Article Frontmatter was published on April 1, 2021 in the journal Georgian Mathematical Journal (volume 28, issue 2).

In this study, given two crossed complexes 𝒞{\mathcal{C}} and 𝒟{\mathcal{D}} of associative R -algebras and a crossed complex morphism f:𝒞→𝒟{f\colon\mathcal{C}\to\mathcal{D}}, we construct a homotopy as a pair (H,f){(H,f\/)}, where H=(Hn){H=(H_{n})} is a sequence of R -linear maps Hn:Cn→Dn+1{H_{n}\colon C_{n}\to D_{n+1}}. Then we show that for a fixed pair 𝒞{\mathcal{C}} and 𝒟{\mathcal{D}} of

A quasilinearization algorithm is developed for boundary value problems at resonance. To do so, a standard monotonicity condition is assumed to obtain the uniqueness of solutions for the boundary value problem at resonance. Then the method of upper and lower solutions and the shift method are applied to obtain the existence of solutions. A quasilinearization algorithm is developed and sequences of

For certain classes of functions of Λ-bounded variation on [-π,π]{[-\pi,\pi]}, Waterman and D’Antonio constructed a sequence of summability kernels such that the corresponding integral means converge for any function from the class, but not for any function from a broader Λ′⁢BV{\Lambda^{\prime}\mathrm{BV}} class. In our paper, we extend this result to the case of functions of two variables.

Let ℛ{\mathcal{R}} be a prime ring with center Z⁢(ℛ){Z(\mathcal{R})} and *{*} an involution of ℛ{\mathcal{R}}. Suppose that ℛ{\mathcal{R}} admits generalized derivations F , G and H associated with a nonzero derivation f , g and h of ℛ{\mathcal{R}}, respectively. In the present paper, we investigate the commutativity of a prime ring ℛ{\mathcal{R}} satisfying any of the following identities: (i) [F⁢(x)

We consider explicit sufficient conditions for all solutions of a first-order linear difference equation with several variable delays and non-negative coefficients to be oscillatory. The conditions have the form of inequalities bounding below the upper and lower limits of the sums of coefficients over a subset of the discrete semiaxis. Our main results are oscillation tests based on a new principle

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{\mathcal{C}} in the non-classical setting when solutions are sought in the Bessel potential spaces ℍps⁢(𝒞){\mathbb{H}^{s}_{p}(\mathcal{C})}, 1p

In this paper, by means of the generalized Olsen type inequality, related to the Riesz potential and its commutators, the author establishes the interior estimates on the generalized Morrey space for Schrödinger type elliptic equations with potentials satisfying the reverse Hölder condition.

Based on a variational principle for smooth functionals defined on reflexive Banach spaces, the existence of at least one weak solution for a non-homogeneous Neumann problem in an appropriate Orlicz–Sobolev space is discussed.

We extend some numerical radius inequalities for adjointable operators on Hilbert C*{C^{*}}-modules. A new refinement of a numerical radius inequality for some Hilbert space operators is given. More precisely, we prove that if T∈ℬ⁢(ℋ){T\in\mathcal{B}(\mathcal{H})} is an invertible operator, then ∥T∥2≤∥T∥2+1∥T-1∥22≤ω⁢(T).\frac{\|T\|}{2}\leq\frac{\sqrt{\|T\|^{2}+\frac{1}{\|T^{-1}\|^{2}}}}{2}\leq% \omega(T)

In this article, an operational definition, generating function, explicit summation formula, determinant definition and recurrence relations of the generalized families of Hermite–Appell polynomials are derived by using integral transforms and some known operational rules. An analogous study of these results is also carried out for the generalized forms of the Hermite–Bernoulli and Hermite–Euler polynomials

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

In this paper we first classify left-invariant generalized Ricci solitons on four-dimensional hypercomplex Lie groups equipped with three families of left-invariant Lorentzian metrics. Then, on these Lorentzian spaces, we explicitly calculate the energy of an arbitrary left-invariant vector field X and determine the exact form of all left-invariant unit time-like vector fields which are spatially harmonic

It is known that if f⁢(z)=zp+∑n=p+1∞an⁢zn{f(z)=z^{p}+\sum_{n=p+1}^{\infty}a_{n}z^{n}} and it is analytic in a convex domain D⊂ℂ{D\subset\mathbb{C}} and, for some real α, we have ℜ⁢𝔢⁡{exp⁡(i⁢α)⁢f(p)⁢(z)}>0{\operatorname{\mathfrak{Re}}\{\exp(i\alpha)f^{(p)}(z)\}>0}, z∈D{z\in D}, then f⁢(z){f(z)} is at most p -valent in D . This Ozaki condition is a generalization of the well-known Noshiro–Warschawski

The procedure of positioning of pipes, for example for the automobile exhaust construction, by the use of 3D CAD systems is difficult and time consuming because of obstacles in an engine compartment. There are also further strong technical requirements and restrictions. An automatic generation procedure based on the level set method and the numerical solution of the Eikonal equation is proposed. The

In this paper, we study the nonlinear Volterra integral equation satisfied by the early exercise boundary of the American put option in the one-dimensional diffusion model for a stock price with constant interest rate and constant dividend yield and with a local volatility depending on the current time t and the current stock price S . In the classical Black–Sholes model for a stock price, Theorem

In this paper, we propose the concept of a weighted pseudo δ-almost automorphic function under the matched space for time scales and we present some properties. Also, we obtain sufficient conditions for the existence of weighted pseudo δ-almost automorphic mild solutions to a class of semilinear dynamic equations under the matched spaces for time scales.

Article Frontmatter was published on February 1, 2021 in the journal Georgian Mathematical Journal (volume 28, issue 1).

A progressive iterative approximation technique for rational least squares fitting curves is developed. The format is interesting in CAGD (Computer Aided Geometric Design) and improves the recent algorithms. An improved chord method for the root finding based on rational operators is also presented.

In this paper, we study (non-Abelian) extensions of a given hom-Lie color algebra and provide a geometrical interpretation of extensions. In particular, we characterize an extension of a hom-Lie color algebra 𝔤{\mathfrak{g}} by another hom-Lie color algebra 𝔥{\mathfrak{h}} and discuss the case where 𝔥{\mathfrak{h}} has no center. We also deal with the setting of covariant exterior derivatives, Chevalley

Effective sufficient conditions are given for the unique solvability of the Cauchy problem for linear systems of generalized ordinary differential equations with singularities.

Sufficient conditions for the existence and uniqueness of a classical solution to a nonlocal problem for a system of Sobolev-type differential equations with integral condition are established. By introducing a new unknown function, we reduce the considered problem to an equivalent problem consisting of a nonlocal problem for the system of hyperbolic equations of second order with a functional parameter

We prove that a completely regular locale L is realcompact if and only if the “remainder” β⁢L∖L{\beta L\smallsetminus L} is the join of the zero-sublocales of β⁢L{\beta L} that miss L . This extends a result of Mrówka which characterizes realcompact spaces in terms of their remainders in Stone–Čech compactifications. We prove that β⁢L∖L{\beta L\smallsetminus L} is Lindelöf if and only if L is of countable

The purpose of the present paper is to obtain the degree of approximation in terms of a Lipschitz type maximal function for the Kantorovich type modification of Jakimovski–Leviatan operators based on multiple Appell polynomials. Also, we study the rate of approximation of these operators in a weighted space of polynomial growth and for functions having a derivative of bounded variation. A Voronvskaja

In this paper, we show some refinements of generalized numerical radius inequalities involving the Young and Heinz inequality. In particular, we present wpp⁢(A1*⁢T1⁢B1,…,An*⁢Tn⁢Bn)≤n1-1r21r⁢∥∑i=1n[Bi*⁢f2⁢(|Ti|)⁢Bi]r⁢p+[Ai*⁢g2⁢(|Ti*|)⁢Ai]r⁢p∥1r-inf∥x∥=1⁡η⁢(x),w_{p}^{p}(A_{1}^{*}T_{1}B_{1},\dots,A_{n}^{*}T_{n}B_{n})\leq\frac{n^{1-\frac{1% }{r}}}{2^{\frac{1}{r}}}\bigg{\|}\sum_{i=1}^{n}[B_{i}^{*}f^{2}(|T_{i}|)B_{i}]^{%

By using a purely probabilistic argumentation, two theorems are proved. They simplify the existing methods of analysis for the M/G/1{M/G/1} queuing system by means of the supplementary variables method.

In the paper, we derive a formula of integration by parts for the left-sided Hilfer derivative. Moreover, we present a formula of such a type for the right-sided Hilfer derivative. These results generalize the well-known rules of integration by parts for the Riemann–Liouville and Caputo derivatives. Finally, we use these rules to investigate some boundary value problems involving mixed fractional derivatives

The present paper deals with the Lie triple systems via Leibniz algebras. A perfect Lie algebra as a perfect Leibniz algebra and as a perfect Lie triple system is considered and the appropriate universal central extensions are studied. Using properties of Leibniz algebras, it is shown that the Lie triple system universal central extension is either the universal central extension of the Leibniz algebra

In this paper, we define accessibility on an iterated function system (IFS) and show that it provides a sufficient condition for the transitivity of this system and its corresponding skew product. Then, by means of a certain tool, we obtain the topologically mixing property. We also give some results about the ergodicity and stability of accessibility and, further, illustrate accessibility by some

In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers and several integer sequences related to them have been studied. In the present paper, new sets of Bell–Sheffer polynomials are introduced. Connections with Bell numbers are shown.

The main goal of the present paper is to study some results concerning generalized derivations of prime rings with involution. Moreover, we provide examples to show that the assumed restriction cannot be relaxed.

The nonlinear fractional differential equation with nonlocal fractional integro-differential boundary conditions in Banach spaces is studied, an existence result is obtained by using the method associated with the technique of measures of noncompactness and an appropriate fixed point theorem. An example is given to illustrate the theory.

Gabardo and Nashed studied nonuniform wavelets by using the theory of spectral pairs for which the translation set Λ={0,r/N}+2⁢ℤ{\Lambda=\{0,r/N\}+2\mathbb{Z}} is no longer a discrete subgroup of ℝ{\mathbb{R}} but a spectrum associated with a certain one-dimensional spectral pair. In this paper, we establish three sufficient conditions for the nonuniform wavelet system {ψj,λ⁢(x)=(2⁢N)j/2⁢ψ⁢((2⁢N)j⁢x-λ)

We give some new characterizations of quasi-Frobenius rings. Namely, we prove that for a ring R , the following statements are equivalent: (1) R is a quasi-Frobenius ring, (2) M2⁢(R){M_{2}(R)} is right Johns and every closed left ideal of R is cyclic, (3) R is a left 2-simple injective left Kasch ring with ACC on left annihilators, (4) R is a left 2-injective semilocal ring such that R/Sl{R/S_{l}}

In this paper, we study a conformally flat 3-space 𝔽3 which is an Euclidean 3-space with a conformally flat metric with the conformal factor 1F2, where F⁢(x)=e-x12-x22 for x=(x1,x2,x3)∈ℝ3. In particular, we construct all helicoidal surfaces in 𝔽3 by solving the second-order non-linear ODE with extrinsic curvature and mean curvature functions. As a result, we give classification of minimal helicoidal

The basic problem of satisfaction of boundary conditions is considered when the generalized stress vector is given on the surfaces of elastic plates and shells. This problem has so far remained open for both refined theories in a wide sense and hierarchic type models. In the linear case, it was formulated by I. N. Vekua for hierarchic models. In the nonlinear case, bending and compression-expansion

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

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

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