type 0 estimator for four selected values of γ. Note that the values of the tuning parameters in the fours frames of this figure correspond to those inFigure 3. Comparing these two figures, it seems that the influence functions of both estimators take quite similar values when both tuning parameters are the same.
First consider the case (i) in which finite samples are generated from the von Mises-Fisher distribution without contamination. Random samples of sizes n = 10, 20, 30, 50 and 100 from the von Mises-Fisher distributions vM p (ξ) with p = 2 and ξ = (2.37, 0) ′ and p = 3 and ξ = (3.99, 0, 0) ′ are generated. For each combination of n and ξ, 2000 simulation samples are gathered. We discuss the performance of the estimators in terms of the mean squared error. The estimate of the mean squared error is given by 2000 j=1 ξ j − ξ 2 /2000, where theξ j 's (j = 1, . . . , 2000) are the estimates of ξ for jth simulation sample. * * *
(b) and (c) exhibit the values of CV(f ξ , α) for the types 1 and 0 estimators, respectively, for α = h/100 (h = 1, . . . , 100). The curves of the frames show somewhat similar behaviours when the tuning parameters take values between 0 and 0.45, while they look different if the tuning parameters are greater 0.45. These frames suggest that, for the Basu et al. divergence, the values of CV are more stable than the Jones et al. divergence for the tuning parameter greater than 0.45. * * *
ε
a) p = 2, ξ = (0.52, 0) ′ and p = 3, ξ = (0.78, 0, 0) ′ , (b) p = 2, ξ = (1.16, 0) ′ and p = 3, ξ = (1.80, 0, 0) ′ , (c) p = 2, ξ = (2.37, 0) ′ and p = 3, ξ = (3.99, 0, 0) ′ , and (d) p = 2, ξ = (10.27, 0) ′ and p = 3, ξ = (20.0, 0, 0) ′ , for each ε = 0.02, 0.05, 0.1 and 0.2. (a) p = 2 and ξ = (0.52, 0) ′ and p = 3 and ξ = (= 0.02 ε = 0.05 ε = 0.1 ε = 0.2 ε = 0.02 ε = 0.05 ε = 0.1 ε = 0.2
ε
(ζ) with (i) p = 2, ξ = (2.37, 0) ′ and ζ = (−100, 0) ′ and (ii) p = 3, ξ = (3.99, 0, 0) ′ and ζ = (−199, 0, 0) ′ for each ε = 0.05, 0.1, 0.2, and 0.3. = 0.02 ε = 0.05 ε = 0.1 ε = 0.2 ε = 0.02 ε = 0.05 ε = 0.1 ε = 0.2
Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :
12345Influence functions (4) of maximum likelihood estimators for (a) ξ = (0.10, 0) ′ , (b) ξ = (0.52, 0) ′ , (c) ξ = (1.16, 0) ′ and (d) ξ = (2.37, 0) ′ for the vM 2 (ξ) model. For convenience, the norms of the influence functions are divided by four. In each frame, the white dot denotes the origin, while the black one denotes A 2 ( ξ ) ξ/ ξ . (a) Plot of the von Mises-Fisher density vM 2 {(5, 0) ′ } (solid), the unit circle (dashed), the disc N (dotted) and the area Ar 2 (bold and solid) with α = 0.05 and (b) plot of δ satisfying Equation (6) with α = 0.05 and p = 2 (solid), 3 (dashed), 4 (dot-dashed) and 5 (dotted) as a function of ξ . Influence functions (13) of the type 1 estimator for the vM 2 {(2.37, 0) ′ } model with (a) β = 0.05, (b) β = 0.1, (c) β = 0.25 and (d) β = 0.5. For convenience, the norms of the influence functions are divided by four. The white dot denotes the origin and the dotted line represents the vM 2 {(2.37, 0) ′ } density. Influence functions (19) of the type 0 estimator for the vM 2 {(2.37, 0) ′ } model with (a) γ = 0.05, (b) γ = 0.1, (c) γ = 0.25 and (d) γ = 0.5. For convenience, the norms of the influence functions are divided by four. The white dot denotes the origin and the dotted line represents the vM 2 {(2.37, 0) ′ } density. (a) Plot of measurements of resultant directions of 22 sea stars after 11 days of movement, plots of values of CV (21) for 100 selected values of tuning parameters between 0 and 1 for (b) type 1 and (c) type 0 estimators, and Q-Q plots for the data excluding one outlier for (d) type 1 estimator and (e) type 0 estimator where quantiles of the estimators (x-axis) and of the empirical distribution (y-axis) are plotted.
be the von Mises-Fisher vM p (ξ) density. Then the Basu et al. divergence is
Table 1 .
1Estimates of the relative mean squared errors of the minimum divergence estima-tors with respect to maximum likelihood estimator for some selected sample sizes and tun-
ing parameters for the von Mises-Fisher distribution vM p (ξ) with p = 2 and ξ = (2.37, 0) ′
and p = 3 and ξ = (3.99, 0, 0) ′ .
p = 2
p = 3
n = 10
n = 20
n = 30
n = 50 n = 100
n = 10
n = 20
n = 30
n = 50 n = 100
Type 1 β = 0.02
0.994
0.996
0.996
1.000
0.999
0.991
0.996
0.996
0.999
1.000
β = 0.05
0.993
0.995
0.995
1.003
1.001
0.985
0.997
0.996
1.003
1.003
β = 0.1
1.016
1.012
1.008
1.021
1.012
1.006
1.019
1.008
1.021
1.019
β = 0.25
5.464
1.231
1.154
1.151
1.104
1.465
1.213
1.132
1.146
1.126
β = 0.5
14.836
2.357
1.711
1.559
1.393
6.512
1.969
1.568
1.515
1.418
β = 0.75
21.547
4.314
2.735
2.232
1.723
19.290
3.028
2.159
1.994
1.752
Type 0 γ = 0.02
0.994
0.996
0.996
1.000
0.999
0.991
0.996
0.996
0.999
1.000
γ = 0.05
0.993
0.995
0.995
1.003
1.001
0.985
0.997
0.996
1.003
1.003
γ = 0.1
1.016
1.013
1.008
1.021
1.012
1.007
1.019
1.008
1.021
1.019
γ = 0.25
6.179
1.247
1.164
1.158
1.109
1.618
1.234
1.143
1.156
1.133
γ = 0.5
81.673
3.501
2.036
1.702
1.479
458.822
3.312
1.860
1.697
1.530
γ = 0.75 737.175 58.871 137.763
4.240
2.135
2892.802 820.520 178.742
4.206
2.285
Table 2 .
2Estimates of the relative mean squared errors of the minimum divergence es-timators with respect to maximum likelihood estimator for some selected contamination
sizes and tuning parameters for a mixture of the von Mises-Fisher and the uniform dis-
(b) p = 2 and ξ = (1.16, 0) ′ and p = 3 and ξ = (1.80, 0, 0) ′ ε = 0.02 ε = 0.05 ε = 0.1 ε = 0.2 ε = 0.02 ε = 0.05 ε = 0.1 ε = 0.2 ε = 0.02 ε = 0.05 ε = 0.1 ε = 0.2 ε = 0.02 ε = 0.05 ε = 0.1 ε = 0.2 (d) p = 2 and ξ = (10.27, 0) ′ and p = 3 and ξ = (20.00, 0, 0) ′ ε = 0.02 ε = 0.05 ε = 0.1 ε = 0.2 ε = 0.02 ε = 0.05 ε = 0.1 ε = 0.2Type 1 β = 0.02
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
β = 0.05
1.001
1.001
1.000
1.000
1.000
1.001
1.000
0.999
β = 0.1
1.002
1.002
1.001
1.001
1.001
1.002
1.001
0.999
β = 0.25
1.009
1.009
1.006
1.004
1.011
1.012
1.008
1.001
β = 0.5
1.030
1.030
1.021
1.013
1.047
1.047
1.035
1.014
β = 0.75
1.064
1.062
1.046
1.028
1.107
1.106
1.081
1.040
Type 0 γ = 0.02
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
γ = 0.05
1.001
1.001
1.000
1.000
1.000
1.001
1.000
0.999
γ = 0.1
1.002
1.002
1.001
1.001
1.001
1.002
1.001
0.999
γ = 0.25
1.009
1.009
1.006
1.004
1.011
1.012
1.008
1.001
γ = 0.5
1.031
1.031
1.022
1.013
1.049
1.050
1.037
1.015
γ = 0.75
1.071
1.070
1.051
1.030
1.122
1.122
1.093
1.046
p = 2
p = 3
Type 1 β = 0.02
1.000
1.000
0.999
0.998
1.000
0.999
0.996
0.994
β = 0.05
1.001
0.999
0.997
0.995
1.001
0.998
0.991
0.985
β = 0.1
1.004
1.000
0.995
0.990
1.006
0.998
0.983
0.970
β = 0.25
1.024
1.012
0.994
0.977
1.040
1.016
0.973
0.930
β = 0.5
1.098
1.065
1.015
0.960
1.161
1.100
1.002
0.879
β = 0.75
1.213
1.155
1.063
0.952
1.337
1.232
1.079
0.855
Type 0 γ = 0.02
1.000
1.000
0.999
0.998
1.000
0.999
0.996
0.994
γ = 0.05
1.001
0.999
0.997
0.995
1.001
0.998
0.991
0.985
γ = 0.1
1.004
1.000
0.995
0.990
1.006
0.998
0.983
0.970
γ = 0.25
1.025
1.013
0.994
0.976
1.042
1.017
0.974
0.929
γ = 0.5
1.111
1.074
1.020
0.959
1.190
1.120
1.012
0.874
γ = 0.75
1.296
1.216
1.103
0.960
1.515
1.355
1.173
0.854
(c) p = 2 and ξ = (2.37, 0) ′ and p = 3 and ξ = (3.99, 0, 0) ′
p = 2
p = 3
Type 1 β = 0.02
0.999
0.991
0.986
0.989
0.990
0.967
0.962
0.974
β = 0.05
1.001
0.980
0.964
0.972
0.982
0.922
0.907
0.935
β = 0.1
1.011
0.965
0.930
0.943
0.980
0.862
0.822
0.869
β = 0.25
1.097
0.961
0.840
0.857
1.058
0.777
0.628
0.682
β = 0.5
1.386
1.084
0.761
0.726
1.325
0.838
0.508
0.469
β = 0.75
1.736
1.289
0.760
0.640
1.616
0.968
0.511
0.391
Type 0 γ = 0.02
0.999
0.991
0.986
0.989
0.990
0.967
0.962
0.974
γ = 0.05
1.001
0.980
0.964
0.972
0.982
0.922
0.907
0.935
γ = 0.1
1.011
0.965
0.929
0.943
0.980
0.861
0.821
0.869
γ = 0.25
1.102
0.962
0.837
0.854
1.065
0.776
0.619
0.673
γ = 0.5
1.480
1.141
0.760
0.705
1.446
0.900
0.502
0.421
γ = 0.75
2.245
1.660
0.851
0.619
2.158
1.308
0.616
0.342
p = 2
p = 3
Type 1 β = 0.02
0.841
0.891
0.945
0.983
0.691
0.792
0.894
0.967
β = 0.05
0.635
0.723
0.851
0.953
0.372
0.486
0.700
0.903
β = 0.1
0.412
0.461
0.665
0.891
0.194
0.172
0.332
0.733
β = 0.25
0.297
0.157
0.198
0.535
0.173
0.071
0.054
0.114
β = 0.5
0.365
0.153
0.109
0.164
0.220
0.081
0.049
0.065
β = 0.75
0.452
0.180
0.115
0.144
0.271
0.095
0.057
0.075
Type 0 γ = 0.02
0.841
0.891
0.945
0.983
0.691
0.792
0.894
0.967
γ = 0.05
0.634
0.723
0.851
0.953
0.371
0.485
0.699
0.903
γ = 0.1
0.411
0.459
0.663
0.890
0.193
0.168
0.326
0.730
γ = 0.25
0.300
0.154
0.186
0.508
0.177
0.070
0.048
0.081
γ = 0.5
0.400
0.169
0.109
0.114
0.254
0.094
0.053
0.042
γ = 0.75
0.593
0.245
0.145
0.111
0.407
0.146
0.085
0.065
Table 3 .
3Estimates of the relative mean squared errors of the minimum divergence estimators with respect to maximum likelihood estimator for some selected sample sizes and tuning parameters for a mixture of the von Mises-Fisher distributions (1 − ε) vM p (ξ) +ε vM p
Table 4 .
4Estimates of the parameters and tuning parameters for the maximum likelihood estimators and two minimum divergence estimators. The maximum likelihood estimators are obtained for some subsets of the data excluding no samples, one at 2.57 and ones at 2.57 and 5.20. The parametersμ andκ are defined byμ = Arg(cosξ 1 + i sinξ 2 ) and , respectively, whereξ = (ξ 1 ,ξ 2 ) ′ .κ = (ξ 2
1 +ξ 2
2 ) 1/2 MLE
MLE (with one
MLE (with two
Type 1 Type 0
sample excluded) samples excluded)
tuning parameter
−
−
−
0.59
0.48
µ
0.0541
0.0232
0.0712
0.0377 0.0380
κ
3.30
5.74
7.66
5.86
5.98
AcknowledgementFinancial support for the research of the first author was provided, in part, by the Ministry of Education, Culture, Sport, Science and Technology in Japan under a Grant-in-Aid for Young Scientists (B) (20740065).A Proof of Theorem 1From M -estimation theory(Hampel et al., 1986, Section 4.2c), the influence function of the maximum likelihood estimator (4) is of the formAfter some algebra, it follows that
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| []
|
[
"Learning to Locomote with Deep Neural-Network and CPG-based Control in a Soft Snake Robot",
"Learning to Locomote with Deep Neural-Network and CPG-based Control in a Soft Snake Robot"
]
| [
"Xuan Liu ",
"Renato Gasoto ",
"Cagdas Onal ",
"Jie Fu "
]
| []
| []
| In this paper, we present a new locomotion control method for soft robot snakes. Inspired by biological snakes, our control architecture is composed of two key modules: A deep reinforcement learning (RL) module for achieving adaptive goal-reaching behaviors with changing goals, and a central pattern generator (CPG) system with Matsuoka oscillators for generating stable and diverse behavior patterns. The two modules are interconnected into a closed-loop system: The RL module, acting as the "brain", regulates the input of the CPG system based on state feedback from the robot. The output of the CPG system is then translated into pressure inputs to pneumatic actuators of a soft snake robot. Since the oscillation frequency and wave amplitude of the Matsuoka oscillator can be independently controlled under different time scales, we adapt the option-critic framework to improve the learning performance measured by optimality and data efficiency. We verify the performance of the proposed control method in experiments with both simulated and real snake robots. | null | [
"https://arxiv.org/pdf/2001.04059v2.pdf"
]
| 210,164,489 | 2001.04059 | 0bb850f408a842bbf1b459d37f06b8d6e62b3dd6 |
Learning to Locomote with Deep Neural-Network and CPG-based Control in a Soft Snake Robot
Xuan Liu
Renato Gasoto
Cagdas Onal
Jie Fu
Learning to Locomote with Deep Neural-Network and CPG-based Control in a Soft Snake Robot
In this paper, we present a new locomotion control method for soft robot snakes. Inspired by biological snakes, our control architecture is composed of two key modules: A deep reinforcement learning (RL) module for achieving adaptive goal-reaching behaviors with changing goals, and a central pattern generator (CPG) system with Matsuoka oscillators for generating stable and diverse behavior patterns. The two modules are interconnected into a closed-loop system: The RL module, acting as the "brain", regulates the input of the CPG system based on state feedback from the robot. The output of the CPG system is then translated into pressure inputs to pneumatic actuators of a soft snake robot. Since the oscillation frequency and wave amplitude of the Matsuoka oscillator can be independently controlled under different time scales, we adapt the option-critic framework to improve the learning performance measured by optimality and data efficiency. We verify the performance of the proposed control method in experiments with both simulated and real snake robots.
I. INTRODUCTION
Soft continuum robots, featured with flexible geometric shapes and resiliently deformable materials, are gradually exhibiting great potentials in tasks under dangerous and cluttered environments [1]. Such properties also allow them to closely mimic animal behaviors, which are widely studied in bio-inspired robotics [2], [3]. However, due to the infinitely many degrees of freedom and hard-to-control dynamics of soft robots, planning and control of soft robots remains a challenging problem.
In this paper, we are motivated to study bio-inspired approaches for locomotion control of a soft snake robot, designed and fabricated in our lab [4]. The crucial observation from nature is that most animals with soft bodies and elastic actuators can learn and adapt to various new motion skills with just a few trials. This is mainly attributed to the prior information encoded into their vertebrate neural circuits [5] and the evolving operations by the primary motor cortex (M1) in the brain. Such a mechanism allows animals to learn, from a reasonable number of trials and errors, to regulate the inputs to neural circuits for achieving desired motion [6].
We propose a control design for soft snake robots with two key features: First, we use model-free deep reinforcement learning [7], [8] in adaptive control to mitigate the uncertainties in the dynamic responses of soft actuators; Second, we employ a Central Pattern Generator (CPG) network consisting Polytechnic Institute. 2 Renato Gasoto is affiliated with NVIDIA. xliu9, rggasoto, cdonal, [email protected] This work was supported in part by the National Science Foundation under grant #1728412, and by NVIDIA. of Matsuoka oscillators as low-level motion controller to ensure the stability of the learning-based control system and to improve the diversity of locomotion behaviors in the soft snake robot. Figure 2 shows the final structure of our controller composed of the deep policy Neural Networks (NNs) for the Proximal Policy Optimization Option-Critics (PPOC) algorithm and the Matsuoka CPG net. These two modules -a reinforcement learning (RL) module and a CPG net -are integrated as follows: The RL module learns to control neural stimuli inputs and frequency ratio to a CPG network given state feedback from the soft snake robot and the control objective. The neural stimuli input regulates the amplitudes and phases of the outputs in the CPG net in realtime, which steers the snake to the desired direction. While the frequency ratio has a long term effect that changes the oscillating frequency of the CPG net, resulting in changes in the velocity of the robot. The output of the CPG net is directly transformed as actuation inputs to the robot. The system is closed-loop and Bounded-input, Bounded-Output (BIBO) stable due to the stability of the CPG net [9, Thm.1].
A. Related work
In literature, bio-inspired control methods, including bipedal [10]- [13] and serpentine locomotion [14]- [19], have been studied for the control design of robotic locomotion.
The key concept of bio-inspired control is to generate motion trajectories mimicking animals' behaviors and then to track these trajectories with a closed-loop control design. The CPG, also known as a neural oscillator, is a classical nonlinear system that models the neuron circuits and their firing mechanisms that control organ contractions and body movements in animals [5], [20]. In [21], the authors developed a trajectory generator for a rigid salamander robot using Kuramoto CPGs and used low-level PD controllers to track the desired motion trajectories generated by the oscillator. In [16], the authors improved the synchronization property of the CPG by adapting its frequency parameter with additional linear dynamics. In [18], the authors introduced a control loop that adjusts the frequency and amplitude of the oscillation for adapting to changes in the terrain. The most recent work [19] employed Spiking Neural Net (SNN) under the Reward-Modulated Spike-Timing-Dependent Plasticity (R-STDP) scheme to learn operations of a Kuramoto CPG net that drives a rigid snake robot towards the goal object using visual information. Despite the success of bio-inspired control on those rigid robots, it may be infeasible to use the existing control scheme on soft robots. These rigid robot controllers depend on high-performance encoders for tracking the CPG trajectories with a small error. Such tracking performance is hard to achieve given the unstable properties of soft materials during contact, and the nonlinear and stochastic dynamical response from the soft actuators.
B. Main contributions
Generally speaking, the proposed control design leverages adaptability and optimality in deep reinforcement learning and stability and diversity in behavior patterns in the CPG system. The insight from the CPG system guides the architecture design of deep NNs to improve data efficiency and learning rate using hierarchical RL, in particular, the option-critic framework [22]. Moreover, this paper leverages the high-fidelity of a physics-based simulator and domain randomization techniques [23], the trained policy in simulator is directly applied to the real soft snake robot, resulting in desirable and intelligent tracking of moving targets.
II. SYSTEM OVERVIEW
Each soft link of the robot is made of Ecoflex™ 00-30 silicone rubber. The links are connected through rigid bodies enclosing the electronic components that are necessary to control the snake robot. In addition, the rigid body components have a pair of wheels, that model the anisotropic friction that a real snake would have from its scales. Only one chamber on each link is active (pressurized) at a time. The simulator made by Gasoto, et al. [24] models the inflation and deflation of the air chamber, as well as the resulting deformation of the soft bodies with tetrahedral finite elements. Simulating a full snake with 4 links takes less than 12ms per step, allowing real-time performance on a sampling frequency og 60 Hz. The link curvature error between simulator and real robot is within 3% corresponding to the input pressure ranged from 0 to 8 psi. Figure 1 shows the notation of state space configuration of the robot. At time t, state h(t) ∈ R 2 is the planar Cartesian position of the snake head, ρ g (t) ∈ R is the distance from h(t) to the goal position, d g (t) ∈ R is the distance travelled along the goal direction from the initial head position h(0), v(t) ∈ R is the instantaneous planar velocity of the robot, and v g (t) ∈ R is the projection of this velocity vector to the goal direction, θ g (t) is the angular deviation between the goal direction and the velocity direction of the snake robot.
According to [25], the bending curvature of each body link at time t is computed by κ i (t) = δi(t) li(t) , for i = 1, . . . , 4, where δ i (t) and l i (t) are the relative bending angle and the length of the middle line of the i th soft body link.
III. THE MATSUOKA OSCILLATOR NETWORK
Neural oscillators, including Matsuoka oscillator [9] and Hindmarsh-Rose neuron model [26] are driven by spiking or constant stimulus signals. These CPGs are less commonly used in bio-inspired robot control because the wave patterns generated by these CPGs are often difficult to predict and not directly operable. However, their complex bifurcation structures provide richer trajectory patterns, including limit cycles, equilibrium points, and even chaos. Based on this, we use Matsuoka oscillator as a lower level motion controller for the our soft snake robot.
A. Primitive Matsuoka CPG
A primitive Matsuoka CPG consists of two mutually inhibited neurons named extensor and flexor. We use subscripts e and f to represent variables related to extensor neuron and flexor neuron, respectively. Such structure can be represented by the following dynamical system,
K f τ rẋ e i = −x e i − w f e i z f i − by e i − N j=1 w ji y e j + u e i (1) K f τ aẏ e i = z e i − y e i K f τ rẋ f i = −x f i − w ef i z e i − by f i − N j=1 w ji y f j + u f i K f τ aẏ f i = z f i − y f i ψ i = A(z e i − z f i ),
where the tuple x q i , y q i , q ∈ {e, f } represents the activation state and self-inhibitory state of i-th neuron respectively, z q i = max(0, x q i ) is the output of i-th neuron, b ∈ R is a weight parameter, and u e i , u f i are the tonic inputs to the oscillator. There are two time constants in the system, τ r ∈ R is the rate of discharge and τ a ∈ R is the adaptation rate of the system. Both of them are multiplied by a frequency ratio K f ∈ R. The parameters w f e i ∈ R and w ef i ∈ R are mutual inhibition weights between flexor and extensor in a primitive oscillator. Since the symmetricity condition in the previous work [9], [27] suggests w f e i = w ef i , a new symbol α i will be used to represent the equivalent mutual inhibition weights of i th primitive oscillator in the rest part of this paper. The parameter w ji ∈ R is the inhibition weight of the coupling signal from the output of another coupled primitive oscillator. In our system, all couplings terms are inhibiting signals (negatively weighted), and only the tonic inputs are activating signals (positively weighted).
B. Configuration of the Matsuoka CPG network on soft snake robot
We first construct a coupled CPG system for the soft snake robot. In Fig. 2, the CPG Net section shows the structure of our CPG Network. It is an inverted, double-sized version of Network VIII introduced in Matsuoka's paper [27]. There are overall eight nodes in the network. Each pair of nodes (e.g., the two nodes colored green) in a row represents a primitive Matsuoka oscillator (1). The edges are corresponding to the coupling relations among the nodes. In this graph, all the edges with hollowed endpoints are positive activating signals, while the others with solid endpoints are negative inhibiting signals. The network includes four linearly coupled primitive Matsuoka oscillators. The oscillators are numbered 1 to 4 from head to tail of the robot. The outputs ψ = {ψ 1 , ψ 2 , ψ 3 , ψ 4 } from the primitive oscillators are the normalized input ratio to the maximum safe pressure input for the pneumatic actuators of each chamber (ψ i = 1 stands for a full inflation of the i-th extensor actuator and zero inflation of the i-th flexor actuator, and vice versa for ψ i = −1). As the safe input pressure for each soft actuator in our robot is 8 psi [24], the actual pressure input to the i-th chamber is 8ψ i psi. The primitive oscillator with green nodes on the head link stands for the pace-maker oscillator of the entire CPG net. The pace-maker works as an initiator in the oscillating system, followed by the rest parts oscillating under certain phase difference in sequence. Fig. 2 also shows that all tonic inputs are activating signals. For simplicity, we'll use a vector
u = [u e 1 , u f 1 , u e 2 , u f 2 , u e 3 , u f 3 , u e 4 , u f 4 ] T (2)
to represent all tonic inputs to the CPG net. However, a raw configuration is not enough to guarantee stable and synchronized oscillation for the whole system. To avoid configuring a non-rhythmic system, the following constraint must be satisfied [9]:
(τ a − τ r ) 2 < 4τ r τ a b.(3)
Given this constraint, Genetic Programming (GP) algorithms can be implemented to configure a set of parameters in the system that optimizes the performance of serpentine locomotion controlled by the CPG system [28]. For the goal-reaching task and serpentine locomotion scenario, we define the fitness function-the optimization criteria-in GP as follows:
F (T ) = a 1 |v g (t)| − a 2 |θ g (t)| + a 3 |d g (t)|),
where parameters a 1 , a 2 , a 3 , T ∈ R + are constant coefficients 1 . The function is a weighted sum over the instantaneous velocity towards the goal, its instantaneous angular deviation, and distance traveled on the goal direction at termination time t = T . A better fitted configuration should stay at a large |v g (T )| to show the existence of oscillation after a period of time T , which is treated as an evidence for sustained oscillation in this problem. This is important since only a sustained Matsuoka oscillator can generate stable limit cycles [9]. In addition, it is also required to have less angular deviation from the goal direction (with a small |θ g (T )|), and with more distance travelled on the goal direction (|d g (T )|).
After 300 generations, GP converges to a parameter configuration of the CPG net, shown in Table. I in the Appendix. We use the Salesman tournament algorithm for the selection step, a Gaussian sampler for the mutation step, and cross two point method for the crossover step. The GP algorithm is implemented in Deap 1.2.2 [29], a python library for evolutionary algorithms.
IV. MANEUVERABILITY OF THE MATSUOKA CPG NET Maneuvering a CPG system to generate various trajectory patterns is nontrivial due to its high dimensionality and nonlinearity. Thus, we aim to understand and leverage the dynamical properties of the CPG system in learning-based control design for improving the computational and data efficiency. Inspired by the mechanism of vehicle driving, we explore the tonic input as a throttle, stable equilibrium as braking, frequency ratio as a gearbox, and the amplitude bias and duty cycle of the tonic input as two mechanisms for steering control. We summarize the key properties of the CPG net into two categories -steering control and speed control of the soft snake robot.
A. Key properties of Matsuoka Oscillator for gait generation a) Biased tonic inputs for steering: It has been pointed out that the steering pattern of the snake robot can be realized by making joint angle trajectories asymmetric [30]. Previous methods realize steering by either directly adding a displacement [21] to the output of the CPG system, or using a secondary system such as a deep neural network to manipulate the output from multiple CPG systems [10]. In addition to these methods, we observe two different types of tonic inputs that can generate imbalanced output waveforms in the Matsuoka oscillator, which are: (1) applying biased amplitudes and (2) biased duty cycles to the tonic inputs. This observation provides more flexibility in steering control of snake robots.
(a) (b) (c) (d)
The asymmetric amplitude can be achieved by applying biased values to a specific pair of tonic inputs of the system. In experiments, we observe that imbalanced tonic inputs between flexor and extensor can cause asymmetry in the output of the CPG system, resulting in the similar effect of a biased output amplitude. An example is shown in Fig. 3a, showing a clockwise turning. The corresponding joint trajectory is given in Fig. 3c, where one CPG output ψ 1 (blue curve) has an noticeable bias to the negative amplitude axis. This shows that the flexor chamber of the first link is bending more to the right-hand side of the robot.
From the time perspective, the asymmetry is related to the difference in time duration between the actuation of extensor and flexor. To be more specific, such a time difference is determined by the duty cycle of each CPG output ψ i . The duty cycle can be controlled by switching different modes between limit cycles and equilibrium points of the system. Instead of setting a fixed tonic input vector u, we can use different tonic input vectors switched periodically. When the duty cycle of actuating signals are different between flexors and extensors, an asymmetric oscillating pattern will occur.
We construct two tonic input vectors u 1 and u 2 , with u 1 = [1, 0, 1, 0, 1, 0, 1, 0] T and u 2 = [0, 1, 0, 1, 0, 1, 0, 1] T . As Fig. 3d shows, when we set the duty cycle of u 1 to be 1/12 in one oscillating period, the rest 11/12 of time slot for will be filled with u 2 . The CPG output on each link shows an imbalanced oscillation with longer time duration on the negative amplitude axis, indicating longer bending time on the flexor. As a result, the robot makes a clockwise (right) turn, with a circle trajectory presented in Fig. 3b. b) Frequency and amplitude for speed regulation: In [31], Matsuoka has provided a method to estimate the amplitude A and frequency ω of Matsuoka oscillator. We denote the estimated frequency asω and estimated amplitude asÂ. Based on [31, eq. (5), eq. (6)], since K f and u q i are the only changing parameters, the mapping relations can be concluded aŝ
ω(K f ) ∝ 1 K f ,(4)A(u q i ) ∝ u q i ∀ q ∈ {e, f }, i ∈ {1, 2, 3, 4}
. It becomes clear that the frequency and amplitude of the Matsuoka CPG system can be controlled independently by the frequency ratio K f and the tonic inputs u q i , for q ∈ {e, f }. Moreover, since the frequency ratio K f only influences the strength but not the direction of the vector field of the Matsuoka CPG system, manipulating K f will not affect the stability. Figure 4 shows the distribution of locomotion velocity over different amplitudes and frequencies by taking 10000 uniform samples within the region u q i ∈ [0.5, 1.0] for q ∈ {e, f }, i ∈ {1, 2, 3, 4} with all u q i to be the same in one sample, and K f ∈ [0.25, 1.25]. What we can observe from Fig. 4 is that, given fixed tonic input, the average velocity increase nearly monotonically with the frequency ratio K f . While the amplitude of tonic input does not affect the velocity that much, especially when K f is low. Thus, we may use K f primarily for speed control for the robot. The Matsuoka CPG network is a BIBO stable system [9, Thm.1]. Given bounded tonic inputs, the state trajectories of the system will converge to either a stable limit cycle, or an asymptotically stable equilibrium, or a chaotic attracting region. It is noted that a Matsuoka oscillator without tonic inputs, i.e., u = 0, directly turns into a non-harmonic damped system [9], [27], it is possible to start and stop the oscillation by tuning the tonic inputs with specific values.
B. Equilibrium stability for the termination of oscillation
Besides letting u = 0, we observe that when the tonic input vector u follows an exclusive form, that is, [u e i , u f i ] ∈ {[0, 1], [1, 0]} for i = 1, . . . , 4, the CPG system converges to a stable equilibrium. As the example in Fig. 5 shows, when u is switched from u = 1 to u = [1, 0, 0, 1, 1, 0, 0, 1] T , the system converges to a equilibrium. Noticed that the equilibrium point is not at the origin point, but fixed at some nonzero constant value, which leads to a constant air pressure input to the snake chambers. This means that the exclusive equilibrium can also bend the body of the snake to a certain turning direction before completely stopping the oscillation.
In conclusion, the key properties of maneuverability of the Matsuoka CPG net are summarized as follows:
• The value of tonic input contributes mainly to the oscillating amplitude. Biased tonic inputs will result in unsymmetrical oscillation and thus turning in the robot. A non-zero tonic input can be used for either initiating the oscillation or stopping the oscillation. • Oscillating frequency has a great impact on the locomotion speed of the snake robot. In our system, lower frequency leads to higher speed. Further, the speed control by changing the frequency ratio can be made independent of steering control by changing the tonic inputs.
V. LEARNING HIERARCHICAL PPOC-CPG NET FOR SET POINT TRACKING
We employ a model-free RL algorithm, proximal policy optimization (PPO) [8], as the 'brain' of the CPG network. The algorithm is expected to learn the optimal policy that takes state feedback from the robot and outputs tonic inputs and frequency ratio of the CPG net to generate proper oscillating waveforms for goal-reaching locomotion.
A. Preliminary: Proximal Policy Optimization
We use reinforcement learning (RL) to learn an approximately optimal controller for soft snake locomotion. In RL, the control problem is modeled as an Markov Decision Process (MDP) M = (S, A, γ, s 0 , R, P ) with state space S, action space A, a discounting factor γ, an initial state s 0 , a reward function R and a probabilistic transition function P . Given a policy π : S → ∆(A) and an initial state s 0 ∼ d(s 0 ), the value function V π (s t ) = E π [ t k=0 γ k R(s k , a k )|s 0 ], is the total expected return given policy π in the MDP.
Policy search [7] in model-free RL approximates the policy with parameterized functions and performs gradient descent to find the optimal policy function parameters. Given the total expected return
L(θ) = E ξ∼π θ (ξ) [r(ξ)],
where θ is the vector of parameters in the policy/value function approximation, ξ is a trajectory sampled in the Markov chain induced by the policy on the MDP, r(ξ) is the accumulated discounted reward along the path. Based on this a gradient estimator is deducted as follows [7],
∂L(θ) ∂θ = E ξ∼π θ (ξ) [ ∂ log π(a t |s t ) ∂θ A π (s t , a t )],
where A π (s t , a t ) = Q π (s t , a t ) − V π (s t ) is the advantage function for reducing the variance of the gradient estimator. Hierarchical RL is introduced to improve the learning performance by exploring an action space consisting of lowlevel actions and high-level sub-policies called options. Each option o =< I, π o , β o > includes a set I ⊆ S of states at which the option can be initiated, π o : S → ∆(A) is an intra-option policy, and β o : S → [0, 1] is the termination condition-β o (s) outputs the probability that the option o should terminated at the state s. (parameterized on ϑ). After augmenting the action space with options, the policy in the MDP learns simultaneously a high-level policy π : S → A ∪ O and the set of options defined by the intra-option policies and termination functions. The optioncritic framework [22] approximates the high-level policy and options (intra-option policies and termination functions) using function approximations and search for the optimal parameters in these approximators. In [32], PPO is extended with the option-critic framework, referred to as the Proximal Policy Optimization Option-Critic (PPOC) method.
B. PPOC-CPG net
In the CPG net, the change in frequency ratio K f will not affect the stability of the CPG net but can regulate the amplitude and frequency of the output directly (4), resulting in speed control of the robot. Thus, we use the option-critic framework [33], [34] to learn the optimal controller with the tonic inputs (low-level primitive actions) and frequency ratio (high-level actions) of the CPG net.
Modeling the soft snake robot as an MDP, the state vector is given by s = [ρ g ,ρ g , θ g ,θ g , κ 1 , κ 2 , κ 3 , κ 4 ] T ∈ R 8 (see Fig. 1). The primitive actions are a subset of continuous tonic input vectors. In particular, we define a four dimensional action vector a = [a 1 , a 2 , a 3 , a 4 ] T ∈ R 4 and map a to tonic input vector u as follows,
u e i = 1 1 + e −ai , and u f i = 1 − u e i , for i = 1, . . . , 4. (5)
This mapping bounds the tonic input within [0, 1], as only positive inputs make sense in the CPG net, and reduces the dimension of the input space from 8 to 4. Thus, it enables more efficient policy search. Furthermore, the action space after dimension reduction still includes tonic input vectors necessary for braking and for turning (see Section IV).
The set of options is defined by the domain of frequency ratios K f . We do not include K f in the action space because the frequency ratio needs not to change as frequently as the tonic inputs. Thus, we treat K f as options, which can be switched from one to another infrequently given the outputs of termination functions. Specifically, each option is defined by I, π y : S → {y}×A, β y where I = S allows the option to be available at any state in the MDP, y ∈ K f is a frequency ratio, and β y is the termination function. The options share the same NNs for their intro-option policies and the same NNs for termination functions. However, these NNss takes different frequency ratios as part of the inputs. The set of parameters to be learned by policy search include parameters for intra-option policy function approximation, parameters for termination function approximation, and parameters for high-level policy function approximation (for determining the next option). PPOC in the openAI Baselines [35] is employed as the policy search in the RL module. Now, referring back to Figure 2, we have a Multi-layer perceptron (MLP) neural network with two hidden layers to approximate the optimal control policy that controls the inputs of the CPG net in (1). The output layer of MLP is composed of action a (green nodes), option in terms of frequency ratio (pink node) and the terminating probability (blue node) for that option. The input layer of MLP consists of state vector (yellow nodes) and the last time step outputs. The inclusion of previous outputs as part of the observation inputs for the option-critic network is observed in several previous works [10], [36], [37]. The purpose of this setup is to let the actor network learn its consequential dynamics by tracking the past actions in one or multiple steps. Due to the BIBO stability of the CPG net and that of the soft snake robots, we ensure that the closed-loop robot system with the PPOC-CPG controller is BIBO stable.
C. Curriculum and reward design for goal-reaching
We aim to learn the goal-reaching controller for the soft snake robot. Inspired by previous work that uses curriculum teaching [38] accelerate motor skills learning, we design different levels of tasks as follows: At each task level, the center of goal is sampled based on the current location and head direction of the robot. Each sampled target comes with a red circle indicating the accepting region of the goal. The sampling distribution is a uniform distribution in the fan area determined by the range of angle θ i and distance bound [ρ l i , ρ u i ] in polar coordinate given by the predefined curriculum.
As shown in Fig. 6, when the task level increases, we have
r i < r i−1 , θ i > θ i−1 , ρ u i > ρ u i−1 , and ρ u i − ρ l i < ρ u i−1 − ρ l i−1
; that is, the robot has to be closer to the goal in order to be success, the goal is sampled in a range further from the initial position of the robot. We select discrete sets of {r i }, {θ i }, [ρ l i , ρ u i ] and determine a curriculum table. The task level will be automatically upgraded if the training performance reach the desired success rate σ out of n trials, i.e σ i = 0.9 indicates at least 90 successful goal-reaching tasks out of n = 100 at level i.
The reward function at time t is defined as
R(t) = c v |v g (t)| + c g cos(θ g (t)) i k=0 1 r k I(l g (t) < r k ),
where c v , c g ∈ R + are constant weights, r k is the goal radius given by the curriculum levels from level 0 current level i, l g is the linear distance between the head of the robot and the goal, and I(l g (t) < r k ) is an indicator function to tell if current l g is within the acceptance radius for the k-th level. This reward can be explained by two objectives. The first part multiplied by c v is a potential-based reward for encouraging movement toward the goal; While the second part multiplied by c g evaluates the level of skill the current agent has for the goal-reaching task. For every task, if the robot enters an accepting region with radius r i defined by i-th level curriculum, it will receive a summation of rewards 1/r k for all k ≤ i (the closer to the goal the higher this summation), shaped by the approach angle θ g (the straighter the direction towards the goal, the higher the reward). If the agent reaches the goal defined by the current level, a new goal is randomly sampled in the current or next level. There are two failing situations, where the desired goal will be re-sampled and updated. The first situation is starving, which happens when the robot gets stuck with locomotion speed under the given threshold for a certain length of time. The second case is missing the goal, which happens when the robot keeps moving towards the wrong direction to the goal region for a certain amount of time.
VI. EXPERIMENT VALIDATION
We used a four-layered neural net configuration with 128× 128 hidden layer neurons. There are 16 inputs consisting of a state vector in R 8 , a vector containing previous actions in R 4 , and the previous option and the terminating probability. The outputs are in R 6 , with current action in R 4 , and current option and terminating probability. The action vector a is decoded by (5) to generate tonic input vector u. At every step, the algorithm samples the current termination function to decide whether to terminate the current option and obtain a new frequency ratio K f or keep the previous option. The KLdivergence was set to 0.02, while γ = 0.99 and λ = 0.96 (See [22], [32] for details about parameters). The backpropagation of the critic net was done with Adam Optimizer and a step size of 5e − 4. The starvation time for failing condition is 60 ms. The missing goal criterion is determined by whenever v g (t) stays negative for over 30 time steps.
A. Policy training
For goal-reaching tasks with increasing difficulty, Figure 7a shows improving performance over different levels versus the number of learning episodes. Figure 7b shows the corresponding total reward with respect to the number of learning episodes. As shown in Fig. 7a, we first train the policy net with fixed options (at this moment, the termination probability is always 0, and a fixed frequency ratio K f = 1.0 is used). When both the task level and the reward cannot increase anymore (at about 3857 episodes), we allow the learning algorithm to change K f along with termination function β, and keep training the policy until the highest level in the curriculum is passed. In this experiment, the learning algorithm equipped with stochastic gradient descent converges to a near-optimal policy after 6400 episodes of training. The whole process takes about 12 hours with 4 snakes training in parallel on a workstation equipped with an Intel Core i7 5820K, 32GB of RAM, and one NVIDIA GTX1080 ti GPU. In order to compensate for simulation inaccuracies, most notably friction coefficients, we employed a domain randomization technique [23], in which a subset of physical parameters are sampled from a distribution with mean on the measured value. The Domain Randomization (DR) parameters used for training are on Table II, on the Appendix. Figure 8 shows a sampled trajectory in the simulated snake robot controlled by the learned PPOC-CPG policy. Below the trajectory plot in Fig. 8 is the recorded pressure input trajectory to the first chamber. In the picture, green circles indicate the goals reached successfully, and the red circle represents a new goal to be reached next. The colors on the path show the reward of the snake state, with a color gradient from red to green, indicating the reward value from low to high. Several maneuvering behaviors discussed in Section IV are exhibited by the policy. First, as the higher frequency can result in lower locomotion speed, the trained policy presents a specific two-phase behavior -(1) The robot starts from a low oscillation frequency to achieve a high speed when it is far from the goal; (2) then it switches to higher oscillation frequency using different options when it is getting closer to the goal. This allows it to stay close on the moving direction straight towards the goal. If this still does not work, the tonic inputs will be used to force stopping the oscillation with the whole snake bending to the desired direction (see IV-B), and then restart the oscillation to acquire a larger turning angle.
B. Experiments with the real robot Fig. 9: Trajectory of the sim and real snake robot in goal-reaching task using learned controller.
We apply the learned policy directly to the real robot. The policy is tested on goal-reaching tasks guided by a mobile robot with an accuracy radius of r = 0.175 meter (The robot base has a 0.16 meter radius). The learning policy obtains the mobile robot position using the Mocap system in 60Hz, and send the control commands to the robot through a low-latency wireless transmitter. An example trajectory is shown in Fig. 9. The real robot tracked the goals set by the mobile robot with an average speed of about 0.04m/s. Though trained on fixed goals only, it can also follow the slow-moving target in the test. This result shows the feasibility of the learned policy on the real robot. In the future, we plan to perform more experiments to quantify the performance loss from the simulated to the real robot.
VII. CONCLUSION
The contribution of this paper is two folds: First, we investigate the properties of Matsuoka oscillator for generating diverse gait patterns for rhythmic and even non-rhythmic locomotion in snake-like robots. Second, we construct a PPOC-CPG net that uses a CPG net to actuate the soft snake robot, and a reinforcement learning algorithm to learn a closed-loop near-optimal control policy that utilizes different oscillation patterns in the CPG net. This learning-based control method shows promising results on goal-reaching and tracking behaviors in soft snake robots. This control architecture may be extended to motion control of other robotic systems, including bipedal and soft manipulators. Our next step is to extend the proposed control framework to a three-dimensional soft snake robot and to realize more complex motion and force control in soft snake robots using distributed sensors and visual feedback.
APPENDIX
Fig. 1 :
1Notation of the state space configuration of the robot.
Fig. 2 :
2Illustrating the input-output connection of the PPOC-CPG net.
Fig. 3 :
3(a) Locomotion trajectory with biased amplitude of tonic inputs u = [T (2) to the CPG net. (b) Locomotion trajectory with biased duty cycle of tonic inputs. And plots of CPG outputs corresponding to tonic inputs with (c) biased amplitude and (d) biased duty ratios.
Fig. 4 :
4Relating oscillating frequency and amplitude to the average linear velocity of serpentine locomotion.
Fig. 5 :
5Switch from stable limit cycle to stable equilibrium.
Fig. 6 :
6Task difficulty upgrade from level i − 1 to level i. As the curriculum level increases, goals are sampled at a narrower distance and wider angle, and acceptance area gets smaller.
Fig. 7 :
7(a) Learning progress of task level and (b) total learning score in the goal-reaching task.
Fig. 8 :
8A sample trajectory and the corresponding control events on the first link CPG output ψ 1 .
TABLE I :
IParameter Configuration of Matsuoka CPG Net Controller for the Soft Snake Robot.Parameters
Symbols
Values
Amplitude
A
4.4044
* Self inhibition weight
b
10.8939
* Discharge rate
τr
0.1869
* Adaptation rate
τa
0.4555
Period ratio
Kf
1.0
Mutual inhibition weights
α1
2.1669
α2
3.1948
α3
5.3696
α4
9.5222
Coupling weights
w12
4.1244
w23
5.0448
w34
8.5053
w21
8.3042
w32
8.1086
w43
6.2195
TABLE II :
IIDomain randomization parametersParameter
Low
High
Ground Friction
0.1
1.5
Wheel Friction
0.8
1.2
Rigid Body mass (kg)
0.035
0.075
Tail mass (kg)
0.065
0.085
Head mass (kg)
0.075
0.125
Max link pressure (psi)
5
12
Gravity angle (rad)
-0.001
0.001
Renato Gasoto, Xuan Liu are PhD students under the supervision of Jie Fu and Cagdas Onal in the Robotics Engineering program at Worcester
In experiments, the following parameters are used: a 1 = 40.0, a 2 = 100.0, a 3 = 50.0, and T = 6.4 sec.
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]
| []
| Advances in the emerging field of coherent quantum feedback control (CQFC) have led to the development of new capabilities in the areas of quantum control and quantum engineering, with a particular impact on the theory and applications of quantum optical networks. We consider a CQFC network consisting of two coupled optical parametric oscillators (OPOs) and study the squeezing spectrum of its output field. The performance of this network as a squeezed-light source with desired spectral characteristics is optimized by searching over the space of model parameters with experimentally motivated bounds. We use the QNET package to model the network's dynamics and the PyGMO package of global optimization algorithms to maximize the degree of squeezing at a selected sideband frequency or the average degree of squeezing over a selected bandwidth. The use of global search methods is critical for identifying the best possible performance of the CQFC network, especially for squeezing at higher-frequency sidebands and higher bandwidths. The results demonstrate that the CQFC network of two coupled OPOs makes it possible to vary the squeezing spectrum, effectively utilize the available pump power, and overall significantly outperform a single OPO. Additionally, the Hessian eigenvalue analysis shows that the squeezing generation performance of the optimally operated CQFC network is robust to small variations of phase parameters. | null | [
"https://arxiv.org/pdf/1701.04242v1.pdf"
]
| 119,464,006 | 1701.04242 | 71296b45b6c12b868e2fcd4d55f206d9031def8e |
Optimizing squeezing in a coherent quantum feedback network of optical parametric oscillators
(Dated: October 30, 2018)
Constantin Brif
Sandia National Laboratories
94550LivermoreCAUSA
Mohan Sarovar
Sandia National Laboratories
94550LivermoreCAUSA
Daniel B S Soh
Sandia National Laboratories
94550LivermoreCAUSA
Ginzton Laboratory
Stanford University
94305StanfordCAUSA
David R Farley
Sandia National Laboratories
94550LivermoreCAUSA
Scott E Bisson
Sandia National Laboratories
94550LivermoreCAUSA
Optimizing squeezing in a coherent quantum feedback network of optical parametric oscillators
(Dated: October 30, 2018)
Advances in the emerging field of coherent quantum feedback control (CQFC) have led to the development of new capabilities in the areas of quantum control and quantum engineering, with a particular impact on the theory and applications of quantum optical networks. We consider a CQFC network consisting of two coupled optical parametric oscillators (OPOs) and study the squeezing spectrum of its output field. The performance of this network as a squeezed-light source with desired spectral characteristics is optimized by searching over the space of model parameters with experimentally motivated bounds. We use the QNET package to model the network's dynamics and the PyGMO package of global optimization algorithms to maximize the degree of squeezing at a selected sideband frequency or the average degree of squeezing over a selected bandwidth. The use of global search methods is critical for identifying the best possible performance of the CQFC network, especially for squeezing at higher-frequency sidebands and higher bandwidths. The results demonstrate that the CQFC network of two coupled OPOs makes it possible to vary the squeezing spectrum, effectively utilize the available pump power, and overall significantly outperform a single OPO. Additionally, the Hessian eigenvalue analysis shows that the squeezing generation performance of the optimally operated CQFC network is robust to small variations of phase parameters.
I. INTRODUCTION
Feedback control is ubiquitous in classical engineering. However, its extension to the quantum realm has been challenging due to the unique character of the quantum measurement, which requires coupling of the observed quantum system to a classical measurement apparatus. Consequently, measurement-based quantum control has to deal with the fundamental effect of stochastic measurement back action on the quantum system, along with the need to amplify quantum signals up to macroscopic levels and high latency of classical controllers in comparison to typical quantum dynamic time scales [1,2]. An alternative approach that has attracted significant interest in the last decade is coherent quantum feedback control (CQFC) [3][4][5], which considers networks where the quantum system of interest (called the plant) is controlled via coupling (either direct or, more often, through intermediate quantum fields) to an auxiliary quantum system (called the controller). CQFC schemes utilize coherent quantum signals circulating between the plant and controller, thus avoiding the need for signal amplification and associated excess noise. Also, both plant and controller can evolve on the same time scale, which eliminates the latency issues. Due to these advantages, CQFC makes it possible to engineer quantum networks with new and unique characteristics [4][5][6][7].
The theoretical foundation of CQFC is a powerful framework based on input-output theory, which is used for modeling networks of open quantum systems connected by electromagnetic fields [8][9][10][11] (see also [3][4][5] for reviews). Moreover, recent developments, including the SLH formalism [12][13][14], the quantum hardware description language (QHDL) [15], and the QNET software package [16], have added important capabilities for, respectively, modular analysis, specification, and simulation of such quantum optical networks. Together, the existing theoretical tools enable efficient and automated design and modeling of CQFC networks.
Proposed and experimentally demonstrated applications of CQFC include the development of autonomous devices for preparation, manipulation, and stabilization of quantum states [17][18][19][20][21][22], disturbance rejection by a dynamic compensator [23], linear-optics implementation of a modular quantum memory [24], generation of optical squeezing [25][26][27][28], generation of quantum entanglement between optical field modes [29][30][31][32][33][34], coherent estimation of open quantum systems [35,36], and ultra-low-power optical processing elements for optical switching [37][38][39] and analog computing [40,41]. In addition to tabletop bulk-optics implementations, CQFC networks have been also implemented using integrated silicon photonics [42] and superconducting microwave devices [43,44]. Squeezed states of light [45][46][47][48] have found numerous applications in quantum metrology and quantum information sciences, including interferometric detection of gravitational waves [49,50], continuous-variable quantum key distribution (CV-QKD) [51][52][53][54][55][56], generation of Gaussian entanglement [55][56][57], and quantum computing with continuousvariable cluster states [58][59][60][61]. Different applications require squeezed states with different properties. For example, detectable gravitational waves are expected to have frequencies in the range from 10 Hz to 10 kHz, and, consequently, quadrature squeezed states used to increase the measurement sensitivity in interferometric detectors should have a high degree of squeezing at sideband frequencies in this range. On the other hand, in CV-QKD the secure key rate is proportional to the bandwidth of squeezing, and hence it would be useful to generate states with squeezing bandwidth extending to 100 MHz or even higher. It would be also of interest to extend the maximum of squeezing to high sideband frequencies.
In recent years, there have been remarkable advances in the generation of squeezed states [62][63][64][65][66][67][68][69][70][71][72][73][74], however, achieving significant control over the squeezing spectrum still remains an ongoing effort. In 2009, Gough and Wildfeuer [25] proposed to enhance squeezing in the output field of a degenerate optical parametric oscillator (OPO) by incorporating the OPO into a CQFC network, where a part of the output beam is split off and then fed back into the OPO. Iida et al. [26] reported an experimental demonstration of this scheme, while Német and Parkins [28] proposed to modify it by including a time delay into the feedback loop. Another significant modification of this scheme was proposed and experimentally demonstrated by Crisafulli et al. [27], who included a second OPO to act as the controller, with the plant OPO and the controller OPO coupled by two fields propagating between them in opposite directions. Due to the presence of quantum-limited gains in both arms of the feedback loop, this CQFC network has a very rich dynamics. In particular, by tuning the network's parameters it is possible to significantly vary the squeezing spectrum of its output field, for example, shift the maximum of squeezing from the resonance to a high-frequency sideband [27].
The full range of performance of the CQFC network of two coupled OPOs as a squeezed-light source, however, still remains to be explored. In this paper, we study the limits of the network's performance by performing two types of optimizations: (1) maximizing the degree of squeezing at a chosen sideband frequency and (2) maximizing the average degree of squeezing over a chosen bandwidth; in both cases, the searches are executed over the space of network parameters with experimentally motivated bounds. To maximize the chances of finding a globally optimal solution, we use the PyGMO package of global optimization algorithms [75] and employ a hybrid strategy which executes in parallel eight searches (using seven different global algorithms). Before each optimization is completed, the searches are repeated multiple times, and intermediate solutions are exchanged between them after each repetition. This strategy enabled us to discover that the CQFC network, when optimally operated, is capable of achieving a remarkably high degree of squeezing at sideband frequencies and bandwidths as high as 100 MHz, with a very effective utilization of the available pump power. We also find that the obtained optimal solutions are quite robust to small variations of phase parameters.
II. BACKGROUND
The derivations in this section largely follow those in Refs. [25,27], with some additional details and modifications.
A. Input-output model of a quantum optical network Consider a network of coupled linear and bilinear optical elements such as mirrors, beam-splitters, phase-shifters, lasers, and degenerate OPOs. The quantum theory of such a network considers quantized cavity field modes which are coupled through cavity mirrors to external (input and output) quantum fields [9][10][11]. Let n be the number of the network's input ports (equal to the number of output ports) and m be the number of cavities (in this model, we assume that each cavity supports one internal field mode). Let a, a in , and a out denote vectors of boson annihilation operators for, respectively, the cavity modes, the input fields, and the output fields:
a = a 1 . . . a m , a in = a in,1 . . . a in,n , a out = a out,1 . . . a out,n . (1)
Assuming that all input fields are in the vacuum state, the network is fully described by the (S, L, H) model (also called the SLH model) [12][13][14], which includes the n × n matrix S that describes the scattering of external fields, the n-dimensional vector L that describes the coupling of cavity modes and external fields, and the Hamiltonian H that describes the intracavity dynamics. For the model considered here, elements {S ij } of S are c-numbers, while H and elements {L i } of L are operators on the combined Hilbert space of all cavity modes in the network. The Heisenberg equations of motion (also known as quantum Langevin equations) for the cavity mode operators {a (t)} are ( = 1)
da dt = −i[a , H] + L L [a ] + Γ l , = 1, . . . , m.(2)
Here, L L is the Lindblad superoperator:
L L [a ] = n i=1 L † i a L i − 1 2 L † i L i a − 1 2 a L † i L i ,(3)
and Γ l is the noise operator:
Γ l = a † in S † [a , L] + [L † , a ]Sa in ,(4)
where a † in = [a † in,1 , . . . , a † in,n ] and L † = [L † 1 , . . . , L † n ] are row vectors of respective Hermitian conjugate operators. The generalized boundary condition for the network is
a out = Sa in + L.(5)
For the type of networks that we consider, elements of L are linear in annihilation operators of the cavity modes, i.e.,
L = Ka,(6)
where K is an n × m complex matrix with elements {K i = [L i , a † ]}, and the Hamiltonian has the bilinear form:
H = a † Ωa + i 2 a † Wa ‡ − i 2 a T W † a,(7)
where a † = [a † 1 , . . . , a † m ] and a ‡ = a †T are, respectively, row and column vectors of boson creation operators for the cavity modes, Ω is an m × m Hermitian matrix, and W is an m × m complex matrix. With such L and H, the Heisenberg equations of motion (2) take the form:
da dt = Va + Wa ‡ + Ya in ,(8)where V = − 1 2 K † K − iΩ is an m × m complex matrix and Y = −K † S is an m × n complex matrix.
To obtain the transfer-matrix function from input to output fields, we seek the solution of Eq. (8) in the frequency domain. Using the Fourier transform, we define:
b(t) = 1 √ 2π ∞ −∞ dω b(ω)e −iωt , (9a) b † (t) = 1 √ 2π ∞ −∞ dω b † (−ω)e −iωt ,(9b)
where b(t) stands for any element of a(t), a in (t), and a out (t).
The field operators are in the interaction frame, and therefore ω is the sideband frequency (relative to the carrier frequency). We also use the double-length column vectors of the form:
b(ω) = b(ω) b ‡ (−ω) ,(10)
where b(ω) stands for either of a(ω), a in (ω), and a out (ω).
With this notation, Eq. (8) together with its Hermitian conjugate can be transformed into one matrix equation and solved forȃ(ω) in the frequency domain:
a(ω) = (Ȃ + iωI 2m ) −1K †Sȃ in (ω).(11)
Here, I 2m is the 2m × 2m identity matrix,Ȃ = ∆(V, W), K = ∆(K, 0),S = ∆(S, 0), and we use the notation:
∆(A, B) = A B B * A * .
Analogously, the boundary condition of Eq. (5) together with its Hermitian conjugate can be transformed into one matrix equation in the frequency domain:
a out (ω) =Sȃ in (ω) +Kȃ(ω).(12)
In Eqs. (11) and (12),ȃ(ω) is a 2m-dimensional vector, a in (ω) andȃ out (ω) are 2n-dimensional vectors,Ȃ is a 2m × 2m matrix,K is a 2n × 2m matrix, andS is a 2n × 2n matrix. By substituting Eq. (11) into Eq. (12), one obtains the quantum input-output relations in the matrix form:
a out (ω) =Z(ω)ȃ in (ω),(13)
whereZ
(ω) = I 2n +K(Ȃ + iωI 2m ) −1K † S(14)
is the network's transfer-matrix function. The 2n × 2n matrix Z(ω) can be decomposed into the block form:
Z(ω) = Z − (ω) Z + (ω) Z + (−ω) * Z − (−ω) * ,(15)
where Z − (ω) and Z + (ω) are n × n matrices. Correspondingly, input-output relations of Eq. (13) can be expressed for each of the output fields (i = 1, . . . , n) as:
a out,i (ω) = n j=1 Z − ij (ω)a in,j (ω) + Z + ij (ω)a † in,j (−ω) ,(16a)a † out,i (−ω) = n j=1 Z + ij (−ω) * a in,j (ω) + Z − ij (−ω) * a † in,j (−ω) .(16b)
B. Squeezing spectrum
Consider the quadrature of the ith output field in time and frequency domains:
X i (t, θ) = a out,i (t)e −iθ + a † out,i (t)e iθ ,(17a)X i (ω, θ) = a out,i (ω)e −iθ + a † out,i (−ω)e iθ ,(17b)
where θ is the homodyne phase. The power spectral density of the quadrature's quantum noise (commonly referred to as the squeezing spectrum) is [45,46]:
P i (ω, θ) = 1 + ∞ −∞ dω : X i (ω, θ), X i (ω , θ) : ,(18)
where : : denotes the normal ordering of boson operators and x, y = xy − x y . Since all input fields are in the vacuum state, X i (ω, θ) = X i (ω , θ) = 0, and one obtains:
P i (ω, θ) = 1 + N i (ω) + N i (−ω) + M i (ω)e −2iθ + M i (ω) * e 2iθ ,(19)
where
N i (ω) = ∞ −∞ dω a † out,i (−ω )a out,i (ω) ,(20a)M i (ω) = ∞ −∞ dω a out,i (ω)a out,i (ω ) .(20b)
By substituting Eqs. (16) into Eqs. (20) and evaluating expectation values for vacuum input fields, one obtains:
N i (ω) = n j=1 Z + ij (ω) 2 ,(21a)M i (ω) = n j=1 Z − ij (ω)Z + ij (−ω).(21b)
In this work, we are only concerned with squeezing properties of the field at one of the output ports. We will designate this port as corresponding to i = 1 and denote the squeezing spectrum of this output field as P(ω, θ) = P 1 (ω, θ). In squeezing generation, the figure of merit is the quantum noise change relative to the vacuum level, measured in decibels, and since P vac (ω, θ) = 1, the corresponding spectral quantity is Q(ω, θ) = 10 log 10 P(ω, θ).
Negative values of Q correspond to quantum noise reduction below the vacuum level (i.e., squeezing of the quadrature uncertainty). The maximum degree of squeezing corresponds to the minimum value of Q. The maximum and minimum of P(ω, θ) as a function of θ,
P + (ω) = max θ P(ω, θ), P − (ω) = min θ P(ω, θ),(23)
are power spectral densities of the quantum noise in antisqueezed and squeezed quadrature, respectively. Analogously to Eq. (22), logarithmic spectral measures of antisqueezing and squeezing for the two quadratures are defined as Q ± (ω) = 10 log 10 P ± (ω), respectively. Expressing M(ω) as M(ω) = |M(ω)|e iθ M (ω) and using Eq. (19), it is easy to find (we omit the subscript i = 1 for simplicity):
P ± (ω) = 1 + N (ω) + N (−ω) ± 2|M(ω)|,(24)
with anti-squeezed and squeezed quadrature corresponding to θ = θ M (ω)/2 and θ = [θ M (ω) − π]/2, respectively. Note that, in general, these optimum values of the homodyne phase θ depend on the sideband frequency ω, so, for example, if the goal is to maximize the degree of squeezing at a particular sideband frequency ω opt , then the optimum phase value θ opt = [θ M (ω opt ) − π]/2 should be selected accordingly.
III. SQUEEZING FROM A SINGLE OPO
A network that produces squeezed light by means of a single degenerate OPO [47] is schematically shown in Fig. 1. The OPO consists of a nonlinear crystal enclosed in a Fabry-Pérot cavity. The pump field for the OPO is assumed to be classical and not shown in the scheme. Each partially transparent mirror in the network (including cavity mirrors and a beamsplitter) has two input ports and two output ports. A vacuum field enters into each input port. The OPO cavity has a fictitious third mirror to model intracavity losses (mainly due to absorption in the crystal as well as scattering and Fresnel reflection at the crystal's facets). The beamsplitter B models losses in the output transmission line (e.g., due to coupling into a fiber) and inefficiencies in the homodyne detector (not shown) used to measure the squeezing spectrum of the output field. Taking into account all optical elements, the network is modeled as having four input ports, four output ports, and one cavity mode (n = 4, m = 1). Parameters of the single OPO network are described in Table I. With ξ = |ξ|e iθ ξ , there is a total of seven real parameters. Note that we use angular frequencies throughout this paper. For each cavity mirror, the leakage rate is where T i is the power transmittance of the ith mirror, c is the speed of light, and l eff is the effective cavity length (taking into account the length and refractive index of the crystal). To simplify the notation, we also use alternative parameters:
κ i = cT i 2l eff , i = 1, 2, 3,(25)γ = κ 1 + κ 2 + κ 3 ,(26)
to denote the total leakage rate (including losses) from the cavity, and
t B = cos(θ B ), r B = sin(θ B ),(27)
to denote, respectively, the transmittivity and reflectivity of the beamsplitter. The QNET package [16] is used to derive the (S, L, H) model of the network, and the resulting components of the model are
S = 0 t B 0 −r B 1 0 0 0 0 0 1 0 0 r B 0 t B , L = √ κ 2 t B a √ κ 1 a √ κ 3 a √ κ 2 r B a , H = ω 0 a † a + i 2 ξa †2 − i 2 ξ * a 2 ,
where a is the annihilation operator of the cavity field mode. Using the formalism of Sec. II A, we obtain: Ω = ω 0 , W = ξ,
K = [ √ κ 2 t B , √ κ 1 , √ κ 3 , √ κ 2 r B ] T , V = −η, Y = − [ √ κ 1 , √ κ 2 , √ κ 3 , 0] , A = −η ξ ξ * −η * , (Ȃ + iωI 2 ) −1 = − 1 λ(ω) η * − iω ξ ξ * η − iω ,
where we defined auxiliary parameters:
η = 1 2 γ + iω 0 , λ(ω) = (η * − iω)(η − iω) − |ξ| 2 .
These results make it straightforward to analytically compute the transfer-matrix functionZ(ω) of Eq. (14). Since we are only interested in squeezing properties of the field at the output port 1, it is sufficient to use only the respective rows of matrices Z − (ω) and Z + (ω), i.e.,
Z − 1 (ω) = √ κ 2 t B (η * − iω) λ(ω) Y + [0, t B , 0, −r B ] ,(28a)Z + 1 (ω) = √ κ 2 t B ξ λ(ω) Y.(28b)
By substituting elements of Z − 1 (ω) and Z + 1 (ω) into Eqs. (21), we obtain:
N 1 (ω) = γκ 2 T B |ξ| 2 |λ(ω)| 2 ,(29a)M 1 (ω) = γ(η * − iω) − λ(ω) |λ(ω)| 2 κ 2 T B ξ,(29b)
where T B = t 2 B is the power transmittance of the beam splitter. Using Eq. (19), the resulting squeezing spectrum is
P(ω, θ) = 1 + 2κ 2 T B |ξ| γ|ξ| + µ(ω) cos ϕ + γω 0 sin ϕ |λ(ω)| 2 ,(30)where µ(ω) = 1 4 γ 2 + |ξ| 2 + ω 2 − ω 2 0 and ϕ = θ ξ − 2θ.
The spectra for anti-squeezed and squeezed quadrature are obtained as the maximum and minimum (cf. Eq. (23)) of P(ω, θ) in Eq. (30) for ϕ = tan −1 [γω 0 /µ(ω)] and ϕ = tan −1 [γω 0 /µ(ω)] + π, respectively, and are given by
P ± (ω) = 1 ± 2κ 2 T B |ξ| µ 2 (ω) + γ 2 ω 2 0 ± γ|ξ| |λ(ω)| 2 .(31)
In order to compare the theoretical spectra with experimental data, it is common to express the pump amplitude as
|ξ| = 1 2 γx, x = P/P th ,(32)
where P is the OPO pump power and P th is its threshold value. Analogously to the scaled pump amplitude x = 2|ξ|/γ, it is convenient to use scaled frequencies Ω = 2ω/γ and Ω 0 = 2ω 0 /γ. With this notation, Eq. (31) takes the form:
P ± (ω) = 1 ± 4T B ρx (1 + y 2 ) 2 + 4Ω 2 0 ± 2x (1 − y 2 ) 2 + 4Ω 2 ,(33)
where ρ = κ 2 /γ = T 2 /(T 1 + T 2 + L) is the escape efficiency of the cavity, L = T 3 denotes the intracavity power loss, and y 2 = x 2 + Ω 2 − Ω 2 0 . In the case of zero detuning, ω 0 = 0, the squeezing spectrum of Eq. (30) becomes
P(ω, θ) = 1 + 2κ 2 T B |ξ| γ|ξ| + ( 1 4 γ 2 + |ξ| 2 + ω 2 ) cos ϕ ( 1 4 γ 2 − |ξ| 2 − ω 2 ) 2 + γ 2 ω 2 .(34)
The corresponding spectra for anti-squeezed and squeezed quadrature are obtained for ϕ = 0 and ϕ = π, respectively. They can be expressed by taking Ω 0 = 0 in Eq. (33), which reproduces the familiar result [46,47]:
P ± (ω) = 1 ± T B ρ 4x (1 ∓ x) 2 + Ω 2 .(35)
The spectra of Eq. (35) have Lorentzian shapes with maximum (for anti-squeezing) and minimum (for squeezing) at the resonance (zero sideband frequency), and with the degree of squeezing rapidly decreasing as the sideband frequency increases. For applications such as CV-QKD, it would be valuable to significantly extend the squeezing bandwidth. It would Noise power (dB)
Anti-squeezing
Squeezing
ω 0 /2π =0 MHz ω 0 /2π =25 MHz ω 0 /2π =50 MHz FIG. 2.
Squeezing spectra of the output light field from a single OPO network with different values of the cavity's frequency detuning ω0/2π (given in the legend). Logarithmic power spectral densities of the quantum noise in anti-squeezed and squeezed quadrature, Q ± (ω) = 10 log 10 P ± (ω), are shown versus the sideband frequency ω/2π for P ± (ω) of Eq. (33). The values of network parameters are listed in the text.
be also of interest to achieve a maximum degree of squeezing (i.e., a minimum value of P − ) at a high-frequency sideband. Therefore, we investigate whether such modifications of the squeezing spectrum are possible by using a nonzero value of the cavity's frequency detuning. Consider a single OPO with a set of experimentally motivated parameters: pump power P = 1.5 W, pump wavelength λ p = 775 nm, and signal wavelength λ s = 1550 nm; an MgO:PPLN crystal with length l c = 20 mm, refractive index (at λ s ) n s = 2.1, and effective nonlinear coefficient d eff = 14 pm/V; a Fabry-Pérot cavity with effective length l eff = 87 mm, left mirror reflectance R 1 = 0.98 (T 1 = 0.02, κ 1 /2π ≈ 5.484 MHz), right mirror reflectance R 2 = 0.85 (T 2 = 0.15, κ 2 /2π ≈ 41.132 MHz), intracavity loss L = 0.02 (κ 3 /2π ≈ 5.484 MHz), and total leakage rate γ/2π ≈ 52.1 MHz; output transmission line loss L tl = R B = 0 (T B = 1). These parameters correspond to OPO's threshold power P th ≈ 14.86 W and scaled pump amplitude x = P/P th ≈ 0.318. Using these parameters, we compute the squeezing spectra P ± (ω) of Eq. (33) for three detuning values: ω 0 /2π = {0, 25, 50} MHz. The resulting logarithmic spectra Q ± (ω) = 10 log 10 P ± (ω) for antisqueezed and squeezed quadrature are shown in Fig. 2. These results indicate that, while the use of nonzero detuning can increase the degree of squeezing at higher-frequency sidebands as compared to the case of ω 0 = 0, this increase is very small. Also, no improvement in the squeezing bandwidth (quantified as the average degree of squeezing over a selected bandwidth) is achieved through the use of nonzero detuning. These observations motivate us to explore the use of the CQFC network with two coupled OPOs as a light source with the potential to generate a widely tunable squeezing spectrum. Real Phase shift of the first phase shifter φ2
In1 Out1 In2 Out2 In3 Out3 OPO1 alpha=xi_p kappa_1=kappa_p_1 kappa_2=kappa_p_2 kappa_3=kappa_p_3 Delta=omega_p In1 Out1 In2 Out2 In3 Out3 OPO2 alpha=xi_c kappa_1=kappa_c_1 kappa_2=kappa_c_2 kappa_3=kappa_c_3 Delta=omega_c Plant Controller In1 In2 Out1 Out2 (−) B1 theta=theta_1 In1 In2 Out1 Out2 (−) B2 theta=theta_2 In1 In2 Out1 Out2 (−) B3
Real Phase shift of the second phase shifter θ1
Real Rotation angle of the first beamsplitter θ2
Real Rotation angle of the second beamsplitter θ3
Real Rotation angle of the third beamsplitter
IV. SQUEEZING FROM A NETWORK OF TWO COUPLED OPOS
The CQFC network that includes two coupled degenerate OPOs [27] is schematically shown in Fig. 3. Each OPO consists of a nonlinear crystal enclosed in a Fabry-Pérot cavity.
Pump fields for both OPOs are assumed to be classical and not shown in the scheme. From the control theory perspective, OPO1 is considered to be the plant and OPO2 the (quantum) controller. Each partially transparent mirror in the network (including cavity mirrors and beamsplitters) has two input ports and two output ports. A vacuum field enters into each input port, except for two input ports of cavity mirrors used for the feedback loop between the plant and controller. Each OPO cavity has a fictitious third mirror to model intracavity losses. Beamsplitters B1 and B2 represent the light diverted to lock the cavities as well as losses in optical transmission lines between the OPO cavities. Beamsplitter B3 represents losses in the output transmission line (e.g., due to coupling into a fiber) and inefficiencies in the homodyne detector (not shown) used to measure the squeezing spectrum of the output field. Phase shifters P1 and P2 are inserted into transmission lines between the OPOs to manipulate the interference underlying the CQFC control. Taking into account the feedback loop between the plant and controller, the network is modeled as having seven input ports, seven output ports, and two cavity modes (n = 7, m = 2).
Parameters of the network of two coupled OPOs are listed in Table II. With ξ p = |ξ p |e iθp and ξ c = |ξ c |e iθc , there is a total of 17 real parameters. The relationship between leakage rate and power transmittance of a cavity mirror is given, similarly to Eq. (25), by
κ pi = cT pi 2l p,eff , κ ci = cT ci 2l c,eff , i = 1, 2, 3,(36)
where T pi (T ci ) is the power transmittance of the ith mirror and l p,eff (l c,eff ) is the effective cavity length for the plant (controller). To simplify the notation, we also use alternative parameters:
γ p = κ p1 + κ p2 + κ p3 , γ c = κ c1 + κ c2 + κ c3(37)
to denote the total leakage rate (including losses) from, respectively, the plant and controller cavities,
t i = cos(θ i ), r i = sin(θ i ), i = 1, 2, 3(38)
to denote, respectively, the transmittivity and reflectivity of each beamsplitter, and
φ = φ 1 + φ 2(39)
to denote the total phase shift for the feedback roundtrip path. Similarly to Eq. (32), we also define the scaled pump amplitudes x p and x c for the plant and controller OPOs, respectively:
x p = 2|ξ p | γ p = P p P p,th , x c = 2|ξ c | γ c = P c P c,th ,(40)
where P p (P c ) is the OPO pump power and P p,th (P c,th ) is its threshold value for the plant (controller). The QNET package [16] is used to derive the (S, L, H) model of the network, and the resulting components of the model are
S = t 1 t 2 t 3 e iφ −r 1 t 2 t 3 e iφ −r 2 t 3 e iφ2 −r 3 0 0 0 r 1 t 1 0 0 0 0 0 t 1 r 2 e iφ1 −r 1 r 2 e iφ1 t 2 0 0 0 0 t 1 t 2 r 3 e iφ −r 1 t 2 r 3 e iφ −r 2 r 3 e iφ2 t 3 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 ,(41)L = t 3 √ κ p1 t 1 t 2 e iφ + √ κ p2 a p + √ κ c2 t 2 t 3 e iφ2 a c √ κ p1 r 1 a p √ κ p1 t 1 r 2 e iφ1 a p + √ κ c2 r 2 a c r 3 √ κ p1 t 1 t 2 e iφ + √ κ p2 a p + √ κ c2 t 2 r 3 e iφ2 a c √ κ c1 a c √ κ p3 a p √ κ c3 a c ,(42)H = (ω p + Im ν) a † p a p + ω c a † c a c + i 2 ν 12 a † p a c + H.c. + i 2 ξ p a †2 p + ξ c a †2 c + H.c. ,(43)
where a p and a c denote, respectively, the annihilation operators of the plant's and controller's cavity field modes, and we defined auxiliary parameters:
ν 1 = √ κ c2 κ p1 t 1 e iφ1 , ν 2 = √ κ c2 κ p2 t 2 e iφ2 , ν 12 = ν * 1 − ν 2 , ν = √ κ p1 κ p2 t 1 t 2 e iφ .
By comparing Eq. (43) to the corresponding Hamiltonian without feedback:
H nf = ω p a † p a p + ω c a † c a c + i 2 ξ p a †2 p + ξ c a †2 c + H.c. ,(44)
we observe that two main effects induced by feedback are (1) the appearance of an effective interaction between the plant's and controller's cavity modes, governed by the term i 2 ν 12 a † p a c + H.c., and (2) the modification of the plant detuning by Im ν which is proportional to sin φ.
Using the formalism of Sec. II A, we obtain:
Ω = ω p + Im ν i 2 ν 12 − i 2 ν * 12 ω c , W = ξ p 0 0 ξ c , K = t 3 √ κ p1 t 1 t 2 e iφ + √ κ p2 √ κ c2 t 2 t 3 e iφ2 √ κ p1 r 1 0 √ κ p1 t 1 r 2 e iφ1 √ κ c2 r 2 r 3 √ κ p1 t 1 t 2 e iφ + √ κ p2 √ κ c2 t 2 r 3 e iφ2 0 √ κ c1 √ κ p3 0 0 √ κ c3 , V = − η p ν 2 ν 1 η c , Y = − √ κ p1 − √ κ p2 t 1 t 2 e iφ − √ κ c2 t 1 e iφ1 √ κ p2 r 1 t 2 e iφ √ κ c2 r 1 e iφ1 √ κ p2 r 2 e iφ2 0 0 0 0 − √ κ c1 − √ κ p3 0 0 − √ κ c3 T , A = −η p −ν 2 ξ p 0 −ν 1 −η c 0 ξ c ξ * p 0 −η * p −ν * 2 0 ξ * c −ν * 1 −η * c ,
where we used additional auxiliary parameters:
η p = 1 2 γ p + iω p + ν, η c = 1 2 γ c + iω c .
It is possible to analytically invert the matrixȂ + iωI 4 in order to obtain the transfer-matrix functionZ(ω) and squeezing spectrum P(ω, θ) in analytic form. However, the resulting expressions are too complicated and visually uninformative to be shown here. For practical purposes, it is more efficient to numerically evaluateZ(ω) and P(ω, θ) for any given set of parameter values.
V. SQUEEZING OPTIMIZATION PROCEDURE
A. Objective function
In order to quantitatively investigate the tunability of the squeezing spectrum in the CQFC network of two coupled OPOs, we numerically optimize the degree of squeezing at various sideband frequencies. Specifically, we minimize the objective function of the form:
J = P − (ω opt ) + gP − (ω opt )P + (ω opt ),(45)
where ω opt is the selected sideband frequency. The first term in Eq. (45) is the minimum of the squeezing spectrum at ω opt , while the second term is the uncertainty product times the weight parameter g. This second term is included in order to eliminate solutions with a very large uncertainty of the antisqueezed quadrature. In all optimization results shown below, the weight parameter is g = 0.001. With such a small value of g, the difference between the values of J and P − is always insignificant, and therefore, for the sake of simplicity, we refer to the problem of minimizing J as squeezing optimization.
All solutions encountered during a search are checked to satisfy the Routh-Hurwitz stability criterion [76], i.e., that all eigenvalues of the matrixȂ in Eq. (14) have negative real parts. Any unstable solution is eliminated from the consideration by assigning to it a very large objective value (J = 10 6 ).
B. Optimization variables
For a given ω opt , the objective J is a function of the network parameters -seven real parameters for the single OPO network:
{T 1 , T 2 , L, ω 0 , x, θ ξ , L tl },(46)
and 17 real parameters for the CQFC network of two coupled OPOs:
{T p1 , T p2 , L p , ω p , x p , θ p , T c1 , T c2 , L c , ω c , x c , θ c , φ 1 , φ 2 , L 1 , L 2 , L 3 } .(47)
Recall that, for the single OPO network, L = T 3 is the intracavity power loss and L tl = R B is the power loss in the output transmission line. Similarly, for the CQFC network of two coupled OPOs, L p = T p3 and L c = T c3 are the intracavity power losses for the plant and controller OPOs, respectively, and L i = r 2 i (i = 1, 2, 3) are power losses in the transmission lines. In cases where the two intracavity loss values are equal, we denote L in = L p = L c , and where the three transmission line loss values are equal, we denote L out = L 1 = L 2 = L 3 .
Numerical simulations demonstrate that an increase in any of the losses always leads to a deterioration of squeezing, and therefore if a loss parameter can vary in a specified interval [L l , L u ], an optimization will always converge to the lower bound L l . Therefore, it makes sense to to exclude the loss parameters from the optimization variables, i.e., to execute each optimization with all loss parameters having preassigned fixed values (of course, these values can vary from one optimization run to another to explore various experimentally relevant regimes). Consequently, there remain five optimization variables for the single OPO network:
{T 1 , T 2 , ω 0 , x, θ ξ },(48)
and 12 optimization variables for the CQFC network of two coupled OPOs:
{T p1 , T p2 , ω p , x p , θ p , T c1 , T c2 , ω c , x c , θ c , φ 1 , φ 2 }.(49)
Each optimization variable z can vary in an interval [z l , z u ] (where z l is the lower bound and z u is the upper bound). The bound intervals are
• [0, 2π] for all phase variables (θ ξ , θ p , θ c , φ 1 , φ 2 ); • [−ω u , ω u ] for
C. Optimization methodology
Preliminary optimization runs using local algorithms (e.g., Sequential Least Squares Programming) demonstrated that different choices of initial parameter values resulted in different solutions of varying quality. These results mean that the fitness landscape contains multiple local optima. In order to reach a solution of very high quality, we decided to use global search methods. Specifically, we used PyGMO, a suite of global (stochastic) algorithms [75]. Since these global algorithms are heuristic in nature, they do not guarantee the convergence to a global optimum; in fact, as shown in Table III, while multiple global methods are capable of finding high-quality solutions, the performance varies between different algorithms as well as between optimizations with different values of ω opt for the same algorithm.
To maximize the chances of finding a globally optimal solution, we employed a hybrid strategy, where each optimization executes in parallel eight searches (using seven different global algorithms), with a fully connected topology of solution exchanges between them. These eight searches include two instances of Artificial Bee Colony and one instance of each: Covariance Matrix Adaptation Evolution Strategy, Differential Evolution variant 1220, Differential Evolution with p-best crossover, Improved Harmony Search, Particle Swarm Optimization variant 5, and Compass Search guided by Monotonic Basin Hopping. Each optimization uses the population size of N pop = 30 for each of the global searches, and the evolutions are repeated N ev = 30 times (with solution exchanges between the searches after the completion of each evolution except the last one); the algorithm parameters (the number of no improvements before halting the optimization, N stop , the number of generations, N gen , and the number of iterations, N iter ) used in the searches are the same as those shown in Table III for individual algorithms. As indicated by the results in Table III, this hybrid strategy consistently finds the best solution, as compared to any individual algorithm. Multiple trials with larger values of N pop , N ev , N gen , and N iter did not typically result in an improvement of the solution quality, and thus did not warrant the increased run time.
VI. SQUEEZING OPTIMIZATION RESULTS
First of all, we would like to compare the performance of the CQFC network of two coupled OPOs versus that of the single OPO network, in terms of the maximum degree of squeezing achievable under comparable conditions. Figures 4 and 5 show the optimized degree of squeezing, Q − (ω opt ), at ω opt /2π = 100 MHz, for both networks, versus the upper limits on various network parameters (T u and x u in Fig. 4, and ω u and x u in Fig. 5), with constant loss values: L = L in = 0.01, L tl = L out = 0.1. We observe that the CQFC network of two coupled OPOs generates stronger squeezing than the single OPO network, even as total losses in transmission lines in the former are three times larger than those in the latter (30% versus 10%). In both networks, the maximum degree of squeezing increases with both T u (more light is allowed to leave the cavities) and x u (higher pump power), with these increases being roughly linear for the single OPO network and faster than linear in the CQFC network of two coupled OPOs. These results demonstrate that the feedback makes it possible to more effectively utilize the available pump power. Figure 5 also shows that, for both networks, the maximum degree of squeezing is independent of the upper limit ω u on the cavity detuning frequency; furthermore, we found that in most cases the maximum degree of squeezing is actually achieved with zero detuning. In all results shown below, optimizations used the upper limit value ω u /2π = 100 MHz.
We also investigate the dependence of the maximum degree of squeezing, Q − (ω opt ), on the sideband frequency ω opt at which it is optimized. This dependence is shown in Fig. 6, for both networks, for different values of transmission line losses and pump amplitude bound. We observe that the CQFC network of two coupled OPOs not only generates stronger squeezing than the single OPO network, but that the degradation of squeezing associated with the increase of ω opt is substantially slower in the former than in the latter. The capability of the CQFC network to moderate the degradation of squeezing at higher values of ω opt is associated with a rather abrupt change in the regime of network operation, which is manifested by a rapid change in the slope of the curves in subplots (a) and (c) of Fig. 6.
To explore further the emergence of this new operation regime, we focus on the CQFC network of two coupled OPOs, with Fig. 7 showing the dependence of the maximum degree of squeezing on ω opt for more values of transmission line losses. We observe that the value of ω opt at which the operation regime switches, increases with both x u and L out . The difference between the curve slopes in the low-ω opt and highω opt regimes decreases as L out increases.
To understand the physical differences between operations of the CQFC network in the low-ω opt and high-ω opt regimes, we consider the dependence of the optimal values of power transmittances of cavity mirrors, T p1 , T p2 , T c1 , and T c2 , on ω opt . This dependence is shown in Fig. 8 for optimizations with T u = 0.9, L in = 0.01, and various values of L out and x u . First, we see that the optimal values of T p2 and T c1 are constant over the entire range of ω opt values; specifically, T p2 = 0.9 is at the upper bound, which corresponds to the maximum flow from the plant cavity to the 1st output field (the one whose squeezing properties are measured), and T c1 = 0 is at the lower bound, which corresponds to the minimum flow from the controller cavity to the 5th output field (the one which is not used for either squeezing measurement or feedback). In contrast to this simple behavior of the optimal values of T p2 and T c1 , the optimal values of T p1 and T c2 , which regulate the feedback between the plant and controller OPOs, demonstrate much more intricate dependence on ω opt . The optimal value of T p1 and especially that of T c2 undergo a substantial and rather abrupt change at the critical ω opt value at which the network's operation switches between the low-ω opt and high-ω opt regimes. As ω opt increases through the critical point, T p1 changes from a lower to a higher value, while T c2 decreases from the upper bound T c2 = 0.9 to a much lower value. In other words, the low-ω opt optimal regime is characterized by the maximum flow of light from the controller to the plant and a much lower flow in the opposite direction, while the high-ω opt optimal regime is characterized by roughly similar flows of light in both directions. These patterns characterizing the regimes of optimal network operation, their dependencies on pump and loss parameters, and the rapid switch between the regimes, are quite non-intuitive, and finding them would be rather unlikely without the use of a stochastic global search that explores vast areas of the fitness landscape. The sharp change of the optimal value of T c2 associated with the regime switch makes it easy to identify the sideband frequency ω opt , at which the high-ω opt regime commences (the precision of determining the ω opt /2π values is limited by the sampling interval, which is 2 MHz in our data). The values of ω opt /2π are shown in Table IV for T u = 0.9, L in = 0.01, and various values of L out and x u . We see that ω opt increases monotonously with both L out and x u .
We also note that the optimal values of the scaled pump amplitudes, x p and x c , are almost always at (or very close to) the upper bound x u , i.e., in either regime the optimally operated CQFC network usually uses all the pump power it can get. The maximum use of the pump power is also observed for the optimal operation of the single OPO network. Indeed, as seen in Fig. 9, for both networks, the optimal values of the pump power are virtually independent of ω opt and losses, while they scale quadratically with x u . Due to the rapid growth of the optimal pump power with x u , only values x u ≤ 3 should be considered realistic for the optimal operation with a typical tabletop experimental setup considered in this paper.
Next, we investigate the squeezing spectrum Q − (ω) gen-erated under the optimal operation of either network for various values of ω opt , x u , and transmission line losses. Figure 10 shows Q − (ω) for both networks for various values of ω opt , L tl = L out , and x u . We see that the optimally operated single OPO network generates exactly the same Lorentzian squeezing spectrum for any choice of ω opt . In contrast, the CQFC network of two coupled OPOs is capable of generating diverse squeezing spectra, with the specific spectral shape varying to fit the selected value of ω opt , and overall generates much stronger squeezing over a major portion of the spectrum (especially, at frequencies around ω opt ). Interestingly, the capability of the CQFC network to generate a squeezing spectrum Q − (ω) that has the minimum at ω = ω opt is attained only if the selected value of ω opt is within the highω opt regime of optimal network operation, i.e., ω opt ≥ ω opt (for a given set of bound and loss values). Conversely, as seen for ω opt /2π = 5 MHz in all subplots of Fig. 10 and for ω opt /2π = 25 MHz in subplots (h) and (i) of Fig. 10, the squeezing spectrum has the minimum at ω = 0 if ω opt is within the low-ω opt regime of optimal network operation. Finally, we explore further the dependence of the optimized degree of squeezing, Q − (ω opt ), on the intracavity and transmission line losses for the CQFC network of two coupled OPOs. Figure 11 shows Q − (ω opt ) versus (a) L in and L 3 (with L 1 = L 2 = 0.1), and (b) L in and L out . The situation when L 3 = L 1 = L 2 is practically relevant since L 3 includes, in addition to losses in the output transmission line, inefficiencies in the homodyne detector used to measure the squeezing spectrum of the output field. The results shown in Fig. 11 confirm that any increase in losses is detrimental to squeezing and quantify this relationship.
VII. CORRELATIONS BETWEEN OPTIMAL VALUES OF PHASE VARIABLES
Since the phase parameters play a significant role in tuning the quantum interference that governs the CQFC network's performance, an interesting question is whether their optimal values are correlated. Optimal values of a parameter can be cast as a vector each element of which corresponds to a distinct value of ω opt , and correlations can be computed between pairs of such vectors. Specifically, we computed the Pearson correlation coefficient for all six pairs of four phase variables (θ p , θ c , φ 1 , φ 2 ), and found that substantial correlations only exist between φ 1 and φ 2 . Table V shows the values of the Pearson correlation coefficient r(φ 1 , φ 2 ), computed for T u = 0.9, L in = 0.01, and various values of L out and x u . The correlation in Table V generally decreases as L out and x u increase. This trend can be compared to the one observed in Table IV where the sideband frequency ω opt at which the high-ω opt regime commences increases as L out and x u increase. Since the vectors φ 1 (ω opt ) and φ 2 (ω opt ) contain elements corresponding to both operation regimes, the trend observed in Table V implies that the correlation r(φ 1 , φ 2 ) generally decreases as the number of vector components corresponding to the high-ω opt regime decreases. A plausible explanation of this behavior is that the correlation between the two phase variables is higher in the high-ω opt regime. To test this hypothesis, we computed the Pearson correlation coefficient r(φ 1 , φ 2 ) for the pair of vectors φ 1 = φ 1 (ω opt ≥ ω opt ) and φ 2 = φ 2 (ω opt ≥ ω opt ) that include only elements corresponding to the high-ω opt regime. The values of r(φ 1 , φ 2 ) are shown in Table VI for T u = 0.9, L in = 0.01, and various values of L out and x u . The correlations in Table VI are consistently larger than 0.5, and, furthermore, we find that r(sin φ 1 , sin φ 2 ) = 1.0 and r(cos φ 1 , cos φ 2 ) = −1.0 (up to numerical precision) for all considered values of L out and x u . These findings indicate a significant degree of concerted action in how the CQFC network of two coupled OPOs operates in the high-ω opt regime.
VIII. ROBUSTNESS OF OPTIMAL SOLUTIONS
Any practical implementation of a quantum optical network inevitably involves imprecisions and imperfections, which may affect the desired performance. This issue is of especial importance in a CQFC network, which relies on a precise quantum interference between the pump and controller fields to manipulate the properties of the output field (see, for example, the superposition of a p and a c in the first element of the L vector in Eq. (42)). This interference depends on the values of phase variables, and a key question is how robust is an optimal solution to small variations in these values. To analyze this robustness, we computed the Hessian of the objective function J with respect to the phase variables, for a variety of optimal sets of network parameters.
For the single OPO network, J depends on one phase variable θ ξ , and the Hessian H has one element ∂ 2 J/∂θ The numerical analysis shows that the Hessian is zero (up to numerical precision) for all of these optimal solutions. Therefore, small fluctuations in the value of the pump phase θ ξ should have no effect on the optimized degree of squeezing.
For the CQFC network of two coupled OPOs, J depends on four phase variables (θ p , θ c , φ 1 , φ 2 ), and the Hessian H is a 4 × 4 matrix of second-order derivatives. We The numerical analysis shows that two of the Hessian eigenvalues (h 3 and h 4 ) are zero (up to numerical precision) for all of these optimal solutions. Therefore, robustness to small phase variations is determined by two nonzero Hessian eigenvalues (h 1 and h 2 ). Figures 12-14 show these nonzero Hessian eigenvalues as functions of ω opt and x u (Fig. 12), ω opt and L out (Fig. 13), and x u and L out (Fig. 14). We see that h 1 is typically much larger than h 2 , and hence the magnitude of h 1 is the main factor determining the robustness properties of the optimal solutions. The dependence of h 1 on ω opt , seen in Figs. 12 and 13, demonstrates a significant difference in robustness properties between the low-ω opt and high-ω opt regimes. The low-ω opt regime is intrinsically robust for a broad range of parameter values. In the high-ω opt regime, a reasonable degree of robustness is achieved for x u ≥ 0.2 (i.e., for pump powers above 4 W for the OPO parameters considered here). Larger losses in transmission lines (L out ≥ 0.1) also enhance robustness.
The four components of the Hessian eigenvector e 1 (which corresponds to the largest eigenvalue h 1 ) are shown in Fig. 15 versus ω opt . They also exhibit an abrupt change associated with the switch of the optimal operation regime at ω opt . In the low-ω opt regime, the eigenvector component corresponding to φ 2 has the largest value and the rest of the components have smaller absolute values, but none is negligible. In the highω opt regime, the components corresponding to φ 1 and φ 2 have similar values, while the components corresponding to θ p and θ c are close to zero. These results are consistent with the find- ings that the low-ω opt regime is characterized by the maximum flow of light passing through the phase shifter P2 (from the controller to the plant), while the high-ω opt regime is characterized by roughly similar flows of light passing through the phase shifters P1 and P2 (in both directions). The decrease of the optimized degree of squeezing due to small variations of phase parameters can be quantified using the computed Hessian eigenvalues or, alternatively, via direct Monte Carlo averaging over a random distribution of phase variable values. Figure 16 shows the optimized degree of squeezing, Q − (ω opt ), for the CQFC network of two coupled OPOs (with ω opt /2π = 100 MHz, x u = 0.2, T u = 0.9, and various values of L out ), versus the standard deviation of phase uncertainty, σ phase (for simplicity, we assume a normal distribution with zero mean and the same value of σ phase for uncertainty in each of the four phase variables). We see a good agreement between the Hessian-based and Monte Carlo computations for σ phase ≤ 0.1 (and even for σ phase ≤ 0.2 for L out ≥ 0.1). We also see that the deterioration of squeezing induced by phase variations is quite tolerable for σ phase ≤ 0.1 (especially, for L out ≥ 0.1). Note that our squeezing optimization procedure does not explicitly include a robustness requirement, and hence the observed high level of robustness might be surprising, but it is likely related to the natural tendency of stochastic optimization algorithms to eliminate solutions that are very sensitive to small parameter variations.
IX. SQUEEZING BANDWIDTH OPTIMIZATION
In CV-QKD with squeezed states, the secure key rate is proportional to the bandwidth of squeezing. Therefore, we also explored the capability of the CQFC network of two coupled OPOs to generate output states with high squeezing bandwidth, by optimizing the average degree of squeezing over a frequency interval [0, ω B ], for various values of ω B . Specifically, the objective function for these optimizations is (50) Here, N B = ω B /h B (i.e., N B + 1 is the number of sampling points), ω k = kh B , and h B is the sampling interval. Except for the different choice of the objective function, the rest of the optimization procedure is the same as that described in Sec. V. In optimization runs that minimized J B , we considered four bandwidth values ω B /2π = {25, 50, 75, 100} MHz and used the fixed sampling interval h B /2π = 1 MHz.
J B = P − (ω B ) ≡ P − (ω) = 1 N B + 1 NB k=0 P − (ω k ).
For illustration purposes, we use a logarithmic measure of average squeezing, Q − (ω B ) = 10 log 10 P − (ω B ),
however, note that Q − (ω B ) = Q − (ω) . Table VII shows the best values of Q − (ω B ) for ω B /2π = 100 MHz, for both the CQFC network of two coupled OPOs and the single OPO network, obtained in optimizations with T u = 0.9, L = L in = 0.01, and various values of L tl = L out and x u . We see that the CQFC network of two coupled OPOs significantly outperforms the single OPO network in terms of the average squeezing generated over the 100 MHz bandwidth, especially for lower values of transmission line losses. It is also interesting to examine the squeezing spectrum Q − (ω) generated under the optimal operation of either network when we minimize J B = P − (ω B ). Figure 17 shows Q − (ω) for both networks for various values of ω B , L tl = L out , and x u . Similarly to the results shown in Sec. VI (cf. Fig. 10), we find that the optimally operated single OPO network generates exactly the same Lorentzian squeezing spectrum for any choice of ω B . In contrast, the CQFC network of two coupled OPOs is capable of adapting the generated squeezing spectrum depending on the selected value of ω B and overall produces much higher squeezing bandwidth.
X. CONCLUSIONS
We modeled the squeezing spectrum of the output field of the CQFC network of two coupled OPOs and used a suite of global optimization methods to examine the limits to which this spectrum can be varied under conditions typical for tabletop experiments. We found that, in contrast to a single OPO, the CQFC network can utilize the interference between the fields in the plant OPO and the controller OPO to significantly modify the squeezing spectrum of the output field in response to the selected optimization objective. In particular, when the objective is to maximize the degree of squeezing at a highfrequency sideband ω opt , the CQFC network can operate in an optimal regime characterized by a high degree of cooperativity between the plant OPO and the controller OPO, as quantified by the flows of light between them and the correlation between the phase shifts φ 1 and φ 2 . In this operation regime, the optimized squeezing spectrum Q − (ω) of the CQFC network of two coupled OPOs has the minimum at ω = ω opt , while the minimum of the optimized spectrum of the single OPO network is always at zero sideband frequency.
For both types of optimization objectives considered in this work (maximizing the degree of squeezing at a selected sideband frequency and maximizing the average degree of squeezing over a selected bandwidth), the CQFC network of two coupled OPOs significantly outperforms a single OPO in terms of squeezing achieved under similar conditions, even with higher losses in the CQFC network due to additional components and transmission lines. Also, the CQFC network is more effective in terms of converting a higher pump power into a stronger squeezing. While this superior performance of the CQFC network of two coupled OPOs relies on a phasesensitive interference between multiple fields, we discovered, perhaps surprisingly, that squeezing generated by the optimally operated CQFC network is rather robust to small variations of phase parameters. This robustness can be attributed to the tendency of global optimization algorithms to avoid solutions that are overly sensitive to small parameter variations, but the fact that such robust network configurations do actually exist is quite remarkable.
Overall, our results strongly indicate that CQFC networks provide a very effective tool for engineering quantum optical systems with new properties and unprecedented levels of performance. This work also demonstrates the usefulness of advanced optimization methods for analyzing and improving the performance of such networks.
FIG. 3 .
3A schematic depiction of the CQFC network of two coupled OPOs.
all cavity detuning frequencies (ω 0 , ω p , ω c );• [0, T u ] for all power transmittances of actual cavity mirrors (T 1 , T 2 , T p1 , T p2 , T c1 , T c2 );• [0, x u ] for all scaled pump amplitudes (x, x p , x c ).The values of upper bounds ω u , T u and x u are specified (along with the values of losses) for each optimization run. In all optimizations, the fixed physical parameters are selected the same for all OPOs: pump wavelength λ p = 775 nm, signal wavelength λ s = 1550 nm; an MgO:PPLN crystal with length l c = 20 mm, refractive index (at the signal wavelength) n s = 2.1, and effective nonlinear coefficient d eff = 14 pm/V; a Fabry-Pérot cavity with effective length l eff = 87 mm. These values are characteristic for a typical tabletop experiment with bulk-optics components.
III. Performance of different algorithms for squeezing optimization in the CQFC network of two coupled OPOs. The table shows the best degree of squeezing, Q − (ωopt) = 10 log 10 P − (ωopt) (in dB), found using various algorithms, for Lin = 0.01, Lout = 0.05, ωu/2π = 100.0 MHz, xu = 0.3, Tu = 0.9, and five different ωopt values: ωopt/2π = {5, 25, 50, 100, 200} MHz. Optimizations for each individual algorithm execute four parallel searches with the population sizes of Npop = 30, and the evolutions are repeated Nev = 30 times (with solution exchanges between the searches after the completion of each evolution except the last one). Algorithm parameters such as the number of no improvements before halting the optimization, Nstop, the number of generations, Ngen, and the number of iterations, Niter, are indicated in the
FIG. 4 .FIG. 5 .
45The optimized degree of squeezing, Q − (ωopt), for (a) the CQFC network of two coupled OPOs and (b) the single OPO network, versus the upper limits on the power transmittance of cavity mirrors, Tu, and the scaled pump amplitude, xu. Other parameters are ωopt/2π = 100 MHz, ωu/2π = 100 MHz, L = Lin = 0.01, L tl = Lout = 0The optimized degree of squeezing, Q − (ωopt), for (a) the CQFC network of two coupled OPOs and (b) the single OPO network, versus the upper limits on the cavity detuning frequency, ωu, and the scaled pump amplitude, xu. Other parameters are ωopt/2π = 100 MHz, Tu = 0.9, L = Lin = 0.01, L tl = Lout = 0.1.
FIG. 6 .FIG. 7 .
67The optimized degree of squeezing, Q − (ωopt), versus ωopt, for the CQFC network of two coupled OPOs (subplots (a) and (c)) and the single OPO network (subplots (b) and (d)). The transmission line losses are L tl = Lout = 0.01 in subplots (a) and (b), and L tl = Lout = 0.1 in subplots (c) and (d). Each subplot shows four curves corresponding to different values of xu (xu = {0.1, 0.2, 0.3, 0.4}), as indicated in the legend. Other parameters are Tu = 0.9, L = Lin = 0.01. The optimized degree of squeezing, Q − (ωopt), versus ωopt, for the CQFC network of two coupled OPOs. The values of xu are: (a) xu = 0.1, (b) xu = 0.2, (c) xu = 0.3, and (d) xu = 0.4. Each subplot shows six curves corresponding to different values of transmission line losses: Lout = {0.01, 0.05, 0.1, 0.15, 0.2, 0.25}, as indicated in the legend. Other parameters are Tu = 0.9, Lin = 0.01.
FIG. 8 .
8The optimal values of power transmittances of cavity mirrors, Tp1 (subplots (a), (b), (c)), Tp2 (subplots (d), (e), (f)), Tc1 (subplots (g), (h), (i)), and Tc2 (subplots (j), (k), (l)), versus ωopt, for the CQFC network of two coupled OPOs. The transmission line losses are Lout = 0.01 (subplots (a), (d), (g), (j)), Lout = 0.1 (subplots (b), (e), (h), (k)), and Lout = 0.2 (subplots (c), (f), (i), (l)). Each subplot shows four curves corresponding to different values of xu (xu = {0.1, 0.2, 0.3, 0.4}), as indicated in the legend. Other parameters are Tu = 0.9, Lin = 0.01.
FIG. 9 .FIG. 10 .
910The optimal values of the pump power for the plant OPO in the CQFC network of two coupled OPOs (subplots (a) and (c)) and for the single OPO (subplots (b) and (d)), versus ωopt.The transmission line losses are L tl = Lout = 0.01 in subplots (a) and (b), and L tl = Lout = 0.1 in subplots (c) and (d). Each subplot shows four curves corresponding to different values of xu (xu = {0.1, 0.2, 0.3, 0.4}), as indicated in the legend. Other parameters are Tu = 0.9, L = Lin = 0.01. The squeezing spectrum Q − (ω) for the optimal operation of both networks. Each subplot shows four curves corresponding to the optimally operated CQFC network of two coupled OPOs for different values of ωopt (ωopt/2π = {5, 25, 50, 100} MHz), along with a curve corresponding to the optimally operated single OPO network for any value of ωopt, as indicated in the legend. The transmission line losses are L tl = Lout = 0.01 (subplots (a), (b), (c)), L tl = Lout = 0.05 (subplots (d), (e), (f)), and L tl = Lout = 0.1 (subplots (g), (h), (i)). The upper limit on the scaled pump amplitude is xu = 0.1 (subplots (a), (d), (g)), xu = 0.2 (subplots (b), (e), (h)), and xu = 0.3 (subplots (c), (f), (i)). Other parameters are Tu = 0.9, L = Lin = 0.01.
FIG. 11 .
11The optimized degree of squeezing, Q − (ωopt), for the CQFC network of two coupled OPOs, versus (a) Lin and L3 (with L1 = L2 = 0.1), and (b) Lin and Lout. Other parameters are ωopt/2π = 100 MHz, xu = 0.2, Tu = 0.9.
2 ξ . H was computed for 3500 optimal solutions (all combinations of ω opt /2π = {2, 4, . . . , 100} MHz, x u = {0.1, 0.2, . . . , 0.5}, T u = {0.5, 0.9}, and L tl = {0.01, 0.05, 0.1, . . . , 0.3}, with L = 0.01).
computed the eigenvalues {h 1 , . . . , h 4 } and eigenvectors {e 1 , . . . , e 4 } of the Hessian H for 3500 optimal solutions (all combinations of ω opt /2π = {2, 4, . . . , 100} MHz, x u = {0.1, 0.2, . . . , 0.5}, T u = {0.5, 0.9}, and L out = {0.01, 0.05, 0.1, . . . , 0.3}, with L in = 0.01).
FIG. 12 .
12The first (a) and second (b) Hessian eigenvalues for the CQFC network of two coupled OPOs, versus ωopt and xu. Other parameters are Lout = 0.1, Tu = 0.9, Lin = 0.01.
FIG. 13 .FIG. 14 .
1314The first (a) and second (b) Hessian eigenvalues for the CQFC network of two coupled OPOs, versus ωopt and Lout. Other parameters are xu = 0.2, Tu = 0.9, Lin = 0.01. The first (a) and second (b) Hessian eigenvalues for the CQFC network of two coupled OPOs, versus xu and Lout. Other parameters are ωopt/2π = 100 MHz, Tu = 0.9, Lin = 0.01.
FIG. 15 .
15Components of the first Hessian eigenvector for the CQFC network of two coupled OPOs, versus ωopt. The four curves show components corresponding to the phase variables (θp, θc, φ1, φ2), as indicated in the legend. The parameters are xu = 0.2, Tu = 0.9, Lin = 0.01, Lout = 0.1.
FIG. 16 .
16The optimized degree of squeezing, Q − (ωopt), for the CQFC network of two coupled OPOs, versus the standard deviation of phase uncertainty, σ phase . The four curves correspond to different values of Lout (Lout = {0.05, 0.10, 0.15, 0.20}), as indicated in the legend. Other parameters are ωopt/2π = 100 MHz, xu = 0.2, Tu = 0.9, Lin = 0.01. For each value of Lout, the plot shows the results computed using the Hessian eigenvalues (lines) along with the data computed via Monte Carlo averaging over a random distribution of phase values (circles).
FIG. 17 .
17The squeezing spectrum Q − (ω) for the optimal operation of both networks under the minimization of JB = P − (ωB) of Eq. (50). Each subplot shows four curves corresponding to the optimally operated CQFC network of two coupled OPOs for different values of ωB (ωB/2π = {25, 75, 50, 100} MHz), along with a curve corresponding to the optimally operated single OPO network for any value of ωB, as indicated in the legend. The transmission line losses are L tl = Lout = 0.01 (subplots (a), (b), (c)), L tl = Lout = 0.05 (subplots (d), (e), (f)), and L tl = Lout = 0.1 (subplots (g), (h), (i)). The upper limit on the scaled pump amplitude is xu = 0.1 (subplots (a), (d), (g)), xu = 0.2 (subplots (b), (e), (h)), and xu = 0.3 (subplots (c), (f), (i)). Other parameters are Tu = 0.9, L = Lin = 0.01.
TABLE I .
IParameters of the single OPO network.Parameter
Type
Description
κ1
Positive
Leakage rate for the left cavity mirror
κ2
Positive
Leakage rate for the right cavity mirror
κ3
Positive
Leakage rate for intracavity losses
ω0
Real
Frequency detuning of the cavity
ξ
Complex
Pump amplitude of the OPO
θB
Real
Rotation angle of the beamsplitter
params=xi_p:complex;xi_c:complex;kappa_p_1:real;kappa_p_2:real;kappa_p_3:real;kappa_c_1:real;kappa_c_2:real;kappa_c_3:real;omega_p:real;omega_c:real;theta_1:real;theta_2:real;theta_3:real;phi_1:real;phi_2:realtheta=theta_3
In1
Out1
P1
phi=phi_1
In1
Out1
P2
phi=phi_2
vac
in5
vac
in7
vac
in6
vac
in1
vac
in2
vac
in3
vac
in4
lock
out5
lock
out2
lock
out3
loss
out7
loss
out6
loss
out4
output
out1
device=OPO_CQFN_1
module−name=OPO_CQFN_1
TABLE II .
IIParameters of the CQFC network of two coupled OPOs.Parameter
Type
Description
κp1
Positive
Leakage rate for the left mirror of the plant OPO cavity
κp2
Positive
Leakage rate for the right mirror of the plant OPO cavity
κp3
Positive
Leakage rate for losses in the plant OPO cavity
ωp
Real
Frequency detuning of the plant OPO cavity
ξp
Complex
Pump amplitude of the plant OPO
κc1
Positive
Leakage rate for the left mirror of the controller OPO cavity
κc2
Positive
Leakage rate for the right mirror of the controller OPO cavity
κc3
Positive
Leakage rate for losses in the controller OPO cavity
ωc
Real
Frequency detuning of the controller OPO cavity
ξc
Complex
Pump amplitude of the controller OPO
φ1
TABLE
table. The hybrid strategy (eight parallel searches using seven global algorithms) is described in the text.ωopt/2π
Algorithm
5 MHz
25 MHz
50 MHz
100 MHz
200 MHz
Sequential Least SQuares Programming (local only)
−4.270
−4.021
−3.396
−2.676
−1.809
Compass Search (local only)
−8.824
−7.540
−8.274
−8.113
−2.527
Compass Search guided by Monotonic Basin Hopping (Nstop = 5)
−9.105
−7.611
−7.037
−8.255
−7.540
Artificial Bee Colony (Ngen = 200)
−9.791
−8.945
−8.788
−8.427
−7.811
Covariance Matrix Adaptation Evolution Strategy (Ngen = 500)
−9.798
−8.869
−8.806
−8.423
−7.811
Differential Evolution, variant 1220 (Ngen = 800)
−9.805
−8.626
−8.809
−8.429
−7.813
Differential Evolution with p-best crossover (Ngen = 1000)
−9.805
−8.953
−8.808
−8.429
−7.813
Improved Harmony Search (Niter = 1000)
−9.805
−8.949
−8.808
−8.429
−7.813
Particle Swarm Optimization, variant 5 (Ngen = 1)
−9.219
−8.623
−7.090
−8.332
−7.617
Particle Swarm Optimization, variant 6 (Ngen = 1)
−8.811
−7.936
−7.403
−7.536
−5.932
Simple Genetic Algorithm (Ngen = 1000)
−9.805
−7.665
−8.809
−8.429
−7.813
Corana's Simulated Annealing (Niter = 20000)
−7.432
−5.015
−4.893
−6.110
−4.754
Hybrid strategy (eight parallel searches using seven global algorithms)
−9.805
−8.953
−8.809
−8.429
−7.813
TABLE IV .
IVThe sideband frequency ω opt /2π (in MHz), at which the high-ωopt regime commences, for the CQFC network of two coupled OPOs with Tu = 0.9, Lin = 0.01, and various values of Lout and xu. The accuracy of the reported values is limited by the sampling interval of 2 MHz.Lout
xu
0.01
0.05
0.10
0.15
0.20
0.25
0.30
0.1
8
16
22
28
34
42
48
0.2
10
18
26
32
40
48
56
0.3
14
24
30
38
46
56
66
0.4
20
30
36
44
54
68
90
TABLE V .
VThe Pearson correlation coefficient r(φ1, φ2), for the CQFC network of two coupled OPOs with Tu = 0.9, Lin = 0.01, and various values of Lout and xu.Lout
xu
0.01
0.05
0.10
0.15
0.20
0.25
0.1
0.575
0.438
0.436
0.354
0.275
0.223
0.2
0.381
0.474
0.471
0.244
0.169
-0.041
0.3
0.588
0.322
0.316
0.165
0.132
-0.121
0.4
0.423
0.373
0.189
0.172
-0.057
-0.002
TABLE VI .
VIThe Pearson correlation coefficient r(φ 1 , φ 2 ), for the CQFC network of two coupled OPOs with Tu = 0.9, Lin = 0.01, and various values of Lout and xu.Lout
xu
0.01
0.05
0.10
0.15
0.20
0.25
0.1
0.620
0.546
0.592
0.629
0.532
0.599
0.2
0.610
0.604
0.597
0.578
0.650
0.679
0.3
0.649
0.671
0.559
0.644
0.602
0.606
0.4
0.555
0.646
0.540
0.687
0.604
0.582
TABLE VII .
VIIThe best values of Q − (ωB) for ωB/2π = 100 MHz, for the CQFC network of two coupled OPOs and the single OPO network, obtained in optimizations with Tu = 0.9, L = Lin = 0.01, and various values of L tl = Lout and xu.CQFC network of two coupled OPOs
Lout
xu
0.01
0.05
0.10
0.15
0.20
0.25
0.1
-3.382
-2.688
-2.408
-2.157
-1.930
-1.724
0.2
-5.773
-4.937
-4.339
-3.829
-3.385
-2.994
0.3
-7.850
-6.857
-5.886
-5.109
-4.463
-3.913
0.4
-9.994
-8.441
-7.073
-6.049
-5.234
-4.559
Single OPO
L tl
xu
0.01
0.05
0.10
0.15
0.20
0.25
0.1
-1.428
-1.361
-1.277
-1.196
-1.115
-1.037
0.2
-2.843
-2.684
-2.493
-2.310
-2.134
-1.965
0.3
-4.248
-3.966
-3.638
-3.332
-3.047
-2.779
0.4
-5.637
-5.193
-4.696
-4.249
-3.845
-3.475
ACKNOWLEDGMENTSThis work was supported by the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia is a multi-mission laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the United States Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.
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"Thuy-Duong Vuong [email protected] \nStanford University\n\n",
"Elizabeth Yang [email protected] \nUC Berkeley\n\n"
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| We present a framework for speeding up the time it takes to sample from discrete distributions µ defined over subsets of size k of a ground set of n elements, in the regime where k is much smaller than n. We show that if one has access to estimates of marginals P S∼µ [i ∈ S], then the task of sampling from µ can be reduced to sampling from related distributions ν supported on size k subsets of a ground set of only n 1−α · poly(k) elements. Here, 1/α ∈ [1, k] is the parameter of entropic independence for µ. Further, our algorithm only requires sparsified distributions ν that are obtained by applying a sparse (mostly 0) external field to µ, an operation that for many distributions µ of interest, retains algorithmic tractability of sampling from ν. This phenomenon, which we dub domain sparsification, allows us to pay a one-time cost of estimating the marginals of µ, and in return reduce the amortized cost needed to produce many samples from the distribution µ, as is often needed in upstream tasks such as counting and inference.For a wide range of distributions where α = Ω(1), our result reduces the domain size, and as a corollary, the cost-per-sample, by a poly(n) factor. Examples include monomers in a monomer-dimer system, non-symmetric determinantal point processes, and partitionconstrained Strongly Rayleigh measures. Our work significantly extends the reach of prior work of Anari and Dereziński who obtained domain sparsification for distributions with a logconcave generating polynomial (corresponding to α = 1). As a corollary of our new analysis techniques, we also obtain a less stringent requirement on the accuracy of marginal estimates even for the case of log-concave polynomials; roughly speaking, we show that constant-factor approximation is enough for domain sparsification, improving over O(1/k) relative error established in prior work. | 10.4230/lipics.itcs.2022.5 | [
"https://arxiv.org/pdf/2109.06442v2.pdf"
]
| 237,502,872 | 2109.06442 | cd001fa12f73935784f964069c0f812efa70ab03 |
Domain Sparsification of Discrete Distributions using Entropic Independence
15 Sep 2021
Nima Anari [email protected]
Stanford University
Michał Dereziński
UC Berkeley
Thuy-Duong Vuong [email protected]
Stanford University
Elizabeth Yang [email protected]
UC Berkeley
Domain Sparsification of Discrete Distributions using Entropic Independence
15 Sep 2021
We present a framework for speeding up the time it takes to sample from discrete distributions µ defined over subsets of size k of a ground set of n elements, in the regime where k is much smaller than n. We show that if one has access to estimates of marginals P S∼µ [i ∈ S], then the task of sampling from µ can be reduced to sampling from related distributions ν supported on size k subsets of a ground set of only n 1−α · poly(k) elements. Here, 1/α ∈ [1, k] is the parameter of entropic independence for µ. Further, our algorithm only requires sparsified distributions ν that are obtained by applying a sparse (mostly 0) external field to µ, an operation that for many distributions µ of interest, retains algorithmic tractability of sampling from ν. This phenomenon, which we dub domain sparsification, allows us to pay a one-time cost of estimating the marginals of µ, and in return reduce the amortized cost needed to produce many samples from the distribution µ, as is often needed in upstream tasks such as counting and inference.For a wide range of distributions where α = Ω(1), our result reduces the domain size, and as a corollary, the cost-per-sample, by a poly(n) factor. Examples include monomers in a monomer-dimer system, non-symmetric determinantal point processes, and partitionconstrained Strongly Rayleigh measures. Our work significantly extends the reach of prior work of Anari and Dereziński who obtained domain sparsification for distributions with a logconcave generating polynomial (corresponding to α = 1). As a corollary of our new analysis techniques, we also obtain a less stringent requirement on the accuracy of marginal estimates even for the case of log-concave polynomials; roughly speaking, we show that constant-factor approximation is enough for domain sparsification, improving over O(1/k) relative error established in prior work.
Introduction
Sparsification has been a crucial idea in designing many fast algorithms; famous examples include cut or spectral graph sparsifiers [ST11] and dimension reduction using sparsified Johnson-Lindenstrauss transforms [DKS10]. In this work, we address the question of sparsifying discrete distributions, with the goal of speeding up the fundamental task associated with distributions: sampling from them.
As an illustrative example building towards our notion of sparsification for distributions, consider the task of sampling a (uniformly) random edge in a graph. Suppose that we have a graph on n non-isolated vertices, and we are allowed to make adjacency queries. How many of the n vertices do we have to "look at" before we observe both endpoints of some edge? The answer to this question depends on the structure of the graph; for a star graph, where a single vertex is connected to the other n − 1, we would have to find this central vertex to have any chance of observing an edge; so no amount of smart guessing can result in looking at ≪ n vertices. However, for regular graphs, where every vertex has the same degree, because of the Birthday Paradox phenomenon, it is enough to look at a sample of O( √ n) vertices picked uniformly at random to observe an edge between two of them with overwhelming probability. The bound of O( √ n) is indeed the best possible, since in a random perfect matching (degree 1 regular graph), the best and only sensible strategy is to pick vertices at random. This curious phenomenon generalizes to hypergraphs as well. On a k-uniform hypergraph, with hyperedges representing sets of k vertices, to observe a hyperedge one has to generally look at ≃ n vertices in the worst case. But on regular hypergraphs, a substantially smaller sample, namely ≃ n 1−1/k many vertices, will contain a hyperedge with high probability [see, e.g., Suz+06, for forms of Birthday Paradox related to k-sets and k-collisions]. Notice that this improvement quickly deteriorates as k gets large, and becomes meaningless as soon as k ≃ log n.
When can the bound of n 1−1/k for regular hypergraphs be improved, ideally to a polynomially small fraction of the n vertices, even for k ≫ 1? Moreover, suppose that instead of desiring just one of the hyperedges, we want to extract an (approximately) uniformly random hyperedge. Can we still produce a hyperedge following this distribution, by only looking at a small subset of vertices? In a nutshell, how small of a (random) vertex set can we look at in order to have a distributionally representative "sparsification" of the entire hypergraph?
In this work, we tie the answer to these questions to notions of high-dimensional expansion, specifically the notion of entropic independence introduced by Anari, Jain, Koehler, Pham, and Vuong [Ana+21]. To every measure, a.k.a. weighted hypergraph, on size k subsets of {1, . . . , n} denoted by µ : ( [n] k ) → R ≥0 , one can associate a parameter of entropic independence 1/α ∈ [1, k], defined formally in Section 2. A larger α corresponds to better high-dimensional expansion. We show that the hypergraph defined by µ, while being truly k-uniform, behaves almost as if it was (1/α)-uniform: informally, we can "sparsify" this hypergraph by looking at only n 1−α · poly(k) vertices, under some "regularity assumptions." To avoid confusion with other classical concepts of graph and hypergraph sparsification, which primarily keep the vertex set while deleting a subset of the edges, we call this type of sparsification domain sparsification.
A long line of recent works have obtained breakthroughs in sampling and counting by viewing combinatorial distributions as (weighted) hypergraphs and studying notions of high-dimensional expansion for them [Ana+19; CGM19; AL20; ALO20; CLV20; GM20; Ana+20; Che+21b; Fen+21; Ali+21; Liu21; Bla+21; JPV21; Ana+21; ALO21; Che+21a]. A central theme in all of the aforementioned works is the establishment of some form of high-dimensional expansion for a hypergraph encoding the probability distribution of interest. At a high-level, these notions can be viewed as measures of proximity to independent/product distributions. Sampling from distributions extremely close to product distributions is roughly as easy as sampling i.i.d. from the marginal distribution over single elements; it is no surprise then, that for distributions with limited correlations, knowledge of marginals can boost sampling time. This is what we formally establish in this work.
Our main result applies to distributions that have entropic independence [Ana+21], a notion stronger than spectral independence [ALO20], but weaker than fractional log-concavity and sector-stability [Ali+21]. Roughly speaking, a background measures µ over k-sized sets is entropically independent, if for any (randomly chosen) set S, the relative entropy of a uniformly random element of S is at most 1/αk fraction of the relative entropy of S, where we usually take α = Ω(1). The main intuition leading to our results is that high correlations in such distributions must be limited to small groups of elements. By sampling enough many elements from the domain, we cover these correlated groups.
Similar to graph sparsification, in domain sparsification we need to reweigh the sparsified object. This is achieved by the standard operation of applying an external field. For a weight vector λ ∈ R n ≥0 , the λ-external field applied to µ is the distribution λ ⋆ µ defined by
λ ⋆ µ(S) ∝ µ(S) ∏ i∈S λ i . Theorem 1 (Informal). Let µ : ( [n] k ) → R ≥0 be (1/α)-entropically independent.
Suppose that we have access to estimates p 1 , . . . , p n of the marginals and an oracle that can produce i.i.d. samples i ∈ [n] with P[i] ∝ p i ; suppose that our estimates satisfy p 1 + · · · + p n = k and
p i ≥ Ω(P[i ∈ S])
for all i. Then we can produce a random sparse external field λ ∈ R n ≥0 with at most n 1−α · poly(k) nonzero entries, in time n 1−α · poly(k), such that a random sample S of λ ⋆ µ approximately follows the distribution defined by µ.
Theorem 1 follows directly from Propositions 24 and 25 and Lemma 26. We outline our techniques in Section 1.3.
Notice that we do not need degree-regularity of µ, which would be equivalent to P S∼µ [i ∈ S] being exactly the same for all i. Instead, it is enough to just have an estimate of these marginals, a weaker condition than regularity. This is because instead of sampling a subset of vertices uniformly at random, we can sample a biased subset of vertices, with probability biases defined by the marginals, and domain sparsification will still work.
Prior to our work, domain sparsification was known for distributions with log-concave generating polynomials (the case of α = 1) [AD20], based on techniques inspired by earlier algorithms for sampling from determinantal point processes (an even narrower class) [Der19; DCV19]. All of these distributions satisfy forms of negative dependence [BBL09; AD20] that were crucial in obtaining domain sparsification for them. Our work significantly extends the reach of domain sparsification beyond these classes; as we will see, for any distribution µ, we have α ≥ 1/k, and as a simple corollary we get nontrivial domain sparsification for any distribution µ as long as k = O(1), a result which appears to be nontrivial on its own.
The main application of domain sparsification is in accelerating the time it takes to produce multiple samples from a distribution µ. Suppose that an algorithm A can produce (approximate) samples from a distribution µ and any distribution obtained from it by an external field, in time T(n, k), which usually depends polynomially on n. 1 Then after a preprocessing step, where we use A to estimate the marginals of µ, we can produce new samples in time T(n 1−α · poly(k), k) per sample, which is polynomially smaller than T(n, k), as long as k is smaller than some poly(n) threshold. Notice that the preprocessing step has to be done only once, and its cost gets amortized when we are interested in obtaining multiple samples from µ. A careful implementation, directly adapted from what was done for log-concave polynomials by Anari and Dereziński [AD20], can bootstrap domain sparsification with estimation of marginals to complete the preprocessing step in roughly ≃ T(n, k) + n · poly(k, log n) · T(n 1−α · poly(k), k) time.
Corollary 2 (Informal, adapted from [AD20]). Suppose that we have an algorithm A that can produce approximate samples from any external field λ applied to µ in time T(m, k), where m is the sparsity of λ. Then we can produce the marginal estimates p i and the i.i.d. sampling oracle required in Theorem 1 in time O T(n, k) + n · poly(k, log n) · T(n 1−α · poly(k), k) .
Further, for any desired t, we can produce t i.i.d. approximate samples from µ in time
O T(n, k) + max {t, n · poly(k, log n)} · T(n 1−α · poly(k), k) .
Sampling is often used to solve the problem of approximate counting, that is computing the partition function ∑ S µ(S).
To obtain an ǫ-relative error approximation, known reductions between counting and sampling [JVV86] introduce at least a multiplicative factor of 1/ǫ 2 to the sampling time. Directly adapting the same technique for log-concave polynomials [AD20] and combining with our new domain sparsification result, we obtain an ǫ-relative error of the counts in time ≃ T(n, k) + max {n, 1/ǫ 2 } · poly(k, log n) · T(n 1−α · poly(k), k). Notice that here 1/ǫ 2 is multiplied by the term T(n 1−α · poly(k), k) that can be substantially smaller than T(n, k); as a result, we can get a substantially improved running time for the high-precision regime where ǫ is inverse-polynomially small.
Corollary 3 (Informal, adapted from [AD20]). Suppose that we have an algorithm A that can produce approximate samples from any external field λ applied to µ in time T(m, k), where m is the sparsity of λ. Then we can compute an ǫ relative error approximation of ∑ S µ(S) in time O T(n, k) + max {n, 1/ǫ 2 } · poly(k, log n)T(n 1−α · poly(k), k) .
Remark 4. For many applications, we can derive entropic independence of µ from a stronger property called fractional log-concavity [Ali+21; Ana+21]. For an α-fractionally log-concave distribution, recent work of Anari, Jain, Koehler, Pham, and Vuong [Ana+21] established Modified Log-Sobolev Inequalities for natural (multi-step) down-up random walks. For simplicity of exposition, assume that 1/α ∈ Z and α = Ω(1). Then, these random walks produce approximate samples from µ in the following number of steps:
O k 1/α · log log 1 P µ [S 0 ] ,
where S 0 is the starting point of the random walk. Further, each step of the random walk requires querying µ at n 1/α points, leading to a total runtime of
O (kn) 1/α · log log 1 P µ [S 0 ] .
In most settings, such as when the bit-complexity of µ is bounded by poly(n), the extra log log(1/ P µ [S 0 ]) can be safely ignored, as long as we make sure S 0 is in the support. So one can think of this Markov chain as an algorithm A that, up to this initial step of finding a suitable starting point, satisfies T(n, k) =Õ (nk) 1/α .
Note that for this choice of the algorithm A and running time T, the bounds in Corollaries 2 and 3 simplify as n · poly(k, log n) · T(n 1−α · poly(k), k) ≃ poly(k, log n) · T(n, k).
However, our results apply to any choice of a base sampling algorithm A.
A challenging part of obtaining our results is the lack of negative dependence inequalities, which were used by the prior work of Anari and Dereziński [AD20]. These negative dependence inequalities result in domain sparsification with sparsified domain size solely depending on k, with no dependence on n. We show in Section 4 that our analysis of our domain sparsification scheme is tight. An intriguing question is if we can find other domain sparsification schemes, perhaps using higher-order marginals, that sparsify domains to size poly(k, log n)? We make the following conjecture.
Conjecture 5 (Informal). Let µ be an α-fractionally-log-concave distribution for some α = Ω(1). Given access to estimates for high-order marginals of the form P S∼µ [T ⊆ S] for all T of size ℓ ≃ 1/α, and an oracle that produces i.i.d. samples from these marginals, there is a domain sparsification scheme for µ which reduces the domain size to only poly(k).
Despite the attractiveness of a bound independent of n, we give evidence that obtaining these domain sparsification schemes requires entirely new ideas; we show in Section 4 that if we replace fractional log-concavity by entropic independence (which is sufficient for our main result, Theorem 1) in the above conjecture, the conjecture becomes false.
Applications
Here we mention examples of distributions to which our results can be applied beyond those covered by prior work of Anari and Dereziński [AD20]. Our examples satisfy fractional logconcavity [Ali+21] which entails both entropic independence [Ana+21], and the existence of the base sampling algorithm A.
For a distribution µ : ( [n] k ) → R ≥0 we define the generating polynomial g µ to be
g µ (z 1 , . . . , z n ) := ∑ S µ(S) ∏ i∈S z i .
We say that the distribution µ or the polynomial g µ is α-fractionally-log-concave for some parameter α ≤ 1 if log g µ (z α 1 , . . . , z α n ) is concave as a function over the positive orthant (z 1 , . . . , z n ) ∈ R n ≥0 . Notice that any multi-affine homogeneous polynomial with nonnegative coefficients is the generating polynomial of a distribution µ. Throughout the paper, we often equate these polynomials with the distributions they represent, and freely talk about fractional log-concavity of either the generating polynomial or the distribution. For more details, see [Ali+21].
Example 6. If g is a degree-k homogeneous multi-affine polynomial, then it is 1 k -log-concave. Every monomial ∏ i∈S z 1/k i is concave, since by Hölder's inequality
∏ i∈S (λz i + (1 − λ)y i ) ≥ λ ∏ i∈S z 1/k i + (1 − λ) ∏ i∈S y 1/k i k . Now, g(z 1/k 1 , . . . , z 1/k n ) is concave as (weighted)-sum of concave functions ∏ i∈S z 1/k i .
Thus is also log-concave, as log is a monotone and concave function.
Example 7. We present another toy class of α-fractionally-log-concave polynomials that provides some intuition despite not having many applications. Let µ be an α-fractionally-log-concave polynomial over the variables z 1 , . . . , z n . If we replace each z i with the monomial ∏ m j=1 z [Ali+21]. For example, if the starting distribution µ is the uniform distribution over bases of a matroid, then α = 1, and the resulting distribution will be 1/m-fractionally log-concave.
(j) i , we obtain a degree mk, α m -fractionally-log-concave polynomial over the variables {z (j) i | i ∈ [n], j ∈ [m]}
Notice that if we normalize this blown-up polynomial to convert it into to a distribution, for any i ∈ [n], the elements i (1) , . . . , i (m) are all perfectly correlated. On the other hand, if i = j, any two elements i (m i ) , j (m j ) inherit the correlations from the log-concave distribution.
Example 8. Let G be a graph and k ∈ N. For each set S ⊆ ( V 2k ), set µ(S) to be proportional to the number of perfect matchings on S. Sampling from µ allow us to approximately count the number of k-matching, i.e., matchings using k edges. [Ali+21] proved that for any value of k, this distribution is fractionally log-concave with α ≥ 1/4. Not all choices of G result in efficient sampling algorithms. The implementation of each iteration of the Markov chain involves counting perfect matchings over S ⊆ V, and we do not have a poly(k) time algorithm for counting matchings in general graphs. We thus only consider downward closed graph families with an FPRAS for counting perfect matchings, e.g., bipartite graphs [JSV04], planar graphs [Kas67], certain minor-free graphs [EV19], and small genus graphs [GL99]. Our main results imply that as long as we estimate the probability of every vertex being part of a random k-matching, we can reduce the task of sampling k-matchings on an n vertex graph to graphs with only n 3/4 · poly(k) many vertices.
Example 9. Let L be a nonsymmetric positive semidefinite matrix, i.e., an n × n matrix L that satisfies L + L ⊺ 0. Then, the nonsymmetric k-determinantal point process [see, e.g., Gar+19; Gar+20; AV21] with kernel L, defined by
µ(S) = det(L S,S ) for all S ∈ ( [n] k ) is fractionally log-concave with α ≥ 1/4 [Ali+21].
Example 10. Suppose that we start with a measure µ 0 on ( [n] k ) that is Strongly Rayleigh [see BBL09, for definition], such as a (symmetric) determinant point process, or the uniform distribution over spanning trees of a graph. Suppose that we partition the ground set into a constant number
c = O(1) of parts: [n] = A 1 ∪ A 2 ∪ ·A c , and fix cardinalities k 1 , . . . , k c ∈ Z ≥0 , with k 1 + · · · + k c = k.
Then the partition-constrained version of µ 0 can be defined as
µ(S) ∝ µ 0 (S) · 1 [|S ∩ A i | = k i for i = 1, . . . , c] .
As long as c = O(1), this distribution µ will be Ω(1)-fractionally-log-concave [Ali+21]. For some discussion of partition-constrained Strongly Rayleigh measures, see [Cel+16].
Related Work
Log-Concavity
Log-concavity has been a well-studied concept in continuous sampling since it captures many common distributions like uniform distributions over convex bodies, and Gaussian distributions. Discrete notions of log-concavity we work with in this paper have been introduced by [Gur09; AOV18; BH19; Ana+19]. The formulation of [Ana+19] is what we refer to in this paper as "logconcave," and is motivated by examples such as the uniform distribution over bases of a matroid and the special subcase of the uniform distribution over spanning trees.
We have a nearly complete picture for MCMC-based sampling algorithms for homogeneous log-concave distributions. For degree-k distributions, Anari, Liu, Oveis Gharan, and Vinzant [Ana+19] analyzed the down-up walk, which occurs between sets of size k and sets of size (k − 1); in the case of matroid bases, this walk is also known as a form of the "basis exchange walk." Furthermore, Cryan, Guo, and Mousa [CGM19] proved a Modified Log-Sobolev Inequality (MLSI) for this walk, and Anari, Liu, Oveis Gharan, Vinzant, and Vuong [Ana+20] further reduced the runtime of sampling by analyzing a warm start to the down-up walk algorithm. Most recently, Anari and Dereziński [AD20] devised an algorithm for sampling a log-concave distribution when we are given the single-element marginals; we will elaborate upon their contributions more in Section 3.3, where we compare their algorithm to ours.
Intermediate Sampling and Determinantal Point Processes
A class of domain sparsification algorithms, related to the algorithms we used here, called intermediate sampling was first proposed by [DWH18;Der19] CDV20], and the approach was extended to DPPs over continuous domains by [DWH19]. Crucially, these algorithms take advantage of the additional structure in DPPs, to enable distortion-free intermediate sampling: instead of using a Markov chain, this uses rejection sampling to draw exactly from the target distribution. This approach is not possible more generally, since µ typically does not have a tractable partition function. However, [AD20] showed that the original analysis of distortion-free intermediate sampling can largely be retained for distributions with log-concave generating polynomials, as long as we switch to a Markov chain implementation.
On the other hand, in this work, we largely abandon the original analysis in favor of a new one which is specific to the Markov chain and requires less precision in marginal estimates. As a result, we show that the preprocessing cost for Markov chain intermediate sampling is substantially smaller than for distortion-free intermediate sampling. This leads to significant improvements in time complexity even for DPPs, e.g., by reducing the preprocessing cost in [DCV19] fromÕ(nk 6 + k 9 ) toÕ(nk 2 + k 3 ), whereÕ hides polylogarithmic terms.
Overview of Techniques
Given an entropically independent distribution µ : ( [n] k ) → R ≥0 , we first preprocess it using isotropic transformation, which is further detailed in Section 3.1. This converts our distribution µ into a related distribution µ ′ whose single element marginals P S ′ ∼µ ′ [i ∈ S ′ ] are approximately uniform. We prove various properties of µ ′ in Proposition 24, including the fact that µ ′ has ground set size linear in n. This preprocessing step may be of independent interest for other discrete sampling problems outside of entropically independent and fractionally log-concave distributions.
After this step, we may assume our distribution µ has already undergone isotropic transformation. Next, we design a Markov chain M t µ that has µ as its stationary distribution, where taking a step requires sampling from a sparsified distribution ν. We refer to this algorithm as Markov Chain Intermediate Sampling. The benefit of this Markov chain over other natural Markov chains (e.g., down-up random walks [see, e.g., Ali+21]) is that each step requires paying attention only to a subset of elements as opposed to all. We show that n 1−α · poly(k) size is sufficient to ensure that M t µ mixes rapidly. Specifically, in Lemma 26, we prove that a single step of M t µ from any S ∈ supp(µ) satisfies P(S, ·) − µ ≤ 1 4 . The mixing time analysis of M t µ is novel and improves upon the methods used in [AD20]. The improvement is discussed and concretely illustrated with an example distribution in Section 3.3. The proof of Lemma 26 relies heavily on new negative dependence inequalities that are "average-case" rather than "worst-case", since the worst-case inequalities simply do not hold for fractionally-logconcave distributions. We show that these "average-case" inequalities suffice for fast mixing of M t µ , thus opening the intermediate sampling framework to wider families of distributions.
Acknowledgements
Preliminaries
We use [n] to denote the set {1, . . . , n}. For a set S, we use ( S k ) to denote the family of subsets of S of size k. For a distribution µ, we use X ∼ µ to denote that X is a random variable distributed according to µ. For a set U, we abuse notation and let X ∼ U denote X following the uniform distribution over U.
For a distribution µ over size k sets and a set T of size potentially larger than k, we abuse notation and use µ(T) to denote:
µ(T) := ∑ S⊆T µ(S).
Markov Chains and Mixing Time
Definition 11. Let µ, ν be two discrete probability distributions over the same event space Ω. The total variation distance, or TV-distance, between µ and ν is given by
µ − ν TV = 1 2 ∑ ω∈Ω |µ(ω) − ν(ω)|
Definition 12. Let P be an ergodic Markov chain on a finite state space Ω and let µ denote its (unique) stationary distribution. For any probability distribution ν on Ω and ǫ ∈ (0, 1), we define
t mix (P, ν, ǫ) = min {t ≥ 0 | νP t − µ TV ≤ ǫ},
and
t mix (P, ǫ) = max {t mix (P, 1 x , ǫ) | x ∈ Ω} ,
where 1 x is the point mass distribution supported on x.
We will drop P and ν if they are clear from context. Moreover, if we do not specify ǫ, then it is set to 1/4. This is because the growth of t mix (P, ǫ) is at most logarithmic in 1/ǫ (cf. [LP17]).
The modified log-Sobolev constant of a Markov chain, defined next, provides control on its mixing time. For a detailed coverage see [LP17].
Definition 13. Let P denote the transition matrix of an ergodic, reversible Markov chain on Ω with stationary distribution µ.
• The Dirichlet form of P is defined for f , g ∈ Ω → R by E P ( f , g) = f , (I − P)g µ = (I − P) f , g µ .
• The modified log-Sobolev constant of P is defined to be
ρ 0 (P) = inf E P ( f , log f ) 2 · Ent µ [ f ] : f : Ω → R ≥0 , Ent µ [ f ] = 0 , where Ent µ [ f ] = E µ [ f log f ] − E µ [ f ] log E µ [ f ].
Note that, by rescaling, the infimum may be restricted to functions f :
Ω → R ≥0 satisfying Ent µ [ f ] = 0 and E µ [ f ] = 1.
The relationship between the modified log-Sobolev constant and mixing times is captured by the following well-known lemma.
Lemma 14 (cf. [BT06]). Let P denote the transition matrix of an ergodic, reversible Markov chain on Ω with stationary distribution µ and let ρ 0 (P) denote its modified log-Sobolev constant. Then, for any probability distribution ν on Ω and for any ǫ ∈ (0, 1)
t mix (P, ν, ǫ) ≤ ρ 0 (P) −1 · max x∈Ω log log ν(x) µ(x) + log 1 2ǫ 2 .
In particular, t mix (P, ǫ) ≤ ρ 0 (P) −1 · log log 1 min x∈Ω µ(x) + log 1 2ǫ 2 .
Theorem 15 (cf. [LP17]). If an irreducible aperiodic Markov chain with stationary distribution µ and transition matrix P satisfies P t (S, ·) − µ TV ≤ 1/4 for all S ∈ supp(µ) and some t ≥ 1, then for any ǫ ∈ (0, 1/4], t mix (P, ǫ) ≤ t log(1/ǫ).
Fractional Log-Concavity and Entropic Independence
We recall the notion of fractional log-concavity [Ali+21] and entropic independence [Ana+21].
Definition 16 ([Ali+21]). A probability distribution µ : ( [n] k ) → R ≥0 is α-fractionally-log-concave if g µ (z α 1 , . . . , z α n )
is log-concave for z 1 , . . . , z n ∈ R n ≥0 . If α = 1, we say µ is log-concave. To define entropic independence we need the definition of the "down" operator.
Definition 17 (Down Operator). For ℓ ≤ k define the row-stochastic matrix D k→ℓ ∈ R
( [n] k )×( [n] ℓ ) ≥0 by D k→ℓ (S, T) = 0 if T ⊆ S 1 ( k ℓ )
otherwise.
Note that for a distribution µ on size k sets, µD k→ℓ will be a distribution on size ℓ sets. In particular, µD k→1 will be the vector of normalized marginals of µ: (P[i ∈ S]/k) i∈ [n] .
Definition 18 ([Ana+21, Definition 2, Theorem 3]). A probability distribution µ :
( [n] k ) → R ≥0 is (1/α)-entropically-independent for α ∈ (0, 1], if for all probability distributions ν on ( [n] k ), D KL (νD k→1 µD k→1 ) ≤ 1 αk D KL (ν µ).
Or equivalently,
∀(z 1 , . . . , z n ) ∈ R n ≥0 : g µ (z α 1 , . . . , z α n ) 1/kα ≤ n ∑ i=1 p i z i ,(1)
where p = (p 1 , . . . , p n ) := µD k→1 .
We note that α-fractionally log concavity implies (1/α)-entropic independence [Ana+21, Theorem 3].
An important part of our sparsification scheme is a process to transform distributions into a "nearisotropic" position (defined as having roughly equal marginals) by subdividing the elements of the ground set. More precisely, let µ be a distribution generated by g µ (z 1 , . . . , z n ), then the distribution µ ′ obtained by subdividing z i into t i copies has generating polynomial
g µ ′ (z (1) 1 , . . . , z (t n ) n ) = g µ z (1) 1 + . . . + z (t 1 ) 1 t 1 , . . . , z (1) n + . . . + z (t n ) n t n .
Subdivision preserves both entropic independence and fractional log-concavity.
Proposition 19. If µ is (1/α)-entropically-independent distribution then µ ′ is also (1/α)-entropic independence.
Proposition 20. If µ is α-fractionally-log-concave distribution then µ ′ is also α-fractionally-log-concave.
We leave the proofs to Section 5.
Intermediate Sampling Algorithm
Isotropic Transformation
We define, similar to [AD20], a distribution µ to be isotropic if for all i ∈ [n], the marginal probability P S∼µ [i ∈ S] is k n . We remark that this is only similar in name and spirit, but different in nature, to the analogous notion of isotropy for continuous distributions; the latter is defined based on the covariance matrix of the distribution, while the former is defined based on marginals. In this paper, isotropy captures "uniformity" over the elements of [n] in their marginal probabilities. Below we discuss a subdivision process [AD20] that transforms an arbitrary distribution µ over
( [n]
k ) into a distribution µ ′ that is nearly-isotropic. Definition 21. Let µ : ( n k ) → R ≥0 be an arbitrary probability distribution, and assume that we have estimates p 1 , . . . , p n of the marginals with p 1 + · · · + p n = k and p i ≥ Ω(P S∼µ [i ∈ S]) for all i. Let t i := ⌈ n k p i ⌉. We will create a new distribution out of µ: For each i ∈ [n], create t i copies of the element i and let the collection of all these copies be the new ground set:
U = n i=1 {i (1) , . . . , i (t i ) }. Define the following distribution µ ′ : ( U k ) → R ≥0 from µ: µ ′ i (j 1 ) 1 , . . . , i (j k ) k := µ({i 1 , . . . , i k }) t 1 · · · t k .
We call µ ′ the isotropic transformation of µ. Another way we can think of µ ′ is that to produce a sample from it, we can first generate a sample {i 1 , . . . , i k } from µ, and then choose a copy i (j m ) m for each element i m uniformly at random.
Remark 22. We note that subdivision or isotropic transformation and external fields behave well together. In particular, a sample from an external field λ applied to µ ′ can be obtained by first applying an appropriate external field (summing the field values over duplicate elements) to µ and then replacing each element with a copy of it with probability proportional to λ. In fact, subdivision is mostly a tool for analysis. In our algorithms, we never have to formally perform subdivision, and we can just sample from distributions defined as λ ⋆ µ for appropriate external fields λ.
Remark 23. To obtain the estimates {p i } for all i, we can apply the proof of [AD20, Lem. 23], with ǫ constant, rather than ǫ = O( 1 k ). This provides a running time reduction for our preprocessing step even in the case of log-concave polynomials.
There are three desirable properties of µ ′ we need to establish for subdivision to be an effective preprocessing step. The first is that subdivision preserves (1/α)-entropic independence, which is shown in Proposition 19. The next is for the marginals P S∼µ ′ [i (j) ∈ S] to all be close to k |U| for all i (j) ∈ U; in other words, µ ′ is actually close to being isotropic. The last is for |U| ≤ O(n), so if we ran a sampling algorithm on µ ′ , the increased size of our ground set does not accidentally inflate our desired asymptotic running times. We remark however, that this last concern can be avoided by simply not running the sampling algorithm on the subdivided distribution, but rather on λ ⋆ µ for an appropriate external field λ.
Proposition 24. Let µ : ( n k ) → R ≥0 , and let µ ′ : ( U k ) → R ≥0 be the subdivided distribution from Definition 21. The following hold for µ ′ :
1. Near-isotropy: For all i (j) ∈ U, the marginal P S∼µ ′ [i (j) ∈ S] ≤ O(k/ |U|).
Linear ground set size: The number of elements |U| ≤ O(n).
Proof. First, we verify that |U| is at most O(n):
|U| = n ∑ i=1 t i ≤ n ∑ i=1 1 + n k p i = n + n k n ∑ i=1 p i = 2n.
Next, we check that for any i (j) , the marginal probabilities P S∼µ ′ [i (j) ∈ S] are at most O(k/ |U|).
Here, we interpret the sampling from µ ′ as first sampling from µ, and then choosing a copy for each element.
P S∼µ ′ [i (j) ∈ S] = ∑ S∋i P[we chose copy j | we sampled S from µ] · P[we sampled S from µ] = ∑ S∋i 1 t i · µ(S) = 1 t i ∑ S∋i µ(S) = 1 t i · P S∼µ [i ∈ S].
Since
t i ≥ n k p i ≥ n k · Ω(P S∼µ [i ∈ S]), we get that P S∼µ ′ [i (j) ∈ S] ≤ O P S∼µ [i ∈ S] n k · P S∼µ [i ∈ S] = O(k/n) ≤ O(k/ |U|).
Domain Sparsification via Markov Chain Intermediate Sampling
Here, we first describe, for any general distributions µ, a Markov chain based on generating intermediate samples T ⊆ [n], that mixes to µ. Then, in Lemma 26 and Proposition 27, we state our main result that for distributions µ which are (1/α)-entropically independent and nearlyisotropic, the size of T only needs to be n 1−α · poly(k) for the mixing to occur in one step.
Take distribution µ : ( [n] k ) → R ≥0 , and consider the following Markov chain M t µ defined for any positive integer t, with the state space supp(µ). Starting from S 0 ∈ supp(µ), one step of the chain is given by:
1. Sample T ∼ ( [n]\S 0 t−k ). 2. Downsample S 1 ∼ µ S 0 ∪T , where µ S 0 ∪T is µ restricted to S 0 ∪ T, a.
k.a. 1 S 0 ∪T ⋆ µ, and update S 0 to be S 1 .
We note that the requirement S 0 ∈ supp(µ) is not strictly necessary for this step to be defined.
Proposition 25. For any distribution µ : ( [n]
k ) → R ≥0 , the chain M t µ for t ≥ 2k is irreducible, aperiodic and has stationary distribution µ.
Proof. Let P denote the transition probability matrix of M t µ . Since t ≥ 2k, for any S, S ′ ∈ supp(µ), there is a positive probability that we sample T ⊇ S ∪ S ′ . Thus, we have P(S, S ′ ) > 0, and P is both irreducible and aperiodic.
To derive the stationary distribution, suppose that we perform one step of the chain starting from S 0 ∼ µ. We first derive the distribution of the intermediate set R := S 0 ∪ T.
For anyR ∈ ( [n]
t ), the probability of samplingR for the intermediate set R is
P[R =R] = ∑ S 0 ∈(R k ) µ(S 0 ) · P[T =R \ S 0 ] = 1 ( n−k t−k ) · µ(R)
For anyS 1 ∈ supp(µ), the probability of samplingS 1 is
P[S 1 =S 1 ] = ∑ R∈( [n] r ) P[S 1 =S 1 | R =R] P[R =R] = ∑ R∈( [n] t ):R⊇S 1 µ(S 1 ) µ(R) · 1 ( n−k t−k ) · µ(R) = µ(S 1 ) ∑ (R\S 1 )∈( [n]\S 1 t−k ) 1 ( n−k t−k ) = µ(S 1 )
Above, we summed over allR that contain the target setS 1 .
The following lemma is the key to analyzing the sampling algorithm, since it quantifies the decrease in TV distance after running one step of M t µ . It will be proven in Section 3.4. Proof. The bound on TV distance follows via
Lemma 26.
P(S 0 , ·) − µ TV = ∑ S∈( [n] k ):P[S 1 =S]<µ(S) (µ(S) − P[S 1 = S]) ≤ ǫ ∑ S∈( [n] k ):P[S 1 =S]<µ(S) µ(S) ≤ ǫ
The mixing time bound follows from Theorem 15.
We have shown that M t µ is fast mixing (in fact, mixing in one step for appropriately large t). Next, we show that for a wide class of distributions, namely, the class of α-fractionally-log-concave distributions with α = Ω(1) [see Ali+21, for examples], each step of M t µ can be implemented in poly(n, k) time via a local Markov chain, i.e., the (muti-step) down-up random walk [Ali+21, Def. 1].
Remark 28 (Runtime analysis). Suppose µ is α-fractionally-log-concave, and we start with S (0) 0 such that µ(S (0) 0 ) ≥ 2 −n c for some constant c > 1 and we run the chain for τ steps. Then with probability ≥ 1 − τ2 −n , for all 0 ≤ i ≤ τ, the i th -step starting point, denoted by S +2ni) . This can be shown via induction on i. Conditioned on µ(S
(i) 0 , satisfies µ(S (i) 0 ) ≥ 2 −(n c(i) 0 ) ≥ 2 −n c −2ni , we have P[µ(S (i+1) 0 ) ≤ 2 −(n c +2(i+1)n) ] = µ(S (0) 0 ∪ T) −1 ∑ S⊆(S (0) 0 ∪T):µ(S)≤2 −(n c +2(i+1)n) µ(S) ≤ (1) 2 −(n c +2(i+1)n) · 2 n µ(S (i) 0 ) ≤ 2 −n where in (1) we use the crude bound S ⊆ (S (0) 0 ∪ T) : µ(S) ≤ 2 −(n c +2(i+1)n) ≤ 2 n . Suppose that this good event happens, i.e. ∀i ∈ [0, τ] : µ(S (i) 0 ) ≥ 2 −n c −2ni
We observe that α-fractional-log-concavity is preserved by subdividing Proposition 20 and restricting to a subset of the ground set [Ali+21]. In the down-sampling step, we run the (multistep) down-up walk starting at S (i) 0 , and use [Ana+21, Thm. 4] to bound the runtime. To this end, we need to bound
E T∼( [n]\S (i) 0 t−k ) log 1 + log µ(S (i) 0 ∪ T) µ(S (i) 0 ) ≤ (1) log 1 + log E T∼( [n]\S (i) 0 t−k ) [ µ(S (i) 0 ∪ T) µ(S (i) 0 ) ] = log 1 + log 1 P[S 1 = S (i) 0 ] ≤ (2) log 1 + log 1 µ(S 1 = S (i) 0 )(1 − ǫ)
≤ (3) c log n + log τ + log log 1 1 − ǫ where (1) follows from Jensen's inequality for concave function f (x) = log(1 + log(x)) on [1, ∞), (2) from Lemma 26 and (3) from lower bound on µ(S (i) 0 ). The down-sampling then costs O (t − k) ⌈1/α⌉ k 1/α c log n + log τ + log log 1 1 − ǫ and the total runtime is
O τ(t − k) ⌈1/α⌉ k 1/α c log n + log τ + log log 1 1 − ǫ .
As a slight optimization, we can replace (t − k) ⌈1/α⌉ k 1/α with k ⌈1/α⌉ (t − k) 1/α when both µ and its complement µ comp are α-fractionally-log concave, by down-sampling from µ comp S 0 ∪T then output the complement as S 1 , where µ comp :
( [n] n−k ) → R ≥0 is the complement of µ, defined by µ comp ([n] \ S) = µ(S)∀S ∈ ( [n]
k ). In all important instances of α-fractionally-log-concavity, 1 α ∈ N and this optimization is unnecessary. The bound on total runtime can be simplified into O(n 1/α−1 poly(k, log 1 ǫ )).
Advantage Over Rejection Sampling
While we use a similar intermediate sampling framework as [AD20], our novel analysis of Markov chain intermediate sampling improves the runtime and applies to wider families of distributions. In order to fully understand the advantages realized by our intermediate sampling framework, we first need an overview of a rejection sampling-based implementation of intermediate sampling [Der19], which inspired the analysis of [AD20]. We then provide an example of Let S 0 ∈ supp(µ). One step of rejection sampling is given by:
1. Sample T ∼ ( [n]\S 0 t−k ). 2. Accept the set S 0 ∪ T with probability µ(S 0 ∪ T) max T ′ ∈( [n]\S 0 t−k ) µ(S 0 ∪ T ′ ) 3. Downsample S 1 ∼ µ S 0 ∪T .
The key difference between rejection sampling and our Markov chain intermediate sampling algorithm is the rejection step, which is necessary if we want our chain to mix to the correct stationary distribution µ. In order to have a sufficiently large acceptance probability, and assuming µ is isotropic, we require that for all T,
µ(T) ≤ t n k · (1 + ǫ) k
Here, ǫ is a parameter related to the guarantee on P(S 0 , ·) − µ TV . Using this bound, we can ensure that the expected acceptance probability is 1 − O(ǫk).
This inequality describes a "worst-case" condition on T. This "worst-case" type analysis originated from earlier works that introduced intermediate sampling for Determinantal Point Processes [Der19]. The proof of our worst-case inequality on µ(T) relies heavily on the fact that the KL divergence between a log-concave distribution µ and an arbitrary distribution ν contracts by a precise amount when applying the down operator D k→m .
D KL (νD k→ℓ µD k→ℓ ) ≤ ℓ k · D KL (ν µ)
This contraction is well-known for log-concave distributions [CGM19], but does not hold with the factor ℓ/k for α-fractionally-log-concave distributions. On the other hand, the inequality we need to show (from the proof Lemma 26) is "average-case" in nature, and when µ is isotropic, it takes the form:
E T∼( [n] t ) [µ(T)] ≤ t n k · 1 1 − ǫ
To concretely illustrate the advantage of Markov chain intermediate sampling, let us consider an example where the worst-case inequality fails to hold. Suppose that k = 2, n is even, and µ samples a set from {1, n 2 + 1}, {2, n 2 + 2},... uniformly at random, so that µ({i, n 2 + i}) = 2 n .
This distribution is isotropic, 1 2 -sector stable [Ali+21], and 1 2 -fractionally-log-concave, and yet, according to the worst-case analysis, it does not yield enough acceptance probability when t = o(n). For any set T, we have that
µ(T) ≤ t 2 · 2 n = t n
Equality is achieved by selecting a subset T that contains as many pairs of the form {i, n 2 + i} as possible, i.e., at least (t − 1)/2. Thus, the worst-case analysis would suggest that no non-trivial intermediate sampling is possible for the distribution µ; this is because t/n ≫ (t/n) 2 for small values of t.
However, our relaxed average-case analysis captures the fact that realistically, not every element of T will be paired up. In fact, we expect only a constant number of pairs when t = O( √ n), so for this example, we have:
E T∼( [n] t ) [µ(T)] ≤ C · 2 n ≤ O 1 n = O t 2 n 2
Proof of Lemma 26
In this section, we will prove Lemma 26.
Lemma 29. Let U, V be a sets of size u, v ≤ k respectively with U ∩ V = ∅. We have:
t − (u + v) n − (u + v) u ≤ P T∈( [n]\V t−v ) [U ⊆ T] ≤ t − v n − v u
Proof. Since we are sampling the elements of T uniformly at random from [n],
P T∈( [n]\V t−v ) [U ⊆ T] = ( n−v−u t−v−u ) ( n−v t−v ) = (t − v)(t − v − 1) · · · (t − v − u + 1) (n − v)(n − v − 1) · · · (n − v − u + 1) ≤ t − v n − v u
Similarly, we also have:
P T∈( [n]\V t−v ) [U ⊆ T] ≥ t − v − u + 1 n − v − u + 1 u ≥ t − (u + v) n − (u + v) u
Proof of Lemma 26. Let R = S 0 ∪ S, and let r = |S 0 ∪ S|. Note that |S \ S 0 | = r − k and
P[S 1 = S] = E T∼( [n]\S 0 t−k ) [ µ(S) ∑ S ′ ⊆(S 0 ∪T) µ(S ′ ) S ⊆ T] · P T∼( [n]\S 0 t−k ) [(S \ S 0 ) ⊆ T] = E T ′ ∼( [n]\R t−r ) [ µ(S) ∑ S ′ ⊆(R∪T ′ ) µ(S ′ ) ] · P T∼( [n]\S 0 t−k ) [(S \ S 0 ) ⊆ T] ≥ (1) µ(S) E T ′ ∼( [n]\R t−r ) [∑ S ′ ⊆(R∪T ′ ) µ(S ′ )] · P T∼( [n]\S 0 t−k ) [(S \ S 0 ) ⊆ T] ≥ (2) µ(S) E T ′ ∼( [n]\R t−r ) [∑ S ′ ⊆(R∪T ′ ) µ(S ′ )] · ( t − r n − r ) r−k
Inequality (1) is an application of Jensen's inequality to the function f (x) = c x , which is convex when x > 0. Inequality (2) use Lemma 29 with U = S \ S 0 and V = S 0 .
Now if we bound E T ′ ∼( [n]\R t−r ) [∑ S ′ ⊆(R∪T ′ ) µ(S ′ )] by ( t−r n−r ) r−k 1 1−ǫ then we are done. E T ′ ∼( [n]\R t−r ) [ ∑ S ′ ⊂(T ′ ∪R) µ(S ′ )] = ∑ S ′ ∈( [n] k ) µ(S ′ ) · P T ′ ∼( [n]\R t−r ) [(S ′ \ R) ⊆ T ′ ]] ≤ ∑ S ′ ∈( [n] k ) µ(S ′ ) · t − r n − r |S ′ \R|
In the very last line, we applied Lemma 29 with U = |S ′ \ R| and V = R.
If we set z i = ( n−r t−r ) 1/α if i ∈ (S 0 ∪ S) 1 else , then we can rewrite ∑ S ′ ∈( [n] k ) µ(S ′ ) · t − r n − r |S ′ \(S 0 ∪S)| = t − r n − r k ∑ S ′ ∈( [n] k ) µ(S ′ ) · n − r t − r |S ′ ∩(S 0 ∪S)| = t − r n − r k · g µ (z α 1 , . . . , z α n )
Applying Eq.
(1) to g µ (z α 1 , . . . , z α n ) and noting that p i =
P S∼µ [i∈S] k , we obtain g µ (z α 1 , . . . , z α n ) ≤ n ∑ i=1 P S∼µ [i ∈ S] k · z i kα log g µ (z α 1 , . . . , z α n ) ≤ kα log n ∑ i=1 P S∼µ [i ∈ S] k · z i ≤ (1) kα −1 + n ∑ i=1 P S∼µ [i ∈ S] k · z i = (2) kα n ∑ i=1 P S∼µ [i ∈ S] k · (z i − 1) = α n ∑ i=1 P S∼µ [i ∈ S] · (z i − 1) where in (1) we use log x ≤ x − 1 for x ∈ (0, ∞) and in (2) we use ∑ n i=1 P S∼µ [i∈S] k = 1.
Substitute z i as specified above into the final inequality, we get
log g µ (z α 1 , . . . , z α n ) ≤ α ∑ i∈(S 0 ∪S) Ck n n − r t − r 1/α = Cαkr n n − r t − r 1/α ≤ 2Ck 2 n n − r t − r 1/α .
Lower Bound on Intermediate Sampling
We first show that the dependence of our sparsification analysis on n is optimal. Consider the uniform distribution µ 0 over singletons of a ground set of n/k elements. Any distribution on singletons is log-concave as the generating polynomial is linear and thus log-concave. Now apply the construction of Example 7 with m = k to µ 0 in order to obtain a new distribution µ on ( [n] k ). This distribution is uniform over parts of a particular partition of the ground set [n] into n/k sets S 1 , . . . , S n/k . As such, the distribution is also isotropic. Note that this distribution is also 1/k-entropically independent.
If we sample a uniformly random set T of size t, then the chance that S i is contained in T can be upperbounded by
n − k t − k / n t ≃ (t/n) k .
Thus the chance that any of the S i are contained in T can be upperbounded (via a union bound) by roughly n · (t/n) k . Thus, as long as t ≪ n 1−1/k , the above is negligible. Without having any S i in the support with high probability, we obviously cannot faithfully produce a sample of µ from subsets of T.
Next we construct an example showing that even higher-order marginals cannot remove the dependence on n for entropically independent distributions (in sharp contrast with Conjecture 5). Our construction is based on Reed-Solomon codes.
Lemma 30. Let q be a prime number and F q the finite field of size q. Fix k points {x 1 , . . . , x k } ⊆ F q where k is a constant and choose a set of k random permutations from F q to F q and call them π 1 , . . . , π k . Let µ : ( Ω k ) → R ≥0 be the uniform distribution over sets {(x i , y i ) | i ∈ [k]} s.t. p(x i ) = π i (y i ) for some polynomial g of deg(p) ≤ d. Note that the ground set Ω is {x 1 , . . . , x k } × F q . Then 1. µ satisfies (1/α)-entropic independence with α = d+1 k . 2. Any domain sparsification scheme to sample from µ requires t =Ω(n 1−α ), even when we are allowed to sample higher order marginals.
Proof. The distribution µD k→(d+1) is uniform over {(x j , y j ) : j ∈ J ⊆ [k], |J| = (d + 1)}, because for any such set, there exists a unique polynomial p of degree at most d such that p(x j ) = π j (y j ) for all j ∈ J. Thus, high-order marginals, up to order d + 1, are independent of the choice of permutations π 1 , . . . , π k .
The support of µD k→(d+1) forms the basis of a partition matroid: for each x = x i , we have a block consisting of all points {(x, y) : y ∈ F q }, and for each set in the support of µD k→(d+1) , we have at most one element per block. Since we have a uniform distribution over matroid bases, µD k→(d+1) is log-concave, and thus it satisfies 1-entropic independence. We use this to upper bound D KL (νD k→1 µD k→1 ), and from here, conclude d+1 k -entropic independence of µ: D KL (νD k→1 µD k→1 ) = D KL ((νD k→(d+1) )D (d+1)→1 (µD k→(d+1) )D (d+1)→1 )
≤ 1 d + 1 · D KL (νD k→(d+1) µD k→(d+1) ) ≤ 1 d + 1 · D KL (ν µ) = 1 d+1 k · k · D KL (ν µ)
The second line follows from µD k→(d+1) satisfying 1-entropic independence, and the third line comes from the data-processing inequality.
We now prove that for all t ≤ o n 1−α , no domain sparsification scheme exists, even with access to higher order marginals. For d ′ ≤ d + 1, the distribution µD k→d ′ is uniform over the size d ′ independent sets of the partition matroid defined above. One consequence of the independence of high-order marginals from the choice of permutations π 1 , . . . , π k is that the higher order marginals do not provide any information about the identity of the distribution µ.
Suppose that we choose our sample in domain sparsification from a distribution whose ground set is the sparse subset T. We want an upper bound on the probability (over the choice of permutations) that T contains some S ∈ supp(µ).
In order for T to contain a valid S, there must be some subset in S ∈ ( T k ) associated to a degree ≤ d polynomial p satisfying p(x i ) = π i (y i ). However it is easy to see that for any particular set S the probability (over the choice of permutations) that S is in the support of µ is ≃ 1/q k−d−1 .
We can upper bound P[S ⊆ T for some S ∈ supp(µ)] by a union bound as follows:
t k · 1 q k−d−1 ≤ t k q k−d−1
This implies that for any t ≤ o(q (k−d−1)/k ) ≤ o(n (k−d−1)/k ), the probability of containing a set in the support is negligible. Note that we have α = d+1 k , so 1 − α = k−d−1 k , which completes the lower bound.
Missing Proofs
In this section we prove Propositions 19 and 20. To complete the proofs, we need to understand a few results concerning the correlation matrix of a distribution µ to get an alternate characterization of α-fractional-log-concavity (Theorem 32).
Definition 31. For distribution µ : ( [n]
k ) → R ≥0 , let the correlation matrix Ψ cor µ ∈ R n×n be defined by
Ψ cor µ (i, j) = 1 − P[i] if j = i P[j | i] − P[j] else where P[j | i] = P T∼µ [j ∈ T | i ∈ T], P[j] = P T∼µ [j ∈ T].
For a distribution µ on ( [n] k ) and λ = (λ 1 , . . . , λ n ) ∈ R n >0 , the λ-external field applied to µ is also a distribution on ( [n] k ), denoted by λ * µ, given by
P λ * µ [S] ∝ µ(S) · ∏ i∈S λ i .
Note that for any (z 1 , . . . , z n ) ∈ R n ≥0 , g λ * µ (z 1 , . . . , z n ) ∝ g µ (λ 1 z 1 , . . . , λ n z n ).
Theorem 32 ([Ali+21, Lemma 67, Remark 68]). A polynomial g is α-fractionally-log-concave if and only if the largest eigenvalue of λ * µ's correlation matrix satisfies λ max (Ψ cor λ * µ ) ≤ 1 α for all external fields λ ∈ R n ≥0 .
Now that we understand an alternate characterization of α-fractional-log-concavity, we can use it to complete the proof of Proposition 20.
Proof. By induction, we only need to prove the statement when one element is subdivided. So we need to show that if g := g µ is fractionally log-concave, then h(z (1) 1 , . . . , z (k 1 ) 1 , z 2 , . . . , z n ) := g(z (1) 1 + . . . + z (k 1 ) 1 , z 2 , . . . , z n ) is α-fractionally log-concave By Theorem 32, this is equivalent to
λ max (Ψ cor λ ′ * µ ′ ) ≤ 1 α ∀ λ = (λ (1) 1 , . . . , λ (k 1 ) 1 , λ 2 , . . . , λ n ) ∈ R n+k 1 −1 ≥0 ,
where µ ′ is the distribution generated by h. Without loss of generality, we assume ∑ k 1 j=1 λ (j) 1 = 1. Let Ψ and Ψ ′ be the correlation matrix of λ * µ and λ ′ * µ ′ , where λ = (1, λ 2 , . . . , λ 2 ) ∈ R n ≥0 . We want to relate the eigenvalues of Ψ ′ to Ψ, thereby showing that Ψ ′ ≤ 1 α . Note that P λ ′ * µ ′ [1 (i 1 ) ] = λ (i 1 ) 1 P µ [1] and P λ ′ * µ ′ [1 (i 1 ) | 1 (i 2 ) ] = 0 ∀j = 1 : P λ ′ * µ ′ [1 (i 1 ) | j] = λ (i 1 ) 1 P λ * µ [1 | j] and P λ ′ * µ ′ [j] = P λ * µ [j] and P λ ′ * µ ′ [j | 1 (i 1 ) ] = P λ * µ [j | 1]
Let v 1 , . . . , v n be the orthogonal basis ·, diag(P λ * µ [i (j i ) ]) of eigenvectors of Ψ with corresponding eigenvalues ρ 1 ≥ ρ 2 ≥ · · · ≥ ρ n . It is easy to check that for each v i , the vector ) associated with eigenvalue 1, where u i k 1 −1 i=1 forms an orthogonal basis wrt ·, diag(P λ * µ [i]) of the vector space u ∈ R k 1 u ⊥ (λ (i)
1 ) k 1 i=1 . Observe that w i n+k 1 −1 i=1
forms an orthogonal basis wrt ·, diag(P λ ′ * µ ′ [i (j i ) ]) of eigenvectors of Ψ ′ , thus the spectrum of Ψ ′ is {ρ 1 , . . . , ρ n } ∪ 1 (k 1 −1)
Thus, λ max (Ψ ′ ) ≤ max(ρ 1 , 1) ≤ 1 α , by Theorem 32 applied to µ.
Next we prove Proposition 19.
Proof. We have a characterization of entropic independence as i + · · · + z t i i )/t i for z i in the above ineuality.
Conclusion
A natural direction to extend our research is to better understand the trade-offs between the runtime of approximate sampling algorithms and how much information we are given in the form of marginals for α-fractionally-log-concave distributions. Recall that α-fractional-log-concavity implies 1/α-entropic independence, but still captures many interesting distributions (see Section 1.1), so while we exhibit a lower bound against domain sparsification for 1/α-entropically independent distributions, even with access to higher-order marginals, the stronger assumption of fractional log-concavity may avoid this barrier.
For a k-uniform distribution, the entire distribution itself is indeed captured by the order-k marginals. So if one is to believe that α-fractionally log-concave distributions behave like distributions over 1/α-sized sets, does that mean order-1/α marginals are sufficient to get rid of all dependence on n in domain sparsification? For α = 1, the answer is affirmative as was shown by Anari and Dereziński [AD20]. A fascinating open question is whether we can extend this to fractionally log-concave polynomials as suggested by Conjecture 5.
-concave distributions where Markov chain intermediate sampling succeeds using a smaller intermediate sample size than what is required for rejection sampling.
eigenvector of Ψ ′ associated with eigenvalue ρ i . In addition, Ψ ′ also has eigenvectors w n+i := (u i 1 , . . . , u i k 1 , 0, . . . , 0 n−1 times
g µ (z α 1 , . . . , z α n ) 1/kα ≤ ∑ i∈S p i z i ,where p i = P S∼µ i ∈ S/k are the normalized marginals of µ. Subdivision replaces the generating polynomial by a new polynomial, where z i is replaced by an average of t i new variables. It is easy to see that the normalized marginal of each of the new variable is simply p i /t i . Thus z i in the r.h.s. of the above expression also gets replaced by the same average of the t i new variables. So the proof is just a matter of plugging in (z (1)
in the context of sampling from Determinantal Point Processes (DPPs, [DM21]), also known as Volume Sampling [Des+06; DR10; GS12]. DPPs are a family of distributions (a small, but important, subset of distributions with logconcave generating polynomials) which arise for instance when sampling random spanning trees [Gué83], as well as in randomized linear algebra [DW17; Der+19], machine learning [KT11; KT12; DKM20], optimization [NST19; Der+20; MDK20], and other areas [Mac75; Hou+06; Bar+17]. The complexity of intermediate sampling for DPPs was further improved by [DCV19;
Recall that if we used the marginal estimates required by Theorem 1, then by applying Proposition 24 we get an equivalent distribution µ ′ on a ground set of size O(n) that satisfies the above assumption of P S∼µ ′ [i ∈ S] ≤ Ck/n for some C = O(1) (see Lemma 26).Let µ : ( [n]
k ) be a 1/α-entropically independent distribution. Suppose that for all i ∈ [n], we
have P S∼µ [i ∈ S] ≤ Ck
n . Then, for any constant ǫ ∈ (0, 1
4 ], and t = Ω n 1−α (Ck 2 log 1
1−ǫ ) α , the output
of a single step of the Markov chain M t
µ starting from S 0 satisfies
∀S ∈
[n]
k
: P[S 1 = S] ≥ µ(S)(1 − ǫ)
Proposition 27. Let µ : ( [n]
k ) be a 1/α-entropically independent distribution, and let ǫ ∈ (0, 1
4 ] be a
constant. Suppose P S∼µ [i ∈ S] ≤ Ck
n for all i. Choose the intermediate sample size t according to
Lemma 26. Then
P(S 0 , ·) − µ TV ≤ ǫ
and the Markov chain M t
µ mixes to a distribution that has TV distance ǫ ′ < ǫ from µ in O(log(1/ǫ ′ ))
steps.
Typically the running time has logarithmic dependencies on the approximation error and potentially magnitude of external fields, but for simplicity of exposition we hide them here.
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| []
|
[
"Journal of Geophysical Research: Space Physics On the Persistent Shape and Coherence of Pulsating Auroral Patches",
"Journal of Geophysical Research: Space Physics On the Persistent Shape and Coherence of Pulsating Auroral Patches"
]
| [
"B K Humberset \nDepartment of Physics and Technology\nBirkeland Centre for Space Science\nUniversity of Bergen\nBergenNorway\n",
"J W Gjerloev \nDepartment of Physics and Technology\nBirkeland Centre for Space Science\nUniversity of Bergen\nBergenNorway\n\nThe Johns Hopkins University Applied Physics Laboratory\nLaurelMDUSA\n",
"I R Mann \nDepartment of Physics\nUniversity of Alberta\nEdmontonAlbertaCanada\n",
"R G Michell \nDepartment of Astronomy\nUniversity of Maryland\nCollege ParkMDUSA\n\nNASA Goddard Space Flight Center\nGreenbeltMDUSA\n",
"M Samara \nNASA Goddard Space Flight Center\nGreenbeltMDUSA\n"
]
| [
"Department of Physics and Technology\nBirkeland Centre for Space Science\nUniversity of Bergen\nBergenNorway",
"Department of Physics and Technology\nBirkeland Centre for Space Science\nUniversity of Bergen\nBergenNorway",
"The Johns Hopkins University Applied Physics Laboratory\nLaurelMDUSA",
"Department of Physics\nUniversity of Alberta\nEdmontonAlbertaCanada",
"Department of Astronomy\nUniversity of Maryland\nCollege ParkMDUSA",
"NASA Goddard Space Flight Center\nGreenbeltMDUSA",
"NASA Goddard Space Flight Center\nGreenbeltMDUSA"
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| The pulsating aurora covers a broad range of fluctuating shapes that are poorly characterized.The purpose of this paper is therefore to provide objective and quantitative measures of the extent to which pulsating auroral patches maintain their shape, drift and fluctuate in a coherent fashion. We present results from a careful analysis of pulsating auroral patches using all-sky cameras. We have identified four well-defined individual patches that we follow in the patch frame of reference. In this way we avoid the space-time ambiguity which complicates rocket and satellite measurements. We find that the shape of the patches is remarkably persistent with 85-100% of the patch being repeated for 4.5-8.5 min. Each of the three largest patches has a temporal correlation with a negative dependence on distance, and thus does not fluctuate in a coherent fashion. A time-delayed response within the patches indicates that the so-called streaming mode might explain the incoherency. The patches appear to drift differently from the SuperDARN-determined ⃗ E × ⃗ B convection velocity. However, in a nonrotating reference frame the patches drift with 230-287 m/s in a north eastward direction, which is what typically could be expected for the convection return flow. | 10.1029/2017ja024405 | [
"https://arxiv.org/pdf/1806.09428v1.pdf"
]
| 49,309,059 | 1806.09428 | 9ae1dcceadd9e16189f08342c0dc1efd90be3fe3 |
Journal of Geophysical Research: Space Physics On the Persistent Shape and Coherence of Pulsating Auroral Patches
B K Humberset
Department of Physics and Technology
Birkeland Centre for Space Science
University of Bergen
BergenNorway
J W Gjerloev
Department of Physics and Technology
Birkeland Centre for Space Science
University of Bergen
BergenNorway
The Johns Hopkins University Applied Physics Laboratory
LaurelMDUSA
I R Mann
Department of Physics
University of Alberta
EdmontonAlbertaCanada
R G Michell
Department of Astronomy
University of Maryland
College ParkMDUSA
NASA Goddard Space Flight Center
GreenbeltMDUSA
M Samara
NASA Goddard Space Flight Center
GreenbeltMDUSA
Journal of Geophysical Research: Space Physics On the Persistent Shape and Coherence of Pulsating Auroral Patches
10.1029/2017JA024405
The pulsating aurora covers a broad range of fluctuating shapes that are poorly characterized.The purpose of this paper is therefore to provide objective and quantitative measures of the extent to which pulsating auroral patches maintain their shape, drift and fluctuate in a coherent fashion. We present results from a careful analysis of pulsating auroral patches using all-sky cameras. We have identified four well-defined individual patches that we follow in the patch frame of reference. In this way we avoid the space-time ambiguity which complicates rocket and satellite measurements. We find that the shape of the patches is remarkably persistent with 85-100% of the patch being repeated for 4.5-8.5 min. Each of the three largest patches has a temporal correlation with a negative dependence on distance, and thus does not fluctuate in a coherent fashion. A time-delayed response within the patches indicates that the so-called streaming mode might explain the incoherency. The patches appear to drift differently from the SuperDARN-determined ⃗ E × ⃗ B convection velocity. However, in a nonrotating reference frame the patches drift with 230-287 m/s in a north eastward direction, which is what typically could be expected for the convection return flow.
Introduction
The broad definition of pulsating aurora covers low-intensity aurora that undergoes repetitive, quasiperiodic, or occasionally periodic intensity fluctuations on time scales ranging from less than 1 s to several tens of seconds (Royrvik & Davis, 1977).
Patch Drift
One of the outstanding issues associated with fluctuating patches is whether they drift with the ⃗ E × ⃗ B velocity. Answering this apparently simple question is, however, difficult due to observational complexities. The Super Dual Auroral Radar Network (SuperDARN) provides observations with a spatial resolution of some 50 km by 50 km for the measured line-of-sight backscatter, but the resolution of the final convection solution is far coarser as it requires the inclusion of a statistical fill-in model. Spaceborne measurements are also complicated due to their inability in separating spatial and temporal variations. It takes some 7 s for a low-Earth-orbit (LEO) satellite to traverse a patch of 50 km, which is comparable to the on time of the patch, and thus it is impossible to separate spatial and temporal variations. It should also be mentioned that single-satellite measurements only provide observations along the satellite trajectory and thus the 2-D convection distribution cannot be unambiguously inferred.
It has been suggested that imaging of pulsating auroral patches can be used to remote sense magnetospheric convection (e.g., Nakamura & Oguti, 1987;Yang et al., 2015;. However, the assumption here is that all monitored fluctuating auroral patches move with the plasma convection ⃗ E × ⃗ B velocity. The drift speed has been consistently measured to be on the order of 1 km/s in the dawn sector, presumably at the ⃗ E × ⃗ B velocity (Davis, 1978;Scourfield et al., 1983), but there have been studies which found indications that the drift of the pulsating aurora can be different from the plasma convection (Swift & Gurnett, 1973;Wescott et al., 1976). It is therefore important to establish if all patches are drifting solely with the plasma convection ⃗ E × ⃗ B velocity. Humberset et al. (2016) provided the temporal characteristics of six pulsating auroral patches. They found that the patches were highly variable in all measurable parameters and thus suggested the term "fluctuating aurora" instead of the commonly used term "pulsating aurora." The problem with pulsating is that it implies the patch to vary regularly. None of the proposed mechanisms appears to explain the observational constraints set by these patches in a convincing manner. In this paper we therefore provide spatial characteristics for the same event.
Purpose of the Paper
The purpose of this paper is to provide objective and quantitative measurements of the extent to which pulsating auroral patches maintain their morphology and fluctuate in a coherent fashion. Does the patch maintain its shape? Does the entire patch vary with the same fluctuation and phase? Do the patches drift with the plasma convection ⃗ E × ⃗ B velocity?
We use a ground-based all-sky imager that provided two-dimensional coverage of an electrodynamic parameter, the 557.7 nm emissions. This has the distinct advantage of allowing for a separation of spatial and temporal variations by translating the emissions to a nonrotating reference frame. To be specific, we trace the patch in its frame of reference, assuming that the drift of the patch is in the nonrotating frame of reference. In this way we avoid the space-time ambiguity, which complicates rocket and satellite measurements. Whereas it would take a LEO satellite less than one fluctuation to cross a patch, a rocket takes a few on-off cycles, for which the observed variation is both temporal and spatial. With all-sky imagers we can follow a pulsating auroral patch over its lifetime. In section 2 we describe the data and event; section 3 outlines the technique and methodology; section 4 shows the pulsating auroral patches and their quantitative characteristics; in section 5 we discuss our results; and finally in section 6 we summarize and draw conclusions.
Data and Event
The all-sky imager (ASI) utilized is located at Poker Flat Research Range in Alaska at −147.4 ∘ geographic longitude, 65.1 ∘ geographic latitude. The ASI uses a narrow band filter to capture the green line 557.7 nm emissions from atomic oxygen. It produces frames with 512 by 512 pixels resolution at a frame rate of 3.31 Hz.
We utilize the same event and observations as described in Humberset et al. (2016). Their findings and other published results have shown that the pulsating aurora is not regularly periodic, but rather erratic, leading them to suggest that the name "fluctuating aurora" is more appropriate. This is therefore the term used in the following sections. Below we give a short description of the event.
Our event was recorded on 1 March 2012. Figure 1 shows the SuperMAG data set of indices and time-shifted interplanetary magnetic field (IMF; Gjerloev, 2012;Newell & Gjerloev, 2011
Technique
Prior to the analysis we correct for the distorted fish-eye view of the ASI frames by performing a projection of the emissions onto a Cartesian grid with uniform spatial resolution as described in Humberset et al. (2016). We use a thin shell approximation assuming an altitude of 110 km for the auroral emissions. The Cartesian grid has a spatial resolution of 1.0 by 1.0 km and is organized geographically with north on top and west to the right to facilitate a correction of the ASI rotating with the Earth. The most distorted limb pixels are avoided by only including patches that are located within the center 400 by 400 km FOV. To determine the characteristics of the fluctuating auroral patches, we perform the analysis explained in the following subsections.
Journal of Geophysical
Patch Identification
The individual patches are manually identified from the movie in the supporting information. However, due to the erratic nature of the fluctuating aurora, this is not straightforward. To determine start and end times of the analysis, we develop position-time diagrams like the ones showed in Figure 2. To make the position-time diagrams, we sample emissions from the same line of pixels for all individual images and plot them successively. The diagrams allow a determination of whether the patch is fluctuating and for how long we can follow it before it either disappears or, for example, joins together with an adjacent patch. More importantly, we use position-time diagrams to decide whether the patch is an individual patch with a different train of fluctuations than the adjacent patches. irregularities in the fluctuations across patches 2 and 4. These and the relation to adjacent patches will further be described in section 4.1.
Fixed Point Analysis
The advantage of studying fluctuating auroral patches from ground-based ASI is the ability to untangle the space-time ambiguity by removing the apparent velocities of the patches in the ASI frame of reference. The velocity of a patch in the ASI frame of reference is due to a combination of the ASI rotating with the Earth (moving the FOV across the night sky) and the drift of the patch in an inertial frame [effectively Geocentric Solar Ecliptic (GSE) coordinates]. To determine the apparent drift of the patch, we make an outline around the patch and find the velocity that best follows the patch through its lifetime. This technique assumes that the velocity is constant during the lifetime of the patch (∼10 min), which none of the patches appear to violate. Using the apparent drift velocity, a new movie is made where the patch is centered at all times.
Contour and Extract the Patch
The complication of adjacent patches render automated techniques for patch boundary determination difficult. A manual determination therefore proved to be the more reliable technique. The extracted patch is expected to vary during the train of intensity fluctuations between off/dim and on/bright. A new contour of the patch is therefore determined at the peak intensity of each fluctuation as determined from the median of a square box within the patch. The contours are manually identified by drawing a line around the patch. In this way we can consistently avoid capturing adjacent patches. The brain is brilliant at identifying patterns, but exactly where the contour lies can of course be somewhat subjective. The contours are therefore validated by an independent control of where the contours are drawn. Figure 3 shows three example contours for each of the patches (1-4) from the beginning, center, and last part of the time interval analyzed. For each of the contours the time from start of the analysis (Δt) is given so that it easily can be compared to the position-time diagrams in Figure 2.
Quantitative Characteristics
Following the above selection, the patch can easily be extracted as all pixels within the closed contour. Each contour is considered valid over the fluctuation. If we are not able to draw a convincing contour, for example, because adjacent patches are on/bright, the previous and following contours are interpolated using a square function. In this way we can easily find the patch intensity defined as the total emissions within the patch
Journal of Geophysical Research: Space Physics
10.1029/2017JA024405
(counts can be converted to kR using the relationship: R = (counts − 350) × 9.16, where 350 is the imager dark counts).
The fluctuations are found from local minima after applying a boxcar filter and a fluctuation threshold (see Humberset et al., 2016, for a more thorough explanation).
The drift of the patch in an inertial frame (effectively GSE coordinates) is found by correcting the velocity of the patch in the ASI frame of reference for moving with Earth's rotation (see section 3.2). The velocity of Earth's rotation is easily calculated from the Earth's rotational period and distance at the geographic latitude and altitude of the patches.
The characteristics of the time variation and energy deposition of the patches are provided in an earlier study by Humberset et al. (2016). However, note that the more accurate manual contouring of this study allows us to extend the interval of analysis and to use the total emissions instead of the median emissions within the patch.
Results
To answer the stated science objective, we first give an overview of the patches before we show the persistence of patch shape and whether the entire patch fluctuates in a coherent fashion. At last we compare the patch drift to the ⃗ E × ⃗ B velocity.
Overview of Patches
In this study we analyze four patches with a considerable size ranging from ∼1,000 to 5,000 km 2 (assuming an altitude of 110 km). All are relatively well defined with a sharp boundary to the dark sky or diffuse background emissions. However, the fluctuating auroral display is very complex with adjacent patches, possible spatiotemporal variations/streaming within the patch and relative movement between the patches. Even patches that at first seem like relatively well-separated individual patches are difficult to trace over a considerable part of their lifetime. We therefore start with an overview of each of the four patches.
Patch 1
Patch 1 is a large patch (∼4,800 km 2 ) that we analyze for about 8.5 min (15:19:03-15:27:30 UT). Patch 1 can be identified in the movie about 15 min earlier, but it is obscured by adjacent patches. The shape persists for some time after we end the analysis, but it is increasingly difficult to decide if Patch 1 has merged with adjacent patches or if it has split into several patches. Figure 4 shows an overview of Patch 1 over the 8.5 min during which we can confidently contour and extract it as an individual patch. It displays eight example images, where Patch 1 is alternating between either on/bright or off/dim, and the train of fluctuations in intensity (total emissions within the patch). The dashed lines indicate the time and patch intensity of the example images. In the first example image ( Figure 4a) the patch is on/bright and surrounded by several nearby and adjacent patches that also are on. The adjacent patches to the south (geographically) are on/bright, while the adjacent patches to the north-east and north are off/dim. In the second image ( Figure 4b) the patch is off/dim, while the adjacent patches to the south remain, and the adjacent patch to the north-east is relatively bright. In the next images (Figures 4e and 4f ), Patch 1 has moved away from the southern patches and it is easier to draw the contour. In the last example image ( Figure 4h) the patch has entirely disappeared. Note that the patch typically is slightly brighter than the surrounding background even when it is off. This can be seen in the other example images (Figures 4b, 4d, and 4f ), where especially the eastern part of the patch remains with slightly higher emission than the surrounding background. As can be seen from the train of intensity fluctuations, they are in general highly irregular both in duration and intensity. This led Humberset et al. (2016) to suggest that pulsating aurora is not an appropriate term and suggested "fluctuating aurora" instead. The patch shape may be the only parameter that remains relatively constant.
Patch 2
Patch 2 (∼2,600 km 2 ) is analyzed for 7.25 min (15:19:00-15:26:14 UT), as shown in Figure 5. It can be identified about 16 min earlier, but it starts out as two patches that are difficult to separate from adjacent patches. In the first example images (Figures 5a and 5c), the patch is surrounded by several adjacent patches, which makes it difficult to decide where to draw the contour. The adjacent patches to the south (geographically) are on/bright, while the adjacent patches to the north-east and north are off/dim. These adjacent patches disappear after the first minutes, while two new adjacent patches appear later in the interval. The first is south (bright in image e) of the patch, while the second is east (visible but relatively dim in image g) of the patch. These are, however, easy to avoid when drawing the contours.
a) b) c) d) e) f) g) h)
Patch 2 also has some intriguing irregularities in the fluctuations across the patch, where one side of the patch is bright while the other side of the patch is dim/off. This can be seen around 1.8, 2.4 and 3.5 min on the time axis of panel 2 in Figure 2.
We stop the analysis when Patch 2 drifts close to Patch 4 (west/right in example image g). The patch emissions are in general highly erratic both in duration and intensity, while the patch shape stretches in the east-west direction and slightly rotates.
Patch 3
Patch 3 is a relative small patch (∼1,000 km 2 ) that we follow for about 10 min (15:17:05-15:27:12 UT). The patch emission and example images, where the patch is considered either bright/on or dim/off, are shown in Figure 6. At first glance it seems to be related to the smeared out southern patches (south in the example images). However, a visual inspection of the movie and several position-time diagrams leaves no doubt that Patch 3 is an individual patch from about 15:17 UT. We choose to start the analysis when it is still very close to the southern patches (example images a-c), which of course makes the drawing of the first contours a little uncertain. The relative movement then causes the patch to split from the southern patches and get closer to the other nearby patches 1 (to the east) and 4 (to the northwest). Throughout the interval the patch emissions show a complex fluctuating behavior, while the patch shape appears to slowly rotate clockwise. Toward the end of the interval the patch emissions are dimming. The patch occurs a few more times after we stop the analysis but the fluctuations are very dim.
a) b) c) d) e) f) g) h)
Patch 4
Patch 4 (∼3,300 km 2 ) starts out as a complex patch system. We start the analysis when it has become well defined and follow it for the ∼9 min (15:17:14-15:26:02 UT) that is displayed in Figure 7.
Patch 4 is the toughest of the patches to contour, mainly due to two features. The first feature is the smeared-out eastern edge of the patch where the emissions falls off gradually. This makes it difficult to decide where the contour should be drawn, especially in the end of the interval when the nearby patches have come closer (see example image g in Figure 7). The second feature is the northern part of the patch, which has a difficult shape to contour and where an adjacent patch fills the gap between Patch 4 and the nearby patch to the north (see image a). At 6.0-6.7 min into the time interval, the northern adjacent patch fluctuates in the same manner as Patch 4 (see image e) before it fades away. However, when it brightens again about 30 s later, it has joined together with the northern part of Patch 4 and starts to fluctuate in a different manner to the main part of Patch 4 (see images g and h). We continue to follow Patch 4, where we leave out the northern part which has separated. This is evident in the drop in the patch emissions in Figure 7.
Patch 4 has some intriguing irregularities in the fluctuations across the patch, and a core which fluctuates less than the rest of the patch, as can be seen in the position-time diagram of Patch 4 ( Figure 2). There is however no obvious pattern. The patch can vary or retract and stream out (southeastward and northward) several times before the whole patch is dimming. When it brightens, the core sometimes turns on first, while at other times the southeastward patch brightens first. Another interesting irregularity appears about 5.5 min into the time interval, where the edges of the patch (east-west) continue to fluctuate but are very dim compared to the bright core. This can be seen in panel 4 of Figure 3, where the contour at ΔT = 7.82 min from start looks like it is drawn too wide because we include the dim edges. Similar to the other patches, also the Patch 4 emissions are in general highly complex both in duration and intensity. It is apparent that during the off-time, the patch still has emissions slightly higher than the surrounding background. Except for the detaching of the northern part, the patch shape remains relatively constant.
a) b) c) d) e) f) g) h)
Persistence of Patch Shape
To what extent does a patch maintain its shape? The fluctuating auroral patches are often described as persistent. Exactly how persistent is often loosely described, and there are no quantitative measures to how the shape of a patch evolves. We therefore provide a quantitative and qualitative estimate of the persistency of the patch shape. Figure 8 gives a quantitative estimate of how persistent the patch shape is. It shows the temporal evolution of the shape in the patch frame of reference as the percentage of the patch (pixels within the contour) that overlaps with the patch at the first fluctuation of the analysis interval. The shape of Patch 1 is remarkably constant (85-100%) throughout the analysis interval of about 8.5 min. Patch 2 gradually drops to about 70% overlap. Patch 3 drops to 60-70% within a minute, and then the overlap drops gradually for the next 9 min until it reach as low as 40% overlap. Patch 4 also has a remarkably constant shape (90-80% until about 7 min when part of the patch detach and there is a small drop in the overlap. Figure 9 vividly illustrates how remarkably persistent the shape of each patch (1-4) is. It shows 20 of the contours, including the first and last contours, in the patch frame of reference. The temporal color scale visualizes the evolution of the patch shape. The shape of Patch 1 stretches in the magnetic east-west direction, but remains remarkably similar over the ∼8.5 min we follow it. Patch 2 also stretches in the geographic east-west direction and rotates slightly from the geographic to the magnetic east-west direction (counterclockwise). Patch 3 rotates clockwise from geographic north-south and shrinks in size the last minute before it fades away. The shape of Patch 4 evolves into a more narrow shape and the curl of the northern part stretches until it detaches for the last ∼1.5 min.
Fluctuations Within the Patch
Does the entire patch fluctuate in a coherent fashion? If the answer is yes, that would imply that all pixels within the patch are highly correlated with each other and, specifically, that this correlation is independent on distance between the pixels. Figure 10 shows the results from a straightforward analysis aimed at answering this simple question. We pick a number of pixels along a line that is within the patch at all times. For each pixel we determine the time series and calculate the correlation coefficients between pixel at the beginning of the arrow and the other pixels along the colored line. Figure 10 shows the average of the correlation coefficients for all fluctuations in the analysis interval as a function of the distance between the pixels.
As the figure clearly shows, the correlation coefficient has a pronounced dependence on the spatial distance, except for the relatively small Patch 3. If we consider coherency between fluctuations to have a correlation
Patch Drift
It has been suggested that the fluctuating auroral patches are drifting with the ionospheric ⃗ E × ⃗ B drift (e.g., Scourfield et al., 1983;Yang et al., 2015Yang et al., , 2017. The velocities of the patches are found in the geographical oriented Cartesian grid and rotated to the Altitude Adjusted Corrected Geomagnetic (AACGM) coordinate system. The drifts of the patches in a reference frame not corotating with the Earth are in the range of 230-287 m/s in a north-eastward direction.
As an estimate of the ⃗ E × ⃗ B velocity we use the SuperDARN (Greenwald et al., 1995) data provided by Mr. Robin Barnes at the Johns Hopkins University Applied Physics Laboratory using the technique described by Ruohoniemi and Baker (1998). The resulting ⃗ E × ⃗ B convection velocities, also called fitted velocities, are found in a reference frame corotating with Earth. The ⃗ E × ⃗ B velocities are therefore compared to the patch drift velocities that are not corrected for the ASI rotating with Earth.
In Figure 11 we show the patch velocities and the median of the ⃗ E × ⃗ B (SuperDARN) velocities at the location of each of the patches. We find that all patches move in a north and slightly eastward direction with a speed of 53-104 m/s. According to the SuperDARN results the convection is mostly eastward with a speed of about 140 m/s. The direction of the patches and the ionospheric plasma differs by 51 ∘ to 73 ∘ . Assuming that the SuperDARN drifts are correct, this result indicates that the fluctuating patches drift independently from the ionospheric plasma.
Discussion
We first address the inherent limitations of the ASI data set and technique, and then we discuss to what extent the fluctuating auroral patches maintain their shape and fluctuate in a coherent fashion, and if the patches always drift with the ⃗ E × ⃗ B velocity.
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10.1029/2017JA024405 Figure 10. For each of the patches (1-4): The average correlation between all fluctuations of the first pixel to the next pixels as a function of the distance between them. The standard error is shown as error bars. The pixels are sampled along the line which is highlighted in color on the example image to the right. The arrow indicates the increasing distance and thus the first pixel.
Inherent Limitations
The main limitations are as follows: (1) observations are limited to four patches, and the analysis is limited by (2) the camera capabilities, and (3) the fact that we only study the 557.7 nm emissions.
Limited Number of Patches
The fluctuating auroras are observed under a wide variety of conditions (e.g., Jones et al., 2011), while this study is limited to effectively one set of conditions. Of most importance, the patches are located in the dawn sector at ∼4 MLT during substorm activity. The patchy type of fluctuating auroras are most frequently observed toward dawn (Cresswell, 1972). The limited observations of course result in limited statistics, and we cannot address how the characteristics depend on local time, geomagnetic conditions, and so on. At the same time this means that any variations between the four patches are not due to such dependencies.
Camera Capabilities
The camera is an all-sky imager sampling images at a time rate of 3.31 Hz. An all-sky imager has a large FOV providing good coverage of the auroral display, but it has some considerable distortions. The main issue is that ASI obtains the column integration of auroral emissions that originate from a range of altitudes and magnetic field lines. Additionally, there are distortions due to optics and elevation (look direction). The question is whether smearing affects our ability to answer the stated science objective.
The common height assumption of 110 km of the 557.7 nm aurora does not have any considerable effects on the Earth's rotation velocity or on the ASI pixel size, which is much more affected by the elevation angle. There is no robust way to remove smearing. Therefore, we mediate the smearing by evaluating patches within the center 400 by 400 km FOV. The resampled image has a pixel size of 1.0 km. Based on the size of the largest pixels in the fish-eye view that are projected onto the center FOV of the resampled grid, we determine that the smearing is less than ∼2.5 km.
The 3.31-Hz time resolution of the ASI data is limited by the Nyqvist frequency of 1.65 Hz. We can therefore not distinguish rapid temporal behavior such as ∼3 Hz modulations often found to be superimposed on the slower on-off fluctuations. Running the all-sky camera at a higher cadence is possible, but the fluctuating aurora that occurs with higher-frequency fluctuations, such as the ∼3 Hz modulations, likely occurs at small spatial scales on top of the large-scale fluctuations (Nishiyama et al., 2016). A narrow field of view imager is therefore more appropriate for investigating the spatiotemporal variations of the higher frequencies, as was done by Samara and Michell (2010), which reported fluctuations up to 10-15 Hz. It is possible that these higher-frequency fluctuations are caused by completely different mechanisms, and therefore, it would make sense to investigate them independently.
The 557.7 nm Emissions
The 557.7 nm emissions are limited by chemical effects. Its green-light is the brightest line in the visible spectrum in most auroras, and also for fluctuating aurora. It is however limited by a mean effective lifetime of 0.3 to 0.59 s of the O( 1 S) excitation, the highest values found in observations of sharp-edged fluctuating patches ). This is due to the ∼0.75 s lifetime of the direct O( 1 S) excitation, quenching of the excited state below 100 km and additional indirect excitation processes (e.g., Brekke, 2013). The result is a time lag to the prompt emissions (e.g., the blue 427.8 nm) and a temporal smoothing over the same time scale, meaning that the 557.7 nm filter used could smooth out possible impulsive behavior. Also the O( 1 S) excitations resulting from the highest-energy electrons (>20 keV) that penetrates to altitudes below 100 km could be quenched and therefore become less visible in the 557.7 nm emissions. We are however investigating sharp-edged fluctuating patches. So if the observations by can be generalized, the high values of mean effective lifetime of the O( 1 S) excitation can indicate that we capture most of the fluctuating precipitation.
In addition, impulsive behavior from a source region far from Earth can be smoothed by the energy-time dispersion due to the energy-dependent transport times along the field line connecting the magnetosphere and the ionosphere. This was discussed by Humberset et al. (2016) that found the most probable time lag in the beam of energetic electron precipitation to be 0.6 s or less. It is therefore likely that the spatiotemporal characteristics we observe are due to the behavior of the underlying source mechanism and not a dispersion effect.
To What Extent Does the Patch Maintain Its Shape?
The fluctuating auroral patches are often described as persistent (e.g., Johnstone, 1978;Oguti, 1976). We found a quantitive measure (Figure 8) to the persistency of the patch shape in the form of percentage overlapping pixels. Patches 1, 2, and 4 all have a remarkably persistent shape with above 85% overlap for 4.5-8.5 min. The exception is Patch 3 for which the overlap drops to ∼60% in about a minute. However, Patch 3 might go through a transition and then remain relatively constant throughout the rest of the interval. In Figure 12 test this by showing the percentage overlap compared to the preceding patch shape. The percentage of overlapping pixels no longer drops to a low value but instead appears to vary between 85 and 100%. This means that Patch 3 indeed goes through a transition in the first minutes, and a closer inspection of the contours in Figure 9 reveals that the transition is due to a change to the northern part of the patch (the before relatively dim northwestern part fades away). All of the patches can therefore be considered to have remarkably persistent shapes with above 85% overlap over periods ranging from 4.5 to 8.5 min.
Qualitative Evolution of Patch Shape
In our analysis we have not accounted for any rotation of the patch. If we adjusted patches 2 and 4 for rotation, their shapes could be considered as persistent over a longer time than what is evident from Figure 8. As can be seen in Figure 9, the patches mostly undergo gradual changes and largely maintain their shapes over the time we follow them. Either they get slightly more elongated and narrow, rotate, or a combination of those. We also note that three of the patches (1-3) rotate and elongate toward the magnetic east-west direction, while one of the patches (4) does not seem to obey the same trend.
Implications
This is the first study with a quantitative description of patch shape evolution. It is suggested that the shape of fluctuating patches is governed by the wave resonance/cold plasma region at the magnetospheric equator, and that cold plasma of ionospheric origin acts to keep the region stable (e.g., Li et al., 2012;Liang et al., 2015;Oguti, 1976). Alternatively, it is suggested that conductivity gradients in the ionosphere due to the energetic electron precipitation can modify the shapes of fluctuating auroral patches (Hosokawa et al., 2010). There are however no detailed predictions of patch evolution from these processes.
A remarkably stable patch shape implies two strict requirements: (1) The region of the underlying source mechanism must be stable, and (2) the mapping of the magnetic field lines connecting the patch to the magnetosphere must also be stable. With respect to the latter, this is baffling since these patches occurred during substorm conditions when the magnetosphere undergoes a large-scale reconfiguration. We speculate that the mechanism responsible for the visual part of the fluctuating patch is located not in the distant magnetosphere but much closer to the Earth (Sato et al., 2004), and thus is insensitive to changes in the magnetospheric field line morphology.
How Often Do the Patches Merge or Detach?
It happens that patches merge with one or several nearby patches. Exactly when and why this happens is not known, and neither is it clear what this implies for the underlying cause of the patch. We therefore give a summary of the merging and detaching that the above patches underwent.
Patch 2 starts out as at least two patches. It started to gradually merge into one individual patch from about 15:14 UT (5 min before we start the analysis). However, as described above, the two parts fluctuate in a slightly irregular manner 1.5-3.5 min into the time interval. Patch 2 moves relatively closer to Patch 4 and they merge at about 15:27 UT, a few tens of seconds after we stop the analyses. Also, about 2.5 min earlier the northern part of Patch 4 merges with an adjacent patch and detaches to form a new individual patch. We have not registered any merging and detaching of patches 1 and 3. In only 13 min two of the four patches participate in as much as three events of merging and detaching.
Does the Patch Fluctuate in a Coherent Fashion?
It is known that fluctuating aurora can occur in different spatiotemporal modes (see section 1.3). However, it is not known why the shapes sometimes fluctuate coherently and sometimes not, and the underlying cause of the spatiotemporal variations remains to be determined. We therefore ask whether the patches fluctuate in a coherent fashion. As seen in Figure 10, the three largest patches have temporal correlation coefficients that show negative dependencies on distance. Thus, only one of four patches fluctuates in a coherent fashion.
Time-Delayed Response
As evident from Figure 10, the patches are clearly spatially variable, meaning that the intensity in one part of the patch might have higher/lower intensity and fluctuate less or stronger. To further investigate whether the spatial variation we see is due to so-called expanding or streaming mode fluctuating aurora, where the luminosity moves inside the patch due to a time-delayed response, we find the cross correlation and calculate the phase speed from the distance between the pixels and the time lag of the best fit of their fluctuations. To avoid the possibility that positive and negative phase speeds can cancel each other out, the averages are found separately for the positive and negative phase speeds. The results are shown in Figure 13. The correlation shows the same trend of negative dependence on distance as in Figure 10, but it is much less pronounced as the lowest correlation values have increased from 0.2-0.5 to 0.55-0.6. Thus, by shifting the time series we find a better fit. This is reflected in the phase speeds along the patch reaching as high as −150 km/s to 80 km/s for Patch 1, −90 km/s to 20 km/s for Patch 2, −50 km/s to 30 km/s for Patch 4. Even for Patch 3, which initially is considered relatively coherent, we find a better fit of the fluctuations by shifting them, resulting in negative phase speed of up to 50 km/s. The trend of increasing phase speed per distance is likely a combination of two effects: (1) For small distances the speed is underestimated by our 0.3 s temporal resolution. (2) From the cross correlation we expect the fluctuations to be different, meaning that the best fit can result in a small shift that is not necessarily connected to a time-delayed response, but results in a relatively large phase speed. This can result in an overestimation of the average phase speed at large distances. The phase speeds we find are in agreement with earlier measured expansion speeds of ∼25-250 km/s usually uniformly in all directions, sometimes in a preferred direction
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10.1029/2017JA024405 (Kosch & Scourfield, 1992;. It is clear that the luminosity moves inside the patch due to a time-delayed response and that the patches possibly can be defined as streaming/expanding mode fluctuating aurora.
Slower Varying Background Emissions
The fluctuations of the patches are superposed onto a background of slower varying emissions. This supports the description of Cresswell (1972) that the patchy type of fluctuating auroras generally undergo incomplete intensity fluctuations. The background of slower varying emissions than the intensity fluctuations are therefore treated as an offset and removed. The offset is found from the off/dim values to visualize the spatiotemporal variation of each fluctuation (build up and decay). Movies of the patches corrected for the offset (see the supporting information and Movies S2-S5) show the resulting spatiotemporal variation within the patch. We repeated the cross-correlation analysis (not shown) and do not find any significant changes from the results in Figure 13, and conclude that the background can be treated as mere offset. However, we notice that the offset corrected for the background emissions in the patch surroundings are of comparable brightness to the fluctuation. For example, for Patch 1 the median patch emissions is about 2.7 kR and the median offset is 2.5 kR, while Patch 2 has median patch emissions of about 3.8 kR and a median offset of 1.8 kR.
Together with the observation that the background is not entirely removed between the fluctuations, this might support the suggestion that the background is not merely diffuse emissions but part of the mechanism (Dahlgren et al., 2017).
Implications
It is not clear what causes the variation in fluctuation within the patch. The expanding/streaming spatial fluctuations are, for example, suggested to be accounted for by atmospheric waves (Luhmann, 1979) or by the pitch angle scattering moving to adjacent regions with lower densities of cold plasma as the energy of the waves grows (Tagirov et al., 1999). However, the auroral observational consequences are not quantified. The more recent studies are linking spatial structuring of fluctuating aurora to the higher-frequency fluctuations (>3 Hz) commonly found superimposed on the main emission fluctuation (Nishiyama et al., 2012;Samara & Michell, 2010;). For example, Samara and Michell (2010) found that higher temporal frequencies exist when there are smaller-scale structures present, and Nishiyama et al. (2016) found that the rapid fluctuations were highly localized in substructures of the main form (propagating/moving mode). The spatial structuring was for various reasons found in agreement with chorus rising tone elements, but it is not clear if the rapid fluctuations are caused by completely different mechanisms than those causing the main on-off fluctuation. Neither is it clear whether the rapid fluctuations can explain the observed variation within the patch.
A closer inspection of the patch movies (see the supporting information and Movies S2-S5) show that the spatiotemporal variation is highly irregular. From one fluctuation to the next it varies what part of the patch turns on, or turns on first and in which order, and if there is expansion and/or subtraction and/or streaming. During one on-off fluctuation there can be several smaller fluctuations within a smaller area of the patch (sometimes the whole patch) that does not have sufficient a build-up or decay to be considered an individual fluctuation. We also find individual fluctuations that comprise only part of the patch, for example where one part brightens as another part fades. This supports Kosch and Scourfield (1992), which found no pattern in the occurrence of various modes. They rather found that one form could experience both pure intensity fluctuations and pure spatial fluctuations within 20 s of observation, and that parts of a form can show different fluctuating modes. It is however not yet clear if the above suggestions can explain such irregular spatiotemporal characteristics.
In summary, the intensity of the individual auroral patches does not fluctuate with the same increases and decreases across the patch. The patches' shapes, however, are remarkably persistent. This means that the fluctuating auroral patches are incoherent on scale sizes smaller than the individual patch, and at the same time coherent on the scale of the overall patch. This supports the suggestions that there are separate processes controlling the scale size of the whole patch and the subscales inside the patch.
Do Patches Always Drift With ⃗
E × ⃗ B Velocity? In a nonrotating reference frame (effectively GSE coordinates) the patches drift with 230-287 m/s in a north-eastward direction, which is what typically could be expected for the convection return flow. However, when compared to the SuperDARN convection velocities (corotating) in Figure 11, they are on average only about 50% of the SuperDARN drifts and not in the same direction. Assuming that the SuperDARN drifts are correct, this indicates that the fluctuating patches do not drift with the ⃗ E × ⃗ B velocity.
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Uncertainty in the SuperDARN Drifts
The SuperDARN drifts can, however, be uncertain. The line-of-sight Doppler velocity estimates by SuperDARN are often found to be smaller than the concurrent ion drifts as measured by LEO satellites (Drayton et al., 2005) and velocity measurements by the EISCAT incoherent scatter radar, likely because the high-frequency waves are scattered by small-scale dense structures with refractive indices well below those that are assumed (Gillies et al., 2010). Another issue that can cause uncertainty are the assumptions in deriving the convection solution.
The convection solution (so-called fitted velocities) are partially controlled by the radar line-of-sight measurements where these are available, and empirical data. Our patches are located between radar backscatter at ∼4 MLT (see supporting information Movie S6 of the SuperDARN polar plots that indicate the location of the measured backscatter). Regardless, the temporal and spatial resolution of SuperDARN (2 min and 50 km) is borderline for the patches (lifetimes of some minutes and scale sizes of some 10 km). For these reasons we cannot eliminate the possibility that the uncertainty of the SuperDARN drifts can explain that our patches do not drift with the ⃗ E × ⃗ B velocity.
Patches Drift Relative to Each Other
An example of the complex drift patterns of the patches is that they drift relative to each other. For example, Patches 2 and 4 start out separated by about 40 km and about 8 min later they are adjacent and eventually merge into one patch, suggesting a relative drift speed of roughly 80 m/s, which is comparable to the velocities we measured in the ASI frame of reference corotating with Earth, and thus considerable. Our interpretation of this finding is that the electric field pattern is structured and dynamic and thus not necessarily in conflict with ⃗ E × ⃗ B drift.
Past Findings
The few studies that have focused on finding the drift of fluctuating aurora argues that it moves with the ⃗ E × ⃗ B velocity, in possible disagreement with our results. Scourfield et al. (1983) found the average drift velocities of fluctuating auroral forms to be in excellent agreement with the radar velocities for a few minutes, after which they started to deviate in direction of up to 55 ∘ (magnitudes do not differ by more than 25%) the following 40 min of the event. Nakamura and Oguti (1987) found the drifts of fluctuating auroral patches and arc fragments from time gradients in position-time diagrams (so-called keograms and ewograms) of all-sky TV data. They then displayed the global drift pattern of the two events and found that they resembled the general ionospheric convection pattern measured by radars. Yang et al. (2015) used time-gradients in ewograms to find the eastward velocity component of five fluctuating patches from three events. They found the eastward patch drifts in the range of 156-550 m/s to be slightly larger but in good agreement with the localized eastward convection velocities from SuperDARN. The northward velocity components are not compared. Further, Yang et al. (2017) used the same technique to identify patch east-west velocities from 357 hrs of fluctuating auroral patches events and found velocities ranging from tens of several hundreds m/s in the corotating frame of reference. They argue that the patches are governed by the convection mainly because they mostly move eastward after midnight and westward before midnight. The conclusions of the above mentioned studies are loosely drawn on basis of the east-west velocity component alone and on the general and statistical global velocity patterns.
Implications
If we assume that the SuperDARN drifts are correct, this indicates that the fluctuating patches drift slower than the ⃗ E × ⃗ B velocity. This implies that the underlying mechanism also must drift slower than the ⃗ E × ⃗ B velocity. If the underlying mechanism is controlled by a region of cold plasma at the magnetic equator, this could indicate that the cold plasma does not originate from the ionosphere. If there were to be another source, it would point toward the corotating plasmasphere. The plasmasphere is found to have a rather structured boundary, especially during moderate geomagnetic activity (Sandel et al., 2003), and a plasmaspheric plume has been linked to a subauroral proton arc (Spasojević et al., 2004). However, in the dawn sector auroral oval the occurrence probability of plumes are low (Darrouzet et al., 2008), and the observed patch drifts are found to have a small drift relative to the corotating frame of reference (see Figure 11). Therefore there are no strong alternatives to explain that the underlying mechanism would move differently from the ⃗ E × ⃗ B velocity.
Implied Inconsistency
Above we concluded, on one hand, that the patch shape is maintained to a very high degree and interpreted this as indicative of the mechanism being located at low altitudes and not in the plasma sheet. On the other hand, we found that the patches do not appear to drift with the SuperDARN-determined ⃗ E × ⃗ B velocity and
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concluded that this was in conflict with the frozen-in condition, and thus that the mechanism must be located at higher altitudes. These two findings are clearly in conflict. While the determination of patch shape and drift velocities are highly reliable, it is possible that the SuperDARN drift velocities are uncertain. This would actually be in line with the previous published papers discussed in section 5.5.3, which found that the patches likely drift with the ⃗ E × ⃗ B velocity.
Summary and Conclusions
We have provided objective and quantitative measures of the extent to which pulsating auroral patches maintain their shape, drift, and fluctuate in a coherent fashion. We use ground-based all-sky imager observations that provide good spatial and temporal resolution (3.31 Hz) of fluctuating patches that allow for a separation of spatial and temporal variations. We traced four individual fluctuating patches using a manual contouring technique and found the characteristics of shape evolution, within patch coherency and drift of the patches in a nonrotating reference frame. The characteristics of four fluctuating auroral patches from a single event do not allow for general conclusions. Our patches are located in dawn sector at ∼4 MLT during substorm activity. On the basis of these four patches we conclude the following:
1. For all of the patches their shape can be considered remarkably persistent with 85-100% of the patch being repeated for 4.5-8.5 min. 2. For the three largest patches the temporal correlation coefficient show a negative dependence on distance.
A time-delayed response within all of the patches indicate that the so-called streaming spatiotemporal mode can explain part of the variability. Thus, only one of four patches fluctuates in a coherent fashion. 3. The patches appear to drift differently from the SuperDARN-determined ⃗ E × ⃗ B convection velocity. However, in a nonrotating reference frame the patches drift with 230-287 m/s in a north-eastward direction, which is what typically could be expected for the convection return flow. 4. The patches drift relative to each other. Our interpretation of this finding is that the electric field pattern is structured and dynamic and thus not necessarily in conflict with ⃗ E × ⃗ B drift.
Our interpretation of the findings is that the mechanism is located at lower altitudes and not in the plasma sheet and that the patches likely drift with the ⃗ E × ⃗ B velocity.
Our findings and the findings of Humberset et al. (2016) show that the only parameter that appears to be consistent for fluctuating auroral patches, is their shape. The patches do not fluctuate in a coherent fashion, and the energy deposition is highly variable from one fluctuation to the next. The on-time varies wildly and does not show any correlation to the preceding off-time, nor the peak intensity. This supports the suggestion made by Humberset et al. (2016) that pulsating aurora is a misnomer and that the name fluctuating aurora is more appropriate. This study was supported by the Research Council of Norway under contract 223252. I. R. M. is supported by a Discovery grant from Canadian NSERC. The authors acknowledge the use of SuperDARN data, a collection of radars funded by national scientific funding agencies of Australia, Canada, China, France, Japan, South Africa, United Kingdom, and the United States of America. A special thanks to Robin Barnes at the Johns Hopkins University Applied Physics Laboratory for providing the SuperDARN data, which are also freely available from vt.superdarn.org. The authors acknowledge the use of SuperMAG indices and all-sky imager data from the Multi-spectral Observatory of Sensitive EMCCDs (MOOSE, moose.space.swri.edu). The SuperMAG indices were obtained freely from supermag.jhuapl.com. We greatly acknowledge James Weygand for the ACE solar wind data. MOOSE all-sky imager data can be obtained from Robert G. Michell and Marilia Samara. The data analyzed in this study are available in the data set Humberset et al. (2018).
Figure 1 .
1Geomagnetic indices (top) IMF, (middle) SML/SMU (SuperMAG equivalent of the AE indices) maximum westward/eastward auroral electrojet strength, and (bottom) SMR (SuperMAG equivalent of the SYMH index) symmetric ring current index. The time of the event is highlighted in gray. We use the SuperMAG data set of indices and ACE IMF data which is propagated to the front of the magnetosphere (courtesy of Dr. James Weygand).
Figure 2 .
2For each of the patches (1-4): Evolution, over the time interval analyzed, of a line of pixels crossing the patch (position-time diagrams). The line of pixels are highlighted in color, and the dashed arrows indicate the direction of the sampling. The time of the example image is indicated by a colored tick mark. In step 1 of the technique position-time diagrams are used to decide if the patch is fluctuating and can be considered as one individual and well-defined patch.
Figure 2 Figure 3 .
23shows the evolution for each of the patches (1-4). The line of pixels are highlighted in color and the dashed arrows indicate the direction of the sampling. The time of the example image is indicated by a colored tick mark. The patches are clearly fluctuating in intensity and for each of the patches the variation is convincingly similar across the patch to consider it as an individual patch. However, there is also some intriguing Journal of Geophysical Research: For each of the patches (1-4): Examples of contours that are manually identified by drawing a line around the patch (from top to bottom), where Δt is the time from start of the analysis (relative to the time axis onFigure 2). The patch can then be extracted as all pixels within the closed contour.
Figure 4 .
4Overview of Patch 1. The train of fluctuations in intensity (total emissions within the patch) over the time interval that is analyzed and eight example images where the patch is either considered on/bright or off/dim. The dashed lines indicate the time and patch intensity of the example images.
Figure 5 .Figure 6 .
56Overview of Patch 2 displaying the same parameters asFigure 4. Overview of Patch 3 displaying the same parameters asFigure 4.
Figure 7 .Figure 8 .
78Overview of Patch 4 displaying the same parameters asFigure 4. Percentage of patch (pixels within the contour) that overlaps with the patch at the first fluctuation.
Figure 9 .
9For each of the patches (1-4): Twenty of the contours, including the first and last contours, in the patch frame of reference. The temporal color scale visualizes the evolution of the patch shape. coefficient above 0.5, Patch 3 fluctuates in a coherent fashion, while the fluctuations of patches 1 and 4 could be considered coherent for about 15 km and Patch 2 for about 50 km. However, regardless of correlation value the patches 1, 2, and 4 appear to indicate a nearly linear dependence on distance. Thus, for only one of four patches do we find that the entire patch fluctuates in a coherent fashion.
Figure 11 .
11The drift velocities of the patches compared to the ⃗ E × ⃗ B velocity (SuperDARN) shown in the AACGM magnetic coordinate system (corotating). SuperDARN = Super Dual Auroral Radar Network; AACGM = Altitude Adjusted Corrected Geomagnetic.
Figure 12 .
12For each of the patches (1-4): Percentage of patch (pixels within the contour) that overlaps with the patch at the peak of the latter fluctuation.
Figure 13 .
13For each of the patches (1-4): The average of the best fit cross correlations and phase speeds between all fluctuations of the first pixel to the pixels sampled along the patch as shown inFigure 10. To avoid the possibility that positive and negative phase speeds can cancel each other out, the average of the positive phase speeds are found separately from the average of the negative phase speeds. The standard error is shown as error bars.
). The geomagnetic conditions are moderately disturbed, with a moderately southward IMF (top panel) and a SML index (SuperMAG equivalent of the AL index, middle panel) of the maximum westward auroral electrojet strength showing many hours of almost continuous substorm activity prior to our event. Our event (highlighted in gray) is during the second substorm. Before 15:00 UT the fluctuating aurora covers the southern part of the sky with large east-west structures, which seem to fluctuate in a propagating and/or streaming fashion. At about 15:00 UT the fluctuating aurora starts to break up into more well-defined patches. The four selected patches are located at ∼4 magnetic local time (MLT) and ∼65 ∘ magnetic latitude. Altogether, the data set includes hundreds of fluctuations recorded from about 15:17 to 15:28 UT. The research data are available in the data setHumberset et al. (2018).
Patch emissions [kR]0
2
4
6
Time from start [min]
1.5×10 4
2.0×10 4
2.5×10 4
3.0×10 4
Patch emissions [kR]0
2
4
6
8
Time from start [min]
2.0×10 4
2.5×10 4
3.0×10 4
3.5×10 4
4.0×10 4
4.5×10 4
Acknowledgments
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| []
|
[
"Parameterized Problems Complete for Nondeterministic FPT time and Logarithmic Space *",
"Parameterized Problems Complete for Nondeterministic FPT time and Logarithmic Space *"
]
| [
"Hans L Bodlaender \nDepartment of Information and Computing Sciences\nUtrecht University\nthe Netherlands\n",
"Carla Groenland \nDepartment of Information and Computing Sciences\nUtrecht University\nthe Netherlands\n",
"Jesper Nederlof \nDepartment of Information and Computing Sciences\nUtrecht University\nthe Netherlands\n",
"Céline M F Swennenhuis \nDepartment of Mathematics and Computer Science\nEindhoven University of Technology\nThe Netherlands\n"
]
| [
"Department of Information and Computing Sciences\nUtrecht University\nthe Netherlands",
"Department of Information and Computing Sciences\nUtrecht University\nthe Netherlands",
"Department of Information and Computing Sciences\nUtrecht University\nthe Netherlands",
"Department of Mathematics and Computer Science\nEindhoven University of Technology\nThe Netherlands"
]
| []
| Let XNLP be the class of parameterized problems such that an instance of size n with parameter k can be solved nondeterministically in time f (k)n O(1) and space f (k) log(n) (for some computable function f ). We give a wide variety of XNLP-complete problems, such as List Coloring and Precoloring Extension with pathwidth as parameter, Scheduling of Jobs with Precedence Constraints, with both number of machines and partial order width as parameter, Bandwidth and variants of Weighted CNF-Satisfiability and reconfiguration problems. In particular, this implies that all these problems are W [t]-hard for all t. This also answers a long standing question on the parameterized complexity of the Bandwidth problem. | 10.1109/focs52979.2021.00027 | [
"https://arxiv.org/pdf/2105.14882v1.pdf"
]
| 235,254,493 | 2105.14882 | 6baab48436a892e831b30364604773ed745f8326 |
Parameterized Problems Complete for Nondeterministic FPT time and Logarithmic Space *
Hans L Bodlaender
Department of Information and Computing Sciences
Utrecht University
the Netherlands
Carla Groenland
Department of Information and Computing Sciences
Utrecht University
the Netherlands
Jesper Nederlof
Department of Information and Computing Sciences
Utrecht University
the Netherlands
Céline M F Swennenhuis
Department of Mathematics and Computer Science
Eindhoven University of Technology
The Netherlands
Parameterized Problems Complete for Nondeterministic FPT time and Logarithmic Space *
Let XNLP be the class of parameterized problems such that an instance of size n with parameter k can be solved nondeterministically in time f (k)n O(1) and space f (k) log(n) (for some computable function f ). We give a wide variety of XNLP-complete problems, such as List Coloring and Precoloring Extension with pathwidth as parameter, Scheduling of Jobs with Precedence Constraints, with both number of machines and partial order width as parameter, Bandwidth and variants of Weighted CNF-Satisfiability and reconfiguration problems. In particular, this implies that all these problems are W [t]-hard for all t. This also answers a long standing question on the parameterized complexity of the Bandwidth problem.
Introduction
Already since the 1970's, an important paradigm in classical complexity theory has been that an increased number of alternations of existential and universal quantifiers increases the complexity of search problems: This led to the central definition of the polynomial hierarchy [42], whose study resulted in cornerstone results in complexity theory such as Toda's theorem and lower bounds for Our contribution We show that the class XNLP (i.e., N [f poly, f log]) can play an important role in establishing the parameterized complexity of a large collection of well studied problems, ranging from abstract problems on different types of automata (see e.g. [21] or later in this paper), logic, graph theory, scheduling, and more. In this paper, we give a number of different examples of problems that are complete for XNLP. These include Bandwidth, thus indirectly answering a question that was posed over 25 years ago.
Problem
Source Longest Common Subsequence + [21] Timed Non-determ. Cellular Automaton + [21], see Subsection 2.5 Chained CNF-Satisfiability + Subsection 3. [46]; Appendix A Table 1: An overview of XNLP-complete problems is given. For problems marked with +, the source also gives an XNLP-hardness or completeness proofs for variants of the stated problem. We use the abbreviations CL = Clique, IS = Independent Set, DS = Dominating Set, pw = parameterized by pathwidth.
In Table 1, we list the problems shown to be XNLP-complete in either this paper or by Elberfeld et al. [21]. Figure 1 shows for the problems from which problem the reduction starts to show XNLP-hardness.
Often, membership in XNLP can be seen by looking at the algorithm that establishes membership in XP. Many problems in XNLP typically have a dynamic programming algorithm that sequentially builds tables, with each individual table entry expressible with O(f (k) log n) bits. We then get membership in XNLP by instead of tabulating all entries of a table, guessing one entry of the next table -the step resembles the text-book transformation between a deterministic and non-deterministic finite automaton.
Interestingly, hardness for the class XNLP also has consequences for the use of memory of deterministic parameterized algorithms. Pilipczuk and Wrochna [39] conjecture that Longest Common Subsequence (variant 1) has no XP algorithm that runs in f (k)n c space, for a computable function f and constant c; if this conjecture holds, then no XNLP-hard problem has such an algorithm. See Section 5 for more details.
When a problem is XNLP-hard, it is also hard for each class W [t] (see Lemma 2.2). Thus, XNLP-hardness proofs are also a tool to show hardness for W [t] for all t. In this sense, our results strengthen existing results from the literature: for example, List Coloring and Precoloring Extension parameterized by pathwidth (or treewidth) were known to be W [1]-hard [22], and Precedence Constraint K-processor Scheduling parameterized by the number of processors K was known to be W [2]-hard [8]. Our XNLP-hardness proofs imply hardness for W [t] for all t. Moreover, our XNLP-hardness proofs are often simpler than the existing proofs that problems are hard for W [t] for all t.
Related to the class XNLP is the class XNL: the parameterized problems that can be solved by a nondeterministic algorithm that uses f (k) log n space. There is no explicit time bound, but we can freely add a time bound of 2 f (k) log n , and thus XNL is a subset of XP. XNL can be seen as the parameterized counterpart of NL. Amongst others, XNL was studied by Chen et al. [13], who showed that Compact Turing Machine Computation is complete for XNL.
Hardness for a class is always defined with respect to a class of reductions. In our proofs, we use parameterized logspace reductions (or, in short, pl-reductions). A brief discussion of other reductions can be found in Subsection 5.2.
Paper overview. In Section 2, we give a number of preliminary definitions and results. In Section 3 we introduce three new problems and prove that they are XNLP-complete. In Section 4 we then use these problems as building blocks, to prove other problems to be either XNLP-complete or XNLP hard. For each of the problems, its background and a short literature review specific to it will be given inside its relevant subsection. Some related variants have been moved to the appendix. Final comments and open problems are given in Section 5.
Preliminaries
In this section we formally define the class XNLP and give some preliminary results.
The section is organized as follows: first we introduce some basic notions in Subsection 2.1, next we formally define the class XNLP in Subsection 2.2. In Subsection 2.3 we then introduce the type of reductions that will be used in this paper and in Subsection 2.4 we go over some preliminary results. Subsection 2.5 ends the section with a discussion of Cellular automata, for which Elberfeld et al. [21] already established it was XNLP complete. From this problem we will (indirectly) derive the XNLP-hardness for all other XNLP-hard problems in this paper.
Basic notions
We assume the reader to be familiar with a number of notions from complexity theory, parameterized algorithms, and graph theory. A few of these are reviewed below, along with some new and less well-known notions.
We use interval notation for sets of integers, i.e., [a, b] = {i ∈ Z | a ≤ i ≤ b}. All logarithms in this paper have base 2. N denotes the set of the natural numbers {0, 1, 2, . . .}, and Z + denotes the set of the positive natural numbers {1, 2, . . .}.
Definition of the class XNLP
In this paper, we study parameterized decision problems, which are subsets of Σ * × N, for a finite alphabet Σ. The following notation is used, also by e.g. [21], to denote classes of (non-)deterministic parameterized decision problems with a bound on the used time and space. Here, we use the following notations: poly for a polynomial function in the input size; log for O(log n); n for the input size; f for a computable function of the parameter; ∞ if there is no a priory bound for the resource. Let D[t, s] denote the class of parameterized decision problems that can be solved by a deterministic algorithm in t time and s space and let N [t, s] be analogously defined for non-deterministic algorithms. Thus, A special role in this paper is played by the class N [f poly, f log]: the parameterized decision problems that can be solved by a non-deterministic algorithm that simultaneously uses at most f (k)n c time and at most f (k) log n space, on an input (x, k), where x can be denoted with n bits, f a computable function, and c a constant. Because of the special role of this class, we use the shorter notation XNLP.
XNLP is a subclass of the class XNL, which was studied by Chen et al. [13]. XNL is the class of problems solvable with a non-deterministic algorithm in f (k) log n space (f , k, n as above), i.e, XNL is the class N [∞, f log].
We assume that the reader to be familiar with notions from parameterized complexity, such as XP, W [1], W [2], . . . , W [P ] (see e.g. [17]). For classes of parameterized problems, we can often make a distinction between non-uniform (a separate algorithm for each parameter value), and uniform. Throughout this paper, we look at the uniform variant of the classes, but we also will assume that functions f of the parameter in time and resource bounds are computable -this is called strongly uniform by Downey and Fellows [17].
Reductions
Hardness for a class is defined in terms of reductions. We mainly use parameterized logspace reductions, which are a special case of fixed parameter tractable reductions. Both are defined below; the definitions are based upon the formulations in [21]. Two other types of reductions are briefly discussed in the Conclusion (Section 5.2.)
• A parameterized reduction from a parameterized problem Q 1 ⊆ Σ * 1 × N to a parameterized problem Q 2 ⊆ Σ * 2 × N is a function f : Σ * 1 × N → Σ * 2 × N, such that the following holds.
1. For all (x, k) ∈ Σ * 1 × N, (x, k) ∈ Q 1 if and only if f ((x, k)) ∈ Q 2 . 2. There is a computable function g, such that for all (x, k) ∈ Σ * 1 × N, if f ((x, k)) = (y, k ), then k ≤ g(k).
• A parameterized logspace reduction or pl-reduction is a parameterized reduction for which there is an algorithm that computes f ((x, k)) in space O(g(k) + log n), with g a computable function and n = |x| the number of bits to denote x.
• A fixed parameter tractable reduction or fpt-reduction is a parameterized reduction for which there is an algorithm that computes f ((x, k)) in time O(g(k)n c ), with g a computable function, n = |x| the number of bits to denote x and c a constant.
In the remainder of the paper, unless stated otherwise, completeness for XNLP is with respect to pl-reductions.
Preliminary results on XNLP
We give some easy observations that relate XNLP to other notions from parameterized complexity. The following easy observation can be seen as a special case of the fact that N [∞, S(n)] ⊆ D [2 S(n) , ∞], see [2,Theorem 4.3]. Proof. Using standard techniques, we can transform the non-deterministic algorithm to a deterministic algorithm that employs dynamic programming: tabulate all reachable configurations of the machine (a configuration is a tuple, consisting of the contents of the work tape, the state of the machine, and the position of the two headers). From a configuration, we can compute all configurations that can be reached in one step, and thus we can check if a configuration that has an accepting state can be reached.
The number of such configurations is bounded by the product of a single exponential of the size of the work tape (i.e., at most 2 f (k) log n = n f (k) for some computable function f ), the constant number of states of the machine, and the O(f (k) log n) · n number of possible pairs of locations of the heads, and thus bounded by a function of the form n g(k) with g a computable function.
Lemma 2.2. If a parameterized problem Q is XNLP-hard, then it is hard for each class W [t] for all t ∈ Z + .
Proof. Observe that the W [t]-complete problem Weighted t-Normalized Satisfiability belongs to XNLP. (In Weighted t-Normalized Satisfiability, we have a Boolean formula with parenthesis-depth t and ask if we can satisfy it by setting exactly k variables to true and all others to false; we can non-deterministically guess which of the k Boolean variables are true; verifying whether this setting satisfies the formula can be done with O(t + k log n) bits of space, see e.g. [17].)
Each problem in W [t] has an fpt-reduction to Weighted t-Normalized Satisfiability, and the latter has a pl-reduction (which is also an fpt-reduction) to any XNLP-hard problem Q. The transitivity of fpt-reductions implies that Q is then also hard for W [t]. Lemma 2.3 (Chen et al. [13]). If N L = P , then there are parameterized problems in FPT that do not belong to XNL (and hence also not to XNLP).
Proof. Take a problem Q that belongs to P, but not to NL. Consider the parameterized problem Q with (x, k) ∈ Q if and only if x ∈ Q. (We just ignore the parameter part of the input.) Then Q belongs to FPT, since the polynomial time algorithm for Q also solves Q . If Q is in XNL, then there is an algorithm that solves Q in (non-deterministic) logarithmic space, a contradiction. So Q belongs to FPT but not to XNL.
Chen et al. [13] introduce the following problem. Theorem 2.4 (Chen et al. [13]). CNTMC is XNL-complete under pl-reductions.
It is possible to show XNLP-completeness for a 'timed' variant of this problem.
Timed CNTMC
Input: the encoding of a non-deterministic Turing Machine M ; the encoding of a string x over the alphabet of the machine; an integer T given in unary. Parameter: k. Question: Is there an accepting computation of M on input x that visits at most k cells of the work tape and uses at most T time?
The fact that the time that the machine uses is given in unary, is needed to show membership in XNLP.
Theorem 2.5. Timed CNTMC is XNLP-complete.
We state the result without proof, as the proof is similar to the proof of Theorem 2.4 from [13], and we do not build upon the result. We instead start with a problem on cellular automata which was shown to be complete for XNLP by Elberfeld et al. [21]. We discuss this problem in the next subsection. Elberfeld et al. [21] show a number of other problems to be XNLP-complete, including a timed version of the acceptance of multihead automata, and the Longest Common Subsequence problem, parameterized by the number of strings. The latter result is discussed in the Conclusion, Section 5.1.
Cellular automata
In this subsection, we discuss one of the results by Elberfeld et al. [21]. Amongst the problems that are shown to be complete for XNLP by Elberfeld et al. [21], of central importance to us is the Timed Non-deterministic Cellular Automaton problem. We use the hardness of this problem to show the hardness of Chained CNF-Satisfiability in Subsection 3.1.
In this subsection, we describe the Timed Non-deterministic Cellular Automaton problem, and a variant. We are given a linear cellular automaton, a time bound t given in unary, and a starting configuration for the automaton, and ask if after t time steps, at least one cell is in an accepting state.
More precisely, we have a set of states S. We assume there are two special states s L and s R which are used for the leftmost and rightmost cell. A configuration is a function c : {1, . . . , q} → S, with c(1) = s L , c(q) = s R and for i ∈ [2, q − 1], c(i) ∈ S \ {s L , s R }. We say that we have q cells, and in configuration c, cell i has state c(i). The machine is further described by a collection of
4-tuples T in S × (S \ {s L , s R }) × S × (S \ {s L , s R }).
At each time step, each cell i ∈ [2, q] reads the 3-tuple (s 1 , s 2 , s 3 ) of states given by the current states of the cells i − 1, i and i + 1 (in that order). If there is no 4-tuple of the form (s 1 , s 2 , s 3 , s 4 ) for some s 4 ∈ S, then the machine halts and rejects; otherwise, the cell selects an s 4 ∈ S with (s 1 , s 2 , s 3 , s 4 ) ∈ T and moves in this time step to state s 4 . (In a non-deterministic machine, there can be multiple such states s 4 and a non-deterministic step is done. For a deterministic cellular automaton, for each 3-tuple (s 1 , s 2 , s 3 ) there is at most one 4-tuple (s 1 , s 2 , s 3 , s 4 ) ∈ T .) Note that the leftmost and rightmost cell never change state: their states are used to mark the ends of the tape of the automaton.
Timed Non-deterministic Cellular Automaton
Input: Cellular automaton with set of states S and set of transitions T ; configuration c on q cells; integer in unary t; subset A ⊆ S of accepting states. Parameter: q. Question: Is there an execution of the machine for exactly t steps with initial configuration c, such that at time t at least one cell of the automaton is in A?
We will build on the following result.
Theorem 2.6 (Elberfeld et al. [21]). Timed Non-deterministic Cellular Automaton is XNLP-complete.
We recall that the class, denoted by XNLP in the current paper, is called N [f poly, f log] in [21].
Elberfeld et al. [21] state that asking that all cells are in an accepting state does not make a difference, i.e., if we modify the Timed Non-deterministic Cellular Automaton problem by asking if all cells are in an accepting state at time t, then we also have an XNLP-complete problem.
We also discuss a variant that can possibly be useful as another starting point for reductions.
Timed Non-halting Non-deterministic Cellular Automaton Input: Cellular automaton with set of states S and set of transitions T ; configuration c on q cells; integer in unary t; subset A ⊆ S of accepting states. Parameter: q. Question: Is there an execution of the machine for exactly t steps with initial configuration c, such that the machine does not halt before time t?
Corollary 2.7. Timed Non-halting Non-deterministic Cellular Automaton is XNLP-complete.
Proof. Membership in XNLP follows in the same way as for Timed Nondeterministic Cellular Automaton, see [21]; observe that we can store a configuration using q log |S| bits. Hardness follows by modifying the automaton as follows. We take an automaton that accepts, if and only if at time t all cells are in an accepting state. Now, we enlarge the set of states as follows: for each time step t ∈ [0, t], and each state s ∈ S \ {s L , s R }, we create a state s t . The initial configuration c is modified to c by setting c (i) = s 0 for i ∈ [2, q − 1] when c(i) = s. We enlarge the set of transitions as follows. For each t ∈ [0, t − 1] and (s 1 , s 2 , s 3 , s 4 ) ∈ T , we create a transition (s t 1 , s t 2 , s t 3 , s t +1
4
) in the new set of transitions. In this way, each state of the machine also codes the time: at time t all cells except the first and last will have a state of the form s t .
We run the machine for one additional step, i.e., we increase t by one. We create one additional accepting state s a . For each accepting state s ∈ A, we make transitions (x, s t , y, s a ) for all possible values x and y can take. When s ∈ A, then there are no x, y, z for which there is a transition of the form (x, s t , y, z). This ensures that a cell has a possible transition at time t if and only if it is in an accepting state. In particular, when all states are accepting, all cells have a possible transition at time t; if there is a state that is not accepting at time t, then the machine halts.
Building Blocks
In this section, we introduce three new problems and prove that they are XNLPcomplete, namely Chained CNF-Satisfiability, Chained Multicolored Clique and Accepting NNCCM. These problems are called building blocks, as their main use is proving XNLP-hardness for many other problems (see Figure 1).
Note that even though the XNLP-hardness of many problems in this paper are also (indirectly) reduced from List Coloring, this problem will be discussed in Section 4, as it is a well-studied problem with applications.
Chained CNF-Satisfiability
In this subsection we give a useful starting point for our transformations: a variation of Satisfiability which we call Chained Weighted CNF-Satisfiability. The problem can be seen as a generalization of the W [1]-hard problem Weighted CNF-Satisfiability [15].
Chained Weighted CNF-Satisfiability Input: r disjoint sets of Boolean variables X 1 , X 2 , . . . X r , each of size q; integer k ∈ N; Boolean formulas F 1 , F 2 , . . . , F r−1 , where each F i is an expression in conjunctive normal form on variables X i ∪ X i+1 . Parameter: k. Question: Is it possible to satisfy the formula F 1 ∧ F 2 ∧ · · · ∧ F r by setting exactly k variables to true from each set X i and all others to false?
Our main result in this subsection is the following. We will also prove a number of variations later. Proof. Membership in XNLP is easy to see. Indeed, for i from 1 to r, we non-deterministically guess which variables in each X i are true, and keep the indexes of the true variables in memory for the two sets X i , X i+1 . Verifying F i (X i , X i+1 ) can easily be done in logarithmic space and linear time.
To show hardness, we transform from Timed Non-deterministic Cellular Automaton.
For each time step t , each cell r ∈ [1, q], and each state s ∈ S, we have a Boolean variable x t ,r,s with x t ,r,s denoting whether the rth cell of the automaton at time t is in state s. For each time step t ∈ [1, t − 1], each cell r ∈ [2, q], and each transition z ∈ T we have a variable y t ,r,z that expresses that cell r uses transition z at time t .
We will build a Boolean expression and partitions of the variables, such that the expression is satisfiable by setting exactly k variables to true from each set in the partition if and only if the machine can reach time step t with at least one cell in accepting state starting from the initial configuration. The partition is based on the time of the automaton: for each time step t , the set X t consists of all variables of the form x t ,r,s and y t ,r,s . For each set X t we require that exactly 2q − 2 variables are set to true.
The formula has the following ingredients.
• At each step in time, each cell has exactly one state. Moreover, it uses a transition (unless at the first or final cell). To encode that we have at least one state, we use the expression s∈S x t ,r,s for each t ∈ [1, t − 1] and r ∈ [1, q]. To encode that there is at least one transition, we use the clause z∈T y t ,r,z for all t ∈ [1, t − 1] and r ∈ [2, q − 1]. Since we need to set exactly 2q − 2 variables to true from X t , and this is the total allowed amount, the pigeonhole principle shows that for each step in time t and each cell r, at most (and hence, exactly) one state s exists for which variable x t ,r,s is true. Similarly, for all cells apart from the first and last, there is exactly one transition z for which y t ,r,z is true.
• We start in the initial configuration. We encode this using clauses with one literal x 0,r,s whenever cell r has state s in the initial configuration.
• We end in an accepting state. This is encoded by s∈A r
x t,r,s .
• Left and right cells do not change. We add clauses x t ,1,s L and x t ,q,s R with one literal for all time steps t .
• If a cell has a value at a time t > 0, then there was a transition that caused it. This is encoded by
x t ,r,s ⇒ (s1,s2,s3,s)∈T y t −1,r,(s1,s2,s3,s) .
This is expressed in conjunctive normal form as ¬(x t ,r,s ) ∨ (s1,s2,s3,s)∈T y t −1,r,(s1,s2,s3,s) .
• If a transition is followed, then the cell and its neighbors had the corresponding states. For each time step t ∈ [1, t − 1], cell r ∈ [2, q − 1] and transition z = (s 1 , s 2 , s 3 , s 4 ) ∈ T , we express this as
y t ,r,z ⇒ (x t ,r−1,s1 ∧ x t ,r,s2 ∧ x t ,r+1,s3 ) .
We can rewrite this to the three clauses ¬(y t ,r,z )∨x t ,r−1,s1 , and ¬(y t ,r,z )∨ x t ,r,s2 , and ¬(y t ,r,z ) ∨ x t ,r+1,s3 . Since each 'inner' cell has y t,
The last two steps ensure that the transition chosen from the y-variables agrees with the states chosen from x-variables. It is not hard to see that we can build the formula with parameterized logarithmic space and in polynomial time, and that the formula is of the required shape.
A special case of the problem is when all literals that appear in the formulas F i are positive, i.e., we have no negations. We call this special case Chained Weighted Positive CNF-Satisfiability. Proof. We modify the proof of the previous result. Note that we can replace each negative literal by the disjunction of all other literals from a set where exactly one is true, i.e., we may replace ¬(x t ,r,s ) and ¬(y t ,r,z ) by s =s x t ,r,s and z =z y t ,r,z respectively. The modification can be carried out in logarithmic space and polynomial time, and gives an equivalent formula. Thus, the result follows.
A closer look at the proof of Theorem 3.1 shows that F 2 = F 3 = · · · = F r−2 , and more specifically, we have a condition on X 1 , a condition on X r , and identical conditions on all pairs X i ∪ X i+1 with i from 1 to r − 1. Thus, we also have XNLP-completeness of the following special case:
Regular Chained Weighted CNF-Satisfiability Input: r sets of Boolean variables X 1 , X 2 , . . . X r , each of size q; an integer k ∈ N; Boolean formulas F 0 , F 1 , F 2 in conjunctive normal form, where F 0 and F 2 are expressions on q variables, and F 1 is an expression on 2q variables. Parameter: k. Question: Is it possible to satisfy the formula
F 0 (X 1 ) ∧ 1≤i≤r−1 F 1 (X i , X i+1 ) ∧ F 2 (X r )
by setting exactly k variables to true from each set X i and all others to false? Also, the argument in the proof of Theorem 3.2 can be applied, and thus Regular Chained Weighted Positive CNF-Satisfiability (the variant of the problem above where all literals in F 0 , F 1 , and F 2 are positive) is XNLPcomplete.
For a further simplification of our later proofs, we obtain completeness for a regular variant without the first and last of the three formulas.
Regular Chained Weighted CNF-Satisfiability -II Given: r sets of Boolean variables X 1 , X 2 , . . . X r , each of size q; integer k ∈ N ; Boolean formula F 1 , which is in conjunctive normal form and an expression on 2q variables. Parameter: k. Question: Is it possible to satisfy the formula
1≤i≤r−1 F 1 (X i , X i+1 )
by setting exactly k variables to true from each set X i and all others to false?
Theorem 3.3. Regular Chained Weighted CNF-Satisfiability -II is XNLP-complete.
Proof. The idea of the proof is to add the constraints from F 0 and F 2 to F 1 , but to ensure that they are only 'verified' at the start and at the end of the chain.
To achieve this, we add variables t i,j for i ∈ [1, r] and j ∈ [1, r], with t i,j part of X i . We increase the parameter k by one. The construction is such that t i,j is true, if and only if i = j; t i,1 implies all constraints from F 0 , and t i,r implies all constraints from F 2 . The details are as follows.
• We ensure that for all i ∈ [1, r], exactly one t i,j is true. This can be done by adding a clause 1≤j≤r t i,j .
As the number of disjoint sets of variables that each have at least one true variable still equals k (as we increased both the number of these sets and k by one), we cannot have more than one true variable in the set.
• For all i ∈ [1, r] and j ∈ [1, r], we enforce the constraint t i,j ⇔ t i+1,j+1 by adding the clauses ¬t i,j ∨ t i+1,j+1 and t i,j ∨ ¬t i+1,j+1 .
• We add a constraint that ensures that t i,1 is false for all i ∈ [2, r]. This can be done by adding the clause with one literal ¬t i+1,1 to the formula F 1 (X i , X i+1 ), i.e., we have a condition on a variable that is an element of the set given as second parameter. Together with the previous set of constraints, this ensures that t i,1 is true then i = 1.
• Similarly, we add a constraint that ensures that t i,r is false for i < r. This is done by adding a clause with one literal ¬t i,r to F 1 (X i , X i+1 ).
• We add a constraint of the form t 1 → F 0 (X) to F 1 ; for all i, the variable t 1 is substituted by t i,1 and X by X i .
• We add a constraint of the form t r → F 2 (X) to F 1 ; for all i, the variable t 1 is substituted by t i,1 and X by X i .
The first four additional constraints given above ensure that for all i ∈ [1, r] and j ∈ [1, r], t i,j is true if and only if i = j. Thus, the fifth constraint enforces F 0 (X 1 ) (since t 1,1 has to be true); for i > 1, this constraint has no effect. Likewise, the sixth constraint enforces F 2 (X r ). Hence, the new set of constraints is equivalent to the constraints for the first version of Regular Chained Weighted CNF-Satisfiability.
Using standard logic operations, the constraints can be transformed to conjunctive normal form; one easily can verify the time and space bounds.
Again, with a proof identical to that of Theorem 3.2, we can show that the variant with only positive literals (Regular Chained Weighted Positive CNF-Satisfiability -II) is XNLP-complete.
From the proofs above, we note that each set of variables X i can be partitioned into k subsets, and a solution has exactly one true variable for each subset, e.g., for each t , exactly one x t ,r,s is true in the proof of Theorem 3.1. This still holds after the modification in the proofs of the later results. We define the following variant.
Partitioned Regular Chained Weighted CNF-Satisfiability Input: r sets of Boolean variables X 1 , X 2 , . . . X r , each of size q; an integer k ∈ N ; Boolean formula F 1 , which is in conjunctive normal form and an expression on 2q variables; for each i, a partition of X i into X i,1 , . . . , X i,k with for all i 1 , i 2 , j: |X i1,j | = |X i2,j |. Parameter: k. Question: Is it possible to satisfy the formula
1≤i≤r−1 F 1 (X i , X i+1 )
by setting from each set X i,j exactly 1 variable to true and all others to false?
Chained Multicolored Clique
The Multicolored Clique problem is an important tool to prove fixed parameter intractability of various parameterized problems. It was independently introduced by Pietrzak [38] (under the name Partitioned Clique) and by Fellows et al. [23].
In this paper, we introduce a chained variant of Multicolored Clique. In this variant, we ask to find a sequence of cliques, that are overlapping with the previous and next clique in the chain.
Chained Multicolored Clique
Input: Graph G = (V, E); partition of V into sets V 1 , . . . , V r , such that for each edge {v, w} ∈ E, if v ∈ V i and w ∈ V j , then |i − j| ≤ 1; function f : V → {1, 2, . . . , k}. Parameter: k. Question: Is there a subset W ⊆ V such that for each i ∈ [1, r], W ∩ (V i ∪ V i+1 ) is a clique, and for each i ∈ [1, r] and each j ∈ [1, k], there is a vertex w ∈ V i ∩ W with f (w) = j?
Thus, we have a clique with 2k vertices in V i ∪ V i+1 for each i ∈ [1, r − 1], with for each color a vertex with that color in V i and a vertex with that color in V i+1 . Importantly, the same vertices in V i are chosen in the clique for
V i−1 ∪ V i as for V i ∪ V i+1 for each i ∈ [2, r − 1]
. Below, we call such a set a chained multicolored clique.
Theorem 3.5. Chained Multicolored Clique is XNLP-complete.
Proof. Membership in XNLP is easy to see: iteratively guess for each V i which vertices belong to the clique. We only need to keep the clique vertices in V i−1 and V i in memory. We now prove hardness via a transformation from List Coloring parameterized by pathwidth (which is proved XNLP-complete in Subsection 4.1). Suppose that we are given a graph G = (V, E), a path decomposition (X 1 , . . . , X r ) of width k for G and a list of colors L v for each vertex v ∈ V .
First, we build an equivalent instance of List Coloring parameterized by pathwidth with the property that all the sets X i have size exactly k + 1. This can be done by adding k + 1 − |X i | new isolated vertices to each set X i with |X i | < k + 1, which obtain the color list {1}.
From this, we build an instance of Chained Multicolored Clique as follows. For each X i , we choose a bijection f i :
X i → {1, . . . , k + 1}. For each vertex v ∈ X i and each color c ∈ S v , we create a vertex w i,v,c in V i and color it f (w i,v,c ) = f i (v).
Thus, for each vertex v, we have created a vertex for each combination (c, i) of a possible color and possible bag X i for the vertex to appear. We have the following edges.
• We have an edge {w i,v,c , w i,v ,c } if c = c or {v, v } ∈ E, for all v, v ∈ X i with v = v , c ∈ S v , and c ∈ S v . • We have an edge {w i,v,c , w i+1,v,c } for all v ∈ X i ∩ X i+1 . • We have an edge {w i,v,c , w i+1,v ,c } if c = c or {v, v } ∈ E, for all v ∈ X i , v ∈ X i+1 with v = v , c ∈ S v , and c ∈ S v .
Let H = (V 1 , . . . , V r ) be the resulting graph (with coloring f ).
Claim 3.6. H has a chained multicolored clique if and only if G has a list coloring.
Proof. Suppose that G has a list coloring c.
For each i ∈ [1, r] and v ∈ X i , select the vertex w i,v,c(v)
. It is easy to verify that the resulting set is a chained multicolored clique. Suppose now that H has a chained multicolored clique S. First, note that for each i ∈ [1, r] and v ∈ X i , there is exactly one vertex of the form w i,v,γ in S. There can be at most one such vertex because {w i,v,γ , w i,v,γ } is not an edge of H, and there has to be at least one since all vertices of color
f i (v) in V i are of the form w i,v,γ . Consider a vertex v ∈ X i ∩ X i+1 . Let γ, γ be such that w i,v,γ , w i+1,v,γ ∈ S. If γ = γ , then w i,v,γ is not adjacent to w i+1,v,cγ . So we can define c(v) as the unique γ for which w i,v,γ ∈ S for i such that v ∈ X i . Note that c(v) is in the list of v.
It remains to show that the function c is a proper coloring. Consider an edge {v, v } ∈ E. By the definition of path decomposition, there is a bag X i with v, v ∈ X i . By the discussion above, both w i,v,c(v) ∈ S and w i,v ,c(v ) ∈ S, and so these vertices are adjacent. As {v, v } ∈ E, the construction of H shows that we must have c(v) = c(v ). This implies that c is a proper coloring.
It is not hard to see that H and f can be constructed in logarithmic space and fpt time; the result follows.
A simple variation is the following. We are given a graph G = (V, E), a partition of V into sets V 1 , . . . , V r with the property that for each edge {v, w} ∈ E, if v ∈ V i and w ∈ V j then |i−j| ≤ 1, and a coloring function f : V → {1, 2, . . . , k}. A chained multicolored independent set is an independent set S with the property that for each i ∈ [1, r] and each color j ∈ [1, k], the set S contains exactly one vertex v ∈ V i of color f (v) = j. The Chained Multicolored Independent Set problem asks for the existence of such a chained multicolored independent set, with the number of colors k as parameter. We have the following simple corollary.
Corollary 3.7. Chained Multicolored Independent Set is XNLP-complete.
Proof. This follows directly from Theorem 3.5, by observing that the following 'partial complement' of a partitioned graph G = (V 1 ∪ · · · V r , E) can be constructed in logarithmic space: create an edge {v, w} if and only if there is an i
with v ∈ V i and w ∈ V i ∪ V i+1 , v = w and {v, w} ∈ E.
Non-decreasing counter machines
In this subsection, we introduce a new simple machine model, which can also capture the computational power of XNLP (see Theorem 3.8). This model will be a useful stepping stone when proving XNLP-hardness reductions in Section 4.
A Nondeterministic Nonincreasing Checking Counter Machine (or: NNCCM) is described by a 3-tuple (k, n, s), with k and n positive integers, and s = (s 1 , . . . , s r ) a sequence of 4-tuples (called checks). For each i ∈ {1, . . . , r}, the 4-tuple s i is of the form (c 1 , c 2 , r 1 , r 2 ) with c 1 , c 2 ∈ {1, 2, . . . , k} positive integers and r 1 , r 2 ∈ {0, 1, 2, . . . , n} non-negative integers. These model the indices of the counters and their values respectively.
An NNCCM (k, n, s) with s = (s 1 , . . . , s r ) works as follows. The machine has k counters that are initially 0. For i from 1 to r, the machine first sets each of the counters to any integer that is at least its current value and at most n. After this, the machine performs the ith check s i = (c 1 , c 2 , r 1 , r 2 ): if the value of the c 1 th counter equals r 1 and the value of the c 2 th counter equals r 2 , then we say the ith check rejects and the machine halts and rejects. When the machine has not rejected after all r checks, the machine accepts.
The nondeterministic steps can be also described as follows. Denote the value of the cth counter when the ith check is done by c(i). We define c(0) = 0.
For each i ∈ {1, . . . , r}, c(i) is an integer that is nondeterministically chosen from [c(i − 1), n].
We consider the following computational problem.
Accepting NNCCM
Given: An NNCCM (k, n, s) with all integers given in unary.
Parameter: The number of counters k. Question: Does the machine accept?
Theorem 3.8. Accepting NNCCM is XNLP-complete.
Proof. We first argue that Accepting NNCCM belongs to XNLP. We simulate the execution of the machine. At any point, we store k integers from [0, n] that give the current values of our counters, as well as the index i ∈ [1, r] of the check that we are performing. This takes only O(k log n + log r) bits. When we perform the check s i = (c 1 , c 2 , r 1 , r 2 ), we store the values c 1 , c 2 , r 1 , r 2 in order to perform the check using a further O(log n) bits. So we can simulate the machine using O(k log n + log r) space. The running time is upper bounded by some function of the form f (k) · poly(n, r). We now prove hardness via a transformation from Chained Multicolored Clique. We are given a k-colored graph with vertex sets V 1 , . . . , V r . By adding isolated vertices if needed, we may assume that, for each i ∈ [1, r], the set V i contains exactly m vertices of each color. We will assume that r is even; the proof is very similar for odd r. We set n = mr.
We create 4k counters: for each colour i ∈ [1, k], there are counters c i,1,+ , c i,1,− , c i,0,+ , c i,0,− . We use the counters c i,1,± for selecting vertices from sets V j with j odd and the counters c i,0,± for selecting vertices from sets V j with j even.
The intuition is the following. We increase the counters in stages, where in stage j we model the selection of the vertices from V j . Say j is even. We increase the counters c i,0,+ , c i,0,− to values within [jm + 1, (j + 1)m]. Since counters may only move up, there can be at most one ∈ [1, m] for which the counters at some point take the values c i,0,+ = jm + and c i,0,− = (j + 1)m + 1 − . We enforce that such an exists and interpret this as placing the th vertex of color i in V j into the chained multicolored clique.
We use the short-cut (c 1 , c 2 , R 1 , R 2 ) for the sequence of checks ((c 1 , c 2 , r 1 , r 2 ) :
r 1 ∈ R 1 , r 2 ∈ R 2 ) performed in lexicographical order, e.g. for R 1 = {1, 2} and R 2 = {1, 2, 3}, the order is (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3).
For each j ∈ [1, r], the jth vertex selection check confirms that for each
i ∈ [1, k], (c i,par,+ , c i,par,− ) is of the form (jm + , (j + 1)m + 1 − ) for some ∈ [1, m],
where par denotes the parity of j. For each i ∈ [1, k], we set c 1 = c i,par,+ and c 2 = c i,par,− , and perform the following checks in order.
1. (c 1 , c 2 , [0, jm − 1], [0, n]); 2. (c 1 , c 2 , [0, n], [0, jm − 1]); 3. for ∈ [1, m], (c 1 , c 2 , {jm + }, [jm, (j + 1)m] \ {(j + 1)m + 1 − }); 4. (c 1 , c 2 , [(j + 1)m + 1, n], [0, n]); 5. (c 1 , c 2 , [0, n], [(j + 1)m + 1, n]).
Suppose all the checks succeed. After the second check, c 1 and c 2 are both at least jm, and before the last two checks, c 1 and c 2 are both at most (j+1)m. The middle set of checks ensure that there is some for which the c 1 th counter and c 2 th counter have been simultaneously at the values jm + and (j + 1)m + 1 − respectively. We say the check chooses for c 1 and c 2 . Since the counters can only move up, this is unique. This is used as a subroutine below, where we create a collection of checks such that the corresponding NNCCM accepts if and only if the k-colored graph has a chained multicolored clique. For j = 1 to r, we do the following.
• Let par ≡ j mod 2 denote the parity of j and let par ∈ {0, 1} denote the opposite parity.
• We perform a jth vertex selection check.
• We verify that all selected vertices in V j are adjacent. Let uu be a nonedge with u, u ∈ V j . Let , i, , i with i = i be such that u is the th vertex of color i in V j and u is the th vertex of color i . We add the check (c i,par,+ , c i ,par,+ , jm + , jm + ). This ensures that we do not put both u and u in the clique of V j .
• If j > 1, then we verify that all selected vertices in V j are adjacent to all selected vertices in V j−1 . Let uu be a non-edge with u ∈ V j and u ∈ V j−1 and let , i, , i be as above. We add the check (c i,par,+ , c i ,par ,+ , jm + , jm + ) to ensure that we do not put both u and u into the clique.
• We finish with another jth vertex selection check. If j > 1, we also do a (j − 1)th vertex selection check. This ensures our counters are still 'selecting vertices' from V j−1 and V j .
We now argue that the set of checks created above accepts if and only if the graph contains a chained multicolored clique. Suppose first that such a set W ⊆ V exists for which W ∩ (V j ∪ V j+1 ) forms a clique for all j and W contains at least one vertex from V j of each color. We may assume that W j = W ∩ V j is of size k for each j. Let j ∈ [1, r] be given of parity par and for each i ∈ [k], let the f (i)th vertex of V j of color i be in W j . Before the first jth vertex selection check, we move the counters (c i,par,+ , c i,par,− ) to (jm+f (i), (j +1)m−1−f (i)), and these will be left there until the first (j + 2)th vertex selection check. This will ensure that all the checks accept. Suppose now that all the checks accept. We first make an important observation. Let two counters c 1 and c 2 be given. If in an iteration above, the first vertex selection check chooses for c 1 and c 2 and the second vertex selection check chooses , then it must be the case that = . Indeed, we cannot increase c 1 or c 2 beyond (j + 1)m before the last vertex selection check, and they need to be above jm due to the first. If the first vertex selection check selects , then the c 1 the counter is at least jm + , so in order for it to be jm + in the second check, we must have ≥ . Considering the value of the c 2 th counter, we also find ≥ and hence = . In particular, the counters cannot have moved between the two vertex selection checks.
It is hence well-defined to, for j ∈ [1, r] of parity par, let W j be the set of vertices that are for some color i ∈ [1, k] the th vertex of color i in V j , for the unique value that is selected by a jth vertex selection check for c i,par,+ and c i,par,− . We claim that W = ∪ r j=1 W j is our desired multicolored chained clique. It contains exactly one vertex per color from V j . Suppose u ∈ W j and u ∈ W j−1 ∪ W j are not adjacent and distinct. Let i, par, be such that u is the th color of parity par in V j for j of parity par, and similar for i , par , . Then at the jth iteration, the check (c i,par,+ , c i ,par,+ , jm + , jm + ) has been performed (because uu is a non-edge). Since u, u ∈ W and this check is done between vertex selection checks, the counters have to be at those values. This shows that there is a check that rejects, a contradiction. So W must be a chained multicolored clique.
The Accepting NNCCM problem appears to be a very useful tool for giving XNLP-hardness proofs. Note that one step where the k counters can be increased to values at most n can be replaced by kn steps where counters can be increased by one, or possibly a larger number, again to at most n. This modification is used in some of the proofs in the following section.
Applications
In this section, we consider several problem, which we prove to be XNLPcomplete.
We start by proving XNLP-completeness for List Coloring parameterized by pathwidth in Subsection 4.1 and Dominating Set parameterized by logarithmic pathwidth in Subsection 4.2. We continue to prove XNLP-completeness for a well-studied scheduling problem, Scheduling with Precedence Constraints parameterized by the number of machines and the partial order width in Subsection 4.3. Next, in Subsection 4.4 we discuss Uniform Emulations of Weighted Paths and use it to prove XNLP-completeness for Bandwidth in Subsection 4.5. We end the section with proving Timed Dominating Set Reconfiguration to be XNLP-complete in Subsection 4.6.
Note that some variants of these problems are discussed in Appendix B and Appendix C.
List Coloring Parameterized by Pathwidth
In this subsection, we consider the List Coloring and Precoloring Extension problems, parameterized by treewidth or pathwidth. These problems belong to XP [32]. Fellows et al. [22] have shown that List Coloring and Precoloring Extension parameterized by treewidth are W [1]-hard. (The transformation in their paper also works for parameterization by pathwidth.) We strengthen this result by showing that List Coloring and Precoloring Extension parameterized by pathwidth are XNLP-complete. Note that this result also implies W [t]-hardness of the problems for all integers t.
Given
a graph G = (V, E) with lists L v for vertex v ∈ V (G), a list coloring for G is a choice of color f (v) ∈ L v for each vertex v such that f (v) = f (w) when {v, w} ∈ E.
A path decomposition of a graph G = (V, E) is a sequence (X 1 , X 2 , . . . , X r ) of subsets of V with the following properties.
1. 1≤i≤r X i = V . 2. For all {v, w} ∈ E, there is an i ∈ I with v, w ∈ X i . 3. For all 1 ≤ i 0 < i 1 < i 2 ≤ r, X i0 ∩ X i2 ⊆ X i1 .
The width of a path decomposition (X 1 , X 2 , . . . , X r ) equals max 1≤i≤r |X i | − 1, and the pathwidth of a graph G is the minimum width of a path decomposition of G.
In this subsection, we assume that a path decomposition of width at most k is given as part of the input. Proof. The proof that the problem is in XP for bounded treewidth by Jansen and Scheffler [32] can easily be transferred to a proof that the problem is in XNLP. Intuitively, we guess which of the table entries in the DP table for each bag in the path decomposition we take; we store one table entry in O(pw log n) bits. (Each table entry for a bag gives the choice of colors for the vertices in that bag.)
To show hardness, we transform from Monotone Partitioned Chained CNF-Satisfiability.
Suppose we have an instance of Monotone Partitioned Chained CNF-Satisfiability, say, variable sets X i,j with i ∈ [1, r] and j ∈ [1, k], each of size q and a formula F (on 2kq Boolean variables), and we ask if we can set exactly one variable from each set X i,j to true, such that
r−1 i=1 F (X i,1 , . . . X i,k , X i+1,1 , . . . , X i+1,k )
holds, with F in conjunctive normal form with only positive literals.
We build a graph G, a path decomposition of width 2k + 1 of G, and an assignment of colors to vertices of G as follows.
First, for every set X i,j we create a vertex v i,j , and for every variable in X i,j we create a color in the list of colors for the vertex v i,j . The ith bag in the path decomposition consists of all the vertices v i,j and v i+1,j (for all j). Some additional vertices and bags will now be added in order to model the clauses.
Fix i ∈ [1, r − 1]
. Consider a clause c, which is a conjunction of positive literals, on the set of variables S i = j X i,j ∪ X i+1,j . Let x ∈ X i,j be such that x does not appear in c. We create a new vertex w c,x adjacent to v i,j with color list {x, n i,j }; the colors n i,j are new. We create one further new vertex z c for the clause c that is adjacent to all vertices w c,x and whose color list contains the colors n i,j and n i+1,j for all j. Proof. Suppose that a satisfying truth assignment exists. Color a vertex v i,j with the color of the variable from the set X i,j which is set to true. Consider a vertex w c,x with c a clause and x ∈ X i,j a literal that does not appear in the clause c. We color w c,x with the color x if x is set to false, and with the color n i,j if x is set to true. This avoids conflicts with the color of its neighbor v i,j .
For each vertex z c , there must be a literal that satisfies c, say x ∈ X i,j . This implies that all x ∈ X i,j \ {x} are set to false, and thus no neighbor of z c is colored with n i,j . Thus, we can color z c with n i,j .
Conversely, suppose that a list coloring of G exists. We define a truth assignment by setting a variable x ∈ X i,j to true, if and only if the color of v i,j is x. This selects one true variable per set X i,j . It remains to verify that each clause is satisfied.
Consider a clause c. Let n i,j be the color of z c and let x ∈ X i,j be the color of v i,j . It must be that x ∈ c: if x does not appear in c, then the vertex w c,x exists and cannot be colored, since the color x has already been used for its neighbor v i,j and the color n i,j has already been used for its neighbor z c . As x ∈ c and the color of v i,j is x, we find that c is satisfied by the truth assignment. This shows that each clause is satisfied, and finishes the proof of the claim.
We now discuss how to build a path decomposition of width 2k + 1 for G. The procedure has three nested loops -for each, we can use O(log n) bits to denote where in the loop we are. The resulting path decomposition is the concatenation of the bags, generated in the order implied by the loops.
The outermost loop goes through the 'time steps': we loop i from 1 to r − 1. The middle loop goes through the clauses for step i, i.e., all clauses c with variable set j (X i,j ∪ X i+1,j ). The innermost loop deals with one clause c with variables from the set j (X i,j ∪X i+1,j ). For each variable x ∈ j (X i,j ∪X i+1,j ) that does not appear in c, we take a bag with vertex set
{z c , w c,x } ∪ k j=1 {v i,j , v i+1,j }.
Each bag has size at most 2k + 2; moreover, it is easy to see that the procedure gives a path decomposition.
As we can construct the graph G, the color lists and the path decomposition using O(k log n) bits in memory, the result now follows.
We deduce the following result from a well-known transformation from List Coloring. Proof. Consider an instance of List Coloring with graph G = (V, E) and color lists L v for all v ∈ V . Let C = v∈V L v be the set of all colors. For each vertex v with color list L v , for each color γ ∈ C \ L v , add a new vertex that is only adjacent to v and is precolored with γ. The original vertices are not precolored. It is easy to see that we can extend the precoloring of the resulting graph to a proper coloring, if and only if the instance of List Coloring has a solution.
The procedure increases the pathwidth by at most one. Indeed, if we are given a path decomposition of G, then we can build the path decomposition of the resulting graph as follows. We iteratively visit all the bags X i of the path decomposition. Let v ∈ X i be a vertex for which X i is the first bag it appears in (so i = 1 or v ∈ X i−1 ). We create a new bag X i ∪ {w} for each new neighbor w of v and place this bag somewhere in between X i and X i+1 . This transformation can be easily executed with O(k log n) bits of memory.
Note that we assume that a path decomposition of the input graph G is given as part of the input. We conjecture that such a path decomposition can be found with a non-deterministic algorithm using logarithmic space and 'fpt' time, but the details need a careful study. Elberfeld et al. [20] show that for each fixed k, determining if the treewidth is at most k, and if so, finding a tree decomposition of width at most k belongs to L.
From Theorem 4.1 and Corollary 4.3, we can also directly conclude that List Coloring and Precoloring Extension are XNLP-hard when parameterized by the treewidth. However, we leave membership of these problems when parameterized by treewidth as an open problem.
Logarithmic Pathwidth
There are several well known problems that can be solved in time O(c k n) for a constant c on graphs of pathwidth or treewidth at most k. Classic examples are Independent Set and Dominating Set (see e.g., [14,Chapter 7.3]), but there are many others, e.g., [5,44].
In this subsection, rather than bounding the pathwidth by a constant, we allow the pathwidth to be linear in the logarithm of the number of vertices of the graph. We consider the following problem.
Log-Pathwidth Dominating Set
Input: Graph G = (V, E), path decomposition of G of width , integer K. Parameter: / log |V | . Question: Does G have a dominating set of size at most K?
In Appendix C, we study the independent set and clique variants of this problem. Proof. It is easy to see that the problem is in XNLP -run the standard dynamic programming algorithm for Dominating Set on graphs with bounded pathwidth, but instead of computing full tables, guess the table entry for each bag.
Hardness for XNLP is shown with help of a reduction from Partitioned Regular Chained Weighted CNF-Satisfiability.
Suppose we are given F 1 , X 1 , . . . , X r , q, k, X i,j (1 ≤ i ≤ r, 1 ≤ j ≤ k) as instance of the Partitioned Regular Chained Weighted CNF-Satisfiability problem.
We assume that each set X i,j is of size 2 t for some integer t; otherwise, add dummy variables to the sets and a clause for each X i,j that expresses that a non-dummy variable from the set is true. Let c be the number of clauses in F 1 . We now describe in a number of steps the construction of G.
Variable choice gadget For each set X i,j we have a variable choice gadget. The gadget consists of t copies of a K 3 . Each of these K 3 's has one vertex marked with 0, one vertex marked with 1, and one vertex of degree two, i.e., this latter vertex has no other neighbors in the graph.
The intuition is the following. We represent each element in X i,j by a unique t-bit bitstring (recall that we ensured that |X i,j | = 2 t ). Each K 3 represents one bit, and together these bits describe one element of X i,j , namely the variable that we set to be true. Clause checking gadget Consider a clause φ on X i ∪ X i+1 . We perform following construction separately for each such clause, and in particular the additional vertices defined below are not shared among clauses. For each j ∈ [1, k], we add 2 t + 2 additional vertices for both X i,j and X i+1,j , as follows.
• We create a variable representing vertex v x for each variable x ∈ X i,j ∪ X i+1,j . We connect v x to the each vertex that represents a bit in the complement of the bit string representation of x. We call this vertex a .
• We add two new vertices for X i,j and also for X i+1,j (see z 1 and z 2 in Figure 2). These vertices are incident to all variable representing vertices for variables from the corresponding set.
We call this construction a variable set gadget. The clause checking gadget for φ consists of 2k variable set gadgets (for each j ∈ [1, k], one for each set of the form X i,j and X i+1,j ), and one additional clause vertex. The clause vertex is incident to all vertices in the current gadget that represent a variable that satisfies that clause. In total, we add 2 · k · (2 t + 2) + 1 variables per clause.
The intuition is the following. From each triangle, we place either the vertex marked with 0 or with 1 in the dominating set. This encodes a variable for each X i,j as follows. The vertices in the triangles dominate all but one of the variable representing vertices; the one that is not dominated is precisely the encoded variable. This vertex is also placed in the dominating set (we need to pick at least one variable representing vertex in order to dominate the vertices z 1 and z 2 private to X i,j )). The clause is satisfied exactly when the clause vertex has a neighbor in the dominating set.
The graph G is formed by all variable choice and clause checking gadgets, for all variables and clauses. Recall that c is the number of clauses in F 1 . Proof. Suppose we have a solution for the latter. Select in the variable choice gadgets the bits that encode the true variables and put these in the dominating set S. As we have rk sets of the form X i,j , and each is represented by t triangles of which we place one vertex per triangle in the dominating set, we already used rkt vertices.
For each clause, we have 2k sets of the form X i,j . For each, take the vertex that represents the true variable of the set and place it in S. Note that all vertices that represent other variables are dominated by vertices from the variable choice gadget. For each set X i,j , the chosen variable representing vertex dominates the vertices marked z 1 and z 2 . As the clause is satisfied, the clause vertex is dominated by the vertex that represents the variable representing vertex for the variable that satisfies it. This shows S is a dominating set. We have (r − 1)c clauses in total, and for each we have 2k sets of the form X i,j . We placed one variable representing vertex for each of these (r − 1)c · 2k sets in the dominating set. In total, |S| = rkt + 2kc(r − 1) as desired.
For the other implication, suppose there is a dominating set S of size rkt + 2kc(r − 1).
We must contain one vertex from each triangle in a variable choice gadget (as the vertex with degree two must be dominated). Thus, these gadgets contain at least rkt vertices from the dominating set.
Each of the vertices marked z 1 and z 2 must be dominated. So we we must choose at least one variable choice vertex per variable set gadget. This contributes at least 2kc(r − 1) vertices.
It follows that we must place exactly those vertices and no more: the set S contains exactly one vertex from each triangle in a variable choice gadget and exactly one variable selection vertex per variable set gadget. Note that we cannot replace a variable selection vertex by a vertex marked z 1 or z 2 : either it is already dominated, or we also must choose its twin, but we do not have enough vertices for this.
From each triangle in a variable choice gadget, we must choose either the vertex marked 0 or 1, since otherwise one of the variable selection vertices will not be dominated. This dominates all but one of the variable selection vertices, which must be the variable selection vertex that is in the dominating set. We claim that we satisfy the formula by setting exactly these variables to true and all others to false. (Note also that we set exactly one variable per set X i,j to true this way.) Indeed, the clause vertex needs to be dominated by one of the variable selection vertices, and by definition this implies that the corresponding chosen variable satisfies the clause.
The path decomposition can be constructed as follows. Let V i for i ∈ [1, r−1] be the set of all vertices in the variable choice gadgets of all sets X i,j and X i+1,j with j ∈ [1, k]. For any given i, we create a path decomposition that covers all vertices of V i and all clause gadgets connected to V i . We do this by sorting the clauses in any order, and then one by one traversing the clause checking gadgets. Each time, we begin by adding the z-vertices and the clause vertex of the gadget to the current bag. Then, iteratively we add and remove all the variable representing vertices one by one. After this, we remove the z-vertices and clause vertex from the bag and continue to the next clause. At any point, there are at most 6k · t + 4k + 2 vertices in a bag: 6k · t from the set V i , 4k z-variables, one clause vertex and one variable representing vertex. We then continue the path decomposition by removing all vertices related to variable choice gadgets of the set X i and adding all vertices related to variable choice gadgets of the set X i+2 .
As the construction can be executed in logarithmic space, the result follows.
Scheduling with precedence constraints
In 1995, Bodlaender and Fellows [8] showed that the problem to schedule a number of jobs of unit length with precedence constraints on K machines, minimizing the makespan, is W [2]-hard, with the number of machines as parameter.
A closer inspection of their proof shows that W [2]-hardness also applies when we take the number of machines and the width of the partial order as combined parameter. In this subsection, we strengthen this result, showing that the problem is XNLP-complete (and thus also hard for all classes W [t], t ∈ Z + .) In the notation used in scheduling literature to characterize scheduling problems, the problem is known as P |prec, p j = 1|C max , or, equivalently, P |prec, p j = p|C max .
Scheduling with Precedence Constraints
Given: K, D positive integers; T set of tasks; ≺ partial order on T of width w. Parameter: K + w. Question: Is there a schedule f :
T → [1, D] with |f −1 {i}| ≤ K for all i ∈ [1, D] such that t ≺ t implies f (t) < f (t ).
In other words, we parametrise P |prec, p j = 1|C max by the number of machines and the width of the partial order.
R i = {t ∈ T | t ≺ s for all s ∈ S i−1 }.
By definition, for t ∈ R i , we find that t ≺ s for all s ∈ f −1 [1, i − 1]. In order to update S i given S i−1 , we add all selected elements (which must be maximal) and remove the elements s ∈ S i−1 for which s ≺ t for some t ∈ f −1 {i}. After computing S i and outputting f −1 {i}, we remove S i−1 and f −1 {i} from our memory. At any point in time, we store at most O((K + w) log |T |) bits.
To prove hardness, we transform from the Accepting NNCCM problem, with k given counters with values in [0, n] and set of checks (s 1 , . . . , s r ). We adapt the proof and notation of Bodlaender and Fellows [8].
We create K = 2k + 1 machines and a set of tasks T with a poset structure on this of width at most 3(k + 1). Let c = (kn + 1)(n + 1). The deadline is set to be D = cr + n + 1. We will construct one sequence of tasks for each of the k counters, with kn + 1 repetitions of n + 1 time slots for each of s 1 , . . . , s r (one for each value in [0, n] that it might like to check), and then n extra time slots for 'increasing' counters. Each t ∈ [1, D] can be written in the form t = (j − 1)c + α(n + 1) + y for some y ∈ [1, n + 1], j ∈ [1, r], α ∈ [0, kn]. We define j(t) and α(t) to be the unique values of j and α respectively for which this is possible. Our construction has the following components.
• Time sequence. We create a sequence a 1 ≺ a 2 ≺ . . . ≺ a D of tasks that represents the time line.
• Time indicators. By adding a task b with a t−1 ≺ b ≺ a t+1 , we ensure that at time i one less machine is available. We place k − 1 such tasks at special indicator times. That is, we set I = {(j − 1)c + α(n + 1) + n + 1 | j ∈ [1, r], α ∈ [0, kn]} and create a task b
(x) t with a t−1 ≺ b (x) t ≺ a t+1
for each x ∈ [1, k − 1] and t ∈ I.
• Counter sequences. For each i ∈ [k], we create a sequence c
(i) 1 ≺ c (i) 2 ≺ . . . ≺ c (i) D−n of tasks. If c (i)
t− is planned in at the same moment as time vertex a t (for some ∈ [0, n] and t ∈ [n + 1, D]) then this is interpreted as counter i taking the value at that time.
• Check tasks. Let t = (j − 1)c + α(n + 1) + n + 1 ∈ I be the indicator time for j ∈ [1, r] and α ∈ [0, kn]. If s j = (i 1 , i 2 , r 1 , r 2 ), then for x ∈ {1, 2}, we add a check task d
(ix) t−rx 'parallel' to c (ix) t−rx , that is, we add the precedence constraints c (ix) t−rx−1 ≺ d (ix) t−rx ≺ c (ix) t−rx+1 .
Intuitively, the repetitions allow us to assume that concurrent with each time task are exactly k tasks of the form c y to be scheduled simultaneously (since we have 2k machines available). However, when t ∈ I then we only have access to k + 1 machines, so we may have at most one d happens exactly if s j rejects (due to the i x th counter being at position r x for both x ∈ {1, 2}). The width w of the defined partial order is at most 3(k + 1). We now claim that there is a schedule if and only if the given machine accepts.
Suppose that the machine accepts. Let (p 1 , . . . , p r ) with p j = (p
j ∈ [1, r]. We schedule for t ∈ [1, D], x ∈ [1, k − 1], i ∈ [1, k], when defined, f (a t ) = t, f (b x t ) = t, f (c (i) t ) = t + p (i) j(t) , f (d (i) t ) = t + p (i) j(t) .
Recall that j(t) is the unique value j ∈ [1, r] for which we can write t = (j − 1)c + α(n + 1) + y for some α ∈ j and s j = (i 1 , i 2 , r 1 , r 2 ) with i x = i, r x = p (i) j for some x ∈ {1, 2}. Since the machine accepts, there can be at most one such i. Hence f −1 {t} contains at most one check task when t ∈ I. We again find
|f −1 {t}| ≤ 1 at + (k − 1) b (x) t + k c (i) t + 1 d (i) t ≤ 2k + 1.
Suppose now that a schedule f : T → [1, D] exists. By the precedence constraints, we must have f (a t ) = t for all t ∈ [1, D] and f (b x t ) = t for all t ∈ I and x ∈ [1, k − 1]. Let j ∈ [1, r] be given. Let
C = {c (i) t | j(t) = j, i ∈ [1, k]}.
For at least one α ∈ [0, kn], it must be the case that |f −1 {t} ∩ C| = k for all t with α(t) = α and j(t) = j. This is because for each of the k counter sequences, there are at most n places where we may 'skip'; this is why we repeated the same thing kn + 1 times. We select the smallest such α. Note that for all t ∈ T with α(t) = α and j(t) = j, we also find that f (c
(i) t ) = f (d (i) t ) for all i ∈ [1, k].
There is a unique t ∈ I with α(t) = α and j(t) = j; for this choice of t, and for all i ∈ [1, k], we set p (i) j to be the value for which c
(i) t−p (i) j ∈ f −1 {t}.
This defines a vector p j ∈ [0, n] k of positions which we set the counters in before s j is checked. Note that the values of the counters are non-decreasing, and that s j does not fail since f −1 {t} contains at most one d (i) t (due to its size constraints), which implies that s j accepts.
We remark that it is unclear if the problem is in XNLP when we take only the number of machines as parameter. In fact, it is a longstanding open problem whether Scheduling with Precedence Constraints is NP-hard when there are three machines (see e.g., [26,40]).
Uniform Emulation of Weighted Paths
In this subsection, we give a proof that the Uniform Emulation of Weighted Paths problem is XNLP-complete. The result is a stepping stone for the result that Bandwidth is XNLP-complete even for caterpillars with hair length at most three (see the discussion in Subsection 4.5).
The notion of (uniform) emulation of graphs on graphs was originally introduced by Fishburn and Finkel [25] as a model for the simulation of computer networks on smaller computer networks. Bodlaender [3] studied the complexity of determining for a given graph G and path P m if there is a uniform emulation of G on P m . In this subsection, we study a weighted variant, and show that already determining whether there is a uniform emulation of a weighted path on a path is hard.
An emulation of a graph G = (V, E) on a graph H = (W, F ) is a mapping f : V → W such that for all edges {v, w} ∈ E, f (v) = f (w) or {f (v), f (w)} ∈ F . We say that an emulation is uniform if there is an integer c, such that |f −1 {w}| = |{v | f (v) = w}| = c for all w ∈ W . We call c the emulation factor.
Determining whether there is a uniform emulation of a graph H on a path P m is NP-complete, even for emulation factor 2, if we allow H to be disconnected. The problem to determine for a given connected graph H if there is a uniform emulation of H on a path P m belongs to XP with the emulation factor c as parameter [3].
Recently, Bodlaender [4] looked at the weighted variant of uniform emulation on paths, for the case that H is a path. Now, we have a path P n and a path P m , a weight function w : It is not hard to see that the problem, given n, m, and weight function w : [1, n] → Z + , to determine if there is a uniform emulation of P n on P m 29 with emulation factor c is in XP, with the emulation factor c as parameter; the dynamic programming algorithm from [3] can easily be adapted.
As an intermediate step for a hardness proof for Bandwidth, Bodlaender [4] showed that Uniform Emulation of Weighted Paths is hard for all classes W [t], t ∈ Z + . In the current subsection, we give a stronger result, and show the same problem to be XNLP-complete. Our proof is actually simpler than the proof in [4] -by using Accepting NNCCM as starting problem, we avoid a number of technicalities. Proof. We first briefly sketch a variant of the dynamic programming algorithm (see [3]) that shows membership in XNLP. For i from 1 to m, guess which vertices are mapped to i. Keep this set, and the sets for i − 1 and i − 2 in memory. We reject when neighbors of vertices mapped to i − 1 are not mapped to {i − 2, i − 1, i} (or not to {1, 2} or {m − 1, m} for i = 1 or i = m respectively) or when the total weight of vertices mapped to i does not equal c. We also check whether cm = n i=1 w(i). Since P n is connected, and f (i) is defined for some i ∈ [1, n], we find that f (i) is defined for all i ∈ [1, n] (since our check ensures that f (j) is defined for each neighbor j of i in P n , and then also for all neighbors of j etcetera). We have O(c) vertices from P n in memory, and thus O(c log n) bits.
Uniform Emulation of Weighted Paths
To show that the problem is XNLP-hard, we use a transformation from Accepting NNCCM. Suppose that we are given an NNCCM(k, n, s), with s a sequence of r checks. We build an instance of Uniform Emulation of Weighted Paths as follows.
First, we define a number of constants:
• d 1 = 3k + 2, • d 2 = k · d 1 + 1, • d 3 = k · d 2 + 1,
• c = 2k · d 3 + 1,
• n 0 = 3n + 1,
• M = 1 + (r + 1) · n 0 .
We construct a weighted path P N for some value N whose value follows from our construction below, and ask for a uniform emulation of P N to P M , with emulation factor c.
The path P N has three subpaths with different functions. We concatenate these in order to obtain P N .
• The floor. We ensure that the ith vertex of the floor must be mapped to the ith vertex of P M . We will use this to be able to assume that the counter components always 'run from left to right'.
• The k counter components. Each counter component models the values for one of the counters from the NNCCM and the goal is to ensure the emulation of this part is possible if and only if all the checks succeed.
• The filler path. This is a technical addition aimed to ensure that a total weight of c gets mapped to j for all j ∈ [1, M ].
The floor consists of the following M (weighted) vertices.
• A vertex of weight c − k · d 2 .
• A path with M − 2 vertices. The ith vertex of P N (and hence the (i − 1)th of this part) has weight c − 2d 1 + 1 if i = n 0 · j + 1 for some j and weight c − 3d 1 otherwise. The positions of the higher weight of c − 2d 1 + 1 are called test positions.
• A vertex of weight c − k · d 3 − 1.
We have k counter components, that give after the floor, the successive parts of the 'large path'. The qth counter component has the following successive parts.
• We start with M − 2 vertices of weight 1.
• We then have a vertex of weight d 2 ; this vertex is called the left turning point.
• Then we have M − 2 + n vertices of weight either 1 or d 1 . The ith vertex has weight d 1 if and only if i = n 0 · j + α and the check s j verifies whether counter q is equal to α. The ith vertex has weight 1 otherwise. We call this the main path of the counter component.
• A vertex of weight d 3 . This vertex is called the right turning point of the counter component.
Assuming the left and right turning points are mapped to 1 and M , the main path of the gadget will be mapped between 2 and M − 1. These main paths will tell us what the values of the counters are at any moment. The number of shifts then stands for the number a counter gadget represents, i.e. if the ith vertex of the main path is mapped to i − α + 1, then α is the value of the counter at that moment. In this way, assuming that left and right turning points are mapped to 1 and M , the value of counters can only increase. We add a 'filler path' to ensure the total weight of all vertices of P M is at least M c, by adding a path of min{γ − M c, 0} vertices of weight 1. • Set j to be the smallest integer with p ≤ n 0 · j + 1; if j ≤ r, then n 0 · j + 1 is the first test position (defined in the floor gadget) after p.
• If i−(p−1) < q(j), and p is in the interval [n 0 ·(j −1)+1+n+1, n 0 ·j −n], then we map both the ith and the (i + 1)th vertex of the main path to p. We increase i by 2 and p by 1; the increase of i − (p − 1) by one represents the increase of the counter by one. Otherwise, we map the ith vertex of the main path to p, and increase both i and p by one.
Since q(r + 1) = n, in the end we will map M − 2 + n vertices to M − 2 positions. Note that the final step above does not decrease the value of i − (p − 1). The interval [n 0 ·(j−1)+1+n+1, n 0 ·j−n] contains at least n ≥ q(j) integers, and thus we may map two vertices to the same position until we obtain i − (p − 1) ≥ q(j). We then stop increasing the value i − p + 1, so i − p + 1 = q(j) at the jth test position p = n 0 · j + 1, and map a single vertex to this.
What vertex gets mapped to the jth test position? If the qth counter does not participate in the jth check, then the vertex will have weight one. Otherwise, suppose that check s j verifies whether counter q is equal to the value α. The vertex i mapped to the jth test position has weight d 1 if and only if i = n 0 ·j +α, which is equivalent to q(j) = α since i − (p − 1) = q(j) and p = n 0 · j + 1. As we were considering an accepting run for the NNCCM, there is at most one counter component that maps a vertex of weight d 1 (rather than 1) to the jth test position.
After we have placed the main path, we send the right turning point to M and continue with the next counter component or, if q = k, the filler path. At this point, 1 and M have received weight exactly c (from one floor vertex and k turning points). For all test positions, we have a floor vertex of weight c − 2d 1 + 1, at most one heavy vertex of weight d 1 , and at most 2k vertices of weight one (at most two per counter component). For a vertex on P M that is not 1, M or a test position, we have a floor vertex of weight c − 3d 1 , at most two heavy vertices of weight d 1 and at most 3k vertices of weight one (per counter component, it obtains at most two vertices for the main path and one for path going left). In all cases, we have a weight less than c.
We now use the filler path to make the total weight equal to c everywhere. (By definition, the filler path has enough weight 1 vertices available for this.)
Conversely, suppose that we have a uniform emulation f . We cannot map two floor vertices to the same vertex as each has a weight larger than c/2. As we have M floor vertices, one of the following two statements holds:
1. For all i, the ith floor vertex is mapped to i.
For all i, the ith floor vertex is mapped to
M − i + 1.
Without loss of generality, we assume that the ith floor vertex is mapped to i.
We first show that each right turning point is mapped to M and each left turning point is mapped to 1. Since M is the only vertex where the available weight (which is at most c minus the weight of the floor vertex) is at least the weight of a right turning point, all right turning points must be mapped to M . After this, there is no more weight available on this vertex, and the only vertex where a left turning point can fit is 1.
We now use the test positions to define values for the counters. Because the main path of the counter component is from a left turning point (mapped to 1) to a right turning point (mapped to M ), at least one vertex of the main path is mapped to the jth test position n 0 · j + 1. If the ith vertex of the main path is mapped to the test position, then we set q(j) = i − n 0 · j. (If multiple vertices are mapped to the test position, we choose arbitrarily one of those values.) We find that the values we defined for the counters are non-decreasing since the difference between i = i(j) and n 0 · j can only become larger for larger j.
We are left to prove that this defines an accepting sequence. Assume towards a contradiction that the check s j = (q, q , α q , α q ) fails because of q(j) = α q and q (j) = α q . That would imply both the (n 0 · j + q(j))th vertex from the qth counter component and the (n 0 · j + q (j)) vertex from the q th counter component were mapped to test position n 0 · j + 1. Both these vertices have weight d 1 , by the construction of the main path of counter components. Since we already have a total weight of c − 2d 1 + 1 from the floor gadget, the total weight mapped to the jth test position is > c. This is a contradiction. Hence the given solution for the NNCCM instance is a valid one.
Bandwidth
In this subsection, we discuss the XNLP-completeness problem of the Bandwidth problem. The question where the parameterized complexity of Band-width lies was actually the starting point for the investigations whose outcome is reported in this paper; with the main result of this subsection (Corollary 4.10) we answer a question that was asked over a quarter of a century ago.
In the Bandwidth problem, we are given a graph G = (V, E) and an integer k and ask if there is a bijection f :
V → [1, |V |] such that for all {v, w} ∈ E: |f (v) − f (w)| ≤ k.
The problem models the question to permute rows and columns of a symmetric matrix, such that all non-zero entries are at a small 'band' along the main diagonal. Already in 1976, the problem was shown to be NP-complete by Papadimitriou [37]. Later, several special cases were shown to be hard; these include caterpillars with hairs of length at most three [35]. A caterpillar is a tree where all vertices of degree at least three are on a common path; the hairs are the paths attached to this main path.
We are interested in the parameterized variant of the problem, where the target bandwidth is the parameter:
Bandwidth Given: Integer k, undirected graph G = (V, E) Parameter: k Question: Is there a bijection f : V → [1, |V |] such that for all edges {v, w} ∈ E: |f (v) − f (w)| ≤ k?
Bandwidth belongs to XP. In 1980, Saxe [41] showed that Bandwidth can be solved in O(n k+1 ) time; this was later improved to O(n k ) by Gurari and Sudborough [27]. In 1994, Bodlaender and et. [9] reported that Bandwidth for trees is W [t]-hard for all t ∈ Z + -the proof of that fact was published 26 years later [4]. A sketch of the proof appears in the monograph by Downey and Fellows [17]. More recently, Dregi and Lokshtanov [19] showed that Bandwidth is W [1]-hard for trees of pathwidth at most two. In addition, they showed that there is no algorithm for Bandwidth on trees of pathwidth at most two with running time of the form f (k)n o(k) assuming the Exponential Time Hypothesis. Recently, Bodlaender [4] published a proof that Bandwidth is W [t]-hard for all t, even for caterpillars with hairs of length at most three. That result is obtained by first showing that Uniform Emulation of Weighted Paths is W [t] hard for all t, and then giving a transformation from that problem to Bandwidth for caterpillars with maximum hair length tree. The latter transformation uses gadgets from the NP-completeness proof by Monien [35]. Proof. This can be seen in different ways: one can look at the XP algorithms for Bandwidth from [41] or [27], and observe that when instead of making full tables in the dynamic programming algorithm, we guess from each table one entry, one obtains an algorithm in XNLP.
Alternatively, we loop over all connected components W of G, compute its size and for each, guess for i from 1 to |W | the ith vertex in the linear ordering, i.e., f −1 (i). Keep the last 2k + 1 guessed vertices in memory, and verify that all neighbors of
f −1 (i − k) belong to f −1 [i − 2k, i].
The main result of this subsection is just a corollary of earlier results. Corollary 4.10. Bandwidth for caterpillars with hairs of length at most three is XNLP-complete.
Proof. Membership was argued above in Lemma 4.9.
Hardness follows directly from the hardness of Uniform Emulation of Weighted Paths (Theorem 4.7), and the transformation in [4] from Uniform Emulation of Weighted Paths to Bandwidth for caterpillars with hairs of length at most three, as a closer inspection of that proof gives that the reduction is also a pl-reduction.
Timed Reconfiguration
We now consider yet another very different setting: reconfiguration. We first discuss dominating set reconfiguration; in Appendix B, we discuss reconfiguration of cliques and independent sets.
Given a graph G and dominating sets S and S , we wish to know whether we can go from one dominating set to the other via a sequence of dominating sets. All dominating sets are of the same size k (which is our parameter) and can be visualised by placing k 'tokens' on the vertices of the graph. The following two rules that are commonly considered for when dominating sets S 1 and S 2 are adjacent (that is, can be consecutive in the sequence).
• Token Jumping (TJ). We may 'jump' a single token, that is, the dominating sets S 1 and S 2 of size k are adjacent if S 1 = S 2 \ {u} ∪ {v} for some u, v ∈ V (G). This rule is equivalent 2 to Token Addition and Removal (TAR), in which a token can be either added or removed, as long as the total number of tokens does not go above k + 1.
• Token Sliding (TS). We may 'slide' a single token along an edge, that is, the dominating sets S 1 and S 2 of size k are adjacent if S 1 = S 2 \ {u} ∪ {v} for some uv ∈ E(G).
The TJ/TAR rule has been most widely studied, and the problem is known to be PSPACE-complete (even for simple graph classes such as planar graphs and classes of bounded bandwidth) [28] and W [2]-hard with parameter k + , for k the number of tokens and the length of the reconfiguration sequence [36]. Under TS, the problem is also known to be PSPACE-complete even for various restricted graph classes [12]. It turns out that these problems are XNLP-complete when we bound the number of steps in the reconfiguration sequences as follows. We first consider the token sliding rule and then explain how to adjust the reduction for the token jumping rule.
Theorem 4.11. Timed TS-Dominating Set Reconfiguration is XNLPcomplete.
Proof. The problem is in XNLP follows since we can 'explore' all dominating sets that can be reached from S within T steps nondeterministically, storing the current positions of the k counters and the number of steps taken so far using O(k log n) bits, accepting if the current position encodes S and halting after T steps. We prove hardness via a reduction from Partitioned Regular Chained Weighted Positive CNF-Satisfiability. We obtain r sets of Boolean variables X 1 , . . . , X r , a Boolean formula F 1 , and a partition of each set X i into sets X i,1 , . . . , X i,k , each of size q. We will assume that r is even. We will create a graph and dominating sets S, S of size 2k + 2 such that S can be reconfigured to S in T = 5 2 r − 2 steps if and only if ∧ 1≤i≤r−1 F 1 (X i , X i+1 ) can be satisfied by setting exactly one variable to true from each set X i,j .
The idea of the construction is to create a 'timer' of 2r−2 steps, of which each dominating set in the sequence must contain exactly one element, and enforce that the first dominating set S contains the start vertex t 0 whereas the second dominating set S contains the ending vertex t 2r−3 . This means the token on t 0 needs to slide over the timeline, and we use the time constraint to enforce this can happen in a single fashion, using the vertices t 2i to 'move k tokens from X i to X i+2 ' and the vertex t 2i−1 to 'verify whether the formula F 1 (X i , X i+1 ) holds'.
Our construction has the following parts:
• A 'timer': a path with vertices t 0 , t 1 , . . . , t 2r−3 , along with two time guardians g t,1 , g t,2 that are adjacent to t i for all i ∈ [0, 2r − 3] and nothing else. In order to dominate the two time guardians using only a single vertex, we need to choose one of the t i . This part of the construction is illustrated in Figure 4.
• For each variable x a vertex v x .
t 0 t 1 t 2 t 2r−3 g t,1
g t,2
Figure 4: In the time line, at least one vertex needs to get chosen in order to dominate the time guardian vertices g t,1 and g t,2 .
• A 'dominator' vertex d with a pendant vertex p d (which is only adjacent to d and enforces that we must always use one token for d or p d ). We use d to dominate all the t i and all the variable vertices.
• For each clause φ in the Boolean formula
F 1 , a vertex w φ,i for all i ∈ [1, r − 1]. The vertex w φ,i is adjacent to vertex v x for x ∈ (X i ∪ X i+1 ) ∩ φ (that is, setting x to true satisfies φ). Moreover, w φ,i is adjacent to all t j with j = 2i − 1.
• Vertices m i,j for i ∈ [1, r], j ∈ [1, k]. These will ensure that we move tokens from variables in X i to variables in X i+2 at certain time steps.
• Vertices g oe,0,j,a and g oe,1,j,a for all j ∈ [1, k] and a ∈ {1, 2}. For a ∈ {1, 2} and j ∈ [1, k], the vertex g oe,0,j,a (respectively g oe,1,j,a ) is adjacent to all vertices v x for x ∈ X i,j with i even (respectively odd) and nothing else; these ensure that for each 'variable group' we always have at least one token on a vertex corresponding to this group.
• Start vertices s 1 , . . . , s 2k to add to the initial dominating set S. The vertices s 1 , . . . , s k are adjacent to v x for all x ∈ X 1 . The vertices s k+1 , . . . , s 2k are adjacent to v x for all x ∈ X 2 .
• End vertices s 1 , . . . , s 2k to add to the final dominating set S . The vertices s 1 , . . . , s k are adjacent to v x for all x ∈ X r−1 . The vertices s k+1 , . . . , s 2k are adjacent to v x for all x ∈ X r .
There are some further edges as explained later. An outline of the construction is depicted in Figure 5. Due to the dominator vertex and its pendant vertex, we do not have to worry about dominating the vertices t i and the variable vertices.
The guardian vertices enforce that we need to place a minimum number of tokens on certain vertex sets in order to have a dominating set. One token is always on the dominator vertex; we will call this the dominator token. We need 2k tokens on various variable sets, which we refer to as variable tokens and one token on one of the t i , which we refer to as the time token. Since we only have 2k + 2 tokens, there are no further tokens available and we need to place exactly For i ∈ [1, r − 1], the clause vertices w φ,i are non-adjacent to t j if j = 2i − 1; this ensures that when the time token is at t 2i−1 , the variable tokens need to take care of dominating w φ,i (which happens if and only if they are placed in a satisfying assignment).
For i ∈ [1, r − 2], we make v x adjacent to v y for all x ∈ X i and y ∈ X i+2 . This allows us to 'slide' variable tokens from X 1 to X 3 to X 5 etcetera. For i ∈ [3, r], we ensure that variable tokens are moved from X i−2 to X i when the time token is at t 2(i−2) using the move vertices m i, ; these are adjacent to t j for all j = 2(i − 2) + 1 and to all v x for x ∈ X i, . For i ∈ [1,2], the move vertices m i, are adjacent to v x for all x ∈ X i, and to all t j for j = 1. Before the time token can be moved from t 0 to t 1 , the tokens on the start vertices need to be moved from S to the 2k variable sets X i, for i ∈ {1, 2}, ∈ [1, k] in order to dominate the m i, .
To move tokens from S to S , we need to move the 2k tokens via the variable vertices. This takes at least r/2 + 1 moves, as each token can go through either the odd or the even variable sets. The token on t 0 has to move to t 2r−3 , which takes at least 2r − 3 moves. We set the time bound to T = r/2 + 1 + 2r − 3 so that we are enforced to exactly take these steps. In particular, for each i ∈ [1, r] and ∈ [1, k], there is a unique x ∈ X i, for which a variable token gets placed on v x for some x ∈ X i, in a valid reconfiguration sequence.
In conclusion, there is a reconfiguration sequence from S to S within T steps if and only if we can select one variable per X i, so that when these are set to true, the formula ∧ 1≤i≤r−1 F 1 (X i , X i+1 ) is satisfied.
The construction for the token jumping rule is similar, so we only sketch it below.
t 0 t 1 t 2 t 3 t 4 t 0 t 1 t 2 t 3 t 4 g t,1
g t,2 Sketch. We expand on the construction for token sliding as depicted in Figure 6.
We create a second path t 0 , . . . , t 2r−3 . For each i ∈ [0, 2r − 3], we create a guardian vertex g tj ,i adjacent to t j for all j = i and to t i−1 , t i (if these exist).
The t i are also dominated by the dominator vertex d, and also have two private guardian vertices g t,1 , g t,2 . We will increase the number of tokens by one to 2r + 3, add t 0 to S and t 2r−3 to S and increase the allowed time T by 2r − 3 in order to ensure we can exactly slide the second timer token from t 0 to t 2r−3 along the time line.
Suppose we have placed 2k + 3 tokens in a way that dominates all vertices. Then there is exactly one token on some t i and exactly one on some t i . Since g tj ,i is also dominated, we find i ∈ {i, i − 1}. Hence in order to move t 0 to t 2r−3 and t 0 to t 2r−3 , we can need to pass through the configurations
(t 0 , t 0 ) → (t 1 , t 0 ) → (t 1 , t 1 ) → (t 2 , t 1 ) → (t 2 , t 2 ) → . . .
and this enables us to use the same construction as for token sliding.
Similar to Timed Dominating Set Reconfiguration, we can define Timed Independent Set Reconfiguration and Timed Clique Reconfiguration. In Appendix B, we give a proof that these are XNLP-complete, again assuming the number of steps are given in unary. We remark that many other solution concepts have been studied for reconfiguration, such as satisfiability and coloring versions, and we refer the reader to the survey [45] for more information.
Conclusion
We end the paper with some discussions and open problems. We start by discussing a conjecture on the space usage of XNLP-hard problems, then discuss the type of reductions we use, and then give a number of open problems.
Space efficiency of XNLP-hard problems
Pilipczuk and Wrochna [39] made the following conjecture. In the Longest Common Subsequence problem, we are given k strings s 1 , . . . , s k over an alphabet Σ and an integer r and ask if there is a string t of length r that is a subsequence of each s i , i ∈ [1, k].
Conjecture 5.1 (Pilipczuk and Wrochna [39]). The Longest Common Subsequence problem has no algorithm that runs in n f (k) time and f (k)n c space, for a computable function f and constant c, with k the number of strings, and n the total input size.
Interestingly, this conjecture leads to similar conjectures for a large collection of problems. As Longest Commom Subsequence with the number of strings k as parameter is XNLP-complete [21], Conjecture 5.1 is equivalent to the following conjecture. Recall that the class XNLP is the same as the class N [f poly, f log n]. If Conjecture 5.1 holds, then no XNLP-hard problem has an algorithm that uses XP time and simultaneously 'FPT' space (i.e., space bounded by the product of a computable function of the parameter and a polynomial of the input size). Thus, XNLP-hardness proofs yield conjectures about the space usage of XP algorithms, and Conjecture 5.1 is equivalent to the same conjecture for Bandwidth, List Coloring parameterized by pathwidth, Chained CNF-Satisfiability, etc.
Reductions
In this paper, we mainly used parameterized logspace reductions (pl-reductions), i.e., parameterized reductions that run in f (k) + O(log n) space, with f a computable function.
Elberfeld et al [21] use a stronger form of reductions, namely parameterized first-order reductions or pFO-reductions, where the reduction can be computed by a logarithmic time-uniform para AC O -circuit family. In [21], it is shown that Timed Non-Deterministic Cellular Automaton and Longest Common Subsequence (with the number of strings as parameter) are XNLP-complete under pFO-reductions. We have chosen to use the easier to handle notion of logspace reductions throughout the paper, and not to distinguish which steps can be done with pl-reductions and which not.
One might want to use the least restricted form of reductions, under which XNLP remains closed, and that are transitive, in order to be able to show hardness for XNLP for as many problems as possible. Instead of using O(f (k)+ log n) space, one may want to use O(f (k) · log n) space -thus allowing to use a number of counters and pointers that depends on the parameter, instead of being bounded by a fixed constant. However, it is not clear that XNLP is closed under parameterized reductions with a O(f (k) · log n) space bound, as the reduction may use O(n f (k) ) time.
To remedy this, we can simultaneously bound the time and space of the reduction. A parameterized tractable logspace reduction (ptl-reductions) is a parameterized reduction that simultenously uses O(f (k) · log n) space and O(g(k) · n c ) time, with f and g computable functions, k the parameter, and n the input size. One can observe that the same argument ('repeatedly recomputing input bits when needed') that shows transitivity of L-reductions (see [2,Lemma 4.15]) can be used to show transitivity of parameterized tractable logspace reductions (and of parameterized logspace reductions).
We currently are unaware of a problem where we would use ptl-reductions instead of pl-reductions. However, the situation reminds of a phenomenon that also shows up for hardness proofs for classes in the W-hierarchy. Pl-reductions allow us to use time that grows faster than polynomial in the parameter value. If we have an fpt-reduction that uses O(f (k)n c ) time with c a constant, and f a polynomial function, then this reduction is also a many-to-one reduction, and could be used in an NP-hardness proof for the unparameterized version of the problem. Most but not all fpt-reductions from the literature have such a polynomial time bound. However, in the published hardness proofs, the distinction is usually not made explicit.
Candidate XNLP-complete problems
In this paper, we introduced a new parameterized complexity class, and showed a number of parameterized problems complete for the class, including Bandwidth. We expect that there are more problems complete for XNLP. Typical candidates are problems that are known to be hard for W [t] for all integers t. Possibly, in some cases, only small modifications of proofs may be needed, but in other cases, new proofs have to be invented. A number of such candidates are the following:
• Linear graph ordering problems, like Colored Cutwidth (and variants), Feasible Register Allocation, Triangulating Colored Graphs (see [10]), Topological Bandwidth.
• Domino Treewidth, see [7]. (We conjecture that XNLP-hardness can be proved with help of Accepting NNCCM; membership in XNLP is unclear due to the tree-like structure of positive instances.)
• Shortest Common Supersequence as mentioned in [17].
• Restricted Completion to a Proper Interval Graph with Bounded Clique Size, see [33],
• Problems parameterized by treewidth or pathwidth that are known to be in XP but not in FPT. Membership in XNLP for several problems parameterized by treewidth is also open (including Dominating Set and Independent Set). See e.g. [43] for some candidates for problems that might be XNLP-hard when parameterized by treewidth or pathwidth.
Complexity theoretic questions
We end this paper with a few related complexity theoretic open problems.
• Is XNLP closed under complementation? (This is the case for NL but probably not for NP.)
• Is XNLP unequal to XNL, under some well established assumptions?
• Can we get a problem not allowing 'best of both worlds', i.e. that can be solved in either log-space or in fpt time, but not both simultaneously?
parameterized by the number of machines k or by the combination of the number of machines k and the size of the alphabet Σ; the result is due to Hallett, but has not been published. More recently, the problem and many variations were studied by Wehar [46]. Amongst others, he showed that Finite State Automata Intersection with the number of machines as parameter is XNL-complete. He also considered the variant where the automata are acyclic.
Acyclic Finite State Automata Intersection
Input: k deterministic finite state automata on an alphabet Σ for which the underlying graphs are acyclic (except for self-loops at an accepting or rejecting state). Parameter: k. Question: Is there a string s ∈ Σ * that is accepted by each of the automata?
Wehar [46,Chapter 5] showed that Acyclic Finite State Automata Intersection is equivalent under LBL-reductions (parameterized reductions that do not change the parameter) to a version of Timed CNTMC (see Section 2.4) where the given time bound T is linear. The proof technique of Wehar [46] can also be used to show that Acyclic Finite State Automata Intersection is XNLP-complete. Instead, we give below a different simple reduction from Longest Common Subsequence, which is XNLP-complete [21]. Suppose that we are given k strings s 1 , . . . , s k ∈ Σ * and an integer m. We consider k + 1 automata on alphabet Σ: one for each string and one that checks the length of the solution.
Longest Common Subsequence
The length automaton has m + 1 states q 0 , . . . , q m , where q 0 is the starting state and q m is the accepting state. For each i ∈ [0, m − 1], and each σ ∈ Σ we have a transition from q i to q i+1 labeled with σ. We also have for each σ ∈ Σ a transition (self-loop) from q m to q m . The following claim is easy to observe.
Claim A.2. The length automaton accepts a string s ∈ Σ * if and only if the length of s is at least m.
For each i ∈ [1, k], we have a subsequence automaton for string s i . Suppose that its length is |s i | = t. The string automaton for s i has the following t + 2 states: q 0 , . . . , q t , q R . All states except for the rejecting state q R are accepting states, and q 0 is the starting state. The automaton for s i has the following transitions. For each j ∈ [0, t − 1] and σ ∈ Σ, if the substring s i j+1 · · · s i t contains the symbol σ, then let s i j be the first occurrence of σ in this substring, and take a transition from q j to q j labeled with σ. (Recall that for each state q and each σ ∈ Σ, if no transition out of q labeled with σ was defined in the previous step, then we take a transition from q to q R labeled with σ.) The name 'subsequence automaton' is explained by the following claim. Proof. Note that after reading a character, the automaton moves to the index of the next occurrence of this character after the current index. So, when we read a subsequence s, we are in state q j when j is the smallest integer with s a subsequence of the substring s i 1 . . . s i j . When there is no such next character, then s is not a subsequence and we move to the rejecting state.
From the claims above, it follows that a string s ∈ Σ * is a subsequence of s 1 , . . . , s k of length at least m if and only if s is accepted by both the length automaton and the subsequence automata of s 1 , . . . , s k . We can compute the automata in O(f (k) + log n) space, given the strings s 1 , . . . , s k and m (with n the number of bits needed to describe these).
It is also not difficult to obtain XNLP-completeness for the restriction to a binary alphabet. First, add dummy symbols to Σ to ensure that the size of the alphabet Σ is a power of two. Each transition labeled with a dummy symbol leads to the rejecting state q R , i.e., we accept the same set of strings. Now, encode each element of Σ with a unique string in {0, 1} log |Σ| . For each state q in the automata, replace the outgoing transitions by a complete binary tree of depth log |Σ| with the left branches labeled by 0 and the right branches labeled by 1. For a leaf of this tree, look at the string z ∈ {0, 1} log |Σ| formed by the labels when one follows the path from q to this leaf. This codes a character in Σ; now let this leaf have a transition to the state reached from q when this character is read. This straightforward transformation gives an automaton that precisely accepts the strings when we replace each character by its code in {0, 1} log |Σ| . If we apply the same transformation to all automata, we obtain an equivalent instance but with Σ = {0, 1}. We can conclude the following result.
Corollary A.4. Acyclic Finite State Automata Intersection is XNLPcomplete for automata with a binary alphabet.
B Reconfiguration
In Subsection 4.6, we showed that Timed Dominating Set Reconfiguration is XNLP-complete. Here, we give a similar result for Timed Independent Set Reconfiguration and Timed Clique Reconfiguration, again assuming that the number of steps is given in unary. The problem descriptions are the same as for Timed Dominating Set Reconfiguration; we spell out the definition for independent set below for the convenience of the reader. Independent set reconfiguration has been widely studied and is known to be PSPACE-complete for both TJ [30] and TS [29], and W [1]-hard for TJ when parameterized by the number of tokens [31]. We remark that token jumping with k tokens is equivalent to the token addition-removal rule where the solution always needs to contain at least k − 1 vertices. On the other hand, token sliding and token jumping are not equivalent. If complements cannot be taken efficiently, specific graph classes are studied or the token sliding rule is used, the clique reconfiguration and the independent set reconfiguration could have different complexities.
In this section, we show that both the timed variants of both clique and independent set reconfiguration are XNLP-complete with parameter the number of tokens. The proofs are very similar, and since the idea of the proof is most naturally phrased for Timed TJ-Clique Reconfiguration, we present it first for that and then deduce the other variants from it.
Theorem B.1. Timed TJ-Clique Reconfiguration is XNLP-complete, with the number of steps given in unary.
Proof. Membership in XNLP is easy: one can guess the sequence of moves, and keep the current positions of the tokens in memory.
For the hardness, we transform from Chained Multicolored Clique. Suppose that we are given an integer k and a graph G = (V, E) with a partition of V into sets V 1 , . . . , V r . Let f : V → [1, k] be a given vertex coloring. We look for a chained multicolored clique W ⊆ V , that is, for each
i ∈ [r], |W ∩ V i | = k, for each i ∈ [1, r − 1], W ∩ (V i ∪ V i+1
) is a clique, and for each i ∈ [1, r] and j ∈ [1, k], W ∩ V i contains exactly one vertex of color j.
We may assume that for each edge {v, w} ∈ E, if v ∈ V i and w ∈ V i , then |i − i | ≤ 1 and if i = i , then f (v) = f (w). (Edges that do not fulfill these properties will never contribute to a chained multicolored clique, and thus we can remove them. A transducer exists that removes all edges that do not fulfill the property using logarithmic space.)
We build a graph H as follows.
• Start with G.
• Add vertex sets V −1 , V 0 , V r+1 , V r+2 , each of size k. Extend f as follows: for each color j ∈ [1, k], and each of the sets V −1 , V 0 , V r+1 , V r+2 , take one vertex of the set and color it with j.
• For all v ∈ V i and w ∈ V i , v = w, add an edge only if at least one of the following holds:
-i ∈ {−1, 0} and i ∈ {−1, 0}, i.e. V −1 ∪ V 0 is a clique, -i = 0 and i = 1,
i ∈ {r + 1, r + 2} and i ∈ {r + 1, r + 2}, i.e. V r+1 ∪ V r+2 is a clique, i = r and i = r + 1,
-|i − i | = 2, and f (v) = f (w): a vertex is adjacent to all vertices 'two sets away', except those with the same color.
We reconfigure a clique with 2k vertices. The initial configuration is V −1 ∪V 0 and the final configuration is V r+1 ∪ V r+2 .
Claim B.2. Suppose that G has a chained multicolored clique with k colors. Then we can reconfigure V −1 ∪ V 0 to V r+1 ∪ V r+2 with k · (r + 2) jumping moves in H, with each intermediate configuration a clique.
Proof. Let W be the chained multicolored clique with k colors.
Write W = W ∪ V −1 ∪ V 0 ∪ V r+1 ∪ V r+2 . The starting configuration is S = V −1 ∪ V 0 .
Take the following sequence: for i from −1 to r, for j from 1 to k, let v be the vertex in W ∩ V i with f (v) = j, and w be the vertex in W ∩ V i+2 with f (v) = j and update S to S ∪ {w} \ {v}.
It is easy to verify that at each step S is a clique; the total number of moves equals k · (r + 2). Claim B.3. Suppose that we can reconfigure V −1 ∪ V 0 to V r+1 ∪ V r+2 with k · (r + 2) jumping moves in H, with each intermediate configuration a clique. Then G has a chained multicolored clique with k colors.
Proof. The level of a vertex v ∈ V (G) is the unique i ∈ [1, r] for which v ∈ V i . Note that for each edge {v, w} in H, if v ∈ V i and w ∈ V i , then |i − i | ≤ 2. Thus, each clique can contain vertices of at most three consecutive levels. Also, vertices in the same level with the same color are not adjacent, and vertices two levels apart with the same color are not adjacent. Thus, if H has a clique with 2k vertices, then this clique is a subset of V i ∪ V i+1 ∪ V i+2 for some i ∈ [−1, r], with the property that for each color j ∈ [1, k], there is one vertex with this color in V i ∪ V i+2 and one vertex with this color in V i+1 .
Denote the initial, final and intermediate clique configurations by S. To each such set S, we associate a value. The potential Φ(S) is the sum of all levels of the vertices in a clique S.
We now argue that each move of a token can increase Φ(S) by at most two. For any clique S, there is an
i with S ⊆ V i ∪ V i+1 ∪ V i+2 and S ∩ V i = ∅. If |S ∩ V i | ≥ 2,
then after a move of one vertex, there still will be at least one vertex in S ∩ V i . This means that we can only move to vertices in V i with
i ≤ i + 2. If |S ∩ V i | = 1, then suppose S ∩ V i = {v} with f (v) = j. Now, if we
move v then we can only move v to a vertex in V i ∪ V i+2 with color j. In both cases, the potential of S increases by at most two.
The initial configuration S 0 has Φ(S 0 ) = −k and the final configuration S k(r+2) has Φ(S) = k · (2r + 3). Thus, each of the k · (r + 2) moves must increase the potential by exactly two. Thus, each move must take a vertex and move it to a vertex exactly two levels later. As each clique configuration spans at most three levels, we have that after each kth move, the configuration spans two levels, and necessarily is a clique with for each color, one vertex from each of these two levels. The collection of these cliques, except the first and last two, forms a chained multicolored clique in G.
To finish the hardness proof, we combine the two claims above and observe that H can be constructed in logarithmic space. Proof. Taking the complement of a graph can be done in logarithmic space; thus, the result follows directly from Theorem B.1, noting that we only 'jump' over non-edges in the proof of Theorem B.1, so that in the complement the reconfiguration sequence adheres to the token sliding rules as well.
For the clique variant, a slight alteration to the current construction works.
Corollary B.5. Timed TS-Clique Reconfiguration is XNLP-complete.
Sketch. We follow the proof of Theorem B.1. We build the graph H in a similar fashion, apart from the edges between pairs of sets (
V i , V i+2 ). For each i ∈ [−1, r], we place an edge from v ∈ V i to w ∈ V i+2 if and only if f (v) ≥ f (w).
Inductively, one can show that any clique must still span at most three consecutive layers and that each clique contains exactly one vertex of each color from the even and odd levels at any point. Indeed, the vertex of color k in V −1 can only slide to a vertex of color k in V 1 , and since the vertices of color k in V 1 form an independent set, this forces the vertex of color k − 1 in V −1 to slide to a vertex of color k − 1 in V 1 etcetera.
The same argument can be used for turning a clique into a configuration sequence; however, now it has become important that we slide the vertices in the right order, that is, the first token to be moved from V i to V i+2 needs to be the one on a vertex of color 1 (to ensure we remain a clique at any point).
C Log pathwidth
We study the complexity of the independent set and clique analogues of Log-Pathwidth Dominating Set of Subsection 4.2. The independent set variant is defined as follows; the clique variant is defined analogously.
Log-Pathwidth Independent Set
Input: Graph G = (V, E), path decomposition of G of width , integer K. Parameter: / log |V | . Question: Does G have an independent set of size at least K?
Theorem C.1. Log-Pathwidth Independent Set is XNLP-complete.
Proof. As with Log-Pathwidth Dominating Set, the problem is in XNLP as we can use the standard dynamic programming algorithm on graphs with bounded pathwidth, but guess the table entry for each bag.
We show XNLP-hardness with a reduction from Paritioned Regular Chained Weighted CNF-Satisfiability. The reduction is similar to the reduction to Log-Pathwidth Dominating Set, but uses different gadgets specific to Independent Set. The clause gadget, which will be defined later, is a direct copy of a gadget from [34].
Suppose that F 1 , q, X 1 , . . . , X r , k, X i,j (1 ≤ i ≤ r, 1 ≤ j ≤ k) is the given instance of the Paritioned Regular Chained Weighted CNF-Satisfiability problem. Let F 1 = {C 1 , ..., C m } and let |C i | be the number of variables in clause C i . We assume |C i | to be even for all i ∈ [1, m]; if this number is odd, then we can add an additional copy of one of the variables to the clause. Enlarging |X i,j | by at most a factor of 2 (by adding dummy variables to X i ), we may assume that t = log |X i,j | is an integer. (Recall that log has base 2 in this paper.)
Variable choice gadget This gadget consists of t copies of a K 2 (a single edge), with one vertex marked with a 0 and the other with a 1. Each element in X i,j can be represented by a unique q-bit bitstring. One K 2 describes one such bit and together these t bits describe one element of X i,j , which is going to be the variable that we assume to be true. We will ensure that in a maximal independent set, we need to choose exactly one vertex per copy of K 2 , and hence always represent a variable. Clause checking gadget The clause checking gadget is a direct copy of a gadget from [34] and is depicted in Figure 7. The construction is as follows. Let C = x 1 ∨ · · · ∨ x be a clause on X i ∪ X i+1 (for some i ∈ [1, r]). We create two paths p 0 , . . . , p +1 and p 1 , . . . , p and add an edge {p j , p j } for j ∈ [1, ]. We then add the variable representing vertices v 1 , . . . , v . For j ∈ [1, ], the vertex v j is connected to p j and p j and to the vertices representing the complement of the respective bits in the representation of x j . This ensures that v j can be chosen if and only if the vertices chosen from the variable choice gadget represent x j . For each clause, we add such a clause checking gadget on 3 + 2 new vertices. The pathwidth of this gadget is 4.
Recall that we assume to be even for all clauses. Each clause checking gadget has the property that it admits an independent set of size + 2 (which is then maximum) if and only if at least one of the v-type vertices is in the independent set, as proved in [34]. Otherwise its maximum independent set is of size + 1. We try to model the clause being satisfied by whether or not the independent set contains + 2 vertices from this gadget.
The graph G is formed by combining the variable choice and clause checking gadgets.
Claim C.2. There is an independent set in G of size rkt+(r −1) Proof. Suppose that there is a valid assignment of the variables that satisfies the clauses. We select the bits that encode the true variables in the variable choice gadgets and put these in the independent set. This forms an independent set because only one element per variable choice gadget is set to true by a valid assignment. As we have rk sets of the form X i,j , and each is represented by t times two vertices of which we place one in the independent set, we now have rkt vertices in the independent set.
For each clause checking gadget for a clause on variables, we add a further + 2 vertices to the independent set as follows. Let x i be a variable satisfying the clause. We add the variable representing vertex v i (corresponding to x i ) into the independent set. Since v i is chosen in the independent set, an additional + 1 vertices can be added from the corresponding clause checking gadget. In total we place rkt + (r − 1) m i=1 (|C i | + 2) vertices in the independent set. To prove the other direction, let S be an independent set of size rkt + (r − 1) m i=1 (|C i | + 2). Any independent set can contain at most t vertices from any variable choice gadget (one per K 2 ), and at most |C| + 2 vertices per clause checking gadget corresponding to clause C. Hence, S contains exactly t vertices per variable choice gadget and exactly |C| + 2 vertices per clause gadget.
Consider a variable choice gadget. Since S contains exactly t vertices (one per K 2 ), a variable from the set X i,j is encoded. It remains to show that we satisfy the formula by setting exactly these variables to true and all others to false. Consider a clause C and its related clause checking gadget. As |C| + 2 vertices from its gadget are in S, this means that at least one of its vertices of the form v i is also in S. None of the neighbors of v i can be in S, so the corresponding variable x i must be set to true. Thus the clause is indeed satisfied.
A path decomposition can be constructed as follows. For i = 1, . . . , r, let V i be the set of all vertices in the variable choice gadgets of the sets X i,j and X i+1,j (for all j ∈ [1, k]). We create a sequence of bags that contains V i as well as a constant number of vertices from clause checking gadgets. We transverse the clause checking gadgets in any order. For a given clause checking gadget of a clause on variables, we first create a bag containing V i and p 0 , p 1 , p 1 and v 1 . Then for s = 2, . . . , , we add p s , p s and v s to the bag and remove p s−2 , p s−2 and v s−2 (if these exist). At the final step, we remove p −1 , p −1 and v −1 from the bag and add p +1 . We then continue to the next clause checking gadget. Each bag contains at most 4k · t from the set V i and at most 6 from any clause gadget.
This yields a path decomposition of width at most 4kt + 6 ≤ 10k log |X i,j | which is of the form g(k) log(n) for n the number of vertices of G and g(k) a function of the problem we reduced from, as desired. Since G can be constructed in logarithmic space, the result follows.
We now briefly discuss the Log-Pathwidth Clique problem. We are given a graph G = (V, E) with a path decomposition of width , and integer K and ask if there is a clique in G with at least K vertices. Let k = / log |V | .
The problem appears to be significantly easier than the corresponding versions of Dominating Set and Independent Set, mainly because of the property that for each clique W , there must be a bag of the path decomposition that contains all vertices of W (see e.g., [11].) Thus, the problem reverts to solving O(n) instances of Clique on graphs with O(k log n) vertices. This problem is related to a problem called Mini-Clique where the input has a graph with a description size that is at most k log n. Mini-Clique is M [1]-complete under FPT Turing reductions (see [18,Corollary 29.5.1]), and is a subproblem of our problem, and thus Log-Pathwidth Clique is M [1]-hard. However, instances of Log-Pathwidth Clique can have description sizes of Ω(log 2 n). The result below shows that it is unlikely that Log-Pathwidth Clique is XNLP-hard.
Proposition C.3. Log-Pathwidth Clique is in W [2].
Proof. Suppose that we are given a graph G = (V, E) and path decomposition (X 1 , . . . , X r ) of G of width at most k log n − 1. Let K be a given integer for which we want to know whether G has a clique of size K. We give a Boolean expression F of polynomial size that can be satisfied with exactly 2k+2 variables set to true, if and only if G has a clique of size K. (This shows membership in W [2].) First, we add variables b 1 , . . . , b r that are used for selecting a bag; we add the clause 1≤i≤r b i .
For each bag X i , we build an expression that states that G[X i ] has a clique with r vertices as follows. Partition the vertices of X i in k groups of at most log n vertices each. For each group S ⊆ X i , we have a variable c i,S for each subset S of vertices of the group that induces a clique in G. (We thus have at most kn such variables per bag.) The idea is that when c i,S is true, then the vertices in S are part of the clique, and the other vertices in the group are not. The variable c i,S is false if the bag X i is not selected (we add the clause b i ∨ ¬c i,S ). For each group S, we add a clause that is the disjunction over all cliques S ⊆ S of c i,S and ¬b i . This encodes that if the bag X i is selected, then we need to choose a subset S from the group S.
With the b-and c-variables, we specified a bag and a subset of the vertices of the bag. It remains to add clauses that verify that the total size of all selected subsets equals K, and that the union of all selected subsets is a clique. For the latter, we add a clause ¬c i,S ∨ ¬c i,S for each pair S , S of subsets of different groups for which S ∪ S does not induce a clique. For the former, we number the groups 1, 2, . . . , k, and create variables t j,q for j ∈ [1, k] and q ∈ [0, K]. The variable t j,q expresses that we have chosen q vertices in the clique from the first j groups of the chosen bag. For each j ∈ [1, k], we add the clause q t j,q . To enforce that the t-variables indeed give the correct clique sizes, we add a large (but polynomial) collection of clauses. We add a variable t 0,0 that must be true (using a one-literal clause). For the jth group with vertex set S ⊆ X i , each clique S ⊆ S, and each q, q ∈ [0, r] (with q = 0 when j = 1), whenever q + |S | = q , we have a clause ¬t j−1,q ∨ ¬c i,S ∨ ¬t j,q . Finally, we require (with a one-literal clause) that t k,K holds. If there is a satisfying assignment, then the union of all sets S for which c i,S is true, forms a clique of size K in the graph.
We allow 2k + 2 variables to be set to true: one variable b i to select a bag, one for t 0,0 , one for a c i,S -variable in each of the k groups and one t j,q -variable for each j ∈ [1, k] (where for j = k, q = K is enforced).
Figure 1 :
1Reductions between XNLP-hard problems from this paper. Several variants of problems are not shown.
FPT can be denoted by D[f poly, ∞]; we can denote XP by D[n f , ∞], NP by N [poly, ∞], L by D[∞, log], etcetera.
Lemma 2 . 1 .
21XNLP is a subset of XP.
CNTMC
Input: the encoding of a non-deterministic Turing Machine M ; the encoding of a string x over the alphabet of the machine. Parameter: k. Question: Is there an accepting computation of M on input x that visits at most k cells of the work tape?
Theorem 3 . 1 .
31Chained Weighted CNF-Satisfiability is XNLP-complete.
Theorem 3 . 2 .
32Chained Weighted Positive CNF-Satisfiability is XNLPcomplete.
Theorem 4. 1 .
1List Coloring parameterized by pathwidth is XNLP-complete.
Claim 4 . 2 .
42There is a satisfying truth assignment fulfilling the conditions of the Monotone Partitioned Chained CNF-Satisfiability problem, if and only if there is a proper coloring where each vertex has a color from its list.
Corollary 4 . 3 .
43Pre-coloring extension parameterized by pathwidth is XNLPcomplete.
Theorem 4. 4 .
4Log-Pathwidth Dominating Set is XNLP-complete.
Figure 2 :
2Example of clause gadget. Here, t = 2 and only the variable with index 00 satisfies clause c
Claim 4. 5 .
5There is a dominating set in G of size rkt + 2kc(r − 1), if and only if the instance of Partitioned Regular Chained Weighted CNF-Satisfiability has a solution.
Theorem 4. 6 .
6Scheduling with Precedence Constraints is XNLPcomplete.Proof. To see that the problem is in XNLP, we definef −1 {1}, f −1 {2}, . . . , f −1 {D} in order as follows. For i ∈ [0, D − 1], we temporarily store a set S i ⊆ T containing the maximal elements of f −1[1, i], initialising S 0 = ∅. Each such set S i forms an antichain and hence has size at most w, the width of the partial order on T . Once S i−1 is defined for some i ∈ [1, D], we define f −1 {i} by selecting up to K tasks to schedule next, taking a subset of the minimal elements of the set of 'remaining tasks'
For most t ∈ [1, D], we are happy for a potential d
Figure 3 :
3this time; if t 'corresponds' to s j = (i 1 , i 2 , r 1 , r 2 ), then only d(i1) t−r1 , d (i2)t−r2 can potentially be planned simultaneously with a t , which Illustration for a time sequence with time indicators. E.g., in the example, 3 ∈ I.
n] k be the positions of the k counters before s j is checked, for
[ 0 ,
0kn] and y ∈ [1, n + 1]. At each t ∈ [1, D], f −1 {t} contains exactly one a t and, because the counters are non-decreasing, at most one c (i) t and at most one d (i) t for each i ∈ [1, k]. So when t ∈ I (and hence b x t is not defined), |f −1 {t}| ≤ 2k + 1 as desired. Let t ∈ I and j = j(t) ∈ [1, r], α = α(t) ∈ [0, kn]. If d (i) t ∈ f −1 {t} for some i ∈ [1, k], then t = t − p (i)
[1, n] → Z + and ask for an emulation f : [1, n] → [1, m], such that there is a constant c with i∈f −1 {j} w(i) = c for all j ∈ [1, m]. Again, we call c the emulation factor.
Input: Positive integers n, m, c, weight function w : [1, n] → [1, c]. Parameter: c. Question: Is there a function f : [1, n] → [1, m], such that f is a uniform emulation of P n on P m with emulation factor c, i.e., |f (i) − f (i + 1)| ≤ 1 for all i ∈ [1, n − 1] and i∈f −1 {j} w(i) = c for all j ∈ [1, m]? Theorem 4.7. Uniform Emulation of Weighted Paths is XNLP-complete.
Claim 4. 8 .
8The NNCCM described by (k, n, s) accepts if and only if P N with the defined vertex weights has a uniform emulation on P M , Proof. If the NNCCM accepts, then we can build a uniform emulation as follows. Fix some accepting run and let q(j) be the value of counter q at the jth check in this run. Map the ith vertex of the floor to i. Now, successively map each counter component as follows. We start by 'moving back to 1', i.e., we map the M − 2 vertices of weight 1 to M − 1, M − 2, . . . , 2 and map the left turning point to 1. To map the vertices on the main path of this counter component, we use the following procedure. For convenience, we define q(r + 1) = n. • Set i = 1, p = 2. We see p ∈ [1, M ] as a position on P M , and i ∈ [1, M + n] as the number of the vertex from the main path we are currently looking at. The number i − (p − 1) represents the value of the counter.
Theorem 4.7 now follows from Claim 4.8 and observing the resources (logarithmic space, fpt-time) by the transformation.
Lemma 4. 9 .
9Bandwidth is in XNLP.
Timed TS-Dominating Set ReconfigurationGiven: Graph G = (V, E); dominating sets S, S of size k; integer T given in unary.Parameter: k. Question: Does there exist a sequence S = S 1 , S 2 , . . . , S T = S of dominating sets of size k, with for all i ∈ [2, T ], S i = S i−1 \ {u} ∪ {v} for some uv ∈ E(G)? Timed TJ-Dominating Set Reconfiguration Given: Graph G = (V, E); dominating sets S, S of size k; integer T given in unary. Parameter: k. Question: Does there exist a sequence S = S 1 , S 2 , . . . , S T = S of dominating sets of size k, with for all i ∈ [2, T ], S i = S i−1 \ {u} ∪ {v} for some u, v ∈ V (G)?
Figure 5 :
5An overview of the construction is given. these tokens at each time step. In particular, at each time step we can either move the timer token within the timer, or move the variable tokens within their respective sets.We set S = {t 0 , d, s 1 , . . . , s 2k } and set S = {t 2r−3 , d, s 1 , . . . , s 2k }.
Theorem 4 . 12 .
412Timed TJ-Dominating Set Reconfiguration is XNLPcomplete.
Figure 6 :
6This 'timer' mimicks token sliding under the token jumping rule.
Conjecture 5.2. N [f poly, f log n] ⊆ D[n f , f poly(n)].
Input: k strings s 1 , . . . , s k over an alphabet Σ; integer m. Parameter: k. Question: Does there exist a string s ∈ Σ * of length at least m that is a subsequence of s i for all i ∈ [1, k]? Theorem A.1. Acyclic Finite State Automata Intersection is XNLPcomplete.Proof. For XNLP-membership of Acyclic Finite State Automata Intersection, the argument from [46, Proposition 5.1] can be used: guess the solution string one character at the time, and keep track of the states of the machines during these guesses. To see this does not take too much time, let n be the largest number of vertices in a graph underlying one of the machines. Since the graphs are acyclic, all machines enter an accepting or rejecting state within n transitions. The simple transformation from Longest Common Subsequence parameterized by the number of strings k to Acyclic Finite State Automata Intersection is given below.
Claim A. 3 .
3The subsequence automaton for string s i accepts a string s ∈ Σ * if and only if s is a subsequence of s i .
Timed TS-Independent Set ReconfigurationGiven: Graph G = (V, E); independent sets S, S of size k; integer T given in unary.Parameter: k. Question: Does there exist a sequence S = S 1 , S 2 , . . . , S T = S of independent sets of size k, with for all i ∈ [2, T ], S i = S i−1 \ {u} ∪ {v} for some uv ∈ E(G)? Timed TJ-Independent Set Reconfiguration Given: Graph G = (V, E); independent sets S, S of size k; integer T given in unary. Parameter: k. Question: Does there exist a sequence S = S 1 , S 2 , . . . , S T = S of independent sets of size k, with for all i ∈ [2, T ], S i = S i−1 \ {u} ∪ {v} for some u, v ∈ V (G)?
Corollary B. 4 .
4Timed TJ-Independent Set Reconfiguration and Timed TS-Independent Set Reconfiguration are XNLP-complete.
Figure 7 :
7An example of a clause gadget is drawn with two variable gadgets. The vertex v 1 represents a variable which is encoded by bit string 10, where v 2 represents a variable encoded by 00.
m i=1 (
i=1|C i |+2) if and only if the instance of Partitioned Regular Chained Weighted CNF-Satisfiability has a solution.
For example the naïve algorithm O(n k+1 ) time algorithm for finding cliques on k vertices on n-vertex graphs can be improved to run in n 0.8k time, but similar run times for Dominating Set refute the Strong Exponential Time Hypothesis.
Formally, there is a reconfiguration sequence for TJ with k tokens between S 1 and S 2 if and only if there is one for TAR with upper bound k + 1.
A Acyclic Finite State Automata IntersectionIn this section, we briefly discuss the Finite State Automata Intersection problem, and a variant for acyclic automata. In this problem, we are given k deterministic finite state automata on an alphabet Σ and ask if there is a string s ∈ Σ * that is accepted by each of the automata.In the overview of the parameterized complexity of various problems in[17], the problem is mentioned to be hard for all classes W [t], t ∈ Z + , when either
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| []
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[
"FORMATION, SURVIVAL, AND DESTRUCTION OF VORTICES IN ACCRETION DISKS",
"FORMATION, SURVIVAL, AND DESTRUCTION OF VORTICES IN ACCRETION DISKS"
]
| [
"Yoram Lithwick "
]
| []
| []
| Two dimensional hydrodynamical disks are nonlinearly unstable to the formation of vortices. Once formed, these vortices essentially survive forever. What happens in three dimensions? We show with incompressible shearing box simulations that in 3D a vortex in a short box forms and survives just as in 2D. But a vortex in a tall box is unstable and is destroyed. In our simulation, the unstable vortex decays into a transient turbulent-like state that transports angular momentum outward at a nearly constant rate for hundreds of orbital times. The 3D instability that destroys vortices is a generalization of the 2D instability that forms them. We derive the conditions for these nonlinear instabilities to act by calculating the coupling between linear modes, and thereby derive the criterion for a vortex to survive in 3D as it does in 2D: the azimuthal extent of the vortex must be larger than the scale height of the accretion disk. When this criterion is violated, the vortex is unstable and decays. Because vortices are longer in azimuthal than in radial extent by a factor that is inversely proportional to their excess vorticity, a vortex with given radial extent will only survive in a 3D disk if it is sufficiently weak. This counterintuitive result explains why previous 3D simulations always yielded decaying vortices: their vortices were too strong. Weak vortices behave two-dimensionally even if their width is much less than their height because they are stabilized by rotation, and behave as Taylor-Proudman columns. We conclude that in protoplanetary disks weak vortices can trap dust and serve as the nurseries of planet formation. Decaying strong vortices might be responsible for the outwards transport of angular momentum that is required to make accretion disks accrete. Subject headings: accretion, accretion disks -instabilities -solar system: formation -turbulence the flow, i.e. that the velocity field must have an inflection point. Lovelace et al. (1999) generalize Rayleigh's inflection point theorem to compressible and nonhomentropic disks. | 10.1088/0004-637x/693/1/85 | [
"https://arxiv.org/pdf/0710.3868v2.pdf"
]
| 16,581,644 | 0710.3868 | 588215a22bf06cca2c077e6d77e96870cc838a5d |
FORMATION, SURVIVAL, AND DESTRUCTION OF VORTICES IN ACCRETION DISKS
19 Feb 2009 Draft version February 19, 2009 February 19, 2009
Yoram Lithwick
FORMATION, SURVIVAL, AND DESTRUCTION OF VORTICES IN ACCRETION DISKS
19 Feb 2009 Draft version February 19, 2009 February 19, 2009Preprint typeset using L A T E X style emulateapj v. 10/09/06 Draft versionSubject headings: accretion, accretion disks -instabilities -solar system: formation -turbulence
Two dimensional hydrodynamical disks are nonlinearly unstable to the formation of vortices. Once formed, these vortices essentially survive forever. What happens in three dimensions? We show with incompressible shearing box simulations that in 3D a vortex in a short box forms and survives just as in 2D. But a vortex in a tall box is unstable and is destroyed. In our simulation, the unstable vortex decays into a transient turbulent-like state that transports angular momentum outward at a nearly constant rate for hundreds of orbital times. The 3D instability that destroys vortices is a generalization of the 2D instability that forms them. We derive the conditions for these nonlinear instabilities to act by calculating the coupling between linear modes, and thereby derive the criterion for a vortex to survive in 3D as it does in 2D: the azimuthal extent of the vortex must be larger than the scale height of the accretion disk. When this criterion is violated, the vortex is unstable and decays. Because vortices are longer in azimuthal than in radial extent by a factor that is inversely proportional to their excess vorticity, a vortex with given radial extent will only survive in a 3D disk if it is sufficiently weak. This counterintuitive result explains why previous 3D simulations always yielded decaying vortices: their vortices were too strong. Weak vortices behave two-dimensionally even if their width is much less than their height because they are stabilized by rotation, and behave as Taylor-Proudman columns. We conclude that in protoplanetary disks weak vortices can trap dust and serve as the nurseries of planet formation. Decaying strong vortices might be responsible for the outwards transport of angular momentum that is required to make accretion disks accrete. Subject headings: accretion, accretion disks -instabilities -solar system: formation -turbulence the flow, i.e. that the velocity field must have an inflection point. Lovelace et al. (1999) generalize Rayleigh's inflection point theorem to compressible and nonhomentropic disks.
INTRODUCTION
Matter accretes onto a wide variety of objects, such as young stars, black holes, and white dwarfs, through accretion disks. In highly ionized disks magnetic fields are important, and they trigger turbulence via the magnetorotational instability (Balbus & Hawley 1998). However, many disks, such as those around young stars or dwarf novae, are nearly neutral (e.g., Sano et al. 2000;Gammie & Menou 1998). In these disks, the fluid motions are well described by hydrodynamics.
Numerical simulations of hydrodynamical disks in two-dimensions-in the plane of the disk-often produce long-lived vortices (Godon & Livio 1999;Umurhan & Regev 2004;Johnson & Gammie 2005). If vortices really exist in accretion disks, they can have important consequences.
First and foremost, they might generate turbulence. Since turbulence naturally transports angular momentum outwards 2 , as is required for mass to fall inwards, it might be vortices that cause accretion disks to accrete. Second, in disks around young stars, long-lived vortices can trap solid particles and initiate the formation of planets (Barge & Sommeria 1995).
Why do vortices naturally form in 2D simulations? Hydrodynamical disks are stable to linear perturbations. However, they are nonlinearly unstable, despite some claims to the contrary in the astrophysical literature.
In two dimensions, the incompressible hydrodynamical equations of a disk are equivalent to those of a nonrotating linear shear flow (e.g., Lithwick 2007, hereafter L07). And it has long been known that such flows are nonlinearly unstable (Gill 1965;Lerner & Knobloch 1988;L07). This nonlinear instability is just a special case of the Kelvin-Helmholtz instability. Consider a linear shear flow extending throughout the x-y plane with velocity profile v = −qxŷ, where q > 0 is the constant shear rate, so that −q is the flow's vorticity. (In the equivalent accretion disk, the local angular speed is Ω = 2q/3.) This shear flow is linearly stable to infinitesimal perturbations. But if the shear profile is altered by a small amount, the alteration can itself be unstable to infinitesimal perturbations. To be specific, let the alteration be confined within a band of width ∆x, and let it have vorticity ω = ω(x) (with |ω| q), so that it induces a velocity field in excess of the linear shear with components u y ∼ ω∆x and u x = 0. Then this band is unstable to infinitesimal nonaxisymmetric (i.e. nonstream-aligned) perturbations provided roughly that
|k y | 1 q |ω| ∆x ⇒ 2D instability(1)
where k y is the wavenumber of the nonaxisymmetric perturbation. 3 For any value of |ω| and ∆x, the band is al-3 More precisely, the necessary and sufficient condition for instability in the limit |ω| ≪ q is that |ky| < 1 2q
R ∞ −∞ dω/dx x−x 0
dx, where x 0 is any value of x at which dω/dx = 0 (Gill 1965;Lerner & Knobloch 1988, L07). For arbitrarily large ω, Rayleigh's inflection point theorem and Fjørtoft's theorem give necessary (though insufficient) criteria for instability (Drazin & Reid 2004). The former states that for instability, it is required that dω/dx = 0 somewhere in ways unstable to perturbations with long enough wavelength. Remarkably, instability even occurs when |ω| is infinitesimal. Hence we may regard this as a true nonlinear instability. Balbus & Hawley (2006) assert that detailed numerical simulations have not shown evidence for nonlinear instability. The reason many simulations fail to see it is that their boxes are not long enough in the y-direction to encompass a small enough non-zero |k y |.
In two dimensions, the outcome of this instability is a long-lived vortex (e.g., L07). A vortex that has been studied in detail is the Moore-Saffman vortex, which is a localized patch of spatially constant vorticity superimposed on a linear shear flow (Saffman 1995). When |ω| q, where ω here refers to the spatially constant excess vorticity within the patch, and when the vorticity within the patch (ω − q) is stronger than that of the background shear, then the patch forms a stable vortex that is elongated in y relative to x by the factor
∆y ∆x ∼ q |ω| .(2)
This relation applies not only to Moore-Saffman vortices, but also to vortices whose ω is not spatially constant. It may be understood as follows. A patch with characteristic excess vorticity ∼ ω and with ∆y ≫ ∆x induces a velocity field in the x-direction with amplitude u x ∼ |ω|∆x, independent of the value of ∆y (e.g., §6 in L07). As long as |ω| q, the y-velocity within the vortex is predominantly due to the background shear, and is ∼ q∆x. Therefore the time to cross the width of the vortex is t x ∼ ∆x/u x ∼ 1/|ω|, and the time to cross its length is t y ∼ ∆y/(q∆x). Since these times must be comparable in a vortex, equation (2) follows. Equation (2) is very similar to equation (1). The 2D instability naturally forms into a 2D vortex. Futhermore, the exponential growth rate of the instability is ∼ |ω|, which is comparable to the rate at which fluid circulates around the vortex. More generally, an arbitrary axisymmetric profile of ω(x) tends to evolve into a distinctive banded structure. Roughly speaking, bands where ω < 0 contain vortices, and these are interspersed with bands where ω > 0, which contain no vortices. (Recall that we take the background vorticity to be negative; otherwise, the converse would hold.) The reason for this is that only regions that have ω < 0 can be unstable, as may be inferred either from the integral criterion for instability given in footnote 3, or from Fjørtoft's theorem. For more detail on vortex dynamics in shear flows, see the review by Marcus (1993).
What happens in three dimensions? To date, numerical simulations of vortices in 3D disks have been reported in two papers. Barranco & Marcus (2005) initialized their simulation with a Moore-Saffman vortex, and solved the anelastic equations in a stratified disk. They found that this vortex decayed. As it decayed, new vortices were formed in the disk's atmosphere, two scale heights above the midplane. The new vortices survived for the duration of the simulation. Shen et al. (2006) performed both 2D and 3D simulations of the compressible hydrodynamical equations in an unstratified disk, initialized with large random fluctuations. They found that whereas the 2D simulations produced long-lived vortices, in three dimensions vortices rapidly decayed.
Intuitively, it seems clear that a vortex in a very thin disk will behave as it does in 2D. And from the 3D simulations described above it may be inferred that placing this vortex in a very thick disk will induce its decay. Our main goal in this paper is to understand these two behaviors, and the transition between them. A crude explanation of our final result is that vortices decay when the 2D vortex motion couples resonantly to 3D modes, i.e., to modes that have vertical wavenumber k z = 0. As described above, a vortex with excess vorticity |ω| has circulation frequency ∼ |ω|, and k y /k x ∼ |ω|/q, where k x and k y are its "typical" wavenumbers. Furthermore, it is well-known that the frequency of axisymmetric (k y = 0) inertial waves is Ωk z / k 2 x + k 2 z (see eq.
[41]). Equating the two frequencies, and taking the k x of the 3D mode to be comparable to the k x of the vortex, as well as setting q = 3Ω/2 for a Keplerian disk, we find k z ∼ k y (3) as the condition for resonance. Therefore a vortex with length ∆y will survive in a box with height ∆z ∆y, because in such a box all 3D modes have too high a frequency to couple with the vortex, i.e., all nonzero k z exceed the characteristic k y ∼ 1/∆y. But when ∆z ∆y, there exist k z in the box that satisfy the resonance condition (3), leading to the vortex's destruction. This conclusion suggests that vortices live indefinitely in disks with scale height less than their length (h ∆y) because in such disks all 3D modes have too high a frequency for resonant coupling. This conclusion is also consistent with the simulations of Barranco & Marcus (2005) and Shen et al. (2006). Both of these works initialized their simulations with strong excess vorticity |ω| ∼ q, corresponding to nearly circular vortices. Both had vertical domains that were comparable to the vortices' width. Therefore both saw that their vortices decayed. Had they initialized their simulations with smaller |ω|, and increased the box length L y to encompass the resulting elongated vortices, both would have found long-lived 3D vortices. Barranco & Marcus's discovery of long-lived vortices in the disk's atmosphere is simple to understand because the local scale height is reduced in inverse proportion to the height above the midplane. Therefore higher up in the atmosphere the dynamics becomes more two-dimensional, and a given vortex is better able to survive the higher it is. 4
Organization of the Paper
In §2 we introduce the equations of motion, and in §3 we present two pseudospectral simulations. One illustrates the formation and survival of a vortex in a short box, and the other illustrates the destruction of a vortex in a tall box.
In § §4-5 we develop a theory explaining this behavior. The reader who is satisfied by the qualitative description leading to equation (3) may skip those two sections. The theory that we develop is indirectly related to the transient amplification scenario for the generation of turbulence. Even though hydrodynamical disks are linearly stable, linear perturbations can be transiently amplified before they decay, often by a large factor. It has been proposed that sufficiently amplified modes might couple nonlinearly, leading to turbulence (e.g., Chagelishvili et al. 2003;Yecko 2004;Afshordi et al. 2005). However, to make this proposal more concrete, one must work out how modes couple nonlinearly. In L07, we did that in two dimensions. We showed that the 2D nonlinear instability of equation (1) is a consequence of the coupling of an axisymmetric mode with a transiently amplified mode, which may be called a "swinging mode" because its phasefronts are swung around by the background shear. In §5 we show that the 3D instability responsible for the destruction of vortices is a generalization of this 2D instability. It may be understood by examining the coupling of a 3D swinging mode with an axisymmetric one. 3D modes become increasingly unstable as |k z | decreases, and in the limit that k z → 0, the 3D instability matches smoothly onto the 2D one. Thicker disks are more prone to 3D instability because they encompass smaller |k z |.
EQUATIONS OF MOTION
We solve the "shearing box" equations, which approximate the dynamics in an accretion disk on lengthscales much smaller than the distance to the disk's center. We assume incompressibility, which is a good approximation when relative motions are subsonic. We also neglect vertical gravity, and hence stratification and buoyancy, which is an oversimplification. To fully understand vortices in astrophysical disks, one must consider the effects of vertical gravity as well as of shear and rotation. In this paper, we consider only two pieces of this puzzle-shear and rotation. Adding the third piece-vertical gravityis a topic that we leave for future investigations. See also the Conclusions for some speculations.
An unperturbed Keplerian disk has angular velocity profile Ω(r) ∝ r −3/2 . In a reference frame rotating at constant angular speed Ω 0 ≡ Ω(r 0 ), where r 0 is a fiducial radius, the incompressible shearing box equations of motion read
∂ t v + v · ∇v = −2Ω 0ẑ ×v + 2qΩ 0 xx − ∇P/ρ , (4) ∇ · v = 0(5)
adopting Cartesian coordinates x, y, z, which are related to the disk's cylindrical r, θ via x ≡ r − r 0 and y ≡ r 0 (θ − Ω 0 t);x andẑ are unit vectors, and
q ≡ − dΩ d ln r r0 = 3 2 Ω 0 ,(6)
We retain q and Ω 0 as independent parameters because they parameterize different effects: shear and rotation, respectively. The first term on the right-hand side of equation (4) is the Coriolis force, and the second is what remains after adding centrifugal and gravitational forces. Decomposing the velocity into v = −qxŷ + u ,
where the first term is the shear flow of the unperturbed disk, yields
(∂ t − qx∂ y ) u + u · ∇u = qu xŷ − 2Ωẑ×u − ∇P/ρ(8) ∇ · u = 0 ,(9)
dropping the subscript from Ω 0 , as we shall do in the remainder of this paper. An unperturbed disk has u = 0. In addition to the above "velocity-pressure" formulation, an alternative "velocity-vorticity" formulation will prove convenient. It is given by the curl of equation (8),
(∂ t −qx∂ y )ω = −qŷω x +(2Ω−q)∂ z u+∇×(u × ω) (10) where ω ≡ ∇ × u(11)
is the vorticity of u. Equation (10), together with the inverse of equation (11)
u = −∇ −2 ∇ × ω ,(12)
form a complete set. Equation (10) implies that the total vorticity field is frozen into the fluid, because it is equivalent to
∂ t ω tot = ∇×(v × ω tot ) ,(13)
where
ω tot ≡ (2Ω − q)ẑ + ω(14)
is the total vorticity; note that −qẑ is the vorticity of the unperturbed shear flow in the rotating frame, and hence (2Ω − q)ẑ is the unperturbed vorticity in the nonrotating frame. The vorticity-velocity picture is similar to MHD, where it is the magnetic field that is frozen-in because it satisfies equation (13) in place of ω tot . However, in MHD the velocity field has its own dynamical equation, whereas in incompressible hydrodynamics it is determined directly from the vorticity field via equation (12).
TWO PSEUDOSPECTRAL SIMULATIONS
The pseudospectral code is described in detail in the Appendix of L07. It solves the velocity-pressure equations of motion with an explicit viscous term
ν∇ 2 u(15)
added to equation (8). (In L07, we did not include this term because we only considered inviscid flows.) The equations are solved in Fourier-space by decomposing fields into spatial Fourier modes whose wavevectors are advected by the background flow −qxŷ. As a result, the boundary conditions are periodic in the y and z dimension, and "shearing periodic" in x. Most of our techniques are standard (e.g., Maron & Goldreich 2001;Rogallo 1981;Barranco & Marcus 2006). One exception is our method for remapping highly trailing wavevectors into highly leading ones, which is both simpler and more accurate than the usual method. In addition, our remapping does not introduce power into leading modes, because a mode's amplitude has always been set to zero before the remap. The code was extensively tested on 2D flows in L07. A number of rather stringent 3D tests are performed in this paper. We shall show that the code correctly reproduces the linear evolution of 3D modes ( §4), as well as the nonlinear coupling between them ( §5). We also show in the present section that it tracks the various contributions to the energy budget, and that the sum of the contributions vanishes to high accuracy. Figures 1-4 show results from two pseudospectral simulations. One simulation illustrates the formation and survival of a vortex, and the other illustrates vortex destruction. In the first (the "short box"), the number of The initial state is unstable to vertically symmetric (kz = 0) perturbations, and forms into a vortex. But it is stable to 3D (kz = 0) perturbations, and the evolution remains two-dimensional. The bottom panels show horizontal slices through the boxes in the upper panels, midway through the boxes. At time=150, the vortex has already formed. Only fluid with ωz < −0.08 is shown in the middle panels to highlight the vortex, and to illustrate that surfaces of constant ωz remain purely vertical. At time=500, the vortex still survives. Its amplitude is slowly decaying by viscosity, which acts on timescale=1130. We set Ω = 1 and q = 3/2. The number of modes in the simulation is nx × ny × nz = 64 × 64 × 32, and the size of the simulation box is (Lx, Ly, Lz) = ( 1 15 , 1, 1 2 ). In this figure, Lz is to scale relative to Ly, but Lx has been expanded by a factor of 5 for clarity. Figure 1, except that the height Lz has been increased by a factor of 4, so that it now exceeds Ly. The resulting evolution is dramatically different. The initial state is now unstable not only to 2D perturbations, but to 3D ones as well. In the middle panels, surfaces of constant ωz are warped, and the evolution is no longer vertically symmetric. In the right panels, the flow looks turbulent. (20)-(22). E u 2 initially decays, and then rises to a peak near t ∼ 200 as nonaxisymmetric perturbations turn the axisymmetric mode into a vortex. Subsequently, the vortex decays due to viscosity. The spikiness of the evolution is due to the boundary conditions, as explained in the text. Also shown in the bottom panel is the error due to numerical effects, ∆Eerror (eq. [24]). It is unlabelled because it is mostly obscured by ∆E shear . But it is nearly equal to zero everywhere, showing that the code accurately tracks the components of the energy budget.
Fourier modes used is n x × n y × n z = 64 × 64 × 32, and the simulation box has dimensions L x = 1 15 , L y = 1, and L z = 1 2 . In the second (the "tall box"), the setup is identical, except that it has L z = 2 instead of 1/2. Both simulations are initialized by setting
ω z t=0 = −0.1 cos 2π L x x .(16)
In addition, small perturbations are added to longwavelength modes. Specifically, labelling the wavevectors as
(k x , k y , k z ) = 2π j x L x , j y L y , j z L z ,(17)
with integers (j x , j y , j z ), we select all modes that satisfy |j x | ≤ 3, |j y | ≤ 3, and |j z | ≤ 3, and set the Fourier amplitude of their ω z to 10 −4 e iφ , where φ is a random phase. But we exclude the (j x , j y , j z ) = (0, 0, 0) mode, as well as (j x , j y , j z ) = (±1, 0, 0), which is given by equation (16). Finally, we set Ω = 1, q = 3/2, ν = 10 −7 , and integration timestep dt = 1/30. With our chosen initial conditions, the mode given by equation (16) is nonlinearly unstable to vertically symmetric (k z = 0) perturbations, and hence it tends to wrap up into a vortex. From the approximate criterion for instability (eq. [1]), we see that to illustrate the wrapping up into a vortex of a mode with a small amplitude, The initial evolution is almost the same as that seen in the short box (Fig. 3). But 3D perturbations are unstable and force the destruction of the vortex. In the time interval 300 t 600, while the initial axisymmetric disturbance decays in a turbulent-like state, the value of E u 2 is significantly larger than its initial value, and ∆E shear rises nearly linearly in time, corresponding to nearly constant outwards transport of angular momentum in a disk. The contribution of numerical errors to the time-integrated energy budget, ∆Eerror (eq. [24]), remains small throughout.
one must make the simulation box elongated in the ydirection relative to the x-scale of the mode in equation (16).
In the short box ( Fig. 1), the evolution proceeds just as it would in two dimensions. The initial mode indeed wraps up into a vortex, and the evolution remains vertically symmetric throughout. Once formed, the vortex can live for ever in the absence of viscosity. But in our simulation, there is a slow viscous decay. The timescale for viscous decay across the width of the vortex is ∼ 1/νk 2 x = 1130, taking k x = 2π/L x . In the tall box (Fig. 2), the evolution is dramatically different. In this case, the initial state is unstable not only to 2D perturbations, but to 3D (k z = 0) ones as well. In the middle panel of that figure, we see that instead of forming a vertically symmetric vortex as in the short box, surfaces of constant ω z are warped. By the third panel, the flow looks turbulent. Figures 3-4 shows the evolution of the energy in these simulations. Projecting u onto the Navier-Stokes equation (eq. [8] with viscosity included), and spatially averaging, we arrive at the energy equation
d dt u 2 2 = q u x u y + ν u·∇ 2 u ,(18)
after applying the shearing-box boundary conditions, where angled brackets denote a spatial average. The time integral of this equation is
E u 2 − E u 2 | t=0 = ∆E shear + ∆E visc ,(19)
where
E u 2 ≡ u 2 2 (20) ∆E shear ≡ q t 0 u x u y dt ′ (21) ∆E visc ≡ ν t 0 u·∇ 2 u dt ′ .(22)
The pseudospectral code records each of these terms, and Figure 3 shows the result in the short box simulation. At very early times, E u 2 decays from its initial value due to viscosity. At the same time, the small vertically symmetric perturbations are growing exponentially, and they start to give order unity perturbations by t ∼ 150, by which time a vortex has been formed ( Fig. 1). As time evolves, E u 2 gradually decays due to viscosity on the viscous timescale = 1130. The evolution is very spiky.
We defer a discussion of this spikiness to the end of this section. Figure 4 shows the result in the tall box. The early evolution of E u 2 is similar to that seen in the short box. Both start with the same E u 2 , and an initial period of viscous decay is interrupted by exponentially growing perturbations. But in the tall box, not only are vertically symmetric modes growing, but modes with k z = 0 are growing as well. By t ∼ 150, there is a distorted vortex that subsequently decays into a turbulent-like state. The energy E u 2 rises to a value significantly larger than its initial one, and it continues to rise until t ∼ 600, when it starts to decay. Throughout the time interval 300 t 600, ∆E shear rises nearly linearly in time, showing that u x u y is positive and nearly constant.
It is intriguing that u x u y is positive for hundreds of orbits, because it suggests that decaying vortices might transport angular momentum outwards in disks and hence drive accretion. Understanding the level of the turbulence, its lifetime, and its nature are topics for future work. Here we merely address the sign of u x u y . The quantity u x u y is the flux of y-momentum in the +x-direction (per unit mass and spatially averaged). It corresponds to the flux of angular momentum in a disk. A positive u x u y implies an outwards flux of angular momentum, as is required to drive matter inwards in an accretion disk. (Even though the shearing box cannot distinguish inwards from outwards, the sign of the angular momentum within a box depends on which side of the shearing box one calls inwards. Therefore, outwards transport of (positive) angular momentum is welldefined in a shearing box.) In the shearing box, any force that tends to diminish the background shear flow −qxŷ necessarily transports y-momentum in the +x-direction. Hence the fact that u x u y > 0 in Figure 4 shows that the turbulence exerts forces that resist the background shear, as one might expect on physical grounds. One can also understand why u x u y > 0 from energy considerations. Since ∆E visc < 0, as may be seen explicitly by an integration by parts, i.e. u·∇ 2 u = − i,j (∂ j u i ) 2 , equation (19) may be rearranged to read
∆E shear = |∆E visc | + E u 2 − E u 2 | t=0 .(23)
If the turbulence reaches a steady state-as it approximately does in Figure 4 during the time interval 300 t 600-then the last two terms in the above equation are nearly constant, whereas |∆E visc | increases linearly with time. Hence ∆E shear must also increase. The fact that energy dissipation implies outwards transport of angular momentum is a general property of accretion disks (e.g., Lynden-Bell & Pringle 1974). Since turbulence always dissipates energy, it must also transport angular momentum outwards. However, this argument can be violated if an external energy source drives the turbulence, in which case one would have to add this energy to the left-hand side of equation (23). For example, the simulations of Stone & Balbus (1996) show that convective disks can transport angular momentum inwards when an externally imposed heat source drives the convection. Also shown in the bottom panels of Figures 3-4 is the integrated energy error
∆E error ≡ ∆E shear + ∆E visc + E u 2 | t=0 − E u 2(24)
due to numerical effects, which is seen to be small. (In Figure 3, ∆E error is not labelled because the curve is mostly obscured by ∆E shear ; it can be seen near t ∼ 200, and is everywhere very nearly equal to zero.) The fact that ∆E error nearly vanishes throughout the simulations is not guaranteed by the pseudospectral algorithm. Rather, we have chosen ν to be large enough that the algorithm introduces negligible error into the energy budget. To be more precise, at each timestep in the pseudospectral code, modes that have |j x | > n x /3 or |j y | > n y /3 or |j z | > n z /3, where j x,y,z are defined via equation (17), have their amplitudes set to zero ("dealiased"). This introduces an error that is analogous to grid error in grid-based codes. By choosing ν to be sufficiently large, it is the explicit viscosity that forces modes with large k to have small amplitudes, in which case the dealiasing procedure has little effect on the dynamics. Increasing the resolution n x × n y × n z would allow a smaller ν to be chosen-implying a larger effective Reynolds number-while keeping the energy error small. The curves of E u 2 show sharp narrow spikes every time interval ∆t = 10, with width ∼ 1. Similar spikes have been seen in other simulations (Umurhan & Regev 2004;Shen et al. 2006), but they are stronger and narrower in our simulations because our simulation box is elongated. These spikes are due to the shearing-periodic boundary conditions. It is perhaps simplest to understand them by following the evolution in k-space, as we shall do in §5 (see also L07). But for now, we explain their origin in real-space. By the nature of shearing-periodic boundary conditions, associated with the simulation box centered at x = 0 are "imaginary boxes" centered at x = jL x with integer j = ±1, ±2, · · ·. These imaginary boxes completely tile the x − y plane, and each contains a virtual copy of the conditions inside the simulation box. The boxes move relative to the simulation box in the y-direction, with the speed of the mean shear at the center of each box, −qjL x . Therefore, in the time interval ∆t = L y /(qL x ) = 10, all the boxes line up. When this happens, the velocity field u that is induced by the vorticity within all the boxes (via eq. [12]) becomes large, because all the boxes reinforce each other, and therefore E u 2 exhibits a spike. Even though the shearingperiodic boundary conditions that we use are somewhat artificial, we are confident that using more realistic open boundary conditions would not affect the main results of this paper-and particularly not the stability of axisymmetric modes to 3D perturbations. In L07, where we considered 2D dynamics, we investigated both open and shearing-periodic boundary conditions, and showed explicitly that both give similar results. We also feel that the boundary conditions likely do not affect the level and persistence of the "turbulence" seen in Figure 4. However, this is less certain. Future investigations should more carefully address the role of boundary conditions.
LINEAR EVOLUTION
In the remainder of this paper, we develop a theory explaining the stability of vortices seen in the above numerical simulations. We first consider the linear evolution of individual modes, and then proceed to show how nonlinear coupling between linear modes can explain vortex stability.
The linear evolution has been considered previously (Afshordi et al. 2005;Johnson & Gammie 2005;Balbus & Hawley 2006). Only two aspects of our treatment are new. First, we give the solution in terms of variables that allow the simple reconstruction of the full vectors ω and u. And second, we give the analytic expression for matching a leading mode onto a trailing mode that is valid for all k y and k z ,
The linearized equation of motion is (eq. [10])
(∂ t − qx∂ y )ω = −qŷω x + (2Ω − q)∂ z u .(25)
A single mode may be written as
ω(x, t) =ω(k 0 , t)e i[k(k 0 ,t)]·x ,(26)
where k 0 is a constant vector that denotes the wavevector at time t = 0, and the wavevector k = k(k 0 , t) has components
k y = k 0y = const (27) k z = k 0z = const (28) k x = k 0x + qtk y = const ,(29)
so that upon insertion into equation (25), the timederivative of the exponential cancels the term −qx∂ y ω. The velocity field induced by such a mode is (eq. [12]) Figure 5 sketches the evolution of wavevectors. Axisymmetric modes (k y = 0) do not move in k-space, as depicted by the sphere at (1, 0, 0) in Figure 5. "Swinging modes" have k y = 0, and their k x is time-dependent. Their fronts of constant phase are advected by the background shear. Swinging modes with k x /k y < 0, as depicted by the two spheres near (1, −1, 0) and (1, −1, 1) in Figure 5, have phasefronts tilted into the background shear, i.e., they are leading modes. As time evolves, the shear first swings their k x through k x = 0, at which point their phasefronts are radially symmetric. Subsequently, they become trailing modes (k x /k y > 0), and approach alignment with the azimuthal direction (k x /k y → ∞).
u(x, t) =û(k 0 , t)e ik[(k 0 ,t)]·x(30)whereû = i k ×ω k 2(31)
We turn now to the evolution of the Fourier amplitudes. In the remainder of this paper we drop the hatŝ
ω → ω ,û → u .(32)
To distinguish real-space fields, we shall explicitly write their spatial dependence, e.g. ω(x).
Because ω(x) is divergenceless, ω only has two degrees of freedom, which we select to be ω x and
ω yz ≡x · (k × ω) k yz = −ω y , if k y = 0 ω z , if k z = 0 (33) where k yz ≡ k 2 y + k 2 z(34)
Our variable ω yz is proportional to the variable U of Balbus & Hawley (2006). Adopting ω x as the second degree of freedom enables the full vectors to be reconstructed as
ω = −ω x k×(k ×x) k 2 yz − ω yz k ×x k yz (35) u = −iω yz k×(k ×x) k 2 k yz + iω x k ×x k 2 yz(36)
The linearized equation (25) is expressed in terms of these degrees of freedom as
k yz qk y d dt ω x ω yz = β Ω κ 0 − 1 2 κ 2 Ω 2 1 1+τ 2 2 0 ω x ω yz ,(37)
after introducing the epicyclic frequency,
κ ≡ 2Ω(2Ω − q) ,(38)
with κ = Ω in a Keplerian disks, and
τ ≡ k x k yz (39) β ≡ κ q k z k y .(40)
As long as k y = 0, τ varies in time through its dependence on k x = k 0x + qtk y . For axisymmetric modes (k y = 0), τ is constant and
d 2 dt 2 ω yz + κ 2 k 2 z k 2 x + k 2 z ω yz = 0 ,(41)
the solution of which is sinusoidal with frequency κk z / k 2 x + k 2 z . Axisymmetric modes with phasefronts aligned with the plane of the disk (k x = k y = 0) have inplane fluid velocities, and they oscillate at the epicyclic frequency of a free test particle, κ. But axisymmetric modes with tilted phasefronts have slower frequencies, because fluid pressure causes deviations from free epicycles. In the limit of vertical axisymmetric phasefronts (k z = k y = 0), the effects of rotation disappear entirely, and this zero-frequency mode merely alters the mean shear flow's velocity profile.
For swinging modes (k y = 0), it is convenient to employ τ as the time variable. Since k yz qk y
d dt = d dτ ,(42)
we have d 2 dτ 2 ω yz + β 2 1 + τ 2 ω yz = 0 .
(43) (Balbus & Hawley 2006). Figure 6 plots numerical solutions of this equation, and shows that it matches the The three spheres depict modes that play important roles in nonlinear instability. The mode at (1, 0, 0) does not move in k-space. The other two modes are swinging modes that are depicted in the leading phase of their swing. They will become trailing after crossing through the radially symmetric plane. The mode crossing through (1, −1, 0) is responsible for 2D instability that forms vortices. The one crossing through (1, −1, 1) is responsible for 3D instability that destroys vortices. output from the pseudospectral code, as well as the analytic theory described below. Given ω yz , it is trivial to construct ω and u from
ω x = κ 2βΩ dω yz dτ(44)
and equations (35) and (36). For highly leading or trailing modes (|τ | ≫ 1), equation (43) has simple power-law solutions,
ω yz = ω A |τ | 1−δ 2 + ω B |τ | 1+δ 2
, |τ | ≫ 1 (45) (Balbus & Hawley 2006), where ω A and ω B are constants that we shall call the "normal-mode" amplitudes, and
δ ≡ 1 − 4β 2 ,(46)
which is imaginary for |β| > 1/2. As a mode's wavevector evolves along a line in k-space, its amplitude is oscillatory if this line is much closer to the k z axis than to the k y one, and non-oscillatory if the converse is true. The transition occurs at |β| = 1/2. This behavior may be understood as a competition between shear and epicyclic oscillations. The timescale for k x to change by an order-unity factor due to the shear is t shear ∼ |k x /k x | = |k x /qk y |, and the timescale for epicyclic oscillations of axisymmetric modes is t epi ∼ κ −1 |k x /k z | for |k x | ≫ |k z |. Therefore |β| ∼ t shear /t epi , and when |β| ≫ 1 the epicyclic time is shorter and so the mode's amplitude oscillates as its wavevector is slowly advected by the shear. But when |β| ≪ 1 the shear changes the wavevector faster than the amplitude can oscillate. The solution (45) breaks down in mid-swing. As a swinging wave changes from leading to trailing, its "normal-mode amplitudes" change abruptly on the timescale that τ changes from ±1 to ∓1 via
ω A ω B trail = T AA T AB T BA T BB ω A ω B lead(47)
where the transition matrix has components
T AA = −T BB = csc (δπ/2)(48)T BA = − cot(δπ/2) 2 T AB = −2 δ+1 1 δ 1 − δ 1 + δ Γ(1 + δ/2) 2 Γ(1/2 + δ/2) 2(49)
and determinant = −1, and hence is its own inverse. The components are complex when |β| > 1/2. To derive these components, we took advantage of the fact that equation (43) [45]) with constant ω A and ω B = 0 for τ > 0; while for τ < 0, the normal mode amplitudes are set to different constants that are given by equation (47). We exclude the domain |τ | < 1 from the dashed curve, because the analytic approximation does not apply there. Circles show output from the pseudospectral code, integrated with a timestep dt = 1/15 and with the viscosity set to zero.
2005; Balbus & Hawley 2006)
, and matched these onto the normal-mode solution given above. We omit the unenlightening details.
NONLINEAR EVOLUTION: FORMATION AND DESTRUCTION OF VORTICES
Qualitative Description
The instability that destroys vortices is a generalization of the one that forms them. We review here how vortices form, before describing the instability that destroys them. In §5.2, we make this description quantitative.
Vortices form out of a nonlinear instability that involves vertically symmetric (k z = 0) modes. (See L07 for more details of the 2D dynamics than are presented here.) Consider the two vertically symmetric modes shown in Figure 5: the "mother" mode at (1, 0, 0) and the "father" mode that is depicted crossing through (1, −1, 0). Triplets of integers (j x , j y , j z ) label values of wavevectors (k x , k y , k z ) (for example, via [17]). The mother is both axi-and vertically-symmetric, and the father is a leading swinging mode.
As the father swings through radial symmetry, i.e. as it crosses through the point (0, −1, 0), its velocity field is strongly amplified by the background shear. This can be seen from §4, which shows that swinging modes with k z = 0 have ω yz =const., and hence u x = i(ω yz /k yz )/(1 + τ 2 ), which becomes largest when τ crosses through 0. When the father is near the peak of its transient amplification (|τ | 1), it couples most strongly with the mother, and they produce a "son" near (1, −1, 0) = (1, 0, 0) + (0, −1, 0). The son will then swing through radial symmetry where it will couple (oedipally) with the mother to produce a grandson near (1, −1, 0), which can repeat the cycle. We summarize this 2D instability feedback loop as linear amplification :
(1, −1, 0) → (0, −1, 0) nonlinear coupling : (0, −1, 0) + (1, 0, 0) → (1, −1, 0) The criterion for instability is simply that the amplitude of the son's ω yz be larger than that of the father. As shown in L07, if instability is triggered, its nonlinear outcome in two dimensions is a long-lived vortex.
The three-dimensional instability that is responsible for destroying vortices is a straightforward generalization. The mother mode is still at (1, 0, 0), but now the father mode starts near (1, −1, 1). Symbolically, the feedback loop is linear amplification :
(1, −1, 1) → (0, −1, 1) nonlinear coupling : (0, −1, 1) + (1, 0, 0) → (1, −1, 1)
The 2D instability described above is just a special case of this 3D one in the limit that k z = 0. In general, the stability of a mother mode at (1, 0, 0) with given k x =k x and ω =ω depends on the k y and k z of the fathermode perturbations (as well as on the parameters q and Ω). Which k y and k z are accessible in turn depends on the dimensions L y × L z of the simulation box-or equivalently, on the circumferential distance around a disk and the scale-height. In §5.2, we map out quantitatively the region in the k y − k z plane that leads to instability. For now, it suffices to note that the unstable region has |k y | |k xω |/q and |k z | |k y |. We conclude that a given mother mode suffers one of three possible fates, depending on L y and L z .
1. If L y is less than a critical value (∼ q/|ωk x |), then the mother mode is stable to all perturbations.
2. If L y is larger than this critical value, then the mother mode is unstable to vertically symmetric (k z = 0) perturbations; if in addition L z is sufficiently small that all modes with k z = 0 are stable, then the mother mode turns into a long-lived vortex (Figure 1).
3. If both L y and L z are sufficiently large, the mother mode is unstable both to vertically symmetric and to 3D perturbations. When this happens, the mother starts to form a vortex, but this vortex is 3D-unstable. The result is turbulence (Figure 2).
There is also a possibility that is intermediate between numbers 2 and 3: if the conditions described in number (50) and (52), as well as the grandson's equation. Also shown as circles are the output from a pseudospectral simulation, showing excellent agreement with the "exact" solutions. The following parameters have been chosen: Ω = 1, q = 3/2,ω = 0.005, kx = 2π · 15, ky = −2π/30, kz = 0.45|ky| ⇒ β = −0.3. The small disagreement between pseudospectral and exact solutions for ω ′ x at τ < −2τ is due to the conjugate modes that, for simplicity, we have not included in equations (50) and (52); see footnote 5. The two right panels show the mode amplitudes, defined via equation (A1) for the son, and similarly for the father and grandson. Although these two right panels contain the same information as the left ones, they are helpful in constructing the analytic form of the growth factor χ (see Appendix). With the parameters chosen for this figure, equation (54) predicts χ = −2.2 for the amplification factor between successive generations, in agreement with that seen in the figure.
The Stability Criterion
To quantify the previous discussion, we choose an initial state as in Figure 5, with the mother mode at (1, 0, 0) and the father a leading mode crossing through (1, −1, 1). The son mode, not depicted in the figure, is initially crossing through the point (2, −1, 1). We set its initial vorticity-as well as the initial vorticity of all modes other than the mother and father-to zero. 5 The father's wavevector and Fourier amplitude are labelled as in §4, 5 We ignore the complex conjugate modes for simplicity. Since ω(x) is real-valued, each mode with wavevector and amplitude (k, ω) is accompanied by a conjugate mode that has (−k, ω * ). In our initial state, there are really four modes with non-zero amplitudes: the mother at (1, 0, 0) and its conjugate at (−1, 0, 0), and the father and its conjugate. We may ignore the conjugate modes because they do not affect the instability described here. As shown in L07 for the 2D case, their main effect is that when the son swings through (0, −1, 1), not only does it couple with the mother at (1, 0, 0) to produce a grandson at (1, −1, 1), but it also couples with the conjugate mother at (−1, 0, 0) to partially kill its father, which is then at (−1, −1, 1) (bringing to mind the story of Oedipus). But since the father is a trailing mode at this time, it no longer participates in the instability. Nonetheless, the conjugate modes do play a role in the nonlinear outcome of the instability. and the mother's and son's are labelled with bars and primes:
father : k , ω mother :k ≡kx ,ω ≡ωẑ son : k ′ ≡kx + k , ω ′ Note thatk=const., andω x = 0 because the vorticity must be transverse to the wavevector. We also setω y = 0; otherwiseū z = 0, which corresponds to a mean flow out the top of the box and in through the bottom.
At early times, the father mode swings through the point (0, −1, 1). Since the only other nonvanishing mode at this time is the mother, there are no mode couplings that can nonlinearly change the father's amplitude. Therefore its amplitude is governed by the linear equation (37), which we reproduce here as d dτ
ω x ω yz = β Ω κ 0 − 1 2 κ 2 Ω 2 1 1+τ 2 2 0 ω x ω yz .(50)
During its swing, it couples with the mother to change the amplitude of the son. The linear part of the son's evolution is given by the above equation with primed vorticity and wavevector in place of unprimed. The nonlinear part is given by
d dt ω ′ nonlin = ik ′ × (ū × ω + u ×ω)(51)
(eq. [10]) whereū = −i(ω/k)ŷ and u = ik × ω/k 2 (eq. [31]). Adding the linear and nonlinear parts, and re-expressing in terms of our chosen degrees of freedom, we find d dτ
ω ′ x ω ′ yz = β Ω κ 0 − 1 2 κ 2 Ω 2 1 1+(τ +τ) 2 2 0 ω ′ x ω ′ yz −ω q 1 τ βq κ 1 1+τ 2 0 1 τ −τ 1+τ 2 ω x ω yz ,(52)
where the dimensionless constant τ ≡k x k yz (53) depends on both the mother's and father's wavevectors. It is the father's τ ≡ k x /k yz that is being used as the time-coordinate for evolving the son's amplitude. The grandson's equation is the obvious extension: denoting the grandson's amplitudes with double primes, one need only make the following replacements in equation (52): ω ′ → ω ′′ , ω → ω ′ , and τ → τ +τ . Subsequent generations evolve analogously. The father's equation (50) is easily solved, as shown in §4. Inserting this solution into equation (52) produces a linear inhomogeneous equation for the son's amplitude, and similarly for the grandson's. Figure 7 plots numerical solutions of these equations. Also shown as circles are output from a pseudospectral simulation, showing excellent agreement.
In the Appendix, we solve equation (52) analytically to derive the amplification factor χ, which is the ratio of the son's amplitude at any point in its evolution (e.g., when it is radially symmetric), to the father's amplitude at the same point in its evolution. We find We set Ω = 1 and q = 3/2. To make the solid lines in these plots (the "exact solutions"), we repeated the integrations that produced the lines in Figure 7, but varying kz for each ky until perturbations neither grew nor decayed. The dashed line in the left panel shows that the analytic approximation of equation (54) agrees reasonably well with the exact solution. We do not show equation (54) in the right panel because the agreement is poorer there. Right panel shows two X's for the values of the smallest non-zero |ky| and |kz| in the simulations of Figures 1-2, i.e., |ky/k| = Lx/Ly = 0.067 for both simulations, and |kz/k| = Lx/Lz = 0.13 for the short box and = 0.033 for the tall box. The tall box contains a 3D-unstable mode that leads to the destruction of the vortex into a turbulent-like state. The short box contains no such mode, and is stable to 3D perturbations.
χ = −ω qτ δ √ π 1 + δ δ 2 1 + qΩ κ 2 (1 − δ) Γ(1 + δ/2) Γ(1/2 + δ/2)(54)
where δ = 1 − 4β 2 . Equation (54) is applicable in the limit |ω|/q ≪ 1. For 2D modes (β = 0 ⇒ δ = 1), it recovers equation 42 of L07 (see also eq. [1] of this paper):
χ 2D = −πω qk k y .(55)
Marginally stable modes have |χ| = 1. Figure 8 plots curves of marginal stability. The left panel is for the caseω = 0.005, as in Figure 7, and the right panel is for ω = 0.05, as in the pseudospectral simulations presented at the outset of this paper (eq. [16]; Figs. 1-2). The left panel shows that equation (54) gives a fair reproduction of the exact curve. We do not show equation (54) in the right panel, because it gives poorer agreement there (since |ω|/q is too large). In the right panel we also plot X's for the values of the smallest nonvanishing 3D wavenumbers in the simulations of Figures 1-2. In the short-box simulation, all 3D modes lie in the stable zone. Therefore the dynamics remains two-dimensional. But in the 3D box, there is a 3D mode in the unstable zone that destroys the vortex and gives rise to turbulent-like behavior.
It is interesting to consider briefly how the instability described here connects with the Rayleigh-unstable case, which occurs when κ 2 < 0 . At small |k y |, the marginally stable curves in Figure 8 are given by |β| = 1/2, where β = (κ/q)(k z /k y ). Hence if one decreases κ from its Keplerian value Ω, the marginally stable curve becomes steeper in the k z − k y plane, and an increasing number of 3D modes become unstable. As κ → 0, if a 2D mode with some k y is unstable, then so are all 3D modes with the same k y . Therefore any 2D-unstable state is also 3D-unstable, and any forming vortex would decay into turbulence.
CONCLUSIONS
Our main result follows from Figure 8, which maps out the stability of a "mother mode" (i.e., a mode with wavevectorkx and amplitudeω) to nonaxisymmetric 3D perturbations. A mother mode is unstable provided that the k y and k z of the nonaxisymmetric perturbations satisfy both |k y | kω /q and |k z | |k y |, dropping orderunity constants. Based on this result, we may understand the formation, survival, and destruction of vortices. Vortices form out of mother modes that are unstable to 2D (k z = 0) perturbations. Mother modes that are unstable to 2D modes but stable to 3D (k z = 0) ones, form into long-lived vortices. Mother modes that are unstable to both 2D and 3D modes are destroyed. Therefore a mother mode with givenk andω will form into a vortex if the disk has a sufficiently large circumferential extent and a sufficiently small scale height, i.e., if r k−1 q/ω and h k−1 q/ω, where r is the distance to the center of the disk, and h is the scale height. Alternatively, the mother mode will be destroyed in a turbulent-like state if both r and h are sufficiently large (r k−1 q/ω and h k−1 q/ω).
Our result has a number of astrophysical consequences. In protoplanetary disks that do not contain any vortices, solid particles drift inward. Gas disks orbit at sub-Keplerian speeds, v gas ∼ Ωr(1 − η), where Ωr is the Keplerian speed and η ∼ (c s /Ωr) 2 , with c s the sound speed. Since solid particles would orbit at the Keplerian speed in the absence of gas, the mismatch of speeds between solids and gas produces a drag on the solid particles, removing their angular momentum and causing them to fall into the star. For example, in the minimum mass solar nebula, meter-sized particles fall in from 1 AU in around a hundred years. This rapid infall presents a serious problem for theories of planet formation, since it is difficult to produce planets out of dust in under a hundred years. Vortices can solve this problem (Barge & Sommeria 1995). A vortex that has excess vorticity −ω and radial width 1/k can halt the infall of particles provided thatω/k (Ωr)η, because the gas speed induced by such a vortex more than compensates for the sub-Keplerian speed induced by gas pressure. 6 Previous simulations implied that 3D vortices rapidly decay, and so cannot prevent the rapid infall of solid particles (Barranco & Marcus 2005;Shen et al. 2006). Our result shows that vortices can survive within disks, and so restores the viability of vortices as a solution to the infall problem.
A more important-and more speculativeapplication of our result is to the transport of angular momentum within neutral accretion disks.
In our simulation of a vortex in a tall box, we found that as the vortex decayed it transported angular momentum outward at a nearly constant rate for hundreds of orbital times. If decaying vortices transport a significant amount of angular momentum in disks, they would resolve one of the most important outstanding questions in astrophysics today: what causes hydrodynamical accretion disks to accrete? To make this speculation more concrete, one must understand the amplitude and duration of the "turbulence" that results from decaying vortices. This is a topic for future research.
In this paper, we considered only the effects of rotation and shear on the stability of vortices, while we neglected the effect of vertical gravity. There has been a lot of research in the geophysical community on the dynamics of fluids in the presence of vertical gravity, since stably stratified fluids are very common on Earth-in the atmosphere, oceans, and lakes. In numerical and laboratory experiments of strongly stratified flows, thin horizontal "pancake vortices" often form, and fully developed turbulence is characterized by thin horizontal layers. (e.g., Brethouwer et al. 2007). Pancake vortices are stabilized by vertical gravity, in contrast to the vortices studied in this paper which are stabilized by rotation. Gravity inhibits vertical motions because of buoyancy: it costs gravitational energy for fluid to move vertically. The resulting quasi-two-dimensional flow can form into a vortex. 7 We may speculate that in an astrophysical disk vertical gravity provides an additional means to stabilize vortices, in addition to rotation. But to make this speculation concrete, the theory presented in this paper should be extended to include vertical gravity.
We have not addressed in this paper the origin of the axisymmetric structure (the mother modes) that give rise to surviving or decaying vortices. One possibility is that decaying vortices can produce more axisymmetric structure, and therefore they can lead to self-sustaining turbulence. This seems to us unlikely. We have not seen evidence for it in our simulations, but this could be because of the modest resolution of our simulations. Other possibilities for the generation of axisymmetric structure include thermal instabilities, such as the baroclinic instability, or convection, or stirring by planets. This, too, is a topic for future research.
Fig. 1 .
1-Vortex Formation and Survival in a Short Box: Color depicts ωz.
Fig. 2 .
2-Vortex Destruction in a Tall Box: The setup is identical to the short-box simulation of
Fig. 3 .
3-Energy in the Short Box: The three contributions to the energy budget, E u 2 , ∆E shear , and ∆E visc , are defined in equations
Fig. 4 .
4-Energy in the Tall Box:
Fig. 5 .
5-Evolution of Wavevectors: Modes have constant ky and kz, and kx = qtky+const.
has hypergeometric solutions (Johnson & Gammie
Fig. 6 .
6-Linear Evolution of Mode Amplitudes for Three Values of β: Time runs from right to left. Solid curves show the exact, numerically integrated solution of equation (43). The initial value of dωyz/dτ was chosen so that ω B = 0 initially (eq. [45]). Dashed lines show the analytic solution (eq.
Fig. 7 .
7-Nonlinear Evolution of 3D Instability: Time runs from right to left. In the two left panels, lines show numerical solutions of equations
Fig. 8 .
8-Curves of Marginal Stability for a Mother Mode Withω = 0.005 (left panel) andω = 0.05 (right panel): Left panel corresponds to Figure 7 and right panel corresponds to the pseudospectral simulations of Figures 1-2.
CITA. Toronto, Ontario, Canada; [email protected] 2 Energy conservation implies that turbulence transports angular momentum outwards; see §3. Nonetheless, if an external energy source (e.g., the radiative energy from the central star) drives the turbulence, then angular momentum could in principle be transported inwards.
However,Barranco & Marcus (2005) also include buoyancy forces in their simulations, which we ignore here. How buoyancy affects the stability of vortices is a topic for future work.
hold, the essentially 2D dynamics that results can nonlinearly produce new mother modes that are unstable to 3D perturbations. In this paper, we shall not consider this possibility further, since it did not occur in the pseudospectral simulations of §3. We merely note that in our simulations of this possibility (not presented in this paper), we found that when the new mother modes decayed, they also destroyed the original mother mode.
We implicitly assume here that the stopping time of the particle due to gas drag is comparable to the orbital time, which is true for meter-sized particles at 1 AU in the minimum mass solar nebula. A more careful treatment shows that a vortex can stop a particle with stopping time ts provided thatω/k (Ωr)(Ωts)η(Youdin 2008).7 Billant & Chomaz (2000) show that a vertically uniform vortex column in a stratified (and non-rotating and non-shearing) fluid suffers an instability (the "zigzag instability") that is characterized by a typical vertical lengthscale λz ∼ U/N , where U is the horizontal speed induced by the vortex, N is the Brunt-Väisälä
frequency, and the horizontal lengthscale of the vortex L h is assumed to be much greater that λz (hence the pancake structure). We may understand Billant & Chomaz's result in a crude fashion with an argument similar to that employed in the introduction to explain the destruction of rotation-stabilized vortices: since the frequency of buoyancy waves is N kx/kz (when |kx| ≪ |kz|), and since the frequency at which fluid circulates around a vortex is U/L h ∼ kxU , there is a resonance between these two frequencies for vertical lengthscale 1/kz ∼ U/N .APPENDIXANALYTIC EXPRESSION FOR GROWTH FACTOR χIn this Appendix, we derive equation (54) by analytically integrating equation (52) for the son's vorticity, given the father's vorticity as a function of time ( §4), and taking the mother's vorticityω to be constant, which is valid when the father's amplitude is small relative to the mother's. The numerical integral of equation(52)is shown inFigure 7. Recall that initially τ =τ > 0 and τ decreases in time, and typicallyτ ≫ 1. It simplifies the analysis to work with the son's "normal-mode" amplitudes ωwhere τ ′ ≡ τ +τ (A2) Substituting this into equation(52), the time derivative of the above matrix cancels the homogeneous term in that equation if we approximate 1 + τ ′ 2 ≃ τ ′ 2 , which holds until just before the time that τ = −τ . The inhomogeneous term producesSince ω x and ω yz are known ( §4), a straightforward integration yields ω ′ A just before τ = −τ . To perform this integral, we resort to some approximations, guided by the solution shown inFigure 7. For the first term, we needwhere ǫ is a parameter that satisfies 1 ≫ ǫ ≫ 1/τ . In the first line, we used equation(43), and we discarded the ω A mode from the second integral because the ω B mode increases faster with increasing |τ |. FromFigure 7, ω ′ A nearly vanishes until τ ≃ 0. Therefore, in the second line we approximated the first integral as −(τ 1+δ 2 /β 2 )dω yz /dτ | −ǫ . The third line holds in the limit of small ǫ. The other three terms in equation (A3) are integrated similarly, yieldingjust before the time when τ = −τ , i.e., just before the son is radially symmetric. At this time,Figure 7shows that ω ′ B very nearly vanishes. Therefore, just after the son is radially symmetric, it will have ω ′ B = T BA ω ′ A (eq. [47]), with ω ′ A given by equation(A7). This gives the amplification factor χ ≡ ω ′ B /ω B that is displayed in equation(54).
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"STORM: A Model for Sustainably Onboarding Software Testers",
"STORM: A Model for Sustainably Onboarding Software Testers"
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"Tobias Lorey [email protected] ",
"Stefan Mohacsi [email protected] ",
"Armin Beer ",
"Michael Felderer [email protected] ",
"\nAtos IT Solutions and Services GmbH Vienna\nUniversity of Innsbruck Innsbruck\nAustria, Austria\n",
"\nBeer Test Consulting\nBadenAustria\n",
"\nUniversity of Innsbruck Innsbruck\nAustria\n"
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"Atos IT Solutions and Services GmbH Vienna\nUniversity of Innsbruck Innsbruck\nAustria, Austria",
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]
| []
| AbstractÐRecruiting and onboarding software testing professionals are complex and cost intensive activities. Whether onboarding is successful and sustainable depends on both the employee as well as the organization and is influenced by a number of often highly individual factors. Therefore, we propose the Software Testing Onboarding Model (STORM) for sustainably onboarding software testing professionals based on existing frameworks and models taking into account onboarding processes, sustainability, and test processes. We provide detailed instructions on how to use the model and apply it to real-world onboarding processes in two industrial case studies. | 10.48550/arxiv.2206.01020 | [
"https://arxiv.org/pdf/2206.01020v1.pdf"
]
| 249,282,407 | 2206.01020 | 4d91820c08aaf8c9231dafdb524d80ee6375c55d |
STORM: A Model for Sustainably Onboarding Software Testers
Tobias Lorey [email protected]
Stefan Mohacsi [email protected]
Armin Beer
Michael Felderer [email protected]
Atos IT Solutions and Services GmbH Vienna
University of Innsbruck Innsbruck
Austria, Austria
Beer Test Consulting
BadenAustria
University of Innsbruck Innsbruck
Austria
STORM: A Model for Sustainably Onboarding Software Testers
AbstractÐRecruiting and onboarding software testing professionals are complex and cost intensive activities. Whether onboarding is successful and sustainable depends on both the employee as well as the organization and is influenced by a number of often highly individual factors. Therefore, we propose the Software Testing Onboarding Model (STORM) for sustainably onboarding software testing professionals based on existing frameworks and models taking into account onboarding processes, sustainability, and test processes. We provide detailed instructions on how to use the model and apply it to real-world onboarding processes in two industrial case studies.
I. INTRODUCTION
A trend towards integrating environmental protection, social equity, and economic development, also referred to as sustainability [1], has emerged in many industries and academic disciplines in recent years. Examples of this development include sustainable finance [2], corporate sustainability [3], sustainable tourism [4], and sustainable agriculture [5]. Software engineering and software testing are no exceptions to the sustainability trend [6], [7].
Despite being studied in several software engineering domains [8], [9], [10] sustainability has not been used extensively in the context of onboarding software testing professionals. Onboarding is a procedure whereby employees moving from team outsiders to becoming team members [11]. It is a cost and labor-intensive activity with the goal to have a newcomer being accepted by the team and begin to work productively.
Experiences from large-scale projects indicate the following problems in onboarding software professionals:
• Onboarding new team members virtually is common practice today, which has even accelerated during the COVID-19 pandemic. However, new team members may miss opportunities to ask questions and establish a closer relationship with their colleagues.
• Employee and company expectations often differ concerning the content and duration of the onboarding phase.
• Onboarding of new employees may be inhibited by not adequately creating a structured onboarding plan.
• Knowledge networking, formal training, and certifications are often not viewed as long-term investments by an organization and lack funding. As a consequence, the objective of setting up a more sustainable onboarding process has to be considered. Reasons to introduce sustainability in the onboarding process are (1) sustainable and green initiatives gaining interest in industry, (2) increase employee satisfaction and productivity, (3) decrease costs by reducing employee turnover, and (4) increase public reputation as a good place to work.
From the employees' perspective, work-life balance, career opportunities, and job satisfaction play an increasingly important role in their work lives. Such factors are directly related to sustainability. Activities in the onboarding process have to promote these goals to increase employee satisfaction and reduce employee turnover [12].
However, from a company perspective, onboarding activities are limited by budget constraints and the duration of the onboarding phase, given that software testing professionals often need to work productively as soon as possible. Therefore, an organization must find the right balance to incorporate sustainability into existing onboarding processes characterized by organizational and financial aspects and the need to support the test process early on.
Multiple factors impact the sustainability of onboarding new employees into an organization and a team in the context of software testing. Therefore, we see the need to connect the concepts of onboarding and sustainability in the context of software testing professionals. We propose the Software Testing Onboarding Model (STORM) for increasing sustainability in the onboarding process of new software testing professionals, which is structurally based on the floodlight model [13] from the domain of requirements engineering.
This article is structured as follows: In Section 2, we present background and related work on sustainability, onboarding in software engineering and software testing, test processes, and the floodlight model. We describe the proposed Software Testing Onboarding Model and its application in Section 3. Section 4 evaluates the model in two case studies from industry. We discuss the model's application and its implications for industry in Section 5 and provide a conclusion in Section 6.
II. BACKGROUND AND RELATED WORK
In this section we provide an overview of the concepts of sustainability in software engineering, onboarding new employees, software testing processes, and the floodlight model.
A. Sustainability in Software Engineering
Penzenstadler & Femmer [14] propose a generic sustainability framework for software engineering that consists of three different levels to improve sustainability in organizations. There are five distinct sustainability dimensions at the top: individual-, environmental-, economic-, technical-, and social sustainability. The middle-level consists of values, which are morals indicative of each dimension, indicators, and regulations. The bottom-level consists of activities, which are specific steps to be taken to achieve the specified goal in the context of sustainability. The framework's practical application consists of two distinct phases: 1) analysis and 2) application & assessment. The analysis phase consists of instantiating the generic sustainability model, and the application & assessment phase consists of specifying responsibilities and monitoring sustainability using previously defined indicators and metrics. Condori-Fernandez et al. [10] investigate how quality requirements contribute to sustainability dimensions. They found that quality attributes like availability and operability are related to technical sustainability. Kern et al. [15] introduce a causal model for analyzing various sustainability criteria of software. Their focus is on resource and energy efficiency.
B. Onboarding in Software Engineering and Software Testing
Our work is related to recruiting and onboarding software professionals in organizations and IT projects and incorporating sustainability in the onboarding procedure. We have identified a variety of existing literature on this topic. Begel and Simon [16] focus on the reduction of stress and anxiety during onboarding by fostering social networking for newcomers. They regard strategies like mentoring and pair programming as potential success factors for graduates starting their first jobs. Buchan et al. [11] find that mentoring, online communities, peer support, and team socializing are considered to be the most important onboarding techniques named by professionals in software development companies. They categorize onboarding techniques into the following categories: (1) working with people, (2) working with artifacts, and (3) undertaking an activity. Gregory at al. [17] answer the question of how newcomers should be integrated into an agile project team. They apply Bauer's onboarding framework because it is generic and empirically based. Caldwell and Peters [18] propose a ten-step model for quality onboarding, observing the organizational impact and employee perception. The goal is to identify ethical implications. Pham et al. [19] found that the lack of testing skills of inexperienced new hires is a problem for software development companies that requires different coping strategies. Florea and Stray [12] investigate educational backgrounds and skill acquisition of software testing professionals. They find that software testers often need to demonstrate curiosity and skills increase with experience. Sharma and Stol [20] explore the relationship between onboarding of new employees and turnover intention. Their research model relates the onboarding activities orientation, training and support to onboarding success, which impacts job satisfaction and workplace quality required to reduce turnover intention. The strongest relationship was found to be the impact of support on onboarding success. Brito et al. [21] focus on onboarding activities in large-scale globally distributed projects taking Bauer's model into account. The newcomer's performance is observed in an exploratory case study by evaluating the productivity of the newcomers.
Bauer [22] provides a comprehensive framework for onboarding new employees to an organization. The framework consists of six separate phases. The onboarding process proposed by Bauer is initiated with the (1) recruiting of a new employee. After the selection of a candidate the onboarding continues with the (2) orientation of the newcomer. The newcomer is introduced to the use of (3) processes and tools. (4) Coaching and support has to be in place in order to assure efficient onboarding. (5) Training aims to improve task performance by practice-based learning, certification and career development. Sensible (6) feedback culture and the integration of the newcomer into a team facilitates the onboarding process.
C. Docker's Floodlight Model
Docker [13] proposed a model to highlight the interactions between requirements and acceptance criteria in the context of requirements engineering. The floodlight model explains how requirements define one or more possible solutions to a problem. These potential solutions are then filtered and limited by predefined acceptance criteria and other constraints until acceptable solutions are found. The name floodlight model is used to visualize how various acceptance criteria cast floodlights on possible solutions. Its terminology is inspired by floodlights used on a theater stage. Having multiple floodlights overlap on a solution implies that the criteria are fulfilled.
III. STORM: SOFTWARE TESTING ONBOARDING MODEL
This section describes the Software Testing Onboarding Model (STORM) which takes into account emerging sustainability factors. We divide this section into a static and dynamic view. The former describes the model's concepts and their relationships among each other while the latter provides guidance on the application steps.
A. Constructs
The model's structure is based on the floodlight model by Docker [13]. We adapt and amend Docker's floodlight model to the context of onboarding software testers to identify suitable onboarding solutions which comply with the onboarding process, sustainability dimensions, the test process, and further relevant criteria of companies and projects regarding organizational and financial aspects.
The following constructs are part of the framework as shown in Figure 1.
Onboarding Process: The onboarding process defines the problem space of our model. We refer to the phases of a standard onboarding process [22] (recruiting, orientation, processes and tools, coaching and support, training, and feedback)
Onboarding Process
Sustainability Dimensions
Financial & Organizational Aspects S1 S2
S3 S4
Solution Space
Possible onboarding process solutions (S1, S2, ...)
Limitation of solutions based on sustainability dimensions
Specification of further limitations (e.g., cost, regulations, organizational culture) which represent sources of problems that must be solved for sustainably onboarding software testing professionals.
Test Process
Conditions and requirements relevant to the test process
Solution Space: A set of possible onboarding solutions (S1, S2, S3, . . . ) is defined to address identified challenges in the onboarding process of a software testing professional. Potential onboarding solutions are subsequently analyzed regarding their sustainability fit, test process fit, organizational fit, and financial aspects. To visualize the interaction between these aspects and solutions, each of the floodlights cast a light on acceptable onboarding solutions. Overlapping floodlights indicate acceptable solutions regarding multiple acceptance criteria. The solution space is similar to the solutions as introduced by Docker [13] in the original floodlight model in the context of requirements engineering but are of a more proactive nature that reflects the onboarding process consisting of phases and activities as well.
Sustainability Dimensions: Potential onboarding process solutions and activities are screened for their sustainability fit. Sustainability dimensions of interest in the software engineering domain include individual, economic, social, environmental, and technical sustainability [14]. Depending on the company's goals, some sustainability dimensions may be considered more important than others or may even be obsolete. These dimensions can be relevant for the company, its employees and stakeholders, the environment, society as a whole or a community, or any combination of the above.
Test Process: Onboarding solutions for software testers must support the current or desired test process. They must, therefore, fit the organization's test process in regard to aspects such as training and certifications, agile practices, project structure, and testing tools. Solutions can be analyzed and filtered based on the existing test process or a target test process. A standard ISTQB [23] test process includes the phases test planning, test monitoring and control, test analysis, test design, test implementation, test execution, and test completion and can be used to identify whether onboarding solutions, e.g., certifications and trainings, fit the test process.
Financial & Organizational Aspects: Solutions are further examined in regard to their organizational and financial aspects, e.g., implementation cost, legal and regulatory requirements, organizational processes and culture.
B. Model Application
This section describes the application of the onboarding model as shown in Figure 2. The model is designed to be incorporated into existing onboarding processes and serves as a tool for both designing and analyzing onboarding processes focused on the software testing domain.
We divide the onboarding process into three distinct phases: preparation and recruiting phase, application of the Software Testing Onboarding Model and execution and outcome of the onboarding process. The preparation phase consists of initiating the recruiting of a suitable software testing professional, as well as defining specific onboarding goals from which problems are derived. Solutions need to be proposed to solve these onboarding challenges. The onboarding model is applied at this stage to facilitate analysis of solutions and filter acceptable ones. Each potential solution is analyzed step-bystep in regard to (1) sustainability, (2) testing processes, and (3) organizational constraints such as financial aspects. Each step of the analysis classifies a solution whether it is positive, neutral, or negative in the respective category. More granular scales, e.g., a Likert scale, may be used if required. Depending on the outcome of the analyses, a set of solutions and activities will be applied in the onboarding process. Selected acceptable solutions will be implemented during the onboarding phase of a new software testing employee. The outcome should be closely monitored in regard to the actual impact of the executed solutions.
IV. INDUSTRIAL CASES We will now apply the onboarding model described in Section 3 to two industrial case studies in an Austrian social insurance institution and a global IT services company (Atos). The two presented companies are developing software applications. Both organizations require skilled software testing professionals to work in IT projects and face various challenges during the recruiting and onboarding phase of newcomers.
A. Industrial Case 1: Social Insurance Institution
The Austrian social insurance institution offers a wide range of medical services to its insurants. It consists mainly of a department for developing internal software and business units for managing the medical care of about one million insurants. Software development is performed co-located by internal and external personnel. This case study focuses on hiring and onboarding a test automation engineer. The onboarding model was applied by test management experts of the social insurance institution. Problem statements and possible solutions are summarized in Figure 3.
Onboarding Goals: Onboarding a new testing professional in the social insurance institution closely interacts with the company's environmental, economic and organizational aspects. The career of a new software testing employee depends on the individual skill level: A junior tester must have the ability to perform the assigned work packages according to the guidelines and defined processes. A junior tester should hold a Foundation Level certificate of the ISTQB. By designing test cases based on the requirements specification, the tester must communicate with analysts, project managers, and software developers in a project. The tester should also be able to pass a software testing curriculum. Requirements for a senior tester are more focused and include technical competence, soft skills, and in-depth knowledge of requirements. A senior testing professional can identify problems and suggest improvements in test case design, development, and release management.
Onboarding problem 1: Which person shall I hire? In all projects, iterative and incremental development and a standard test process were used. Automated functional and performance tests were implemented in the majority of projects. However, the requirements coverage was too low, and the effort and duration of maintaining the test cases and test environment made the process inefficient. Therefore, the social insurance institution decided to hire new testers to satisfy the demand for new or ongoing projects. The typical path of integrating a novice tester into a project begins by building testing knowledge and continues with the domain knowledge of the business unit. In addition, soft skills are needed, for instance, to report test results to the stakeholders.
Possible solutions and results:
• Solution 1 ± A manual tester with the skills of a junior is cheaper than a test automation engineer. Experience from different projects in the social insurance institution showed that their skills are not sufficient to develop an economically efficient test environment.
• Solution 2 ± An employee of a business unit with sound domain knowledge can be trained in workshops to learn systematic test case design methods. The newcomer will be classified as a junior tester in the organization's career path. Nevertheless, additional time during onboarding needs to be provided. This is a viable and economically sustainable solution.
• Solution 3 ± Hiring an external test automation specialist is expensive but recommended to put a framework into operation and enable organizational learning in the team. However, it must be guaranteed that the specialist will be available for the entire project duration of three years to ensure sustainability. Onboarding problem 2 ± How could I alleviate the task of a test automation engineer? According to the ISTQB glossary, the role of a test automation engineer is a person responsible for the design, implementation, and maintenance of a test automation architecture as well as the technical evolution of the resulting test automation solutions [23]. At the social insurance institution, the critical goals of a new test automation engineer were defining a test automation strategy and sharing assets and methods to ensure their consistent implementation across the organization.
Transitions of co-located teams into virtual teams are nowadays common in industry [24]. In the social insurance institution, relationships among team members and spontaneous opportunities to learn skills on the job were important. How-ever, the spatial separation made it more difficult to innovate testing procedures and follow improvements.
Possible solutions and results:
• Solution 1 ± Professional certification is viewed as the basis for domain and technical knowledge and the career path of a new employee. A new test automation engineer should attain at least the ISTQB Foundation Level certificate, one or more Advanced Level certificates (most notably the Technical Test Analyst), and possibly even the Expert Level of test automation. The application of this solution depends on the budget for workshops and certification exams.
• Solution 2 ± Test automation benefits from the high testability of the requirements. A newcomer should review the testability of the requirements to get an insight into his tasks [25]. Improving the requirements specifications pays off concerning increased efficiency in release planning and better maintainability of test cases despite its cost.
• Solution 3 ± Another option is to prioritize and reduce the number of automated test cases. As part of the iterativeincremental development process, the question of which new features should be incorporated in a release is vital for product success. Quality-driven resource planning must be applied for all available resources [26]. To prioritize and reduce the number of automated test cases is no option as it would substantially lower test efficiency and effectiveness.
• Solution 4 ± Internal testing software training by employees of a successful project and domain experts is a cost-efficient way to train a new employee and training sessions can be conducted both on premise as well as virtually.
• Solution 5 ± Training and consulting by the testing tool manufacturer is another way of know-how transfer for a test automation engineer. However, experience from previous workshops in the social insurance institution indicates that this solution is not sustainable, especially concerning its cost.
B. Industrial Case 2: Large IT Service Provider (Atos)
Atos is a global IT services company with over 100,000 employees across 72 countries. Atos is offering a wide range of services including test consulting, test management, and test automation. The following use case was performed by test experts of the Austrian Digital Assurance department who applied the Floodlight Model to a couple of non-trivial problems. Problem statements and possible solutions are summarized in Figure 4.
Onboarding Goals: When onboarding new testers, Atos focuses on technical, social, and economic sustainability. Firstly, the hiring strategy is not limited to tools and technologies that are currently in use but also considers upcoming trends. Another important goal is integrating newcomers quickly into the team, which gives them the opportunity to establish social ties and raises productivity.
Onboarding problem 1 ± Which testing skills are the most relevant for a job candidate? Depending on the customer and the circumstances of the project, different testing approaches are being used. If the technical prerequisites for test automation are not fulfilled or the customer process is too chaotic (e.g., frequent changes of the user interface), manual test execution is still applied.
For all other projects, test automation is strongly encouraged. However, the question whether to use a traditional, well-established automation technique or to introduce a new, innovative approach needs to be resolved.
The decision is also crucial for the hiring strategy. The skill set for new testers has to correspond to the testing approach in the upcoming projects.
Possible solutions and results:
• Solution 1 ± Manual testers are usually much cheaper than test automation engineers. Learning the theoretical background of SW testing theory and obtaining the ISTQB Foundation Level certification is only a matter of a few weeks and requires little technical skills or previous experience. On the other hand, hiring manual testers is less economically sustainable because they cannot be used for other projects in which test automation is applied.
• Solution 2 ± The existing test automation approach focuses on traditional GUI-based test automation using commercial tools that simulate user actions and verify the outcome on the screen. This method is wellestablished and integrated into the existing test process. A test automation engineer who is familiar with this approach is more expensive than a manual tester but also more flexible due to their technical background. However, there remains the question if the traditional automation approach is fit for the future.
• Solution 3 ± Innovative test automation approaches such as AI-based testing and comprehensive RPA with tools such as UiPath have been emerging in recent years.
Hiring test automation engineers who are familiar with these new techniques is tempting, but first the additional cost and effort for adapting the test process, buying new tool licenses etc. need to be considered. In the long term, investing in new technologies is more sustainable, but in our opinion the transition should be done incrementally rather than with a radical disruption. Thus, we decided to try out some innovative approaches in the scope of pilot projects while sticking to the well-established method in most other projects. The hiring was done accordingly ± a number of young test automation engineers who already had some experience with the innovative methods was hired as well as some engineers who were familiar with the traditional approach.
Onboarding problem 2 ± How can new employees be integrated in a team despite COVID-19 restrictions? Integrating a new employee into an existing project team can be difficult at the best of times but has been particularly challenging during the COVID-19 crisis. With personal meetings prohibited or severely restricted, most employees had to work exclusively from their home office. A get-together with other team members was only possible in the form of online meetings, which despite the use of video cameras cannot fully replace a personal meeting.
Nonetheless, the switch to exclusive home office went very smoothly at Atos. Most employees were already used to occasionally working in home office and already had the technical equipment in place. Also, at the beginning of the pandemic there was a notable wave of solidarity and team spirit amongst the employees who had realized that they could only master the crisis together. The team members already knew each other well and social contacts between them were well-established. However, this was not the case for new employees who had to be integrated into the existing teams. The need for new hirings was rather intensified than abated by the pandemic due to an increased number of demands by government agencies. But how can new employees be welcomed into a team and social sustainability be achieved if personal meetings are impossible?
Possible solutions and results:
• Solution 1 ± The simplest option is directly assigning new employees to a customer project and hoping that they will get to know the other colleagues in the course of time. This will generate no additional costs but might make the collaboration with other team members more difficult. If we put the focus on social sustainability, this option can be clearly dismissed due to the danger of new employees feeling uncomfortable because they have to work with people they've never met.
• Solution 2 ± An online team building event in which the new employees are introduced to the other team members and get the opportunity to establish social ties can improve social sustainability and future collaboration. On the other hand, extra time and budget is required. Thus, we decided to organize occasional events for welcoming several new employees at a time, but not a separate event for every new staff member.
• Solution 3 ± In agile projects, daily stand-up meetings have been a key part of the culture even before the pandemic. During the COVID-19 crisis, these meetings proved invaluable to maintain social contacts between the team members and avoid loneliness. Apart from discussing professional topics, the regular online meetings at 9am gave old and new employees alike the opportunity to talk about their feelings of uncertainty, grief, and fear that were caused by the pandemic. In our experience, connecting with people on an emotional basis has been the most effective stimulus for social sustainability.
• Solution 4 ± Being introduced by a mentor to the other team members and into the project is a good way to facilitate collaboration and improve social sustainability. Some extra time and budget are required for mentor and mentee, but we found that this investment usually pays off very quickly.
V. DISCUSSION
We successfully applied the Software Testing Onboarding Model (STORM) to two industrial cases. The onboarding model enabled company-specific suggestions for relevant onboarding problems for which we received positive feedback from the experts who evaluated the Software Testing Onboarding Model.
If a test manager in the social insurance institution considers hiring a new employee, the software testing onboarding model facilitates the effective integration into a team according to the expert. It promotes career paths for new employees as well as pathways towards sustainability and overall test process improvement. The same is valid for the IT services company Atos. Our model enabled the expert to analyze a total of seven possible solutions to two relevant onboarding problems. For instance, daily online team meetings fit the agile culture of the company, improve social sustainability through regular discussions, and can be introduced at no additional cost.
While the onboarding problems were specific to the two organizations, we believe that these onboarding problems can be generalized to other companies in the IT sector as many face similar situations.
VI. CONCLUSION
In this article, we proposed a model for sustainable onboarding in the context of software testing. We described the structure of the model and the relationship among its constructs. The potential benefits of the implementation of the presented onboarding model STORM in an organization are:
• contribute an onboarding solution that integrates into the onboarding and software lifecycle of different types of companies and projects,
• gain a better understanding of the organizational environment, infrastructure, and test process related to sustainable onboarding,
• design and develop (new or modified) onboarding solutions that meet sustainability needs for both employees and companies,
• assess and mitigate risks during the onboarding of new employees,
• verify and improve onboarding solutions. We demonstrated its applicability in two industrial case studies. In the future we plan to refine the proposed onboarding model and distinguish between onboarding in co-located, distributed, hybrid, and on-premise organizations.
Fig. 2 .
2Onboarding Model: Application
Fig. 3 .
3STORM application in industrial case 1: Social insurance institution
Fig. 4 .
4STORM application in industrial case 2: IT services company (Atos)
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| []
|
[
"Non-supersymmetric Asymmetric Orbifolds with Vanishing Cosmological Constant",
"Non-supersymmetric Asymmetric Orbifolds with Vanishing Cosmological Constant"
]
| [
"Yuji Satoh *[email protected]†[email protected] ",
"Yuji Sugawara ",
"Taiki Wada ",
"\nInstitute of Physics\nDepartment of Physical Sciences\nCollege of Science and Engineering\nUniversity of Tsukuba\n305-8571IbarakiJapan\n",
"\nRitsumeikan University\n525-8577ShigaJapan\n",
"\nIntroduction\n\n"
]
| [
"Institute of Physics\nDepartment of Physical Sciences\nCollege of Science and Engineering\nUniversity of Tsukuba\n305-8571IbarakiJapan",
"Ritsumeikan University\n525-8577ShigaJapan",
"Introduction\n"
]
| []
| We study type II string vacua defined by torus compactifications accompanied by Tduality twists. We realize the string vacua, specifically, by means of the asymmetric orbifolding associated to the chiral reflections combined with a shift, which are interpreted as describing the compactification on 'T-folds'. We discuss possible consistent actions of the chiral reflection on the Ramond-sector of the world-sheet fermions, and explicitly construct non-supersymmetric as well as supersymmetric vacua. Above all, we demonstrate a simple realization of non-supersymmetric vacua with vanishing cosmological constant at one loop. Our orbifold group is generated only by a single element, which results in simpler models than those with such property known previously. | 10.1007/jhep02(2016)184 | [
"https://arxiv.org/pdf/1512.05155v4.pdf"
]
| 53,130,109 | 1512.05155 | 6f138d018e1adaad8935d04d563e3e794208c5e9 |
Non-supersymmetric Asymmetric Orbifolds with Vanishing Cosmological Constant
22 Mar 2016 December, 2015
Yuji Satoh *[email protected]†[email protected]
Yuji Sugawara
Taiki Wada
Institute of Physics
Department of Physical Sciences
College of Science and Engineering
University of Tsukuba
305-8571IbarakiJapan
Ritsumeikan University
525-8577ShigaJapan
Introduction
Non-supersymmetric Asymmetric Orbifolds with Vanishing Cosmological Constant
22 Mar 2016 December, 2015arXiv:1512.05155v4 [hep-th] UTHEP-679
We study type II string vacua defined by torus compactifications accompanied by Tduality twists. We realize the string vacua, specifically, by means of the asymmetric orbifolding associated to the chiral reflections combined with a shift, which are interpreted as describing the compactification on 'T-folds'. We discuss possible consistent actions of the chiral reflection on the Ramond-sector of the world-sheet fermions, and explicitly construct non-supersymmetric as well as supersymmetric vacua. Above all, we demonstrate a simple realization of non-supersymmetric vacua with vanishing cosmological constant at one loop. Our orbifold group is generated only by a single element, which results in simpler models than those with such property known previously.
Introduction
Compactifications on non-geometric backgrounds have been receiving increasing attention in superstring theory. A particularly interesting class of such backgrounds is formulated as the fibrations of which the transition functions involve the duality transformations in string theory [1,2,3]. For T-duality, for instance, one then has 'T-folds' [4]. Another interesting class is the backgrounds with non-geometric fluxes that do not have naive geometrical origins in higher dimensional theories. In some cases, these are reduced to geometric ones by dualities, but are truly non-geometric in general [5,6,7].
These string vacua on non-geometric backgrounds are described by the world-sheet conformal field theory (CFT) on the same footing as geometric ones. We should emphasize that many of such vacua are well-defined only at particular points on the moduli space, at which enhanced symmetries emerge and the α ′ -corrections become important. The world-sheet CFT approach would provide reliable descriptions of strings even in such backgrounds. In this respect, a simple and important class of non-geometric backgrounds is realized as asymmetric orbifolds [8], in which the left-and the right-movers of strings propagate on different geometries. Especially, as typical T-duality twists are identified with chiral reflections, simple examples of T-folds are realized as the orbifolds by the chiral reflection combined with the shift in the base circle. These types of string vacua have been studied based on the world-sheet CFT e.g. in [9,10,11,12,13,14,15,16]. 1 In this paper, we study type II string vacua defined by torus compactifications twisted by T-duality transformations in the above sense. We carefully discuss possible consistent actions of the chiral reflection on the Ramond sector of the world-sheet fermions, and explicitly construct non-supersymmetric as well as supersymmetric (SUSY) vacua. 2 Among others, we present a simple realization of non-SUSY vacua with vanishing cosmological constant at the one-loop level, at least. Namely, we construct the string vacua realizing the bose-fermi cancellation despite the absence of any supercharges in space-time. Previous constructions of such string vacua are found e.g. in [21,22,23,24,25,26,27]. 3 A novel feature, as well as an advantage, in our construction is that we only have to utilize a cyclic orbifold, in which the orbifold group is generated by a single element, and hence the construction looks rather simpler than previous ones given in those papers. It would be notable that one can achieve (nearly) vanishing cosmological constant without SUSY in a fairly simple way in the framework of non-geometric string compactifications. Our construction suggests that they would provide useful grounds also for the cosmological constant problem.
To be more precise, we first analyze in some detail the asymmetric orbifolds representing T-folds, where the partition sums from each sector in the total partition function are combined according to the windings around the 'base' circle. It turns out that the consistent action of the chiral reflections therein is not unique, from which a variety of supersymmetric T-fold vacua can be derived. As general for asymmetric orbifolds, the moduli of the internal ('fiber') tori are fixed for consistency, while a continuous radius of the base circle remains. The supersymmetry is broken by further implementing the Scherk-Schwarz type boundary condition for the worldsheet fermions [33,34] along the base circle. In the case where the chiral reflections act as Z 4 transformations in a fermionic sector, the resultant world-sheet torus partition function and hence the one-loop cosmological constant vanish: if the partition sum for the left-moving fermions is non-vanishing in a winding sector, that for the right-moving fermions vanishes, and vice versa. It is crucial here that the chiral partition sums for the fermions depend on the winding numbers in an asymmetric way. We see that all the ingredients in our setup, i.e., T-folds (asymmetric orbifolds, base winding), careful treatment of the chiral reflections and the Scherk-Schwarz twist, cooperate in this mechanism. Although we focus on specific examples in this paper, our construction would be more general. It provides a systematic way to find string vacua of T-folds, and a novel mechanism for non-supersymmetric string vacua with vanishing cosmological constant at one-loop. This paper is organized as follows: In section 2, which is a preliminary section, we survey the building blocks (partition sums) for the modular invariant partition functions of the asymmetric orbifolds discussed later, specifying how to achieve the modular covariance in relevant sectors. Though this part might be slightly technical, the results, especially those for the fermionic sector presented in subsection 2.2, are important in the later discussion both on T-fold vacua and on vanishing cosmological constant. The readers may refer only to the definitions of the building blocks, if they are interested mostly in the physical consequences.
In section 3, we begin our main analysis of type II string vacua compactified on asymmetric orbifolds/T-folds. We first consider the supersymmetric ones. The SUSY breaking is then discussed by further incorporating the Scherk-Schwarz twist, which leads us to the non-SUSY vacua implementing the bose-fermi cancellation. In section 4, we analyze the spectra of the physical states and check the unitarity, mainly focusing on the case of the non-SUSY vacua. We also demonstrate the absence of the instability caused by the winding tachyons, which would be typically possible for the Scherk-Schwarz compactification. We conclude with a summary and a discussion for possible future directions in section 5.
Preliminaries: Building Blocks for Asymmetric Orbifolds
In this paper, we would like to study the type II string vacua constructed from asymmetric orbifolds of the 10-dimensional flat background given by
M 4 × S 1 × R base × T 4 fiber , (2.1) where M 4 (X 0,1,2,3 -directions) is the 4-dimensional Minkowski space-time.
Intending the twisted compactification of the 'base space' R base (X 5 -direction), we consider the orbifolding defined by the twist operator T 2πR ⊗ σ : T 2πR is the translation along the base direction by 2πR, and σ denotes an automorphism acting on the 'fiber sector' T 4 fiber (X 6,7,8,9 ), which is specified in detail later. We especially focus on the cases where σ acts as the 'chiral reflection', or the T-duality 7,8,9).
transformation, σ : (X i L , X i R ) −→ (X i L , −X i R ), (i = 6,
(2.
2)
The S 1 -factor (X 4 -direction) in (2.1) is not important in our arguments. We begin our analysis by specifying the relevant bosonic and fermionic sectors and their chiral blocks that compose the modular invariants for our asymmetric orbifolds.
Bosonic T 4 fiber Sector
In the bosonic sector, let us first consider the 4-dimensional torus with the SO(8)-symmetry enhancement which we denote as T 4 [SO (8)], in order that the relevant asymmetric orbifold action (chiral reflection) is well-defined. The torus partition function of this system is
Z T 4 [SO(8)] (τ,τ ) = 1 2 θ 3 η 8 + θ 4 η 8 + θ 2 η 8 . (2.3)
Another system that is compatible with our asymmetric orbifolding and of our interests is the product of the 2-dimensional tori with the SO(4)-symmetry, T 2 [SO(4)] × T 2 [SO(4)], the partition function of which is given by
Z T 2 [SO(4)]×T 2 [SO(4)]] (τ,τ ) = 1 4 θ 3 η 4 + θ 4 η 4 + θ 2 η 4 2 . (2.4)
It is useful to note the equivalence
T 2 [SO(4)] × T 2 [SO(4)] ∼ = T 4 [SO(8)]/Z 2 ∼ = S 1 [SU(2)] 4 ,(2.5)
where S 1 [SU(2)] expresses the circle of the self-dual radius R = 1. 4 Namely, while both of X 6,7 and X 8,9 are compactified on the 2-torus T 2 [SO(4)] at the fermionic point with radius √ 2, the following four compact bosons have the self-dual radius,
Y 1 ± := 1 √ 2 X 6 ± iX 7 , Y 2 ± := 1 √ 2 X 8 ± iX 9 .
(2.6)
The equivalence (2.5) is confirmed by the simple identities (B.3). We then consider the action of the automorphism σ for T 4 [SO (8)] and T 2 [SO(4)]×T 2 [SO(4)].
Since relative phases for the left and the right movers are generally possible in asymmetric orbifolding, in addition to the action without phases, we consider an action with phases according to [10] for T 2 [SO(4)] × T 2 [SO (4)]. In total, we consider the following three cases as models relevant to our construction of string vacua given in section 3. This means that the moduli of T 4 fiber need be restricted to the particular points given here, while the radius of S 1 base can be freely chosen. We particularly elaborate on the derivation of the building blocks for the case of T 4 [SO (8)], and mention on other cases briefly. The explicit forms of the relevant building blocks are summarized in Appendix B. The case with phases for T 4 [SO (8)] can be similarly discussed following [10,16], although we do not work on it in this paper.
Chiral reflection in T 4 [SO(8)]
We start with T 4 [SO (8)]. In this case, the orbifold action is defined by the chiral reflection (2.2) acting only on the right-moving components. We simply assume σ acts as the identity operator on any states in the left-mover, and also that σ 2 acts as the identity over the Hilbert space of the untwisted sector of the orbifolds of our interest. 5 We note that the action of σ 2 on the twisted sectors should be determined so that it preserves the modular invariance of the total system, and does not necessarily coincide with the identity. This is a characteristic feature of asymmetric orbifolds. See for example [35].
Let us evaluate the building blocks in this sector of the torus partition function. These are schematically written as
F T 4 [SO(8)] (a,b) (τ,τ ) := Tr σ a -twisted sector σ b q L 0 − c 24qL 0 − c 24 .
(2.7)
Here, we allow a, b to be any integers despite a periodicity, which is at most of order 4 as seen below, since we later identify them as the winding numbers along the base circle S 1 base . We can 4 Throughout this paper, we use the α ′ = 1 convention. 5 This assumption is not necessarily obvious. Actually, if we fermionize the string coordinates along T 4 [SO (8)],
we can also realize more general situations as in our discussion given in subsection 2.2. We do not study these cases for simplicity in this paper.
most easily determine the building blocks F
T 4 [SO(8)] (a,b)
by requiring the modular covariance,
F T 4 [SO(8)] (a,b) (τ,τ )| S = F T 4 [SO(8)] (b,−a) (τ,τ ), F T 4 [SO(8)] (a,b) (τ,τ )| T = F T 4 [SO(8)] (a,a+b) (τ,τ ), (2.8)
together with the trace over the untwisted sector,
F T 4 [SO(8)] (0,b) (τ,τ ) = θ 3 θ 4 η 2 2 · 1 2 θ 3 η 4 + θ 4 η 4 , ( ∀ b ∈ 2Z + 1). (2.9)
Then, the desired building blocks are found to be
F T 4 [SO(8)] (a,b) (τ,τ ) = (−1) a 2 θ 3 θ 4 η 2 2 · 1 2 θ 3 η 4 + θ 4 η 4 (a ∈ 2Z, b ∈ 2Z + 1), (−1) b 2 θ 2 θ 3 η 2 2 · 1 2 θ 3 η 4 + θ 2 η 4 (a ∈ 2Z + 1, b ∈ 2Z), e − iπ 2 ab θ 4 θ 2 η 2 2 · 1 2 θ 4 η 4 − θ 2 η 4 (a ∈ 2Z + 1, b ∈ 2Z + 1), 1 2 θ 3 η 8 + θ 4 η 8 + θ 2 η 8 (a ∈ 2Z, b ∈ 2Z).
(2.10)
One can confirm that they indeed satisfy the modular covariance relations (2.8).
Chiral reflection in
T 2 [SO(4)] × T 2 [SO(4)]
In the first case of T 2 [SO(4)] × T 2 [SO(4)] or the Z 2 -orbifold of T 4 [SO (8)], we may consider the same orbifold action σ as given in case 1, Namely, it acts as the identity on the left-mover, and assumes σ 2 = 1 in the untwisted sector. The modular covariant building blocks of the torus partition function are just determined in the same way as above. We present them in (B.2) in Appendix B.
Chiral reflection in T 2 [SO(4)] × T 2 [SO(4)] with a phase factor
In the second case of T 2 [SO(4)] ×T 2 [SO(4)], we include the phase factors for the Fock vacua when defining σ, while the action of the chiral reflection (2.2) is kept unchanged. To be more specific, recalling the equivalence (2.5), let us introduce 4 copies of the SU(2)-current algebra of level 1 whose third components are identified as
J 3, (i) = i∂Y 1 + , i∂Y 1 − , i∂Y 2 + , i∂Y 2 − , (i = 1, . . . , 4), (2.11)
where Y 's are the compact bosons in (2.6). With these currents, σ is now explicitly defined according to [10] by
σ := 4 i=1 e iπJ 3, (i) L,0 ⊗ e iπJ 1, (i) R,0 .
(2.12)
We then obtain the building blocks according to the same procedure : the blocks for the (0, b)sectors with ∀ b ∈ 2Z + 1 are computed first, and then those for other sectors are obtained by requiring the modular covariance. It turns out that these are eventually equal to the building blocks of the symmetric Z 2 -orbifold defined by 7,8,9).
(X i L , X i R ) −→ (−X i L , −X i R ), ( ∀ i = 6,
(2.13)
Of course, this fact is not surprising since (2.12) is equivalent to the symmetric one
4 i=1 e iπJ 1, (i) L,0 ⊗ e iπJ 1, (i) R,0 ,
(2.14)
by an automorphism of SU(2) ⊗4 , as was pointed out in [11]. We exhibit the building blocks in this case in (B.4).
Fermionic Sector
We next consider the fermionic sector. The orbifold action should act on the world-sheet
fermions as σ : (ψ i L , ψ i R ) −→ (ψ i L , −ψ i R ), (i = 6, 7, 8, 9), (2.15)
to preserve the world-sheet superconformal symmetry. (2.15) uniquely determines the action on the Hilbert space of the NS-sector. However, it is not on the R-sector, and as is discussed in the next section, we obtain different string vacua according to its choice. The fermionic part is thus crucial in our analysis. In the following, we include the fermions ψ i (i = 2, 3, 4, 5) in other transverse part from M 4 × S 1 × S 1 base , on which σ acts trivially. If retaining the Poincare symmetry in 4 dimensions, we then have two possibilities, which can be understood from the point of view of bosonization as follows:
(i) Z 2 action on the untwisted R-sector
In this case, we bosonize ψ i R (i = 2, . . . , 9) as
ψ 2 R ± iψ 3 R ≡ √ 2e ±iH 0,R , ψ 4 R ± iψ 5 R ≡ √ 2e ±iH 1,R , ψ 6 R ± iψ 7 R ≡ √ 2e ±iH 2,R , ψ 8 R ± iψ 9 R ≡ √ 2e ±iH 3,R , (2.16)
and define the spin fields for SO(8) as
S ǫ 0 ,ǫ 1 ǫ 2 ǫ 3 ,R ≡ e i 2 3 i=0 ǫ i H i,R , (ǫ i = ±1). (2.17)
Then, (2.15) translates into
σ : (H 0,R , H 1,R , H 2,R , H 3,R ) −→ (H 0,R , H 1,R , H 2,R + π, H 3,R + π),
(2. 18) and thus, we find σ 2 = 1 for all the states in the NS and R-sectors in the untwisted sector.
This type of twisting preserves half of the space-time SUSY. In fact, the Ramond vacua that are generated by the spin fields (2.17) survive the σ-projection when ǫ 2 + ǫ 3 = 0.
(ii) Z 4 action on the untwisted R-sector
In this case, we bosonize ψ i R (i = 2, . . . , 9) as
H ′ 0,R ≡ H 0,R , ψ 4 R ± iψ 6 R ≡ √ 2e ±iH ′ 1,R , ψ 5 R ± iψ 7 R ≡ √ 2e ±iH ′ 2,R , H ′ 3,R ≡ H 3,R , (2.19)
and define the spin fields for SO (8) as
S ′ ǫ 0 ,ǫ 1 ǫ 2 ǫ 3 ,R ≡ e i 2 3 i=0 ǫ i H ′ i,R , (ǫ i = ±1). (2.20)
This time, (2.15) translates into
σ : (H ′ 0,R , H ′ 1,R , H ′ 2,R , H ′ 3,R ) −→ (H ′ 0,R , −H ′ 1,R , −H ′ 2,R , H ′ 3,R + π).
(2.21)
Then, σ 2 = −1 for the R-sector, while σ 2 = 1 still holds for the NS sector. In other words, we have found in this second case that
σ 2 = (−1) F R , (2.22)
where F R denotes the 'space-time fermion number' (mod 2) from the right-mover. The operator (−1) F R acts as the sign flip on all the states belonging to the right-moving R-sector.
As long as the M 4 part or ψ 2,3 are kept intact, other possibilities essentially reduce to one of these two. The chiral blocks of the right-moving fermions in the eight-dimensional transverse part are then determined in the same way as in the bosonic T 4 sector: we first evaluate the trace over the untwisted sector with the insertion of σ b , and next require the modular covariance.
For case (i), we then have the desired chiral blocks f (a,b) (τ ) with
f (a,b) (τ ) = (−1) a 2 θ 3 η 2 θ 4 η 2 − θ 4 η 2 θ 3 η 2 + 0 (a ∈ 2Z, b ∈ 2Z + 1), (−1) b 2 θ 3 η 2 θ 2 η 2 + 0 − θ 2 η 2 θ 3 η 2 (a ∈ 2Z + 1, b ∈ 2Z), −e iπ 2 ab 0 + θ 2 η 2 θ 4 η 2 − θ 4 η 2 θ 2 η 2 (a ∈ 2Z + 1, b ∈ 2Z + 1), θ 3 η 4 − θ 4 η 4 − θ 2 η 4 (a ∈ 2Z, b ∈ 2Z).
(2.23)
Each term from the left to the right corresponds to the NS, NS, and R sector, respectively, where the ' NS' denotes the NS-sector with (−1) f inserted (f is the world-sheet fermion number). These trivially vanish as expected from the space-time SUSY. We note that in the fermionic sectors the modular covariance means 6 :
f a,b (τ )| S = f (b,−a) (τ ), f (a,b) (τ )| T = −e −2πi 1 6 f (a,a+b) (τ ), (2.24)
with the phase for the T-transformation. Since the total blocks for the transverse fermions consist of f (a,b) (τ ) and the left-moving part,
J (τ ) ≡ θ 3 η 4 − θ 4 η 4 − θ 2 η 4 ,
(2.25)
(2.24) indeed assures the proper modular covariance:
J (τ )f (a,b) (τ ) S = J (τ )f (b,−a) (τ ), J (τ )f (a,b) (τ ) T = J (τ )f (a,a+b) (τ ). (2.26)
We next consider the chiral blocks for case (ii), which we denote by f (a,b) (τ ). In this case, the treatment of the R-sector needs a little more care. First, from (2. 21) we find that
f (0,b) (τ ) = f (0,b) (τ ), ( ∀ b ∈ 2Z + 1), (2.27)
which are vanishing. On the other hand, the blocks for the sectors of a, b ∈ 2Z are non-trivially modified due to (2.22). Again it is easy to evaluate the trace over the (0, b)-sector, and by requiring the modular covariance (in the sense of (2.24) or (B.8)), we finally obtain
f (a,b) (τ ) = f (a,b) (τ ), (a ∈ 2Z + 1 or b ∈ 2Z + 1), (2.28) and f (a,b) (τ ) = θ 3 η 4 − θ 4 η 4 − θ 2 η 4 (a ∈ 4Z, b ∈ 4Z), θ 3 η 4 − θ 4 η 4 + θ 2 η 4 (a ∈ 4Z, b ∈ 4Z + 2), θ 3 η 4 + θ 4 η 4 − θ 2 η 4 (a ∈ 4Z + 2, b ∈ 4Z), − θ 3 η 4 + θ 4 η 4 + θ 2 η 4 (a ∈ 4Z + 2, b ∈ 4Z + 2).
(2. 29) In contrast to f (a,b) , these f (a,b) are in general non-vanishing, which signals the SUSY breaking in the right-moving sector. This completes our construction of the chiral building blocks. These are used in the following sections.
String Vacua on T-folds
Now we construct type II string vacua by combining the building blocks derived in the previous section. They are interpretable as describing the compactification on T-folds.
First, to describe the 'base sector' for S 1 base , we introduce the following notation,
Z R,(w,m) (τ,τ ) := R √ τ 2 |η(τ )| 2 e − πR 2 τ 2 |wτ +m| 2 , (w, m ∈ Z),(3.1)
where R is the radius of the compactification and the integers w, m are identified as the spatial and temporal winding numbers. In terms of these, we find 7
Tr base (T 2πR ) m q L 0 − c 24 qL 0 − c 24 = Z R,(0,m) (τ,τ ),(3.2)
and the torus partition function of a free compact boson with radius R reads
Z R (τ,τ ) = w,m∈Z Z R,(w,m) (τ,τ ). (3.3)
To calculate the total partition function, we proceed as follows: First, we evaluate
Z (0,m) (τ,τ ) ≡ Tr w=0 sector (T 2πR ⊗ σ) m q L 0 − c 24 qL 0 − c 24 = Z R,(0,m) (τ,τ ) Tr untwisted σ m q L 0 − c 24 qL 0 − c 24 . (3.4)
Second, we extend (3.4) to the partition function of the general winding sector Z (w,m) (τ,τ ) by requiring the modular covariance. It is straightforward to perform this, given the relevant building blocks in the previous section. These two steps are also in parallel with the previous section. Finally, we obtain the total partition function by summing over the winding numbers w, m ∈ Z along the base circle as
Z(τ,τ ) = w,m∈Z Z (w,m) (τ,τ ). (3.5)
Supersymmetric Vacua
In this way, we can construct string vacua, depending on the combination of the bosonic T 4 sector (1-3) in section 2.1 and the transverse fermionic sector (i, ii) in section 2.2. All these are supersymmetric. 7 Here we adopt the conventional normalization of the trace for the CFT for R base ,
Tr base q L0− c 24 qL 0− c 24 = R √ τ 2 |η| 2 .
This means that we start with S 1 N R for the base CFT with an arbitrary integer N , and regard the insertion of the shift operators (T 2πR ) m as implementing the Z N -orbifolding.
As the first example, we consider T 4 [SO (8)] in the background (2.1). Choosing case (i) for the fermionic sector, we obtain the torus partition function as
Z(τ,τ ) = 1 4 Z tr M 4 ×S 1 (τ,τ ) w,m∈Z Z R,(w,m) (τ,τ ) F T 4 [SO(8)] (w,m) (τ,τ ) J (τ ) f (w,m) (τ ), (3.6)
where Z tr M 4 ×S 1 (τ,τ ) denotes the bosonic partition function for the transverse part of M 4 × S 1sector. J (τ ) is the contribution from the left-moving free fermions defined in (2.25), and the overall factor 1/4 is due to the chiral GSO projections. This is manifestly modular invariant by construction and defines a superstring vacuum, which preserves 3/4 of the space-time SUSY, that is, 16 supercharges from the left-mover and 8 supercharges from the right-mover.
For case (ii), we replace f (w,m) (τ ) in (3.6) with f (w,m) (τ ) given in (2.28), (2. 29), and obtain the torus partition function
Z(τ,τ ) = 1 4 Z tr M 4 ×S 1 (τ,τ ) w,m∈Z Z R,(w,m) (τ,τ ) F T 4 [SO(8)] (w,m) (τ,τ ) J (τ ) f (w,m) (τ ). (3.7)
This time, we are left with the 1/2 space-time SUSY that originates only from the left-mover. 8 It is straightforward to construct the string vacua in other four cases based on T 2 [SO(4)] × T 2 [SO(4)]: one has only to replace the bosonic building blocks F
Non-SUSY String Vacua with Vanishing Cosmological Constant
An interesting modification of the half SUSY vacuum represented by (3.7) is to replace the base circle along the X 5 -direction with the Scherk-Schwarz one [33,34]. This means that we implement the orbifolding of the background (2.1) by the twist operator 9
g := T 2πR ⊗ (−1) F L ⊗ σ,(3.8)
where (−1) F L acts as the sign flip on any states of the left-moving Ramond sector. Again σ denotes the chiral reflection for the T 4 -sector and is assumed to satisfy σ 2 = (−1) F R as for (3.7). The action of the twist operators g n is summarized in Table 1.
base (X 5 ) T 4 (X 6,7,8,9 ) left-moving fermions right-moving fermions This modification leads to the following torus partition function,
g 4n T 2π(4n)R 1 1 1 g 4n+1 T 2π(4n+1)R σ (−1) F L σ g 4n+2 T 2π(4n+2)R 1 1 (−1) F R g 4n+3 T 2π(4n+3)R σ (−1) F L (−1) F R σZ(τ,τ ) = 1 4 Z tr M 4 ×S 1 (τ,τ ) w,m∈Z Z R,(w,m) (τ,τ ) F T 4 [SO(8)] (w,m) (τ,τ ) f (2w,2m) (τ ) f (w,m) (τ ). (3.9)
Here, the chiral blocks for left-moving fermions have been replaced with f (2w,2m) (τ ) as in (3.7) due to the extra twisting (−1) F L . One can confirm that this partition function vanishes for each winding sector, similarly to usual supersymmetric string vacua. Indeed, f (w,m) (τ ) = 0 for ∀ w ∈ 2Z + 1 or ∀ m ∈ 2Z + 1, while f (2w,2m) (τ ) = 0 for ∀ w, m ∈ 2Z. Then, we see a bose-fermi cancellation at each mass level of the string spectrum, after making the Poisson resummation with respect to the temporal winding m in a standard fashion. We will observe this aspect explicitly in section 4. Thus, the vacuum energy or the cosmological constant in space-time vanishes at the one-loop level.
A remarkable fact here is that the space-time SUSY is nonetheless completely broken:
• For w = 0, only the supercharges commuting with the orbifold projection 1 4 n∈Z 4 g n | fermion would be preserved. However, since the relevant projection includes both (−1) F L and (−1) F R , all the supercharges in the unorbifolded theory cannot commute with it. This implies that all the supercharges from this sector are projected out.
• For w = 0, if we had a supercharge, we would observe a bose-fermi cancellation between two sectors with winding numbers w ′ and w ′ + w for ∀ w ′ ∈ Z, which would imply
Z (NS,NS) w ′ (τ,τ ) + Z (R,R) w ′ (τ,τ ) = − Z (NS,R) w ′ +w (τ,τ ) + Z (R,NS) w ′ +w (τ,τ ) .
(3.10)
However, we explicitly confirm, as expected, in section 4.2 that such relations never hold for the partition function (3.9) due to the factor Z R,(w,m) (τ,τ ) from the base circle.
Here, it would be worthwhile to emphasize a crucial role of the shift operator T 2πR | base in the above argument. Obviously, one has a vanishing partition function even without Z R,(w,m) (τ,τ ):
Z(τ,τ ) = 1 4 · 4 Z tr M 4 ×S 1 ×S 1 (τ,τ ) a,b∈Z 4 F T 4 [SO(8)] (a,b) (τ,τ ) f (2a,2b) (τ ) f (a,b) (τ ). (3.11)
For the untwisted sector with a = 0, all the supercharges are projected out in the same way as above. However, new Ramond vacua can appear from the a = 0 sectors in this case, 10 and the space-time SUSY revives eventually. The inclusion of T 2πR | base was a very simple way to exclude such a possibility, since supercharges cannot carry winding charges generically, as pointed out above. This is also in accord with an intuition that in the twisted sectors masses are lifted up by the winding charges.
Asymmetric/Generalized Orbifolds and T-folds
We have explicitly constructed the non-geometric superstring vacua/partition functions, (3.6),
(3.7), (3.9) for the asymmetric orbifolds associated with the chiral reflection. In this subsection, we would like to comment on the relation to the construction of T-folds in [10,16]. In these works, the T-duality twists are accompanied by extra phases, so that the full operator product expansion (OPE), not only the chiral one, respects the invariance under the twist: supposed that two vertex operators including both the left and right movers are invariant, their OPE yields invariant operators. This is in accord with the ordinary principle of orbifolding by symmetries. The construction of (B.4) includes such phases and the resultant models represent the T-folds in this sense. Asymmetric orbifolding, however, generally respects the chiral OPE only, and belongs to a different class.
Here, we recall that, from the CFT point of view, T-duality is in general an isomorphism between different Hilbert spaces, which keeps the form of the Hamiltonian invariant. At the self-dual point, it acts within a single Hilbert space, but is not yet an ordinary symmetry, since the transformation to the dual fields is non-local. Thus, it may not be obvious if the OPE should fully respect the invariance under the T-duality twists. Indeed, in the case of the critical Ising model, the OPE of the order and disorder fields, which are non-local to each other, reads σ(z,z)µ(0, 0) ∼ |z| −1/4 ωz 1/2 ψ(0) +ωz 1/2ψ (0) , (3.12) where ψ,ψ are the free fermions, ω = 1 √ 2 e iπ 4 andω is its complex conjugate. Under the Kramers-Wannier duality (T-duality), these fields are mapped as (σ, µ, ψ,ψ) → (µ, σ, ψ, −ψ). One then finds that the OPE of two invariant fields (σ + µ)(z,z)(σ + µ)(0, 0) yields non-invariant fields, since the diagonal part σσ + µµ yields invariant ones.
In addition, we note that sensible CFTs may be obtained from the twists by transformations which are not the full symmetries. We refer to such CFTs as "generalized orbifold" CFTs, according to [37,38,39] where such CFTs are studied in the context of the topological conformal interfaces [40,41,42,43]. An application to non-geometric backgrounds has been discussed in [15]. Even though the twists are not necessarily by the full symmetries, the transformations may need to commute with the Hamiltonian, since the position of the twist operators matters otherwise. In this terminology, general asymmetric orbifold models and hence ours based on the twists without the extra phases belong to this class. In any case, our resultant models are consistent in that they are modular invariant and, as shown in the next section, have sensible spectra.
Taking these into account, we expect that the world-sheet CFTs for T-folds are generally given by the asymmetric/generalized orbifold CFTs, and that our asymmetric orbifolds without, as well as with, the extra phases also represent T-folds, as we have assumed so far (see also [1,2,3,7,9,13,14]). It would be an interesting issue if all these non-geometric models have the corresponding supergravity description as low-energy effective theory of T-folds. As is discussed shortly, the difference of the spectra due to the phases typically appear in the massive sector. However, the massless spectra can also differ, for example, at special points of the moduli, and thus supergravity may distinguish them.
Regarding the interpretation as T-folds, we also note that the chiral reflections both for tions in the untwisted bosonic sector as expected, whereas they do not generally in other sectors, for example, in the fermionic sectors (see also [10,35]). This, however, is not a contradiction: that means that such sectors are in different representations.
Analysis on Spectra
Massless Spectra in the Untwisted Sectors
(R, R) |ǫ 0 , ǫ 1 , ǫ 2 , ǫ 3 ⊗ |ǫ 0 ,ǫ 1 ,ǫ 2 ,ǫ 3 8 vectors, (ǫ 2 +ǫ 3 = 0) 16 (pseudo) scalars (R, NS) |ǫ 0 , ǫ 1 , ǫ 2 , ǫ 3 ⊗ψ ν −1/2 |0 8 gravitini, (ν = 2, ..., 5) 8 Weyl fermions (NS, R) ψ µ −1/2 |0 ⊗ |ǫ 0 ,ǫ 1 ,ǫ 2 ,ǫ 3 4 gravitini,
(µ = 2, ..., 9) (ǫ 2 +ǫ 3 = 0) 12 Weyl fermions Table 3: Massless spectrum of the SUSY vacuum (3.6) way, one can easily write down the massless spectrum. We exhibit it in Table 2. 11 Since our background includes the S 1 -factor (X 4 -direction) that is kept intact under the orbifolding, it is evident by considering T-duality that the type IIA and type IIB vacua lead us to the same massless spectra in 4 dimensions. Thus, we do not specify here which we are working on.
It is evident from Table 2 that we have the same number of the massless bosonic and fermionic degrees of freedom. Nevertheless, there are no 4-dimensional gravitini, reflecting the absence of the space-time SUSY.
For comparison, it would be useful to exhibit the massless spectra in the untwisted sector for the 3/4-SUSY vacuum (3.6) and the 1/2-SUSY vacuum (3.7). We present them in Table 3 and Table 4.
Unitarity
The torus partition functions we constructed in the previous section include the non-trivial phase factors which originate from the requirement of the modular covariance and depend on the winding numbers along the base circle. Thus, it may not be so obvious whether the spectrum is unitary in each vacuum, though that is evident in the untwisted sector with w = 0 by construction.
An explicit way to check the unitarity is to examine the string spectrum by the Poisson resummation of the relevant partition function with respect to the temporal winding m along the base circle. To this end, we decompose the partition functions with respect to the spatial winding w and the spin structures, and factor out the component of Z tr M 4 ×S 1 :
Z(τ,τ ) = 1 4 Z tr M 4 ×S 1 (τ,τ ) s,s w∈Z Z (s,s) w (τ,τ ),(4.1)
where s,s = NS, R denote the left and right-moving spin structures. For instance, let us pick up the non-SUSY vacuum built from T 4 [SO(8)] given by (3.9). Making the Poisson resummation, we find that each function Z (s,s) w (τ,τ ) with fixed w becomes as follows:
• w ∈ 4Z ; Z (NS,NS) w (τ,τ ) = −Z (R,NS) w (τ,τ ) = 1 4 n∈Z q 1 4 ( n 2R −Rw) 2 q 1 4 ( n 2R +Rw) 2 × θ 3 η 8 + θ 4 η 8 + θ 2 η 8 θ 3 η 4 − θ 4 η 4 2 , (4.2) Z (R,R) w (τ,τ ) = −Z (NS,R) w (τ,τ ) = 1 4 n∈Z+ 1 2 q 1 4 ( n 2R −Rw) 2 q 1 4 ( n 2R +Rw) 2 × θ 3 η 8 + θ 4 η 8 + θ 2 η 8 θ 2 η 8 .
(4.3)
• w ∈ 4Z + 2 ;
Z (NS,NS) w (τ,τ ) = −Z (R,NS) w (τ,τ ) = 1 4 n∈Z+ 1 2 q 1 4 ( n 2R −Rw) 2 q 1 4 ( n 2R +Rw) 2 × θ 3 η 8 + θ 4 η 8 + θ 2 η 8 θ 3 η 4 + θ 4 η 4 θ 3 η 4 − θ 4 η 4 , (4.4) Z (R,R) w (τ,τ ) = −Z (NS,R) w (τ,τ ) = 1 4 n∈Z q 1 4 ( n 2R −Rw) 2 q 1 4 ( n 2R +Rw) 2 × θ 3 η 8 + θ 4 η 8 + θ 2 η 8 θ 2 η 8 .
(4.5)
• w ∈ 2Z + 1 ;
Z (NS,NS) w (τ,τ ) = −Z (NS,R) w (τ,τ ) = 1 4 r∈Z 2 n∈Z q 1 4 ( n 2R −Rw) 2 q 1 4 ( n 2R +Rw) 2 ×(−1) rn θ 2 θ 3 ( r 2 ) η 2 4 (−1) r θ 3 ( r 2 ) η 4 + θ 2 η 4 θ 3 η 4 + θ 4 η 4 , (4.6) Z (R,R) w (τ,τ ) = −Z (R,NS) w (τ,τ ) = 1 4 r∈Z 2 n∈Z q 1 4 ( n 2R −Rw) 2 q 1 4 ( n 2R +Rw) 2 ×(−1) rn θ 2 θ 3 ( r 2 ) η 2 4 θ 3 ( r 2 ) η 4 + (−1) r θ 2 η 4 θ 2 η 4 .
(4.7)
Here, we denoted θ i ≡ θ i (τ, 0), and θ 3 ( r 2 ) ≡ θ 3 (τ, r 2 ), and made use of the identity θ 4 3 − θ 4 4 − θ 4 2 = 0.
As expected, all of these partition functions are suitably q-expanded so as to be consistent with the unitarity. On the other hand, for the 1/2-SUSY vacuum (3.7), we find the following:
• w ∈ 4Z ;
Absence of Winding Tachyons
Recall that our non-SUSY string vacuum (3.9) from T 4 [SO (8)] has been constructed by including the Sherk-Schwarz type modification. Therefore, we would potentially face an issue of the instability caused by the winding tachyons that are typical in the Sherk-Schwarz compactification. That would be implied by the 'wrong GSO projections' observed in (4.4), (4.6). 12 However, the spectrum is in fact free from the winding tachyons.
To show this, we first note that potentially dangerous states come from the winding sectors with w ∈ 4Z + 2 or w ∈ 2Z + 1, which are anticipated from the wrong GSO projections. Among them, we further focus on the NS-NS sector, since the spectrum is lifted in the R-R sector due to the θ 2 -factors, and the partition functions in the NS-R and R-NS sectors are the same as for the NS-NS or the R-R sector up to sign. From the partition functions (4.4), (4.6), we then find the following:
• For w ∈ 4Z + 2, the wrong GSO states are in the right-mover. The lightest excitations appear in the sectors of w = ±2, the conformal weights of which read
h L = 1 2 + 1 4 n 2R ± 2R 2 , h R = 1 4 n 2R ∓ 2R 2 , (4.15)
with the KK momenta n ∈ Z + 1 2 . Their minima for the physical states are achieved by setting n = ∓ 1 2 , to give
h L = h R = 1 2 + 1 4 1 4R − 2R 2 ≥ 1 2 . (4.16)
This means that the winding states from these sectors are always massive except at the special radius R = 1 2 √ 2 of the base circle, where extra massless excitations appear.
• For w ∈ 2Z + 1, the wrong GSO states are in the left-mover. The lightest excitations appear in the sectors of w = ±1, and the leading contribution from the θ-part comes from θ 3 ( r 2 ) = 1 + (−1) r q 1 2 + · · · . The summation over r ∈ Z 2 then projects the KK momenta onto n ∈ 2Z + 1, and the conformal weights read
h L = 1 4 n 2R ± R 2 , h R = 1 2 + 1 4 n 2R ∓ R 2 , (n ∈ 2Z + 1). (4.17)
Their minima for the physical states are achieved by setting n = ±1, to give This means that the winding states from these sectors are always massive except at the special radius R = 1 √ 2 , where extra massless excitations appear.
h L = h R = 1 2 + 1 4 1 2R − R 2 ≥ 1 2 .
These demonstrate that no winding tachyons emerge in the non-SUSY vacuum (3.9).
The non-SUSY vacua associated with F . In this case, the conformal weights of the w = ±1 sectors become
h L = 1 4 + 1 4 n 2R ± R 2 , h R = 1 2 + 1 4 n 2R ∓ R 2 , (n ∈ Z + 1 2 ). (4.19)
Here, h L also acquires the twisted energy from the extra θ 2 -factor. The KK momenta are shifted by one half due to the absence of the phase factors depending on the temporal winding m (see (B.4)). Consequently, the lightest excitations lie in the sectors with w = ±1, n = ± 1 2 , giving
h L = h R = 1 2 + 1 4 1 4R − R 2 ≥ 1 2 . (4.20)
Again these are always massive except at the massless point R = 1 2 . Also, for the F
Summary and Discussions
In this paper, we have studied type II string vacua which are defined by the asymmetric orbifolding based on the chiral reflections/T-duality twists in T 4 combined with the shift in the base circle, in such a way that the modular invariance is kept manifest. They represent the non-geometric string vacua for T-folds, supposed that the world-sheet description of T-folds is generally given by asymmetric/generalized orbifolds. Including appropriate phases as in (B.4), the full OPE also respects the invariance under the T-duality twists in accord with [10]. As the main result, we have presented simple examples of the non-SUSY vacua with vanishing cosmological constant at one loop. We summarize the points to be emphasized as follows:
• Our non-SUSY vacuum (3.9) has been defined by a cyclic orbifold which is generated by a single element g in (3.8). Thus, it provides a simpler model than the previous ones [21,22,23,24,25,26,27]. In this construction, taking both the asymmetric orbifold action with σ 2 = (−1) F R and the Scherk-Schwarz compactification (orbifolding by (−1) F L ⊗ T 2πR base ) at the same time is truly crucial in order to make the SUSY-breaking compatible with the bose-fermi cancellation. Indeed, it is important that the left and right-moving non-SUSY chiral blocks f ( * , * ) (τ ), f ( * , * ) (τ ), which originate from the SUSY-breaking twists (−1) F L , (−1) F R , depend on the winding numbers along the Scherk-Schwarz circle in an asymmetric way.
• The modular invariant partition function given in (3.9) is q-expanded so as to be compatible with unitarity, as shown in subsection 4.2. Curiously, it turns out that the left-moving bose-fermi cancellation occurs in the even winding sectors, while we have the right-moving bose-fermi cancellation in the odd winding sectors. This aspect is in sharp contrast with any SUSY vacua.
• Despite the absence of the space-time SUSY and adopting the Scherk-Schwarz type compactification, we are free from the tachyonic instability at any radius of the Scherk-Schwarz circle.
To conclude, we would like to make a few comments on possible future studies. First of all, it would indeed be an interesting issue whether our non-SUSY vacuum (3.9) has vanishing cosmological constant at higher loops. Since the orbifold structure of this vacuum is simpler than those of the previous ones quoted above, it would be worthwhile to examine especially the two-loop case by following the analysis in [28].
Secondly, in order to search a more broad class of such vacua, one may extend the construction in this paper to other toroidal models of asymmetric orbifolds. Furthermore, toward more realistic models, it would also be important to consider the non-geometric string vacua from SCFTs other than the toroidal ones. For previous attempts based on the N = 2 SCFTs, see e.g. [12]. A challenging direction in this respect, and along [15], would be to construct such vacua based on the generalized orbifolds through the topological interfaces, which are wrapped around the cycles of the world-sheet torus in correlation with the shift operators. 13 The point here would be how to organize the world-sheet chiral sectors depending on the winding numbers along the Scherk-Schwarz like circle, so that the bose-fermi cancellation does occur. We expect that the novel feature of the cancellation, which is remarked at the end of subsection 4.2, would be observed only in the non-geometric backgrounds.
Thirdly, one may also extend this work so as to include the open string sectors, namely, D-branes. Possibilities of the bose-fermi cancellation in the open string Hilbert space have been investigated [36] under particular SUSY breaking configurations of D-branes. Closely related studies of D-branes in asymmetric orbifolds by the T-duality twists have been presented e.g. in 13 For applications of the world-sheet conformal interfaces to string theory, see e.g. [44,45,46,47] Appendix A: Summary of Conventions and Useful Formulas Theta functions: Here, we have set q := e 2πiτ , y := e 2πiz ( ∀ τ ∈ H + , ∀ z ∈ C), and used abbreviations, θ i (τ ) ≡ θ i (τ, 0) (θ 1 (τ ) ≡ 0), Θ m,k (τ ) ≡ Θ m,k (τ, 0). It is straightforward to prove the following identities:
Θ 0,1 (τ ) η(τ ) = 2η(τ ) θ 2 (τ ) , Θ 1/2,1 (τ ) η(τ ) = η(τ ) θ 4 (τ ) , Θ 1/2,1 (τ ) η(τ ) = η(τ ) θ 3 (τ ) . (A.8)
Poisson resummation formula: (a ∈ 4Z + 2, b ∈ 4Z + 2).
(B.10)
or (B.4) without any changes in other sectors.
T 4 [
4SO(8)] and T 2 [SO(4)]×T 2 [SO(4)] are indeed realized as self-dual O(4, 4, Z) transformations which leave background geometries invariant. The elements of O(4, 4, Z) act as Z 2 transforma-
clarify the physical content of the non-SUSY vacuum with the bose-fermi cancellation (3.9), let us examine the massless spectrum in the untwisted sector (w = 0) that survives in the low energy physics. The massless states from the twisted sectors (w = 0) can appear only at the special radius R (see subsection 4.3).We first note the fact that all the right-moving Ramond vacua are projected out by the orbifold action g; recall σ 2 = (−1) F R for the world-sheet fermions. Therefore, the candidates of the bosonic and fermionic massless states only reside in the (NS, NS) and (R, NS)-sectors, respectively. It is thus enough to search the (NS, NS) and (R, NS) massless states invariant under the action of (−1) F L ⊗ σ within the Hilbert space of the unorbifolded theory.
Z
(NS,NS) w (τ,τ ) = Z (R,R) w (τ,τ ) = −Z (R,NS) w (τ,τ ) = −Z (NS,R) w (τ,τ ) = (4.7).(4.9)
12
In the T-fold vacuum (3.7), despite the existence of the space-time SUSY, we still find the wrong GSO fermions in the right-mover (with no SUSY), since Z (NS,NS) w = −Z (R,NS) w coincides with the partition function
(4. 4 )
4for w ∈ 4Z + 2. Of course, one can confirm the absence of tachyonic modes in this model by a similar argument given here.
can be examined in a parallel way, and we obtain almost the same spectra of the winding excitations. However, there is a slight difference for the sectors of w ∈ 2Z + 1 in the model fromF T 2 [SO(4)]×T 2 [SO(4)] ( * , * )
T 2 [
2SO(4)]×T 2 [SO(4)] ( * , * ) model, we find that both of (4.18) and (4.20) emerge as light excitations with w = ±1, which get massless at R = 1 √ 2 and R = 1 2 respectively.
( 1
1− q m )(1 − yq m )(1 − y −1 q m ), n q k(n+ m 2k ) 2 y k(n+ m 2k ) ,
1
1a,b) (τ, ǫ)| T ≡ f (a,b) (τ + 1, ǫ) = −e −2πi 1 6 f (a,a+b) (τ, ǫ). (B.8) (ii) For the case of σ 2 L = (−1) F L : f (a,b) (τ ) = f (a,b) (τ ), (a ∈ 2Z + 1 or b ∈ 2Z +
Table 1 :
1Action of the twist operators g n
Table 2 :
2Massless spectrum of the non-SUSY vacuum (3.9).
spin structure
left
right
4D fields
(NS, NS)
ψ µ
−1/2 |0
⊗ψ ν
−1/2 |0
graviton, 8 vectors,
(µ = 2, ..., 9)
(ν = 2, ..., 5) 14 (pseudo) scalars
Table 4 :
4Massless spectrum of the SUSY vacuum (3.7)
For the aspects of non-commutativity in non-geometric backgrounds, see e.g.[17,18,19,13,20] 2 For non-supersymmetric orbifolds in heterotic string theory, see e.g.[29,30,31,32] and references therein.3 In the papers[21,22,23], the authors further conjectured that the cosmological constant remains vanishing at two and higher loops. However, a careful world-sheet analysis[28] shows that it does not actually vanish at two loops in those models, at least pointwise on the moduli space.
Since f (a,b) (τ ) vanish, (2.24) may appear to be subtle. Hence, we present a more rigid interpretation of modular covariance in Appendix B.
See the discussions given in section 3.2 for the counting of unbroken supercharges in more detail.9 If following the notion of the original Scherk-Schwarz compactification, it would be better to introduceg ′ := T 2πR ⊗ (−1) F S ⊗ σ ≡ T 2πR ⊗ (−1) FL ⊗ σ(−1) FR ,instead of (3.8), where F S ≡ F L + F R is the space-time fermion number. However, the argument given here is almost unchanged even in that case, and especially, we end up with the same torus partition function(3.9).
In fact, the orbifolding by (−1) FL (or (−1) FR ) acts as the 'chirality flip' of the Ramond sector, which transfers the type IIA (IIB) vacua to the type IIB (IIA) ones similarly to T-duality. See e.g.[36].
Here, the '14 (pseudo) scalars' include the dilaton and the 4-dimensional axion field (dual of B µν ), which universally exist.
T 2 [SO(4)]×T 2 [SO(4)] (a,b)
AcknowledgmentsWe would like to thank K. Aoki for a useful conversation. This work is supported in part by JSPS KAKENHI Grant Number 24540248 and 23540322 from Japan Society for the Promotion of Science (JSPS).• w ∈ 2Z + 1 ;These analyses can be extended to other vacua built from Fin (B.4). In each case, we obtain the unitary q-expansion in a parallel way as above.We remark that the above results (4.8) and (4.9) suggest that there are supercharges both from the left and right movers for the SUSY T-fold(3.6). Similarly, (4.10), (4.11) and(4.12)are consistent with the existence of the chiral SUSY that originates only from the left-mover. Then, how about the non-SUSY vacuum (3.9)? We note that, for instance,for w ∈ 2Z. These relations of the bose-fermi cancellation look as if we had left-moving SUSY, in spite that no supercharges exist in the left-mover in fact. On the other hand, we findfor w ∈ 2Z + 1, which would appear to be consistent with right-moving SUSY. We emphasize that any supercharges can never be compatible with both (4.13) and (4.14) at the same time.It may be an interesting issue whether such a curious feature is common to the vacua showing the bose-fermi cancellation without SUSY.We also point out that the bose-fermi cancellation in (3.10) among different winding sectors does not happen (for arbitrary w ′ ), as is clear from the explicit forms of the partition functions presented above. Even at a special radius, the cancellation for arbitrary winding in (3.10) is not possible.Appendix B: Summary of Building BlocksIn Appendix B, we summarize the notations of relevant building blocks to construct the torus partition functions used in the main text.Bulidng Blocks for the Bosonic T 4 -secotor:1. Chiral reflection in T 4 [SO(8)]:
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| []
|
[
"Towards Applicative Relational Programming",
"Towards Applicative Relational Programming"
]
| [
"H Ibrahim ",
"M H Van Emden "
]
| []
| []
| Functional programming comes in two flavours: one where "functions are first-class citizens" (we call this applicative) and one which is based on equations (we call this declarative). In relational programming clauses play the role of equations. Hence Prolog is declarative. The purpose of this paper is to provide in relational programming a mathematical basis for the relational analog of applicative functional programming. We use the cylindric semantics of first-order logic due to Tarski and provide a new notation for the required cylinders that we call tables. We define the Table/Relation Algebra with operators sufficient to translate Horn clauses into algebraic form. We establish basic mathematical properties of these operators. We show how relations can be first-class citizens, and devise mechanisms for modularity, for local scoping of predicates, and for exporting/importing relations between programs.Applicative versus declarative definitionsSome functional programming systems are applicative; others declarative. Relational programming, on the other hand, only exists in declarative form. In this section we explain how we use this terminology, and argue that relational programming should also have both declarative and applicative forms. | null | [
"https://arxiv.org/pdf/cs/0602099v1.pdf"
]
| 1,739,811 | cs/0602099 | 576b658baf324787e5f28ea6983154e112497625 |
Towards Applicative Relational Programming
28 Feb 2006 16 March 1992
H Ibrahim
M H Van Emden
Towards Applicative Relational Programming
28 Feb 2006 16 March 1992
Functional programming comes in two flavours: one where "functions are first-class citizens" (we call this applicative) and one which is based on equations (we call this declarative). In relational programming clauses play the role of equations. Hence Prolog is declarative. The purpose of this paper is to provide in relational programming a mathematical basis for the relational analog of applicative functional programming. We use the cylindric semantics of first-order logic due to Tarski and provide a new notation for the required cylinders that we call tables. We define the Table/Relation Algebra with operators sufficient to translate Horn clauses into algebraic form. We establish basic mathematical properties of these operators. We show how relations can be first-class citizens, and devise mechanisms for modularity, for local scoping of predicates, and for exporting/importing relations between programs.Applicative versus declarative definitionsSome functional programming systems are applicative; others declarative. Relational programming, on the other hand, only exists in declarative form. In this section we explain how we use this terminology, and argue that relational programming should also have both declarative and applicative forms.
In functional programming
There are two ways for defining functions in functional programming, one using λ-expressions and the other using equations. Consider for example the higher-order function "twice". It can be denoted by the λ-expression "λf.λx.f (f x)"; we call this an applicative definition. This function can also be defined in a declarative style, that is, by asserting as true certain equations, as follows:
twice F = g(F) g(F) X = F (F X)
The characteristics of each style can be summarized in the first two columns of table 1. In the applicative style, functions can be results of functions, can be bound to variables, and so on. Hence they are much like other types of values. This is not the case in the declarative style. As a result, it is sometimes said that in the applicative style, functions are "first-class citizens." Furthermore, it can be shown that the two styles have complementary strengths, namely local scoping and modularity in the applicative one, and a natural expression of recursion and selection in the declarative one. Thus it is valuable for a programmer to have both available and to be able to switch effortlessly between the two. In functional programming, the theory of such a combination has been developed and practical applications have been reported [1,7] 1.2 In relational programming "Relational programming" is a natural counterpart for functional programming, with relations instead of function as basic entities. Logic programming, the best developed form of relational programming, is exclusively based on Horn clauses, which constitute a declarative paradigm. As the third column of table 1 suggests, a major problem in relational programming is the absence of an applicative language in which Characteristics − P aradigms f unctional relational implicit element at a time equations definite clauses declarative explicit whole function/relation λ − calculus ? applicative Table 1: Classifying functional and relational definitions.
relations can be first-class citizens. This paper develops the basis for such a language, keeping in mind the importance of it being first-order, and easily intertranslatable.
The meaning of Horn clauses
As basis for applicative relational programming, we use an algebraic view of the meaning of Horn clauses. The following example serves as introduction. Consider the clause for relational composition: p(X, Z) ← q(X, Y ), r(Y, Z). Relation p is defined by means of operations on q, r, and the tuples of variables. We make these operations explicit by appealing to cylindric set algebra [4,5]. Accordingly, a Horn clause can be interpreted as a relational inclusion where the right-hand side is a projection of the intersection of cylinders on the relations in the condition part. Consider, for example, the geometric interpretation of the clause as illustrated in figure 1. This can be expressed by a formula using the operators of T RA introduced in this paper. As a preview, we list the formula here: p ⊃ (X, Z)/(q : (X, Y ) ∩ r : (Y, Z)), where ":" is relational application yielding a cylinder, "∩" is intersection, and "/" is relational projection. To allows local scoping through λ-binding, we want to have p, q, and r as relational variables. This also allows minimization operators, such as µ [3], to apply as well. The result will be relation-valued expressions in a first-order setting.
Relational semantics for first-order logic.
Truth-functional semantics for first-order logic is an assignment of a truth value to a closed formula, where this assignment is relative to a given interpretation that assigns meanings to constants, function symbols, and predicate symbols. Relational semantics assigns a meaning to a formula that may have free variables; i.e. an open formula. It gives as meaning the relation consisting of the tuples of individuals that, if assigned to the free variables in the formula, would give a true closed formula according to truth-functional semantics.
For example, the meaning with respect to an interpretation I of the formula q(X, Y ) under relational semantics is the binary relation r = {(α, β)|q(α, β) is variable-free and true in I} Note that the meaning of a relation under relational semantics is invariant under renaming of free variables. As a result, relational semantics is not a homeomorphism between the algebra of formulas and the algebra of relations. For example, if it were a homeomorphism, then the meaning of q(X, Y ) ∧ q(Y, Z) would be the intersection of the meanings of the conjuncts. Hence it would be r ∩ r = r. However, it should be the composition of r with itself according to the algebra of binary relations.
Therefore we need something else: cylindric semantics, introduced in the following.
Tarski's cylinders. Tarski introduced a device that leads to a semantics for formulas that is a homeomorphism. Consider relations that are subsets of D 1 × · · · × D n (call this the domain product) and consider a relation b (call it the base) that is a subset of D j1 × · · · × D j k where the selector, J = {j 1 , . . . , j k } is a subset of {1, . . . , n}.
Then π −1 J (b) = {(x 1 , . . . , x n )|x i ∈ D i , i = 1, .
. . , n and (x j1 , . . . , x j k ) ∈ b} is the cylinder with base b, domain product D 1 × · · · × D n , and selector J. A cylinder, being a set of tuples of the same length, is always a relation. But a relation is not always a cylinder.
For example, take a domain product with n equal to 3, D 1 = D 2 = D 3 = {a, b, c}, with selector equal to {1, 2}, and with base r = {(a, b), (b, c), (c, a)}. Then the cylinder specified by these properties is
{ (a, b, a), (a, b, b), (a, b, c), (b, c, a), (b, c, b), (b, c, c),(1)
(c, a, a), (c, a, b), (c, a, c) } Rather than to assign to a formula as meaning a relation according to relational semantics, Tarski assigns a cylinder on this relation. He assumes an enumeration of all variables in the language. This gives a set J = {j 1 , . . . , j k } of integers for a formula with k free variables. The semantics according to Tarski then assigns to the formula as meaning the cylinder determined by J as selector on the base that is the relation obtained by relational semantics. Such a semantics is a homeomorphism where conjunction corresponds to intersection. Other syntactic constructs correspond to other operations in what Tarski calls "cylindric set algebra."
To return to our example, suppose X, Y , and Z are the only variables in the language (hence n = 3), and are enumerated in this order. Then the free variables of q(X, Y ) correspond to the selector J = {1, 2}. According to Tarski, the meaning of q(X, Y ) is then cylinder 1. The free variables of q(Y, Z) correspond to the selector J = {2, 3}. According to cylindric semantics, the meaning of q(Y, Z) is then
{ (a, a, b), (b, a, b), (c, a, b), (a, b, c), (b, b, c), (c, b, c), (2) (a, c, a), (b, c, a), (c, c, a) }
The meaning of q(X, Y )∧q(Y, Z) is the intersection of these two cylinders, that is, {(a, b, c), (b, c, a), (c, a, b)}. According to Tarski, the meaning of ∃Y.q(X, Y ) ∧ q(Y, Z) is obtained by projecting this with selector
J = {1, 3}. The result is {(a, c), (b, a), (c, b)}, which is the relational composition of r with itself.
In this paper we define an algebra operating on cylinders and relations that allows us to translate to algebraic form the definite clauses of Prolog. We call it TRA, for Table/Relation Algebra. The current example serves to give a preview of this translation. Let a relational composition be defined by the clause
p(X, Z) ← q(X, Y ), q(Y, Z).
In classical syntax this is
(∃Y.q(X, Y ) ∧ q(Y, Z)) ⊃ p(X, Z).
As we just observed, the condition has as meaning the projection on (X,Z) of the intersection of the cylinders denoted by q(X, Y ) and q(Y, Z). In general, we shall see that a definite clause asserts, interpreted in terms of TRA, that the meaning of the conclusion includes the projection (determined by the variables in the conclusion) of the intersection of the cylinders that are the meanings of the conditions.
Cylinders defined as tables.
It is rare to be able to list, as a set of tuples, a cylinder that is a meaning of a formula; the language usually has infinitely many variables, so that the tuples are very long. Moreover, there are often infinitely many tuples in the cylinder. But there is no problem specifying such a cylinder, as we only need to specify the base relation and the selector (the domain product is usually implicitly understood). For example, cylinder 1 can be specified by the set of substitutions
{{X = a, Y = b}, {X = b, Y = c}, {X = c, Y = a}},
as this specifies that the base relation of the cylinder is {(a, b), (b, c), (c, a)} and that the selector is {1, 2}, which corresponds to the variables X and Y in the enumeration. Sets of substitutions of this type, where the substituted variables in each are the same, play a central role in this paper. Observe, in the first place, that a table is a natural notation for such a set of substitutions. The right-hand sides make up the lines of the table, while the left-hand sides need not be repeated, hence can be the headings of the table's columns; see table 2. Because of this, we call such sets of substitutions "tables." In the second place, observe that the set of answers to a Prolog query is a table. For example, suppose that the logic program P is the set of facts {q(a, b), q(b, c), q(c, a)}. The query ?-q(X, Y ) produces table 2 as set of answer substitutions.
X Y a b b c c a
The Table/Relation Algebra (TRA)
TRA is an algebra of operations on relations, tables, queries, and logic programs. It facilitates translation to and from definite clause form, and also facilitates scoping and modularity.
We define an n-ary relation over a Herbrand universe H to be a set of n-tuples of elements of H. Prolog's answer substitutions for the set of variables in a query can be regarded as a set of equations in solved form; i.e. all left-hand sides are variables that occur only there [6]. Restricting the solved forms to have a common set of variables as left-hand sides makes it natural to present a set of answer substitutions as a table. Hence we define an n-ary table to be a set of sets of equations in solved form, where each set of equations in the table has the same set of n variables as left-hand sides.
The reason for calling the concept just defined "table" is that these sets of equations can be most economically represented in print as a table where the left-hand sides are the headings of the columns (analogous to the attributes of relational data models) and the right-hand sides are the entries of the table. A difference with the relational data model is that in our concept of table the entries can be terms of any complexity containing variables without any restriction.
Tables as queries on logic programs.
Tables, as defined above, can be obtained, in logic programming, as follows:
Definition 1 Let P be a logic program, let Q be a query, and let T be an SLD-tree for P and Q. Then the expression (Q where P ) has as value the table of the answer substitutions associated with all the success leaves of T .
When in Prolog a query Q fails for a logic program P , there are no answer substitutions and the table (Q where P ) is empty. We use the symbol ⊥ (bottom) for all of the empty tables {}. We use another special symbol ⊤ (top) for the table consisting only of the empty answer substitution {{}}, which results from a successful SLD-derivation starting in a query with no variables.
The following lemmas follow from the definition of "where".
Lemma 1 θ ∈ (Q where P ) implies that θ is a correct answer substitution for {Q} ∪ P.
Lemma 2
For every correct answer substitution θ for {Q} ∪ P , ∃ η ∈ (Q where P ) such that θ is an instance (with respect to the Herbrand universe) of η.
Lemma 3
The value of (Q where P ) does not depend on the SLD-tree T in definition 1.
Intersection of tables
Tables are a convenient notation for cylinders. As the intersection of cylinders is important, we need to define intersection between tables. Let θ 1 and θ 2 be answer substitutions to the queries G 1 and G 2 respectively, and consider the query ?-G 1 , G 2 . When G 1 and G 2 have a common variable, θ 1 ∪ θ 2 is not in solved form and may not be solvable. However, as Colmerauer [2] observed, the solved form of θ 1 ∪ θ 2 , if it exists, is an answer substitution for the query ?-G 1 , G 2 . Hence:
Definition 2 The intersection operation of tables S and T is
S ∩ T def = {φ(s ∪ t) | s ∈ S, t ∈ T, and ψ(s ∪ t)}.
Here ψ(s) means that its argument set of equations is solvable; φ(s) is only defined when s is solvable and then it denotes the solved form of s. For example, if relation q, as defined in a program P , is The reason for choosing the name "intersection" and the symbol "∩" is given by the following
{(a, b), (b, c), (c, d), (d, e)}, then the result of the query ((← q(X, Y )) where P ) ∩ ((← q(Y, Z)) where P ) is table 3. X Y Z a b c b c d c d e
Theorem 1 For all tables S and T, S ∩ T is the set intersection of S and T regarded as cylinders (hence relations, hence sets).
It is easy to verify the following properties about the intersection of tables:
Theorem 2 ∩ is associative and commutative. ∩ has a unique null element, which is ⊥ and a unique unit element, which is ⊤.
Theorem 3 For any program P and goal statements G 1 , G 2 and G which consists of all goals in G 1 and
G 2 , we have (G 1 where P ) ∩ (G 2 where P ) = (G where P ).
Modularity The previous theorem suggests using the combination of where and ∩ as a modular compositional tool, as in the following example:
((← F 1 , F 2 ) where P ) ∩ ((← G) where Q).
Thus we see that within the same table it can be specified of each goal with respect to which program it is defined.
From relations to tables
We often need to get a table out of a relation, rather than from a program by posing a query. For this we define the relational application operator ":". We introduce its definition through a heuristic development.
Let r be the n-ary relation {(a 1 , . . . , a n ), (b 1 , . . . , b n ), (c 1 , . . . , c n )}. A table "most like" r can be easily constructed by adding as a heading a set of n distinct variables, say X 1 , . . . , X n , as in table 4. This table X 1 · · · X n a 1 · · · a n b 1 · · · b n c 1 · · · c n can then be identified using an arbitrary predicate symbol, say p, as follows:
(← p(X 1 , . . . , X n )) where {p(e 1 , . . . , e n ) | (e 1 , . . . , e n ) ∈ r}.
The obvious generalization of allowing any terms t 1 , . . . , t n instead of the distinct variables suggests Though correct, its arbitrary and auxiliary predicate p is undesirable. It is easily verifiable that this definition is equivalent, operationally, to:
From tables to relations
Just as we defined the application operator to get tables from relations, we define a projection operation to get relations back from tables. This operation should not merely discard the table's "headings" 2 , as suggested in [9]. Instead, we present another heuristic development to define it properly. Let T be the previous table 4. Now, {(X 1 θ, . . . , X n θ) | θ ∈ T } is the relation resulting from discarding the heading; i.e. r above. To generalize, we first let {j 1 , . . . , j k } be a subset of {1, . . . , n}. Then, {(X j1 θ, . . . , X j k θ) | θ ∈ T } is a projection of r over the (j 1 , . . . , j k ) columns. Hence, projection is a way of getting a relation from a table. Secondly, allow any tuple (t 1 , . . . , t k ) of terms instead of the variables (X j1 , . . . , X j k ) and consider {(t 1 θ, . . . , t k θ) | θ ∈ T }. Since for an arbitrary table T any θ ∈ T may contain variables, the result must further be grounded in order to obtain a relation. Hence the final definition:
Definition 4 The projection operation (denoted /) of a table T over a tuple of terms
(t 1 , . . . , t n ) is (t 1 , . . . , t n )/T def = ξ({(t 1 θ, . . . , t n θ) | θ ∈ T }),
where ξ(x) is the set of variable-free instances of the expression x.
Are project and apply inverses?
Now that we have operations from tables to relations and vice versa, one may wonder whether these are each other's inverses. The short answer is, in general, "no", because ((t 1 , . . . , t n )/T ) : (t 1 , . . . , t n )
is not always the table T . Take, for example, the case that t 1 , . . . , t n have no variables. Then the above expression is ⊤ whenever T is not ⊥. But the absence of variables in t 1 , . . . , t n is a rather pathological case. When we add restrictions, we can say that, in a sense, "/" and ":" are each other's inverses, as shown by the following theorems.
Translation of definite clauses to TRA
It should be clear now that the operators of TRA correspond closely to the operations hidden in definite Horn clauses, according to cylindric semantics. This simplifies the translation of definite clauses to TRA, and we therefore omit an explicit description of it. The principle of the translation is that each definite clause states that the relation denoted by the predicate symbol in the conclusion includes the projection (on the tuple of the terms in the conclusion) of the intersection of the tables denoted by the conditions. The operations of TRA are general enough to translate the definite clauses of pure Prolog.
Example. We illustrate the translation by an example that is a typical Prolog program. It gives a quicksort program from an ordinary list to a difference list.
The Prolog program is: qsort is a relational variable and is not part of the language of clausal logic. But all terms, and the atomic formula that is the first argument of where, are in clausal logic.
The entire expression states an inclusion between relations and may or may not be satisfied, depending on the value of qsort. It may be shown that there is a least relation as value for qsort that satisfies the inclusion. The inclusion serves as definition of this least relation.
The applicative relational program has the following advantages:
1. The where expression has a table as value and can be replaced by any other expression with the same value; see item 3. In particular, the identifier partition is local to the where expression. The declarative Horn clause formalism does not provide locality for predicate names.
2. The identifier qsort is a variable and is not a predicate symbol. Hence a minimization operator, like µ of [3], can be applied to the entire inclusion with respect to qsort. The result is a relation-valued expression that can, for example, be the operator argument of the relational application ":". The identifier qsort is then local to this operator. In this way any number of levels of locality can be built, just as in applicative functional programming.
3. Combining a selection/minimization operator with the functional λ-abstraction and application operators yields an import/export facility for relations between program modules. As an example, replace P above with (λorder.P )(νleq.Orderings),
where order is a relational variable in P used to define the order of partitioning, Orderings is a program that defines different orders, and "ν" is the minimization operator that selects one of the relations (leq) defined in a program module (Orderings). The effect of all this is exporting the leq relation from program Orderings, and importing it into program P as the order of partitioning, yielding an ascending, or descending, qsort relation.
Related work
This paper is a continuation of a line of research going back as far as the work of Peirce and of Schroeder in the 19th century on algebra of relations. These algebras were not adequate to serve as basis for a semantics for full first-order predicate logic. In the 1930's Tarski provided cylindric set algebra, which remedied this shortcoming. The simplicity of definite clauses, although covered by Tarski's work, suggest the independent treatment given in this paper. After Tarski's work, the next most important step was the paper by de Bakker and de Roever [3]. By restricting themselves to binary relations, they took their starting point before Tarski. They elucidated the mechanisms of defining binary relations, especially the use of the minimization operator. TRA facilitates extending of the definition techniques of de Bakker and de Roever to include the type of algebra introduced by Tarski.
There are some relations to recent work that do not seem to be part of a grand design. Codd's relational calculus is declarative in spirit, whereas his relational algebra is applicative in spirit. Both seem to us to be improved upon by the earlier work by Tarski. However, TRA can be viewed as a relational algebra counterpart for the Datalog query language, when substituted for Codd's relational calculus. Our where expressions were inspired by Nait-Abdallah's ions. The main difference is that Nait-Abdallah's work is syntactic, concentrating on rewriting rules.
Our work is also related to higher-order logic programming by having the same objectives, but a different approach; see the concluding remarks. Finally, applicative programming constitutes important related work [8], as mentioned in the introduction.
Conclusions
We have characterized distinctions between the applicative and declarative paradigms in functional programming. We have noted that relational programming exists only in declarative form. We have developed a relational algebra (TRA) useful as a mathematical basis for applicative relational programming. The operations of TRA are chosen in such a way that the definite clauses of Prolog have a simple translation.
Relations can be first-class citizens. One of the most powerful paradigms in computing is "functions as first-class citizens" [8]. This in reaction to languages where functions are more restricted in their use than, say, numbers. Certain functional programming languages have demonstrated the advantages of having functions as first-class citizens. Prolog, the only relational programming language, does not have relation-valued expressions of any kind. We have shown that relations can be first-class citizens.
Modular logic programming. We have shown how to support modularity and local scoping for relations, which is a deficiency in declarative logic programming. We have devised a mechanism that facilitates the exportation and importation of relations between programs.
Higher-order logic is not needed, at least not to provide the desirable programming features implied by having functions and relations as first-class citizens. In functional programming, a function is said to be higher-order when it takes a function as argument or produces one as result. A logic is said to be higherorder when one can quantify over function or predicate symbols. These two senses of "higher-order" do not coincide. This is proved by the existence of formalizations of λ-calculus in first-order logic. A constant of logic can denote any individual, including a function. Therefore, a first-order variable, one ranging over individuals, can range over functions. In that sense, our expressions contain first-order variables that range over relations. This is an important point, as first-order logic is more tractable, theoretically and practically, than higher-order logic.
Figure 1 :
1A geometric interpretation of relational composition: p contains the projection of the intersection of the cylinders on q and r.
For the benefit of further examples, suppose that the universe of discourse is {a, b, c} 1 and that r= {(a, b), (b, c), (c, a)}.
r
: (t 1 , . . . , t n ) def = (← p(t 1 , . . . , t n )) where {p(e 1 ,.. . , e n ) | (e 1 , . . . , e n ) ∈ r}.
Definition 3
3The application operation of an n-ary relation r to a tuple (t 1 , . . . , t n ) of terms is r : (t 1 , . . . , t n ) def = {φ({t 1 = e 1 , . . . , t n = e n }) | (e 1 , . . . , e n ) ∈ r and ψ({t 1 = e 1 , . . . , t n = e n })}.
Theorem 4
4For all tables T and all terms t 1 , . . . , t n in which all the variables, and no other ones, in T 's heading occur, we have ((t 1 , . . . , t n )/T ) : (t 1 , . . . , t n ) = ξ(T ). For an inverse in the other direction, compare the n-ary relation r with (t 1 , . . . , t n )/(r : (t 1 , . . . , t n )). That this expression does not always equal r is shown by (c, d)/({(a, b)} : (c, d)) = {}, where a, b, c and d are constants. This example suggests: Theorem 5 For all n-ary relations r and all terms t 1 , . . . , t n , we have (t 1 , . . . , t n )/(r : (t 1 , . . . , t n )) ⊆ r.However, by strengthening the restrictions, we can have equality instead of inclusion.Theorem 6 For all n-ary relations r and all distinct variables x 1 , . . . , x n , we have (x 1 , . . . , x n )/(r : (x 1 , . . . , x n )) = r.
qsort([ ],U-U). qsort([X|Xs],U-W) :partition(X,Xs,Y1,Y2),qsort(Y1,U-[X|V]),qsort(Y2,V-W). The TRA version is: qsort ⊇ (([ ], U − U )/⊤) ∪ (([X|Xs], U − W ) / (((← partition(X, Xs, Y 1, Y 2)) where P ) ∩(qsort : (Y 1, U − [X|V ])) ∩(qsort : (Y 2, V − W )))).
Table 2 :
2A table for cylinder 1.
Table 3 :
3A table from intersection of two tables.
Table 4 :
4A table "most like" r.
The above footnote applies here also.
One reason is, as mentioned before, that our table's contents include variables, whereas the elements of a relation's tuples are variable-free.
AcknowledgmentsThanks to J.H.M. Lee for his careful reading of an earlier version of this work. Generous support was provided by the British Columbia Advanced Systems Institute, the Institute of Robotics and Intelligent Systems, the Canadian Institute for Advanced Research, the Laboratory for Automation, Communication and Information Systems Research, and the Natural Science and Engineering Research Council of Canada.
Lambda-equational Logic Programming. M H M Cheng, University of WaterlooPhD thesisM.H.M. Cheng. Lambda-equational Logic Programming. PhD thesis, University of Waterloo, 1987.
Prolog and infinite trees. Alain Colmerauer, Logic Programming. Academic PressAlain Colmerauer. Prolog and infinite trees. In Logic Programming, pages 231-251. Academic Press, 1982.
A calculus for recursive program schemes. J W De Bakker, W P De Roever, Automata, Languages, and Programming. M. NivatJ.W. de Bakker and W.P. de Roever. A calculus for recursive program schemes. In M. Nivat, editor, Automata, Languages, and Programming, 1973.
Cylindric Set Algebras. L Henkin, J D Monk, A Tarski, H Andréka, I Németi, Springer Lecture Notes in Mathematics. 883Springer-VerlagL. Henkin, J. D. Monk, A. Tarski, H. Andréka, and I. Németi. Cylindric Set Algebras, volume 883 of Springer Lecture Notes in Mathematics. Springer-Verlag, 1981.
Cylindric Algebras, Parts I,II. Leon Henkin, J Donald Monk, Alfred Tarski, Studies in Logic and the Foundations of Mathematics. North-HollandLeon Henkin, J. Donald Monk, and Alfred Tarski. Cylindric Algebras, Parts I,II. Studies in Logic and the Foundations of Mathematics. North-Holland, 1985.
An efficient unification algorithm. Alberto Martelli, Ugo Montanari, ACM Transactions of Programming Languages and Systems. 4Alberto Martelli and Ugo Montanari. An efficient unification algorithm. ACM Transactions of Pro- gramming Languages and Systems, 4:258-282, 1982.
Contributions to functional programming in logic. B E Richards, University of VictoriaMaster's thesisB.E. Richards. Contributions to functional programming in logic. Master's thesis, University of Victoria, 1990.
Denotational Semantics: The Scott-Strachey approach to Programming Language Theory. Joseph E Stoy, MIT PressJoseph E. Stoy. Denotational Semantics: The Scott-Strachey approach to Programming Language The- ory. MIT Press, 1977.
Principles of Database and Knowledge-Base Systems. Jeffrey D Ullman, Computer Science PressJeffrey D. Ullman. Principles of Database and Knowledge-Base Systems. Computer Science Press, 1988.
| []
|
[
"Root Systems and Purely Elastic S-Matrices II",
"Root Systems and Purely Elastic S-Matrices II"
]
| [
"Patrick Dorey [email protected] \nService de Physique Théorique de Saclay\n\n\n91191Gif-sur-Yvette cedexFrance\n\nLaboratoire de la Direction des Sciences de la Matière du Commissariatà l'Energie Atomique\n\n"
]
| [
"Service de Physique Théorique de Saclay\n",
"91191Gif-sur-Yvette cedexFrance",
"Laboratoire de la Direction des Sciences de la Matière du Commissariatà l'Energie Atomique\n"
]
| []
| Starting from a recently-proposed general formula, various properties of the ADE series of purely elastic S-matrices are rederived in a universal way. In particular, the relationship between the pole structure and the bootstrap equations is shown to follow from properties of root systems. The discussion leads to a formula for the signs of the three-point couplings in the simply-laced affine Toda theories, and a simple proof of a result due to Klassen and Melzer of relevance to Thermodynamic Bethe Ansatz calculations.August 1991 † | 10.1016/0550-3213(92)90407-3 | [
"https://arxiv.org/pdf/hep-th/9110058v1.pdf"
]
| 14,663,708 | hep-th/9110058 | 93ed18d16846e4dcfbfcf0738769cfd7921de409 |
Root Systems and Purely Elastic S-Matrices II
Patrick Dorey [email protected]
Service de Physique Théorique de Saclay
91191Gif-sur-Yvette cedexFrance
Laboratoire de la Direction des Sciences de la Matière du Commissariatà l'Energie Atomique
Root Systems and Purely Elastic S-Matrices II
arXiv:hep-th/9110058v1
Starting from a recently-proposed general formula, various properties of the ADE series of purely elastic S-matrices are rederived in a universal way. In particular, the relationship between the pole structure and the bootstrap equations is shown to follow from properties of root systems. The discussion leads to a formula for the signs of the three-point couplings in the simply-laced affine Toda theories, and a simple proof of a result due to Klassen and Melzer of relevance to Thermodynamic Bethe Ansatz calculations.August 1991 †
Introduction
Occasionally, an integrable perturbation of a conformal field theory results in a massive scattering theory which is purely elastic, in that the S-matrix is diagonal in a suitable basis [1]. This observation has lead to some work on such S-matrices as interesting objects in their own right. A series of examples connected with the ADE series of Lie algebras has been uncovered, both directly in the context of perturbed conformal field theory [2], and also via the study of affine Toda field theories [3][4][5][6][7][8][9] (in fact, there are slight differences between the S-matrices found in the two contexts; these will be mentioned where relevant).
As is often the case in the study of integrable quantum field theories, the proposed Smatrices have not (at least so far) been derived from first principles, but rather deduced on the basis of certain assumptions and consistency requirements. However, for a theory with a diagonal S-matrix, the Yang-Baxter equation -often a very powerful tool in the study of the S-matrices of integrable theories [10] -is trivially satisfied, and so gives no information. There does remain the possibility that two particles in the theory may fuse to form a third as a bound state [11]. As emphasised by Zamolodchikov [1], for purely elastic scattering the resulting bootstrap equations are sufficiently simple to provide a useful set of consistency conditions, constraining both the conserved charges and the S-matrix.
The nature of the bootstrap solutions for the ADE theories is in fact closely linked to properties of the corresponding root systems, and in particular the action on these root systems of the Coxeter element of the Weyl group [12]. The aim of this paper is to explore this connection a little further, with particular emphasis on the implications for the structure of the S-matrix elements. By refining the notations used in [12], it turns out to be possible to streamline the discussion considerably. After a description of some necessary formulae, section two outlines how this goes. While this section contains no new results, it does give a proof of the S-matrix bootstrap equations which simplifies and clarifies that given previously. Section three is devoted to a discussion of the pole structure of S-matrix, and shows how this is exactly in accordance with the bound-state structure predicted by the fusings. Various empirically observed features of the purely elastic S-matrices turn out to be simple consequences of the properties of root systems. An application for some of the calculations in section three is given in section four, giving a description of one set of signs for the three-point couplings of the simply-laced affine Toda theories, in terms of roots and weights. Finally, section five gives a universal proof of an elegant formula due to Klassen and Melzer [9], and section six contains some concluding remarks.
Preliminaries
Since this paper is a direct sequel to [12], the reader is referred back to that paper for details of the motivations and many of the original formulae. As mentioned in the introduction, elements of the discussion given there become rather clearer if the notation (in particular, the labelling of the orbits of the Coxeter element), is changed slightly. For this, some results to be found in a paper by Kostant [13] will be needed, and this section starts with a brief review of the relevant material.
First though, note that various of the definitions and results to be given in this section work equally well for simply-laced and non simply-laced root systems. For example, the discussion in [13] makes no distinction between the two cases. It is also worth noting that the discussion in [12] of the conserved charge bootstrap goes through essentially unchanged for the non simply-laced root systems. However, the S-matrix formulae given in that paper seem to be hard to generalise beyond the ADE series, possibly reflecting the difficulties that were found in the (quantum) problem of finding S-matrices for the non simply-laced affine Toda theories [4,7]. For this paper, then, attention will be restricted to the alreadyknown purely elastic scattering theories, that is to those associated with the simply-laced Lie algebras. The discussion will be based on a simply-laced root system Φ, of rank r, with {α i } a set of simple roots. Letting w i denote the Weyl reflection corresponding to
the simple root α i (so w i (x) = x − 2 α 2 i (α i , x)α i ), set w = w 1 w 2 . . . w r ,
so that w is a Coxeter element. Also, let w be the subgroup of W , the Weyl group, generated by w. Finally, for i = 1 . . . r define a root φ i by
φ i = w r w r−1 . . . w i+1 (γ i ). (2.1)
Then the following results are given in [13]:
(i) With the definition of positive and negative roots implied by the given choice of simple roots, φ i > 0 and w(φ i ) < 0.
(ii) If α is a root such that α > 0 and w(α) < 0, then α is one of the φ i .
(iii) Let Γ i be the orbit of φ i under w . Then the Γ i 's are disjoint, each has h elements, and thus their union is all of Φ.
The set {φ i } possesses one further useful property, which can be found for example in [14].
If λ i is the fundamental weight corresponding to the simple root α i , then
φ i = (1 − w −1 )λ i . (2.2)
(To prove this result, note that w i (λ j ) = λ j − δ ij α j , which follows since the fundamental weights are dual to the simple co-roots α ∨ i ≡ 2
α 2 i α i .
For the simply-laced cases of interest here, α 2 i = 2 and the λ i are dual to the simple roots themselves.) This relation can be inverted, w having no eigenvalue equal to one. Writing
(1 − w −1 ) −1 = R, (2.3)
it is easily checked that
R = 1 h h 1 pw p = − 1 h h−1 p=0 pw −p . (2.4)
This mapping is not orthogonal, but rather satisfies
(Rα, β) + (α, Rβ) = (α, β). (2.5)
Other identities, such as (Rα, β) = −(wα, Rβ), can also be found but only (2.5) will be used below. Projectors onto the various eigenspaces of w are given by
P s = 1 h h−1 p=0 ω −sp w p = 1 h h−1 p=0 ω sp w −p ,(2.6)
with corresponding eigenvalues ω s = e 2πis/h . If s is not an exponent of the algebra, then the spin s eigenspace is null, and P s = 0.
It is often useful to focus on a particular ordering of the simple roots, linked to a twocolouring of the Dynkin diagram. This ordering is such that {α i } splits into two subsets, each of which contains only mutually orthogonal roots: (α i , α j ) = 0 if i and j have the same colour. By a small abuse of notation, the symbols •, • ′ , •, • ′ and so on will occasionally be used to denote arbitrary indices taken from the corresponding (black or white) subset. A slightly different notation was used in [12]: black simple roots were called 'type alpha', white ones 'type beta'.
{α 1 , α 2 , . . . α r } = {α 1 , . . . α k } ∪ {α k+1 ,
In this ordering,
w = w {•} w {•} with w {•} = i∈• w i , w {•} = j∈• w j .
The internal orthogonality of the two subsets of the simple roots implies that the reflections for simple roots of the same colour commute, and that a reflection for a given simple root will leave invariant all other simple roots of the same colour. It follows that
φ • = w {•} (α • ), φ • = α • . (2.8)
In [12], the coset representatives were α • and −α • ; since both w {•} and −1 induce charge conjugation on the cosets, this means that, strictly speaking, the assignment of cosets to simple roots induced by the φ i differs by an overall charge conjugation from that used in the earlier paper. However this is merely a matter of labelling convention -for example, one can interchange the values of the conserved charges on particle and antiparticle simply by negating the normalisations of the even spin charges -and so can be ignored.
The ADE S-matrices will be built as products of functional 'building blocks'. In [12], the blocks used were
{x} + = x − 1 + x + 1 + (perturbed conformal) x − 1 + x + 1 + x − 1 + B + x + 1 − B + (affine Toda) (2.9) where x + = sinh θ 2 + iπx 2h (2.10)
and B(β) = 1 2π
β 2 1 + β 2 /4π . (2.11)
The block for an affine Toda theory contains an extra coupling-constant dependent part over the 'minimal' version suitable for perturbed conformal theories. This turns out to have no effect on the physical pole structure (β being real, and B(β) therefore between 0 and 2), and so it is reasonable to use the same notation for both cases. In fact, the discussion of pole structure will be a little more transparent if this block is swapped for another, namely
{x} − = −{−x} −1 + (perturbed conformal) {−x} −1 + (affine Toda) (2.12)
Of course, any formula involving {x} − can be immediately rewritten in terms of {x} + .
One further piece of notation will be needed. For any pair of roots α, β ∈ Φ, an integer u(α, β) can be defined modulo 2h via the following relations:
u(α, β) = −u(β, α) u(wα, β) = u(α, β) + 2 u(φ • , φ • ′ ) = u(φ • , φ • ′ ) = 0 u(φ • , φ • ) = 1. (2.13)
For the coset representatives φ i , the abbreviated notation u ij ≡ u(φ i , φ j ) will often be used.
The definition is natural in that πs h u(α, β) is the (signed) angle between the projections of the roots α and β into the ω s eigenspace of w. Now the three-point couplings are described by the following fusing rule [12] (also relevant in other contexts [15]):
C ijk = 0 iff ∃ roots α (i) ∈ Γ i , α (j) ∈ Γ j , α (k) ∈ Γ k with α (i) + α (j) + α (k) = 0. (2.14)
(Note the use here of a convention that will be adhered to for the rest of this paper: α (i) , for example, is used for any root that lies in G i , the orbit of the root φ i . It is important not to confuse this with the simple root α i -in particular, even though the labelling of the orbits ultimately derives from the choice of simple roots, via (2.1), there are cases where the simple root α i does not lie in the orbit Γ i .) Since the fusing angles U k ij are the relative angles of projections into the ω 1 eigenspace, they are related to the u(α (i) , α (j) ) by
U k ij = π h |u(α (i) , α (j) )|, where α (i) + α (j) ∈ Γk. (2.15)
Being signed angles, the u(α, β)'s also satisfy u(α, β) + u(β, γ) + u(γ, α) = 0 mod 2h (2. 16) for (any) three roots α, β and γ.
Armed with these conventions, the expression given in [12] for the two-particle Smatrix can be rewritten in a compact way:
S ij = h−1 p=0 {2p + 1 + u ij } (λ i ,w −p φ j ) ± . (2.17)
Note, the expression is identical in form whether it is written in terms of {x} + or {x} − . This is equivalent to unitarity (S ij (θ)S ij (−θ) = 1) and follows from
(λ i , w −p φ j ) = −(λ i , w p+1+u ij φ j ). (2.18a)
Symmetry (S ij = S ji ) can also be checked, using
(λ i , w −p φ j ) = (λ j , w −p−u ij φ i ). (2.18b) (A 'mixed' equality, (λ i , w −p φ j ) = −(λ j , w p+1 φ i ), follows directly from (2.
2), independently of the special root ordering (2.7).) Equivalent formulae were given in [12], but in a less compact way.
Equation (2.17) can be put into a perhaps more suggestive form by noting from (2.13)
that 2p + u ij is just u(φ i , w −p φ j ). Hence S ij = α (j) ∈Γ j {u(φ i , α (j) ) + 1} (λ i ,α (j) ) ± . (2.19)
The unitarity and symmetry of this formula can be checked directly by rewriting (2.18)
as (λ i , α (j) ) = −(λ i , w u(φ i ,α (j) )+1 α (j) ), (2.20a) (λ i , α (j) ) = (λ j , w (u(φ j ,α (i) )−u(φ i ,α (j) ))/2 α (i) ). (2.20b)
Note, the exponent of w in (2.20b) is an integer, since u(φ j , α (i) ) and u(φ i , α (j) ) are always either both even or both odd. The important case for the symmetry of (2
.19) is (λ i , α (j) ) = (λ j , α (i) ) if u(φ j , α (i) ) = u(φ i , α (j) ).
The S-matrix bootstrap equations [1] can be checked very simply from (2.19). First these equations are rewritten, for each particle species l and each nonvanishing three-point
coupling C ijk , as S li (θ)S lj (θ + iU k ij )S lk (θ − iU j ik ) = 1. (2.21)
(This alternative bootstrap equation, obtained from the more usual one via the equations of unitarity (S ij (θ)S ij (−θ) = 1) and crossing (S ij (θ) = S i (iπ − θ)), is analogous to the symmetrical version of the conserved charge bootstrap equation used in [12].) The structure of this equation is clarified if a shift operator T y is introduced, defined by
T y f (θ) = f (θ + iπy h ) (2.22)
and acting on the blocks as
T y {x} ± = {x ± y} ± . (2.23)
Recalling from (2.14) that C ijk = 0 implies the existence of a root triangle
{α (i) , α (j) , α (k) },
the relation (2.15) can be used to write equation (2.21) as
S li T u(α (i) ,α (j) ) S lj T u(α (i) ,α (k) ) S lk = 1. (2.24)
Note how the fact that u(α, β) is a signed angle takes care of the relative minus sign between the two shifts in the earlier equation, (2.21). Of course, depending on the orientation of the projection of the root triangle into the s = 1 subspace, there could be an overall negation of the shifts in (2.24) compared to (2.21). In fact, root triangles projecting to both orientations always exist -this will be discussed in more detail in section four -but this is not a problem since (2.21) also holds with the labels j and k exchanged, an operation which itself has the effect of negating the two shifts.
Finally, acting on both sides with T u(φ l ,α (i) ) and using (2.16) gives
T u(φ l ,α (i) ) S li T u(φ l ,α (j) ) S lj T u(φ l ,α (k) ) S lk = 1 (α (i) + α (j) + α (k) = 0). (2.25)
To verify that the bootstrap equations in this form are satisfied by (2.19) is almost immediate. The left hand side of (2.25) involves three roots running through the orbits Γ i , Γ j and Γ k for the S-matrix elements S li , S lj and S lk respectively. If these orbits are labelled starting at α (i) , α (j) and α (k) (instead of the roots φ i , φ j and φ k that would have been used had (2.17) been the starting point), then via (2.23) the left hand side is
h−1 p=0 {u(φ l , w −p α (i) ) ± u(φ l , α (i) ) + 1} (λ l ,w −p α (i) ) ± × {u(φ l , w −p α (j) ) ± u(φ l , α (j) ) + 1} (λ l ,w −p α (j) ) ± × {u(φ l , w −p α (k) ) ± u(φ l , α (k) ) + 1} (λ l ,w −p α (k) ) ± .
That this is equal to one is apparent if the {x} − blocks have been used ({x} + being appropriate for the equivalent version of (2.25) with all shifts negated). From (2.13), for each value of p the three blocks involved are then equal to the same function, namely
{2p + 1} − . The total power to which this is raised is λ i , w −p (α (i) + α (j) + α (k) ) . But this is zero, since α (i) , α (j) and α (k) form a root triangle for the coupling C ijk . Hence the whole expression is equal to one, as required.
Section five will give a general proof of a result for the minimal scattering theories due to Klassen and Melzer, for which it will be helpful to have an expression for the minimal S-matrix elements in terms of the unitary blocks
x = x + / −x + .(2.S ij = h−1 p=0 2p + u ij (λ i ,w −p φ j ) = α (j) ∈Γ j u(φ i , α (j) ) (λ i ,α (j) ) . (2.27)
Pole structure
This section will involve a detailed examination of the physical pole structure of Smatrices described by (2.17). The nature of the residues will be studied, for which a little more information on the building block {x} − will be needed. This is a 2πi periodic function, real for imaginary θ. It has simple poles at (x − 1)πi/h and (x + 1)πi/h, with residues positive multiples of −i, +i respectively. Outside the interval between these two poles (or its repetition modulo 2πi), the function is positive on the imaginary axis. These facts hold equally well for the perturbed conformal or the affine Toda blocks.
It is helpful to introduce a pictorial notation in which the S-matrix element is represented by a 'wall' of rectangles, stacked along the imaginary axis. The building block {x} − is depicted by a single rectangle above the imaginary axis, stretching from (x − 1)πi/h to (x + 1)πi/h:
{x} − ≡ −i +i (x−1)πi/h (x+1)πi/h
The residues of the poles at (x ± 1)πi/h, up to some real, positive constant, are shown above the block. A product of blocks making up an S-matrix element is represented by stacking the rectangles to make a wall. For example,
{3} 2 − {5} − ≡ 2πi/h 4πi/h 6πi/h
Poles occur at the ends of the blocks, of higher order where the ends of blocks coincide.
In this example there is a pole of order three at θ = 4πi/h, coming from the two blocks to the left and one to the right. To represent a full S-matrix element, a wall of length 2π is sufficient, since {x+2h} − = {x} − . In fact, the unitarity constraint imposes that the height for {−x} − must be exactly the negative of that for {x} − , so a stretch of length π will do; it is convenient to let it straddle the physical strip, running from 0 to iπ. To give a couple of examples, here are two of the S-matrix elements from the E 8 -related scattering theories:
S 35 : 0 • • • • • iπ S 88 : 0 • • • • • iπ
The rest of each picture, a stretch of wall running from (say) iπ to 2iπ, can be obtained by reflecting the piece shown about the line Im(θ) = iπ, and then negating all the heights.
Note that the heights are positive inside the physical strip, and negative outside -a general phenomenon that will be commented on at the end of this section. The positions of expected forward-channel poles (found from the three-point couplings and the masses) are shown by the symbols • below the axis -they occur precisely at the 'downhill' sections of wall (reading left to right). This apparent coincidence exemplifies a well-established relationship between the fusing structure and the nature of the pole residues, and will now be discussed.
The nature of the residue of any pole is easy to find from the corresponding picture 2 (the term residue being used somewhat loosely to mean the coefficient of the most singular part). Blocks not contributing directly to a pole simply multiply its residue by a positive real number, and can be ignored (this is the reason for the minus sign in (2.12)). Furthermore, the contribution from two directly abutting blocks (one to the left and one to the right of the pole) is a positive multiple of (+i).(−i) = 1, so these can also be ignored. Thus the nature of the residue is determined solely by the difference in the number of blocks immediately to the left and right of the pole, that is by the change in height of the wall of blocks at the position of the pole. The residues for the three simplest possibilities are The mechanism by which this rule is reproduced, even for the third-order poles, is quite complicated [5].)
∼ −i , (3.1a) ∼ +1 , (3.1b) ∼ +i .
Another 'empirical' observation can be made: apart from the occasional exception at the very edge of the physical strip (for example, at 0 and iπ in S 88 above) the wall height never changes by more than ±1. In the exceptional cases, the height change is always from −1 to 1 or back. Wall segments of negative height have zeroes instead of poles, so there is a cancellation and S-matrix is analytic at these points (and in fact is, by unitarity, forced to be equal to ±1). Hence (3.1) turns out to cover all possibilities for S-matrix poles. In particular, even-order poles always have positive real residues.
It might be expected that all the observations described above should have a universal explanation in the context of root systems. In fact, this can be achieved using only the most elementary properties of the simply-laced roots.
Referring back to (2.19), consider the pole in S ij at relative rapidity πi h u(φ i , α (j) ) (with 0 < u(φ i , α (j) ) < h for the physical strip). The two blocks contributing to this pole involve α (j) and wα (j) , that involving wα (j) being to the left (recall from (2.13) that u(φ i , wα (j) ) = u(φ i , α (j) ) − 2). Thus the change δh in wall height at this pole is given by
δh = (λ i , α (j) ) − (λ i , wα (j) ). (3.2)
Using (2.2) for the second equality, this simplifies:
δh = (1 − w −1 )λ i , α (j) = (φ i , α (j) ). (3.3)
Being the inner product of two roots (of a simply-laced algebra), it is now clear that the change in wall height, if not zero, can only be ±1 or ±2. These possibilities can be examined in turn.
A change of −1 should correspond to a bound state. But (φ i , α (j) ) = −1 implies that φ i + α (j) is a root, −α (k) say, since it is just the Weyl reflection of φ i with respect to α (j) . This gives a root triangle {φ i , α (j) , α (k) } and from the fusing rule a non-zero threepoint coupling C ijk . The fusing angle for thek bound state is, by (2.15), π h u(φ i , α (j) )exactly that corresponding to the pole under discussion. Conversely if there is a threepoint coupling such that a bound state at the relevant fusing angle is a possibility, then δh = −1 is forced (since φ i + α (j) = −α (k) is then a root, and so δh = (φ i , α (j) ) = 1 2 (α 2 (k) − φ 2 i − α 2 (j) ) = −1). The story for δh = 1 is similar, with the conclusion that δh = 1 if and only if there is a bound state in the crossed channel.
These two cases have taken up both forward and crossed channels, so an even-order pole can never have an associated single-particle bound state. It only remains to remark that δh = ±2 implies φ i = ±α (j) , and a relative rapidity for the putative pole of 0 or iπ, the first to be found in ii scattering, the second in iī. As already mentioned, such 'poles' disappear by unitarity.
This then provides a universal explanation for the interplay between the bootstrap equations and the pole structure of the purely elastic S-matrices. In particular it elucidates the 'internal' consistency of these S-matrices, obeying as they do the very equations that they imply via their pole structure.
To close this section, a comment on a slightly simpler feature of (2.17) and (2.19), which also has an interpretation in terms of root systems. Over the full range from 0 to 2πi, the wall height can be both positive and negative, and indeed must be so to satisfy the unitarity constraint. But on physical grounds, the height had better not be negative for blocks in the physical strip: in such an eventuality, a perturbed conformal theory Smatrix would have zeroes in the physical strip, while the affine Toda theory would gain physical-strip poles with coupling-constant dependent positions. Now this height is given as the inner product of some root with a fundamental weight, and so is positive or negative according to whether the relevant root is positive or negative with respect to the given set of simple roots (to say this in another way, the wall height around πi h u(φ i , α (j) ) + 1 is just one 'component' of the height of the root α (j) , namely that piece due to the simple root α i ). Referring in particular back to (2.17), it is clear that the physical requirement reduces, traversing each orbit Γ j of the Coxeter element starting at the special root φ j , to a discussion of which roots are positive and which negative. Although the details will be omitted here, it is straightforward to use results (i), (ii) and (iii) from the beginning of section two, together with the implication from unitarity that approximately half 3 of the roots in a given orbit are positive, to see that the desired property of the wall heights does indeed follow from general theory.
The signs of the three-point couplings
In the perturbative treatment of the affine Toda theories, expansion of the potential to order β results in a set of three-point couplings C ijk which, if non-vanishing, obey the 'area rule':
C ijk = σ ijk 4β √ h ∆ ijk ,
where ∆ ijk is the area of a triangle of sides m i , m j and m k (the particle masses) and σ ijk is a phase of unit modulus, which given the hermiticity of the original Lagrangian can be taken to be plus or minus one. The vanishing/non-vanishing of the coupling is described by the rule (2.14). This, together with the normalisation of C ijk , has now been derived in a general way [16]. However the signs σ ijk are a little more subtle. Clearly they can be changed around by negating some of the fields, but this does not mean that they can all be set to 1. Indeed, cancellations necessary for perturbation theory to be compatible with integrability often depend crucially on the presence of relative phases between different terms, which would not be present if all the σ's were equal to 1 (for some examples of this, see [5,6]).
Despite the apparent arbitrariness involved, there is a special choice of normalisations which connects with the root system data already described. Recall that any three-point coupling C ijk results in a bound-state pole in ij scattering of odd order 2m + 1, and note that the field normalisations used in [4] can be altered so that in every case,
σ ijk = (−1) m . (4.1)
This is actually only a small increase in information over a formula given in [5], itself a consequence of a formula found by Braden and Sasaki [6]. Their result reads:
σ ijk = −σ ilm σ jmn σ knl , (4.2)
holding in this form whenever the triangle ∆ ijk is tiled internally by the three other mass/coupling triangles, ∆ ilm , ∆ jmn and ∆ knl . Note, (4.2) does not change with changes to the field normalisations. The consequence of this, remarked in [5], is that if ∆ ijk is tiled Now higher poles are also associated with tilings by mass triangles [5]. For an oddorder pole of order 2m + 1, with ij producing a bound statek, the triangle ∆ ijk has a 'maximal' nested tiling (in fact, many such) by 2m + 1 triangles, maximal in the sense that each constituent triangle cannot be further tiled. The number 2m + 1 was called the depth of the coupling triangle ∆ ijk in [5]. Applying (4.3) then gives the phases for all coupling triangles as products of the phases for triangles of depth 1. It is then only necessary to check that all the unit depth phases can all be set to 1 to deduce (4.1) from (4.3). Now (4.1), together with the discussion of the last section, can be used to give an expression for the signs in terms of the roots and weights. The physical strip pole for thē k bound state in ij scattering will be at πi h u(φ i , α o (j) ) for some α o (j) ∈ Γ j (cf the discussion preceding (3.2)). Note, {φ i , α o (j) , α o (k) }, the corresponding root triangle for C ijk , is oriented such that 0 < u(φ i , α o (j) ) < h. The order of the pole is
2m + 1 = (λ i , α o (j) ) + (λ i , wα o (j) ). (4.4)
Now the change δh in the wall height at this point is −1, as the pole is forward channel; so combining (4.4) with (3.2) gives
m = (λ i , α o (j) ) (4.5)
and
σ ijk = (−1) (λ i ,α o (j)
) . This expression for σ ijk involves the choice of one particular root triangle, and furthermore its symmetry in i, j and k is not at all obvious. To remedy these defects, the idea of the orientation of a root triangle will have to be made a little more precise. Let {α (i) , α (j) , α (k) } be any root triangle implying the nonvanishing of the coupling C ijk ; the orientation ǫ(α (i) , α (j) , α (k) ) is then defined to be +1 if the projection into the s = 1 subspace has a clockwise sense (going from i to j to k), and −1 if anticlockwise. Since
ǫ(α (i) , α (j) , α (k) ) = +1 if and only if 0 < u(α (i) , α (j) ) < h, the root triangle {φ i , α o (j) , α o (k)
} used above has orientation +1. It will be helpful to define another quantity for (general) root triangles, namely f (α (i) , α (j) , α (k) ) = 1 2 + (Rα (i) , α (j) ), (4.6) where R, given by (2.3), has the important property that Rφ i = λ i . Finally, set
m(α (i) , α (j) , α (k) ) = ǫ(α (i) , α (j) , α (k) )f (α (i) , α (j) , α (k) ) − 1 2 . (4.7)
For the original root triangle {φ i , α o (j) , α o (k) }, this coincides with (4.5) (this is the reason for the − 1 2 ). But m(α (i) , α (j) , α (k) ) is the same for any triangle of roots for C ijk , and furthermore is symmetrical in i, j and k. To establish these properties, some more information on the set of root triangles is needed. There are in fact 2h ordered triplets {α (i) , α (j) , α (k) } for each non-zero three-point coupling (the ordering being needed to count correctly the cases when, say, i = j). Of these, h can be found simply by acting 'diagonally' with powers of w on an initial triangle. These all have the same orientation. To see that there are exactly h more, consider the action of w ≡ −w {•} (one could equally well set w = w {•} ; all the identities to be given below would be unchanged). As mentioned in [12], on the cosets w has the effect of two successive charge conjugations, that is no effect at all: wΓ i = Γ i . But when acting on triangles, w reverses the orientation: ǫ( wα (i) , wα (j) , wα (k) ) = −ǫ(α (i) , α (j) , α (k) ). (4.8) Applying powers of w to { wα (i) , wα (j) , wα (k) } then gives h more possibilities, and the geometry of the projection into the s = 1 subspace shows that there can be no more. (To be more precise, since U k ij is fixed, (2.15) shows that once the root α (i) has been chosen there are only two possible directions for the s = 1 projection of α (j) . But the roots in a single orbit all project to different directions for s = 1, so there are at most two roots in Γ j which can form a triangle involving α (i) and a root from Γ k . Letting α (i) run through its orbit then gives a maximum of 2h triangles.) An important consequence of the above is that all triangles for a given coupling can be obtained from an initial one by acting with w and w. Now the effect of these two operations on the orientation has already been given: w leaves ǫ invariant, while w negates it. It is also clear that w leaves f unchanged, w being an orthogonal transformation which commutes with R. The action of w is a little harder to see, but the identity wR w = 1 − R together with the fact that the inner product of α (i) and α (j) must be −1 (their sum being another root) implies f ( wα (i) , wα (j) , wα (k) ) = −f (α (i) , α (j) , α (k) ).
(4.9)
Comparing with (4.8) establishes the invariance of m(α (i) , α (j) , α (k) ) under the diagonal action of both w and w, and hence its insensitivity to the choice of root triangle.
There remains the symmetry of (4.7) between i, j and k. The orientation is completely antisymmetric in its arguments, so it will be enough to demonstrate that the same is true of f . First consider swapping α (i) and α (j) in (4.6). From (2.5), and using again that the inner product of α (i) and α (j) must be −1, this sends f to −f . Now consider a cyclic permutation of α (i) , α (j) and α (k) . The three roots sum to zero, so (Rα (j) , α (k) ) = (Rα (j) , −α (i) − α (j) ) = (Rα (i) , α (j) ), using (2.5) and also the fact that (Rα (j) , α (j) ) = 1.
Thus f is unchanged by a cyclic permutation of its arguments, and this is enough to establish antisymmetry.
This completes the proof of the claims following equation (4.7). The expression can now be used in (4.1), it already having been mentioned that for one particular triangle (and hence for all) (4.7) coincides with the previously-derived (4.5). This can be used in a complete specification of the three-point couplings for the simply-laced affine Toda theories. With a simple rewriting of the resulting expression for (−1) m , the three-point coupling data can be summarised as follows:
• C ijk = 0 iff ∃ α (i) + α (j) + α (k) = 0 (where α (i) ∈ Γ i , α (j) ∈ Γ j , α (k) ∈ Γ k ).
• There is a normalisation of the fields such that in these cases
C ijk = ǫ(α (i) , α (j) , α (k) )(−1) (Rα (i) ,α (j) ) 4β √ h ∆ ijk .
A formula due to Klassen and Melzer
In their investigations of the Thermodynamic Bethe Ansatz, Klassen and Melzer [9] observed an interesting universal feature of the minimal purely elastic S-matrices. If the matrix N ij is defined by
N ij = − 1 2πi lnS ij (θ) θ=∞ θ=−∞ (5.1)
then for the minimal (perturbed conformal field theory) S-matrices,
N = 2C −1 − I, (5.2)
where C is the relevant (non-affine) Cartan matrix, and I the unit matrix. Using this result, Klassen and Melzer were able to calculate the central charges of the ultra-violet limits of these theories, finding agreement with the idea that these S-matrices do indeed describe perturbations of certain conformal field theories. Such issues will not be the concern here; the purpose of this section is merely to give a simple and universal proof of (5.2) starting from the general S-matrix expression of [12].
Klassen and Melzer used unitary blocks f α (θ) ≡ hα (θ), in terms of which the Smatrix element S ij was written
S ij = α∈A ij f α (θ).
The parameter α was taken to satisfy −1 < α ≤ 1, and (5.1) reduced to
N ij = α∈A ij (1 − |α|)sgn(α),(5.3)
with the convention that sgn(0) = 0 (since f 0 ≡ 1). It will be more convenient below to choose a different range for α, namely 0 ≤ α < 2. Using f α = f α+2 to relabel the negatively-indexed blocks, together with the fact that (1 − |α|)sgn(α) = 1 − (α + 2) for −1 < α < 0, equation (5.3) becomes
N ij = α∈A ij (1 − α) − |A ij ∩ {0}|, (5.4)
where now 0 ≤ α < 2, and the second term undoes the overcounting in the first whenever α = 0.
To evaluate (5.4), it is simplest to use the first expression of (2.27). Each unitary block 2p + u ij corresponds to an α ∈ A ij equal to (2p + u ij )/h, while |A ij ∩ {0}| is simply the number of blocks 0 . Such blocks will only be found if u ij = 0, and so |A ij ∩ {0}| = 0 if i and j have different colours. Otherwise, 0 is raised to the power (λ i , φ j ). For the the ordering (2.7) of the simple roots, the full set of these inner products is
(λ • , φ • ′ ) = δ •• ′ (λ • , φ • ′ ) = −C •• ′ (λ • , φ • ′ ) = 0 (λ • , φ • ′ ) = δ •• ′ (5.5)
Hence the general result is |A ij ∩ {0}| = δ ij , and N ij is given by
N ij = h−1 p=0 (1 − (2p + u ij )/h)(λ i , w −p φ j ) − δ ij . (5.6)
Strictly speaking, the fact that u •• = −1 means that if i is of type • and j of type •, the first block counted by (5.6) corresponds to α = −1/h, outside the desired range. This can be ignored as the power to which this block is raised, (λ i , φ j ), is zero here by (5.5).
Making use of equations (2.4) and (2.6),
N ij = λ i , (h − u ij )P 0 + 2R φ j − δ ij = 2(λ i , λ j ) − δ ij ,
the results P 0 = 0 (0 is never an exponent) and Rφ j = λ j giving the second equality. Since (λ i , λ j ) is exactly the inverse of the Cartan matrix, equation (5.2) now follows immediately.
Conclusions
Various previously-observed features of the ADE purely elastic scattering theories are now known to follow from general principles, especially given recent work on affine Toda perturbation theory [16] and the Clebsch-Gordan rule for fusings [17]. With regards to the S-matrices, a notable feature of the treatment given above, as compared to that in [12], is that the splitting of the roots according to type (colour) has become much less important.
Essentially, once the notations (2.13) have been set up, this distinction can be forgotten.
All that is needed is equation (2.15), relating the quantities u(α, β) to the fusing angles.
However, it should not be forgotten that the splitting of the particles into two sets does express a geometrical property of the projections of their orbits, reflected in the fact [12] that two particles of the same type always fuse at an even multiple of π/h, two particles of opposite type at an odd multiple of π/h. Hence the split still has physical implications, even if these are best left hidden for most calculations.
As regards future work, there are two obvious questions to ask. Within the context of theories with diagonal S-matrices, are there any other physically reasonable possibilities beyond those associated with the ADE series? To formulate this question properly, it must be decided exactly what is meant by 'physically reasonable'. For example, whenever a subalgebra of the fusing algebra can be formed (such as emerges when a twisted folding is considered [4][7]), a self-consistent set of S-matrix elements, obeying the bootstrap equations that they imply via their odd-order poles, can be obtained simply by taking the corresponding submatrix of the 'parent' S-matrix. However, in such cases there are always higher-order physical poles which are inexplicable without the full set of particles of the parent theory, forcing their re-inclusion and returning the theory to the ADE set. (A similar phenomenon allows the solitons of the sine-Gordon theory to be inferred from the S-matrix elements of the breathers alone [18].) The only other purely elastic S-matrices that seem to have been discussed so far [19,20] all obey rather different self-consistency requirements, in that the prescription for assigning forward and crossed channels to the odd-order poles is changed. Such conditions appear to be appropriate for theories with non-hermitian lagrangians [19], and so these S-matrices should in any case fall outside an initial (perhaps ADE) classification of unitary purely elastic scattering theories. Nevertheless, it would be interesting if they could also be given an interpretation in terms of root systems.
The second question concerns the relevance of any of the above to theories with multiplets and non-diagonal S-matrices. This would require some similarity in structure between the bootstraps for the purely elastic scattering theories and those for at least a subset of the more complicated models. At least at the level of the fusings, the same structure has emerged in the study of perturbations of N =2 supersymmetric conformal theories [15] (although in fact not for all the ADE series). Other possible candidates for inclusion in the subset include the principal chiral models [21]. S-matrices with the same scalar (CDD) part have been proposed for perturbations of certain conformal field theories [22]. There are many coincidences (for example, of mass spectra) and even some explicit calculations [23] suggesting that a connection may indeed be found, but the greatlyincreased complexity of the bootstrap equations makes a complete analysis difficult.
a and c, the total number of blocks to left and right of the pole is odd, and so the pole itself is of odd order. These odd order poles always have an interpretation in terms of the production of a bound state. Examination of the ADE scattering theories on a case-by-case basis has shown that case c, the downhill pole with a +i residue, is always forward channel, while case a, uphill, is crossed channel. The pictures give a simple 'uphill/downhill' mnemonic by which to decide if an odd-order pole corresponds to a forward channel bound state, in agreement with what has already been observed in the two E 8 examples. (Note, though, that it remains unclear from the point of view of perturbation theory why the +i residue should be forward channel for the higher odd-order poles.
(
in a 'nested' fashion by 2q + 1 other triangles {∆ A }, A = 1, . . . 2q + 1 (with phase factors {σ A }), then σ ijk = (The tiling of a triangle is nested if it is tiled by three other triangles each of which is either untiled, or is itself tiled in a nested way. This allows (4.1) to be used inductively to derive (4.3).)
26 )
26(This block, unitary in the sense that it individually satisfies the unitarity constraint mentioned above, was used in[4], along with the larger unitary block {x} = {x} + /{−x} + .)Via (2.9) and (2.26), the formulae (2.17) and (2.19) become, in the minimal cases,
For the affine Toda S-matrices, this sort of block notation also allows the value of this residue, to leading order in β, to be identified; this was used in[5].
in fact exactly half, except for A 2n for which h is odd and the situation is marginally more complicated.
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| []
|
[
"Sphere tangencies, line incidences, and Lie's line-sphere correspondence",
"Sphere tangencies, line incidences, and Lie's line-sphere correspondence"
]
| [
"Joshua Zahl "
]
| []
| []
| Two spheres with centers p and q and signed radii r and s are said to be in contact if |p − q| 2 = (r − s) 2 . Using Lie's line-sphere correspondence, we show that if F is a field in which −1 is not a square, then there is an isomorphism between the set of spheres in F 3 and the set of lines in a suitably constructed Heisenberg group that is embedded in (F [i]) 3 ; under this isomorphism, contact between spheres translates to incidences between lines.In the past decade there has been significant progress in understanding the incidence geometry of lines in three space. The contact-incidence isomorphism allows us to translate statements about the incidence geometry of lines into statements about the contact geometry of spheres. This leads to new bounds for Erdős' repeated distances problem in F 3 , and improved bounds for the number of point-sphere incidences in three dimensions. These new bounds are sharp for certain ranges of parameters. | 10.1017/s0305004121000256 | [
"https://arxiv.org/pdf/2002.11677v1.pdf"
]
| 211,505,994 | 2002.11677 | e8943a077597ad9b9cd8ede8d55d5b68c02f029a |
Sphere tangencies, line incidences, and Lie's line-sphere correspondence
February 27, 2020
Joshua Zahl
Sphere tangencies, line incidences, and Lie's line-sphere correspondence
February 27, 2020
Two spheres with centers p and q and signed radii r and s are said to be in contact if |p − q| 2 = (r − s) 2 . Using Lie's line-sphere correspondence, we show that if F is a field in which −1 is not a square, then there is an isomorphism between the set of spheres in F 3 and the set of lines in a suitably constructed Heisenberg group that is embedded in (F [i]) 3 ; under this isomorphism, contact between spheres translates to incidences between lines.In the past decade there has been significant progress in understanding the incidence geometry of lines in three space. The contact-incidence isomorphism allows us to translate statements about the incidence geometry of lines into statements about the contact geometry of spheres. This leads to new bounds for Erdős' repeated distances problem in F 3 , and improved bounds for the number of point-sphere incidences in three dimensions. These new bounds are sharp for certain ranges of parameters.
Introduction
Let F be a field in which −1 is not a square. For each quadruple (x 1 , y 1 , z 1 , r 1 ) ∈ F 4 , we associate the (oriented) sphere S 1 ⊂ F 3 described by the equation (x − x 1 ) 2 + (y − y 1 ) 2 + (z − z 1 ) 2 = r 2 1 . We say two oriented spheres S 1 , S 2 are in "contact" if (x 1 − x 2 ) 2 + (y 1 − y 2 ) 2 + (z 1 − z 2 ) 2 = (r 1 − r 2 ) 2 .
(1.1)
If F = R, then this has the following geometric interpretation:
• If r 1 and r 2 are non-zero and have the same sign, then (1.1) describes internal tangency.
• If r 1 and r 2 are non-zero and have opposite signs, then (1.1) describes external tangency.
In this paper we will explore the following type of extremal problem in combinatorial geometry: Let n be a large integer. If S is a set of n oriented spheres in F 3 (possibly with some additional restrictions), how many pairs of spheres can be in contact? This question will be answered precisely in Theorem 1.8 below.
When F = R, variants of this problem have been studied extensively in the literature [1,4,13,22]. For example, Erdős' repeated distances conjecture in R 3 [8] asserts that n spheres in R 3 of the same radius must determine O(n 4/3 ) tangencies. The current best-known bound is O(n 295/197+ε ) in R 3 [23]. In Theorem 1.9 we will establish the weaker bound O(n 3/2 ) which is valid in all fields for which −1 is not a square.
Before discussing this problem further, we will introduce some additional terminology. Let i be a solution to x 2 + 1 = 0 in F and let E = F [i]. Each element ω ∈ E can be written uniquely as ω = a + bi, with a, b ∈ F . We define the involutionω = a − bi, and we define Re(ω) = 1 2 (ω +ω), Im(ω) = 1 2 (ω −ω).
We define the Heisenberg group H = {(x, y, z) ∈ E 3 : Im(z) = Im(xȳ)}.
(1.
2)
The Heisenberg group contains a four-parameter family of lines. In particular, if a, b, c, d ∈ F , then H contains the line
{(0, c + di, a) + (1, b, c − di)t : t ∈ E},(1.3)
and every line contained in H that is not parallel to the xy plane is of this form. If a 1 , b 1 , c 1 , d 1 and a 2 , b 2 , c 2 , d 2 are elements of F , then the corresponding lines are coplanar (i.e. they either intersect or are collinear) precisely when
(c 1 + d 1 i − c 2 − d 2 i, a 1 − a 2 ) ∧ (b 1 − b 2 , c 1 − d 1 i − c 2 + d 2 i) = 0. (1.4)
The Heisenberg group H has played an important role in studying the Kakeya problem [15,16,19]. More recently, it has emerged as an important object in incidence geometry [7,10]. Our study of contact problems for spheres in F 3 begins by observing that the contact geometry of (oriented) spheres in F 3 is isomorphic to the incidence geometry of complex lines in H that are not parallel to the xy plane. Concretely, to each oriented sphere S 1 ⊂ F 3 centered at (x 1 , y 1 , z 1 ) with radius r 1 , we can associate a line of the form (1.3), with a = −z 1 − r 1 , b = −z 1 + r, c = −x 1 , d = −y 1 . Two oriented spheres S 1 and S 2 are in contact if and only if the corresponding lines 1 and 2 are coplanar. We will discuss this isomorphism and its implications in Section 2. This isomorphism is not new-it is known classically as Lie's line-sphere correspondence. However, we are not aware of this isomorphism previously being used in the context of combinatorial geometry.
The isomorphism is interesting for the following reason. In the past decade there has been significant progress in understanding the incidence geometry of lines in E 3 . This line of inquiry began with Dvir's proof of the finite field Kakeya problem [5] and Guth and Katz's proof of the joins conjecture [11], as well as subsequent simplifications and generalizations of their proof by Quilodrán and independently by Kaplan, Sharir, and Shustin [14]. More recently, and of direct relevance to the problems at hand, Guth and Katz resolved the Erdős distinct distances problem in R 2 (up to the endpoint) by developing new techniques for understanding the incidence geometry of lines in R 3 . Some of these techniques were extended to all fields by Kollár [17] and by Guth and the author [12]. The isomorphism described above allows us to translate these results about the incidence geometry of lines into statements about the contact geometry of spheres. We will describe a number of concrete statements below.
To begin exploring the contact geometry of oriented spheres, we should ask: what arrangements of oriented spheres in F 3 have many pairs of spheres that are in contact? The next example shows that there are sets of spheres in F 3 so that every pair of oriented spheres is in contact.
Example 1.1. Let (x, y, z, r) ∈ F 4 and let (u, v, w) ∈ F 3 with u 2 + v 2 + w 2 = s 2 for some nonzero s ∈ F . Consider the set of oriented spheres P = {(x + ut, y + vt, z + wt, r + st) : t ∈ F }. Every pair of spheres from this set is in contact. See Figure 1. Definition 1.2. Let P be a set of spheres, every pair of which is in contact. If P is maximal (in the sense that no additional spheres can be added to P while maintaining this property), then P is called a "pencil of contacting spheres." If S is a set of oriented spheres and P is a pencil of contacting spheres, we say that P is k-rich (with respect to S) if at least k spheres from S are contained in P. We say P is exactly k-rich if exactly k spheres from S are contained in P. Remark 1.3. If we identify each sphere in a pencil of contacting spheres with its coordinates (x, y, z, r), then the corresponding points form a line in F 4 . If we identify each sphere in a pencil of contacting spheres with its corresponding line in H, then the resulting family of lines are all coplanar and pass through a common point (possibly at infinity 1 ). This will be discussed further in Section 2.4.
The next result says that any two elements from a pencil of contacting spheres uniquely determine that pencil. Lemma 1.4. Let S 1 and S 2 be distinct oriented spheres that are in contact. Then the set of oriented spheres that are in contact with S 1 and S 2 is a pencil of contacting spheres.
We will defer the proof of Lemma 1.4 to Section 2.4. Example 1.1 suggests that rather than asking how many spheres from S are in contact, we should instead ask how many k-rich pencils can be determined by S-each pencil that is exactly k rich determines k 2 pairs of contacting spheres. We begin with the case k = 2. Since each pair of spheres can determine at most one pencil, a set S of oriented spheres determines at most |S| 2 2-rich pencils of contacting spheres. The next example shows that in general we cannot substantially improve this estimate, because there exist configurations of n oriented spheres that determine n 2 4 2-rich pencils. Example 1.5. Let S 1 , S 2 and S 3 be three spheres in F 3 , no two of which are in contact. Suppose that at least three spheres are in contact with each of S 1 , S 2 , and S 3 , and denote this set of spheres by C. Let C be the set of all spheres that are in contact with every sphere from C. Then every sphere from C is in contact with every sphere from C and vice-versa. Furthermore, Lemma 1.4 implies that no two spheres from C are in contact, and no two spheres from C are in contact, so the spheres in C ∪ C do not determine any 3-rich pencils of contacting spheres. This means that if S ⊂ C and S ⊂ C , then S ∪ S determines |S| |S | 2-rich pencils of contacting spheres. See Figure 2.
Definition 1.6. Let C and C be two sets of oriented spheres, each of cardinality at least three, with the property that each sphere from C is in contact with each sphere from C , and no two spheres from the same set are in contact. If C and C are maximal (in the sense that no additional spheres can be added to C or C while maintaining this property), then C and C are called a "pair of complimentary conic sections." Figure 2: A pair of complimentary conic sections. The sets C and C are shown individually on the left and center, respectively, and C ∪ C is shown on the right. If F = R, then (in this example) all of the spheres in C have positive radius, and all of the spheres in C have negative radius. In particular, while certain pairs of spheres in C (resp. C ) are (externally) tangent, no pairs of spheres are in contact in the sense of (1.1). Note that while the centers of the spheres in C are contained in a line, the corresponding quadruples (x 1 , y 1 , z 1 , r 1 ) are contained in an irreducible degree-two curve.
Remark 1.7. If we identify each sphere in a pair of complimentary conic sections with its coordinates (x, y, z, r), then the corresponding points are precisely the F -points on a pair of conic sections in F 4 , neither of which are lines. If we identify each sphere in a pair of complimentary conic sections with its corresponding line in H, then the resulting families of lines are contained in two rulings of a doubly-ruled surface in E 3 . Note, however, that the two rulings of this doubly-ruled surface contain additional lines that do not come from the pair of complimentary conic sections, since not all lines in the rulings will be contained in H. This will be discussed further in Section 2.5.
We are now ready to state our main result. Informally, it asserts that the configurations described in Examples 1.1 and 1.5 are the only way that many pairs of spheres in F 3 can be in contact.
Theorem 1.8. Let F be a field in which −1 is not a square. Let S be a set of n oriented spheres in F 3 , with n ≤ (char F ) 2 (if F has characteristic zero then we impose no constraints on n). Then for each 3 ≤ k ≤ n, S determines O(n 3/2 k −3/2 ) k-rich pencils of contacting spheres. Furthermore, at least one of the following two things must occur:
• There is a pair of complimentary conic sections C, C so that |C ∩ S| ≥ √ n and |C ∩ S| ≥ √ n.
• S determines O(n 3/2 ) 2-rich pencils of contacting spheres.
Theorem 1.8 leads to new bounds for Erdős' repeated distances problem in F 3 . Theorem 1.9. Let F be a field in which −1 is not a square. Let r ∈ F \{0} and let P ⊂ F 3 be a set of n points in F 3 , with n ≤ (char F ) 2 (if F has characteristic zero then we impose no constraints on n). Then there are O(n 3/2 ) pairs (x 1 , y 1 , z 1 ), (x 2 , y 2 , z 2 ) ∈ P satisfying
(x 1 − x 2 ) 2 + (y 1 − y 2 ) 2 + (z 1 − z 2 ) 2 = r 2 .
(1.5)
As discussed above, when F = R the conjectured bound is O(n 4/3 ) and the current best known bound is O(n 3 2 − 1 394 +ε ) [23]. For general fields in which −1 is not a square, the previous best-known bound was O(n 5/3 ).
Remark 1.10. The bound O(n 3/2 ) given above cannot be improved without further restrictions on n. Indeed, if we select P to be a set of p 2 points in a plane in F 3 p , then by pigeonholing there exists an element r ∈ F p so that there are roughly p 3 solutions to (1.5). When n is much smaller than p 2 (e.g. if n ≤ p) it seems plausible that there should be O(n 4/3 ) solutions to (1.5).
Remark 1.11. The requirement that −1 not be a square is essential, since if −1 is a square in F then for each r ∈ F \{0}, the sphere {(x, y, z) ∈ F 3 : x 2 + y 2 + z 2 = r 2 } is doubly-ruled by lines (see e.g. [20,Lemma 6]). It is thus possible to find an arrangement of n/2 spheres of radius r, all of which contain a common line . Let P = P 1 ∪ P 2 , where P 1 is the set of centers of the n/2 spheres described above and P 2 is a set of n/2 points on . Then P has n 2 /4 pairs of points that satisfy (1.5). Theorem 1.9 can also be used to prove new results for the incidence geometry of points and spheres in F 3 . In general, it is possible for n points and n spheres in F 3 to determine n 2 pointssphere incidences. For example, we can place n points on a circle in F 3 and select n spheres which contain that circle. The following definition, which is originally due to Elekes and Tóth [6] in the context of hyperplanes, will help us quantify the extent to which this type of situation occurs. Definition 1.12. Let F be a field, let P ⊂ F 3 , and let η > 0 be a real number. A sphere S ⊂ F 3 is said to be η-non-degenerate (with respect to P) if for each plane H ⊂ F 3 we have |P ∩ S ∩ H| ≤ η|P ∩ S|. Theorem 1.13 (Point-sphere incidences). Let F be a field in which −1 is not a square. Let S be a set of n spheres (of nonzero radius) in F 3 and let P be a set of n points in F 3 , with n ≤ (char F ) 2 (if F has characteristic zero then we impose no constraints on n). Let η > 0 and suppose the spheres in S are η-non-degenerate with respect to P. Then there are O(n 3/2 ) incidences between the points in P and the spheres in S, where the implicit constant depends only on η.
In [1], Apfelbaum and Sharir proved that m points and n η-non-degenerate spheres in R 3 determine O * (m 8/11 n 9/11 + mn 1/2 ) incidences, where the notation O * (·) suppresses sub-polynomial factors. When m = n, this bound simplifies to O * (n 17/11 ). Thus in the special case m = n, Theorem 1.13 both strengthens the incidence bound of Apfelbaum and Sharir and extends the result from R to fields in which −1 is not a square.
When F = R, additional tools from incidence geometry become available, and we can say more.
Theorem 1.14. Let S a set of n oriented spheres in R 3 . Then for each 3 ≤ k ≤ n, S determines O(n 3/2 k −5/2 + nk −1 ) k-rich pencils of contacting spheres.
Note that a k-rich pencil determines k 2 pairs of contacting spheres. Since the quantity k 2 n 3/2 k −5/2 is dyadically summable in k, Theorem 1.14 allows us to bound the number of pairs of contacting spheres, provided not too many spheres are contained in a common pencil or pair of complimentary conic sections. Corollary 1.15. Let S a set of n oriented spheres in R 3 . Suppose that no pencil of contacting spheres is √ n rich. Then at least one of the following two things must occur
• There is a pair of complimentary conic sections C, C so that
|C ∩ S| ≥ √ n and |C ∩ S| ≥ √ n. (1.6)
• There are O(n 3/2 ) pairs of contacting spheres.
The following (rather uninteresting) example shows that Theorem 1.14 can be sharp when there are pairs of complimentary conic sections that contain almost √ n spheres from S. A similar example shows that Theorem 1.14 can be sharp when there are pencils that contain almost √ n spheres from S. More interestingly, the following "grid" construction shows that Theorem 1.14 can be sharp even when S does not contain many spheres in a pencil or many spheres in complimentary conic sections. . We can verify that the equation
(a − a ) 2 + (b − b ) 2 + (c − c ) 2 = (d − d ) 2 , a, a , b, b , c, c , d, d ∈ [m] (1.7)
has roughly m 6 = n 3/2 solutions. No linear pencil of contacting spheres contains more than m = n 1/4 spheres from S. This means that S determines roughly n 3/2 2-rich pencils, and also determines roughly n 3/2 pairs of contacting spheres. On the other hand, if we fix three non-collinear points (a j , b j , c j , d j ) ∈ [m] 4 , j = 1, 2, 3, then the set of points (a , b , c , d ) ∈ [m] 4 satisfying (1.7) for each index j = 1, 2, 3 must be contained in an irreducible degree two curve in R 4 . Bombieri and Pila [2] showed that such a curve contains O(n 1/8+ε ) points from [m] 4 . We conclude that every pair of complimentary conic sections contains O(n 1/8+ε ) spheres from S.
Thanks
The author would like to thank Kevin Hughes and Jozsef Solymosi for helpful discussions, and would like to thank Gilles Castel for assistance creating Figures 1 and 2. The author was partially funded by a NSERC discovery grant.
Lie's line-sphere correspondence
In this section we will discuss the contact-incidence isomorphism introduced in Section 1. Throughout this section, F will be a field in which −1 is not a square, and E = F [i] is a degree-two extension of F , where i 2 = −1.
Lines and the Klein quadric
In this section we will be concerned with points in E 6 and its projectivization EP 5 . We will write elements of E 6 using the slightly strange index set (p 14 , p 24 , p 34 , p 23 , p 31 , p 12 ), and elements of EP 5 will be denoted [p 14 : p 24 : p 34 : p 23 : p 31 : p 12 ]. Let K(·, ·) be the symmetric bilinear form on E 6 given by K (p 14 , p 24 , p 34 , p 23 , p 31 , p 12 ),(p 14 , p 24 , p 34 , p 23 , p 31 , p 12 ) = p 14 p 23 + p 23 p 14 + p 24 p 31 + p 31 p 24 + p 34 p 12 + p 12 p 34 .
We define the Klein quadric to be the set {p = [p 14 : p 24 : p 34 : p 23 : p 31 : p 12 ] ∈ EP 5 : K(p, p) = 0}.
Since the polynomial Q K (p) = K(p, p) is homogeneous, the above set is well-defined. The equation K(p, p) = 0 is called the Plücker relation, and since char(F ) = 2, it can also be written as
p 14 p 23 + p 24 p 31 + p 34 p 12 = 0. (2.1)
There is a one-to-one correspondence between projective lines in EP 3 and points in the Klein quadric. For our purposes, however, it will be more useful for us to identify a certain class of (affine) lines in E 3 with a large subset of the Klein quadric. Concretely, the (affine) line (0, s, t) + E (1, u, v) can be identified with the point
Oriented spheres and the Lie quadric
In this section we will recall some basic facts from Lie sphere geometry. The primary reference is [3], especially Chapter 2, and [18]. Lie sphere geometry studies objects called "Lie spheres," which unify the notion of a sphere, point, and plane (the latter two objects can be thought of as spheres that have zero and infinite radius, respectively). We will restrict attention to points and spheres.
Let L(·, ·) be the symmetric bilinear form on F 6 defined by
L (a, b, c, d, e, f ), (a , b , c , d , e , f ) = 2bb + 2cc + 2dd − ae − ea − 2f f .
We define the Lie quadric to be the set
{q = [a : b : c : d : e : f ] ∈ F P 5 : L(q, q) = 0}.
Since the polynomial Q L (q) = L(q, q) is homogeneous, the above set is well-defined. We will refer to the equation L(q, q) = 0 as as the Lie relation. It can also be written as
b 2 + c 2 + d 2 − ae − f 2 = 0. (2.3)
For each point q ∈ F P 5 with L(q, q) = 0, we define the set
S q = {(x, y, z) ∈ F 3 : a(x 2 + y 2 + z 2 ) + 2bx + 2cy + 2dz + e = 0}. (2.4)
When a = 0, S q is the sphere defined by the equation
(x + b/a) 2 + (y + c/a) 2 + (z + d/a) 2 = (f /a) 2 . (2.5)
In particular, the oriented sphere S centered at the point (x 1 , y 1 , z 1 ) with radius r 1 can be identified with the point
q S = [ a : b : c : d : e : f ] = [ 1 : −x 1 : −y 1 : −z 0 : x 2 1 + y 2 1 + z 2 1 − r 2 1 : r 1 ].
(2.6)
A set of the form S q , with q in the Lie Quadric will be referred to as a Lie sphere. We say that two Lie-spheres S and S are in "contact" if their corresponding Lie points q and q satisfy the relation L(q, q ) = 0, or equivalently
(b − b ) 2 + (c − c ) 2 + (d − d ) 2 = (f − f ) 2 .
(2.7)
Indeed, examining (1.1), (2.6), and (2.7), we see that two oriented spheres S and S are in contact in the sense of (1.1) precisely when their corresponding Lie points satisfy L(q, q ) = 0.
Line-sphere correspondence
Consider the following map φ from the Lie quadric (which is a subset of F P 5 ) to the Klein quadric (which is a subset of EP 5 ). The point q = [a : b : c : d : e : f ] is mapped to φ(q) = [ p 14 : p 24 : p 34 : p 23 : p 31 :
p 12 ] = [ a : d + f : −b + ic : −e : d − f : −b − ic ].
(2.8)
If q, q ∈ F P 5 , then K(φ(q), φ(q )) = p 14 p 23 + p 23 p 14 + p 24 p 31 + p 31 p 24 + p 34 p 12 + p 12 p 34
= (a)(−e ) + (−e)(a ) + (d + f )(d − f ) + (d − f )(d + f ) + (−b + ic)(−b − ic ) + (−b − ic)(−b + ic ) = −ae − ea + 2dd − 2f f + 2bb + 2cc = L(q, q ).
(2.9)
Setting q = q , we see that if q is an element of the Lie Quadric then φ(q) is an element of the Klein Quadric. Furthermore, if q and q are distinct elements in the Lie Quadric, then the Lie spheres corresponding to q and q are in contact if and only if their images under φ are coplanar. If S is the oriented sphere centered at the point (x 1 , y 1 , z 1 ) ∈ F 3 with radius r 1 ∈ F , then by combining (2.6) and (2.8), we see that S is mapped to the line in E 3 with Plücker coordinates [ p 14 :
p 24 : p 34 : p 23 : p 31 :
p 12 ] = [ a : d + f : −b + ic : −e : d − f : −b − ic ] = [ 1 : −z 1 + r 1 : x 1 − iy 1 : r 2 1 − x 2 1 − y 2 1 − z 2 1 : −z 1 − r 1 : x 1 + iy 1 ].
(2.10)
This corresponds to the line (0, ω, t)+E(1, u,ω), where t = −z 1 −r 1 , u = −z 1 +r 1 , and ω = −x 1 −iy 1 . As we saw in Section 1, lines of this form are contained in the Heisenberg group H.
Pencils of contacting spheres
In this section we will explore the structure of pencils of contacting spheres in K 3 , the Lie quadric, the Heisenberg group. We will begin by proving Lemma 1.4. For the reader's convenience we will restate it here.
Lemma 1.4. Let S 1 and S 2 be distinct oriented spheres that are in contact. Then the set of oriented spheres that are in contact with S 1 and S 2 is a pencil of contacting spheres.
Proof. We will prove the following equivalent statement: Let 1 and 2 be coplanar lines in the Heisenberg group that are not parallel to the xy plane. Let L be the set of lines in the Heisenberg group not parallel to the xy plane that are coplanar with both 1 and 2 (so in particular 1 , 2 ∈ L). Then all of the lines in L are contained in a common plane in E 3 and all pass through a common point (possibly at infinity). Furthermore, L is maximal in the sense that no additional lines can be added to L while maintaining the property that each pair of lines is coplanar. First, observe that every line in H that is not parallel to the xy plane points in a direction v of the form v = (1, b, c − di) ∈ E 3 , with b, c, d ∈ F . Every such line that points in the direction v must be contained in the plane
Π v = {(x, y, z) ∈ E 3 : bx + y = c + di}.
(2.11)
In particular, this implies that any set of lines in H that all point in the same direction and are not parallel to the xy plane must be contained in a common plane in E 3 . Next, observe that if w = (x 1 , y 1 , z 1 ) ∈ H, then every line contained in H passing through w must be contained in the plane
Π w = {(x, y, z) ∈ E 3 : iy 1 x − ix 1 y − iz = −iz 1 }.
(2.12)
If w and w are distinct points in H, then the corresponding planes Π w and Π w are also distinct, and thus either are disjoint or intersect in a line. In particular, this implies that if two distinct lines contained in H intersect at a point w ∈ H, then the unique plane in E 3 containing both lines is precisely Π w . Now, let 1 , 2 , and 3 be three lines in H that are pairwise coplanar. The following argument will show that 1 , 2 , and 3 must be contained in a common plane and must pass through a common point (possibly at infinity).
First, if all three lines are parallel, then we have already shown that they are contained in a common plane and we are done. If not all three lines are parallel, then we can suppose that 1 and 2 intersect at a point w ∈ H, so 1 and 2 must be contained in Π w . If 3 is parallel to one of these lines, then without loss of generality we can suppose it is parallel to 1 , and both lines point in direction v. Then 2 intersects two distinct lines contained in Π v , so 2 must also be contained in Π v . But if w = 2 ∩ 3 , then this implies Π w = Π w = Π v , which is impossible since w = w . Thus we may suppose that 3 is not parallel to either 1 or 2 . Suppose 3 intersects 1 and 2 at distinct points. Then 3 must also be contained in Π w . But at least one of the points 1 ∩ 2 and 1 ∩ 3 must differ from w, which implies there is a point w = w with Π w = Π w ; this is a contradiction. We conclude that 3 passes through 1 ∩ 2 . Since all lines in H passing through w must be contained in Π w , we conclude that 1 , 2 , and 3 are coplanar and coincident.
It now immediately follows that if every line in L is coplanar with both 1 and 2 , then each of these lines must either be parallel with 1 and 2 (if 1 and 2 are parallel), or must pass through their common intersection point (if 1 and 2 are not parallel). Furthermore, each line in L must be contained in the plane spanned by 1 and 2 . In particular, all of the lines in L are coplanar and pass through a common point (possibly at infinity). By construction the set L is maximal, i.e. no additional lines can be added to L while maintaining the property that all pairs of lines in L are coplanar.
If w = (x 1 , y 1 , z 1 ) ∈ H, then the set of lines in H that are not parallel to the xy plane containing w are of the form (0, c + id, a) + E (1, b, c − id), where
c + b Re x 1 = Re y 1 , d + b Im x 1 = Im y 1 , a + c Re x 1 + d Im x 1 = Re z 1 . (2.13)
The set of quadruples (a, b, c, d) satisfying (2.13) belong to a line in K 4 ; we will call this line V w .
Similarly, if v = (1, b, c − di) ∈ E 3 , with b, c, d ∈ F , the set of lines in H pointing in direction v are of the form (0, c + id, a) + E(1, b, c − id) with a ∈ F . We will call this line V v .
Remark 2.1. The above discussion implies that the spheres in a pencil of contacting spheres must either all have the same (signed) radius, or must all have different radii.
Complimentary conics
In this section we will discuss the structure of sets of spheres that form complimentary conics. The next lemma says that pencils of spheres and and pairs of complimentary conics are the only configurations allowing many spheres to be in contact.
Lemma 2.2. Let S and S be sets of oriented spheres, each of cardinality at least three, so that every sphere in S is in contact with every sphere in S and vice-versa. Then either S ∪ S is contained in a pencil of contacting spheres (in which case every sphere in S ∪ S is in contact with every other sphere), or S and S are contained in complimentary conic sections.
Proof. First, suppose that two spheres S 1 , S 2 ∈ S are in contact. Then by Proposition 1.4, the spheres in {S 1 ∪ S 2 } ∪ S are contained in a pencil P of contacting spheres. Since |S| ≥ 3, we have that |P ∩ S | ≥ 3. Proposition 1.4 now implies that every sphere from S is contained in P. We conclude that S ∪ S is contained in a pencil of contacting spheres. An identical argument applies if two spheres S 1 , S 2 ∈ S are in contact. Thus we can suppose that no two spheres from S are in contact, and no two spheres from S are in contact. Let C be the set of spheres in contact with each sphere from S, and let C be the set of spheres that are in contact with each sphere from C ; we have that C and C are complimentary conics containing S and S , respectively.
We will now consider the structure of a pair of complimentary conic sections. Let q 1 , q 2 , q 3 be elements of the Lie quadric, no two of which are in contact. Let 1 , 2 and 3 be the lines in E 3 corresponding to the images of q 1 , q 2 , and q 3 under φ. Let F be the algebraic closure of F (so in particular E ⊂ F ), and let i be the Zariski closure of i in F 3 . Then 1 , 2 and 3 are skew lines in F 3 , and the set of lines in F 3 that are coplanar with each i form a ruling of a doubly ruled surface in F 3 . Let R be the set of lines in the ruling that contains the lines 1 , 2 and 3 and let R be the set of lines in the other ruling. The sets R and R are irreducible conic curves in the variety {p ∈ F P 5 : K(p, p) = 0}. We then have C = φ −1 (R) and C = φ −1 (R ). Since φ is linear, this implies that C and C are precisely the F -points of an irreducible curve in F P 5 that is contained in the Lie quadric (or more precisely, contained in the variety {q ∈ F P 5 : L(q, q) = 0}). Recalling the identification (2.5) of points in the Lie quadric with spheres in F 3 , we see that C and C correspond to sets of oriented spheres whose (x, y, z, r) coordinates are contained in conic sections, neither of which are lines. Note that for some triples q 1 , q 2 , q 3 it is possible that no elements of the Lie quadric will be in contact with each q i . This can happen, for example, if F = R; q 1 and q 2 correspond to disjoint spheres; and q 3 corresponds to a sphere contained inside q 1 . In this situation the complimentary conic sections C and C are well defined, but C does not have any R-points.
Incidence geometry in the Heisenberg group
In this section we will explore the incidence geometry of lines in the Heisenberg group. In particular, we will prove Theorems 1.8, 1.9, and 1.13. Our main tool is the following structure theorem for sets of lines in three space that determine many 2-rich points. The version stated here is Theorem 3.8 from [12]; a similar statement can also be found in [17].
Theorem 3.1. There are absolute constants c > 0 and C so that the following holds. Let L be a set of n lines in E 3 , with n ≤ c(char E) 2 (if E has characteristic zero then we impose no constraints on n). Then for each A ≥ C √ n, either L determines at most CAn 2-rich points, or there is a plane or doubly-ruled surface in E 3 that contains at least A lines from L.
Recall that lines contained in the Heisenberg group are somewhat special. In particular, Lemma 1.4 says that any set of pairwise coplanar lines must also pass through a common point (possibly at infinity). The next lemma is a version of Theorem 3.1 adapted to lines in the Heisenberg group. The hypotheses and conclusions of the theorem have also been slightly tweaked to better fit our needs for proving Theorems 1.8, 1.9, and 1.13.
Lemma 3.2.
There is an absolute constant C 1 so that the following holds. Let L be a set of n lines in the Heisenberg group that are not parallel to the xy plane, with n ≤ (char E) 2 (if E has characteristic zero then we impose no constraints on n). Suppose that each line in L is coloured either red or blue. Then for each A ≥ C 1 √ n, either there are at most C 1 An points that are incident to at least one red and one blue line, or there is a doubly-ruled surface with one ruling that contains at least A red lines and a second ruling that contains at least A blue lines.
Proof. Suppose that there are more than C 1 An points that are incident to at least one red and one blue line (we will call such points bichromatic 2-rich points). We will show that if C 1 is chosen sufficiently large then there is a doubly-ruled surface with one ruling that contains at least A red lines and a second ruling that contains at least A blue lines.
Observe that Lemma 3.2 assumes that n ≤ (char E) 2 , while Theorem 3.1 places the more stringent requirement n ≤ c(char E) 2 . Our first task will be to reduce the size of L slightly. Without loss of generality we can assume that 0 < c ≤ 1, since if c > 1 then we can replace c with 1 and Theorem 3.1 remains true. Let L ⊂ L be a set obtained by randomly keeping each element in L with probability c/10. With high probability, a set of this form will have cardinality at most c(char E) 2 and will will determine at least C 2 An bichromatic 2-rich points, with C 2 = C 1 c 2 1000 . In particular, we can assume that L has both of these properties.
Our next task is to prune the set L slightly so that all of the unpruned lines contain many bichromatic 2-rich points. Define L 0 = L . For each index j = 0, 1, . . ., let L j+1 be obtained by removing a line from L j that contains at most 2A bichromatic 2-rich points. If no such line exists, then halt. Observe that L j has cardinality |L | − j and determines at least C 2 An − 2Aj bichromatic 2-rich points. If C 2 is sufficiently large then this process must halt for some index j with |L j | > 0.
Let L be the resulting set; we have that L determines at least C 3 An bichromatic 2-rich points, with C 3 = C 2 /2, and every line determines at least 2A bichromatic 2-rich points.
Apply Theorem 3.1 to L . If C 1 (and thus C 3 ) is chosen sufficiently large then there exists a plane or doubly-ruled surface Z that contains at least 2A lines from L ; Let L Z be the set of lines from L that are contained in Z. We claim that Z cannot be a plane; indeed if Z was a plane then by Lemma 1.4 the lines in L Z must all intersect at a common point (possibly at infinity). In particular, each of the lines in L Z contain at least 2A 2-rich points, and at least 2A − 1 of these 2-rich points must come from lines in L \L Z . Thus the lines in L \L Z must intersect Z in at least (2A)(2A − 1) > n distinct points. Since each line in L \L Z intersects Z in at most one point, this is impossible.
We conclude that Z is a doubly ruled surface. Since each line in L that is not contained in Z can intersect Z in at most two points, by pigeonholing there exists a line ∈ L Z that contains ≥ 2A bichromatic 2-rich points, and at least A of these points come from lines contained in the dual ruling of Z. Without loss of generality the line is red; this means that the ruling dual to contains at least A blue lines. Again by pigonholing, at least one of these blue lines contains ≥ 2A bichromatic 2-rich points, and at least A of these points come from lines contained in the ruling of Z dual to (i.e. the ruling that contains ). We conclude that the ruling containing must contain at least A red lines. Corollary 3.3. Let L be a set of n lines in the Heisenberg group that are not parallel to the xy plane, with n ≤ (char E) 2 (if E has characteristic zero then we impose no constraints on n). Then either L determines O(n 3/2 ) 2-rich points, or there is a doubly-ruled surface, each of whose rulings contain at least √ n lines from L.
Proof. Suppose that L determines C 2 n 3/2 2-rich points. Randomly colour each line in L either red or blue. With high probability L determines at least C 2 3 n 3/2 bichromatic 2-rich points. If C 2 is sufficiently large, then Lemma 3.3 implies that there is a doubly-ruled surface, each of whose rulings contain at least √ n lines from L.
Lemma 3.2 allows us to understand configurations of lines in E 3 that contain many bichromatic 2-rich points. The next lemma concerns configurations of lines in E 3 that contain many k-rich points for k ≥ 3. Since the proof of the next lemma is very similar to that of Lemma 3.2 (in particular it uses the same ideas of random sampling and refinement), we will just provide a brief sketch that highlights where the proofs differ.
Lemma 3.4. Let L be a set of n lines in the Heisenberg group that are not parallel to the xy plane, with n ≤ (char E) 2 (if E has characteristic zero then we impose no constraints on n). Let k ≥ 3. Then L determines O(n 3/2 k −3/2 ) k-rich points.
Proof sketch. Let L ⊂ L be obtained by randomly selecting each line ∈ L with probability 2/k. Then with positive probability we have that |L | ≤ 100|L|/k, and the number of 3-rich points determined by L is at least half the number of k-rich points determined by |L|. We now argue as in the proof of Corollary 3.3; if L determines ≥ C 1 (n/k) 3/2 3-rich points, then there must exist a doubly-ruled surface Z that contains ≥ (n/k) 1/2 lines from L , and each of these lines must contain at least 3 √ n 3-rich points. In particular, there must be at least 3n 3-rich points contained in Z. Since Z is doubly (not triply!) ruled, this means that each of these 3-rich points must be incident to a line from L that is not contained in Z. Since each line in L not contained in Z can intersect Z in at most 2 points, the number of 3-rich points created in this way is at most 2n, which is a contradiction.
With these preliminary results established, we are now ready to prove the main results of this section.
Proof of Theorem 1.8. First, using the line-sphere correspondence described in Section 2.3, Theorem 1.8 is implied by the following statement about lines in the Heisenberg group:
Let L be a set of n lines in the Heisenberg group that are not parallel to the xy plane, with n ≤ (char E) 2 (if E has characteristic zero then we impose no constraints on n). Then for each 3 ≤ k ≤ n, there are O(n 3/2 k −3/2 ) k-rich points. Furthermore, at least one of the following two things must occur:
• There is a doubly-ruled surface, each of whose ruling contain at least √ n lines from L.
• L determines O(n 3/2 ) 2-rich points.
The theorem now follows immediately from Corollary 3.3 and Lemma 3.4.
Our proof of Theorem 1.9 will follow a similar strategy to the proof of Theorem 1.8. The main thing to verify is that not too many lines can be contained in a doubly-ruled surface.
Proof of Theorem 1.9. Let (x i , y i , z i ) ∈ F 3 , i = 1, 2, 3 be three points. Consider the set of points (x, y, z) ∈ F 3 satisfying
(x − x 1 ) 2 + (y − y 1 ) 2 + (z − z 1 ) 2 = r 2 , (3.1) (x − x 2 ) 2 + (y − y 2 ) 2 + (z − z 2 ) 2 = r 2 , (3.2) (x − x 3 ) 2 + (y − y 3 ) 2 + (z − z 3 ) 2 = r 2 . (3.3)
Note that these three equations are satisfied precisely when
(x i , y i , z i ) ∈ S(x, y, z), i = 1, 2, 3, where S(x, y, z) = {(x , y , z ) ∈ F 3 : (x − x ) 2 + (y − y ) 2 + (z − z ) 2 = r 2 }.2(x, y, z) · (x 2 − x 1 , y 2 − y 1 , z 2 − z 1 ) = x 2 1 + y 2 1 + z 2 1 − x 2 2 − y 2 2 − z 2 2 , (3.5) 2(x, y, z) · (x 3 − x 1 , y 3 − y 1 , z 3 − z 1 ) = x 2 1 + y 2 1 + z 2 1 − x 2 3 − y 2 3 − z 2 3 . (3.6)
Note that the vectors (x 2 − x 1 , y 2 − y 1 , z 2 − z 1 ) and (x 3 − x 1 , y 3 − y 1 , z 3 − z 1 ) are parallel precisely if the points (x 1 , y 1 , z 1 ), (x 2 , y 2 , z 2 ), and (x 3 , y 3 , z 3 ) are collinear (and thus there are no solutions to (3.1),(3.2), and (3.3)).
If (x 2 − x 1 , y 2 − y 1 , z 2 − z 1 ) and (x 3 − x 1 , y 3 − y 1 , z 3 − z 1 ) are not parallel, then the set of points satisfying (3.1), (3.2), and (3.3) is precisely the set of points satisfying (3.1), (3.5), and (3.6); this is the intersection of a line and a sphere of radius r 2 . Again, since a sphere of radius r 2 cannot contain any lines, this intersection consists of at most 2 points. To summarize, |{(x, y, z) ∈ F 3 : (x, y, z) satisfies (3.1), (3.2), and (3.3)}| ≤ 2.
(3.7)
Let L 1 = {φ(p) : p ∈ P}, where φ is the map defined in Section 2.3, and we identify a point p ∈ P with the corresponding sphere of zero radius. Let L 2 = {φ(S p ) : p ∈ P}, where S p is the sphere with center p and radius r.
Apply Lemma 3.2 to L 1 L 2 , where the first set of lines are coloured red and the second set is coloured blue. The bound (3.7) shows that there cannot exist a doubly-ruled surface with three red lines in one ruling and three blue lines in the other ruling. We conclude that L 1 L 2 determines O(n 3/2 ) bichromatic 2-rich points. Thus there are O(n 3/2 ) pairs of points from P that satisfy (1.5).
Finally, we will prove Theorem 1.13. The main observation is that while a pencil that is exactly k-rich determines k 2 pairs of tangent spheres, this pencil only determines k − 1 or fewer pointsphere incidences (i.e. at most k − 1 tangencies between a sphere of zero radius and a sphere of nonzero radius).
Proof of Theorem 1.13. Let L 1 be the set of lines associated to the points from P (i.e. spheres of radius 0), and let L 2 be the set of lines associated to the spheres from S. Observe that by Remark 2.1, each k-rich pencil can contribute at most k − 1 point-sphere incidences. Applying Lemma 3.4 to L 1 L 2 , we see that for each k ≥ 3 the set of pencils of richness between k and 2k can contribute at most 2kO(n 3/2 k −3/2 ) = O(n 3/2 k −1/2 ) incidences. Summing dyadically in k, we conclude that the set of pencils of richness at least 3 can contribute O(n 3/2 ) incidences.
It remains to control the number of incidences arising from 2-rich points. Let A = 2η −1 √ n . With this choice of A, apply Lemma 3.2 to L 1 L 2 , where the first set of lines are coloured red and the second set is coloured blue. If L 1 L 2 contains O(An) bichromatic 2-rich points then we are done. If not, then there is a doubly-ruled surface Z with one ruling that contains at least A red lines and a second ruling that contains at least A blue lines. Let L 2 ⊂ L 2 be the set of blue lines from one of these rulings (recall that blue lines correspond to spheres from S). Recall that each red line in L 1 that is not contained in Z intersects Z in at most two points. Thus since A ≥ 2η −1 √ n, by pigeonholing there must exist a line ∈ L 2 that is incident to fewer than ηA lines from L 1 that are not contained in Z. This implies that the corresponding sphere S ∈ S is not η-non-degenerate, which contradicts the assumption that all of the spheres in S are η-non-degenerate.
Improvements over R
In this section we will show how Theorem 1.8 can be improved when F = R. The main tool will be the following polynomial partitioning theorem due to Guth [9]. While the definition of the dimension of a real algebraic variety is slightly technical, we will only use the elementary facts that points in R d are algebraic varieties of dimension 0 and lines are algebraic varieties of dimension 1. Applying Theorem 4.1 to a set of points and to a set of lines in R 3 (with parameter D/2 in each case) and taking the product of the resulting partitioning polynomials, we obtain a polynomial whose zero-set efficiently partitions a set of points and a set of lines simultaneously. We will record this observation below. Note that while Corollary 4.2 guarantees that few points are contained in each cell and few lines meet each cell, it is possible that many points and lines are contained in the "boundary" Z(P ) of the cells. The next lemma helps us understand what configurations of points and lines are possible inside Z(P ). Proof. Statements of this form appear frequently in the literature (see e.g. Theorem 1.9 from [21]), and follow directly from the machinery of Guth and the author developed in [12]. We will briefly outline the proof here. We say a point p ∈ Z(P ) is 3-flecnodal if there are at least three distinct lines that contain p and are contained in Z(P ). If a line ⊂ Z(P ) contains more than CD 3-flecnodal lines for some absolute constant C, then all but finitely many points of must be 3-flecnodal (i.e. a Zariski-dense subset of is 3-flecnodal). If there are ≥ CD 2 lines ⊂ Z(P ), each of which is 3-flecnodal at all but finitely many points, then there is a Zariski-dense subset O ⊂ Z(P ) so that for each point p ∈ O there are at least three lines containing p and contained in Z(P ). But since P is irreducible, this immediately implies that Z(P ) is a plane. The next two lemmas will help us understand the incidence geometry of configurations of coplanar lines coming from the Heisenberg group. Recall that throughout this section, F = R and E = C. Proof. Let p, q ∈ P. Define V p = p∈ ⊂H , and define V q similarly. Then V p (resp. V q ) is precisely the intersection of H with the tangent plane of H at p (resp. q). In particular, V p ∩ V q is contained in a complex line, and P\{p, q} ⊂ V p ∩ V q . Since the choice of p and q was arbitrary, we conclude that P is a set of at least 5 points, and every subset of cardinality |P| − 2 is collinear. This implies that all the points of P are collinear. Proof. By Corollary 3.4, for each k ≥ 3 the number of k-rich points determined by L is O(n 3/2 k −3/2 ). Since a k-rich point contributes k incidences, we conclude that the number of incidences coming from points with richness ≥ 3 is O(n 3/2 ). The number of incidences coming from points with richness ≤ 2 is at most 2m.
We are now ready to prove the main result of this section. where C is an absolute constant Proof. Our proof is closely modeled on the techniques of Sharir and Zlydenko from [21]. We will prove the result by induction on m; the base case for our induction will be when m ≤ m 0 , where m 0 is an absolute constant to be specified below. Observe that if m 2 ≥ cn 3 for a fixed constant c > 0, then then the result follows from Henceforth we will assume that m 2 ≤ cn 3 ,
(4.3)
where c is a constant that will be specified below.
For each line ∈ L of the form (0, ω, t) + C(1, u,ω), let q ∈ R 4 be the point (t, u, Re ω, Im ω).
Define Q = {q : ∈ L}.
For each point p ∈ P, let W p ⊂ R 4 be the line described by (2.13). Define
W = {W p : p ∈ P}.
By Lemma 4.5, for any set of points Q ⊂ Q and any set of lines W ⊂ W, we have the incidence bound I(Q , W ) = O(|Q | 3/2 + |W |). Let A : R 4 → R 3 be a surjective linear map; we will choose this map so that the image of every line in W remains a line, and no additional incidences are added. We will abuse notation slightly by replacing Q with the set {A(q) : q ∈ Q} (so now Q ⊂ R 3 ) and replacing W with the set {A(W ) : W ∈ W} (so now W is a set of lines in R 3 ). Note that (4.4) remains true with these new definitions of Q and W.
Define D = c min(n 3/5 m −2/5 , m 1/2 ) , (4.5) where c > 0 is the same constant as in (4.3). In the analysis that follows, "Case 1" will refer to the situation where n 3/5 m −2/5 ≥ m 1/2 , and "Case 2" will refer to the situation where 3/5 m −2/5 < m 1/2 . Observe that D ≤ cn 1/3 . Indeed, in Case 1 we have m ≥ n 2/3 and thus D ≤ n 3/5 m −2/5 ≤ cn 1/3 . In Case 2 we have m ≤ n 2/3 , and again D ≤ cm 1/2 ≤ cn 1/3 . If we select m 0 ≥ c −2 , then we have ensured that cm 1/2 > 1. Inequality (4.3) implies that cn 3/5 m −2/5 ≥ 1; thus we can assume that D ≥ 1. In summary, we have 1 ≤ D ≤ cn 1/3 . (4.9)
Our next task is to control the number of incidences formed by lines in W . Factor P into irreducible components P 1 · · · P h · P h+1 · · · P k , where the polynomials P 1 , . . . , P h each have degree at least two and the polynomials p h+1 . . . p k have degree one (it is possible that all polynomials have degree at least two or no polynomials have degree at least two. In the former case we set k = h and in the latter case we set h = −1). For each index i, define
Q i = Q ∩ Z(P i ) \ 1≤j<i Q j .
We have Q ∩ Z(P ) = k i=1 Q i . Similarly, let W i be the set of lines that are contained in Z(P i ) and are not contained in Z(P j ) for any index j < i.
If q ∈ for some q ∈ Q i and ∈ W j with i = j, then the line must intersect Z(P i ), and cannot be contained in Z(P i ). Since can intersect Z(P i ) in at most deg(P i ) points, we can bound the number of such "cross-index" incidences as follows.
|{(q, ) : q ∈ , q ∈ Q i , ∈ W j for some index j = i}| It remains to control incidences p ∈ where p ∈ Q i and ∈ W i . Let Q rich i consist of those points q ∈ Q i that are incident to at least 3 lines from W i . Define Q poor i = Q i \Q rich i . By Lemma 4.3, for each index 1 ≤ i ≤ h there are ≤ C 1 deg(P i ) 2 "bad" lines for some absolute constant C 1 . For each index 1 ≤ i ≤ h, let W bad i ⊂ W i be the set of bad lines associated to P i and let W good
i = W i \W bad i .
If we choose the constant c from (4.5) sufficiently small, then
h i=1 |W bad i | ≤ h i=1 C 1 deg(P i ) 2 ≤ C 1 D 2 ≤ m/2.
Thus by the induction hypothesis we have
h i=1 I(Q rich i , W bad i ) ≤ I h i=1 Q rich i , h i=1 W bad i ≤ C m 2 3/5 n 3/5 + h i=1 W bad i + h i=1 Q rich i .
If ∈ W good i then can be incident to at most C 2 D bad points, for some absolute constant C 2 . Since If q ∈ Q poor i then q can be incident to at most two lines from W i . Thus as long as C ≥ 2 we have
h i=1 I(Q i , W i ) ≤ C m 2 3/5 n 3/5 + h i=1 |W i | + h i=1
|Q i | + O(m 3/5 n 3/5 ). (4.11)
It remains to bound I(Q i , W i ) for h + 1 ≤ i ≤ k. By Lemma 4.4, if |W i | ≥ 5, then the lines in W i correspond to a set of points in H that are collinear. In particular, this implies
I(Q i , W i ) ≤ |Q i | + |W i |.
On the other hand, if |W i | ≤ 4 then
I(Q i , W i ) ≤ 4|Q i |.
Combining these bounds with (4.11) and summing in i, we conclude that if C ≥ 4 then
k i=1 I(Q i , W i ) ≤ C m 2 3/5 n 3/5 + h i=1 |W i | + h i=1 |Q i | + C k i=h+1 |W i | + k i=h+1
|Q i | ≤ C m 2 3/5 n 3/5 + m + n .
(4.12)
Combining (4.7), (4.8), (4.9), (4.10), and (4.12), we conclude that there is an absolute constant C 2 so that I(Q, W) ≤ C m 2 3/5 n 3/5 + m + n + C 2 m 3/5 n 3/5 . If the constant C is chosen sufficiently large so that C ≥ 2 3/5 C 2 , then we obtain (4.2) and the induction closes.
Proof of Theorem 1.14. Let L = {φ(S) : S ∈ S} and let k ≥ 3. We need to prove that L determines O(n 3/2 k −5/2 + n/k) k-rich points. When k is small the result follows from Theorem 1.8, so we can assume that k ≥ C, for some absolute constant C to be determined later. Let P ⊂ H be the set of k-rich points determined by L, and let m = |P|. We clearly have I(P, L) ≥ km. On the other hand, Proposition 4.6 implies that I(P, L) = O(m 3/5 n 3/5 + m + n).
(4.14)
Thus if the constant C is selected sufficiently large compared to the implicit constant in (4.14), then m = O(n 3/2 k −5/2 + nk −1 ), as desired.
Figure 1 :
1A pencil of contacting spheres. The left and right images are two different perspectives of the same set. Note that there is precisely one sphere of radius 0. If F = R, then each sphere on one side of the center point has positive radius, while each sphere on the other side has negative radius.
Example 1 . 16 .
116Let n = 2m 2 and let S be a disjoint union S = m j=1 (S j ∪ S j ), where for each index j, each of S j and S j is a set of m spheres contained in complimentary conic sections. The spheres in S determine m 3 = (n/2) 3/2 2-rich pencils, and no pair of complimentary conic sections satisfy (1.6).
Example 1 . 17 .
117Let n = m 4 , let [m] = {1, . . . , m}, and let S consist of all oriented spheres centered at (a, b, c) of radius d, where a, b, c, d ∈ [m]
[
p 14 : p 24 : p 34 : p 23 : p 31 : p 12 ] = [ 1 : u : v : sv − tu : t : −s ]. (2.2) We will call this point the Plücker coordinates of the line. Conversely, the point with Plücker coordinates [1 : p 24 : p 34 : p 23 : p 31 : p 12 ] can be identified with the line (0, −p 12 , p 31 )+E(1, p 24 , p 34 ). Two lines and˜ are coplanar (and thus either intersect or are parallel) if and only if their respective Plücker points p and p satisfy the relation K(p, p ) = 0.
, since −1 is not a square in F , the set S(x, y, z) does not contain any lines (see e.g. [20, Lemma 6]), so (3.1), (3.2), and (3.3) have no solutions when the points (x i , y i , z i ), i = 1, 2, 3 are collinear. Subtracting (3.2) from (3.1), and subtracting (3.3) from (3.1), we obtain the equations
Theorem 4. 1 .
1Let V be a set of real algebraic varieties in R d , each of which has dimension e and is defined by a polynomial of degree at most C. Then for each D ≥ 1, there is a d-variate "partitioning" polynomial P of degree at most D so that R d \Z(P ) is a disjoint union of O(D d ) "cells" (open connected sets), and each of these cells intersect O(|V|D e−d ) varieties from V. The implicit constant depends on d and C, but (crucially) is independent of D and |V|.
Corollary 4 . 2 .
42Let P be a set of points in R 3 and let W be a set of lines in R 3 . Then for each D ≥ 1, there is a polynomial P of degree at most D so that R 3 \Z(P ) is a disjoint union of O(D 3 ) cells; each cell contains O(|P|D −3 ) points from P; and each cell intersects O(|V|D −2 ) lines from W.
Lemma 4 . 3 .
43Let P ∈ R[x, y, z] be irreducible and let D = deg(P ). If Z(P ) is not a plane, then there are at most O(D 2 ) "bad" lines contained in Z(P ). If ⊂ Z(P ) is not a bad line, then there are O(D) "bad" points p ∈ . If q ∈ is not a bad point, then it is incident to at most one additional line that is contained in Z(P ).
Lemma 4 . 4 .
44Let P ⊂ H be a set of points with |P| ≥ 5. Suppose that every pair of points is connected by a line in the Heisenberg group. Then P is contained in a (complex) line.
Lemma 4. 5 .
5Let P be a set of m points and let L be a set of n lines in the Heisenberg group that are not parallel to the xy plane. Then I(P, L) = O(n 3/2 + m).
Proposition 4. 6 .
6Let P be a set of m points and let L be a set of n lines in the Heisenberg group that are not parallel to the xy plane.
Lemma 4.5; indeed, if m 2 ≥ cn 3 then n 3/2 + m = O(m) and thus I(P, L) = O(n 3/2 + m) = O(m 3/5 n 3/5 + m + n).
4.2 to Q and W with this choice of D; we obtain a partitioning polynomial P ∈ R[x, y, z]. R 3 \Z(P ) is a union of O(D 3 ) cells, each of which contains O(n/D 3 ) points from Q and each of which intersects O(m/D 2 ) line from W. In Case 1, we use (4.4) to control the number of point-line incidences inside the cells. If we index the cells O 1 , . . . , O t with t = O(D 3 ) and define W i to be the set of lines from W that intersect O
= W W , where W consists of the lines not contained in Z(P ) and W consists of the lines contained in Z(P ). Since each line in W can intersect Z(P ) at most D times, we have I(Q ∩ Z(P ), W ) ≤ Dm ≤ cm 3/5 n 3/5 .
If two lines are parallel, then we say these lines pass through a common point at infinity.
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Incidences of not too degenerate hyperplanes. G Elekes, C D Tóth, Proc. 21st Annu. ACM Sympos. 21st Annu. ACM SymposG. Elekes and C. D. Tóth. Incidences of not too degenerate hyperplanes. In Proc. 21st Annu. ACM Sympos. Comput. Geom., pages 16-21, 2005.
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On sets of distances of n points in Euclidean space. P Erdős, Magyar Tudományos Akadémia Matematikai Kutató Intézet Közleményei. 5P. Erdős. On sets of distances of n points in Euclidean space. Magyar Tudományos Akadémia Matematikai Kutató Intézet Közleményei, 5:165-169, 1960.
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. L Guth, Polynomial Methods in Combinatorics. AMS. L. Guth. Polynomial Methods in Combinatorics. AMS, 2018.
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Unit distances in three dimensions. H Kaplan, J Matoušek, Z Safernová, M Sharir, Combin. Prob. Comput. 21H. Kaplan, J. Matoušek, Z. Safernová, and M. Sharir. Unit distances in three dimensions. Combin. Prob. Comput., 21:597-610, 2012.
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An improved bound on the Minkowski dimension of Besicovitch sets in R 3. N Katz, I Laba, T Tao, Ann. of Math. 152N. Katz, I. Laba, and T. Tao. An improved bound on the Minkowski dimension of Besicovitch sets in R 3 . Ann. of Math., 152:383-446, 2000.
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Point-plane incidences and some applications in positive characteristic. M Rudnev, Combinatorics and Finite Fields: Difference Sets, Polynomials, Pseudorandomness and Applications. K. Schmidt and A. WinterhofDe GruyterM. Rudnev. Point-plane incidences and some applications in positive characteristic. In K. Schmidt and A. Winterhof, editors, Combinatorics and Finite Fields: Difference Sets, Polynomials, Pseudorandomness and Applications, pages 211-240. De Gruyter, 2019.
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| []
|
[
"Extractive Summarization of Related Bug-fixing Comments in Support of Bug Repair",
"Extractive Summarization of Related Bug-fixing Comments in Support of Bug Repair"
]
| [
"Rrezarta Krasniqi [email protected] \nDept. of Computer Science and Engineering\nUniversity of Notre Dame Notre Dame\nINUSA\n"
]
| [
"Dept. of Computer Science and Engineering\nUniversity of Notre Dame Notre Dame\nINUSA"
]
| []
| When developers investigate a new bug report, they search for similar previously fixed bug reports and discussion threads attached to them. These discussion threads convey important information about the behavior of the bug including relevant bug-fixing comments. Often times, these discussion threads become extensively lengthy due to the severity of the reported bug. This adds another layer of complexity, especially if relevant bug-fixing comments intermingle with seemingly unrelated comments. To manually detect these relevant comments among various cross-cutting discussion threads can become a daunting task when dealing with high volume of bug reports. To automate this process, our focus is to initially extract and detect comments in the context of query relevance, the use of positive language, and semantic relevance. Then, we merge these comments in the form of a summary for easy understanding. Specifically, we combine Sentiment Analysis, and the TextRank Model with the baseline Vector Space Model (VSM). Preliminary findings indicate that bug-fixing comments tend to be positive and there exists a semantic relevance with comments from other cross-cutting discussion threads. The results also indicate that our combined approach improves overall ranking performance against the baseline VSM. | 10.1109/apr52552.2021.00014 | [
"https://arxiv.org/pdf/2103.15211v2.pdf"
]
| 232,404,697 | 2103.15211 | 383ddde3639c84384342fa03e77eee64745b9af9 |
Extractive Summarization of Related Bug-fixing Comments in Support of Bug Repair
25 May 2021
Rrezarta Krasniqi [email protected]
Dept. of Computer Science and Engineering
University of Notre Dame Notre Dame
INUSA
Extractive Summarization of Related Bug-fixing Comments in Support of Bug Repair
25 May 20212021 IEEE/ACM International Workshop on Automated Program Repair (APR)
When developers investigate a new bug report, they search for similar previously fixed bug reports and discussion threads attached to them. These discussion threads convey important information about the behavior of the bug including relevant bug-fixing comments. Often times, these discussion threads become extensively lengthy due to the severity of the reported bug. This adds another layer of complexity, especially if relevant bug-fixing comments intermingle with seemingly unrelated comments. To manually detect these relevant comments among various cross-cutting discussion threads can become a daunting task when dealing with high volume of bug reports. To automate this process, our focus is to initially extract and detect comments in the context of query relevance, the use of positive language, and semantic relevance. Then, we merge these comments in the form of a summary for easy understanding. Specifically, we combine Sentiment Analysis, and the TextRank Model with the baseline Vector Space Model (VSM). Preliminary findings indicate that bug-fixing comments tend to be positive and there exists a semantic relevance with comments from other cross-cutting discussion threads. The results also indicate that our combined approach improves overall ranking performance against the baseline VSM.
I. INTRODUCTION
A "bug fixing" comment is a comment in a conversation thread attached to a bug report that describes the bug fix. When developers report bugs via issue tracking systems, the conversations between the end-users and developers are presented as a collection of threads. When a fix is communicated, the person who has made the fix or recommends a fix often concludes the conversation with a more-positive language that summarizes the problems and the repairs. Recent studies have indicated that sentiment patterns were noticeable in opensource bug report discussion threads [4,10]. We also observed, that positive sentiment patterns were noticeable when bug reports were concluded as 'fixed'. Our broader objective is to automate this process suitably for large-scale open-source platforms. However, there are two emerging factors that need to be taken into account when dealing with open-source platforms [5,6,8]. The most pressing ones span across two directions, primarily time dimension and content dimension. When dealing with the former one, studies showed that opensource platforms were plagued with thousands of bug reports daily [1,13]. Due to this sheer volume of bugs, it becomes prohibitively time consuming for the developers to fix these bugs efficiently and effectively [11]. When dealing with the latter one, a recent qualitative study conducted by Arya et. al [2] showed that some of these issue discussion threads over time become extremely lengthy and complex to comprehend because both developers and end-users carry different backgrounds. Consequently, this communication scenario leads to lengthy discussions. In hindsight, the content factor poses a challenge to developers who must reason which information hidden within these lengthy discussion threads is relevant and useful for finding the root cause for the new bugs. This adds another layer of complexity, especially if relevant bug-fixing comments intermingle with seemingly unrelated comments. Motivated by these observations, this work presents a three-pronged approach that recommends bug-fixing comments extracted from crosscutting issue discussion threads that are beneficial for recommending a bug fix. These bug-fixing comments are analysed and extracted by considering user query relevance, positive language, and semantic relevance. Specifically, we combine Sentiment Analysis (SA), and the TextRank Model (TR) with the baseline Vector Space Model (VSM) as employed by Zhou et al. [16]. We use SA to detect positive sentiments in the comments. We then use the TR heuristic to build the semantic connections amongst similar related comments from those cross-cutting discussion threads.
II. OVERVIEW OF APPROACH Our approach recommends comments extracted from issuetracking cross-cutting discussion threads in three steps. First, it uses the baseline VSM to retrieve user query-related comments with initial keyword scores. Second, it uses Sentiment Analysis (SA) to compute sentiment keyword scores based on lookup bonus and penalty opinion lexicon, and re-rank comments written in a more positive style from step-1. Finally, it uses meta-heuristic technique such as TextRank (TR) to re-rank keywords based on their co-occurrence and their order of importance from step-2 that are semantically connected. In sum, the above steps represent a combined linear weighted function where keyword weights of each comment are influenced from the combined scores obtained by VSM, SA, and TR.
A. RetroRank Tool
We have built a GUI-based tool that we refer to as Retro-Rank. In a nutshell, RetroRank takes as an input a user query (e.g., a set of keywords extracted from bug's title or bug's long description) and returns as an output a list of recommended bug-fixing comments extracted from cross-cutting discussion threads. RetroRank relies on MySQL database where it stores all OSS bug reports and bug repositories. RetroRank builds upon two existing packages. Tkinter is a standard GUI library that is used to build the GUI framework. Sklearn is another library that is used to build the underlying functionalities of our approach. Broadly speaking, RetroRank is designed to adapt to any project that supports modern development practices. The inclusion of 'pre-merge' mechanism is an example where relevant comments can be used as part of code review [7,12].
III. EVALUATION STRATEGY
A. Preliminary Synthetic Study
In the synthetic study, we analyzed 25 real bugs to compare our approach against several configurations. Our evaluation strategy employed four steps. First, we manually identified the goldset of bugs and bug-fixing comments. Second, we recruited 12 developers to create 'search queries' for new bugs. Third, we used those 'user created queries" from the previous step to run each configuration. Finally, based on the results retrieved from each configuration, we counted and marked the ranking position where the goldset (i.e., bug-fixing comments from past resolved bugs) appeared in each configuration's retrieved list. Due to space limitation, we only reported the significance of ranking performed of each configuration as denoted in Table I, where µ denotes the average ranking obtained by each configuration. We plan to extend this work by fully reporting its findings.
B. Preliminary Results: Overall Ranking Performance
Preliminary results indicate that VSM+SA+TR achieved the highest performance, followed by VSM+SA, VSM+TR, and the baseline VSM as employed by Zhou et. al [16]. Findings in Table I showed that the average ranking position µ of the recommended comment for VSM+SA+TR was 1.8, whereas for the VSM was 9.1. This indicates, VSM+SA+TR placed the recommended comment on average around first position, versus VSM which placed it around the ninth position. We also noted that both VSM+SA and VSM+TR ranked in a higher position than the baseline VSM. Overall, results showed that differences in ranking position were statistically significant as reported in Table I, H 5 (3.0E−11) with p-value of 0.05 and a confidence level of 0.95. Student's Paired t-Test [14] was used as a primary metric as done in other studies [3,9,15].
IV. CONCLUSION AND EVALUATION PLANS
In this position paper, we presented a novel idea of how issue tracking cross-cutting discussion threads can be used as the core content in support of bug repair. As part of the evaluation plan, we will conduct a larger user study where we will empirically compare the best-performing approach to the baseline VSM. Our objective is to also measure comment(s) relevance returned by each approach. Additionally, we will perform a qualitative user-study where the rationale behind our work will be explained in-depth.
TABLE I :
IStatistical summary of the results. Value n denotes the number of bugs evaluated in terms of ranking performance. µ denotes average ranking position obtained by each configuration. The rest represent Student's Paired t-Test metrics.Metric
Approach
n
µ
p
t
t crit
Decision
Config-1
VSM+SA+TR
VSM+SA
25
1.8
3.4
2.8E-05
-4.812
2.0301
Reject
Config-2
VSM+SA+TR
VSM+TR
25
1.8
3.7
2.9E-06
-5.557
2.0301
Reject
Config-3
VSM
VSM+TR
25
9.1
3.7
1.3E-09
8.146
2.0638
Reject
Config-4
VSM
VSM+SA
25
9.1
3.4
2.0E-09
8.004
2.0638
Reject
Config-5
VSM
VSM+SA+TR
25
9.1
1.8
3.0E-11
9.513
2.0638
Reject
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| []
|
[
"Growth models on the Bethe lattice",
"Growth models on the Bethe lattice"
]
| [
"Abbas Ali Saberi \nDepartment of Physics\nInstitut für Theoretische Physik\nUniversity of Tehran\nPost Office Box 14395-547TehranIran\n\nUniversität zu Köln\nZülpicher Str. 7750937KölnGermany\n\nSchool of Particles and Accelerators\nInstitute for Research in Fundamental Sciences (IPM)\nPost Office Box19395-5531TehranIran\n"
]
| [
"Department of Physics\nInstitut für Theoretische Physik\nUniversity of Tehran\nPost Office Box 14395-547TehranIran",
"Universität zu Köln\nZülpicher Str. 7750937KölnGermany",
"School of Particles and Accelerators\nInstitute for Research in Fundamental Sciences (IPM)\nPost Office Box19395-5531TehranIran"
]
| []
| I report on an extensive numerical investigation of various discrete growth models describing equilibrium and nonequilibrium interfaces on a substrate of a finite Bethe lattice. An unusual logarithmic scaling behavior is observed for the nonequilibrium models describing the scaling structure of the infinite dimensional limit of the models in the Kardar-Parisi-Zhang (KPZ) class. This gives rise to the classification of different growing processes on the Bethe lattice in terms of logarithmic scaling exponents which depend on both the model and the coordination number of the underlying lattice. The equilibrium growth model also exhibits a logarithmic temporal scaling but with an ordinary power law scaling behavior with respect to the appropriately defined lattice size. The results may imply that no finite upper critical dimension exists for the KPZ equation. | 10.1209/0295-5075/103/10005 | [
"https://arxiv.org/pdf/1307.5661v1.pdf"
]
| 119,274,949 | 1307.5661 | 4705f468e58329a19e9a9c552a98e49e2b4b4bad |
Growth models on the Bethe lattice
22 Jul 2013
Abbas Ali Saberi
Department of Physics
Institut für Theoretische Physik
University of Tehran
Post Office Box 14395-547TehranIran
Universität zu Köln
Zülpicher Str. 7750937KölnGermany
School of Particles and Accelerators
Institute for Research in Fundamental Sciences (IPM)
Post Office Box19395-5531TehranIran
Growth models on the Bethe lattice
22 Jul 2013(Dated: May 11, 2014)
I report on an extensive numerical investigation of various discrete growth models describing equilibrium and nonequilibrium interfaces on a substrate of a finite Bethe lattice. An unusual logarithmic scaling behavior is observed for the nonequilibrium models describing the scaling structure of the infinite dimensional limit of the models in the Kardar-Parisi-Zhang (KPZ) class. This gives rise to the classification of different growing processes on the Bethe lattice in terms of logarithmic scaling exponents which depend on both the model and the coordination number of the underlying lattice. The equilibrium growth model also exhibits a logarithmic temporal scaling but with an ordinary power law scaling behavior with respect to the appropriately defined lattice size. The results may imply that no finite upper critical dimension exists for the KPZ equation.
The Kardar-Parisi-Zhang (KPZ) equation [1] is a simple nonlinear Langevin equation that describes the macroscopic properties of a wide variety of nonequilibrium growth processes [2,3]. This equation is also related to many other important physical problems such as the Burgers equation [4], dissipative transport in the driven-diffusion equation [5] and directed polymers in a random medium [6][7][8]. The KPZ equation for a stochastically growing interface described by a single valued height function h(x, t) on a d-dimensional substrate x, is
∂ t h(x, t) = ν∇ 2 h + λ 2 (∇h) 2 + η(x, t),(1)
where the first term represents relaxation of the interface caused by a surface tension ν, the second describes the nonlinear growth locally normal to the surface, and the last is an uncorrelated Gaussian white noise in both space and time with zero average η(x, t) = 0 and η(x, t)η(x ′ , t ′ ) = 2Dδ d (x − x ′ )δ(t − t ′ ), mimicking the stochastic nature of the growth process. The steady state interface profile is usually described in terms of the roughness: w = h 2 (x, t) − h(x, t) 2 which for a system of size L behaves like L α f (t/L α/β ), where f (x) → const as x → ∞ and f (x) ∼ x β as x → 0, so that w grows with time like t β until it saturates to L α when t ∼ L α/β . α and β are the roughness and the growth exponents, respectively, whose exact values are known only for the special case d = 1 as α = 1/2 and β = 1/3. The ratiō z = α/β is called dynamic exponent. A scaling relation α +z = 2 follows from the invariance of Eq. (1) to an infinitesimal tilting of the surface which retains only one independent exponent, say α, in the KPZ dynamics.
It is well known that for dimensions d ≤ 2 the surface is always rough, while for d > 2, the equation (1) * Electronic address: [email protected] shows two different regimes in terms of the dimensionless strength of the nonlinearity coefficient whose critical value λ c separates flat and rough surface phases. In the weak coupling (flat) regime (λ < λ c ) the nonlinear term is irrelevant and the behavior is governed by the λ = 0 fixed point i.e., the linear Edward-Wilkinson (EW) equation [9], for which the exponents are known exactly: α = (2 − d)/2 and β = (2 − d)/4. In the more challenging strong-coupling (rough) regime (λ > λ c ), where the nonlinear term is relevant, the behavior of the KPZ equation is quite controversial and characterized by anomalous exponents. There is, however, a longstanding controversy concerning the existence and the value of an upper critical dimension d c above which, regardless of the strength of the nonlinearity, the surface remains flat.
At odds with many theoretical discussions supporting the existence of a finite upper critical dimension [10] between three and four [11,12], and an analytical evidence that d c is bounded from above by four [13], or many others suggesting d c ≈ 2.5 [14] or d c = 4 [15,16], there is nevertheless a long list of evidence questioning these suggestions [17][18][19][20][21][22][23], some of which concluded that no finite upper critical dimension exists at all (for the most recent study, see [23]).
Here I study the infinite dimensional properties of growth models from two different KPZ and EW classes and compare them to realize whether the nonlinear term in (1) is relevant in this limit. It is inspired by the fact that if the nonlinear term is irrelevant in infinite dimensions, then one would expect the same statistical behavior for the models coming from each of the two classes. The result would shed a light on the existence of the upper critical dimension for the KPZ equation. To this aim, I investigate two discrete nonequilibrium models, ballistic deposition (BD) and restricted solid-on-solid (RSOS) models which are believed to be in the KPZ class [2,[24][25][26][27][28], as well as an equilibrium model, random deposition with surface relaxation (RDSR) [29], which belongs to For a given finite lattice of fixed size k, one lattice site is randomly chosen at each step and a particle is added to that site which can either increase the height according to the standard rules of BD and RSOS models, or it can diffuse through the neighboring edges until it finds the column with a local minima in the searched area according to the RDSR model. the EW class, all defined on the Bethe lattice, an effectively infinite dimensional lattice. I find that the models from different universality classes correspond to different statistical growth properties and scaling behavior, the evidence that questions the existence of a finite upper critical dimension for the KPZ equation.
Due to its distinctive topological structure, several statistical models involving interactions defined on the Bethe lattice [30] are exactly solvable and computationally inexpensive [31]. Various systems including magnetic models [30], percolation [32][33][34][35], nonlinear conduction [36], localization [32,37], random aggregates [38,39] and diffusion processes [40][41][42] have been studied on the Bethe lattice whose analytic results gave important physical insights to subsequent developments of the corresponding research fields. The Bethe lattice is defined as a graph of infinite points each connected to z neighbors (the coordination number) such that no closed loops exist in the geometry (see Fig. 1). A finite type of the graph with boundary is also known as a Cayley tree and possesses the features of both one and infinite dimensions: since N k , the total number of sites in a Bethe lattice with k shells, is given as
N k = [z(z −1) k −2]/(z −2), the lattice dimension defined by d = lim k→∞ [ln N k / ln k] is infinite.
It is therefore often mentioned in the literature that the Bethe lattice describes the infinite-dimensional limit of a hypercubic lattice. As the lattice grows the number of sites in the surface, or the last shell, grows exponen-tially z(z − 1) k−1 . Therefore, as the number of shells tends to infinity, the proportion of surface sites tends to (z − 2)/(z − 1). By surface boundary we mean the set of sites of coordination number unity, the interior sites all have a coordination number z. Thus the vertices of a Bethe lattice can be grouped into shells as functions of the distances k from the central vertex. Here k is the number of bonds of a path between the shell and the central site and will be used as a measure of lattice size.
I have carried out extensive simulations of the BD, RSOS and RDSR models on a finite Bethe lattice of different size k and different coordination number z (Fig. 1). I will first compute the surface width w(t,k) as a function of time t and examine its various scaling properties. For a given lattice size k, each Monte Carlo time step is defined as the time required for N k particles to deposit on the surface. I show that the surface widths for the models feature a normal behavior as for a typical growth model on a regular lattice: w increases fast and finally saturates to a fixed value w s . Nevertheless, the best fit to our data at early time before saturation shows that w does not increase algebraically with time (w ∼ t β ), as is usually observed for growth models on ordinary lattices. Rather I find a logarithmic scaling behavior w ∼ ln(t) β [45] for all considered models. I also find that the saturated width w s of the interface for two BD and RSOS models behaves like a logarithmic scaling law w s ∼ ln(k) α , while for the equilibrium RDSR model, it shows an ordinary power law behavior with the lattice size, w s ∼ k α . The model-dependent exponents are also found to be functions of the coordination number of the underlying lattice, α i (z) and β i (z), where i= * , ⋆ and • denotes for BD, RSOS and RDSR models, respectively. Let me call α i (z) and β i (z), roughness and growth exponents, respectively. This different scaling form with respect to the finite-dimensional case can be associated to the exponential (instead of polynomial) growth of the volume of a shell as a function of its radius on the Bethe lattice.
I first consider the BD model on a lattice with z = 3. Fig. 2 shows the surface width w(t,k) as a function of logarithm of time, for the seventeen different sizes, from the 4th to the 20th generation. At early times before saturation, the data falls onto a straight line in a log-log scale indicating that the surface width initially increases algebraically with the logarithm of time as w ∼ ln(t) β * , with β * (z = 3) ≃ 0.75 (2). The best fit to the saturated width w s as a function of different lattice size, gives a scaling relation w s ∼ ln(k) α * , with α * (z = 3) ≃ 0.825 (10). As shown in the inset of Fig. 2, by standard rescaling of the parameteres, all curves collapse onto a single function.
In order to see how these exponents depend on z, series of extensive simulations were performed for the BD model on a Bethe lattice with different coordination number z = 3, 4, 5, 6 and 7. For each z, the surface width w(t,k) was measured for different lattice size k. Fig. 3 illustrates the saturated surface widths as functions of the lattice size for each coordination number. The solid lines in the figure show the best logarithmic fits of form w s ∼ ln(k) α * (z) to the data, assigning a z-dependent roughness exponent α * to each data set. I also find that the growth exponent β * is dependent on the coordination number of the substrate lattice. α * (z) and β * (z) are plotted in the inset of Fig. 3. As can be seen, α * decreases, while β * increases almost linearly with z.
To see whether such a logarithmic scaling behavior is a characteristic feature of the nonequilibrium growth models on the Bethe lattice, I have also measured the surface width for the RSOS model, for the seventeen different sizes, from the k = 4th to the 20th generation, and for z = 3. The results are shown in Fig. 4. These suggest the same scaling behavior but with different estimated roughness and growth exponents α ⋆ (z = 3) ≃ 0.90 (1) and β ⋆ (z = 3) ≃ 0.57(2), respectively. The exponents for this model depend again on the coordination number but both are decreasing with z. For z = 4, the exponents are estimated as α ⋆ (z = 4) ≃ 0.68(1) and β ⋆ (z = 4) ≃ 0.47(2) (see inset of Fig. 4).
In random deposition with surface relaxation (RDSR) [29,43], each particle is randomly dropped onto the surface, and it is allowed to diffuse around on the surface within a prescribed region about the deposited column, until it finds the column with a local minima in the searched area. The corresponding continuum model is the EW equation [9]. Fig. 5 summarizes the results obtained from implementing the RDSR model on a finite Bethe lattice of different size from the k = 4th to the 16th generation with z = 3. The first remarkable observation is that the crossover time to the steady state is quite larger than that needed for the above discussed nonequilibrium models, growing exponentially with k in this case. Therefore, the CPU time required for simulations to reach the desired accuracy is orders of magnitude higher. As shown in Fig. 5, the temporal logarithmic scaling behavior for this model again holds. I find the scaling relation w ∼ ln(t) β • (z) , with β • (z = 3) ≃ 0.51 (2). The most remarkable scaling feature observed in the RDSR model is that, unlike the two BD and RSOS models, the average saturated width w s has no longer a power law relation with the logarithm of the size, but with the size k itself. The resulting data is plotted in the inset of Fig. 5. I find that w s ∼ k α • (z) , with α • (z = 3) ≃ 0.60(1). The roughness and growth exponents again depend on the coordination number. Simulations for the RDSR model on a finite Bethe lattice of different size from k = 3th to the 14th generation for z = 4, provide a satisfactory estimation of the exponents: α • (z = 4) ≃ 0.562(10) (see inset of Fig. 5) and β • (z = 4) ≃ 0.46 (2).
To summarize, I have studied three different growth models on a substrate of a finite Bethe lattice with different coordination number. A different scaling behavior is seen with respect to the same models on the ordinary lattice or those on the fractal substrates [44]. Two models i.e., the BD and RSOS models, are chosen from the nonequilibrium growth processes in the KPZ universality class, and the third i.e., the RDSR model, is an equilibrium model from EW class. For all considered models, the surface width grows with a power law scaling relation with the logarithm of the time before saturation. The initial growth is characterized by an exponent which depends on the coordination number of the underlying Bethe lattice as well as on the model in question. In the steady state regime, the scaling behavior distinguishes between equilibrium and nonequilibrium models. The average saturated width for the nonequilibrium models has a power law scaling relationship with the logarithm of the lattice size, while for the equilibrium RDSR model, it shows a usual power law behavior with the lattice size (instead of the logarithm of the size).
If we admit that the Bethe lattice, as a substrate of a growth model, reflects the infinite-dimensional limit properties of the models, the present results would then imply that the nonlinear term in the KPZ equation has a relevant contribution at this limit, consequently questioning the existence of a finite upper critical dimension for the KPZ equation.
I would like to thank J. Krug and M. Sahimi for their useful comments, and H. Dashti-Naserabadi for his help with programming. Supports from the Deutsche Forschungsgemeinschaft via SFB/TR 12, and the Humboldt research fellowship are gratefully acknowledged. I also acknowledge partial financial supports by the research council of the University of Tehran and INSF, Iran.
FIG. 1 :
1Part of a Bethe lattice with coordination number z = 3 embedded in the plane which is considered here as a substrate of different growth models. The vertically incident particles can land at the top of the lattice sites represented by open small circles at different shells k = 0, 1, 2, · · · .
FIG. 2 :
2(Color online) Surface width w(t,k) for the BD model on a finite Bethe lattice of coordination number z = 3, as a function of logarithm of time, for the seventeen different sizes, from the k = 4th to the 20th generation. Inset: Data collapse for the same data with k > 6. The time is rescaled by ln(k)z * (having assumedz * = α * /β * = 0.825/0.75 = 1.1), and the width is rescaled by ln(k) α * with α * = 0.825. online) Main: saturated surface widths as functions of lattice size k, for the BD model on the finite Bethe lattices of different coordination number z = 3, 4, 5, 6 and 7. The solid lines show the best logarithmic fits of form ws ∼ ln(k) α * (z) to the data. Inset: the growth α * and roughness β * exponents as functions of z.
FIG
. 4: (Color online) Main: surface width w(t,k) for the RSOS model on a finite Bethe lattice of coordination number z = 3, as a function of logarithm of time, for the seventeen different sizes, from the k = 4th to the 20th generation. Inset: saturated surface widths as functions of lattice size k, for two different coordination numbers z = 3 and 4.FIG. 5: (Color online) Main: surface width w(t,k) for the equilibrium RDSR model on a finite Bethe lattice of coordination number z = 3, as a function of logarithm of time, for the thirteen different sizes, from the k = 4th to the 16th generation. Inset: saturated surface widths as functions of lattice size k, for two different coordination numbers z = 3 and 4. Unlike the nonequilibrium BD and RSOS models, ws here shows a power law scaling behavior ws ∼ k α • (z) , with the lattice size k.
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For the sake of simplicity, I use the same symbols for the exponents α and β, in the logarithmic scaling laws. For the sake of simplicity, I use the same symbols for the exponents α and β, in the logarithmic scaling laws.
| []
|
[
"A NEW DIAGNOSTIC DIAGRAM OF IONIZATION SOURCE FOR HIGH REDSHIFT EMISSION LINE GALAXIES",
"A NEW DIAGNOSTIC DIAGRAM OF IONIZATION SOURCE FOR HIGH REDSHIFT EMISSION LINE GALAXIES"
]
| [
"Kai Zhang ",
"Lei Hao "
]
| []
| []
| We propose a new diagram, the Kinematic-Excitation diagram (KEx diagram), which uses the [O III]/Hβ line ratio and the [O III] λ5007 emission line width (σ [O III] ) to diagnose the ionization source and physical properties of the Active Galactic Nuclei (AGNs) and the star-forming galaxies (SFGs). The KEx diagram is a suitable tool to classify emission-line galaxies (ELGs) at intermediate redshift because it uses only the [O III] λ5007 and Hβ emission lines. We use the SDSS DR7 main galaxy sample and the Baldwin−Phillips−Terlevich (BPT) diagnostic to calibrate the diagram at low redshift. We find that the diagram can be divided into 3 regions: one occupied mainly by the pure AGNs (KEx-AGN region), one dominated by composite galaxies (KEx-composite region), and one contains mostly SFGs (KEx-SFG region). AGNs are separated from SFGs in this diagram mainly because they preferentially reside in luminous and massive galaxies and have high [O III]/Hβ. The separation of AGN from star-forming galaxies is even cleaner thanks to the additional 0.15/0.12 dex offset in σ [O III] at fixed luminosity/stellar mass.We apply the KEx diagram to 7,866 galaxies at 0.3 < z < 1 in the DEEP2 Galaxy Redshift Survey, and compare it to an independent X-ray classification scheme using Chandra observation. X-ray AGNs are mostly located in the KEx-AGN region while X-ray SFGs are mostly located in the KEx-SFG region. Almost all of Type1 AGNs lie in the KEx-AGN region. These confirm the reliability of this classification diagram for emission line galaxies at intermediate redshift. At z∼2, the demarcation line between star-forming galaxies and AGNs should shift 0.3 dex higher in σ [O III] to account for evolution. | 10.3847/1538-4357/aab207 | [
"https://arxiv.org/pdf/1610.03495v1.pdf"
]
| 119,275,264 | 1610.03495 | dec6178df1e8e6bcf70872d12b64b90d9e4d9b35 |
A NEW DIAGNOSTIC DIAGRAM OF IONIZATION SOURCE FOR HIGH REDSHIFT EMISSION LINE GALAXIES
11 Oct 2016 Draft version July 19, 2018 July 19, 2018
Kai Zhang
Lei Hao
A NEW DIAGNOSTIC DIAGRAM OF IONIZATION SOURCE FOR HIGH REDSHIFT EMISSION LINE GALAXIES
11 Oct 2016 Draft version July 19, 2018 July 19, 2018Preprint typeset using L A T E X style emulateapj v. 5/2/11 Draft versionSubject headings: galaxies: active-galaxies:Seyfert-(galaxies:) quasars: emission lines
We propose a new diagram, the Kinematic-Excitation diagram (KEx diagram), which uses the [O III]/Hβ line ratio and the [O III] λ5007 emission line width (σ [O III] ) to diagnose the ionization source and physical properties of the Active Galactic Nuclei (AGNs) and the star-forming galaxies (SFGs). The KEx diagram is a suitable tool to classify emission-line galaxies (ELGs) at intermediate redshift because it uses only the [O III] λ5007 and Hβ emission lines. We use the SDSS DR7 main galaxy sample and the Baldwin−Phillips−Terlevich (BPT) diagnostic to calibrate the diagram at low redshift. We find that the diagram can be divided into 3 regions: one occupied mainly by the pure AGNs (KEx-AGN region), one dominated by composite galaxies (KEx-composite region), and one contains mostly SFGs (KEx-SFG region). AGNs are separated from SFGs in this diagram mainly because they preferentially reside in luminous and massive galaxies and have high [O III]/Hβ. The separation of AGN from star-forming galaxies is even cleaner thanks to the additional 0.15/0.12 dex offset in σ [O III] at fixed luminosity/stellar mass.We apply the KEx diagram to 7,866 galaxies at 0.3 < z < 1 in the DEEP2 Galaxy Redshift Survey, and compare it to an independent X-ray classification scheme using Chandra observation. X-ray AGNs are mostly located in the KEx-AGN region while X-ray SFGs are mostly located in the KEx-SFG region. Almost all of Type1 AGNs lie in the KEx-AGN region. These confirm the reliability of this classification diagram for emission line galaxies at intermediate redshift. At z∼2, the demarcation line between star-forming galaxies and AGNs should shift 0.3 dex higher in σ [O III] to account for evolution.
INTRODUCTION
The diagnostic diagrams are major tools to understand the nature of galaxies. They are crucial in evaluating the galaxy evolution scenarios such as the cosmic accretion and star-formation histories. The most widely-used diagram is the BPT (Baldwin, Philips & Terlevich 1981) or VO87 diagram (Veilleux & Osterbrock 1987). The advent of spectroscopic sky surveys like the SDSS and the photoionization models (Ferland et al. 1998) make the classification observationally constrained and theoretically understood. Kewley et al. (2001) used a variety of H II region photoionization models to give a theoretical star-forming galaxy boundary on the BPT diagram. The sources above this line are unlikely to be ionized by stars. Kauffmann et al. (2003) used the SDSS main galaxies sample to map their detailed distribution on the BPT diagram, and proposed that the right branch of the seagull shape distribution are all AGNs. The sources lie between the two dividing lines are called composite galaxies because their gas may be ionized by AGN and SF at the same time. Kewley et al. (2006) further proposed low ionization line criteria for separating Seyfert2s and LINERs. Other refinement of the classification are proposed by many authors (e.g., Stasińska et al. 2006;Cid Fernandes et al. 2010.
Our understanding of the BPT diagram is very comprehensive. The y-axis of the BPT diagram reflects mainly the ionization parameter while the xaxis is mostly determined by the metallicity (Storchi-Bergmann, Calzetti, & Kinney 1994, Raimann et al. 2000, Denicoló et al. 2002Pettini & Pagel 2004;Stasińska et al. 2006;Groves et al. 2004a,b;Groves et al. 2006;Kewley & Ellison 2008). The distinguishing power of the BPT diagram relies on the fact that the AGN radiation is harder than star-forming galaxies at similar stellar mass, AGNs have higher ionization parameters, and AGNs reside exclusively in massive metal-rich galaxies (Kauffmann et al. 2003.
The BPT diagram, however, suffers a few limitations. It needs at least 4 lines ([O III] λ5007, Hβ, [N II] λ6583, Hα) to make a classification. When the strength of these 4 lines are similar, the more signal-to-noise ratio cuts imposed, the more sources are missed. The [N II] λ6583 and Hα emission line will shift out of the optical wavelength range when the redshift is greater than 0.4, making the classification diagram futile for higher redshift sources with optical spectrum only.
With spectroscopic sky surveys pushing to higher redshift and fainter luminosities, the need for a good classification diagram for emission line galaxies (ELGs) at higher redshift is compelling. Some efforts have been made to develop diagnostic diagrams with spectral features in narrower wavelength range. Tresse et al. (1996) and Rola et al. (1997) (Weiner et al. 2006), [O II]/Hβ (Lamareille 2010), U-B color (Yan et al. 2011), or stellar mass (Juneau et al. 2011. Marocco et al. (2011) also used D n (4000) vs [O III]/Hβ for high-z galaxies classification. These methods take advantage of the fact that AGNs reside in massive, red galaxies in local universe, and are in general efficient in separating pure AGNs and star-forming galaxies. The composite galaxies, however, are mixed with Seyfert2s or star-forming galaxies on these diagrams.
The emission line velocity dispersion (σ) may trace the kinematics of different components in AGNs and starforming galaxies. [O III] in AGN comes from the narrow line region, which better traces the bulge kinematics (e.g. Ho 2009). [O III] in star-forming galaxies mainly comes from the H II regions, which locate mainly in the disk. The kinematic of the bulge/disk is expected to be different. Catinella et al. (2010) showed that the velocity dispersion is different for bulge and disk dominated galaxies at given baryonic mass. Besides, emission lines of AGN have extra broadening due to outflows (Greene & Ho 2005;Zhang et al. 2011). In principle, we could use the width of the narrow emission lines as a proxy of the influence of bulge potential for AGNs/star-forming galaxies classification. Following the idea of simplifying the BPT diagram as introduced in last paragraph (Weiner et al. 2006;Lamareille 2010;Yan et al. 2011;Juneau et al. 2011Juneau et al. , 2013, and the idea of different kinematics of AGN and star-forming galaxies, we propose to replace the [N II]/Hα in the BPT diagram with σ [O III] (or σ gas in general) to separate AGNs from star-forming galaxies at high redshift. In Section 2, we give descriptions of the data we use. In Section 3, a new diagnostic diagram: the Kinematics-Excitation Diagram (KEx diagram) is proposed, and we explain why it works and calibrate it at z<0.3. Section 4 gives the calibration of the KEx diagram at 0.3<z<1, and Section 5 gives the calibration at z∼2. Discussion is given in Section 6. We use a cosmology with H 0 = 70 km s −1 Mpc −1 , Ω m = 0.3, and Ω Λ = 0.7 throughout this paper.
SAMPLE AND MEASUREMENTS
2.1. Low-redshift data We start from the main galaxy sample (Strauss et al. 2002) of the Sloan Digitial Sky Survey Data Release 7 (Abazajian et al. 2009). The sample is complete in rband Petrosian magnitude between 15 and 17.77 over 9380 deg 2 . We limit the redshift range to z < 0.33, and there are 835,410 spectroscopic galaxies. To properly measure the emission lines, we use the scheme developed in Hao et al. (2005) to subtract the stellar absorptions.They used several hundreds SDSS low redshift pure absorption galaxies to construct the PCA eignspec-tra, and used the first 8 eignspectra to fit the continuum. An A-star template is added to represent young stellar population. A power law is also added when fitting AGN spectra. The continuum-subtracted line emissions are left for refined line fitting. We use 1-gaussian function to fit the [O II] λ6548 is fixed to 3 and their profile and center are tied to be the same. In addition, we fit Hα and Hβ a second time adding one broad gaussian to account for possible Hα and Hβ broad lines. The lower limit of σ of the broad component is 400km s −1 . The typical FWHM of Hα broad component of Type1 AGNs is larger than 1200km s −1 (Hao et al. 2005). We regard the broad Hα component to be prominent if a F-test suggests the improvement is significant at 3σ level. The intrinsic velocity dispersion σ int of the emission lines is obtained by subtracting the instrument resolution of ∼ 56km/s using σ 2 int = σ 2 obs − σ 2 Instrument . The errors of the σ and line strength are obtained by the MPFIT package which only includes the fitting errors (Markwardt 2009). We perform a simple test to check how well we can measure the line width. We add a gaussian to a continuum with a given equivalent width (EW). The σ of the gaussian is the combination of emission line intrinsic width 10 1.8 = 63.1km s −1 (typical star-forming galaxy) and the instrumental resolution of 56km s −1 . Random errors are added according to S/N =3, 5, 7, 10. We fit the emission line using the MPFIT package, and measure the error of σ by comparing the measured value with the input one. The simulation is run for 500 times. At EW=3 (typical value for −0.5 < log[O III]/Hβ < 1 star-forming galaxy) and S/N =3, 5, 7, 10, the errors in emission line width: σ are 0.19 dex, 0.08 dex, 0.06 dex, and 0.04 dex. For the worst case: EW=3, S/N=3, the error in σ is 0.19 dex. For sources with higher [O III]/Hβ and higher emission line width, the measurement are more reliable. Davis et al. 2003;Newman et al. 2013). 3 The DEEP2 survey has a limiting magnitude of R AB = 24.1, and it covers 3.2 deg 2 spanning 4 separate fields on the sky. The spectra span a wavelength range of 6500-9100Å at a spectral resolution of R∼5000. The DEEP2 DR4 include 52,989 galaxies. For our study, we limit the redshift range to be 0.32 < z < 0.82, to ensure the detection of Hβ and [O III] in the DEEP2 wavelength coverage. The sample size is cut to 12,739 galaxies. The spectra are obtained with the DEIMOS spectrograph at the Keck Observatory and reduced with the pipeline 4 developed by the DEEP2 team at the University of California Berkeley. All the DEEP2 footprints are observed by Chandra Advanced CCD Imaging Spectrometer (ACIS-I) with total exposures across all four XDEEP2 fields range from ∼10 ks to 1.1 Ms (Goulding et al. 2012;Laird et al. 2009;Nandra et al. 2005 Yan et al (2011) and Juneau et al (2011).
The dividing lines proposed by Kewley et al. (2001Kewley et al. ( , 2006 and Kauffmann et al. (2003) are used to classify the emission line galaxies into star-forming galaxies, composite galaxies, Seyfert2s, and LINERS, as shown in Figure 1. We require the signal to noise ratio to be greater than 3 for Hβ, Figure 2 separately. From left to right are star-forming galaxies, composite galaxies, LINERs and Seyfert2s. In panel (a), the star-forming galaxies cluster around the lower-left corner on the diagram, the boundary of the star-forming galaxies is clear and sharp. We derive an empirical curve to follow the boundary:
log[O III]/Hβ = −2 × log σ [O III] + 4.2(1)
This curve can be used to separate AGNs from starforming galaxies. The detailed distribution of BPTclassified galaxies on the KEx diagram is given in Table 1. 97% (5674/5860) of the BPT-classified Seyfert2s (above Kewley01 line) and 35% (5587/16003) of the BPT-classified composite sources lie above the new classification line and will be classified as AGNs by the KEx diagram. 98.8% of BPT-classifed SFGs are classified as KEx-SFGs. 81% of the KEx-classifed AGNs are BPT-classifed AGNs (above Kewley01 line), 90% of the KEx-classifed SFGs are BPT-classifed SFGs. For all the sources on the upper side of the line, 7.7% are classified as star-forming galaxies in traditional BPT diagram. 65.1% (10,416/16,003) of BPT composites are in the KEx-SFG region. 46.9% ((10,416+116+186)/(16003+998+5860)) of non-SF galaxies are on the KEx-SFG side. This means the new diagram is very efficient for selecting AGNs above Kewley01 line with high completeness and low contamination rate.
One may notice that the composite galaxies cluster near the SFG-AGN dividing line and clearly separate from Seyfert2s. We draw a horizontal line: We can see in Figure 2 and Table 1 that 116/998 LIN-ERs are classified as star-forming galaxies on the KEx diagram, 426/998 are KEx-composites, and 458 are KEx-AGNs. LINER like emission could be produce by Lowluminosity AGNs (Ferland & Netzer 1983;Halpern & Steiner 1983;Groves et al. 2004b;Ho 2008), post-AGB stars (Binette et al. 1994;Yan & Blanton 2012;Singh et al. 2013), fast shocks (Dopita & Sutherland 1995), photoionization by the hot X-ray-emitting gas (Voit & Donahue 1990;Donahue & Voit 1991), or thermal conduction from the hot gas (Sparks et al. 1989). Despite the ionization origins are diverse, the host galaxies of LINERs are massive, making them only weakly overlap with the star-forming galaxies on the KEx diagram. Since our KEx diagram does not include the information of lowionization lines, the LINERs are not well separated from AGNs.
log[O III]/Hβ = 0.3(2)
We note that only 1/3 of the BPT-classified composites galaxies are in the KEx-composite region, while most of the remaining 2/3 are in the KEx-SFGs region. Only a small fraction are in the KEx-AGN region. This may be becausethe BPT-classified composite galaxies have a diverse origin too. They could be relatively weak AGNs (Kauffmann et al. 2003;Yuan et al. 2010;Ellison et al. 2011), shock heated (e.g., Rich et al. 2014. ), or H II region with a special physical condition (Kewley et al. 2001). Trouille et al (2011) showed that the composites are most similar to AGNs in their TBT diagram and show not only photoionization properties like AGNs but also an excess X-ray emission relative to the infrared emission, indicating non stellar processes. This is a more likely scenario than shocks or varying H II region conditions. In the case of shocks, even galaxies with a lot of regions locally dominated by shocks have overall line ratios that place them in the BPT-SFG region instead of the composite region (Rich et al 2011). Some composites may have an intrinsically less luminous AGN, and that the NLR gas is moving slower, both in rotation and in outflow. This means our KEx diagram, which is successful in separating strong AGN from SFGs, may not have enough diagnostic power to pick out weak AGNs who have low contrast in both the [O III]/Hβ ratio and the kinematics relative to SFGs. KEx diagram also suffers from the mix of composites with other populations, especially the KEx-SFG, but to a lesser degree than some of the alternative diagrams.
In Figure 3, we plot KEx-AGN, KEx-composites, and KEx-SFGs on the BPT diagram. The KEx-composites cluster around the composite region on the BPT diagram. It is possible that the KEx diagram can be used to further diagnose the real nature of the composite galaxies. Panel (b) in Figure 2 also suggest that the [O III] line width could potentially be used to further constrain the nature of composite galaxies. For example, broad emission line width is usually regarded as a tracer of shock (e.g., Rich et al. 2011Rich et al. ,2014. The different line width be- tween sub-population of composites could potentially be used to constrain the relative importance of AGN and SF processes. We leave the exploration of the power of the KEx diagram to diagnose sub-classes for future studies.
Why the KEx diagram works?
Comparing to the widely used BPT diagram, the KEx diagram uses σ [O III] instead of the [N II]/Hα ratio as the horizontal axis to diagnose the ionizing source and physical properties of emission line galaxies. We showed that it gives very consistent result with the BPT diagram.
Here we discuss the physical reasons behind the diagram and why it works. σ [O III] in principle traces the motion of the gas. To first order, this motion is determined by the gravity of the galaxy. The emission line width correlate well with the stellar velocity dispersion in almost all types of emission line galaxies (e.g, Nelson 2000; Wang & Lu 2001;Bian et al. 2006;Chen et al. 2008;Komossa et al. 2007Komossa et al. , 2008Dumas et al. 2007;Greene & Ho 2005;Ho 2009). In AGNs, several additional sources of line broadening may at work in addition to the stellar kinematics (Greene & Ho 2005). So the basic principle behind the KEx diagram is the different kinematics of emitting gas in AGNs and star-forming galaxies.
The boundary between star-forming galaxies and AGNs on the KEx indicates there is a maximum σ [O III] for star-forming galaxies. In local star-forming galaxies, [O III] comes from the H II region, which is more correlated with the kinematics of the disk. It was found that the width of narrow emission lines of star-forming galaxies could efficiently trace the maximum rotation velocity of a galaxy (Rix et al. 1997;Mallén-Ornelas et al. 1999;Weiner et al. 2006;Mocz et al. 2012), so the line width reflects how fast the disk is rotating. The rotation speed is directly linked to the luminosity of the galaxy through the Tully-Fisher relation (TFR, Tully & Fisher 1977). The most luminous spiral galaxies also rotate the fastest. A reasonable hypothesis is that the galaxies near the maximum σ [O III] boundary in the KEx diagram are the most luminous ones. We plot all the star-forming galaxies in our sample on the [O III]/Hβ vs. R band absolute magnitude in Figure 4. The star-forming galaxies cluster in the left-down corner of the diagram, and show a clear boundary. We draw a magenta boundary curve to define the maximum luminosity a star-forming galaxy can reach at a given [O III]/Hβ. Using the Tully-Fisher relation obtained by Mocz et al. (2012), we convert the luminosity in the curve to σ gas , and obtain a [O III]/Hβ vs. σ gas curve, which is shown as the dashed line in Figure 2 (a). We can see that the dashed line covers the regions where SFGs and Composites locate, separating Seyfert2s from SFGs and Composites. This illustrates that SFGs and Composites are consistent with the TFR relation prediction, while AGNs are located outside the Figure 4. This is consistent with previous findings that Seyfert2s reside exclusively in massive, luminous galaxies (e.g., Kauffmann et al. 2003). Even though the host of Seyfert2s are luminous and massive, the overlap between Seyfert2s and star-forming galaxies on these two plots is large compared with that on the KEx diagram Figure 4. The fraction of BPT-classified Seyferts that are mis-classified as SFGs are higher on these two diagrams than on the KEx diagram.
There is an additional enhancement in σ [O III] at fixed luminosity or stellar mass for AGNs relative to SFGs. This enhancement helps AGNs separate further from SFGs on the KEx diagram. In Figure 5, we plot the median σ gas derived from five emission lines against M r and stellar mass in the left and right panels. We focus on the σ [O III] (blue line) first and discuss other emission lines later. The errors are calculated using the bootstrap method. We can see that at a given luminosity where both AGNs and SFGs cover, the σ [O III] of Seyfert2s is on average 0.15 dex higher than star-forming galaxies. And at a given stellar mass, Seyfert2s are 0.12 dex higher than star-forming galaxies. These offsets are critical to the clean separation between Seyfert2s and star-forming galaxies on the KEx diagram. Other emission lines gave similar results.
One reason of the difference in σ [O III] may be that the emission lines in Seyfert2s are produced by gas in the narrow line region which extend into the bulge and the emission line in star-forming galaxies are emitted by gas in the H II region in the disk. The matter distribution and kinematics are very different in the disk and bulge of a galaxy It is shown in Catinella et al. (2012) that at a given luminosity or baryon mass, the disk dominated galaxy show 0.1 dex smaller σ * (measured from the SDSS 3" fiber) than the bulge dominated galaxies. At a given mass, the bulge dominated galaxies, which are more concentrated, show a higher σ * than the disk dominated galaxies. Besides the difference in host galaxies, several other physical reasons may be involved in the fact that Seyfert2s have broader emission lines. Greene & Ho (2005) Komossa et al. 2008). This supports that the [O III] core is produced by ionized gas in bulge while another source of broadening related to AGN (such as possibly winds or outflows from accretion disk) is at work. Radio jet may play a part in broadening the emission line too (Mullaney et al. 2013).
The differences in σ gas and σ * between AGNs and SFGs are more pronounced against M r than against M*. In particular, σ * is significantly higher in AGN hosts at a fixed M r but not so different at a fixed stellar mass. This means that AGN hosts have higher mass-to-light ratios, which can be interpreted as having more important bulge components. More massive bulges host more massive black holes, and therefore be detectable as Seyfert 2s down to lower Eddington ratios. Conversely, lower mass AGNs are only identified as Seyfert 2s for comparatively higher Eddington ratios and their [O III] line width excess could be more pronounced relative to other lines than for higher mass AGNs if the Eddington ratio is the driving factor for additional broadening (e.g., Greene & Ho 2005;Ho 2008).
In summary, the boundary between star-forming galaxies/composites and AGNs on the KEx diagram is defined by the Tully-Fisher relation of the most luminous and massive galaxies. The AGNs reside in luminous and massive galaxies and at a given luminosity/stellar mass, their σ [O III] are 0.15/0.12 dex higher than the star-forming galaxies. These effects make the KEx diagram an efficient classification tool for emission line galaxies.
KEX DIAGRAM CALIBRATION AT 0.3<Z<1
The main purpose of the KEx diagram is for emission line galaxies classification at high redshift. We have demonstrated the KEx diagram could successfully separate emission line galaxies in local universe, mainly due to AGN occur in massive galaxies with high bugle-to-disk ratio and AGN have extra broadening due to outflow . At high redshift, the properties of galaxy and AGN host are different. σ [ O III] could be higher because galaxies were more gas rich, there were more unstable disks with high "σ/V" (Papovich et al. 2005;Reddy et al. 2006;Tacconi et al. 2010;Shim et al. 2011) and AGN were more luminous so had higher Eddington ratios. Besides, the [O III]/Hβ of the star-forming galaxies would be higher too. Galaxies at z ∼ 1.5 have typically higher [O III]/Hβ ratios than z<0.3 galaxies (e.g., Liu et al. 2008;Brinchmann et al. 2008;Trump et al. , 2012Kewley et al. 2013a,b). The physical properties in H II region and AGN NLR could be different from the local universe as well (Kewley et al. 2013a,b). There is also evidence for AGN in relatively low mass hosts at higher redshifts .
To calibrate, we need a sample of AGNs and starforming galaxies that are already classified . There are several methods for calibration: Firstly, X-ray identification may act as an independent reference for classification calibration (Yan et al. 2011;Juneau et al. 2011Juneau et al. , 2013 even though with some drawbacks. The X-ray AGNs and optical AGNs may not be the same population (Hickox et al. 2009) and the sources with reliable deep X-ray data are limited. Besides, X-rays surveys are less sensitive to moderate-luminosity AGNs in galaxies of lower stellar masses (Aird et al. 2012) or heavily obscured Compton-thick AGNs. Yan et al. (2011) estimated at L bol > 10 44 ergs −1 , about 2/3 of the emission-line AGNs with 0.3<z<0.8 and I AB < 22 will not be detected in the 2-7 keV band in the 200 ks Chandra images due to ab-sorption and/or scattering of the X-rays in the EGS field. We use DEEP2 data and X-ray identification for KEx calibration in Section 4.2. DEEP2 contain both starforming galaxies and AGNs, mostly star-forming galaxies. This can help us constrain our calibration on the SF side. Secondly, If NIR spectra are available, the [N II] and Hα emission lines could be used for optical classification using the BPT diagram as a corner stone. The use of the BPT at higher redshift remains potentially valid, but that it needs to be further verified and the sample size is limited as well (Trump et al. 2012, Kewley et al. 2013bJuneau et al. 2014). At z∼1, Type2 AGN sample is very limited, because Type2 AGNs can only be identified using BPT diagram to z∼0.4 using only optical spectrum, and the NIR spectroscopy sample which enable the BPT diagram to extend to z∼1 is limited. Thirdly, there are many Type1 AGNs from SDSS in the 0.3<z<1 range, because the broad lines can be identified using bluer wavelength range, and the volume covered by SDSS is large. Type 1 AGNs can be identified by their blue color or/and broad emission lines. We use them as independently identified AGNs to perform a sanity check of the KEx classification when considering only the narrow line components in Section 4.1. We make the assumption that the Type1 and Type2 sources have identical narrow line features in the frame of unification model (Antonucci et al. 1993). Some differences in narrow lines indeed exist due to NLR stratification, outflow, or narrow line Baldwin effect (Veilleux et al. 1991;Zhang et al. 2008Zhang et al. , 2011Stern & Laor 2013), but the assumption is valid grossly.
Calibration using Type1 AGNs Our KEx diagram uses only Hβ narrow component and [O III] emission lines so it is straightforward to use type1
AGNs to calibrate our KEx diagram. We first use a sample of low-z(z<0.3) type1 AGNs from SDSS DR7 main galaxy sample for testing purpose. These sources have Hα broad component with significance greater than 3σ as described in Section 2.1. We plot these sources in the KEx diagram in the Panel (a) of Figure 6 We further plot a sample of intermediate-z Type1 AGNs selected from SDSS DR4 QSO catalog with 0<z<0.8 on the KEx diagram. The sources in this sample have small contamination from host galaxies in their optical light, and their properties and data reduction are described in Dong et al. (2011). The detailed analysis of narrow line properties, especially the [O III] line, could be found in Zhang et al. (2011Zhang et al. ( , 2013b. The bolometric luminosity range of these sources is 10 44 ∼ 10 47 ergs −1 . In left panel of Figure 6, we plot this sample on KEx diagram in orange. Almost all(96%) of the sources lie in the KEx-AGN region, and a small fraction of the Type1 sources(4%) lie in the KEx-composite region. Only 13(0.3%) Type1 sources are classified as KEx-SFGs. According to unification model, if these Type1 sources are viewed edge-on, almost all of them would be rightly classified as AGNs. There are few points at z>0.3 and that they may be consistent with either no shift or a small shift of 0.1 dex.
One may notice that some Type1 AGNs lie outside the low-z locus. Some sources have higher [O III]/Hβ line ratio and some have larger σ [O III] . These could be partly understood by the orientation effect. The Type1 AGNs are found to have higher ionization state than Type2 AGNs (Veilleux et al. 1991c;Schmitt et al. 2003a,b), and this is because that high-ionization lines arise from regions closer to the nuclei thus more likely to be blocked when viewed edge-on. The inclination effect may play a role in the higher width of [O III] emission here. [O III] emission line is known to show blue-wing asymmetric profile (Heckman et al. 1981;Zhang et al. 2011) and this is believed to be due to narrow line region outflows. When the outflows are viewed in a face-on orientation, we would see larger overall outflow velocity and this would lead to larger line width at the same time.
Calibration using the DEEP2 Survey
Our intermediate-redshift galaxy sample is based on observations from the DEEP2 Galaxy Redshift Survey. Most of the galaxies in DEEP2 surveys are star-forming galaxies, as indicated in X-ray study (Goulding et al. 2012;Laird et al. 2009;Nandra et al. 2005), and there are also many X-ray AGNs in this sample. Even though using the X-ray data to calibrate the KEx suffer some problems as discussed in previous subsection and in other works (Hickox et al. 2009;Yan et al. 2011;Juneau et al. 2011Juneau et al. , 2013, it is interesting to check if the X-ray and the KEx classification are consistent and what causes the differences An X-ray luminosity threshold: L 2−10keV > 10 42 ergs −1 is adopted. There is no star-forming galaxies with X-ray luminosity higher than this value in local universe.The sensitivity of the X-ray data will not result in mis-classification of AGNs and star-forming galaxies, even though faint sources are missed in shallow areas. However, weak AGNs with L 2−10keV < 10 42 ergs −1 exist even though they are more ambiguous to differentiate from star-forming or starbursting galaxies with X-ray emission without additional information. For DEEP2 X-ray data, the sensitivity of the shallowest data could ensure the detection of luminous X-ray sources (L 2−10keV > 10 42 ergs −1 ).
The [O III]/Hβ vs σ [O III] for DEEP2 are plotted in green triangles in right panel of Figure 6. We use the same method described in Section 3 for spectral fitting. We apply a S/N cut of 3 to Hβ and [O III] λ5007. 7,866 sources satisfy this criteria. We convert the hard 2-8 keV X-ray flux to rest-frame 2-10keV luminosities (L X (2 − 10keV )) by assuming a power-law spectrum with photon index (γ = 1.8). The sources with L X (2 − 10keV ) > 10 42 ergs −1 are classified as X-ray AGNs, and sources with L X (2 − 10keV ) < 10 42 ergs −1 are classified as star-forming galaxies. We caution L X (2 − 10keV ) > 10 42 ergs −1 sources may be star-forming galaxies at highz, due to higher SF activity in the early universe. Many L X (2 − 10keV ) < 10 42 ergs −1 may be AGNs but dim intrinsically or due to obscuration. The X-ray sources are plotted in pink and purple triangles in right panel of Figure 6. We can see that most of the X-ray AGNs/starforming galaxies are consistently classified as optical AGNs/star-forming galaxies. Out of the 93 X-ray AGNs, 48 (52%) are classified as 18(19%) are KExcomposite, and 26 (29%) are KEx-SFGs. The pink dots are X-ray AGNs, selected using log L X (2 − 10keV ) > 42 erg/s. The purple dots are galaxies with log L X (2 − 10keV ) < 42 erg/s, X-ray star-forming galaxies according to our criteria. X-ray AGNs are located mainly in the KEx-AGN and KEx-Composite regions, while X-ray SFGs are located in the KEx-SFG region. At z<1, the KEx diagram does not need re-calibration. The blue, dark green, green, yellow, orange, and red lines are median [O III]/Hβ at given σ [O III] for z ∼ 0.1 (from SDSS), 0.3 < z < 0.4, 0.4 < z < 0.45, 0.5 < z < 0.6, 0.6 < z < 0.7, 0.7 < z < 0.8 (from DEEP2) KEx-SFGs respectively. The seemingly offset between z∼0.1 and higher redshift is consistent with aperture effect.
Out of the 83 X-ray starbursts in our sample, 19 are KEx-AGNs, 14 are KEx-composite, and 49 (59%) are KEx-SFGs. In Juneau et al. (2011), for the X-ray starbursts, 50% (8/16) are classified as MEx-SFGs, while 19% (3/16) are in the intermediate region and the remaining 31% (5/16) reside in the AGN region. The two results are consistent.
On the other hand, we notice that some sources are optically classified as star-forming galaxies but have very powerful X-ray emission, indicating harboring an active nuclei. 29% of X-ray AGNs are classified as starforming galaxies on the KEx diagram. In Juneau et al. (2011), 20% of their X-ray AGNs are classified as MEx-intermediate, and 15% are MEx-SFGs. Considering the intermediate region is mixed with star-forming region on MEx diagram, our result is consistent with theirs. Yan et al. (2011) found 25% of their X-ray AGNs reside in star-forming region of their optical classification diagram which replaces the [N II]/Hα in the BPT diagram with rest-frame U-B color. This is consistent with our result too. Castello-Mor et al. (2012) studied the sources with L X (2 − 10keV ) > 10 42 ergs −1 but classified as star-forming galaxies on the BPT diagram. These sources have large thickness parameter
(T = F X /F [O III] )
, large X-ray to optical flux ratio (X/O > 0.1), broad Hβ line width, steep X-ray spectra, and display soft excess. These mis-matches illustrate neither X-ray or optical classification are complete. Different classification schemes are complementary to each other. At z<1, the evolution in σ [O III] and [O III]/Hβ is not large so we don't need to shift the dividing line.
CALIBRATION AT Z∼2
In this section, we discuss extrapolating the KEx diagram to redshift greater than 2, and use z>2 emission line galaxies to test the validation of the diagram. The 2 < z < 4 epoch is critical to galaxy formation and evolution. During this epoch, the Hubble sequence was not fully established (Kriek et al. 2009;Förster Schreiber et al. 2006, 2009, but the bimodality was in place (Kriek et al. 2009). It is at this redshift that the star-formation density and AGN activity peak (e.g., Barger et al. 2001;Elbaz & Cesarsky 2003;Di Matteo et al. 2005;Hopkins 2004;Hopkins & Beacom 2006). There are thousands of emission line galaxies discovered at this redshift range, and most of them are detected using the dropouts techniques (e.g., Steidel et al. 1996Steidel et al. , 1998Steidel et al. , 2003Steidel et al. , 2004Pettini et al. 1998Pettini et al. , 2001, or colorselection (Franx et al. 2003;Daddi et al. 2004;Kong et al. 2006 ).
At z<1, the KEx can be used to successfully separate AGNs and star-forming galaxies mainly due to 3 reasons: First, AGNs reside in the most luminous and most massive galaxies. Second, the Tully-Fisher relation which define the boundary between star-forming galaxies/composites and AGNs is valid. Third, the AGNs have systematic higher σ gas than star-forming galaxies at fixed luminosity and stellar mass. The first condition is likely to be true for AGNs up to z∼3 (Xue et al. 2010;Mullaney et al. 2012) because the moderately luminous AGN fraction depends strongly on stellar mass but only weakly on redshift. The TFR, however, is likely to evolve with redshift.
Unlike local disk galaxies who is totally rotationsupported with V rot /σ = 10 ∼ 20(e.g. Dib, Bell & Burkert 2006), a large fraction of high redshift star- forming galaxies have velocity dispersion comparable or even larger than rotation velocity (Förster Schreiber et al. 2006, 2009Wright et al. 2007;Law et al. 2007;Genzel et al. 2008;Cresci et al. 2009;Vergani et al. 2012). Since we use the integrated emission line profile, the increase in velocity dispersion further broaden the emission line.
Even after we have extracted the rotation velocity using the Integrated Field Spectroscopy (IFS) for these high-redshift galaxies, the derived Tully-Fisher relation is still different from the local well-defined relation. The rotation speed of z∼2 galaxies is ∼0.2 dex higher than galaxies of similar stellar mass in local galaxies (Cresci et al. 2009;Gnerucci et al. 2011). Meanwhile, the [O III]/Hβ of star-forming galaxies is known to be higher at high redshift (Shapley et al. 2005;Erb 2006a;Groves et al. 2006;Liu et al. 2008;Brinchmann et al. 2008;Shirazi et al. 2014). Thus, we expect that the KEx diagram, particularly the separation boundary of AGN and star-forming galaxies, evolve to higher σ gas or higher [O III]/Hβ with redshift because the TFR and [O III]/Hβ redshift evolution, and galaxies at high redshift have larger velocity dispersions, To explore these effects in detail and test the application of the KEx diagram at z∼2, we compile a sample of Lyman-break galaxies (LBGs) and color-selected BzK galaxies at z > 2 who have [O III] and Hβ emission line ratio and gas velocity dispersion measurements from literature. Our sample include 10 galaxies (3 are AGNs) from Kriek et al. (2007), 1 LBG from Teplitz et al. (2000), 6 LBGs from Pettini et al. (1998Pettini et al. ( , 2001, and 2 gravitationally-lensed star-forming galaxies from Hainline et al. (2009). The AGN fraction of this sample is very limited because LBGs have very low AGN fraction (about 3-5%, Erb et al., 2006b, Reddy et al., 2005, Steidel et al., 2002. Figure 7 shows the z > 2 LBGs are not confined to the KEx-SFGs region of local galaxies. Most of them lie in the KEx-AGN or KEx-composite region. If we shift the dividing line 0.3 dex to the right, most of the galaxies are rightly classified. Out of the 0.3 shift, the evolution of TFR could contribute 0.2 dex (Cresci et al. 2009;Gnerucci et al. 2011;Vergani et al. 2012;Buitrago et al. 2013). Besides, the high redshift star-forming galaxies have higher velocity dispersion, which is not included in the evolution of TFR.
To test if AGNs still show different kinematic from star-forming galaxies at this redshift, we compile radio galaxies that are confirmed to be AGNs and see how they distribute in the KEx diagram. We use 4 radio AGNs from Nesvadba et al. (2008) and CDFS-695: A shock or/and AGN from van Dokkum et al. (2005. They clearly separate from the star-forming galaxies on the KEx diagram after shifting the boundary by 0.3 dex. We further check the [O III] width distribution of a sample of z∼2 QSOs from Netzer et al. (2004) and plot the histogram on lower panel of Figure 7. Only one source has [O III] width less than 100km/s, and most of the QSOs have σ [O III] ∼ 300km/s. There is no doubt that these QSOs would have high [O III]/Hβ ratio even though we do not have their detailed values. Therefore, they are likely to separate from the z∼2 star-forming galaxies. Strong outflow driven by AGN is reported in the Radio Galaxies (Nesvadba et al. 2008(Nesvadba et al. , 2011, so the large line width in [O III] is at least partly due to the outflow as discussed in Section 3.2. Judging from our empirical results, the KEx diagram is likely to work after shifting the dividing line 0.3 dex to the right at z∼2. More data is needed to better constrain the boundary.
6. OTHER RELEVANT ISSUES 6.1. Comparison with previous classification diagrams Tresse et al. (1996) and Rola et al. (1997) (DEW diagram) to select pure AGNs with z<1.3 using only optical spectrum. But different types of galaxies overlap with each other severely on these classification diagrams. Trouille et al. (2011) proposed to use g − z, [Ne III], and [O II] to clearly separate AGNs from star-forming galaxies. This method is very efficient in separating different types of galaxies, but the [Ne III] emission line is weak even in AGNs. Thus this diagram requires high signal-to-noise ratio spectra of galaxies for reliable classification. This is particular hard for high redshift objects which are usually faint.
Our KEx diagram has a similar logic as the Color-Excitation (CEx) and Mass-Excitation (MEx) diagrams proposed by Yan et al. (2011) andJuneau et al. (2011). The CEx diagram makes use of the fact that AGNs reside in red or green galaxies in local galaxies. But this is likely to be wrong at higher redshift (Trump et al. 2012). The MEx diagram and the one using rest-frame H-band magnitude (Weiner et al. 2006) is based on the fact that AGNs are harbored by massive galaxies. This is more robust at high redshift due to the AGN downsizing effect. Trump et al. (2012) tested the validation of these two diagrams at z∼1.5 and found that the MEx remains effective at z>1 but CEx needs a new calibration. As discussed in Section 3.2, we can separate AGNs and star-forming galaxies because AGN reside in massive galaxies, and have σ [O III] 0.12 dex higher than starforming galaxies of similar stellar mass as an enhancement. This enhancement makes the KEx diagram more efficient at separating AGNs from star-forming galaxies. One advantage of the KEx is the requirements for only a spectrum that covers a small spectral range to obtain all required quantities. The KEx diagram only requires [O III] and Hβ lines for a robust classification. Figure 5 we plot the median value of σ against R band absolute magnitude and stellar mass for different narrow lines using galaxies from SDSS DR7. We plot the stellar velocity dispersion (σ * ) in red line for reference. The σ * is stored in the SDSS spectrum file header. 32 K and G giant stars in M67 are used as stellar templates. These stellar templates are convolved with the velocity dispersion to fit the rest-frame wavelength range 4000-7000Å by minimizing χ 2 . The final estimation is the mean value of the estimates given by the "Fourier-fitting" and "Direct-fitting" methods. We found that the SFGs show similar sigma for all emission lines. The Seyfert2s, however, show some systematics in line width. The [O III] line is broader than other low ionization lines and recombination lines. This is expected, because the [O III] emitting region is more concentrated due to its high ionization potential (Veilleux et al. 1991;Trump et al. 2012). Unexpectedly, Hβ show the smallest line width, and the discrepancy is largest at the high luminosity high stellar mass end. One possibility for this discrepancy is the Balmer absorption fitting is not perfect. The incorrect absorption fitting affects resulting emission line flux and profile. Groves et al. (2012) found significant discrepancy in Hβ when using CB07 for SDSS DR7 and BC03 for SDSS DR4. The Hα, which should arises from the same emitting region of Hβ, does not follow the behavior of Hβ but show the same trend as low-ionization lines because the absorption correction is much milder.
When comparing the line width of emission lines and σ * , we found that the difference between σ * in starforming galaxies and Seyfert2s of similar stellar mass is very small. At high stellar mass end, galaxies tend to have bulges, and those would contribute to increase the measured σ * . However, if the ionized gas in SFGs still comes from the disk, then this component does not get the additional dispersion contribution from the bulge. Also, one can expect σ gas < σ * because the gas is more dissipative than the stars and can slow down dynamically (Ho 2009). We leave this topic for future studies.
[O III]/Hβ evolution
It is found that the [O III]/Hβ in star-forming galaxies gets higher at z∼1-2 (Shapley et al. 2005;Erb 2006a;Liu et al. 2008;Brinchmann et al. 2008;Hainline et al. 2009;Wright et al. 2010;Shirazi et al. 2014). The reason for this trend is under debate. Brinchmann et al. (2008) found that the location of starforming galaxies in the [O III]/Hβ versus [N II]/Hα diagnostic diagram highly depends on their excess specific star formation rate relative to galaxies of similar mass. They infer that an elevated ionization parameter U is responsible for this effect, and propose that this is also the cause of higher [O III]/Hβ in high-redshift star-forming galaxies in the BPT diagram Shirazi et al. 2014) . Liu et al. (2008) argue that the high [O III]/Hβ sources have higher electron densities and temperatures. It is also possible that AGN or shock increase the [O III]/Hβ ratio Wright et al. 2010). It is interesting to check how [O III]/Hβ evolves with redshift at given σ [O III] in the KEx diagram.
In the right panel of Figure 6, we plot the median [O III]/Hβ at given σ [O III] for the SDSS z<0.33 galaxies and the DEEP2 galaxies on the KEx diagram. The red, orange, yellow, green, blue and purple lines are median [O III]/Hβ at given σ [O III] for z ∼ 0.1, 0.3 < z < 0.4, 0.4 < z < 0.45, 0.5 < z < 0.6, 0.6 < z < 0.7, 0.7 < z < 0.8 KEx-SFGs respectively. We can see in Figure 6 that the [O III]/Hβ vs. σ [O III] relation does not evolve from z∼0.3 to z∼0.8, but these galaxies have on average 0.2 dex higher [O III]/Hβ than the local galaxies at given σ [O III] . However, we note that SDSS has a fixed aperture of 3" which acquires the light from the center of the galaxy , while DEEP2 spectra are obtained through long-slits, which enable them to include light from the outskirt of the galaxy. The outskirt of the galaxy have lower metallicity and larger rotation speed than the center. and Hβ emission lines, thus it is a suitable tool to classify emission-line galaxies (ELGs) at higher redshift than more traditional line ratio diagnostics because it does not require the use of the [N II]/Hα ratio. Using the SDSS DR7 main galaxy sample and the BPT diagnostic, we calibrate the diagram at low redshift. We find that the diagram can be divided into 3 regions: one occupied mainly by the pure AGNs (KEx-AGN region), one dominated by composite galaxies (KEx-composite region), and one contains mostly SFGs (KEx-SFG region). The new diagram is very efficient for selecting AGNs with high completeness and low contamination rate. We further apply the KEx diagram to 7,866 galaxies at 0.3 < z < 1 in the DEEP2 Galaxy Redshift Survey, and compare the KEx classification to an independent X-ray classification using Chandra observation. Almost all Type1 AGNs at z<0.8 lie in the KEx-AGN region, confirming the reliability of this classification diagram for emission line galaxies at intermediate redshift. At z∼2, the demarcation line between star-forming galaxies and AGNs should be shifted to 0.3 dex higher σ [O III] due to evolution AGNs are separated from SFGs in this diagram mainly because in addition to that they preferentially reside in luminous and massive galaxies, they show 0.15/0.12 dex higher σ [O III] than star-forming galaxies at given luminosities/stellar masses. Higher σ [O III] also arise from AGN-driven broadening effects (such as winds or out-flows). When we push to higher redshift, the evolution of [O III]/Hβ and the TFR result in the shift of dividing line between AGNs and SFGs. KEx needs high enough spectral resolution to measure σ [O III] , and this diagnostic diagram is purely empirical now because it is hard to link ionization and kinematics theoretically. Despite the caveats, it provides a robust diagnostic of ionization source when only [O III] and Hβ are available. Zhang, K., Wang, T.-G., Gaskell, C. M., & Dong, X.-B. 2013, ApJ, 762, 51 Zhang, K., Wang, T.-G., Yan, L., & Dong, X.-B. 2013
[O III], [O I], [N II], Hα and [S II] lines to ensure classification on the BPT diagram into sub-types. The [O I]/Hαdiagram is shown for reference and not taken into account in the classification. How the different types of galaxies populate the KEx diagram is shown in Panel (d) of Figure 1. We plot different types of emission line galaxies on [O III]/Hβ vs σ [O III] plot in
Fig. 1 .
1-Pandel (a)-(c): The BPT diagram for SDSS low redshift emission line galaxies classification. The blue dots are star-forming galaxies, the green dots are composites, the red dots are Seyfert2s, and the magenta dots are LINERs. The classification and color scheme are based on a combination of Panels (a) and (b). The solid and dashed lines in Panel (a) are demarcation lines fromKauffmann et al. (2003) andKewley et al. (2001). The solid line in Panel (b) is fromKewley et al. (2006). Panel (d): The KEx diagram for all emission line galaxies from SDSS DR7 main galaxy sample. The legends are the same as Panel (a). KEx diagram is efficient in separating AGNs from SFGs.
Fig. 2 .
2-Panel (a): The blue contours are the distribution of BPT-classified star-forming galaxies on the KEx diagram. The lowest level of the contour is 97 percentile The solid lines are the dividing lines to classify emission line galaxies into KEx-AGNs, KEx-composites, and KEx-SFGs. The dashed line is the Tully-Fisher Relation prediction of the most luminous galaxies, as shown in Figure 5. Panel (b)-(d): The distribution of BPT-classified composites, LINERs and Seyfert2s on the KEx diagram. The solid lines are the same as Panel (a).
Fig. 3 .
3-The distribution of KEx-AGN (pink), KEx-composites (green) and KEx-SFGs (purple) on the BPT diagram.
Fig. 4 .
4-Left panel: [O III]/Hβ vs. R band absolute magnitude. The blue contours are star-forming galaxies and the red dots are AGNs. The lowest level of the contour is 90 percentile. The dash line is the boundary line we draw around the distribution. Its corresponding line on the KEx diagram after Tully-Fisher Relation transformation is shown in left panel of Figure 2. Right panel: [O III]/Hβ vs. stellar mass. The blue contours are star-forming galaxies and the red dots are AGNs. The lowest levels is 90 percentile. The solid and dashed lines are the demarcation lines of MEx diagram (Juneau et al. 2011). AGNs and SFGs are not separated as far on these 2 diagrams as on the KEx diagram.
Fig. 5 .
5-Left panel: Widths of different narrow lines against R band absolute magnitude. The solid lines and dash-dotted lines are median value of σ at given R band absolute magnitude for Sy2s and star-forming galaxies respectively. Sy2s are shown as the solid line and star-forming galaxies are represented by the dash-dotted line. Different color represent different emission lines and the description of the legends are shown at the left-up corner. Right panel: The median σ of different narrow lines against stellar mass. The legend is the same as the left panel. At fixed R band luminosity/ Stellar mass, AGNs show 0.15/0.12 dex higher σ [O III] than SFGs. This enhancement helps AGNs separate further from SFGs on the KEx diagram.
the distribution of Seyfert2s (red dots) on the [O III]/Hβ vs. M r (absolute petrosian magnitude) and [O III]/Hβ vs. M * (stellar mass) diagrams (MEx Diagram, Juneau et al. 2011) in Figure 4. The magenta lines in the right panel are the MEx dividing lines. The sources above the upper curve are MEx-AGN and the region between the solid upper curve and dashed line is the MEx-composite region. The contours are 5, 40, 70, 95 percentiles. The stellar mass is drawn from the MPA-JHU catalog. Seyfert2s occupy the bright and massive end of
. 4 .
42% (246/5860) of Sy2s are on the MEx-SF side, while 3.2% (186/5860) of Sy2s are on the KEx-SF side. On the [O III]/Hβ vs M R plot, 7.7% (450/5960) of Sy2s lie on the SFG side of the dividing curve shown in the left panel of
. The [O III]/Hβ only include the narrow component of Hβ. These sources (4624) reside mostly in the KEx-AGN and KEx-composite regions, while only 414(9%) of them are in the KEx-SFG region.
Fig. 6 .
6-Left panel: Left panel: The purple dots and contours are sources with broad Hα emission line from SDSS DR7 main galaxy sample. The orange dots and contours are the Type1 AGNs from Dong et al. (2011). The solid lines are KEx demarcation lines. Type1 AGNs are located mainly in the KEx-AGN and KEx-Composite regions. Right panel: The gray dots are intermediate redshift galaxies from DEEP2 survey.
Fig. 7 .
7-z∼2 galaxies from literature in the KEx diagram. The references where the data come from is shown at the right-up corner. The purple color denotes the sources are star-forming galaxies, while the red color denotes they are AGNs. Upper limits are shown with downward arrows. The red lines are the same demarcation lines as Figure 2. The dashed line is the KEx-SFGs dividing line shifted 0.3 dex right-ward. The solid purple line is the [O III]/Hβ vs σ [O III] relationship for z ∼ 0.1 KEx-SFGs from DEEP2, and the dashed purple line is derived by shifting the relationship for z ∼ 0.1 KEx-SFGs 0.3 dex right ward. The [O III] width distribution of z∼2 QSOs from Netzer et al. (2004) is shown in the lower panel. The demarcation lines needs to be shifted due to the evolution of kinematics and [O III]/Hβ at z∼2.
proposed to use the of EW([O II] λ3727) (equivalent width of [O II] λ3727), EW([O III] λ5007) and EW(Hβ) for galaxy classification. Stasińska et al. (2006) studied using [O II] λ3727 for galaxy classification, and proposed a method that uses 4000Å break: D n (4000), EW([O II] λ3727), and EW([Ne III] λ3870 ) (DEW diagram) to select pure AGNs with z<1.3 using only the optical spectra. Trouille et al. (2011) proposed to use g − z, [Ne III], and [O II] to clearly separate AGNs from star-forming galaxies at intermediate redshift. A fruitful way to push to high redshift is to retain the [O III]/Hβ while replacing the [N II]/Hα with other quantities like H band absolute magnitude
λ3727, [O I]λ6300, Hβ, [O III] λ5007, [N II] λ6548, Hα, [N II] λ6583, and [S II] λλ6717, 6731 (Hereafter [O II], [O I], Hβ, [O III], [N II] λ6548, Hα, [N II] λ6583, and [S II]) respectively. The σ (line width, 1/2.35 Full Width at Half Maximum in km s −1 ) of the gaussian profile for [O III] λ5007 is denoted as σ [O III] . The line ratio of [N II] λ6583/[N II]
2.2. Intermediate-redshift data Our intermediate-redshift galaxy sample is based on observations from the DEEP2 Galaxy Redshift Survey (hereafter DEEP2;
). The intermediate-redshift data is used for calibration of the new KEx diagnostic diagram at z<1. 3. CLASSIFICATION USING SDSS MAIN GALAXY SAMPLE 3.1. Classification Since the BPT diagram needs at least [O III], Hβ, [N II] and Hα for a classification, it is not applicable to sources at z>0.4 with optical spectrum alone. We propose a new diagnostic diagram: [O III]/Hβ vs σ [O III] to diagnose the ionization source and physical properties of emission line galaxies. We call it the Kinematics-Excitation Diagram (KEx diagram) hereafter. This approach shares similar logic to the work of
found that the excess [O III] line width relative to the stellar or lower ionization lines kinematics is about 30-40% (0.11-0.15 dex), with small variations depending on AGN luminosity, AGN Eddington ratio, SFR, etc. When considering only the core of [O III], the excess in [O III] line width goes away (Green & Ho 2005,
proposed to use EW([O II]), EW([O III]) and EW(Hβ) for galaxy classification at high redshift. Stasińska et al. (2006) studied using [O II] for galaxy classification, and even proposed a method that uses D n (4000), EW([O II]), and EW([Ne III])
6.2. Comparison of different narrow line widths In KEx diagram, we use [O III] emission line width for diagnostic. In order to check if different lines have different width, in
This would shift the [O III]/Hβ vs. σ [O III] relation in the right-up (higher [O III]/Hβ, higher σ [O III] ) direction as we see . So our result are consistent with no evolution in the [O III]/Hβ vs σ [O III] relation from z=0-0.8. 7. SUMMARY AND CONCLUSION We propose a new diagram, the Kinematic-Excitation diagram (KEx diagram), using the [O III]/Hβ line ratio and the [O III] λ5007 emission line width (σ [O III] ) to diagnose the emissions of the AGNs and the star-forming galaxies. The KEx diagram uses only the [O III] λ5007
TABLE 1
1The statistic of galaxy classification in the KEx diagram
http://deep.ps.uci.edu/ 4 http://astro.berkeley.edu/∼cooper/deep/spec2d/
ACKNOWLEDGEMENTSWe thank Lisa Kewley, Renbin Yan, Shude Mao, Junqiang Ge, Jong-Hak Woo, Chun Lyu for helpful discussions and suggestions. We thank an anonymous referee for helpful suggestions that improve the paper significantly.TypeStar-Forming Galaxies Composites LINERs Seyfert2s Seyfert1s Quasars DEEP2 (DR7 galaxy sample) (DR4 quasar sample) (1)(2)
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| []
|
[
"THREE-DIMENSIONAL RELATIVISTIC PAIR PLASMA RECONNECTION WITH RADIATIVE FEEDBACK IN THE CRAB NEBULA",
"THREE-DIMENSIONAL RELATIVISTIC PAIR PLASMA RECONNECTION WITH RADIATIVE FEEDBACK IN THE CRAB NEBULA"
]
| [
"B Cerutti [email protected] \nDepartment of Astrophysical Sciences\nPrinceton University\n08544PrincetonNJUSA\n\nCenter for Integrated Plasma Studies, Physics Department\nUniversity of Colorado\nUCB 39080309-0390BoulderCOUSA\n",
"G R Werner [email protected] \nCenter for Integrated Plasma Studies, Physics Department\nUniversity of Colorado\nUCB 39080309-0390BoulderCOUSA\n",
"D A Uzdensky [email protected] \nCenter for Integrated Plasma Studies, Physics Department\nUniversity of Colorado\nUCB 39080309-0390BoulderCOUSA\n",
"M C Begelman \nJILA\nUniversity of Colorado and National Institute of Standards and Technology\nUCB 44080309-0440BoulderCOUSA\n\nDepartment of Astrophysical and Planetary Sciences\nUniversity of Colorado\nUCB 39180309-0391BoulderCOUSA\n",
"Lyman Spitzer Jr",
"Fellow "
]
| [
"Department of Astrophysical Sciences\nPrinceton University\n08544PrincetonNJUSA",
"Center for Integrated Plasma Studies, Physics Department\nUniversity of Colorado\nUCB 39080309-0390BoulderCOUSA",
"Center for Integrated Plasma Studies, Physics Department\nUniversity of Colorado\nUCB 39080309-0390BoulderCOUSA",
"Center for Integrated Plasma Studies, Physics Department\nUniversity of Colorado\nUCB 39080309-0390BoulderCOUSA",
"JILA\nUniversity of Colorado and National Institute of Standards and Technology\nUCB 44080309-0440BoulderCOUSA",
"Department of Astrophysical and Planetary Sciences\nUniversity of Colorado\nUCB 39180309-0391BoulderCOUSA"
]
| [
"DRAFT VERSION JANUARY"
]
| The discovery of rapid synchrotron gamma-ray flares above 100 MeV from the Crab Nebula has attracted new interest in alternative particle acceleration mechanisms in pulsar wind nebulae. Diffuse shock-acceleration fails to explain the flares because particle acceleration and emission occur during a single or even sub-Larmor timescale. In this regime, the synchrotron energy losses induce a drag force on the particle motion that balances the electric acceleration and prevents the emission of synchrotron radiation above 160 MeV. Previous analytical studies and 2D particle-in-cell (PIC) simulations indicate that relativistic reconnection is a viable mechanism to circumvent the above difficulties. The reconnection electric field localized at X-points linearly accelerates particles with little radiative energy losses. In this paper, we check whether this mechanism survives in 3D, using a set of large PIC simulations with radiation reaction force and with a guide field. In agreement with earlier works, we find that the relativistic drift kink instability deforms and then disrupts the layer, resulting in significant plasma heating but few non-thermal particles. A moderate guide field stabilizes the layer and enables particle acceleration. We report that 3D magnetic reconnection can accelerate particles above the standard radiation reaction limit, although the effect is less pronounced than in 2D with no guide field. We confirm that the highest energy particles form compact bunches within magnetic flux ropes, and a beam tightly confined within the reconnection layer, which could result in the observed Crab flares when, by chance, the beam crosses our line of sight. | 10.1088/0004-637x/782/2/104 | [
"https://arxiv.org/pdf/1311.2605v2.pdf"
]
| 119,277,966 | 1311.2605 | f3b18d3ee32222b058da32de7486a9d8cf950acd |
THREE-DIMENSIONAL RELATIVISTIC PAIR PLASMA RECONNECTION WITH RADIATIVE FEEDBACK IN THE CRAB NEBULA
2014
B Cerutti [email protected]
Department of Astrophysical Sciences
Princeton University
08544PrincetonNJUSA
Center for Integrated Plasma Studies, Physics Department
University of Colorado
UCB 39080309-0390BoulderCOUSA
G R Werner [email protected]
Center for Integrated Plasma Studies, Physics Department
University of Colorado
UCB 39080309-0390BoulderCOUSA
D A Uzdensky [email protected]
Center for Integrated Plasma Studies, Physics Department
University of Colorado
UCB 39080309-0390BoulderCOUSA
M C Begelman
JILA
University of Colorado and National Institute of Standards and Technology
UCB 44080309-0440BoulderCOUSA
Department of Astrophysical and Planetary Sciences
University of Colorado
UCB 39180309-0391BoulderCOUSA
Lyman Spitzer Jr
Fellow
THREE-DIMENSIONAL RELATIVISTIC PAIR PLASMA RECONNECTION WITH RADIATIVE FEEDBACK IN THE CRAB NEBULA
DRAFT VERSION JANUARY
142014Draft version January 14, 2014Preprint typeset using L A T E X style emulateapj v. 5/2/11Subject headings: Acceleration of particles -Magnetic reconnection -Radiation mechanisms: non-thermal -ISM: individual (Crab Nebula)
The discovery of rapid synchrotron gamma-ray flares above 100 MeV from the Crab Nebula has attracted new interest in alternative particle acceleration mechanisms in pulsar wind nebulae. Diffuse shock-acceleration fails to explain the flares because particle acceleration and emission occur during a single or even sub-Larmor timescale. In this regime, the synchrotron energy losses induce a drag force on the particle motion that balances the electric acceleration and prevents the emission of synchrotron radiation above 160 MeV. Previous analytical studies and 2D particle-in-cell (PIC) simulations indicate that relativistic reconnection is a viable mechanism to circumvent the above difficulties. The reconnection electric field localized at X-points linearly accelerates particles with little radiative energy losses. In this paper, we check whether this mechanism survives in 3D, using a set of large PIC simulations with radiation reaction force and with a guide field. In agreement with earlier works, we find that the relativistic drift kink instability deforms and then disrupts the layer, resulting in significant plasma heating but few non-thermal particles. A moderate guide field stabilizes the layer and enables particle acceleration. We report that 3D magnetic reconnection can accelerate particles above the standard radiation reaction limit, although the effect is less pronounced than in 2D with no guide field. We confirm that the highest energy particles form compact bunches within magnetic flux ropes, and a beam tightly confined within the reconnection layer, which could result in the observed Crab flares when, by chance, the beam crosses our line of sight.
INTRODUCTION
The non-thermal radiation emitted in pulsar wind nebulae is commonly associated with ultra-relativistic electron-positron pairs injected by the pulsar and accelerated at the termination shock. In the Crab Nebula, the particle spectrum above ∼ 1 TeV responsible for the X-ray to gamma-ray synchrotron emission is well modeled by a single power-law distribution of index −2.2, which is usually associated with first-order Fermi acceleration at the shock front (see e.g., Kirk et al. 2009). Since the detections of the first flares of high-energy gamma rays in 2010 (Abdo et al. 2011;Tavani et al. 2011;Balbo et al. 2011) and the following ones detected since then (Striani et al. 2011;Buehler et al. 2012;Striani et al. 2013;Mayer et al. 2013;Buson et al. 2013), we know that the Crab Nebula occasionally accelerates particles up to a few 10 15 eV (see reviews by Arons 2012 andBlandford 2013). This discovery is very puzzling because the particles are accelerated to such energies within a few days, which corresponds to their Larmor gyration time in the Nebula. This is far too fast for Fermi-type acceleration mechanisms which operate over multiple crossings of the particles through the shock (e.g., Blandford & Eichler 1987). In addition, the observed particle spectrum is very hard, which is not compatible with the steep power-law 2 expected with diffuse shock-acceleration (Buehler et al. 2012). Even more surprising, the particles emit synchrotron radiation above the wellestablished radiation reaction limit photon energy of 160 MeV (Guilbert et al. 1983;de Jager et al. 1996;Lyutikov 2010;. It implies that the particles must be subject to extreme synchrotron cooling over a sub-Larmor timescale. Hence, in principle, synchrotron cooling should prevent the acceleration of pairs to such high energies in the first place.
Fortunately, there is a simple way to circumvent these tight constraints on particle acceleration if there is a region of strong coherent electric field associated with a low magnetic field perpendicular to the particle motion, i.e., if E > B ⊥ . This supposes that a non-ideal, dissipative magnetohydrodynamic process is at work somewhere in the Nebula. Using a simple semi-analytical approach, and Cerutti et al. (2012a) showed that such extreme particle acceleration can occur within a Sweet-Parker-like reconnection layer (Parker 1957;Sweet 1958;Zweibel & Yamada 2009), where the reversing reconnecting magnetic field traps and confines the highest energy particles deep inside the layer where E > B ⊥ (Speiser 1965;Kirk 2004;Contopoulos 2007). The reconnection electric field accelerates the particles almost linearly along a few light-day long layer. This solution solves the sub-Larmor acceleration problem at the same time. Twodimensional (2D) particle-in-cell (PIC) simulations of relativistic pair plasma reconnection with radiation reaction force have confirmed and strengthened the viability of this scenario . These simulations can also explain the observed rapid intra-flare time variability of the > 160 MeV synchrotron flux, the apparent photon spectral shape, as well as the flux/cutoff energy correlation (Buehler et al. 2012).
Although these 2D PIC simulations provide a fairly complete assessment of extreme particle acceleration in reconnection layers, it is still a simplified picture of a truly threedimensional process. We know from previous 3D reconnection studies (Zenitani & Hoshino 2005, 2008; Daughton et al. 2011;Liu et al. 2011;Sironi & Spitkovsky 2011;Kagan et al. 2013;Markidis et al. 2013) that the reconnection layer is unstable to the relativistic tearing and kink modes, and a combination of these two into oblique modes. The kink (and oblique) instabilities, which cannot arise in the 2D simulations of Cerutti et al. (2013), can lead to significant deformation or even disruption of the reconnection layer in 3D simulations, subsequently suppressing particle acceleration. However, a moderate guide magnetic field can stabilize the layer (Zenitani & Hoshino 2005, 2008Sironi & Spitkovsky 2011).
In this work, we extend the previous 2D study of Cerutti et al. (2013) by performing large 3D PIC simulations of pair plasma reconnection with radiative feedback and with guide field, in the context of the Crab flares. In the next section, we first present the numerical techniques and the setup of the simulations chosen for this study. Then, we investigate separately the effect of the tearing and the kink instabilities on the efficiency of particle acceleration, using a set of 2D simulations in Section 3. In Section 4, we establish the conditions for particle acceleration above the radiation reaction limit and emission of > 160 MeV synchrotron radiation in 3D reconnection. In addition, we report in this section on strong anisotropy and inhomogeneity of the highest-energy particles in 3D reconnection consistent with 2D results, and their important role in explaining the observed Fermi-LAT gamma-ray flux of the Crab flares. We summarize and discuss the results of this work in Section 5.
2. NUMERICAL APPROACH AND SIMULATION SETUP 2.1. Numerical techniques All the simulations presented in this work were performed with Zeltron 1 , a parallel three-dimensional electromagnetic PIC code . Zeltron solves self-consistently Maxwell's equations using the Yee finite-difference time-domain (FDTD) algorithm (Yee 1966), and Newton's equation following the Boris FDTD algorithm (Birdsall & Langdon 2005). Unlike most PIC codes, Zeltron includes the effect of the radiation reaction force in Newton's equation (or the so-called "Lorentz-Abraham-Dirac equation") induced by the emission of radiation by the particles (see also, e.g., Jaroschek & Hoshino 2009;Tamburini et al. 2010;Capdessus et al. 2012). In the ultra-relativistic regime, the radiation reaction force, g, is akin to a continuous friction force, proportional to the radiative power and opposite to the particle's direction of motion (e.g., Landau & Lifshitz 1975;Tamburini et al. 2010;Cerutti et al. 2012a). The expression of the radiation reaction force used in Zeltron is given by
g = − 2 3 r 2 e γ E + u × B γ 2 − u · E γ 2 u,(1)
where r e ≈ 2.82 × 10 −13 cm is the classical radius of the electron, γ is the Lorentz factor of the particle, E and B are the electric and magnetic fields, and u = γv/c is the four-velocity divided by the speed of light. This formulation is valid if -Initial simulation setup and geometry. The computational domain is a rectangular box of volume Lx × Ly × Lz with periodic boundary conditions in all directions. The box initially contains two flat, anti-parallel, relativistic Harris layers in the xz-plane centered at y = Ly/4 and y = 3Ly/4, of some thickness 2δ. The magnetic field structure is composed of the reconnecting field, B 0 , along the x-direction, which reverses across the layers, and a uniform guide field, Bz = αB 0 , along the z-direction.
γB/B QED 1, where B QED = 4.4 × 10 13 G is the quantum critical magnetic field. Because of the relativistic effects, the typical frequency of the expected radiation is ∼ γ 3 1 times the relativistic cyclotron frequency. Hence, the radiation is not resolved by the grid and time step of the simulation. It must be calculated separately. Zeltron computes the emitted optically thin radiation (spectrum, and angular distributions) assuming it is pure synchrotron radiation. This is valid if the change of the particle energy is small, ∆γ/γ 1, during the formation length of a synchrotron photon, given by the relativistic Larmor radius divided by γ. We checked a posteriori that this assumption is indeed correct. The code also models the inverse Compton drag force on the particle motion in an imposed photon field, but this effect is negligible in the context of the Crab flares (Cerutti et al. 2012a), hence this capability will not be utilized in the following. To perform the large 3D simulations presented in this paper, Zeltron ran on 97,200 cores on the Kraken supercomputer 2 with nearly perfect scaling.
Simulation setup
The simulation setup chosen here is almost identical to our previous two-dimensional pair plasma reconnection simulations with radiation reaction force in Cerutti et al. (2013). The computational domain is a rectangular box of dimensions L x , L y and L z , respectively along the x-, yand z-directions, with periodic boundary conditions in all directions. We set up the simulation with two flat anti-parallel relativistic Harris current layers (Kirk & Skjaeraasen 2003) in the xz-plane located at y = L y /4 and y = 3L y /4 ( Figure 1). Having two current sheets is only a convenient numerical artifact that allows us to use periodic boundary conditions along the y-direction, but it does not have a physical meaning in the context of our model of the Crab flares where only one reconnection layer is involved. The electric current, J z , flows in the ±z-directions, and is NOTE. -There are three distinct subsets of simulations. The first subset comprises 5 2D simulations of reconnection in the xy-plane, designed to study the effect of the guide field strength α on the dynamics of reconnection and particle acceleration. The second subset comprises 7 2D simulations of the reconnection layer in the yz-plane, in order to study the development of the kink instability as a function of the guide field strength α. The last set of simulations is chosen to test particle acceleration beyond the radiation reaction limit in 3D, in the best and in the worst cases identified in the 2D subsets.
supported by electrons counter-streaming with positrons at a mildly relativistic drift velocity (relative to the speed of light) β drift = 0.6. The plasma (electrons and positrons) is spatially distributed throughout the domain, with the following density profile
n = n 0 cosh y−Ly/4 δ −2 + 0.1n 0 if y < L y /2 n 0 cosh y−3Ly/4 δ −2 + 0.1n 0 if y > L y /2 . (2)
The first term is the density of the drifting pairs carrying the initial current, concentrated within the layer half-thickness δ = λ D /β drift , where λ D is the relativistic Debye length (Kirk & Skjaeraasen 2003). This population is modeled with a uniform and isotropic (in the co-moving frame) distribution of macro-particles with variable weights to account for the density profile and to decrease the numerical noise in low-density regions. The second term is a uniform and isotropic background pair plasma at rest in the laboratory frame with a density chosen to be 10 times lower than at the center of the layers (i.e., 0.1n 0 ). The drifting and the background particles are distributed in energy according to a relativistic Maxwellian with the same temperature θ 0 ≡ kT /m e c 2 = 10 8 , where k is the Boltzmann constant and m e is the rest mass of the electron. The temperature of the drifting particles is defined in the co-moving frame. This temperature models the ultrarelativistic plasma already present in the Crab Nebula, prior to reconnection, whose particles could have been accelerated at the wind termination shock or even by other reconnection events throughout the nebula. However, observations show that in reality the background plasma is distributed according to a broad and steep power-law, extending roughly between γ min = 10 6 and γ max = 10 9 (responsible for the UV to 100 MeV synchrotron spectrum). This large dynamic range of particle energies translates directly into an equally large dynamic range of relativistic Larmor radii and hence of length scales that must be resolved in the simulation, which is beyond the reach of our numerical capabilities.
The initial electromagnetic field configuration is
B = −B 0 tanh y−Ly/4 δ e x + αB 0 e z if y < L y /2 B 0 tanh y−3Ly/4 δ e x + αB 0 e z if y > L y /2 ,(3)E = 0 ,(4)
where e x , e z are unit vectors along the xand z-directions. B 0 is the upstream reconnecting magnetic field and α is a dimensionless parameter of the simulation that quantifies the strength of the guide field component B z in units of B 0 (Figure 1). Observations constrain the magnetic field in the emitting region to about 1 mG, which is much higher than the expected average quiescent field of order 100-200 µG (Meyer et al. 2010). In this work, we choose B 0 = 5 mG to be consistent with our previous studies of the flares (Cerutti et al. 2012a. Hence, the energy scale at which the radiation reaction force equals the electric force, assuming that E = B 0 = 5 mG, is
γ rad m e c 2 = 3em 2 e c 4 2r 2 e B 0 ≈ 1.3 × 10 9 m e c 2 ,(5)
where e is the fundamental electric charge. Below, we express lengths in units of the typical initial Larmor radius of the particles in the simulations, i.e., ρ 0 = θ 0 m e c 2 /eB 0 ≈ 3.4 × 10 13 cm. In all the simulations, the layer half-thickness is then δ/ρ 0 ≈ 2.7 and the relativistic collisionless electron skin-depth d e ≡ θ 0 m e c 2 /4πn 0 e 2 ≈ 1.8ρ 0 . Similarly, timescales are given in units of the gyration time of the bulk of the particles in the plasma, i.e., ω −1 0 ≡ ρ 0 /c ≈ 1140 s. The initial distribution of fields and plasma results in a low plasma-β or high magnetization of the upstream plasma (i.e., outside the layers). Here, the magnetization parameter is σ ≡ B 2 0 /4π(0.1n 0 )θ 0 m e c 2 ≈ 16. The system is initially set at an equilibrium, i.e., there is a force balance across the reconnection layers between the magnetic pressure and the drifting particle pressure. This equilibrium is unstable to two competing instabilities, namely the relativistic tearing and kink instabilities, as well as oblique modes that combine tearing and kink modes (Zenitani & Hoshino 2005, 2008 2013). In contrast to Cerutti et al. (2013), we choose here not to apply any initial perturbation in order to avoid any artificial enhancement of one type of instability over the other. Instabilities are seeded with the numerical noise only. This choice has a direct computational cost because the lack of perturbation significantly delays the onset of reconnection (See Sections 3.1, 4.1), but it enables a fair comparison between the growth rates of both instabilities (Sections 3.2, 4.2). Another important consequence of this choice specific to this study is the significant radiative cooling of the particles before reconnection can accelerate them (Section 3.3).
Set of simulations
In this work, we performed a series of 14 simulations. This set includes 12 2D simulations, and 2 3D simulations. The 2D simulations are designed to study the effect of the guide field strength, α = 0, 0.25, 0.5, 0.75 and 1 on the developments of instabilities (kink and tearing) and on particle acceleration/emission. To analyze the developments both instabilities separately, we follow the same approach as Zenitani & Hoshino (2007, 2008, i.e., we consider the dynamics of the Harris current layers in the xy-plane where the tearing modes alone develop similarly to our previous study in Cerutti et al. (2013), and in the yz-plane where the kink modes alone develop (the kink and the tearing modes are perpendicular to each other). The box is square of size L x × L y = (200ρ 0 ) 2 and L y × L z = (200ρ 0 ) 2 with 1440 2 cells and 16 particles per cell (all species together). The spatial resolution is ρ 0 /∆x ≈ 7.2, where ∆x is the grid spacing in the x-direction (∆x = ∆y = ∆z), which ensures the conservation of the total energy to within 1% error throughout the simulation. From this 2D scan, we identify the best/worst conditions for efficient particle acceleration and emission above the radiation reaction limit in 3D. The 3D box is cubical of size L x × L y × L z = (200ρ 0 ) 3 with 1440 3 grid cells and 16 particles per cell (all species together). The simulation time step is set at 0.3 times the critical Courant-Friedrichs-Lewy time step, ∆t = 0.3∆t CFL ≈ 0.029ω −1 0 in 2D and ≈ 0.024ω −1 0 in 3D, in order to maintain satisfactory total energy conservation in the presence of strong radiative damping. Table 1 enumerates all the simulations presented here.
RESULTS OF THE 2D RUNS
In this section, we present and discuss the results of the 2D runs listed in Table 1. After describing the overall time evolution of the reconnection layers in the xyand yz-planes (Section 3.1), we present a Fourier analysis of the tearing and kink instabilities as a function of the guide field strength (Section 3.2). Then, we deduce from the particle and photon spectra the most/least favorable conditions for particle accelera- tion beyond γ rad and synchrotron emission > 160 MeV in 3D (Section 3.3).
3.1. Description of the time evolution Figure 2 shows the time evolution of the total plasma density and field lines at four characteristic stages of 2D magnetic reconnection in the xy-plane with α = 0 (run 2DXY0). Because there is no initial perturbation, the layers remain static until tω 0 ≈ 120 when the layer tears apart into about 7 plasmoids per layer separated by X-points where field lines reconnect. The noise of the macro-particles in the PIC code is sufficient to seed the tearing instability. The reconnection electric field E z is maximum at X-points and is responsible for most of particle acceleration. The high magnetic tension of freshly reconnected field lines pushes the plasma towards the ±x-directions and drives the large scale reconnection outflow that forces magnetic islands to merge with each other. Reconnection proceeds until there is only one big island per layer remaining in the box. At the end of the simulation (tω 0 = 353), about 70% of the initial magnetic energy is dissipated in the form of particle kinetic energy. All the energy gained by the particles is then lost via the emission of synchrotron radiation. Adding a guide field does not suppress the tearing instability, but it creates a charge separation across the layer that induces a strong E y electric field (see also Zenitani & Hoshino 2008;Cerutti et al. 2013). Figure 3 presents the time evolution of the 2D simulation in the yz-plane with no guide field (run 2DYZ0). The initial setup of fields and particles is identical to run 2DXY0, except that the reconnecting field (B x ) is now perpendicular to the simulation plane. Hence, reconnection and tearing modes cannot be captured by this simulation. Instead, we observe the development of the kink instability as early as tω 0 ≈ 100 in the form of a small sinusoidal deformation of the current sheets with respect to the initial layer mid-plane. The sinusoidal deformation proceeds along the z-direction, with the deformation amplitude in the y-direction increasing rapidly up to about a quarter of the simulation box size (about 50ρ 0 ). At this stage, the folded current layers are disrupted, leading to fast and efficient magnetic dissipation. About 55% of the total magnetic energy is dissipated by the end of the simulation. The guide field has a dramatic influence on the stability of the layers. The amplitude of the deformation as well as the magnetic energy dissipated decreases with increasing guide field. For α 0.75, the layers remain flat during the entire duration of the simulation and no magnetic energy is dissipated. In this case, the only noticeable time evolution is a slight decrease of the layer thickness due to synchrotron cooling. To maintain pressure balance across the layers with the unchanged upstream magnetic field, the layer must com-press to compensate for the radiative energy losses (Uzdensky & McKinney 2011). We observed also a compression of the reconnection layer in the xy-reconnection simulations.
Fourier analysis of unstable modes
To compare the relative strength of the tearing instability versus the kink instability, we perform a spectral analysis of the fastest growing modes that develop in the simulations. To study the kink instability, we do a fast Fourier transform (FFT) along the z-direction of the small variations of the reconnecting magnetic field in the bottom layer midplane, δB x (z,t) = B x (y = L y /4, z,t) − B x (y = L y /4, z, 0), during the early phase of the 2D simulations in the yz-plane. For the tearing modes, we follow the same procedure for the fluctuations in the reconnected field along the x-direction, δB y (x,t) = B y (x, y = L y /4,t) − B x (y = L y /4, z, 0), in the 2D simulations in the xy-plane.
We present in Figure 4 the time evolution of the fastest growing modes as well as the dispersion relations for the tearing and kink modes, with no guide field. In the linear regime (tω 0 125), we infer the growth rates by fitting the amplitude of each mode with |FFT(δB x,y /B 0 )| ∝ exp γ gr (k)t , where γ gr is the growth rate of the mode of wave-number k. We find that the fastest growing tearing mode is at k x δ ≈ 0.58, which coincides with the analytical expectation of k x δ = 1/ √ 3 (Zelenyi & Krasnoselskikh 1979) as found by Zenitani & Hoshino (2007). The wavelength of this mode is
L x /λ x = L x /2π √ (3)δ ≈ 7;
this explains the number of plasmoids formed in the early stages of reconnection (see top right panel in Figure 2). The fastest growing kink mode has a wavelength L z /λ z ≈ 8 (or k z δ ≈ 0.67) which is consistent with the deformation of the current layers observed in Figure 3, top right panel. The corresponding growth rate is γ KI ω −1 0 ≈ 0.055, which is comparable with the fastest tearing growth rate, γ TI ω −1 0 ≈ 0.045 (Figure 4, bottom panel). This is expected for an ultra-relativistic plasma (kT m e c 2 ) with a drift velocity β drift = 0.6 (Zenitani & Hoshino 2007). It is worth noting that the dispersion relations for both the kink and the tearing instabilities are not sharply peaked around the fastest growing modes; a broad range of low-frequency modes is almost equally unstable (i.e., for 0 < k x,z δ 1).
In agreement with Zenitani & Hoshino (2008) and as pointed out in Section 3.1, we find that the kink instability depends sensitively on the guide field strength. Figure 5 shows that the fastest growth rate decreases rapidly between α = 0.25 and α = 0.75 from γ KI ω −1 0 ≈ 0.055 to undetectable levels. Thus, the guide field stabilizes the layer along the zdirection. In contrast, as mentioned earlier in Section 3.1, the tearing growth rate depends only mildly on α; we note a decrease from γ TI ω −1 0 ≈ 0.045 for α = 0 to γ TI ω −1 0 ≈ 0.025 for α = 1. For α 0.5, the tearing instability dominates over the kink.
Particle and photon spectra
The critical quantities of interest here are the particle energy distributions, γ 2 dN/dγ, and the instantaneous optically thin synchrotron radiation spectral energy distribution (SED) emitted by the particles, νF ν ≡ E 2 dN ph /dtdE, where E is the photon energy. Figure 6 shows the time evolution of the total particle spectra at different times with no guide field in the xy-plane (top panel) and in the yz-plane (bottom panel). In the early stage (tω 0 132), both simulations are subject to pure synchrotron cooling (i.e., with no acceleration or heating) of the plasma that results in a decrease of the typical Lorentz factor of the particles from γ/γ rad ≈ 0.3 at tω 0 = 0 to γ/γ rad ≈ 0.08 at tω 0 = 132. The decrease of the mean particle energy within the layer explains the shrinking of the layer thickness described in Section 3.1.
At tω 0 132, the instabilities trigger magnetic dissipation and particles are energized, but the particle spectra differ significantly in both cases. In run 2DXY0, where the tearing instability drives reconnection, the particle spectrum extends to higher and higher energy with time until the end of the simulation, where the maximum energy reaches γ max /γ rad ≈ 2.5, i.e., well above the nominal radiation reaction limit. The spectrum above γ/γ rad = 0.1 cannot be simply modeled with a single power-law, but it is well contained between two steep power laws of index −2 and −3. We know from our previous study that the high-energy particles are accelerated via the reconnection electric field at X-points and follow relativistic Speiser orbits . The maximum energy is then given by the electric potential drop along the z-direction (neglecting radiative losses), i.e.,
γ max ∼ eE z L x m e c 2 = eβ rec B 0 L x m e c 2 ≈ 3γ rad ,(6)
for a dimensionless reconnection rate β rec ≈ 0.2. Particles above the radiation reaction limit (γ > γ rad ) account for about 5% of the total energy of the plasma at tω 0 = 318 (Figure 7, top panel), and are responsible for the emission of synchrotron radiation above 160 MeV. Figure 7 (bottom panel) shows the resulting isotropic synchrotron radiation SED at tω 0 = 318, where about 11% of the radiative power is > 160 MeV. The SED peaks at E = 10 MeV and extends with a power-law of index −0.42 up to about 300-400 MeV before cutting off exponentially.
In contrast, in run 2DYZ0, where the kink instability drives the annihilation of the magnetic field, the particles are heated up to a typical energy γ/γ rad ≈ 0.3. The particle spectrum is composed of a Maxwellian-like distribution on top of a cooled distribution of particles formed at tω 0 132 (Figures 6, 8). The mean energy of the hot particles corresponds to a nearly uniform redistribution of the total dissipated magnetic energy to kinetic energy of background particles, i.e.,
γ ∼ 0.55 × B 2 0 /8π (0.1n 0 )m e c 2 = 0.55 × σθ 0 2 ≈ 0.34γ rad ,(7)
where the numerical factor 0.55 accounts for the fraction of the total magnetic energy dissipated at the end of the simulation. Hence, the development of the kink prevents the acceleration of particles above γ rad and the emission of synchrotron photons above 160 MeV. Figure 8 (bottom panel) shows that the total synchrotron radiation SED peaks and cuts off at E = 10 MeV, far below the desired energies > 160 MeV. Because a moderate guide field suppresses the effect of the kink instability, hence magnetic dissipation, the particles are not heated for α 0.5, and the initial spectrum continues cooling until the end of the yz-plane simulation where the particles radiate low-energy (∼ 1 MeV) synchrotron radiation (Figure 8). In the xy-plane reconnection simulations, the guide field tends to decrease the maximum energy of the particles and of the emitted radiation (Figure 7, see also Cerutti are averaged over all directions. The particle Lorentz factor is normalized to the nominal radiation reaction limit γ rad ≈ 1.3 × 10 9 . et al. 2013). The guide field deflects the particles outside the layer, reducing the time spent by the particle within the accelerating region.
RESULTS OF THE 3D RUNS
From the previous section, we find that the tearing and kink modes grow at a similar rate and wavelength in our setup. Both instabilities lead to fast dissipation of the magnetic energy in the form of thermal and non-thermal particles. The kink instability tends to disrupt the layer, which prevents nonthermal particle acceleration and emission above the standard radiation reaction limit. It is desirable to impose a moderate guide field to diminish the negative effect of the kink on particle acceleration, but too strong a guide field is not advantageous either, as it decreases the maximum energy reached by the particles and radiation. Hence, we decided to run a 3D simulation with an α = 0.5 guide field (run 3D050, see Table 1), which appears to be a good compromise. For comparison, we also performed a 3D simulation without guide field (run 3D0). In this section, we first describe the time evolution of 3D reconnection in the two runs (Section 4.1). Then, we provide a quantitative analysis of the most unstable modes in the (k x × k z )-plane in the linear regime (Sec- tion 4.2). In addition, we address below the question of particle acceleration, emission (Section 4.3), particle and radiation anisotropies (Section 4.4), the expected radiative signatures (i.e., spectra and lightcurves) and comparison with the Fermi-LAT observations of the Crab flares (Sections 4.5, 4.6). Figure 9 (left panels) shows the time evolution of the plasma density 3 in the zero-guide field simulation at tω 0 = 0, 173, 211 and 269. The initial stage where the layer remains apparently static lasts for about tω 0 = 144, i.e., half of the whole simulation time. At tω 0 144, overdensities appear in the layers in the form of 7-8 tubes (flux ropes) elongated along the z-direction. These structures are generated by the tearing instability and are the 3D generalization of the magnetic islands observed in 2D reconnection. As the simulation proceeds into the non-linear regime, the flux ropes merge with each other creating bigger ones, as magnetic islands do in 2D γ/γ rad reconnection. However, in 3D this process does not happen at the same time everywhere along the z-direction, which results in the formation of a network of interconnected flux ropes at intermediate times (173 tω 0 211). In parallel to this process, the kink instability deforms the two layers along the z-direction in the form of sine-like translation of the layers' mid-planes in the ±y-directions. During the most active period of reconnection (tω 0 173), the kink instability takes over and eventually destroys the flux ropes formed by the tearing modes (see left bottom panel in Figure 9). Only a few coherent structures survive at the end of the simulation (tω 0 = 269). In particular the reconnection electric field, which is strongest along the X-lines between two flux ropes, loses its initial coherence. This results in efficient particle heating but poor particle acceleration (see below, Section 4.3). At the end of this run about 52% of the total magnetic energy is dissipated, although the simulation does not reach the fully saturated state.
Plasma time evolution
The right panels in Figure 9 shows the time evolution of the plasma density for α = 0.5 guide field. One sees immediately that the guide field effectively suppresses the kink deformations of the layers in the ±y−directions, as expected from the 2D simulations in the yz-plane (See Section 3) and from Zenitani & Hoshino (2008). In contrast, the tearing instability seems undisturbed and breaks the layer into a network of 8 flux tubes. Towards the end of the simulation, there are about 3 well-defined flux ropes containing almost all the plasma that went through reconnection. At this point in time, 20% of the total magnetic energy (i.e., including the reconnecting and the guide field energy) has dissipated, in agreement with the 2D run 2DXY050.
Fourier analysis of unstable modes
Following the analysis presented in Section 3.2, we perform a Fourier decomposition of the magnetic fluctuations in the bottom layer mid-plane, (x, y = L y /4, z), to study the most unstable modes that develop in the 3D simulations. Figure 10 presents the growth rate of each modes in the (k x × k z )plane estimated from the variations of B x (left panel) and B y (right panel), for α = 0. As pointed out in Section 3.2 and by Zenitani & Hoshino (2008) and Kagan et al. (2013), we find that the reconnecting field B x effectively captures the kinklike modes along k z whereas the reconnected field B y is most sensitive to tearing-like modes along k x . The dispersion relations show that pure kink (along k z for k x = 0) and pure tearing (along k x for k z = 0) modes grow at rates in very good agreement with the corresponding 2D simulations. With a growth rate γ GR ≈ 0.06ω 0 , the fastest growing mode in the simulation is a pure kink mode of wavenumber k z δ ≈ 0.7, or L z /λ z ≈ 8 consistent with the deformation of the layer observed in the earlier stage of reconnection (Figure 9, left panels) and with the 2D run 2DYZ0. The fastest tearing mode has a growth rate γ GR ≈ 0.045ω 0 at k x δ ≈ 0.5 and generates the ≈ 7 ini- tial flux ropes obtained in the simulation. The (k x × k z )-plane is also filled with oblique modes, i.e., waves with a non-zero k x -and k z -component, with growth rates comparable to the fastest tearing and kink modes. The existence of these modes is reflected by the flux ropes being slightly tilted in the xzplane. Adding an α = 0.5 guide field decreases the amplitude of the low-frequency (k z δ 1) growth rates of the kink modes ( Figure 11). In particular, the growth rate of the fastest mode for α = 0, k z δ = 0.7, decreases from 0.06ω 0 to 0.03ω 0 . As a result, the fastest growing kink mode is now at k z δ = 0.8 with a rate ≈ 0.04ω 0 , while the fastest growing tearing modes is approximatively unchanged, in excellent agreement with the 2D runs ( Figure 11). Figure 12 presents the particle and photon energy distributions averaged over all directions at tω 0 = 265, with no guide field. The distributions are remarkably similar to the 2D run 2DYZ0 ones, and differ significantly from run 2DXY0. The high-energy part of the particle spectrum peaks at γ/γ rad = 0.3 which is the signature of particle heating via magnetic dissipation rather than particle acceleration through tearingdominated reconnection (Section 3.3, Eq. 7). We note that the spectrum extends to higher energy than the pure magnetic dissipation scenario, slightly above γ rad , suggesting that there is a non-thermal component as well. On the contrary, in the α = 0.5 guide field case (run 3DG050, Figure 13), things are closer to the pure tearing reconnection case of run 2DXY050. The particle energy distribution is almost flat in the 0.04 γ/γ rad 0.4 range, but barely reaches above γ rad as in the zero-guide field case. Nevertheless, the synchrotron emission > 160 MeV is more intense than in the zero guide field case. Even though there is clear evidence for particle acceleration above the radiation reaction limit, the effect remains slightly weaker than in 2D with no guide field. A bigger box size would help to improve the significance of this result.
Particle and photon spectra
-150°-120°-90°-60°-30°0°30°60°90°120°150°- 75°- 60°- 45°- 30°- 15°0°1 5°3 0°4 5°6 0°7 5°E =0.1 [MeV] 0
Particle and photon anisotropies
The angular distribution of the particles is of critical interest for determining the apparent isotropic radiation flux seen by a distant observer who probes one direction only. In 2D reconnection, we expect a pronounced beaming of the particles that increases rapidly with their energy (Cerutti et al. 2012b. We confirm here that this phenomenon exists also in 3D, even with a finite guide field. Figure 14 presents energyresolved maps of the angular distribution of the positrons (left panels) and their optically thin synchrotron radiation (right panel) in run 3D050. The direction of motion of the particles is measured with two angles: the latitude, φ, varying between −90 • and 90 • , defined as
φ = sin −1 u y u 2 x + u 2 y + u 2 z ,(8)
and the longitude, λ, defined between −180 • and 180 • given by
λ = cos −1 uz √ u 2 x +u 2 z if sin λ > 0 − cos −1 uz √ u 2 x +u 2 z if sin λ < 0 ,(9)
where u x , u y , and u z are the components of the particle 4velocity vector. We find that the low-energy particles (γ/γ rad 0.1) nearly conserve the initially imposed isotropy, because they are still upstream and have not been energized by reconnection. In contrast, the high-energy particles (γ/γ rad 0.1) are significantly beamed along the reconnection plane (at X-lines and with flux ropes) within φ = ±15 • and λ = ±60 • . The λ = ±60 • angle is of special interest here because it coincides with the direction of the undisturbed magnetic field lines outside the reconnection layers for a α = 0.5 guide field (λ 0 = ± tan −1 (1/α) ≈ ±63 • ). The particles are accelerated along the z-direction by the reconnection electric field, and move back and forth across the layer mid-plane following relativistic Speiser orbits . At the same time, the particles are deflected away by the reconnected field and the guide field creating a characteristic "S" shape in the angular maps. To a lesser extent, the zero-guide field case also presents some degree of anisotropy, but the deformation and then the disruption of the layer by the kink instability effectively broaden the beams.
The synchrotron angular distribution closely follows the particle one, essentially because relativistic particles radiate along their direction of motion within a cone of semi-aperture angle ∼ 1/γ 1. However, there is a noticeable offset between the distribution of the highest-energy particles with γ γ rad and the radiation above 100 MeV. This discrepancy is due to the different zones where particles accelerate and
Isotropic Anisotropic
FIG. 15.-Top: Comparison between the isotropically averaged particle energy distribution (dashed line) and the apparent isotropic distribution in the φ = −9.2 • , λ = 34.5 • direction (solid line). Bottom: Comparison between the isotropically averaged synchrotron SED (dashed line), the apparent isotropic SED in the φ = 5.5 • , λ = −63.6 • direction at tω 0 = 288, for α = 0.5 (solid line), and the observed Fermi-LAT spectra (data points) during the flares in February 2009, September 2010, and in April 2011, as well as the average quiescent spectrum from 1 MeV to 10 GeV (Abdo et al. 2011;Buehler et al. 2012). Observed fluxes are converted into isotropic luminosities, assuming that the nebula is at 2 kpc from Earth.
where particles radiate. In the accelerating zone, the electric field is intense and leads to linear particle acceleration along the z-axis, whereas the perpendicular magnetic field, B ⊥ , is weak deep inside the reconnection layers, yielding little synchrotron radiation. These high-energy particles then radiate 100 MeV emission abruptly, i.e., within a fraction of a Larmor gyration, only when they are deflected outside the layer where B ⊥ ∼ B 0 . The beam dump is well localized at λ = ±60 • , i.e., along the upstream magnetic field lines (see hot-spots in Figure 14, bottom-right panel). 4.5. Apparent spectra and comparison with observations As a consequence of this strong anisotropy, the observed spectra of particles and radiation depend sensitively on the viewing angle. Figure 15 compares the isotropic particle (top panel) and photon energy distributions (apparent intrinsic isotropic luminosities, νL ν , bottom panel) with the distributions along one of the directions dominated by the highest energy particles (φ = −9.2 • , λ = 34.5 • ) and radiation (φ = 5.5 • , λ = −63.6 • ) as it appears to a distant observer at Fluxes are given in ph/s/cm 2 , assuming a distance of 2 kpc between the layer and the observer. The horizontal gray band shows the average observed flux of the Crab Nebula above 100 MeV measured by the Fermi-LAT, equal to 6.1 ± 0.2 × 10 −7 ph/s/cm 2 (Buehler et al. 2012). tω 0 = 288, for α = 0.5 (see Section 4.4 and Figure 14). The bottom panel in Figure 15 also compares the Fermi-LAT measurements of the gamma-ray spectra of the February 2009, September 2010 and April 2011 flares and the average quiescent spectrum of the Crab Nebula (Abdo et al. 2011;Buehler et al. 2012) with the simulated gamma-ray spectra. The particle spectrum in the (φ = −9.2 • , λ = 34.5 • ) direction is very hard, and peaks at γ/γ rad ≈ 0.6 with a total apparent isotropic energy ≈ 7 × 10 40 erg, which corresponds to about half the total magnetic energy dissipated by the end of the simulation, or about 10 times more energy than in the isotropic distribution. The photon spectrum in the (φ = 5.5 • , λ = −63.6 • ) direction peaks at about 100 MeV but extends up to ∼ 1 GeV. The resulting gamma-ray luminosity above 100 MeV is L γ ≈ 3 × 10 35 erg/s, which is about 40 times brighter than the isotropically averaged flux from the layer, and about 3 times brighter than the observed average (quiescent) luminosity of the Crab Nebula (L Crab γ ≈ 10 35 erg/s, assuming a distance of 2 kpc between the nebula and the observer). Thanks to beaming, the level of gamma rays expected from the model is consistent with the moderately bright flares, such as the February 2009 or September 2010 ones (see Figure 15, bottom panel), but the simulation cannot reproduce the flux of the brightest flares, such as the April 2011 event. Presumably, increasing the box size would be enough to increase the density of the emitting particles > 100 MeV and account for the most intense flares.
Lightcurves
The beam of high-energy radiation is also time variable, both in direction and in intensity. Figure 16 presents the computed time evolution of the synchrotron photon flux integrated above 100 MeV in the directions defined by φ = 5.5 • , λ = −63.6 • and φ = −5.5 • , λ = 49.1 • , as well as the computed time evolution of the isotropically-averaged flux and the observed average flux in the Crab Nebula measured by the Fermi-LAT (Buehler et al. 2012) for comparison. This calculation assumes that all the photons emitted at a given instant throughout the box reach the observer at the same time, i.e., it ignores the time delay between photons emitted in different regions with respect to the observer. Along the directions probed here, the > 100 MeV flux doubling timescale is of order 10 − 20ω −1 0 or 3-6 hours, for both the rising and the decaying time, which is compatible with the observations of the Crab flares (Buehler et al. 2012;Mayer et al. 2013), as well as with our previous 2D simulations in Cerutti et al. (2013). Shorter variability timescales may still exist in the simulation but our measurement is limited by the data dumping period set at T dump ≈ 10ω −1 0 . These synthetic lightcurves also clearly illustrate the effect of the particle beaming on the observed flux discussed in Section 4.5. Along the direction of the beam, the > 100 MeV flux can be 10 times more intense than the isotropically-averaged one, and even exceeds the measured quiescent gamma-ray flux of the Crab Nebula by a factor ≈ 2-3 at the peak of the lightcurves 3.5 to 4 days after the beginning of the simulation. Each time the beam crosses our line of sight, we see a rapid bright flare of the most energetic radiation emitted in the simulation.
The high-energy particles are strongly bunched within the magnetic flux ropes (within magnetic islands in 2D, see Cerutti et al. 2012bCerutti et al. , 2013. As a result, the typical size of the emitting regions is comparable to the dimensions of the flux ropes, i.e., of order L x /10 ≈ 20ρ 0 along the xand y-directions (Figure 9), which corresponds to about 6 light-hours. We conclude that particle bunching is at the origin of the ultrashort time variability (3-6 hours) found in the reconstructed lightcurve ( Figure 16). Particle bunching and anisotropy help to alleviate the severe energetic constraints imposed by the Crab flares.
CONCLUSION
We found that, unlike classical models of particle acceleration, 3D relativistic pair plasma reconnection can accelerate particles above the standard radiation reaction limit in the Crab Nebula. We also confirm the existence of a strong energy-dependent anisotropy of the particles and their radiation, resulting in an apparent boosting of the high-energy radiation observed when the beam crosses our line of sight. In this case, the simulated gamma-ray flux > 100 MeV exceeds the measured quiescent flux from the nebula by a factor 2-3, and reproduces well the flux of moderately bright flares, such as the February 2009 or the September 2010 events. Simulating brighter flares (e.g., the April 2011 flare) may be achieved with a larger box size. In addition, the bunching of the energetic particles within the magnetic flux ropes results in rapid time variations of the observed gamma-ray flux ( 6 hours). The results are consistent with observations of the Crab flares and with our previous 2D simulations , although this extreme acceleration is less pronounced than in 2D due to the deformation of the layer by the kink instability in 3D. If there is no guide field, we found that the kink instability grows faster than the tearing instability, resulting in the disruption of the reconnection layers and significant particle heating rather than reconnection and non-thermal particle acceleration. In agreement with Zenitani & Hoshino (2007, 2008, we observe that a moderate guide field (α ∼ 0.5) is enough to reduce the negative effect of the kink on the acceleration of particles. However, a strong guide field (i.e., α 1) quenches particle acceleration and the emission of high-energy emission because it deflects the particles away from the X-lines too rapidly.
Applying a guide field is probably not the only way to suppress kink instability. In the Harris configuration, an initial
FIG. 1.-Initial simulation setup and geometry. The computational domain is a rectangular box of volume Lx × Ly × Lz with periodic boundary conditions in all directions. The box initially contains two flat, anti-parallel, relativistic Harris layers in the xz-plane centered at y = Ly/4 and y = 3Ly/4, of some thickness 2δ. The magnetic field structure is composed of the reconnecting field, B 0 , along the x-direction, which reverses across the layers, and a uniform guide field, Bz = αB 0 , along the z-direction.
; Daughton et al. 2011; Kagan et al. FIG. 2.-Snapshots of the plasma density at tω 0 = 0 (top left), 132 (top right), 265 (bottom left), and 353 (bottom right) of the 2D simulation 2DXY0 in the xy-plane (with no guide field, α = 0). Magnetic field lines are represented by solid white lines. In this simulation, the development of the tearing instability forms multiple plasmoids separated by X-points which facilitates fast magnetic reconnection. Reconnection dissipates about 70% of the magnetic energy in the form of energetic particles and radiation (see Figures 6-7).
FIG. 3 .
3-Snapshot of the plasma density at tω 0 = 0 (top left), 132 (top right), 176 (bottom left), and 265 (bottom right) of the 2D simulation 2DYZ0 in the yz-plane (with no guide field, α = 0). Although this simulation cannot capture magnetic reconnection that proceeds in the xy-plane, it shows that the layers rapidly destabilize along the z-direction due to the kink instability. The layers are deformed and eventually completely disrupted, leading to efficient dissipation of the magnetic energy (about 55%), mostly in the form of heat (seeFigures 6-8).
FIG. 4 .
4-Top: Time evolution of the fastest growing tearing (red solid line, kxδ ≈ 0.58) and kink (blue dashed line, kzδ ≈ 0.67) modes, in the simulation 2DXY0 and 2DYZ0 where α = 0. The duration of the linear phase is tω 0 ≈ 125 (delimited by the vertical dotted line) and is about the same in both simulations. Bottom: Dispersion relations of the tearing (red solid line) and kink (blue dashed line) instabilities during the linear stage. This plot shows only the region of small wavenumber kz,x where the most unstable modes are found. The vertical black dotted lines mark the fastest growing modes.
FIG. 5 .
5-Linear growth rates of the fastest growing modes for the tearing (for γ TI (kx = 0.58), red solid line) and the kink (for γ KI (kz = 0.67), blue dashed line) instabilities multiplied by ω −1 0 , as a function of the guide field strength α. Each dot represents one simulation. The analysis of the kink/tearing instability was performed using the set of 2D simulations in the yz-/xy-plane.
FIG. 6 .
6-Particle energy distribution normalized to the total number of particles (γ 2 (1/N)dN/dγ) of the 2D simulations in the xy-plane (top, run 2DXY0) and yz-plane (bottom, run 2DYZ0) for α = 0. The spectra are obtained at time tω 0 = 0 (dotted line), 132, 176, 265 and 353 (dashed line) and
FIG
. 7.-Particle energy distribution normalized to the total number of particles (γ 2 (1/N)dN/dγ, top) and synchrotron radiation spectral energy distribution normalized by the total (frequency-integrated) photon flux ((1/F)νFν , bottom) of the 2D simulations in the xy-plane at tω 0 = 318, averaged over all directions. The spectra are obtained for α = 0, 0.25, 0.5, 0.75 and 1. The particle Lorentz factor in the top panel is normalized to the nominal radiation reaction limit γ rad ≈ 1.3 × 10 9 . In the bottom panel, the blue dashed line is a power-law fit of index ≈ −0.42 of the α = 0 SED between E = 20 MeV and E = 350 MeV.
FIG. 8 .
8-Same as inFigure 7, but for the 2D simulations in the yz-plane.
FIG. 9 .
9-Time evolution of the plasma density (color-coded isosurfaces) in the bottom half of the simulation box at tω 0 = 0, 173, 211, and 269 (from top to bottom), for α = 0 (left panels, run 3D0) and α = 0.5 (right panels, run 3D050). Low-density isosurfaces (blue) are transparent in order to see the high-density regions (red) nested in the flux tubes. The time is given in units of ω −1 0 , and spatial coordinates are in units of ρ 0 .FIG. 10.-Linear growth rates in run 3D0 γ GR times ω −1 0 in the (kx × kz)-plane (color-coded plots) using the fluctuations in Bx (left panel) which are most sensitive to kink-like modes, and in By (right panel) which are most sensitive to tearing-like modes. The blue solid lines in each subplots give the growth rates along the kx-axis for kz = 0 (bottom subplots) and along the kz-axis for kx = 0 (left subplots). The red dashed lines show the dispersion relation for the pure kink and tearing modes obtained in Section 3.2 for comparison.FIG. 11.-Same as inFig. 10with an α = 0.5 guide field (run 3D050).
FIG
. 12.-Isotropically averaged particle energy distribution (top panel) and SED (bottom panel) obtained in 2D (xy-plane, blue dotted line, and in the yz-plane, green dashed line) and 3D (red solid line) with no guide field.
FIG
. 13.-Same as inFigure 12, but with an α = 0.5 guide field.
νF ν )/dΩdE[Normalized] FIG. 14.-Positron angular distributions (dN/dΩdγ, left panels) and their synchrotron radiation angular distribution (d(νFν )/dΩdE, right panels) in run 3D050 (α = 0.5) at tω 0 = 291. Each panel is at a different energy bin: γ/γ rad = 0.01 (top left), 0.3 (middle left) and 1 (bottom left) for the particles and E = 0.1 MeV (top right) 10 MeV (middle right) and 100 MeV (bottom right) for the photons. The color-coded scale is linear and normalized to the maximum value in each energy bin. The angular distribution is shown in the Aitoff projection, where the horizontal-axis is the longitude, λ, varying between ±180 • (−zaxis) and the vertical axis is the latitude, φ, varying between −90 • (−y-direction) to +90 • (+y-direction). The origin of the plot corresponds to the +z-direction.
16.-Instantaneous synchrotron photon flux integrated above 100 MeV as a function of time in the directions φ = 5.5 • , λ = −63.6 • (blue solid line), φ = −5.5 • , λ = 49.1 • (green dot-dashed line), and isotropically averaged (red dotted line) in run 3D050.
1 http://benoit.cerutti.free.fr/zeltron.html
TABLE 1 COMPLETE
1LIST OF THE NUMERICAL SIMULATIONS PRESENTED IN THIS PAPERName simulation
Lx/ρ 0
Ly/ρ 0
Lz/ρ 0
Grid cells
# particles
α
2DXY0
200
200
-
1440 2
3.32 × 10 7
0
2DXY025
200
200
-
1440 2
3.32 × 10 7
0.25
2DXY050
200
200
-
1440 2
3.32 × 10 7
0.5
2DXY075
200
200
-
1440 2
3.32 × 10 7
0.75
2DXY1
200
200
-
1440 2
3.32 × 10 7
1
2DYZ0
-
200
200
1440 2
3.32 × 10 7
0
2DYZ025
-
200
200
1440 2
3.32 × 10 7
0.25
2DYZ040
-
200
200
1440 2
3.32 × 10 7
0.4
2DYZ050
-
200
200
1440 2
3.32 × 10 7
0.5
2DYZ060
-
200
200
1440 2
3.32 × 10 7
0.6
2DYZ075
-
200
200
1440 2
3.32 × 10 7
0.75
2DYZ1
-
200
200
1440 2
3.32 × 10 7
1
3D0
200
200
200
1440 3
4.78 × 10 10
0
3D050
200
200
200
1440 3
4.78 × 10 10
0.5
tω 0 =288 , α =0.510 -1
10 0
γ/γ rad
10 38
10 39
10 40
10 41
γ 2
dN/dγ
[erg]
γ =γ rad
tω 0 =288 , α =0.5
Isotropic
Anisotropic
10 0
10 1
10 2
10 3
10 4
E [MeV]
10 33
10 34
10 35
10 36
10 37
νL
ν [erg/s]
160 MeV
Average
Apr. 11
Sep. 10
Feb. 09
National Institute for Computational Sciences (www.nics. tennessee.edu/).
Movies are available at this URL: http://benoit.cerutti. free.fr/movies/Reconnection_Crab3D/. 10 -1 10 0
light-cylinder in pulsars offers another interesting environment for studying relativistic reconnection subject to strong synchrotron cooling. Pairs energized by reconnection may be at the origin of the GeV pulsed emission in gamma-ray pulsars(Lyubarskii 1996;Pétri 2012;Uzdensky & Spitkovsky 2012;Arka & Dubus 2013). We speculate that synchrotron radiation from the particles could even account for the recently reported > 100 GeV pulsed emission from the Crab (see, e.g., VERI-TAS Collaboration et al. 2011;Aleksić et al. 2012) if particle acceleration above the radiation reaction limit operates in this context. BC thanks G. Lesur for discussions about the linear analysis of the unstable modes in the simulations. The authors thank the referee for his/her useful comments. BC acknowledges support from the Lyman Spitzer Jr. Fellowship awarded by the Department of Astrophysical Sciences at Princeton University, and the Max-Planck/Princeton Center for Plasma Physics. This work was also supported by NSF grant PHY-0903851, DOE Grants DE-SC0008409 and DE-SC0008655, NASA grant NNX12AP17G through the Fermi Guest Investigator Program. Numerical simulations were performed on the local CIPS computer cluster Verus and on Kraken at the National Institute for Computational Sciences (www.nics.tennessee.edu/). This work also utilized the Janus supercomputer, which is supported by the National Science Foundation (award number CNS-0821794), the University of Colorado Boulder, the University of Colorado Denver, and the National Center for Atmospheric Research. The Janus supercomputer is operated by the University of Colorado Boulder. The figures published in this work were created with the matplotlib library(Hunter 2007)and the 3D visualization with Mayavi2(Ramachandran & Varoquaux 2011).
In real systems, the reconnection layer is not likely to be smooth, flat, undisturbed, and in equilibrium. A small perturbation in the field lines, like a pre-existing X-point, can favor fast reconnection before the kink instability has time to grow. Hence, while we have shown one set of conditions emitting > 160 MeV radiation, there may be other conditions allowing reconnection to produce similar results. The reconnection model could be refined if future multiwavelength observations can pin down the location of the flares in the Crab Nebula (so far there is nothing obvious. Liu, Alternatively, starting with an out-of-equilibrium layer could also drive a fast onset of reconnection. BegelmanLyubarsky. Porth et al. 2013b). In addition, theoretical studies. numerical simulations (Mizuno et altearing-dominated reconnection (Zenitani & Hoshino 2007) as observed by Liu et al. (2011). Alternatively, starting with an out-of-equilibrium layer could also drive a fast onset of reconnection (Kagan et al. 2013). In real systems, the reconnection layer is not likely to be smooth, flat, undis- turbed, and in equilibrium. A small perturbation in the field lines, like a pre-existing X-point, can favor fast reconnection before the kink instability has time to grow. Hence, while we have shown one set of conditions emitting > 160 MeV radia- tion, there may be other conditions allowing reconnection to produce similar results. The reconnection model could be refined if future multi- wavelength observations can pin down the location of the flares in the Crab Nebula (so far there is nothing obvious, see e.g., Weisskopf et al. 2013). One promising location for reconnection-powered flares could be within the jets in the polar regions where the plasma is expected to be highly magnetized (i.e., σ 1) with stronger magnetic field close to the pulsar rotational axis (Uzdensky et al. 2011; Cerutti et al. 2012a; Lyubarsky 2012; Komissarov 2013; Mignone et al. 2013; Porth et al. 2013b). In addition, theoretical stud- ies (Begelman 1998), numerical simulations (Mizuno et al.
) indicate that the jets are unstable to kink instabilities. The non-linear development of these instabilities could lead to the formation of current sheets, presumably with a non-zero guide field, and then to magnetic dissipation in the Crab Nebula in the form of powerful gamma-ray flares. Porth, and possibly X-ray observations. Rees & Gunnwhich may contribute to solving the "σ-problem" in pulsar wind nebulaePorth et al. 2013a,b; Mignone et al. 2013), and pos- sibly X-ray observations (Weisskopf 2011) indicate that the jets are unstable to kink instabilities. The non-linear devel- opment of these instabilities could lead to the formation of current sheets, presumably with a non-zero guide field, and then to magnetic dissipation in the Crab Nebula in the form of powerful gamma-ray flares, which may contribute to solving the "σ-problem" in pulsar wind nebulae (Rees & Gunn 1974;
. & Kennel, Coroniti, Kennel & Coroniti 1984).
Relativistic reconnection may also be at the origin of other astrophysical flares. Most notably, TeV gamma-ray flares observed in blazars. Aharonian, Relativistic reconnection may also be at the origin of other astrophysical flares. Most notably, TeV gamma-ray flares ob- served in blazars (e.g., Aharonian et al. 2007; Albert et al.
2011) present challenges similar to the Crab flares (e.g., ultra-short time variability, problematic energetics) that could be solved by invoking relativistic reconnection in a highly magnetized jet. Aleksić , Aleksić et al. 2011) present challenges similar to the Crab flares (e.g., ultra-short time variability, problematic en- ergetics) that could be solved by invoking relativistic recon- nection in a highly magnetized jet (Giannios et al. 2009, 2010;
The physical conditions in blazar jets are quite different than in the Crab Nebula (e.g., composition, inverse Compton drag, pair creation), which may change the dynamics of reconnection. Nalewajko, Giannios. The current sheet that forms beyond theNalewajko et al. 2011, 2012; Cerutti et al. 2012b; Giannios 2013). The physical conditions in blazar jets are quite dif- ferent than in the Crab Nebula (e.g., composition, inverse Compton drag, pair creation), which may change the dynam- ics of reconnection. The current sheet that forms beyond the
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| []
|
[
"Three component fermion pairing in two dimensions",
"Three component fermion pairing in two dimensions"
]
| [
"Theja N De Silva \nDepartment of Physics, Applied Physics and Astronomy\nThe State University of New York at Binghamton\n13902BinghamtonNew YorkUSA\n"
]
| [
"Department of Physics, Applied Physics and Astronomy\nThe State University of New York at Binghamton\n13902BinghamtonNew YorkUSA"
]
| []
| We study pairing of an interacting three component Fermi gas in two dimensions. By using a mean field theory to decouple the interactions between different pairs of Fermi components, we study the free energy landscapes as a function of various system parameters including chemical potentials, binding energies, and temperature. We find that the s-wave pairing channel is determined by both chemical potentials and the interaction strengths between the three available channels. We find a second order thermal phase transition and a series of first order quantum phase transitions for a homogenous system as we change the parameters. In particular, for symmetric parameters, we find the simultaneous existence of three superfluid orders as well as re-entrant quantum phase transitions as we tune the parameters. | 10.1103/physreva.80.013620 | [
"https://arxiv.org/pdf/0901.3770v3.pdf"
]
| 119,259,375 | 0901.3770 | 8a533dbdccd775e06ac93d9e9ecb7ec0cb6fcd0a |
Three component fermion pairing in two dimensions
Theja N De Silva
Department of Physics, Applied Physics and Astronomy
The State University of New York at Binghamton
13902BinghamtonNew YorkUSA
Three component fermion pairing in two dimensions
We study pairing of an interacting three component Fermi gas in two dimensions. By using a mean field theory to decouple the interactions between different pairs of Fermi components, we study the free energy landscapes as a function of various system parameters including chemical potentials, binding energies, and temperature. We find that the s-wave pairing channel is determined by both chemical potentials and the interaction strengths between the three available channels. We find a second order thermal phase transition and a series of first order quantum phase transitions for a homogenous system as we change the parameters. In particular, for symmetric parameters, we find the simultaneous existence of three superfluid orders as well as re-entrant quantum phase transitions as we tune the parameters.
I. INTRODUCTION
Recent experimental progress achieved in ultra-cold atomic gases allows one to set up test beds for controlled study of many body physics. By tuning three dimensional two-body scattering length between atoms in different hyperfine spin states of a dilute system at low temperatures, interaction strength between atoms can be controlled very precisely [1]. This can be done by using magnetically tuned Feshbach resonance [2]. Moreover, interaction and spatial dimensionality can be effectively controlled by applying an optical lattice. An effective two-dimensional system can be created by applying a relatively strong one-dimensional optical lattice to an ordinary three-dimensional system. As there are always two body bound states exist for attractive two body potentials in two dimensions [3], different pairs of Fermi atoms can undergo Bose Einstein condensation and form superfluidity at low temperatures. The two dimensional bound state energies can be controlled by tuning either three dimensional scattering length or the laser intensity which used to create one dimensional lattice to accommodate 2D layers.
In this paper we study three component Fermi gases in two dimensions. A mixture of 6 Li atoms which has favorable collisional properties among its lowest three hyperfine spin states will be an ideal system to explore novel three component superfluidity. Three component 6 Li mixture in three dimensions have already been trapped and manipulated experimentally [4,5]. In Ref. [4], using radio frequency spectroscopic data and a quantum scattering model, scattering lengths and the Feshbach resonance positions in the lowest three channels of 6 Li atoms have been determined. As there are three broad s-wave Feshbach resonances, one can prepare the system at various interaction strengths between each pairs. In Ref. [5], collisional stability of the lowest three channels of 6 Li atoms has been studied. As the spin relaxation time is large compared to the other time scales in the experiments, experimentalists were able to maintain a fixed spin population throughout the experiments. As a physically accessible system, a three-component ultracold atomic system can be used to study the physics of nuclear matter. Three-component Fermi pairing is believed to occur in the interior of neutron stars and in heavy-ion collisions [6]. Nevertheless, this system can be used to understand the competition between quantum phases and re-entrant phase transitions.
Properties of three-component Fermi systems have been extensively studied in recent past [7,8,9,10,11,12]. However, all these studies are carried out in a three dimensional or one dimensional environments. Further, authors in all these references except Ref. [8] have restricted their parameter space either by assuming equal interaction strengths or equal chemical potentials or by neglecting the interaction between some hyperfine spin components. In ref. [8], the authors have studied the properties of a harmonically trapped three component gas in three dimensions.
In this paper, we neglect the possibility of three body bound states in two dimensions and consider only two body pairing states. This is reasonable, because of the system we are considering is dilute and the atomic interactions are short range in nature. Therefore, it is unlikely to have many atoms interacting in the same region of space. Further, we neglect the harmonic confinement and consider the system as spatially homogenous. Twocomponent Fermi gases in two dimensions have already been studied in theory [13,14]. The purpose of this paper is to investigate how pairing will occur when a third spin component is added to such a two-component gas. More precisely, we study the competition of individual components to form Bose condensed pairs by investigating the landscapes of the free energy as a function of various parameters which include the temperature, chemical potentials of the hyperfine spin components, and the interaction strengths between different pairs of fermions components. For appropriately chosen parameters, we find that the system undergoes a second order thermal phase transition from normal state to a superfluid state as one lowers the temperature. At low temperatures, we find a series of first order quantum phase transitions as we change the chemical potentials or the interactions between hyperfine spin components. At low temperatures, we find simultaneous existence of three types of superfluid phases (at symmetric parameters corresponding to different pairing channels) and normal phases in this novel three-component Fermi system. Interestingly, we find that the system can undergo re-entrant phase transitions as we simultaneously tune the chemical potentials and interactions.
The paper is organized as follows. In the following section, we introduce the theoretical model and use a mean field approximation to decouple the interaction terms. Then using a canonical transformation, we diagonalize the Hamiltonian to derive the free energy of the system. In section III, we present our results with a discussion. Finally, our summary and conclusions are given in section IV.
II. FORMALISM
We consider an interacting three-component Fermi atomic gas trapped in two dimensions. We take the model Hamiltonian of the system as
H = d 2 r n ψ † n (r)[−h 2 ∇ 2 2D 2m − µ n ]ψ n (r)(1)+ 1 2 n =n U nn ψ † n (r)ψ † n (r)ψ n (r)ψ n (r)
where r 2 = x 2 + y 2 , ∇ 2D is the 2D gradient operator and U nn is the 2D interaction strength between component n and n . The operator ψ † n (r) creates a fermion of mass m with hyperfine spin n = 1, 2, 3 at position r = (x, y). The chemical potential of the n'th component is µ n . Notice that we have neglected the interaction between the same components. This is reasonable as we are considering a dilute atomic system, and the interactions are short-range in nature, s-wave scattering channel is dominated over the other scattering channels. By using a mean field decoupling of the interacting terms, the mean field Hamiltonian in the momentum space can be written as,
H M F = ij ψ † i A ij ψ j + 1 2 ij (ψ † i B ij ψ † j + h.c) − i =j |∆ ij | U ij(2)
where i = k, n and j = −k, n. Here we defined the superfluid order parameters U ij ψ i ψ j = ∆ ij and two matrices A and B,
A = 1 0 0 0 2 0 0 0 3 (3) B = 0 ∆ 3 −∆ 2 −∆ 3 0 ∆ 1 ∆ 2 −∆ 1 0 (4)
where n =h 2 k 2 /(2m) − µ n and ∆ ij = ijk ∆ k . As the mean field Hamiltonian is quadratic in Fermi operators, it can be diagonalized with a canonical transformation to get
H M F = n,k Λ n η † n η n + 1 2 n,k ( n − Λ n ) − n |∆ n | U n(5)
where we use the notation U ij = | ijk |U k . The Fermi operators η n represent the quasi particles in the system with n = 1, 2, 3. The quasi particle energies Λ n = √ λ n are given by [16]. Here we defined
λ n = 2 √ −Q cos{[θ + (n − 1)2π]/3} − A k /3. The parameter θ = arccos[R/ −Q 3 ] with Q = (3B k − A 2 k )/9 and R = (9A k B k − 27C k − 2A 3 k )/54A k = − n 2 n − 2 ∆ 2 n , B k = n ∆ 4 n + 2 n ∆ 2 n 2 n + n =m ∆ 2 n ∆ 2 m + 1/2 n =mC k = −( n ∆ 2 n n + 1 2 3 ) 2 . The grand potential of the system Ω = −1/(β) ln[Z G ] with Z G = tr{e [−βH M H ] } is then given by Ω = −1/(β) n,k [ln(1 + e −βΛn )](6)+ 1 2 n,k ( n − Λ n ) − n |∆ n | U n where β = 1/(k B T )
is the inverse temperature and k B is the Boltzmann constant. As the short range nature of the interaction, the grand potential is diverging so that regularization must be done in standards way by writing −|∆ n | 2 /U n = k |∆ n | 2 /(h 2 k 2 /m + E Bn ). Here E Bij = | ijk |E Bk is the binding energy between two hyperfine spin components i and j. Notice that we use the same notation for E Bn as we used for ∆ ij = ijk ∆ k and U ij = | ijk |U k . For two dimensions, converting the k into integral d 2 k/(2π) 2 and then by changing the variable by k 2 = z, the grand potential can be converted into an one dimensional integral. We numerically perform this integral and numerically minimize the grand potential for the seven parameter space (three chemical potentials, three binding energies and the temperature).
III. RESULTS AND DISCUSSION
In Fig. 1, we plot the free energy landscapes for different temperatures at a selected set of parameters. We choose the binding energy in the 2 − 3 channel to be small (E B1 = 0.1µ 1 ) so that pairing is not possible in this channel. As a result, we find a second order thermal phase transition as we lower the temperature. As can be seen, at high temperature [ Fig. 1-(a)], free energy is minimum when both paring order parameters (∆ 2 and ∆ 3 ) in channels 1 − 3 and 1 − 2 are zero. As we lower the temperature, channel 1 − 2 undergoes pairing and form Bose condensation. This is because the binding energy and average chemical potential in this channel is larger than those of channel 1 − 3 and 2 − 3. Further lowering the temperature results more atom pairing and condensation in channel 1 − 2 giving larger superfluid order parameter ∆ 3 . In principle, it is possible to have a sequence of second order thermal and first order quantum phase transitions at three different critical temperatures, if one change the interactions or the chemical potentials together with temperature. The reason for this sequence of phase transition is that there are many ways of pairing when various favorable channels are available.
In Fig. 2, we plot the temperature dependence of the superfluid order parameter (∆ 3 ) in channel 1 − 2 for chosen values of parameters. We choose the parameters such that pairing is possible only in channel 1 − 2 so that a single minimum is available in the free energy. As can be seen, the superfluid order parameter continuously increases as one lower the temperature, showing a second order thermal phase transition.
By varying the average chemical potentials and binding energies of the pairing channels at low temperatures, one can control the first order quantum phase transitions from one superfluid phase to another. As a demonstration, we plot the free energy landscapes in Fig. 3 for a selected set of parameters. Again, we chose the parameters such that the pairing in channel 2 − 3 is very weak and the corresponding superfluid order parameter (∆ 1 ) is zero. When the binding energy in channel 1 − 2 is larger than that of the channel 1−3, the free energy minimum is at a non zero value of ∆ 3 but zero value of ∆ 2 at the same chemical potentials. However, when the binding energies are equal at equal average chemical potentials, both superfluid order parameters are non zero and free energy gives many stable stationary points as seen in Fig. 3-(b) (now the global minimum is not a point, but a quarter of a circle). The reason for this line of global minimum is the symmetry of the parameter space. By increasing the binding energy in channel 1−3 over the channel 1−2, the minimum free energy pass to the non zero ∆ 2 but zero ∆ 3 . Similar first order quantum phase transitions can be seen by controlling the average chemical potentials at fixed and equal binding energies. More generally, by controlling the chemical potentials and binding energies, one can observe not only a series of quantum phase transition, but a phase with multi-component superfluid order (simultaneous existence of three superfluid order parameters). Notice that one can have a re-entrant quantum phase transition by changing both chemical potentials and binding energies simultaneously.
In Fig. 4, we plot superfluid order parameters as a function of the chemical potential of the third component. The binding energies are fixed to be the same for all three channels. As can be seen in figure, fermions pairing occurs in channel 1 − 2 at smaller µ 3 . This is because the average chemical potential in this channel is the largest for µ 3 < E B . Further, as the average chemical potential is constant, superfluid order parameter ∆ 3 is constant. For µ 3 > E B , average chemical potential in channel 2−3 is the largest and increasing with increasing µ 3 . As a result, superfluid order parameter ∆ 1 increases with µ 3 . In the entire range of µ 3 , average chemical potential in channel 1 − 3 is smaller than that of the other channels so that the pairing in this channel is not favorable. As seen in Fig. 4, the superfluid order parameters have large discontinuity which represents a sharp first order quantum phase transition.
At the same chemical potentials and the same interaction strengths of the channels, free energy gives many stable stationary points at which the condition ∆ 2 1 + ∆ 2 2 + ∆ 2 3 = C is satisfied. The constant C depends on both the chemical potentials and the interaction strengths (binding energies). As shown in Fig. 5, when the free energy has a minimum, the order parameters represent a surface in order parameter space.
Within our mean field description, we were able to handle only two-body correlations. One needs to go beyond mean field theory to understand the role of threebody correlations in a three-component system. If three atoms can overlap in the same region of space, then the three-body correlations can play a role giving Thomas effect [17] and Efimov effect [18]. Thomas effect is the collapse of a three body system due to the overlap of atoms. Atom loss in a trap is undoubtedly related to the Thomas effect. In a three component atomic system, collision between a condensed pair and a third species atom can support the Thomas effect. Efimov effect is the accumulation of three-body bound states at strongly interacting limit. These effects are forbidden in two component gases. For sufficiently low densities, these effects are forbidden even in three component systems so that our results are applicable to dilute ultra-cold atomic gases. We discussed the pure 2D limit in this paper, however one can generalize the theory to include the weak atom tunneling between layers as done in Ref. [14] for two component gases.
Our results in two dimensions look qualitatively similar to the ones obtained in three dimensions in Ref. [7]. However, we find that the superfluidity is more sensitive to the parameters in two dimensions than three dimensional systems. As we have seen above, superfluidity is very sensitive to both chemical potentials and the interactions between different pairs. In current experimental setups, Feshbach resonance allows one to control the interactions to different values by tuning the scattering lengths. However, the scattering lengths between different pairs of fermions cannot be controlled independently. Therefore, chemical potential is the suitable parameter to drive the quantum phase transitions. Typically, chemical potentials can be controlled by changing the atomic population in both two dimensions and three dimensions. However, in order to change the chemical potentials this way, one has to start the experiment all over with a different atomic sample. The advantage of using quasi two dimensional system is that one can change the effective chemical potential by controlling the tunneling between layers. This tunneling can be controlled by the laser in-FIG. 5: Superfluid order parameters for symmetric parameters. We use the same chemical potentials for the three species µ1 = µ2 = µ3 = µ and same interaction strengths for the three channels EB1 = EB2 = EB3 = EB. The three surfaces (∆ 2 1 + ∆ 2 2 + ∆ 2 3 = constant) shown in the order parameter space are for EB = 0.5µ, 1.2µ, and 2.0µ. We fix the temperature to be kBT = µ/50. tensity of the optical lattice. As shown in Ref. [15], the first order quantum phase transitions in two dimensional systems can easily be controlled by the laser intensity.
IV. SUMMARY AND CONCLUSIONS
We studied Fermi superfluidity of an interacting three component system in two dimensions. We used a mean field theory to investigate the behavior of free energy as a function of chemical potentials, binding energies, and the temperature. Depending on the chemical potentials and binding energies, we find a second order thermal phase transition as we lower the temperature. At low temperatures, by controlling the parameters, on can have a series of first order quantum phase transitions and re-entrant phase transitions. These series of phase transitions are associated with pairing between different hyperfine spin components of fermions.
The possible pairing is determined by both average chemicals potentials and the interaction strengths of the paring channels. The channel which has largest paring strength forms the superfluid, while the unpaired component form a Fermi sea. At low temperature, first order quantum phase transition can be induced by increasing the average chemical potential or the interaction strength of one channel over the other. If the pairing strengths are equal in all three channels, then it is possible to have three superfluid phases simultaneously. As we have not considered the interaction between condensed pairs, we do not expect the phase separation of superfluid phases in spatially homogenous environments [9]. However in trapped systems, it has been shown that the phase separation of superfluid phases is possible in three dimensions [8].
We speculate that the simultaneous existence of three types of superfluid phases in trapped systems can be detected by standard experimental methods. For example, superfluidity can be demonstrated by the creation of vortices [19] and then distinguished them by probes coupling to each atom types. Alternatively, one can measure the energy gap using radio frequency spectroscopy [20] or measure condensate fraction using the pair projection method [21].
arXiv:0901.3770v3 [cond-mat.other] 3 Aug 2009 FIG. 1: Free energy contours showing a second order thermal phase transition for the parameters µ2 = 0.8µ1, µ3 = 0.75µ1, EB1 = 0.1µ1, EB2 = µ1 and EB3 = 0.99µ1. From (a) to (d) temperature varies as kBT = µ1/1.00, kBT = µ1/1.11,kBT = µ1/1.20, and kBT = µ1/50. The global minimas in figures (a) to (d) are at (∆1/µ1, ∆2/µ1, ∆3/µ1) =
2 n 2 mFIG. 2 :
22+ n =m =l ∆ 2 n m l and Superfluid order parameters ∆3 as a function of temperature. We fixed the binding energies EB1 = 0.1µ1, EB2 = 0.5µ1, EB3 = µ1 and chemical potentials µ2 = 0.8µ1 and µ3 = 0.75µ1.
FIG. 3 :
3Low temperature free energy contours showing first order quantum phase transitions for the parameters µ1 = µ2 = µ3 = µ, EB1 = 0.1µ, EB3 = 0.6µ and kBT = µ/50. From panel (a) to (c) EB2 varies as 0.4µ, 0.6µ, 0.7µ respectively. In panel (a) and (c) the minimas are given at (∆1/µ, ∆2/µ, ∆3/µ) = (0, 0, 1.25) and (0, 1.37, 0) respectively. In panel (b), free energy gives many stationary points where the global minimum is a quarter of a circle. In panel (d), we use EB2 = EB3 = 0.05µ and the minimum is given at (0.46, 0, 0).
FIG. 4 :
4Superfluid order parameters ∆1/EB (black line) and ∆3/EB (gray line) as a function of µ3/EB. We fixed the binding energies EB1 = EB2 = EB3 = EB and the temperature kBT = EB/50. The chemical potentials are µ1 = EB and µ2 = 1.2EB. In the entire range of µ3, ∆2 is zero.
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| []
|
[
"Mixed Integer Reformulations of Integer Programs and the Affine TU-dimension of a Matrix",
"Mixed Integer Reformulations of Integer Programs and the Affine TU-dimension of a Matrix"
]
| [
"Jörg Bader \nInstitute for Operations Research\nETH Zürich\nSwitzerland\n",
"Robert Hildebrand \nIBM T.J. Watson Research Center\n\n",
"⋆ ",
"Robert Weismantel \nInstitute for Operations Research\nETH Zürich\nSwitzerland\n",
"Rico Zenklusen \nInstitute for Operations Research\nETH Zürich\nSwitzerland\n"
]
| [
"Institute for Operations Research\nETH Zürich\nSwitzerland",
"IBM T.J. Watson Research Center\n",
"Institute for Operations Research\nETH Zürich\nSwitzerland",
"Institute for Operations Research\nETH Zürich\nSwitzerland"
]
| []
| We study the reformulation of integer linear programs by means of a mixed integer linear program with fewer integer variables. Such reformulations can be solved efficiently with mixed integer linear programming techniques. We exhibit examples that demonstrate how integer programs can be reformulated using far fewer integer variables. To this end, we introduce a generalization of total unimodularity called the affine TUdimension of a matrix and study related theory and algorithms for determining the affine TU-dimension of a matrix. We also present bounds on the number of integer variables needed to represent certain integer hulls. | 10.1007/s10107-017-1147-2 | [
"https://arxiv.org/pdf/1508.02940v3.pdf"
]
| 13,670,434 | 1508.02940 | 42937fb121bf95b5b2ac7cfb6454bb5708419b40 |
Mixed Integer Reformulations of Integer Programs and the Affine TU-dimension of a Matrix
13 Apr 2017 April 14, 2017
Jörg Bader
Institute for Operations Research
ETH Zürich
Switzerland
Robert Hildebrand
IBM T.J. Watson Research Center
⋆
Robert Weismantel
Institute for Operations Research
ETH Zürich
Switzerland
Rico Zenklusen
Institute for Operations Research
ETH Zürich
Switzerland
Mixed Integer Reformulations of Integer Programs and the Affine TU-dimension of a Matrix
13 Apr 2017 April 14, 2017arXiv:1508.02940v3 [math.OC]integer programmingmaster knapsack problemtotal unimodularity
We study the reformulation of integer linear programs by means of a mixed integer linear program with fewer integer variables. Such reformulations can be solved efficiently with mixed integer linear programming techniques. We exhibit examples that demonstrate how integer programs can be reformulated using far fewer integer variables. To this end, we introduce a generalization of total unimodularity called the affine TUdimension of a matrix and study related theory and algorithms for determining the affine TU-dimension of a matrix. We also present bounds on the number of integer variables needed to represent certain integer hulls.
Introduction
Reformulations of integer programs with linear constraints are common in the integer programming literature. The main motivation behind these reformulations is that linear programs can be rapidly solved in both theory and practice. Often integer programs are reformulated such that the linear programming relaxation is improved, hence creating better bounds for branch-and-bound based algorithms [22]. These reformulations include Lagrangian relaxation, Dantzig-Wolfe reformulation, and cutting planes. In a much stronger sense, many combinatorial optimization problems can be formulated exactly using linear inequalities. These formulations typically involve totally unimodular (TU) matrices or totally dual integral systems. Since the feasible set of the linear relaxation exactly describes its integer hull in these settings, the optimization problem can be solved by solving the linear relaxation. See, for instance, [19].
The aim of this work is to instead reformulate integer linear programs by means of a mixed integer linear program with few integer variables. We focus on reformulating the feasible region of the integer linear program by mixed integer constraints. For a polyhedron P = {x ∈ R n | Ax ≤ b}, we wish to find a matrix W ∈ Z k×n such that conv(P ∩ Z n ) = conv({x ∈ P | W x ∈ Z k }).
(1)
Since W is an integral matrix, the inclusion ⊆ in property (1) is always fulfilled. Furthermore, if W is the n × n identity matrix, the property (1) is trivially satisfied. The objective here is to find an integer matrix W with few rows k that can model the integer hull of the feasible region as in (1).
With such a reformulation at hand, the underlying optimization problem can then be solved using mixed integer linear programming techniques. From a theoretical point of view, Lenstra [14] presented an algorithm to solve mixed integer linear programs in polynomial time when the number of integer variables is fixed. Also in practice, mixed integer linear programs with few integer variables can lead to algorithms with improved running time [15].
As a centerpiece of this paper, we study the following generalization of total unimodularity that admits reformulations of P for all integral right hand sides b.
Definition 1 We say a matrix A ∈ Z m×n admits a k-row affine TU decomposition if there exist matrices U ∈ Z m×k and W ∈ {0, ±1} k×n such that à W is totally unimodular, wherẽ
A ∈ Z m×n is the unique matrix satisfying A =Ã + U W . The minimum k such that A admits a k-row affine TU decomposition is called the affine TU-dimension of A.
Observe that A has affine TU-dimension 0 if and only if A is TU. Also, the affine TUdimension of any matrix is at most n sinceà = 0 m×n , U = A and W = I n produces a valid affine TU decomposition. Affine TU decompositions with k rows can admit a model of the feasibility region of related polyhedra with only k integer variables.
Theorem 2 Let A =Ã + U W ∈ Z m×n with W ∈ {0, ±1} k×n be an affine TU decomposition, b ∈ Z m and ℓ ∈ (Z ∪ {−∞}) n , u ∈ (Z ∪ {∞}) n with ℓ ≤ u.
Then conv x ∈ R n | ℓ ≤ x ≤ u, Ax ≤ b, W x ∈ Z k is an integral polyhedron. In particular, P = {x ∈ R n | ℓ ≤ x ≤ u, Ax ≤ b} and W satisfy property (1).
As an example of the convenience of Theorem 2, the parity polytope has a simple description using one integer variable.
Example 3 (Parity polytope) The n-dimensional (even) parity polytope is the convex hull of all 0-1 vectors in R n that have an even cardinality support. This polytope has exponentially many inequalities [10], but can be described by an extended formulation with only 4n − 1 inequalities [3,11]. Alternatively, it can be described using one integrality constraint as P = conv x ∈ [0, 1] n 1 2 n i=1 x i ∈ Z . P is the projection of the polytope
Q = conv x z ∈ [0, 1] n × R n i=1 x i + 2z = 0, z ∈ Z = conv x z ∈ [0, 1] n × R A x z ≤ b, W x z ∈ Z with A = 1 T n 2 −1 T n −2 , b = 0 0 and W = [0 T n , 1].
The matrix A admits an affine TU decom-
position as A = 1 T n 0 −1 T n 0 + 2 −2 W .
By Theorem 2, Q is an integral polyhedron, thus P is also an integral polyhedron.
In Section 2, we expand upon the theory of affine TU decompositions: we investigate structure of affine TU decompositions and prove Theorem 2 and related results.
In Section 3, we give various examples of how understanding the affine TU-dimension can create a mixed integer model with few integer variables.
In Section 4, we focus on computational issues in connection to affine TU decompositions and study the complexity of determining the affine TU-dimension of a matrix. In particular, we show that it is NP -Hard to decide if the (affine) TU-dimension of A ∈ Z m×n is less than n. When k and the number m of rows of A is fixed, we give a polynomial time algorithm to determine if the affine TU-dimension of A is equal to k.
In Section 5, we study mixed integer reformulations for knapsack polytopes. We prove a general lower bound of n 2 integrality constraints necessary to achieve property (1) for the linear relaxation of knapsack polytopes. We then give a nonconstructive proof that in every 0-1 knapsack polytope we can replace the integrality on all the n variables by at most n − 2 integrality constraints to achieve property (1). Apart from the added integrality constraints, we use only the original knapsack inequality and the 0-1 bounds on the variables, but no additional inequalities. In a final example we show the potential power of the addition of linear inequalities in the mixed integer model. We present a class of knapsack polytopes having an exponential sized polyhedral description. For these polytopes, by introducing linearly many additional linear inequality constraints one can replace the integrality constraints on all the variables by a single joint integrality constraint.
Related Work
The matrices with affine TU-dimension 1 have been called nearly totally unimodular matrices in [6]. One of several applications of matrices with affine TU-dimension 1 is edge coloring in nearly bipartite graphs [6]. An undirected graph G is called nearly bipartite if it is not bipartite but one can obtain a bipartite graph by deleting one vertex of G [5]. The incidence matrix of a nearly bipartite graph has affine TU-dimension 1. By Theorem 2, integer hulls described by nearly totally unimodular matrices can be captured using just one integrality constraint, and hence integer linear optimization on these problems can be done in polynomial time using a mixed integer linear program. This, however, is not mentioned in [6].
Mixed integer reformulations were studied by Martin [16] under the name variable redefinition. Martin showed how to reformulate problems that can be solved using dynamic programs. The main motivation here was to create tighter linear programming relaxations that improve bounds in a branch and bound algorithm.
Many polynomial time algorithms in combinatorial optimization rely on the method of guessing the value of certain problem-related variables. Depending on the situation, guessing can for instance be done by polynomial enumeration of all possible combinations of values. Often one can interpret this approach as a polyhedral problem with an underlying mixed integer property like in (1). As an example, Hassin et al. [7] gave an efficient polynomial time approximation scheme for the constrained minimum spanning tree problem. Their algorithm relies on a partition of the edge set into (logarithmically many) buckets T i . Then it guesses the number of chosen edges e∈Ti x e in each bucket T i , and constructs an approximate solution from this information. The guessing can be interpreted as solution of the linear relaxation of the polyhedral problem with additional integral constraints e∈Ti x e ∈ Z for all i.
Another example of guessing integral values of certain variables is presented in Oriolo et al. [17]. They provide an exact efficient algorithm for a special case of a network design problem on rings which relies on guessing the right value for an integral variable b. Since their integral variable could have an exponential range, a complete enumeration is not efficient. They thus provide another argument, leading to an efficient procedure. The introduction of the integrality constraint b ∈ Z gives an alternative conceptually simple way to obtain a polynomial time algorithm.
We would like to mention the close connection of mixed integer reformulations to extended formulations. Consider the special case that the number of possible values d = W x ∈ Z k in (1) is polynomial in n. Then one can rewrite
conv({x ∈ P | W x ∈ Z k }) = conv d∈Z k {x ∈ P | W x = d}
as a linear program of polynomial size in an extended space with a method of Balas [1]. In this special case, a formulation as in (1) can serve as a compact certificate for the existence of such an extended formulation.
Lastly, we note that the set x ∈ R n | Ax ≤ b, W x ∈ Z k is a projection from a n + k dimensional space. When W is unimodular, as a convenient feature, we can find a change of variables that allows the integrality constraints W x ∈ Z k to be modeled easily using integral variables without increasing the total number of variables. To see this, consider the Hermite normal form transformation of a unimodular matrix W ∈ Z k×n with full row rank k, that is W · L = [I k 0 k×(n−k) ] with L ∈ Z n×n unimodular. Then by the bijective linear transformation x = Ly, we have
max{c T x | Ax ≤ b, W x ∈ Z k } = max{(c T L)y | ALy ≤ b, y 1 , . . . , y k ∈ Z}.
Using this transformation prevents the need to write the problem in a higher dimension to model the integrality constraints.
Properties of TU decompositions
We will begin with a homogeneous version of affine TU decompositions that is easier to study.
Definition 4
We say a matrix A ∈ Z m×n admits a k-row TU decomposition if there exist matrices U ∈ Z m×k and W ∈ {0, ±1} k×n with A = U W such that W is totally unimodular. The minimum k such that A admits a k-row TU decomposition is called the TU-dimension of A.
An obvious lower bound on the TU-dimension of a matrix is given by its rank, therefore even a TU matrix can have a large TU-dimension. The affine version of TU decompositions can rule out this artifact.
Following Theorem 2, for a given matrix A, we are interested in finding a k-row affine TU decomposition A =Ã + U W with small k (the number of rows of W ). With this purpose in mind, U can be restricted to not contain an all-zero column. Otherwise, we could delete this r-th column of U together with the r-th row of W and still obtain a valid affine TU decomposition.
Furthermore, we may assume without loss of generality that W ∈ {0, ±1} k×n has full rank k. This will allow us to simplify some proofs in what follows, but as the following remark shows it is not a real limitation.
Remark 5 Let A ∈ Z m×n and U ∈ Z m×k such that A =à + U W where à W is TU and W ∈ {0, ±1} k×n has rank k ′ < k.
For any I ⊆ {1, . . . , k}, let W I · be the matrix consisting of the rows of W with indices from I, and U · I be the matrix consisting of the columns of U with indices from I. Now fix some I ⊆ {1, . . . , k}, with |I| = k ′ such that the TU matrix W I · has rank k ′ . For any j ∈ I c := {1, . . . , k} \ I, consider W j · , the j-th row of W . Then (W j · ) T = (W I · ) T r j ∈ Z n for some r j ∈ R k ′ . Since W is TU, there exists such a solution r j with r j ∈ Z k ′ .
Then W I c · = RW I · where R ∈ Z (k−k ′ )×k ′ is the matrix composed of rows (r j ) T for j ∈ I c . Since à W I · is TU, A =à + U W =à + U · I W I · + U · I c W I c · =à + (U · I + U · I c R)W I · is a k ′ -row affine TU decomposition of A.
We now prove Theorem 2. A polyhedron P ⊆ R n is called integral if P = conv(P ∩ Z n ).
Proof (of Theorem 2). Let I 1 be the r 1 × n submatrix of the identity matrix I n with rows corresponding to the finite indices ℓ i ∈ Z, and let I 2 be the r 2 × n submatrix of I n with rows corresponding to the finite indices u i ∈ Z.
Notice that any affine TU decomposition A =Ã+U W gives rise to an affine TU decomposition
A := A −I 1 I 2 = Ã −I 1 I 2 + U 0 r1×k 0 r2×k W. Moreover, withb := b −l I 1 u I 2 one has conv x ∈ R n | ℓ ≤ x ≤ u, Ax ≤ b, W x ∈ Z k = conv x ∈ R n |Âx ≤ b, W x ∈ Z k .
Thus, it suffices to prove that for any affine TU decomposition
A =Ã + U W , the set S = conv x ∈ R n | Ax ≤ b, W x ∈ Z k is an integral polyhedron.
Observe first that S is a polyhedron. To see this, let
S = conv (x, d) ∈ R n × Z k | Ax ≤ b, W x − d = 0 k .
ThenŜ is a polyhedron since it is the mixed integer hull of a rational polyhedron (see, e.g. [19,Section 16.7]). Since projections of polyhedra are also polyhedra, the projection S = proj x (Ŝ) ofŜ onto the x variables is also a polyhedron. We now show that S is integral. Consider a decomposition of A as stated. For every fixed d ∈ Z k , b − U d is an integral vector, which implies that
P d := x ∈ R n |Ãx ≤ b − U d, W x = dS = conv d∈Z k P d = conv d∈Z k conv(P d ∩ Z n ) = conv d∈Z k (P d ∩ Z n ) = conv (S ∩ Z n ) ,
and thus, S is an integral polyhedron. The second equality follows from the fact that P d is an integral polyhedron and the last inequality follows since
S ∩ Z n = ∪ d∈Z k P d ∩ Z n . This is clear since for any x ∈ P ∩ Z n , W x ∈ Z k by integrality of W , and hence x ∈ P d for some d ∈ Z k . Therefore, conv x ∈ R n | Ax ≤ b, W x ∈ Z k is an integral polyhedron. From the argu- ments outlined before, also conv x ∈ R n | ℓ ≤ x ≤ u, Ax ≤ b, W x ∈ Z k is an integral poly- hedron. ⊓ ⊔
In a slightly restricted setting also a converse of Theorem 2 holds. For this, let us first link affine TU decompositions to affine decompositions containing general unimodular matrices. Recall that an integral (not necessarily square) r × s matrix is called unimodular if it has full row rank r and each nonsingular r × r submatrix has determinant ±1. An integral matrix M ∈ Z r×s of rank r is unimodular if and only if the
polyhedron {x ∈ R s | M x = v, x ≥ 0} is integral for each v ∈ Z m (see, e.g. [19, Theorem 19.2]). Now, let A I m W 0 k×m ∈ Z (m+k)×(n+m) have rank m + k. Notice that A I m W 0 k×m is unimodular if and only if {(x, y) ∈ R n+m | Ax + y = b, W x = d, x ≥ 0, y ≥ 0} is integral for all b ∈ Z m and all d ∈ Z k . The latter is true if and only if {x ∈ R n | Ax ≤ b, W x = d, x ≥ 0} is integral for all b ∈ Z m and all d ∈ Z k .
For reference we sum up this insight as a remark.
Remark 6 Let A =Ã + U W ∈ Z m×n be a decomposition of A with only integral matrices. (i) The matrix A I m W 0 k×m is unimodular if and only if {x ∈ R n | Ax ≤ b, W x = d, x ≥ 0}
is an integral polyhedron for all b ∈ Z m and all d ∈ Z k .
(ii) Moreover, if A I m W 0 k×m is unimodular then conv x ∈ R n | Ax ≤ b, W x ∈ Z k , x ≥ 0 is an integral polyhedron for each b ∈ Z m .
An anonymous reviewer pointed out to us that the equivalence in (i) above does not hold anymore if one drops the nonnegativity assumption on x.
Theorem 7 Let A ∈ Z m×n , and let W ∈ Z k×n have rank k such that the polyhedron
{x ∈ R n | Ax ≤ b, W x = d, x ≥ 0} is integral for all b ∈ Z m and for all d ∈ Z k . Then there exist matrices U ∈ Z m×k and W ′ ∈ {0, ±1} k×n such that A =Ã + U W ′ is an affine TU decomposition. Moreover, for every b ∈ Z m , conv x ∈ R n | Ax ≤ b, W x ∈ Z k , x ≥ 0 = conv x ∈ R n | Ax ≤ b, W ′ x ∈ Z k , x ≥ 0 .
Proof. Since W has rank k, we have n ≥ k. By Remark 6 we know that A I m W 0 k×m is a unimodular matrix. We claim that also W is unimodular. For this, consider any nonsingular
k × k submatrix W J of W , where J with |J| = k is the set of columns chosen. Since B = A J I m W J 0 k×m has rank m + k, we have det(W J ) = ± det(B) ∈ {±1}. Thus, W is unimodular. Let W = [W 1 W 2 ]
where we assume without loss of generality that the last k columns of W are the full rank matrix W 2 . Then we have
A I m W 0 k×m = A 1 A 2 I m W 1 W 2 0 k×m = I m A 2 0 k×m W 2 · A 1 − A 2 W −1 2 W 1 0 m×k I m W −1 2 W 1 I k 0 k×m . The matrix I m A 2 0 k×m W 2
is unimodular since it is (up to permutation of the columns) a full row rank submatrix of the unimodular matrix A I m W 0 k×m . As a product of unimodular matrices,
A 1 − A 2 W −1 2 W 1 0 m×k I m W −1 2 W 1 I k 0 k×m isA 1 − A 2 W −1 2 W 1 W −1 2 W 1 is TU. Trivially, the matrix A 1 − A 2 W −1 2 W 1 0 m×k W −1 2 W 1 I k is TU as well.
In particular,
A = [A 1 −A 2 W −1 2 W 1 0 m×k ]+A 2 [W −1 2 W 1 I k ] is an affine TU decomposition. Moreover, W x ∈ Z k if and only if [W −1 2 W 1 I k ]x ∈ Z k . ⊓ ⊔
Examples of TU decompositions
Given a matrix, in order to find (affine) TU decompositions of it having few rows, one can exploit certain relations between its columns. We first show that any master knapsack polytope
P Master = conv({x ∈ {0, 1} n | n i=1 ix i ≤ b})
can be modeled with only O( √ n) many integrality constraints. Although there is a wellknown polynomial time algorithm for integer linear optimization over P Master using dynamic programming (see, e.g. [19,Section 18.5]), this characterization shows that the structure of the resulting problem is less complicated than that of general integer programs.
Example 8 The affine TU-dimension of a T = [1, 2, . . . , n] is Θ( √ n).
It follows from Theorem 2 that P Master can be modeled with O( √ n) many integrality constraints.
Without loss of generality, suppose that n = ℓ 2 for some ℓ ∈ Z + . This is without loss of generality since the affine TU-dimension of a T as a function of n is monotonically increasing and since Θ( √ n), Θ(⌈ √ n⌉), and Θ(⌊ √ n⌋) are equivalent.
We begin by exhibiting a matrix W to provide an upper bound. By defining
W = I ℓ I ℓ I ℓ . . . I ℓ 0 T ℓ 1 T ℓ 0 T ℓ . . . 0 T ℓ 0 T ℓ 0 T ℓ 1 T ℓ 0 T ℓ . . . 1 T ℓ ∈ {0, 1} (2ℓ−1)×ℓ 2 and u T = 1 2 . . . ℓ ℓ 2ℓ . . . ℓ 2 − ℓ
one has a (2ℓ − 1)-row TU decomposition a T = u T W . Thus a T has TU-dimension at most 2ℓ − 1. Since the first entry of the vector u is 1, we can setã to be the first row of W , delete this row in W and the first entry in u, and get a (2ℓ − 2)-row affine TU decomposition. We next explain the lower bound. Consider any k-row TU decomposition a T = u T W with u ∈ Z k , and W ∈ {0, ±1} k×n TU. Since the vector a has distinct entries, W needs to have distinct columns. As shown in [8,Theorem 4.2], using slightly different terminology, a TU matrix with k rows has at most k 2 + k + 1 distinct columns (for a more modern approach, see also [19,Section 21.3]). Since W has n distinct columns, k 2 + k + 1 ≥ n. In particular, k ≥ √ n − 1. A simple argument shows that any k ′ -row affine TU decomposition of a T gives rise to a (k ′ + 1)-row TU decomposition of a T . Hence, any affine TU decomposition for the master knapsack constraint vector a T = [1, 2, . . . , n] needs to have at least √ n − 2 rows.
Example 9
The affine TU-dimension of a T = [2, 2 2 , . . . , 2 n ] is n. Consider any TU decomposition of a T as a T = u T W where W has k rows and full row rank and u has no 0 entries. Since W is TU, it can be shown that there exist r 1 , . . . , r n−k ∈ {0, ±1} n that span the kernel of W (see Lemma 12 in Section 4). By the decomposition above, a T r i = 0 for all i = 1, . . . , n − k. But by the structure of a T , observe that a T r = 0 for all r ∈ {0, ±1} n . Thus, we must have k = n. It follows that W is unimodular (and square) and that W −1 ∈ Z n×n . Thus we can write u = (W −1 ) T a. Since the greatest common divisor of the entries in a T is 2, u T has no ±1 entry. Now, consider any affine TU decomposition of a T as a T =ã T +ū TW and suppose the number of rows ofW is less than n. Choosing u T = [1,ū T ] and W = ã T W . But then u T and W provide a TU decomposition of a T with at most n rows in W . This is a contradiction with the above arguments since the first entry of u T is 1. Thus, the affine TU dimension of a T is at least n. Since W can be chosen as the identity matrix, the affine TU dimension of a T is exactly n.
Next we give an affine TU decomposition for a class of matrices and show how this can be used to model the problem of finding r-flows in directed graphs.
Example 10 (Block TU structure)
(i) Let A ī A i be TU for i = 1, . . . , r, where A i ∈ Z mi×ni ,Ā i ∈ Z ki×ni with M = m 1 + . . . + m r . Let U i ∈ Z k×ki for i = 1, . . . , r and set A = A 1 A 2 . . . A r U 1Ā1 U 2Ā2 . . . U rĀr .
Then A has a (k 1 + . . . + k r )-row affine TU decomposition with
U = 0 M×k1 0 M×k2 . . . 0 M×kr U 1 U 2 . . . U r and W = Ā 1Ā 2 . . .Ā r .
(ii) Consider the task of finding r flows in a directed graph G = (V, E) subject to certain joint capacity constraints as follows. Each flow f i : E → Z (for i = 1, . . . , r) is allowed to use the arcs E i ⊆ E to satisfy a demand d i to be transported from s i to t i . For each i we demand flow conservation with respect to f i at all vertices in V \ {s i , t i }. Some of the arcs in E are allowed to be shared by several of the flow problems. For each arc e ∈ E, define I(e) = {i = 1, . . . , r | e ∈ E i } to be the set of flow problems involving the arc e. We assume that for each arc e ∈ E we are given a maximum capacity c e ∈ Z that acts as a joint bound i∈I(e) f i (e) ≤ c e . This task is a variant of the multi-commodity flow problem. In contrast to our variant, in the multi-commodity flow problem one assumes that I(e) = {1, . . . , r} for each e ∈ E. As is standard for multi-commodity flow problems (see, e.g. [13, Chapter 19]), the above task can be modeled by using integral variables f i (e) ∈ Z for each e ∈ E and each i = 1, . . . , r. Linear constraints Af ≤ b ensure feasibility of the flows and implement the capacity constraints. Notice that the set of capacity constraints decomposes into the ones for the arcs e ∈ E with |I(e)| = 1, and the ones for e with |I(e)| ≥ 2.
One can rearrange the rows of A in order to get a matrix A as in part (i). For every i = 1, . . . , r, the matrix A i corresponds to the flow conservation constraints of the flow f i , together with the capacity constraints for e ∈ E i with |I(e)| = 1. Moreover, the U i are identity matrices, and the capacity constraints corresponding to e ∈ E with |I(e)| ≥ 2 form the matrix [Ā 1 . . .Ā r ] where all theĀ i are subsets of the rows of identity matrices.
Since A ī A i is the constraint matrix for flow i, this matrix is TU. Applying the affine TU decomposition from part (i), we only need integral variables for the arcs that are allowed to be shared by multiple flows. These are e∈E:|I(e)|≥2 |I(e)| ≤ r · |{e ∈ E | |I(e)| ≥ 2}| integral variables.
Next we relate the class of almost totally unimodular matrices to the matrices with affine TU-dimension 1 (called nearly TU matrices by [6]).
A square matrix A is called almost totally unimodular if A is not TU but every proper submatrix of A is TU. Almost totally unimodular matrices were introduced by Padberg [18] and are studied as a building block for k-balanced matrices [4, and references therein].
Example 11 Assume A = [aĀ] ∈ {0, ±1} n×n is almost totally unimodular. Then one can write A = [0 nĀ ] + a[1, 0, . . . , 0]. This is a 1-row affine TU decomposition, since every (n − 1) × (n − 1) submatrix ofĀ is TU. Thus, every almost totally unimodular matrix has affine TU-dimension 1.
This example shows that integer linear programs described with an almost totally unimodular constraint matrix can be solved in polynomial time by solving a mixed integer linear program with only one integer variable. The authors of [23] study a more general class of matrices called almost unimodular matrices. Through studying the lattice width of polyhedra with constraint matrices that have small subdeterminants, they show that integer linear programs described by almost unimodular matrices can also be solved in polynomial time.
We conclude this section with a discussion of the so-called integer decomposition property. A polyhedron P has the integer decomposition property, if for any positive integer k every integral vector in kP is the sum of k integral vectors from P . A matrix A ∈ Z m×n is TU (has affine TU-dimension 0) if and only if {x ∈ R n | Ax ≤ b, x ≥ 0 n } has the integer decomposition property for each b ∈ R m [2]. It is straightforward from the proof in [2] that for a TU matrix A and b ∈ Z m , also P = {x ∈ R n | Ax ≤ b} = conv({x ∈ Z n | Ax ≤ b}) has the integer decomposition property. Gijswijt [6] showed that for A ∈ {0, ±1} m×n with affine TU-dimension 1 and b ∈ Z m , the polyhedron P A,b := conv({x ∈ Z n | Ax ≤ b}) has the integer decomposition property, too.
The integer decomposition property does not hold in general for polytopes P A,b where the matrix A has affine TU-dimension 2 and b ∈ Z m . As a counterexample, consider the parity polytope P = conv {(0, 0, 0) T , (0, 1, 1) T , (1, 0, 1) T , (1, 1, 0) T } . It does not have the integer decomposition property, since (1, 1, 1) T ∈ 2P ∩ Z 3 cannot be written as the sum of two points in P ∩ Z 3 (see, e.g., [9]). The polytope P has a natural inequality description as
P = {x ∈ R 3 | Ax ≤ b} with A = 1 1 1 −1 1 1 1 −1 1 1 1 −1 and b = 2 0 0 0 .
A has the 2-row affine TU decomposition
1 1 1 −1 1 1 1 −1 1 1 1 −1 = 1 0 0 −1 0 0 1 0 0 1 0 0 + 1 1 1 1 −1 1 1 −1 0 1 0 0 0 1 .
Since P does not have the integer decomposition property, there exists no 1-row affine TU decomposition for A. Thus, A has affine TU-dimension 2. In particular, not every P A,b arising from a matrix A with affine TU-dimension 2 has the integer decomposition property.
Determining the TU-dimension
Seymour [20] gave a decomposition procedure to recognize if a matrix is TU. A careful implementation of this decomposition procedure results in an algorithm that decides in polynomial time if a matrix is TU, see Truemper [21]. Thus one can decide in polynomial time if a matrix has affine TU-dimension 0. Given any specific value k, we would like to decide if A has affine TU-dimension k.
For B ∈ R m×n , let im(B) := {By ∈ R m | y ∈ R n } be the column space of B and ker(B) = {x ∈ R n | Bx = 0 m } be the kernel of B. A TU decomposition of a matrix A ∈ Z m×n is strongly related to the existence of ±1 combinations of column vectors of A generating the vector 0 m , i.e. vectors r ∈ ker(A) ∩ {0, ±1} n \ {0 n }.
We will need the following fact, a proof of which can for example be found in [12].
Lemma 12 (Totally unimodular matrices have totally unimodular kernels) Let A ∈ {0, ±1} (n−k)×n be TU and with rank n − k. Then one can find a TU matrix W ∈ {0, ±1} k×n such that ker(A) = im(W T ).
We now show that it is NP -hard to decide if a matrix has affine TU-dimension n.
Theorem 13 It is an NP-hard problem to decide if a matrix A ∈ Z m×n has affine TUdimension equal to n, even when we restrict to positive matrix entries and m = 1.
Proof. It was shown in [24] that the equal-sum-subsets problem is NP -complete. Given b ∈ Z n + , this problem is to decide if there exists a nonzero r ∈ {0, ±1} n such that b T r = 0.
We reduce the equal-sum-subsets problem to deciding if the transpose of the vector 2n · b admits a (n − 1)-row affine TU decomposition. In other words, for any vector b ∈ Z n + , a T = 2n · b T , we show that b is a yes-instance to the equal-sum-subset problem if and only if there exists a decomposition a T =ã T + u T W with u ∈ Z n−1 , W ∈ {0, ±1} (n−1)×n ,ã ∈ {0, ±1} n , and W together with the row vectorã T is TU.
If there exists an affine TU decomposition for a T as above with W TU having rank k ≤ n−1, then by Lemma 12 there exists a TU matrix R ∈ {0, ±1} (n−k)×n having rank n − k ≥ 1 such that
{x ∈ R n | W x = 0 n−1 } = {R T t | t ∈ R n−k }.
In particular R has only nonzero rows. Consider its first row r ∈ {0, ±1} n , which, as all other rows of W , satisfies W r = 0 n−1 . Then a T r =ã T r + u T W r =ã T r, thus
0 = a T r −ã T r = 2n · b T r −ã T r .
Sinceã, r ∈ {0, ±1} n ,ã T r ∈ {−n, . . . , n}. Furthermore 2n · b T r ∈ 2nZ, thereforeã T r = 0 and b T r = 0 for the nonzero r ∈ {0, ±1} n . Now assume there is a solution to the equal-sum-subsets problem, a nonzero r ∈ {0, ±1} n such that b T r = 0. By Lemma 12 there exists a TU matrix W ∈ {0, ±1} (n−1)×n such that
{x ∈ R n | r T x = 0} = {W T u | u ∈ R n−1 }.
Thus b = W T u has a solution u ∈ R n−1 , and since W is TU we may choose u to be integral. Then a T = 2n · b T admits the (n − 1)-row affine TU decomposition a T = 0 T n + (2n · u) T W , where W together with the row 0 T n is TU. Therefore the problem to decide if a given matrix has affine TU-dimension n is NP -hard.
⊓ ⊔
In view of the hardness result, it is an open question if there exists a polynomial time algorithm to decide if A has affine TU-dimension k when k is fixed. Below we provide polynomial time algorithms for other certain special cases of the task. We first present some auxiliary facts. A proof of the first one can be found in [19,Section 19.3/19.4].
Lemma 14 For a nonsingular matrix E, B D E C is TU if and only if BE −1 D − BE −1 C −E −1 E −1 C is TU.
Lemma 15 Suppose A ∈ Z m×n admits an affine TU decomposition with W ∈ {0, ±1} k×n . Then there exists a matrix W ′ ∈ {0, ±1} k×n that contains I k as a submatrix such that A also admits an affine TU decomposition with W ′ .
Furthermore, there exists a unimodular matrix T ∈ Z k×k such that W ′ = T W . The matrices T and W ′ can be computed in polynomial time.
Proof. Without loss of generality, assume that W has rank k. By reordering the columns of A and W with the same column permutation we may assume that W 1 , the first k columns of W = [W 1 W 2 ], have column rank k.
Since W 1 is unimodular,
W −1 1 ∈ Z k×k is also unimodular. Then we have W ′ := W −1 1 W = [I k W ′ 2 ] with W ′ 2 = W −1 1 W 2 ∈ Z k×(n−k)
. Consider an affine TU decomposition A =à + U W , where we defineà = [à 1Ã2 ] with A 1 being the first k columns ofÃ. In particular, the matrix à 1Ã2 W 1 W 2 is TU. Applying
Lemma 14 with E = W 1 shows that à 1 W −1 1à 2 −à 1 W −1 1 W 2 −W −1 1 W −1 1 W 2
is TU as well, and so is the
matrix 0 m×kÃ2 −Ã 1 W −1 1 W 2 I k W −1 1 W 2 .
In particular, also W ′ is TU. Since we have
A = Ã 1Ã2 + U W 1 W 2 = 0 m×kÃ2 −Ã 1 W −1 1 W 2 + (Ã 1 + U W 1 ) I k W −1 1 W 2 ,
the latter is also an affine TU decomposition of A with the TU matrix W ′ . This completes the proof of existence for T := W −1 1 . The matrices T and W ′ can be computed in polynomial time using basic linear algebra techniques.
⊓ ⊔ Theorem 16 Let n > k, A 1 ∈ Z m×k , A 2 ∈ Z m×(n−k) , and W 2 ∈ {0, ±1} k×(n−k) TU.
W 2 ] = A 1 − U A 2 − U W 2 I k W 2 with E = I k implies that A 1 − U A 2 − A 1 W 2 −I k W 2
is TU as well. In particular,
A 2 − A 1 W 2 W 2 is TU.
For the if part, defineà = [0 m×k A 2 − A 1 W 2 ] ∈ Z m×n and U = A 1 ∈ Z m×k . Then
à [I k W 2 ] = 0 m×k A 2 − A 1 W 2 I k W 2 is
Observation 17
Assume that n is fixed and A ∈ Z m×n is given.
For an affine TU decomposition one has W ∈ {0, ±1} k×n and in view of Remark 5 without loss of generality W has rank k ≤ n. Moreover, by Lemma 15 one can limit the search to matrices W that contain I k as a submatrix. Enumerating these polynomially many matrices for all k ≤ n, and applying for each of them the efficient algorithm from Theorem 16, one can find a matrix W having fewest number of rows such that A admits an affine TU decomposition together with W .
Thus, one can compute the affine TU-dimension of A in polynomial time for fixed n.
Next we show that for fixed k, the number of rows of W , and fixed m, the number of rows of A, we can decide efficiently if there exist U and W such that A admits an affine TU decomposition.
Theorem 18 Suppose A ∈ Z m×n . Then in polynomial time we can decide if A has affine TU-dimension k, provided that m and k are fixed.
Proof. Let us first reduce the statement to the case that A has distinct columns.
For this, without loss of generality assume the columns of A to be ordered as A = [A 1 A 2 ], where A 1 has distinct columns and A 2 has columns that are repeats of A 1 . Let A 1 have r columns. Consider any affine TU decomposition of A =Ã + U W with a TU matrix [Ã W ]. Let us divide the latter matrix into its first r columns, and the remaining ones
à W = à 1Ã2 W 1 W 2 .
Then we can we can choose columns
à ′ 2 W ′ 2
as repeats of columns of à 1 W 1 such that with
A ′ = [Ã 1Ã ′ 2 ] and W ′ = [W 1 W ′ 2 ]
we get the (possibly different) affine TU decomposition A =Ã ′ + U W ′ . In particular, A and A 1 have the same affine TU-dimension.
From now on assume that A ∈ Z m×n has distinct columns and gives rise to an affine TU decomposition A =à + U W . Then the TU matrix à W with m + k rows also has distinct columns. A TU matrix with r rows and distinct columns has at most r 2 + r + 1 columns (for a reference, see Remark 8 above). Thus, n ≤ (m + k) 2 + m + k + 1. Now with k and m, also n is constant. By Observation 17, we thus can compute the affine TU-dimension of A in polynomial time.
⊓ ⊔
We close this section with a conjecture on the complexity of determining the affine TUdimension of a matrix.
Conjecture 19
Suppose A ∈ Z m×n . Then in polynomial time we can decide if A has affine TU-dimension k, provided that k is fixed.
The above theorems and algorithms can be easily adapted to the consideration of TU decompositions instead of affine TU decompositions. In particular, the proof of Theorem 13 does also show the NP -hardness of recognizing TU-dimension n.
Observation 20 It is NP-hard to decide if a matrix A ∈ Z m×n has TU-dimension equal to n, even when we restrict to positive matrix entries and m = 1.
Reformulations for knapsack polytopes
In this section we investigate the size of reformulations specific to both A and b. In particular, we focus on bounds for the minimum number of integrality constraints needed to model a general 0-1 knapsack polytope. We then provide an example that demonstrates how adding even a single integrality constraint can vastly reduce the number of inequalities needed to describe the integer hull of a knapsack polytope.
Lemma 21 (Lower bound) Let m ∈ Z + and consider the knapsack polytope
P = x y ∈ [0, 1] 2m 2(x 1 + y 1 ) + 2 2 (x 2 + y 2 ) + · · · + 2 m (x m + y m ) ≤ 2 m+1 − 1 in dimension n = 2m.
Let W, W ′ ∈ Z k×m be integral matrices and let
Q = conv P ∩ x y [W W ′ ] x y ∈ Z k .
If k < m, then Q is not an integral polyhedron.
Proof. Since k < m, dim(ker(W )) ≥ 1. Let r ∈ ker(W ) \ {0 m } such that a T r ≥ 0 where a T := (2, 2 2 , . . . , 2 m ). Let
x 0 ∈ {0, 1} m with x 0 i = 0 if r i ≥ 0 1 if r i < 0 .
Then there exists aλ > 0 with x 0 + λr ∈ [0, 1] m for all 0 < λ <λ. Let x 0 be the complement of x 0 , that is,
x 0 ∈ {0, 1} m and x 0 + x 0 = 1 m . Observe that x 0 x 0 ∈ P ∩ {0, 1} 2m since a T x 0 + a T x 0 = a T 1 m = 2 m+1 − 2 < 2 m+1 − 1.
Let 0 <λ < min{λ, 1 a T r } where if a T r = 0 we set 1 a T r to ∞. Now, suppose that Q is an integral polytope. This implies that Q∩ x y y = x 0 is an integral polytope, and hence, its projection onto the first m variables, Q x 0 := x x x 0 ∈ Q ⊆ [0, 1] m is also an integral polytope. By the choice of r andλ, we have a T x 0 ≤ a T (x 0 +λr) and
a T (x 0 +λr) + a T x 0 = m i=1 2 i +λa T r < 2 m − 1, thus x 0 +λr ∈ Q x 0 . Consider the optimizers G = arg max{a T x | x ∈ Q x 0 } and F = arg max{a T x | x ∈ Q x 0 ∩ {0, 1} m }. Since Q x 0 is integral, G = conv(F )
By choice of a, the value a T x is distinct for all x ∈ {0, 1} m and by construction of Q x 0 , F = {x 0 } and G = conv(F ) = {x 0 }. But since a T x 0 ≤ a T (x 0 +λr), this implies that x 0 +λr ∈ G and hence x 0 +λr = x 0 , which is a contradiction with the fact thatλ > 0 and r = 0 m .
Thus, we conclude that Q was not integral.
⊓ ⊔
We next provide a positive result for modeling knapsack polytopes and show that every 0-1 knapsack polytope can be modeled using at most n − 2 integrality constraints together with the upper and lower bounds on the variables. We first prove a theorem about separation of disjoint polytopes.
Theorem 22
Let P, Q ⊆ R n be polytopes such that P ∩ Q = ∅ and dim(conv(P ∪ Q)) = n.
Then there exist h ∈ R n , α 1 , α 2 ∈ R with α 1 < α 2 such that P ⊆ {x ∈ R n | h T x ≤ α 1 }, Q ⊆ {x ∈ R n | h T x ≥ α 2 }, and max{|{x ∈ vert(P ) | h T x = α 1 }|, |{x ∈ vert(Q) | h T x = α 2 }|} ≥ ⌈ n+1 2 ⌉.
Proof. If either P or Q is empty, then the theorem is immediate. Assuming Q = ∅, one may take a valid inequality h T x ≤ α 1 for which {x ∈ P | h T x = α 1 } is a facet of P , and then set α 2 > α 1 . So suppose P, Q = ∅. Letx ∈ rel int(P ),ŷ ∈ rel int(Q). LetP = P + t andQ = Q + t where t =x − 2ŷ. Letx =x + t andȳ =ŷ + t. Thenx ∈ rel int(P ),ȳ ∈ rel int(Q) andx = 2ȳ. We will prove the result forP andQ, which implies the result for P and Q.
SinceP ∩Q = ∅, by the (strict) hyperplane separation theorem, there exists a pair (h,ᾱ) such thath T x <ᾱ for all x ∈P andh T y >ᾱ for all y ∈Q. Define
H = {(h, α) ∈ R n × R | h T x ≤ᾱ, h T y ≥ α, α ≥ᾱ ∀ x ∈ vert(P ), y ∈ vert(Q)}.
Since (h,ᾱ) was a strict separator and since vert(P ) and vert(Q) are finite, there exists an ǫ > 0 sufficiently small such that the point (h,ᾱ + ǫ) strictly satisfies all inequalities of H. Hence, (h,ᾱ + ǫ) ∈ int(H), and therefore H is full-dimensional.
We claim that H is a bounded polyhedron. Indeed, let (r,β) ∈ rec(H) where rec(H) = {(r, β) ∈ R n × R | r T x ≤ 0, r T y ≥ β, β ≥ 0 ∀ x ∈ vert(P ), y ∈ vert(Q)} denotes the recession cone of H. By convexity,r Tx ≤ 0 andr Tȳ ≥β ≥ 0. Sincex = 2ȳ, this implies thatr Tx =r Tȳ = 0 andβ = 0. Sincex ∈ rel int(P ) andȳ ∈ rel int(Q), we have thatr T x = 0 for all x ∈P andr T y = 0 for all y ∈Q. Since dim(conv(P ∪Q)) = n, there exist z 1 , z 2 ∈ conv(P ∪Q) and a λ > 0 such that λr = z 1 − z 2 . By definition of conv(P ∪Q), there exist x 1 , x 2 ∈P , y 1 , y 2 ∈Q, µ 1 , µ 2 ∈ [0, 1] such that z i = µ i x i + (1 − µ i )y i for i = 1, 2.
Combining this, we see that λ r 2 2 = λr Tr = λr T (µ 1 x 1 + µ 2 x 2 + (1 − µ 1 )y 1 + (1 − µ 2 )y 2 ) = 0, where the last equality comes from the fact thatr T x = 0 for all x ∈P andr T y = 0 for all y ∈Q. Hencer = 0 n and rec(H) = {(0 n , 0)}. Therefore H is bounded (see, e.g., [19, Section 8.2 (5) (iii)]). Finally, since H is a bounded, full dimensional polytope, there exists a vertex (ĥ,α) of H such that the inequality α ≥ᾱ is not tight. Since this vertex is defined by n+1 tight inequalities from H that are not the inequality α ≥ᾱ, there exists a set of n + 1 points corresponding to the tight inequalities, each of them being a vertex of eitherP orQ. By pigeonhole principle, there must be at least ⌈ n+1 2 ⌉ such vertices in eitherP orQ. This completes the proof.
⊓ ⊔
Consider such a face F of P respectively Q spanned by at least n+1 2 vertices. We want to find a TU matrix W ∈ {0, ±1} k×n such that W x = d with the same d ∈ Z k for all x ∈ F . Below we prove that for all 0-1 knapsack polytopes this is possible for n ≥ 4 with k = n − 2, where P = conv x ∈ {0, 1} n | a T x ≤ b and Q = conv({0, 1} n \ P ). Thus, for all 0-1 knapsack polytopes, it suffices to introduce n − 2 integrality constraints to satisfy property (1).
Theorem 23 Let n ≥ 4, a ∈ Z n , b ∈ Z and P = x ∈ [0, 1] n | a T x ≤ b . There exists W ∈ Z (n−2)×n such that (1) holds.
Proof. Let S = P ∩ {0, 1} n and S c = {0, 1} n \ S. Without loss of generality we may assume S = ∅ and S c = ∅.
Since the polytopes conv(S) and conv(S c ) are disjoint and conv(S ∪ S c ) = [0, 1] n , by Theorem 22 there exists h ∈ R n , α 1 , α 2 with α 1 < α 2 such that conv(S) ⊆ {x ∈ R n | h T x ≤ α 1 }, conv(S c ) ⊆ {x ∈ R n | h T x ≥ α 2 }, and |{x ∈ S | h T x = α 1 }| ≥ ⌈ n+1 2 ⌉ or |{x ∈ S c | h T x = α 2 }| ≥ ⌈ n+1 2 ⌉. Since n ≥ 4, ⌈ n+1 2 ⌉ ≥ 3. In case that |{x ∈ S | h T x = α 1 }| ≥ ⌈ n+1 2 ⌉, let x 1 , x 2 , x 3 ∈ {x ∈ S | h T x = α 1 } be disjoint vertices of S. Otherwise, choose x 1 , x 2 , x 3 ∈ {x ∈ S c | h T x = α 2 } to be disjoint vertices of S c . Consider
A = (x 2 − x 1 ) (x 3 − x 1 ) ∈ {0, ±1} 2×n .
Since where the rows of W span the kernel of the TU matrix A. By Lemma 12 we may choose W ∈ Z (n−2)×n to be TU. Since W it TU, the set Q z = {x ∈ [0, 1] n | W x = z} is an integral polytope for every z ∈ Z n−2 . By construction of W , for every z ∈ Z n−2 there exists an α ∈ R such that Q z ⊆ {x ∈ R n | h T x = α}. Thus either Q z ⊆ conv(S) or Q z ⊆ conv(S c ). Therefore P ∩ Q z = conv(P ∩ Z n ) ∩ Q z and conv(P ∩ Z n ) = conv z∈Z n−2 P ∩ Q z = conv({x ∈ P | W x ∈ Z n−2 }).
⊓ ⊔
It is an interesting open question to determine the minimum number of integrality constraints needed to describe any knapsack polytope. We now know this number lies between n
is an integral polyhedron by total unimodularity of à W [19, Theorem 19.1]. Then
x 1 , x 2 , x 3 ∈ {0, 1} n , the columns of A are from the set {[0, 0] T , [0, ±1] T , [±1, 0] T , ±[1, 1] T }. Hence, A is TU. Furthermore, the affine hull aff(x 1 , x 2 , x 3 ) satisfies aff(x 1 , x 2 , x 3 ) = {x 1 + A T y | y ∈ R 2 } = {x 1 + x | x ∈ R n , W x = 0 n−2 },
unimodular as well. Moreover, it is well known that a matrix C ∈ Z s×r is TU if and only if [C I s ] is unimodular (see, e.g. [19, Section 19.1]),thus
=Ã + U W is an affine TU decomposition, if it exists. Proof. For the only if part, assume there is a matrix U such that [A 1 A 2 ] =Ã+ U [I k WThen
[A 1 A 2 ] admits a k-row affine TU decomposition with U ∈ Z m×k and W = [I k W 2 ] if and
only if
A 2 − A 1 W 2
W 2
is TU.
Moreover, given A and W = [I k W 2 ], we can find in polynomial time U andà such that
A 2 ] and
Ã
[I k W 2 ]
is TU. Applying Lemma 14 on the TU matrix
Ã
[I k
is TU. If this is the case, we have an affine TU decomposition A =à + U W withà = [0 m×k A 2 − A 1 W 2 ] and U = A 1 . Otherwise, there does not exist such a decomposition . ⊓ ⊔TU by assumption, and [A 1 A 2 ] =à + U [I k W 2 ] by construction.
Thus, in order to find an affine TU decomposition with given A and W = [I k W 2 ] in poly-
nomial time, we apply Truemper's polynomial time algorithm [21] to decide if
A 2 − A 1 W 2
W 2
and n − 2.Allowing the addition of polynomially many linear inequalities to the mixed integer reformulation is natural and likely advantageous in many scenarios. In this vein, the parity polytope in Section 1 has been formulated using a single integrality condition together with the bounds on the variables and one additional inequality. We conclude this section with a family of knapsack examples in dimension n, where the polyhedral description consists of Θ(n k ) linear constraints. Here, k is a parameter controlling the weight distribution of the instances. Each of these knapsack polytopes can be described by only O(n) linear constraints together with one integrality constraint.
AcknowledgementWe thank Santanu S. Dey for discussing his idea for the lower bound in Example 8. We owe thanks to Shmuel Onn who made us aware of a much simplified version of the proof of Theorem 18.We also want to express our gratitude to two anonymous reviewers. Their detailed comments and suggestions on an earlier version of the manuscript led to enhancements on the general structure of our paper, as well as greatly improved the paper in many ways.Example 24 (Knapsack with big and small weights) Let n ∈ Z, k ∈ Z, k ≥ 3 and b ∈ R + . Consider weights a i ∈ R + where b k+1 < a i ≤ b k for all i ∈ S = {1, . . . , s} and k−1 k+1 b < a i ≤ b for all i ∈ B = {s + 1, . . . , n}. To simplify notation, assume thatOne can show that the facet description of the polytope conv(P ∩ Z n ) is given bywhere i (R) is the index of an item from R having smallest weight a i . It follows easily that conv(P ∩ Z n ) is the convex hull of all x ∈ [0, 1] n that satisfy s j=i x j + j∈B: aj >b−aiNotice that the latter mixed integer description in fact is an affine TU decomposition. However, the above polyhedral description for the considered knapsack polytope is specific for the right-hand-side b and the fixed k.
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[
"ESSENTIAL DIMENSION OF PROJECTIVE ORTHOGONAL AND SYMPLECTIC GROUPS OF SMALL DEGREE",
"ESSENTIAL DIMENSION OF PROJECTIVE ORTHOGONAL AND SYMPLECTIC GROUPS OF SMALL DEGREE"
]
| [
"Sanghoon Baek "
]
| []
| []
| In this paper, we study the essential dimension of classes of central simple algebras with involutions of index less or equal to 4. Using structural theorems for simple algebras with involutions, we obtain the essential dimension of projective and symplectic groups of small degree. | 10.1080/00927872.2013.847948 | [
"https://arxiv.org/pdf/1111.4697v1.pdf"
]
| 14,577,034 | 1111.4697 | ab06bf2fbca22002ae2548b5bab7be2518f573a6 |
ESSENTIAL DIMENSION OF PROJECTIVE ORTHOGONAL AND SYMPLECTIC GROUPS OF SMALL DEGREE
20 Nov 2011
Sanghoon Baek
ESSENTIAL DIMENSION OF PROJECTIVE ORTHOGONAL AND SYMPLECTIC GROUPS OF SMALL DEGREE
20 Nov 2011
In this paper, we study the essential dimension of classes of central simple algebras with involutions of index less or equal to 4. Using structural theorems for simple algebras with involutions, we obtain the essential dimension of projective and symplectic groups of small degree.
Introduction
Let F be a field, A a central simple F -algebra, and (σ, f ) a quadratic pair on A (see [5,5.B]). A morphism of algebras with quadratic pair φ :
(A, σ, f ) → (A ′ , σ ′ , f ′ ) is an F -algebra morphism φ : A → A ′ such that σ ′ • φ = φ • σ and f • φ = f ′ . For any field extension K/F , we write (A, σ, f ) K for (A ⊗ F K, σ ⊗ Id K , f K ), where f K : Sym(A K , σ K ) → K.
For n ≥ 2, let D n denote the category of central simple F -algebras of degree 2n with quadratic pair, where the morphisms are the F -algebra homomorphisms which preserve the quadratic pairs and let A 2 1 denote the category of quaternion algebras over anétale quadratic extension of F , where the morphisms are the F -algebra isomorphism. Then, there is an equivalence of groupoids:
(1) D 2 ≡ A 2 1 ; see [5,Theorem 15.7].
Moreover, if we consider the full subgroupoid 1 A 2 1 of A 2 1 whose objects are F -algebras of the form Q×Q ′ , where Q and Q ′ are quaternion algebras over F , and the full subgroupoid 1 D n of D n whose objects are central simple algebras over F with quadratic pair of trivial discriminant, then the equivalence in (1) specialize to the following equivalence of subgroupoids: [5,Corollary 15.12].
(2) 1 D 2 ≡ 1 A 2 1 ; see
For n ≥ 1, we denote by C n be the category of central simple F -algebras of degree 2n with symplectic involution, where the morphisms are the F -algebra isomorphism which preserve the involutions.
By Galois cohomology, there are canonical bijections (see [5, §29.D,F])
(3) D n ←→ H 1 (F, PGO 2n )
The work has been partially supported from the Fields Institute and from Zainoulline's NSERC Discovery grant 385795-2010.
and (4)
C n ←→ H 1 (F, PGSp 2n ).
Let T : Fields/F → Sets be a functor from the category Fields/F of field extensions over F to the category Sets of sets and let p be a prime. We denote by ed(T ) and ed p (T ) the essential dimension and essential p-dimension of T , respectively. We refer to [4,Def. 1.2] and [8, Sec.1] for their definitions. Let G be an algebraic group over F . The essential dimension ed(G) (respectively, essential p-dimension ed p (G)) of G is defined to be ed(H 1 (−, G)) (respectively, ed p (H 1 (−, G))), where H 1 (E, G) is the nonabelian cohomology set with respect to the finitely generated faithfully flat topology (equivalently, the set of isomorphism classes of G-torsors) over a field extension E of F .
A morphism S → T from Fields/F to Sets is called p-surjective if for any E ∈ Fields/F and any α ∈ T (E), there is a finite field extension L/E of degree prime to p such that α L ∈ Im(S(L) → T (L)). A morphism of functors S → T from Fields/F to Sets is called surjective if for any E ∈ Fields/F , S(E) → T (E) is surjective. Obviously, any surjective morphism is p-surjective for any prime p. Such a surjective morphism gives an upper bound for the essential (p)-dimension of T and a lower bound for the essential (p)-dimension of S,
2 m → C 1 defined by (x, y) → ((x, y), γ),
where γ is the canonical involution. As this morphism is surjective, by (5) we have ed(C 1 ) ≤ 2, thus ed(PGSp 2 ) = 2. This can be recovered from the exceptional isomorphism PGSp 2 ≃ O + 3 . Acknowledgements: I am grateful to A. Merkurjev for useful discussions. I am also grateful to J. P. Tignol and S. Garibaldi for helpful comments.
2. Essential dimension of projective orthogonal and symplectic groups associated to central simple algebras of index ≤ 4
First, we compute upper bounds for the essential dimension of certain classes of simple algebras with involutions of index less or equal to 2.
Let (Q, γ) be a pair of a quaternion over a field F and the canonical involution. For any field extension K/F and any integer n ≥ 3, we write QH + n (K) (respectively, QH − n (K)) for the set of isomorphism classes of (M n (Q), σ h ), where σ h is the adjoint involution on M n (Q) with respect to a hermitian form (respectively, skew-hermitian form) h (with respect to γ). If n is odd, we write 1 QH − n (K) for the set of isomorphism classes of (M n (Q), σ h ), where σ h is the adjoint involution on M n (Q) with respect to a skew-hermitian form h with disc(σ h ) = 1.
Lemma 2.1. Let F be a field and n ≥ 3 any integer. Then
(1) ed(QH + n ) ≤ n + 1. (2) ed(QH − n ) ≤ 3n − 3 if char(F ) = 2. (3) ed( 1 QH − n ) ≤ 3n − 4 if char(F ) = 2. Proof. (1) If h is a hermitian form, then h = t 1 , t 2 , · · · , t n for some t i ∈ F .
We consider the affine variety
X = G 2 m ×A n−1 F if char(F ) = 2, G m ×A n F if char(F ) = 2,
and define a morphism X(K) → QH + n (K) by
(a, b, t 1 , · · · , t n−1 ) → ((a, b) ⊗ M n (K), σ 1,t 1 ,··· ,t n−1 ) if char(F ) = 2, ([a, b) ⊗ M n (K), σ 1,t 1 ,··· ,t n−1 ), if char(F ) = 2.
As a scalar multiplication does not change the adjoint involution, this morphism is surjective. Therefore, by (5), we have ed(QH + n ) ≤ n + 1. (2) From now we assume that char(F ) = 2. If h is a skew-hermitian form, then h = q 1 , q 2 , · · · , q n for some pure quaternions q i ∈ Q. We consider the affine variety Y = G 2 m ×A 1 × A 3(n−2) with coordinates (a, b, c, t 1 , · · · , t 3n−6 ) and the conditions
ac 2 + b = 0, at 2 1+3(k−1) + bt 2 2+3(k−1) − abt 2 3k = 0 for all 1 ≤ k ≤ n − 2. Define a morphism φ K : Y (K) → QH − n (K) by (a, b, c, t 1 , · · · , t 3n−6 ) → ((a, b) ⊗ M n (K), σ h ), where p = i, q = ci + j, r k = t 1+3(k−1) i + t 2+3(k−1) j + t 3k ij for 1 ≤ k ≤ n − 2,
and h = p, q, r 1 , · · · , r 3n−2 .
We show that Y is a classifying variety for QH − n . Suppose that we are given a quaternion K-algebra Q = (a, b) and a skew hermitian form h = p, q, r 1 , · · · , r n−2 for some pure quaternions p, q, r k . We can find a scalar c ∈ K such that p and q − cp anticommute, thus we have Q ≃ (p 2 , (q − cp) 2 ). For 1 ≤ k ≤ n − 2, let
(6) r k = t 1+3(k−1) p + t 2+3(k−1) (q − cp) + t 3k p(q − cp)
with t 1 , t 2 , · · · , t 3n−6 ∈ K. Then (M n (Q), σ h ) ≃ (M n ((p 2 , (q − cp) 2 )), σ h ) is the image of φ K . Therefore, by (5), we have ed(QH − n ) ≤ 3n − 3.
(3) Assume that n = 2m + 1 for m ≥ 1. We consider the variety Y in (2) with an additional condition −a(ac 2 +b)
m−1 k=1 at 2 1+3(k−1) +bt 2 2+3(k−1) −abt 2 3k = n−2 k=m at 2 1+3(k−1) +bt 2 2+3(k−1) −abt 2 3k .
We show that this variety with the same morphism φ K in (2) is a classifying variety for 1 QH − n . Suppose that we are given a quaternion K-algebra Q = (a, b) and a skew hermitian form h = p, q, r 1 , · · · , r n−2 for some pure quaternions p, q, r k . We do the same procedure as in (1), so that we have (M n (Q), σ h ) ≃ (M n ((p 2 , (q − cp) 2 )), σ h ) and (6). As disc(σ h ) = 1, there is a scalar d ∈ K × such that −p 2 q 2 r 2 1 · · · r 2 m−1 = ( d r 2 m · · · r 2 n−2 ) 2 r 2 m · · · r 2 n−2 .
We set f = d/r 2 m · · · r 2 n−2 . As a scalar multiplication does not change the adjoint involution, we can modify h by the scalar f . As (
p 2 , (q − cp) 2 ) ≃ (f 2 p 2 , f 2 (q − cp) 2 ), (M n (Q), σ h ) is the image of φ K . Therefore, by (5), we have ed( 1 QH − n ) ≤ 3n − 4.
Remark 2.2. The main idea of the proof of the case where h is a skewhermitian form is from Merkurjev's work on algebras of degree 4 in his private note.
Assume that n is odd. Then we have QH + n = C n , QH − n = D n , and 1 QH − n = 1 D n . Hence, by [5,Theorem 4.2] and the exceptional isomorphism PGO 6 ≃ PGU 4 , we have Corollary 2.3. Assume that n ≥ 3 is odd. Then
(1) ed(PGSp 2n ) ≤ n + 1. 2 ) ≤ ed(PGL ×n 2 ) ≤ n · ed(PGL 2 ) = 2n. On the other hand, the natural morphism
(7) H 1 (−, PGL ×n 2 ) → Dec 2 n (−)
is surjective, where Dec 2 n (K) is the set of all decomposable algebras of degree 2 n over a field extension K/F , hence, by (5), we have 2n = ed 2 (Dec 2 n ) ≤ ed 2 (PGL ×n 2 ).
Lemma 2.6. Let F be a field of characteristic different from 2. Then
ed 2 (PGO 2 r ), ed 2 (PGSp 2 r ) ≥ 2 if r = 1, 4 if r = 2, (r − 1)2 r−1 if r ≥ 3.
Proof. Consider the forgetful functors
(8) H 1 (−, PGO 2 r ) → Alg 2 r ,2 and (9) H 1 (−, PGSp 2 r ) → Alg 2 r ,2 ,
where Alg 2 r ,2 (K) is the set of isomorphism classes of simple algebras of degree 2 n and exponent dividing 2 over a field extension K/F . These functors are surjective by a theorem of Albert. It is well known that ed 2 (Alg 2,2 ) = 2, ed 2 (Alg 4,2 ) = 4. For r ≥ 3, we have ed 2 (Alg 2 r ,2 ) ≥ (r − 1)2 r−1 by [3,Theorem]. Therefore, by (5), we have the above lower bound for ed 2 (PGO 2 r ) and ed 2 (PGSp 2 r ).
The following Lemma 2.7(1) was proved by Rowen in [10, Theorem B] (see also [5,Proposition 16.16]) and Lemma 2.7(2) was proved by Serhir and Tignol in [11,Proposition]. We shall need the explicit forms of involutions on the decomposed quaternions as in (1): Lemma 2.7. Let F be a field of characteristic different from 2. Let (A, σ) be a central simple F -algebra of degree 4 with a symplectic involution σ.
(1) If A is a division algebra, then we have
(A, σ) ≃ (Q, σ| Q ) ⊗ (Q ′ , γ),
where σ| Q is an orthogonal involution defined by σ| Q (x 0 + x 1 i + x 2 j + x 3 k) = x 0 + x 1 i + x 2 j − x 3 k with a quaternion basis (1, i, j, k) for Q and γ is the canonical involution on a quaternion algebra Q ′ . (2) If A is not a division algebra, then we have
(A, σ) ≃ (M 2 (F ), ad q ) ⊗ (Q ′ , γ),
where q is a 2-dimensional quadratic form, ad q is the adjoint involution on M 2 (F ), and γ is the canonical involution on a quaternion algebra Q ′ .
Proof. (1) By [10,Proposition 5.3], we can choose a i ∈ A\F such that σ(i) = i and [F (i) : F ] = 2. Let φ be the nontrivial automorphism of F (i) over F . By [10,Proposition 5.4], there is a j ∈ A\F such that σ(j) = j and ji = φ(i)j. Then i and j generate a quaternion algebra Q, σ| Q (i) = i, σ| Q (j) = j, and σ| Q (k) = −k with k = ij. Hence, σ| Q is an orthogonal involution on Q. By the double centralizer theorem, we have A ≃ Q ⊗ C A (Q), where C A (Q) is the centralizer of Q ⊂ A and is isomorphic to quaternion algebra Q ′ over F . By [5,Proposition 2.23], the restriction of σ on Q ′ is the canonical involution γ.
(2) See [11,Proposition]. Proof. (1) By the exceptional isomorphism (2), we have
PGO + 4 = PGL 2 × PGL 2 .
The proof follows from Lemma 2.5 with n = 2.
(2) By Lemma 2.6, we have ed 2 (PGO 4 ) ≥ 4. For the opposite inequality, we consider the affine variety X defined in A 4 F with the coordinates (a, b, c, e) by e(a 2 − b 2 e)(c 2 − e) = 0. Define a morphism X → A 2 1 by
(a, b, c, e) → (a + b √ e, c + √ e) if F ( √ e) is a quadratic field extension, (a, b) × (c, √ e) otherwise,
We show that X is a classifying variety for A 2 1 . Let Q = (a + b √ e, c + d √ e) be a quaternion algebra over a quadratic extension L = F ( √ e). If b = d = 0, we can modify c by a norm of L( √ a)/L, hence we may assume that d = 0.
Similarly, we can assume that d = 1, replacing e by ed 2 . Thus, the morphism X → A 2 1 is surjective. By (5), ed(A 2 1 ) ≤ 4. Hence, the opposite inequality ed(PGO 4 ) ≤ 4 comes from the exceptional isomorphism (1) and the canonical bijection (3).
(3) By Lemma 2.6, we have ed 2 (PGSp 4 ) ≥ 4. For the opposite inequality, we define a morphism G 4 m → C 2 by (x, y, z, w) → ((x, y), σ) ⊗ ((z, w), γ) if x = 1, (M 2 (F ), ad q ) ⊗ ((z, w), γ) if x = 1, where σ is an involution defined by σ(x 0 +x 1 i+x 2 j +x 3 k) = x 0 +x 1 i+x 2 j −x 3 k with a quaternion basis (1, i, j, k) of the quaternion algebra (x, y), q = 1, y is a quadratic form, and γ is the canonical involution on the quaternion algebra (z, w). Note that multiplying any two dimensional quadratic form by a scalar does not change the adjoint involution. By Lemma 2.7, this morphism is surjective. Therefore, by (5), we have ed(C 2 ) ≤ 4, hence the result follows from the canonical bijection (4).
Remark 2.9.
(1) Assume that F is a field of characteristic 2. By [1, Corollary 2.2], we have ed 2 (Alg 4,2 ) = ed 2 (Dec 4 ) ≥ 3. As the morphisms (7) and (8)
T ) ≤ ed(S) and ed p (T ) ≤ ed p (S); see [4, Lemma 1.9] and [8, Proposition 1.3].Example 1.1. Let (M 2 (F ), γ) ∈ C 1 , where γ is the canonical involution on M 2 (F ). As (M 2 (K), γ K ) ≃ (M 2 (F ), γ) ⊗ F K for any field extension K/F , we have ed((M 2 (F ), γ)) = 0.Assume that char(F ) = 2. The exact sequence1 → µ 2 → Sp 2 → PGSp 2 → 1induces the connecting morphism ∂ : H 1 (−, PGSp 2 ) → Br 2 (−) which sends a pair (Q, γ) of a quaternion algebra with canonical involution to the Brauer class[Q]. As this morphism is nontrivial, by [4, Corollary 3.6] we have ed(PGSp 2 ) ≥ 2 (or by Lemma 2.6). Consider the morphism G
( 2 )
2ed(PGO 2n ) ≤ 3n − 3 if char(F ) = 2. In particular, ed(PGU 4 ) ≤ 6. (3) ed(PGO + 2n ) ≤ 3n − 4 if char(F ) = 2. Remark 2.4. In fact, ed 2 (PGSp 2n ) = ed(PGSp 2n ) = n+1 for n ≥ 3 odd and char(F ) = 2. The lower bound was obtained by Chernousov and Serre in [6, Theorem 1] and the exact value was obtained by Macdonald in [7, Proposition 5.1]. Lemma 2.5. [2, Section 2.6] Let F be a field of characteristic different from 2. Then ed 2 (PGL ×n 2 ) = ed(PGL ×n 2 ) = 2n. Proof. By [4, Lemma 1.11], we have ed 2 (PGL ×n
Proposition 2. 8 .
8Let F be a field of characteristic different from 2. Then (1) ed 2 (PGO + 4 ) = ed(PGO + 4 ) = 4. (2) ed 2 (PGO 4 ) = ed(PGO 4 ) = 4. (3) ed 2 (PGSp 4 ) = ed(PGSp 4 ) = 4.
are surjective, we get ed 2 (PGO + 4 ) ≥ 3 and ed 2 (PGO 4 ) ≥ 3, respectively. On the other hand, the upper bounds in Proposition 2.8 (1) and (2) still hold, hence 3 ≤ ed(PGO + 4 ), ed(PGO 4 ) ≤ 4. (2) As ed(O + 5 ) = 4 by [9, Theorem 10.3], Proposition 2.8 (3) can be recovered from the exceptional isomorphism PGSp 4 ≃ O + 5 .
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[
"Self Organizing Nebulous Growths for Robust and Incremental Data Visualization",
"Self Organizing Nebulous Growths for Robust and Incremental Data Visualization"
]
| [
"Damith A Senanayake ",
"Wei Wang ",
"Shalin H Naik ",
"Saman Halgamuge "
]
| []
| []
| Non-parametric dimensionality reduction techniques, such as t-SNE and UMAP, are proficient in providing visualizations for fixed or static datasets, but they cannot incrementally map and insert new data points into existing data visualizations. We present Self-Organizing Nebulous Growths (SONG), a parametric nonlinear dimensionality reduction technique that supports incremental data visualization, i.e., incremental addition of new data while preserving the structure of the existing visualization. In addition, SONG is capable of handling new data increments no matter whether they are similar or heterogeneous to the existing observations in distribution. We test SONG on a variety of real and simulated datasets. The results show that SONG is superior to Parametric t-SNE, t-SNE and UMAP in incremental data visualization. Specifically, for heterogeneous increments, SONG improves over Parametric t-SNE by 14.98 % on the Fashion MNIST dataset and 49.73% on the MNIST dataset regarding the cluster quality measured by the Adjusted Mutual Information scores. On similar or homogeneous increments, the improvements are 8.36% and 42.26% respectively. Furthermore, even in static cases, SONG performs better or comparable to UMAP, and superior to t-SNE. We also demonstrate that the algorithmic foundations of SONG render it more tolerant to noise compared to UMAP and t-SNE, thus providing greater utility for data with high variance or high mixing of clusters or noise. | 10.1109/tnnls.2020.3023941 | [
"https://arxiv.org/pdf/1912.04896v1.pdf"
]
| 209,202,398 | 1912.04896 | a7719ececf7454a150ab3c790bfa1a5dbb82d4a3 |
Self Organizing Nebulous Growths for Robust and Incremental Data Visualization
Damith A Senanayake
Wei Wang
Shalin H Naik
Saman Halgamuge
Self Organizing Nebulous Growths for Robust and Incremental Data Visualization
1Index Terms-t-SNEUMAPSONGVector QuantizationNonlinear Dimensionality ReductionHeterogeneous Incremental Learning
Non-parametric dimensionality reduction techniques, such as t-SNE and UMAP, are proficient in providing visualizations for fixed or static datasets, but they cannot incrementally map and insert new data points into existing data visualizations. We present Self-Organizing Nebulous Growths (SONG), a parametric nonlinear dimensionality reduction technique that supports incremental data visualization, i.e., incremental addition of new data while preserving the structure of the existing visualization. In addition, SONG is capable of handling new data increments no matter whether they are similar or heterogeneous to the existing observations in distribution. We test SONG on a variety of real and simulated datasets. The results show that SONG is superior to Parametric t-SNE, t-SNE and UMAP in incremental data visualization. Specifically, for heterogeneous increments, SONG improves over Parametric t-SNE by 14.98 % on the Fashion MNIST dataset and 49.73% on the MNIST dataset regarding the cluster quality measured by the Adjusted Mutual Information scores. On similar or homogeneous increments, the improvements are 8.36% and 42.26% respectively. Furthermore, even in static cases, SONG performs better or comparable to UMAP, and superior to t-SNE. We also demonstrate that the algorithmic foundations of SONG render it more tolerant to noise compared to UMAP and t-SNE, thus providing greater utility for data with high variance or high mixing of clusters or noise.
Abstract-Non-parametric dimensionality reduction techniques, such as t-SNE and UMAP, are proficient in providing visualizations for fixed or static datasets, but they cannot incrementally map and insert new data points into existing data visualizations. We present Self-Organizing Nebulous Growths (SONG), a parametric nonlinear dimensionality reduction technique that supports incremental data visualization, i.e., incremental addition of new data while preserving the structure of the existing visualization. In addition, SONG is capable of handling new data increments no matter whether they are similar or heterogeneous to the existing observations in distribution. We test SONG on a variety of real and simulated datasets. The results show that SONG is superior to Parametric t-SNE, t-SNE and UMAP in incremental data visualization. Specifically, for heterogeneous increments, SONG improves over Parametric t-SNE by 14.98 % on the Fashion MNIST dataset and 49.73% on the MNIST dataset regarding the cluster quality measured by the Adjusted Mutual Information scores. On similar or homogeneous increments, the improvements are 8.36% and 42.26% respectively. Furthermore, even in static cases, SONG performs better or comparable to UMAP, and superior to t-SNE. We also demonstrate that the algorithmic foundations of SONG render it more tolerant to noise compared to UMAP and t-SNE, thus providing greater utility for data with high variance or high mixing of clusters or noise.
Index Terms-t-SNE, UMAP, SONG, Vector Quantization, Nonlinear Dimensionality Reduction, Heterogeneous Incremental Learning
I. INTRODUCTION
I N data analysis today, we often encounter high-dimensional datasets with each dimension representing a variable or feature. For example, in experimental biology, one typically records the expression levels of thousands of genes per cell [1] or per population [2]. When analysing such datasets, reducing the data dimensionality is highly useful to gain insights into the structure of the data. Visualization of high-dimensional data is achieved by reducing the data down to 2 or 3 dimensions.
In practice, we often assume static data visualization, i.e. the data are presented to the dimensionality reduction methods all at once. However, with the advent of big data, the data may be presented incrementally for the following two main reasons. First, the dataset may be extremely large and has to be divided and processed sequentially [3]. Second, there are scenarios where data is incrementally acquired through a series of experiments, such as the continuous acquisition of Geographical Data [4] or data gathered by mining social media [5]. In Fig. 1 with either homogeneous data (new data that has a structure similar to the already observed structure) or heterogeneous data (new data that has a structure unlike the already observed structure). In real-world situations, both scenarios may be present indistinguishably, and these necessitate incremental data visualization, where we either directly use or continually train a pre-trained model to visualize the data increments. In addition, it is often required that the visualization on existing data do not change drastically after the new data is added for consistency in data interpretation. In the following, we explore the applicability of existing dimensionality reduction techniques for incremental data visualization.
In static data visualization scenarios, t-distributed Stochastic Nonlinear Embedding (t-SNE) [6] and Uniform Manifold Approximation and Projection (UMAP) [7] are two state-of-theart methods frequently used for dimensionality reduction. Both t-SNE and UMAP first create a graph in the input space, where the vertices are the input points and the edge-weights represent the pseudo-probability of two inputs being in the same local neighborhood. t-SNE calculates the edge-weights by assuming Gaussian distributions in local neighborhoods. These pseudoprobabilities are then replicated on an output map of lowdimensionality (typically 2 or 3). However, to allow for clear cluster separation, t-SNE assumes the output probabilities to be sampled from a student's t-distribution. UMAP follows a different strategy to calculate the edge weights in the input space, by assuming that the local neighborhoods lie on a Riemannian manifold. Using this assumption, UMAP normalizes the local pairwise distances to obtain a fuzzy simplicial set that represents a weighted graph similar to that of t-SNE. UMAP then uses a suitable rational quadratic kernel function in lowdimensional output space to approximate the edge probabilities of the weighted graph.
Being non-parametric models, t-SNE and UMAP do not retain a mapping from the inputs to the outputs, and thus cannot be directly applied for incremental data visualization. Instead, t-SNE and UMAP need to be reinitialized and retrained at each increment of data. In UMAP, the heuristic initialization using spectral embedding (Laplacian Eigenmaps) [8] provides some degree of stability in visualizations of datasets from the same distribution as similar datasets have similar nearest neighbor graphs that provide similar graph laplacians. However, it remains to be answered how well such heuristic initialization performs when the new data have heterogeneous structure to the existing data.
Previous studies on t-SNE and UMAP have attempted to retain a parametric model for new data. One approach is to train a neural network regressor on visualizations obtained by t-SNE or UMAP. However, to our best knowledge, there is no existing work with a generally applicable version of this approach. Another approach is to use deep Latent Variable Models, e.g., Variational Autoencoders [9] and parametric t-SNE [10], to represent the low-dimensional visualizations. However, it is questionable whether such approaches provide visualizations with comparable quality of cluster separation to the non-parametric counterparts of the respective algorithms. Additionally, these methods suffer from issues commonly associated with deep neural networks such as requiring a large amount of training data [11], high computational complexity [12] and lack of model interpretability [13].
Self Organizing Map (SOM) [14] and its variants are arguably the only dimensionality reduction methods that retain a parametric graph on the input space to approximate the input data distribution locally. SOMs obtain the graph by vector encoding, i.e., SOMs partition the input space into Voronoi regions by mapping each input to the closest element in a set of representative vectors called Coding Vectors. These Coding Vectors, now representing the centroids of the voronoi regions, are mapped onto a low-dimensional (typically 2 or 3) uniform output grid. Such a uniform (or regular) grid can have either a square, triangular or hexagonal topology of locally connected output vectors. The topology preservation of SOMs is achieved by moving the set of coding vectors in the input space such that the coding vectors corresponding to neighbors in the lowdimensional output grid are placed close together.
There are two main problems associated with the visualizations provided by SOMs. First, the SOM visualizations have poor cluster separation possibly due to the uniform nature of the output grid. Since all edges in the output graph is of equal length, the difference between the edge lengths in the input space are not preserved in the output grid [15]. Second, the size of the map (the number of coding vectors ) needs to be known a priori. Growing Cell Structures (GCS), presented by Fritzke [15], uses a non-uniform triangulation (where the coding vectors represent the vertices of the triangles) of the input space to tackle these two problems. However, the graph inferred in GCS is planar thus non-planar graph structures in local neighborhoods cannot be preserved. Fritzke's later work of Growing Neural Gas (GNG) [16] approximates non-planar graph structures on the coding vectors (by having more densely connected local subgraphs than GCS) in local neighborhoods. However, due to this non-planarity, GNG cannot provide a visualization of the inferred graph by triangulating a set of low-dimensional points as GCS does as such triangulation relies on the planarity of the graph. A possible approach for visualizing the coding vectors of a static Neural Gas using a Cross Entropy measure (NG-CE) has been investigated by Estevez et al. [17], however NG-CE has not been extended to support the dynamically growing nature of GNGs. Furthermore, NG-CE relies having a fully trained set of coding vectors prior to the visualization, and does not support simultaneous approximation of the input topology and the simultaneous projection of the said topology to a low dimensional embedding. Growing SOM (GSOM) [18] provides another approach to overcome SOM's problem with unknown map size by using a uniform but progressively growing grid. This is different to the GNGs where the grids were non-uniform and non-planar. However, similar to SOM, GSOM still suffers from poor cluster separation due to the uniform output grid . Additionally the topology of localities in the output map of GSOM is limited to be 2 or 3 dimensional.
Inspired by GNG and NG-CE, we propose Self Organizing Nebulous Growths (SONG), a nonlinear dimensionality reduction algorithm with two main advantages over t-SNE, UMAP and their variants. First, SONG retains a parametric model for incremental data visualization, while being significantly more efficient than the parametric variants of t-SNE and UMAP. Second, SONG solves the visualization problems of SOM and produces comparable or better visualizations than t-SNE and UMAP in the static data visualization scenarios. In the following section, we describe our proposed method in detail.
II. METHOD
In the proposed SONG, we use a set of coding vectors C = {c ∈ R D } to partition and represent the input dataset X = {x ∈ R D }. For an input x i ∈ X, we define an index set I (k) = {i l |l = 1, ..., k} for a user-defined k, where c i l is the l-th closest coding vector to x i . Moreover, we define a set of directional edges between coding vectors C and an adjacency matrix E, such that if a coding vector c m is one of the closest neighbors to another coding vector c l , they are connected by an edge with edge strength E(l, m) > 0. We organize the graph {C, E} to approximate the input topology.
We also define a set of low-dimensional vectors Y = {y ∈ R d }, d << D which has a bijective correspondence with the set of coding vectors C. When d = 2 or 3, Y represents the visualization of the input space, i.e., the input x i ∈ X is visualized as y i1 ∈ Y. We preserve the topology of C given by E in Y by positioning Y such that, if E(l, m) > 0 or E(m, l) > 0 , y l and y m will be close to each other in the visualization. Typically, the number of c ∈ C and corresponding low dimensional vectors y ∈ Y is far less than the number of input data points X. By retaining the parameters C, E and Y, SONG obtains a parametric mapping from input data to visualization.
We initialize a SONG model by randomly placing d + 1 coding vectors C in the input space, since d + 1 is the minimum number for coding vectors to obtain a topology preserving visualization in a d-dimensional visualization space (see Supplement Section 1.1 for proof). No edge connection is assumed at initialization (i.e. E = 0). The corresponding Y are also randomly placed in the d-dimensional output space. Next, we approximate local topology of any given x ∈ X using C, and project this approximated topology to Y in the visualization space. To be specific, SONG randomly samples an input point x i ∈ X and performs the following steps at each iteration until terminated:
1) Updating the Directional Edges in E between Coding
Vectors C based on x i : This step modifies the adjacency matrix E to add or remove the edges between coding vectors based on local density information at x i . Eventually, if no edge is added or removed for repeated sampling of all input x i s (i.e. the whole dataset is sampled with no addition or removal of edges), the graph is considered stable and SONG has finished its training. We describe this step in detail in Section II-A. 2) Self Organization of Coding Vectors C: This step moves x i 's closest coding vector c i1 closer to x i , along with any coding vectors c j if c i1 and c j are connected by an edge as indicated by E(i 1 , j) > 0. This movement enforces the closeness of coding vectors connected by edges. We describe this in detail in Section II-B.
3) Topology Preservation of the Low-dimensional Points
Y: Given that c i1 encodes x i and corresponds to y i1 in the output space, we organize low-dimensional points y j ∈ Y in the locality of y i1 such that the coding vector topology at c i1 is preserved in the output space. This step is described in Section II-C. 4) Growing C and Y to Refine the Inferred Topology:
There may be cases where c i1 and its neighboring coding vectors are insufficient to capture the local fine topology at x i , e.g., inputs from multiple subclusters may have the same c i1 . In such cases, we place new coding vectors close to c i1 , and new corresponding lowdimensional vectors close to y i1 , without reinitializing the parametric model {X, C, E}. Details of this are in Section II-D.
For a given epoch, we randomly sample (without replacement) a new input x i ∈ X and repeat the four steps, until all x i ∈ X are sampled. The algorithm is terminated if the graph becomes stable in Step 1) or we have executed the maximum number of epochs. When new data X are presented, we simply allow x i to be sampled from X at the next iteration, and continue training without reinitializing the parameters C, Y and E.
A. Updating the Directional Edges in E between Coding Vectors based on x i
For each input x i randomly sampled from X, we conduct three operations to any edge-strength e i1j = E(i 1 , j) at current iteration t:
• Renewal: we reset e t i1j to 1 if j ∈ I (k) . • Decay: if e t−1
i1j > 0, we decay it by a constant multiplier , i.e., e t i1j = e t−1 i1j · and ∈ (0, 1). • Pruning: we set e t i1j to 0 if e t−1 i1j < e min . This helps to obtain a sparse graph. Note that e min is predefined to obtain the desired degree of sparseness in the graph. The edge strength e i1j reflects the rate the edge is renewed and is proportional to p i1j , the probability of i 1 and j being close neighbors to the input x i . Since the edge strengths e i1j ∈ [0, 1], larger edge strengths can be interpreted as c i1 and neighboring coding vectors representing finer topologies (shorter distances between C). Conversely, the smaller the edge strengths, the coarser the topology represented by such edges. Note that here we define finer and coarser topologies in their conventional sense [19].
Note that edge strength e i1j obtained above is directional and thus the adjacency matrix E is asymmetric. We observe faster convergence in subsequent optimization with a symmetric adjacency matrix, which is simply calculated as:
E s = E + E T 2(1)
Next, we use the coding vector graph to approximate the topology of the input through self organization.
B. Self-Organization of Coding Vectors C
To ensure the coding vectors C are located at the centers of input regions with high probability densities (such as cluster centers), we move the coding vectors c i1 towards x i by a small amount to minimize the following Quantization Error (QE):
QE(x) = 1 2 x i − c i1 2(2)
However, moving c i1 independent of other coding vectors may cause the coding vectors sharing an edge with c i1 to be no longer close to c i1 , which disorganizes the graph. To avoid this, we also move c i1 's neighboring coding vectors c j (as indicated by E s (i 1 , j) > 0) towards x i by a smaller amount than that of c i1 . Moreover, the more distant c j is from c i1 , the smaller the movement of c j should be. This ensures the organization of distant neighbors is proportionately preserved by this movement. Therefore, we define a loss function that monotonically decreases when the distance from the coding vectors to x i increases. In addition, to penalize large neighborhoods (and thereby large edge lengths), we scale the loss function by x i −c i k 2 . Note that we treat the distance to the k th coding vector from x as a constant for the considered neighborhood, and the gradient of x i − c i k w.r.t. c i k is not calculated. The final loss function is:
L (x i ) = − x i − c i k 2 2 cj ∈Ni 1 exp(− x i − c j 2 x i − c i k 2 ) (3) where N i1 = {c j | E s (j, i 1 ) > 0}
is the set of c i1 's neighboring coding vectors. Using stochastic gradient descent to minimize this loss, we calculate the partial derivatives of the loss w.r.t. a given c for the sampled x i as:
∂L (x i ) ∂c = (x i − c) × exp (− x i − c 2 x i − c i k 2 )(4)
Next, we describe how we optimize the output embedding (the placement of Y) to reflect the topology inferred in the input space.
C. Topology Preservation of the Low-dimensional Points Y Similar to UMAP and inspired by NG-CE, SONG optimizes the embedding Y by minimizing the Cross Entropy (CE) between the probability distribution p in the input space and a predefined low-dimensional probability distribution q in the output space. We define the local cross entropy for a given x i ∈ X as:
CE(x i ) = ∀j −p i1j log(q i1j ) − (1 − p i1j ) log(1 − q i1j ) (5)
where q i1j is the probability that output points y i1 and y j are located close together, and is calculated using the following rational quadratic function:
q i1j = 1 1 + a y i1 − y j 2b(6)
See Supplement Section 2 for how to calculate the hyperparameters a and b.
The cross entropy (see Eq. 5) can be interpreted as two sub-components: an attraction component CE attr = −p i1j log(q i1j ) that attracts y i1 towards y j , and an re-
pulsion component CE rep = −(1 − p i1j ) log(1 − q i1j
) that repulses y i1 from y j . The attraction component heavily influences local arrangement at y i1 , since distant y j results in p i1j = 0. The repulsion component, on the other hand, influences the global arrangement of neighborhoods, as 1−p i1j = 1 for such distant y j . Due to the difference of influences, we derive the gradients of these two components separately. The attraction component has a gradient as defined by:
∂CE attr ∂y = (y i1 − y j ) · 2ab · p ij · y j − y i1 2b−2 1 + y j − y i1 2b(7)
The gradient for the repulsion component is:
∂CE rep ∂y = (y i1 − y j ) · 2b · (1 − p ij ) y j − y i1 2 (1 + y j − y i1 2b )(8)
We use stochastic gradient descent to minimize CE attr and CE rep w.r.t y i1 , y j ∈ Y. We select stochastic gradient descent over batch gradient descent to avoid convergence on sub-optimal organizations [20]. Moreover, since the edge renewal rate is proportional to the p i1j , we propose to use the symmetric edge strengthsê i1j ∈ E s as an approximation of p i1j in CE attr . This avoids the explicit assumptions on p i1j as made by t-SNE and UMAP. In a similar fashion, we use the negative sampling of edges (i.e. sampling of j where E s (i 1 , j) = 0) to approximate (1 − p i1j ) in CE rep . In this negative sampling, for a very large dataset, we randomly sample a set of non-edges, such that for each x i , the number of sampled non-edges n ns equals the number of edges connected to c i1 (n(e i )) multiplied by a constant rate. Similar ideas have been used in Word2Vec [21] and UMAP [7]. The algorithmic summary of this step is shown in Algorithm 3.
D. Growing C and Y to Refine the Inferred Topology By iterating the above three steps from Sections II-B -II-C, the topology of x can be approximated using {C, E} and preserved onto Y in the visualization space. However, since the optimal number of coding vectors C is unknown a priori, SONG starts with a small number of C which may be insufficient to capture all the structures in X such as clusters and sub-clusters. Therefore, we grow the sizes of C and Y as needed during training. Additionally, such growth can accommodate structural changes, e.g., addition of new clusters, when new data are presented in the incremental data visualization scenarios. Note that this growth of coding vectors and low-dimensional vectors is conditional, so this step may not be done for certain iterations.
Inspired by the GNG, we define a Growth Error associated with c i1 as:
G i1 (t) ← G i1 (t − 1) + x i − c i1(9)
where t is the index of current iteration. When any G i1 (t) exceeds a predefined threshold, we place a new coding vector c at the centroid between x i and its k nearest coding vectors, so that the regions that have high Growth Error get more populated with coding vectors. Due to the stochastic sampling, the current x i and its neighboring data may not be sampled in the next iterations, thus the newly created coding vector may not be duly connected in subsequent repetitions of Step 1 and it may eventually drift away. To avoid this, at the current iteration, we add new edges from the newly added coding vectors to all neighbors of the c i1 . The placement of new y is conducted similarly, and by placing the y close to the neighborhood of y i1 , the convergence of the new y to a suitable position is made faster, than by placing the new y randomly on the output map. We summarize this step in Algorithm 4. These four steps form a complete iteration of SONG algorithm, which we summarize in Algorithm 1. In the next section, we evaluate the performance of the SONG algorithm.
III. EXPERIMENTS AND RESULTS
In this section, we compare SONG against Parametric t-SNE and non-parametric methods: t-SNE [6] and UMAP [7] on a series of data visualization tasks. First, we consider incremental data visualization with heterogeneous increments of data in Section III-A and homogeneous increments of data in Section III-B. It is noteworthy that the former is more likely the case to be assumed in real problems with incremental data Algorithm 1: SONG Algorithm with Decaying Learning Rate 1 t ← Iteration index, initialized as 0; 2 t max ← Maximum number of iterations ; 3 d ← Output dimensionality; usually 2 or 3; 4 k ← Number of neighbors to consider at a given locality, k ≥ d + 1; 5 α ← Learning rate starting at α 0 ; 6 C ← Random matrix of size (d + 1) × D; 7 E ← Edges on C, from each c to other cs, all initialized as non-edges (0); 8 Y ← Random matrix of size(d + 1) × d; 9 r ← Number of negative edges to select, per positive edge for negative sampling; 10 a, b ← Appropriate parameters to get desired spread and tightness as per Eq. 6; 11 θ g ← User defined growth threshold; 12 while t < t max do 13 for x i ∈ X do 14 Update E s as per Section II-A; 15 Update I (k) ; 16 ns = r · n(ê i1 ), here n(ê i1 ) is the number of edges from or to c i1 ; 17 Record the neighbors of c i1 as Update Y as per Algorithm 3;
N t−1 i1 ← {j | E s (i 1 , j) > 0} ;28 G i1 ← G i1 + x i − c i1 ; 29
if G i1 > θ g then 30 Grow C and Y as per Algorithm 4;
t ← t + 1; 34 α ← α 0 × (1 − t tmax ); 35 end
streams due to lack of the ground-truth. Then we evaluate the visualization quality of SONG in static data visualization scenarios: we assess the SONG's robustness to noisy and highly mixed clusters in Section III-C, and we evaluate SONG's capability in preservation of topologies in Section III-D. In our analysis, we define model-retaining methods as methods that reuse a pretrained model and refine it when presented with new data, while model-reinitializing methods reinitialize and retrain a model from scratch. Therefore SONG and Parametric t-SNE are model-retaining methods, and t-SNE and UMAP are
j ∈ N t i1 do 3 y j ← y j + α · (y i1 − y j ) · 2ab·êi 1 j · yj −yi 1 2b−2 1+} with E s (i 1 , j) = 0 ; 7 for j ∈ J do 8 y j ← y j − α · (y i1 − y j ) · 2b yj −yi 1 2 (1+ yj −yi 1 2b ) 9 end
model-reinitializing methods. However, for fair comparison, we introduce a model-reinitializing version of SONG called SONG + Reinit. The 'incremental visualizations' can only be fairly assessed with model-retaining methods, but for the sake of completeness, we extend this comparison to the modelreinitializing methods as well.
We use the hyper-parameters in Table I for each method. For t-SNE [6] and UMAP [7], the recommended hyper-parameters in original papers were used as we did not observe any improvement in results by tuning these parameters. Similarly, for parametric t-SNE, we used the set of parameters provided by the GitHub implementation 1 . The tuned hyper-parameters for SONG are noted in Table I I HYPERPARAMETERS USED FOR SONG, PARAMETRIC T-SNE, T-SNE AND UMAP THROUGHOUT
A. Visualization of Data with Heterogenous Increments
We first evaluate SONG presented with heterogeneous increments, where new clusters or classes may be added to the existing datasets.
Setup: Three datasets are used: the Wong dataset [22], MNIST hand-written digit dataset [23] and the Fashion MNIST dataset [24]. Wong dataset has over 327k single human T-cells measured for expression levels for 39 different surface markers (i.e., 39 dimensions) such as CCR7 surface marker. There are many types of cells present in this dataset, such as lymphoid cells, naive T-cells, B-Cell Follides, NK T cells etc, which we expect to be clustered separately. However, there may be some cell types that cannot be clearly separated as clusters in visualizations [25]. Since we have no ground-truth labels and UMAP provides superior qualitative cluster separation on this dataset [25], we assume that the clusters visible in the UMAP visualization of the dataset represent different cell types. This assumption allows us to do a sampling of 20k, 50k, 100k and 327k samples such that each time we add one or several cell types to the data. However, due to the lack of ground truth, we only conduct qualitative analysis on the cluster quality for each method. In addition, we conduct 'logicle transformation' [26] to normalize the Wong dataset as a preprocessing step. On the other hand, Fashion MNIST dataset is a collection of 60k images of fashion items belonging to 10 classes, each image having 28 × 28 pixels, therefore 784 pixel intensity levels (dimensions), associated with a known ground-truth label. Similar to Fashion -MNIST dataset, MNIST dataset is a collection of 60k images of hand written digits, each with 784 pixels and an associated label from 0 to 9. Since both MNIST and Fashion MNIST datasets have known ground-truth labels, we start with two randomly selected classes and present two more classes to the algorithm at each increment. We ran a K-Means clustering on the visualizations provided for MNIST and Fashion MNIST by each method, and calculated the Adjusted Mutual Information (AMI) [27] scores against the ground-truth labels. The AMI scores were averaged over five iterations with random initializations. Additionally, both MNIST and Fashion MNIST datasets are reduced to 20 dimensions using Principal Components Analysis (PCA) as a preprocessing step in order to reduce the running time of our experiments. We assume that the first 20 principle components capture most of the variance in the datasets [28]. Each of the intermediate and incrementally growing datasets of Wong, MNIST and Fashion MNIST datasets is visualized using SONG, SONG + Reinit, Parametric t-SNE, t-SNE and UMAP.
Results: For all three datasets: Wong (Fig. 2), Fashion MNIST (Fig. 3) and MNIST (Fig. 4), SONG shows the most stable placement of clusters when new data are presented, compared to parametric t-SNE and model-reinitalized methods, UMAP, t-SNE and SONG + Reinit. For the Wong dataset in Fig. 2, both SONG and Parametric t-SNE show similar cluster placements in the first two visualizations. However, Parametric t-SNE visualizations become distorted as more data are presented. In contrast, SONG provides consistently stable visualizations. On MNIST and Fashion MNIST datasets, the cluster placements provided by SONG have a noticeable change for the first two increments, but become more stable in their later increments. Parametric t-SNE shows a high level of cluster mixing in visualizations, which becomes more evident in later increments. Although UMAP shows similar relative placement of clusters at later increments for the MNIST and Fashion MNIST datasets, arbitrary rotations of the complete map are visible even for such visualizations. This may be due to UMAP using Spectral Embedding as the heuristic initialization instead of random initialization. Table II summarizes the AMI Scores for the Fashion MNIST and MNIST datasets in heterogeneous incremental visualization scenarios. Note that in Table II, we have highlighted the best scores for each increment in the model-retaining methods. Since for the model-reinitializing methods such incremental visualization is not directly comparable with the model-retaining methods, we have only highlighted the winner for the complete dataset out of the model-reinitializing methods. In Table II, SONG provides superior cluster purity than Parametric t-SNE, confirming our observations on the level of cluster mixing present in visualizations by Parametric t-SNE. Here, SONG shows an average improvement of 14.98% for Fashion MNIST and 49.73% for MNIST in AMI compared to Parametric t-SNE. We observe that out of the non-parametric algorithms, SONG + Reinit is comparable but slightly inferior to UMAP, and superior to t-SNE.
For the Wong dataset (see Fig. 2), SONG + Reinit, Parametric t-SNE , t-SNE and UMAP all have drastic movement of clusters in consecutive visualizations. In t-SNE, we see a set of Gaussian blobs (possibly due to the Gaussian distribution assumption), with no discernible structure of cluster placement as visible in SONG and UMAP. Parametric t-SNE shows stable placement of clusters in the first two visualizations. However, when more data are presented, we see a high level of mixed clusters in the visualization.
In the Fashion MNIST visualizations (Fig.3), we see drastic re-arrangement of placement when using SONG + Reinit, Parametric t-SNE , t-SNE and UMAP. We emphasize that SONG does not show rotations, as seen in visualizations provided by UMAP.
In Fig. 4, the hierarchy of clusters is more preserved in SONG and UMAP than t-SNE and Parametric t-SNE for the MNIST dataset. We expect in low-dimensional embedding space, the distances between clusters should vary as not all pairs of clusters are equally similar to each other, e.g., "1" should be more similar to "7" than to "3" or to "5". In the results of t-SNE, however, the clusters are separated by similar distances, thereby the results do not provide information about the varying degrees of similarity between clusters. For both UMAP and SONG, the distances separating the clusters vary as expected. While parametric t-SNE shows similar placement of clusters with rotations or flips in the last two visualizations, the level of cluster mixing is relatively high.
B. Visualization of Data with Homogeneous Increments
In this section, we further examine how each method performs when the incrementally added data proportionally represent all classes and clusters, using the same three datasets: Wong, Fashion-MNIST and MNIST.
Setup: For Wong dataset, we pick four different numbers of random samples, 10k, 20k, 50k and 327k. We select these numbers to investigate if the incremental inference of topology can be achieved starting from a small number of samples. For Fashion MNIST and MNIST datasets, we randomly sample four batches of data: 12k, 24k, 48k and 60k images at a time. Since Fashion-MNIST and MNIST have known ground-truths, for each visualization, we again conduct a k-means clustering to investigate the separability of clusters in the visualization, and use AMI to evaluate the clusters quality.
Furthermore, for Fashion MNIST and MNIST datasets, we develop a metric called the consecutive displacement of Y (CDY), to quantify the preservation of cluster placement in two consecutive incremental visualizations. CDY is defined as follows. Initially, we apply each algorithm to 6000 randomly sampled images, and iteratively add 6000 more to the existing III AMI SCORES OF THE VISUALIZED HOMOGENEOUS INCREMENTS OF THE FASHION MNIST AND MNIST DATASETS. IN SONG AND PARAMETRIC T-SNE, A TRAINED MODEL FROM ONE INTERMEDIATE DATASET IS UPDATED AND USED TO VISUALIZE THE NEXT DATASET. SONG + REINIT, T-SNE, AND UMAP ARE REINITIALIZED AND RETRAINED visualization, until we have presented all images in a dataset. At the t-th iteration, we record the visualizations of the existing data (without the newly added data) before and after the training with the 6000 new images, namely Y (t−1) and Y (t) . Next we calculate the CDY of a point y i in the existing visualization Y (t−1) as:
CDY(y i ) = y (t) i − y (t−1) i
We record the average and standard deviations of CDY calculated for all points in the visualization. We note that the lack of a ground-truth which gives us information about an accurate placement of clusters renders a similar analysis for the Wong dataset prohibitive.
Result: Among compared methods, SONG shows the highest stability in cluster placement when new data are presented, as shown in Fig.5 for the Wong dataset, and Fig. 6 for the MNIST and Fashion-MNIST datasets. Furthermore, SONG shows good quality in the clusters inferred in the output embedding as per Table III. In Table III On Wong dataset, UMAP and SONG have more similar cluster placement in the homogeneous increment cases (see Fig. 5) than that of the heterogeneous increment cases (see Fig. 2). This increased similarity may possibly be due to the initial intermediate data samples in homogeneous cases more accurately representative of the global structure of the data than that of heterogeneous cases. However, SONG shows no rotations in the visualization when data is augmented; in contrast, UMAP shows different orientations of similar cluster placements. Table III shows that the AMI scores for SONG on the MNIST dataset have increased as we present more data to the SONG algorithm. For the Fashion MNIST dataset, the AMI scores for SONG remain relatively low throughout the increments. This difference of trends may be due to the higher level of mixing of classes in the Fashion MNIST dataset than the MNIST, which makes it more difficult to separate the classes in Fashion MNIST into distinct clusters despite having more data. Table III further shows that SONG provides visualizations of comparable quality to UMAP and superior to t-SNE and parametric t-SNE. We note that SONG generally produces lower AMIs than SONG + Reinit, possibly because that SONG attempts to preserve the placement of points in existing visualizations which may cause some structural changes caused by new data to be neglected. Neglecting such changes may explain the slight drop of performance in the incremental visualization vs the visualization of the complete dataset. However, we see that the incremental scores of SONG are not considerably worse than that of SONG + Reinit where SONG is trained on existing data and increment from scratch.
In Fig. 6, SONG has the lowest CDY values for both MNIST and Fashion MNIST, throughout the increments. SONG also shows small standard deviations, showing that the CDYs for all points are indeed limited. In contrast, t-SNE has the largest displacements and standard deviations. Surprisingly, the average CDYs for each increment in Parametric t-SNE is relatively higher than heuristically reinitialized UMAP. We note that parametric t-SNE has a low standard deviation of displacement compared to t-SNE. Given a completely random re-arrangement of clusters would cause high standard deviation, this implies that parametric t-SNE produces translational or rotational displacements while keeping the cluster structure intact. Notably, UMAP does fairly well compared to other methods in terms of cluster displacement by having the second lowest average displacement and standard deviation. However, we see that in addition to having a smaller movement of points, SONG shows a strict decrease in displacement when more data are presented. SONG + Reinit, shows comparatively large average CDYs, as well as large standard deviations of CDYs, implying large movements between consecutive visualizations.
C. Tolerance to Noisy and Highly Mixed Clusters
We explore how well SONG performs in the presence of noisy data, which we simulated as a series of datasets with high levels of cluster mixing and large clster standard deviations.
Setup: We compare SONG against UMAP and t-SNE on a collection of 32 randomly generated Gaussian Blobs datasets using 8 different cluster standard deviations (4,8,10,12,14,16,18,20) and 4 different numbers of clusters (10,20,50,100) for each standard deviation. These datasets have a dimensionality of 60. In addition, for each algorithm, we calculate the Adjusted Mutual Information (AMI) score for the visualizations provided against the known labels of the Gaussian clusters. Result: Table IV shows that SONG has the highest accuracy for discerning mixing clusters in all cases. The resulting visualizations for one of these datasets for SONG, UMAP and t-SNE are provided in Fig.7. We observe that the cluster representations in the visualizations by SONG are more concentrated than that of both UMAP and t-SNE.
We refer to the Supplement Section 3.1 for an extended set of visualizations, where we additionally changed the dimensionality to observe how it affects these observations. In this extended study we test the visualization performance of the three algorithms on an additional 125 datasets. These datasets are generated by simulating datasets corresponding to 5 cluster standard deviations (1, 2, 3, 4, 10), 5 numbers of clusters (3, 4, 20, 50, 100) and 5 numbers of dimensions (3,15,45,60,120). In these visualizations, consistent to our observations in Table IV, we observe that SONG has better separation of clusters while UMAP and t-SNE show fuzzy cluster boundaries when the level of cluster mixing increases.
D. Qualitative Topology Preservation of SONG
To qualitatively examine the capability of SONG to preserve specific topologies in the input data, we used the COIL-20 dataset [29], which is frequently used to assess the topology preservation of visualization methods [6] [7].
Setup: We compare SONG with UMAP and t-SNE on visualizing the COIL-20 dataset, which has 20 different objects, each photographed at pose intervals of 5-degrees, resulting in 1440 images in total. Each image has 4096 pixels. As preprocessing, we reduced the COIL-20 dataset down to its first 300 principle components. Because of the rotating pose angles, we expect to see 20 circular clusters in our visualization, where each cluster represents a different object.
Result: The separation of circular clusters in SONG is similar to UMAP as shown in Fig. 8. For highly inseparable clusters, SONG and UMAP preserves the circular topologies better than t-SNE, where t-SNE shows an arch-like shape instead of circular structures.
E. Running Time Comparison with t-SNE and UMAP
In our Supplement Section 3.2, we show that SONG is faster for the same dataset configurations than t-SNE. However, as SONG needs to recalculate the pairwise distances between two sets of high-dimensional vectors (X and C) multiple times for self organization, SONG has a performance bottleneck which renders it slower than UMAP.
IV. CONCLUSIONS AND FUTURE WORK
In this work, we have presented a parametric nonlinear dimensionality reduction method that can provide topologypreserving visualizations of high-dimensional data, while allowing new data to be mapped into existing visualizations without complete reinitialization. In our experiments, we presented SONG with both heterogeneous (Section III-A) and homogeneous (Section III-B) data increments, and observed that in both cases, SONG is superior than parametric t-SNE in preserving cluster placements when incorporating new data. We further observed in the aforementioned Sections that the visualizations provided by SONG has cluster quality on par with UMAP, and superior to non-parametric t-SNE. We also conclude that SONG is robust to noisy and highly mixed clusters (Section III-C), and that SONG is capable of preserving specific topologies (Section III-D) inferred from the input.
In addition, SONG has a few merits compared to existing methods. First of all, although originated from SOM, SONG is a general dimensionality reduction method. SOM relies on a uniform grid of 2 or 3 dimensions to obtain the self organization of coding vectors, which renders SOM with limited utility in providing representations of more than 3 dimensions. Since SONG uses a generic graph independent of output dimensionality, SONG can be a viable preprocessing technique to reduce the dimensionality down to more than three dimensions.
Second, SONG is especially useful for visualization of large datasets where considerable heterogeneity is present. This heterogeneity may be the product of undesired batch effects [30], or genuine variation of populations of the data. Consequently, SONG may be a promising tool for large-scale benchmarking projects that require coordination and curation of highly heterogeneous data, such as the Human Cell Atlas [31].
However, SONG has a few limitations to be addressed in future work. First, SONG has a higher computational complexity than UMAP because in SONG, the high-dimensional parametric graph in the input space need to be recalculated several times, whereas in UMAP, the high-dimensional KNN graph is constructed only once. The number of recalculations required to obtain a stable graph influences how fast SONG works. We have empirically determined that 8-10 recalculations of the graph is sufficient to provide a comparable approximation with UMAP. One possible direction of minimizing this graph reconstruction bottle-neck is to use batch gradient descent instead of stochastic gradient descent at later stages of learning. In our implementation, we chose stochastic gradient descent in an attempt to obtain an optimal visualization quality as the batch versions of Self Organization algorithms are prone to sub-optimal solutions [20]. This may be viable at later stages when the graph is relatively stable and unchanging compared to the earlier stages. It should be noted that SONG still is less complex than t-SNE because SONG uses a negative sampling trick where we do not compute pairwise embedding distances globally (see Supplement Section 3.2).
Second, the current version of SONG cannot adjust the trade-off between two aspects of cluster placement: 1) preserving the already inferred topological representations and 2) adapting to represent new data which potentially alter the existing topology in high-dimensional space. This trade-off is indicated by the discrepancy between the AMI scores of SONG and SONG + Reinit calculated on the incremental data visualizations. Besides, in the Wong dataset ( Fig. 2 and Fig. 5), the topology of the visualizations obtained using SONG shows differences from that using SONG + Reinit, which may be explained by SONG's preference to preserve the topology in existing visualizations. Future work would explore the introduction of an 'agility' parameter that can regulate the aforementioned trade-off.
At last, throughout our manuscript, we discuss SONG as an unsupervised learning method. Another research direction is to incorporate known or partially known labels of data to enhance cluster quality and separation of clusters.
we show how data can be augmented D. Senanayake, W. Wang and S. Halgamuge are all with the department of Mechanical Engineering of the University of Melbourne, Victoria, Australia S. Naik is with the Walter and Eliza Hall Institute for Medical Research, Melbourne, Victoria, Australia
Fig. 1 .
1Illustration of incremental data visualization: a) the visualization of the initially available data contain three clusters; b) the initial data are augmented with homogeneous (similar) data, where clusters become denser in the visualization; c) the initial data are augmented with heterogeneous (dissimilar) data; new clusters are added to the visualization.
j ← c j + α · ∂L (xi)
Fig. 2 .Fig. 3 .
23The Wong dataset visualized by SONG, SONG + Reinit, Parametric t-SNE and UMAP. The colors represent the CCR7 expression levels following the visualizations provided in[25], where Light Green represents high CCR7 expression and Dark Purple represents low CCR7 expressions. Incremental visualization of the Fashion MNIST dataset using SONG, SONG + Reinit, Parametric t-SNE, t-SNE and UMAP, where two classes are added at a time.
Fig. 4 .
4Incremental visualization of the MNIST dataset using SONG, SONG + Reinit, Parametric t-SNE, t-SNE and UMAP, where two classes are added at a time.
Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .
5678Random samples of varying sizes from the Wong dataset presented to SONG, Parametric t-SNE, SONG + Reinit , UMAP and t-SNE incrementally. The Euclidean Displacement of each established point after subsequent presentation of 6000 images to each algorithm. The visualizations of a dataset having 100 clusters, each cluster having a standard deviation of 10, and 60 dimensions using the three methods SONG, COIL-20 dataset when reduced using the three algorithms a)t-SNE, b)UMAP and c)SONG. Both UMAP and SONG preserve the circular topologies, even in clusters where the classes are not well separated, to a greater degree than t-SNE.
Self Organizing Nebulous Growths for Robust and Incremental Data Visualization Damith A. Senanayake, Wei Wang, Shalin H. Naik, Saman Halgamuge
Select ns random samples J = {j 1 , ... ,j nsyj −yi 1
2b
4 end
5 /* Negative Sampling for Repulsion */;
6
. https://github.com/jsilter/parametric tsne Algorithm 4: Growing C and Y to Refine the Inferred Topology1 1 create new coding vector such that
w n ← 1
k
l={1...k}
w i l ;
2 create new low-dimensional vector such that
y n ← 1
k
l={1...k}
y i l ;
3 for j ∈ {i 1 , ..., i k } do
4
E(j, n) = 1;
5 end
TABLE
TABLE II AMI
IISCORES ON HETEROGENEOUS INCREMENTS OF FASHION MNIST AND MNIST DATASETS. FOR FAIR COMPARISON MODEL-RETAINING METHODS (SONG AND PARAMETRIC T-SNE) AND MODEL-REINITIALIZING METHODS (SONG + REINIT, T-SNE AND UMAP) ARE SEPARATED. THE BEST AMI SCORES ARE HIGHLIGHTED.Fashion MNIST
MNIST
No. Classes
2
4
6
8
10 2
4
6
8
10
SONG
70.9 86.1 84 71.2 61.5 88.4 88.0 79.2 75.0 81.0
Parametric
t-SNE
50.3 80.6 76.1 60.2 57.8 39.2 58.8 60.8 56.8 59.3
SONG + Reinit 70.9 76.8 78.4 69.1 61.0 88.4 86.4 77.8 75.3 81.0
t-SNE
14.2 56.0 59.9 57.3 56.3 89.3 67.0 72.1 71.2 73.8
UMAP
25.2 77.3 79.7 67.5 59.1 92.2 92.0 81.8 81.8 84.9
TABLE
AT EACH INCREMENT.Fashion MNIST
MNIST
12k
24k
48k
60k
12k
24k
48k
60k
SONG
59.6 60.1 58.1 59.5 74.8 79.4
80.7 81.9
Parametric t-SNE 51.2 55.8 57.2 54.8 44.9 53.3 57
61.2
SONG-Reinit
59.6 58.8 61.2 61
74.8 79.8 80.9 84
t-SNE
58.9 58
59.4 53.5 77.5 74.1 78.6 77.4
UMAP
60.1 59.5 58.7 59
76.6 80.8 83.2 84.9
, we have highlighted the best scores for each increment in the model-retaining methods. However, for the model-reinitializing methods, we have highlighted the winner for the complete dataset. Compared to Parametric t-SNE, SONG has an average improvement of accuracy by 8.36% on Fashion MNIST and 42.26% on MNIST. Out of three model-reinitialized methods, UMAP has similar placements of clusters in consecutive visualizations, but shows complete rotations in early to mid intermediate representations on the Wong dataset. UMAP stabilizes towards the last increments. However, for the three datasets, this stabilization happens at different stages. t-SNE shows arbitrary placement of clusters at each intermediate representation, making t-SNE not as good as UMAP or SONG for incremental visualizations.
TABLE IV THE
IVAMI SCORES FOR DIFFERENT GAUSSIAN BLOBS CONFIGURATIONS IN 60 DIMENSIONS. EACH CONFIGURATION OF GAUSSIAN BLOBS HAVE DIFFERENT NUMBERS OF CLUSTERS AND DIFFERENT CLUSTER STANDARD DEVIATIONS. HAVING A LARGE NUMBER OF CLUSTERS AND A LARGE STANDARD DEVIATION INCREASES THE PROBABILITY OF MIXING OF CLUSTERS.SONG
UMAP
t-SNE
No. Clusters 10
20
50
100
10
20
50
100
10
20
50
100
Cluster Std. Deviation
4
100
100
100
100
100
100
100
100
100
100
100
100
8
99.9 99.6 99.2 99.1 99.9 99.2 98.4 97.1 99.7 99.2 98.0 96.0
10
97.6 95.4
91.4 89.2 96.8 92.8 84.4 67.9 95.7 90.4 80.3 68.4
12
90.0 84.1 75.2 65.8 86.5 75.4 51.0 33.2 82.7 68.5 48.5 34.5
14
77.6 66.6 52.2 39.7 70.9 51.3 24.8 19.6 65.8 43.5 23.0 21.3
16
62.8 49.2 31.2 21.2 54.2 30.3 13.4 15.8 46.7 23.4 12.8 17.5
18
50.0 35.2 16.7 16.0 36.8 17.2 8.7
14.5 32.9 13.7 8.7
15.6
20
38.1 22.6 10.2 14.6 25.7 10.3 6.84 13.6 22.7 9.1
7.0
14.5
ACKNOWLEDGEMENTSThe authors thank Mr. Rajith Vidanaarachchi and Ms. Tamasha Malepathirana for proof-reading the manuscript. This work is partially funded by ARC DP150103512.Damith Senanayake Damith Senanayake is a PhD candidate at the University of Melbourne. He completed his undergraduate degree in Computer Science and Engineering from the University of Moratuwa, Sri Lanka in 2015. Currently his research research focuses on developing analysis methods for highly dimensional data using neural networks and computational and algebraic geometry and topology. He further focuses on applications in bioinformatics.Wang Wei, PhD Wei Wang received the BE degree in Mechatronic Engineering from Zhejiang University in 2013 and PhD degree at the Department of Mechanical Engineering, University of Melbourne in 2019. His research has been focused on deep generative models, Bayesian statistics and image processing, with a current interest in exploring compact and highly performant deep neural network structures.
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| [
"https://github.com/jsilter/parametric"
]
|
[
"Conformal metrics on R 2m with constant Q-curvature",
"Conformal metrics on R 2m with constant Q-curvature"
]
| [
"Luca Martinazzi "
]
| []
| []
| We study the conformal metrics on R 2m with constant Q-curvature Q ∈ R having finite volume, particularly in the case Q ≤ 0. We show that when Q < 0 such metrics exist in R 2m if and only if m > 1. Moreover we study their asymptotic behavior at infinity, in analogy with the case Q > 0, which we treated in a recent paper. When Q = 0, we show that such metrics have the form e 2p g R 2m , where p is a polynomial such that 2 ≤ deg p ≤ 2m − 2 and sup R 2m p < +∞. In dimension 4, such metrics are exactly the polynomials p of degree 2 with lim |x|→+∞ p(x) = −∞. | 10.4171/rlm/525 | [
"https://arxiv.org/pdf/0805.0749v1.pdf"
]
| 12,187,589 | 1401.0944 | 0e84b08ca21af616a6fc2286992f9d6fbf7b4a4f |
Conformal metrics on R 2m with constant Q-curvature
6 May 2008 April 29, 2008
Luca Martinazzi
Conformal metrics on R 2m with constant Q-curvature
6 May 2008 April 29, 2008
We study the conformal metrics on R 2m with constant Q-curvature Q ∈ R having finite volume, particularly in the case Q ≤ 0. We show that when Q < 0 such metrics exist in R 2m if and only if m > 1. Moreover we study their asymptotic behavior at infinity, in analogy with the case Q > 0, which we treated in a recent paper. When Q = 0, we show that such metrics have the form e 2p g R 2m , where p is a polynomial such that 2 ≤ deg p ≤ 2m − 2 and sup R 2m p < +∞. In dimension 4, such metrics are exactly the polynomials p of degree 2 with lim |x|→+∞ p(x) = −∞.
Introduction and statement of the main theorems
Given a constant Q ∈ R, we consider the solutions to the equation
(−∆) m u = Qe 2mu on R 2m ,(1)
satisfying α := 1 |S 2m | R 2m e 2mu(x) dx < +∞.
Geometrically, if u solves (1) and (2), then the conformal metric g := e 2u g R 2m has Q-curvature Q 2m g ≡ Q and volume α|S 2m |. For the definition of the Qcurvature and related remarks, we refer to [Mar1]. Notice that given a solution u to (1) and λ > 0, the function v := u − 1 2m log λ solves
(−∆) m v = λQe 2mv in R 2m ,
hence what matters is just the sign of Q, and we can assume without loss of generality that Q ∈ {0, ±(2m − 1)!}. Every solution to (1) is smooth. When Q = 0, that follows from standard elliptic estimates; when Q = 0 the proof is a bit more subtle, see [Mar1,Corollary 8].
For Q ≥ 0, some explicit solutions to (1) are known. For instance every polynomial of degree at most 2m − 2 satisfies (1) with Q = 0, and the function u(x) = log 2 1+|x| 2 satisfies (1) with Q = (2m − 1)! and α = 1. This latter solution has the property that e 2u g R 2m = (π −1 ) * g S 2m , where π : S 2m → R 2m is the stereographic projection.
For the negative case, we notice that the function w(x) = log 2 1−|x| 2 solves (−∆) m w = −(2m − 1)!e 2mw on the unit ball B 1 ⊂ R 2m (in dimension 2 this corresponds to the Poincaré metric on the disk). However, no explicit entire solution to (1) with Q < 0 is known, hence one can ask whether such solutions actually exist. In dimension 2 (m = 1) it is easy to see that the answer is negative, but quite surprisingly the situation is different in dimension 4 and higher and we have:
Theorem 1 Fix Q < 0. For m = 1 there is no solution to (1)-(2). For every m ≥ 2, there exist (several) radially symmetric solutions to (1)-(2).
Having now an existence result, we turn to the study of the asymptotic behavior at infinity of solutions to (1)-(2) when m ≥ 2, Q < 0, having in mind applications to concentration-compactness problems in conformal geometry. To this end, given a solution u to (1)-(2), we define the auxiliary function
v(x) := − (2m − 1)! γ m R 2m log |y| |x − y| e 2mu(y) dy,(3)
where γ m := ω 2m 2 2m−2 [(m − 1)!] 2 is characterized by the following property:
(−∆) m 1 γ m log 1 |x| = δ 0 in R 2m . Then (−∆) m v = −(2m − 1)!e 2mu .
We prove Theorem 2 Let u be a solution of (1)-(2) with Q = −(2m − 1)!. Then
u(x) = v(x) + p(x),(4)
where p is a non-constant polynomial of even degree at most 2m − 2. Moreover there exist a constant a = 0, an integer 1 ≤ j ≤ m − 1 and a closed set Z ⊂ S 2m−1 of Hausdorff dimension at most 2m − 2 such that for every compact subset K ⊂ S 2m−1 \Z we have
lim t→+∞ ∆ ℓ v(tξ) = 0, ℓ = 1, . . . , m − 1, v(tξ) = 2α log t + o(log t), as t → +∞, lim t→+∞ ∆ j u(tξ) = a,(5)
for every ξ ∈ K uniformly in ξ. If m = 2, then Z = ∅ and sup R 2m u < +∞.
Finally lim inf |x|→+∞ R gu (x) = −∞,(6)
where R gu is the scalar curvature of g u := e 2u g R 2m .
Following the proof of Theorem 1, it can be shown that the estimate on the degree of the polynomial is sharp. Recently J. Wei and D. Ye [WY] showed the existence of solutions to ∆ 2 u = 6e 4u in R 4 with R 4 e 4u dx < +∞ which are not radially symmetric. It is plausible that also in the negative case non-radially symmetric solutions exist.
For the case Q = 0 we have Theorem 3 When Q = 0, any solution to (1)-(2) is a polynomial p with 2 ≤ deg p ≤ 2m − 2 and with sup R 2m p < +∞.
In particular in dimension 2 (case m = 1), there are no solutions. In dimension 4 the solutions are exactly the polynomials of degree 2 with lim |x|→∞ p(x) = −∞.
Finally, there exist 1 ≤ j ≤ m − 1 and a < 0 such that
lim |x|→∞ ∆ j p(x) = a.(7)
The case when Q > 0, say Q = (2m − 1)!, has been exhaustively treated. The problem
(−∆) m u = (2m − 1)!e 2mu on R 2m , R 2m e 2mu dx < +∞(8)
admits standard solutions, i.e. solutions of the form u(x) := log 2λ 1+λ 2 |x−x0| 2 , λ > 0, x 0 ∈ R 2m that arise from the stereographic projection and the action of the Möbius group of conformal diffeomorphisms on S 2m . In dimension 2 W. Chen and C. Li [CL] showed that every solution to (8) is standard. Already in dimension 4, however, as shown by A. Chang and W. Chen [CC], (8) admits non-standard solutions. In dimension 4 C-S. Lin [Lin] classified all solutions u to (8) and gave precise conditions in order for u to be a standard solution in terms of its asymptotic behavior at infinity.
In arbitrary even dimension, A. Chang and P. Yang [CY] proved that solutions of the form u(x) = log 2 1 + |x| 2 + ξ(π −1 (x)) are standard, where π : S 2m → R 2m is the stereographic projection and ξ is a smooth function on S 2m . J. Wei and X. Xu [WX] showed that any solution u to (8) is standard under the weaker assumption that u(x) = o(|x| 2 ) as |x| → ∞, see also [Xu]. We recently treated the general case, see [Mar1], generalizing the work of C-S. Lin. In particular we proved a decomposition u = p + v as in Theorem 2 and gave various analytic and geometric conditions which are equivalent to u being standard.
The classification of the solutions to (8) has been applied in concentrationcompactness problems, see e.g. [LS], [RS], [Mal], [MS], [DR], [Str1], [Str2], [Ndi]. There is an interesting geometric consequence of Theorems 2 and 3, with applications in concentration-compactness: In the case of a closed manifold, metrics of equibounded volumes and prescribed Q-curvatures of possibly varying sign cannot concentrate at points of negative or zero Q-curvature. For instance we shall prove in a forthcoming paper [Mar2] Theorem 4 Let (M, g) be a 2m-dimensional closed Riemannian manifold with Paneitz operator P 2m g satisfying ker P 2m g = {const}, and let u k : M → R be a sequence of solutions of
P 2m g u k + Q 2m g = Q k e 2mu k ,(9)
where Q 2m g is the Q-curvature of g (see e.g. [Cha]), and where the Q k 's are given continuous functions with
Q k → Q 0 in C 0 . Assume also that there is a Λ > 0 such that M e 2mu k dvol g ≤ Λ,(10)
for all k. Then one of the following is true.
(i) For every 0 ≤ α < 1, a subsequence is converging in C 2m−1,α (M ). (ii) There exists a finite set S = {x (i) : 1 ≤ i ≤ I} such that u k → −∞ in L ∞ loc (M \S). Moreover M Q g dvol g = I(2m − 1)!|S 2m |,(11)
and
Q k e 2mu k dvol g ⇀ I i=1 (2m − 1)!|S 2m |δ x (i) ,(12)
in the sense of measures.
Finally Q 0 (x (i) ) > 0 for 1 ≤ i ≤ I.
In sharp contrast with Theorem 4, on an open domain Ω ⊂ R 2m (or a manifold with boundary), m > 1, concentration is possible at points of negative or zero curvature. Indeed, take any solution u of (1)-(2) with Q ≤ 0, whose existence is given by Theorem 1, and consider the sequence
u k (x) := u(k(x − x 0 )) + log k, for x ∈ Ω
for some fixed x 0 ∈ Ω. Then (−∆) m u k = Qe 2mu k and u k concentrates at x 0 in the sense that as k → ∞ we have u k (x 0 ) → +∞, u k → −∞ a.e. in Ω and e 2mu k dx ⇀ α|S 2m |δ x0 in the sense of measures.
The 2 dimensional case (m = 1) is different and concentration at points of non-positive curvature can be ruled out on open domains too, because otherwise a standard blowing-up procedure would yield a solution to (1)-(2) with Q ≤ 0, contradicting with Theorem 1.
An immediate consequence of Theorem 4 and the Gauss-Bonnet-Chern formula, is the following compactness result (see [Mar2]):
Corollary 5 In the hypothesis of Theorem 4 assume that either
1. χ(M ) ≤ 0 and dim M ∈ {2, 4}, or 2. χ(M ) ≤ 0, dim M ≥ 6 and (M, g) is locally conformally flat, where χ(M ) is the Euler-Poincaré characteristic of M . Then only case (i) in Theorem 4 occurs.
The paper is organized as follows. The proof of Theorems 1, 2 and 3 is given in the following three sections; in the last section we collect some open questions. In the following, the letter C denotes a generic constant, which may change from line to line and even within the same line.
Proof of Theorem 1
Theorem 1 follows from Propositions 6 and 8 below.
Proposition 6 For m = 1, Q < 0 there are no solutions to (1)-(2).
Proof. Assume that such a solution u exists. Then, by the maximum principle, and Jensen's inequality,
∂BR udσ ≥ u(0), ∂BR e 2u dσ ≥ 2πRe 2u(0) . Integrating in R on [1, +∞), we get R 2 e 2u dx = +∞, contradiction.
Lemma 7 Let u(r) be a smooth radial function on R n , n ≥ 1. Then there are positive constants b m depending only on n such that
∆ m u(0) = b m u (2m) (0),(13)u (2m) := ∂ 2m u ∂r 2m . In particular ∆ m u(0) has the sign of u (2m) (0)
. For a proof see [Mar1].
Proposition 8 For m ≥ 2, Q < 0 there exist radial solutions to (1)-(2).
Proof. We consider separately the cases when m is even and when m is odd. Case 1: m even. Let u = u(r) be the unique solution of the following ODE:
∆ m u(r) = −(2m − 1)!e 2mu(r) u (2j+1) (0) = 0 0 ≤ j ≤ m − 1 u (2j) (0) = α j ≤ 0 0 ≤ j ≤ m − 1,
where α 0 = 0 and α 1 < 0. We claim that the solution exists for all r ≥ 0. To see that, we shall use barriers, compare [CC,Theorem 2]. Let us define w + (r) = α 1 2 r 2 , g + := w + − u.
Then ∆ m g + ≥ 0. By the divergence theorem,
BR ∆ j g + dx = ∂BR d∆ j−1 g + dr dσ.
Moreover, from Lemma 7, we infer
∆ j g + (0) ≥ 0 for 0 ≤ j ≤ m − 1,
hence we see inductively that ∆ j g + (r) ≥ 0 for every r such that g + (r) is defined and for 0 ≤ j ≤ m − 1. In particular g + ≥ 0 as long as it exists.
Let us now define
w − (r) := m−1 i=0 β i r 2i − A log 2 1 + r 2 , g − := u − w − ,
where the β i 's and A will be chosen later. Notice that
∆ m w − (r) = ∆ m − A log 2 1 + r 2 = −(2m − 1)!A 2 1 + r 2 2m .
Since α 1 < 0, lim r→+∞ 2 1+r 2 2m e mα1r 2 = +∞, and taking into account that u ≤ w + , we can choose A large enough, so that
∆ m g − (r) = (2m − 1)! A 2 1 + r 2 2m − e 2mu(r) ≥ (2m − 1)! A 2 1 + r 2 2m − e mα1r 2 ≥ 0.
We now choose each β i so that
∆ j g − (0) ≥ 0, 0 ≤ j ≤ m − 1,
and proceed by induction as above to prove that g − ≥ 0. Hence w − (r) ≤ u(r) ≤ w + (r) as long as u exists, and by standard ODE theory, that implies that u(r) exists for all r ≥ 0. Finally
R 2m e 2mu(|x|) dx ≤ R 2m e mα1|x| 2 dx < +∞. Case 2: m ≥ 3 odd. Let u = u(r) solve ∆ m u(r) = (2m − 1)!e 2mu(r) u (2j+1) (0) = 0 0 ≤ j ≤ m − 1 u (2j) (0) = α j ≤ 0 0 ≤ j ≤ m − 1,
where the α i 's have to be chosen. Set
w + (r) := β − r 2 − log 2 1 + r 2 , g + := w + − u,
where β < 0 is such that e −r 2 +β ≤ 2 1+r 2 2 , hence 2 1 + r 2 − 1 + r 2 2 e −r 2 +β ≥ 0 for all r > 0.
Then, as long as g + ≥ 0, we have
∆ m g + (r) = (2m − 1)! 2 1 + r 2 2m − e 2mu(r) ≥ (2m − 1)! 2 1 + r 2 2m − e 2mw+(r) ≥ 0 Choose now the α i 's so that, u (2i) (0) < w (2i) + (0), for 0 ≤ i ≤ m − 1. From Lemma 7, we infer that ∆ i g + (0) ≥ 0, 0 ≤ i ≤ m − 1,
and we see by induction that g + ≥ 0 as long as it is defined. As lower barrier, define
w − (r) = m−1 i=0 β i r 2i , g − := u − w − ,
where the β i 's are chosen so that ∆ i g − (0) ≥ 0. Then, observing that ∆ m g − (r) = (2m − 1)!e 2mu(r) > 0, as long as u is defined, we conclude as before that g − ≥ 0 as long as it is defined. Then u is defined for all times.
Let R > 0 be such that, for every r ≥ R, w + (r) ≤ − r 2 2 . Then
R 2m e 2mu(|x|) dx ≤ BR e 2mu(|x|) dx + R 2m \BR e −m|x| 2 dx < +∞.
Proof of Theorem 2
The proof of Theorem 2 is divided in several lemmas. The following Liouvilletype theorem will prove very useful.
Theorem 9 Consider h : R n → R with ∆ m h = 0 and h ≤ u − v, where e pu ∈ L 1 (R n ) for some p > 0, (−v) + ∈ L 1 (R n ). Then h is a polynomial of degree at most 2m − 2.
Proof. As in [Mar1,Theorem 5], for any x ∈ R 2m we have
|D 2m−1 h(x)| ≤ C R 2m−1 BR(x) |h(y)|dy = − C R 2m−1 BR(x) h(y)dy + 2C R 2m−1 BR(x) h + dy(14)
and BR(x) h(y)dy = O(R 2m−2 ), as R → ∞.
Then
BR(x) h + dy ≤ BR(x) u + dy + C BR(x) (−v) + dy ≤ 1 p BR(x) e pu dy + C R 2m ,
and both terms in (14) divided by R 2m−1 go to 0 as R → ∞.
Lemma 10 Let u be a solution of (1)-(2). Then, for |x| ≥ 4
v(x) ≤ 2α log |x| + C.
Proof. As in [Mar1,Lemma 9], changing v with −v.
Lemma 11 For any ε > 0, there is
R > 0 such that for |x| ≥ R, v(x) ≥ 2α − ε 2 log |x| + (2m − 1)! γ m B1(x) log |x − y|e 2mu(y) dy.(16)
Moreover
(−v) + ∈ L 1 (R 2m ).(17)
Proof. To prove (16) we follow [Lin], Lemma 2.4. Choose R 0 > 0 such that
1 |S 2m | BR 0 e 2mu dx ≥ α − ε 16 , and decompose R 2m = B R0 ∪ A 1 ∪ A 2 , A 1 := {y ∈ R 2m : 2|x − y| ≤ |x|, |y| ≥ R 0 }, A 2 := {y ∈ R 2m : 2|x − y| > |x|, |y| ≥ R 0 }.
Next choose R ≥ 2 such that for |x| > R and |y| ≤ R 0 , we have log |x−y| |y| ≥ log |x| − ε. Then, observing that (2m−1)!|S 2m | γm = 2, we have for |x| > R
(2m − 1)! γ m BR 0 log |x − y| |y| e 2mu(y) dy ≥ log |x| − ε 16 (2m − 1)! γ m BR 0 e 2mu dy ≥ 2α − ε 8 log |x| − Cε.(18)
Observing that log |x − y| ≥ 0 for y / ∈ B 1 (x), log |y| ≤ log(2|x|) for y ∈ A 1 , A1 e 2mu dy ≤ ε|S 2m | 16 and log(2|x|) ≤ 2 log |x| for |x| ≥ R, we infer
A1 log |x − y| |y| e 2mu(y) dy = A1 log |x − y|e 2mu(y) dy − A1 log |y|e 2mu(y) dy ≥ B1(x) log |x − y|e 2mu(y) dy − log(2|x|) A1 e 2mu dy ≥ B1(x) log |x − y|e 2mu(y) dy − log |x| ε|S 2m | 8 .(19)
Finally, for y ∈ A 2 , |x| > R we have that |x−y| |y| ≥ 1 4 , hence
A2 log |x − y| |y| e 2mu(y) dy ≥ − log(4) A2 e 2mu dy ≥ −Cε.(20)
Putting together (18), (19) and (20), and possibly taking R even larger, we obtain (16). From (16) and Fubini's theorem
R 2m \BR (−v) + dx ≤ C R 2m R 2m χ |x−y|<1 log 1 |x − y| e 2mu(y) dydx = C R 2m e 2mu(y) B1(y) log 1 |x − y| dxdy ≤ C R 2m
e 2mu(y) dy < ∞.
Since v ∈ C ∞ (R 2m ), we conclude that BR (−v) + dx < ∞ and (17) follows.
Lemma 12 Let u be a solution of (1)- (2), with m ≥ 2. Then u = v + p, where p is a polynomial of degree at most 2m − 2.
Proof. Let p := u − v. Then ∆ m p = 0. Apply (17) and Theorem 9.
Lemma 13 Let p be the polynomial of Lemma 12. Then if m = 2, there exists δ > 0 such that
p(x) ≤ −δ|x| 2 + C.(21)
In particular lim |x|→∞ p(x) = −∞ and deg p = 2. For m ≥ 3 there is a (possibly empty) closed set Z ⊂ S 2m−1 of Hausdorff dimension dim H (Z) ≤ 2m − 2 such that for every K ⊂ S 2m−1 \Z closed, there exists δ = δ(K) > 0 such that
p(x) ≤ −δ|x| 2 + C for x |x| ∈ K.(22)
Consequently deg p is even.
Proof. From (17), we infer that there is a set A 0 of finite measure such that
v(x) ≥ −C in R 2m \A 0 .(23)
Case m = 2. Up to a rotation, we can write
p(x) = 4 i=1 (b i x 2 i + c i x i ) + b 0 .
Assume that b i0 ≥ 0 for some 1 ≤ i 0 ≤ 4. Then on the set
A 1 := {x ∈ R 4 : |x i | ≤ 1 for i = i 0 , c i0 x i0 ≥ 0}
we have p(x) ≥ −C. Moreover |A 1 | = +∞. Then, from (23) we infer
R 4 e 4u dx ≥ A1\A0 e 4(v+p) dx ≥ C|A 1 \A 0 | = +∞,(24)
contradicting (2). Therefore b i < 0 for every i and (21) follows at once.
Case m ≥ 3. From (2) and (23) we infer that p cannot be constant. Write
p(tξ) = d i=0 a i (ξ)t i , d := deg p,
where for each 0 ≤ i ≤ d, a i is a homogeneous polynomial of degree i or a i ≡ 0. With a computation similar to (24), (2) and (23) imply that a d (ξ) ≤ 0 for each ξ ∈ S 2m−1 . Moreover d is even, otherwise a d (ξ) = −a d (−ξ) ≤ 0 for every ξ ∈ S 2m−1 , which would imply a d ≡ 0. Set
Z = {ξ ⊂ S 2m−1 : a d (ξ) = 0}.
We claim that dim H (Z) ≤ 2m − 2. To see that, set
V := {x ∈ R 2m : a d (x) = 0} = {tξ : t ≥ 0, ξ ∈ Z}.
Since V is a cone and Z = V ∩ S 2m−1 , we only need to show that dim H (V ) ≤ 2m − 1. Set
V i := {x ∈ R 2m : a d (x) = . . . = ∇ i a d (x) = 0, ∇ i+1 a d (x) = 0}. Noticing that V i = ∅ for i ≥ d (otherwise a d ≡ 0), we find V = ∪ d−1 i=0 V i .
By the implicit function theorem, dim H (V i ) ≤ 2m − 1 for every i ≥ 0 and the claim is proved.
Finally, for every compact set K ⊂ S 2m−1 \Z, there is a constant δ > 0 such that a d (ξ) ≤ − δ 2 , and since d ≥ 2, (22) follows.
Corollary 14 Any solution u of (1)-(2) with m = 2, Q < 0 is bounded from above.
Proof. Indeed u = v + p and, for some δ > 0, v(x) ≤ 2α log |x| + C, p(x) ≤ −δ|x| 2 + C.
Lemma 15
Let v : R 2m → R be defined as in (3) and Z as in Lemma 13. Then for every K ⊂ S 2m−1 \Z compact we have
lim t→+∞ ∆ m−j v(tξ) = 0, j = 1, . . . , m − 1 (25) for every ξ ∈ K uniformly in ξ; for every ε > 0 there is R = R(ε, K) > 0 such that, for t > R, ξ ∈ K, v(tξ) ≥ (2α − ε) log t(26)
Proof. Fix K ∈ S 2m−1 \Z compact and set C K := {tξ : t ≥ 0, ξ ∈ K}. For any σ > 0, 1 ≤ j ≤ 2m − 1,
R 2m \Bσ(x) e 2mu(y) |x − y| 2j dy → 0 as |x| → ∞(27)
by dominated convergence. Choose a compact set K ⊂ S 2m−1 \Z such that K ⊂ int( K) ⊂ S 2m−1 . Since u ≤ C( K) on C e K by Lemma 10 and Lemma 13, we can choose σ = σ(ε) > 0 so small that
Bσ(x) e 2mu |x − y| 2j dy ≤ C( K) Bσ (x) 1 |x − y| 2j dy ≤ C( K)ε, for x ∈ C K , |x| large, where |x| is so large that B σ (x) ⊂ C e K . Therefore (−1) j+1 ∆ j v(x) = C R 2m e 2mu |x − y| 2j dy → 0, for x ∈ C K , as |x| → ∞,
We have seen in Lemma 11, that for any ε > 0 there is R > 0 such that for
|x| ≥ R v(x) ≥ 2α − ε 2 log |x| + (2m − 1)! γ m B1(x) log |x − y|e 2mu(y) dy,(28)
and (26) follows easily by choosing K as above and observing that u ≤ C( K) on C e K , hence on B 1 (x) for x ∈ C K with |x| large enough.
Proof of Theorem 2. The decomposition u = v + p and the properties of v and p follow at once from Lemmas 10, 12, 13 and 15; (6) follow as in [Mar1,Theorem 2]. As for (5), let j be the largest integer such that ∆ j p ≡ 0. Then ∆ j+1 p ≡ 0 and from Theorem 9 we infer that deg p = 2j, hence ∆ j p ≡ a = 0. u < +∞, and, since u cannot be constant, we infer that deg u ≥ 2 is even. The proof of (7) is analogous to the case Q < 0, as long as we do not care about the sign of a. To show that a < 0, one proceeds as in [Mar1,Theorem 2]. For the case m = 2 one proceeds as in Lemma 13, setting v ≡ 0 and A 0 = ∅.
Example. One might believe that every polynomial p on R 2m of degree at most 2m − 2 with R 2m e 2mp dx < ∞ satisfies lim |x|→∞ p(x) = −∞, as in the case m = 2. Consider on R 2m , m ≥ 3 the polynomial u(x) = −(1 + x 2 1 )| x| 2 , where x = (x 2 , . . . , x 2m ). Then ∆ m u ≡ 0 and
R 2m e 2mu dx = R R 2m−1 e −2m(1+x 2 1 )|e x| 2 d xdx 1 = R dx 1 (1 + x 2 1 ) 2m−1 2 · R 2m−1 e −2m|e y| 2 d y < +∞.
On the other hand, lim sup |x|→∞ u(x) = 0.
Open questions
Open Question 1 Does the claim of Corollary 14 hold for m > 2? In other words, is any solution u to (1)-(2) with Q < 0 bounded from above?
This is an important regularity issue, in particular with regard to the behavior at infinity of the function v defined in (3). If sup R 2m u < +∞, then one can take Z = ∅ in Theorem 2, as in the case Q > 0, see [Mar1,Theorem 1].
Definition 16 Let P 2m 0 be the set of polynomials p of degree at most 2m − 2 on R 2m such that e 2mp ∈ L 1 (R 2m ). Let P 2m + be the set of polynomials p of degree at most 2m − 2 on R 2m such that there exists a solution u = v + p to (1)-(2) with Q > 0. Similarly for P 2m − with Q < 0. Related to the first question is the following Open Question 2 What are the sets P 2m 0 , P 2m ± ? Is it true that P 2m 0 ⊂ P 2m + and P 2m 0 ⊂ P 2m − ? J. Wei and D. Ye [WY] proved that P 4 0 ⊂ P 4 + (and actually more). Consider now on R 2m , m ≥ 3, the polynomial p(x) = −(1 + x 2 1 )| x| 2 , x = (x 2 , . . . , x 2m ).
As seen above, e 2mp ∈ L 1 (R 2m ), hence p ∈ P 2m 0 . Assume that p ∈ P 2m − as well, i.e. there is a function u = v + p satisfying (1)-(2) and Q < 0. Then we claim that sup R 2m u = ∞. Assume by contradiction that u is bounded from above. Then (15) and (16) Open Question 3 Even in the case that u is not bounded from above, is it true that one can take Z = ∅ in Theorem 2 for m ≥ 3 also?
For instance, in order to show that v(x) = 2α log |x| + o(log |x|) as |x| → +∞, thanks to (16), it is enough to show that B1(x) log |x − y|e 2mu(y) dy = o(log |x|), as |x| → +∞, which is true if sup R 2m u < ∞, but it might also be true if sup R 2m u = ∞.
Open Question 4 What values can the α given by (1)-(2) assume for a fixed Q?
As usual, it is enough to consider Q ∈ {0, ±(2m − 1)!}. When m = 1, Q = 1, then α = 1, see [CL]. When m = 2, Q = 6, then α can take any value in (0, 1], as shown in [CC]. Moreover α cannot be greater than 1 and the case α = 1 corresponds to standard solutions, as proved in [Lin]. For the trivial case Q = 0, α can take any positive value, and for the other cases we have no answer.
imply that v(x) = 2α log |x| + o(log |x|), as |x| → ∞.
Proof of Theorem 3. From Theorem 9, with v ≡ 0, we have that u is a polynomial of degree at most 2m − 2. Then, as in[Mar1, Lemma 11], we have4 The case Q = 0
sup
R 2m
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Struwe Q-curvature flow on S 4. A Malchiodi, M , J. Diff. Geom. 73A. Malchiodi, M. Struwe Q-curvature flow on S 4 , J. Diff. Geom. 73 (2006), 1-44.
Classifications of solutions to the higher order Liouville's equation in R 2m. L Martinazzi, preprintL. Martinazzi Classifications of solutions to the higher order Liou- ville's equation in R 2m , preprint (2007).
Concentration-compactness phenomena in higher order Liouville's equation. L Martinazzi, in preparationL. Martinazzi Concentration-compactness phenomena in higher order Liouville's equation, in preparation.
Ndiaye Constant Q-curvature metrics in arbitrary dimension. C B , J. Func. Analysis. 2511C. B. Ndiaye Constant Q-curvature metrics in arbitrary dimension, J. Func. Analysis 251 no.1 (2007), 1-58.
Struwe Asymptotic profile for a fourth order PDE with critical exponential growth in dimension four. F Robert, M , Adv. Nonlin. Stud. 4F. Robert, M. Struwe Asymptotic profile for a fourth order PDE with critical exponential growth in dimension four, Adv. Nonlin. Stud. 4 (2004), 397-415.
A flow approach to Nirenberg's problem. M Struwe, Duke Math. J. 1281M. Struwe A flow approach to Nirenberg's problem, Duke Math. J. 128(1) (2005), 19-64.
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| []
|
[
"Chebyshev polynomial coefficient bounds for a subclass of bi-univalent functions",
"Chebyshev polynomial coefficient bounds for a subclass of bi-univalent functions"
]
| [
"Şahsene Altınkaya \nDepartment of Mathematics\nFaculty of Arts and Science\nUludag University\nBursaTurkey\n",
"Sibel Yalçın \nDepartment of Mathematics\nFaculty of Arts and Science\nUludag University\nBursaTurkey\n"
]
| [
"Department of Mathematics\nFaculty of Arts and Science\nUludag University\nBursaTurkey",
"Department of Mathematics\nFaculty of Arts and Science\nUludag University\nBursaTurkey"
]
| []
| In this work, considering a general subclass of bi-univalent functions and using the Chebyshev polynomials, we obtain coefficient expansions for functions in this class. | null | [
"https://arxiv.org/pdf/1605.08224v2.pdf"
]
| 119,174,637 | 1605.08224 | 1009e644a660b1f6b67907c59543a81a3f405cc7 |
Chebyshev polynomial coefficient bounds for a subclass of bi-univalent functions
9 Feb 2017 February 10, 2017
Şahsene Altınkaya
Department of Mathematics
Faculty of Arts and Science
Uludag University
BursaTurkey
Sibel Yalçın
Department of Mathematics
Faculty of Arts and Science
Uludag University
BursaTurkey
Chebyshev polynomial coefficient bounds for a subclass of bi-univalent functions
9 Feb 2017 February 10, 2017Chebyshev polynomialsbi-univalent functionscoefficient boundssubordination Mathematics Subject Classification, 2010: 30C45, 30C50
In this work, considering a general subclass of bi-univalent functions and using the Chebyshev polynomials, we obtain coefficient expansions for functions in this class.
Introduction and Definitions
Let D be the unit disk {z : z ∈ C and |z| < 1} , A be the class of functions analytic in D, satisfying the conditions f (0) = 0 and f ′ (0) = 1.
Then each function f in A has the Taylor expansion f (z) = z + ∞ n=2 a n z n .
(1)
Further, by S we shall denote the class of all functions in A which are univalent in D.
The Koebe one-quarter theorem [5] ensures that the image of D under every function f from S contains a disk of radius 1 4 . Thus every such univalent function has an inverse f −1 which satisfies
f −1 (f (z)) = z , (z ∈ D) and f f −1 (w) = w , |w| < r 0 (f ) , r 0 (f ) ≥ 1 4 , where f −1 (w) = w − a 2 w 2 + 2a 2 2 − a 3 w 3 − 5a 3 2 − 5a 2 a 3 + a 4 w 4 + · · · .
A function f ∈ A is said to be bi-univalent in D if both f and f −1 are univalent in D. Let Σ represent the class of bi-univalent functions in D given by (1). For some intriguing examples of functions and characterization of the class Σ, one could refer Srivastava et al. [14], and the references stated therein (see also, [12]). Recently there has been triggering , interest to study the bi-univalent functions class Σ (see [1], [3], [9], [10], [13]) and obtain non-sharp estimates on the first two Taylor-Maclaurin coefficients |a 2 | and |a 3 |. Not much is known about the bounds on the general coefficient |a n | for n ≥ 4. In the literature, there are only a few works determining the general coefficient bounds |a n | for the analytic bi-univalent functions ( [2], [7], [8]). The coefficient estimate problem for each of |a n | ( n ∈ N\ {1, 2} ; N = {1, 2, 3, ...}) is still an open problem.
Chebyshev polynomials have become increasingly important in numerical analysis, from both theoretical and practical points of view. There are four kinds of Chebyshev polynomials. The majority of books and research papers dealing with specific orthogonal polynomials of Chebyshev family, contain mainly results of Chebyshev polynomials of first and second kinds T n (t) and U n (t) and their numerous uses in different applications, see for example, Doha [6] and Mason [11].
The Chebyshev polynomials of the first and second kinds are well known. In the case of a real variable x on (−1, 1), they are defined by T n (t) = cos nθ,
U n (t) = sin(n + 1)θ sin θ ,
where the subscript n denotes the polynomial degree and where t = cos θ.
If the functions f and g are analytic in D, then f is said to be subordinate to g, written as
f (z) ≺ g (z) , (z ∈ D)
if there exists a Schwarz function w (z) , analytic in D, with
w (0) = 0 and |w (z)| < 1 (z ∈ D) such that f (z) = g (w (z)) (z ∈ D)
.
Definition 1 A function f ∈ Σ is said to be in the class H Σ (λ, t) , λ ≥ 0 and t ∈ √ 2 2 , 1 , if the following subordination hold (1 − λ) zf ′ (z) f (z) + λ 1 + zf ′′ (z) f ′ (z) ≺ H(z, t) = 1 1 − 2tz + z 2 (z ∈ D) (2)
and
(1 − λ) wg ′ (w) g (w) + λ 1 + wg ′′ (w) g ′ (w) ≺ H(w, t) = 1 1 − 2tw + w 2 (w ∈ D) (3) where g (w) = f −1 (w) . We note that if t = cos α, α ∈ − π 3 , π 3 , then H(z, t) = 1 1 − 2tz + z 2 = 1 + ∞ n=1 sin(n + 1)α sin α z n (z ∈ D).
Thus H(z, t) = 1 + 2 cos αz + (3 cos 2 α − sin 2 α)z 2 + · · · (z ∈ D).
Following see, we write N) are the Chebyshev polynomials of the second kind. Also it is known that
H(z, t) = 1 + U 1 (t)z + U 2 (t)z 2 + · · · (z ∈ D, t ∈ (−1, 1)), where U n−1 = sin(n arccos t) √ 1 − t 2 (n ∈U n (t) = 2tU n−1 (t) − U n−2 (t),
and U 1 (t) = 2t,
U 2 (t) = 4t 2 − 1, U 3 (t) = 8t 3 − 4t, . . .(4)
The Chebyshev polynomials T n (t), t ∈ [−1, 1], of the first kind have the generating function of the form
∞ n=0 T n (t)z n = 1 − tz 1 − 2tz + z 2 (z ∈ D).
However, the Chebyshev polynomials of the first kind T n (t) and the second kind U n (t) are well connected by the following relationships
dT n (t) dt = nU n−1 (t), T n (t) = U n (t) − tU n−1 (t),
2T n (t) = U n (t) − U n−2 (t).
In this paper, motivated by the earlier work of Dziok et al. [4], we use the Chebyshev polynomial expansions to provide estimates for the initial coefficients of bi-univalent functions in H Σ (λ, t). We also solve Fekete-Szegö problem for functions in this class.
2 Coefficient bounds for the function class H Σ (λ, t) Theorem 2 Let the function f (z) given by (1) be in the class H Σ (λ, t) . Then
|a 2 | ≤ 2t √ 2t (1 + λ) 2 − 4 λ + λ 2 t 2 and |a 3 | ≤ 4t 2 (1 + λ) 2 + t 1 + 2λ .
Proof. Let f ∈ H Σ (λ, t) . From (2) and (3), we have
(1 − λ) zf ′ (z) f (z) + λ 1 + zf ′′ (z) f ′ (z) = 1 + U 1 (t)w(z) + U 2 (t)w 2 (z) + · · · ,(5)
and
(1 − λ) wg ′ (w) g (w) + λ 1 + wg ′′ (w) g ′ (w) = 1 + U 1 (t)v(w) + U 2 (t)v 2 (w) + · · · ,(6)
for some analytic functions w, v such that w(0) = v(0) = 0 and |w(z)| < 1, |v(w)| < 1 for all z ∈ D. From the equalities (5) and (6), we obtain that
(1 − λ) zf ′ (z) f (z) + λ 1 + zf ′′ (z) f ′ (z) = 1 + U 1 (t)c 1 z + U 1 (t)c 2 + U 2 (t)c 2 1 z 2 + · · · ,(7)
and
(1 − λ) wg ′ (w) g (w) +λ 1 + wg ′′ (w) g ′ (w) = 1+U 1 (t)d 1 w+ U 1 (t)d 2 + U 2 (t)d 2 1 w 2 +· · · .(8)
It is fairly well-known that if |w(z)| = c 1 z + c 2 z 2 + c 3 z 3 + · · · < 1 and |v(w)| = d 1 w + d 2 w 2 + d 3 w 3 + · · · < 1, z, w ∈ D, then
|c j | ≤ 1, ∀j ∈ N.
It follows from (7) and (8) that
(1 + λ) a 2 = U 1 (t)c 1 ,(9)2 (1 + 2λ) a 3 − (1 + 3λ) a 2 2 = U 1 (t)c 2 + U 2 (t)c 2 1 ,(10)
and
− (1 + λ) a 2 = U 1 (t)d 1 ,(11)2 (1 + 2λ) 2a 2 2 − a 3 − (1 + 3λ) a 2 2 = U 1 (t)d 2 + U 2 (t)d 2 1 .(12)
From (9) and (11) we obtain
c 1 = −d 1(13)
and
2 (1 + λ) 2 a 2 2 = U 2 1 (t) c 2 1 + d 2 1 .(14)
By adding (10) to (12), we get
[4 (1 + 2λ) − 2 (1 + 3λ)] a 2 2 = U 1 (t) (c 2 + d 2 ) + U 2 (t) c 2 1 + d 2 1 .(15)
By using (14) in equality (15), we have
2 (1 + λ) − 2U 2 (t) U 2 1 (t) (1 + λ) 2 a 2 2 = U 1 (t) (c 2 + d 2 ) .(16)
From (4) and (16) we get
|a 2 | ≤ 2t √ 2t (1 + λ) 2 − 4 λ + λ 2 t 2 .
Next, in order to find the bound on |a 3 | , by subtracting (12) from (10), we obtain
4 (1 + 2λ) a 3 − 4 (1 + 2λ) a 2 2 = U 1 (t) (c 2 − d 2 ) + U 2 (t) c 2 1 − d 2 1 .(17)
Then, in view of (13) and (14) , we have from (17)
a 3 = U 2 1 (t) 2 (1 + λ) 2 c 2 1 + d 2 1 + U 1 (t) 4 (1 + 2λ) (c 2 − d 2 ) .
Notice that (4), we get
|a 3 | ≤ 4t 2 (1 + λ) 2 + t 1 + 2λ .
3 Fekete-Szegö inequalities for the function class H Σ (λ, t)
Theorem 3 Let f given by (1) be in the class H Σ (λ, t) and µ ∈ R. Then
a 3 − µa 2 2 ≤ t 1 + 2λ ; f or |µ − 1| ≤ 1 8(1+2λ) (1+λ) 2 t 2 − 4λ(1 + λ) 8 |1 − µ| t 3 4(1 + λ)t 2 − (1 + λ) 2 (4t 2 − 1) ; f or |µ − 1| ≥ 1 8(1+2λ) (1+λ) 2 t 2 − 4λ(1 + λ)
.
Proof. From (16) and (17)
a 3 − µa 2 2 = (1 − µ) U 3 1 (t) (c 2 + d 2 ) 2 (1 + λ) U 2 1 (t) − 2U 2 (t) (1 + λ) 2 + U 1 (t) 4(1 + 2λ) (c 2 − d 2 ) = U 1 (t) h (µ) + 1 4(1+2λ) c 2 + h (µ) − 1 4(1+2λ) d 2 where h (µ) = U 2 1 (t) (1 − µ) 2 (1 + λ) U 2 1 (t) − U 2 (t) (1 + λ) 2 .
Then, in view of (4), we conclude that
a 3 − µa 2 2 ≤ t 1 + 2λ
0 ≤ |h (µ)| ≤ 1 4 (1 + 2λ) 4t |h (µ)| |h (µ)| ≥ 1 4 (1 + 2λ) .
Taking µ = 1 we get
Corollary 4 If f ∈ H Σ (λ, t) , then a 3 − a 2 2 ≤ t 1 + 2λ .
Initial coefficient bounds for a general class of bi-univalent functions. Ş Altınkaya, S Yalçın, Int. J. Anal. 867871ppArticle IDŞ. Altınkaya and S. Yalçın, Initial coefficient bounds for a general class of bi-univalent functions, Int. J. Anal., Article ID 867871, (2014), 4 pp.
Coefficient bounds for a subclass of bi-univalent functions. Ş Altınkaya, S Yalçın, TWMS J. Pure Appl. Math. 62Ş. Altınkaya and S. Yalçın, Coefficient bounds for a subclass of bi-univalent functions, TWMS J. Pure Appl. Math., 6 no. 2 (2015), 180-185.
Coefficient estimates certain subclasses of bi-univalent functions. O Crişan, Gen. Math. Notes. 162O. Crişan, Coefficient estimates certain subclasses of bi-univalent functions, Gen. Math. Notes, 16 no. 2 (2013), 93-1002.
Application of Chebyshev polynomials to classes of analytic functions. J Dziok, R K Raina, J Sokol, C. R. Acad. Sci. Paris, Ser. I. 353J. Dziok, R. K. Raina and J. Sokol, Application of Chebyshev polynomials to classes of analytic functions, C. R. Acad. Sci. Paris, Ser. I, 353 (2015), 433-438.
P L Duren, Grundlehren der Mathematischen Wissenschaften. New York, USASpringerUnivalent FunctionsP. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wis- senschaften, Springer, New York, USA, 1983.
The first and second kind Chebyshev coefficients of the moments of the general-order derivative of an infinitely differentiable function, Intern. E H Doha, J. Comput. Math. 51E. H. Doha, The first and second kind Chebyshev coefficients of the mo- ments of the general-order derivative of an infinitely differentiable function, Intern. J. Comput. Math., 51 (1994), 21-35.
Faber polynomial coefficient estimates for analytic bi-close-to-convex functions. S G Hamidi, J M Jahangiri, C. R. Acad. Sci. Paris, Ser.I. 3521S. G. Hamidi and J. M. Jahangiri, Faber polynomial coefficient estimates for analytic bi-close-to-convex functions, C. R. Acad. Sci. Paris, Ser.I, 352 no. 1 (2014), 17-20.
Coefficient estimates for certain classes of bi-univalent functions. J M Jahangiri, S G Hamidi, Int. J. Math. Math. Sci. 4ppJ. M. Jahangiri and S. G. Hamidi, Coefficient estimates for certain classes of bi-univalent functions, Int. J. Math. Math. Sci., ArticleID 190560, (2013), 4 pp.
Coefficient inequality for certain new subclasses of analytic bi-univalent functions. B S Keerthi, B Raja, Theoretical Mathematics and Applications. 1B. S. Keerthi and B. Raja, Coefficient inequality for certain new subclasses of analytic bi-univalent functions, Theoretical Mathematics and Applica- tions, 3 no. 1 (2013), 1-10.
Coefficient bounds for a certain subclass of biunivalent functions. N Magesh, J Yamini, Int. Math. Forum. 827N. Magesh and J. Yamini, Coefficient bounds for a certain subclass of bi- univalent functions, Int. Math. Forum, 8 no. 27 (2013), 1337-1344.
Chebyshev polynomials approximations for the L-membrane eigenvalue problem. J C Mason, SIAM J. Appl. Math. 15J. C. Mason, Chebyshev polynomials approximations for the L-membrane eigenvalue problem, SIAM J. Appl. Math., 15 (1967), 172-186.
Coefficient bounds for bi-univalent functions. T Hayami, S Owa, Pan Amer. Math. J. 224T. Hayami and S. Owa, Coefficient bounds for bi-univalent functions, Pan Amer. Math. J., 22 no. 4 (2012), 15-26.
On a new subclass of bi-univalent functions. S Porwal, M Darus, J. Egypt. Math. Soc. 213S. Porwal and M. Darus, On a new subclass of bi-univalent functions, J. Egypt. Math. Soc., 21 no. 3 (2013), 190-193.
Certain subclasses of analytic and bi-univalent functions. H M Srivastava, A K Mishra, P Gochhayat, Appl. Math. Lett. 23H. M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23 (2010), 1188- 1192.
| []
|
[
"Realizing spin-dependent gauge field with biaxial metamaterials",
"Realizing spin-dependent gauge field with biaxial metamaterials"
]
| [
"Fu Liu \nSchool of Physics and Astronomy\nUniversity of Birmingham\nB15 2TTBirminghamUnited Kingdom\n",
"Tao Xu \nCollege of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology\nSoochow University\n215006SuzhouChina\n",
"Saisai Wang \nCollege of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology\nSoochow University\n215006SuzhouChina\n",
"ZhiHong Hang \nCollege of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology\nSoochow University\n215006SuzhouChina\n",
"Jensen Li \nSchool of Physics and Astronomy\nUniversity of Birmingham\nB15 2TTBirminghamUnited Kingdom\n"
]
| [
"School of Physics and Astronomy\nUniversity of Birmingham\nB15 2TTBirminghamUnited Kingdom",
"College of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology\nSoochow University\n215006SuzhouChina",
"College of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology\nSoochow University\n215006SuzhouChina",
"College of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology\nSoochow University\n215006SuzhouChina",
"School of Physics and Astronomy\nUniversity of Birmingham\nB15 2TTBirminghamUnited Kingdom"
]
| []
| + These authors contributed equally to this workArtificial magnetic field in electromagnetism is becoming an emerging way as a robust control of light based on its geometric and topological nature. Other than demonstrating topological photonics properties in the diffractive regime using photonic crystals or arrays of waveguides, it will be of great interest if similar manipulations can be done simply in the long wavelength limit, in which only a few optical parameters can be used to describe the system, making the future optical component design much easier. Here, by designing and fabricating a metamaterial with split dispersion surface, we provide a straight-forward experimental realization of spin-dependent gauge field in the real space using a biaxial material. A "magnetic force bending" for light of desired pseudospins is visualized experimentally by such a gauge field as a manifestation of optical spin Hall effect. Such a demonstration is potentially useful to develop pseudospin optics, topological components and spinenabled transformation optical devices. | 10.1002/adom.201801582 | [
"https://arxiv.org/pdf/1803.04594v1.pdf"
]
| 119,429,467 | 1803.04594 | cf11ba3340c1b650db5109da0cd0019a7154454f |
Realizing spin-dependent gauge field with biaxial metamaterials
Fu Liu
School of Physics and Astronomy
University of Birmingham
B15 2TTBirminghamUnited Kingdom
Tao Xu
College of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology
Soochow University
215006SuzhouChina
Saisai Wang
College of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology
Soochow University
215006SuzhouChina
ZhiHong Hang
College of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology
Soochow University
215006SuzhouChina
Jensen Li
School of Physics and Astronomy
University of Birmingham
B15 2TTBirminghamUnited Kingdom
Realizing spin-dependent gauge field with biaxial metamaterials
1
+ These authors contributed equally to this workArtificial magnetic field in electromagnetism is becoming an emerging way as a robust control of light based on its geometric and topological nature. Other than demonstrating topological photonics properties in the diffractive regime using photonic crystals or arrays of waveguides, it will be of great interest if similar manipulations can be done simply in the long wavelength limit, in which only a few optical parameters can be used to describe the system, making the future optical component design much easier. Here, by designing and fabricating a metamaterial with split dispersion surface, we provide a straight-forward experimental realization of spin-dependent gauge field in the real space using a biaxial material. A "magnetic force bending" for light of desired pseudospins is visualized experimentally by such a gauge field as a manifestation of optical spin Hall effect. Such a demonstration is potentially useful to develop pseudospin optics, topological components and spinenabled transformation optical devices.
The concept of a photonic gauge field and its associated artificial magnetic field is an emerging way to manipulate light by exploring the geometrical or topological properties of light [1][2][3][4][5][6][7][8][9][10][11]. More robust operations or more intuitive design principles can thus be triggered comparing to conventional approaches. Currently, it has become central to various optical devices from geometric-phase metasurfaces for beam structuring , when the concept is applied to the polarization space, to topological photonics for generating oneway edge states, when the concept is applied to the reciprocal space [12,13]. For the case of a photonic gauge field in the real space, it can be regarded as the analogy of a vector potential acting on electron motion, whose quantum-mechanical effect is historically considered as the Aharonov-Bohm effect.
While a light is bent conventionally through a spatially varying profile of dispersion surfaces with varying sizes or shapes, a photonic gauge field in the real space essentially bends light through the shifting of the local dispersion surfaces [14][15][16][17]. It enables us to demonstrate various peculiar wave phenomena such as negative refraction, gauge-field enabled waveguiding, one-way edge state and asymmetric wave transport [1][2][3][4][5][6][7][8][9][10][11]. Realspace photonic gauge field realizations are proposed, while most of them are in a network of resonators or waveguides with direction-dependent coupling, achievable by different optical paths [1][2][3], dynamic modulation [4][5][6][7][8][9], magneto-optical effect [10], or an inhomogeneously strained structural profile [11]. If a real-space photonic gauge field can be "materialized" to work in the effective medium regime and a few effective medium parameters are enough to capture its physical description, we shall be able to apply the corresponding design principle to a simpler model to promote the development of more compact optical components. In fact, the materialization approach has been applied to design topological metamaterials, which is relevant in considering gauge field and artificial magnetic field in the reciprocal space [18][19][20]. On the other hand, a tilted anisotropic medium, by considering a field transformation approach [21][22][23], is theoretically suggested to generate a real-space photonic gauge field, but the stringent requirement of matched magnetic and electric response complicates experimental realization.
Here, we experimentally demonstrate a real space photonic gauge field using a biaxial medium with only electric response. We simply refer it as a gauge-field medium. In the current scheme, as long as the different permittivities along the three principal axes satisfy a geometric-mean relationship, a gauge field exists for waves traveling on a particular chosen working plane. This gauge-field pushes the originally degenerated circular dispersion surfaces on the working plane towards opposite directions, depending on which decoupled polarization, or called pseudospin, is employed. Pseudospin-dependent beam splitting and propagation near the associated singular (diabolical) points are demonstrated.
Our work provides a feasible route to incorporate gauge field using metamaterials.
Furthermore, there is actually no fundamental limitation to push the current working principle to the optical regime as the required mean relationship is discussed. Such a biaxial medium approach may also allow future generalization to more complicated gauge-field media, for example, to realize a non-Abelian gauge field using biaxial media [24].
We start from the gauge field medium, which is represented by permittivity and permeability tensors:
̿ = [ 0 0 0 0 − − ] , ̿ = [ 0 0 − 0 0 − ],(1)
where 0 and are the common principal values and the U(1) gauge field =̂+ is the tilted anisotropy terms with the same magnitude but opposite signs in the two tensors. It is defined for wave propagation on the − plane, satisfying a wave equation using and as basis [12]. The eigenmodes of the medium are governed by the secular equation
[ 2 − 0 2 ( 0 − 2 ) −2 0 ⋅ 2 0 ⋅ 2 − 0 2 ( 0 − 2 ) ] [ ] = 0,
where =̂+̂ is the wavevector, 0 is the wavenumber and = | |. Two shifted circular dispersions are attained. Each of them corresponds to a constant and decoupled eigen-polarization with either = , defined as pseudo spin-up + , or = − , defined as pseudo spin-down − . The size of the shifting of the dispersion surfaces in the reciprocal space is given by ∓ 0 for the two pseudo-spins (Fig. 1).
The requirement of responses in matched values and the additional tilted anisotropy terms in both tensors are relaxed in two steps. The so-called reduced parameter approximation technique is often employed for metamaterials requiring both electric and magnetic responses [25], which is valid when impedance mismatch is negligible or when the material profile is slowly varying in the scale of wavelength. First, by exploiting the fact that the information of the in-plane fields is lost in establishing the secular equation, one can prove that the following medium
̿ = [ 0 0 2 0 0 −2 2 −2 + 3 2 / 0 ] , ̿ = [ 0 0 0 0 0 0 0 0 − 2 / 0 ],(2)
gives exactly the same secular matrix, hence the same dispersion surfaces (shape and size) and constant polarizations for the two modes. It allows us to lump the tilted anisotropy terms from ̿ to ̿ only. Second, we confine to a particular case = 0 + 2 / 0 , making ̿ to be isotropic. A conventional reduced parameter approximation is further applied by dividing ̿ and multiplying ̿ by 0 at the same time. The final medium becomes
̿ = [ 0 2 0 2 0 0 0 2 −2 0 2 0 −2 0 0 2 + 4 2 ],(3)
with ̿ being the identity matrix (i.e. simply the free-space). We recognize that the resultant medium in Eq. (3) is simply a biaxial dielectric medium.
If we write =̂ where ̂ is a unit vector lying on the -plane, a clockwise rotation of the medium by an angle = acot( / 0 ) /2 about ̂ can diagonalize ̿ to its principal values = 0 2 , = 0 2 tan 2 and = 0 2 cot 2 along ̂, ̂×̂ and ̂ respectively. In other words, suppose we have a biaxial dielectric medium ̿ =̂̂+̂̂+̂̂ satisfying a geometric-mean relationship on its permittivities along the three orthogonal directions:
2 = .(4)
We reach a simple recipe to generate a gauge field
= ±̂(√ − √ )/2 (5)
in such a medium, for waves propagating on the -plane, which is now defined as the plane containing ̂ and ̂cos −̂sin . The "±" sign corresponds to the two choices of with the same magnitude but opposite signs.
The biaxial medium satisfying Eq. (4) has the same dispersion surfaces of the ideal gauge field medium and the decoupled pseudospins are only subject to a redefinition to 0 = ± . Resultant from the criteria, the polarization remains constant and ensures that when the wave changes direction in the medium, the two pseudospins stay decoupled with each other.
We adopt a biaxial metamaterial for realization, with a unit cell shown in Fig. 1(a).
Printed circuit boards (PCBs) with a square array of metallic fractal structures are periodically stacked along the direction to construct a three-dimensional (3D) metamaterial. We assume > > . Figure 1(b) shows a typical dispersion surface of such a biaxial metamaterial in the 3D reciprocal space. For clarity, the cross-section on the − plane is shown with thick black lines. For the polarization ( , , ), a circular dispersion surface of radius 0 = √ is expected due to the lack of magnetic response.
There is also an ellipse with semi-minor / major axis of index 1 = √ / 2 = √ for the polarization ( , , ) . The two dispersion surfaces intersect with each other at 4 diabolical points. A plane containing the -axis can then be drawn to pass through a pair of these diabolical points, defined as the ′ − , as shown in Fig. 1(b). The ′ − plane is our working plane for in-plane wave propagation. The dispersion surfaces on the working plane are zoomed in as solid red curves with the principal indices indicated in Fig. 1(c). They look like two split circles, suggesting a gauge field can be defined to indicate the size of splitting, which is the same as in Eq. (5), now with = , = , = and ̂=̂. The "±" sign corresponds to the two choices of the plane to pass through the diabolical points. In the current biaxial metamaterial with arbitrary principal values, the working plane has an orientation (rotated by from the z-axis) governed by
tan 2 = − − ,(6)
where the geometric-mean criteria with = cot 2 = tan 2 is covered but not limited to. It indicates that the gauge field medium derived in previous section is not a singular case but rather a case that can be approached smoothly using the above consideration of biaxial medium. This adds tolerance in favor to experimental realization.
A small deviation from the geometric-mean criteria only leads to a less circular dispersion surfaces and the a less independent polarization on the reciprocal space. Therefore, for a biaxial medium where the difference between the principal permittivities are not too big, we are expecting a broad working regime for the gauge field, as long as one of the principal permittivities is situated roughly in the middle between the other two principal values. In such a case, is around ±45°. Based on this analysis, we move on to our fabricated biaxial metamaterial working in the microwave regime, as shown in Fig. 2(a). The inset shows one unit cell with periodicity Fig. 2(b)) due to a much smaller thickness of PCB than . On the other hand, > 1 > is achieved by inducing electric resonances of the fractal structure. Scattering S-parameters are measured for microwaves, with polarization in either x or z direction, impinging the sample at normal incidence. / is then retrieved from the S-parameters
and is plotted as solid black / red symbols in Fig. 2(b), with excellent agreement with corresponding retrieved results from full-wave simulations plotted in solid lines. Due to a resonating mode below 10 GHz, is between 0 and 1 from 11.5 to 14 GHz. For , as its resonance is much higher than 14GHz, its value stays around 2 from 10 to 14 GHz. The frequency dispersion embedded in our fabricated biaxial metamaterial actually helps to scan and pick the working frequency to satisfy the mean criteria. The working frequency with material parameters matching Eq. (4) is found at around 12.3 GHz, indicated by a vertical gray line in Fig. 2(b), where = 1.965 and = 0.504.
A key feature of a gauge field is its ability to shift dispersion surfaces. Here, we measure the dispersion surfaces on the ′ − plane. To have generality among various frequencies for comparison, the plane is set at = 45° as an approximation discussed above. The sample is now rotated by 45° in the x-y plane. The S-parameters at different incident angles are then measured. The wavevector stays on the ′ − plane, equivalently the ′ − plane in the laboratory frame. Since the near-field coupling between neighboring layers is negligible for our structure, only one layer of the fractal structure is employed to avoid the necessity of index branch selection from S-parameter retrieval. The results are shown in Fig. 2(c)-(e), for three slightly different frequencies 12.3, 12.8, and 13.5 GHz in order to follow gap closing and opening. The dispersion surfaces obtained from the retrieved effective material parameters at normal incidence ( Fig. 2(b)) are plotted in red color where a good agreement can be found to experimental results at various incident angles indicated by symbols. From the results, two shifted circles with diabolical points at = 0 are obtained at 12.8 GHz, which is slightly shifted from the predicted 12.3 GHz due to the deviation of from its ideal value. It can also be understood by plugging 45° into Eq. (6), which suggests the geometric-mean criteria should be modified to a harmonic-mean version: 1/ = (1/ + 1/ )/2 . As long as the differences in the principal permittivities are small, the various ways to take mean value do not differ significantly from each other and we expect the gauge field should still work with physics driven by split circular dispersions and decoupled pseudo-spin propagations. Beam splitting occurs for two pseudospins (black and blue) and the abrupt artificial magnetic field will induce a L beam shifting of both spins. c, simulated ′ and ′ field distribution with ′ Gaussian beam incidence from left. d, The measured ′ and ′ field profile 30 mm away from the exit surface. L=17.5 mm beam shifting is found to be of best fit to analytically obtained profiles. material profile, it bends photon using an artificial magnetic field, which relates to the gauge field by = ∇ × . When we have a step change of gauge field in this work, such a "magnetic force" bending can be understood as a surface effect. Similar interpretation applies to a pseudo-electric field from gradient index ~∇ . If the refractive index is gradually changing, there is a layer with bulk pseudo-electric field, an incident beam will change direction by the pseudo-electric field, with a lateral shift across the interface. For a step change of index, all these effects are lumped to the surface, there is no lateral shift but a change of direction across the step, related to change of index Δ . In the current case, a step change of gauge field changes the incident beam's direction abruptly on the interface without lateral shift. On the other hand, at the exit interface that the step change is reversed, the artificial magnetic field also reverses sign to bend light back to its original direction, now with a lateral beam shifting in position due to the change of propagation direction within the medium.
The experimental setup to observe beam shifting can be found in Fig. 3(a). Sixty-one layers of structured PCBs with periodicity 5 mm along direction is used to construct the bulk sample. The fractal structures are now rotated by 45° to make the working plane being ′ − and foam (in white, with measured relative permittivity 1.002) is used to fill the gaps between the layers. An empirical Gaussian-like microwave beam impinges on the sample and the transmitted beam profile can be measured by a probe antenna mounted on a translational stage. We first consider linear polarized incidence at 12.8GHz with electric field polarized along ′ axis, which is a superposition of the two pseudospins + (with ′ = ′ ) and − (with ′ = − ′ ). Please be reminded that 0 = √~1 . As illustrated in Fig. 3(b), after ′ polarized beam (in red) impinges on the sample (in blue), beam shifting occurs as + ( − ) will bend upward (downward) in the y direction. Numerical simulation results with ′ incidence from left can be found in Fig. 3(c), with a Gaussian beam width of 120 mm to reflect the experimental setup. The position of the bulk gauge field medium is between the two solid lines whose material parameters = 1.965, = 1.016, = 0.504 as experimentally retrieved at 12.3 GHz. ′ field component emerges after propagating through the bulk sample and the exit ′ beam width is slightly widened.
Moreover, it can be recognized that the upper exit ′ beam has a π phase difference to the lower one. The exit beam profile is a superposition of spatially separated pseudospins: + and − . ′ ( ′ ) field will be added up in (out of) phase. A different Δ will induce a different beam splitting L and a different exit beam profile of ′ / ′ component will be induced. In experiment, we measured a line field profile along y direction 30 mm away from the sample exit surface and the experimental results are shown in circles in Fig. 3(d).
The ′ beam profile without sample at the same position can also be obtained for reference.
As discussed previously, the original ′ incidence will split into two pseudospins while its beam profile remains. Thus we have the knowledge of beam profiles of the pseudospin beams and their ′ and ′ components: identical to that of ′ beam without sample. As illustrated in the right panel of Fig. 3(b), we can construct the new analytical exit beam profiles at different beam splitting L. The best match between the analytical and the experimentally measured beam profiles is with L=17.5 mm, where least square method is applied whose results are shown as squares in Fig. 4(d). In the present work, we have experimentally demonstrated how a spin-dependent 2D gauge field in the real-space can be realized using a metamaterial with biaxial permittivity.
Benefitting from the established reduced material parameters approximation, in this work, we relax the requirement of both electric and magnetic responses of an ideal gauge field medium. Only tilted anisotropy from a biaxial permittivity tensor is required, making the realization of a gauge field medium straight-forward. A biaxial metamaterial with fractal geometry in PCB working in the microwave frequency regime is then designed to demonstrate the gauge field. The dispersion surfaces on the working plane are two shifted circles induced by gauge field whose eigen-polarizations are nearly constant as pseudospins, which are defined here as a linear combination of electric and magnetic fields.
The beam shifting from a surface pseudo-magnetic field constructed by the designed metamaterial is verified in experiments. We hope that by proposing a simple scheme to achieve gauge field media using biaxial materials, we pave the road to pseudospin optics and future topology related optical components.
Fig. 1
1Obtaining gauge field from biaxial dielectric medium. a, Cubic lattice with unit cell design containing a two-level fractal structure. Principal axes are along the Cartesian axes. b, 3D dispersion surface. Only three-quarters of it is shown with cutting plane on the − plane and the ′ − plane. The direction of ′ -axis is defined to pass through the degeneracy point between the circle of index 0 = √ and the ellipse with principal axes indices 1 = √ , and 2 = √ on the − plane. c, Dispersion surface on the ′ − plane ( ′ = 0). These are two shifted ellipses with opposite shifting by a gauge field of =̂.
Fig. 2
2Spin-split circular dispersion surfaces. a, Fabricated metamaterial to realize the gauge field. Copper fractal structure printed on PCB with geometry parameters 1 = 2.3mm 2 = 2.4 mm 3 = 1.8 mm 4 = 1.2 mm, and copper width = 0.2 mm. Copper layer has a thickness of 0.033 mm. The unit cell has a periodicity = 5 mm in all directions. b, Effective medium and retrieved from simulation (solid lines) and experiment (symbols). The dashed line is analytically calculated by averaging the permittivity indirection. The vertical gray line indicates the working frequency 12.4 GHz at which the permittivity satisfies Eq. (4). c-e, Two dimensional dispersion surfaces at 12.3, 12.8, and 13.5 GHz respectively on the ′ − plane. Symbols are from experiment measurement while the red curves are the analytic dispersion from the effective medium. At 12.8 GHz, the dispersion surfaces for the two polarizations are degenerated at a point on the ′ axis while gaps are opened at the other two frequencies.
= 5
5(in all three directions) and the geometric parameters with values specified in figure caption. Copper fractal structures are printed on FR4 PCB with thickness 0.115 and relative permittivity 3.3. The permittivity is very close to one (1.016 after averaging, dashed line in
Fig 3
3Experiment Beam shift measurement. a, experimental setup. b, Schematic with linear ′ polarization incidence.
Fig. 4
4Frequency spectrum on beam shift with pseudo spin-up incidence. a-b, show the measured and theoretical spectrum for co-spin ( + ) detection. c-d， show the measured and theoretical spectrum for the cross-polarization ( − ) detection. Now we pursue with dispersive effects. A pseudospin source can be constructed by rotating the horn antenna by 45° from ′ . + has its electric field along the ̂′ +̂ direction while − has its electric field along the ̂′ −̂ direction. With + incidence, we measure both the + and − field profiles along direction for different frequencies.The power spectra with normalization to the maximal value of total power (| + | 2 + | − | 2 ) for each frequency are shown in Figs. 4(a) and (c). For comparison, Figs. 4(b) and (d) shows the counterpart transmission spectra obtained from simulations and similar behavior in the spectrum is observed. The working frequency is found at about 12.6GHz in experiment (12.8GHz in simulation) where the transmitted beam has smallest − conversion, as indicated by the horizontal dashed lines. At the working frequency, the transmitted | + | 2 beam shift about 18 mm (17 mm) in the positive direction from experimental (simulation) results compare to the beam center = 0 without sample, as shown by the vertical dashed lines. At another frequency 12.1GHz in experiment (12.2GHz in simulation), the incident + beam is largely converted into its cross-polarization − , where the dispersion surfaces are way far away from two circles with two constant decoupled polarizations From the results, we observe that the working frequencies with minimal cross-polarization conversion (and with expected beam shift induced by the gage field) is from around 12.4 to 12.8 GHz, roughly corresponding to the two frequencies satisfying the geometric-mean and harmonic-mean criteria.
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| []
|
[
"Universality of tunnelling particles in Hawking radiation",
"Universality of tunnelling particles in Hawking radiation"
]
| [
"Harold Erbin *[email protected]†[email protected] \nArnold Sommerfeld Center for Theoretical Physics\nLudwig-Maximilians-Universität München\nTheresienstraße 3780333MünchenGermany\n",
"Vincent Lahoche \nLabri\nUniversité de Bordeaux\nUmr 580033405TalenceFrance\n"
]
| [
"Arnold Sommerfeld Center for Theoretical Physics\nLudwig-Maximilians-Universität München\nTheresienstraße 3780333MünchenGermany",
"Labri\nUniversité de Bordeaux\nUmr 580033405TalenceFrance"
]
| []
| The complex path (or Hamilton-Jacobi) approach to Hawking radiation corresponds to the intuitive picture of particles tunnelling through the horizon and forming a thermal radiation. This method computes the tunnelling rate of a given particle from its equation of motion and equates it to the Boltzmann distribution of the radiation from which the Hawking temperature is identified. In agreement with the original derivation by Hawking and the other approaches, it has been checked case by case that the temperature is indeed universal for a number of backgrounds and the tunnelling of particles from spin 0 to 1 (and in some cases with spin 3/2 and 2). In this letter we give a general proof that the temperature is indeed equal for all (massless and massive) particles with spin from 0 to 2 on an arbitrary background (limited to be Einstein for spin greater than 1) in any number of dimensions. Moreover we propose a general argument to extend this result to any spin greater than 2. | 10.1103/physrevd.98.104001 | [
"https://arxiv.org/pdf/1708.00661v2.pdf"
]
| 118,987,420 | 1708.00661 | b8d321238e89b1679734aeb16163b1ff010d573f |
Universality of tunnelling particles in Hawking radiation
3rd August 2017 2 Aug 2017
Harold Erbin *[email protected]†[email protected]
Arnold Sommerfeld Center for Theoretical Physics
Ludwig-Maximilians-Universität München
Theresienstraße 3780333MünchenGermany
Vincent Lahoche
Labri
Université de Bordeaux
Umr 580033405TalenceFrance
Universality of tunnelling particles in Hawking radiation
3rd August 2017 2 Aug 20171
The complex path (or Hamilton-Jacobi) approach to Hawking radiation corresponds to the intuitive picture of particles tunnelling through the horizon and forming a thermal radiation. This method computes the tunnelling rate of a given particle from its equation of motion and equates it to the Boltzmann distribution of the radiation from which the Hawking temperature is identified. In agreement with the original derivation by Hawking and the other approaches, it has been checked case by case that the temperature is indeed universal for a number of backgrounds and the tunnelling of particles from spin 0 to 1 (and in some cases with spin 3/2 and 2). In this letter we give a general proof that the temperature is indeed equal for all (massless and massive) particles with spin from 0 to 2 on an arbitrary background (limited to be Einstein for spin greater than 1) in any number of dimensions. Moreover we propose a general argument to extend this result to any spin greater than 2.
Introduction
In his seminal paper [1] Hawking proved that black holes emit a thermal radiation at a temperature T due to quantum mechanical effects. The intuitive picture of this radiation is the following: pairs of virtual particles created near a black hole horizon through vacuum fluctuations become real once one of them cross the horizon while the other extracts energy from the black hole. This idea has lead to two different approaches of the Hawking radiation: the complex path (or Hamilton-Jacobi) method due to Shankaranarayanan-Srinivasan-Padmanabhan [2][3][4] (see also [5]), and the null geodesic method (or Parikh-Wilczek) method [6] (see [7] for a review). Both methods are not restricted to black hole radiation but can also be applied for any black hole with a thermal horizon for other background which can be thermal, such as the Rindler or de Sitter spaces. Moreover they can also be used to define the Hawking temperature in situations where the traditional methods are not defined [7].
The first method computes the tunnelling rate of a particle of a given spin s by solving its equations of motion in the black hole background through a WKB approximation, and then equates this rates to the probability given by the Boltzmann distribution at temperature T . The main drawback of this method is that the computations depend on the spin and the intermediate steps do not show any reason for the temperature to be universal. Nonetheless it has been checked explicitly case by case for many backgrounds that the tunnelling of particles with different spins (mostly s = 0, 1/2, 1, but also s = 3/2, 2 in some instances) always yields the same temperature, see [8][9][10][11][12][13][14][15][16] for a selected sample and references therein for more details.
The goal of this letter is to prove that the Hawking temperature is universal for the tunnelling of neutral massless and massive particles with spin ranging from 0 to 2 for a generic background in any dimension (restricted to be Einstein for s = 3/2, 2). This is achieved by showing that, in the WKB approximation, the equation of motion for a spin s ≤ 2 particle reduces to the Hamilton-Jacobi equation of a scalar field. We then give a general argument to extend this result to massive particle of any spin s > 2. Moreover we stress that our proof is fully covariant, in contrast with the former computations which were not explicitly covariant since the fields and the background metric were written in components.
The limitation on the background for spins s > 1 and the need of non-minimal coupling are related to the well-known difficulty propagating higher-spin particles on a curved background [17][18][19]. An interesting question would be to analyse the subleading quantum corrections and the deviation from thermality due to the backreaction of the radiation and to see whether they differ for the different types of particles. The generalization of our argument to background with gauge fields under which the particles are charged is expected to be straightforward, even if one can expect difficulties already for s = 1 due to inconsistencies in the coupling of spin s ≥ 1 to electromagnetic fields [20,21].
In section 2 we review the complex path method for a scalar field and we show in section 3 how the higher-spin cases reduce to this case.
Complex path method for the scalar field
One considers a background (case of interests being black holes, the Rindler space, etc.) in d dimensions described by a fixed metric g µν . For definiteness the background metric is taken to be a solution of the Einstein equations with a cosmological constant
R µν − 1 2 g µν R + Λg µν = 0 ,(1)
where Λ is the cosmological constant, R µν = R ρ ρµ ν is the Ricci tensor obtained by contracting the Riemann tensor, and R = g µν R µν is the Ricci scalar. As we will see, this restriction to Einstein spacetimes for deriving the Hawking temperature only concerns spins higher than 1. The Laplacian on this background is defined by
∆ = g µν ∇ µ ∇ ν (2)
where ∇ µ is the covariant derivative with the Levi-Civita connection. The tunnelling rate Γ for a particle is given by
Γ = P out P in = |φ out | 2 |φ in | 2 (3)
where P in (P out ) is the tunnelling probability for an ingoing (outgoing) particle and φ in (φ out ) is the associated solution to the equation of motion. Assuming that the radiation is thermal 1 this rate can be equated to the Boltzmann distribution through the detailed balance
Γ = e − E tot T (4)
where E tot is the total energy (including kinetic, rotational, electromagnetic, etc.) carried by the particle tunnelling, and measured by a freely falling observer in the vicinity of the external horizon. From this point we consider a free (massive or massless) spin 0 scalar field φ. The equation of motion for a scalar field in a curved background with non-minimal coupling
−∆ + m 2 2 + ξR φ = 0 (5)
where m 2 can be zero. This equation can be solved at leading order in through the WKB approximation
φ = φ 0 e iS/ ,(6)
where φ 0 is a constant wave function. Inserting this ansatz into (5) provides, at leading order in , the Hamilton-Jacobi equation on curved space
g µν ∂ µ S∂ ν S + m 2 = 0 ,(7)
and allows to identify S with the classical action and one can note that the non-minimal coupling term is subleading (such terms are also present for higher spins and will not contribute at leading order). It can be solved quite generically with the following ansatz [5,10]
S out = −Et + W (r 0 ) + F (x i ) + K, S in = −Et − W (r 0 ) + F (x i ) + K ,(8)
where t is the time, r 0 the radial location of the horizon and x i denotes any other coordinate, and K is a complex constant. One needs to ensure that the ingoing probability is one in the classical limit because the horizon necessarily absorbs the particle. This manifests differently depending on the choice of coordinates. 2 It may occur that the inverse of the radial velocity has no pole for an ingoing classical particle, implying that this imaginary part vanishes. If this is not the case then one needs to find the relation between the ingoing and outgoing actions such that this condition holds. Both situations amount to setting Im K = Im W (r 0 ) and one finally obtains the tunnelling rate
Γ = e −Etot/T = e −2(Im Sout−Im Sin)/ = e −4 Im W (r0)/ ,(9)
which yields the temperature:
T = E tot 4 Im W (r 0 ) .(10)
In order to make contact with the well-known formula of the Hawking radiation, one can show for general rotating black holes (including the Schwarzschild back hole as a limiting case) that the expression for W (r 0 ) is proportional to the surface gravity κ:
Im W (r 0 ) = πE tot 2κ ,(11)
and the final result agrees with the well know Hawking temperature formula [1] T = κ 2π
.
The reason is that W (r 0 ) is defined by an integral over r with a pole at the horizon due to the presence of the metric components in the denominator: evaluating the integral yields a residue (imaginary) proportional to the surface gravity. In the case of Schwarzschild one finds
f (r) = 1 − 2M r , r 0 = 2M =⇒ κ = f (r 0 ) 2 = 1 4M , T = 8πM .(13)
Tunnelling of higher-spins
In this section -which contains our new results -we will extend the results of the previous section and show that the equations of motion for higher-spin particles reduce to the Hamilton-Jacobi equation (7) of a scalar field in the leading order of the WKB approximation. This is sufficient to establish that the temperature will be given by (10) and thus identical for all spins. 3
Spin 1/2 The equation of motion for a spin 1/2 fermion ψ is
/ ∇ − m ψ = 0(14)
where / ∇ ≡ γ µ ∇ µ and γ µ are the Dirac matrices. The multiplication of (14) with / ∇ gives the second order partial derivative equation:
− ∆ψ + 1 4 Rψ + m 2 2 ψ = 0.(15)
As for the scalar field, the WKB approximation for this equation can be investigated using the following ansatz
ψ = ψ 0 e iS/ ,(16)
where ψ 0 is constant. Putting this ansatz in (15), we deduce an equation for S, and keeping only the leading order terms in , it reduces to the the scalar Hamilton-Jacobi equation (7).
Spin 1 The equation of motion for a massive vector field
A µ can be derived from the standard Proca Lagrangian, and may be written in the form
0 = −∆A µ + ∇ ν ∇ µ A ν + m 2 2 A ν . (17a)
Up to straightforward manipulations, this equation is equivalent to
− ∆A µ + R µν A ν + m 2 2 A µ = 0(18)
together with the constraint ∇ µ A µ = 0 (19) which can be imposed at the dynamical level as a consequence of the equation of motion for m 2 = 0, or through a gauge transformation
δA µ = ∇ µ α,(20)
for vanishing mass, the scalar field α being the gauge parameter. Remember that this constraint is necessary for ensuring that the correct degrees of freedom propagate (the spin 1) while the extraneous ones are removed (the spin 0 part). In the leading order of the WKB approximation
A µ = A 0µ e iS/(21)
the equation (18) corresponds to the scalar Hamilton-Jacobi equation (7).
Note that for the two previous cases it was not necessary to use the fact that the background metric is a solution of the Einstein equation (1). Hence the universality of Hawking temperature for spin s = 0, 1/2, 1 is valid for any background, irrespective of the theory of gravity or the matter content under consideration, with the exception of gauge couplings. 4
Spin 3/2 The massive Rarita-Schwinger field is described by a (bi-)spinor-valued vector field ψ µ whose equation of motion is:
γ µνρ ∇ ν ψ ρ − m γ µν ψ ν = 0.(22)
Some lengthy but simple manipulations [26] show that ψ µ obeys the Dirac equation
/ ∇ − m ψ µ = 0 ,(23)
together with the condition / ψ = 0 .
Note that these conditions result from the equation of motion (22) if
m 2 = 0, m 2 0 , m 2 0 ≡ d − 2 2(d − 1) 2 Λ ,(25)
or from the gauge invariance under the following transformation otherwise:
δψ µ = ∇ µ − m 0 (d − 2) γ µ ,(26)
where is a spinor-valued gauge parameter. As discussed for the spin 1, the constraint ensures that only the spin 3/2 part of the field propagates. However it can be imposed only if the background satisfies the Einstein equation (1) for the spin 3/2. Finally the equation (23) can be multiplied with / ∇ which leads to
− ∆ψ ρ + γ µν R σ µν ρ ψ σ + R 4 ψ ρ + m 2 2 ψ ρ = 0 ,(27)
and inserting the WKB ansatz ψ µ = ψ 0µ e iS/ (28) inside the equation (27) brings it to the form of (7) at the leading order in .
Spin 2 The massive spin 2 field is usually described by a symmetric tensor of rank 2, h µν , whose equation of motion may be written as [27] −
∆h µν + g µν ∆h − ∇ µ ∇ ν h − g µν ∇ ρ ∇ σ h ρσ + ∇ ρ ∇ µ h νρ + ∇ ρ ∇ ν h µρ − 2ξ d R h µν − 1 − 2ξ d R h g µν + m 2 2 h µν − hg µν = 0 (29)
where ξ is an arbitrary parameter parametrizing the non-minimal coupling (the latter is necessary in order to get the correct constraints on the propagating degrees of freedom below). Then the equation (29) can be simplified to
− ∆h µν − 2R ρ σ µ ν h ρσ − 2(ξ − 1) d Rh µν + m 2 2 h µν = 0(30)
together with the constraints h = 0,
∇ µ h µν = 0(31)
if the background satisfies the Einstein equation (1). In the case where the condition
m 2 = m 2 0 ≡ − 4 2 (1 − ξ) d − 2 Λ(32)
holds then the constraints (31) result from the equation of motion (29) [27]. Otherwise if m 2 = m 2 0 then they can be imposed through a gauge transformation
δh µν = ∇ µ ζ ν + ∇ ν ζ µ .(33)
Note that this include the case of the graviton propagating on a curved space which corresponds to m 2 = 0 and ξ = 1 [28]. 5 In the WKB approximation
h µν = h 0µν e iS/(34)
the equation (29) is again equivalent to (7).
Higher spins More generally one can consider a massive particle of arbitrary integer spin s > 2 (the case of half-integer is a straightforward extension) represented by a field φ µ1···µs symmetric in all indices for which the equation of motion is
− ∆φ µ1···µs + f (R) ν1···νs µ1···µs φ ν1···νs + m 2 2 φ µ1···µs = 0 (35) after elimination of the auxiliary fields and imposing the constraints [29][30][31] ∇ µ φ µµ2···µs = 0, g µν φ µνµ3···µs = 0,
where f (R) is a function of the Riemann tensor and its contractions, arising both from anticommutation of covariant derivatives and from non-minimal coupling terms (which ensures causality and unitarity [32]). Introducing the WKB ansatz φ µ1···µs = φ 0,µ1···µs e iS/ (37) yields the Hamilton-Jacobi equation (7). The reason is that curvature terms cannot have factors of because they do not contain derivatives as the Laplacian or built-in factors as the mass term. Any other term would be eliminated by the constraints (which are necessary for the theory to exist and be consistent).
This hypothesis is not strictly correct due to backreaction of the radiation on the geometry[6], but we will ignore this effect for our purpose.
This point involves different subtleties and making a precise statement is very coordinate-dependent. We refer the reader to the literature for more details[5,[22][23][24][25].3 For this it is important that the evaluation of the action in (10) depends only on the properties of the background and not on the type of the particle.
Indeed the coupling to the gauge field in the covariant derivative comes with a factor −1 . On the other coupling to other scalar and fermions fields can come only with positive power of , implying that these terms do not contribute at the leading order of the WKB approximation.
To our knowledge the gauge transformation (33) has not been discussed elsewhere for generic ξ.
AcknowledgmentsWe are grateful to Eric Huguet and Vincent Rivasseau for reading the draft of the manuscript. H.E. acknowledges support from the Cnrs and the Lptens during the realization of this project. The work of H.E. is conducted under a Carl Friedrich von Siemens Research Fellowship of the Alexander von Humboldt Foundation for postdoctoral researchers.
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| []
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[
"Heegaard Floer Homology and Triple Cup Products",
"Heegaard Floer Homology and Triple Cup Products"
]
| [
"Tye Lidman "
]
| []
| []
| We give a complete calculation of HF ∞ (Y, s) with mod 2 coefficients for all three-manifolds Y and torsion Spin c structures s. The computation agrees with the conjectured calculation of Ozsváth and Szabó in [10]. This therefore establishes an isomorphism with Mark's cup homology, HC ∞ (Y ), mod 2 [7].Review of the Link Surgery FormulaWe assume familiarity with Heegaard Floer homology of three-manifolds and links, as in[12],[13]. We now give a brief overview of the link surgery formula of Manolescu and Ozsváth[6]. | null | [
"https://arxiv.org/pdf/1011.4277v1.pdf"
]
| 118,987,634 | 1011.4277 | c49323172a486323f9c64815a1eec2763efe5f2e |
Heegaard Floer Homology and Triple Cup Products
18 Nov 2010
Tye Lidman
Heegaard Floer Homology and Triple Cup Products
18 Nov 2010arXiv:1011.4277v1 [math.GT]
We give a complete calculation of HF ∞ (Y, s) with mod 2 coefficients for all three-manifolds Y and torsion Spin c structures s. The computation agrees with the conjectured calculation of Ozsváth and Szabó in [10]. This therefore establishes an isomorphism with Mark's cup homology, HC ∞ (Y ), mod 2 [7].Review of the Link Surgery FormulaWe assume familiarity with Heegaard Floer homology of three-manifolds and links, as in[12],[13]. We now give a brief overview of the link surgery formula of Manolescu and Ozsváth[6].
Introduction
Throughout the last decade, Heegaard Floer homology has been a ubiquitous tool in low-dimensional topology. Variants of this theory exist for nearly any object one can find a use for and have been studied in a wide range of contexts, such as embeddings of symplectic surfaces in fourmanifolds, lens space surgery obstructions, and the classification theory of tight contact structures. In many cases, these invariants can be calculated completely combinatorially, making them accessible and desirable to utilize.
For closed three-manifolds, the Heegaard Floer chain complexes come in many flavors, CF , CF + , CF − , and CF ∞ . These flavors are all in fact derived from CF ∞ by some formal construction on the chain level. Therefore, having an understanding of the homology, HF ∞ , provides foundational information for the other flavors; for example, the Z-rank of HF (Y ) is always bounded below by the Z[U, U −1 ]-rank of HF ∞ (Y ).
In [11], Ozsváth and Szabó calculate HF ∞ (Y, s) for all Y with b1(Y ) ≤ 2 and all Spin c structures s. It is also shown that there is a universal coefficients spectral sequence for torsion Spin c structures, with E3 term given by Λ * (H 1 (Y ; Z)) ⊗ Z[U, U −1 ], which converges to HF ∞ . They conjecture in [10] that the d3 differential is given by contraction via the integral triple cup product form, µY , and that all higher differentials vanish. This agrees with our previous calculations of HF ∞ extended to b1(Y ) ≤ 4 for coefficients in F = Z/2Z [5]. These computations immediately extend to the case of HF ∞ , where HF ∞ is defined to be HF ∞ with mod 2 coefficients, completed with respect to the variable U . Throughout this paper, all Floer homologies will be calculated with mod 2 coefficients, unless noted otherwise.
In [7], Mark constructs a complex over Z[U, U −1 ], C ∞ * (Y ), with chain group Λ * (H 1 (Y ; Z))⊗Z[U, U −1 ]; in fact, C ∞ * (Y ) is exactly the conjectured complex (E3, d3) as mentioned above. For compatibility, we will work with F coefficients for C ∞ * as well, where triple cup products are taken in integral cohomology and then reduced mod 2.
In this paper, we use the link surgery formula of Manolescu and Ozsváth (Theorem 1.1 of [6]) to calculate HF ∞ (Y, s) for all Y and torsion s and compare the result with HC ∞ . Theorem 1.1. Let Y be a three-manifold equipped with a torsion Spin c structure s. The relatively-graded F[U, U −1 ]-modules HF ∞ * (Y, s) and HC ∞ * (Y ) are isomorphic. Therefore, HF ∞ (Y, s) agrees with Conjecture 4.10 of [10] mod 2.
Remark 1.2. It is also known in monopole Floer homology that for torsion Spin c structures, HC ∞ (Y ) ∼ = HM(Y, s) over Q (see Section IX of [3]); furthermore the announced Main Theorem of Kutluhan, Lee, and Taubes [4] shows HM (Y, s, Z) = HF ∞ (Y, s, Z). Thus, Theorem 1.1 is already known with Q-coefficients. To illustrate the importance of coefficients, we point out that the three-manifolds M2n−1 and M2n defined in Example 3.3 of [2] have isomorphic HC ∞ (·, Q) for n ≥ 1, but different HC ∞ (·, F) for their unique torsion Spin c structures [5]. In fact, this observation combined with the fact that HC ∞ (M2n−1, F) and HC ∞ (M2n−1, Q) have equal rank proves that there is always 2-torsion in HF ∞ (M2n, Z) when n ≥ 1.
In Section 2 of [6] it is shown that HF ∞ vanishes for non-torsion Spin c structures. Since the link surgery formula is proved for HF ∞ instead of HF ∞ , the methods of this paper cannot be used to calculate HF ∞ for non-torsion Spin c structures. However, it is pointed out in the same section that for torsion Spin c structures, HF ∞ is completely determined by HF ∞ ; in other words, HF ∞ i ∼ = HF ∞ i . Therefore, for the purposes of this paper, we are content to work with HF ∞ when studying torsion s. From now on, the coefficients of C ∞ will also be F[[U, U −1 ].
Instead of working with the universal coefficients spectral sequence, a different approach is taken to calculate HF ∞ . We will express HF ∞ (Y, s0) as the homology of a hypercube of chain complexes by [6]; this will be the structure on which we construct a different spectral sequence, this time coming from a filtration more reminiscent of the quantum grading in Khovanov homology.
Given a framed link (L, Λ) in S 3 , Manolescu and Ozsváth construct the link surgery formula: an infinite product of hypercubes of chain complexes over F[[U, U −1 ] whose total complex has homology isomorphic to HF ∞ (S 3 Λ (L)) [6]. This complex is built up from a generalization of the mapping cone construction for integral surgeries of [14], but applied for each sublink of L.
By the work of Section 2 in [5], we only need to calculate HF ∞ in the case that Y is 0-surgery on certain links with all pairwise linking zero; we call such a link homologically split. For 0-surgery on a homologically split link with ℓ components, the associated hypercubes of chain complexes for a fixed Spin c structure naturally take the shape of a single hypercube, {0, 1} ℓ , in the sense that there is only a single Heegaard Floer complex associated to each vertex; therefore, the complex is finite dimensional over F[[U, U −1 ]. We place a filtration on the complex based entirely on the location in the cube. This will induce the spectral sequence we would like to study.
For the unique torsion Spin c structure on such a Y , denote the complex for 0-surgery on L coming from the surgery formula by C ∞ (L, Λ0, 0) (see Section 7 of [6]), where Λ0 represents the 0-framing. We establish the following, analogous to Conjecture 4.10 of [10]. Theorem 1.3. Consider a homologically split link L ⊂ S 3 . There is a filtration on C ∞ (L, Λ0, 0) such that the induced spectral sequence has the following properties. The first two pages have vanishing differentials
and E3 ∼ = Λ * (H 1 (Y ; Z)) ⊗ F[[U, U −1 ]. Furthermore, via this identification d3 : Λ i (H 1 (Y ; Z) ⊗ F) ⊗ U j → Λ i−3 (H 1 (Y ; Z) ⊗ F) ⊗ U j−1 is given by φ k 1 ∧ . . . ∧ φ k i → ιµ Y (φ k 1 ∧ . . . ∧ φ k i ),(1)
where µY is the integral triple cup product form,
µY (φ k 1 ∧ φ k 2 ∧ φ k 3 ) = φ k 1 ⌣ φ k 2 ⌣ φ k 3 , [Y ]
, thought of as a 3-form on H 1 (Y ; Z). Finally, the higher differentials vanish.
The reader familiar with cup homology will immediately recognize the complex (E3, d3). Definition 1.4. (Definition 8 of [7]) The relatively-graded chain complex C ∞ * (S 3 0 (L)) is exactly (E3, d3) under the identifications of Theorem 1.3. The first three differentials in Theorem 1.3 will be calculated by applying our understanding of b1(Y ) ≤ 3 from Theorem 10.1 of [11] and Theorem 1.3 of [5]. It will be easy to show that this gives upper bounds on the rank of HF ∞ for arbitrary Y . The rest of the paper is then devoted to proving inductively that the higher differentials in the above spectral sequence vanish. The idea is to choose a component K in L and replace it by K1#K2, where 0-surgery on each (L − K) ∪ Ki yields a simpler cup product structure. We will study the link surgery formula for L as a combination of the complexes with each Ki in place of K. This will enable us to show that the higher differentials vanish based on knowledge of their vanishing for the simpler links. From this, we can easily deduce Theorem 1.1 for all three-manifolds. Remark 1.5. It seems likely that if the link surgery formula could be proven over Z, then the arguments of this paper could be used to prove Theorem 1.1 for Z-coefficients as well.
Their machine takes as input a framed link (L, Λ) in S 3 and outputs a special type of chain complex with homology isomorphic to HF ∞ (YΛ(L)). Recall that HF ∞ comes from the complex
CF ∞ = CF ∞ ⊗ F[[U, U −1 ].
While we only work with HF ∞ in this paper, their surgery formula is done for all flavors of Heegaard Floer homology.
In order to explain the link surgery formula, there is a very large amount of notation and formalism required simply to state the theorem. Therefore, we will first give a complete description in the case that L is a knot to give a more concrete set-up. Then we will give the general framework, but with slightly less details. Everything here will be presented for HF ∞ , but the same framework applies to the other flavors as well. Finally, for notation, we use x ∨ y to denote max{x, y}.
Surgery Formula for Knots
We begin with an oriented knot, K ⊂ S 3 . We will restate (without proof) a well-known formula for HF ∞ (S 3 n (K)) (compare with Theorem 1.1 of [14]). Begin with a doubly-pointed Heegaard diagram for K, H K = (Σ, α, β, z, w), and a Heegaard diagram for S 3 , H ∅ = (Σ, α ′ , β ′ , w ′ ), so they have the same underlying surface. Let's suppose for simplicity that after removing the basepoint z, now a Heegaard diagram for S 3 , that we can relate H K to H ∅ by a sequence of handleslides and isotopies (this means the isotopies of curves do not cross the basepoint and there are no (de)stabilizations). Construct a finite sequence of Heegaard diagrams, H K,+K , which begins at H K with z removed and follows the sequence of isotopies and handleslides, terminating at H ∅ . We define H K,−K analogously for the removal of w.
First, define H(K) = Z and H(K) = H(K) ∪ {−∞, +∞}; also, let H(∅) = 0 and H(∅) = 0 ∪ {−∞, +∞}. Note that there are two sublinks of K, namely K and ∅. If we use K or +K, we will mean that it has the induced orientation; −K will refer to the reversed orientation. Fix s ∈ H(K) and an oriented sublink, M ⊂ K. Define
p M (s) = +∞ if M = +K, −∞ if M = −K, s if M = ∅.
Similarly, define ψ M (s) = +∞ if M = K and set ψ ∅ (s) = s. We want to construct two complexes for each s ∈ H(K), namely one for H ∅ and one for H K . The first complex is given by
A ∞ (H ∅ , p K (s)) = A ∞ (H ∅ , +∞) = CF ∞ (H ∅ ).
The complex for H K will be more complicated and will actually depend on s. Recall that there is an absolute Alexander grading on Tα ∩ T β coming from H K which satisfies a homological symmetry about 0 and A(x) − A(y) = nz(φ) − nw(φ) for any φ ∈ π2(x, y). With this, we define the complex A ∞ (H K , s) to have the same chain groups as CF ∞ (H K ), the free module over F[[U, U −1 ] generated by Tα ∩ T β . It has the differential
∂(x) = D ∅ (x) = y∈Tα∩T β φ∈π 2 (x,y),µ(φ)=1 #(M(φ)/R) · U Es(φ) y, where Es(φ) = (A(x) − s) ∨ 0 − (A(y) − s) ∨ 0 + nw(φ).
Notice that as s becomes very positive (respectively negative), ∂ is only counting w (respectively z).
We would like a way to relate these complexes. Define the inclusions,
I ±K s : A ∞ (H K , s) → A ∞ (H K , p ±K (s)) by I ±K s (x) = U (±(A(x)−s))∨0 x.
This essentially corresponds to removing z or w from the Heegaard diagram, since s is sent to ±∞. We set I ∅ s to simply be the identity. Recall that we had sequences of Heegaard diagrams, H K,±K , relating H K with z or w removed to H ∅ . Each isotopy or handleslide induces a chain map between Floer complexes by counting triangles [12]. Composing these induced maps along the sequence results in the destabilization maps
D ±K p ±K (s) : A ∞ (H K , p ±K (s)) → A ∞ (H ∅ , ψ ±K (p ±K (s))). The composition D M p M (s) • I M s is denoted Φ M s , and thus Φ ∅ s (x) = ∂(x)
. Consider the following complex composed of all the smaller complexes we have built up:
C ∞ (H, n) = s∈H(K) (A ∞ (H K , s) ⊕ A ∞ (H ∅ , ψ K (s)))
with differential given by
D ∞ (s, x) = (s + n, Φ −K ψ M (s) (x)) + (s, Φ +K ψ M (s) (x)) + (s, Φ ∅ ψ M (s) (x)), for x ∈ A ∞ (H K−
Spin c Structures
We now generalize the construction above to arbitrary framed links. For simplicity, we will assume that Heegaard diagrams for links have exactly one z basepoint for each component, but may (and will) have additional w basepoints in the diagram not on any component of the link. Also, we will ignore all details about admissibility of the Heegaard diagrams as well (see Section 4 of [6]). The starting point will be an oriented link L in S 3 with components K1, . . . , Kn and a framing Λ telling us how to perform surgery on L. The framing Λ will be given as the linking matrix of the resulting 3manifold after surgery; diagonal entries are the surgery coefficients and the off-diagonal entries are the pairwise linking numbers of the components. Note that we may think of the row-vectors Λi as elements in H1(S 3 − L). When we are considering oriented sublinks, M will refer to an arbitrary orientation, while M with no vector decoration will indicate that M has the orientation induced from L.
As in other Heegaard Floer homologies, we want to see where the Spin c structures appear in our theory. It will be necessary to also relate the relative Spin c structures defined on S 3 − L to those on S 3 − M for sublinks M ⊂ L.
Define the affine space
H(L) = n i=1 H(L)i, where H(L)i = lk(Ki, L − Ki) 2 + Z.
It is not hard to see that as lattices Spin c (S 3 Λ (L)) ∼ = H(L)/Λ (where /Λ means quotienting out by the action of each row-vector of Λ, Λi, on the lattice); it turns out that such an identification can be made explicitly. Therefore, we will often refer to Spin c structures as equivalence classes With this we can define a new Heegaard Floer complex for each choice of s. Begin with a Heegaard diagram for L, H L . Recall that there is an Alexander grading for each component of L on Tα ∩ T β , again given by making absolute the relative grading Ai(x) − Ai(y) = nz i (φ) − nw i (φ) (we require the Alexander grading of link Floer homology to be symmetric about 0).
For each s0 ∈ H(L) and each M ⊂ L, we will define the complex A ∞ (H L−M , ψ M (s0)). For notation, set s = ψ M (s0). The chain groups will all be the same, freely generated over F[[U1, . . . , Un, U −1 1 , . . . , U −1 n ] by Tα ∩ T β ; here n is the number of w basepoints. The differential will be defined by D 0 = ∂ :
A ∞ (H L−M , ψ M (s0)) → A ∞ (H L−M , ψ M (s0))
, which is given by
∂(x) = y∈Tα∩T β φ∈π 2 (x,y),µ(φ)=1 #(M(φ)/R) · U E 1 s 1 (φ) 1 . . . U E n sn (φ) n y, where E i s i (φ) = (Ai(x) − si) ∨ 0 − (Ai(y) − s) ∨ 0 + nw i (φ).
If si is very positive (or negative), then these counts are again just nw i (φ) (or nz i (φ)); also, we must use the convention that ∞ − ∞ = 0 so this is consistent when si = −∞. Therefore, setting some si to +∞ is the same thing as forgetting the ith component of the link and having an additional basepoint wi.
Complete Systems of Hyperboxes
(C ε * , D 0 ε ) for ε ∈ E(d) equipped with additional operators D ε ′ ε : C ε * → C ε+ε ′ * + ε ′ −1 , for ε ′ = 0 in {0, 1} n ;
the operators are 0 if ε + ε ′ is no longer in the hyperbox. These operators are required to satisfy the following relation for all ε ′ ∈ {0, 1} n :
γ≤ε ′ D ε ′ −γ ε+γ • D γ ε = 0
The way to think of this is that the D ε ′ are chain maps when ε ′ = 1 and chain homotopies for ε ′ = 2. The higher maps are chain homotopies of chain homotopies, etc.
This can be made into a total complex as
(C * = ε C ε * + ε , D = ε,ε ′ D ε ε ′ )
We will omit the subscript notation from the D from now on, where it will just be assumed that the map is 0 if the relevant domains don't match up. Furthermore, ∂ will denote D 0 at any vertex of the hyperbox.
Let (Σ, α, z, w) be a Heegaard diagram for a handlebody, with basepoints z = {z1, . . . , zn} and w = {w1, . . . , wn} on Σ − α.
We will assume that all bipartition functions send everything to β, so we will not worry about defining α-hyperboxes or keeping track of bipartition functions. We will ultimately work with a basic system, so this assumption will not be a problem (see Section 6.7 of [6]).
Definition 2.4. An empty β-hyperbox of size d, H, is a collection of isotopic sets of β-curves on Σ−z−w, {β ε } ε∈E(d) . A filling of H is a choice of elements Θ ε,ε ′ ∈ A ∞ (T βε , T β ε ′ , 0)
for any neighbors ε < ε ′ . These are required to satisfy equation (50) in [6], namely summing over the polygon maps associated to each possible sequence Θε 1 ,ε 2 , Θε 2 ,ε 3 , . . . Θε l−1 ,ε l in the Heegaard multiple (Σ, α, β ε 1 , . . . , β ε l ), is identically 0. If ε − ε ′ = 1, Θ ε,ε ′ must also correspond to a cycle generating the top-dimensional homology group of A(T βε , T β ε ′ , 0) (where A is given by setting one Ui to 0). Therefore, for us a hyperbox of Heegaard diagrams for L is simply a set of α-curves and an empty β-hyperbox equipped with a choice of filling such that each (Σ, α, β ε , z, w) is a Heegaard diagram for L. Remark 2.5. Given a fixed s ∈ H(L), we can create a hyperbox of chain complexes from a hyperbox of Heegaard diagrams as follows: for each
ε ∈ E(d) we set (C ε(M ) s , D 0 ) to be A ∞ (Tα, T β ε(M ) , ψ M (s)). If ε ′ − ε = 1, then the chain map D ε ′ −ε ε
consists of counting triangles in the Heegaard triple (α, β ε , β ε ′ ) with fixed generator Θ ε,ε ′ . The higher homotopies are defined similarly; we sum up the corresponding holomorphic polygon counts over a specified sequence of the Θ elements in the Heegaard multiple (Σ, α, β ε , . . . , β ε ′ ).
It is a lemma of Manolescu and Ozsváth that any empty β-hyperbox admits a filling and thus every empty β-hyperbox can be made into a hyperbox of Heegaard diagrams.
Given an m-component sublink M ⊂ L ′ ⊂ L and a hyperbox of Heegaard diagrams H for L ′ , we construct a new hyperbox r M (H). This is defined as follows. Remove the zi on components of I+( L, M ) from each Heegaard diagram in H; remove the wi that correspond to components of I−( L, M ) and relabel them as zi. Note that this is now a hyperbox for L ′ − M . Let's study some special cases. If M = ∅, then a hyperbox for the pair ( L ′ , ∅) is a single Heegaard diagram, which we denote by H L ′ . If M is a single component K, then we have H L ′ ,±K is a one-dimensional hyperbox, or in other words, a finite sequence of Heegaard diagrams. For the integer surgeries formula, this related H K with z or w removed to H ∅ ; this is exactly the idea that we would like to keep in mind. This box is going to tell us how to define the maps analogous to D ±K .
Given a sublink,
M ′ ⊂ M , there is a hyperbox for ( L − M ′ , M − M ′ ) inside of the size d H L, M . This hyperbox, H L, M (M ′ , M )
, is given by the sub-hyperbox with specified corners d · ε(M ′ ) and d · ε(M ) (here we are doing componentwise multiplication).
For knots, we simply pointed out that for large |s|, A ∞ (H K , s) behaves as though there is either no z or no w basepoint and can be compared to A ∞ (H ∅ , ψ K (s)). We now state the analogous requirement for comparing hyperboxes with certain basepoints removed. Say that two hyperboxes are
compatible if H L, M (∅, M ′ ) ∼ = r M −M ′ (H L, M ′ ) for M ′ a sublink consistently oriented with M ⊂ L. Similarly, H L, M and H L−M ′ , M −M ′ are compatible if H L, M (M ′ , M ) ∼ = H L−M ′ , M −M ′ .
Here, the relation ' ∼ =' means that the hyperboxes of Heegaard diagrams are related by a single isotopy. In other words, there is a single isotopy of Σ not passing any curves over basepoints, independent of ε, which takes the Heegaard diagrams at vertex ε on one hyperbox to the Heegaard diagram at vertex ε on the other. Manolescu and Ozsváth construct complete systems of hyperboxes for any oriented link in S 3 . Remark 2.8. There is an additional technical condition that must be satisfied to be a complete system in the sense of Manolescu and Ozsváth (Definitions 6.25 and 6.26 in [6]): it essentially says that the paths traced out by the basepoints on the Heegaard surfaces while passing between the different isotopies of diagrams in the hyperboxes must be nullhomotopic. This will not be a problem with the special types of complete systems we will work with, so we do not mention this anymore.
The Surgery Complex
Given a complete system of hyperboxes of Heegaard diagrams for an ncomponent link L, H, we would like to turn
C ∞ (H, Λ) = s∈H(L) M ⊂L A ∞ (H L−M , ψ M (s))
into an n-dimensional hypercube of chain complexes analogous to the case for knots. We will set the chain complex at the vertex ε(M ) to be
C ε(M ) = s∈H(L) A ∞ (H L−M , ψ M (s))
with the differential given by the product of the component-wise differentials. While these chain complexes do not depend on Λ, the D ε that we will ultimately construct will depend heavily on this choice.
We now want to define the analogue for the maps Φ ±K relating the A ∞ complexes. We will construct a map from
A ∞ (H M , s) to A ∞ (H M ′ , ψ M −M ′ (s))
for each M ′ ⊂ M and s ∈ H(M ). The first step is to remove the z or w basepoints in H M that do not correspond to components of M ′ to get a Heegaard diagram for M ′ ; this corresponds to I in the integer surgery formula for knots.
We can define the general inclusions I M s :
A ∞ (H L ′ , s) → A ∞ (H L ′ , p M (s)) by I M s (x) = i∈I + ( L, M ) U (A i (x)−s i )∨0 i j∈I − ( L, M ) U (s j −A j (x))∨0 j x.
Note that this is only defined if si is not ±∞ when i ∈ I∓( L, M ); this issue will not arise when we define the total complex. We would now like to define the destabilizations,
D M p M (s) : A ∞ (H L ′ , p M (s)) → A ∞ (H L ′ −M , ψ M (p M(
s))) analogous to the case of D ±K . We first identify
r M (H L ′ ) with its corresponding vertex in H L ′ , M , H L ′ , M
(0,...,0) , by compatibility. This induces a map on A ∞ by counting triangles, but this must count basepoints with Es(φ) instead of nw(φ).
For simplicity, we first assume that H L ′ , M is in fact a hypercube. The idea is that each way of traversing the edges of the hypercube gives a sequence of isotopies and handleslides from H L ′ , M (0,...,0) to H L ′ , M (1,...,1) ; D M will measure the failure for the induced triangle maps to commute. Recall that the filling gives a map from A ∞ (Tα, T β (0,...,0) , 0) to A ∞ (Tα, T β (1,...,1) , 0) by counts of holomorphic polygons. We can twist this map by counting basepoints according to the E i s (φ) as opposed to the usual nw(φ). This in fact gives the desired map
A ∞ (H L ′ , M (0,...,0) , p M (s)) → A ∞ (H L ′ , M (1,...,1) , p M (s)). If H L ′ , M is not a hypercube,A ∞ (H L ′ ,±K i j , p ±K (s)) to A ∞ (H L ′ ,±K i j+1 , p ±K i (s)).
In this case we would simply take the mapD ±K i to be the composition of the di triangle-counting maps.
In fact, compression will produce a hypercube with vertices given by the corresponding complexes at the far corners of the hyperbox, namelỹ C ε for ε ∈ {0, 1} n will be given by C d·ε . Also, if ε − ε ′ = 1 and ε ≥ ε ′ in the compressed hypercube, then along an edge,D ε−ε ′ will simply be given by the composition of the edge maps from the original hyperbox. However, in general this does not work (a composition of chain homotopies is not a chain homotopy for the compositions). For illustration, we define the appropriately compressed map for a size (2, 1) hyperbox and refer the interested reader to Section 3 in [6] for the general case. Example 2.9. Consider a hyperbox of chain complexes, C, of size (2, 1). We can turn this into a hypercube of chain complexes,C as follows. Takẽ C ε 1 ,ε 2 = C 2ε 1 ,ε 2 and keepD 0 = D 0 . In other words, the complexes at the vertices are given by the corners of the hyperbox. The mapD 1,0 is given by D 1,0 • D 1,0 , whileD 0,1 = D 0,1 . So far we have not done anything different from above, butD 1,1 will have to be more complicated. A standard exercise in homological algebra shows that the correct choice
forD 1,1 is D 1,0 • D 1,1 + D 1,1 • D 1,0 .
In the Heegaard Floer setting, D 1,0 and D 0,1 are triangle-counting maps, while D 1,1 counts holomorphic rectangles.
Once the correct map from
A ∞ (H L ′ , M 0 , p M (s)) to A ∞ (H L ′ , M d·ε(M ) , p M (s)) is defined,= D M p M (s) • I M s . The differential D ∞ on C ∞ (H, Λ) is given by (s, x) → N⊂L−M (s + Λ L, N , Φ N ψ M (s) (x)).
Here, x ∈ A ∞ (H L−M , ψ M (s)) and Λ L, N = i∈I − ( L, N) Λi. Note that the sum is over all possible oriented sublinks of L − M . Manolescu and Ozsváth prove that this is indeed the total complex of a hypercube of chain complexes. It is important to note that by construction, if s ∈ [s0], then s + Λ L, N ∈ [s0] for any N . Therefore, C ∞ (H, Λ) splits as a sum of complexes corresponding to the Spin c struc-
tures, C ∞ (H, Λ, [s]).
We are finally ready to state the link surgery formula.
(L) corresponding to [s] ∈ H(L)/Λ, there is a relatively-Z/N Z-graded F[[U, U −1 ]-vector space isomorphism HF ∞ * (S 3 Λ (L), s) ∼ = H * (C ∞ (H, Λ, [s]), D ∞ ),
where N is the usual divisibility of the Spin c structure (see, for example, [12]).
Remark 2.11. While the A ∞ complexes are defined over a ring with many formal variables Ui, the theorem implies that they become equal in homology. Remark 2.12. In the case of a torsion Spin c structure, this is a relative Z-grading. We will use the fact that the differential lowers this relative grading by 1; namely we know the exact grading change of any map Φ M . This grading will be the key ingredient for establishing the identification with the complex for cup homology, which is why this argument only works for torsion Spin c structures.
3 Why is the surgery formula special for HF ∞ ?
Let's study some special properties of the link surgery formula which are unique to the infinity flavor. The reason why these properties will not hold for the other flavors is that the inclusion maps I will simply not be quasiisomorphisms, as multiplication by U is not an isomorphism for CF + or CF − . However, the inclusions are quasi-isomorphisms for HF ∞ and thus have the simplest behavior on this flavor; since the inclusions encode the information coming from the link we are performing surgery on (they describe the induced filtrations on CF(S 3 ) similar to CF L), CF ∞ will not retain much information about the choice of individual components. We now make this notion more precise. Fix a complete system H for the framed link (L, Λ). The following is essentially a combination of Proposition 4.2 in [5] and Lemma 7.9 in [6].
A ∞ (H M , s) → A ∞ (H M −K j , ψ ±K j (s))
are quasi-isomorphisms that lower the relative grading by 1.
For notation, C will represent the hypercube of chain complexes C ∞ (H, Λ), or possibly the subcomplex corresponding to a torsion Spin c structure when this will not cause confusion. This is naturally a chain complex, even if it is not a sub-or quotientcomplex. This is because any such face-module is the result of a sequence of subcomplexes of quotient-complexes of subcomplexes etc. Choose a component Kj that is not in LF such that εj = 0 in F . We can construct a new subface complex of the same dimension, Fj , given by changing εj to 1 for each ε and giving it the inherited chain complex structure (this is now a subcomplex).
Lemma 3.3. With the notation as above, CF and CF j are quasi-isomorphic.
Proof. We now study the induced map S +K j = M ⊂L F Φ +K j ∪+M from CF to the CF j , where Kj is not a component of LF (we must take the product over all s). This is a chain map by construction. Consider the filtration on the mapping cone of S +K j given by Fj (x) = − i =j εi. The only components that preserve the filtration level will be ∂ and Φ K j . Since the maps Φ K j are quasi-isomorphisms between each of the corresponding A ∞ complexes, the associated graded is acyclic. Therefore, the entire mapping cone is acyclic. Thus, the two face complexes are quasiisomorphic.
Remark 3.4. This does not imply that all associated face complexes of the same dimension in C ∞ (H, Λ) are quasi-isomorphic. This does, however, tell us how to relate face complexes to the complexes corresponding to surgery on certain sublinks. Proof. For each Kj in L−LF with εj = 0, simply apply S +K j to obtain the desired filtered quasi-isomorphisms to the complex with εj = 1. After exhausting all such Kj , the resulting subcomplex corresponds to S 3 Λ| L F (LF ) after an appropriate restriction of H (see Section 11 of [6]).
Remark 3.6. This lemma will be used repeatedly throughout this paper with the key observation that these quasi-isomorphisms of the faces preserve the filtration F (up to some absolute shift which we will ignore).
The First Two Differentials
In this section, we set the framework to prove Theorem 1.3 by inducing the correct filtration and dispensing with the first two differentials in the spectral sequence.
Consider an arbitrary ℓ-dimensional hypercube of chain complexes, C. We study the filtration FC on the total complex given by FC(x) = ℓ − ε for any x in C ε . This induces a spectral sequence with E1 term the homology of the total complex with respect to only D 0 = ∂, which converges to the homology of the total complex. Now, let H be a complete system of Heegaard diagrams for L = K1 ∪ K2 ∪ . . . ∪ K ℓ , a homologically split link of ℓ components. We will denote by Y the manifold obtained by 0-surgery on each component of L. The framing is Λ0, which is simply the 0-matrix. We can then induce the filtration FC on C ∞ (H, Λ0, 0), the subcomplex corresponding to the unique torsion Spin c structure; we will refer to this as the ε-filtration. Note that this Spin c structure is represented by a single element in H(L). Remark 4.1. Because of this, the complex C ∞ (H, Λ0, 0) has only one A ∞ complex at each vertex of {0, 1} ℓ ; in fact we can see the E1 term must have rank 2 ℓ , by Remark 3.2. Furthermore, we will suppress the orientations of Y and L, as this can be seen to not affect any of the calculations.
We will use heavily the notion of the depth of a filtration; this is the maximal difference in the filtration levels between any two elements. It is clear that all differentials, d k , in the induced spectral sequence from a filtration vanish for k greater than the depth. For the ε-filtration coming from an ℓ-component link, the depth is simply ℓ.
Proposition 4.2. The first two differentials, d1 and d2, in the spectral sequence vanish.
A key ingredient in the proof will be the following theorem due to Ozsváth and Szabó. Note that this can also be quickly proven using the results from Section 3. Proof of Proposition 4.2. We prove this by induction on ℓ. C will now refer to C ∞ (H, Λ0, 0). Since in this subcomplex there can only be one possible value of ψ M (s) (modulo ∞'s) for each sublink, namely 0 ∈ H(L − M ), we will omit this from the notation.
Let us now show that d1 ≡ 0. As a warm-up, if ℓ = 0, then
HF ∞ (Y ) ∼ = F[[U, U −1 ]. Since E1 ∼ = E∞ ∼ = H * (C 0 , ∂) ∼ = F[[U, U −1 ], we must have that d1 = 0. Now, for ℓ = 1, we see that HF ∞ (Y ) ∼ = F[[U, U −1 ] ⊕ F[[U, U −1 ] by Theorem 4.2. Again, E1 ∼ = H * (C 0 , ∂) ⊕ H * (C 1 , ∂) ∼ = F[[U, U −1 ] ⊕ F[[U, U −1 ],
so d1 = 0. Suppose that d1 vanishes for any link with ℓ components. Let L have (ℓ+1)-components and have d i 1 represent the component of the differential d1 which maps from C ε to C ε+τ i , where τi = (0, . . . , 0, 1, 0, . . . , 0) with the 1 in component i. Now, let's consider the subcomplex Cj = ε j =1 C ε , which corresponds to 0-surgery on the ℓ-component sublink L ′ = L − Kj, for some j = i. Note that the inclusion of Cj with its own ε-filtration as an ℓ-component link coming via the identifications of Section 3 into C is a morphism of filtered complexes. Therefore, this induces homomorphisms between E C j n and E C n for all n (see, for example, [8]). It is easy to see that this is an injection on the E1 terms. Thus, we must have that the image of the kernel of the corresponding map d i 1 for the 0-surgery complex for L ′ is exactly the kernel of d i 1 |C j . By assumption, d1 is identically 0 for the complex associated to an ℓ-component sublink,
so d i 1 |C j = 0.
We now want to see that d i 1 is 0 on the quotient complex ε j =0 C ε = C/Cj. Since d i 1 has no nonzero component from C/Cj to Cj, this will show that d i 1 is identically 0 everywhere. We can in fact identify C/Cj with Cj, simply by applying the filtered quasi-isomorphism
S +K j = M ⊂L−K j Φ +K j ∪+M
from Lemma 3.3. Therefore, d i 1 is 0 on C/Cj. Repeating this argument for various i and j, we obtain d1 ≡ 0.
In fact, we can repeat this argument to prove that d2 is identically 0 as well. For ℓ = 0 and 1, this is trivial simply by the depth of the filtration. Thus, we begin our analysis with ℓ = 2. As before, from [11] we have that HF ∞ (Y ) has rank 4 over F[[U, U −1 ]. However, we know that the total rank of the E1 page must in fact be 2 ℓ = 4. Therefore, d2 and the higher differentials vanish. Now, for the induction step, we want to notice that E2 ∼ = E1 by the previous argument that d1 = 0. Therefore, we have the same injectivity properties on the E2 pages coming from the inclusion of the faces Cj. We get that d2 is 0 on Cj again by including the corresponding complex for sublink L − Kj , which has vanishing d2 by induction. For C/Cj, we have a similar statement to the d1 case, which is that
d i 1 ,i 2 2 = 0 for i1, i2 = j, where d i 1 ,i 2 2
is the component of d2 which maps from C ε to C ε+τ i 1 +τ i 2 . This follows by again identifying C/Cj with Cj via a filtered quasi-isomorphism. After doing this for different values of j, we see that d i 1 ,i 2 2 = 0 for all pairs (i1, i2). This shows d2 vanishes.
The Third Differential
We may now identify the E3 page with E1 in a natural way. However, this still does not yet look like an exterior algebra. This next lemma creates the image we desire. To motivate this, recall that the E1 term has rank 2 ℓ , which is exactly the total rank of the exterior algebra for an ℓ-dimensional space. For notational purposes, let Λ * F (Y ) = Λ * (H 1 (Y ; Z)) ⊗ F. In the following lemma, we must take the word 'natural' with a grain of salt. As the link surgery formula only establishes a relative grading on C ∞ (H, Λ0, 0), we simply fix a choice of absolute grading on this complex. For the remainder of the paper, we will keep everything fixed with respect to this choice of absolute grading, in the sense that inclusions of complexes corresponding to sublinks should respect this absolute grading. In this sense the identifications we will make will be 'natural' with respect to these inclusions.
Lemma 5.1. Let L be a homologically split link and consider the εfiltration on C ∞ (H, Λ0, 0). There is a natural association of the E3 page of the induced spectral sequence with Λ
* F (Y ) ⊗ F[[U, U −1 ], such that d3 : Λ i F (Y ) ⊗ U j → Λ i−3 F (Y ) ⊗ U j−1 .
Proof. Because d1 = d2 = 0, we need only make the identification with the exterior algebra for the E1 term. Recall from Remark 3.2 that each vertex of the hypercube has ∂-homology isomorphic to HF ∞ (S 3 ) (there is only one s associated to each C ε by our choice of link and Spin c structure). Therefore, the term E p 1 , filtration level p, of the corresponding spectral sequence will simply be ℓ p copies of HF ∞ (S 3 ). We can identify the E1 page with Λ * F ⊗ F[[U, U −1 ] as follows. Choose a basis {x i } for H 1 such that x i corresponds to the Hom-dual of the meridian of Ki (this can be done since H1 is torsion free). Suppose that ε = ℓ − k. First, identify 1 in Λ * F with the generator of H * (C (1,...,1) , ∂) ∼ = HF ∞ (S 3 ) with fixed absolute grading 0. We can then choose a generator of H * (C ε(M ) , ∂) to be the image of 1 after inverting the corresponding sequence of k destabilizations coming from each Ki ⊂ M , Φ +K i * . We then associate to this element x i 1 ∧. . .∧x i k in Λ k F , where εi m = 0 for 1 ≤ m ≤ k. By the definition of a hyperbox of chain complexes, the induced maps on ∂-homology, Φ +K i * , commute; so, we see the order does not matter in this construction. Each exterior algebra element lives in the filtration level corresponding to the number of times we have applied a (Φ +K * ) −1 . However, since the destabilization maps lower the relative grading on C by 1 (these are components of the differential), we can see that each 'element of H 1 (Y )' has grading 1 and wedge product is additive on grading; furthermore, U still has grading -2. This therefore establishes the relatively-graded isomorphism of E3 with Λ *
F ⊗ F[[U, U −1 ]
. Recall that d3 lowers filtration level by 3, but it only lowers grading by 1. In other words, d3 will take α⊗1, for α ∈ Λ i F , to β ∈ Λ i−3 F ⊗U j for some j. By out identifications, the grading will be lowered by 3 + 2j; therefore, j must be −1. Extending this gives d3 :
Λ i F (Y ) ⊗ U j → Λ i−3 F (Y ) ⊗ U j−1 , as desired.
With these identifications, we are now ready to prove the first half of the main theorem.
Proof of First Three Differentials in Theorem 1.3. We let Y be presented by 0-surgery on a homologically split link L with ℓ components. Since there is a unique torsion Spin c structure on Y , s0, and HF ∞ (Y, s) = 0 for non-torsion s, HF ∞ (Y ) = HF ∞ (Y, s0).
Since there are three-manifolds with b1(Y ) = 3 where HF ∞ does not have rank 8, we cannot repeat our arguments to show that all higher differentials vanish. We will, however, be able to use the same argument to calculate d3; see how it acts on faces and add up the components. Note that we have identified the E3 pages with Λ *
F ⊗ F[[U, U −1 ] by associating H * (C ε , ∂) with span F[[U,U −1 ] {x i 1 ∧ . . . ∧ x i k } where εi m = 0 for 1 ≤ m ≤ k = ε .
We now can easily see how the subcomplexes and quotient complexes given by faces of the hypercube fit into this picture via inclusions/projections of H 1 . Let's prove that d3 is given by (1).
Again, use d i 1 ,i 2 ,i 3 3
to represent the components that map from C ε to C ε+τ i 1 +τ i 2 +τ i 3 and let x i 1 be the Hom-duals of the meridians of Ki in H 1 (these naturally exist in H 1 of 0-surgery on any homologically split link containing Ki). Consider the three-form, µi 1 ,i 2 ,i 3 , on H 1 (Y ) given by
x i 1 ⌣ x i 2 ⌣ x i 3 , [S 3 0 (Ki 1 ∪Ki 2 ∪Ki 3 )
] on x i 1 ∧x i 2 ∧x i 3 and 0 otherwise. The induction arguments with filtered morphisms of the previous section also show that d i 1 ,i 2 ,i 3 3 will be given by interior multiplying µi 1 ,i 2 ,i 3 if it does so for sublinks with ℓ − 1 components; this again follows by the injectivity on the E1 = E3 terms.
We claim that µY =
i 1 <i 2 <i 3 µi 1 ,i 2 ,i 3 .
This is because the value of µY on x i 1 ∧ x i 2 ∧ x i 3 is given by the Milnor invariants of L,μL(i1, i2, i3) (see [16]), and thus µi 1 ,i 2 ,i 3 takes the valuē
µK i 1 ∪K i 2 ∪K i 3 (i1, i2, i3) on x i 1 ∧x i 2 ∧x i 3 .
Since Ki 1 ∪Ki 2 ∪Ki 3 is a sublink of L, their Milnor invariants agree (see, for example, [9]). This follows by first establishing it on Cj by naturality of the inclusions of filtered complexes and then identifying C/Cj with Cj.
Thus, it suffices to establish that for the base case, ℓ = 3, d i 1 ,i 2 ,i 3 3 = d3 is multiplication by some power of U and x 1 ⌣ x 2 ⌣ x 3 , [Y ] . We can use the calculations of [5] to do this without much effort. The E3 page has total rank 8. However, in that paper, HF ∞ is computed to have
dimension 8 − 2 · x 1 ⌣ x 2 ⌣ x 3 , [Y ] .
Since d3 can only be nonzero on E 3 3 by the depth of the filtration, we must study
E 3 3 = Λ 3 F ⊗ F[[U, U −1 ] and E 0 3 = Λ 0 F ⊗ F[[U, U −1 ]
. Each has rank 1, generated by x 1 ∧ x 2 ∧ x 3 and 1 respectively in Λ * F . Therefore, we may conclude that d3 sends
x 1 ∧ x 2 ∧ x 3 ⊗ U j to x 1 ⌣ x 2 ⌣ x 3 , [Y ] · 1 ⊗ U j−1 .
This shows that d3 is given by contraction by the integral triple cup product form for b1 = 3, completing the proof.
Throughout the past two sections, we have only been proving facts about three-manifolds that can be expressed simply as 0-surgery on a homologically split link. We now see that this was sufficient generality.
Corollary 5.2. Let s be a torsion Spin c structure on a closed, connected,
oriented three-manifold Y . Then, dim F[[U,U −1 ] HF ∞ (Y, s) ≤ dim F[[U,U −1 ] HC ∞ (Y ).
Therefore, the rank of HF ∞ (Y, s) is at most that of the rank conjectured by Ozsváth and Szabó.
Proof. Since the homology predicted by Ozsváth and Szabó is isomorphic to HC ∞ (Y ), Theorem 1.3 implies that this inequality of ranks is the case for any three-manifold given by 0-surgery on a homologically split link. This is because we are comparing two spectral sequences agreeing up to (E3, d3), where the higher differentials of one all vanish. Now let Y be arbitrary. Choose a homologically split link, L, with the property that 0-surgery has integral triple cup product form isomorphic to that of Y . By the work of Cochran, Gerges, and Orr [2], such a link exists and HC ∞ (Y ) ∼ = HC ∞ (S 3 0 (L)). Furthermore, we know from [5] that HF ∞ (Y, s) ∼ = HF ∞ (S 3 0 (L)). This proves the result.
The rest of this paper is now devoted to showing the higher differentials in our spectral sequence vanish, or equivalently, proving the opposite inequality:
dim F[[U,U −1 ] HF ∞ ≥ dim F[[U,U −1 ] HC ∞ (+)
Composing Knots and Complexities of Links
In order to prove (+), it is necessary to again proceed inductively. However, we must induct on something more complicated than simply b1. We will assume that the higher differentials vanish when b1 ≤ ℓ − 1 (this is automatic for b1 ≤ 3), but we will need a way to seemingly induct on the set of homologically split links with ℓ components. Recall that two three-manifolds are surgery equivalent if there is a finite sequence of ±1surgeries on nullhomologous knots taking one manifold to the other (see [2]). As mentioned before, we only need to prove (+) for a single representative of each surgery equivalence class of three-manifold with H1(Y ) ∼ = Z ℓ (Section 2 of [5]); we will take this liberty and change our links around to make them more suitable to the link surgery formula. All components will have framing 0, so we will not distinguish between the link and the resulting manifold obtained by 0-surgery, or between b1 and the number of components. Thus, we will make statements like surgery equivalent links to mean that the manifolds obtained by 0-surgery on each link are surgery equivalent. Also, s0 will always refer to the unique torsion Spin c structure on the underlying manifold.
We let L1 L2 indicate that the two links are separated by an embedded 2-sphere (and both links will always be nonempty when using this notation). Begin with an ℓ-component homologically split link L. Order the components K1, . . . , K ℓ ; we refer to the Milnor linking invariants asμL(i, j, k) orμL(K, K ′ , K ′′ ) if the indices are unclear. We always assume that the three indices are all distinct. We will also usually assume i < j < k, but may switch this at the reader's inconvenience to keep notation simple. Note that changing the order of the indices does not changē µ mod 2. Example 6.1. Suppose thatμL(1, 2, 3) = n and allμ vanish for all other triples of indices. Let L ′ = K1 ∪ K2 ∪ K3. Then we know that L is surgery equivalent to a link of the form (L − L ′ ) L ′ [2]. Given (+) for all links with at most ℓ − 1 components, the connect-sum formula will guarantee that HF ∞ (S 3 0 (L)) ∼ = HF ∞ (S 3 0 (L ′ )) ⊗ HF ∞ (S 3 0 (L − L ′ )); since this formula also holds for HC ∞ with F[[U, U −1 ] coefficients (see the proof of Theorem 2 in [7]), this proves Theorem 1.1 for L as well.
With this example in mind, we define a complexity of L with the hope that a reduction in complexity makes the link closer to being split. This is defined as
c(L) = #{(i, j, k) :μL(i, j, k) = 0, 1 ≤ i < j < k ≤ ℓ}(2)
Let's study the links with the simplest c-complexity first. Remark 6.2. If c(L) = 0 or c(L) = 1, then we know how to complete the proof from Example 6.1. If c(L) ≥ 2, then there are two options. Either L is surgery equivalent to some L1 L2 or there exists some component Ki which hasμL(i, j, k) nonzero for at least two different pairs (j, k) (reordering of (i, j, k) possibly necessary). If it splits as L1 L2, then again we are done by the connect-sum formulae.
For a fixed b1 we will induct on the c-complexity. To keep sight of the final goal, the plan for the rest of the paper will be to prove the following theorem similar to the method of composing knots in [5]. We are therefore led to the following proposition. Proposition 6.4. Suppose c(L) ≥ 2 and that L is not surgery equivalent to any L1 L2. Let Kr be a component with at least two different pairs (s, t) such thatμL(r, s, t) are nonzero. Then, there is an ordered ℓ-component linkL with the following two properties. First,μL(i, j, k) = µL(i, j, k) for all i, j, k. Second, there is a knot K ⊂L which we can express as K ′ #K ′′ , where c((L − K) ∪ K ′ ) < c(L) and c((L − K) ∪ K ′′ ) < c(L).
Before proving Proposition 6.4, we now see how this will be applied in conjunction with Theorem 6.3.
Proof of Theorem 1.1. For fixed b1 = ℓ, we induct on c. By Remark 6.2, we need only concern ourselves with the case where c(L) ≥ 2 and L does not decompose as two geometrically split links. Apply the proposition to replace L byL. SinceμL(i, j, k) =μL(i, j, k), L andL will be surgery equivalent. It now suffices to prove (+) onL.
Decompose the component K as K ′ #K ′′ . SinceL−K ∪K ′ andL−K ∪ K ′′ have strictly smaller c-values, (+) holds for each of these. Theorem 6.3 completes the proof of (+) for all b1 = ℓ. Applying Theorem 1.3 shows that for 0-surgeries on homologically split links HF ∞ and HC ∞ have the same rank; the relative gradings also agree simply because the complexes (E3, d3) and C ∞ * agree on relative gradings. Having this for 0-surgeries on homologically split links gives the relatively-graded isomorphism for arbitrary three-manifolds by repeating the arguments from the proof of Corollary 5.2.
We now recall a helpful theorem of Cochran describing theμ-invariants of connect sums. Therefore, in the case of two homologically split links,μ L#L ′ (i, j, k) = µL(i, j, k) +μ L ′ (i, j, k).
Proof of Proposition 6.4. By hypothesis, we may consider two distinct pairs (j1, k1) and (j2, k2) such thatμL(r, j1, k1) andμL(r, j2, k2) are nonzero for L. Construct an ℓ-component homologically split link L ′ with an ordering on the components such that
µ L ′ (a, b, c) = μL(a, b, c) if (a, b, c) = (r, j2, k2), 0 if (a, b, c) = (r, j2, k2).
Such a link can be explicitly constructed by repeated applications of "Borromean braiding" (see Corollary 3.5 of [2] for more details). Next, isotope a small arc from both K ′ j 2 and K ′ k 2 out and away from the rest of the diagram for L ′ , and isotope the arc from K ′ j 2 such that it createsμL(r, j2, k2) twists. We now take an unknot, U , and thread it through the twists of K ′ j 2 and through K ′ k 2 as in Figure 1. This three-component sublink (U, K ′ j 2 , K ′ k 2 ) has Milnor invariants equal toμL(r, j2, k2).
We will choose K ′ to be K ′ r in L ′ and K ′′ = U . From this we can see thatμ (L ′ −K ′ )∪K ′′ (K ′′ , K ′ j 2 , K ′ k 2 ) =μL(r, j2, k2) and all other µ (L ′ −K ′ )∪K ′′ (K ′′ , ·, ·) vanish. We now want to see that the connect sum K = K ′ #K ′′ yields a linkL = (L ′ − K ′ ) ∪ K with allμL(a, b, c) = µL(a, b, c).
To show this, it suffices to prove thatL can be constructed by connecting two geometrically split ℓ-component links by bands between pairs of components which intersect the separating 2-sphere exactly once; furthermore, we require that theμ-invariants for these two links add up tō µL. The result will then follow from the additivity ofμ in Theorem 6.5.
We choose our two links as follows. The first link will be L ′ . The other link is an ℓ-component link, L * , consisting of two split sublinks: a threecomponent homologically split sublink withμL * (r, j2, k2) =μL(r, j2, k2) and an (ℓ − 3)-component unlink, so all other invariants vanish. Clearly the values ofμ add up as expected and Figure 2 demonstrates how we can connect them to obtainL with K = K ′ #K ′′ . Note that L * r is what creates K ′′ inL.
By construction, both c((L − K) ∪ K ′ ), which equals c(L ′ ), and c((L − K) ∪ K ′′ ) are strictly less than c(L). This completes the proof. Figure 2: ExpressingL as the connect-sum of L ′ and L * Remark 6.6. Because L andL produce indistinguishable cup homology and HF ∞ , we will use L to in fact refer toL for the remainder of the paper.
K ′ j 2 K ′ k 2 Ū µL(r, j2, k2) twistsL ′ L * L * r L * j 2 L * k 2 K ′ k 2 K ′ j 2 K ′ j 1 K ′ k 1 K ′ r S 2
Constructing and Chopping Down the Complex
Recall that our goal is to prove Theorem 6.3; this loosely said that knowing the higher differentials vanish for a link with a component replaced by K ′ or K ′′ , then this holds after instead replacing with K ′ #K ′′ . We now construct a complex which contains all of the Heegaard Floer information of K ′ and K ′′ simultaneously, where we have identified K = K ′ #K ′′ as the component to reduce complexity at. With this we will be able to use our inductive knowledge for K ′ and K ′′ to produce the desired result. The way that this is done is via a standard Kirby calculus trick (see, for example, [15]); we express 0-surgery on K as 0-surgery on three components: K ′ , K ′′ , and an unknot U geometrically linking each once as shown in Figure 3. Thus, if our link has ℓ components, then the corresponding link that we would like to study will have ℓ + 2. As further abuse of notation we will now call this link L, since 0-surgery results in the same manifold. The framing Λ will change as well due to the algebraic linking that has been introduced. Reorder the components in such a way that K ′ , K ′′ , and U are the first, second, and third components respectively. For notational purposes, we will relabel these as K1, K2, and K3. This three-component sublink will arise often, so we will refer to it as W . We see that Λ1 = Λ2 = (0, 0, 1, 0, . . . , 0) and Λ3 = (1, 1, 0, . . . , 0). Therefore, the equivalence class in H(L) corresponding to s0 will in fact be a 2-dimensional lattice spanned by Λ1 and Λ3; in fact, s0 = [( 1 2 , 1 2 , 1, 0, . . . , 0)]. By inducing the proper filtrations and removing acyclic complexes, we will significantly cut down the size of C ∞ (H, Λ, [( 1 2 , 1 2 , 1, 0, . . . , 0)]) to a smaller finite-dimensional object. Before continuing, we remark that the reader interested in this proof should first try to follow the calculation of HF − for surgeries on the Hopf link via the link surgery formula (Section 8.1 of [6]), as the arguments will be based on this. Let's also recall the sim-plified notation used in that computation. We let ε1ε2 . . . ε ℓ+2 s represent the complex A ∞ (H L−M , ψ M (s)) where Ki is in M if and only if εi = 1. To shorten notation further in our setting, we will use ε1ε2ε3 * (s 1 ,s 2 ,s 3 ) to denote the hypercube of chain complexes at (s1, s2, s3, 0, . . . , 0) with ε1, ε2, ε3 fixed, but all remaining εi free. We are setting the last components of s to be 0 since this corresponds to choosing the unique torsion Spin c structure on S 3 0 (L − W ). In fact, 111 * (s 1 ,s 2 ,s 3 ) is exactly the complex corresponding to 0-surgery on L − W with the torsion Spin c structure.
A key map that we will study is
Γ ±K i = N⊂L−W Φ ±K i ∪ N .
Here we are summing over all possible orientations of sublinks N , whereas the previous maps S +K i only allowed for N = +N . This complex is acyclic, as the Φ +K 1 are quasi-isomorphisms. After removing this subcomplex, we are left with {s3 ≤ 1}. Since Λ1 = Λ2, we cannot repeat the argument for Φ −K 3 to remove {s3 < 1}. We must tread carefully to chop the remaining complex down further. Consider the filtration, F2(x) = s3 − 2ε1 − i =1 εi. This odd-looking filtration is defined such that Φ −K 1 lowers the filtration level, but Φ −K 2 does not, even though Λ1 = Λ2. We now study the subcomplex {s3 = 1, ε1 = ε2 = 1} ⊕ {s3 = 0, ε1 + ε2 ≥ 1} ⊕ {s3 ≤ −1}. This subcomplex is best seen by the boxed elements in Figure 5.
Γ +K 1 / / Γ +K 2 ) ) S S S S S S S S S S S S S S 101 * ( 1 2 , 1 2 ,1) 001 * ( 1 2 , 1 2 ,0) Γ −K 1 o o Γ −
The associated graded splits into a product of complexes analogous to the ones defined previously. Since Φ −K 2 is a quasi-isomorphism, we may again remove this acyclic complex in our study. We can now see that the remaining complex is the same as the one in the statement of the proposition, except for the fact that all ε3 are 0 instead of 1. However, we can simply apply the map S +K 3 = M ⊂L ′ Φ +K 3 ∪+M to all of the components to obtain a filtered quasi-isomorphism between the complex with ε3 = 0 and the one with ε3 = 1 by Lemma 3.3. This completes the proof.
000
Reshaping the Complex
We have actually reduced the computation to calculating the homology of the complex given by 00 * (0,0) Here we have suppressed the ε3-coordinate by destabilizing this component via S +K 3 and applying Lemma 3.5 (it is easy to see this applies to the truncated complex as well). Because of this, we have applied ψ +K 3 to all s. Since all values of s and ψ M (s) are now all 0, we will suppress this for the remainder of the proof.
Γ +K 1 / / Γ +K 2 ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q 10 * (0,0) 00 * (0,0) Γ −K 1 o o Γ −K 2 v v m
To simplify this problem further, we need some simple linear algebra. Given a mapping complex over a vector space, M (φ : (V, ∂V ) → (W, ∂W )), to determine the rank of H(M (φ)), we only need to know the homologies of V and W and rk φ * . In the case that H(V ) ∼ = H(W ) we have the convenient formula
dim H(M (φ)) = 2 dim H(V ) − 2 rk φ * ,
where this is a statement about the F[[U, U −1 ]-dimensions of the entire homology vector spaces, not at each grading. This follows easily from the long exact sequence associated to H(V ),H(W ), and H(M (φ)). We now apply this to prove a convenient lemma.
Lemma 8.1. Suppose V is a finite-dimensional vector space over a field of characteristic 2. Consider the complex given by V equipped with the differential ∂ ≡ 0. Let F, G, J, K : V → V , and define Θ :
V ⊕ V → V ⊕ V by Θ(v, w) = (F (v) + G(w), J(w) + K(v)).
Furthermore, suppose that J is a quasi-isomorphism (or equivalently, an invertible map). Then, the homology of the mapping cone, M (Θ), has the same dimension as the homology of M (F − GJ −1 K).
Proof. We know that the homology of M (Θ) has rank given by 2 dim(V ⊕ V ) − 2 rk Θ. We study the matrix Θ = F G K J . It is easy to see that this matrix has the same rank as X =
F − GJ −1 K 0 K J . Now, we have dim H(M (Θ)) =2 dim(V ⊕ V ) − 2 rk X =4 dim V − 2(rk(F − GJ −1 K) + dim V ) =2 dim V − 2 rk(F − GJ −1 K) = dim H(M (F − GJ −1 K)).
In order to apply Lemma 8.1 to Proposition 7.1, we must see that one of the maps Γ is a quasi-isomorphism. In fact, it is easy to see each Γ map is a quasi-isomorphism by applying the same filtration arguments as in the proof of Proposition 7.1. Thus, it remains to calculate the rank of the induced map Γ +K 1 *
+ Γ −K 1 * • (Γ −K 2 * ) −1 • Γ +K 2 *
from H * (00 * ) to H * (10 * ). For notational convenience, we will abbreviate this induced map by Ψ K 1 ,K 2 .
The Maps Γ ±K i
In order to study the Γ maps, it is useful to note that the homology of each complex, 00 * , 10 * , or 01 * , is naturally isomorphic to HF ∞ (S 3 0 (L ′ )) by Lemma 3.5, where L ′ is L − W . Recall that L ′ consisted of one less component than the level of b1 that we wanted for Theorem 1.1. Therefore, we may assume that this homology is exactly HC ∞ (S 3 0 (L ′ )). However, we want to study this a little more carefully.
Let x 1 represent the Hom-dual of the class [K3] in H 1 (S 3 0 (L ′ ∪ Ki)). Choose the other basis vectors of H 1 to be given by the meridians of the components of L ′ . Using the ε-filtration restricted to the * -part of the complex, we may think of the E3 term for 00 * as
x 1 ∧ Λ * F (S 3 0 (L ′ )) ⊗ F[[U, U −1 ] and the E3 term of 10 * /01 * as simply Λ * F (S 3 0 (L ′ )) ⊗ F[[U, U −1 ]
. Therefore, we will think of taking homology of 00 * as taking x 1 and wedging with the elements of HC ∞ (S 3 0 (L ′ )). The homology of each complex 00 * /10 * /01 * is actually the homology of E3 with respect to d3 − d K 1 3 thought of as a differential on C ∞ (S 3 0 (L ′ )). We mean by d K 1 3 the components of d3 that correspond to contracting by triple cup products that contain x 1 . Therefore, we can consider the maps
d K i 3 : (x 1 ∧ Λ * F (S 3 0 (L ′ ))) ⊗ U j → Λ * F (S 3 0 (L ′ )) ⊗ U j−1 . However, d K i
interior multiplying by two different trilinear forms corresponds to interior multiplying by the wedge product). Therefore, it makes sense to talk about (d K i 3 ) * . Finally, by choosing a basis with the meridian of K to be x 1 and the other basis vectors as meridians of the components of L ′ , we can identify Λ * F (S 3 0 (L)) with each Λ * F (S 3 0 (L ′ ∪ Ki)). Thus, we can make sense of the statement d K
3 = d K 1 3 + d K 2 3
. Better yet, this statement is actually true by the construction of K = K1#K2 in Proposition 6.4.
What we hope to find is that the map Ψ K 1 ,K 2 is precisely (d K 3 ) * from H * (00 * ) to H * (10 * ), so as to guarantee that the higher differentials must vanish. It turns out that this is not the case, but it will be true up to some terms of higher order.
We now give a brief outline for how the rest of the proof is going to go. First, we will set up the complex so that Γ +K i * will essentially be given by the identity and Γ −K i * by the identity plus a term that corresponds to (d K i 3 ) * . Therefore, Ψ K 1 ,K 2 will be (d K 1 3 ) * + (d K 2 3 ) * + (d K 1 3 ) * • (d K 2 3 ) * . Finally, it will be a simple slight of hand to prove from here that the rank of HF ∞ is at least that of the predicted homology, HC ∞ .
From now on we will use 00 * (and similar complexes) to refer to its homology, as well as for the maps Γ ±K i , unless specified otherwise. A mindful reader may have noticed that while we have been using hyperboxes of Heegaard diagrams to make all of the constructions so far, there have been no restrictions on the choice of complete system. Now is the time where we do so.
The complete system of hyperboxes that we will work with is a basic system of hyperboxes. Instead of recalling the construction, we will review only the properties we will use and refer the reader to Section 6.7 in [6]. Lemma 9.1. Suppose we are working in a basic system. If M has at least two components, one of which is compatibly oriented with L, then Φ M vanishes; in other words, for all K, Φ +K = S +K = Γ +K .
Proof. Let M ′ be a nonempty sublink of L − K and suppose that K is the ith component, consistently oriented. We will show that D +K∪ M ′ vanishes. Since Φ = D • I, this will prove the lemma.
Let's study destabilization maps more carefully. Destabilizing a link of k components is given by compressing the hyperboxes, or in other words, playing the kth standard symphony for some hypercubical collection (see Section 3 of [6]); if one of the edges in the hyperbox that we are summing over has length 0, the sum over algebra elements in the hypercubical collection when playing the song will be empty, if k ≥ 2. This is true because when k ≥ 2, the kth standard symphony contains a harmony with the element i at least once. According to the definition of playing a song, and thus in compression, in order for there to be nonzero terms in the formula, the number of harmonies that contain i must be at most di; however, we have established that this is 0. Therefore, the destabilization for +K ∪ M ′ must be 0.
Lemma 9.2. For a basic system, the map Γ +K i is contraction by the dual of x 1 in H 1 (Y ) * , after the appropriate identifications with HC ∞ .
Proof. We prove this with i = 1. We want to study Γ +K 1 : 00 * → 10 * . Let's return to the chain level for now. First of all, by Lemma 9.1 Γ +K 1 = S +K 1 = Φ +K 1 in a basic system. Now, we may identify the complex 00 * → 10 * with the complex 01 * → 11 * by applying Γ +K 2 = Φ +K 2 , which we can think of as the complex 0 * → 1 * for L ′ ∪ K1.
However, by construction, the generators of the E3 pages in 0 * (exterior elements with x 1 in them) were defined by applying (Φ +K 1 * ) −1 to the corresponding element in 1 * in Lemma 5.1 (with no x 1 ). Therefore, under these identifications on the E3 page, Γ +K 1 is given by simply removing x 1 . Thus, passing to the homology of 0 * and 1 * , the E4 = E∞ page by induction, the induced maps also behave the same way.
By ignoring the "x i ∧" components of 00 * , we will in fact think of Γ +K i as the identity. It turns out that knowing Γ +K i is exactly what we need to understand Γ −K i via our inductive arguments. Lemma 9.3. With the basic system and the corresponding identifications as above, the map Γ −K i is given by Id + (d K i 3 ) * .
Proof. Again, we assume i = 1. By applying S +K 2 = Φ +K 2 as above, we can prove the result on 0 * → 1 * instead. Now, by induction, HF ∞ (S 3 0 (L ′ ∪ K1)) ∼ = HC ∞ (S 3 0 (L ′ ∪ K1)) and therefore the higher differentials after d3 vanish. We consider the entire E3 page for surgery on L ′ ∪ K1. However, we induce a new filtration on this,F(x) = −ε1. Since this filtration has depth 1, to understand the spectral sequence that this filtration gives, it suffices to calculate the homology of the associated graded and then d1.
Let us do a quick algebra review first. Consider a chain map f : C1 → C2. Now, we filter M (f ) by
O(x) = 1 if x ∈ C1, 0 if x ∈ C2.
We can explicitly describe the pages in the spectral sequence arising from O. The E0 term splits as C1 ⊕C2. The E1 term will be given by H * (C1)⊕ H * (C2). Now, d1 will be given by f * . Finally, all higher differentials will vanish. On the chain level, HF ∞ (S 3 0 (L ′ ∪ K1) is quasi-isomorphic to the mapping cone of Γ +K 1 + Γ −K 1 : 0 * → 1 * . Now, split d3 as (d3 − d K 1 3 ) + d K 1 3 . With respect to the spectral sequence coming fromF, we see the only component of the differential that preserves the filtration level is d3 − d K 1 3 . Therefore, the associated graded splits as H * (0 * ) and H * (1 * ). The differential, dF 1 , is thus given by Γ +K 1 * + Γ −K 1 * . However, we also know that (d K 1 3 ) * must be dF 1 . Applying Lemma 9.2 gives the desired result.
M , ψ M (s)). The s in the first component is simply serving as an index. Here we are using the convention that Φ ±K s (x) = 0 if x ∈ A ∞ (H ∅ , ψ K (s)) (in other words, if x is not in the domain). We now have a slightly altered version of the integer surgery formula for knots (see Theorem 7.5 of [6]) Theorem 2.1. (Ozsváth-Szabó) The homology of the complex C ∞ (H, n) is isomorphic to HF ∞ (S 3 n (K)).
[s]. We extend these lattices to H(L)i = H(L)i ∪ {+∞, −∞} and H(L) = ⊕ n i=1 H(L)i. Let I+( L, M ) be the set of indices of components of M which are consistently oriented with L. The remaining components of M form I−( L, M ). We define the maps p M i : H(L)i → H(i ∈ I+( L, M ), −∞ if i ∈ I−( L, M ), si otherwise. We can then apply restrictions p M (s) = (p M 1 (s1), . . . , p M n (sn)). This will allow us to remove the components of M , but still keep track of Spin c structures consistently. By viewing H(L) as an affine space over H1(S 3 − L) we can define the map ψ M : H(L) → H(L − M ) by ψ M (s) = s − [ M ]/2. In other words, we ignore the components of s coming from M , but we must change the remaining components based on their linking with the components of M . We extend this to go from H(L) to H(L − M ) in the obvious way.
Definition 2. 2 .
2An n-dimensional hyperbox of size d = (d1, . . . , dn) ∈ N n is the following subset of N n E(d) = {ε = (ε1, . . . , εn)|0 ≤ εi ≤ di} If d = (1, . . . , 1), then E(d) is a hypercube. The length of ε, ε , is given by i εi. The elements of E(d) are called vertices. Also, there is a natural partial order on E(d) given by ε ≤ ε ′ if and only if εi ≤ ε ′ i for all i. Two vertices in the hyperbox are neighbors if they differ by an element of {0, 1} n . The important example to keep in mind is given by the n-dimensional hypercube determined by the set of sublinks of the n-component link L. We identify the sublinks of L with the vertices of {0, 1} n by setting ε(M )i to be 1 if Ki ⊂ M and 0 otherwise.
Definition 2 . 3 .
23An n-dimensional hyperbox of chain complexes of size d is a collection of chain complexes
Definition 2.6. A hyperbox for the pair ( L ′ , M ), H L ′ , M is an m-dimensional hyperbox of Heegaard diagrams for L ′ − M .
Definition 2.7. A complete system of hyperboxes for L is a collection of hyperboxes for each pair ( L ′ , M ), H L ′ , M , such that for any sublink of M ′ with orientation induced by M , H L ′ , M is compatible with H L ′ −M ′ , M −M ′ and H L ′ , M ′ .
but instead has size d, then applying the above maps, D M , will go to H L ′ , M (1,...,1) instead of H L ′ −M . Therefore, we must do what is called compression to arrive at H L ′ , M d·(1,...,1) = H L ′ −M . If M = ±Ki for a knot Ki, then we would like to go from H L ′ ,±K i 0 to H L ′ ,±K i d (these will be the two vertices in the hypercube). There exist trianglecounting maps from
we simply apply our identification of this final Heegaard diagram with H L ′ −M to get one last triangle counting map, again by compatibility. This final composition is the destabilization D M p M (s) . For an arbitrary sublink, M , define the map, Φ M s
Theorem 2 .
210. (Manolescu-Ozsváth, Theorem 1.1 of [6]) Consider a complete system of hyperboxes,H, for L ⊂ S 3 and a framing Λ. Given a Spin c structure s on S 3 Λ
Proposition 3. 1 .
1Consider s in an equivalence class corresponding to a torsion Spin c structure. For any component Kj, the maps Φ±K j s :
=
A ∞ (H L−M , s). This is quasiisomorphic to A ∞ (H ∅ , +∞) by applying Φ ±K j for all components Kj ⊂ L − M . We can conclude that each C ε s has homology given by F[[U, U −1 ], so it is generated by a single element. However, we can do better than this. Consider a face, F , of any dimension in {0, 1} n . Let LF be the sublink consisting of components such that Φ +K i does not vanish in CF (in other words, both εi = 0 and εi = 1 appear in the face). Define H(L) : (L, Λ|L F ) to be the quotient of the lattice H(L) by the sublattice generated by Λi where Ki is a component of LF . Furthermore, let H(L, Λ|L F ) denote the set of s in H(L) with [s] ∈ H(L) : H(L, Λ|L F ). Let's construct the following module CF = s∈H(L,Λ| L F ) ε(M )∈F A ∞ (H L−M , ψ M (s))
Lemma 3. 5 .
5Consider the filtration F(x) = − ε defined on CF . The face complex CF is filtered quasi-isomorphic (up to an absolute shift in filtration) to the complex C ∞ (H ′ , Λ|L F ) corresponding to S 3 Λ| L F (LF ), where H ′ is a complete system of hyperboxes of Heegaard diagrams for LF coming from restricting H.
Theorem 4. 3 (
3Theorem 10.1 of [11]). Suppose b1(Y ) ≤ 2 and s is torsion. Then, HF ∞ (Y, s) is a free Z[U, U −1 ]-module of rank 2 b 1 (Y ) .
Theorem 6 . 3 .
63Suppose K = K ′ #K ′′ is a component of L. If (+) holds for (L − K) ∪ K ′ and (L − K) ∪ K ′′ , then it will hold for L.
Theorem 6.5. (Theorem 8.13 of [1]) Suppose L and L ′ are ℓ-component links that are separated by an embedded 2-sphere and thatμL(J) =μ L ′ (J) = 0 for multi-indices J of length at most ℓ. Construct L#L ′ by connecting each pair of components Li and L ′ i with a band that passes through the separating sphere exactly once. Thenμ L#L ′ (I) =μL(I) +μ L ′ (I) for any multi-index I of length at most ℓ + 1.
Figure 1 :
1Threading the unknot to recreateμ L (r, j 2 , k 2 )
Figure 3 :
3An equivalent diagram for 0-surgery on K ′ #K ′′
Proposition 7 . 1 .
71The complex for 0-surgery on all components in L with Spin c structure s0, C = C ∞ (H, Λ, [( 1 2 , 1 2 , 1, . . . , 0)]), is quasi-
Figure 4 :
4The complex {s 1 = 1 2 , ε 3 = 0}We can further reduce this complex in a similar way, by collapsing in the Λ1-direction. Consider the filtration,F1(x) = −(s3 + i =1 εi), on the subcomplex {s3 > 1} of {s1 = 1 2 , ε3 = 0}.The associated graded splits as a product of (0, ε2, 0, ε4, . . . , ε l+2 s , ∂)Φ +K 1 / / (1,ε2, 0, ε4, .. . , ε l+2 s , ∂)
Figure 5 :
5The boxed terms form the final acyclic complex
Basic systems have the property that if M ′ has the induced orientation of M for a sublink M ′ ⊂ M , then H M , M ′ consists of a single Heegaard diagram and H M − M ′ is obtained from H M by removing the z basepoints corresponding to components of M ′ . Let K be the ith component of L. By compatibility, a hyperbox H +K∪ M , M ′ has di = 0 (it has 0 in the ith component of the size).
Λ1=Λ2. . .*
100 *
110 *
( 1
2 , 1
2 , 1)
010 *
000 *
100 *
110 *
( 1
2 , 1
2 , 0)
010 *
000 *
100 *
110 * ( 1
2 , 1
2 , −1)
010 *
, 1 2 ,1)
is in fact a chain map with respect to the differential d3 − d K i 3 on these two complexes since ια • ι β = ι α∧β = ι β • ια (mod 2) (the composition of
AcknowledgmentsI would like to thank Ciprian Manolescu for his guidance on the link surgery formula and general advice on this problem.Proof. From now on, a complex {si > r} will refer to all ε1ε2ε3 * (s 1 ,s 2 ,s 3 ) , where si > r; this is regardless of whether the component Ki has been destabilized or what the value of si is under some subsequent ψ M maps. For this reason, we omit the s from the Φ maps.Induce the filtration on C defined by F3(x) = −(s1 + i =3 εi) for x ∈ ε1ε2ε3 * (s 1 ,s 2 ,s 3 ) . The components of the differential that preserve filtration level are given by ∂ and Φ +K 3 . Consider the subcomplex {s1 > 1 2 }. The associated graded with respect to the filtration on the subcomplex splits as a product of complexes of the form (ε1, ε2, 0, ε4, . . . , ε ℓ+2 s , ∂)Since the maps Φ +K 3 are quasi-isomorphisms, we have that the associated graded, and thus all of {s1 > 1 2 }, is acyclic. Therefore, C is quasi-isomorphic to the quotient complex C/{s1 > 1 2 }, which is {s1 ≤ 1 2 }. We then induce a similar filtration, G3(x) = s1 − i εi. The differentials preserving the filtration level will now be ∂ and Φ −K 3 . We consider the subcomplex,. This subcomplex now splits as a product of complexes of the form (ε1, ε2, 0, ε4, . . . , ε l+2 s , ∂)Similarly, since the maps Φ −K 3 are quasi-isomorphisms, C ′ is acyclic. We are content to remove this and study only the remaining terms, namely {s1 = 1 2 , ε3 = 0}. We have essentially collapsed this complex in the Λ3-direction. It is best to visualize the remaining complex viaFigure 4.Recall that we are interested in the calculation ofafter we have taken homology. By our identifications of the previous section, this corresponds to studyingwhere Id secretly means contracting out the x 1 component of the exterior algebra elements.is the differential on C ∞ (S 3 0 (L ′ )), so this squares to 0 as well. Therefore, (d K i 3 ) 2 = 0. It is now easy to see that; thus, the corresponding relation on homology holds.We now want to show that adding this cross-term cannot remove any of the kernel of (d K 3 ) * . This will be enough to calculate HF ∞ . Proposition 10.2. The kernel of (d K 3 ) * is contained in the kernel of (4).Proof. Suppose that (d K 3 ) * (x) is 0. This implies (d K 1 3 ) * (x) = (d K 2 3 ) * (x) by Lemma 10.1. We want to see that (d K13 ) * • (d K 2 3 ) * (x) = 0. Simply calculate,Proof of Theorem 6.3. The key observation is that Proposition 10.2 implies that the rank of HF ∞ (S 3 0 (L)) is at least that of HC ∞ (S 3 0 (L)).
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