Notion Of Convergence Research Articles

DIFFERENCE DOUBLE SEQUENCES OF BI-COMPLEX NUMBERS

In this article, we have introduced the notion of convergence of difference double sequences in Pringsheim’s sense, difference null in Pringsheim’s sense, bounded difference, bounded convergence differ ence, bounded null difference, regular convergence difference and reg ular null difference double sequences of bi-complex numbers. We have proved that these are linear spaces. With the help of the Euclidean norm defined on bi-complex numbers, we have established their differ ent algebraic and topological properties, as well as some of their geo metric properties. Suitable examples have been discussed to support the introduction of these classes of sequences and during the investi gation of their properties for failure cases.

Kähler–Einstein metrics on families of Fano varieties

Abstract Given a one-parameter family of ℚ-Fano varieties such that the central fiber admits a unique Kähler–Einstein metric, we provide an analytic method to show that the neighboring fiber admits a unique Kähler–Einstein metric. Our results go beyond by establishing uniform a priori estimates on the Kähler–Einstein potentials along fully degenerate families of ℚ-Fano varieties. In addition, we show the continuous variation of these Kähler–Einstein currents and establish uniform Moser–Trudinger inequalities and uniform coercivity of the Ding functionals. Central to our article is introducing and studying a notion of convergence for quasi-plurisubharmonic functions within families of normal Kähler varieties. We show that the Monge–Ampère energy is upper semi-continuous with respect to this topology, and we establish a Demailly–Kollár result for functions with full Monge–Ampère mass.

A novel links between the various modes of convergence

The article introduces several notions of convergence of a series from all possible angles (also called modes of convergence). There are several relations among the various modes of convergence ( Pointwise convergence, converges absolutely, uniformly and normally), which are discussed in the present paper and are summarized by a diagrams, we analyzes the genesis of new links between convergence at the present stage. Our idea is based on the classical convergences, during which a new algebraic properties are called. According Theorem 2.1 and the truth table we will find a new links between the various modes of convergence, Pointwise convergence, converges absolutely, uniformly and normally. These are all different kinds of convergence. A series might converge in one sense but not another. Some of these convergence types are ''stronger'' than others and some are ''weaker.'' Here, we would like to provide definitions of different types of convergence and discuss how they are related.

Time-Synchronized Convergence Control for n-DOF Robotic Manipulators with System Uncertainties.

A time-synchronized (TS) convergence control method for robotic manipulators is proposed. Adversely to finite-time control, a notion of time-synchronization convergence is introduced based on the ratio persistence property, which can ensure that all system components converge simultaneously in a finite time. Firstly, a robust disturbance observer is constructed to be compatible with the time-synchronized control framework and precisely estimate system uncertainties. Furthermore, we design a (finite) time-synchronized controller to ensure that all states of the robotic manipulator simultaneously converge to an equilibrium point, irrespective of initial conditions. Stability analysis shows the feasibility of the proposed TS control method. At last, simulations are performed with a two-link rehabilitation robotic system, and the comparison results indicate its superiority.

I-Symmetric Spaces and the π-Images of Metric Spaces

Symmetric spaces and sn-symmetric spaces, as a generalization of metric spaces, have many important properties and have been widely discussed. We consider characterizations and mapping properties of sn-symmetric spaces under ideal convergence. I-symmetric spaces and I-sn-symmetric spaces are defined and studied. These not only generalize some classical results on symmetric spaces but also provide new directions to study generalized metric spaces using the notion of ideal convergence. As an application of I-sn-symmetric spaces, some relevant properties of statistical convergence are obtained. Some unanswered questions in this field are raised.

A New Notion of Convergence Defined by The Fibonacci Sequence: A Novel Framework and Its Tauberian Conditions

The Fibonacci sequence has broad applications in mathematics, where its inherent patterns and properties are utilized to solve various problems. The sequence often emerges in areas involving growth patterns, series, and recursive relationships. It is known for its connection to the golden ratio, which appears in numerous natural phenomena and mathematical constructs. In this research paper, we introduce new concepts of convergence and summability for sequences of real and complex numbers by using Fibonacci sequences, called Δ-Fibonacci statistical convergence, strong Δ-Fibonacci summability, and Δ-Fibonacci statistical summability. And, these new concepts are supported by several significant theorems, properties, and relations in the study. Furthermore, for this type of convergence, we introduce one-sided Tauberian conditions for sequences of real numbers and two-sided Tauberian conditions for sequences of complex numbers.

Homogenization of high-contrast media in finite-strain elastoplasticity

This work is devoted to the analysis of the interplay between internal variables and high-contrast microstructure in inelastic solids. As a concrete case-study, by means of variational techniques, we derive a macroscopic description for an elastoplastic medium. Specifically, we consider a composite obtained by filling the voids of a periodically perforated stiff matrix by soft inclusions. We study the Γ-convergence of the related energy functionals as the periodicity tends to zero, the main challenge being posed by the lack of coercivity brought about by the degeneracy of the material properties in the soft part. We prove that the Γ-limit, which we compute with respect to a suitable notion of convergence, is the sum of the contributions resulting from each of the two components separately. Eventually, convergence of the energy minimizing configurations is obtained.

A new notion of convergence defined by weak Fibonacci lacunary statistical convergence in normed spaces

Abstract The applications of a Fibonacci sequence in mathematics extend far beyond their initial discovery and theoretical significance. The Fibonacci sequence proves to be a versatile tool with real-world implications and the practical utility of manifests in various fields, including optimization algorithms, computer science and finance. In this research paper, we introduce new versions of convergence and summability of sequences in normed spaces with the help of the Fibonacci sequence called weak Fibonacci φ-lacunary statistical convergence and weak Fibonacci φ-lacunary summability, where φ is a modulus function under certain conditions. Furthermore, we obtain some relations related to these concepts in normed spaces.

Statistical Convergence in A-Metric Spaces

In this paper, we will present the notion of statistical convergence in A-metric spaces, which is an important concept in summability theory. We will define statistical convergence in A-metric spaces and investigate its basic properties. Furthermore, we will explore the relationship between strongly p-Cesàro and statistical convergence in A-metric spaces.

Competition Policy Dynamics: the Influence of Domestic-Specific Factors in a Globalized Landscape

The primary purpose of this essay is to unravel the multifaceted dynamics of competition policy and offer insights into the factors driving policy variations worldwide. It aims to contribute to the ongoing discourse on competition policy by bridging the gap between theory and practice, offering empirical support for observed trends, and providing practical implications for policymakers and practitioners. This research employs the literature study. The analysis reveals that, while there is a discernible global convergence toward economic-based competition policy and independent competition authorities, domestic-specific factors such as levels of privatization, democracy, political ideology, legal origin, and types of capitalism continue to exert significant influence, shaping competition policies in distinctive ways. This nuanced understanding challenges the notion of complete global convergence and underscores the importance of context-sensitive policy design. The findings have practical implications for policymakers and practitioners, emphasizing the need to tailor competition policy frameworks to the unique circumstances of each country. In a rapidly changing global landscape, recognizing and accommodating these factors is crucial for the continued relevance and adaptability of competition policy.

A Generalized Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-Convergence Concept for a Class of Equilibrium Problems

A novel generalization of Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-convergence applicable to a class of equilibrium problems is studied. After the introduction of the latter, a variety of its applications is discussed. The existence of equilibria with emphasis on Nash equilibrium problems is investigated. Subsequently, our Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-convergence notion for equilibrium problems is introduced and discussed as well as applied to a class of penalized generalized Nash equilibrium problems and quasi-variational inequalities. The work ends with a comparison of our results to previous generalizations in the literature.

Relative uniform convergence of quantum difference sequence of functions related to the $\ell_p$-space

The notion of relative uniform convergence of a sequence of functions was introduced and investigated by Moore, followed by E. W. Chittenden in the twentieth century. The concept attracted the researcher at the beginning, of the twenty-first century. Jackson F. H. initiated the concept of the notion of the quantum difference operator. We introduced the quantum difference sequence space $m(\phi, ru, \nabla_p), p\geq 0$ of W.L.C. Sargent type. We examine its various properties such as solidity, convergence-free, completeness, etc. Also, some induction results have been established.

Exploring a modification of [formula omitted] convergence

In the work by M. C. Lee, A. Naber, and R. Neumayer (Lee et al., 2023) a beautiful ɛ-regularity theorem is proved under small negative scalar curvature and entropy bounds. In that paper, the dp distance for Riemannian manifolds is introduced and the quantitative stability results are given in terms of this notion of distance, with important examples showing why other existing notions of convergence are not adequate in their setting. Due to the presence of an entropy bound, the possibility of long, thin splines forming along a sequence of Riemannian manifolds whose scalar curvature is becoming almost positive is ruled out. It is important to rule out such examples since the dp distance is not well behaved in the presences of splines that persist in the limit. Since there are many geometric stability conjectures (Gromov, 2023; Sormani, 2023) where we want to allow for the presence of splines that persist in the limit, it is crucial to be able to modify the dp distance to retain its positive qualities and prevent it from being sensitive to splines. In this paper we explore one such modification of the dp distance and give a theorem which allows one to estimate the modified dp distance, which we expect to be useful in practice.

Large-Eddy Simulations of a Supersonic Impinging Jet Using OpenFOAM

Supersonic impinging jets are a versatile configuration that can model the compressible flows of cold-spray manufacturing and vertical take-off-and landing strategy. In this work, rhoCentralFoam, solver of the OpenFOAM framework, and a large-eddy simulation formulation were used to simulate an underexpanded supersonic jet of Mach 1.45 and nozzle pressure ratio of 4, impinging on a flat wall situated at 1.5 nozzle diameters away from the jet outlet. Care was taken in the mesh construction to properly capture the characteristic standoff shock and vortical structures. The grid convergence index was evaluated with three meshes of increasing spatial resolution. All meshes can generally be considered as sufficient in terms of results focused on time-averaged values and mean physical properties such as centerline Mach number profile. However, the highest resolution mesh was found to capture fine shear vortical structures and behaviors that are absent in the coarser cases. Therefore, the notion of adequate grid convergence may differ between analyses of time-averaged and transient information, and so should be determined by the user’s intention for conducting the simulations. To guide the selection of mesh resolution, scaling analyses were performed, for which the current rhoCentralFoam solver displays a good weak scaling performance and maintains a linear strong scaling up to 4096 cores (32 nodes) for an approximately 40 million-cell mesh. Due to the internode communication bottlenecks of OpenFOAM and improvements in central processing units, this work recommends, for future scaling analyses, adopting a “cells-per-node” basis over the conventional “cells-per-core” basis, with particular attention to the interconnect speed and architecture used.

Relative uniform convergence of p-absolutely summable sequence of functions

In this article we introduce the notion of relative uniform convergence of p-absolutely summable sequence of functions. We have investigated its different properties such as solidness, symmetric, convergence-free, monotonic etc. We have established some inclusion relations involving this sequence space.

Wijsman Ideal Convergence Sequence in Neutrosophic Metric Spaces

Objectives: An examination of convergence for a sequence of closed sets in Neutrosophic Metric Spaces is the main goal of this research. Particularly, the focus of this study is directed at Wijsman -convergence ( and Wijsman -convergence . Methods: The various properties of Neutrosophic Metric Spaces are investigated in this study. Using exact mathematical techniques, this study comprehends the features of convergent sequences in these domains. Furthermore, this study expresses particular views on Wijsman -Cauchy and Wijsman -Cauchy sequences. Findings: The convergence behaviour of closed sets in Neutrosophic Metric Spaces is a result of this research. it presents findings and remarks about the sequences of Wijsman - convergence, Wijsman - convergence, Wijsman -Cauchy, and Wijsman -Cauchy. These findings improve the current understanding of convergence in relation to NMS. Novelty: What makes this work unique is that it focuses on convergence notions in Neutrosophic Metric Spaces, specifically on Wijsman and Wijsman - convergence enriches the current body of information in this domain. Keywords: Neutrosophic Metric Spaces, Wijsman I ­convergent and Wijsman I ¿ ­ convergent sequences, Wijsman I ¿ ­Cauchy and Wijman I ­Cauchy sequences

Kantorovich-Stancu type (α,λ,s) - Bernstein operators and their approximation properties

ABSTRACT In this study, we establish a new class of Kantorovich-Stancu type α , λ , s − Bernstein operators via an adaptation of Bézier bases which are formulated with the inclusion of the shape parameters λ ∈ − 1 , 1 , α ∈ 0 , 1 , and a positive parameter s . First, we present a uniform convergence result for these operators and, subsequently, examine the convergence properties by utilizing the weighted B -statistical convergence notion. Furthermore, we estimate the rate of the weighted B -statistical convergence of these operators. We conclude our work by providing a numerical example with explanatory graphs to show their approximation behaviours.

Optimistic Rates: A Unifying Theory for Interpolation Learning and Regularization in Linear Regression

We study a localized notion of uniform convergence known as an “optimistic rate” [ 34 , 39 ] for linear regression with Gaussian data. Our refined analysis avoids the hidden constant and logarithmic factor in existing results, which are known to be crucial in high-dimensional settings, especially for understanding interpolation learning. As a special case, our analysis recovers the guarantee from Koehler et al. [ 21 ], which tightly characterizes the population risk of low-norm interpolators under the benign overfitting conditions. Our optimistic rate bound, though, also analyzes predictors with arbitrary training error. This allows us to recover some classical statistical guarantees for ridge and LASSO regression under random designs and helps us obtain a precise understanding of the excess risk of near-interpolators in the over-parameterized regime.

Variational-like inclusion problems involving A-maximal m-relaxed η-accretive mappings in the sense of s.i.p. and fixed point problems

This paper is devoted to the introduction of a new class of accretive mappings in the framework of q-uniformly smooth Banach spaces called A-maximal m-relaxed η-accretive mappings in the sense of semi-inner product of type (p) (for short, s.i.p.(p)). The resolvent operator associated with such a mapping is defined; its Lipschitz continuity is proved and an estimate of its Lipschitz constant is computed under some suitable conditions. At the same time, the problem of finding a common element of the set of fixed points of a given nearly asymptotically nonexpansive mapping and the set of solutions of a variational-like inclusion problem involving A-maximal m-relaxed η-accretive mappings in the sense of s.i.p.(p) is investigated. To achieve this goal, a new iterative algorithm is constructed. Using the notions of graph convergence and resolvent operator associated with an A-maximal m-relaxed η-accretive mapping in the sense of s.i.p.(p), a new equivalence relationship between the graph convergence and the resolvent operator convergence of a sequence of A-maximal m-relaxed η-accretive mappings in the sense of s.i.p.(p) is established. As an application of this equivalence, the strong convergence of the sequence generated by our proposed iterative algorithm to a common point of the above two sets is demonstrated under some suitable assumptions imposed on the parameters. The results presented in this paper unify, improve and generalize some recent works in this field.

Approximation of Kantorovich-type Generalization of (p,q) - Bernstein type Rational Functions Via Statistical Convergence

In this paper, we use the modulus of continuity to study the rate of A-statistical convergence of the Kantorovich-type (p,q) - analogue of the Balázs–Szabados operators by using the statistical notion of convergence. Mathematics subject classification: Primary 4H6D1; Secondary 4H6R1; 4H6R5.