Summability Research Articles

Origins of the Failure of the Activity Virial Series

We examine the development of the virial equation of state when expressed as a series in the activity with coefficients labeled bn. Using the one-dimensional hard-rod model as a prototype, we consider steps in its development that introduce inaccuracy that leads it to form a divergent series. We discuss the role of volume dependence of the virial coefficients and present expressions and calculations for volume-dependent coefficients bn(V) for the hard-rod model up to n = 200. We examine alternative methods for computing properties from the bn. We recommend that further efforts be made to compute volume-dependent virial coefficients as a means to understand better the virial equation of state and to make it more robust in applications.

Dyson–Schwinger equations in minimal subtraction

We compare the solutions of one-scale Dyson–Schwinger equations (DSEs) in the minimal subtraction (MS) scheme to the solutions in kinematic momentum subtraction (MOM) renormalization schemes. We establish that the MS-solution can be interpreted as a MOM-solution, but with a shifted renormalization point, where the shift itself is a function of the coupling. We derive relations between this shift and various renormalization group functions and counterterms in perturbation theory. As concrete examples, we examine three different one-scale Dyson–Schwinger equations: one based on the 1-loop multiedge graph in D=4 dimensions, one for D=6 dimensions, and one for mathematical toy model. For each of the integral kernels, we examine both the linear and nine different non-linear Dyson–Schwinger equations. For the linear cases, we empirically find exact functional forms of the shift between MOM and MS renormalization points. For the non-linear DSEs, the results for the shift suggest a factorially divergent power series. We determine the leading asymptotic growth parameters and find them in agreement with the ones of the anomalous dimension. Finally, we present a tentative exact solution to one of the non-linear DSEs of the toy model.

Hypergeometric structures in Feynman integrals

For the precision calculations in perturbative Quantum Chromodynamics (QCD) gigantic expressions (several GB in size) in terms of highly complicated divergent multi-loop Feynman integrals have to be calculated analytically to compact expressions in terms of special functions and constants. In this article we derive new symbolic tools to gain large-scale computer understanding in QCD. Here we exploit the fact that hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package Sigma in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code HypSeries transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code solvePartialLDE is designed. Generalized hypergeometric functions, Appell-, Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton–type functions are considered. We illustrate the algorithms by examples.

Wide-angle effects in multi-tracer power spectra with Doppler corrections

We examine the computation of wide-angle corrections to the galaxy power spectrum including redshift-space distortions and relativistic Doppler corrections, and also including multiple tracers with differing clustering, magnification and evolution biases. We show that the inclusion of the relativistic Doppler contribution, as well as radial derivative terms, are crucial for a consistent wide-angle expansion for large-scale surveys, both in the single and multi-tracer cases. We also give for the first time the wide-angle cross-power spectrum associated with the Doppler magnification-galaxy cross correlation, which has been shown to be a new way to test general relativity.In the full-sky power spectrum, the wide-angle expansion allows integrals over products of spherical Bessel functions to be computed analytically as distributional functions, which are then relatively simple to integrate over.We give for the first time a complete discussion and new derivation of the finite part of the divergent integrals of the form ∫0 ∞ drrnjℓ (kr) jℓ' (qr), which are necessary to compute the wide-angle corrections when a general window function is included. This facilitates a novel method for integrating a general analytic function against a pair of spherical Bessel functions.

Certain aspects of ℐ-statistical supremum and ℐ-statistical infimum of real-valued sequences

Abstract Over the last few years, numerous researchers have contributed significantly to summability theory by connecting various notions of convergence concepts of sequences. In this paper, we introduce the concepts of ℐ {\mathcal{I}} -statistical supremum and ℐ {\mathcal{I}} -statistical infimum of a real-valued sequence and study some fundamental features of the newly introduced notions. We also introduce the concept of ℐ {\mathcal{I}} -statistical monotonicity and establish the condition under which an ℐ {\mathcal{I}} -statistical monotonic sequence is ℐ {\mathcal{I}} -statistical convergent. We end up giving a necessary and a sufficient condition for the ℐ {\mathcal{I}} -statistical convergence of a real-valued sequence.

Upper bounds and asymptotic expansion for Macdonald's function and the summability of the Kontorovich-Lebedev integrals

Uniform upper bounds and the asymptotic expansion with an explicit remainder term are established for the Macdonald function . The results can be applied, for instance, to study the summability of the divergent Kontorovich-Lebedev integrals in the sense of Jones. Namely, we answer affirmatively a question (cf. [Ehrenmark U. Summability experiments with a class of divergent inverse Kontorovich-Lebedev transforms. Comput Math Appl. 2018;76(1):141–154.]) whether these integrals converge for even entire functions of the exponential type in a weak sense.

Regularized limit, analytic continuation and finite-part integration

Finite-part integration is a recent method of evaluating a convergent integral in terms of the finite-parts of divergent integrals deliberately induced from the convergent integral itself [E. A. Galapon, The problem of missing terms in term by term integration involving divergent integrals, Proc. R. Soc. A 473 (2017) 20160567]. Within the context of finite-part integration of the Stieltjes transform of functions with logarithmic growths at the origin, the relationship is established between the analytic continuation of the Mellin transform and the finite-part of the resulting divergent integral when the Mellin integral is extended beyond its strip of analyticity. It is settled that the analytic continuation and the finite-part integral coincide at the regular points of the analytic continuation. To establish the connection between the two at the isolated singularities of the analytic continuation, the concept of regularized limit is introduced to replace the usual concept of limit due to Cauchy when the later leads to a division by zero. It is then shown that the regularized limit of the analytic continuation at its isolated singularities equals the finite-part integrals at the singularities themselves. The treatment gives the exact evaluation of the Stieltjes transform in terms of finite-part integrals and yields the dominant asymptotic behavior of the transform for arbitrarily small values of the parameter in the presence of arbitrary logarithmic singularities at the origin.

Continued functions and critical exponents: tools for analytical continuation of divergent expressions in phase transition studies

Resummation methods using continued functions are implemented to converge divergent series appearing in perturbation problems related to continuous phase transitions in field theories. In some cases, better convergence properties are obtained using continued functions than diagonal Pade approximants, which are extensively used in literature. We check the reliability of critical exponent estimates derived previously in universality classes of O(n)-symmetric models (classical phase transitions) and Gross-Neveu-Yukawa models (quantum phase transitions) using new methods.

Subtitling strategies of imperative sentence from English into Indonesian in the Divergent Series: Insurgent movie

The topic of this research is subtitling strategies of imperative sentence in The Divergent Series: Insurgent movie. The aims of this research is to find out the kind of subtitling strategies in the movie. This research used a descriptive method. The source data for this research are English subtitle and Indonesian subtitle from The Divergent Series: Insurgent movie. The data in this research is the dialogue in the movie. The researcher used the theory from Gottlieb (1992). The result of this research there are 8 kind of subtitling strategies, there are 73 data classified as Paraphrase, 32 data classified as Transfer, 13 data classified as Expansion, 21 data classified as Imitation, 4 data classified as Condensation, 5 data classified as Decimation, 12 data classified as Deletion, and 5 data classified as Resignation. The most subtitling strategy that used in this research is Paraphrase strategy because there are 73 data classified as Paraphrase strategy.

A novel fractal-fractional analysis of the stellar helium burning network using extended operational matrix method

The second stage, in which the star uses nuclear fuel in its interior, represents the helium burning phase. At that stage, three elements are synthesised: carbon, oxygen, and neon. This paper aims to establish a numerical solution for the helium burning system (HBN) fractal-fractional differential equations (FFDEs). The extended operative matrix method (OM) is employed in the solution of a system of differential equations. The product abundances of the four elements (helium, carbon, oxygen and neon) were obtained in a form of divergent series. These divergent series are then accelerated using Euler-Abell transformation (EUAT) and Pade approximation (EUAT-PA) to obtain more reliable results. Nine fractal-fractional (FF) gas models are calculated, and fractal-fractional parameters’ influence on product abundances is discussed. The findings show that modeling nuclear burning networks with the OM fractal-fractional derivative produces excellent results, establishing it as an accurate, resilient, and trustworthy approach, and the fractional HB models can have a considerable impact on stellar model calculations.

Uniform integrability of sequences of random elements with respect to weak topologies and weak integrals

In probability theory, uniform integrability of families of random variables or random elements plays an important role in the mean convergence. In this paper, we introduce a new version of uniform integrability for sequences in normed spaces in the weak sense. We study the relationship of this new concept with summability theory by considering statistical convergence. We also define a new type of uniform integrability of random elements taking values in topological vector spaces by considering weak integrals. Moreover, we study the connection of summability theory with this new concept as well.

Effective slip length for transverse shear flow over partially invaded grooves

We consider transverse shear flow over a $2L$ -periodic array of air-filled grooves, which are separated by solid slats. We address the scenario where the liquid partially invades the grooves. Following Crowdy's solution of the longitudinal problem (J. Fluid Mech. vol. 925, 2021, R2), we focus upon the simplified geometry of infinitely narrow slats. For flat menisci, the ratio $\lambda$ of the slip length to $L$ depends upon a single geometric parameter, namely the ratio $H$ of the invasion depth to $L$ . We analyse the singular limit $H\ll 1$ using matched asymptotic expansions, with an outer region on the scale of the period and an inner region on the scale of a single slat. In the outer region, we employ the complex-variable formulation of Stokes flows, obtaining the complex velocity as a divergent series. This formal representation is transformed to an admissible closed-form expression. In the inner region, the problem is equivalent to that of Stokes flow past a finite line segment. Asymptotic matching yields $\lambda = [\ln (2/{\rm \pi} H)-1]/{\rm \pi}$ .

Posing problems and solving self-generated problems: the case of convergence and divergence of series

The aim of the study was to examine how students generate problems about the concept of series, and how they evaluate and solve these problems. The difficulties experienced by the students while posing problems and their perceptions of the posing-evaluating-solving approach were also explored. In the first stage, 46 second-year university students posed problems for four tasks. In the second stage, which occurred a week after the first stage, the students evaluated their self-generated problems in terms of appropriateness to the tasks. The students who thought that they did not pose problems suitable for the tasks revised or reposed the problems, and then solved them by specifying the convergence or divergence of the series in the problems. The students also considered the impact of problem-posing implementation on their knowledge of the series. Findings showed that students’ difficulties led to unsuitable and unsolvable problems. The most prominent difficulties were using sequences instead of series and using convergent series instead of divergent series or vice versa. Moreover, providing the students with opportunities to evaluate and solve their self-generated problems at the end of the second stage was effective for the increase in the number of suitable and solvable problems.

Devilish Tricks for Sequence Acceleration

The two most famous quotes about divergent series are Abel's "Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever," and Heaviside's "This series is divergent, therefore we may be able to do something with it." Today a lot more is known about divergent series than in either's day, so we can say now that, on balance, Heaviside wins, and we now have plenty of license to use divergent series. This article talks about some "well-known" methods (that is, well-known to experts) to do so, and in particular talks about some of the devilishly good features of evalf/Sum, long one of my favourite tools in Maple. But Abel had a point, too, and we'll see some "shameful" things, which will give the reader some necessary caution to go along with their license.

A note on a general divergent series

ABSTRACT The aim of this note is to give a general theorem related to the divergence of series. A direct consequence of this theorem is the divergence of harmonic series. Furthermore, we show that this theorem can imply the divergence of some other familiar divergent series.

Conservative Binary Dynamics with a Spinning Black Hole at O(G^{3}) from Scattering Amplitudes.

We compute the conservative two-body Hamiltonian of a compact binary system with a spinning black hole through O(G^{3}) to all orders in velocity, including linear and quadratic spin terms. To obtain our results we calculate the classical limit of the two-loop amplitude for the scattering of a massive scalar particle with a massive spin-1 particle minimally coupled to gravity. We employ modern scattering amplitude and loop integration techniques, in particular numerical unitarity, integration-by-parts identities, and the method of regions. The conservative potential in terms of rest-frame spin vectors is extracted by matching to a nonrelativistic effective field theory. We also apply the Kosower-Maybee-O'Connell (KMOC) formalism to calculate the impulse in the covariant spin formalism directly from the amplitude. We work systematically in conventional dimensional regularization and explicitly evaluate all divergent integrals that appear in full- and effective-theory amplitudes, as well as in the phase-space integrals that arise in the KMOC formalism.

Some Results of Generalized Weighted Norlund-Euler-<img src=image/13428701_01.gif> Statistical Convergence in Non-Archimedean Fields

Non-Archimedean analysis is the study of fields that satisfy the stronger triangular inequality, also known as ultrametric property. The theory of summability has many uses throughout analysis and applied mathematics. The origin of summability methods developed with the study of convergent and divergent series by Euler, Gauss, Cauchy and Abel. There is a good number of special methods of summability such as Abel, Borel, Euler, Taylor, Norlund, Hausdroff in classical Analysis. Norlund, Euler, Taylor and weighted mean methods in Non-Archimedan Analysis have been investigated in detail by Natarajan and Srinivasan. Schoenberg developed some basic properties of statistical convergence and also studied the concept as a summability method. The relationship between the summability theory and statistical convergence has been introduced by Schoenberg. The concept of weighted statistical convergence and its relations of statistical summability were developed by Karakaya and Chishti. Srinivasan introduced some summability methods namely y-method, Norlund method and Weighted mean method in p-adic Fields. The main objective of this work is to explore some important results on statistical convergence and its related concepts in Non-Archimedean fields using summability methods. In this article, Norlund-Euler-<img src=image/13428701_02.gif> statistical convergence, generalized weighted summability using Norlund-Euler-<img src=image/13428701_02.gif> method in an Ultrametric field are defined. The relation between Norlund-Euler-<img src=image/13428701_02.gif> statistical convergence and Statistical Norlund-Euler-<img src=image/13428701_02.gif> summability has been extended to non-Archidemean fields. The notion of Norlund-Euler-<img src=image/13428701_02.gif> statistical convergence and inclusion results of Norlund-Euler statistical convergent sequence has been characterized. Further the relation between Norlund-Euler-<img src=image/13428701_02.gif> statistical convergence of order α & β has been established.

Cauchy's Further Contributions to Mathematics: The Divergent Series and Cauchy's Wrong Theorem

In this article we will first highlight Cauchy's contribution to divergent series: a contribution of fundamental importance for mathematics scholars of all time and then stop at the concept as an incomplete result of Cauchy with respect to the hypotheses relating to the continuity of the sum of a series of functions. Although he was not the only seller, this erroneous result remained famous. It is referred to as Cauchy's incomplete theorem.

Functional Summation of Series

We consider a summation technique which reduces summation of a series to the solution of some linear functional equations. Partial sums of a series satisfy an obvious difference equation. This equation is transformed to the functional equation on the interval [0,1] for the continuous argument. Then this equation is either solved explicitly (to within an arbitrary constant) or an asymptotic expansion of the solution is computed at the origin. The sum of the original series is determined uniquely as a constant needed for the matching of the asymptotic series with partial sums of the original series. The notion of a limit is not involved in this computational technique, which allows summation of divergent series as well.

A Special Form of Slower Divergent Series

A Special Form of Slower Divergent Series