Vol. 31 (2025), no. 1 (Current Issue)

In this article, we have introduced the concepts of statistically monotonic sequences of bi-complex numbers using the two types of partial order relations on the set of bi-complex numbers and have established some results. We have also investigated the concept of statistically monotonic sequence, statistically order convergence and statistically relatively uniform convergence in a bi-complex Riesz space and have studied some of their properties.

This article discusses the structure of expansions that represent non-perturbative solutions of the Cauchy problem for the evolution equation hierarchies for the state and observables of many-particle systems with topological nearest-neighbor interaction. The generating operators for these expansions are derived using a proposed cluster expansion method applied to the groups of operators in the Liouville equations for both states and observables, respectively. The article also introduced the concept of a cumulant representation for distribution functions that describe the state of many particles with topological interactions and constructs a non-perturbative solution to the Cauchy problem for the hierarchy of nonlinear evolution equations for the cumulants of distribution functions. Furthermore, a relationship is established between the constructed solution and the series expansion structures for reduced distribution and correlation functions.

This paper examines four-dimensional matrices in $(\mathcal{L}_{1},\mathcal{L}_{1})$ under standard matrix product. Using established characterizations of $(\mathcal{L}_{1},\mathcal{L}_{1};P)$, we demonstrate that $(\mathcal{L}_{1},\mathcal{L}_{1})$ forms a Banach algebra under standard matrix operations. We prove that $(\mathcal{L}_{1},\mathcal{L}_{1};P)$ is a closed, convex semigroup with identity under matrix product. Finally, we present a Mercerian-type theorem for four-dimensional matrices via matrix product.

Based on the new definition of the $\mathcal{N}^{\alpha}_F$-derivative function introduced by Juan E. Nápoles Valdes et al. (2020) in [1], we give a new definition and some results of the $\mathcal{N}^{\alpha}_F$-fractional semi-groups of operators.

In this paper, we introduce and study new types of star-selection principles, namely semi-neighbourhood semi-star-Rothberger (Menger, Hurewicz, and Lindelöf) and neighbourhood semi-star-Rothberger (Menger, Hurewicz, and Lindelöf) spaces. We establish several properties of these selection principles and investigate their relationships with other selection properties in topological spaces. Furthermore, we present a collection of fundamental theorems and propositions that characterize these spaces.

We introduce generalized Lipschitz classes $\mbox{Lip}^M(\eta)$ and $\mbox{lip}^M(\eta)$ of functions associated with the canonical Sturm-Liouville operator \[L^M:=\frac{\mbox{d}^2}{\mbox{d}x^2}+\left(\frac{A'(x)}{A(x)}-2i\frac{a}{b}x\right)\frac{\mbox{d}}{\mbox{d}x} -\left(\frac{a^2}{b^2}x^2+i\frac{a}{b}x\frac{A'(x)}{A(x)}+i\frac{a}{b}\right),\] where $A$ is a nonnegative function satisfying certain conditions; and we prove two versions of Boas-type theorems for the canonical Sturm-Liouville transform $\mathscr{F}^M$. An application to the canonical Sturm-Liouville multipliers is given. Boas-type results for the canonical Fourier-Bessel transform and the canonical Fourier-Jacobi transform are special cases of this work.

This paper provides a comprehensive study of lacunary weak convergence for double sequences, defined through Orlicz functions. It delves into the examination of significant topological and algebraic properties, such as solidity, symmetry, and monotonicity, within the framework of these spaces. To enhance the theoretical foundation, the study includes a range of illustrative examples that highlight instances where certain conditions fail. Furthermore, the paper investigates and establishes inclusion relationships between the newly defined spaces and other existing spaces in the literature. The findings significantly contribute to the broader understanding of sequence spaces, particularly focusing on their structural and convergence characteristics. These results not only enhance the mathematical framework but also provide a foundation for future research into the applications and implications of lacunary weak convergence in double sequences.