Convergence Of Sequence Research Articles

Exploring a Novel Approach to Deferred Nörlund Statistical Convergence

This study introduces novel concepts of convergence and summability for numerical sequences, grounded in the newly formulated deferred Nörlund density, and explores their intrinsic connections to symmetry in mathematical structures. By leveraging symmetry principles inherent in sequence behavior and employing two distinct modulus functions under varying conditions, profound links between sequence convergence and summability are established. The study further incorporates lacunary refinements, enhancing the understanding of Nörlund statistical convergence and its symmetric properties. Key theorems, properties, and illustrative examples validate the proposed concepts, providing fresh insights into the role of symmetry in shaping broader convergence theories and advancing the understanding of sequence behavior across diverse mathematical frameworks.

New limits of the Lie product formula type in Banach algebras

We find new limits of the Lie product formula type in Banach algebras with unit. Some sample results: Let X, Y, Z be Banach algebras with unit, xn,ynn∈N⊂X×Y convergent sequences with limn→∞xn=x, limn→∞yn=y and T:X×Y→Z a continuous bilinear operator with T1,1=1. Then for all sequences of natural numbers ann∈N with limn→∞an=∞ we have limn→∞T∏k=1nexkank+nk+2n,∏k=1neykank+2nk+3nan=eln43Tx,1+ln2T1,y;limn→∞T∏k=1ncosxkannn+k,∏k=1ncoskyknann2+k2nan2=e-ln2Tx2,1+1-π4T1,y22.

Quick-and-Easy Validation of Protein-Ligand Binding Models Using Fragment-Based Semiempirical Quantum Chemistry.

Electronic structure calculations in enzymes converge very slowly with respect to the size of the model region that is described using quantum mechanics (QM), requiring hundreds of atoms to obtain converged results and exhibiting substantial sensitivity (at least in smaller models) to which amino acids are included in the QM region. As such, there is considerable interest in developing automated procedures to construct a QM model region based on well-defined criteria. However, testing such procedures is burdensome due to the cost of large-scale electronic structure calculations. Here, we show that semiempirical methods can be used as alternatives to density functional theory (DFT) to assess convergence in sequences of models generated by various automated protocols. The cost of these convergence tests is reduced even further by means of a many-body expansion. We use this approach to examine convergence (with respect to model size) of protein-ligand binding energies. Fragment-based semiempirical calculations afford well-converged interaction energies in a tiny fraction of the cost required for DFT calculations. Two-body interactions between the ligand and single-residue amino acid fragments afford a low-cost way to construct a "QM-informed" enzyme model of reduced size, furnishing an automatable active-site model-building procedure. This provides a streamlined, user-friendly approach for constructing ligand binding-site models that requires neither a priori information nor manual adjustments. Extension to model-building for thermochemical calculations should be straightforward.

On (λ,η)(f)-statistical convergence of double sequences in 2-normed linear spaces

On (λ,η)(f)-statistical convergence of double sequences in 2-normed linear spaces

On convergence in G_Q metric spaces

This paper reviews some basic aspects and proposes and explores the idea of convergence for triple sequences in G_Q-metric space. Furthermore, a thorough analysis and definition of statistical convergence in this context are provided. This link is explored and its ramifications discussed in the last part, which focuses on the relationship between strong summability and the statistical convergence of G_Q-metric spaces.

\(\mathcal{I}\)-STATISTICAL CONVERGENCE OF COMPLEX UNCERTAIN SEQUENCES IN MEASURE

The main aim of this paper is to present and explore some of properties of the concept of \(\mathcal{I}\)-statistical convergence in measure of complex uncertain sequences. Furthermore, we introduce the concept of \(\mathcal{I}\)-statistical Cauchy sequence in measure and study the relationships between different types of convergencies. We observe that, in complex uncertain space, every \(\mathcal{I}\)-statistically convergent sequence in measure is \(\mathcal{I}\)-statistically Cauchy sequence in measure, but the converse is not necessarily true.

Statistical Soft Wijsman Convergence

One effective approach to addressing uncertainties in a parametric manner is soft set theory, which generalizes classical set theory. Building on this foundation, we introduce the concepts of statistical and lacunary statistical convergence for sequences of closed sets within the framework of soft metric spaces. We establish fundamental properties associated with these types of convergence and explore their interrelationships, resulting in several inclusion results. Furthermore, we utilize a generalized version of the natural density function, referred to as the ϕ-weighted density function, to strengthen our findings.

Quasi statistically rough convergence of sequences in gradual normed linear spaces

In the present article, we set forth with the new notion of quasi statistically rough convergence in the gradual normed linear spaces. We establish significant results that present several fundamental properties of this new notion. We also introduce the notion of $st_{q}^{r}(\mathcal{G})$-limit set and prove that it is gradually closed, convex, and plays an important role for the quasi statistically boundedness of a sequence in a gradual normed linear space.

I*-CONVERGENCE OF FUNCTION SEQUENCES IN ASYMMETRIC METRIC SPACE

In this paper, by considering ideal which is special subfamily of power set of natural numbers I∗-convergence of sequence of functions in asymmetric metric spaces is defined and some results about new concept are given.Obtained results is supported some examples to show differences by the classical ones.

Numerical investigation of an orthotropic finite elasticity problem using a constrained minimization theory to prevent material overlapping

We consider the problem of an elastic annular disk with uniform thickness in equilibrium in the absence of body force. The disk is fixed on its inner surface and compressed by a uniform pressure on its outer surface. The disk is made of a cylindrically orthotropic material that has a constitutive response that is stiffer in the radial direction than in the tangential direction. Material properties of this type are found in carbon fibers with radial microstructure and some kinds of wood.In the context of the classical linear elasticity theory, the solution of this problem predicts material overlapping for a large enough pressure, which is not physically acceptable. An approach to prevent this anomalous behavior consists of minimizing the total potential energy functional subject to the constraint that the determinant of the deformation gradient, J, be positive. We have used this approach to eliminate material overlapping, yielding solutions with J not close to one, which contradicts the basic assumption of infinitesimal strains upon which the classical linear theory is founded.In this work, we extend our investigation to the nonlinear elasticity theory. Necessary conditions for a deformation field to be a minimizer were found elsewhere. It includes a nonzero Lagrange multiplier that represents a reaction pressure to prevent material overlapping. We use a finite element formulation to find that, differently from its linearly elastic counterpart, the Lagrange multiplier field associated with the local injectivity constraint remains bounded. In addition, we obtain convergent sequences of approximate solutions of the disk problem formulated with both this constrained minimization theory and a compressible and orthotropic Mooney-Rivlin material. This material has certain growth conditions on the deformation field that prevents material overlapping in classical problems of mechanics. Both sequences yield convergent solutions that are in very good agreement with each other.

An inertial conjugate gradient projection method for large-scale nonlinear equations and its application in the image restoration problems

Based on the acceleration effect of the inertial extrapolation technique on the convergence of iterative sequences, the number of algorithms incorporating this technique has gradually increased in recent years. Currently, there is a relative paucity of studies focusing on the Polak-Ribière-Polyak (PRP) conjugate gradient algorithm that integrate the inertial extrapolation technique. In this article, we introduce an inertial three-term PRP conjugate gradient projection method by incorporating an inertial extrapolation step into the three-term PRP algorithm, where the search direction exhibits sufficient descent and trust region characteristics. The search rule employs a derivative-free technique. Under suitable hypotheses, the proposed algorithm demonstrates global convergence. Numerical results indicate the superiority and competitiveness of this innovative method. Furthermore, its effectiveness in addressing image restoration problems underscores the practicality of this algorithm.

Fast Proxy Centers for the Jeffreys Centroid: The Jeffreys-Fisher-Rao Center and the Gauss-Bregman Inductive Center.

The symmetric Kullback-Leibler centroid, also called the Jeffreys centroid, of a set of mutually absolutely continuous probability distributions on a measure space provides a notion of centrality which has proven useful in many tasks, including information retrieval, information fusion, and clustering. However, the Jeffreys centroid is not available in closed form for sets of categorical or multivariate normal distributions, two widely used statistical models, and thus needs to be approximated numerically in practice. In this paper, we first propose the new Jeffreys-Fisher-Rao center defined as the Fisher-Rao midpoint of the sided Kullback-Leibler centroids as a plug-in replacement of the Jeffreys centroid. This Jeffreys-Fisher-Rao center admits a generic formula for uni-parameter exponential family distributions and a closed-form formula for categorical and multivariate normal distributions; it matches exactly the Jeffreys centroid for same-mean normal distributions and is experimentally observed in practice to be close to the Jeffreys centroid. Second, we define a new type of inductive center generalizing the principle of the Gauss arithmetic-geometric double sequence mean for pairs of densities of any given exponential family. This new Gauss-Bregman center is shown experimentally to approximate very well the Jeffreys centroid and is suggested to be used as a replacement for the Jeffreys centroid when the Jeffreys-Fisher-Rao center is not available in closed form. Furthermore, this inductive center always converges and matches the Jeffreys centroid for sets of same-mean normal distributions. We report on our experiments, which first demonstrate how well the closed-form formula of the Jeffreys-Fisher-Rao center for categorical distributions approximates the costly numerical Jeffreys centroid, which relies on the Lambert W function, and second show the fast convergence of the Gauss-Bregman double sequences, which can approximate closely the Jeffreys centroid when truncated to a first few iterations. Finally, we conclude this work by reinterpreting these fast proxy Jeffreys-Fisher-Rao and Gauss-Bregman centers of Jeffreys centroids under the lens of dually flat spaces in information geometry.

Some Topological Properties on The Cartesian Product of Fuzzy b-Metric Space

The aim of this paper is to introduce the concept of the Cartesian product of two fuzzy b-metric spaces with related properties. We have introduced the property that the Cartesian product of two complete fuzzy b-metric spaces is a complete fuzzy b-metric space in terms of the product t-norm (.), as well as the minimum t-norm (Λ). The properties of convergent and Cauchy sequence are introduced by using this concept.

On the Generalized Weighted Statistical Convergence

Statistical convergence and summability represent a significant generalization of traditional convergence for sequences of real or complex values, allowing for a broader interpretation of convergence phenomena. This concept has been extensively examined by numerous researchers using various mathematical tools and applied to different mathematical structures over time, revealing its relevance across multiple disciplines. In the present study, a generalized definition of the concepts of statistical convergence and summability, termed (△_v^m )_u-generalized weighted statistical convergence and (△_v^m )_u-generalized weighted by [¯N_t ]-summability for real sequences, is introduced using the weighted density and generalized difference operator. Based on this definition, several fundamental properties and inclusion results, obtained by differentiating the components used in the definitions, are provided.

Analytical solution of time-fractional Emden-Fowler equations arising in astronomy and mathematical physics

The time-fractional Emden-Fowler model is an interesting mathematical model because it is widely used and important in mathematics and applied sciences. The current investigation compiles the Elzaki transform homotopy perturbation method (ET-HPM) and the Laplace transform homotopy perturbation method (LT-HPM) for the analytical solution of the considered model. In their developments, we show that fractional orders are simple to compute, and the resulting series lead to accurate results after a few iterations. We consider the fractional orders in the context of the Caputo form. The primary aim of these approaches is to compare the analytical results of the models under consideration. These schemes do not impose any constraints on the parameters or make any assumptions that could complicate the process of obtaining the analytical results. We show that the derived outcomes are in convergent sequence with a rapid iterative formula. The iterative series and the graphical illustrations in various fractional orders confirm the efficiency of the proposed approaches. The derived results are also compared to the precise solutions, which show that our suggested approaches are simple, effective, and accurate.

Existence of normalized solutions for a fractional Kirchhoff system

In this paper, we investigate the following fractional Kirchhoff system in R N : { ( 1 + b ∫ R N | ( − Δ ) s 2 u | 2 d x ) ( − Δ ) s u + V 1 ( x ) u + λ 1 u = μ 1 | u | p 1 − 2 u + r 1 β | u | r 1 − 2 u | v | r 2 , ( 1 + b ∫ R N | ( − Δ ) s 2 v | 2 d x ) ( − Δ ) s v + V 2 ( x ) v + λ 2 v = μ 2 | v | p 2 − 2 v + r 2 β | u | r 1 | v | r 2 − 2 v , under the constraints ∫ R N | u | 2 d x = a 1 2 , ∫ R N | v | 2 d x = a 2 2 , where 0<s<1, 2s<N<4s, b, μ 1 , μ 2 , β are all positive numbers and r 1 , r 2 ≥ 1 . V 1 ( x ) , V 2 ( x ) ≥ 0 are the potential functions and a 1 , 0 $ ]]> a 2 > 0 are prescribed constants. In L 2 -critical case, we first obtain the existence of non-negative solutions through a constrained minimization problem. Then under some further assumptions on r i and V i ( x ) for i = 1, 2, we conclude the solution is Hölder continuous and decays like | x | − ( N + 2 s ) at infinity. Finally, the positivity of solutions is verified. In L 2 -supercritical case, the normalized solutions are obtained by a minimax theorem. Precisely, the theorem allows us to obtain a bounded Palais-Smale sequence satisfying the Pohozaev identity in limit sense. Further, the signs of Lagrange multipliers are determined to ensure the strong convergence of the approximate critical point sequence in H s × H s .

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.

Some Results on Neutrosophic I-Statistically Convergence

This paper introduces the neutrosophic -statistical convergent difference sequence spaces defined through a modulus function. Additionally, we establish new topological spaces and examine various topological properties within these neutrosophic -statistical convergent difference sequence spaces.

Some Convergence Types of Function Sequences and Their Relations

In this paper, we investigate various types of convergence for sequences of functions and examine the relationships among these types. Our findings contribute to a deeper understanding of the structural properties of function sequences and their convergence behaviors.

A stochastic Bregman golden ratio algorithm for non-Lipschitz stochastic mixed variational inequalities with application to resource share problems

In the study of stochastic mixed variational inequalities(SMVIs), Lipschitz is an indispensable assumption for the convergence analysis. However, practical applications may not satisfy this assumption. In this paper, we propose a stochastic Bregman golden ratio algorithm for solving non-Lipschitz SMVIs. Since our algorithm only requires to calculate one stochastic approximation of the expected mapping per iteration, the computations can be reduced. Under some moderate conditions, we prove the almost surely convergence of the iteration sequence and the O(1/K) convergence rate, where K denotes the maximum iteration. Furthermore, we derive the probabilities of large deviation results, which provide a high probability guarantee for the convergence of the proposed algorithm. Numerical experiments on Logistic regression problems and modified entropy regularized LP boosting problems show that our algorithm is competitive compared with some existing algorithms. Finally, we apply our algorithm to solve a non-Lipschitz resource sharing problem.