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Makar Plakhotnyk

Publications and source records attributed to Makar Plakhotnyk.

14 recordsLinked to original sources
arXiv2021

The cone of quasi-semimetrics and exponent matrices of tiled orders

Finite quasi semimetrics on $n$ can be thought of as nonnegative valuations on the edges of a complete directed graph on $n$ vertices satisfying all possible triangle inequalities. They comprise a polyhedral cone whose symmetry groups were studied for small $n$ by Deza, Dutour and Panteleeva. We show that the symmetry and combinatorial symmetry groups are as they conjectured. Integral quasi semimetrics have apecial place in the theory of tiled orders, being known as exponent matrices, and can be viewed as monoids under componentwise maximum; we provide a novel derivation of the automorphism group of that monoid. Some of these results follow from more general consideration of polyhedral cones that are closed under componentwise maximum.

math.CO↗
arXiv2019

Classification of commutative pairs of surjective maps of interval, one of which is unimodal

We introduce here a classification of unimodal maps $[0, 1]\rightarrow [0, 1]$, which commute with piecewise linear surjective maps $[0, 1]\rightarrow [0, 1]$. Remind that if continuous piecewise linear unimodal map $g$ commutes with a non-constant piecewise linear map $ψ$, which is not an iteration of $g$, then $g$ is topologically conjugated with the tent map by piecewise linear conjugacy. We use the obtained classification to illustrate the mentioned fact.

math.DS↗
arXiv2019

Piecewise linear unimodal maps with non-trivial continuous piecewise linear commutator

Let $g:\, [0, 1]\rightarrow [0, 1]$ be piecewise linear unimodal map. We say that $g$ has non-trivial piecewise linear commutator, if there is a continuous piecewise linear $ψ:\, [0, 1]\rightarrow [0, 1]$ such that $g\circ ψ= ψ\circ g$, and, moreover, $ψ$ is neither an iteration of $g$, not a constant map. We prove that if $g$ has a non-trivial piecewise linear commutator, then $g$ is topologically conjugated with the tent map by a piecewise linear conjugacy.

math.DS↗
arXiv2018

On one generalization of skew tent maps

We generalize in this work the properties of the conjugacy of skew tent maps. It is known that the conjugacy $h$ from a skew tent map $g_1$ to $g_2$ is differentiable at a point $x^*$ if and only if there exists left and right limits $\lim\limits_{n\rightarrow \infty}h_n'(x^*-)$ and $\lim\limits_{n\rightarrow \infty}h_n'(x^*+)$, where $h_n$ is a piecewise linear function, which coincides with $h$ at $g_1^{-n}(0)$, and all whose kinks belong to $g_1^{-n}(0)$. The attempts to generalize this result to some reacher class of unimodal maps is natural. For this reason we introduce the class of piecewise linear maps, all whose kinks are in the complete pre-image of $0$ and study the relation of the differential properties of their conjugacy with ones of the mentioned approximation~$(h_n)_{n\geq 1}$.

math.DS↗
arXiv2017

The derivative of the conjugacy for the pair of tent-like maps from an interval into itself

We consider in this article the properties of the topological conjugacy of the piecewise linear unimodal maps $g:\, [0,\, 1]\rightarrow [0,\, 1]$, all whose kinks belong to the complete pre-image of $0$. We call such maps firm carcass maps. We prove that every firm carcass maps $g_1$ and $g_2$ are topologically conjugated. For the conjugacy $h$ such that $h\circ g_1 = g_2\circ h$ we denote $\{ h_n, n\geq 1\}$ the piecewise linear approximations of $h$, whose graphs connect the points $\{ (x, h(x)),\ g_1^n(x)=0\}$. For any $x\in [0,\, 1]$ we reduce the question about the value of $h'(x)$ to the properties of the sequence $\{h_n'(x),\, n\geq 1\}$. We prove that each conjugacy of firm carcass maps either has the length 2, or is piecewise linear.

math.DS↗
arXiv2017

Self-semiconjugation of piecewise linear unimodal maps

We devote this work to the functional equation $ψ\circ g = g\circ ψ$, where $ψ$ is an unknown function and $g$ is piecewise linear unimodal map, which is topologically conjugated to the tent map. We will call such $ψ$ self-semiconjugations of $g$. Our the main results are the following: 1. Suppose that there is a self-semiconjugation of $g$, whose tangent at $0$ is not a power of $2$, and suppose that all the kinks of $g$ are in the complete pre-image of $0$. Then all the self-semiconjugations of $g$ are piecewise linear. 2. Suppose that all self-semiconjugations of $g$ are piecewise linear. Then the conjugacy of $g$ and the tent map is piecewise linear.

math.DS↗
arXiv2017

Self semi conjugations of Ulam's Tent-map

We study the self-semiconjugations of the Tent-map $f:\, x\mapsto 1-|2x-1|$ for $x\in [0,\, 1]$. We prove that each of these semi-conjugations $ξ$ is piecewise linear. For any $n\in \mathbb{N}$ we denote $A_n = f^{-n}(0)$ and describe the maps $ψ:\, A_n\rightarrow [0,\, 1]$ such that $ψ\circ f = f\circ ψ$. Also we describe all possible restrictions, of self-semiconjugations of the Tent-map onto $A_n$ and prove that for any $α\in A_n\setminus A_{n-1}$ a restriction is completely determined by its value at $α$.

math.DS↗
arXiv2017

The max-plus algebra of exponent matrices of tiled orders

An exponent matrix is an $n\times n$ matrix $A=(a_{ij})$ over ${\mathbb N}^0$ satisfying (1) $a_{ii}=0$ for all $i=1,\ldots, n$ and (2) $a_{ij}+a_{jk}\geq a_{ik}$ for all pairwise distinct $i,j,k\in\{1,\dots, n\}$. In the present paper we study the set ${\mathcal E}_n$ of all non-negative $n\times n$ exponent matrices as an algebra with the operations $\oplus$ of component-wise maximum and $\odot$ of component-wise addition. We provide a basis of the algebra $({\mathcal E}_n, \oplus, \odot,0)$ and give a row and a column decompositions of a matrix $A\in {\mathcal E}_n$ with respect to this basis. This structure result determines all $n\times n$ tiled orders over a fixed discrete valuation ring. We also study automorphisms of ${\mathcal E}_n$ with respect to each of the operations $\oplus$ and $\odot$ and prove that ${\rm Aut}(\mathcal{E}_n,\, \odot ) = {\rm Aut}(\mathcal{E}_n,\, \oplus ) = {\rm Aut}(\mathcal{E}_n,\, \odot ,\oplus ,0) \simeq {\mathcal{S}}_n \times C_2,$$n>2.$

math.RA↗
arXiv2017

Kepler's laws with introduction to differential calculus

We explain the solution of the following two problems: obtaining of Kepler's laws from Newton's laws (so called two bodies problem) and obtaining the fourth Newton's law (the formula for gravitation) as a corollary of Kepler's laws. This small book is devoted to the scholars, who are interested in physics and mathematics. We also make a series of digressions, where explain some technique of the higher mathematics, which are used in the proofs.

math.HO↗
arXiv2016

On the history of the Ulam's Conjugacy

We show the results on the history of the invention of the conjugacy $h(x)=\frac{2}π\arcsin\sqrt{x}$ of one-dimensional $[0,\, 1]\rightarrow [0,\, 1]$ maps $f(x)=4x(1-x)$ and $g(x)=1-|1-2x|$.

math.HO↗
arXiv2016

Topological conjugation of one dimensional maps

Topological conjugateness of one dimensional unimodal dynamical systems, which are generated by interval [0, 1] into itself maps are studied. We study the smoothness and differentiability of the conjugacy of symmetrical and non-symmetrical tent maps. Also weprove the extremal property of the length of the graph of the conjugacy of symmetrical and non-symmetrical tent maps.

math.DS↗