Convergent Series Research Articles

Two-sided estimates of Lebesgue constants for spaces of functions of generalized bounded variation and Hölder spaces

A proof of two-sided estimates of Lebesgue constants for spaces Hω⋂BV(X), where Hω is the Hölder space and BV(X) is the space of functions of bounded variation constructed from the symmetric space of sequences X. Choosing different spaces of sequences as X, we obtained new criteria for the convergence of classical Fourier series.

Generalized Approach to Optimal Polylinearization for Smart Sensors and Internet of Things Devices

This study introduces an innovative numerical approach for polylinear approximation (polylinearization) of non-self-intersecting compact sensor characteristics (transfer functions) specified either pointwise or analytically. The goal is to partition the sensor characteristic optimally, i.e., to select the vertices of the approximating polyline (approximant) along with their positions, on the sensor characteristics so that the distance (i.e., the separation) between the approximant and the characteristic is rendered below a certain problem-specific tolerance. To achieve this goal, two alternative nonlinear optimization problems are solved, which differ in the adopted quantitative measure of the separation between the transfer function and the approximant. In the first problem, which relates to absolutely integrable sensor characteristics (their energy is not necessarily finite, but they can be represented in terms of convergent Fourier series), the polylinearization is constructed by the numerical minimization of the L1-metric (a distance-based separation measure), concerning the number of polyline vertices and their locations. In the second problem, which covers the quadratically integrable sensor characteristics (whose energy is finite, but they do not necessarily admit a representation in terms of convergent Fourier series), the polylinearization is constructed by numerically minimizing the L2-metric (area- or energy-based separation measure) for the same set of optimization variables—the locations and the number of polyline vertices.

Comparison of Approximate Analytical and Numerical Solutions of the Allen Cahn Equation

Allen Cahn (AC) equation is highly nonlinear due to the presence of cubic term and also very stiff; therefore, it is not easy to find its exact analytical solution in the closed form. In the present work, an approximate analytical solution of the AC equation has been investigated. Here, we used the variational iteration method (VIM) to find approximate analytical solution for AC equation. The obtained results are compared with the hyperbolic function solution and traveling wave solution. Results are also compared with the numerical solution obtained by using the finite difference method (FDM). Absolute error analysis tables are used to validate the series solution. A convergent series solution obtained by VIM is found to be in a good agreement with the analytical and numerical solutions.

A neural network approach to explore bioelectromagnetics aspects of blood circulation conveying tetra-hybrid nanoparticles and microbes in a ciliary artery with an endoscopy span

Bioelectromagnetics is gaining remarkableness due to its wide-ranging applications in medical and biochemical industries. Our proposed model delves into the bioelectromagnetics aspects of blood circulation in a scenario involving tetra-hybrid nanoparticles and gyrotactic microbes within a ciliated arterial annulus, particularly one mimicking an endoscopic environment. Our model takes into account influential factors, including cilia beating, buoyancy force, heat source, viscous and Joule warming, arterial wall properties, chemical reaction, and bioconvection. Different geometric shapes of suspended tetra-hybrid nanoparticles (spheres, bricks, cylinders, and platelets) are incorporated into the model formation. The close-form solution of the Poisson equation is found in terms of Bessel’s functions. The homotopy perturbation methodology (HPM) is implemented to track down the rapidly convergent series solutions of the highly non-linear coupled flow equations arising. The hemodynamical attributes of distinct evolving factors on the different dimensionless hemodynamic profiles and entities of interest are elucidated evocatively via a sort of graphs and charts. Our results prove that the factors related to electroosmosis have a dual effect on the blood flow pattern. The bloodstream rapidity is alleviated or boosted for the assisting or opposing electroosmosis process. Cooling of blood within the annulus is achieved with a larger cilia length. The Peclet number positively correlates with the gyrotactic microbes’ density. Blood infused with tetra-hybrid nanoparticles exhibits a superior heat transmission rate across the arterial wall compared to others. Additionally, an artificial neural network (ANN) model, developed using HPM-derived data, accurately predicts shearing stress (SS), achieving high accuracy rates of 99.98% in testing and 100% in validation for MB flow. The model’s new findings could be noteworthy for engineers and medical device designers to develop innovative medical devices and technologies based on bioelectromagnetic principles.

The force-force correlator at the hard thermal scale of hot QCD

High-energy particles traversing the Quark-Gluon plasma experience modified (massive) dispersion, although their vacuum mass is negligible compared to the kinetic energy. Due to poor convergence of the perturbative series in the regime of soft loop momenta, a more precise determination of this effective mass is needed. This paper continues our investigation on the factorisation between strongly-coupled infrared classical and perturbative ultraviolet behavior. The former has been studied non-perturbatively within EQCD by determining a non-local operator on the lattice. By computing the temperature-scale contribution to the same operator in 4D QCD at next-to-leading order (NLO), we remove the ultraviolet divergence of the EQCD calculation with an opposite infrared divergence from the hard thermal scale. The result is a consistent, regulator-independent determination of the classical contribution where the emergence of new divergences signals sensitivities to new regions of phase space. We address the numerical impact of the classical and NLO thermal corrections on the convergence of the factorised approach and on the partial applicability of our results to calculations of transport coefficients.

On the convergence of series of fractional Hasse derivative bases in Fréchet spaces

This paper introduces a generalization of the Hasse derivative in the sense of the complex conformable derivative. The definition coincides with the classical version of the Hasse derivative of order . Accordingly, a new base, named complex conformable Hasse derivative bases (CCHDBs), is defined. We investigate the existence of expansions of analytic functions in a series of CCHDBs in Fréchet space on closed and open disks, open regions surrounding closed disks, for all entire functions and at the origin. Moreover, an upper bound for the order and type of the CCHDBs is obtained and proved to be attainable. The ‐property of CCHDBs is also discussed. Our results improve and extend the analog results in the complex analysis related to the classical complex derivative of a base of polynomials (BPs). The obtained results clarify several implications for the CCHDBs of special functions such as Euler, Bernoulli, Chebyshev, Bessel, and Gontcharaff polynomials.

Cauchy's Criterion for Non-negative Hyperbolic Series

Hyperbolic number plane constitutes a significant area of algebraic research, with their properties finding wide-ranging applications across various fields. Meanwhile, Cauchy's criterion, as an effective tool for determining series convergence, has also been extensively utilized in the study of hyperbolic numbers within mathematical analysis and practical problem-solving, offering valuable insights and practical applications. The paper conducted an in-depth exploration of the non-negative hyperbolic plane, applying the Comparison test and further investigating the Cauchy's criterion on the non-negative hyperbolic plane. Through the application of the comparison test, the paper have arrived at an important conclusion: when considering two numbers, and , in the non-negative hyperbolic complex plane, and , the convergence of implies the convergence of , and divergence of implies the divergence of . Furthermore, through a detailed analysis of Cauchy’s criterion, the paper also discovered a key finding: when a certain number r in the non-negative hyperbolic plane satisfies , if , the corresponding infinite series converges, whereas if , the series diverges. The paper not only pushes the frontiers of mathematical theory but also unlocks fresh avenues for interdisciplinary exploration. By delving into uncharted territories, it expands our understanding of mathematical principles and their applicability across diverse fields. Moreover, it fortifies mathematics' central role in addressing real-world challenges, providing essential tools and insights. As a cornerstone for future research, this work paves the way for innovative mathematical investigations and their practical applications, fostering a deeper symbiosis between theory and the real world.

Study of multi-dimensional problems arising in wave propagation using a hybrid scheme

Many scientific phenomena are linked to wave problems. This paper presents an effective and suitable technique for generating approximation solutions to multi-dimensional problems associated with wave propagation. We adopt a new iterative strategy to reduce the numerical work with minimum time efficiency compared to existing techniques such as the variational iteration method (VIM) and homotopy analysis method (HAM) have some limitations and constraints within the development of recurrence relation. To overcome this drawback, we present a Sawi integral transform (S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {S}$$\\end{document}T) for constructing a suitable recurrence relation. This recurrence relation is solved to determine the coefficients of the homotopy perturbation strategy (HPS) that leads to the convergence series of the precise solution. This strategy derives the results in algebraic form that are independent of any discretization. To demonstrate the performance of this scheme, several mathematical frameworks and visual depictions are shown.

Olsson.wl & ROC2.wl: Mathematica packages for transformations of multivariable hypergeometric functions & regions of convergence for their series representations in the two variables case

We present the Olsson.wlMathematica package which aims to find transformations for some classes of multivariable hypergeometric functions. It is based on a well-known method developed by P. O. M. Olsson [1] (1964) to derive the analytic continuations of the Appell F1 double hypergeometric series using the linear transformations of the Gauss F12 hypergeometric function. We provide a brief description of the method of Olsson and demonstrate the use of the commands of the Olsson.wl package using some examples that are presented in the text and in some ancillary Mathematica notebooks. In particular, we reproduce various results of the literature on multivariable hypergeometric functions and show practical applications of this package in the derivation of novel formulas. In the context of high energy physics, we also demonstrate how it can be used to disentangle some known results about the analytic continuation of some series representations of the one-loop pentagon in multi-Regge kinematics and D=6−2ϵ. We also provide a companion package, called ROC2.wl, which is dedicated to the derivation of the regions of convergence of double hypergeometric series. This package can be used independently of Olsson.wl. Program summaryProgram Title:Olsson.wlCPC Library link to program files:https://doi.org/10.17632/gc63xzwzz5.1Licensing provisions: GNU General Public License v3.0.Programming language: Wolfram Mathematica version 11.3 and beyond.Nature of problem: To find the transformation formulas of multivariable hypergeometric series appearing in Feynman integral calculus.Solution method: Mathematica implementation of the method of Olsson [1]. The method uses the transformation theory of lower variable hypergeometric functions to find the transformation formulas of higher variable hypergeometric functions.Companion package:ROC2.wl, which is a Mathematica package used inside Olsson.wl and can also be used as a standalone package. The package is used to find the region of convergence of double hypergeometric functions, using Horn's theorem.

Unconditional Convergence of Sub-Gaussian Random Series

Unconditional Convergence of Sub-Gaussian Random Series

Exact and explicit analytical solution for the Sakiadis boundary layer

It has recently been established [Naghshineh et al. IMA J. of Appl. Math., 88, 1 (2023)] that a convergent series solution may be obtained for the Sakiadis boundary layer problem once key parameters are determined iteratively using the series itself. Here, we provide exact and explicit analytical expressions for these parameters, including that associated with wall shear, thus completing the exact analytical solution. The resulting solution to the Sakiadis problem is summarized here for direct use.

Mathematical definition of the fine-structure constant: A clue for fundamental couplings in astrophysics

Astrophysical tests of the stability—or not—of fundamental couplings (e.g., can the numerical value ∼1/137 of the fine-structure constant α = e2/ℏc vary with astronomical time?) are a very active area of observational research. Using a specific α-free non-relativistic and nonlinear isotropic quantum model compatible with its quantum electrodynamics (QED) counterpart yields the 99% accurate solution α = 7.364 × 10−3 vs its experimental value 7.297 × 10−3. The ∼1% error is due to the deliberate use of mean-field Hartree approximation involving lowest-order QED in the calculations. The present theory has been checked by changing the geometry of the model. Moreover, it fits the mathematical solution of the original nonlinear integro-differential Hartree system by use of a rapidly convergent series of nonlinear eigenstates [G. Reinisch, Phys. Lett. A 498, 129347 (2024)]. These results strongly suggest the mathematical transcendental nature—e.g., like for π or e—of α’s numerical value of ∼1/137 and, hence, its astrophysical as well as its cosmological stability.

Q-homotopy analysis method for time-fractional Newell–Whitehead equation and time-fractional generalized Hirota–Satsuma coupled KdV system

In this paper, two types of fractional nonlinear equations in Caputo sense, time-fractional Newell–Whitehead equation (FNWE) and time-fractional generalized Hirota–Satsuma coupled KdV system (HS-cKdVS), are investigated by means of the q-homotopy analysis method (q-HAM). The approximate solutions of the proposed equations are constructed in the form of a convergent series and are compared with the corresponding exact solutions. Due to the presence of the auxiliary parameter h in this method, just a few terms of the series solution are required in order to obtain better approximation. For the sake of visualization, the numerical results obtained in this paper are graphically displayed with the help of Maple.

Approximate‐analytical iterative approach to time‐fractional Bloch equation with Mittag–Leffler type kernel

The paper aims to analyze the fractional Bloch equation with Caputo and Atangana–Baleanu–Caputo (ABC) derivative having a nonsingular kernel. The fixed point theorem is used to prove the existence and uniqueness of the proposed equation. Furthermore, the approximate‐analytical solution of the proposed equation is obtained by Aboodh transform iterative method (ATIM) in the form of a convergent series. The technique (ATIM) combines the new iterative method with the Aboodh transform. The convergence analysis of the approximate solution is discussed using the Cauchy convergence theorem. The validity of the ATIM is shown by numerical simulation and graphs. Also, the preference for nonsingular kernel over singular kernel is shown with the help of tables and graphs. For integer order, the obtained solutions from Caputo and ABC derivative are compared with the exact solutions and published work.

Turbulence ingestion noise generation in rotating blades

The interaction between ingested turbulence and rotating blades is a key source of broadband noise in engineering applications. In this paper, a far-field noise model accounting for source correlation across the span of the blade and between blades is developed and applied to the study of homogeneous isotropic turbulence ingestion by a model cooling fan, wind turbine and aircraft propeller. The widely used theory of Amiet is revisited and it is shown that previous works produce conflicting results when attempting to account for blade-to-blade correlation. Central to the model is the calculation of the time between blade chops of the same turbulent eddy as heard by the observer. In this paper it is shown that, when derived correctly, Amiet's theory accounts for correlated sources between blades and, thus, can predict haystacking tones. Comparisons with the new rotational formulation and with experimental data enable us to show that Amiet's theory can be used to accurately predict turbulence ingestion noise from open rotors. In particular, it is found that the infinite-span assumption in strip theory and the neglect of correlation effects across the span do not undermine the accuracy of this theory. This is of great importance because, unlike Amiet's theory, models which treat rotational effects and source correlation exactly are expensive to evaluate routinely at high frequencies due to the slow convergence of infinite series with Bessel functions.

Neutrino amplitude decomposition, S matrix rephasing invariance, and reparametrization symmetry

The S matrix rephasing invariance is one of the fundamental principles of quantum mechanics that originates in its probabilistic interpretation. For a given S matrix which describes neutrino oscillation, one can define the two different rephased amplitudes SαβReph−1≡eiλ1/2ExSαβ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {S}_{\\alpha \\beta}^{\ extrm{Reph}-1}\\equiv {e}^{i\\left({\\lambda}_1/2E\\right)x}{S}_{\\alpha \\beta} $$\\end{document} and SαβReph−2≡eiλ2/2ExSαβ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {S}_{\\alpha \\beta}^{\ extrm{Reph}-2}\\equiv {e}^{i\\left({\\lambda}_2/2E\\right)x}{S}_{\\alpha \\beta} $$\\end{document}, which are physically equivalent to each other, where λk/2E denotes the energy eigenvalue of the k-th mass eigenstate. We point out that the transformation of the reparametrization (Rep) symmetry obtained with “Symmetry Finder” maps SαβReph−1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {S}_{\\alpha \\beta}^{\ extrm{Reph}-1} $$\\end{document} to SαβReph−2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {S}_{\\alpha \\beta}^{\ extrm{Reph}-2} $$\\end{document}, and vice versa, providing a local and manifest realization of the S matrix rephasing invariance by the Rep symmetry of the 1–2 state exchange type. It is strongly indicative of quantum mechanical nature of the Rep symmetry. The rephasing and Rep symmetry relation, though its all-order treatment remains incomplete, is shown to imply absence of the pure 1–3 exchange symmetry in Denton et al. perturbation theory. It then triggers a study of convergence of perturbation series.

A New Method for Solving a Neutral Functional-Differential Equation with Proportional Delays

This study presents and implements a new hybrid technique that combines the Sawi transform (ST) and Homotopy perturbation method (HPM) to solve neutral functional-differential equations with proportional delays. Some of the important properties of the method are established and validated. We start the method by first applying ST to obtain the recurrence relation. We, next, implement HPM to find convergent series solutions of the recurrence relation. The series is free of assumptions and restrictions, highlighting its adaptability and robustness. Moreover, the convergence of the method is established through convincing proof. To demonstrate its effectiveness and applicability, we provide five examples. The method yields accurate approximate solutions, or in some cases exact solutions, with a few number of iterations, reinforcing its reliability and validity. Moreover, the performance of the method is compared with some available methods and demonstrates its superiority and efficiency.

Extending the Scope of Conceptual Density Functional Theory with Second Order Analytical Methodologies.

In the context of the growing impact of conceptual density functional theory (DFT) as one of the most successful chemical reactivity theories, response functions up to second order have now been widely applied; in recent years, among others, particular attention has been focused on the linear response function and also extensions to higher order have been put forward. As the larger part of these studies have been carried using a finite difference approach to compute these concepts, we now embarked on (an extension of) an analytical approach to conceptual DFT. With the ultimate aim of providing a complete set of analytically computable second order properties, including the softness and hardness kernels, the hardness as the simplest second order response function is scrutinized again with numerical results highlighting the difference in nature between the analytical hardness (referred to as hardness condition) and the Parr-Pearson absolute chemical hardness. The hardness condition is investigated for its capability to gauge the (de)localization error of density functional approximations (DFAs). The analytical Fukui function, besides overcoming the difficulties in the finite difference approach in treating negatively charged systems, also showcases the errors of deviating from the straight-line behavior using fractional occupation number calculations. Subsequently, the softness kernel and its atom-condensed inverse, the hardness matrix, are accessed through the Berkowitz-Parr relation. Revisiting the softness kernel confirms and extends previous discussions on how Kohn's Nearsightedness of Electronic Matter principle can be retrieved and identified as the physicist's version of the chemist's "transferability of functional groups" concept. The accurate, analytical hardness matrix evaluation on the other hand provides further support for the basics of Nalewajski's charge sensitivity analysis. Based on Parr and Liu's functional expansion of the energy functional, a new energy decomposition is introduced with an order of magnitude analysis of the different terms for a series of simple molecules both at their equilibrium geometry and upon variation in bond length and dihedral angle. Finally, for the first time, the perturbation expansion of the energy functional is studied numerically up to second order now that all response functions and integration techniques are at hand. The perturbation expansion energies are in excellent agreement with those obtained directly from DFA calculations giving confidence in the convergence of the perturbation series and its use in judging the importance of the different terms in reactivity investigations.

Block-Supersymmetric Polynomials on Spaces of Absolutely Convergent Series

In this paper, we consider a supersymmetric version of block-symmetric polynomials on a Banach space of two-sided absolutely summing series of vectors in Cs for some positive integer s>1. We describe some sequences of generators of the algebra of block-supersymmetric polynomials and algebraic relations between the generators for the finite-dimensional case and construct algebraic bases of block-supersymmetric polynomials in the infinite-dimensional case. Furthermore, we propose some consequences for algebras of block-supersymmetric analytic functions of bounded type and their spectra. Finally, we consider some special derivatives in algebras of block-symmetric and block-supersymmetric analytic functions and find related Appell-type sequences of polynomials.

Rapidly convergent series and closed-form expressions for a class of integrals involving products of spherical Bessel functions

Rapidly convergent series are derived to efficiently evaluate a class of integrals involving the product of spherical Bessel functions of the first kind occurring in acoustic and electromagnetic scattering from circular disks and apertures. Depending on the involved parameters, the series can be further reduced to closed-form expressions in terms of generalized hypergeometric functions or Meijer G-functions, which can immediately be evaluated through current mathematical toolboxes. Numerical results are provided showing that the accuracy of the series representation can easily be controlled and that the proposed solutions are at least 1000 times faster than specific quadrature schemes which, in general, have to deal with irregularly oscillating and slowly decaying functions.