Summability Research Articles

Banach limits: some new thoughts and perspectives

The existence of a Banach limit as a translation invariant positive continuous linear functional on the space of bounded scalar sequences which is equal to 1 at the constant sequence $$(1,1,\ldots ,1,\ldots )$$ is proved in a first course on functional analysis as a consequence of the Hahn Banach extension theorem. Whereas its use as an important tool in classical summability theory together with its application in the existence of certain invariant measures on compact (metric) spaces is well known, a renewed interest in the theory of Banach limits has led to certain applications which have opened new vistas in the structure of Banach spaces. The paper is devoted to a discussion of certain developments, both classical and recent, surrounding the theory of Banach limits including the structure of the set of Banach limits with special emphasis on certain aspects of their applications to the existence of certain invariant measures, vector valued analogues of Banach limits, functional equations and in the structure theory of Banach spaces involving the existence of selectors of certain multi-valued mappings into the metric space of non-empty, convex, closed and bounded subsets of a Banach space with respect to the Hausdorff metric. The paper shall conclude with a brief description of some recent results of the author on the study of ‘simultaneous continuous linear’ operators (linear selections) involving Hahn Banach extensions on spaces of Lipschitz functions on (subspaces of) Banach spaces. Some open problems that naturally arise in this circle of ideas shall also be included at their appropriate places.

Image theory for a sphere with negative permittivity

An image system for a point charge outside a dielectric sphere is presented for all complex values of relative permittivity ϵ = ϵ′ + iϵ″. The standard image integral solution of a point charge outside a dielectric sphere involving an image point charge plus a line source is shown to diverge for ϵ′ < −1, and a correction is proposed for this case, involving image multipoles of infinite magnitude which regularize the divergent line integral. The number of these multipoles depends on the position of ϵ relative to the resonant values ϵ = −1 − 1/n for positive integer n. The internal potential and dipole sources are also considered.

Alternative Analytic and Friendly Solution of the Bose-Einstein Integral

A study to find an analytic, and fairly friendly, solution of Bose-Einstein Integral has been conducted, by employing various common mathematical relations, mainly the Fourier Series. The result obtained was without approximation, and did not involve any divergent series. This study was carried out with motivation to obtain analytic solutions from Integral Bose-Einstein, which is very important in the study of Planck Distribution of blackbody radiation, and in Bose-Einstein statistic; and to show that the Bose-Einstein Integral can, in fact, be solved without involving complicated special functions. The Bose-Einstein Integral, surely, had been solved by large number of authors, generally, by utilizing various special functions, such as, by using the Riemann-Zeta function [1][2][6][7]. This study produced the same results as previously obtained through various ways, that is π4 / 15, which guarantees that the steps we conducted were correct mathematically.

Construction of new generalizations of Wynn’s epsilon and rho algorithm by solving finite difference equations in the transformation order

We construct new sequence transformations based on Wynn’s epsilon and rho algorithms. The recursions of the new algorithms include the recursions of Wynn’s epsilon and rho algorithm and of Osada’s generalized rho algorithm as special cases. We demonstrate the performance of our algorithms numerically by applying them to some linearly and logarithmically convergent sequences as well as some divergent series.

NNLO final-state quark-pair corrections in four dimensions

We describe how NNLO final state quark-pair corrections are computed in FDR by directly enforcing gauge invariance and unitarity in the definition of the regularized divergent integrals. We give details of our approach and show how virtual and real contributions can be merged together without relying, explicitly or implicitly, on dimensional regularization. As an example, we recompute the { H} rightarrow b {bar{b}} + {jets} and gamma ^*rightarrow {jets} inclusive rates at the NNLO accuracy in the large N_F limit of QCD. This demonstrates, for the first time, that physical results with intermediate infrared singularities can be obtained at NNLO using a fully four-dimensional procedure.

A note on some alternating series involving zeta and multiple zeta values

In this article, we study a class of conditionally convergent alternating series including, as a special case, the famous series ∑n⩾2(−1)nζ(n)n which links Euler's constant γ to special values of the Riemann zeta function at positive integers. We give several new relations of the same kind. Among other things, we show the existence of a similar relation for the Apostol-Vu harmonic zeta function which have never been noticed before. We also highlight a deep connection with the Ramanujan summation of certain divergent series which originally motivated this work.

Effective conductivity of a random suspension of highly conducting spherical particles

• A constructive approach for random 3D composites is developed. • The question of conditionally divergent series for random composites is resolved. • The famous Jeffreys formula for the effective conductivity of suspensions is revised and extended. • A theory of 3D Eisenstein functions is developed. Randomly distributed non-overlapping perfectly conducting spheres are embedded in a conducting matrix with the concentration of inclusions f . Jeffrey (1973) suggested an analytical formula valid up to O ( f 3 ) for macroscopically isotropic random composites. A conditionally convergent sum arose in the spatial averaging . In the present paper, we apply a method of functional equations to random composites and correct Jeffrey’s formula. The main revision concerns the proper investigation of the conditionally convergent sum and correction the f 2 -term. An algorithm for symbolic computations of the effective conductivity tensor is developed and performed up to O ( f 10 3 ) . The obtained formulae explicitly demonstrate the dependence of the effective conductivity tensor on the deterministic and probabilistic distributions of inclusions in the f 2 -term, and in the f 3 -term. This leads to the conclusion that some previous formulae presented as universal, i.e., valid for all random composites, may be actually applied only to dilute or to special composites when interaction between inclusions do not matter.

Categorifying induction formulae via divergent series

We show how to get explicit induction formulae for finite group representations, and more generally for rational Green functors, by summing a divergent series over Dwyer's subgroup and centralizer decomposition spaces. This results in formulae with rational coefficients. The former space yields a well-known induction formula, the latter yields a new one. As essentially immediate corollaries of the existing literature, we get similar formulae in group cohomology and stable splittings of classifying spaces.

The q-Borel Sum of Divergent Basic Hypergeometric Series rφs(a;b;q,x)

We study the divergent basic hypergeometric series which is a $q$-analog of divergent hypergeometric series. This series formally satisfies the linear $q$-difference equation. In this paper, for that equation, we give an actual solution which admits basic hypergeometric series as a $q$-Gevrey asymptotic expansion. Such an actual solution is obtained by using $q$-Borel summability, which is a $q$-analog of Borel summability. Our result shows a $q$-analog of the Stokes phenomenon. Additionally, we show that letting $q\to1$ in our result gives the Borel sum of classical hypergeometric series. The same problem was already considered by Dreyfus, but we note that our result is remarkably different from his one.

On the summability of divergent power series satisfying singular PDEs

The aim of this note is to apply the Borel–Laplace summation method studied by H. Chen, Z. Luo and C. Zhang (Summability of formal solutions of singular PDEs by means of two-dimensional Borel–Laplace method, preprint) to the divergent power series solutions to two families of nonlinear PDEs. The first one contains particularly a two-dimensional version of the so-called Euler equation (ODE), while the second is called totally characteristic type PDE by H. Chen and H. Tahara (On the holomorphic solution of non-linear totally characteristic equations, Math. Nachr. 219 (2000) 85–96).

On universality and convergence of the Fourier series of functions in the disc algebra

We construct functions in the disc algebra whose Fourier series are pointwise universal on countable and dense sets and their sets of divergence contain Gδ and dense sets and have Hausdorff dimension zero. We also see that some classes of closed sets of measure zero do not accept uniformly universal Fourier series, although all such sets accept divergent Fourier series.

On the convergence of the Neumann series for electrostatic fracture response

The feasibility of Neumann-series expansion of Maxwell’s equations in the electrostatic limit is investigated for potentially rapid and approximate subsurface imaging of geologic features proximal to metallic infrastructure in an oilfield environment. Although generally useful for efficient modeling of mild conductivity perturbations in uncluttered settings, we have raised the question of its suitability for situations such as oilfields, in which metallic artifacts are pervasive and, in some cases, in direct electrical contact with the conductivity perturbation on which the Neumann series is computed. Convergence of the Neumann series and its residual error are computed using the hierarchical finite-element framework for a canonical oilfield model consisting of an L-shaped, steel-cased well, energized by a steady-state electrode, and penetrating a small set of mildly conducting fractures near the heel of the well. For a given node spacing [Formula: see text] in the finite-element mesh, we find that the Neumann series is ultimately convergent if the conductivity is small enough — a result consistent with previous presumptions on the necessity of small conductivity perturbations. However, we also determine that the spectral radius of the Neumann series operator grows as approximately [Formula: see text], thus suggesting that in the limit of the continuous problem [Formula: see text], the Neumann series is intrinsically divergent for all conductivity perturbations, regardless of their smallness. The hierarchical finite-element methodology itself is critically analyzed and shown to possess the [Formula: see text] error convergence of traditional linear finite elements, thereby supporting the conclusion of an inescapably divergent Neumann series for this benchmark example. Application of the Neumann series to oilfield problems with metallic clutter should therefore be done with careful consideration to the coupling between infrastructure and geology. The methods used here are demonstrably useful in such circumstances.

(In)Animate Semiotics: Virtuality and Deleuzian Illusion(s) of Life

It is well known that, despite his close engagement with cinema, Gilles Deleuze was less concerned with animated film, being somewhat dismissive of its capabilities. In recent years, however, a number of attempts have been made – most notably by William Schaffer, Thomas Lamarre and Dan Torre – to construct Deleuzian positions in animation theory. This article outlines some of these approaches, whilst engaging critically with Torre’s writings. In particular, it foregrounds Torre’s neglect of the post-structural, political dimension of Deleuzian thought through an examination of the concepts of faciality, the close-up, and relation as they occur in Deleuzian and Deleuzo-Guattarian philosophy. This is in part facilitated through a comparison of Stuart Blackton’s Humorous Phases of Funny Faces (1906) – a work directly addressed by Torre, and Emile Cohl’s Fantasmagorie (1908) – a work which he largely passes by. It is claimed here that, despite a number of apparent similarities, the animations of Cohl and Blackton express a radically divergent series of ontological commitments. Cohl offers the audience an experience of chaotic, mutable, relational complexity that revels in its incoherence, whilst Blackton presents a series of more straightforward set pieces, dwelling for the most part upon object-centric representational form. The tension between representation and becoming that occurs between these works is employed to facilitate a critical engagement with Torre’s process-cognitivism. It is suggested that Torre’s work, though exceptional in its pedagogic value, is likewise expressive of this tension, and that in its effort firstly to combine a series of process-philosophical and cognitivist ideas, and secondly to unpack the radical ideas of Deleuze through the more conservative philosophy of Nicholas Rescher, it runs the risk of falling back into a quasi-Kantian philosophy of generality and representation.

Divergent series of Taylor coefficients on almost all slices

We show that there exists a holomorphic function, continuous to the boundary in a bounded, balanced, strictly pseudoconvex domain $\Omega$ with $C^{2}$ boundary such that almost every slice function has a series of Taylor coefficients divergent with every power $p\in(0,2)$.

Interplay between Approximation Theory and Renormalization Group

The review presents general methods for treating complicated problems that cannot be solved exactly and whose solution encounters two major difficulties. First, there are no small parameters allowing for the safe use of perturbation theory in powers of these parameters, and even when small parameters exist, the related perturbative series are strongly divergent. Second, such perturbative series in powers of these parameters are rather short, so that the standard resummation techniques either yield bad approximations or are not applicable at all. The emphasis in the review is on the methods advanced and developed by the author. One of the general methods is Optimized Perturbation Theory now widely employed in various branches of physics, chemistry, and applied mathematics. The other powerful method is Self-Similar Approximation Theory allowing for quite simple and accurate summation of divergent series. These theories share many common features with the method of renormalization group, which is briefly sketched in order to stress the similarities in their ideas and their mutual interconnection.

Resummation in QFT with Meijer G-functions

We employ a recent resummation method to deal with divergent series, based on the Meijer G-function, which gives access to the non-perturbative regime of any QFT from the first few known coefficients in the perturbative expansion. Using this technique, we consider in detail the ϕ4 model where we estimate the non-perturbative β-function and prove that its asymptotic behavior correctly reproduces instantonic effects calculated using semiclassical methods. After reviewing the emergence of the renormalons in this theory, we also speculate on how one can resum them. Finally, we resum the non-perturbative β-function of abelian and non-abelian gauge-fermion theories and analyze the behavior of these theories as a function of the number of fermion flavors. While in the former no fixed points are found, in the latter, a richer phase diagram is uncovered and illustrated by the regions of confinement, large-distance conformality, and asymptotic safety.

Application of the Cauchy integral formula as a tool of analytic continuation for the resummation of divergent perturbation series

Previous attempts to the resummation of divergent power series by means of analytic continuation are improved applying the Cauchy integral formula for complex functions. The idea is tested on divergent Møller-Plesset perturbation expansions of the electron correlation energy. In particular, the potential curve of the LiH molecule is computed from single reference MPn results which are divergent for bond distances larger than 3.6 Å. Preliminary results for the Hartree-Fock molecule are also tabulated.

Hyperreal Numbers for Infinite Divergent Series

Treating divergent series properly has been an ongoing issue in mathematics. However, many of the problems in divergent series stem from the fact that divergent series were discovered prior to having a number system which could handle them. The infinities that resulted from divergent series led to contradictions within the real number system, but these contradictions are largely alleviated with the hyperreal number system. Hyperreal numbers provide a framework for dealing with divergent series in a more comprehensive and tractable way.

Finite-part integration of the generalized Stieltjes transform and its dominant asymptotic behavior for small values of the parameter. II. Non-integer orders

This paper constitutes the second part on the subject of finite part integration of the generalized Stieltjes transform Sλ[f]=∫0∞f(x)(ω+x)−λdx about ω = 0, where now λ is a non-integer positive real number. Divergent integrals with singularities at the origin are induced by writing (ω + x)−λ as a binomial expansion about ω = 0 and interchanging the order of operations of integration and summation. The prescription of finite part integration is then implemented by interpreting these divergent integrals as finite part integrals which are rigorously represented as complex contour integrals. The same contour is then used to express Sλ[f] itself as a complex contour integral. This led to the recovery of the terms missed by naïve term-wise integration which themselves are finite parts of divergent integrals whose singularity is at the finite upper limit of integration. When the function f(x) has a zero at the origin of order m = 0, 1, … such that m − λ &amp;lt; 0, the correction terms missed out by the naïve term by term integration give the dominant contribution to Sλ[f] as ω → 0. Otherwise, the correction term is sub-dominant to the leading convergent terms in the naïve term by term integration. We apply these results by obtaining exact and asymptotic representations of the Kummer and Gauss hypergeometric functions by evaluating their known Stieltjes integral representations. We then apply the method of finite part integration to obtain the asymptotic behavior of a generalization of the Stieltjes integral which is relevant in the calculation of the effective index of refraction of a shallow potential well.

Dynamic Spectrum Extraction Method Based on Absolute Difference Summation and Statistical Theory

Dynamic spectrum theory has great importance in the field of noninvasive measurement of blood components. To enhance the computational efficiency and data utilization of the existing extraction methods, this paper proposes a new one based on the absolute difference summation (ADS) and statistical theory. The ADS is used to obtain the eigenvalue from a photoplethysmography (PPG) signal. The statistical method is used to obtain the final dynamic spectrum. The experimental data of PPG signal from 133 volunteers were extracted by the new method and the single-trial (ST) extraction method, and the partial least squares model was used to build the calibration models. Compared with the ST extraction, the new method showed better prediction ability. The correlation coefficient of the prediction set increased from 0.85 to 0.92, and the root mean square error of the prediction decreased from 13.49 to 9.86 g/L, which proved that this method can significantly improve the quality of the dynamic spectrum.