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## Entropy and Complexity of Polygonal Billiards with Spy mirrors

Cornell University
,
2015.
No. 1501.04584.

Skripchenko A., Troubetzkoy S.

We prove that a polygonal billiard with one-sided mirrors has zero
topological entropy. In certain cases we show sub exponential and for other
polynomial estimates on the complexity.

Gromov V., Мазайшвили К. В., Заикин П. В. et al., Вестник кибернетики 2022 Т. 45 № 1 С. 72-82

The prevalence of chronic kidney disease is growing every year and is already comparable to such socially significant diseases as hypertension and diabetes mellitus, as well as obesity and metabolic syndrome [1,2]. The standard solution for hemodialysis patients is to create a permanent vascular access in the form of an arteriovenous fistula. However, its use ...

Added: November 14, 2022

Rubchinskiy A., , in : Models, Algorithms, and Technologies for Network Analysis. Springer Proceedings in Mathematics & Statistics. Vol. 197.: Springer, 2017. P. 127-152.

A new approach to network decomposition problems (and, hence, to classification problems, presented in network form) is suggested. Opposite to the conventional approach, consisting in construction of one, “the most correct” decomposition (classification), the suggested approach is focused on construction of a family of classifications. Basing on this family, two numerical indices are introduced and ...

Added: October 20, 2017

Apenko S.M., Physica A: Statistical Mechanics and its Applications 2012 Vol. 391 No. 1-2 P. 62-77

We present a possible approach to the study of the renormalization group (RG) flow based
entirely on the information theory. The average information loss under a single step of
Wilsonian RG transformation is evaluated as a conditional entropy of the fast variables,
which are integrated out, when the slow ones are held fixed. Its positivity results in the
monotonic ...

Added: October 23, 2014

Glutsyuk A., Pacific Journal of Mathematics 2020 Vol. 305 No. 2 P. 577-595

We consider two nested billiards in ℝd, d≥3, with C2-smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the billiards are confocal ellipsoids. This together with the previous analogous result of the author in two dimensions solves completely the Commuting Billiard Conjecture due ...

Added: October 12, 2019

Bufetov A. I., Mkrtchyan S., Scherbina M. et al., Journal of Statistical Physics 2013 Vol. 152 No. 1 P. 1-14

We show that beta ensembles in Random Matrix Theory with generic real analytic potential have the asymptotic equipartition property. In addition, we prove a Central Limit Theorem for the density of the eigenvalues of these ensembles. ...

Added: March 13, 2014

Bezhaeva Z., Oseledets V. I., Theory of Probability and Its Applications 2013 Vol. 57 No. 1 P. 135-144

The formula for calculating the entropy and the Hausdorff dimension of an invariant Erdos measure for the pseudogolden ratio and all values Bernoulli parameter is obtained. This formula make possible calculating the entropy and the Hausdorff dimension with high accuracy. ...

Added: February 15, 2014

Granata D., Behdani B., Pardalos P. M., Journal of Combinatorial Optimization 2012 Vol. 24 No. 4 P. 459-467

We address the complexity class of several problems related to finding a path in a properly colored directed graph. A properly colored graph is defined as a graph G whose vertex set is partitioned into X(G) stable subsets, where X(G) denotes the chromatic number of G. We show that to find a simple path that ...

Added: January 24, 2013

Bufetov A. I., Geometric and Functional Analysis 2012 Vol. 22 No. 4 P. 938-975

Vershik and Kerov conjectured in 1985 that dimensions of irreducible representations of finite symmetric groups, after appropriate normalization, converge to a constant with respect to the Plancherel family of measures on the space of Young diagrams. The statement of the Vershik-Kerov conjecture can be seen as an analogue of the Shannon-McMillan-Breiman Theorem for the non-stationary ...

Added: October 18, 2012

Apenko S.M., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2013 Vol. 87 P. 024101-024101-4

We present a proof of the monotonic entropy growth for a nonlinear discrete-time model of a random market.
This model, based on binary collisions, also may be viewed as a particular case of Ulam’s redistribution of energy
problem. We represent each step of this dynamics as a combination of two processes. The first one is a linear
energy-conserving ...

Added: October 23, 2014

Glutsyuk A., / Cornell University. Series "Working papers by Cornell University". 2018.

We consider two nested billiards in Rd, d >2, with C2-smooth
strictly convex boundaries. We prove that if the corresponding actions
by reflections on the space of oriented lines commute, then the billiards
are confocal ellipsoids. This together with the previous analogous result of the author in two dimensions solves completely the Commuting
Billiard Conjecture due to Sergei Tabachnikov. ...

Added: August 21, 2018

Dubnov Y. A., Информационные технологии и вычислительные системы 2018 № 2 С. 60-69

The paper considers the problem of reducing the dimension of the feature space for describing objects
in data analysis problems using the example of binary classification. The article provides a detailed
overview of existing approaches to solving this problem and proposes several modifications. In which
the dimensionality reduction is considered as the problem of extracting the most relevant ...

Added: July 4, 2018

Blank M., Problems of Information Transmission 2014 Vol. 50 No. 4 P. 350-363

We study functional consequences of the interlacing property consisting in
that a new configuration of “particles” occurs in gaps between elements of
a previous configuration. This property was introduced by I.M. Gelfand in
terms of spectra of sequences of matrices of increasing dimensions and turned
out to be highly needed in many areas of modern mathematics. We examine
conditions under ...

Added: March 20, 2015

Bezhaeva Z., Oseledets V. I., Journal of Dynamical and Control Systems 2013 Vol. 19 No. 2 P. 301-308

Consider a sofic dynamical system. We obtain an explcit formula for the KS-entropy of sofic dynamicsl system of Blackwell's type. ...

Added: August 22, 2013

Glutsyuk A., Journal of the European Mathematical Society 2021 Vol. 23 No. 3 P. 995-1049

We present a solution of the algebraic version of Birkhoff Conjecture on integrable billiards. Namely we show that every polynomially integrable real bounded convex planar billiard with smooth boundary is an ellipse. We extend this result to billiards with piecewise-smooth and not necessarily convex boundary on arbitrary two-dimensional surface of constant curvature: plane, sphere, Lobachevsky ...

Added: October 12, 2019

Rubchinskiy A., / Высшая школа экономики. Series WP7 "Математические методы анализа решений в экономике, бизнесе и политике". 2015. No. WP7/2015/09.

An algorithm of solution of the Automatic Classification (AC for brevity) problem is set forth in the paper. In the AC problem, it is required to find one or several partitions, starting with the given pattern matrix or dissimilarity / similarity matrix. The three-level scheme of the algorithm is suggested. The output of the procedure ...

Added: October 19, 2017

Popkov Y., Popkov A., Dubnov Y. A., Автоматика и телемеханика 2020 № 7 С. 148-172

A randomized forecasting method based on the generation of ensembles of entropy-optimal forecasting trajectories is developed. The latter are generated by randomized dynamic regression models containing random parameters, measurement noises, and a random input. The probability density functions of random parameters and measurement noises are estimated using real data within the randomized machine learning procedure. ...

Added: October 31, 2020

Los A., Вильбоа Н. В., Mironkin V., Обозрение прикладной и промышленной математики 2016 Т. 23 № 1 С. 3-16

The tasks associated with finding some information characteristics (entropy, the distribution of m-grams, syllables and phrases), the using of which allows to build the predictive models for the development of natural languages, are considered. The texts in Russian, English, German, French and Georgian languages, as classified by time intervals and styles, are experimental studied. ...

Added: May 8, 2016

Bezhaeva Z., Oseledets V. I., Journal of Dynamical and Control Systems 2013 Vol. 19 No. 4 P. 569-573

Algorithm is given for computation of the Hausdorff dimension of the support of the Erdos measure for a Pisot number. ...

Added: November 11, 2013

Alekseev V., Zakharova D. V., Malyshev D. et al., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Математика 2012 № 6(1) С. 115-120

Рассматриваются вопросы асимптотического перечисления наследственных классов графов и их структурного описания, исследуется сложность некоторых задач на таких классах. ...

Added: May 17, 2013

49606783, Nazaikinskii V. E., Mathematical notes 2016 Vol. 100 No. 3 P. 421-428

For an arithmetic semigroup (G, ∂), we define entropy as a function on a naturally defined continuous semigroup Ĝ containing G. The construction is based on conditional maximization, which permits us to introduce the conjugate variables and the Lagrangian manifold corresponding to the semigroup (G, ∂). ...

Added: December 10, 2016

Kanovich M., Ban Kirigin T., Nigam V. et al., Computer Languages, Systems & Structures 2014 No. 40 P. 137-154

It is well-known that the Dolev–Yao adversary is a powerful adversary. Besides acting as the network, intercepting, decomposing, composing and sending messages, he can remember as much information as he needs. That is, his memory is unbounded. We recently proposed a weaker Dolev–Yao like adversary, which also acts as the network, but whose memory is ...

Added: March 23, 2015

Karzhemanov I., Zhdanovskiy I., European Journal of Mathematics 2018 Vol. 4 No. 1 P. 326-329

We consider the so-called surjective rational maps. We study how the surjectivity property behaves in families of rational maps. Some (counter) examples are provided and a general result is proved. ...

Added: March 29, 2018

Glutsyuk A., / Cornell University. Series "Working papers by Cornell University". 2019.

For a given closed convex planar curve γ with smooth boundary and a given p>0, the string construction yields a family of curves Γp for which γ is a caustic. The action of the reflection Tp on the tangent lines to γ≃S1 induces its action on the tangency points: a circle diffeomorphism p:γ→γ. We say ...

Added: November 12, 2019

St. Petersburg : The Euler International Mathematical Institute, 2014

The conference Philosophy, Mathematics, Linguistics: Aspects of Interaction 2014 (PhML-2014) is a sequel in the series of conferences intended to provide a forum for philosophers, mathematicians, linguists, logicians, and computer scientists who share an interest in cross-disciplinary research. The conference PhML-2014 is endorsed by the American National Committee of the Division of Logic, Methodology and Philosophy ...

Added: June 23, 2014