Summability Methods Research Articles

Local stress in periodic composites via the Riesz summability method

Abstract Different summation methods of multidimensional trigonometric Fourier series are adopted for approximating the local strain and stress in elastic periodic composites. In fact, approximating the exact solution of a problem with the original partial sums of Fourier series can produce unwanted effects, such as the Gibbs phenomenon at the jump discontinuities that may occur at the interface between the constituents of the composites. Nevertheless, the Fourier coefficients of the original partial sums contain enough information to provide better estimates via the summability methods. In the means provided by the summability methods, the Fourier coefficients are suitably weighted by reducing windows, involving a regularization of the numerical solution. First, a complete solution method is used in order to determine some Fourier coefficients of the local strain in periodic composites. Next, these Fourier coefficients are used to construct suitable summations and means exhibiting better convergence properties than those of the original partial sums of Fourier series. The proposed method, valid for a great variety of the geometry of the constituents embedded in the matrix, is here applied to unidirectional composites with cylindrical fibers. The behavior of the iterated Fejer partial sums, Fejer and Riesz means is investigated in order to improve the convergence, observing that the Riesz means have better convergence properties than those of the original partial sums of Fourier series.

New Definitions about A I -Statistical Convergence with Respect to a Sequence of Modulus Functions and Lacunary Sequences

In this paper, using an infinite matrix of complex numbers, a modulus function and a lacunary sequence, we generalize the concept of I -statistical convergence, which is a recently introduced summability method. The names of our new methods are A I -lacunary statistical convergence and strongly A I -lacunary convergence with respect to a sequence of modulus functions. These spaces are denoted by S θ A I , F and N θ A I , F , respectively. We give some inclusion relations between S A I , F , S θ A I , F and N θ A I , F . We also investigate Cesáro summability for A I and we obtain some basic results between A I -Cesáro summability, strongly A I -Cesáro summability and the spaces mentioned above.

Inductance Modeling of Interconnections in 3-D Stacked-Chip Packaging

This letter explores the inductance of interconnections including through-silicon vias (TSVs) and redistribution layers (RDLs) in 3-D stacked-chip packaging. It is described that the common summing method of partial inductances will result in some deviations from the full inductance. Then, the inductances of TSVs and RDLs are, respectively, calculated and are verified by a commercial electromagnetic simulator. The modified formulas are proposed for the more accurate full inductance model, which is derived from different design physical parameters of the interconnections. The side effect of a general summing method can be reduced a lot by our proposed analytical model.

On factor relations between weighted and Nörlund means

By $\left( X,Y\right) ,$ we denote the set of all sequences $\epsilon =\left( \epsilon _{n}\right) $ such that $\Sigma \epsilon _{n}a_{n}$ is summable $Y$ whenever $\Sigma a_{n}$ is summable $X,$ where $X$ and $Y$ are two summability methods. In this study, we get necessary and sufficient conditions for $\epsilon \in \left( \left\vert N,q_{n},u_{n}\right\vert _{k},\left\vert \bar{N},p_{n}\right\vert \right) $ and $\epsilon \in \left( \left\vert \bar{N},p_{n}\right\vert ,\left\vert N,q_{n},u_{n}\right\vert _{k}\right) $, $k\geq 1,$ using functional analytic tecniques, where $% \left\vert \bar{N},p_{n}\right\vert $ and $\left\vert N,q_{n},u_{n}\right\vert _{k}$ are absolute weighted and N\"{o}rlund summability methods, respectively, \cite{1}, \cite{5}. Thus, in the special case, some well known results are also deduced.

Degree of Approximation by a General Cλ-Summability Method

In the present study, two theorems explaining the degree of approximation of signals belonging to the class Lip(α, p, w) by a more general Cλ-method (Summability method) have been formulated. Improved estimations have been observed in terms of λ(n); where (λ(n))−α ≤ n−α for 0 < α ≤ 1 as compared to previous studies presented in terms of n. These estimations of infinite matrices are very much applicable in solid state physics which further motivates for an investigation of perturbations of matrix valued functions.

A New Criterion for Borel-Euler Summability Method of Fourier Series

A New Criterion for Borel-Euler Summability Method of Fourier Series

A Tauberian theorem for the generalized Nörlund summability method

Abstract Let ( p n ) {(p_{n})} and ( q n ) {(q_{n})} be any two non-negative real sequences, with R n := ∑ k = 0 n p k ⁢ q n - k ≠ 0 {R_{n}:=\sum_{k=0}^{n}{p_{k}q_{n-k}}\neq 0} ( n ∈ ℕ {n\in\mathbb{N}} ). Let ∑ k = 0 ∞ a k {\sum_{k=0}^{\infty}a_{k}} be a series of real or complex numbers with partial sums ( s n ) {(s_{n})} , and set t n p , q := 1 R n ⁢ ∑ k = 0 n p k ⁢ q n - k ⁢ s k {t_{n}^{p,q}:=\frac{1}{R_{n}}\sum_{k=0}^{n}{p_{k}q_{n-k}s_{k}}} for n ∈ ℕ {n\in\mathbb{N}} . In this paper, we present the necessary and sufficient conditions under which the existence of the limit lim n → ∞ ⁡ s n = L {\lim_{n\to\infty}{s_{n}}=L} follows from that of lim n → ∞ ⁡ t n p , q = L {\lim_{n\to\infty}t_{n}^{p,q}=L} . These conditions are one-sided or two-sided if ( s n ) {(s_{n})} is a sequence of real or complex numbers, respectively.

Inclusion and equivalence relations between absolute Norlund and absolute weighted mean summability methods

In this paper, a set of conditions under which the absolute Nörlund summability method include in the absolute weighted mean method have been established. Three non-trivial examples to show that this inclusion holds have been given, and other three examples to show that even if both and are regular, the inclusion fails to holds have been constructed. The paper give two non-trivial examples to show that the equivalence of these two methods may holds. Finally, we give two examples to show that inclusion may holds in only one way without the other.

On the Possibility of Using the Method of Sign-Perturbed Sums for the Processing of Dynamic Test Data

At the present time, the methods for the measurement and prediction of the dynamic strength of materials are complicated and unstandardized. An experimental data processing method based on the incubation time criterion is considered. Only a finite number of measurements containing random errors and limited statistical information are usually available in practice, since dynamic tests are laborious, and every individual test requires a lot of time. This strongly restricts the number of applicable data processing methods unless we are satisfied with approximate and heuristic solutions. The method of sign-perturbed sums (SPS) is used for the estimation of finite-sample confidence regions with a specified confidence probability under the assumption of noise symmetries. It is shown that several experimental points are sufficient to determine the strength parameter with an accuracy acceptable for engineering calculations. The applicability of the proposed method is demonstrated in the processing of a number of experiments on the dynamic fracture of rocks.

Tauberian conditions with controlled oscillatory behavior for statistical convergence

We present new Tauberian conditions in terms of the general logarithmic control modulo of the oscillatory behavior of a real sequence (sn) to obtain lim n?? sn = ? from st - lim n?? sn = ?, where ? is a finite number. We also introduce the statistical (l,m) summability method and extend some Tauberian theorems to this method. The main results improve some well-known Tauberian theorems obtained for the statistical convergence.

Statistical deferred Cesàro summability and its applications to approximation theorems

Statistical (C,1) summability and a Korovkin type approximation theorem has been proved by Mohiuddine et al. [20] (see [S. A. Mohiuddine, A. Alotaibi and M. Mursaleen, Statistical summability (C,1) and a Korovkin type approximation theorem, J. Inequal. Appl. 2012 (2012), Article ID 172, 1-8). In this paper, we apply statistical deferred Ces?ro summability method to prove a Korovkin type approximation theorem for the set of functions 1, e-x and e-2x defined on a Banach space C[0;1) and demonstrate that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. We also establish a result for the rate of statistical deferred Ces?ro summability method. Some interesting examples are also discussed here in support of our definitions and results.

Tauberian theorems for the statistical convergence and the statistical (C,1,1) summability

Every P-convergent double sequence is statistically convergent and every bounded statistically convergent sequence is statistical (C,1,1) summable. The converse of these implications are not always true. Theorems on which conditioned converses are searched are known as Tauberian theorems. Inspired by the convergence to zero of the difference sequence between a sequence and its arithmetic means in the single sequence case, we obtain Tauberian theorems for the statistical convergence and statistical (C,1,1) summability method by imposing some conditions on the difference sequence between a double sequence and its different arithmetic means.

On the Absolute Matrix Summability Factors of Fourier Series

In this paper, two known theorems on |N, p n | k summability methods of Fourier series have been generalized for |A, p n | k summability factors of Fourier series by using different matrix transformations. New results have been obtained dealing with some other summability methods.

Generalized weighted statistical convergence in intuitionistic fuzzy normed linear spaces

In this paper, we introduce a new statistical convergence type, named weighted λ-statistical convergence to generalize the concept of weighted statistical convergence with respect to the intuitionistic fuzzy norm (µ, υ). Moreover, we establish its relation to weighted statistical convergence and a new summability method, named as Nλ, p -summability with respect to the intuitionistic fuzzy norm (µ, υ).

Annamalai’s Computing Model for Algorithmic Geometric Series and Its Mathematical Structures

This paper presents a new mathematical model for formation as well as computation of geometric series and summability in step-by-step procedures. Also, it provides mathematical structures for geometric series-ordered terms. The novel mathematical model uses Annamalai’s computing method of geometric series and summability, which provided a technique to establish the algorithmic geometric series and its formulae in an earlier paper, for further improvement in the scientific research study. This mathematical/computational models of geometric series are widely used in the fields of physics, engineering, biology, medicine, economics, computer science, queueing theory, finance, and management for the purpose of research and development meeting today’s challenges. In an earlier research article, the geometric series along with exponential decay model were used to determine effective medicine dosage. Few specific mathematical formulae had also been discovered by using Annamalai’s algorithmic geometric series and summability. This could be very interesting and informative for current students and researchers.

Stein’s method for nonconventional sums

We obtain almost optimal convergence rate in the central limit theorem for (appropriately normalized) “nonconventional sums of the form $S_N=\sum _{n=1}^N (F(\xi _n,\xi _{2n},...,\xi _{\ell n})-\bar F)$. Here $\{\xi _n: n\geq 0\}$ is a sufficiently fast mixing vector process with some stationarity conditions, $F$ is bounded Holder continuous function and $\bar F$ is a certain centralizing constant. Extensions to more general functions $F$ will be discusses, as well. Our approach here is based on the so called Stein’s method, and the rates obtained in this paper significantly improve the rates in [7]. Our results hold true, for instance, when $\xi _n=(T^nf_i)_{i=1}^\wp $ where $T$ is a topologically mixing subshift of finite type, a hyperbolic diffeomorphism or an expanding transformation taken with a Gibbs invariant measure, as well as in the case when $\{\xi _n: n\geq 0\}$ forms a stationary and exponentially fast $\phi $-mixing sequence, which, for instance, holds true when $\xi _n=(f_i(\Upsilon _n))_{i=1}^\wp $ where $\Upsilon _n$ is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure.

A new note on generalized absolute Cesàro summability factors

Quite recently, in [10], we have proved a theorem dealing with the generalized absolute Ces?ro summability factors of infinite series by using quasi monotone sequences and quasi power increasing sequences. In this paper, we generalize this theorem for the more general summability method. This new theorem also includes some new and known results.

An application of quasi-monotone sequences to absolute matrix summability

Recently, Bor [5] has obtained two main theorems dealing with |?N,pn|k summability factors of infinite series and Fourier series. In the present paper, we have generalized these theorems for |A,?n|k summability method by using quasi-monotone sequences.

Korovkin type approximation theorems in weighted spaces via power series method

In this paper we consider power series method which is also member of the class of all continuous summability methods. We study a Korovkin type approximation theorem for a sequence of positive linear operators acting from a weighted space C 1 into a weighted space B 2 with the use of the power series method which includes both Abel and Borel methods. We also consider the rates of convergence of these operators.

New approach to the concept of aᶥ -statistical convergence with the number of alpha

Following a very recent approach, we generalize recently introduced summability method, namely, I-statisticalconvergence. We use an infinite matrix of complex numbers and an a number to do this generalization. We call this new convergence as A^I statistical convergence of order a with respect to a sequence of modulus functions. We also define two new spaces by using strong convergence and Ces·ro summability and after giving these descriptions, we investigate their relationship and we obtain some result