Summability Methods Research Articles

Mechanics of fiber composites: Forms of loss of stability and fracture of test specimens resulting from three‐point bending tests

AbstractOn the basis of the linearized equations consistent theory of curvilinear bars the buckling problem of rectilinear short and long laminated fiber reinforced specimens under the three‐point bending conditions has formulated. Based on the method of finite sums in the embodiment of integrating matrices numerical method for solving the above problem has developed. It was shown that the failure of the composite specimens under the three‐point bending conditions is due to the implementation of non‐classical shear buckling mode.

On the summability methods of logarithmic type and the Berezin symbol

We prove by means of the Berezin symbols some theorems for the $\left( L\right) $-summability method for sequences and series. Also, we prove a new Tauberian type theorem for $\left( L\right) $-summability.

Weighted Statistical Relative Invariant Mean in Modular Function Spaces with Related approximation Results

The idea of statistical relative convergence on modular spaces has been introduced by Orhan and Demirci. The notion of σ-statistical convergence was introduced by Mursaleen and Edely and further extended based on a fractional order difference operator by Kadak. The concern of this paper is to define two new summability methods for double sequences by combining the concepts of statistical relative convergence and σ-statistical convergence in modular spaces. Furthermore, we give some inclusion relations involving the newly proposed methods and present an illustrative example to show that our methods are nontrivial generalizations of the existing results in the literature. We also prove a Korovkin-type approximation theorem and estimate the rate of convergence by means of the modulus of continuity. Finally, using the bivariate type of Stancu-Schurer-Kantorovich operators, we display an example such that our approximation results are more powerful than the classical, statistical, and relative modular cases of Korovkin-type approximation theorems.

Fast Ewald summation for Green\u2019s functions of Stokes flow in a half-space

Recently, Gimbutas et al. (J Fluid Mech, 2015. https://doi.org/10.1017/jfm.2015.302) derived an elegant representation for the Green’s functions of Stokes flow in a half-space. We present a fast summation method for sums involving these half-space Green’s functions (stokeslets, stresslets and rotlets) that consolidates and builds on the work by Klinteberg et al. (Res Math Sci 4(1):1, 2017. https://doi.org/10.1186/s40687-016-0092-7) for the corresponding free-space Green’s functions. The fast method is based on two main ingredients: The Ewald decomposition and subsequent use of FFTs. The Ewald decomposition recasts the sum into a sum of two exponentially decaying series: one in real space (short-range interactions) and one in Fourier space (long-range interactions) with the convergence of each series controlled by a common parameter. The evaluation of short-range interactions is accelerated by restricting computations to neighbours within a specified distance, while the use of FFTs accelerates the computations in Fourier space thus accelerating the overall sum. We demonstrate that while the method incurs extra costs for the half-space in comparison with the free-space evaluation, greater computational savings is also achieved when compared to their respective direct sums.

Comparison theorems for summability methods of sequences of fuzzy numbers

In this study we compare Ces\`{a}ro and Euler weighted mean methods of summability of sequences of fuzzy numbers with Abel and Borel power series methods of summability of sequences of fuzzy numbers. Also some results dealing with series of fuzzy numbers are obtained.

Tauberian theorems for iterations of weighted mean summable integrals

Let p be a positive weight function on $$A:=[1, \infty )$$ which is integrable in Lebesgue’s sense over every finite interval (1, x) for $$1<x<\infty $$ , in symbol: $$p \in L^{1}_{loc} (A)$$ such that $$P(x)=\int _{1}^{x} p(t) dt\ne 0$$ for each $$x>1$$ , $$P(1)=0$$ and $$P(x) \rightarrow \infty $$ as $$x \rightarrow \infty $$ . For a real-valued function $$f \in L^{1}_{loc} (A)$$ , we set $$s(x):=\int _{1}^{x}f(t)dt$$ and denote $$\begin{aligned} \sigma ^{(0)}_p(x):=s(x), \sigma ^{(k)}_p(x):=\frac{1}{P(x)}\int _1^x \sigma ^{(k-1)}_p(t) p(t)dt\,\,\, (x>1, k=1,2,\ldots ), \end{aligned}$$ provided that $$P(x)>0$$ . If $$\begin{aligned} \lim _{x\rightarrow \infty }\sigma ^{(k)}_p(x)=L, \end{aligned}$$ we say that $$\int _{1}^{\infty }f(x)dx$$ is summable to L by the k-th iteration of weighted mean method determined by the function p(x), or for short, $$({\overline{N}},p,k)$$ integrable to a finite number L and we write $$s(x)\rightarrow L({\overline{N}},p,k)$$ . It is well-known that the existence of the limit $$\lim _{x \rightarrow \infty } s(x)=L$$ implies that of $$\lim _{x \rightarrow \infty } \sigma ^{(k)}_p(x)=L$$ . But the converse of this implication is not true in general. In this paper, we obtain some Tauberian theorems for the weighted mean method of integrals in order that the converse implication holds true. Our results extend and generalize some classical type Tauberian theorems given for Cesaro and logarithmic summability methods of integrals.

Synthesis and crystal structures of new phosphoric triamides: study of intermolecular interactions by semi-empirical calculations and Hirshfeld surface analysis

In the present paper, crystal structures and Hirshfeld surface analyses of two new phosphoric triamides [2,3,6-F3–C6H2C(O)NH]P(O)(X)2 (X=N(CH3)C6H11 and N(C2H5)2) and an improved model of [OH8C4N]3P(O) are investigated. Moreover, the semi-classical density sums (PIXEL) method, which enables the calculation of interaction energies for molecule–molecule pairs, and AIM calculations were used to evaluate intermolecular forces in the studied compounds. The previously reported structure [2,6-F2-C6H3C(O)NH]P(O)[NHC(CH3)3]2 with a [C(O)NH]P(O)[NH(C)]2 segment, which is different than the [C(O)NH]P(O)[N(C)(C)]2 segment in structures [2,3,6-F3–C6H2C(O)NH]P(O)(X)2, is compared to those of the newly determined structures. The Hirshfeld surface method shows that the crystal cohesions of structures [2,3,6-F3–C6H2C(O)NH]P(O)(X)2 are established via H···H, O···H/H···O, C···H/H···C, and F···H/H···F contacts, while for [OH8C4N]3P(O), H···H and O···H/H···O are the dominant contacts. From PIXEL and AIM calculations and the decomposition of the interaction energies for different molecular pairs, it is shown that the donor and acceptor capability of the atoms involved in an interaction introduces the nature and strength of that interaction. The more acidic NCP–H unit in the C(O)NHP(O) segment (compared to the NP–H unit in the P(O)[NH(C)]2 segment) and the higher H-atom acceptor group P=O (compared to C=O) in the studied structures form the strongest NCP–H···O=P intermolecular hydrogen bond.

Landslide susceptibility assessment of the Chure Khola Catchment area of the Siwalik region, Central Nepal

Landslide susceptibility analysis is carried out in the Chure Khola Catchment, between Amlekhganj and the Churia Mai Range of the Bara District, covering area of 20 sq. km. The catchment lies in the Siwalik Hills composing the Siwalik Group of rocks of Middle Miocene to Early Pleistocene age. Owing to the week and fragile geology, the Siwalik Hills are prone to the gully erosion, shallow landslide and debris flow, during the heavy rain storms in monsoon seasons. In the present study, landslide susceptibility assessment was carried out using two methods, rapid field-based assessment and statistical index methods. For the susceptibility mapping of the river bank slopes, field- based method was used. The seven parameters such as slope angle, slope material, reduction to groundwater, effect of drainage, effect of past failure, effect of vegetation cover and effect of land use were used to calculate the factor of safety in the field. The slope areas were classified as highly susceptible (FS&lt;0.7), susceptible (0.7&lt;FS&lt;1), marginally stable (1&lt;FS&lt;1.2) and stable (FS&gt;1.2) categories, and finally, a susceptibility map was prepared. For the total 4.179 sq. km area where rapid field-based assessment was carried out, the areas covered by highly susceptible, susceptible, marginally stable and stable zones are respectively, 21.56%, 22.11%, 17.37% and 38.95%. Among the highly susceptible and susceptible zones identified, 71% sites have experienced recent slope failures. Landslide susceptibility mapping of the whole catchment area was prepared using statistical index method, and considering seven causative parameters such as elevation, slope, slope aspect, curvature, river proximity, stream density and lithology, which were determined and prepared from DEM using Arc GIS. Eighty percent landslides were used as the training sample for the spatial analysis, whereas 20% landslides were used for the validation of the study. The landslide susceptibility map exhibits the areas covered by very high, high, moderate, low and very low susceptibility zones are 47.18%, 25.28%, 19.77%, 3.60% and 4.16%, respectively. Validity of the study was determined using Riemann Sums method. Success Rate Curve shows that 78.04 % of the areas lie under the curve. Evaluating susceptibility in small watershed is important to mitigate shallow landslide related problems and in rehabilitating forest areas in the Churiya Hills of Nepal.

Modularly weighted four dimensional matrix summability with application to Korovkin type approximation theorem

This work is concerned with various weighted four dimensional matrix summability methods in modular function spaces associating with generalized difference operator involving (p,q)-gamma function. Following very recent results of Kadak (2017) [20], we first introduce a new type of difference operator on double function sequences and, also define modularly weighted A-statistical convergence and modularly statistically N˜A-summability by means of weighted four dimensional regular matrix A. We also present some important inclusion relations between newly proposed summability methods. Moreover, based on the concept of modularly statistically N˜A-summability, we prove a Korovkin type approximation theorem for functions of two variables in modular spaces. Finally, in order to show that our proposed method and its applications to approximation results are stronger than the existing methods, we display an illustrative example using bivariate (p,q)-Bernstein Kantorovich operators.

Localization property on absolute matrix summability factors of Fourier series

In this paper, a known theorem dealing with |N¯,pn|k summability methods of Fourier series is generalized to |A,θn|k summability method by taking normal matrices.

Absolute Matrix Summability Factors of Fourier Series with Quasi-f-Power Increasing Sequences

In this paper, we have generalized a main theorem dealing with quasi-f-power increasing sequence to |A,θn|k summability method by using Fourier series.

A matrix application on absolute weighted arithmetic mean summability factors of infinite series

In this present paper, we have generalized a main theorem dealing with $|\bar{N}, p_{n}|_{k}$ summability of non-decreasing sequences to $|A, p_{n}|_{k}$ summability method by using almost increasing sequences and taking normal matrices in place of weighted mean matrices.

Pointwise Convergence of Fourier-Laguerre Series of Integrable Functions

Abstract We extend and improve the some results of Xh. Z. Krasniqi [Int. J. of Anal. and Appl. Vol. 1, 33-39 (2013)], M. L. Mittal and M. V. Singh [Operators, Int. J. of Analysis, Vol. 2015, Article ID 478345, 4 pages] and from many other papers on summability of Fourier-Laguerre series to strong summability proving the estimate of the deviation of the partial sums from considered functions. There also is a remark on summability methods used in cited papers.

Tauberian theorems for the logarithmic summability methods of integrals

A real-valued continuous function f on $$[1, \infty )$$ is said to be summable by the logarithmic summability method of integrals (shortly, $$\ell $$ summable) if $$\begin{aligned} \lim _{x \rightarrow \infty }\frac{1}{\log {x}}\int _1^x\frac{s(t)}{t}dt \end{aligned}$$ exists, where $$s(x)=\int _{1}^{x}f(t)dt$$ . Our goal in this paper is to obtain some Tauberian theorems for the logarithmic summability method of integrals by some new Tauberian conditions. We also introduce the $$\ell ^{(k)}$$ summability method of integrals and give some Tauberian theorems for this method.

Singular Perturbation of Nonlinear Systems with Regular Singularity

We extend Balser-Kostov method of studying summability properties of a singularly perturbed inhomogeneous linear system with regular singularity at origin to nonlinear systems of the form εzf′=Fε,z,f with F a Cν-valued function, holomorphic in a polydisc D-ρ×D-ρ×D-ρν. We show that its unique formal solution in power series of ε, whose coefficients are holomorphic functions of z, is 1-summable under a Siegel-type condition on the eigenvalues of Ff(0,0,0). The estimates employed resemble the ones used in KAM theorem. A simple lemma is applied to tame convolutions that appear in the power series expansion of nonlinear equations. Applications to spherical Bessel functions and probability theory are indicated. The proposed summability method has certain advantages as it may be applied as well to (singularly perturbed) nonlinear partial differential equations of evolution type.

Singularity Analysis for Heavy-Tailed Random Variables

We propose a novel complex-analytic method for sums of i.i.d. random variables that are heavy-tailed and integer-valued. The method combines singularity analysis, Lindel\"of integrals, and bivariate saddle points. As an application, we prove three theorems on precise large and moderate deviations which provide a local variant of a result by S. V. Nagaev (1973). The theorems generalize five theorems by A. V. Nagaev (1968) on stretched exponential laws $p(k) = c\exp( -k^\alpha)$ and apply to logarithmic hazard functions $c\exp( - (\log k)^\beta)$, $\beta>2$; they cover the big jump domain as well as the small steps domain. The analytic proof is complemented by clear probabilistic heuristics. Critical sequences are determined with a non-convex variational problem.

Development of an emergy computation algorithm for complex systems using depth first search and track summing methods

Abstract Although emergy analysis provides a more comprehensive view of sustainability than other methods, it has been criticized from both theoretical and practical aspects due to its complicated approach, especially when dealing with large sized systems and allocating emergy involved with feedbacks or with input streams that have splits and co-products. The aim of this study was to provide a practical sustainability decision-making software tool to cover all the emergy rules by reformulating and improving previous emergy accounting procedures using the MATLAB language algorithm. The code consisted of the two components of (i) searching the emergy pathways, where the algorithm is adapted from the depth first search method, and (ii) the computation of emergy based on emergy algebra, where the track summing method was deployed. The algorithm was validated on three case studies, comprised of two complex systems with feedbacks, splits and co-products and one case of an industrial symbiosis with a closed loop of materials and energy through reuse and recycle. Three emergy size distribution behaviors (symbiosis, serial and parallel) were revealed and classification of the process behavior could be used to reduce the calculation complexity in further emergy applications.

Tauberian theorems for statistical summability methods of sequences of fuzzy numbers

We give Tauberian conditions of slow decrease type under which statistical Cesaro summability and statistical logarithmic summability of sequences of fuzzy numbers imply convergence in fuzzy number space. Besides, we obtain a sufficient condition for statistically convergent sequences of fuzzy numbers to converge in the ordinary sense.

Multivariate approximation in total variation, II: Discrete normal approximation

The paper applies the theory developed in Part I to the discrete normal approximation in total variation of random vectors in ${\mathbb Z}^d$. We illustrate the use of the method for sums of independent integer valued random vectors, and for random vectors exhibiting an exchangeable pair. We conclude with an application to random colourings of regular graphs.

A Tauberian Theorem for the Power-Series Summability Method

We introduce a one-sided Tauberian condition in terms of the weighted general control modulo oscillatory behavior of the integer order m with m ≥ 1 for the power-series summability method.