Convergence Of Sequence Research Articles

On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants

Abstract In this article, we analyze a rate of attraction of poles of an approximated function to poles of incomplete multipoint Padé approximants and use it to derive a sharp bound on the geometric rate of convergence of multipoint Hermite-Padé approximants to a vector of approximated functions in the Montessus de Ballore theorem when a table of interpolation points is Newtonian.

The convergence exponent and the well approximated sets for Lüroth expansion

For any x in (0,1], let [d1(x),d2(x),⋯,dn(x),⋯] be its Lüroth expansion and {pn(x)qn(x)}n≥1 its convergent sequence. In this note, we study the multifractal spectrum of the convergence exponent of the sequence {dn(x)}n≥1 defined byτ(x):=inf⁡{s≥0:∑n≥1dn−s(x)<∞} and determine the Hausdorff dimension of the setE(α,β)={x∈(0,1]:τ(x)=α}⋂Jβ for any α≥0,β≥0, whereJβ={x∈(0,1]:|x−pn(x)qn(x)|<1qn(x)β+1for infinitely manyn≥1}.

Projection-type method with line-search process for solving variational inequalities

We propose an accelerated projection-type method for solving variational inequalities in Hilbert spaces. The method is suitable for pseudomonotone non-Lipschitz continuous operators and obtains strong convergence for iterative sequences under appropriate conditions. Based on the classical double-projection gradient method, we establish a new line-search rule and add an inertia term to improve the convergence speed of the proposed algorithm. Finally, some numerical examples are implemented to compare the proposed algorithm with the existing results, thereby verifying that our method has better performance.

Applications of $$T^r$$-strongly convergent sequences to Fourier series by means of modulus functions

Applications of $$T^r$$-strongly convergent sequences to Fourier series by means of modulus functions

Multiplicative 𝔪-metric space, fixed point theorems with applications in multiplicative integrals equation and numerical results

Abstract The objective of this manuscript is to introduce the notion of multiplicative 𝔪 {\mathfrak{m}} -metric space, inspired by the concepts of multiplicative metric space and 𝔪 {\mathfrak{m}} -metric space. We propose a generalized notion of the distance function known as the multiplicative 𝔪 {\mathfrak{m}} -metric. Firstly, we present the fundamental definitions for the multiplicative 𝔪 {\mathfrak{m}} -metric space, then delve into the topological aspects, including the convergence of sequences and completeness. Additionally, we provide various illustrations to support our discussion. We also generalize some well-known contraction mappings and prove fixed point theorems on the complete multiplicative 𝔪 {\mathfrak{m}} -metric space. To complement our findings, we have included numerical results along with graphs to provide visual support for our conclusions. Furthermore, we explore the potential of utilizing the multiplicative 𝔪 {\mathfrak{m}} -metric to demonstrate the existence of the solution to a multiplicative integral equation.

Rough Convergence of Double Sequences in $n-$Normed Spaces

In this study, we introduced the concepts of rough convergence, rough Cauchy double sequence, and the set of rough limit points of a double sequence, as well as the rough convergence criteria associated with this set in $n$-normed spaces. Later, we proved that this set is both closed and convex. Finally, we presented the relationships between rough convergence and rough Cauchy double sequence in $n$-normed spaces.

Matrix Transforms Into the Set of a-absolutely Convergent Sequences with Speed and the Regularity of Matrices on the Sub-spaces of C

Let α &gt; 1. The α-absolute convergence with speed, where the speed is defined by a monotonically increasing positive sequence µ, has been studied in the present paper. Let l µ α be the set of all α-absolutely µconvergent sequences and X a sequence space defined by another speed λ. Necessary and sufficient conditions for a matrix A (with real or complex entries) to map X into l µ α have been presented. It is proved as an example that the Zweier matrix Z1/2 satisfies these necessary and sufficient conditions for certain speeds λ and µ. The notion of regularity on the subspace X of the set c of converging sequences is defined, and also, necessary and sufficient conditions for a matrix A to be regular on certain X ⊂ c are presented. It has also been shown that there exists an irregular matrix, which is regular on the subspace X of c.

ANALYSIS OF APPROACHES TO THE STUDY OF LIMITS OF SEQUENCES AND FUNCTIONS AND THE USE OF INFORMATION TECHNOLOGIES

Article analyzes approaches to the study of limits of sequences and functions and the use of information technologies. The theorem on the limit of an intermediate sequence is of great importance when investigating the convergence of sequences and when calculating the limits of sequences. It is developed an algorithm for applying this theorem that is illustrated with examples. An example of calculating the limit of a function with a geometric interpretation of the obtained result is considered. The possibility of completing the calculation of limits at infinity, using the program code written by the author in the LaTeX programming language, is considered.

On the q-statistical convergence of double sequences

On the q-statistical convergence of double sequences

Some Results on Statistically Pre-Cauchy Sequences of Fuzzy Variables Defined by Orlicz Functions in Credibility Space

This study introduces the concept of statistically pre-Cauchy sequences of fuzzy variables in five directions of credibility theory: almost surely, in measure, in mean, in distribution, and uniformly almost surely. However, the main focus is kept on statistically pre-Cauchy sequences in measure, in mean, and in distribution. Furthermore, a correlation between statistically pre-Cauchy sequences and statistically convergent sequences is established using fuzzy variables. Additionally, the exploration of statistically pre-Cauchy sequences of fuzzy variables is initiated through the application of Orlicz functions. The objective is to enhance the understanding of the statistical behavior and convergence properties of fuzzy variables in various credibility scenarios.

Lacunary statistical convergence on L - fuzzy normed space

The idea of lacunary statistical convergence sequences, which is a development of statistical convergence, is examined and expanded in this study on L - fuzzy normed spaces, which is a generalization of fuzzy spaces. On L - fuzzy normed spaces, the definitions of lacunary statistical Cauchy and completeness, as well as associated theorems, are provided. The link between lacunary statistical Cauchyness and lacunary statistical boundedness with regard to L - fuzzy norm is also shown.

Analytical Solutions to the Unsteady Poiseuille Flow of a Second Grade Fluid with Slip Boundary Conditions.

This paper deals with an initial-boundary value problem modeling the unidirectional pressure-driven flow of a second grade fluid in a plane channel with impermeable solid walls. On the channel walls, Navier-type slip boundary conditions are stated. Our aim is to investigate the well-posedness of this problem and obtain its analytical solution under weak regularity requirements on a function describing the velocity distribution at initial time. In order to overcome difficulties related to finding classical solutions, we propose the concept of a generalized solution that is defined as the limit of a uniformly convergent sequence of classical solutions with vanishing perturbations in the initial data. We prove the unique solvability of the problem under consideration in the class of generalized solutions. The main ingredients of our proof are a generalized Abel criterion for uniform convergence of function series and the use of an orthonormal basis consisting of eigenfunctions of the related Sturm-Liouville problem. As a result, explicit expressions for the flow velocity and the pressure in the channel are established. The constructed analytical solutions favor a better understanding of the qualitative features of time-dependent flows of polymer fluids and can be applied to the verification of relevant numerical, asymptotic, and approximate analytical methods.

The AA-Viscosity Algorithm for Fixed-Point, Generalized Equilibrium and Variational Inclusion Problems

The aim of this paper is to propose an inertial-type AA-viscosity algorithm for approximating the common solutions of the split variational inclusion problem, the generalized equilibrium problem and the common fixed-point problem of nonexpansive mappings. The strong convergence of an iterative sequence obtained through the proposed method is proved under some mild assumptions. Consequently, approximations of the solution of the split feasibility problem, the relaxed split feasibility problem, the split common null point problem and the split minimization problem are given. The applicability of our proposed algorithm has been illustrated with the help of a numerical example. Our iterative method was then compared graphically with different comparable methods in the existing literature.

Adaptive iterative learning control for a class of parameterized fractional-order systems subjected to backlash-like hysteresis

In this article, a fractional-order adaptive iterative learning control scheme is proposed for a class of parameterized fractional-order systems with unknown control gain and backlash-like hysteresis nonlinearity under disturbance. Based on the sufficient condition for the stability of linear fractional-order systems, a sliding mode surface of tracking errors is constructed to facilitate the controller design and stability analysis. A new boundary layer function is designed by using Mittag-Leffler function to relax the restriction of the identical initial condition of iterative learning control design. The fractional-order differential-type adaptive laws and difference-type learning law are designed to estimate unknown constant parameters and time-varying parameter, respectively. To deal with the influence of backlash-like hysteresis nonlinearity reminder term and unknown bounded external disturbance, a robust control term is designed by employing the hyperbolic tangent function with a convergent series sequence which can guarantee the learning convergence along iteration axis. By constructing Lyapunov-like composite energy function, the stability analysis is presented to prove the convergence of the system output to a small neighborhood of the desired trajectory and the boundedness of all the closed-loop signals. Finally, a simulation example of second-order nonlinear fractional-order system is presented, which demonstrates the effectiveness of the proposed fractional-order adaptive iterative learning control scheme.

Characterization of lacunary ℐ-convergent sequences in credibility space

Abstract This research paper aims to introduce the concept of lacunary ideal Cauchy sequences of fuzzy variables in a credibility space. We establish the interrelationships between this notion with lacunary ideal convergent sequences in the same structure from several aspects of credibility. Furthermore, we explore the concepts of strongly lacunary Cauchy, strongly ℐ {\mathcal{I}} -lacunary Cauchy, and strongly ℐ ∗ {\mathcal{I}^{\ast}} -lacunary Cauchy sequences of fuzzy variables within the context of credibility. We also examine the interconnections between these concepts and analyze their relationships.

Thermodynamic limit of spin systems on random graphs

We utilize the graphon—a continuous mathematical object which represents the limit of convergent sequences of dense graphs—to formulate a general, continuous description of quantum spin systems in thermal equilibrium when the average coordination number grows extensively in the system size. Specifically, we derive a closed set of coupled nonlinear Fredholm integral equations which governs the properties of the system. The graphon forms the kernel of these equations, and their solution yields exact expressions for the macroscopic observables in the system in the thermodynamic limit. We analyze these equations for both quantum and classical spin systems, recovering known results and providing analytical solutions for a range of more complex cases. We supplement this with controlled, finite-size numerical calculations using Monte Carlo and tensor network methods, showing their convergence towards our analytical results with increasing system size. Published by the American Physical Society 2024

Ideal Relative Uniform Convergence of Double Sequence of Positive Linear Functions

In this article, we look into the concept of ideal relative uniform convergence of a double sequence of functions. In addition, we define ideal relative uniform Cauchy and ideal regular relative uniform convergence of a double sequence of positive linear functions defined on a compact domain D with respect to the scale function σ(x) defined on D. We also introduced several classes of ideal relative uniform convergent double sequences of functions and investigated their algebraic and topological properties.

New kind of vector space via Ş-Proximit structure

This paper describes the basic aim of Ş -proximit space for non-empty sets. The relationship between approach space and Ş-proximit space has been clarified. This paper demonstrates that all metric spaces are Ş-proximit spaces. But the opposite is not always true. We define SA-contraction and present various features and the results. In addition, we define new Ş-proximit normed space. We demonstrate that any proximit normed space is normed. But the opposite is not always true. We define the concepts of the Ş -proximit semigroup, Ş-proximit group, and Ş-proximit vector space via instances of problem-solving techniques. We also discussed the definitions of Cauchy sequence, cluster point, Ş -convergent sequence, and Ş -complete.

Relation Between Convergence and Almost Convergence of Complex Uncertain Sequences

In this paper, we introduce a new type of almost convergent complex uncertain sequence with respect to uniformly almost surely. We characterize the notion of almost convergence of sequences of complex uncertain variables further. We establish the interconnection between convergent complex uncertain sequence, bounded complex uncertain sequence and almost convergent complex uncertain sequence in all five aspects of uncertainty.

A MULTIPOINT BOUNDARY VALUE PROBLEM IN TIME FOR A $2B$-PARABOLIC EQUATION WITH DEGENERACY

One of the most important issues in the general theory of differential equations with partial derivatives is establishing the solvability of boundary value problems. Among the boundary value problems for equations with partial derivatives, problems with nonlocal boundary conditions occupy an important place. Such interest in such problems is caused both by their rich practical application (the process of diffusion, moisture distortion in soils, plasma physics, etc.), and by the needs of the general theory of boundary value problems. A general multipoint boundary value problem for nonuniformly $2b$-parabolic equations with degeneracy is studied. The coefficients of parabolic equations and boundary conditions allow power degeneracy of arbitrary order in terms of time variable and spatial variables at some set of points. To solve the given multipoint boundary value problem, solutions of problems with smooth coefficients in Hölder spaces with the appropriate norm are studied. With the help of interpolation inequalities and a priori estimates, estimates of the solution of auxiliary problems and their derivatives in special Gelder spaces are established. Using the theorems of Ross and Archel, a convergent sequence is distinguished from the compact sequence of solutions of the auxiliary problems, the limiting value of which is the solution of the multipoint boundary value problem in time for the $2b$-parabolic equation with degeneracy. Estimates of the solution of the given problem are established in Hölder spaces with power-law weights. The order of the power weight is determined by the order of features of the coefficients of the equations and the boundary conditions. With certain restrictions on the right-hand side of the equation and boundary conditions, an integral image of the given problem is obtained.