Modified Bessel Functions Research Articles

Fast Estimation of Metallic Pipe Properties Using Simplified Analytical Solution in Eddy-Current Testing

The eddy-current testing technique has been extensively explored for the measurement of metallic pipes in various industrial processes. In this study, a simplified analytical model of metallic pipes encircled by a coaxial sensor is derived according to the asymptotic forms of modified Bessel functions. The original Dodd and Deeds (DD) model is thus approximated by an elementary function describing the relationship between the measured inductance spectrum and the pipe properties. Due to the simple nature of the elementary function, the Jacobian matrix can be evaluated much more efficiently. Accordingly, a fast Newton–Raphson method-based optimization approach is developed. The proposed method has been evaluated by comparing it with the numerical simulation of the finite-element method (FEM) and experimental measurements which indicate its effectiveness for the measurement of metallic pipe properties.

New developments in fractional integral inequalities via convexity with applications

<abstract><p>The main objective of this article is to build up a new integral equality related to Riemann Liouville fractional (RLF) operator. Based on this integral equality, we show numerous new inequalities for differentiable convex as well as concave functions which are similar to celebrated Hermite-Hadamard and Simpson's integral inequalities. The present outcomes of this paper are a unification and generalization of the comparable results in the literature on Hermite-Hadamard and Simpson's integral inequalities. Furthermore as applications in numerical analysis, we find some means, q-digamma function and modified Bessel function type inequalities.</p></abstract>

Q-Gaussian Tsallis Functions and Egelstaff-Schofield Spectral Line Shapes

In this article we will discuss the Egelstaff-Schofield line shapes, as used in Raman spectroscopy, and their fit by means of q-Gaussian Tsallis functions. q-Gaussians are probability distributions having their origin in the framework of Tsallis statistics. A continuous real parameter q is characterizing them so that, in the range 1 < q < 3, q-functions pass from the usual Gaussian form, for q close to 1, to that of a heavy tailed distribution, at q close to 3. The value q=2 corresponds to the Cauchy-Lorentzian distribution. This behavior allows the q-Gaussian function to properly mimicking the Egelstaff-Schofield line shape, which has been introduced to fit the bands of first-order Raman scattering in ionic liquids. This line shape is based on a modified Bessel function of the second kind. Moreover, since the Fourier transform of the Egelstaff-Schofield line shape is given by a simple analytical expression, we can use this expression as an easy substitute for the Fourier transform of the q-Gaussian function.

Performance analysis of a fluid flow system modulated by a single server queue prone to catastrophic failures and repairs

&lt;p style='text-indent:20px;'&gt;This article deals with a new class of fluid flow queue regulated by an &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ M/M/1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; queueing system in the presence of active catastrophic failures and subsequent repairs of the server. The distribution of the stationary buffer-content is determined in terms of modified Bessel function of first kind. Besides, an explicit closed-form analytical expression for the probability density function of the fluid level in the stationary regime is derived in an elegant manner. Some important and essential performance measures are obtained. Finally, numerical results are provided to illustrate the behaviour of the buffer content distribution and the performance measures for the proposed fluid flow queueing model.&lt;/p&gt;

On the characterization properties of certain hypergeometric functions in the open unit disk

Our purpose in the present investigation is to study certain geometric properties such as the close-to-convexity, convexity, and starlikeness of the hypergeometric function z{}_{1}F_{2}(a;b,c;z) in the open unit disk. The usefulness of the main results for some familiar special functions like the modified Sturve function, the modified Lommel function, the modified Bessel function, and the {}_{0}F_{1}(-;c;z) function are also mentioned. We further consider a boundedness property of the function _{1}F_{2}(a;b,c;z) in the Hardy space of analytic functions. Several corollaries and special cases of the main results are also pointed out.

Ostrowski-type inequalities pertaining to Atangana–Baleanu fractional operators and applications containing special functions

The objective of this article is to incorporate the concept of the Ostrowski inequality with the Atangana–Baleanu fractional integral operator. A novel integral identity for twice-differentiable functions is established after a rigorous investigation of several basic definitions and existing ideas related to inequalities and fractional calculus. Following that, numerous Ostrowski-type inequalities are provided based on this identity, which uses Mittag–Leffler as its kernel structure. Some specific applications, such as q-digamma functions and modified Bessel functions, are also investigated. Choosing s=1, we also analyze new results for convex functions as special cases. Our findings corroborate some well-documented inequalities.

A Euclidean Fourier‐analytic approach to vertical projections in the Heisenberg group

AbstractAn improved almost everywhere lower bound is given for Hausdorff dimension under vertical projections in the first Heisenberg group, with respect to the Carnot‐Carathéodory metric. This improves the known lower bound, and answers a question of Fässler and Hovila. The approach uses the Euclidean Fourier transform, Basset's integral formula, and modified Bessel functions of the second kind.

The effect of pressure gradient on MHD flow of a tri-hybrid Newtonian nanofluid in a circular channel

The novel theoretical ternary hybrid nanofluid (HNF) model presented in this research focuses on the enhancement of heat transport. In this study, tri-hybrid nanoparticles made of Au, Ag, and TiO2 are dissolved in the blood, which serves as the base fluid. The traditional Tiwari and Das HNF model has been extended to modify it for the situation of tri-HNF. As a result, A geometrical prototype is depicted utilizing proportion. We created the regulating equations for the analytic solution to this problem. The Bessel and modified Bessel functions are used to solving the velocity, temperature, and pressure fields. The outcomes are computed for each velocity, heat, pressure, as well as stress gradient. Graphicoutcomesfor the computational amountsof the immerged variables, notably: Hartmann quantity (M), the solid fractional sizeof the nanomaterial (ϕ), Reynolds quantity(Reβ), and the pulsation variable focused on the periodic pressure gradient, are examined for pressure difference, frictional forces, rapidity outline, temperature outline, crux, streamlines, and vorticity phenomena. The tri-hybrid nanofluid (THNF) has better thermal conductivity than the HNF, according to the graphed results. It has also been revealed that the outside magnetic force may regulate velocity, which influences temperature outlines and consequently heat transmission, which can be improved or decreased by managing the magnetized flux.

Some perpetual integral functionals of the three-dimensional Bessel process

We compute the Laplace transforms of some integral functionals of the three-dimensional Bessel process in terms of modified Bessel functions, Gauss’ hypergeometric functions, and confluent hypergeometric functions. Some new results are obtained, and several established results, such as Dufresne’s perpetuity and a particular case of its translated version, are recovered. In particular, we derive the Laplace transform of the two-dimensional random variable [Formula: see text] where [Formula: see text] is a three-dimensional Bessel process and [Formula: see text]. A crucial step of the derivation is to construct a martingale, as performed in the Bass solution of the Skorokhod embedding problem, with respect to an enlarged filtration.

Slater Tipi Orbitaller Bazında Bir- ve İki-Merkezli Coulomb İntegrallerinin Özellikleri

In this study, two-electron one- and two-center Coulomb integrals with the same and different screening parameters are investigated numerically in the real Slater type orbital (STO) basis using Fourier transform method. In momentum space firstly, for atomic, i.e. one-center, Coulomb integrals are calculated, and analytical expressions are obtained in terms of binomial coefficients. Then, the solutions of the two-center Coulomb integrals are made with the modified Bessel function of second kind and the results are expressed in terms of binomial and Gaunt coefficients, irregular solid harmonics, and finite sum of STOs. A computer program is written in the MATHEMATICA language to determine the accuracy of the analytical expressions that are highly suitable for programming. The numerical results obtained from the program are given in the tables, and it is shown that the results agree with the literature.

Few-parameter noncylindrical paraxial optical beam described by the modified Bessel function

A few-parameter expression for a light beam is found as a solution of the paraxial Helmholtz equation. It is achieved by exploiting appropriately chosen complex variables which entail the separability of the equation. Next, the expression for the beam is obtained independently by superimposing shifted Gaussian beams, whereby the shift can be made either by a real vector (in which case the foci of the Gaussian beams are located on a circle) or by a complex one. The solutions found depend on several parameters, the specific choice of which allows to obtain beams with quite different properties. For several selected parameter values figures are drawn, demonstrating the spatial distribution of the energy density and phase. In special cases, the effect of a shift of the intensity peak from one branch to another and phase singularities are observed.

Microscopic-macroscopic level densities for low excitation energies

Level density ρ(E,Q) is derived within the micro-macroscopic approximation (MMA) for a system of strongly interacting Fermi particles with the energy E and additional integrals of motion Q, in line with several topics of the universal and fruitful activity of A. S. Davydov. Within the extended Thomas Fermi and semiclassical periodic orbit theory beyond the Fermi-gas saddle-point method, we obtain ρ∝Iν(S)/Sν, where Iν (S) is the modified Bessel function of the entropy S. For small shell-structure contribution, one finds ν = κ/2 + 1, where κ is the number of additional integrals of motion. This integer number is a dimension of Q, Q = {N, Z, …} for the case of two-component atomic nuclei, where N and Z are the numbers of neutrons and protons, respectively. For much larger shell structure contributions, one obtains ν = κ /2 + 2. The MMA level density ρ reaches the well-known Fermi gas asymptote for large excitation energies and the finite micro-canonical combinatoric limit for low excitation energies. The additional integrals of motion can also be the projection of the angular momentum of a nuclear system for nuclear rotations of deformed nuclei, number of excitons for collective dynamics, and so on. Fitting the MMA total level density ρ(E, Q) for a set of the integrals of motion Q = {N, Z}, to experimental data on a long nuclear isotope chain for low excitation energies, one obtains the results for the inverse level-density parameter K, which differs significantly from those of neutron resonances due to shell, isotopic asymmetry, and pairing effects.

Extended weighted Simpson-like type inequalities for preinvex functions and their use in physical system

The main aim of this investigation is to establish the weighted Simpson-like type identity and related variants for a mapping for which the power of the absolute of the first derivative is s-preinvex. By considering this identity, numerous novel weighted Simpson’s like type and related estimation type results for bounded first order differentiable functions are apprehended. Several notable results can be obtained as consequences for the suitable selection of n and ω. Meanwhile, the results are illustrated with two special functions involving modified Bessel function and q-digamma function to obtain the efficiency and supremacy of the proposed technique for many problems of wave propagation and static potentials.

MHD Pulsatile Flow of Blood-Based Silver and Gold Nanoparticles between Two Concentric Cylinders

Pulsatory movements appear in a variety of fascinating applications involving periodic flow propagation and control. Pulsing encourages mixing and, as a result, mass and heat exchange with the boundaries. Pulsing also helps to decrease surface fouling by allowing solid particles to migrate. An exact solution of the Navier–Stokes equations for the transport of an incompressible viscous fluid in a channel with arbitrary pressure distribution is described in this study. The flow is defined by two primary parameters: the pulsation parameter, which is determined by the periodic pressure gradient, and the kinetic Reynolds number, which is determined by the pulsation frequency. The purpose of employing hybrid nanofluid (HNF) is to increase the base fluid’s thermal conductivity. We regard Ag and Au as nanoparticles (NPs) and blood as a base fluid for this phenomenon. Broadening this reveals that the consideration of nanoparticles has impressively extended the warm movement at the parcels of both turbulent and laminar frameworks. Attention is paid to the slope of speed, temperature, and voltage. The geometric model is therefore described using a symmetry technique. We developed the governing equation for this problem’s analytical solutions. The velocity and temperature fields solution is given in the form of the Bessel and modified Bessel functions. Graph results show the mathematical benefits of the current limits: for instance, Hartmann number M, solid volume part of nanoparticles ϕ, Reynolds number Reβ, Prandtl number Pr, intermittent slob limit, etc. The strain angles introduced in the stress contrast, frictional force, velocity profile, and temperature profile were obtained, and the characteristics of the vortex were investigated. Resources at various boundaries of the perceptual flow are examined. As with the final essence, the smoothest results are analyzed and recorded. It has also been discovered that the velocity may be regulated by the external magnetic field, which affects the temperature profiles and hence the heat transfer, which can be enhanced or lowered by mastering the magnetic field.

Simultaneous Approximation by Gauss–Weierstrass–Wachnicki Operators

In this note, we spotlight a generalization of Weierstrass integral operators introduced by Eugeniusz Wachnicki. The construction involves modified Bessel functions. The operators are correlated with diffusion equation. Our main result consists in obtaining the asymptotic expansion of derivatives of any order of Wachnicki’s operators. All coefficients are explicitly calculated, and distinct expressions are provided for analytical functions.

New midpoint type Hermite-Hadamard-Mercer inequalities pertaining to Caputo-Fabrizio fractional operators

The objective of the current article is to incorporate the concepts of convexity and Jensen-Mercer inequality with the Caputo-Fabrizio fractional integral operator. Moreover, we present new midpoint versions of Hermite-Hadamard-Mercer (H-H-M) type inequalities for convex functions and the product of two convex functions on fractional integrals. Also, we consider a new identity for differentiable mappings in the context of the Caputo-Fabrizio fractional integral operators. Then, considering this identity as an auxiliary result, some new related H-H-M type inequality with the assistance of Hölder, power-mean, Young, and Hölder-İşcan inequality are presented. Finally, we give some applications of modified Bessel functions and matrices, and we also discuss some future scopes.

Multiple-symbol noncoherent learning detection of coded QAM signals in IEEE 802.15.3 Wireless Multi-media Networks

We consider the noncoherent deep learning problem for coded signal detection under the phase noncoherent channels for remote home healthcare applications with high data rate. In particular, a multiple-symbol noncoherent learning detection (MNLD) scheme based on neural networks is proposed for low-density parity-check (LDPC) coded noncoherent quadrature amplitude modulation (QAM) signals in IEEE 802.15.3 Wireless Multi-media Networks. Our derivation shows that extensive operations for the first kind zero-order modified Bessel function is unavoidable for the implementation of the optimal bit log-likelihood ratio (LLR) for decoding in traditional multiple-symbol detection (MSD) scheme. The perfect estimation of the channel state information (CSI), i.e., a priori information about the variance of the additive white Gaussian noise (AWGN), is also required for the receiver. This is clearly not computationally practical for Wireless Multi-media Networks. Consequently, we developed an improved approach based on feed-forward neural networks to accurately calculate the bit LLR. Furthermore, to decrease the generation size of training set and thus increase the training speed of the proposed neural networks, we uniformly quantize the continuous carrier phase offset (CPO), which is random and unknown, into discrete status. Our simulation results verify the learning efficiency of this simplified training-set generation configuration. The decoding convergence is successfully accelerated and much performance gain is finally achieved when compared with traditional decoding using the perfect bit LLR. This is clearly critical for high reliable transmission of home healthcare information.

Empirical Rule of Fruit Rind Fragmentation in Muskmelon Netting

The suberized netting tissue on the surface of muskmelons is a remnant of the rind fragmentation caused by the mechanical imbalance between the expanding flesh meat and the solidified epidermis during growth. The appearance of this netting pattern is an important indicator for assessing the quality of muskmelons; however, very few quantitative studies have been conducted on the geometry of the netting. This study statistically analyzed the areas of fine epidermis fragments surrounded by the net using image analysis and explored a common rule governing the probability distribution of the fragment area. We found that the fragment area distribution follows a universal law by the modified Bessel function, which is consistent with the theoretical argument for the fracture behavior of brittle materials.

Two general series identities involving modified Bessel functions and a class of arithmetical functions

AbstractWe consider two sequences $a(n)$ and $b(n)$ , $1\leq n&lt;\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 $\chi (n)$ .

Nonanalytic term in the effective potential at finite temperature for a scalar field on compactified space

We study non-analytic terms, which cannot be written in the form of any positive integer power of field-dependent mass squared, in effective potential at finite temperature in one-loop approximation for a real scalar field on the $D$-dimensional space-time, $S_{\tau}^1\times R^{D-(p+1)}\times\prod_{i=1}^pS_i^1$. The effective potential can be recast into the integral form in the complex plane by using the integral representation for the modified Bessel function of the second kind and the analytical extension for multiple mode summations. The pole structure of the mode summations is clarified and all the non-analytic terms are obtained by the residue theorem. We find that the effective potential has a non-analytic term when the dimension of the flat Euclidean space, $D-(p+1)$ is odd. There appears only one non-analytic term for the given values of $D$ and $p$, for which the non-analytic term exists.