Korean Journal of Mathematics

Generalized pseudo $B$-Gabor frames on finite abelian groups

2024-02-06T16:01:44+09:00

We seek for an invertible map $B$ from $L^2(\Gamma)$ to $L^2(G)$, where $G$ is a finite abelian group and $\Gamma$ is the direct product of finite cyclic groups which is isomorphic to $G$, so that any Gabor frame in $L^2(G)$, is a generalized pseudo $B$-Gabor frame.

Common fixed point theorems for three mappings in generalized modular metric spaces

2023-10-04T18:00:31+09:00

In this paper, we obtain common fixed point theorems for three mappings of contractive type in the setting of generalized modular metric spaces. Our results generalize many results available in the literature including common fixed point theorems.

Generalized crossed modules over generalized group-groupoids

2023-06-30T09:44:02+09:00

In this paper we define generalized double group-groupoids and crossed modules over generalized group-groupoids. Also we prove that these algebraic structures are categorically equivalent.

VC-dimension and distance chains in $\mathbb{F}_q^d$

2023-01-02T11:19:20+09:00

Given a domain $X$ and a collection $\mathcal{H}$ of functions $h:X\to \{0,1\}$, the Vapnik-Chervonenkis (VC) dimension of $\mathcal{H}$ measures its complexity in an appropriate sense. In particular, the fundamental theorem of statistical learning says that a hypothesis class with finite VC-dimension is PAC learnable. Recent work by Fitzpatrick, Wyman, the fourth and seventh named authors studied the VC-dimension of a natural family of functions $\mathcal{H}_t^{'2}(E): \mathbb{F}_q^2\to \{0,1\}$, corresponding to indicator functions of circles centered at points in a subset $E\subseteq \mathbb{F}_q^2$. They showed that when $|E|$ is large enough, the VC-dimension of $\mathcal{H}_t^{'2}(E)$ is the same as in the case that $E = \mathbb F_q^2$. We study a related hypothesis class, $\mathcal{H}_t^d(E)$, corresponding to intersections of spheres in $\mathbb{F}_q^d$, and ask how large $E\subseteq \mathbb{F}_q^d$ needs to be to ensure the maximum possible VC-dimension. We resolve this problem in all dimensions, proving that whenever $|E|\geq C_dq^{d-1/(d-1)}$ for $d\geq 3$, the VC-dimension of $\mathcal{H}_t^d(E)$ is as large as possible. We get a slightly stronger result if $d=3$: this result holds as long as $|E|\geq C_3 q^{7/3}$. Furthermore, when $d=2$ the result holds when $|E|\geq C_2 q^{7/4}$.

Applications of fixed point theory in Hilbert spaces

2023-12-20T13:10:43+09:00

In the presented paper, the first section contains strong convergence and demiclosedness property of a sequence generated by Karakaya et al. iteration scheme in a Hilbert space for quasi-nonexpansive mappings and also the comparison between the iteration scheme given by Karakaya et al. with well-known iteration schemes for the convergence rate. The second section contains some applications of the fixed point theory in solution of different mathematical problems.

Certain subclass of strongly meromorphic close to convex functions

2023-10-04T11:03:32+09:00

The purpose of this paper is to introduce a new subclass of strongly meromorphic close-to-convex functions by subordinating to generalized Janowski function. We investigate several properties for this class such as coefficient estimates, inclusion relationship, distortion property, argument property and radius of meromorphic convexity. Various earlier known results follow as particular cases.

A new quarternionic dirac operator on symplectic submanifold of a product symplectic manifold

2023-10-30T11:18:28+09:00

The Quaternionic Dirac operator proves instrumental in tackling various challenges within spectral geometry processing and shape analysis. This work involves the introduction of the quaternionic Dirac operator on a symplectic submanifold of an exact symplectic product manifold. The self adjointness of the symplectic quaternionic Dirac operator is observed. This operator is verified for spin $\frac{1}{2}$ particles. It factorizes the Hodge Laplace operator on the symplectic submanifold of an exact symplectic product manifold. For achieving this a new complex structure and an almost quaternionic structure are formulated on this exact symplectic product manifold.

Factorization properties on the composite Hurwitz rings

2023-10-31T17:10:58+09:00

Let $A \subseteq B$ be an extension of integral domains with characteristic zero. Let $H(A,B)$ and $h(A,B)$ be rings of composite Hurwitz series and composite Hurwitz polynomials, respectively. We simply call $H(A,B)$ and $h(A,B)$ composite Hurwitz rings of $A$ and $B$. In this paper, we study when $H(A,B)$ and $h(A,B)$ are unique factorization domains (resp., GCD-domains, finite factorization domains, bounded factorization domains).

On lacunary $\Delta^{m}$-statistical convergence in g-metric space

2024-01-29T02:04:03+09:00

The aim of this research is to describe lacunary $\Delta^{m}$-statistically convergent sequences with respect to metrics on generalised metric spaces (g-metric spaces) and to look into the fundamental characteristics of this statistical form of convergence. Also, the relationship between strong summability and lacunary $\Delta^{m}$-statistical convergence in g-metric space is established at the end.

Certain aspects of rough ideal statistical convergence on neutrosophic normed spaces

2024-01-11T03:05:06+09:00

In this paper, we have presented rough ideal statistical convergence of sequence on neutrosophic normed spaces as a significant convergence criterion. As neutrosophication can handle partially dependent components, partially independent components and even independent components involved in real-world problems. By examining some properties related to rough ideal convergence in these spaces we have established some equivalent conditions on the set of ideal statistical limit points for rough ideal statistically convergent sequences.

Approximation of solutions through the Fibonacci wavelets and measure of noncompactness to nonlinear Volterra-Fredholm fractional integral equations

2024-01-13T01:33:01+09:00

This paper consists of two significant aims. The first aim of this paper is to establish the criteria for the existence of solutions to nonlinear Volterra-Fredholm (V-F) fractional integral equations on $[0, L]$, where $0<L<\infty$. The fractional integral is described here in the sense of the Katugampola fractional integral of order $\lambda>0$ and with the parameter $\beta>0$. The concepts of the fixed point theorem and the measure of noncompactness are used as the main tools to prove the existence of solutions. The second aim of this paper is to introduce a computational method to obtain approximate numerical solutions to the considered problem. This method is based on the Fibonacci wavelets with collocation technique. Besides, the results of the error analysis and discussions of the accuracy of the solutions are also presented. To the best knowledge of the authors, this is the first computational method for this generalized problem to obtain approximate solutions. Finally, two examples are discussed with the computational tables and convergence graphs to interpret the efficiency and applicability of the presented method.

Classification of four dimensional baric algebras satisfying polynomial identity of degree six

2023-12-21T17:11:42+09:00

In this paper, we proceeded to the classification of four dimensional baric algebras strictly satisfying a polynomial identity of degree six. After some results on the structure of such algebras, we show that the type of an algebra of the studied class is an invariant under change of idempotent in the Peirce decomposition. This last result plays a major role in our classification.

Applications of the Gaussian hypergeometric function to some subclasses of analytic functions

2024-01-11T03:24:54+09:00

In this paper, we derive the necessary and sufficient conditions for the Gaussian hypergeometric function to be in some subclasses of analytic functions.

Application of Gegenbauer polynomials to certain classes of bi-univalent functions of order $\nu+i\varsigma$

2024-01-07T01:33:17+09:00

In this paper, a new class of bi-univalent functions that are described by Gegenbauer polynomials is presented. We obtain the estimates of the Taylor-Maclaurin coefficients $ \left\vert m_{2}\right\vert $ and $\left\vert m_{3}\right\vert $ for each function in this class of bi-univalent functions. In addition, the Fekete--Szeg\"{o} problems function new are also studied.