Symmetries in Entropy Space
|
|
- Stephen O’Brien’
- 5 years ago
- Views:
Transcription
1 1 Symmetries in Entropy Space Jayant Apte, Qi Chen, John MacLaren Walsh Department of Electrical and Computer Engineering Drexel University Philadelphia, PA Thanks to NSF CCF
2 Co-Authors & Software Advertisement Jayant Apte (Ph.D) Defended 10-Aug-2016 Dissertation presents symmetry exploiting algorithms for proving network coding capacity regions. Try our software: ITAP & ITCP Qi Chen, Ph.D. Ph.D. CUHK w/ R. W. Yeung Drexel Fall, 2015 INC, CUHK, 2016 On the Job Market! NEW IT&NC SOFTWARE TOOL RELEASES!!!! (ASPITR homepage & ithub) Information Theoretic Achievability Prover (ITAP): For a given network, determine entire rate region associated w/ a class of linear codes! Also, can determine achievability of a specified rate point, bounds for linear secret sharing, and test where a specified polymatroid is linear over a specified field. Information Theoretic Converse Prover (ITCP): For a given network coding problem, computes polyhedral cones outer bounding its capacity region using a custom new symmetry exploiting polyhedral projection algorithm! Both Shannon & non-shannon outer bounds are supported. Also works with secret sharing.
3 Outline 1. Entropy Region N & Shannon Outer Bound N 2. Portion Symmetric Under roup, and 3. Power-Set Orbit Structures & Hierarchical Results 4.? for N 4, 5 (exhaustive) and N 6.
4 Entropy Region N & Shannon Outer Bound 1. X (X 1,...,X N ) N discrete RVs 2. every subset X A (X i,i 2 A) A {1,...,N} [N] has joint entropy h(x A ). 3. h (h(x A ) A [N]) 2 R 2N 1 entropic vector Example: for N 3, h (h 1,h 2,h 3,h 12,h 13,h 23,h 123 ). 4. a h o 2 R 2N 1 is entropic if 9 joint PMF p X s.t. h(p X )h o. 5. Region of entropic vectors N 6. Closure N is a convex cone h 1 h 12 N is an unknown non-polyhedral convex cone for N 4. N h 2 Shannon bound N : h 2 R 2N 1 s.t. h A + h B h A[B + h A\B (1) h A[B h A (2) i.e. submodular & non.-dec. for N 2{2, 3}, n N, but for N 4, N ( N. determining capacity regions of all networks under network coding, complete characterization of N
5 Entropy Region N WhyCare? All fundamental laws of information theory Network Coding Capacity Region (Streaming, Distributed Storage, & Caching) Implications among conditional independences (raphical Models & Machine Learning) Fundamental Limits for Secret Sharing Schemes More (inequalities for subgroup sizes, matrix rank inequalities, Kolmogorov Complexities, etc.)
6 roup Action & Symmetries h 12 (12) (0,1,1) (1,0,1) h 2 h 1 Any permutation 2 S n, : {1,...,n}!{1,...,n} a bijection, is a symmetry of n. Under, thervs(x 1,...,X n ) 7! (X (1),...,X (n) ), so h A 7! h (A),with (A) :{ (i) i 2A}. Example: (12), ([h 1,h 2,h 12,h 3,h 13,h 23,h 123 ] T )[h 2,h 1,h 12,h 3,h 23,h 13,h 123 ] T. Clearly, if h 2 n or n, (h). (definition is insensitive to ordering of RVs)
7 Key Question:? h 12 Fix S2 S 2 S2 h 2 h 1 What about those h 2 n fixed under a group of problem symmetries S n? Define Fix ( n ) and Fix ( n). Fix (H) :{h 2H (h) h 8 2 } (3) For what types of problem symmetries can we expect Shannon-type inequalities to be su cient? i.e., for which does?
8 Orbits in the Power Set N {1,...,n} Fix,andhence and on depend on through 2 N // theorbits in the power set. (h) h 8 2 () h A h (A) 8 2, 8A N. O (A) :{ (A) 2 }. h A h B 8B 2 O (A) O O (A) A 2 2 N,thepower set orbits. (Partitions 2 N ) Note: Multiple groups can yield the same power set orbits. i.e. can have O O 0 for 6 0.
9 Power Set Orbits, N 4 {N} {N} {N} {N} O(134) O(14) O(14) O(34) O(3) (a) O S4 O A4 (b) O D4 O C4 (c) O V h(12)(34),(13)(24)i (d) O S 2 4 h(12)(34)i Figure 1: Indecomposable power set orbits on N {1, 2, 3, 4} Power Set Orbits under di erent groups are also ordered by refinement of associated partition of the power set.
10 Power Set Orbits, N 5 {N} {N} O(1234) O(1234) O(124) (a) O S5 O A5 O A(1,5) (b) O D5 O C5 Figure 2: Indecomposable orbit structures on N {1, 2, 3, 4, 5} Observe that while a poset, need not be a lattice.
11 ? Some Implications If apple 0,thenO apple O 0, i.e. partition O refines O 0 (extra group elements can force more equivalences) Thm. 2: IfO apple O 0,and, then 0 0. Equiv., if O apple O 0,and 0 ( 0,then (.
12 Chen & Yeung Partition Symmetrical Entropy Functions:? for S n1 S n2 S nk Thm. 1 (Qi Chen & R. W. Yeung): Let p {N 1, N 2,...,N t } be a t-partition of N, and p S N1 S N2 S Nt.For N 4, p p if and only if p {N} or {{i},n \{i}} for some i 2 N. What about general??remainingcases? For any, letp N // {{ (i) 2 } i 2N},then apple p Thm. 1: ( unless N // or N // {i, N\{i}}, i 2N. N // : is transitive. N // {i, N \}: S 1 0, fixes some i, then 0 on remainder is transitive on N\{i}. Priority (Pessimistic) Maximal Transitive Subgroups Priority (Optimistic) Minimal Transitive Subgroups
13 ? Another Implication Thm. 3: Let S 1 act on N[{n +1} (fix n +1, acts on N ). Then S 1 S 1 ). Proof: orbits of O S1 eachorbitino repeated twice: once w/o n +1, and once w/ n +1. proj N S 1 : First2 N 1 coordinates are same subsets obeying same inequalities, must also have the remaining coordinates exist obeying more inequalities. : Takeh 2, extendtoh 0 on N[{n +1} via h 0 A h A\N,then h 0 2 S 1. proj N S1 : TakeX 1,...,X n from X 1,...,X n+1 realizing h 0 2 S1.Realizesa h 2. : Extend as previous, X n+1 0,deterministic.
14 ? Complete Power Set Orbits, N 4 Set Orbits, N 4 Answer for N 4 O(14) power set orbits on N O(14) {1, 2, 3, 4} 14) O(34) s are also ordered by refinement of associated O(3) OV h(12)(34),(13)(24)i (d) OS 2 h(12)(34)i 4 O(134) O(14) Symmetric & Alternating: O(34) S4 : (1234), (12) A : (123), O(3) (12)(34) 4 O(134) 2) O(14) Dihedral & Cyclic: O(14) 6 (a) OS4 OA4 (b) OD4 OC4 (c) O (d) O O(34) 2 D4 : (1234), (13) V h(12)(34),(13)(24)i S4 h(12)(34)i ts, N 4 (2, 2)-Partition: O(3) C4 : (1234) S2 2, 3,S4} 2 : (12), (34) Figure 1: Indecomposable power set orbits on N {1, Normal Klein 4-group: Power Set Orbits under di erent groups are also ordered by refinement of associated (c) OV h(12)(34),(13)(24)i (d) OS 2 h(12)(34)iv : (12)(34), (13)(24) 4 partition of the power set. O(134) (a) OS4 OA4 6 Trivial: () (b) O (1, 3)-Partition:Figure 1: Inde S1 S3 : (234), (23) S1 Power A3 : Set (234), (243) Orbits under di partition of the power set. Double Transp.: S42 : (12)(34) 6 (1, 1, 2)-Partition: S1 S1 S2 : (34) 6
15 ? Complete Answer for N 5 Symmetric, Alternating, & en. Affine: S5 : (12345), (12), A5 : (12345), (123) A(1, 5) : (12345), (2345) 6 (2, 3)-Partition: S2 S3 : (12), (345), (34) S2 A3 : (12), (345) S53 : (12)(45), (345) 6 (1, 1, 3)-partition: S1 S1 S3 : (345), (34) S1 S1 A3 : (345) 6 6 Fix 1, Rest Cyclic or Dihedral: S1 C4 : (2345) S1 D4 : (2345), (24) 6 S1 S42 O(1234) O(124) 6 O O S 5 4-group: A5 (a) Fix 1, Rest Klein S1 V : (23)(45), (24)(35) 6 (1, 2, 2)-partition: S1 S2 S2 : (23), (45) (1, 1, 1, 2)-partition: S1 S1 S1 S2 : (45) Power Set Orbits, N 5 O(1234) (1, 4)-Partition: Cyclic & Dihedral: S1 S4 : (2345), (23) C5 : (12345) S1 A4 : (2345), (23)(45) D5 : (12345), (25)(34) : (23)(45) OA(1,5) Figure 2: Indecomposable orbit structures on N {1, 2, 3, 4, 5} Observe that while a poset, need not be a lattice. 6 trivial: () (b) OD5 OC5
16 ? Some Answers for N 6 Thm. 5: For n 6, Cn ( C n, Dn ( D n, S1 C n 1 ( S 1 C n 1, & S1 D n 1 ( S 1 D n 1. Proof shows that the ray with coordinates matching Vámos projection lies in C n is cut o by Zhang-Yeung non-shannon inequality. but
17 Summary Fig. 1(a) Symmetric & Alternating: S4 : (1234), (12) A4 : (123), (12)(34) n4: Fig. 1(b) Dihedral & Cyclic: 6 D4 : (1234), (13) (2, 2)-Partition: C4 : (1234) Fig. 1(c) S2 S2 : (12), (34) Normal Klein 4-group: V : (12)(34), (13)(24) (1, 3)-Partition: S1 S3 : (234), (23) S1 A3 : (234), (243) Double Transp.: S42 : (12)(34) Fig. 1(d) 6 n 6: Sn, Sn {1,..., n}// not a (1, n 6 S1 Cn, Cn 1 6 Dn 6 S1 Cn S1 Sn S1 Sn S1 Dn 6 (1, 4)-Partition: S1 S4 : (2345), (23) S1 A4 : (2345), (23)(45) 6 6 Fix 1, Rest Klein 4-group: S1 V : (23)(45), (24)(35) S1 S42 : (23)(45) trivial: () 6 Cyclic & Dihedral: C5 : (12345) D5 : (12345), (25)(34) Fig. 2(b) 6 Fix 1, Rest Cyclic or Dihedral: S1 C4 : (2345) S1 D4 : (2345), (24) 1 6 (1, 2, 2)-partition: S1 S2 S2 : (23), (45) (1, 1, 1, 2)-partition: S1 S1 S1 S2 : (45) Dn S1 Dn 6 Symmetric, Alternating, & en. Affine: S5 : (12345), (12), A5 : (12345), (123) A(1, 5) : (12345), (2345) 6 (1, 1, 3)-partition: S1 S1 S3 : (345), (34) S1 S1 A3 : (345) (1, 1, 2)-Partition: S1 S1 S2 : (34) 1) or (n) partition ), 1 Trivial: () Cn Fig. 2(a) 6 Fig. 3 (2, 3)-Partition: S2 S3 : (12), (345), (34) S2 A3 : (12), (345) S53 : (12)(45), (345) 6 n5:
18 The Way Forward 1. For n 6, answer? for other maximal transitive groups. 2. Permutations of the ground set form just one set of interesting symmetries for N, others include Combinatorial Symmetry roup: permutations of the extreme rays which leave the face lattice intact. (Huge Subgroup of S M,w/M # of extreme rays) A ne Symmetry roup: Those Combinatorial symmetries whose ray permutation can be generated by multiplying by a 2 N 1 2 N 1 invertible matrix. (Also a large group) Restricted A ne Symmetry roup: Those a ne symmetries associated with vectors representing the rays of fixed length. (can be computed w/ sympol) 3. Which of the latter are also symmetries of N? 4. What sort of dimensionality reduction can be achieved by exploiting the a ne symmetries when calculating rate regions for network coding, storage repair tradeo s, caching regions, etc?
Symmetries in the Entropy Space
Symmetries in the Entropy Space Jayant Apte, Qi Chen, John MacLaren Walsh Abstract This paper investigates when Shannon-type inequalities completely characterize the part of the closure of the entropy
More informationPartition Symmetrical Entropy Functions*
Partition Symmetrical Entropy Functions* Jayant Apte ASPITRG *Chen, Qi; Yeung, Raymond W., Partition Symmetrical Entropy Functions, arxiv:1407.7405v1 [cs.it] Outline Sneak Peak Background: Basic facts
More informationNon-isomorphic Distribution Supports for Calculating Entropic Vectors
Non-isomorphic Distribution Supports for Calculating Entropic Vectors Yunshu Liu & John MacLaren Walsh Adaptive Signal Processing and Information Theory Research Group Department of Electrical and Computer
More informationSymmetry in Network Coding
Symmetry in Network Coding Formalization, Graph-theoretic Characterization, and Computation Jayant Apte John Walsh Department of Electrical and Computer Engineering Drexel University, Philadelphia ISIT,
More informationExploiting Symmetry in Computing Polyhedral Bounds on Network Coding Rate Regions
Exploiting Symmetry in Computing Polyhedral Bounds on Network Coding Rate Regions Jayant Apte John Walsh Department of Electrical and Computer Engineering Drexel University, Philadelphia NetCod, 205 NSF
More informationEntropic Vectors: Polyhedral Computation & Information Geometry
Entropic Vectors: Polyhedral Computation & Information Geometry John MacLaren Walsh Department of Electrical and Computer Engineering Drexel University Philadelphia, PA jwalsh@ece.drexel.edu Thanks to
More informationMultiterminal Networks: Rate Regions, Codes, Computations, & Forbidden Minors
Multiterminal etworks: Rate Regions, Codes, Computations, & Forbidden Minors Ph D Thesis Proposal Congduan Li ASPITRG & MAL Drexel University congduanli@gmailcom October 5, 204 C Li (ASPITRG & MAL) Thesis
More informationA computational approach for determining rate regions and codes using entropic vector bounds
1 A computational approach for determining rate regions and codes using entropic vector bounds Congduan Li, John MacLaren Walsh, Steven Weber Drexel University, Dept. of ECE, Philadelphia, PA 19104, USA
More informationNetwork Combination Operations Preserving the Sufficiency of Linear Network Codes
Network Combination Operations Preserving the Sufficiency of Linear Network Codes Congduan Li, Steven Weber, John MacLaren Walsh ECE Department, Drexel University Philadelphia, PA 904 Abstract Operations
More informationBounding the Entropic Region via Information Geometry
Bounding the ntropic Region via Information Geometry Yunshu Liu John MacLaren Walsh Dept. of C, Drexel University, Philadelphia, PA 19104, USA yunshu.liu@drexel.edu jwalsh@coe.drexel.edu Abstract This
More informationAlgorithms for Computing Network Coding Rate Regions via Single Element Extensions of Matroids
Algorithms for Computing Network Coding Rate Regions via Single Element Extensions of Matroids Jayant Apte, Congduan Li, John MacLaren Walsh Drexel University, Dept. of ECE, Philadelphia, PA 19104, USA
More informationCharacterizing the Region of Entropic Vectors via Information Geometry
Characterizing the Region of Entropic Vectors via Information Geometry John MacLaren Walsh Department of Electrical and Computer Engineering Drexel University Philadelphia, PA jwalsh@ece.drexel.edu Thanks
More information. Relationships Among Bounds for the Region of Entropic Vectors in Four Variables
. Relationships Among Bounds for the Region of Entropic Vectors in Four Variables John MacLaren Walsh and Steven Weber Department of Electrical and Computer Engineering, Drexel University, Philadelphia,
More informationExtremal Entropy: Information Geometry, Numerical Entropy Mapping, and Machine Learning Application of Associated Conditional Independences.
Extremal Entropy: Information Geometry, Numerical Entropy Mapping, and Machine Learning Application of Associated Conditional Independences A Thesis Submitted to the Faculty of Drexel University by Yunshu
More informationSymmetry in Network Coding
Symmetry in Network Coding Jayant Apte, John MacLaren Walsh Drexel University, Dept. of ECE, Philadelphia, PA 19104, USA jsa46@drexel.edu, jwalsh@coe.drexel.edu Abstract We establish connections between
More informationMatroid Bounds on the Region of Entropic Vectors
Matroid Bounds on the Region of Entropic Vectors Congduan Li, John MacLaren Walsh, Steven Weber Drexel University, Dept of ECE, Philadelphia, PA 19104, USA congduanli@drexeledu, jwalsh@coedrexeledu, sweber@coedrexeledu
More informationA new computational approach for determining rate regions and optimal codes for coded networks
A new computational approach for determining rate regions and optimal codes for coded networks Congduan Li, Jayant Apte, John MacLaren Walsh, Steven Weber Drexel University, Dept. of ECE, Philadelphia,
More informationComputing and Communications 2. Information Theory -Entropy
1896 1920 1987 2006 Computing and Communications 2. Information Theory -Entropy Ying Cui Department of Electronic Engineering Shanghai Jiao Tong University, China 2017, Autumn 1 Outline Entropy Joint entropy
More informationAlgebraic matroids are almost entropic
accepted to Proceedings of the AMS June 28, 2017 Algebraic matroids are almost entropic František Matúš Abstract. Algebraic matroids capture properties of the algebraic dependence among elements of extension
More informationEntropy Vectors and Network Information Theory
Entropy Vectors and Network Information Theory Sormeh Shadbakht and Babak Hassibi Department of Electrical Engineering Caltech Lee Center Workshop May 25, 2007 1 Sormeh Shadbakht and Babak Hassibi Entropy
More informationExtremal Entropy: Information Geometry, Numerical Entropy Mapping, and Machine Learning Application of Associated Conditional Independences
Extremal Entropy: Information Geometry, Numerical Entropy Mapping, and Machine Learning Application of Associated Conditional Independences Ph.D. Dissertation Defense Yunshu Liu Adaptive Signal Processing
More informationLP Duality: outline. Duality theory for Linear Programming. alternatives. optimization I Idea: polyhedra
LP Duality: outline I Motivation and definition of a dual LP I Weak duality I Separating hyperplane theorem and theorems of the alternatives I Strong duality and complementary slackness I Using duality
More informationOn Multi-source Multi-Sink Hyperedge Networks: Enumeration, Rate Region. Computation, and Hierarchy. A Thesis. Submitted to the Faculty
On Multi-source Multi-Sink Hyperedge Networks: Enumeration, Rate Region Computation, and Hierarchy A Thesis Submitted to the Faculty of Drexel University by Congduan Li in partial fulfillment of the requirements
More informationReceived: 1 September 2018; Accepted: 10 October 2018; Published: 12 October 2018
entropy Article Entropy Inequalities for Lattices Peter Harremoës Copenhagen Business College, Nørre Voldgade 34, 1358 Copenhagen K, Denmark; harremoes@ieee.org; Tel.: +45-39-56-41-71 Current address:
More informationLinearly Representable Entropy Vectors and their Relation to Network Coding Solutions
2009 IEEE Information Theory Workshop Linearly Representable Entropy Vectors and their Relation to Network Coding Solutions Asaf Cohen, Michelle Effros, Salman Avestimehr and Ralf Koetter Abstract In this
More informationLecture 1: The Multiple Access Channel. Copyright G. Caire 12
Lecture 1: The Multiple Access Channel Copyright G. Caire 12 Outline Two-user MAC. The Gaussian case. The K-user case. Polymatroid structure and resource allocation problems. Copyright G. Caire 13 Two-user
More informationBelief Propagation, Information Projections, and Dykstra s Algorithm
Belief Propagation, Information Projections, and Dykstra s Algorithm John MacLaren Walsh, PhD Department of Electrical and Computer Engineering Drexel University Philadelphia, PA jwalsh@ece.drexel.edu
More information0 Sets and Induction. Sets
0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set
More informationOn Secret Sharing Schemes, Matroids and Polymatroids
On Secret Sharing Schemes, Matroids and Polymatroids Jaume Martí-Farré, Carles Padró Dep. de Matemàtica Aplicada 4, Universitat Politècnica de Catalunya, Barcelona, Spain {jaumem,cpadro}@ma4.upc.edu June
More informationNetwork Coding on Directed Acyclic Graphs
Network Coding on Directed Acyclic Graphs John MacLaren Walsh, Ph.D. Multiterminal Information Theory, Spring Quarter, 0 Reference These notes are directly derived from Chapter of R. W. Yeung s Information
More informationOn Successive Lumping of Large Scale Systems
On Successive Lumping of Large Scale Systems Laurens Smit Rutgers University Ph.D. Dissertation supervised by Michael Katehakis, Rutgers University and Flora Spieksma, Leiden University April 18, 2014
More information5 Mutual Information and Channel Capacity
5 Mutual Information and Channel Capacity In Section 2, we have seen the use of a quantity called entropy to measure the amount of randomness in a random variable. In this section, we introduce several
More informationCapacity Region of the Permutation Channel
Capacity Region of the Permutation Channel John MacLaren Walsh and Steven Weber Abstract We discuss the capacity region of a degraded broadcast channel (DBC) formed from a channel that randomly permutes
More informationInverse and Implicit Mapping Theorems (Sections III.3-III.4)
MATH W8 Daily Notes Preamble As an executive decision, I am skipping Section III.2. It is something like an 8-page lemma, with a number of concepts and results introduced at this stage mainly for the purpose
More informationFundamental rate delay tradeoffs in multipath routed and network coded networks
Fundamental rate delay tradeoffs in multipath routed and network coded networks John Walsh and Steven Weber Drexel University, Dept of ECE Philadelphia, PA 94 {jwalsh,sweber}@ecedrexeledu IP networks subject
More information1 Ensemble of three-level systems
PHYS 607: Statistical Physics. Fall 2015. Home Assignment 2 Entropy, Energy, Heat Capacity Katrina Colletti September 23, 2015 1 Ensemble of three-level systems We have an ensemble of N atoms, with N 1,
More informationCombinatorics of the Cell Decomposition of Affine Springer Fibers
Combinatorics of the Cell Decomposition of Affine Springer Fibers Michael Lugo Virginia Tech Under the Advising of Mark Shimozono 28 April 2015 Contents 1 Affine Springer Fiber 2 Affine Symmetric Group
More informationSubmodularity in Machine Learning
Saifuddin Syed MLRG Summer 2016 1 / 39 What are submodular functions Outline 1 What are submodular functions Motivation Submodularity and Concavity Examples 2 Properties of submodular functions Submodularity
More informationACO Comprehensive Exam October 18 and 19, Analysis of Algorithms
Consider the following two graph problems: 1. Analysis of Algorithms Graph coloring: Given a graph G = (V,E) and an integer c 0, a c-coloring is a function f : V {1,,...,c} such that f(u) f(v) for all
More informationSecret Sharing. Qi Chen. December 14, 2015
Secret Sharing Qi Chen December 14, 2015 What is secret sharing? A dealer: know the secret S and distribute the shares of S to each party A set of n parties P n {p 1,, p n }: each party owns a share Authorized
More informationVC-DENSITY FOR TREES
VC-DENSITY FOR TREES ANTON BOBKOV Abstract. We show that for the theory of infinite trees we have vc(n) = n for all n. VC density was introduced in [1] by Aschenbrenner, Dolich, Haskell, MacPherson, and
More informationEndotrivial modules. Nadia Mazza. June Lancaster University
Endotrivial modules Nadia Mazza Lancaster University June 2010 Contents Endotrivial modules : background Analysis An exotic turn Final remarks Set-up p a prime; k = k a field of characteristic p; G a finite
More informationClass Equation & Conjugacy in Groups
Subject: ALEBRA - V Lesson: Class Equation & Conjugacy in roups Lesson Developer: Shweta andhi Department / College: Department of Mathematics, Miranda House, University of Delhi Institute of Lifelong
More informationOrbitopes. Marc Pfetsch. joint work with Volker Kaibel. Zuse Institute Berlin
Orbitopes Marc Pfetsch joint work with Volker Kaibel Zuse Institute Berlin What this talk is about We introduce orbitopes. A polyhedral way to break symmetries in integer programs. Introduction 2 Orbitopes
More informationhal , version 1-12 Jun 2014
Applied Probability Trust (6 June 2014) NEW RESULTS ON A GENERALIZED COUPON COLLECTOR PROBLEM USING MARKOV CHAINS EMMANUELLE ANCEAUME, CNRS YANN BUSNEL, Univ. of Nantes BRUNO SERICOLA, INRIA Abstract We
More informationComputational Challenges in Perfect form theory
Computational Challenges in Perfect form theory Mathieu Dutour Sikirić Rudjer Bošković Institute, Zagreb Croatia April 24, 2018 I. Enumerating Perfect forms Notations We define S n the space of symmetric
More informationA multiplicative deformation of the Möbius function for the poset of partitions of a multiset
Contemporary Mathematics A multiplicative deformation of the Möbius function for the poset of partitions of a multiset Patricia Hersh and Robert Kleinberg Abstract. The Möbius function of a partially ordered
More informationThis document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore.
This document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore. Title Finite nilpotent metacyclic groups never violate the Ingleton inequality Author(s) Stancu, Radu; Oggier,
More informationU = γkt N. is indeed an exact one form. Thus there exists a function U(S, V, N) so that. du = T ds pdv. U S = T, U V = p
A short introduction to statistical mechanics Thermodynamics consists of the application of two principles, the first law which says that heat is energy hence the law of conservation of energy must apply
More informationSymmetry, Outer Bounds, and Code Constructions: A Computer-Aided Investigation on the Fundamental Limits of Caching
entropy Article Symmetry, Outer Bounds, and Code Constructions: A Computer-Aided Investigation on the Fundamental Limits of Caching Chao Tian Department of Electrical and Computer Engineering, Texas A&M
More informationFundamental Domains for Integer Programs with Symmetries
Fundamental Domains for Integer Programs with Symmetries Eric J. Friedman Cornell University, Ithaca, NY 14850, ejf27@cornell.edu, WWW home page: http://www.people.cornell.edu/pages/ejf27/ Abstract. We
More information18. Counting Patterns
18.1 The Problem of Counting Patterns 18. Counting Patterns For this discussion, consider a collection of objects and a group of permutation symmetries (G) that can act on the objects. An object is not
More informationRate region for a class of delay mitigating codes and P2P networks
Rate region for a class of delay mitigating codes and P2P networks Steven Weber, Congduan Li, John MacLaren Walsh Drexel University, Dept of ECE, Philadelphia, PA 19104 Abstract This paper identifies the
More informationOrbit coherence in permutation groups
Orbit coherence in permutation groups John R. Britnell Department of Mathematics Imperial College London j.britnell@imperial.ac.uk Groups St Andrews 2013 Joint work with Mark Wildon (RHUL) Orbit partitions
More informationRational Catalan Combinatorics: Intro
Rational Catalan Combinatorics: Intro Vic Reiner Univ. of Minnesota reiner@math.umn.edu AIM workshop Dec. 17-21, 2012 Goals of the workshop 1 Reinforce existing connections and forge new connections between
More informationThe Inflation Technique for Causal Inference with Latent Variables
The Inflation Technique for Causal Inference with Latent Variables arxiv:1609.00672 (Elie Wolfe, Robert W. Spekkens, Tobias Fritz) September 2016 Introduction Given some correlations between the vocabulary
More informationCounting chains in noncrossing partition lattices
Counting chains in noncrossing partition lattices Nathan Reading NC State University NCSU Algebra Seminar, November 16, 2007 1 Counting chains in noncrossing partition lattices Classical noncrossing partitions
More informationCopositive matrices and periodic dynamical systems
Extreme copositive matrices and periodic dynamical systems Weierstrass Institute (WIAS), Berlin Optimization without borders Dedicated to Yuri Nesterovs 60th birthday February 11, 2016 and periodic dynamical
More informationSome notes on Coxeter groups
Some notes on Coxeter groups Brooks Roberts November 28, 2017 CONTENTS 1 Contents 1 Sources 2 2 Reflections 3 3 The orthogonal group 7 4 Finite subgroups in two dimensions 9 5 Finite subgroups in three
More informationHodge theory for combinatorial geometries
Hodge theory for combinatorial geometries June Huh with Karim Adiprasito and Eric Katz June Huh 1 / 48 Three fundamental ideas: June Huh 2 / 48 Three fundamental ideas: The idea of Bernd Sturmfels that
More informationMATH 25 CLASS 21 NOTES, NOV Contents. 2. Subgroups 2 3. Isomorphisms 4
MATH 25 CLASS 21 NOTES, NOV 7 2011 Contents 1. Groups: definition 1 2. Subgroups 2 3. Isomorphisms 4 1. Groups: definition Even though we have been learning number theory without using any other parts
More informationChapter 2: Entropy and Mutual Information. University of Illinois at Chicago ECE 534, Natasha Devroye
Chapter 2: Entropy and Mutual Information Chapter 2 outline Definitions Entropy Joint entropy, conditional entropy Relative entropy, mutual information Chain rules Jensen s inequality Log-sum inequality
More informationSpace-Time Coding for Multi-Antenna Systems
Space-Time Coding for Multi-Antenna Systems ECE 559VV Class Project Sreekanth Annapureddy vannapu2@uiuc.edu Dec 3rd 2007 MIMO: Diversity vs Multiplexing Multiplexing Diversity Pictures taken from lectures
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationDimension reduction for semidefinite programming
1 / 22 Dimension reduction for semidefinite programming Pablo A. Parrilo Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute of Technology
More informationAmobile satellite communication system, like Motorola s
I TRANSACTIONS ON INFORMATION THORY, VOL. 45, NO. 4, MAY 1999 1111 Distributed Source Coding for Satellite Communications Raymond W. Yeung, Senior Member, I, Zhen Zhang, Senior Member, I Abstract Inspired
More informationQuick Sort Notes , Spring 2010
Quick Sort Notes 18.310, Spring 2010 0.1 Randomized Median Finding In a previous lecture, we discussed the problem of finding the median of a list of m elements, or more generally the element of rank m.
More informationOn Common Information and the Encoding of Sources that are Not Successively Refinable
On Common Information and the Encoding of Sources that are Not Successively Refinable Kumar Viswanatha, Emrah Akyol, Tejaswi Nanjundaswamy and Kenneth Rose ECE Department, University of California - Santa
More informationCOS597D: Information Theory in Computer Science September 21, Lecture 2
COS597D: Information Theory in Computer Science September 1, 011 Lecture Lecturer: Mark Braverman Scribe: Mark Braverman In the last lecture, we introduced entropy H(X), and conditional entry H(X Y ),
More informationAsymptotics of generating the symmetric and alternating groups
Asymptotics of generating the symmetric and alternating groups John D. Dixon School of Mathematics and Statistics Carleton University, Ottawa, Ontario K2G 0E2 Canada jdixon@math.carleton.ca October 20,
More informationEE595A Submodular functions, their optimization and applications Spring 2011
EE595A Submodular functions, their optimization and applications Spring 2011 Prof. Jeff Bilmes University of Washington, Seattle Department of Electrical Engineering Winter Quarter, 2011 http://ee.washington.edu/class/235/2011wtr/index.html
More informationTheorems and Definitions in Group Theory
Theorems and Definitions in Group Theory Shunan Zhao Contents 1 Basics of a group 3 1.1 Basic Properties of Groups.......................... 3 1.2 Properties of Inverses............................. 3
More information120A LECTURE OUTLINES
120A LECTURE OUTLINES RUI WANG CONTENTS 1. Lecture 1. Introduction 1 2 1.1. An algebraic object to study 2 1.2. Group 2 1.3. Isomorphic binary operations 2 2. Lecture 2. Introduction 2 3 2.1. The multiplication
More informationLecture 4: Orbits. Rajat Mittal. IIT Kanpur
Lecture 4: Orbits Rajat Mittal IIT Kanpur In the beginning of the course we asked a question. How many different necklaces can we form using 2 black beads and 10 white beads? In the question, the numbers
More informationLecture 6: Gaussian Channels. Copyright G. Caire (Sample Lectures) 157
Lecture 6: Gaussian Channels Copyright G. Caire (Sample Lectures) 157 Differential entropy (1) Definition 18. The (joint) differential entropy of a continuous random vector X n p X n(x) over R is: Z h(x
More informationSF2729 GROUPS AND RINGS LECTURE NOTES
SF2729 GROUPS AND RINGS LECTURE NOTES 2011-03-01 MATS BOIJ 6. THE SIXTH LECTURE - GROUP ACTIONS In the sixth lecture we study what happens when groups acts on sets. 1 Recall that we have already when looking
More informationOn Dependence Balance Bounds for Two Way Channels
On Dependence Balance Bounds for Two Way Channels Ravi Tandon Sennur Ulukus Department of Electrical and Computer Engineering University of Maryland, College Park, MD 20742 ravit@umd.edu ulukus@umd.edu
More informationConditional Information Inequalities for Entropic and Almost Entropic Points
1 Conditional Information Ineualities for Entropic and Almost Entropic Points Tarik Kaced and Andrei Romashchenko Abstract We study conditional linear information ineualities, i.e., linear ineualities
More informationBefore you begin read these instructions carefully.
MATHEMATICAL TRIPOS Part IA Tuesday, 4 June, 2013 1:30 pm to 4:30 pm PAPER 3 Before you begin read these instructions carefully. The examination paper is divided into two sections. Each question in Section
More informationImproving the Linear Programming Technique in the Search for Lower Bounds in Secret Sharing
Improving the Linear Programming Technique in the Search for Lower Bounds in Secret Sharing Oriol Farràs 1, Tarik Kaced 2, Sebastià Martín 3, and Carles Padró 3 1 Universitat Rovira i Virgili, Tarragona,
More informationMathematics 222a Quiz 2 CODE 111 November 21, 2002
Student s Name [print] Student Number Mathematics 222a Instructions: Print your name and student number at the top of this question sheet. Print your name and your instructor s name on the answer sheet.
More informationGIT-Equivalence and Semi-Stable Subcategories of Quiver Representations
GIT-Equivalence and Semi-Stable Subcategories of Quiver Representations Valerie Granger Joint work with Calin Chindris November 21, 2016 Notation Q = (Q 0, Q 1, t, h) is a quiver Notation Q = (Q 0, Q 1,
More informationThe extreme rays of the 5 5 copositive cone
The extreme rays of the copositive cone Roland Hildebrand March 8, 0 Abstract We give an explicit characterization of all extreme rays of the cone C of copositive matrices. The results are based on the
More informationLeast squares regularized or constrained by L0: relationship between their global minimizers. Mila Nikolova
Least squares regularized or constrained by L0: relationship between their global minimizers Mila Nikolova CMLA, CNRS, ENS Cachan, Université Paris-Saclay, France nikolova@cmla.ens-cachan.fr SIAM Minisymposium
More informationEntanglement Inequalities
Entanglement Inequalities Hirosi Ooguri Walter Burke Institute for Theoretical Physics, Caltech Kavli IPMU, University of Tokyo Nafplion, Greece, 5 11 July 2015 1/44 Which CFT's have Gravity Duals? Which
More informationEntropies & Information Theory
Entropies & Information Theory LECTURE I Nilanjana Datta University of Cambridge,U.K. See lecture notes on: http://www.qi.damtp.cam.ac.uk/node/223 Quantum Information Theory Born out of Classical Information
More informationRING ELEMENTS AS SUMS OF UNITS
1 RING ELEMENTS AS SUMS OF UNITS CHARLES LANSKI AND ATTILA MARÓTI Abstract. In an Artinian ring R every element of R can be expressed as the sum of two units if and only if R/J(R) does not contain a summand
More informationIntroduction to Information Theory
Introduction to Information Theory Impressive slide presentations Radu Trîmbiţaş UBB October 2012 Radu Trîmbiţaş (UBB) Introduction to Information Theory October 2012 1 / 19 Transmission of information
More informationA Classi cation of Dodecaphonic Groups
A Classi cation of Dodecaphonic Groups Franck Jedrzejewski CEA, Saclay (France) IRCAM, 15 janvier 2010 1/25 Franck Jedrzejewski A Classi cation of Dodecaphonic Groups Contents 1 Twelve-Tone Technique 2
More informationHidden Symmetry Subgroup Problems
1/27 Hidden Symmetry Subgroup Problems Miklos Santha CNRS, Université Paris Diderot, France and Centre for Quantum Technologies, NUS, Singapore joint work with Thomas Decker Gábor Ivanyos Pawel Wocjan
More informationPartial, Total, and Lattice Orders in Group Theory
Partial, Total, and Lattice Orders in Group Theory Hayden Harper Department of Mathematics and Computer Science University of Puget Sound April 23, 2016 Copyright c 2016 Hayden Harper. Permission is granted
More informationMaximum Correlation Analysis of Nonlinear S-boxes in Stream Ciphers
Maximum Correlation Analysis of Nonlinear S-boxes in Stream Ciphers Muxiang Zhang 1 and Agnes Chan 2 1 GTE Laboratories Inc., 40 Sylvan Road LA0MS59, Waltham, MA 02451 mzhang@gte.com 2 College of Computer
More informationA Hilton-Milner-type theorem and an intersection conjecture for signed sets
A Hilton-Milner-type theorem and an intersection conjecture for signed sets Peter Borg Department of Mathematics, University of Malta Msida MSD 2080, Malta p.borg.02@cantab.net Abstract A family A of sets
More informationPattern avoidance in compositions and multiset permutations
Pattern avoidance in compositions and multiset permutations Carla D. Savage North Carolina State University Raleigh, NC 27695-8206 Herbert S. Wilf University of Pennsylvania Philadelphia, PA 19104-6395
More informationCutting planes from extended LP formulations
Cutting planes from extended LP formulations Merve Bodur University of Wisconsin-Madison mbodur@wisc.edu Sanjeeb Dash IBM Research sanjeebd@us.ibm.com March 7, 2016 Oktay Günlük IBM Research gunluk@us.ibm.com
More informationMath 4320, Spring 2011
Math 4320, Spring 2011 Prelim 2 with solutions 1. For n =16, 17, 18, 19 or 20, express Z n (A product can have one or more factors.) as a product of cyclic groups. Solution. For n = 16, G = Z n = {[1],
More informationTopological properties
CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological
More informationReverse Edge Cut-Set Bounds for Secure Network Coding
Reverse Edge Cut-Set Bounds for Secure Network Coding Wentao Huang and Tracey Ho California Institute of Technology Michael Langberg University at Buffalo, SUNY Joerg Kliewer New Jersey Institute of Technology
More informationarxiv: v2 [math.co] 11 Oct 2016
ON SUBSEQUENCES OF QUIDDITY CYCLES AND NICHOLS ALGEBRAS arxiv:1610.043v [math.co] 11 Oct 016 M. CUNTZ Abstract. We provide a tool to obtain local descriptions of quiddity cycles. As an application, we
More informationExtending Brickell-Davenport Theorem to Non-Perfect Secret Sharing Schemes
Extending Brickell-Davenport Theorem to Non-Perfect Secret Sharing Schemes Oriol Farràs 1 and Carles Padró 2 1 Universitat Rovira i Virgili, Tarragona, Catalonia, Spain 2 Nanyang Technological University,
More information