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EESTIMATES OF BEST APPROXIMATIONS OF FUNCTIONS WITH LOGARITHMIC SMOOTHNESS IN THE LORENTZ SPACE WITH ANISOTROPIC NORM

Author: Akishev Gabdolla
Publication venue: Institute of Mathematics and Mechanics
Publication date: 29/07/2020
Field of study

In this paper, we consider the anisotropic Lorentz space

L_{\bar{p}, \bar\theta}^{*}(\mathbb{I}^{m})

of periodic functions of

m

variables. The Besov space

B_{\bar{p}, \bar\theta}^{(0, \alpha, \tau)}

of functions with logarithmic smoothness is defined. The aim of the paper is to find an exact order of the best approximation of functions from the class

B_{\bar{p}, \bar\theta}^{(0, \alpha, \tau)}

by trigonometric polynomials under different relations between the parameters

\bar{p}, \bar\theta,

and

\tau

.The paper consists of an introduction and two sections. In the first section, we establish a sufficient condition for a function

f\in L_{\bar{p}, \bar\theta^{(1)}}^{*}(\mathbb{I}^{m})

to belong to the space

L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})

in the case 1{<\theta^{2}<\theta_{j}^{(1)}},$ ${j=1,\ldots,m}, in terms of the best approximation and prove its unimprovability on the class

E_{\bar{p},\bar{\theta}}^{\lambda}=\{f\in L_{\bar{p},\bar{\theta}}^{*}(\mathbb{I}^{m})\colon{E_{n}(f)_{\bar{p},\bar{\theta}}\leq\lambda_{n},}

{n=0,1,\ldots\},}

where

E_{n}(f)_{\bar{p},\bar{\theta}}

is the best approximation of the function

f \in L_{\bar{p},\bar{\theta}}^{*}(\mathbb{I}^{m})

by trigonometric polynomials of order

n

in each variable

x_{j},

j=1,\ldots,m,

and

\lambda=\{\lambda_{n}\}

is a sequence of positive numbers

\lambda_{n}\downarrow0

n\to+\infty

. In the second section, we establish order-exact estimates for the best approximation of functions from the class

B_{\bar{p}, \bar\theta^{(1)}}^{(0, \alpha, \tau)}

in the space

L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})

ON GENERALIZED EIGHTH ORDER MOCK THETA FUNCTIONS

Author: Rawat Pramod Kumar
Publication venue: Institute of Mathematics and Mechanics
Publication date: 29/07/2020
Field of study

In this paper we have generalized eighth order mock theta functions, recently introduced by Gordon and MacIntosh involving four independent variables. The idea of generalizing was to have four extra parameters, which on specializing give known functions and thus these results hold for those known functions. We have represented these generalized functions as

q

-integral. Thus on specializing we have the classical mock theta functions represented as

q

-integral. The same is true for the multibasic expansion given

INEQUALITIES FOR ALGEBRAIC POLYNOMIALS ON AN ELLIPSE

Author: Nikiforova Tatiana M.
Publication venue: Institute of Mathematics and Mechanics
Publication date: 26/12/2020
Field of study

The paper presents new solutions to two classical problems of approximation theory. The first problem is to find the polynomial that deviates least from zero on an ellipse. The second one is to find the exact upper bound of the uniform norm on an ellipse with foci

\pm 1

of the derivative of an algebraic polynomial with real coefficients normalized on the segment

[- 1,1]

THE LOCAL DENSITY AND THE LOCAL WEAK DENSITY IN THE SPACE OF PERMUTATION DEGREE AND IN HATTORI SPACE

Author: Yuldashev Tursun K.
Mukhamadiev Farhod G.
Publication venue: Institute of Mathematics and Mechanics
Publication date: 26/12/2020
Field of study

In this paper, the local density

(l d)

and the local weak density

(l w d)

in the space of permutation degree as well as the cardinal and topological properties of Hattori spaces are studied. In other words, we study the properties of the functor of permutation degree

S P^{n}

and the subfunctor of permutation degree

S P_{G}^{n}

P

is the cardinal number of topological spaces. Let

X

be an infinite

T_{1}

-space. We prove that the following propositions hold.(1) Let

Y^{n} \subset X^{n}

; (A) if

d\, \left(Y^{n} \right)=d\, \left(X^{n} \right)

, then

d\, \left(S P^{n} Y\right)=d\, \left(SP^{n} X\right)

; (B) if

l w d\, \left(Y^{n} \right)=l w d\, \left(X^{n} \right)

, then

l w d\, \left(S P^{n} Y\right)=l w d\, \left(S P^{n} X\right)

. (2) Let

Y\subset X

; (A) if

l d \,(Y)=l d \,(X)

, then

l d\, \left(S P^{n} Y\right)=l d\, \left(S P^{n} X\right)

; (B) if

w d \,(Y)=w d \,(X)

, then

w d\, \left(S P^{n} Y\right)=w d\, \left(S P^{n} X\right)

.(3) Let

n

be a positive integer, and let

G

be a subgroup of the permutation group

S_{n}

. If

X

is a locally compact

T_{1}

-space, then

S P^{n} X, \, S P_{G}^{n} X

, and

\exp _{n} X

are

k

-spaces.(4) Let

n

be a positive integer, and let

G

be a subgroup of the permutation group

S_{n}

. If

X

is an infinite

T_{1}

-space, then

n \,\pi \,w \left(X\right)=n \, \pi \,w \left(S P^{n} X \right)=n \,\pi \,w \left(S P_{G}^{n} X \right)=n \,\pi \,w \left(\exp _{n} X \right)

.We also have studied that the functors

SP^{n},

SP_{G}^{n} ,

and

\exp _{n}

preserve any

k

-space. The functors

SP^{2}

and

SP_{G}^{3}

do not preserve Hattori spaces on the real line. Besides, it is proved that the density of an infinite

T_{1}

-space

X

coincides with the densities of the spaces

X^{n}

\,S P^{n} X

, and

\exp _{n} X

. It is also shown that the weak density of an infinite

T_{1}

-space

X

coincides with the weak densities of the spaces

X^{n}

\,S P^{n} X

, and

\exp _{n} X

OPEN PACKING NUMBER FOR SOME CLASSES OF PERFECT GRAPHS

Author: Chandrasekar K. Raja
Saravanakumar S.
Publication venue: Institute of Mathematics and Mechanics
Publication date: 26/12/2020
Field of study

Let

G

be a graph with the vertex set

V(G)

. A subset

S

V(G)

is an open packing set of

G

if every pair of vertices in

S

has no common neighbor in

G.

The maximum cardinality of an open packing set of

G

is the open packing number of

G

and it is denoted by

\rho^o(G)

. In this paper, the exact values of the open packing numbers for some classes of perfect graphs, such as split graphs,

\{P_4, C_4\}

-free graphs, the complement of a bipartite graph, the trestled graph of a perfect graph are obtained

THE DYNAMIC DEFORMATION OF THREE-COMPONENT POROUS MEDIA

Author: Polenov Victor S.
Kukarskikh Lyubov A.
Nitsak Dmitry A.
Publication venue: Institute of Mathematics and Mechanics
Publication date: 29/07/2020
Field of study

A mathematical model of the dynamic deformation of three-component elastic media saturated with liquid and gas, given by elastic moduli and coefficients characterizing the porosity and compressibility of the liquid and gas, is considered. Formulas for determining the propagation velocity of monochromatic waves in ternary porous media are obtained. The existence of three longitudinal waves depends on the discriminant of a cubic equation and the velocity ratio

NONLOCAL PROBLEM FOR A MIXED TYPE FOURTH-ORDER DIFFERENTIAL EQUATION WITH HILFER FRACTIONAL OPERATOR

Author: Yuldashev Tursun K.
Kadirkulov Bakhtiyor J.
Publication venue: Institute of Mathematics and Mechanics
Publication date: 29/07/2020
Field of study

In this paper, we consider a non-self-adjoint boundary value problem for a fourth-order differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The mixed type differential equation under consideration is a fourth order differential equation with respect to the second variable. Regarding the first variable, this equation is a fractional differential equation in the positive part of the segment, and is a second-order differential equation with spectral parameter in the negative part of this segment. A rational method of solving a nonlocal problem with respect to the Hilfer operator is proposed. Using the spectral method of separation of variables, the solution of the problem is constructed in the form of Fourier series. Theorems on the existence and uniqueness of the problem are proved for regular values of the spectral parameter. For sufficiently large positive integers in unique determination of the integration constants in solving countable systems of differential equations, the problem of small denominators arises. Therefore, to justify the unique solvability of this problem, it is necessary to show the existence of values of the spectral parameter such that the quantity we need is separated from zero for sufficiently large

n

. For irregular values of the spectral parameter, an infinite number of solutions in the form of Fourier series are constructed. Illustrative examples are provided

DOMINATION AND EDGE DOMINATION IN TREES

Author: Senthilkumar B.
Venkatakrishnan Yanamandram B.
Naresh Kumar H.
Publication venue: Institute of Mathematics and Mechanics
Publication date: 29/07/2020
Field of study

Let

G=(V,E)

be a simple graph. A set

S\subseteq V

is a dominating set if every vertex in

V \setminus S

is adjacent to a vertex in

S

. The domination number of a graph

G

, denoted by

\gamma(G)

is the minimum cardinality of a dominating set of

G

. A set

D \subseteq E

is an edge dominating set if every edge in

E\setminus D

is adjacent to an edge in

D

. The edge domination number of a graph

G

, denoted by

\gamma'(G)

is the minimum cardinality of an edge dominating set of

G

. We characterize trees with domination number equal to twice edge domination number

PURSUIT-EVASION DIFFERENTIAL GAMES WITH THE GRÖNWALL TYPE CONSTRAINTS ON CONTROLS

Author: Samatov Bahrom T.
Ibragimov Gafurjan
Khodjibayeva Iroda V.
Publication venue: Institute of Mathematics and Mechanics
Publication date: 26/12/2020
Field of study

A simple pursuit-evasion differential game of one pursuer and one evader is studied. The players' controls are subject to differential constraints in the form of the integral Grönwall inequality. The pursuit is considered completed if the state of the pursuer coincides with the state of the evader. The main goal of this work is to construct optimal strategies for the players and find the optimal pursuit time. A parallel approach strategy for Grönwall-type constraints is constructed and it is proved that it is the optimal strategy of the pursuer. In addition, the optimal strategy of the evader is constructed and the optimal pursuit time is obtained. The concept of a parallel pursuit strategy (

\Pi

-strategy for short) was introduced and used to solve the quality problem for "life-line" games by L.A.Petrosjan. This work develops and expands the works of Isaacs, Petrosjan, Pshenichnyi, and other researchers, including the authors

HAHN'S PROBLEM WITH RESPECT TO SOME PERTURBATIONS OF THE RAISING OPERATOR $X-c$

Author: Aloui Baghdadi
Souissi Jihad
Publication venue: Institute of Mathematics and Mechanics
Publication date: 26/12/2020
Field of study

In this paper, we study the Hahn's problem with respect to some raising operators perturbed of the operator

X-c

, where

c

is an arbitrary complex number. More precisely, the two following characterizations hold: up to a normalization, the

q

-Hermite (resp. Charlier) polynomial is the only

H_{\alpha,q}

-classical (resp.

\mathcal{S}_{\lambda}

-classical) orthogonal polynomial, where

H_{\alpha, q}:=X+\alpha H_q

and \(\mathcal{S}_{\lambda}:=(X+1)-\lambda\tau_{-1}.\

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