In this work, we investigate a new type of convergence known as relative statistical convergence through the use of the deferred Nörlund and deferred Riesz means. We demonstrate that the idea of deferred Nörlund and deferred Riesz statistically relative uniform convergence is significantly stronger than deferred Nörlund and deferred Riesz statistically uniform convergence. We provide some interesting examples which explain the validity of the theoretical results and effectiveness of constructed sequence spaces. Furthermore, as an application point of view we prove the Korovkin-type approximation theorem in the context of relative equi-statistical convergence for real valued functions and demonstrate that our theorem effectively extends and most of the earlier existing results. Finally, we present an example involving the Meyer-König-Zeller operator of real sequences proving that our theorem is a stronger approach than its classical and statistical version.

This paper introduces $\varsigma$-neutrosophic fuzzy metric spaces ( $\varsigma$-NFMSs), a significant generalization of neutrosophic fuzzy metric spaces (NFMSs). By extending the parameter space from a single dimension $(0,\mathrm{\infty })$ to a multi-dimensional vector space $(0,\mathrm{\infty }{)}^{\varsigma }$, this framework offers enhanced flexibility for modeling complex systems where uncertainty depends on multiple factors simultaneously. The study investigates the topological properties of $\varsigma$-NFMSs, rigorously proving that their topology is first-countable and that the associated space is Hausdorff. Furthermore, a generalized fixed-point theorem is established within this new framework, extending previous results in NFMSs.

We investigated geometric limit sets of double sequences in finite dimensional $2$-normed spaces when negligibility of index sets was measured by a density induced by a nonnegative Robison–Hamilton (RH)-regular family $\mathfrak{B}$ of four-dimensional matrices. First, using the $\mathfrak{B}$-density, we defined $\mathfrak{B}$-statistical limit superior and limit inferior for the scalar reductions generated by the seminorms $x\mapsto \Vert x,u\Vert$, and we studied the associated real cluster behavior. Next, we introduced $\mathfrak{B}$-statistical cluster points and the $\mathfrak{B}$-statistical core of a double sequence as the intersection of all closed convex sets that contained the sequence outside a $\mathfrak{B}$-density zero set. We obtained a disk-type representation of the core via intersections of sets of the form $\{x\in X:\Vert x-z,u\Vert \le r\}$, where the radii were governed by $\mathfrak{B}$-statistical $lim sup$. We also proved a Knopp-type inclusion theorem: for a family of transforms satisfying a natural $\mathfrak{B}$-regularity condition, the Knopp core of each transform was contained in the $\mathfrak{B}$-statistical core of the original sequence. Finally, replacing $\mathfrak{B}$-density zero sets by a strongly admissible ideal ${\mathcal{I}}_2$ on $\mathbb{N}\times \mathbb{N}$, we defined ideal cores, established disk representations, compared the density-based and ideal cores, and identified an explicit condition under which the ${\mathcal{I}}_2$-core and the ${\mathcal{I}}_2^{\ast }$-core coincided.

This paper introduced a novel class of bivariate $\lambda$-Bernstein operators defined on triangular domain, denoted as $B_m^{{\lambda }_1,{\lambda }_2}(f;x,y)$. These operators leverage a new class of bivariate Bézier basis functions defined on triangular domain with shape parameters ${\lambda }_1$ and ${\lambda }_2$. A Korovkin-type approximation theorem for $B_m^{{\lambda }_1,{\lambda }_2}(f;x,y)$ was established, with the convergence rate being characterized by both the complete and partial moduli of continuity. Additionally, a local approximation theorem and a Voronovskaja-type asymptotic formula were derived for $B_m^{{\lambda }_1,{\lambda }_2}(f;x,y)$. Finally, the convergence of $B_m^{{\lambda }_1,{\lambda }_2}(f;x,y)$ to $f(x,y)$ was illustrated through graphical representations and numerical examples, highlighting instances where they surpass the performance of standard bivariate Bernstein operators defined on triangular domain, $B_m(f;x,y)$.