Showing papers on "Bessel function published in 1999"

Asymptotics of Plancherel measures for symmetric groups

Abstract: We consider the asymptotics of the Plancherel measures on partitions of $n$ as $n$ goes to infinity. We prove that the local structure of a Plancherel typical partition (which we identify with a Young diagram) in the middle of the limit shape converges to a determinantal point process with the discrete sine kernel. On the edges of the limit shape, we prove that the joint distribution of suitably scaled 1st, 2nd, and so on rows of a Plancherel typical diagram converges to the corresponding distribution for eigenvalues of random Hermitian matrices (given by the Airy kernel). This proves a conjecture due to Baik, Deift, and Johansson by methods different from the Riemann-Hilbert techniques used in their original papers math.CO/9810105 and math.CO/9901118 and from the combinatorial approach proposed by Okounkov in math.CO/9903176. Our approach is based on an exact determinantal formula for the correlation functions of the poissonized Plancherel measures involving a new kernel on the 1-dimensional lattice. This kernel is expressed in terms of Bessel functions and we obtain it as a degeneration of the hypergeometric kernel from the paper math.RT/9904010 by Borodin and Olshanski. Our asymptotic analysis relies on the classical asymptotic formulas for the Bessel functions and depoissonization techniques.

A Compact Cylindrical Green’s Function Expansion for the Solution of Potential Problems

Abstract: We show that an exact expression for the Green's function in cylindrical coordinates is where χ ≡ [R2 + R + (z - z')2]/(2RR'), and Qm-1/2 is the half-integer degree Legendre function of the second kind. This expression is significantly more compact and easier to evaluate numerically than the more familiar cylindrical Green's function expression, which involves infinite integrals over products of Bessel functions and exponentials. It also contains far fewer terms in its series expansion—and is therefore more amenable to accurate evaluation—than does the familiar expression for |-'|-1 that is given in terms of spherical harmonics. This compact Green's function expression is well suited for the solution of potential problems in a wide variety of astrophysical contexts because it adapts readily to extremely flattened (or extremely elongated), isolated mass distributions.

Efficient calculation of lattice sums for free-space periodic Green's function

Abstract: An efficient method to calculate the lattice sums is presented for a one-dimensional (1-D) periodic array of line sources. The method is based on the recurrence relations for Hankel functions and the Fourier integral representation of the zeroth-order Hankel function. The lattice sums of arbitrary high order are then expressed by an integral of elementary functions, which is easily computed using a simple scheme of numerical integration. The calculated lattice sums are used to evaluate the free-space periodic Green's function. The numerical results show that the proposed method provides a highly accurate evaluation of the Green's function with far less computation time, even when the observation point is located near the plane of the array.

Approximate description for Bessel, Bessel–Gauss, and Gaussian beams with finite aperture

Abstract: An approximate analysis is derived for the propagation of Bessel, Bessel–Gauss, and Gaussian beams with a finite aperture. This treatment is based on the fact that the circ function can be expanded into an approximate sum of complex Gaussian functions, so that these three beams are typically expressed as a combination of a set of infinite-aperture Bessel–Gauss beams. Correspondingly, the evaluation of the diffracted field distribution of the beams is reduced to the summation of Bessel–Gauss functions. From analytical results, the present approach provides a good description of the diffracted beams in the region far (greater than a factor of the Fresnel distance) from the aperture. A possible extension of this method to other apertured beams is also discussed.

Multiple Scattering in Solids

Abstract: 1 Introduction.- 1.1 Basic Characteristics of MST.- 1.2 Electronic Structure Calculations.- 1.3 The Aim of This Book.- References.- 2 Intuitive Approach to MST.- 2.1 Huygens' Principle and MST.- 2.1.1 Informal Discussion: Point Scatterers.- 2.1.2 Formal Presentation.- 2.2 Time-Independent Green Functions.- References.- 3 Single-Potential Scattering.- 3.1 Partial-Wave Analysis of Single Potential Scattering.- 3.2 General Considerations.- 3.3 Spherically Symmetric Potentials.- 3.3.1 Free-Particle Solutions.- 3.3.2 The Radial Equation for Central Potentials..- 3.3.3 The Scattering Amplitude.- 3.3.4 Normalization of the Scattering Wave Function.- 3.3.5 Integral Expressions for the Phase Shifts.- 3.4 Nonspherical Potentials.- 3.4.1 Alternative Forms of the Solution.- 3.4.2 Direct Determination of thet-Matrix(*).- 3.5 Wave Function in the Moon Region.- 3.5.1 Displaced-Center Approach: Convex Cells.- 3.5.2 Displaced-Cell Approach: Convex Cells.- 3.5.3 Numerical Example: Convergence for Square Cell.- 3.5.4 Displaced-Cell Approach: Concave Cells (*).- 3.6 Effect of the Potential in the Moon Region.- 3.7 Convergence of Basis Function Expansions (*).- 3.7.1 First Justification.- 3.7.2 Second Justification.- References.- 4 Formal Development of MST.- 4.1 Scattering Theory for a Single Potential.- 4.1.1 The S-Matrix and the t-Matrix.- 4.1.2 t-Matrices and Green Functions.- 4.2 Two-Potential Scattering.- 4.2.1 An Integral Equation for thet-Matrix.- 4.3 The Equations of Multiple Scattering Theory.- 4.3.1 The Wave Functions of Multiple Scattering Theory.- 4.4 Representations.- 4.4.1 The Coordinate Representation.- 4.4.2 The Angular-Momentum Representation.- 4.4.3 Representability of the Green Function and the Wave Function.- 4.4.4 Example of Representability.- 4.4.5 The Representability Theorem.- 4.5 Muffin-Tin Potentials.- References.- 5 MST for Muffin-Tin Potentials.- 5.1 Multiple Scattering Series.- 5.1.1 The Angular-Momentum Representation.- 5.1.2 Electronic Structure of a Periodic Solid.- 5.2 The Green Function in MST.- 5.3 Impurities in MST.- 5.4 Coherent Potential Approximation.- 5.5 Screened MST.- 5.6 Alternative Derivation of MST.- 5.7 Korringa's Derivation.- 5.8 Relation to Muffin-Tin Orbital Theory.- 5.9 MST for E < 0.- 5.9.1 The Two-Scatterer Problem in Three Dimensions.- 5.9.2 Arbitrary Number of MT Potentials.- 5.9.3 Convergence and Accuracy of MST (*).- 5.10 The Convergence Properties of MST (*).- 5.10.1 Energy Convergence.- 5.10.2 Convergence of the Wave Function.- 5.10.3 Convergence of Single-Center Expansion of the Wave Function.- 5.10.4 Summary.- References.- 6 MST for Space-Filling Cells.- 6.1 Historical Development of Full-Cell MST.- 6.2 Derivations of MST for Space-Filling Cells.- 6.3 Full-Cell MST.- 6.3.1 Outgoing-Wave Boundary Conditions.- 6.3.2 Empty-Lattice Test.- 6.3.3 Note on Convergence.- 6.3.4 Full-Potential Wave Functions.- 6.4 The Green Function and Bloch Function.- 6.4.1 The Green Function.- 6.4.2 Alternative Expressions for the Green Function.- 6.4.3 Bloch Functions for Periodic, Space-Filling Cells.- 6.5 Variational Formalisms.- 6.5.1 Variational Derivation of MST.- 6.5.2 A Variational Principle for MST.- 6.5.3 First Variational Derivation of MST.- 6.6 Second Variational Derivation (*).- 6.6.1 The Secular Equation for Nonspherical MT Potentials.- 6.6.2 Space-Filling Cells of Convex Shape.- 6.6.3 Displaced-Cell Approach: Convex Cells (*).- 6.6.4 Displaced-Cell Approach: Concave Cells(*).- 6.7 Construction of the Wave Function.- 6.8 The Closure of Internal Sums (*).- 6.9 Numerical Results.- 6.10 Square Versus Rectangular Matrices (*).- References.- 7 Augmented MST(*).- 7.1 General Comments.- 7.2 MST with a Truncated Basis Set: MT Potentials.- 7.3 General Potentials.- 7.4 Green Functions and the Lloyd Formula.- 7.4.1 Green Functions.- 7.4.2 The Lloyd Formula.- 7.4.3 Single Scatterer.- 7.4.4 A Collection of Scatterers.- 7.4.5 First Derivation.- 7.4.6 Second Derivation.- 7.4.7 The Effects of Truncation.- 7.5 Numerical Study of Two Muffin-Tin Potentials.- 7.6 Convergence of Electronic Structure Calculations.- References.- 8 Relativistic Formalism.- 8.1 General Comments.- 8.2 Generalized Partial Waves.- 8.2.1 The Free-Particle Propagator in the Presence of Spin.- 8.2.2 The (?, ?) or ? Representation.- 8.3 Generalized Structure Constants.- 8.4 Free-Particle Solutions.- 8.4.1 Free-Particle Solution of the Dirac Equation.- 8.4.2 The Free-Particle Propagator.- 8.5 Relativistic Single-Site Scattering Theory.- 8.5.1 Spherically Symmetric Potentials.- 8.5.2 Spin-Orbit Coupling.- 8.5.3 Scalar Relativistic Expressions.- 8.5.4 Generally Shaped, Scalar Potentials.- 8.6 Relativistic Multiple Scattering Theory.- References.- 9 The Poisson Equation.- 9.1 General Comments.- 9.2 Multipole Moments.- 9.3 Comparison with the Schrodinger Equation.- 9.4 Convex Polyhedral Cells.- 9.4.1 Mathematical Preliminaries.- 9.4.2 Non-MT, Space-Filling Cells of Convex Shape.- 9.5 Numerical Results for Convex Cells.- 9.6 Concave Cells.- 9.6.1 Analytic Continuation.- 9.7 Direct Analogy with MST.- 9.7.1 Single Cell Charges.- 9.7.2 Multiple Scattering Solutions.- 9.7.3 Space-Filling Charges of Arbitrary Shape.- References.- A Time-Dependent Green Functions.- B Time-Independent Green Functions.- C Spherical Functions.- C.1 The Spherical Harmonics.- C.2 The Bessel, Neumann, and Hankel Functions.- C.3 Solutions of the Helmholtz Equation.- References.- D Displacements of Spherical Functions References D.- References.- E The Two-Dimensional Square Cell.- E.1 Numerical Results (*).- References.- F Formal Scattering Theory.- F.1 General Comments.- F.2 Initial Conditions and the Moller Operators.- F.3 The Moller Wave Operators.- F.4 The Lippmann-Schwinger Equation.- References.- G Irregular Solutions to the Schrodinger Equation.- H Displacement of Irregular Solutions.- K Conversion of Volume Integrals.- L Energy Derivatives.- M Convergence of the Secular Matrix.- N Summary of MST.- N.1 General Framework.- N.2 Single Potential.- N.3 Multiple Scattering Theory.

Gaussian fluctuation for the number of particles in Airy, Bessel, sine and other determinantal random point fields

Abstract: We prove the Central Limit Theorem for the number of eigenvalues near the spectrum edge for hermitian ensembles of random matrices. To derive our results, we use a general theorem, essentially due to Costin and Lebowitz, concerning the Gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions. As another corollary of Costin-Lebowitz Theorem we prove CLT for the empirical distribution function of the eigenvalues of random matrices from classical compact groups.

A new approach to bessel beams

Abstract: A new interpretation of the non-diffracting Bessel beams is given as a superposition of two fields described by Hankel functions. Within our picture, supported by the Sommerfeld radiation condition, we find that Bessel beams must be of finite transverse extension. Our approach also easily explains propagation characteristics of Bessel-Gauss beams and, in general, the propagation of amplitude modulated Bessel beams. Consequences are discussed and contrasted with the classic theory of Bessel beams.

Casimir energy of a ball and cylinder in the zeta function technique

Abstract: A simple method is proposed to construct the spectral zeta functions required for calculating the electromagnetic vacuum energy with boundary conditions given on a sphere or on an infinite cylinder When calculating the Casimir energy in this approach no exact divergencies appear and no renormalization is needed The starting point of the consideration is the representation of the zeta functions in terms of contour integral, further the uniform asymptotic expansion of the Bessel function is essentially used After the analytic continuation, needed for calculating the Casimir energy, the zeta functions are presented as infinite series containing the Riemann zeta function with rapidly falling down terms The spectral zeta functions are constructed exactly for a material ball and infinite cylinder placed in a uniform endless medium under the condition that the velocity of light does not change when crossing the interface As a special case, perfectly conducting spherical and cylindrical shells are also consi

The Law of the Maximum of a Bessel Bridge

Abstract: Let $M_d$ be the maximum of a standard Bessel bridge of dimension $d$. A series formula for $P(M_d \le a)$ due to Gikhman and Kiefer for $d = 1,2, \ldots$ is shown to be valid for all real $d >0$. Various other characterizations of the distribution of $M_d$ are given, including formulae for its Mellin transform, which is an entire function. The asymptotic distribution of $M_d$ is described both as $d$ tends to infinity and as $d$ tends to zero.

“Best possible” upper and lower bounds for the zeros of the Bessel function _{}()

Abstract: Let jν,k denote the k-th positive zero of the Bessel function Jν(x). In this paper, we prove that for ν > 0 and k = 1, 2, 3, . . . , ν − ak 21/3 ν < jν,k < ν − ak 21/3 ν + 3 20 ak 21/3 ν1/3 . These bounds coincide with the first few terms of the well-known asymptotic expansion jν,k ∼ ν − ak 21/3 ν + 3 20 ak 21/3 ν1/3 + · · · as ν →∞, k being fixed, where ak is the k-th negative zero of the Airy function Ai(x), and so are “best possible”.

A comparison of speaker identification results using features based on cepstrum and Fourier-Bessel expansion

Abstract: A compact representation of speech is possible using Bessel functions because of the similarity between voiced speech and the Bessel functions. Both voiced speech and the Bessel functions exhibit quasiperiodicity and decaying amplitude with time. This paper presents the results of speaker identification experiments using features obtained from (1) the Fourier-Bessel expansion and (2) the cepstral representation of speech frames. Identification scores of 65% and 76% were achieved using features based on J/sub 1/(t) expansion of air-to-ground speech transmission databases of 143 and 1054 test utterances, respectively. The corresponding scores for the two databases using cepstral coefficients of a comparable size were 80% and 88%. A comparison of the two sets of features indicates that J/sub 1/(t) can be used to model the hearing perception much like the mel cepstral coefficients.

Efficiency of second-harmonic generation with Bessel beams

Abstract: A numerical model is used to investigate the conversion efficiency of second-harmonic generation with ``nondiffracting'' Bessel beams. We experimentally validate the model and show, in contrast to a previous prediction, that the conversion efficiency is always less than for Boyd-Kleinman focused Gaussian beams.

Nondiffracting bulk-acoustic x waves in crystals

Abstract: The concept of nondiffracting waves is generalized to encompass bulk-acoustic waves within crystalline media. We introduce acoustic Bessel beams and generalized X waves for anisotropic elastic materials. Detailed numerical predictions for propagation-invariant bulk-acoustic beams of various orders, and also X pulses, are presented for experimental verification. The material parameters used have been chosen appropriately for quartz, the most important material for acoustic device applications.

Analytic methods for partial differential equations

Abstract: 1. Mathematical Preliminaries.- 1.1 Introduction.- 1.2 Characteristics and Classification.- 1.3 Orthogonal Functions.- 1.4 Sturm-Liouville Boundary Value Problems.- 1.5 Legendre Polynomials.- 1.6 Bessel Functions.- 1.7 Results from Complex Analysis.- 1.8 Generalised Functions and the Delta Function.- 1.8.1 Definition and Properties of a Generalised Function.- 1.8.2 Differentiation Across Discontinuities.- 1.8.3 The Fourier Transform of Generalised Functions.- 1.8.4 Convolution of Generalised Functions.- 1.8.5 The Discrete Representation of the Delta Function.- 2. Separation of the Variables.- 2.1 Introduction.- 2.2 The Wave Equation.- 2.3 The Heat Equation.- 2.4 Laplace's Equation.- 2.5 Homogeneous and Non-homogeneous Boundary Conditions.- 2.6 Separation of variables in other coordinate systems.- 3. First-order Equations and Hyperbolic Second-order Equations.- 3.1 Introduction.- 3.2 First-order equations.- 3.3 Introduction to d'Alembert's Method.- 3.4 d'Alembert's General Solution.- 3.5 Characteristics.- 3.6 Semi-infinite Strings.- 4. Integral Transforms.- 4.1 Introduction.- 4.2 Fourier Integrals.- 4.3 Application to the Heat Equation.- 4.4 Fourier Sine and Cosine Transforms.- 4.5 General Fourier Transforms.- 4.6 Laplace transform.- 4.7 Inverting Laplace Transforms.- 4.8 Standard Transforms.- 4.9 Use of Laplace Transforms to Solve Partial Differential Equations.- 5. Green's Functions.- 5.1 Introduction.- 5.2 Green's Functions for the Time-independent Wave Equation.- 5.3 Green's Function Solution to the Three-dimensional Inhomogeneous Wave Equation.- 5.4 Green's Function Solutions to the Inhomogeneous Helmholtz and Schrodinger Equations: An Introduction to Scattering Theory.- 5.5 Green's Function Solution to Maxwell's Equations and Time-dependent Problems.- 5.6 Green's Functions and Optics: Kirchhoff Diffraction Theory.- 5.7 Approximation Methods and the Born Series.- 5.8 Green's Function Solution to the Diffusion Equation.- 5.9 Green's Function Solution to the Laplace and Poisson Equations.- 5.10 Discussion.- A. Solutions of Exercises.

Active signal restoration for the telegraph equation

Abstract: We compensate by a prefilter the distortion of an input signal along an electric line modeled by the telegraph equation The prefilter is based on the so-called flatness property of the telegraph equation We derive the explicit equation of the filter and illustrate the relevance of our approach by a few simulations

Fast computation of modified Mathieu functions applied to elliptical waveguide problems

Abstract: Modified Mathieu functions of the first kind and integer order are expanded in power series, which allow an accurate and timesaving computation. The recurrence relations for the computation of the expansion coefficients are derived and can be easily implemented in computer code. Compared to the calculation with series of Bessel function products, the method presented here is simpler and faster. The efficiency of the method is demonstrated by the computation of eigenmodes of elliptical waveguides.

An improved refinable rational approximation to the ideal time delay

Abstract: Time delays are an integral part of high-speed circuits and control-system applications. Rational approximations to the Laplace transform of a time delay T/sub d/, i.e., e-T(d/sup s/) have been used in the past. These approximations include Pade, Bessel, and other variations. The disadvantage of such approximations is that the quality of the response can only be improved by increasing the order of approximation. In some cases, this results in unstable systems, e.g., a fifth-order Pade approximation with no zeros and five poles is unstable. In this paper, a new rational approximation for the ideal time delay is developed that offers a greater degree of precision and control over the type of response achievable within the same order. Comparisons of the errors between the step responses of the approximation developed here with that of Pade and Bessel show that the new approximation can be tuned to result in a stable operation and the lowest error.

Generation of Bessel light beams under the conditions of internal conical refraction

Abstract: A new optical effect involving transformation of the order of a Bessel beam from the zeroth to the first during propagation of light in a biaxial crystal was investigated theoretically and experimentally. The beam transformation occurs during propagation of circularly polarised light along the optical axes of the crystal under the conditions of internal conical refraction. It is shown that there is a possibility of the total transformation of the input-field energy into a first-order Bessel beam when the length of the crystal or the cone angle of the incident beam are selected appropriately.

q-Laguerre polynomials and big q-Bessel functions and their orthogonality relations

Abstract: The q-Laguerre polynomials correspond to an indetermined moment problem. For explicit discrete non-N-extremal measures corresponding to Ramanujan's ${}_1\psi_1$-summation we complement the orthogonal q-Laguerre polynomials into an explicit orthogonal basis for the corresponding L^2-space. The dual orthogonal system consists of so-called big q-Bessel functions, which can be obtained as a rigorous limit of the orthogonal system of big q-Jacobi polynomials. Interpretations on the SU(1,1) and E(2) quantum groups are discussed.

Smoothing of Stokes's discontinuity for the generalized Bessel function. II

Abstract: This is a continuation of an earlier paper in which we investigated the superasymptotics and hyperasymptotics of the generalized Bessel function ϕ ( z ) = ∑ l = 0 ∞ z l Γ ( l + 1 ) Γ ( ρ l + β ) . where 0 < φ < X and g may be real or complex. In this paper, we consider the case -1 < φ < 0. The analysis in the two cases is not quite the same. Here we shall see that contributions towards hyperasymptotic expansions not only come from adjacent saddles but also from curves that are not even steepest-descent paths through any saddles.

Abel transform and integrals of Bessel local times

Abstract: We study integrals of the type ∫ 0 t ϕ(s) d L s , where ϕ is a positive locally bounded Borel function and L t denotes the local time at level 0 of a Bessel process of dimension d , 0 .

Analysis of Gaussian beam and Bessel beam driven laser accelerators.

Abstract: This paper presents a comparison of Gaussian and Bessel beam driven laser accelerators. The emphasis is on the vacuum beat wave accelerator (VBWA), employing two laser beams of differing wavelengths to impart a net acceleration to particles. Generation of Bessel beams by means of circular slits, holographic optical elements, and axicons is outlined and the image space fields are determined by making use of Huygens{close_quote} principle. Bessel beams{emdash}like Gaussian beams{emdash}experience a Guoy phase shift in the vicinity of a focal region, resulting in a phase velocity that exceeds {ital c}, the speed of light {ital in vacuo}. In the VBWA, by appropriate choice of parameters, the Guoy phases of the laser beams cancel out and the beat wave phase velocity equals {ital c}. The particle energy gain and beam quality are determined by making use of an analytical model as well as simulations. The analytical model{emdash}including the {bold v}{times}{bold B} interaction{emdash}predicts that for equal laser powers Gaussian and Bessel beams lead to identical energy gains. However, three-dimensional, finite-emittance simulations, allowing for detuning, transverse displacements, and including all the electromagnetic field components, show that the energy gain of a Gaussian beam driven VBWA exceeds that of a Bessel beam driven VBWAmore » by a factor of 2{endash}3. The particle beam emerging from the interaction is azimuthally symmetric and collimated, with a relatively small angular divergence. A table summarizing the ratios of final energies, acceleration lengths, and gradients for a number of acceleration mechanisms is given. {copyright} {ital 1999} {ital The American Physical Society}« less