scholarly journals ADDITIVITY PROPERTIES OF SOFIC ENTROPY AND MEASURES ON MODEL SPACES

2016 ◽  
Vol 4 ◽  
Author(s):  
TIM AUSTIN
Keyword(s):  
Lower Bound ◽  
Probability Distributions ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Entropy Theory ◽  
Reverse Inequality ◽  
Cartesian Products ◽  
Model Spaces ◽  
Sofic Entropy ◽  
Sofic Groups

Sofic entropy is an invariant for probability-preserving actions of sofic groups. It was introduced a few years ago by Lewis Bowen, and shown to extend the classical Kolmogorov–Sinai entropy from the setting of amenable groups. Some parts of Kolmogorov–Sinai entropy theory generalize to sofic entropy, but in other respects this new invariant behaves less regularly. This paper explores conditions under which sofic entropy is additive for Cartesian products of systems. It is always subadditive, but the reverse inequality can fail. We define a new entropy notion in terms of probability distributions on the spaces of good models of an action. Using this, we prove a general lower bound for the sofic entropy of a Cartesian product in terms of separate quantities for the two factor systems involved. We also prove that this lower bound is optimal in a certain sense, and use it to derive some sufficient conditions for the strict additivity of sofic entropy itself. Various other properties of this new entropy notion are also developed.

scholarly journals On Cartesian products with small crossing numbers

2012 ◽  
Vol 28 (1) ◽  
pp. 67-75
Author(s):  
MARIAN KLESC ◽  
◽  
JANA PETRILLOVA ◽  
Keyword(s):  
Cartesian Product ◽  
Sufficient Conditions ◽  
Crossing Number ◽  
Necessary And Sufficient Conditions ◽  
Line Graphs ◽  
Cartesian Products ◽  
Crossing Numbers ◽  
Necessary And Sufficient

Kulli at al. started to characterize line graphs with crossing number one. In this paper, the similar problems were solved for the Cartesian products of two graphs. The necessary and sufficient conditions are given for all pairs of graphs G1 and G2 for which the crossing number of their Cartesian product G1 × G2 is one or two.

scholarly journals Connected domination game played on Cartesian products

2019 ◽  
Vol 17 (1) ◽  
pp. 1269-1280 ◽  
Author(s):  
Csilla Bujtás ◽  
Pakanun Dokyeesun ◽  
Vesna Iršič ◽  
Sandi Klavžar
Keyword(s):  
Lower Bound ◽  
Domination Number ◽  
Cartesian Product ◽  
Connected Subgraph ◽  
Arbitrary Graph ◽  
Cartesian Products ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Game Domination Number ◽  
Domination Game

Abstract The connected domination game on a graph G is played by Dominator and Staller according to the rules of the standard domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of G. If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the connected game domination number of G. Here this invariant is studied on Cartesian product graphs. A general upper bound is proved and demonstrated to be sharp on Cartesian products of stars with paths or cycles. The connected game domination number is determined for Cartesian products of P3 with arbitrary paths or cycles, as well as for Cartesian products of an arbitrary graph with Kk for the cases when k is relatively large. A monotonicity theorem is proved for products with one complete factor. A sharp general lower bound on the connected game domination number of Cartesian products is also established.

Generation of the alternating group by modular additions

2019 ◽  
Vol 29 (5) ◽  
pp. 303-309
Author(s):  
Fedor M. Malyshev
Keyword(s):  
Permutation Group ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Local Action ◽  
Alternating Group ◽  
Permutation Groups ◽  
Cartesian Products ◽  
Strong Connectedness ◽  
Necessary And Sufficient

Abstract The paper is concerned with systems of generators of permutation groups on Cartesian products of residue rings. Each separate permutation from the system of generators is constructed on the basis of additions, is characterized by the local action, and leaves fixed the major parts of the components of the element being transformed. A criterion of 2-transitivity of the generated permutation group is given in the form of the strong connectedness of the digraph which corresponds to the system of generators and which is defined on the set of numbers of residue rings in the Cartesian product. Necessary and sufficient conditions under which this group contains an alternating group are formulated.

scholarly journals The Geometry of Model Spaces for Probability-Preserving Actions of Sofic Groups

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Tim Austin
Keyword(s):  
Metric Spaces ◽  
Metric Geometry ◽  
Bernoulli Shifts ◽  
Additional Information ◽  
Covering Numbers ◽  
Model Spaces ◽  
Connectedness Property ◽  
Sofic Entropy ◽  
Sofic Groups

AbstractBowen’s notion of sofic entropy is a powerful invariant for classifying probability-preserving actions of sofic groups. It can be defined in terms of the covering numbers of certain metric spaces associated to such an action, the ‘model spaces’. The metric geometry of these model spaces can exhibit various interesting features, some of which provide other invariants of the action. This paper explores an approximate connectedness property of the model spaces, and uses it give a new proof that certain groups admit factors of Bernoulli shifts which are not Bernoulli. This was originally proved by Popa. Our proof covers fewer examples than his, but provides additional information about this phenomenon.

scholarly journals Peg Solitaire on Cartesian Products of Graphs

2021 ◽  
Vol 37 (3) ◽  
pp. 907-917
Author(s):  
Martin Kreh ◽  
Jan-Hendrik de Wiljes
Keyword(s):  
Cartesian Product ◽  
Cartesian Products ◽  
Connected Graphs ◽  
Grid Graphs ◽  
Cartesian Products Of Graphs

AbstractIn 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. In the same article, the authors proved some results on the solvability of Cartesian products, given solvable or distance 2-solvable graphs. We extend these results to Cartesian products of certain unsolvable graphs. In particular, we prove that ladders and grid graphs are solvable and, further, even the Cartesian product of two stars, which in a sense are the “most” unsolvable graphs.

On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences

2021 ◽  
Vol 71 (6) ◽  
pp. 1375-1400
Author(s):  
Feyzi Başar ◽  
Hadi Roopaei
Keyword(s):  
Lower Bound ◽  
Sequence Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Transformations ◽  
Infinite Matrix ◽  
The Matrix ◽  
Alpha Beta ◽  
Necessary And Sufficient ◽  
Bounded Sequences

Abstract Let F denote the factorable matrix and X ∈ {ℓp , c 0, c, ℓ ∞}. In this study, we introduce the domains X(F) of the factorable matrix in the spaces X. Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces X(F). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes (ℓ p (F), ℓ ∞), (ℓ p (F), f) and (X, Y(F)) of matrix transformations, where Y denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix F and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix F. Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix.

scholarly journals Invariant Bipartite Random Graphs on Rd

2014 ◽  
Vol 51 (3) ◽  
pp. 769-779
Author(s):  
Fabio Lopes
Keyword(s):  
Point Processes ◽  
Random Number ◽  
Probability Distributions ◽  
Sufficient Conditions ◽  
Stable Marriage ◽  
Simple Point ◽  
Stable Marriage Problem ◽  
Blue Point ◽  
Translation Invariant ◽  
Positive Integers

Suppose that red and blue points occur in Rd according to two simple point processes with finite intensities λR and λB, respectively. Furthermore, let ν and μ be two probability distributions on the strictly positive integers with means ν̅ and μ̅, respectively. Assign independently a random number of stubs (half-edges) to each red (blue) point with law ν (μ). We are interested in translation-invariant schemes for matching stubs between points of different colors in order to obtain random bipartite graphs in which each point has a prescribed degree distribution with law ν or μ depending on its color. For a large class of point processes, we show that such translation-invariant schemes matching almost surely all stubs are possible if and only if λRν̅ = λBμ̅, including the case when ν̅ = μ̅ = ∞ so that both sides are infinite. Furthermore, we study a particular scheme based on the Gale-Shapley stable marriage problem. For this scheme, we give sufficient conditions on ν and μ for the presence and absence of infinite components. These results are two-color versions of those obtained by Deijfen, Holroyd and Häggström.

scholarly journals Graphs With Stability Index One

1976 ◽  
Vol 22 (2) ◽  
pp. 212-220 ◽  
Author(s):  
D. A. Holton ◽  
J. A. Sims
Keyword(s):  
Automorphism Group ◽  
Large Class ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Stability Index ◽  
Arbitrary Graph ◽  
Stability Properties ◽  
The Stability

AbstractWe consider the effect on the stability properties of a graph G, of the presence in the automorphism group of G of automorphisms (uv)h, where u and v are vertices of G, and h is a permutation of vertices of G excluding u and v. We find sufficient conditions for an arbitrary graph and a cartesian product to have stability index one, and conjecture in the latter case that they are necessary. Finally we exhibit explicitly a large class of graphs which have stability index one.

Application of Stieltjes theory for S-fractions to birth and death processes

1983 ◽  
Vol 15 (03) ◽  
pp. 507-530 ◽  
Author(s):  
G. Bordes ◽  
B. Roehner
Keyword(s):  
Asymptotic Behavior ◽  
Lower Bound ◽  
Special Class ◽  
Jacobi Matrix ◽  
General Theorem ◽  
Sufficient Conditions ◽  
Death Process ◽  
Nearest Neighbour ◽  
Birth And Death Process ◽  
Birth And Death Processes

We are interested in obtaining bounds for the spectrum of the infinite Jacobi matrix of a birth and death process or of any process (with nearest-neighbour interactions) defined by a similar Jacobi matrix. To this aim we use some results of Stieltjes theory for S-fractions, after reviewing them. We prove a general theorem giving a lower bound of the spectrum. The theorem also gives sufficient conditions for the spectrum to be discrete. The expression for the lower bound is then worked out explicitly for several, fairly general, classes of birth and death processes. A conjecture about the asymptotic behavior of a special class of birth and death processes is presented.

scholarly journals Multivariate Scale-Mixed Stable Distributions and Related Limit Theorems

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 749 ◽  
Author(s):  
Yury Khokhlov ◽  
Victor Korolev ◽  
Alexander Zeifman
Keyword(s):  
Limit Theorems ◽  
Probability Distributions ◽  
Sufficient Conditions ◽  
Stable Distributions ◽  
Necessary And Sufficient Conditions ◽  
Independent Random Variables ◽  
Random Vectors ◽  
Scale Mixtures ◽  
Elliptically Contoured ◽  
Necessary And Sufficient

In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate stable distributions. Multivariate analogs of the Mittag–Leffler distribution are introduced. Some properties of these distributions are discussed. The main focus is on the representations of the corresponding random vectors as products of independent random variables and vectors. In these products, relations are traced of the distributions of the involved terms with popular probability distributions. As examples of distributions of the class of scale mixtures of multivariate stable distributions, multivariate generalized Linnik distributions and multivariate generalized Mittag–Leffler distributions are considered in detail. Their relations with multivariate ‘ordinary’ Linnik distributions, multivariate normal, stable and Laplace laws as well as with univariate Mittag–Leffler and generalized Mittag–Leffler distributions are discussed. Limit theorems are proved presenting necessary and sufficient conditions for the convergence of the distributions of random sequences with independent random indices (including sums of a random number of random vectors and multivariate statistics constructed from samples with random sizes) to scale mixtures of multivariate elliptically contoured stable distributions. The property of scale-mixed multivariate elliptically contoured stable distributions to be both scale mixtures of a non-trivial multivariate stable distribution and a normal scale mixture is used to obtain necessary and sufficient conditions for the convergence of the distributions of random sums of random vectors with covariance matrices to the multivariate generalized Linnik distribution.

Sign in / Sign up

Export Citation Format

Share Document