Weber Functions Research Articles

Different kinds of accelerated propagation of relativistic electromagnetic plasma wavepackets

Relativistic electromagnetic plasma waves are described by a dynamical equation that can be solved not only in terms of plane waves, but also for several different accelerating wavepacket solutions. Depending on the spatial and temporal dependence of the plasma frequency, different kinds of accelerating solutions can be obtained, for example, in terms of Airy or Weber functions. Also, we show that an arbitrary accelerated wavepacket solution is possible, for example, for a system with a luminal plasma slab.

Solving the Coriolis Vibratory Gyroscope Motion Equations by Means of the Angular Rate B-Spline Approximation

The exact solution of the movement equation of the Coriolis vibratory gyroscope (CVG) with a linear law of variation of the angular rate of rotation of the base is given. The solution is expressed in terms of the Weber functions (the parabolic cylinder functions) and their asymptotic representations. On the basis of the obtained solution, an analytical solution to the equation of the ring dynamics in the case of piecewise linear approximation of an arbitrary angular velocity profile on a time grid is derived. The piecewise linear solution is compared with the more rough piecewise constant solution and the dependence of the error of such approximations on the sampling step in time is estimated numerically. The results obtained make it possible to significantly reduce the number of operations when it is necessary to study long-range dynamics of oscillations of the system, as well as quantitatively and qualitatively control the convergence of finite-difference schemes for solving the movement equations of the Coriolis vibratory gyroscope.

Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles

We consider the 1D nonlinear Schrodinger equation (NLS) with focusing point nonlinearity, \begin{document}$ \begin{equation} i\partial_t\psi + \partial_x^2\psi + \delta|\psi|^{p-1}\psi = 0, \;\;\;\;\;\;(0.1)\end{equation} $\end{document} where \begin{document}$ {\delta} = {\delta}(x) $\end{document} is the delta function supported at the origin. In the \begin{document}$ L^2 $\end{document} supercritical setting \begin{document}$ p>3 $\end{document} , we construct self-similar blow-up solutions belonging to the energy space \begin{document}$ L_x^\infty \cap \dot H_x^1 $\end{document} . This is reduced to finding outgoing solutions of a certain stationary profile equation. All outgoing solutions to the profile equation are obtained by using parabolic cylinder functions (Weber functions) and solving the jump condition at \begin{document}$ x = 0 $\end{document} imposed by the \begin{document}$ \delta $\end{document} term in (0.1). This jump condition is an algebraic condition involving gamma functions, and existence and uniqueness of solutions is obtained using the intermediate value theorem and formulae for the digamma function. We also compute the form of these outgoing solutions in the slightly supercritical case \begin{document}$ 0 using the log Binet formula for the gamma function and steepest descent method in the integral formulae for the parabolic cylinder functions.

On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

Abstract After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

Efficient Simulation of Large Irregular Arrays on a Finite Ground Plane

An efficient method is presented to take into account the radiation pattern deformation due to the presence of a finite ground plane lying under an antenna array and above a layered dielectric medium. The interactions between the antennas and the ground plane are computed using two main algorithms: on one hand, the near-field interactions are carried out using inhomogeneous plane waves, and on the other hand, a new formulation based on an analytical Hankel transform is derived to handle the intermediate-field interactions. This formulation expresses the electric field as a finite series of Hankel functions and associated Anger–Weber functions. The method is validated here for the second version of the square kilometer array (SKA) log-periodic Antenna (SKALA2) and then applied to an SKA low-frequency station (SKA1-LOW).

BEHAVIOR OF THE DIAMETRIC DISTRIBUTION OF ECOLOGICAL GROUPS IN A MIXED OMBROPHILOUS FOREST FRAGMENT

The Mixed Ombrophilous Forest is one of the most threatened ecosystems in Brazil. It is thus necessary to understand its dynamics of growth over time so that sustainable use and conservation management strategies be proposed. The objective of this study is to evaluate the behavior of the diametric distribution of two pioneer and non-pioneer ecological groups in a fragment of Mixed Ombrophilous Forest considering four years of evaluations: from 2002 to 2017. Exponential, Gamma, Weber and Weibull 3P density probability functions were adjusted. The diametric behavior of the forest and the ecological groups analyzed remained negative exponentially, being a characteristic of natural forests. The distributions that best adjusted were Weibull 3P for the forest as a whole and for the group of pioneers, and Weber for non-pioneers. We concluded that ecological groups have the same pattern of diametric distribution as the forest as a whole. However, different forest management strategies are necessary considering the intrinsic characteristics of each group.

Indefinite integrals of products of special functions

ABSTRACTA method is given for deriving indefinite integrals involving squares and other products of functions which are solutions of second-order linear differential equations. Several variations of the method are presented, which applies directly to functions which obey homogeneous differential equations. However, functions which obey inhomogeneous equations can be incorporated into the products and examples are given of integrals involving products of Bessel functions combined with Lommel, Anger and Weber functions. Many new integrals are derived for a selection of special functions, including Bessel functions, associated Legendre functions, and elliptic integrals. A number of integrals of products of Gauss hypergeometric functions are also presented, which seem to be the first integrals of this type. All results presented have been numerically checked with Mathematica.

Turán-type inequalities for Struve, modified Struve and Anger–Weber functions

ABSTRACTThe aim of this paper is to establish the Turán-type inequalities for Struve functions, modified Struve functions, Anger–Weber functions and Hurwitz zeta function, by using a new form of the Cauchy–Bunyakovsky–Schwarz inequality.

Vertex operator algebras with central charge 1/2 and $-68/7$

In this article we study \textit{simple vertex operator algebras} whose spaces of characters are 3-dimensional, and satisfy \textit{3rd order (modular) linear differential equations}. We classify such vertex operator algebras with central charge 1/2 or $-68/7$. One of the main results is that these vertex operator algebras have \textit{conformal weights} $\{0,1/2,1/16\}$ or $\{0,-2/7,-3/7\}$, respectively, and are isomorphic to the minimal models of central charge $c=c_{3,4}=1/2$ and $c_{2,7}=-68/7$. Moreover, we give the explicit formulas representing characters of the minimal models with $c=1/2$, $-68/7$ by using the classical Weber functions.

Three-dimensional nonparaxial accelerating beams from the transverse Whittaker integral

We investigate three-dimensional nonparaxial linear accelerating beams arising from the transverse Whittaker integral. These beams accelerate along a semicircular trajectory, with almost invariant nondiffracting shapes. The transverse patterns of accelerating beams are determined by their angular spectra, which are constructed from the Mathieu functions, Weber functions, and Fresnel integrals. Our results not only enrich the understanding of multidimensional nonparaxial accelerating beams, but also display their real applicative potential —owing to the usefulness of Mathieu and Weber functions, and Fresnel integrals in describing a wealth of wave phenomena in nature.

Vocoder Simulations of Highly Focused Cochlear Stimulation With Limited Dynamic Range and Discriminable Steps

There is recent interest in focused stimulation of the cochlea via modalities such as tripolar electrical and infrared neural stimulation to improve speech in noise comprehension and music perception. The purpose of this work was to use vocoder-based simulations to investigate speech recognition for broad stimulation (standard monopolar paradigm) versus more focused stimulation under a variety of signal-to-noise ratios, dynamic ranges, and numbers of discriminable loudness steps. Vocoder simulations were used to assess the intelligibility of sentences, consonants, and vowels that were noise vocoded and presented to 7 normal-hearing listeners for identification. A novel aspect of the simulations presented here was the use of nonuniform quantization steps within the dynamic range to more closely simulate the Weber functions observed in cochlear implant users. Intelligibility was assessed for the different filter slopes under a variety of signal-to-noise ratio levels, dynamic ranges, and numbers of discriminable steps. Speech processed via vocoder simulations representing focused stimulation was found to be substantially more intelligible than speech processed via a monopolar electric vocoder simulation, with differences of up to 60 percentage points. There were no significant differences, however, seen between the two focused approaches (signal attenuations of 10 and 17 dB/mm) for the conditions investigated. Speech processed via the highly focused vocoder (17 dB/mm) was robust to constraints on small envelope dynamic range and small number of discriminable steps within the dynamic range, as high performance was maintained with at least a 5 dB dynamic range and eight or more discriminable steps. Significant drops in intelligibility were noted when the number of steps fell below eight. Highly focused stimulation-tripolar electrical and infrared neural stimulation-has potential for increased performance in noise compared with monopolar stimulation, but much work remains to bear this potential out and to take full advantage of each modality's strengths.

Infinite integrals involving Bessel functions by contour integration

The approach of contour integration and residue theorem is systematically developed to evaluate infinite integrals involving Bessel functions or closely related ones. Three independent formulations applicable to different types of integrals are presented. The conventional formulation using Hankel functions as auxiliary functions, whose previous development leave considerable room for improvement, is fully developed. The second and third formulations, which appear unnoticed before, uses as auxiliary functions linear combinations of Bessel and Struve functions, and linear combinations of Anger and Weber functions, respectively. These functions play a crucial role as the Hankel functions do in the conventional formulation. Several general formulae are derived, each expressing an infinite integral in terms of residues. The integrands are products of rational functions, powers and Bessel (Neumann, Struve, Anger or Weber) functions, with some relation satisfied between the index of the power and the order of the Bessel or related function. Typical examples are worked out for each type of integral.

Modular equations for some η-products

The classical modular equations involve bivariate polynomials that can be seen to be univariate with coefficients in the modular invariant $j$. Kiepert found modular equations relating some $\eta$-quotients and the Weber functions $\gamma_2$ and $\gamma_3$. In the present work, we extend this idea to double $\eta$-quotients and characterize all the parameters leading to this kind of equation. We give some properties of these equations, explain how to compute them and give numerical examples.

Exact solutions for dynamic response of a periodic spring and mass structure

This paper derives exact analytical solutions for the dynamic response of a periodic structure for which the unit cell consists of one mass and one spring. The solutions cover arbitrary initial conditions and both polynomial and harmonic external forces. They involve Bessel, Lommel, Weber and hypergeometric functions.

Asymptotic expansions of Bessel, Anger and Weber transformations

In this paper we explore higher order numerical quadrature for the integrations of systems containing Bessel, Anger and Weber functions. The method is constructed by finding the approximate solution of the differential equation and truncating the asymptotic series. Numerical examples based on theoretical results are presented to demonstrate the efficiency of the method.

Simple one-dimensional quantum-mechanical model for a particle attached to a surface

We present a simple one-dimensional quantum-mechanical model for a particle attached to a surface. It leads to the Schrödinger equation for a harmonic oscillator bounded on one side that we solve in terms of Weber functions and discuss the behaviour of the eigenvalues and eigenfunctions. We derive the virial theorem and other exact relationships as well as the asymptotic behaviour of the eigenvalues. We calculate the zero-point energy for model parameters corresponding to H adsorbed on Pd(1 0 0). The model is suitable for an advanced undergraduate or graduate course on quantum mechanics.

Tracking time with song and count: Different Weber functions for musicians and nonmusicians

Participants reproduced target intervals of 6-24 sec with finger taps, with target intervals presented before each of 30 reproductions. During the presentation of the target and reproduction, nonmusician participants were asked to count, sing, or do nothing. In another part of the experiment, musicians were asked to count or sing. Segmentation, either by song or count, minimized deviations from targets and reduced the coefficient of variation (CV) to one third of the nonmusicians' CVs in the no-segmentation condition. The mean deviation from target was significantly lower for musicians than for nonmusicians, as were the musicians' CVs (half the value of the nonmusicians). Most importantly, although the CV of nonmusicians was invariant over the range of intervals conforming to a strict scalar property, it decreased significantly for musicians as target intervals increased.

Schläfli modular equations for generalized Weber functions

Sets of appropriately normalized eta quotients, that we call level n Weber functions, are defined, and certain identities generalizing Weber function identities are proved for these functions. Schläfli type modular equations are explicitly obtained for Generalized Weber Functions associated with a Fricke group Γ0(n)+, for n=2,3,5,7,11,13 and 17.

Optimal Selling Rules in a Regime-Switching Exponential Gaussian Diffusion Model

This paper develops optimal selling rules in asset trading using a regime-switching exponential Gaussian diffusion model. The optimization problem is solved by a combined approach of boundary value problems and probabilistic analysis. A system of linear differential equations with variable coefficients and two-point boundary conditions, satisfied by the objective function of the problem, is derived. The existence and uniqueness of the solution are proved. A closed-form solution in terms of Weber functions is obtained for one-dimensional cases. For m-dimensional cases, a stochastic recursive algorithm for numerically searching the optimal value is developed. Numerical results are reported.

Anisotropic phase-mixing in homogeneous turbulence in a rapidly rotating or in a strongly stratified fluid: An analytical study

Angular phase mixing in rapidly rotating or in strongly stratified flows is quantified for single-time single-point energy components, using linear theory. In addition to potential energy, turbulent kinetic energy is more easily analyzed in terms of its toroidal and poloidal components, and then in terms of vertical and horizontal components. Since the axial symmetry around the direction n (which bears both the system angular velocity and the mean density gradient) is consistent with basic dynamical equations, the input of initial anisotropy is investigated in the axisymmetric case. A general way to construct axisymmetric initial data is used, with a classical expansion in terms of scalar spherical harmonics for the 3D spectral density of kinetic energy e, and a modified expansion for the polarization anisotropy Z, which reflects the unbalance in terms of poloidal and toroidal energy components. The expansion involves Legendre polynomials of arbitrary order, P2n0(cosθ), (n=0,1,2,…,N0), in which the term [cosθ=(k∙n)∕∣k∣] characterizes the anisotropy in k-wavespace; two sets of parameters, β2n(e) and β2n(z), separately generate the directional anisotropy and the polarization anisotropy. In the rotating case, the phase mixing results in damping the polarization anisotropy, so that toroidal and poloidal energy components asymptotically equilibrate after transient oscillations. Complete analytical solutions are found in terms of Bessel functions. The envelope of these oscillations decay with time like (ft)−2 (f being the Coriolis parameter), whereas those for the vertical and horizontal components decay like (ft)−3. The long-time limit of the ratio of horizontal component to vertical one depends only on β2(e), which is eventually related to a classical component in structure-based modeling, independently of the degree of the expansion of the initial data. For the stratified case, both the degree of initial anisotropy and the initial unbalance in terms of potential and poloidal (or kinetic gravity wave) energy are investigated. The latter unbalance is characterized by a ratio χ∕2, assuming initial proportionality between the kinetic energy spectrum and the potential energy one. The phase mixing yields asymptotic equipartition in terms of poloidal and potential energy components, and analytical solutions are found in terms of Weber functions. At large time, the damped oscillations for poloidal, potential and vertical components decay with time like (Nt)−1∕2 (N is the buoyancy frequency), while the oscillations for the horizontal component decay with time like (Nt)−3∕2. The long-time limit of the ratio of horizontal component to vertical one depends only on the parameters χ, β2(e), β0(z), β2(z), and β4(z).