What Might “Understand a Function” Mean?

In this appendix we summarize the properties of Bessel functions, spherical Bessel functions and related functions. These functions are required in the solution of the radial Schrodinger equation and in the derivation of formulae for scattering amplitudes and cross sections. They are also important in the discussion of other topics in this monograph including multiphoton processes. For more complete presentations of their properties, reference should be made to specialized monographs on mathematical functions such as Abramowitz and Stegun [1], Watson [956] and Whittaker and Watson [964].

This is the first of a series of papers in which we use analyticity properties of quantum fields propagating on a spacetime to uncover a new multiverse geometry when the classical geometry has horizons and/or singularities. The nature and origin of the multiverse idea presented in this paper, that is shared by the fields in the standard model coupled to gravity, is different from other notions of a multiverse. Via analyticity we are able to establish definite relations among the universes. In this paper we illustrate these properties for the extended Rindler space, while black hole spacetime and the cosmological geometry of mini-superspace (see Appendix B) will appear in later papers. In classical general relativity, extended Rindler space is equivalent to flat Minkowski space; it consists of the union of the four wedges in (u,v) light-cone coordinates as in Fig.(1). In quantum mechanics, the wavefunction is an analytic function of (u,v) that is sensitive to branch points at the horizons u=0 or v=0, with branch cuts attached to them. The wavefunction is uniquely defined by analyticity on an infinite number of sheets in the cut analytic (u,v) spacetime. This structure is naturally interpreted as an infinite stack of identical Minkowski geometries, or universes, connected to each other by analyticity across branch cuts, such that each sheet represents a different Minkowski universe when (u,v) are analytically continued to the real axis on any sheet. We show in this paper that, in the absence of interactions, information doesn't flow from one Rindler sheet to another. By contrast, for an eternal black hole spacetime, which may be viewed as a modification of Rindler that includes gravitational interactions, analyticity shows how information is lost due to a flow to other universes, enabled by an additional branch point and cut due to the black hole singularity.

In the material characterization by a line-focus-beam acoustic microscope, objective of the measurement is the phase velocity and attenuation of leaky surface acoustic waves. This paper investigates how the analytical property, namely poles, zeros and branch cuts, of an acoustic reflection coefficient affects the measurement of the leaky Rayleigh wave and other modes excited on a specimen with layered structure. A pseudo-Sezawa wave is suitable for the accurate measurement, if fd(f: frequency, d: layer thickness) is large. On the other hand, a Rayleigh wave is recommended for the measurement of a specimen with a thin layer. We also find that the pole and zero reach to a close vicinity of the longitudinal branch point when the thickness approaches to zero, and this is responsible for a velocity difference between a bulk longitudinal wave and a leaky surface skimming compressional wave. A specimen with a lfastr layer on a lslowr substrate is also theoretically investigated

The periodic nature of the positive solutions of a nonlinear fuzzy max-difference equation

CHAPTER 17 - Bessel Functions

Dyson-Schwinger equations (DSEs) are a non-perturbative way to express n-point functions in quantum field theory. Working in Euclidean space and in Landau gauge, for example, one can study the quark propagator Dyson-Schwinger equation in the real and complex domain, given that a suitable and tractable truncation has been found. When aiming for solving these equations in the complex domain, that is, for complex external momenta, one has to deform the integration contour of the radial component in the complex plane of the loop momentum expressed in hyper-spherical coordinates. This has to be done in order to avoid poles and branch cuts in the integrand of the self-energy loop. Since the nature of Dyson-Schwinger equations is such, that they have to be solved in a self-consistent way, one cannot analyze the analytic properties of the integrand after every iteration step, as this would not be feasible. In these proceedings, we suggest a machine learning pipeline based on deep learning (DL) approaches to computer vision (CV), as well as deep reinforcement learning (DRL), that could solve this problem autonomously by detecting poles and branch cuts in the numerical integrand after every iteration step and by suggesting suitable integration contour deformations that avoid these obstructions. We sketch out a proof of principle for both of these tasks, that is, the pole and branch cut detection, as well as the contour deformation.

On the modified Taylor's approximation for the solution of linear and nonlinear equations

We study the asymptotic behaviour, when the parameter " tends to 0, of a class of singularly perturbed triangular systems x˙ = f(x, y), y˙ = G(y, "). We assume that all solutions of the second equation tend to zero arbitrarily fast when " tends to 0. We assume that the origin of equation x˙ = f(x, 0) is globally asymptotically stable. Some states of the second equation may peak to very large values, before they rapidly decay to zero. Such peaking states can destabilize the first equation. The paper introduces the concept of instantaneous stability, to measure the fast decay to zero of the solutions of the second equation, and the concept of uniform infinitesimal boundedness to measure the effects of peaking on the first equation. Whe show that all the solutions of the triangular system tend to zero when " ! 0 and t ! +1. Our results are formulated in both classical mathematics and nonstandard analysis. On étudie le comportement asymptotique, lorsque le paramètre " tend vers 0, de systèmes triangulaires singulièrement perturbés de la forme x˙ = f(x, y), y˙ = G(y, "). On suppose que toutes les solutions de la deuxième équation tendent vers zéro arbitrairement rapidement quand " tend vers 0. On suppose que le système x˙ = f(x, 0) admet l’origine comme équilibre globalement asymptotiquement stable. Certaines solutions de la deuxième équation peuvent présenter un transitoire avec un pic très grand avant de décroître rapidement vers zéro. C’est ce phénomène de peaking qui peut déstabiliser la première équation. On introduit le concept de stabilité instantanée, pour mesurer la décroissance rapide vers zéro des solutions de la deuxième équation, et le concept de système uniformément infinitésimalement borné pour mesurer les effets du peaking sur la première équation. On montre que les solutions du système triangulaire tendent vers zéro quand " ! 0 et t ! +1. Nos résultats sont formulés dans le cadre de l’analyse non standard et sont traduits en termes classiques.

Analytic solutions of the fundamental equation of the reference spectrum method for nuclear matter are obtained in the cases of the square well and exponential potentials with hard cores. In the former case analytic wavefunctions are obtained for every partial wave L. These wavefunctions are expressed in terms of spherical Bessel and Hankel functions or in terms of spherical Bessel, Hankel, and Newmann functions of order L. In the latter case an analytic expression is obtained for the s wavefunction only. This is given in terms of the transcendental functions G±v(r) and M±v(r) which are defined as series of well-known functions. Finally an application is made of our results to the calculation of the well depth of a Λ particle in its ground state in nuclear matter.

The functions of the cost curve and transmission losses are existing nonlinear, simultaneous subject to equality and inequality constraints. Mathematically, the solutions of the nonlinear equations of economic dispatch problem may result in multiple solutions, singal solution or no solution. Conventional methods may fail to find all the multiple solutions if the initial guess values are not appropriately located within the convergency region. To overcome this difficulty te homotopy method is suggested for the solution technique. In this paper the homotopy theorem is incorporated with the B-coefficient method to solve the economic dispatch problem. Not only the initial value is easily obtained but also all the multiple solutions can be obtained. So the next exact least cost of economic dispatch can be derived.

Methods of computer algebra become more familiar to a wide audience of theoretical mathematicians and physicists. The environment of computer algebra system leads to a greater acceptance of computer instruments in the mathematical research. Methods of symbolic manipulation provided by computer algebra systems in combination with high-power number crunching abilities of traditional hardware and software open the way to truly large scale computations often needed by mathematicians and physicists. This chapter describes new low complexity (both operational and logical) methods and algorithms for computations of solutions of differential equations and their efficient evaluation, and of solution of algebraic equations. These computations are often basic in applied problems, but even more so in a variety of problems of number theory and algebraic geometry. Power series manipulations over various rings and fields are one of the most important features of advanced computer algebra systems.

The partial-wave unitarity condition is complicated by the presence of overlapping left and right branch cuts when the lowest exchanged mass is small in comparison to the direct-channel mass. A coupled-channel ND−1 method for constructing unitary amplitudes with overlapping cuts is described. The study is motivated in part by the problem of analyzing the ππ−K̄system near the S* resonance.

The article is concerned with the study of asymptotic behavior of solutions of the Burgers equation and its generalizations with initial value — boundary problem on a finite interval, with constant boundary conditions. Since these equations take a dissipation into account, it is naturally to presuppose that any initial profile will evolve to an invariant time-independent solution with the same boundary values. Yet the answer happens to be slightly more complex. There are three possibilities: the initial profile may regularly decay to an invariant solution; or a Heaviside-type gap develops through a dispersive shock and multi-oscillations; or, exotically, an asymptotic limit is a 'frozen multi-oscillation' piecewise-differentiable solution, composed of different smooth invariant solutions.

Effective coefficient asymptotics of multivariate rational functions via semi-numerical algorithms for polynomial systems

The methods in the previous sections provide asymptotic approximations for functions defined by integrals. Many of these functions are solutions of specific differential equations: the Bessel function is an example. In the case of functions defined as the solutions of differential equations which cannot be solved explicitly, these integral methods described above naturally cannot be used. (We group in the class of explicit solutions those given in the form of an integral.) In these circumstances we must resort to differential equation methods to find asymptotic approximations for the functions. The subject of asymptotic methods for solving differential equations is large. In this and the following sections we give a brief introduction to the subject. In the case of ordinary differential equations the book by Wasow (1965) discusses several aspects with detail and rigour. A large part of each of the books by Erdelyi (1956) and Jeffreys (1966) is devoted to ordinary differential equations. In the partial differential equation area the books by Van Dyke (1964) and Cole (1968)† are of importance in the particular area of asymptotic analysis called singular perturbation theory.KeywordsAsymptotic ExpansionAsymptotic SolutionAsymptotic FormLiouville EquationDominant TermThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.