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Georgian Mathematical Journal

Editors-in-Chief: Ivane Kiguradze and Tengiz Buchukuri
Editorial Board: Maia Kvinikadze , O.V. Besov , B. Bojarski , Pavel Drábek , R. Duduchava , Hans-Jürgen Engelbert , Lasha Ephremidze , R. Gamkrelidze , Hvedri Inassaridze , M. Jibladze , T. Kadeishvili , Otto H. Kegel , Alexander Kharazishvili , S. Kharibegashvili , E. Khmaladze , Tariel Kiguradze , S. I. Krushkal , J. Kurzweil , S. Kwapien , Hans Rudolf Lerche , Michael Mania , Jean Mawhin , A. Meskhi , P.E. Ricci , V. Tarieladze , Hans Triebel and Fabio Zanolin

Language: English

First published: February 1, 1994

Publication Frequency: 6 issues per year

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The Georgian Mathematical Journal was founded by the Georgian National Academy of Sciences and A. Razmadze Mathematical Institute, and is jointly produced with De Gruyter. The concern of this international journal is the publication of research articles of best scientific standard in pure and applied mathematics. Special emphasis is put on the presentation of results obtained by Georgian mathematicians.

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Weak-type estimates for variable Riesz potentials with respect to the Hausdorff content over metric measure spaces

June 17, 2025

Toshihide Futamura, Yoshihiro Sawano, Tetsu Shimomura

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We are concerned with weak-type estimates for variable Riesz potentials I α ⁢ ( ⋅ ) , τ ⁢ f {I_{\alpha(\,\cdot\,),\tau}f} with respect to variable dimensional Hausdorff content over bounded metric measure spaces X equipped with lower Ahlfors Q ⁢ ( ⋅ ) {Q(\,\cdot\,)} -regular measures. As an application, we give an exponential integrability for I α ⁢ ( ⋅ ) , τ ⁢ f {I_{\alpha(\,\cdot\,),\tau}f} with respect to variable dimensional Hausdorff content.
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Multiplicative Lie algebra structure on a nilpotent group of class 2

July 9, 2025

Deepak Pal, Amit Kumar, Sumit Kumar Upadhyay

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This paper explores the properties of multiplicative Lie algebra structures on a nilpotent group of class 2.
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Proof of time evolution integration theorems in calculus for moving surfaces

July 18, 2025

David V. Svintradze

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We present a rigorous proof of the time evolution integration theorems that suitably extend differential geometry to the calculus of moving surfaces. This proof demonstrates that the calculus of moving surfaces is a complete extension of differential geometry to the dynamic manifolds.
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The properties of Banach algebras affected by differential identities

July 18, 2025

Mohamed Moumen, Lahcen Taoufiq

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Let ℬ {\mathcal{B}} be a Banach algebra. The interest of this article lies in the study of the commutativity of ℬ {\mathcal{B}} if certain specific algebraic identities hold over a non-empty open subset of ℬ {\mathcal{B}} . The limitations imposed in the hypothesis of our results are justified by some examples.
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Dolbeault’s theorem for sheaves of pseudoanalytic differential forms

August 1, 2025

Irakli Sikharulidze

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A way is presented to extend the notion of pseudoanalyticity of the Bers–Vekua theory to differential forms defined on a complex manifold. A sheaf is defined whose sections over an open subset are differential forms whose component functions with respect to any trivializing sections satisfy in each argument the linear Carleman–Bers–Vekua equation. Such a sheaf can be defined as the kernel sheaf of an appropriate sheaf homomorphism. For it, under certain conditions, an analogue of the Poincaré lemma is valid and, under the same conditions, the sequence of such sheaf homomorphisms forms a complex, which makes is possible to prove an analogue of the Dolbeault’s theorem for these sheaves.
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Iteration method for the Dirichlet problem of the elasticity theory

June 17, 2025

David Natroshvili

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We construct a convergent recurrence scheme for a solution of the three-dimensional Dirichlet boundary value problem of the elasticity theory. By the potential method, the Dirichlet problem is reduced to the uniquely solvable Fredholm integral equation of the first kind with a weakly singular boundary integral operator generated by the single layer potential. First, we construct a sequence of successive approximations which converges to the solution of the boundary integral equation in appropriate Bessel-potential spaces of functions defined on the boundary. Afterwards, using these approximations as densities of the single layer potential, we formulate another iteration which converges to the solution of the Dirichlet boundary value problem in the appropriate Sobolev–Slobodetskii spaces of functions defined in the region occupied by the elastic body.
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Martingale Hardy spaces and some new weighted maximal operators of partial sums of Walsh–Fourier series

June 28, 2025

Davit Baramidze

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In this paper, we introduce some new weighted maximal operators of the partial sums of the Walsh–Fourier series. We prove that for some “optimal” weights these operators indeed are bounded from the martingale Hardy space H p ⁢ ( G ) {H_{p}(G)} to the Lebesgue space L p ⁢ ( G ) {L_{p}(G)} for 0 < p < 1 {0<p<1} . Moreover, we also prove the sharpness of this result. As a consequence, we obtain some new and well-known results.
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Existence of positive weak solutions for fractional Laplacian elliptic systems

June 19, 2025

José Vanterler da Costa Sousa, Leandro S. Tavares, Nguyen Thanh Chung

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In this paper, we focus on studying the existence of positive weak solutions for a class of fractional Laplacian systems with respect to another function ψ ⁢ ( ⋅ ) {\psi(\,\cdot\,)} with multiple parameters. These systems arise in various applications, such as Lévy flights. We establish new conditions for the existence of a positive solution for the considered system using the method of sub-super solutions, which extends and complements previously known results in the literature.
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Existence and multiplicity analysis of a system of nonlinear elliptic equations: Theoretical results and applications

October 31, 2025

Khaled Kefi, Jian Liu, Mohamed Karim Hamdani

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This paper introduces a mathematical model that provides a versatile framework for the investigation of complex phenomena in bounded domains. The model is described by a system of partial differential equations subject to boundary conditions. The equations involve a set of functions, denoted by w i {w_{i}} , which satisfy a coupled system of equations. The model captures various physical processes and phenomena across different domains. The analysis establishes the existence of solutions for the system of equations under consideration. Moreover, it demonstrates the possibility of multiple solutions.
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On properties of weakly locally compact commutative groups

July 18, 2025

Onise Surmanidze

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The paper presents an algebraic characterization of a twist-free weakly locally compact group whose factor group is linearly compact with respect to the zero component. It is shown that the local direct product of weakly locally compact groups is weakly locally compact. Additionally, a necessary and sufficient condition is established for determining when a weakly locally compact group G , containing a given subgroup H , can be decomposed into a local direct product of subgroups of rank one.
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Banach algebra structure in analytic tent spaces

August 29, 2025

Rong Yang, Xiangling Zhu

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In this paper, the Duhamel product is used to endow analytic tent spaces with a Banach algebra structure. The Hadamard–Bergman convolution on these spaces is also described. Additionally, the Deddens algebras of composition operators are studied.
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Enveloping algebra is a Yetter–Drinfeld module algebra over Hopf algebra of regular functions on the automorphism group of a Lie algebra

October 28, 2025

Zoran Škoda, Martina Stojić

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We present an elementary construction of a (highly degenerate) Hopf pairing between the universal enveloping algebra U ⁢ ( 𝔤 ) {U(\mathfrak{g})} of a finite-dimensional Lie algebra 𝔤 {\mathfrak{g}} over arbitrary field 𝒌 {{\boldsymbol{k}}} , and the Hopf algebra 𝒪 ⁢ ( Aut ⁡ ( 𝔤 ) ) {\mathcal{O}(\operatorname{Aut}(\mathfrak{g}))} of regular functions on the automorphism group of 𝔤 {\mathfrak{g}} . This pairing induces a Hopf action of 𝒪 ⁢ ( Aut ⁡ ( 𝔤 ) ) {\mathcal{O}(\operatorname{Aut}(\mathfrak{g}))} on U ⁢ ( 𝔤 ) {U(\mathfrak{g})} , which, together with an explicitly given coaction, makes U ⁢ ( 𝔤 ) {U(\mathfrak{g})} into a braided commutative Yetter–Drinfeld 𝒪 ⁢ ( Aut ⁡ ( 𝔤 ) ) {\mathcal{O}(\operatorname{Aut}(\mathfrak{g}))} -module algebra. From these data one constructs a Hopf algebroid structure on the smash product algebra ♯ ⁡ 𝒪 ⁢ ( Aut ⁡ ( 𝔤 ) ) U ⁢ ( 𝔤 ) {\mathcal{O}(\operatorname{Aut}(\mathfrak{g}))\mathbin{\sharp}U(\mathfrak{g})} , retaining essential features from earlier constructions of a Hopf algebroid structure on infinite-dimensional versions of the Heisenberg double of U ⁢ ( 𝔤 ) {U(\mathfrak{g})} , including a noncommutative phase space of Lie algebra type, while avoiding the need of completed tensor products. We prove a slightly more general result, where the algebra 𝒪 ⁢ ( Aut ⁡ ( 𝔤 ) ) {\mathcal{O}(\operatorname{Aut}(\mathfrak{g}))} is replaced by 𝒪 ⁢ ( Aut ⁡ ( 𝔥 ) ) {\mathcal{O}(\operatorname{Aut}(\mathfrak{h}))} and where 𝔥 {\mathfrak{h}} is any finite-dimensional Leibniz algebra having 𝔤 {\mathfrak{g}} as its maximal Lie algebra quotient.
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Weighted estimates for commutators of multilinear singular integral operators in Dunkl setting

June 28, 2025

Yan Wang, Fanghui Liao, Zongguang Liu

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In this paper, we study multilinear Dunkl–Calderón–Zygmund operators, focusing on both the establishment of endpoint estimates for the operators and the boundedness of two types of commutators generated by these operators and BMO d {\mathrm{BMO}_{d}} functions, as demonstrated through sharp maximal function estimates.
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Generalized EP elements in Banach *-algebras

December 3, 2025

Huanyin Chen, Marjan Sheibani

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We introduce a new generalized inverse, termed the generalized EP element, which naturally extends the concepts of EP and *-DMP elements within the framework of Banach *-algebras. We provide polar-like characterizations for these generalized EP elements and investigate conditions ensuring their additive closure. Finally, we characterize the generalized core-EP orders associated with these elements.
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Approximation by de la Vallée-Poussin-type means of Fourier series in integral and Hölder metrics

October 31, 2025

Boris Golubov, Sergey Volosivets

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We estimate the degree of approximation by de la Vallée-Poussin-type means in L p {L^{p}} , uniform and Hölder metric. Applications to conjugate functions are also given. Our results extend those of Leindler–Meir–Totik and Kim.
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C-Gorenstein projective, injective, flat modules and trivial ring extensions

June 17, 2025

Lixin Mao

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Let C be a semidualizing module over a commutative ring R and let R ⋉ C {R\ltimes C} be the trivial extension of R by C . We prove that (1) ( M , f ) {(M,f)} is a Gorenstein projective R ⋉ C {R\ltimes C} -module if and only if M is a C -Gorenstein projective R -module; (2) [ M , g ] {[M,g]} is a Gorenstein injective R ⋉ C {R\ltimes C} -module if and only if M is a C -Gorenstein injective R -module; (3) if R is a coherent ring, then ( M , f ) {(M,f)} is a Gorenstein flat R ⋉ C {R\ltimes C} -module if and only if M is a C -Gorenstein flat R -module. Some applications of these results are given.
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Second-order logarithmic methods for the summability of Fourier series

October 28, 2025

Xhevat Z. Krasniqi, Péter Kórus

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In this paper, we first present the second-order Riesz and Nörlund logarithmic summation methods, followed by a class of regular summation techniques based on second-order logarithmic averages. We then examine their applications to the pointwise convergence of Fourier series for continuous and integrable functions.
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Dilation estimates for Orlicz amalgam spaces on the affine group

July 9, 2025

Büşra Arıs

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Let 𝔸 {\mathbb{A}} be the affine group and let Φ 1 , Φ 2 {\Phi_{1},\Phi_{2}} be Young functions. We study the Orlicz amalgam spaces W ⁢ ( L Φ 1 ⁢ ( 𝔸 ) , L Φ 2 ⁢ ( 𝔸 ) ) {W(L^{\Phi_{1}}(\mathbb{A}),L^{\Phi_{2}}(\mathbb{A}))} defined on 𝔸 {\mathbb{A}} , where the local and global component spaces are the Orlicz spaces L Φ 1 ⁢ ( 𝔸 ) {L^{\Phi_{1}}(\mathbb{A})} and L Φ 2 ⁢ ( 𝔸 ) {L^{\Phi_{2}}(\mathbb{A})} , respectively. In this paper, we obtain norm estimates of the dilation operator on the amalgam space W ⁢ ( L Φ 1 ⁢ ( 𝔸 ) , L Φ 2 ⁢ ( 𝔸 ) ) {W(L^{\Phi_{1}}(\mathbb{A}),L^{\Phi_{2}}(\mathbb{A}))} using constructions related to the affine group 𝔸 {\mathbb{A}} .
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On generalized bounded variation functions on Vilenkin groups and applications

August 1, 2025

Ushangi Goginava, Gvantsa Shavardenidze, Tadael Mequanent, Yonathan Demeke

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In the present paper, we introduce certain classes of functions of weighted bounded oscillation on bounded Vilenkin groups. For such classes, we employ the summability methods of the theory of double Vilenkin–Fourier series.
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A series representation for the Catalan constant

August 29, 2025

Horst Alzer

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We prove the following series representation for the Catalan constant G : G = 3 10 - 48 ⁢ ∑ n = 0 ∞ 4 ⁢ n 2 + 8 ⁢ n - 9 ( 4 ⁢ n 2 - 4 ⁢ n + 5 ) ⁢ ( 4 ⁢ n 2 + 20 ⁢ n + 29 ) ⁢ S n 3 , G=\frac{3}{10}-48\sum_{n=0}^{\infty}\frac{4n^{2}+8n-9}{(4n^{2}-4n+5)(4n^{2}+20% n+29)}S_{n}^{3}, where S n = ∑ k = 0 n ( - 1 ) k 2 ⁢ k + 1 S_{n}=\sum_{k=0}^{n}\frac{(-1)^{k}}{2k+1} is the n th partial sum of the classical Leibniz series for π 4 {\frac{\pi}{4}} .
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Boundary value problem with Dirichlet, Neumann and Robin conditions on separate parts of the boundary for an inhomogeneous equation of string vibration

July 9, 2025

Otar Jokhadze, Sergo Kharibegashvili

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In this paper, the well-posedness of a boundary value problem for an inhomogeneous equation of string vibration with Dirichlet, Neumann and Robin conditions on separate parts of the boundary is proved. In this case, the solution is given in quadratures. If the Robin condition turns into the Neumann condition, then the problem becomes ill-posed. In particular, it is shown that the corresponding homogeneous problem has an infinite number of linearly independent solutions that can be written explicitly.
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A property of Cauchy–Stieltjes kernel families based on dilation of measures

May 29, 2025

Ibrahim-Elkhalil Ahmed, Ayed R. A. Alanzi, Raouf Fakhfakh

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In this paper, we introduce a property of the inverse Semicircle and the free Gamma laws based on the dilation of measures in the context of Cauchy–Stieltjes Kernel (CSK) families. Assume that the CSK family produced by a non-degenerate probability measure λ on ℝ {\mathbb{R}} with support limited from above is ℱ + ⁢ ( λ ) = { 𝒬 m λ ⁢ ( d ⁢ y ) : m ∈ ( m 1 λ , m + λ ) } {\mathcal{F}_{+}}(\lambda)=\{\mathcal{Q}_{m}^{\lambda}(dy):m\in(m_{1}^{\lambda% },m_{+}^{\lambda})\} . For α ∈ ℝ ∖ 0 {\alpha\in\mathbb{R}\setminus 0} , consider 𝐇 α {\mathbf{H}_{\alpha}} , x ↦ α ⁢ x {x\mapsto\alpha x} and provide the set of measures 𝐇 α ⁢ ( ℱ + ⁢ ( λ ) ) = { 𝐇 α ⁢ ( 𝒬 m λ ⁢ ( d ⁢ y ) ) : m ∈ ( m 1 λ , m + λ ) } . \mathbf{H}_{\alpha}({\mathcal{F}_{+}}(\lambda))=\{\mathbf{H}_{\alpha}(\mathcal% {Q}_{m}^{\lambda}(dy)):m\in(m_{1}^{\lambda},m_{+}^{\lambda})\}. Let us say that α > 0 {\alpha>0} . We demonstrate that if 𝐇 α ⁢ ( ℱ + ⁢ ( λ ) ) {\mathbf{H}_{\alpha}({\mathcal{F}_{+}}(\lambda))} is a re-parametrization of ℱ + ⁢ ( λ ) {{\mathcal{F}_{+}}(\lambda)} (i.e., 𝐇 α ⁢ ( ℱ + ⁢ ( λ ) ) = ℱ + ⁢ ( λ ) {\mathbf{H}_{\alpha}({\mathcal{F}_{+}}(\lambda))={\mathcal{F}_{+}}(\lambda)} ), then λ is either the free Gamma type law or the inverse Semicircle type law up to scaling.
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The variable exponent periodic boundary value problems on the Lyapunov open curve

November 15, 2025

Lin Li, Fuli He

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This study investigates the periodic Riemann boundary value problem on a Lyapunov open curve within the framework of variable exponent spaces. By extending the open curve to a closed one, we demonstrate that this extension preserves the class of the problem. Periodicity is eliminated through a tangent transformation, and a weight function is introduced to handle discontinuous coefficients. The equivalence of the problem is established using weighted variable indices. By analyzing the singularities at the endpoints, we construct the corresponding canonical function, which yields an explicit solution. Additionally, the problem is extended from the single open curve case to the more general scenario involving multiple open curves.
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Convergence analysis of Kantorovich-type operators in variable exponent Sobolev spaces

December 3, 2025

Benali Aharrouch, Mustapha Bouallala

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In this work, we establish convergence results in variable exponent Sobolev spaces W 1 , p ⁢ ( ⋅ ) ⁢ ( R ) W^{1,p(\,{\cdot}\,)}(\mathbb{R}) , where the exponent function p : R → R p\colon\mathbb{R}\to\mathbb{R} satisfies 1 ≤ p − ≤ p ⁢ ( x ) ≤ p + < + ∞ 1\leq p^{-}\leq p(x)\leq p^{+}<+\infty for all x ∈ R x\in\mathbb{R} , for Kantorovich-type operators activated by kernel functions. Furthermore, we prove uniform convergence for functions belonging to appropriate function spaces.
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Nearly cosine series and generalized trigonometric functions

June 17, 2025

Alessandro Curcio, Giuseppe Dattoli, Emanuele Di Palma, Pierpaolo Natalini, Paolo Emilio Ricci

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In this paper, A class of overlooked trigonometric-like functions is explored, along with the relevant applications. We indeed show that Taylor series, resembling that of an ordinary cosine, are representative of wider classes of functions that are naturally suited to problems ranging from molecular to Laser Physics. The article goes through the original motivations of the proposal and studies the relevant properties within the context of an Umbral interpretation. Their use in applications is discussed within the framework of Free Electron Laser theory, Lennard–Jones potentials and Kramers–Kronig causality identities.
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Summability properties under nonstandard growth conditions

October 11, 2025

Ting Shan, Hongya Gao, Qianqian Liu

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This paper investigates summability properties under nonstandard growth conditions. We consider the boundary value problem of the 𝒜 {\cal A} -harmonic equation { - div ⁡ 𝒜 ⁢ ( x , ∇ ⁡ u ⁢ ( x ) ) = f ⁢ ( x ) in ⁢ Ω , u ⁢ ( x ) = u ∗ ⁢ ( x ) on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}{{-}\operatorname{div}\mathcal{A}(x,% \nabla u(x))}&\displaystyle=f(x)&&\displaystyle\mbox{in }\Omega,\\ \displaystyle u(x)&\displaystyle=u_{\ast}(x)&&\displaystyle\mbox{on }{\partial% \Omega},\end{aligned}\right. where f ⁢ ( x ) {f(x)} and D ⁢ u * ⁢ ( x ) {Du_{*}(x)} lie in some Marcinkiewicz spaces. Summability properties of entropy solutions are obtained under nonstandard growth conditions on the operator 𝒜 : Ω × ℝ n → ℝ n {{\cal A}:\Omega\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}} . We also consider T -minima of the integral functional 𝒥 ⁢ ( v ) = ∫ Ω j ⁢ ( x , ∇ ⁡ v ⁢ ( x ) ) ⁢ 𝑑 x - ∫ Ω f ⁢ ( x ) ⁢ v ⁢ ( x ) ⁢ 𝑑 x , {\cal J}(v)=\int_{\Omega}j(x,\nabla v(x))\,dx-\int_{\Omega}f(x)v(x)\,dx, where f ⁢ ( x ) {f(x)} lies in some Marcinkiewicz space. We consider T -minima among functions whose boundary values agree with u * ⁢ ( x ) {u_{*}(x)} , and obtain some summability properties under nonstandard growth conditions of the operator j ⁢ ( x , ξ ) : Ω × ℝ n → ℝ {j(x,\xi):\Omega\times\mathbb{R}^{n}\rightarrow\mathbb{R}} .
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Commutativity of Banach algebras involving endomorphisms

July 25, 2025

Mohammadi El Hamdaoui

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In this article, we present new contributions to the study of endomorphisms within algebraic structures, particularly focusing on prime Banach algebras. We establish that any prime Banach algebra must be commutative if it admits endomorphisms and multipliers satisfying specific algebraic identities. Our main results generalize several known theorems. Additionally, we provide an illustrative example demonstrating that the primeness assumption in several theorems is indeed essential.
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On index divisors of certain sextic number fields defined by quadrinomials

October 11, 2025

Hamid Ben Yakkou, Issam Aghzer, Abdelkarim Boua

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Let K be a number field of degree 6 generated by a root of an irreducible quadrinomial F ⁢ ( x ) = x 6 + a ⁢ x m + b ⁢ x 2 + c ∈ ℤ ⁢ [ x ] F(x)=x^{6}+ax^{m}+bx^{2}+c\in{\mathbb{Z}}[x] with m ∈ { 3 , 4 , 5 } {m\in\{3,4,5\}} , and let i ⁢ ( K ) {i(K)} denote the index of K . In this paper, for p = 2 {p=2} or 3, we give sufficient conditions on a , b {a,b} and c such that p is a common index divisor of K , and we evaluate ν p ⁢ ( i ⁢ ( K ) ) {\nu_{p}(i(K))} , the highest power of p dividing i ⁢ ( K ) {i(K)} . In this way, we provide a partial answer to Problem 22 of [W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, 3rd ed., Springer Monogr. Math., Springer, Berlin, 2004] for these families of number fields. As an application of our results, we identify new classes of sextic number fields having no power integral basis. Further, several corollaries and examples are provided which illustrate our results. Our approach is based on a theorem of Ore on the decomposition of primes in number fields. [18, 30, 34].
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Modeling and numerical analysis of a viscoelastic contact problem for locking materials with time-fractional derivatives

August 29, 2025

Jinxia Cen, Mustapha Bouallala, Abdesadik Bendarag

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This article explores a new contact problem involving viscoelastic materials locked with a rigid foundation. The constitutive relation in our study is constructed using a fractional Kelvin–Voigt model to describe displacement behavior. The contact is characterized by a variation of the Signorini condition, while friction is modeled by a nonlocal Coulomb’s friction law. We present the mathematical model for the viscoelastic process, provide the variational formulation, and establish the existence of a solution. We present fully discrete finite element schemes for the variational problem and derive error estimates for the approximations.
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Linear maps preserving products equal to vertices of a path algebra

October 28, 2025

Abderrahim Adrabi, Driss Bennis, Brahim Fahid

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This paper studies bijective linear maps that preserve products equal to the vertices on path algebras over a field. These maps are fully characterized on path algebras associated with connected finite acyclic quivers.
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Approximation of Kantorovich-type max-product q-Bernstein operators

May 29, 2025

Sezin Cit, Carmen Violeta Muraru, Ogün Dogru

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In this article, we show that the approximation order of Kantorovich-type max-product Bernstein operators can be improved. We are able to obtain a better approximation order for q -Bernstein operators. In addition, we present a Kantorovich-type generalization of max-product operators based on q -integers. In particular, we find the approximation order (via the classical modulus of continuity) of the Kantorovich-type max-product q -Bernstein operator.
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An introduction to fractal Lebesgue integral

August 29, 2025

Hemanta Kalita, Alireza K. Golmankhanehand

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This manuscript explores various characteristics of generalized fractal measures. We expand the concept of fractal integrals in relation to step functions and examine their numerous properties. Notably, since all step functions are classified as simple functions, we apply the aforementioned generalized measure to introduce Lebesgue-type integrals, referred to as FL -integrals. Additionally, we demonstrate that all F α {F^{\alpha}} -integrable functions are FL -integrals. Lastly, we address the bounded convergence theorem within the context of fractals.
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On a system of finitely many nonlinear Riemann–Liouville fractional differential equations with fractional anti-periodic boundary conditions

November 1, 2025

Bashir Ahmad, Hafed A. Saeed, Ahmed Alsaedi, Ravi P. Agarwal

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In this paper, we introduce a coupled system of n nonlinear Riemann–Liouville fractional differential equations each of order in ( 2 , 3 ) {(2,3)} and equipped with fractional anti-periodic boundary conditions. We explore the existence and uniqueness criteria for solutions of the given problem by means of Leray–Schauder’s alternative and Banach’s contraction mapping principle in a weighted space. The Ulam–Hyers stability is also studied for the problem at hand. Examples are presented to illustrate the main results.
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Trudinger-type inequalities in Musielak–Orlicz spaces over metric measure spaces

June 18, 2025

Takao Ohno, Tetsu Shimomura

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Our aim in this note is to deal with Trudinger-type inequalities for variable Riesz potentials of functions in Musielak–Orlicz spaces over bounded metric measure spaces as an extension of the previous paper by Hurri-Syrjänen and the authors ( Complex Var. Elliptic Equ. 68 (2023), no. 10, 1694–1714). As an application, we prove Trudinger-type inequalities for double phase functionals.
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On power series strong and statistical convergences for double sequences via Orlicz functions

August 29, 2025

Nilay Şahin Bayram, Sevda Yıldız

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The present paper introduces the notions of strong P p 2 {P_{p}^{2}} -convergence and P p 2 {P_{p}^{2}} -statistical convergence for double sequences with respect to an Orlicz function. Based on these new notions of convergence, new double sequence spaces and some of their properties are given. Furthermore, certain inclusion relations are analyzed. Finally, we observe that, under proper choices, the analogous results can be obtained for modulus function.
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On a fourth-order Leray–Lions problem involving s(x)-Hardy potential and singular nonlocal source term

July 9, 2025

Mohamed Karim Hamdani, Khaled Kefi, Romulo Diaz Carlos, Zeineb Klai

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This paper investigates the existence of at least two distinct weak solutions using Bonanno’s theorem [G. Bonanno, Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal. 1 (2012), no. 3, 205–220, Theorem 3.2], and establishes the existence of no fewer than two nontrivial weak solutions through the Bonanno–D’Aguì theorem [G. Bonanno and G. D’Aguì, Two nonzero solutions for elliptic Dirichlet problems, Z. Anal. Anwend. 35 (2016), no. 4, 449–464, Theorem 2.1] applied to a general class of fourth-order Leray–Lions problems that include an s ⁢ ( x ) {s(x)} -Hardy potential and a nonlocal singular term.
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The binary products of algebras with genetic realization

October 28, 2025

Dali Zangurashvili

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We replace the assumption in Proposition 2.4 of our article by a milder condition.
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Highly accurate compact difference schemes for multidimensional delay Schrödinger equations

June 17, 2025

Allaberen Ashyralyev, Deniz Agirseven, Baris Erköse

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In present paper, the second-order accurate stable compact difference schemes (DSs) for the delay Schrödinger-type partial differential equation (DSPDE) in a Hilbert space are constructed. The stability of these DSs is established. As applications, stability estimates (SEs) for the solutions of DSs for two types of DSPDEs are derived. A numerical method is proposed for solving one and two-dimensional DSPDEs.
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New improvements on numerical radius bounds via the Moore–Penrose inverse

October 11, 2025

Mohamed Amine Ighachane, Fuad Kittaneh, Yonghui Ren

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In this paper, we establish several inner product inequalities that characterize the positivity of block operator matrices. By examining specific blocks of positive operator matrices involving the Moore–Penrose inverse, we derive new inner product inequalities that refine recently established bounds. As an application, we further refine certain numerical radius inequalities from the literature. Our findings generalize and extend several well-known results in this field, contributing to ongoing advancements in operator inequalities.
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Weighted Hardy-type inequality in two dimensions

October 1, 2025

Rohit Chauhan, Suket Kumar

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Weighted Hardy-type inequality in two dimensions has been characterized for pairs of weight functions ω 1 ⁢ ( α 1 , α 2 ) {\omega_{1}(\alpha_{1},\alpha_{2})} and ω 2 ⁢ ( α 1 , α 2 ) {\omega_{2}(\alpha_{1},\alpha_{2})} defined on ℝ + 2 {\mathbb{R}_{+}^{2}} , for which the sum of two-dimensional Hardy-type integral operators is bounded from a weighted Lebesgue space L ω 1 β ⁢ ( ℝ + 2 ) {L_{\omega_{1}}^{\beta}(\mathbb{R}_{+}^{2})} to L ω 2 γ ⁢ ( ℝ + 2 ) {L_{\omega_{2}}^{\gamma}(\mathbb{R}_{+}^{2})} for 1 < β < γ < ∞ {1<\beta<\gamma<\infty} .
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Finite groups in which every maximal invariant subgroup of order divisible by p is nilpotent

July 9, 2025

Jiangtao Shi, Mengjiao Shan, Fanjie Xu

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Let A and G be finite groups such that A acts coprimely on G by automorphisms. For any fixed prime divisor p of | G | {|G|} , we provide a complete characterization of the structure of a group G in which every maximal A -invariant subgroup of order divisible by p is nilpotent.
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Mixed partial differential equation: Forward problem linked with the wave-diffusion process

October 11, 2025

Erkinjon Karimov, Niyaz Tokmagambetov, Muzaffar Toshpulatov

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The main target of the present investigation is a mixed partial differential equation involving the Prabhakar integral-differential operator in the time variable. We begin by briefly describing the several physical processes in which mixed partial differential equations play a key role. Then we formulate an initial-boundary problem for the considered equation linked with the wave-diffusion process. Our primary objective is to demonstrate the unique solvability of the formulated problem under specific conditions for the given data. First, we present the explicit solution of the Cauchy problem for an ordinary differential equation with the regularized Prabhakar fractional derivative. We also present important statements on the bivariate Mittag-Leffler function, namely, Euler-type integral representations and certain estimations for the bi-variate Mittag-Leffler-type function E 2 ⁢ ( x , y ) {E_{2}(x,y)} . Since the key tool of investigation is the method of separation of variables, most of our evaluations are linked with the proof of uniform convergence of infinite series. We impose certain conditions on given functions to provide uniform convergence of infinite series corresponding to the solution of the formulated problem.
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Determinant, inertia, and inverse of a class of irreducible tridiagonal matrices

October 28, 2025

Cristian Conde, Ezequiel Dratman, Luciano N. Grippo

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In this paper, we study a class of irreducible tridiagonal matrices and analyze their fundamental properties. We derive explicit formulas for their determinant, providing insights into their spectral behavior. Additionally, we establish results concerning their inertia and obtain closed-form expressions for their inverse. These findings contribute to the broader understanding of structured matrices and have potential applications in various fields of mathematics and other sciences.
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Generalization of some regularity criteria for 3D Boussinesq equations in homogeneous Besov and Triebel–Lizorkin spaces

October 28, 2025

Mohamed Benallia

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In this paper, we generalize some regularity criteria for weak solutions of three-dimensional (3D) Boussinesq equations in homogeneous Besov spaces B ˙ ( ℝ 3 ) p , q s {\dot{{B}}{{}_{p,q}^{s}}(\mathbb{R}^{3})} and homogeneous Triebel–Lizorkin spaces F ˙ ( ℝ 3 ) p , q s {\dot{{F}}{{}_{p,q}^{s}}(\mathbb{R}^{3})} . We also deduce them for the homogeneous Sobolev spaces W ˙ p m ⁢ ( ℝ 3 ) {\dot{W}^{m}_{p}(\mathbb{R}^{3})} in a certain sense. We show that the weak solution ( ω , ϕ ) {(\omega,\phi)} is regular on ] 0 , T ] {]0,T]} for all T > 0 {T>0} if the velocity ω satisfies ω ∈ L β ( 0 , T ; B ˙ ( ℝ 3 ) p , q s ) , ω ∈ L β ( 0 , T ; F ˙ ( ℝ 3 ) p , q s ) , and ω ∈ L β ( 0 , T ; W ˙ p m ( ℝ 3 ) ) , \omega\in L^{\beta}\big{(}0,T;\dot{{B}}{{}_{p,q}^{s}}(\mathbb{R}^{3})\big{)},% \quad\omega\in L^{\beta}\big{(}0,T;\dot{{F}}{{}_{p,q}^{s}}(\mathbb{R}^{3})\big% {)},\quad\mbox{and}\quad\omega\in L^{\beta}\big{(}0,T;\dot{W}^{m}_{p}(\mathbb{% R}^{3})\big{)}, under some conditions on the parameters s , p , q and m .
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A Cauchy–Kowalevski theorem for ψ-bimonogenic functions

May 29, 2025

José Luis Serrano Ricardo, Ricardo Abreu Blaya

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In this paper we prove a Cauchy–Kowalevski theorem for the solutions of the system ∂ x ψ ∂ x ψ f = 0 {{}^{\psi}\partial_{x}{}^{\psi}\partial_{x}f=0} , where ∂ x ψ {{}^{\psi}\partial_{x}} stands for the Cauchy–Riemann operator, associated to the orthonormal basis ψ of ℝ m + 1 {{\mathbb{R}}^{m+1}} .
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On certain new characterizations of weights in higher order Wirtinger-type inequalities via time scales calculus

June 17, 2025

Mahmoud M. Osman, Mario Krnić, Ravi P. Agarwal, Samir H. Saker

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In this paper, we establish a new characterization of weights in dynamic inequality of Wirtinger’s type involving higher order delta derivatives on time scales. More precisely, conditions on nonnegative rd-continuous weighted functions u and υ are given which ensure that a mixed norm dynamic inequality of the form ∥ ψ σ ∥ 𝕃 u q q ⁢ ( [ a , ∞ ) 𝕋 ) ≤ C ⁢ ∥ ψ Δ n ∥ 𝕃 υ p p ⁢ ( [ a , ∞ ) 𝕋 ) , \|\psi^{\sigma}\|_{\mathbb{L}_{u^{q}}^{q}([a,\infty)_{\mathbb{T}})}\leq C\big{% \|}\psi^{\Delta^{n}}\big{\|}_{\mathbb{L}_{\upsilon^{p}}^{p}([a,\infty)_{% \mathbb{T}})}, holds for 1 < p ≤ q ≤ p ′ {1<p\leq q\leq p^{\prime}} , p ′ = p p - 1 {p^{\prime}=\frac{p}{p-1}} and ψ ∈ C r ⁢ d ( n ) ⁢ ( [ a , ∞ ) 𝕋 , ℝ ) {\psi\in C_{rd}^{(n)}([a,\infty)_{\mathbb{T}},\mathbb{R})} . As a special case, these results encompass the results obtained for integral inequalities when 𝕋 = ℝ {\mathbb{T=R}} , for the discrete inequalities when 𝕋 = ℕ {\mathbb{T=N}} , and for the quantum inequalities of quantum series when 𝕋 = q ℕ {\mathbb{T=}q^{\mathbb{N}}} .
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Generalized Volterra integral operators on weighted Fock spaces induced by A ∞ -type weights

July 18, 2025

Chunxu Xu

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In this paper, we study the generalized Volterra integral operators T g n , m ⁢ f ⁢ ( z ) = ∫ 0 z f ( n ) ⁢ ( w ) ⁢ g ( m ) ⁢ ( w ) ⁢ 𝑑 w T_{g}^{n,m}f(z)=\int_{0}^{z}f^{(n)}(w)g^{(m)}(w)\,dw acting on the weighted Fock spaces F α , ω p {F_{\alpha,\omega}^{p}} , where ω is a weight satisfying certain restricted A ∞ {A_{\infty}} -conditions. Using a unified approach, the boundedness, compactness, and essential norm of the generalized Volterra integral operators T g n , m : F α , ω 1 p → F β , ω 2 q {T_{g}^{n,m}:F_{\alpha,\omega_{1}}^{p}\rightarrow F_{\beta,\omega_{2}}^{q}} for all 0 < p , q < ∞ {0<p,q<\infty} are completely characterized. As an application, we generalize our results to the Fock–Sobolev-type spaces.
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Singular value and norm bounds for accretive-dissipative matrices

August 29, 2025

Wasim Audeh, Hamid Reza Moradi, Mohammad Sababheh

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This paper discusses possible new bounds for the singular values and unitarily invariant norms of accretive-dissipative matrices. The obtained bounds are possible generalizations for certain known results from the context of positive matrices to the context of accretive-dissipative ones.
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Oscillatory singular integrals with radial rough kernel and fewnomials phases

June 17, 2025

Boyu Jiang

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In this paper, we study oscillatory singular integrals with radial rough kernels: T Q , h ⁢ f ⁢ ( x ) = p.v. ⁢ ∫ ℝ n f ⁢ ( x - y ) ⁢ Ω ⁢ ( y ) ⁢ h ⁢ ( | y | ) | y | n ⁢ e i ⁢ Q ⁢ ( | y | ) ⁢ d y , T_{Q,h}f(x)=\text{ p.v. }\int_{\mathbb{R}^{n}}f(x-y)\frac{\Omega(y)h(|y|)}{|y|% ^{n}}e^{iQ(|y|)}\,\mathrm{d}y, where Q ⁢ ( t ) {Q(t)} is a real polynomial on ℝ {\mathbb{R}} , Ω ∈ H ⁢ ( S n - 1 ) {\Omega\in H(S^{n-1})} is a homogeneous function of degree zero on the unit sphere S n - 1 {S^{n-1}} with vanishing mean value, and h satisfies the following conditions: 1 R ⁢ ∫ 0 R | h ⁢ ( t ) | q ⁢ d t ≤ C , | 1 p - 1 2 | < min ⁡ { 1 2 , 1 q ′ } , \frac{1}{R}\int_{0}^{R}|h(t)|^{q}\,\mathrm{d}t\leq C,\quad\bigg{|}\frac{1}{p}-% \frac{1}{2}\bigg{|}<\min\biggl{\{}\frac{1}{2},\frac{1}{q^{\prime}}\bigg{\}}, where C is a constant independent of R > 0 {R>0} , and h is the radial rough kernel. Under the assumption Ω ∈ H 1 ⁢ ( S n - 1 ) {\Omega\in H^{1}(S^{n-1})} , we prove that T Q , h {T_{Q,h}} is bounded on the Lebesgue space L p {L^{p}} for 1 < p < ∞ {1<p<\infty} , provided that h satisfies aforementioned condition. The methods we mainly use are created by S. Guo.
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Semigroup real characters generated by quasicharacters

August 29, 2025

Gastão Bettencourt, Sérgio Mendes

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Let S be an infinite, finitely generated semigroup, endowed with a probability measure. Based on the work [A. Erschler and A. Karlsson, Homomorphisms to ℝ \mathbb{R} constructed from random walks, Ann. Inst. Fourier (Grenoble) 60 2010, 6, 2095–2113], we construct a real character of S using a random walk approach. In our construction, the word length used in Erschler and Karlsson’s work is replaced by a quasicharacter.
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A q-analog of Schröder sequence spaces and compact operators

July 9, 2025

Muhammet Çihat Dağli, Taja Yaying, Mohammed Mursaleen

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We introduce q -analog of Schröder matrix and discuss its domains in the spaces c and c 0 {c_{0}} . Also, the Schauder basis and α-, β-, γ-duals are given and certain classes of matrix mappings are characterized on these spaces. Moreover, the characterizations of certain compact operators are established via Hausdorff measure of noncompactness.

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Editor-in-Chief
Ivan Kiguradze
A. Razmadze Mathematical Institute
I. Javakhishvili Tbilisi State University
2, Merab Aleksidze II Lane
0193 Tbilisi, Georgia
Email: gmj-rmi@tsu.ge

Executive Editor
Tengiz Buchukuri
A. Razmadze Mathematical Institute
I. Javakhishvili Tbilisi State University
2, Merab Aleksidze II Lane
0193 Tbilisi, Georgia
E-mail: t_buchukuri@yahoo.com

Editorial Secretary
Maia Kvinikadze
A. Razmadze Mathematical Institute
I. Javakhishvili Tbilisi State University
2, Merab Aleksidze II Lane
0193 Tbilisi, Georgia
Email: maia.kvinikadze@tsu.ge

Editorial Board
Hans-Joachim Baues
Max-Planck-Institut für Mathematik
53072 Bonn, Germany
Email: baues@mpim-bonn.mpg.de

Oleg V. Besov
Steklov Mathematical Institute
Russian Acad. Sciences
119991 Moscow, Russia

Pavel Drábek
Department of Mathematics and NTIS
Faculty of Applied Sciences
University of West Bohemia
Univerzitní 8, 301 00 Plzeň
Czech Republic
Email: pdrabek@kma.zcu.cz

Roland Duduchava
A. Razmadze Mathematical Institute
I. Javakhishvili Tbilisi State University
2, Merab Aleksidze II Lane
0193 Tbilisi, Georgia
Email: dudu@rmi.ge

Lasha Ephremidze
A. Razmadze Mathematical Institute
I. Javakhishvili Tbilisi State University
2, Merab Aleksidze II Lane
0193 Tbilisi, Georgia
Email: lasha@rmi.ge

Revaz Gamkrelidze
Steklov Mathematical Institute
Russian Acad. Sciences
119991 Moscow, Russia
Email: gam@ips.ras.ru

Hvedri Inassaridze
A. Razmadze Mathematical Institute
I. Javakhishvili Tbilisi State University
2, Merab Aleksidze II Lane
0193 Tbilisi, Georgia
Email: hvedri@rmi.ge

Mamuka Jibladze
A. Razmadze Mathematical Institute
I. Javakhishvili Tbilisi State University
2, Merab Aleksidze II Lane
0193 Tbilisi, Georgia
Email: jib@rmi.ge

Tornike Kadeishvili
A. Razmadze Mathematical Institute
I. Javakhishvili Tbilisi State University
2, Merab Aleksidze II Lane
0193 Tbilisi, Georgia

Email: kade@rmi.ge

Otto Helmut Kegel
Institut für Mathematik
Albert-Ludwigs-Universität
79104 Freiburg, Germany
Email: otto.h.kegel@math.uni-freiburg.de

Alexander Kharazishvili
A. Razmadze Mathematical Institute
I. Javakhishvili Tbilisi State University
2, Merab Aleksidze II Lane
0193 Tbilisi, Georgia
Email: kharaz2@yahoo.com

Sergo Kharibegashvili
A. Razmadze Mathematical Institute
I. Javakhishvili Tbilisi State University
2, Merab Aleksidze II Lane
0193 Tbilisi, Georgia
Email: kharibegashvili@yahoo.com

Estate Khmaladze
School of Mathematics, Statistics and Operations Research
Victoria University
Wellington 6140, New Zealand
Email: estate.khmaladze@msor.vuw.ac.nz

Tariel Kiguradze
Department of Mathematical Sciences
Florida Institute of Technology
Melbourne, FL 32901-6975, U.S.A.
Email: tkigurad@fit.edu

Samuel Krushkal
Dept. of Mathematics and Computer Science
Bar-Ilan University 52900
Ramat Gan, Israel
Email: krushkal@macs.biu.ac.il

Stanislaw Kwapien
Institute of Mathematics
Warsaw University
02-097 Warsaw, Poland
Email: kwapstan@mimuw.edu.pl

Hans Rudolf Lerche
Institut f. Math. Stochastik
Albert-Ludwigs-Universität
79104 Freiburg, Germany
Email: lerche@stochastik.uni-freiburg.de

Michael Mania
A. Razmadze Mathematical Institute
I. Javakhishvili Tbilisi State University
2, Merab Aleksidze II Lane
0193 Tbilisi, Georgia
Email: misha.mania@gmail.com

Jean Mawhin
Institut de Mathematique Pure et Appliquee
Universite Catholique de Louvain
1348 Louvain-la-Neuve, Belgium
Email: jean.mawhin@uclouvain.be

Alexander Meskhi
A. Razmadze Mathematical Institute
I. Javakhishvili Tbilisi State University
2, Merab Aleksidze II Lane
0193 Tbilisi, Georgia

Paolo Emilio Ricci
Dipartimento di Matematica "Guido Castelnuovo"
Universita degli Studi di Roma "La Sapienza"
00185 Roma, Italy
Email: paoloemilioricci@gmail.com

Vaja Tarieladze
N. Muskhelishvili Inst. Comput. Math.
Georgian Technical University
0171 Tbilisi, Georgia
Email: vajatarieladze@yahoo.com

Hans Triebel
Mathematisches Institut
Fr.-Schiller-Universität Jena
07743 Jena, Germany
Email: hans.triebel@uni-jena.de

Fabio Zanolin
Dept. of Mathematics and Computer Science
University of Udine
Via delle Science 6
33100 Udine, Italy
Email: fabio.zanolin@dimi.uniud.it

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Mathematics General Mathematics

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eISSN:

1572-9176

ISSN:

1072-947X

Language:

English

Publisher:

De Gruyter

Additional information

First published:

February 1, 1994

Publication Frequency:

6 issues per year

eISSN: 1572-9176

ISSN: 1072-947X

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researchers in the field of mathematics

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