A General Analytic Approximation Technique for the Modified Bessel Functions with Fractional Order Iv (x), Applied to I1/6(x) and I1/7(x)

A general technique to obtain simple analytic approximations for the first kind of modified Bessel functions. The general procedure is shown, and the parameter determination is explained through the applications to this particular case I1/6(x)and I1/7(x). In this way, it shows how to apply the technique to any particular orderν, in order to obtain an approximation valid for any positive value of the variable x. In the present method power series and asymptotic expansion are simultaneously used. The technique is an extension of the multipoint quasirational approximation method, MPQA. The main idea is to look for a bridge function between the power and asymptotic expansion of the I1/6(x), and similar procedure for I1/7(x). To perform this, rational functions are combined with hyperbolic ones and fractional powers. The number of parameters to be determined for each case is four. The maximum relative errors are 0.0049 for ν=1/6, and 0.0047 for ν=7. However, these relative errors decrease outside of the small region of the variables, wherein the maximum relative errors are reached. There is a clear advantage of this procedure compared with any other ones.

Duality between Bessel Functions and Chebyshev Polynomials in Expansions of Functions

In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.

THIRD-ORDER DIFFERENTIAL SUBORDINATION RESULTS FOR ANALYTIC FUNCTIONS INVOLVING THE GENERALIZED BESSEL FUNCTIONS

In the present paper, we derive some third-order differential subordination results for analytic functions in the open unit disk, using the operator Bcκf by means of normalized form of the generalized Bessel functions of the first kind, which is defined as z（Bκ＋1^c f（z））′= κBκ^c f（z）-（κ- 1）Bκ＋1^c f（z）,where b, c, p ∈ C and κ = p ＋（b ＋ 1）/2 ∈ C ／ Z0^-（Z0^-= {0,-1,-2, … }）. The results are obtained by considering suitable classes of admissible functions. Various known or new special cases of our main results are also pointed out.

SIMPLIFICATION OF FREQUENCY EQUATION OF MULTILAYERED CYLINDERS AND SOME RECURSION FORMULAE OF BESSEL FUNCTIONS

In this paper the axially symmetric, radial vibration frequency equation of multilayered cylinders made of the same materials in plain strain state is studied. It is proved in this paper that the frequency equation of:several contact hallow cylinders can be replaced by the frequency equation of a single hollow cylinder, so that the solving process is greatly simplified. Some recursion formulae of Bessel functions are derived from such practical problems as well.

Elastodynamic Infinite Elements with Modified Bessel Shape Functions for Dynamic Soil-Structure Interaction

The paper is devoted to formulations of decay and mapped elastodynamic infinite elements, based on modified Bessel shape functions. These elements are appropriate for Soil-Structure Interaction （SSI） problems, solved in time or frequency domain and can be treated as a new form of the recently proposed Elastodynamic Infinite Elements with United Shape Functions （EIEUSF） infinite elements. The formulation of 2D Horizontal type Infinite Elements （HIE） is demonstrated here, but by similar techniques 2D Vertical （VIE） and 2D Comer （CIE） Infinite Elements can also be formulated. Using elastodynamic infinite elements is the easier and appropriate way to achieve an adequate simulation including basic aspects of Soil-Structure Interaction. Continuity along the artificial boundary （the line between finite and infinite elements） is discussed as well and the application of the proposed elastodynamic infinite elements in the Finite Element Method （FEM） is explained in brief. Finally, a numerical example shows the computational efficiency of the proposed infinite elements.

ON THE ZEROS OF QUOTIENTS OF BESSEL FUNCTIONS

The paper studies the zeros of 2mIm/Im-1- (m+1)I1 /I0, where Im are the Bessel functions.

Some Generalized q-Bessel Type Wavelets and Associated Transforms

In this paper wavelet functions are introduced in the context of q-theory. We precisely extend the case of Bessel and q-Bessel wavelets to the generalized q-Bessel wavelets. Starting from the （q,v）-extension （v = （α,β）） of the q-case, associated generalized q-wavelets and generalized q-wavelet transforms are developed for the new context. Reconstruction and Placherel type formulas are proved.

A Probabilistic Trust Model and Control Algorithm to Protect 6G Networks against Malicious Data Injection Attacks in Edge Computing Environments

Future 6G communications are envisioned to enable a large catalogue of pioneering applications.These will range from networked Cyber-Physical Systems to edge computing devices,establishing real-time feedback control loops critical for managing Industry 5.0 deployments,digital agriculture systems,and essential infrastructures.The provision of extensive machine-type communications through 6G will render many of these innovative systems autonomous and unsupervised.While full automation will enhance industrial efficiency significantly,it concurrently introduces new cyber risks and vulnerabilities.In particular,unattended systems are highly susceptible to trust issues:malicious nodes and false information can be easily introduced into control loops.Additionally,Denialof-Service attacks can be executed by inundating the network with valueless noise.Current anomaly detection schemes require the entire transformation of the control software to integrate new steps and can only mitigate anomalies that conform to predefined mathematical models.Solutions based on an exhaustive data collection to detect anomalies are precise but extremely slow.Standard models,with their limited understanding of mobile networks,can achieve precision rates no higher than 75%.Therefore,more general and transversal protection mechanisms are needed to detect malicious behaviors transparently.This paper introduces a probabilistic trust model and control algorithm designed to address this gap.The model determines the probability of any node to be trustworthy.Communication channels are pruned for those nodes whose probability is below a given threshold.The trust control algorithmcomprises three primary phases,which feed themodel with three different probabilities,which are weighted and combined.Initially,anomalous nodes are identified using Gaussian mixture models and clustering technologies.Next,traffic patterns are studied using digital Bessel functions and the functional scalar product.Finally,the information coherence and content are analyzed.The noise content and abnormal information sequences are detected using a Volterra filter and a bank of Finite Impulse Response filters.An experimental validation based on simulation tools and environments was carried out.Results show the proposed solution can successfully detect up to 92%of malicious data injection attacks.

The Jackson Inequality for the Best L^2-Approximation of Functions on [0,1] with the Weight x

Let L^2（[0, 1], x） be the space of the real valued, measurable, square summable functions on [0, 1] with weight x, and let n be the subspace of L2（[0, 1], x） defined by a linear combination of Jo（μkX）, where Jo is the Bessel function of order 0 and {μk} is the strictly increasing sequence of all positive zeros of Jo. For f ∈ L^2（[0, 1], x）, let E（f, n） be the error of the best L2（[0, 1], x）, i.e., approximation of f by elements of n. The shift operator off at point x ∈[0, 1] with step t ∈[0, 1] is defined by T（t）f（x）=1/π∫0^π f（√x^2 ＋t^2-2xtcosO）dθ The differences （I- T（t））^r/2f = ∑j=0^∞（-1）^j（j^r/2）T^j（t）f of order r ∈ （0, ∞） and the L^2（[0, 1],x）- modulus of continuity ωr（f,τ） = sup{｜｜（I- T（t））^r/2f｜｜：0≤ t ≤τ] of order r are defined in the standard way, where T^0（t） = I is the identity operator. In this paper, we establish the sharp Jackson inequality between E（f, n） and ωr（f, τ） for some cases of r and τ. More precisely, we will find the smallest constant n（τ, r） which depends only on n, r, and % such that the inequality E（f, n）≤ n（τ, r）ωr（f, τ） is valid.

Measurement of Optical Fiber Geometry Parameters with Bessel Function Fitting Method

Geometry parameters of optical fiber are crucial in evaluating the quality of the optical fiber.Near⁃field light distribution method is recommended in GB15972.20-2008 for the measurement of geometry parameters.To distinguish the boundary between fiber core and cladding,it is necessary to illuminate the fiber.The end face of the core is a bright spot with unclear edge,so the true edge of the core and the cladding cannot be accurately judged.A method is proposed in this paper to measure the geometry parameters of optical fiber by Bessel function fitting.Theoretically,the solution to the electromagnetic vector of mode field satisfies Bessel function,and the boundary between the core and the cladding can be precisely extracted by Bessel function fitting.Edges of the fiber were fitted by elliptical curves,and the geometry parameters of the fibers could be calculated.Results show that the maximum deviations of the diameters and the average differences of the fibers were decreased under normal and abnormal conditions respectively.The proposed method is an efficient way to obtain edge data and can improve the accuracy and stability of geometry parameters of optical fibers.

Determinantal Expressions and Recursive Relations for the Bessel Zeta Function and for a Sequence Originating from a Series Expansion of the Power of Modified Bessel Function of the First Kind

In the paper,by virtue of a general formula for any derivative of the ratio of two differentiable functions,with the aid of a recursive property of the Hessenberg determinants,the authors establish determinantal expressions and recursive relations for the Bessel zeta function and for a sequence originating from a series expansion of the power of modified Bessel function of the first kind.

Study Hankel Transforms and Properties of Bessel Function via Entangled State Representation Transformation in Quantum Mechanics

In Phys. Lett. A 313 （2003） 343 we have found that the self-recipràcal Hankel transformation （HT） is embodied in quantum mechanics by a transform between two entangled state representations of continuum variables. In this work we study Hankel transforms and properties of Bessel function via entangled state representations＇ transformation in quantum mechanics.

REPRESENTATION OF GENERALIZED TRANSLATIONS OF THE PRODUCT OF TWO FUNCTIONS ON SIGNED HYPERGROUPS ON THE REAL LINE

In this paper, we consider the generalized translations associated with the Dunkl and the Jacobi-Dunkl differential-difference operators on the real line which provide the structure of signed hrpergroups on R. Especially, we study the representation of the gener- alized translations of the product of two functions for these signed hypergroups.

CHEBYSHEV APPROXIMATION OF THE SECOND KIND OF MODIFIED BESSEL FUNCTION OF ORDER ZERO

The second kind of modified Bessel function of order zero is the solutions of many problems in engineering. Modified Bessel equation is transformed by exponential transformation and expanded by J.P.Boyd's rational Chebyshev basis.

Eigenfunctions for a Quantum Wire on a Single Electron at Its Surface and in the Quantum Well with Beaded Fractional Quantized States for the Fractional Charges

We developed energy profiles for the fractional quantized states both on the surface of electron due to overwhelming centrifugal potentials and inside the electron at different locations of the quantum well due to overwhelming attractive electrodynamic potentials. The charge as a physical constant and single entity is taken as density and segments on their respective sub-quanta (floats on sub quanta) and hence the fractional charge quantiz at in. There is an integrated oscillatory effect which ties all fractional quantized states both on the surface and in the interior of the volume of an electron. The eigenfunctions, i.e., the energy profiles for the electron show the shape of a string or a quantum wire in which fractional quantized states are beaded. We followed an entirely different approach and indeed thesis to reproducing the eigenfunctions for the fractional quantized states for a single electron. We produced very fascinating mathematical formulas for all such cases by using Hermite and Laguerre polynomials, spherical based and Neumann functions and indeed asymptotic behavior of Bessel and Neumann functions. Our quantization theory is dealt in the momentum space.

A REMARK ON LARGE TIME ASYMTOTICS FOR SOLUTIONS OF A NONHOMOGENEOUS VISCOUS BURGERS EQUATION

The existence, uniqueness and regularity of solutions to the Cauchy problem posed for a nonhomogeneous viscous Burger's equation were shown in Chung, Kim and Slemrod [1] by assuming suitable conditions on initial data. Moreover, they derived the asymptotic behaviour of solutions of the Cauchy problem by imposing additional conditions on initial data. In this article, we obtain the same asymptotic behaviour of solutions to the Cauchy problem without imposing additional condition on initial data.

Vibration of an infinite inhomogeneous transversely isotropic viscoelastic medium with cylindrical hole

This paper investigates the influences of higher order viscoelasticity and the inhomogeneities of the transversely isotropic elastic parameters on the disturbances in an infinite medium, caused by the presence of a transient radial force or twist on the surface of a cylindrical hole with circular cross section. Following Voigt＇s model for higher order viscoelasticity, the nonvanishing stress components valid for a transversely isotropic and higher order viscoelastic solid medium have been deduced in terms of radial displacement component. Considering the power law variation of elastic and viscoelastic parameters, the stress equation of motion has been developed. Solving this equation under suitable boundary conditions, due to transient forces and twists, radial displacement and relevant stress components have been determined in terms of modified Bessel functions. The problem for the presence of transient radial force has been numerically analysed. Modulations of displacement and stresses due to different order of viscoelasticity and inhomogeneity have been graphically depicted. The numerical study of the disturbance caused by the presence of twist on the surface may be similarly done but is not pursued in this paper.

EFFICIENT COMPUTATIONAL METHOD FOR TRANSIENT EM RESPONSE OF LARGE-LOOP SOURCE OVER LAYERED EARTH

Calculations of the ElectroMagnetic(EM) response produced by a large horizontal loop placed over layered medium are rather complex because its integral expression contains the product of two Bessel functions and has a divergent term. In this paper, an improved fast Hankel and Gaver-Stehfest transforms are introduced to solve the strong-oscillation and slow-decay properties of the integrand, where one Bessel function in the product is substituted by a carefully chosen polynomial of high accuracy and the other used as the digital filter coefficients in the convolution integral. Comparisons prove the validity and the efficiency of the proposed method.

An Analytical Formulation for Pollutant Dispersion Simulation in the Atmospheric Boundary Layer

In this work we present the solution of the two-dimensional advection-diffusion equation by the GILTT method. The GILTT approach uses, in the series expansion, eigenfunctions given in terms of cosine functions. Here, a different expansion for the solution of the advection-diffusion equation will be explored. In other words, a Sturm-Liouville problem carrying more information of the original problem is considered, given by Bessel functions. Numerical simulations and comparisons with experimental data are presented.

DYNAMIC BEHAVIORS OF CONDUCTIVE CIRCULAR PLATE IN TIME-VARYING MAGNETIC FIELDS

In this study, we proposed an analytical solution for eddy currents as well as electromagnetic forces of a conductive circular plate in a time varying magnetic field. Specifically, an analytical series solution for eddy currents in a circular plate subjected to an axisymmetrie time varying magnetic field has been proposed based on the T-method that has been widely used in the eddy current analysis of conductive and superconductive structures. Accordingly, the dynamic response, the dynamic instability and the magnetic damping of a circular plate in a transverse transient magnetic field as well as a stationary in-plane magnetic field have also been obtained. The analytical series solution proposed in this work as well as the subsequent numerical analysis not only confirmed the emergence of dynamic instability of a circular plate in a strong transverse magnetic field, but also demonstrated the existence of magneto-damping of a circular conductive plate in an in-plane magnetic field. The method developed in this paper provides a potential new possible way by which the analysis of the electromagnetic coupling problems of conductive structures can be simplified.