Two dynamical systems in the space of triangles

Kochetkov Yury

Working paper

Two dynamical systems in the space of triangles

Cornell university preprint series. Arxiv. math. Cornell university, 2021. No. 2101.03734.

Kochetkov Y.

Let M be the space of triangles, defined up to shifts, rotations and dilations. We define two maps f:M->M and g:M->M. The map f corresponds to a triangle of perimeter pi the triangle with angles numerically equal to edges of the initial triangle. The map g corresponds to a triangle of perimeter 2pi the triangle with exterior angles numerically equal to edges of the initial triangle. For p in M the sequence {p,f(p),f(f(p)),...} converges to the equilateral triangle and the sequence {p,g(p),g(g(p)),...} converges to the "degenerate triangle" with angles (0,0,pi). In Supplement an analogous problem about inscribed-circumscribed quadrangles is discussed.

Research target: Mathematics

Language: English

Full text (PDF, 144 Kb)

Keywords: dynamical system

Contemporary Mathematics 744 Dynamics: Topology and Numbers (2020)

Edited by: P. Moree, A. Pohl, L. Snoha et al. United States of America: American Mathematical Society, 2020.

Added: Nov 7, 2020

Optimal Impulse Control of Dynamical Systems

Tumanov M., Piunovskly A., Plakhov A. Optimal Control Applications and Methods. 2019. P. 1-21.

Using the tools of the Markov Decision Processes, we justify the dynamic programming approach to the optimal impulse control of deterministic dynamical systems. We prove the equivalence of the integral and diﬀerential forms of the optimality equation. The theory is illustrated by an example from mathematical epidemiology. The developed methods can be also useful for the study of piecewise deterministic Markov processes.

Added: Feb 27, 2020

Modeling of Microblogging Social Networks: Dynamical System vs. Random Dynamical System

Dmitriev A., Tsukanova O. A., Maltseva S. V. In bk.: Procedia Computer Science. Vol. 122: 5th International Conference on Information Technology and Quantitative Management. Elsevier, 2017. P. 812-819.

Building of adequate dynamical models of microblogging social networks is a topical task that is of interest from both theoretical and practical aspects. Experimental and theoretical results of studies related to choice of the adequate model are presented. The choice was made between two models: a nonlinear dynamical system and a nonlinear random dynamical system. By results of the fractal analysis of observable network time series and defining their probability density function it was established that the nonlinear random dynamical system was more adequate than the nonlinear dynamical system. The character of the observable time series was also explored. The possibility that microblogging social networks can be analyzed by means of Tsallis entropy and self-organized criticality is examined.

Added: Oct 3, 2018

Ergodic Properties of tame dynamical systems

A.V. Romanov. arxiv.org. math. Cornell University, 2018. No. 1806.09132.

We study the problem on the weak-star decomposability of a topological N0-dynamical system (Ω, '), where ' is an endomorphism of a metric compact set Ω, into ergodic components in terms of the associated enveloping semigroups. In the tame case (where the Ellis semigroup E(Ω,') consists of B1-transformations Ω → Ω), we show that (i) the desired decomposition exists for an appropriate choice of the generalized sequential averaging method; (ii) every sequence of weighted ergodic means for the shift operator x → x ◦ ', x ∈ C(Ω), contains a pointwise convergent subsequence. We also discuss the relationship between the statistical properties of (Ω, ') and the mutual structure of minimal sets and ergodic measures.

Added: Jul 3, 2018

Деловой климат в оптовой торговле в I квартале 2012 года

Лола И. С., Лукашина Ж. Е., Пахомова О. П. М.: НИУ ВШЭ, 2012.

Added: May 21, 2012

Model for organizing cargo transportation with an initial station of departure and a final station of cargo distribution

Khachatryan N., Akopov A. S. Business Informatics. 2017. No. 1(39). P. 25-35.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Added: Apr 18, 2017

Nullstellensatz over quasi-fields

Trushin D. Russian Mathematical Surveys. 2010. Vol. 65. No. 1. P. 186-187.

Added: Feb 20, 2018

Laminations from the Main Cubioid

Timorin V., Blokh A., Oversteegen L. et al. arxiv.org. math. Cornell University, 2013. No. 1305.5788.

According to a recent paper \cite{bopt13}, polynomials from the closure $\ol{\phd}_3$ of the {\em Principal Hyperbolic Domain} ${\rm PHD}_3$ of the cubic connectedness locus have a few specific properties. The family $\cu$ of all polynomials with these properties is called the \emph{Main Cubioid}. In this paper we describe the set $\cu^c$ of laminations which can be associated to polynomials from $\cu$.

Added: Oct 6, 2013

Entropy and the Shannon-McMillan-Breiman theorem for beta random matrix ensembles

Bufetov A. I., Mkrtchyan S., Scherbina M. et al. arxiv.org. math. Cornell University, 2013. No. 1301.0342.

Added: Feb 21, 2013

Математическая статистика

Энатская Н. Ю., Хакимуллин Е. Р. М.: МИЭМ, 2011.

Added: Apr 26, 2012

Деловой климат в оптовой торговле во II квартале 2014 года и ожидания на III квартал

Лола И. С., Остапкович Г. В. Современная торговля. 2014. № 10.

Added: Jan 29, 2015

Введение в математическую статистику

Ивченко Г. И., Медведев Ю. И. М.: ЛКИ, 2010.

Added: Apr 12, 2012

Bounded limit cycles of polynomial foliations of ℂP²

Goncharuk N. B., Kudryashov Y. arxiv.org. math. Cornell University, 2015. No. 1504.03313.

In this article we prove in a new way that a generic polynomial vector field in ℂ² possesses countably many homologically independent limit cycles. The new proof needs no estimates on integrals, provides thinner exceptional set for quadratic vector fields, and provides limit cycles that stay in a bounded domain.

Added: Apr 15, 2015