Calculating exit times for series Jackson networks

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.

This paper explores a novel application of the Laplace transform in cryptographic schemes by employing a linear combination of Bessel functions and exponential functions. By leveraging the transform’s properties, we develop an encryption-decryption framework that is computationally efficient and highly secure against certain types of cryptographic attacks. The exponential decay observed in Laplace-transformed domains and the complex behaviour of Bessel functions are powerful tools for data encoding and concealment. Analytical and numerical results validate the method’s effectiveness, showcasing its potential for modern cryptographic systems. Cryptography has long been a cornerstone of secure communication, with its methods evolving in tandem with technological advancements. While traditional schemes rely on number theory or algebraic structures, this paper explores an alternative approach that combines Laplace transforms with special functions. Specifically, we focus on employing Bessel functions in conjunction with exponential functions to construct an encryption algorithm. The Laplace transform’s unique properties, such as linearity, time-shifting, and frequency domain representation, offer a fertile ground for developing innovative cryptographic methods. Then the algorithm described in this study generates a dynamic cipher by processing plaintext using polynomial, exponential, and factorial-based operations. The four fundamental steps comprising the computational process are text preprocessing, dynamic coefficient calculation, encryption packaging, and reverse analysis. The computational cost of each step varies depending on the number of characters processed. The symbol for this character quantity is n. The cryptographic analysis of the problem is discussed in detail.

A novel class of special functions for electromagnetics is presented. Formed by the incomplete Hankel and modified Bessel functions, this class allows solving electromagnetics problems concerning truncated cylindrical structures. The differential and recurrence equations of these functions feature additional terms with respect to the classical theory of the Hankel and Bessel functions. The general properties, the most important analytical characteristics, and the large argument asymptotic approximations of the incomplete functions are derived using the steepest descent path (SDP) technique, showing that each special function splits into two terms. The first one has a discontinuous character and is linked to the saddle-point(s) contribution(s), while the second one, arising from the integral end-point contribution(s), compensates exactly the said discontinuity. In the solution of electromagnetic problems, the first term describes the geometrical optics (GO) field, the diffracted field being described by the second one. The general theory is employed to find the closed form analytical solution of the field radiated from a uniform line current source. Using the properties of the incomplete Hankel functions, it is demonstrated that this source excites cylindrical fields having optical character. Finally, the shape of the spatial regions where the GO solution cannot be applied is determined and discussed in details.

A class of identities relating Whittaker and Bessel functions

The theory of Bloch-electron dynamics for carriers in a homogeneous electric field of arbitrary time dependence is developed in consideration of the properties of multiphoton absorption (MPA) in semiconductors and insulators. The general approach is to utilize the accelerated Bloch-state representation (ABR) as a basis thereby treating the electric field exactly; also, the electric field is described in the vector potential gauge. In developing the ABR, the instantaneous eigenstates for the central Hamiltonian are obtained as the accelerated Bloch states and utilized to find the time-dependent solution to the Schr\"odinger equation. In introducing the Wigner-Weisskopf approximation (WWA), the transition probability amplitude between the states of the system is obtained and used in the application to MPA in semiconductors and insulators. The probability amplitude for MPA is derived for a plane-polarized radiation field. The periodicity of the time-dependent electric field serves as a principle time constant in developing the probability amplitude per period. Utilizing the WWA, the general MPA transition probability is derived for transitions between an arbitrary set of valence and conduction bands for a time-varying electric field in the $x$ direction. Within the WWA, the MPA transition probability, equivalent to the well-known Keldysh transition amplitude, is derived and expressed in terms of infinite-variable generalized Bessel functions and modified Bessel functions. The exact MPA result is analyzed in the small ($\ensuremath{\lesssim}{10}^{6}$ V/cm) and large ($\ensuremath{\gtrsim}{10}^{8}$ V/cm) electric field amplitude limits.

An expanded form of the third-order Markov undulation model is developed which includes undulation of the geoid, vertical deflections, gravity anomaly, and the anomalous gravity gradients. First, the complete set of autocorrelation and cross-correlation functions, as specified by the autocorrelation of undulation, is obtained. Then, a time-domain approach is used to synthesize the corresponding linear system driven by white noise that represents the anomalous gravity field behavior along a straight horizontal track. The resulting nine-state Markov model provides a shaping filter for applying covariance analysis and optimal estimation techniques to inertial navigation and gravity gradiometry systems analysis. ONTINUED mechanical refinement has brought inertial navigation technology to the stage where unmodeled variations in the Earth's gravity field have become significant and sometimes dominant error sources.1 At the same time, inertial navigation system analysts have been facing the task of developing error models to evaluate the impact of gravity uncertainties on navigation system performance. Ad- ditionally, the development of gravity gradiometers has made it necessary to model gravity-gradient signals to specify sensor requirements and analyze gradiometer-aided inertial navigation or gradiometry survey performance. To apply the powerful tools of linear system theory and Kalman filtering, it has become common to consider gravity uncertainties random variables, and to model them as Markov processes. Following the first treatment of vertical deflections as a first-order Markov process,2 a succession of higher order Markov models was proposed culminating with the third- order Markov undulation model3'4 which is fully self- consistent; that is, the auto- and cross-correlation functions of undulation, vertical deflections, and gravity anomaly obey and are derived according to the mathematical relationships of physical geodesy. The motivation for using Markov models for gravity un- certainties is that they can be cast in the form of linear dif- ferential equations driven by white noise. This allows use of the model as a shaping filter in covariance analysis of linear dynamic systems, Kalman filters, and optimal smoothing algorithms. The expanded third-order Markov undulation model described in this paper is derived from the original version, but it has three differences. First, it was found that the original model could easily be extended to include all the anomalous gravity gradients by continued differentiation of auto- and cross-correlation functions. Second, the exact cross-correlation function between undulation and gravity anomaly contains modified Bessel functions.4 Modified Bessel functions are solutions to a class of differential equations whose coefficients are functions of the independent variable. To obtain a Markov shaping filter, it is necessary to substitute an approximate function which is the solution to a linear differential equation with constant coefficients. While the chosen exponential form did not approximate the exact Bessel function as well as at first hoped possible, it provides a suitable representation of the un- dulation-gravity anomaly cross-correlation.

Motivated by recent results by Nadarajah, Srivastava and Gupta (2007) in which the authors introduced a new formula for CDF FK of the McKay Bessel function distribution containing the modified Bessel function we aim to present new formulae for FK, especially for certain values of parameters for which their formula fails to be calculated by in-built numerical routines. Also, several computational series formulae are obtained for the CDF of McKay Bessel distribution which contains modified Bessel function one is given in terms of the Horn confluent hypergeometric function and the others by Kampé de Fériet function Marcum Q-function and modified Bessel functions Horn confluent hypergeometric function and finally in terms of incomplete Gamma function γ and Exton’s double hypergeometric X–function.

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.

A theory of confluent hypergeometric functions is developed, based upon the methods described in the first three papers (I, II and III) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘ basic converging factors ’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of exponential-type integrals, parabolic cylinder functions, modified Bessel functions, and ordinary Bessel functions.

We consider two sequences $a(n)$ and $b(n)$ , $1\leq n<\infty $ , generated by Dirichlet series $$ \begin{align*}\sum_{n=1}^{\infty}\frac{a(n)}{\lambda_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{\mu_n^{s}},\end{align*} $$ satisfying a familiar functional equation involving the gamma function $\Gamma (s)$ . Two general identities are established. The first involves the modified Bessel function $K_{\mu }(z)$ , and can be thought of as a ‘modular’ or ‘theta’ relation wherein modified Bessel functions, instead of exponential functions, appear. Appearing in the second identity are $K_{\mu }(z)$ , the Bessel functions of imaginary argument $I_{\mu }(z)$ , and ordinary hypergeometric functions ${_2F_1}(a,b;c;z)$ . Although certain special cases appear in the literature, the general identities are new. The arithmetical functions appearing in the identities include Ramanujan’s arithmetical function $\tau (n)$ , the number of representations of n as a sum of k squares $r_k(n)$ , and primitive Dirichlet characters $\chi (n)$ .

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.'

The aim of this paper is to make a mathematical model for the temperature distribution in the human body. In this study, we consider the body as a cylinder and analyze data in the radial direction. The modified Bessel's differential equation and modified Bessel function are used to obtain the analytic solution of Pennes's bio-heat equation. We measure the effect of body's temperature distribution from various parameters such as environmental temperature, tissue thermal conductivities, metabolic rates, blood perfusion rates, heat transfer coefficients. The numerical results are used to plot graphs in different cases.

High performance analysis of lateral sloshing response in vertical cylinders with dual circular or arc-shaped porous structures

New index transforms are investigated, which contain as the kernel products of the Bessel and modified Bessel functions. Mapping properties and invertibility in Lebesgue spaces are studied for these operators. Relationships with the Kontorovich–Lebedev and Fourier cosine transforms are established. Inversion theorems are proved. As an application, a solution of the initial value problem for the fourth order partial differential equation, involving the Laplacian is presented.

We establish Turán-type inequalities for Bessel functions, modified Bessel functions, Kr ̈tzel function and Beta function, by using a new form of Cauchy–Bunyakovsky–Schwarz inequality.
 2010 Mathematics subject classification: Primary 26D07; Secondary 33C10.