Applying Laplace transforms to cryptography using Bessel and exponential functions and performing detailed time and memory analysis

We define a new family of special functions that we call lattice Bessel functions. They are indexed by the N-dimensional integer lattice such that they reduce to modified Bessel functions when N = 1, and the exponential function when N = 0. The transition probabilities for an M/M/1 queue going from one state to another before becoming idle (exiting at 0) can be solved in terms of modified Bessel functions. In this paper, we use lattice Bessel functions to solve the analogous problem involving the exit time from the interior of the positive orthant of the N-dimensional lattice for a series Jackson network with N nodes. These special functions allow us to derive asymptotic expansions for the taboo transition probabilities, as well as for the tail of the exit-time distribution.

The conference 'From DNA-Inspired Physics to Physics-Inspired Biology' (1–5 June 2009, International Center for Theoretical Physics, Trieste, Italy) that myself and two former presidents of the American Biophysical Society—Wilma Olson (Rutgers University) and Adrian Parsegian (NIH), with the support of an ICTP team (Ralf Gebauer (Local Organizer) and Doreen Sauleek (Conference Secretary)), have organized was intended to establish stronger links between the biology and physics communities on the DNA front. The relationships between them were never easy. In 1997, Adrian published a paper in Physics Today ('Harness the Hubris') summarizing his thoughts about the main obstacles for a successful collaboration. The bottom line of that article was that physicists must seriously learn biology before exploring it and even having an interpreter, a friend or co-worker, who will be cooperating with you and translating the problems of biology into a physical language, may not be enough. He started his story with a joke about a physicist asking a biologist: 'I want to study the brain. Tell me something about it!' Biologist: 'First, the brain consists of two parts, and..' Physicist: 'Stop. You have told me too much.'

An oscillating infinite series involving product of Bessel function J o(x) and an oscillating infinite series involving trigonometric function sin(x) were evaluated and computed numerically in [1] and [2] respectively. In this paper, an oscillating infinite series involving product of exponential, Bessel and trigonometric functions is evaluated. The series is transformed first into the sum of two infinite integrals by using contour integration and then the infinite integral with oscillating integrand is transformed through some identities into a finite integral containing modified Bessel function K 1(x). Finally, theset two integrals are evaluated numerically without any computational difficulties at all.

Given an entire function f ( z ) f(z) that has only negative zeros, we shall prove that all the functions of type f ( m ) ( x ) / f ( n ) ( x ) , m > n f^{(m)}(x)/f^{(n)}(x),\ m>n are completely monotonic. Examples of this type are given for Laguerre polynomials, ultraspherical polynomials, Jacobi polynomials, Stieltjes-Wigert polynomials, q q -Laguerre polynomials, Askey-Wilson polynomials, Ramanujan function, q q -exponential functions, q q -Bessel functions, Euler’s gamma function, Airy function, modified Bessel functions of the first and the second kind, and the confluent basic hypergeometric series.

The present work deals with the mathematical investigation of some generalizations of Bessel functions. The main motive of this paper is to show that the generating function can be employed efficiently to obtain certain results for special functions. The complex form of Bessel functions is introduced by means of the generating function. Certain enthralling properties for complex Bessel functions are investigated using the generating function method. By considering separately the real and the imaginary part of complex Bessel functions, we get respectively cosine-Bessel functions and sine-Bessel functions for which several novel identities and Jacobi–Anger expansions are established. Also, the generating function of degenerate Bessel functions is investigated and certain novel identities are obtained for them. A hybrid form of degenerate Bessel functions, namely, of degenerate Fubini–Bessel functions, is constructed using the replacement technique. Finally, the explicit forms of the real and the imaginary part of complex Bessel functions are established by a hypergeometric approach.

The Siegel-Shidlovskii method is one of the main methods in the theory of transcendental numbers. By using this method, it is possible to establish the transcendency and algebraic independence of the values of entire functions belonging to a certain class (so-called E-functions) provided that these functions are algebraically independent over C(z). There are many examples of E-functions used in mathematics: exp(z), sinz, cosz, sinhz, coshz, Bessel functions, Kummer functions, incomplete gamma-function, generalized hypergeometric functions, and some other special functions. By using the Siegel-Shidlovskii method, it is possible to obtain the lower estimates for the moduli of polynomials from the values of E-functions. These estimates are called algebraic independence measures of the numbers. They serve as the quantitative characteristics of algebraic independence. The problem of obtaining the estimates of algebraic independence measures has long been dealt with by many researchers. The first estimates of measures for the values of exp(z) were obtained by E. Borel and K. Mahler, and for the values of Bessel functions such estimates were obtained by C. Siegel. Estimates of algebraic independence measures for the values of general E-functions were established by S. Lang, A.I. Galochkin, A.B. Shidlovskii, and Yu.V. Nesterenko. The constant appearing in the estimate of a measure is called effective if it can be expressed in terms of the parameters characterizing the considered functions and points at which their values are taken. An estimate of a measure is considered to be effective if it contains only effective constants. Generally speaking, the algebraic independence condition of functions is not sufficient for obtaining effective estimates of algebraic independence measures. Some additional conditions must be satisfied for this. The case in which the main totality of functions is algebraically dependent over C(z) is the most complicated one. In a number of the author’s works, the estimates of algebraic independence measures of the values of E-functions were improved. After 1995, no papers have been published on this topic, although it still remains of issue.

: A table of Bessel functions whose order and argument are both purely imaginary has been prepared. These Bessel functions are solutions to the Schrodinger equation in which the potential energy is an exponential function. It is found that this function represents the repulsive potential of diatomic molecules quite accurately. (Author)

A common environment in which to place Bessel and circular functions is envisaged. We show, by the use of operational methods, that the Gaussian provides the umbral image of these functions. We emphasize the role of the spherical Bessel functions and a family of associated auxiliary polynomials, as transition elements between these families of functions. The consequences of this point of view and the relevant impact on the study of the properties of special functions is carefully discussed.

Most of the special functions of mathematical physics are connected with the representation of Lie groups. The action of elements D of the associated Lie algebras as linear differential operators gives relations among the functions in a class, for example, their differential recurrence relations. In this paper, we define fractional generalizations Dμ of these operators in the context of Lie theory, determine their formal properties, and illustrate their use in obtaining interesting relations among the functions. We restrict our attention here to the Euclidean group E(2) and the Bessel functions. We show that the two-variable fractional operator relations lead directly to integral representations for the Bessel functions, reproduce known fractional integrals for those functions when reduced to one variable, and contribute to a coherent understanding of the connection of many properties of the functions to the underlying group structure. We extend the analysis to the associated Legendre functions in a following paper.

The Fourier expansions of Eisenstein series for ${\text {GL}}(3,{\mathbf {Z}})$ are obtained by two methods—one analogous to the classical method used by many number theorists, including Weber, in his derivation of the Kronecker limit formula. The other method is analogous to that used by Siegel to obtain Fourier expansions of Eisenstein series for the Siegel modular group. The expansions involve matrix argument $K$-Bessel functions recently studied by Tom Bengtson. These $K$-Bessel functions are natural generalizations of the ordinary $K$-Bessel function which arise when considering harmonic analysis on the symmetric space of the general linear group using a certain system of coordinates.

In this paper, an analysis of free vibrations of shear-type buildings is presented and discussed. The differential equations for the free vibration of shear-type buildings with variably distributed mass and stiffness are established and reduced to Bessel's equations or differential equations with constant coefficients by selecting suitable expressions, such as power functions and exponential functions, for the distribution of shear stiffness and mass. An approach to determining the natural frequencies and mode shapes for shear-type buildings with variably distributed mass and stiffness is proposed. The derived general solutions are expressed in terms of Bessel functions and triangular functions. A numerical example shows that the selected expressions are suitable for describing the distributions of stiffness and mass of typical shear-type buildings and the methods proposed in this paper are convenient to engineering application.

The technological advancements in goods delivery facilities have been increasing year by year in tandem with the growing online trade, which necessitates delivery services to fulfill the transactional process between sellers and buyers. Since 2000, top brand awards have often conducted official survey analyses to provide comparisons of goods or services, one of which includes delivery services. However, the survey rankings based on public opinion are less accurate due to users of delivery services and service companies being unaware of the specific success factors and weaknesses in their services. The aim of this research is to analyze the comparison of text mining using the Indonesian language transformation method, IndoBert. The algorithm utilized to demonstrate analysis performance employs Convolutional Neural Network (CNN) and Bidirectional Long Short-Term Memory (Bi-LSTM). This method is utilized to determine the impact of opinion data from Twitter on the J&T Express expedition delivery service, incorporating both text preprocessing and data without text preprocessing. The IndoBert parameters vary in the learning rate section based on four factors: price, time, returns, and others. The research data consisted of 2525 comments from Twitter users regarding the delivery service spanning from January 1, 2021, to March 31, 2023. The testing showed that Bi-LSTM with text preprocessing performed 2% higher, achieving 79% at a learning rate of 1x10-6, compared to without text preprocessing at the same learning rate, which reached 77%. Additionally, CNN outperformed by 3% with a rate of 83%, compared to 80% without text preprocessing at a learning rate of 1x10-5. The highest accuracy, reaching 83%, was obtained by CNN with parameters set at 1x10-5, and the preprocessing technique was considered superior to Bi-LSTM.

Über ein Turánsches problem für [formula omitted] radiale, positiv definite Funktionen, II

Displacements and stresses due to a vertical point load in an inhomogeneous transversely isotropic half-space

6–7 - Definite Integrals of Special Functions