Ural Mathematical Journal (UMJ)

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D2-SYNCHRONIZATION IN NONDETERMINISTIC AUTOMATA

Author: Shabana Hanan
Publication venue: Institute of Mathematics and Mechanics
Publication date: 29/12/2018
Field of study

We approach the problem of computing a D2-synchronizing word of minimum length for a given nondeterministic automaton via its encoding as an instance of SAT and invoking a SAT solver. In addition, we report some of the experimental results obtained when we had tested our method on randomly generated automata and certain benchmarks

ONE-SIDED $L$ -APPROXIMATION ON A SPHERE OF THE CHARACTERISTIC FUNCTION OF A LAYER

Author: Deikalova Marina V.
Torgashova Anastasiya Yu.
Publication venue: Institute of Mathematics and Mechanics
Publication date: 29/12/2018
Field of study

In the space

L(\mathbb{S}^{m-1})

of functions integrable on the unit sphere

\mathbb{S}^{m-1}

of the Euclidean space

\mathbb{R}^{m}

of dimension

m\ge 3

, we discuss the problem of one-sided approximation to the characteristic function of a spherical layer

\mathbb{G}(J)=\{x=(x_1,x_2,\ldots,x_m)\in \mathbb{S}^{m-1}\colon x_m\in J\},

where

J

is one of the intervals

(a,1],

(a,b),

and

[-1,b),

-1< a<b< 1,

by the set of algebraic polynomials of given degree

n

m

variables. This problem reduces to the one-dimensional problem of one-sided approximation in the space

L^\phi(-1,1)

with the ultraspherical weight

\phi(t)=(1-t^2)^\alpha,\ \alpha=(m-3)/2,

to the characteristic function of the interval

J

. This result gives a solution of the problem of one-sided approximation to the characteristic function of a spherical layer in all cases when a solution of the corresponding one-dimensional problem known. In the present paper, we use results by A.G.Babenko, M.V.Deikalova, and Sz.G.Revesz (2015) and M.V.Deikalova and A.Yu.Torgashova (2018) on the one-sided approximation to the characteristic functions of intervals

ON THE SUMMABILITY OF THE DISCRETE HILBERT TRANSFORM

Author: Aliev Rashid A.
Amrahova Aynur F.
Publication venue: Institute of Mathematics and Mechanics
Publication date: 29/12/2018
Field of study

In this paper, we study the asymptotic behavior of the distribution function of the discrete Hilbert transform of sequences from the class

l_{1}

and find a necessary condition and a sufficient condition for the summability of the discrete Hilbert transform of a sequence from the class

l_{1}

ALTRUISTIC AND AGGRESSIVE TYPES OF BEHAVIOR IN A NON-ANTAGONISTIC DIFFERENTIAL GAME

Author: Kleimenov Anatolii F.
Publication venue: Institute of Mathematics and Mechanics
Publication date: 29/12/2018
Field of study

An example of a non-antagonistic positional (feedback) differential two-person game (NPDG) is considered in which each of two players, in addition to the normal type of behavior, oriented toward maximizing own functional, can use other types of behavior. In particular, it can be altruistic and aggressive types. In the course of the game players can switch their behavior from one type to other. The use by players of types of behavior other than normal can lead to outcomes more preferable for them than in a game with only normal behavior. The example with the dynamics of simple motion on a plane and phase constraints illustrates the procedure of constructing new solutions

EVALUATION OF SOME NON-ELEMENTARY INTEGRALS INVOLVING SINE, COSINE, EXPONENTIAL AND LOGARITHMIC INTEGRALS: PART I

Author: Nijimbere Victor
Publication venue: Institute of Mathematics and Mechanics
Publication date: 27/07/2018
Field of study

The non-elementary integrals

\text{Si}_{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,

\beta\ge1,

\alpha\le\beta+1

and

\text{Ci}_{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,

\beta\ge1,

\alpha\le2\beta+1

, where

\{\beta,\alpha\}\in\mathbb{R}

, are evaluated in terms of the hypergeometric functions

_{1}F_2

and

_{2}F_3

, and their asymptotic expressions for

|x|\gg1

are also derived. The integrals of the form

\int [\sin^n{(\lambda x^\beta)}/(\lambda x^\alpha)] dx

and

\int [\cos^n{(\lambda x^\beta)}/(\lambda x^\alpha)] dx

, where

n

is a positive integer, are expressed in terms

\text{Si}_{\beta,\alpha}

and

\text{Ci}_{\beta,\alpha}

, and then evaluated.

\text{Si}_{\beta,\alpha}

and

\text{Ci}_{\beta,\alpha}

are also evaluated in terms of the hypergeometric function

_{2}F_2

. And so, the hypergeometric functions,

_{1}F_2

and

_{2}F_3

, are expressed in terms of

_{2}F_2

. The exponential integral

\text{Ei}_{\beta,\alpha}=\int (e^{\lambda x^\beta}/x^\alpha) dx

where

\beta\ge1

and

\alpha\le\beta+1

and the logarithmic integral

\text{Li}=\int_{\mu}^{x} dt/\ln{t}

\mu>1

, are also expressed in terms of

_{2}F_2

, and their asymptotic expressions are investigated. For instance, it is found that for

x\gg2

\text{Li}\sim {x}/{\ln{x}}+\ln{\left({\ln{x}}/{\ln{2}}\right)}-2-\ln{2}\hspace{.075cm} _{2}F_{2}(1,1;2,2;\ln{2})

, where the term

\ln{\left({\ln{x}}/{\ln{2}}\right)}-2-\ln{2}\hspace{.075cm} _{2}F_{2}(1,1;2,2;\ln{2})

is added to the known expression in mathematical literature

\text{Li}\sim {x}/{\ln{x}}

. The method used in this paper consists of expanding the integrand as a Taylor and integrating the series term by term, and can be used to evaluate the other cases which are not considered here. This work is motivated by the applications of sine, cosine exponential and logarithmic integrals in Science and Engineering, and some applications are given

SOME REPRESENTATIONS CONNECTED WITH ULTRAFILTERS AND MAXIMAL LINKED SYSTEMS

Author: Chentsov Alexander G.
Publication venue: Institute of Mathematics and Mechanics
Publication date: 25/12/2017
Field of study

Ultrafilters and maximal linked systems (MLS) of a lattice of sets are considered. Two following variants of topological equipment are investigated: the Stone and Wallman topologies. These two variants are used both in the case of ultrafilters and for space of MLS. Under Wallman equipment, an analog of superextension is realized. Namely, the space of MLS with topology of the Wallman type is supercompact topological space. By two above-mentioned equipments a bitopological space is realized

ASYMPTOTIC EXPANSION OF A SOLUTION FOR ONE SINGULARLY PERTURBED OPTIMAL CONTROL PROBLEM IN $\mathbb{R}^n$ WITH A CONVEX INTEGRAL QUALITY INDEX

Author: Shaburov Alexander A.
Publication venue: Institute of Mathematics and Mechanics
Publication date: 14/07/2017
Field of study

The paper deals with the problem of optimal control with a convex integral quality index for a linear steady-state control system in the class of piecewise continuous controls with a smooth control constraints. In a general case, for solving such a problem, the Pontryagin maximum principle is applied as the necessary and sufficient optimum condition. In this work, we deduce an equation to which an initial vector of the conjugate system satisfies. Then, this equation is extended to the optimal control problem with the convex integral quality index for a linear system with a fast and slow variables. It is shown that the solution of the corresponding equation as

\varepsilon\to 0

tends to the solution of an equation corresponding to the limit problem. The results received are applied to study of the problem which describes the motion of a material point in

\mathbb{R}^n

for a fixed period of time. The asymptotics of the initial vector of the conjugate system that defines the type of optimal control is built. It is shown that the asymptotics is a power series of expansion

POSITIVE DEFINITE FUNCTIONS AND SHARP INEQUALITIES FOR PERIODIC FUNCTIONS

Author: Zastavnyi Viktor P.
Publication venue: Institute of Mathematics and Mechanics
Publication date: 25/12/2017
Field of study

Let

\varphi

be a positive definite and continuous function on

\mathbb{R}

, and let

\mu

be the corresponding Bochner measure. For fixed

\varepsilon,\tau\in\mathbb{R}

\varepsilon\ne 0

, we consider a linear operator

A_{\varepsilon,\tau}

generated by the function

\varphi

A_{\varepsilon,\tau}(f)(t):=\int_{\mathbb{R}}e^{-iu\tau} f(t+\varepsilon u)d\mu(u),\quad t\in\mathbb{R},\quad f\in C(\mathbb{T}).

Let

J

be a convex and nondecreasing function on

[0,+\infty)

. In this paper, we prove the inequalities

\| A_{\varepsilon,\tau}(f)\|_p\leqslant \varphi(0)\|f\|_p, \quad \int_{\mathbb{T}}J\left(|A_{\varepsilon,\tau}(f)(t)|\right)\,dt \le \int_{\mathbb{T}}J\left(\varphi(0)|f(t)|\right)\,dt

for

p\in [1,\infty]

and

f\in C(\mathbb{T})

and obtain criteria of extremal function. We study in more detail the case in which

\varepsilon=1/n

n\in\mathbb{N}

\tau=1

, and

\varphi(x)\equiv e^{i\beta x}\psi(x)

, where

\beta\in\mathbb{R}

and the function

\psi

2

-periodic and positive definite. In turn, we consider in more detail the case where the 2-periodic function

\psi

is constructed by means of a finite positive definite function

g

. As a particular case, we obtain the Bernstein–Szegő inequality for the derivative in the Weyl–Nagy sense of trigonometric polynomials. In one of our results, we consider the case of the family of functions

g_{1/n,h}(x):=hg(x)+(1-1/n-h)g(nx)

, where

n\in\mathbb{N}

n\ge 2

-1/n\le h\le 1-1/n

, and the function

g\in C(\mathbb{R})

is even, nonnegative, decreasing, and convex on

(0,+\infty)

with

{\rm supp\,}g\subset[-1,1]

. This case is related to the positive definiteness of piecewise linear functions. We also obtain some general interpolation formulas for periodic functions and trigonometric polynomials which include the known interpolation formulas of M. Riesz, of G. Szegő, and of A.I. Kozko for trigonometric polynomials

AN ALGORITHM FOR COMPUTING BOUNDARY POINTS OF REACHABLE SETS OF CONTROL SYSTEMS UNDER INTEGRAL CONSTRAINTS

Author: Gusev Mikhail I.
Publication venue: Institute of Mathematics and Mechanics
Publication date: 14/07/2017
Field of study

In this paper we consider a reachability problem for a nonlinear affine-control system with integral constraints, which assumed to be quadratic in the control variables. Under controllability assumptions it was proved [8] that any admissible control, that steers the control system to the boundary of its reachable set, is a local solution to an optimal control problem with an integral cost functional and terminal constraints. This results in the Pontriagyn maximum principle for boundary trajectories. We propose here an numerical algorithm for computing the reachable set boundary based on the maximum principle and provide some numerical examples

THE 42\mbox{nd} INTERNATIONAL S.B. STECHKIN’S WORKSHOP-CONFERENCE ON FUNCTION THEORY

Author: Arestov Vitalii V.
Berdyshev Vitalii I.
Publication venue: Institute of Mathematics and Mechanics
Publication date: 25/12/2017
Field of study

The paper is devoted to the description of the history and results of the 42nd International S.B.Stechkin's Workshop on function theory, held in August 2017 in the Ilmen Nature Reserve near the town of Miass, Chelyabinsk region

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