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The theory of periodic Barlow multi-lattices ( X 1 X 2 ... X N ) ∞ with Xi ∈ {A, B, C} and Xi ≠ Xi+1 of stacked two-dimensional hexagonal close-packed layers is presented and used to derive exact lattice sum expressions in terms of fast converging Bessel function expansions for inverse power potentials. We describe in detail the mathematical properties of Barlow sphere packings and demonstrate that only two basic lattice sums are required to describe all periodic packings. For the sticky hard-sphere model with an attractive inverse power law potential, we find a linear correlation between the cohesive energies of different Barlow packings and the face-centered cubic packing fraction. We introduce an efficient algorithm for enumerating the unique periodic Barlow sequences for any given period N. The theory and lattice sums introduced here pave the way for the future treatment of Barlow multi-lattices. [ABSTRACT FROM AUTHOR]

Aspects of both hybrid and fractional calculus are combined in the (Proportional Caputo-Hybrid) Pcap operators, which are helpful in solving differential equations with non-integer orders and modeling a variety of complicated phenomena in science and engineering. In this paper, we establish the Pcap operators via multiplicative calculus which are termed as multiplicative Pcap operators. we initially formulate two H.H (Hermite–Hadamard)-type inequalities applicable to multiplicative (geometric) convex function via multiplicative Pcap operators. Subsequently, by leveraging certain characteristics of multiplicative convex functions, we present novel inequalities related to multiplicative convex function via multiplicative Pcap operators also demonstrating two novel identities applicable to multiplicatively differentiable functions. By leveraging these identities, we then establish inequalities of trapezoid and midpoint types specifically designed for multiplicatively convex functions. Additionally, we explore applications of these findings to special functions and special means. [ABSTRACT FROM AUTHOR]

The inverse problem of the reconstruction of the Sturm–Liouville problem on a half-line from its Weyl function is considered. Given the Weyl function, we obtain the potential in the Schrödinger equation and the boundary condition at the origin. If the boundary condition is known, this problem is equivalent to the inverse scattering problem of the recovery of the potential, which is zero on a half-line, from a given reflection coefficient. We develop a simple and direct method for solving the inverse problem. The method consists of two steps. First, the Jost solution is computed from the Weyl function, by solving a homogeneous Riemann boundary value problem. Second, a system of linear algebraic equations is constructed for the coefficients of series representations of three solutions of the Schrödinger equation. The potential and the boundary condition are then recovered from the first component of the solution vector of the system. A numerical illustration of the functionality of the method is presented. [ABSTRACT FROM AUTHOR]

As typical frame rates allow limited exposure time, camera-captured videos under low-light conditions often suffer from poor contrast and noise. Existing models failed to consider dark and light areas' boundary pixels and varying low-illuminated night videos' weather conditions for removing noise and enhancing contrast. Hence, the video's visual appearance under low-light is improved using BFR-SEQT. Primarily, a video is inputted and converted into frames. Also, colour space is converted from which static and dynamic pixels are detected regarding frame differences. Using LoG-MF and KF algorithms, noise is removed from which foreground and background are separated using SD-FCM. Motion is estimated and features are extracted to enhance contrast. Then, pixel grouping using LC-GOA is done. Lastly, enhanced outputs from both phases are reconstructed and enhanced video is obtained. The proposed model improves video quality by enhancing contrast and removing noise with high PSNR values (27.6589db and 24.5478db), thus outperforming conventional methods. [ABSTRACT FROM AUTHOR]

The purpose of this article is to provide a number of Hermite–Hadamard and Fejér type integral inequalities for a class of h$$ h $$‐superquadratic functions. We then develop the fractional perspective of inequalities of Hermite–Hadamard and Fejér types by use of the Riemann–Liouville fractional integral operators and bring up with few particular cases. Numerical estimations based on specific relevant cases and graphical representations validate the results. Another motivating component of the study is that it is enriched with applications of modified Bessel function of first type, special means, and moment of random variables by defining some new functions in terms of modified Bessel function and considering uniform probability density function. The results in this paper have not been initiated before in the frame of h$$ h $$‐superquadraticity. We are optimistic that this effort will greatly stimulate and encourage additional research. [ABSTRACT FROM AUTHOR]

This article investigates an axisymmetric contact problem involving the interaction between a rigid cylindrical stamp and a poroelastic cylinder of finite dimensions, based on the Cowin–Nunziato theory of media with voids. The stamp is assumed to have a flat base and to be in frictionless contact with the cylinder. The cylinder, in turn, rests on a rigid base without friction, with no normal displacements or tangential stresses on its lateral surface. Under an applied vertical force, the stamp undergoes displacement, compressing the poroelastic cylinder. The mathematical formulation of this problem involves expressing the unknown displacements within the cylinder and the variation in pore volume fraction as a series of Bessel functions. This representation reduces the problem to an integral equation of the first kind, describing the distribution of contact stresses beneath the stamp. The kernel of the integral equation is explicitly provided in its transformed form. The collocation method is employed to solve the integral equation, enabling the determination of contact stresses and the relationship between the indenter's displacement and the applied force. A comparative model parameter analysis is performed to examine the effects of different material porosity parameters and model geometrical characteristics on the results. [ABSTRACT FROM AUTHOR]

In this work, six analytical solutions are given to estimate the energy exchange by transient conduction in pipes with convection conditions. The developed models were adjusted for an interval RI/RE ${R}_{I}/{R}_{E}$ from 0.2 to 0.8, dimensionless Fourier (Fo) and Biot (Bi) numbers, from 0.05 to 50 and 0.005 to 50, respectively. In each case, 616 tests of temperature distributions were computed using the Heisler approximate method (HAM) and the exact models proposed, for different combinations of RI/RE,Bi,and Fo ${R}_{I}/{R}_{E},{Bi},\mathrm{and}\unicode{x02007}{Fo}$. For the comparison made between the analytical solutions and the HAM, 3696 tests were used, detecting that the HAM correlates with the analytical method with an average deviation of ±10% $\pm 10 \% $ and ±20% $\pm 20 \% $ for 71.2%, and 90.4% of the different values of RI/RE,Fo,and Bi ${R}_{I}/{R}_{E},{Fo},\text{and}\unicode{x02007}{Bi}$ combinations evaluated. The best fit was found for Case 5, with a mean deviation of ±10% $\pm 10 \% $ and ±20% $\pm 20 \% $ for 81.1% and 92.3% of the data used, respectively, while the weaker fit was detected for Case 2, with a mean deviation of ±10% $\pm 10 \% $ and ±20% $\pm 20 \% $ for 67.9% and 88.2% of the available tests. [ABSTRACT FROM AUTHOR]

In this study, a method is presented to determine the matching media parameters that maximize the electromagnetic energy penetrating into the human arm modeled as a radially stratified cylinder. In this context, first, the electromagnetic scattering problem related to the layered cylindrical model in question was solved analytically using cylindrical harmonics. Then, based on this solution, a frequency-dependent functional in terms of the electromagnetic parameters of the matching medium was defined, and the parameters that minimize this functional were determined through the graphs of this functional. In this functional, which depends on the permittivity, conductivity and frequency of the matching medium, one parameter was kept constant at every turn while the other two parameters were optimized. The accuracy of the approach was demonstrated by calculating the electric field amplitudes inside and outside the layers for the parameters determined by the proposed method. The numerical results given in this context demonstrate that if a matching medium is used, the penetrating field increases between 1.3 to 13.96 times compared to the case where the matching medium is absent. [ABSTRACT FROM AUTHOR]

To arrive at the dynamic anti-plane characteristics of bi-material structures with an interfacial V-notch, a virtual domain decomposition technique in conjunction with Graf's addition theorem is employed. External force systems are then appropriately solved using the Green's function technique and the "conjunction" method. Thereafter, the displacement amplitude of the interfacial V-notch is analytically derived. In continuing, some parametric studies are conducted and the plotted results are methodically explained and discussed. The obtained results reveal that the dynamic response in the interfacial V-notch structure could become significant for small levels of the shear modulus ratio and the wavenumber ratio. [ABSTRACT FROM AUTHOR]

Summation of infinite series has played a significant role in a broad range of problems in the physical sciences and is of interest in a purely mathematical context. In a prior paper, we found that the Fourier–Legendre series of a Bessel function of the first kind J N k x and modified Bessel functions of the first kind I N k x lead to an infinite set of series involving F 2 1 hypergeometric functions (extracted therefrom) that could be summed, having values that are inverse powers of the eight primes 1 / 2 i 3 j 5 k 7 l 11 m 13 n 17 o 19 p multiplying powers of the coefficient k, for the first 22 terms in each series. The present paper shows how to generate additional, doubly infinite summed series involving F 2 1 hypergeometric functions from Chebyshev polynomial expansions of Bessel functions, and trebly infinite sets of summed series involving F 2 1 hypergeometric functions from Gegenbauer polynomial expansions of Bessel functions. That the parameters in these new cases can be varied at will significantly expands the landscape of applications for which they could provide a solution. [ABSTRACT FROM AUTHOR]

The emergence of localized radial patterns from a Turing instability has been well studied in two- and three-dimensional settings and predicted for higher spatial dimensions. We prove the existence of localized (n+1) -dimensional radial patterns in general two-component reaction–diffusion systems near a Turing instability, where n>0 is taken to be a continuous parameter. We determine the explicit dependence of each pattern's radial profile on the dimension n through the introduction of (n+1) -dimensional Bessel functions, revealing a deep connection between the formation of localized radial patterns in different spatial dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we explore the self-convolution of the kernel function of the aortic fractal operator. Previous research has established a model named "physical fractal", and confirmed that the hemodynamics of the aorta can be inscribed by a fractal operator and that the dominant component of the kernel function of the fractal operator is a weighted first-order Bessel function. These studies primarily focus on solving the fractal operator kernel function and examining the overall properties of the physical fractal. As we began to investigate the internal structure of physical fractals, we discovered that studying the powers of fractal operators is a necessary step. In this paper, we introduce the concept of kernel function self-convolution, establish its connection with the power of the fractal operator, and derive a series of invariant properties for the self-convolution of the aortic operator kernel function. These invariant properties, in turn, are deeply and intrinsically related to the invariant properties of the Bessel functions. The research findings of this paper enrich hemodynamics and biomechanics in physical fractal space and extend the scope of using fractal operators to characterize the dynamics of living organisms. [ABSTRACT FROM AUTHOR]

Bearing in mind an increasing popularity of the fractional calculus the main aim of this paper is to derive several new representation formulae for the cumulative distribution function (cdf) of the McKay I ν Bessel distribution including the Grünwald-Letnikov fractional derivative; also, two connection formulae between cdf of the McKay I ν random variable and the so–called Neumann series of modified Bessel functions of the first kind are established, providing, consequently, a new integral representation for such cdf in terms of a definite integral. Another fashion expression for the given cdf is derived in terms of the Grünwald-Letnikov fractional derivative of the widely applicable Marcum Q–function, which represents a certain simplification of the already existing relationship between McKay I ν random variable and a Marcum Q–functions. The exposition ends with some open questions, drawing the interested reader's attention, among others, to the summation of some Neumann series. [ABSTRACT FROM AUTHOR]

In this article, the Bessel and Tricomi functions are combined with Appell polynomials to introduce the families of Appell–Bessel and Appell–Tricomi functions. The 2-variable 2-parameter Hermite–Bessel and Hermite–Tricomi functions are considered as members of these families, and framed within the representation of the Lie algebra T3. Consequently, the implicit summation formulae for these functions are derived. Certain examples are also considered. The article concludes with the derivation of a relation involving the 2-variable 2-parameter Hermite–Tricomi functions by following the Weisner's approach. [ABSTRACT FROM AUTHOR]

This paper studies the asymptotic behavior of several central objects in Dunkl theory as the dimension of the underlying space grows large. Our starting point is the observation that a recent result from the random matrix theory literature implies a large deviations principle for the hydrodynamic limit of radial Dunkl processes. Using this fact, we prove a variational formula for the large-N asymptotics of generalized Bessel functions, as well as a large deviations principle for the more general family of radial Heckman–Opdam processes. As an application, we prove a theorem on the asymptotic behavior of weight multiplicities of irreducible representations of compact or complex simple Lie algebras in the limit of large rank. The theorems in this paper generalize several known results describing analogous asymptotics for Dyson Brownian motion, spherical matrix integrals, and Kostka numbers. [ABSTRACT FROM AUTHOR]

Properties of the Lommel polynomials $ R_{n,\nu }(z) $ R n , ν (z) which appear in the theory of the Bessel functions, are studied. New functional relations for hypergeometric and Horn functions, including reduction formulas, are derived. [ABSTRACT FROM AUTHOR]

In this paper, we consider a limited-aperture inverse scattering problem for a fast identification of small dielectric objects from two-dimensional Fresnel experimental dataset. To this end, we apply the Kirchhoff migration (KM) imaging technique and design an imaging function from the generated multi-static response matrix. Using the integral equation-based representation formula for the scattered field, we theoretically investigate the applicability of the KM by formulating the imaging function as a uniformly convergent infinite series of integer-order Bessel functions of the first kind. Numerical simulation results using the experimental Fresnel dataset are presented to support the theoretical result. [ABSTRACT FROM AUTHOR]

Many properties of special polynomials, such as recurrence relations, sum formulas, integral transforms and symmetric identities, have been studied in the literature with the help of generating functions and their functional equations. In this paper, we introduce hybrid forms of q-Mittag-Leffler functions. The q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are constructed using a q-symbolic operator. The generating functions, series definitions, q-derivative formulas and q-recurrence formulas for q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are obtained. The N q -transforms and N q -transforms of q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are obtained. These hybrid q-special functions are also studied by plotting their graphs for specific values of the indices and parameters. [ABSTRACT FROM AUTHOR]

In this work, analytical solutions for six different contour conditions are given to calculate the heat transfer by unsteady conduction in pipes with convection. The analytical models are valid for a diameter ratio R I / R E from 0.1 to 0.9, dimensionless Biot (Bi) and Fourier (Fo) numbers, from 0.001 to 50 and 0.01 to 50, respectively. In determining the analytical solutions, the cylindrical functions of Bessel and Neumann were implemented. Using 864 combination values R I / R E ; B i ; F o , the dimensionless temperature distributions were calculated using the corresponding analytical solution and Heisler's approximate method (HAM). In the comparison made between the analytical solutions and HAM, was verified in 5184 tests carried out that the HAM correlates with the analytical solutions on average, finding an average deviation of ± 10% and ± 20% for 73.2% and 92.1% of the points evaluated. The best fit was found for Case 5, with a mean deviation of ± 10% and ± 20% for 80.2% and 95.5% of the data used, respectively, while the weaker fit was detected for the Case 2 with a mean deviation of ± 10% and ± 20% for 69.7% and 89.8% of the data used. [ABSTRACT FROM AUTHOR]

Integral inequalities and the Mittag-Leffler function play a crucial role in many branches of mathematics and applications, including fractional calculus, mathematical physics, and engineering. In this paper, we introduced an extended generalized Mittag-Leffler function that involved several well-known Mittag-Leffler functions as a special case. We also introduced an associated generalized fractional integral to obtain some estimates for fractional integral inequalities of the Hermite-Hadamard and Hermite-Hadamard-Fejér types. This article offered several analytical tools that will be useful to anyone working in this field. To demonstrate the veracity of our findings, we offered a few numerical and graphical examples. A few applications of modified Bessel functions and unitarily invariant norm of matrices were also given. [ABSTRACT FROM AUTHOR]

Analytical solutions for acoustic vortex beams radiated by sources with uniform circular amplitude distributions are derived in the paraxial approximation. Evaluation of the Fresnel diffraction integral in the far field of an unfocused source and in the focal plane of a focused source leads to solutions in terms of an infinite series of Bessel functions for orbital numbers ℓ > − 2. These solutions are reduced to closed forms for 0 ≤ ℓ ≤ 4 , which correspond to orbital numbers commonly used in experiments. A scaling law for the vortex ring radius is derived, and its relevance is characterized using ray theory. [ABSTRACT FROM AUTHOR]

This paper presents a novel approach for recognition of prolonged motor movements from a subject’s electroencephalogram (EEG) using orthogonal functions to model a sequence of sub-gestures. In this approach, an individual’s EEG signals corresponding to physical (or imagery) continuous movement for different gestures are divided into segments associated with their related sub-gestures. Then, a diffusion adaptation approach is introduced to model the interface between the brain neural activity and the corresponding gesture dynamics. In such a formulation, orthogonal Bessel functions are utilized to represent different gestures and used as the target for the adaptation algorithm. This method aims at detecting and evaluating the prolonged motor movements as well as identifying highly complex sub-gestures. This technique can perform satisfactory classification even in the presence of small data sizes while, unlike many regressors, maintaining a low computational cost. The method has been validated using two different publicly available EEG datasets. An inter-subject average validation accuracy of 70% after performing a leave-one-subject-out k-fold cross-validation is obtained for the classification of the smallest dataset when ten sub-gestures are considered.

In this note, we define and study Asai gamma factors over finite fields. We also prove some results about local Asai L-functions over p-adic fields for level zero representations. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze three efficient methods for numerical approximation of oscillatory integrals with products of two Bessel functions. Firstly, the explicit formulas and asymptotic estimates of the generalized moments are derived by using the Meijer G function. Next, we design a Filon-type method by utilizing ordinary Hermite interpolation polynomials. On this basis, we propose a modified Filon-type method based on Taylor interpolation polynomials with two points. In particular, based on special Hermite interpolation polynomials at Clenshaw–Curtis points, we also give a more efficient Clenshaw–Curtis–Filon-type method that can produce more accurate numerical results. Moreover, the recursive relations of the required modified moments are derived. Importantly, we perform rigorous error analysis of the proposed numerical methods in inverse powers of the oscillation frequency by large amount of theoretical analysis. With the increase of the oscillation frequency, the accuracy improves rapidly when both the number of nodes and the multiplicities are fixed. For the fixed oscillation frequency, the accuracy of the obtained approximate values also increases greatly as either the multiplicities or the number of nodes becomes large. Finally, we compare two of these methods at the same computational cost and find that the Clenshaw–Curtis–Filon-type method gives more accurate results. Some preliminary numerical experiments validate our theoretical analysis and verify the efficiency and accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

We study a planar random motion (X (t) , Y (t)) with orthogonal directions which can turn clockwise, turn counterclockwise, and reverse its direction, each with a different probability. The support of the process is given by a time-varying square and the singular distributions on the boundary and the diagonals of the square are obtained explicitly. In the interior of the support, we study the hydrodynamic limit of the distribution. We then investigate the time T(t) spent by the process moving vertically and the joint distribution of (T (t) , Y (t)) . We prove that, in the hydrodynamic limit, the process (X (t) , Y (t)) spends half the time moving vertically. [ABSTRACT FROM AUTHOR]

Zagier, in 1981, conjectured that the constant term of an automorphic function associated to the Ramanujan delta function, i.e. y12∑ n=1∞τ2(n)e−4πny, has a connection with the nontrivial zeros of ζ(s). This conjecture was finally proved by Hafner and Stopple in 2000. Recently, Chakraborty et al. extended this observation for any normalized Hecke eigenform over SL2(ℤ). In this paper, we study the infinite series ∑n=1∞c fℓ(n)nν/2K ν(yn) for ℓ = 1, 2, where cf(n) denotes the nth Fourier coefficient of a normalized Hecke eigenform f(z) and Kν represents the modified Bessel function of the second kind of order ν. We generalize a recent identity of Berndt et al. We also observe that the aforementioned series corresponding to ℓ = 2 has a connection with the nontrivial zeros of ζ(s). [ABSTRACT FROM AUTHOR]

Consider the Sturm–Liouville equation −y′′+q(x)y=ρ2y$$ -{y}^{\prime \prime }+q(x)y={\rho}^2y $$ with a real‐valued potential q∈L1(0,L),ρ∈ℂ,L>0$$ q\in {\mathcal{L}}_1\left(0,L\right),\rho \in \mathrm{\mathbb{C}},L>0 $$. Let u(ρ,x)$$ u\left(\rho, x\right) $$ be its solution satisfying certain initial conditions u(ρk,0)=ak,u′(ρk,0)=bk$$ u\left({\rho}_k,0\right)={a}_k,{u}^{\prime}\left({\rho}_k,0\right)={b}_k $$ for a number of ρk,k=1,2,...,K$$ {\rho}_k,k=1,2,\dots, K $$, where ρk,ak$$ {\rho}_k,{a}_k $$, and bk$$ {b}_k $$ are some complex numbers. Denote ℓk=u′(ρk,L)+Hu(ρk,L)$$ {\ell}_k={u}^{\prime}\left({\rho}_k,L\right)+ Hu\left({\rho}_k,L\right) $$, where H∈ℝ$$ H\in \mathrm{\mathbb{R}} $$. We propose a method for solving the inverse problem of the approximate recovery of the potential q(x)$$ q(x) $$ and number H$$ H $$ from the following data ρk,ak,bk,ℓkk=1K$$ {\left\{{\rho}_k,{a}_k,{b}_k,{\ell}_k\right\}}_{k=1}^K $$. In general, the problem is ill‐posed; however, it finds numerous practical applications. Such inverse problems as the recovery of the potential from a Weyl function or the inverse two‐spectra Sturm–Liouville problem are its special cases. Moreover, the inverse problem of determining the shape of a human vocal tract also reduces to the considered inverse problem. The proposed method is based on special Neumann series of Bessel functions representations for solutions of Sturm–Liouville equations. With their aid the problem is reduced to the classical inverse Sturm–Liouville problem of recovering q(x)$$ q(x) $$ from two spectra, which is solved again with the help of the same representations. The overall approach leads to an efficient numerical algorithm for solving the inverse problem. Its numerical efficiency is illustrated by several examples. [ABSTRACT FROM AUTHOR]

In this paper, we propose a spectral-Galerkin approximation for the quad-curl eigenvalue problem within spherical geometries. Utilizing vector spherical harmonics in conjunction with the Laplace-Beltrami operator, we decompose the quad-curl eigenvalue problem into two distinct categories of fourth-order equations: corresponding to the transverse electric (TE) and transverse magnetic (TM) modes. A thorough analysis is provided for the TE mode. The TM mode, however, is characterized by a system of coupled fourth-order equations that are subject to a divergence-free condition. We develop two separate sets of vector basis functions tailored for the coupled system in both solid spheres and spherical shells. Moreover, we design a parameterized technique aimed at eliminating spurious eigenpairs. Numerical examples are presented to demonstrate the high precision achieved by the proposed method. We also include graphs to illustrate the localization of the eigenfunctions. Furthermore, we employ Bessel functions to analyze the quad-curl problem, revealing the intrinsic connection between the eigenvalues and the zeros of combinations of Bessel functions. [ABSTRACT FROM AUTHOR]

In this article, we give a unified method to derive the classical characteristic functions of all elliptical and related distributions in terms of Bessel functions. The approach is based on the stochastic representation of an elliptical random variable and the characteristic function of uniform distribution on the unit sphere surface in ℝn. In particular, we present the simple closed form of characteristic functions for commonly used distributions such as multivariate t, Pearson Type II, Pearson Type VII, Kotz type, and Bessel distributions. Some extensions are also being investigated. [ABSTRACT FROM AUTHOR]

In this article, the solutions of Bessel's differential equations (DEs) by variable change method are formulated. To do so, we have considered the first and second kind of Bessel's functions which are obtained as solutions of Bessel's equations and it is used to determine the solutions of the lengthening pendulum (LP). To solve the given equations, we have used Frobenius theorem and the gamma function and hence, apply the obtained results to solve the LP. The finding reveals that Bessel's functions establish the solutions of LP equations. The solutions obtained for lengthening the pendulum are illustrated graphically using the computer software of MathLab. The graphical results show that the sinusoidal wave natures are compressed or extended based on the chosen parameter k. Finally, it is concluded that the obtained method gives an effective, efficient, and systematic method. [ABSTRACT FROM AUTHOR]

We prove explicit uniform two-sided bounds for the phase functions of Bessel functions and their derivatives. As a consequence, we obtain new enclosures for the zeros of Bessel functions and their derivatives in terms of inverse values of some elementary functions. These bounds are valid, with a few exceptions, for all zeros and all Bessel functions with nonnegative indices. We provide numerical evidence showing that our bounds either improve or closely match the best previously known ones. [ABSTRACT FROM AUTHOR]

Isolated patches of spatially oscillating pattern have been found to emerge near a pattern-forming instability in a wide variety of experiments and mathematical models. However, there is currently no mathematical theory to explain this emergence or characterize the structure of these patches. We provide a method for formally deriving radial amplitude equations to planar patterns via nonautonomous multiple-scale analysis and convolutional sums of products of Bessel functions. Our novel approach introduces nonautonomous differential operators, which allow for the systematic manipulation of Bessel functions, as well as previously unseen identities involving infinite sums of Bessel functions. Solutions of the amplitude equations describe fully localized patterns with nontrivial angular dependence, where localization occurs in a purely radial direction. Amplitude equations are derived for multiple examples of patterns with dihedral symmetry, including fully localized hexagons and quasipatterns with twelve-fold rotational symmetry. In particular, we show how to apply the asymptotic method to the Swift--Hohenberg equation and general reaction-diffusion systems. [ABSTRACT FROM AUTHOR]

The multilevel fast multipole method (MLFMM) is widely used to accelerate the solutions of acoustic and electromagnetic scattering problems. In the expansions and translation operators of the MLFMM for 3‐D scattering problems, some special functions are used, including spherical Bessel functions, spherical harmonics and Wigner 3j$$ 3j $$ symbol. This makes it difficult to analyze the truncation errors. In this paper, we first give sharp bounds for the truncation errors of the expansions used in the MLFMM, then derive the overall error formula of the MLFMM and estimate its upper bound, the result is finally applied to the cube octree structure. Some numerical examples are performed to validate the proposed results. The method in this paper can also be used to the MLFMM for other 3‐D problems, such as potential problems, elastostatic problems, Stokes flow problems and so on. [ABSTRACT FROM AUTHOR]

The propagation of sound in a uniform flow with rigid body swirl has been considered in the approximation of low swirl Mach number M 2 ≪ 1 ,where the swirl Mach number M = Ω r / c 00 is specified by the angular velocity Ω , stagnation sound speed c 00 and radial distance r; in this case the mass density and sound speed in the mean flow are constant, and the convected wave equation is solved in terms of Bessel functions. In this paper the restriction on swirl Mach number is relaxed from M 2 ≪ 1 to M 4 ≪ 1 , thus keeping O ( M 2 ) terms in the mean flow, mass density and sound speed, that become non-uniform; the acoustic vortical wave equation is no longer the convected wave equation, and is extended to this case and solved in terms of generalized Bessel functions. The radial dependence to second order in the swirl Mach number is specified by a generalized Bessel differential equation for decoupled acoustic or vortical modes and for acoustic-vortical waves to first order in the swirl Mach number. The solutions are obtained: (i) for finite radius in terms of generalized Bessel and Neumann functions, that determine the radial wavenumbers, natural frequencies and normal eigenfunctions for cylindrical or annular ducts with rigid or impedance wall boundary conditions; (ii) asymptotically for large radius in terms of generalized Hankel functions, specifying the growth of amplitude with radius either as a power law or as an exponential of one-half of the square of the swirl Mach number. Thus the compressible uniform mean flow with rigid body rotation is spatially unstable in the radial direction; it is also unstable in time for cut-on modes with real axial wavenumbers and cut-off modes with imaginary axial wavenumbers. Compared with acoustic waves the acoustic-vortical waves have more modes, some with complex rather than real eigenvalues leading to instabilities. [ABSTRACT FROM AUTHOR]

The analytical solution of the steady-state seepage field of a circular cofferdam in nonhomogeneous layered soil of finite depth is derived, including the head function, exit hydraulic gradient formula, and seepage flow formula. The head function is obtained by dividing the circular cofferdam seepage field into regions and then using the separated variable method combined with the Sturm–Liouville theory, and the unknown coefficients in the head function are determined by constructing a system of equations through the integral transformation of the Bessel function. Based on the head function, an analytical equation is also given for the exit hydraulic gradient and seepage flow. The accuracy of the proposed analytical solution is verified by a comparison with numerical results as well as with the results of other methods. The proposed analytical solution is a display analytical solution without singularities and can be used as an effective tool for the analysis of circular cofferdam seepage problems. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behavior of the radii of starlikeness of the normalized Jackson’s second and third q-Bessel functions, focusing on their large orders. To achieve this, we use the Rayleigh sums of positive zeros of both q-Bessel functions, and determine the coefficients of the asymptotic expansions. We derive complete asymptotic expansions for these radii of starlikeness and provide recurrence relations for the coefficients of these expansions. The proofs rely on the notion of Rayleigh sums of positive zeros of q-Bessel functions, Kvitsinsky’s results on spectral zeta functions for q-Bessel functions and asymptotic inversion. Moreover, we derive bounds for the radius of starlikeness of q-Bessel functions by using potential polynomials. [ABSTRACT FROM AUTHOR]

In this paper we develop the parametrix approach for constructing the heat kernel on a graph G. In particular, we highlight two specific cases. First, we consider the case when G is embedded in a Euclidean domain Ω , and we use a heat kernel associated to Ω to obtain a formula for the heat kernel on G. Second, we consider when G is a possibly infinite subgraph of a larger graph G ~ , and we obtain a formula for the heat kernel on G from the heat kernel on G ~ restricted to G. [ABSTRACT FROM AUTHOR]

In the present work we establish for the first time a class of (P , m) -superquadratic functions and look into its features. Using them, we come up with the Jensen and Hermite–Hadamard inequalities, as well as the fractional versions of Hermite–Hadamard inequalities with respect to Riemann–Liouville fractional integral operators. The findings are confirmed by certain numerical calculations and graphical depictions that take a few appropriate examples into account. The study is enhanced by the addition of applications of special means, moments of random variables, and modified Bessel functions of the first kind. This is achieved by considering new functions pertaining to the uniform probability density function and taking the modified Bessel functions of the first kind into consideration. The new results clearly provide extensions and improvements of the work given in the literature. [ABSTRACT FROM AUTHOR]

In this work, we, for the first time, establish a class of multiplicatively (P,m)-superquadratic function and look into its various features. In the light of these features, we come up with the several integer order integral inequalities in the frame of multiplicative calculus. Moreover, we develop the fractional version of Hermite–Hadamard’s type inequalities involving midpoints and end points for multiplicatively (P,m)-superquadratic function with respect to multiplicatively k-Riemann–Liouville fractional integrals. By choosing different values for the parameters of such integral operators, we acquire a simple version of integral inequalities of Hermite–Hadamard’s type as well as its fractional form via multiplicatively Riemann–Liouville fractional integrals for multiplicatively (P,m)-superquadratic function. The findings are confirmed by graphical illustration by taking appropriate examples into account. The study is further enhanced by the addition of applications of special means and first-type modified Bessel functions. The new results clearly provide extensions and improvements of the work available in the literature. [ABSTRACT FROM AUTHOR]

Detecting buried active faults presents the challenge of precisely locating the upper breakpoint, the shallowest point in the Quaternary system where faults occur. Microtremor survey technology, unaffected by urban electromagnetic interference, offers an eco-friendly and efficient method for investigating buried faults and stratigraphic structures in urban areas. This research uses microtremor survey technology to identify the upper breakpoint of the buried Nankou-Sunhe Fault in Changping, Beijing. For data collection, 17 microtremor survey points were deployed across the northern section of the Nankou-Sunhe fault, employing a three-point nested circular array with a point spacing of approximately 200 m to form a profile spanning approximately 320 m. For data analysis, the spatial autocorrelation method was utilized. Each measurement point was divided into 9 sets of radii, ranging from a minimum of approximately 4 m to a maximum of 28 m. The correlation coefficients for each set were calculated, and the dispersion curve for each measurement point was generated by fitting the average coefficients with the Bessel function of the first kind of order zero. The apparent S-wave velocity was determined directly from the dispersion curve using empirical formulas and interpolated to generate the contour cross-section map. Integrating the section and inverted S-wave velocity data can significantly enhance interpretation accuracy, and based on these data, the spatial development characteristics and upper breakpoint locations of the Nankou-Sunhe fault zone were analyzed, and the strata shallower than 100 m were deduced. The results align well with known geological data, such as luminescence dating and 14C dating from boreholes at nearby locations. [ABSTRACT FROM AUTHOR]

It has been demonstrated that the MUltiple SIgnal Classification (MUSIC) algorithm is fast, stable, and effective for localizing small anomalies in microwave imaging. For the successful application of MUSIC, exact values of permittivity, conductivity, and permeability of the background must be known. If one of these values is unknown, it will fail to identify the location of an anomaly. However, to the best of our knowledge, no explanation of this failure has been provided yet. In this paper, we consider the application of MUSIC to the localization of a small anomaly from scattering parameter data when complete information of the background is not available. Thanks to the framework of the integral equation formulation for the scattering parameter data, an analytical expression of the MUSIC-type imaging function in terms of the infinite series of Bessel functions of integer order is derived. Based on the theoretical result, we confirm that the identification of a small anomaly is significantly affected by the applied values of permittivity and conductivity. However, fortunately, it is possible to recognize the anomaly if the applied value of conductivity is small. Simulation results with synthetic data are reported to demonstrate the theoretical result. [ABSTRACT FROM AUTHOR]

In this contribution, we consider an application of the Kirchhoff migration (KM) technique for fast and accurate identification of small dielectric objects from two-dimensional Fresnel experimental dataset. Generally, for successful application of the KM, a complete set of elements from the so-called multi-static response (MSR) matrix must be collected; however, in the Fresnel experimental dataset, many of the elements of an MSR matrix are not measurable. Nevertheless, the existence, location, and outline shape of small objects can be retrieved using the KM by converting unavailable data into the zero constant. However, the theoretical reason behind such conversion has not been confirmed to date. In order to explain this theoretical reason, we convert unavailable measurement data into a constant and demonstrate that the imaging function of the KM can be expressed by an infinite series of the Bessel functions of integer order of the first kind, the object's material properties, and the converted constant. Following the theoretical result, we confirm that converting unknown data into the zero constant guarantees good results and unique determination of the objects. Finally, various numerical simulation results from Fresnel experimental dataset are presented and discussed to validate the theoretical result. [ABSTRACT FROM AUTHOR]

Denote by NN(Ω,λ)$N_{\cal N} (\Omega,\lambda)$ the counting function of the spectrum of the Neumann problem in the domain Ω$\Omega$ on the plane. G. Pólya conjectured that NN(Ω,λ)⩾(4π)−1|Ω|λ$N_{\cal N} (\Omega,\lambda) \geqslant (4\pi)^{-1} |\Omega | \lambda$. We prove that for convex domains NN(Ω,λ)⩾(23j02)−1|Ω|λ$N_{\cal N} (\Omega,\lambda) \geqslant (2 \sqrt 3 \,j_0^2)^{-1} |\Omega | \lambda$. Here j0$j_0$ is the first zero of the Bessel function J0$J_0$. [ABSTRACT FROM AUTHOR]

We extend a certain type of identities on sums of I-Bessel functions on lattices, previously given by G. Chinta, J. Jorgenson, A. Karlsson and M. Neuhauser. Moreover we prove that, with continuum limit, the transformation formulas of theta functions such as the Dedekind eta function can be given by I-Bessel lattice sum identities with characters. We consider analogues of theta functions of lattices coming from linear codes and show that sums of I-Bessel functions defined by linear codes can be expressed by complete weight enumerators. We also prove that I-Bessel lattice sums appear as solutions of heat equations on general lattices. As a further application, we obtain an explicit solution of the heat equation on Z n whose initial condition is given by a linear code. [ABSTRACT FROM AUTHOR]