AROUND THE ERDÖS–GALLAI CRITERION

By an (integer) partition we mean a non-increasing sequence $\lambda=(\lambda_1, \lambda_2, \dots)$ of non-negative integers that contains a finite number of non-zero components. A partition $\lambda$ is said to be graphic if there exists a graph $G$ such that $\lambda = \mathrm{dpt}\,G$, where we denote by  $\mathrm{dpt}\,G$ the degree partition of $G$ composed of the degrees of its vertices, taken in non-increasing order and added with zeros. In this paper, we propose to consider another criterion for a partition to be graphic, the ht-criterion, which, in essence, is a convenient and natural reformulation of the well-known Erdös–Gallai criterion for a sequence to be graphical. The ht-criterion fits well into the general study of lattices of integer partitions and is convenient for applications. The paper shows the equivalence of the Gale–Ryser criterion on the realizability of a pair of partitions by bipartite graphs, the ht-criterion and the Erdös–Gallai criterion. New proofs of the Gale–Ryser criterion and the Erdös–Gallai criterion are given. It is also proved that for any graphical partition there exists a realization that is obtained from some splitable graph in a natural way. A number of information of an overview nature is also given on the results previously obtained by the authors which are close in subject matter to those considered in this paper

CONTROL PROBLEM FOR A PARABOLIC SYSTEM WITH UNCERTAINTIES AND A NON-CONVEX GOAL

We consider the control problem for a parabolic system that describes the heating of a given number of rods. Control is carried out through heat sources that are located at the ends of the rods (only at one end or at both). The density functions of the internal heat sources and exact values of the temperature at the right ends of some rods are unknown, and only the segments of their change are given. The goal of choosing control is to ensure that at a fixed time moment the weighted sum of the average temperatures of the rods belongs to a non-convex terminal set for any admissible unknown functions. After a change of variables, this problem reduces to a one-dimensional differential game. Necessary and sufficient conditions for the game termination are found

ON ONE ZALCMAN PROBLEM FOR THE MEAN VALUE OPERATOR

Let $\mathcal{D}'(\mathbb{R}^n)$ and $\mathcal{E}'(\mathbb{R}^n)$ be the spaces of distributions and compactly supported distributions on $\mathbb{R}^n$, $n\geq 2$ respectively, let $\mathcal{E}'_{\natural}(\mathbb{R}^n)$ be the space of all radial (invariant under rotations of the space $mathbb{R}^n$) distributions in $\mathcal{E}'(\mathbb{R}^n)$, let$\widetilde{T}$ be the spherical transform (Fourier–Bessel transform) of a distribution $T\in\mathcal{E}'_{\natural}(\mathbb{R}^n)$, and let $\mathcal{Z}_{+}(\widetilde{T})$ be the set of all zeros of an even entire function $\widetilde{T}$ lying in the half-plane $\mathrm{Re} \, z\geq 0$ and not belonging to the negative part of the imaginary axis. Let $\sigma_{r}$ be the surface delta function concentrated on the sphere $S_r=\{x\in\mathbb{R}^n: |x|=r\}$. The problem of L. Zalcman on reconstructing a distribution $f\in \mathcal{D}'(\mathbb{R}^n)$ from known convolutions $f\ast \sigma_{r_1}$ and $f\ast \sigma_{r_2}$ is studied. This problem is correctly posed only under the condition $r_1/r_2\notin M_n$, where $M_n$ is the set of all possible ratios of positive zeros of the Bessel function $J_{n/2-1}$. The paper shows that if $r_1/r_2\notin M_n$, then an arbitrary distribution $f\in \mathcal{D}'(\mathbb{R}^n)$ can be expanded into an unconditionally convergent series$$f=\sum\limits_{\lambda\in\mathcal{Z}_{+}(\widetilde{\Omega}_{r_1})}\,\,\, \sum\limits_{\mu\in\mathcal{Z}_+(\widetilde{\Omega}_{r_2})}\frac{4\lambda\mu}{(\lambda^2-\mu^2) \widetilde{\Omega}_{r_1}^{\,\,\,\displaystyle{'}}(\lambda)\widetilde{\Omega}_{r_2}^{\,\,\,\displaystyle{'}}(\mu)}\Big(P_{r_2} (\Delta) \big((f\ast\sigma_{r_2})\ast \Omega_{r_1}^{\lambda}\big)-P_{r_1} (\Delta) \big((f\ast\sigma_{r_1})\ast \Omega_{r_2}^{\mu}\big)\Big)$$in the space $\mathcal{D}'(\mathbb{R}^n)$, where $\Delta$ is the Laplace operator in $\mathbb{R}^n$, $P_r$ is an explicitly given polynomial of degree $[(n+5)/4]$, and $\Omega_{r}$ and $\Omega_{r}^{\lambda}$ are explicitly constructed radial distributions supported in the ball $|x|\leq r$. The proof uses the methods of harmonic analysis, as well as the theory of entire and special functions. By a similar technique, it is possible to obtain inversion formulas for other convolution operators with radial distributions

A NEW CHARACTERIZATION OF SYMMETRIC DUNKL AND qq-DUNKL-CLASSICAL ORTHOGONAL POLYNOMIALS

In this paper, we consider the following $\mathcal{L}$-difference equation$$\Phi(x) \mathcal{L}P_{n+1}(x)=(\xi_nx+\vartheta_n)P_{n+1}(x)+\lambda_nP_{n}(x),\quad n\geq0,$$where $\Phi$ is a monic polynomial (even), $\deg\Phi\leq2$, $\xi_n,\,\vartheta_n,\,\lambda_n,\,n\geq0$, are complex numbers and $\mathcal{L}$ is either the Dunkl operator $T_\mu$ or the the $q$-Dunkl operator $T_{(\theta,q)}$. We show that if $\mathcal{L}=T_\mu$, then the only symmetric orthogonal polynomials satisfying the previous equation are, up a dilation, the generalized Hermite polynomials and the generalized Gegenbauer polynomials and if $\mathcal{L}=T_{(\theta,q)}$, then the $q^2$-analogue of generalized Hermite and the $q^2$-analogue of generalized Gegenbauer polynomials are, up a dilation, the only orthogonal polynomials sequences satisfying the $\mathcal{L}$-difference equation

HETEROGENEOUS SERVER RETRIAL QUEUEING MODEL WITH FEEDBACK AND WORKING VACATION USING ARTIFICIAL BEE COLONY OPTIMIZATION ALGORITHM

This research delves into the dynamics of a retrial queueing system featuring heterogeneous servers with intermittent availability, incorporating feedback and working vacation mechanisms. Employing a matrix geometric approach, this study establishes the steady-state probability distribution for the queue size in this complex heterogeneous service model. Additionally, a range of system performance metrics is developed, alongside the formulation of a cost function to evaluate decision variable optimization within the service system. The Artificial Bee Colony (ABC) optimization algorithm is harnessed to determine service rates that minimize the overall cost. This work includes numerical examples and sensitivity analyses to validate the model's effectiveness. Also, a comparison between the numerical findings and the neuro-fuzzy results has been examined by the adaptive neuro fuzzy interface system (ANFIS)

KERNEL DETERMINATION PROBLEM FOR ONE PARABOLIC EQUATION WITH MEMORY

This paper studies the inverse problem of determining a multidimensional kernel function of an integral term which depends on the time variable $t$ and $(n-1)$-dimensional space variable $x'= \left(x_1,\ldots, x_ {n-1}\right)$ in the $n$-dimensional diffusion equation with a time-variable coefficient at the Laplacian of a direct problem solution. Given a known kernel function, a Cauchy problem is investigated as a direct problem. The integral term in the equation has convolution form: the kernel function is multiplied by a solution of the direct problem's elliptic operator. As an overdetermination condition, the result of the direct question on the hyperplane $x_n = 0$ is used. An inverse question is replaced by an auxiliary one, which is more suitable for further investigation. After that, the last problem is reduced to an equivalent system of Volterra-type integral equations of the second order with respect to unknown functions. Applying the fixed point theorem to this system in Hölder spaces, we prove the main result of the paper, which is a local existence and uniqueness theorem

IK\mathcal{I}^{\mathcal{K}}-SEQUENTIAL TOPOLOGY

In the literature, $\mathcal{I}$-convergence (or convergence in $\mathcal{I}$) was first introduced in [11]. Later related notions of $\mathcal{I}$-sequential topological space and $\mathcal{I}^*$-sequential topological space were introduced and studied. From the definitions it is clear that $\mathcal{I}^*$-sequential topological space is larger(finer) than $\mathcal{I}$-sequential topological space. This rises a question: is there any topology (different from discrete topology) on the topological space $\mathcal{X}$ which is finer than $\mathcal{I}^*$-topological space? In this paper, we tried to find the answer to the question. We define $\mathcal{I}^{\mathcal{K}}$-sequential topology for any ideals $\mathcal{I}$, $\mathcal{K}$ and study main properties of it. First of all, some fundamental results about $\mathcal{I}^{\mathcal{K}}$-convergence of a sequence in a topological space $(\mathcal{X} ,\mathcal{T})$ are derived. After that, $\mathcal{I}^{\mathcal{K}}$-continuity and the subspace of the $\mathcal{I}^{\mathcal{K}}$-sequential topological space are investigated

COMPUTING THE REACHABLE SET BOUNDARY FOR AN ABSTRACT CONTROL SYSTEM: REVISITED

A  control system can be treated as a mapping that maps a control to a trajectory (output) of the system. From this point of view, the reachable set, which consists of the ends of all trajectories at a given time, can be considered an image of the set of admissible controls into the state space under a nonlinear mapping. The paper discusses some properties of such abstract reachable sets. The principal attention is paid to the description of the set boundary

POLYNOMIALS LEAST DEVIATING FROM ZERO IN Lp(−1;1)L^p(-1;1), 0≤p≤∞0 \le p \le \infty , WITH A CONSTRAINT ON THE LOCATION OF THEIR ROOTS

We study Chebyshev's problem on polynomials that deviate least from zero with respect to $L^p$-means on the interval $[-1;1]$ with a constraint on the location of roots of polynomials. More precisely, we consider the problem on the set $\mathcal{P}_n(D_R)$ of polynomials of degree $n$ that have unit leading coefficient and do not vanish in an open disk of radius $R \ge 1$. An exact solution is obtained for the geometric mean (for $p=0$) for all $R \ge 1$; and for $0<p<\infty$ for all $R \ge 1$ in the case of polynomials of even degree. For $0<p<\infty$ and $R\ge 1$, we obtain two-sided estimates of the value of the least deviation

STATISTICAL CONVERGENCE IN A BICOMPLEX VALUED METRIC SPACE

In this paper, we study some basic properties of bicomplex numbers. We introduce two different types of partial order relations on bicomplex numbers, discuss bicomplex valued metric spaces with respect to two different partial orders, and compare them. We also define a hyperbolic valued metric space, the density of natural numbers, the statistical convergence, and the statistical Cauchy property of a sequence of bicomplex numbers and investigate some properties  in a bicomplex metric space and prove that a bicomplex metric space is complete if and only if two complex metric spaces are complete