Several characterizations of Bessel functions and their applications

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.

The problems of stability and controllability of the finite deformation are investigated for a pre-stressed hyperelastic cylindrical tube subjected to the axial compression. The tube is assumed to be composed of an incompressible neo-Hookean material. Based on the theory of small deformation superposed on large elastic deformation, the mathematical model is formulated by a system of incremental equilibrium equations and incremental boundary conditions. Moreover, the general solutions describing the finite deformation of the tube are obtained by the form of Bessel functions. Finally, the system of nonlinear equations governing the stability of the tube are given, the criteria of the stability discussed in terms of the corresponding numerical simulations.

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.

In this paper an advanced anti-jam indoor adaptive GNSS signal acquisition and tracking algorithm is considered. Initially, we were able to determine that double-dwell structure (DDS) reduces the processing time penalty caused by false alarm. Nevertheless, the double-dwell structure is still vulnerable to interference and jamming. In order to determine a suitable advanced anti-jam indoor adaptive GNSS signal acquisition and tracking algorithm for DDS we first perform the Bayesian parameter estimation; i.e., we analytically compute the posterior Bayes probability density function (pdf) and cumulative distribution function (cdf) by applying the Bayes theorem in three steps. First, we compute the complex signal distribution and complex matrix variate signal distribution. This is an original new result never published before. Second, we provide an introduction of the complex Bessel or parabolic function interference distribution due to earlier assumptions of the DDS interference distribution models. We further explain that the process that is required to produce Complex Matrix Variate Bessel Interference Distribution or Complex Matrix Variate Parabolic Function Interference Distribution is an extremely daunting task let alone the Complex Matrix Variate DDS Bayesian posterior density it would seem at the moment nearly an impossible task because it may require the computation of functions such as Kampe de Feriet function and Jack functions of matrix arguments. Third, instead of performing the second step, we relax the assumption of interference to normal distribution. In both the scalar case and complex matrix variate cases we observe that the complex matrix variate Bayesian posterior pdf or cdf is invariant of the observation data or is identical to the prior complex matrix variate signal distribution model. This is an original new and very powerful result never published before. Why this result is so powerful is because up until now we never had a complete theoretical validation of our GNSS receiver design based on either autocorrelation or cross-correlation properties since, the complex matrix variate Bayesian posterior pdf or cdf is invariant of the observation data or is identical to the prior complex matrix variate signal distribution model. Future simulation results will illustrate that an anti-jam indoor adaptive DDS stable detection structure reduces by half the average acquisition time and significantly outperform its predecessor against interference and jamming. Moreover, future simulation results will show further advancements of the Giftet Inc. MATLAB library capability to perform advanced numerical computations based on closed form expressions of the generalized Bessel function distributions and generalized parabolic cylinder function distribution models via Kampe de Feriet function and Jack functions.

This paper presents recent data on differential cross sections for H− formation from collisions of H+ and H0 with Ar in the energy range 1.0 to 4 keV. Experimental data exhibit a sharp maximum at 0° scattering angle as well as an oscillatory structure. The functional form and scaling properties strongly indicate that there is a glory maximum which occurs when the classical deflection function changes over from attractive to repulsive at some finite impact parameter. The oscillations are predicted by the Bessel function and may be said to arise from interference of the contributions from the two branches of the deflection function near to a glory.

In the present investigation, our main aim is to derive some con- ditions on starlikeness, convexity, and close-to-convexity of normalized hyper- Bessel functions. Also we give some similar results for classical Bessel functions by using the relationships between hyper-Bessel and Bessel functions. As a re- sult of the obtained conditions, some examples are also given.

Supply chain stability is a main feature of effective operation. In order to analyze the stability problem of supply chain system under the shock load, Bessel function has been used to simulate the stress wave propagation in supply chain system under the external shock load. The momentum properties of system's wave function are given by the Fourier transform, and it has been shown by superposition of supply chain inherent wave and shock load transfer wave. The supply chain stability conditions have been given by linear continuous system stable condition and discrete dynamic system stable condition. The stability critical state of supply chain system had been proposed by effective management span value through system's simulation analysis.

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].

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.

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.

Spatial variability of sydney air quality by cumulative semivariogram

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.

Convexity properties axe often useful in characterising and finding bounds for special function and their zeros, as well as in questions concerning the existence and uniqueness of zeros in certain intervals. In this survey paper, we describe some work related to the gamma function, the q-gamma function, Bessel and cylinder functions and the Hermite function.

In this paper, we carry out a comprehensive research on the function D ν ( x ) = x − ν ( I ν ( x ) − L ν ( x ) ) for -1/2 $ ]]> ν > − 1 / 2 and x>0, which finds wide applications in various fields, where I ν ( x ) and L ν ( x ) are the modified Bessel function of the first kind and the modified Struve function, respectively. Specifically, we present some basic properties of the function x ↦ D ν ( x ) , along with several useful bounds. Moreover, we establish the monotonicity of the bounded functions x ↦ x D ν ( x ) , x ↦ D μ ( x ) / D ν ( x ) , and x ↦ ( D ν + 1 ( x ) + D ν − 1 ( x ) ) / D ν ( x ) . Finally, numerical computations show that treating D ν ( x ) as a whole can avoid cancellation problems arising from the subtraction of two large numbers when x is extremely large.

After having introduced a suitable extension of the concept of «generalized singular function», a somewhat detailed mathematical investigation of the structure of the integrals involved is made. Among the main results, various integral relationships are obtained, which show the intimate connection between generalized singular functions and Bessel functions. A further elaboration of our formulae, leading to some new multiple integrals involving Bessel functions, will be given in a subsequent paper.