Infinite Sum Research Articles

Linear independence of special values of Jacobi-theta constants

Let m≥2 be any integer and let β>11$$\\end{document}]]> be a real algebraic integer such that all its other Galois conjugates have absolute value less than or equal to 1. Let a1,a2,…,am be distinct positive integers. In this article we prove that the following infinite sums 1,∑n=1∞1βa1n2,∑n=1∞1βa2n2,…,∑n=1∞1βamn2are Q(β)-linearly independent. As a consequence, we prove the linear independence of special values of Jacobi-theta constants.

On the relativistic quantum mechanics of a photon between two electrons in 1+1 dimensions

A Lorentz-covariant system of wave equations is formulated for a quantum-mechanical three-body system in one space dimension, comprised of one photon and two identical massive spin one-half Dirac particles, which can be thought of as two electrons (or alternatively, two positrons). Manifest covariance is achieved using Dirac’s formalism of multi-time wave functions, i.e., wave functions Ψ(xph,xe1,xe2) where xph,xe1,xe2 are generic spacetime events of the photon and two electrons, respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifolds {xph=xe1} and {xph=xe2} compatible with conservation of probability current. The corresponding initial-boundary value problem is shown to be well-posed, and it is shown that the unique solution can be represented by a convergent infinite sum of Feynman-like diagrams, each one corresponding to the photon bouncing between the two electrons a fixed number of times.

Identities for Whitehead products and infinite sums

Identities for Whitehead products and infinite sums

Why do newer degrees of freedom appear in higher-order truncated hydrodynamic theory?

An exact derivation of relativistic hydrodynamics from an underlying microscopic theory has been shown to be an all-order theory. From the relativistic transport equation of kinetic theory, the full expressions of hydrodynamic viscous fluxes have been derived which turn out to include all orders of out-of-equilibrium derivative corrections. It has been shown, that for maintaining causality, it is imperative that the temporal derivatives must include all orders, which can be resummed in non-local, relaxation operator-like forms and finally ‘integrated in’ introducing newer degrees of freedom. The theory can of course be truncated at any higher spatial orders, but the power over the the infinite temporal sum increases correspondingly such that the causality is respected. As a result, the theory truncated at any higher order of spatial gradient, requires newer degrees of freedom for each increasing order.

Range of effectiveness of hydraulic diameter model as an analytical solution for rectangular microchannels.

Spontaneous capillary flow in rectangular microfluidic channels is employed in microfluidic devices for various applications. The exact solution for flow in a rectangular cross-sectional channel has a complex point that contains an infinite sum term. The flow depends on the depth-width ratio of the rectangular channel's cross-section, ε. In previous studies, several approximations from exact solutions were useful for ε values smaller than 1 or 2. In this study, we propose a conversion equation that turns the hydraulic-diameter (HD) model into a rectangular-channel (RC) model. Experiments: Rectangular type flow experiments with ethanol solution were conducted to confirm the difference between the RC and HD models when the analysis was performed based on each model. Findings: depends on . At , the two models coincide. For , deviates by .

Another Look at Dependence: The Most Predictable Aspects of Time Series

Serial dependence and predictability are two sides of the same coin. The literature has considered alternative measures of these two fundamental concepts. In this article, we aim to distill the most predictable aspect of a univariate time series, that is, the one for which predictability is optimized. Our target measure is the mutual information between the past and future of a random process, a broad measure of predictability that takes into account all future forecast horizons, rather than focusing on the one-step-ahead prediction error mean square error. The most predictable aspect is defined as the measurable transformation of the series that maximizes the mutual information between past and future. This transformation arises from the linear combination of a set of basis functions localized at the quantiles of the unconditional distribution of the process. The mutual information is estimated as a function of the sample partial autocorrelations, using a semiparametric method that estimates an infinite sum by a regularized finite sum. The second most predictable aspect can also be defined, subject to suitable orthogonality restrictions. Finally, we illustrate the use of the most predictable aspect for testing the null hypothesis of no predictability and for point and interval prediction of the original time series.

Why Does the GW Approximation Give Accurate Quasiparticle Energies? The Cancellation of Vertex Corrections Quantified.

Hedin's GW approximation to the electronic self-energy has been impressively successful in calculating quasiparticle energies, such as ionization potentials, electron affinities, or electronic band structures. The success of this fairly simple approximation has been ascribed to the cancellation of the so-called vertex corrections that go beyond the GW approximation. This claim is mostly based on past calculations using vertex corrections within the crude local-density approximation. Here, we explore a wide variety of nonlocal vertex corrections in the polarizability and the self-energy, using first-order approximations or infinite summations to all orders. In particular, we use vertices based on statically screened interactions like in the Bethe-Salpeter equation. We demonstrate on realistic molecular systems that the two vertices in Hedin's equation essentially compensate. We further show that consistency between the two vertices is crucial for obtaining realistic electronic properties. We finally consider increasingly large clusters and extrapolate that our conclusions about the compensation of the two vertices would hold for extended systems.

An Ultrametric for Cartesian Differential Categories for Taylor Series Convergence

Cartesian differential categories provide a categorical framework for multivariable differential calculus and also the categorical semantics of the differential $\lambda$-calculus. Taylor series expansion is an important concept for both differential calculus and the differential $\lambda$-calculus. In differential calculus, a function is equal to its Taylor series if its sequence of Taylor polynomials converges to the function in the analytic sense. On the other hand, for the differential $\lambda$-calculus, one works in a setting with an appropriate notion of algebraic infinite sums to formalize Taylor series expansion. In this paper, we provide a formal theory of Taylor series in an arbitrary Cartesian differential category without the need for converging limits or infinite sums. We begin by developing the notion of Taylor polynomials of maps in a Cartesian differential category and then show how comparing Taylor polynomials of maps induces an ultrapseudometric on the homsets. We say that a Cartesian differential category is Taylor if maps are entirely determined by their Taylor polynomials. The main results of this paper are that in a Taylor Cartesian differential category, the induced ultrapseudometrics are ultrametrics and that for every map $f$, its Taylor series converges to $f$ with respect to this ultrametric. This framework recaptures both Taylor series expansion in differential calculus via analytic methods and in categorical models of the differential $\lambda$-calculus (or Differential Linear Logic) via infinite sums.Comment: To be published in the proceedings of MFPS2024

Modifications of Garfield’s Rectangle and Some Applications

Summary We modify Garfield’s rectangle to devise a simple method of generating arctangent identities. We use it to prove an arctangent identity by Robert Simson. We also explore other applications of this method, including an expression for the infinite sum of the arctangents of the inverse of the even indexed Fibonacci numbers in terms of the golden ratio.

Modeling thermal and buckling behavior in electrothermal bimorphs

Abstract Buckling is a structural phenomenon that can induce significant motion with minimal input variation. Electrothermal bimorphs, with their simple input and compact design, can leverage out-of-plane buckling motion for a broad range of applications. This paper presents the development of analytical electrothermal and structural models for such bimorphs. The electrothermal model calculates the temperature distribution within the bimorph caused by electrothermal heating, providing a 2D explicit analytical expression for estimating temperature along the bimorph’s length and cross-section. Nonhomogeneous heating leads to varying strains, which induce axial forces and moments along the bimorph’s neutral plane, varying with thermal expansion. The structural model derives the governing equation of deformation for the bimorph by analyzing internal strains and stresses resulting from deformation, electrothermal heating, and residual stresses. An analytical solution for deflection is obtained, incorporating infinite sums of heating and buckling modes, with closed-form equivalent expressions when possible. The bimorph’s behavior under different scenarios of residual stresses and electrothermal heating is elucidated based on the analytical model. Comparisons with Finite Element simulations demonstrated excellent agreement, highlighting the high accuracy of the proposed models.

고등학교 1학년 학생의 순환소수에 대한 인식과 도형의 무한합에 대한 이해

Objectives This study investigated high school first-year students' awareness of recurring decimals and examined their understanding of infinite sums related to geometric figures and various equivalent expressions of “the same”. Methods To achieve this, we developed survey questions to assess awareness of the recurring decimal 0.333..., and created dynamic materials and worksheets introducing infinite sums of geometric figures. Additionally, we developed worksheets focusing on various equivalent expressions of “the same”. Subsequently, we conducted a survey with 33 first-year students currently enrolled at H High School. After implementing the worksheets in class, we analyzed the results. Results Based on the analysis of survey responses regarding awareness of the recurring decimal 0.333..., students were categorized into three types: Type A showed actual infinite perspectives both algebraically and geometrically, Type B showed actual infinite algebraic perspectives but potential infinite perspectives geometrically, and Type C showed potential infinite perspectives in both algebraic and geometric aspects. Analyzing the responses to assignments on infinite sums of geometric figures based on these types, approximately half of Type A students demonstrated actual infinite perspectives on infinite sums. In contrast, the majority of Type B and C students showed potential infinite perspectives. Furthermore, regarding assignments on various expressions of equivalence based on Types A, B, and C, roughly half of Type A students understood newly presented expressions of equivalence, whereas the majority of Type B and C students did not comprehend these expressions well. Conclusions This research indicates that students' understanding of infinity, which they possess in middle school (2nd year), persists through their high school education. Based on these findings, there is a need for pedagogical efforts in mathematics education across elementary, middle, and high school levels to explicitly address and enhance learning about infinity, which seems implicitly guided by the curriculum currently.

Casimir energy of hyperbolic orbifolds with conical singularities

In this article, we obtain the explicit expression of the Casimir energy for compact hyperbolic orbifold surfaces in terms of the geometrical data of the surfaces with the help of zeta-regularization techniques. The orbifolds may have finitely many conical singularities. In computing the contribution to the energy from a conical singularity, we derive an expression of an elliptic orbital integral as an infinite sum of special functions. We prove that this sum converges exponentially fast. Additionally, we show that under a natural assumption known to hold asymptotically on the growth of the lengths of primitive closed geodesics of the (2, 3, 7)-triangle group orbifold, its Casimir energy is positive (repulsive).

Tunnelling amplitudes through localised external potentials from Feynman diagram summation

Currently there is no general theory of quantum tunnelling of a particle through a potential barrier which is compatible with QFT. We present a complete calculation of tunnelling amplitudes for a scalar field for some simple potentials using quantum field-theoretic methods. Using the perturbative S-matrix formalism, starting with the Klein–Gordon Lagrangian, we show that an infinite summation of Feynman diagrams can recover tunnelling amplitudes consistent with relativistic quantum mechanics. While this work does not include many-particle effects arising from a fully quantised QFT, it is necessary to investigate QFT corrections to tunnelling amplitudes.

Some identities on degenerate harmonic and degenerate higher-order harmonic numbers

The harmonic numbers and higher-order harmonic numbers appear frequently in several areas which are related to combinatorial identities, many expressions involving special functions in analytic number theory, and analysis of algorithms. The aim of this paper is to study the degenerate harmonic and degenerate higher-order harmonic numbers, which are respectively degenerate versions of the harmonic and higher-order harmonic numbers, in connection with the degenerate zeta and degenerate Hurwitz zeta function. Here the degenerate zeta and degenerate Hurwitz zeta function are respectively degenerate versions of the Riemann zeta and Hurwitz zeta function. We show that several infinite sums involving the degenerate higher-order harmonic numbers can be expressed in terms of the degenerate zeta function. Furthermore, we demonstrate that an infinite sum involving finite sums of products of the degenerate harmonic numbers can be represented by using the degenerate Hurwitz zeta function.

PowerPoint as a Tool to Support the Creation of Dynamic Models for Teaching: The Case of the Sum of an Infinite Geometric Sequence

PowerPoint is a learning aid, developed for presentations. PowerPoint has “dynamic” features for creating presentations with dynamic actions. The knowledge of the infinite backward multiplier is difficult to understand. One of the difficulties in understanding this knowledge is infinity, “dynamic”. In this study, we use the dynamic features of PowerPoint to design the sum of an infinite geometric sequence teaching. The study was conducted with 32 students in 11th grade. Students experimented with the dynamic model in PowerPoint. Research results show that the flexible use of PowerPoint’s dynamic features allows the creation of dynamic models of the sum of an infinite geometric sequence. Dynamic models have made it easier for students to build knowledge, creating geometric images of infinite sums. Moreover, with the dynamic features of PowerPoint, creating artistic images, attracts students to actively participate in learning.

Explicit evaluation of a family of Berndt-type integrals

In this paper, we compute a family of infinite integrals involving hyperbolic and trigonometric functions, which we will call Berndt-type integrals. By contour integration, these integrals are first converted to some hyperbolic (infinite) sums, all of which can be calculated in closed forms by two different methods. The first method is indirect and we need to apply some previous results which are computational in nature, and therefore it does not provide general formulas and is inexplicit. For the second method, by comparing the Fourier series expansion and Maclaurin series expansion of one of the Jacobi elliptic functions we are able to directly express the same type of hyperbolic sums more clearly and concisely. When a parameter in these series is set to π, these sums can be expressed as rational polynomials of Γ4(1/4) and 1/π which give rise to the closed formulas of the Berndt-type integrals we are interested in. Moreover, we obtain a structural theorem on the original Berndt integral.

Analysis Of The Asymmetry Factor In Photophoresis Problems With Uniform Polychromatic Fields And Spherical Absorbers

Photophoresis with plane waves and spherical absorbers is traditionally investigated by considering monochromatic radiation. The asymmetry factor, which is proportional to the photophoretic force, can be expressed in using the Mie theory in terms of an infinite sum of terms depending on the wavelength of the impinging light. Here, we analyze an extension of the photophoresis problem for a polychromatic uniform optical field. Such an analysis demands the introduction of a spectrum which describes the weight of each plane wave for the total, polychromatic field. We expect that this work can find applications or pave the way for more advanced investigations in diverse areas of physics and engineering such as optical levitation, optical trap displays and spectroscopy.

Black hole bulk-cone singularities

Lorentzian correlators of local operators exhibit surprising singularities in theories with gravity duals. These are associated with null geodesics in an emergent bulk geometry. We analyze singularities of the thermal response function dual to propagation of waves on the AdS Schwarzschild black hole background. We derive the analytic form of the leading singularity dual to a bulk geodesic that winds around the black hole. Remarkably, it exhibits a boundary group velocity larger than the speed of light, whose dual is the angular velocity of null geodesics at the photon sphere. The strength of the singularity is controlled by the classical Lyapunov exponent associated with the instability of nearly bound photon orbits. In this sense, the bulk-cone singularity can be identified as the universal feature that encodes the ubiquitous black hole photon sphere in a dual holographic CFT. To perform the computation analytically, we express the two-point correlator as an infinite sum over Regge poles, and then evaluate this sum using WKB methods. We also compute the smeared correlator numerically, which in particular allows us to check and support our analytic predictions. We comment on the resolution of black hole bulk-cone singularities by stringy and gravitational effects into black hole bulk-cone “bumps”. We conclude that these bumps are robust, and could serve as a target for simulations of black hole-like geometries in table-top experiments.

Exact results for giant graviton four-point correlators

We study the four-point correlator O2O2DD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left\\langle {\\mathcal{O}}_2{\\mathcal{O}}_2\\mathcal{DD}\\right\\rangle $$\\end{document} in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills theory (SYM) with SU(N) gauge group, where O2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{O}}_2 $$\\end{document} represents the superconformal primary operator with dimension two, while D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{D} $$\\end{document} denotes a determinant operator of dimension N, which is holographically dual to a giant graviton D3-brane extending along S5. We analyse the integrated correlator associated with this observable, obtained after integrating out the spacetime dependence over a supersymmetric invariant measure. Similarly to other classes of integrated correlators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM, this integrated correlator can be computed through supersymmetric localisation on the four-sphere. Employing matrix-model recursive techniques, we demonstrate that the integrated correlator can be reformulated as an infinite sum of protected three-point functions with known coefficients. This insight allows us to circumvent the complexity associated with the dimension-N determinant operator, significantly streamlining the large-N expansion of the integrated correlator. In the planar limit and beyond, we derive exact results for the integrated correlator valid for all values of the ’t Hooft coupling, and investigate the resurgent properties of their strong coupling expansion. Additionally, in the large-N expansion with fixed (complexified) Yang-Mills coupling, we deduce the SL(2, ℤ) completion of these results in terms of the non-holomorphic Eisenstein series. The proposed modular functions are confirmed by explicit instanton calculations from the matrix model, and agree with expectations from the holographic dual picture of known results in type IIB string theory.

Infinite sums of reciprocal quadratics

Infinite sums of reciprocal quadratics