Cross-product Of Bessel Functions Research Articles

Mathematical modelling of fungi-initiated siderophore-iron interactions.

Nearly all life forms require iron to survive and function. Microorganisms utilize a number of mechanisms to acquire iron including the production of siderophores, which are organic compounds that combine with ferric iron into forms that are easily absorbed by the microorganism. There has been significant experimental investigation into the role, distribution and function of siderophores in fungi but until now no predictive tools have been developed to qualify or quantify fungi-initiated siderophore-iron interactions. In this investigation, we construct the first mathematical models of siderophore function related to fungi. Initially, a set of partial differential equations are calibrated and integrated numerically to generate quantitative predictions on the spatio-temporal distributions of siderophores and related populations. This model is then reduced to a simpler set of equations that are solved algebraically giving rise to solutions that predict the distributions of siderophores and resultant compounds. These algebraic results require the calculation of zeros of cross products of Bessel functions and thus new algebraic expansions are derived for a variety of different cases that are in agreement with numerically computed values. The results of the modelling are consistent with experimental data while the analysis provides new quantitative predictions on the time scales involved between siderophore production and iron uptake along with how the total amount of iron acquired by the fungus depends on its environment. The implications to bio-technological applications are briefly discussed.

Cross-product of Bessel functions: Monotonicity patterns and functional inequalities

In this paper we study the Dini functions and the cross-product of Bessel functions. Moreover, we are interested on the monotonicity patterns for the cross-product of Bessel and modified Bessel functions. In addition, we deduce Redheffer-type inequalities, and the interlacing property of the zeros of Dini functions and the cross-product of Bessel and modified Bessel functions. Bounds for logarithmic derivatives of these functions are also derived. The key tools in our proofs are some recently developed infinite product representations for Dini functions and cross-product of Bessel functions.

Radii of Starlikeness and Convexity of a Product and Cross-Product of Bessel Functions

The reality of the zeros of the product and cross-product of Bessel and modified Bessel functions of the first kind is studied. As a consequence the reality of the zeros of two hypergeometric polynomials is obtained together with the number of the Fourier critical points of the normalized forms of the product and cross-product of Bessel functions. Moreover, the interlacing properties of the real zeros of these products of Bessel functions and their derivatives are also obtained. As an application some geometric properties of the normalized forms of the cross-product and product of Bessel and modified Bessel functions of the first kind are studied. For the cross-product and the product three different kind of normalization are investigated and for each of the six functions the radii of starlikeness and convexity are precisely determined by using their Hadamard factorization. For these radii of starlikeness and convexity tight lower and upper bounds are given via Euler-Rayleigh inequalities. Necessary and sufficient conditions are also given for the parameters such that the six normalized functions are starlike and convex in the open unit disk. The properties and the characterization of real entire functions from the Laguerre-P\'{o}lya class via hyperbolic polynomials play an important role in this paper. Some open problems are also stated, which may be of interest for further research.

Zeros of some special entire functions

The real and complex zeros of some special entire functions such as Wright, hyper-Bessel, and a special case of generalized hypergeometric functions are studied by using some classical results of Laguerre, Obreschkhoff, Pólya, and Runckel. The obtained results extend the known theorem of Hurwitz on the exact number of nonreal zeros of Bessel functions of the first kind. Moreover, results on zeros of derivatives of Bessel functions and the cross-product of Bessel functions are also given, which are related to some recent open problems.

The bends on a quantum waveguide and cross-products of Bessel functions

A detailed analysis of the wave-mode structure in a bend and its incorporation into a stable algorithm for calculation of the scattering matrix of the bend is presented. The calculations are based on the modal approach. The stability and precision of the algorithm is numerically and analytically analysed. The algorithm enables precise numerical calculations of scattering across the bend. The reflection is a purely quantum phenomenon and is discussed in more detail over a larger energy interval. The behaviour of the reflection is explained partially by a one-dimensional scattering model and heuristic calculations of the scattering matrix for narrow bends. In the same spirit, we explain the numerical results for the Wigner–Smith delay time in the bend.

Monotonicity of the zeros of a cross product of Bessel functions

Monotonicity of the zeros of a cross product of Bessel functions

Modal characteristics of three-layered optical fiber waveguides: a modified approach

By invoking Debye potentials, we formulate exact eigenvalue equations and the corresponding field distributions for general, three-layered, radially stratified, dielectric, and nonferromagnetic metal, optical fibers. By using cross products of Bessel functions, which may be regarded as the basic functional elements of the eigenvalue equations, a comparison is made between the properties of a three-layer structure and a simple step-index profile, and a simple graphical solution is obtained. The technique is applied to several practical structures, including two-layer fibers having a central index depression in the core, ring-core fibers, W fibers, and progressively stepped three-layered structures. The mathematical procedure is simple, and the results are of interest to optical fiber designers.

Concerning the Zeros of Some Functions Related to Bessel Functions

The real zeros of the Riccati-Bessel functions (12πx)12Jv(x), (12πx)12Yv(x), of their derivatives d[(12πx)12Jv(x)]/dx, d[(12πx)12Yv(x)]/dx,and of their cross products 12πx[Jv(x)Yv(Kx) − Yv(x)Jv(Kx)],ddx[(πx2)12Jv(x)]ddx[(πx2)12Yv(Kx)] − ddx[(πx2)12Yv(x)]ddx[(πx2)12Jv(Kx)]are investigated. Expansions analogous to those provided by McMahon and Olver for the zeros of the Bessel functions are obtained for the zeros of the derivatives of the Riccati-Bessel functions. The analysis of Kalähne for the zeros of the cross product of Bessel functions is considerably expanded and analogous results are obtained for the zeros of the cross product of the derivatives of the Riccati-Bessel functions. Included are derivations of the expansions for large zeros at fixed v, of asymptotic expansions for large v at fixed number of the zero, and also asymptotic expansions for the zeros as K→1 and K→∞. Figures illustrating the behavior of the zeros are provided for v = l + ½, where l is an integer. These zeros correspond to the TE and TM electromagnetic normal modes inside a conducting spherical shell and in the region between two concentric conducting shells.

An Extended Table of Zeros of Cross Products of Bessel Functions

Abstract : The report contains tables of the first five roots of the following transcendental equations: (a) J0(alpha) Y0(alpha) = Y0(alpha) J0(alpha); (b) J1(alpha) Y1(alpha) = Y1(alpha) J1(alpha); (c) J0(alpha) Y1(alpha) = Y0(alpha) J1(alpha) where J0(alpha), Y0(alpha), J1(alpha), Y1(alpha) are Bessel functions of order 0 and 1 respectively. In these equations, alpha is the unknown and k is a parameter which may assume any positive value, other than 0 or 1. Additional tables are included listing an auxiliary quantity gamma which is better suited to interpolation particularly when k is close to unity. (Author)