Summability Research Articles

Eikonal amplitudes on the celestial sphere

Celestial scattering amplitudes for massless particles are Mellin transforms of momentum-space scattering amplitudes with respect to the energies of the external particles, and behave as conformal correlators on the celestial sphere. However, there are few explicit cases of well-defined celestial amplitudes, particularly for gravitational theories: the mixing between low- and high-energy scales induced by the Mellin transform generically yields divergent integrals. In this paper, we argue that the most natural object to consider is the gravitational amplitude dressed by an oscillating phase arising from semi-classical effects known as eikonal exponentiation. This leads to gravitational celestial amplitudes which are analytic, apart from a set of poles at integer negative conformal dimensions, whose degree and residues we characterize. We also study the large conformal dimension limits, and provide an asymptotic series representation for these celestial eikonal amplitudes. Our investigation covers two different frameworks, related by eikonal exponentiation: 2 → 2 scattering of scalars in flat spacetime and 1 → 1 scattering of a probe scalar particle in a curved, stationary spacetime. These provide data which any putative celestial dual for Minkowski, shockwave or black hole spacetimes must reproduce. We also derive dispersion and monodromy relations for these celestial amplitudes and discuss Carrollian eikonal-probe amplitudes in curved spacetimes.

High energy scattering in the Unitary Toy Model

We continue exploring the Unitary Toy Model (UTM) as a playground for high energy collisions in QCD. Our new approach is based on the diagonalization of the evolution Hamiltonian. Part of the spectrum can be identified with intercepts of dressed Pomerons. Analogously to QCD, a multi-Pomeron expansion of the S-matrix is badly divergent asymptotic series. Yet we succeeded to establish resummation procedures resulting in a well behaved S-matrix. In addition the Hamiltonian possesses negative eigenvalues, which dominate the approach of the S-matrix to saturation. We are hopeful that important lessons about BFKL-based Pomeron calculus could be taken from the toy world to real QCD.

Torsional regularization of self-energy and bare mass of electron

Abstract In the presence of spacetime torsion, the momentum components do not commute; therefore, in quantum field theory, summation over the momentum eigenvalues will replace integration over the momentum. In the Einstein–Cartan theory of gravity, in which torsion is coupled to spin, the separation between the eigenvalues increases with the magnitude of the momentum. Consequently, this replacement regularizes divergent integrals in Feynman diagrams with loops by turning them into convergent sums. In this article, we apply torsional regularization to the self-energy of a charged lepton in quantum electrodynamics. We show that torsion eliminates the ultraviolet divergence of the standard self-energy. We also show that the infrared divergence is absent. In the end, we calculate the finite bare masses of the electron, muon, and tau lepton: 0.4329 MeV , 90.95 MeV , and 1543 MeV , respectively. These values constitute about 85% of the observed, re-normalized masses.

On peridynamic acoustics

We consider a nonlocal (peridynamic) version of the classical forced wave equation. This scalar three-dimensional equation contains a weight function (the “micromodulus”) and a length parameter (the “horizon”) that have to be selected. We investigate various properties (the locality limit as the horizon shrinks, plane waves and group velocity), paying attention to how these properties depend on the choice of the micromodulus. We solve the forced peridynamic equation in the static case (avoiding divergent integrals) and in the time-harmonic case (with a radiation condition, when needed).

Smartphone-based drug testing in the hands of patients with substance-use disorder-a usability study.

A clinical study was performed to test the usability of a smartphone eye-scanning app at a needle exchange facility to detect drug use to support therapy. The study recruited 24 subjects who visited the facility one to three times, making a total of 40 visits. During each visit the subjects underwent testing for non-convergence (NC), nystagmus (NY), and pupillary light reflex (PLR) using a smartphone-based eHealth system. The collected eye data were transformed into key features that represent eye characteristics. During each visit, a time-line follow-back interview on recent drug use and a usability questionnaire were completed. Technical usability of the smartphone eye-scanning app was good for PLR and NC, where key features were generated in 82%-91% of the cases. For NY, only 60% succeeded due to cognitive problems to follow instructions. In most cases, subjects were under the influence of drugs when participating in the tests, with an average of 2.4 different drugs ingested within the last 24 h. The key features from PLR could distinguish use of opioids from central stimulants. The usability questionnaire results indicate that 23 of the 24 subjects could perform the eye-scanning by themselves after a short training, even when under severe influence of drugs. The caregiver assessed that 20 out of the 24 challenging subjects could potentially perform these tests in an indoors, home-like environment. Smartphone-based eye-scanning is functional in a patient population with heavy drug use, also when under the influence of drugs. The use of central stimulants can be distinguished from the use of opioids.

On Finite Parts of Divergent Complex Geometric Integrals and Their Dependence on a Choice of Hermitian Metric

Let X be a reduced complex space of pure dimension. We consider divergent integrals of certain forms on X that are singular along a subvariety defined by the zero set of a holomorphic section of some holomorphic vector bundle E→X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E \\rightarrow X$$\\end{document}. Given a choice of Hermitian metric on E we define a finite part of the divergent integral. Our main result is an explicit formula for the dependence on the choice of metric of the finite part.

Divergent series and generalized mixed problem for wave equation

Divergent series and generalized mixed problem for wave equation

Statistical Convergence in A-Metric Spaces

In this paper, we will present the notion of statistical convergence in A-metric spaces, which is an important concept in summability theory. We will define statistical convergence in A-metric spaces and investigate its basic properties. Furthermore, we will explore the relationship between strongly p-Cesàro and statistical convergence in A-metric spaces.

Loops in de Sitter space

We discuss general one and two-loops banana diagrams with arbitrary masses on the de Sitter spacetime by using direct methods of dS quantum field theory in the dimensional regularization approach. In the one-loop case we also compute the effective potential for an O(N) model in d = 4 dimension as an explicit function of the cosmological constant Λ, both exactly and perturbatively up to order Λ. For the two-loop case we show that the calculation is made easy thanks to a remarkable Källén-Lehmann formula that has been in the literature for a while. We discuss the divergent cases at d = 3 using a contiguity formula for generalized hypergeometric functions and we extract the dominant term at d = 4 proving a general formula to deal with a divergent hypergeometric series.

Smoothed asymptotics: From number theory to QFT

Inspired by the method of smoothed asymptotics developed by Terence Tao, we introduce a new ultraviolet regularization scheme for loop integrals in quantum field theory which we call η regularization. This reveals a connection between the elimination of divergences in divergent series of powers and the preservation of gauge invariance in the regularization of loop integrals in quantum field theory. In particular, we note that a method for regularizing the series of natural numbers so that it converges to minus one-twelfth inspires a regularization scheme for non-Abelian gauge theories coupled to Dirac fermions that preserves the Ward identity for the vacuum polarization tensor and other higher point functions. We also comment on a possible connection to Schwinger proper time integrals. Published by the American Physical Society 2024

Hypergeometric Gevrey-0 approximation for the Gevrey-k divergent series with application to eight-loop renormalization group functions of the O(N)-symmetric field model

Mera et al. (Phys Rev Lett 115:143001, 2015) discovered that the hypergeometric function 2F1(a1,a2;b1;ωg)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${}_2F_1(a_1,a_2;b_1; \\omega g)$$\\end{document} can serve as an accurate approximant for a divergent Gevrey-1 type of series with an asymptotic large-order behavior of the form n!nbσn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n! n^b \\sigma ^n$$\\end{document}. What is strange about this approximant is that it has a series expansion with the wrong large-order behavior (Gevrey-0 type). In this work, we extend this discovery to Gevrey-k series where we show that the hypergeometric approximants and its extension to the generalized hypergeometric approximants are not only able to approximate divergent (Gevrey-1) series but also able to approximate strongly-divergent series of Gevrey-k type with k=2,3,…\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k=2,3,\\ldots$$\\end{document}. Moreover, we show that these hypergeometric approximants are able to predict accurate results for the non-perturbative strong-coupling and large-order parameters from weak-coupling data as input. Examples studied here are the ground-state energy for the xn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x^n$$\\end{document} anharmonic oscillators. The hypergeometric approximants are also used to approximate the recent eight-loop series ( g-expansion) of the renormalization group functions for the O(N)-symmetric ϕ4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi ^4$$\\end{document} scalar field model. Form these functions for N=0,1,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=0, 1, 2$$\\end{document}, and 3, critical exponents are extracted which are very competitive to results from more sophisticated approximation techniques.

Lacunary Invariant Statistical Convergence in Fuzzy Normed Spaces

In the study done here regarding the theory of summability, we introduce some new concepts in fuzzy normed spaces. First, at the beginning of the original part of our study, we define the lacunary invariant statistical convergence. Then, we examine some characteristic features like uniqueness, linearity of this new notion and give its important relation with pre-given concepts.

Fast and Reliable Computation of Instantaneous Orbital Collision Probability

Because of the increasing number of objects in low Earth orbit, the fast and reliable estimation of the collision risk is an important challenge for spacecraft owners/operators. Among the available risk indicators, this paper focuses on computing the instantaneous probability of collision, which can be modeled as the integral of a three-dimensional Gaussian probability density function over a Euclidean ball. The authors propose an efficient and accurate method for evaluating this integral. It is based on the combination of two complementary strategies. For the first one, convergent series and numerical error bounds are computed. These bounds provide a tradeoff between the accuracy needed and the number of terms to compute. The second one, using divergent series, approximates the value of the integral with a good accuracy in most cases with only a few terms computed. Based on those two methods, a hybrid algorithm is tested on cases borrowed from the literature and compared against existing methods. Several numerical results and comparisons confirm both the efficiency and robustness of this paper’s approach.

On Derkachov–Manashov R-matrices for the principal series of unitary representations

In 2001–2013 Derkachov and Manashov with coauthors obtained simple and natural expressions of R-matrices for the principal series of representations of the groups SL(2,C), SL(2,R), SL(n,C), SO(1, n). The Yang–Baxter identities for these intertwining operators are kinds of multivariate hypergeometric transformations. Derivations of the identities are based on calculations “of physical level of rigor” with divergent integrals. Our purpose is a formal mathematical justification of these results.

Measurement and analysis of regional water-energy-food nexus resilience with an improved hybrid kernel extreme learning machine model based on a dung beetle optimization algorithm

CONTEXTIn traditional water-energy-food nexus (WEFN) system analysis, the impact of resilience is insufficiently considered, and the accuracy of resilience assessment is low, which can easily lead to system imbalance. Modeling measures can yield optimal evaluation results and provide decision makers with a scientific and efficient basis for management. OBJECTIVEThe objective of this study was to provide a new research paradigm for exploring the sustainable development of WEFN systems by quantitatively and accurately assessing WEFN resilience and identifying its key driving factors. METHODSThe connotation of WEFN resilience was established, and a human-centered conceptual framework for WEFN resilience was constructed. A WEFN resilience evaluation system was scientifically and completely constructed using qualitative and quantitative index screening methods. The hybrid kernel extreme learning machine (HKELM) model was improved based on the dung beetle optimization (DBO) algorithm to measure and analyze the WEFN resilience of the China Beidahuang Group. The CRiteria Importance Through Intercriteria Correlation (CRITIC) method was adopted to calculate the evaluation index weight of the WEFN system of the Beidahuang Group. Model performance verification methods, such as application of the theory of serial number summation, were used to evaluate the performance of the DBO-HKELM model. RESULTS AND CONCLUSIONSThe results showed that over time, the Beidahuang Group's WEFN resilience exhibited a fluctuating upward trend. From 2006 to 2008, WEFN resilience declined to level II because of predatory agricultural production and management practices. From 2008 to 2018, the WEFN resilience increased to level IV due to the protection of uncultivated land, the optimization of irrigation methods, and the increasing maturity of the agricultural modernization development model. Likewise, from 2018 to 2020, the WEFN resilience stabilized at level IV because of the optimization of agricultural planting structures and the reduction in coal consumption. Spatially, the WEFN resilience of various branches of the Beidahuang Group exhibited a trend of “gradually decreasing from north to south.” Differences in certain variables, such as the proportion of area irrigated with electromechanical wells, were the most important factors related to changes in WEFN resilience over time. The DBO-HKELM model has notable advantages over the FA-HKELM and NGO-HKELM models in terms of fitting accuracy, optimization rate, reliability, rationality, and robustness. SIGNIFICANCEThese research results could enrich quantitative assessment methods for determining WEFN system recovery strength and provide scientific support for the safe operation and sustainable development of agricultural economies and societies.

Beyond the Factions: A Psychoanalytic Study of Dream and Trauma in Insurgent

This research builds on the psychoanalytic exploration of Veronica Roth's Divergent Series and focuses on a comparative analysis of the second novel Insurgent and its film adaptation directed by by Robert Schwentke. It looks into the relationship that develops between the characters' mental growth and the author's psychology throughout the book. The main objective is to analyze the psychology of fear as it affects the author and examining how it manifests subconsciously in the depiction of characters. This research includes a detailed study of dream study and its connection between the author and the characters, as well as a trauma study associated with the characters researched in this paper. A detailed qualitative study is conducted find the answer for the research question about whether the author unconsciously influenced the characters' fears examining the use of guilt and different methods of hypnosis to suppress a divergent-minded individual.

Some Results Concerning the Singular Solution of the Conservative Transport Equation in Spherical Geometry

We examine in this work one of the exact solutions of the conservative transport equation for isotropic scattering in spherical geometry, specifically the solution that is singular at the origin and vanishes at infinity. Two representations are known for that solution: one expressed as an infinite divergent series that is derived from the spherical harmonics method and another given by an integral that results from the technique of integration along the particle path and is confirmed here by the method of characteristics. We establish a connection between these representations by showing that the Borel sum of the first reproduces the latter. We also examine computational aspects of the solution expressed in various forms and discuss some standing issues related to it.

Relations between values of arithmetic Gevrey series, and applications to values of the Gamma function

We investigate the relations between the rings E, G and D of values taken at algebraic points by arithmetic Gevrey series of order either −1 (E-functions), 0 (analytic continuations of G-functions) or 1 (renormalization of divergent series solutions at ∞ of E-operators) respectively. We prove in particular that any element of G can be written as multivariate polynomial with algebraic coefficients in elements of E and D, and is the limit at infinity of some E-function along some direction. This prompts to defining and studying the notion of mixed functions, which generalizes simultaneously E-functions and arithmetic Gevrey series of order 1. Using natural conjectures for arithmetic Gevrey series of order 1 and mixed functions (which are analogues of a theorem of André and Beukers for E-functions) and the conjecture D∩E=Q‾ (but not necessarily all these conjectures at the same time), we deduce a number of interesting Diophantine results such as an analogue for mixed functions of Beukers' linear independence theorem for values of E-functions, the transcendence of the values of the Gamma function and its derivatives at all non-integral algebraic numbers, the transcendence of Gompertz constant as well as the fact that Euler's constant is not in E.

A Data Centric Approach Towards Server Time Series Model Compression

Effective model compression plays a pivotal role in mitigating the computational and interpretational challenges inherent in the domain of time series forecasting. In this study, we introduce an innovative data-centric methodology tailored to identify a representative data subset from the entirety of the dataset. This chosen representative segment forms the cornerstone for the training of proficient time series forecasting models. Furthermore, our investigation unveils a compelling outcome of this approach—a substantial reduction in the size of time series forecasting models when trained with this selected representative data segment. This model compression strategy results in a remarkable 56.31% decrease in storage consumption, a discovery of considerable significance for optimizing resources and enhancing scalability in time series forecasting. By distilling the dataset to its fundamental components through our data-centric approach, we aim to enhance both computational efficiency and the interpretability of the resultant models. This paper introduces a pioneering technique to tackle the challenges associated with data volume and model complexity in the field of time series forecasting, offering potential pathways for more efficient and insightful modelling in this domain.

The shape of an axisymmetric meniscus in a static liquid pool: effective implementation of the Euler transformation

Abstract We examine the classical problem of the height of a static liquid interface that forms on the outside of a solid vertical cylinder in an unbounded stagnant pool exposed to air. Gravitational and surface tension forces compete to affect the interface shape as characterized by the Bond number. Here, we provide a convergent power series solution for interface shapes that rise above or fall below the horizontal pool as a function of contact angle and Bond number. We find that the power series solution expressed in terms of the radial distance from the wall is divergent, and thus rewrite the divergent series as a new power series expressed as powers of an Euler transformed variable; this series is modified to match the large distance asymptotic behaviour of the meniscus. The Euler transformation maps non-physical singularities to locations that do not restrict series convergence in the physical domain, while the asymptotic modification increases the rate of convergence of the series overall. We demonstrate that when the divergent series coefficients are used to implement the Euler transformation, finite precision errors are incurred, even for a relatively small number of terms. To avoid such errors, the independent variable in the governing differential equation is changed to that of the Euler transform, and the power series is developed directly without using the divergent series. The resulting power series solution is validated by comparison with a numerical solution of the interface shape and the small Bond number matched asymptotic solution for the height of the interface along the cylinder developed by Lo (1983, J. Fluid Mech., 132, p.65-78). The convergent power series expansion has the ability to exceed the accuracy of the matched asymptotic solution for any Bond number given enough terms, and the recursive nature of the solution makes it straightforward to implement.