Summability Methods Research Articles

A hierarchical matrix approach for computing hydrodynamic interactions

For simulations of large numbers of small, spherical particles in a Stokes flow, the long-range hydrodynamic interactions approximated by the Rotne–Prager–Yamakawa (RPY) kernel can be summed rapidly using, for example, the fast multipole method (FMM) or the particle-mesh Ewald (PME) method. In this paper, we develop new fast methods for computing these sums using the H2 hierarchical matrix representation, for open and for periodic boundary conditions. To the best of our knowledge, the method for infinite periodic sums using the H2 hierarchical matrix representation is the first such method developed. We also consider a more general RPY kernel that handles polydisperse particle radii, and show analytically and experimentally that the proxy surface method for efficiently constructing the H2 hierarchical matrix representation remains effective in this case. Numerical tests demonstrate the well-controlled accuracy of the H2 summation methods and their linear-scaling computation and storage cost. We find that the H2 matrix approach has lower cost for computing the summations compared to FMM and PME, but higher precomputation cost (required for each particle configuration). This precomputation cost can be amortized over several summations when computing Brownian displacements or forces in Brownian and Stokesian dynamics simulations with very large numbers of particles.

A new factor theorem on absolute matrix summability method

Abstract In this paper, we have a new matrix generalization with absolute matrix summability factor of an infinite series by using quasi-β-power increasing sequences. That theorem also includes some new and known results dealing with some basic summability methods

Modeling the effect of sensor failure on the location of counting sensors for origin-destination (OD) estimation

The network sensor location problem (NSLP) for origin–destination (OD) estimation identifies the optimal locations for sensors to estimate the vehicular flow of OD pairs in a road network. Like other measurement apparatuses, these sensors are subject to failure, which can affect the reliability of the OD estimations. In this paper, we propose a novel model that allows us to solve the NSLP for OD demand estimation by identifying the most reliable locations to install sets of sensors with consideration for a nonhomogeneous Poisson process to account for time-dependent sensor failure. The proposed model does not rely on the assumption that true OD demand information is known. We introduce two separate objective functions to minimize the maximum possible information loss (MPIL) associated with OD demand on sensor-equipped links and OD pairs during the lifetimes of the sensors. Both objective functions are formulated to incorporate the possibility of sensor failure into the calculated OD demands. We use stochastic user equilibrium (SUE) to address the stochasticity of traffic route selection. We then employ the weighted sums method (WSM) and an ε-constraint to incorporate the objective functions into an integrated formulation. Two sensor types with different time-dependent failure rates are considered to identify the optimal locations for sets of sensors for OD demand estimation purposes while addressing the available budget constraints. We also address the problem of scheduled/routine maintenance of existing sensors by introducing an additional sensor deployment phase that focuses on maintaining the reliability of information by repairing or replacing failed sensors, installing additional sensors or a combination of both. The numerical results from the proposed model demonstrate how the deployment of more advanced sensors with lower failure rates can effectively improve the reliability of the information obtained from sensors. We also evaluate the use of different weights for the WSM’s objective functions to explore alternative combinations of sensor configurations. The introduction of additional sensors to a network shows that the decision between repairing failed sensors and installing new sensors is highly dependent on the available budget and the failed sensors’ locations.

Failure of Approximation of Odd Functions by Odd Polynomials

We construct a Hilbert holomorphic function space H on the unit disk such that the polynomials are dense in H, but the odd polynomials are not dense in the odd functions in H. As a consequence, there exists a function f in H that lies outside the closed linear span of its Taylor partial sums \(s_n(f)\), so it cannot be approximated by any triangular summability method applied to the \(s_n(f)\). We also show that there exists a function f in H that lies outside the closed linear span of its radial dilates \(f_r, ~r<1\).

A Link between Approximation Theory and Summability Methods via Four-Dimensional Infinite Matrices

In this study, we present a link between approximation theory and summability methods by constructing bivariate Bernstein-Kantorovich type operators on an extended domain with reparametrized knots. We use a statistical convergence type and power series method to obtain certain Korovkin type theorems, and we study certain rates of convergences related to these summability methods. Furthermore, we numerically analyze the theoretical results and provide some computer graphics to emphasize the importance of this study.

MODEL OF EVALUATION OF INVESTMENT ATTRACTIVENESS OF THE ENTERPRISE

In the modern financial and economic conditions of business entities, ensuring the investment attractiveness of the enterprise is the key to its profitability and long-term development. The article considers the essence of investment attractiveness of enterprises. Based on the analysis of literature sources, the concept of investment attractiveness of the enterprise is clarified, which can contribute to a more accurate and clear definition of the processes associated with this activity. Under the "investment attractiveness" as an economic category should be understood as the level of efficiency of the available financial and economic resources of the enterprise, the indicators of which allow potential investors to draw conclusions about the need and feasibility of investing in the enterprise. It is found that the main components of the formation of investment potential of the enterprise are indicators of profitability, liquidity and solvency, business activity, financial stability. In the process of analyzing the literature and conducting a theoretical study identified various methodological approaches to assessing the investment attractiveness of the enterprise, among which are such classic tools as the method of sum of coefficients, the method of complex assessment, the method of sum of places. Their critical analysis is carried out, the advantages and disadvantages are determined. It is proved that the most clear and substantiated assessment of the level of investment attractiveness of the enterprise is considered in the methods based on the definition of the integrated indicator of investment attractiveness of the enterprise. This is the simplest method of analyzing the level of investment attractiveness, which requires the least time, and also allows you to quickly track negative trends in the enterprise, based on the calculation of financial ratios. It is substantiated that the assessment of the level of investment attractiveness of the enterprise should be carried out through rapid assessment, which is based on the calculation of an integrated indicator and provides analysis of the effectiveness of the main coefficients of financial condition based on comparison of marginal (critical and normal) and actual values. The proposed model of rapid assessment allows you to get an objective picture of the financial condition and efficiency of the existing investment attractiveness of the enterprise.

Regular matrix methods of summability and real interpolation

We show that the Banach–Saks property with respect to a regular positive matrix method of summability is inherited by the real interpolation spaces from a space forming the interpolation family and possessing this property. The proof refers to the Galvin–Prikry theorem on Ramsey sets. The results apply to several matrix methods of summability, such as Cesaro, Nørlund or Holder methods.

Statistical Voronoi mean and applications to approximation theorems

In this paper, we give statistical Voronoi mean which is a new statistical summability method, is not need to be regular and positive. We prove a Korovkin type approximation theorem via this method that covers many important summability methods scattered in the literature. Also, we demonstrate that our theorem is stronger than proven by earlier authors with an interesting application. Finally, we establish the rate of convergence.

Noncommutative pointwise convergence of orthogonal expansions of several variables

AbstractIn this paper, the boundedness of the maximal function for an operator‐valued weighted L1 space on the unit sphere of , in which the weight functions are invariant under finite reflection groups, is established. We use it to prove the boundedness of orthogonal expansions in h‐harmonics and this result applies to various methods of summability. Furthermore, we obtain the corresponding pointwise convergence theorems. At last, we give some results in the special reflection group .

Compact and Matrix Operators on the Space $\left\vert \overline N_p^{\phi }\right\vert _{k}$

In this paper, determining the operator norm, we give certain characterizations of matrix transformations from the space $ \left\vert \overline N_p^{\phi }\right\vert _{k}$, the space of all series summable by the absolute weighted mean summability method, to one of the classical sequence spaces $c_{0},c,l_{\infty }.$ Also, we obtain the necessary and sufficient conditions for each matrix in these classes to be compact and establish a number of estimates or identities for the Hausdorff measures of noncompactness of the matrix operators in these classes.

On absolute matrix summability factors of infinite series and Fourier series

Quite recently, Bor (2021) [22] has proved two main theorems dealing with absolute Riesz summability factors of infinite series and Fourier series by using an almost increasing sequence. In this paper, we have generalized these theorems to the |A,pn|k summability method.

Exact reconstruction of sparse non-harmonic signals from their Fourier coefficients

In this paper, we derive a new reconstruction method for real non-harmonic Fourier sums, i.e., real signals which can be represented as sparse exponential sums of the form f(t) = sum _{j=1}^{K} gamma _{j} , cos (2pi a_{j} t + b_{j}), where the frequency parameters a_{j} in {mathbb {R}} (or a_{j} in {mathrm i} {mathbb {R}}) are pairwise different. Our method is based on the recently proposed numerically stable iterative rational approximation algorithm in Nakatsukasa et al. (SIAM J Sci Comput 40(3):A1494–A1522, 2018). For signal reconstruction we use a set of classical Fourier coefficients of f with regard to a fixed interval (0, P) with P>0. Even though all terms of f may be non-P-periodic, our reconstruction method requires at most 2K+2 Fourier coefficients c_{n}(f) to recover all parameters of f. We show that in the case of exact data, the proposed iterative algorithm terminates after at most K+1 steps. The algorithm can also detect the number K of terms of f, if K is a priori unknown and L ge 2K+2 Fourier coefficients are available. Therefore our method provides a new alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony’s method.

On a mean method of summability

Let $p(x)$ be a nondecreasing real-valued continuous function&#x0D; on $R_+:=[0,\infty)$ such that $p(0)=0$ and $p(x) \to \infty$ as $x \to \infty$.&#x0D; Given a real or complex-valued integrable function $f$ in Lebesgue's sense on every bounded interval $(0,x)$&#x0D; for $x&gt;0$, in symbol $f \in L^1_{loc} (R_+)$, we set&#x0D; $$&#x0D; s(x)=\int _{0}^{x}f(u)du&#x0D; $$&#x0D; and&#x0D; $$&#x0D; \sigma _{p}(s(x))=\frac{1}{p(x)}\int_{0}^{x}s(u)dp(u),\,\,\,\,x&gt;0&#x0D; $$&#x0D; provided that $p(x)&gt;0$.&#x0D; &#x0D; A function $s(x)$&#x0D; is said to be summable to $l$ by the weighted mean method determined&#x0D; by the function $p(x)$, in short, $(\overline{N},p)$ summable to $l$,&#x0D; if&#x0D; $$&#x0D; \lim_{x \to \infty}\sigma _{p}(s(x))=l.&#x0D; $$&#x0D; &#x0D; If the limit $\lim _{x \to \infty} s(x)=l$&#x0D; exists, then $\lim _{x \to \infty} \sigma _{p}(s(x))=l$ also exists. However, the converse is not true in general.&#x0D; In this paper, we give an alternative proof a Tauberian theorem stating that convergence follows from summability by weighted mean method on $R_+:=[0,\infty)$ and a Tauberian condition of slowly decreasing type with respect to the weight function due to Karamata. These Tauberian conditions are one-sided or two-sided if $f(x)$ is a real or complex-valued function, respectively. Alternative proofs of some well-known Tauberian theorems given for several important summability methods can be obtained by choosing some particular weight functions.

Nonlinear approximation in N-dimension with the help of summability methods

Nonlinear approximation in N-dimension with the help of summability methods

General methods of convergence and summability

This paper is on general methods of convergence and summability. We first present the general method of convergence described by free filters of mathbb{N} and study the space of convergence associated with the filter. We notice that c(X) is always a space of convergence associated with a filter (the Frechet filter); that if X is finite dimensional, then ell _{infty }(X) is a space of convergence associated with any free ultrafilter of mathbb{N} ; and that if X is not complete, then ell _{infty }(X) is never the space of convergence associated with any free filter of mathbb{N} . Afterwards, we define a new general method of convergence inspired by the Banach limit convergence, that is, described through operators of norm 1 which are an extension of the limit operator. We prove that ell _{infty }(X) is always a space of convergence through a certain class of such operators; that if X is reflexive and 1-injective, then c(X) is a space of convergence through a certain class of such operators; and that if X is not complete, then c(X) is never the space of convergence through any class of such operators. In the meantime, we study the geometric structure of the set mathcal{HB}(lim ):= {Tin mathcal{B} (ell _{infty }(X),X): T|_{c(X)}= lim text{ and }|T|=1} and prove that mathcal{HB}(lim ) is a face of mathsf{B} _{mathcal{L}_{X}^{0}} if X has the Bade property, where mathcal{L}_{X}^{0}:= { Tin mathcal{B} (ell _{infty }(X),X): c_{0}(X) subseteq ker (T) } . Finally, we study the multipliers associated with series for the above methods of convergence.

General Summability Methods in the Approximation by Bernstein–Chlodovsky Operators

In this paper, by using regular summability methods we modify the Bernstein–Chlodovsky operators in order get more general and powerful results than the classical aspects. We study Korovkin-type approximation theory on weighted spaces. As a special case, it is possible to Cesàro approximate (arithmetic mean convergence) to the test function although it fails for the classical Bernstein–Chlodovsky operators. At the end of the paper, we extend our results to the multi-dimensional case.

On the Euler method of summability and concerning Tauberian theorems

For any two regular summability methods (U) and (V), the condition under which V-limx_n=λ implies U-limx_n=λ is called a Tauberian condition and the corresponding theorem is called a Tauberian theorem. Usually in the theory of summability, the case in which the method U is equivalent to the ordinary convergence is taken into consideration. In this paper, we give new Tauberian conditions under which ordinary convergence or Cesàro summability of a sequence follows from its Euler summability by means of the product theorem of Knopp for the Euler and Cesàro summability methods.

Weighted (λ,μ)-statistical convergence and statistical summability methods of double sequences of fuzzy numbers with application to Korovkin-type fuzzy approximation theorem

The paper aims to investigate different types of weighted statistical convergence and weighted statistical summability of double sequences of fuzzy numbers. Relations connecting statistical convergence and statistical summability have been investigated in the environment of the newly defined classes of double sequences of fuzzy numbers. Relevant inclusion relations and related results concerning the new fuzzy summability methods have also been examined in detail. Finally, a Korovkin-type approximation theorem for double sequences of fuzzy positive linear operators has been established using the most generalized version of the summability method and showed its efficiency by means of a concrete example.

Increasing Trends in Air and Sea Surface Temperature in the Central Adriatic Sea (Croatia)

The Adriatic Sea and its coastal region have experienced significant environmental changes in recent decades, aggravated by climate change. The most prominent effects of climate change (namely, an increase in sea surface and air temperature together with changes in the precipitation regime) could have an adverse effect on social and environmental processes. In this study, we analyzed the time series of sea surface temperature and air temperature measured at three meteorological stations in the Croatian part of the Adriatic Sea. To assess the trends and variations in the time series of sea surface and air temperature, different statistical methods were employed, i.e., linear and quadratic regressions, Mann–Kendall test, Rescaled Adjusted Partial Sums method, and autocorrelation. The results evidenced increasing trends in the mean annual sea surface temperature and air temperature; furthermore, sudden variations in values were observed in 1998 and 1992, respectively. Increasing trends in the mean monthly sea surface temperature and air temperature occurred in the warmer parts of the year (from March to August). The results of this study could provide a foundation for stakeholders, decision–makers, and other scientists for developing effective measures to mitigate the negative effects of climate change in the scattered environment of the Adriatic islands and coastal region.

Regular summability methods in the approximation by max-min operators

In this paper, by using nonnegative regular summability methods we improve and generalize the approximation properties of max-min operators which have been investigated systematically in our recent paper published in 2020 in this journal. We also discuss the rate of convergence in the approximation. Applications and concluding remarks at the end of the paper explain why we need such summability methods.