In this paper, by the Faà di Bruno formula and properties of Bell polynomials of the second kind, the authors reconsider the generating functions of Hermite polynomials and their squares, find an explicit formula for higher-order derivatives of the generating function of Hermite polynomials, and derive explicit formulas and recurrence relations for Hermite polynomials and their squares.

In this paper, we define consistent criteria for hypotheses testing, such that the probability of any kind of error is zero for a given criterion. The necessary and sufficient conditions for the existence of such criteria are given.

This paper deals with the existence and regularity of some unilateral problem associated to a nonlinear equation of type -div⁡(a⁢(x,u,∇⁡u))+H⁢(x,u,∇⁡u)=f.

The purpose of this paper is to investigate the behavior of prime rings involving skew derivations with m-potent commutators on Lie ideals. In addition, we provide an example that shows that we cannot expect the same conclusion in case of semiprime rings. Also, we prove some other related results and present some open problems.

It is shown that the cardinality of the continuum is not real-valued measurable if and only if there exists no nonzero σ-finite diffused measure μ on the real line such that all Vitali sets (respectively all Bernstein sets) are μ-measurable.

In this note, we indicate some errors in [S. Ali, N. A. Dar and M. Asci, On derivations and commutativity of prime rings with involution, Georgian Math. J. 23 2016, 1, 9–14] and present the correct versions of the erroneous results.

The paper deals with the three-dimensional Robin type boundary-value problem (BVP) for a second-order strongly elliptic system of partial differential equations in the divergence form with variable coefficients. The problem is studied by the localized parametrix based potential method. By using Green’s representation formula and properties of the localized layer and volume potentials, the BVP under

In this paper, for a discrete hypergroup K and its subhypergroup H, we initiate the related Hecke *-algebra which is an extension of the classical one, and study its basic properties. Especially, we give a necessary and sufficient condition (named (β)) for this algebra to be associative. Also, we show that this new structure is an associative *-algebra if and only if K is a locally compact group.

We prove the following theorem. Let α=a+i⁢b be a nonzero complex number. Then the following statements hold: (i) Let either b≠0 or b=0 and a>0. Let ξ1 and ξ2 be independent complex random variables. Assume that the linear forms L1=ξ1+ξ2 and L2=ξ1+α⁢ξ2 are independent. Then ξj are degenerate random variables. (ii) Let b=0 and a<0. Then there exist complex Gaussian random variables in the wide sense

We establish a necessary and sufficient condition on a non-negative locally integrable function v guaranteeing the (trace) inequality

In the paper, some properties of functions continuous with respect to a density type strong generalized topology are presented. In particular, it is proved that each real function is approximately continuous with respect to this generalized topology almost everywhere. Moreover, some separation axioms for this generalized topological space are investigated.

The quantum groups nowadays attract a considerable interest of mathematicians and physicists. The theory of 𝑞-special functions has received a group-theoretic interpretation using the techniques of quantum groups and quantum algebras. This paper focuses on introducing the 𝑞-Tricomi functions and 2D 𝑞-Tricomi functions through the generating function and series expansion and for the first time establishing

Journal Name: Georgian Mathematical Journal Volume: 27 Issue: 4 Pages: i-iv

In this paper, we introduce an interesting kind of convergence for a double sequence called the uniform convergence at a point. We give an example and demonstrate a Korovkin-type approximation theorem for a double sequence of functions using the uniform convergence at a point. Then we show that our result is stronger than the Korovkin theorem given by Volkov and present several graphs. Finally, in

The purpose of the present research is to investigate a general mixed type boundary value problem for the Laplace–Beltrami equation on a surface with the Lipschitz boundary 𝒞 in the non-classical setting when solutions are sought in the Bessel potential spaces Hps⁢(C), 1p

Journal Name: Georgian Mathematical Journal Volume: 27 Issue: 3 Pages: i-iv

Let (ℌ,μ,α) be a regular Hom-algebra of arbitrary dimension and over an arbitrary base field 𝔽. A basis ℬ={ei}i∈I of ℌ is called multiplicative if for any i,j∈I, we have that μ⁢(ei,ej)∈𝔽⁢ek and α⁢(ei)∈𝔽⁢ep for some k,p∈I. We show that if ℌ admits a multiplicative basis, then it decomposes as the direct sum ℌ=⊕rℑr of well-described ideals admitting each one a multiplicative basis. Also, the minimality

Let ℛ be a prime ring, 𝒬r the right Martindale quotient ring of ℛ and 𝒞 the extended centroid of ℛ. In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m-potent commutators of skew derivations involving multilinear polynomials, i.e.,

This paper relates to a recent trend in complex differential equations which studies solutions in a given domain. The classical settings in complex equations were widely studied for meromorphic solutions in the complex plane. For functions in the complex plane, we have a lot of results of general nature, in particular, the classical value distributions theory describing numbers of a-points. Many of

A new scale of weighted grand mixed norm Lebesgue spaces Lw1p1,φ1⁢(Lw2p2,φ2) is introduced and the boundedness criteria for multiple singular operators are established.

The subject of the paper is suggested by G. Janelidze and motivated by his earlier result giving a positive answer to the question posed by S. MacLane whether the Galois theory of homogeneous linear ordinary differential equations over a differential field (which is Kolchin–Ritt theory and an algebraic version of Picard–Vessiot theory) can be obtained as a particular case of G. Janelidze’s Galois theory

We study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation

We study the capability property of Leibniz algebras via the non-abelian exterior product.

The support of a wavelet transform associated with a square integrable irreducible representation of a homogeneous space is shown to have infinite measure. Assumptions are illustrated and supported by examples. The pointwise homogeneous approximation property for a wavelet transform has been investigated. An analogue of Heisenberg type inequality is also obtained for a wavelet transform on a wavelet

The paper deals with the exploration of those subclasses of the variable exponent Lebesgue space Lp⁢(⋅) with min⁡p⁢(⋅)=1, which are invariant with respect to Cauchy singular integral operators.

Let ℜ be a ring with center Z⁢(ℜ). In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016, 8, 3201–3210], we characterize generalized derivations and related maps that

In this paper, we consider a natural extension in the context of d-orthogonality for asymptotic analysis of orthogonal polynomials. We introduce, for several d-orthogonal polynomials, asymptotic expansions in terms of d-Hermite ones. From these expansions, several limits between d-orthogonal polynomials are obtained.

In this paper, we investigate the absolute convergence of Fourier series of functions in several variables for an odd-dimensional space when these functions have continuous partial derivatives. It should be noted that similar properties for an even-dimensional space were given in [L. D. Gogoladze and V. S. Tsagareishvili, On absolute convergence of multiple Fourier series (in Russian), Izv. Vyssh.

The aim of this paper is to introduce generalized Suzuki type F-contraction fuzzy mappings and to prove the existence of fixed fuzzy points for such mappings in the setup of complete ordered metric spaces. An example is provided to show the validity of our results, followed by couple of remarks about the comparison of obtained results with the existing results in the literature. An application of our

We present the conditions for a block matrix of a ring to have the image-kernel (p,q)-inverse in the generalized Banachiewicz–Schur form. We give representations for the image-kernel inverses of the sum and the product of two block matrices. Some characterizations of the image-kernel (p,q)-inverse in a ring with involution are investigated too.

In this paper, we consider a new class of convex functions, called relative h-preivex functions. Seven new inequalities of Hermite–Hadamard type for relative h-preinvex functions are established using different approaches.

The weighted boundedness properties of multilinear operators associated to singular integral operators with non-smooth kernels for extreme cases are obtained.

It is shown that any function acting from the real line ℝ into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function x→exp⁡(x2) cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs

The aim of this paper is to study a quasistatic frictional contact problem for viscoelastic materials with long-term memory. The contact boundary conditions are governed by Tresca’s law, involving a slip dependent coefficient of friction. We focus our attention on the weak solvability of the problem within the framework of variational inequalities. The existence of a solution is obtained under a smallness

In this work, we investigate the spectrum denoted by Λ for the p⁢(x)-biharmonic operator involving the Hardy term. We prove the existence of at least one non-decreasing sequence of positive eigenvalues of this problem such that sup⁡Λ=+∞. Moreover, we prove that inf⁡Λ>0 if and only if the domain Ω satisfies the p⁢(x)-Hardy inequality.

Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava, A. Çetinkaya and I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 2014, 484–491] by means of the generalized Pochhammer symbol, we introduce here a new extension of the generalized Mittag-Leffler function

The main purpose of this paper is to introduce the concept of strongly ideal lacunary quasi-Cauchyness of sequences of real numbers. Strongly ideal lacunary ward continuity is also investigated. Interesting results are obtained.

In this work, we give a unification and generalization of Laguerre and Hermite polynomials for which the orthogonal property is replaced by d-orthogonality. We state some properties of these new polynomials.

The paper establishes the basic oscillational properties of differential equations of fractional order. It is also shown that the so-called fractional oscillational equation does not have the basic oscillational properties.

In this paper, we determine the forbidden set, introduce an explicit formula for the solutions and discuss the global behavior of solutions of the difference equation

Journal Name: Georgian Mathematical Journal Volume: 27 Issue: 2 Pages: i-iv

Four Ulam type stability concepts for non-instantaneous impulsive fractional differential equations with state dependent delay are introduced. Two different approaches to the interpretation of solutions are investigated. We study the case of an unchangeable lower bound of the Caputo fractional derivative and the case of a lower bound coinciding with the point of jump for the solution. In both cases

In this paper, we investigate the strong summability of two-dimensional Walsh–Fourier series obtained in [F. Weisz, Strong convergence theorems for two-parameter Walsh–Fourier and trigonometric-Fourier series, Studia Math. 117 1996, 2, 173–194] (see Theorem ) and prove the sharpness of this result.

For certain families of topologies, the existence of a common Sierpiński–Zygmund function (of a common Sierpiński–Zygmund function in the strong sense) is established. In this connection, the notion of a Sierpiński–Zygmund space (of a Sierpiński–Zygmund space in the strong sense) is introduced and examined. The behavior of such spaces under some standard topological operations is considered.

The purpose of this paper is to provide a cohomology of n-Hom-Leibniz algebras. Moreover, we study some higher operations on cohomology spaces and deformations.

In this paper, we establish weighted higher order exponential type inequalities in the geodesic space (X,d,μ) by proposing an abstract higher order Poincaré inequality. These are also new in the non-weighted case. As applications, we obtain a weighted Trudinger’s theorem in the geodesic setting and weighted higher order exponential type estimates for functions in Folland–Stein type Sobolev spaces defined

For the nonlinear controlled functional differential equations with several constant delays, the local variation formulas of solutions are proved, in which the effects of the discontinuous initial condition and perturbations of delays and the initial moment are detected.

In this paper we study the boundedness of localization operators associated with the Stockwell transform with symbol in Lp acting on the Wiener amalgam space W⁢(Lp,Lq)⁢(ℝ).

We prove some trigonometric identities involving Chebyshev polynomials of the second kind. The identities were inspired by atomic form factor calculations. Generalizations of these identities, if found, will help to increase the numerical stability of atomic form factor calculations for highly excited states.

The paper is devoted to the construction and study of the additive averaged semi-discrete scheme for one nonlinear multi-dimensional integro-differential equation of parabolic type. The studied equation is based on the well-known Maxwell’s system arising in mathematical simulation of electromagnetic field penetration into a substance.

Consider the first-order linear differential equation with several non-monotone retarded arguments x′⁢(t)+∑i=1mpi⁢(t)⁢x⁢(τi⁢(t))=0, t≥t0, where the functions pi,τi∈C⁢([t0,∞),ℝ+), for every i=1,2,…,m, τi⁢(t)≤t for t≥t0 and limt→∞⁡τi⁢(t)=∞. New oscillation criteria which essentially improve the known results in the literature are established. An example illustrating the results is given.

The purpose of the present research is to investigate model mixed boundary value problems (BVPs) for the Helmholtz equation in a planar angular domain Ωα⊂ℝ2 of magnitude α. These problems are considered in a non-classical setting when a solution is sought in the Bessel potential spaces ℍps⁢(Ωα), s>1p, 1

We prove that if the transpose of every 2×2 idempotent matrix over a division ring D, different from the identity matrix, is not invertible, then D is commutative.

Nonlinear functional partial differential equations with initial conditions are considered on the cone. The weak convergence of a sequence of successive approximations is proved. The proof is given by the duality principle.

In the present paper, we discuss the approximation properties of Jain–Baskakov operators with parameter c. The paper deals with the modified forms of the Baskakov basis functions. Some direct results are established, which include the asymptotic formula, error estimation in terms of the modulus of continuity and weighted approximation. Also, we construct the King modification of these operators, which

In this work, the solvability of a generally nonlocal problem is investigated for a third order linear ordinary differential equation with variable principal coefficient. A novel adjoint problem and Green’s functional are constructed for a completely nonhomogeneous problem. Several illustrative applications for the theoretical results are provided.

In this paper, for 0<γ<1 and b∈Iγ⁢(BMO), the authors give the L2⁢(ℝn) boundedness of μγ;b, the commutator of a fractional differential type Marcinkiewicz integral with rough variable kernel, which is an extension of some known results.

For a locally integrable function f on [0,∞), we define

The purpose of this work is to investigate the existence of solutions for various Neumann boundary value problems associated to the Laplacian-type operators. The main results are obtained using the extension of Mawhin’s continuation theorem.

We consider 3D Navier–Stokes–Voigt equations in smooth bounded domains with homogeneous Dirichlet boundary conditions. First, we study the existence and exponential stability of strong stationary solutions to the problem. Then we show that any unstable steady state can be exponentially stabilized by using either an internal feedback control with a support large enough or a multiplicative Itô noise