Convergence Of Sequence Research Articles

Unbounded Versions of Two Old Summability Theorems

In this note, we obtain extensions of a theorem of Meyer-König and Zeller and a theorem of Wilansky in that the given results do not require a summability matrix to be a bounded operator from the convergent sequences into themselves. The culmination of the results in this note is that a triangle matrix method T with null columns maps a bounded divergent sequence to a null sequence if and only if the range of T is not closed in the null sequences.

A study of fuzzy anti-λ-ideal convergent triple sequence spaces

Abstract A class with ambiguous status of its elements by a membership function that assigns to each element a grade in the close interval [ 0 , 1 ] {[0,1]} ; Lofti Zadeh introduced this idea into theory as fuzzy sets in the year of 1965. A study of fuzzy anti-normed linear spaces by Kočinac on some topological properties motivated us to work on fuzzy anti-normed triple sequence spaces with respect to ideal by using compact linear operator. Further, we prove some theorems, particularly on convergence and completeness.

On uniform statistical convergence

Abstract This paper is a continuation of the so far performed studies on the concept of uniform statistical convergence. We first characterize two inequalities concerning the uniform statistical limit superior that lead to two core inclusion results for bounded real sequences. Using the inverse Fourier transformation, we also give a criterion on uniform statistical convergence, and study some factorization results of the space of uniformly statistically convergent sequences. These results are used to give a Korovkin-type approximation theorem.

On lacunary statistical convergence of double sequences in credibility theory

In 1951, the notion of statistical convergence was a more general concept than the idea of convergence as by presented by Fast. Since then statistical convergence has been was investigated by more and more researchers. On the other hand, in 2006 Li has began the survey of credibility theory and presented some convergence concepts in credibility theory. Because of the esoteric of both credibility and summability theory, the body of literature is limited. As such the only accessible notion to both theories is Pringsheim limits. The goal of this paper is to present a natural multidimensional extension of credibility theory via summability methods. To accomplish this, in this paper we present useful tools to generalize the well-known convergence definition to the concept of lacunary statistical convergence for double sequences in credibility theory. Additionally, some important results and examples will be examined.

Further Results on Lacunary Difference Sequences of Complex Uncertain Variables in 2-Normed Spaces

A novel approach combining the difference operator on sequence spaces and uncertainty theory has been utilized to establish a fresh class of lacunary convergent difference sequences involving complex uncertain variables in 2-normed spaces. This newly introduced class demonstrates remarkable properties related to lacunary convergence. Additionally, the sequence spaces defined in this context have been thoroughly explored, revealing interesting topological characteristics and valuable insights into inclusion relations. This study offers a rejuvenated perspective on lacunary convergence and its associated concepts through the integration of difference operators and uncertainty theory.

Some Observations on g-Metric Spaces in Light of Generalized Statistical Convergence of Double Sequences via Ideals

In this article, we delve into the notions of I2-statistical convergence and I2-lacunary statistical convergence for sequences in general metric spaces, specifically g metric spaces. We thoroughly explore these concepts within the realm of g-metric spaces.

Some Remarks on Rough IK-Statistical Convergence of Sequences in Normed Linear Spaces

This study introduces the concept of rough I*-statistical convergence in a normed linear space, extending the notion of rough I-statistical convergence. Furthermore, we propose the concept of rough IK-statistical convergence in a more comprehensive framework. We examine the properties related to these novel concepts and explore the interconnections among rough I-statistical convergence, rough I*-statistical convergence, and rough IK-statistical convergence. By doing so, we enhance our understanding of these convergence modes and pave the way for their application in various mathematical contexts.

New Types of Convergence of Complex Uncertain Double Sequences

New Types of Convergence of Complex Uncertain Double Sequences

A visual proof: convergence of a hyper power sequence – a generalisation

I provide a visual proof for the Convergence of a Hyper power sequence, which generalises a beautiful result; also proved visually by Azarpanah in 2004.

MORAVA K-THEORY AND FILTRATIONS BY POWERS

Abstract We prove the convergence of the Adams spectral sequence based on Morava K-theory and relate it to the filtration by powers of the maximal ideal in the Lubin–Tate ring through a Miller square. We use the filtration by powers to construct a spectral sequence relating the homology of the K-local sphere to derived functors of completion and express the latter as cohomology of the Morava stabiliser group. As an application, we compute the zeroth limit at all primes and heights.

Telescoping continued fractions for the error term in Stirling’s formula

In this paper, we introduce telescoping continued fractions to find lower bounds for the error term rn in Stirling’s approximation n!=2πnn+1/2e−nern. This improves lower bounds given earlier by Cesàro (1922), Robbins (1955), Nanjundiah (1959), Maria (1965) and Popov (2017). The expression is in terms of a continued fraction, together with an algorithm to find successive terms of this continued fraction. The technique we introduce allows us to experimentally obtain upper and lower bounds for a sequence of convergents of a continued fraction in terms of a difference of two continued fractions.

THE METHOD OF FICTITIOUS EXTREMA LOCALIZATION IN THE PROBLEM OF GLOBAL OPTIMIZATION

The problem of finding the global extremum of a non-negative function on a positive parallelepiped in n-dimensional Euclidean space is considered. A method of fictitious extrema localization in a bounded area near the origin is proposed, which allows to separate the global extremum point from fictitious extrema by discarding it at a significant distance from the localization set of fictitious minima. At the same time, due to the choice of the starting point in the gradient descent method, it is possible to justify the convergence of the iterative sequence to the global extremum of the minimized function.

A study on strong AI-statistical convergence of sequences in probabilistic metric spaces

In this paper using a non-negative regular summability matrix A and a non-trivial admissible ideal I in N we study some basic properties of strong AI-statistical convergence and strong AI-statistical Cauchyness of sequences in probabilistic metric spaces not done earlier. We also introduce the notion of strong AI∗-statistical Cauchyness and study its relationship with strong AI-statistical Cauchyness. Further we study some basic properties of strong AI-statistical limit points and strong AI-statistical cluster points of a sequence in probabilistic metric spaces.

Stability of ground state eigenvalues of non-local Schrödinger operators with respect to potentials and applications

In a first part of this paper we investigate the continuity (stability) of the spectrum of a class of non-local Schrödinger operators on varying the potentials. By imposing conditions of different strength on the convergence of the sequence of potentials, we give either direct proofs to show the strong or norm resolvent convergence of the so-obtained sequence of non-local Schrödinger operators, or via Γ-convergence of the related positive forms for more rough potentials. In a second part we use these results to show via a sequence of suitably constructed approximants that the ground states of massive or massless relativistic Schrödinger operators with spherical potential wells are radially decreasing functions.

Statistical Convergence of Double Sequences in Intuitionistic Fuzzy Metric Spaces

Statistical convergence has been a prominent research area in mathematics since this concept was independently introduced by Fast and Steinhaus in 1951. Afterward, the statistical convergence of double sequences in metric spaces and fuzzy metric spaces has been widely studied. The main goal of the present study is to introduce the concepts of statistical convergence and statistical Cauchy for double sequences in intuitionistic fuzzy metric spaces. Moreover, this study characterizes the statistical convergence of a double sequence by an ordinary convergent of a subsequence of the double sequence. Besides, the current study theoretically contributes to the mentioned concepts and investigates some of their basic properties. Finally, the paper handles whether the aspects should be further investigated.

On Soft Normed Quasilinear Spaces

In this study, we investigate some properties of soft quasi-sequences and present new results. We then study the completeness of soft normed quasilinear space and present an analog of convergence and boundness results of soft quasi sequences in soft normed quasilinear spaces. Moreover, we define regular and singular subspaces of soft quasilinear spaces and draw several conclusions related to these notions. Afterward, we provide examples of these results in soft normed quasilinear spaces generalizing well-known results in soft linear normed spaces. Additionally, we offer new results concerning soft quasi subspaces of soft normed quasilinear spaces. Finally, we discuss the need for further research.

Statistical Convergence of Spliced Sequences in Terms of Power Series on Topological Spaces

In the present paper, $P-$distributional convergence which is defined by power series method has been introduced. We give equivalent expressions for $P-$distributional convergence of spliced sequences. Moreover, convergence of a bounded $\infty$-spliced sequence via power series method is represented in terms of Bochner integral in Banach spaces.

Approximation via statistical relative uniform convergence of sequences of functions at a point with respect to power series method

Approximation via statistical relative uniform convergence of sequences of functions at a point with respect to power series method

A proximal subgradient algorithm with extrapolation for structured nonconvex nonsmooth problems

In this paper, we consider a class of structured nonconvex nonsmooth optimization problems, in which the objective function is formed by the sum of a possibly nonsmooth nonconvex function and a differentiable function with Lipschitz continuous gradient, subtracted by a weakly convex function. This general framework allows us to tackle problems involving nonconvex loss functions and problems with specific nonconvex constraints, and it has many applications such as signal recovery, compressed sensing, and optimal power flow distribution. We develop a proximal subgradient algorithm with extrapolation for solving these problems with guaranteed subsequential convergence to a stationary point. The convergence of the whole sequence generated by our algorithm is also established under the widely used Kurdyka–Łojasiewicz property. To illustrate the promising numerical performance of the proposed algorithm, we conduct numerical experiments on two important nonconvex models. These include a compressed sensing problem with a nonconvex regularization and an optimal power flow problem with distributed energy resources.

King operators which preserve $x^j$

We prove the unique existence of the functions $r_n$ $(n=1,2,\ldots )$ on $[0,1]$ such that the corresponding sequence of King operators approximates each continuous function on $[0,1]$ and preserves the functions $e_0(x)=1$ and $e_j(x)=x^j$, where $j\in\{ 2,3,\ldots\}$ is fixed. We establish the essential properties of $r_n$, and the rate of convergence of the new sequence of King operators will be estimated by the usual modulus of continuity. Finally, we show that the introduced operators are not polynomial and we obtain quantitative Voronovskaja type theorems for these operators.