Integral formulae on foliated symmetric spaces

In this article, we generalize known integral formulae (due to Brito–Langevin–Rosenberg, Ranjan and the second author) for foliations of codimension 1 or unit vector fields and obtain an infinite series of such formulae involving invariants of the Weingarten operator of a unit vector field, of the Jacobi operator in its direction, and their products. We write several such formulae explicitly, on locally symmetric spaces as well as on arbitrary Riemannian manifolds where they involve also covariant derivatives of the Jacobi operator. We work also with foliations of codimension 1 (or vector fields) which admit “good” (in a sense) singularities.     

Spatiospectral concentration of vector fields on a sphere

We construct spherical vector bases that are bandlimited and spatially concentrated, or, alternatively, spacelimited and spectrally concentrated, suitable for the analysis and representation of real-valued vector fields on the surface of the unit sphere, as arises in the natural and biomedical sciences, and engineering. Building on the original approach of Slepian, Landau, and Pollak we concentrate the energy of our function bases into arbitrarily shaped regions of interest on the sphere, and within certain bandlimits in the vector spherical-harmonic domain. As with the concentration problem for scalar functions on the sphere, which has been treated in detail elsewhere, a Slepian vector basis can be constructed by solving a finite-dimensional algebraic eigenvalue problem. The eigenvalue problem decouples into separate problems for the radial and tangential components. For regions with advanced symmetry such as polar caps, the spectral concentration kernel matrix is very easily calculated and block-diagonal, lending itself to efficient diagonalization. The number of spatiospectrally well-concentrated vector fields is well estimated by a Shannon number that only depends on the area of the target region and the maximal spherical-harmonic degree or bandwidth. The spherical Slepian vector basis is doubly orthogonal, both over the entire sphere and over the geographic target region. Like its scalar counterparts it should be a powerful tool in the inversion, approximation and extension of bandlimited fields on the sphere: vector fields such as gravity and magnetism in the earth and planetary sciences, or electromagnetic fields in optics, antenna theory and medical imaging.     

Vectorial Slepian Functions on the Ball

Abstract

Due to the uncertainty principle, a function cannot be simultaneously limited in space as well as in frequency. The idea of Slepian functions, in general, is to find functions that are at least optimally spatio-spectrally localized. Here, we are looking for Slepian functions which are suitable for the representation of real-valued vector fields on a three-dimensional ball. We work with diverse vectorial bases on the ball which all consist of Jacobi polynomials and vector spherical harmonics. Such basis functions occur in the singular value decomposition of some tomographic inverse problems in geophysics and medical imaging. Our aim is to find band-limited vector fields that are well-localized in a part of a cone whose apex is situated in the origin. Following the original approach towards Slepian functions, the optimization problem can be transformed into a finite-dimensional algebraic eigenvalue problem. The entries of the corresponding matrix are treated analytically as far as possible. For the remaining integrals, numerical quadrature formulae have to be applied. The eigenvalue problem decouples into a normal and a tangential problem. The number of well-localized vector fields can be estimated by a Shannon number which mainly depends on the maximal radial and angular degree of the basis functions as well as the size of the localization region. We show numerical examples of vectorial Slepian functions on the ball, which demonstrate the good localization of these functions and the accurate estimate of the Shannon number.     

Vector spherical harmonics in 3-D vector tomography

A method of series expansion in terms of vector spherical harmonics intended for inverting line integrated experimental (Doppler) data is proposed to investigate 3-D vector fields in laboratory plasma in spherical tokamak devices. A number of numerical computations demonstrating 3-D reconstructions of model vector fields have been performed to assess the inversion method proposed.     

On traces for functional spaces related to Maxwell's equations Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications

Hodge decompositions of tangential vector fields defined on piecewise regular manifolds are provided. The first step is the study of L² tangential fields and then the attention is focused on some particular Sobolev spaces of order $‐{1\over 2}$\nopagenumbers\end . In order to reach this goal, it is required to properly define the first order differential operators and to investigate their properties. When the manifold Γ is the boundary of a polyhedron Ω, these spaces are important in the analysis of tangential trace mappings for vector fields in H ( curl , Ω) on the whole boundary or on a part of it. By means of these Hodge decompositions, one can then provide a complete characterization of these trace mappings: general extension theorems, from the boundary, or from a part of it, to the inside; definition of suitable dualities and validity of integration by parts formulae. Copyright © 2001 John Wiley & Sons, Ltd.     

The number of flags in finite vector spaces: asymptotic normality and Mahonian statistics

We study the generalized Galois numbers which count flags of length r in N-dimensional vector spaces over finite fields. We prove that the coefficients of those polynomials are asymptotically Gaussian normally distributed as N becomes large. Furthermore, we interpret the generalized Galois numbers as weighted inversion statistics on the descent classes of the symmetric group on N elements and identify their asymptotic limit as the Mahonian inversion statistic when r approaches ∞. Finally, we apply our statements to derive further statistical aspects of generalized Rogers–Szeg? polynomials, reinterpret the asymptotic behavior of linear q-ary codes and characters of the symmetric group acting on subspaces over finite fields, and discuss implications for affine Demazure modules and joint probability generating functions of descent-inversion statistics.     

Some Calculus Rules for Contingent Epiderivatives

This article presents some calculus rules for contingent epiderivatives of set-valued maps. Among other results the main emphasis is focused on a formula for scalar multiplication, sum formulae and chain rules. The calculus of contingent cones and some inversion theorems are used as a tool. Some applications are also given.     

Derivatives of the Efficient Point Multifunction in Parametric Vector Optimization Problems

In this paper, we deal with the sensitivity analysis in vector optimization. More specifically, formulae for inner and outer evaluating the S-derivative of the efficient point multifunction in parametric vector optimization problems are established. These estimating formulae are presented via the set of efficient/weakly efficient points of the S-derivative of the original multifunction, a composite multifunction of the objective function and the constraint mapping. The elaboration of the formulae in vector optimization problems, having multifunction constraints and semiinfinite constraints, is also undertaken. Furthermore, examples are provided for analyzing and illustrating the obtained results.     

A generalization of Cooke's integral inversion formula with application to remote-sensing theory

The remote sensing of environmental particulate pollutants, particularly their size distribution, frequently leads to the solution of first-kind Fredholm integral equations. The corresponding physical kernel tends to smooth the behavior of the required function for all values of the dependent variable. Thus, the problem is ill posed and needs regularization by the introduction of constraints on the solution (closure condition). However, under physically realistic conditions, the original problem can be transformed so that it presents a unique and stable solution. One such condition is the so-called anomalous-diffraction approximation, for which we provide two alternate inversion formulae. We derive a new inversion formula (see our theorem) which generalizes that of Cooke and which also provides, as a special case, one of Titchmarsh's formulae. We propose a unifying viewpoint for a number of known integral inversion formulae, including those of Fox (his first theorem), Hardy, Hankel, Titchmarsh, Cooke, and our own, along with the mutual interrelationships that exist between them (Fig. 1 and Table 1). One solution to the particulate sounding problem is then obtained from a direct application of our formula [Eq. (25)]. An alternate solution is likewise obtained by applying Titchmarsh's formula (II) [Eq. (27)]. Both solutions can be independently recovered from Fox's first theorem, although under somewhat more restrictive conditions. They are shown to be identical, and to provide the unique solution to the remote sensing problem considered.     

Low-frequency dipolar electromagnetic scattering by a solid ellipsoid in lossless environment

Electromagnetic wave scattering phenomena for target identification are important in many applications related to fundamental science and engineering. Here, we present an analytical formulation for the calculation of the magnetic and electric fields that scatter off a highly conductive ellipsoidal body, located within an otherwise homogeneous and isotropic lossless medium. The primary excitation source assumes a time-harmonic magnetic dipole, precisely fixed and arbitrarily orientated that operates at low frequencies and produces the incident fields. The scattering problem itself is modeled with respect to rigorous expansions of the electromagnetic fields at the low-frequency regime in terms of positive integral powers of the real wave number of the ambient. Obviously, the Rayleigh static term and a few dynamic terms are sufficient for the purpose of the present work, as the additional terms are neglected due to their minor contribution. Therein, the classical Maxwell's theory is suitably modified, leading to intertwined either Laplace's or Poisson's equations, accompanied by the impenetrable boundary conditions for the total fields and the limiting behavior at infinity. On the other hand, the complete spatial anisotropy of the three-dimensional space is secured via the introduction of the genuine ellipsoidal coordinate system, being appropriate for tackling incrementally such scattering boundary value problems. The nonaxisymmetric fields are obtained via infinite series expansions in terms of ellipsoidal harmonic eigenfunctions, providing handy closed-form solutions in a compact fashion, whose validity is verified by a straightforward reduction to simpler geometries of the metal object. The main idea is to demonstrate an efficient methodology, according to which the constructed analytical formulae can offer the appropriate environment for a fast numerical estimation of the scattered electromagnetic fields that could be useful for real data inversion.     

Inversion of Analytic Functions via Canonical Polynomials: A Matrix Approach

An alternative to Lagrange inversion for solving analytic systems is our technique of dual vector fields. We implement this approach using matrix multiplication that provides a fast algorithm for computing the coefficients of the inverse function. Examples include calculating the critical points of the sinc function. Maple procedures are included which can be directly translated for doing numerical computations in Java or C. A preliminary version of this paper has been presented at AISC 2006.     

R^n上广义Radon变换的奇异值分解

当广义Radon变换限制在带权的平方可积函数空间时, 该文构造了一类广义 Radon 变换的奇异值分解,给出了它们的逆变换的一些结果, 从而导出了广义 Radon 变换的反演公式以及值域的特征.     

On generalisations of Stieltjes transform

In this paper two different generalisations of the Stieltjes transform have been given along with their inversion formulae. The results are given in the form of two theorems. These two theorems yield many known generalisations of the transform in question as particular cases on specializing the parameters involved therein.     

Fundamental solution of a multidimensional difference equation with periodical and matrix coefficients

Summary The multidimensional (partial) difference equation with periodical coefficients is transformed into an equation for a vector sequence. Integral formulae for the vector fundamental solution are developed and some results about its asymptotic properties are explained. As an example, the results are used for a simple difference equation on a hexagonal grid.     

Fréchet subdifferentials of efficient point multifunctions in parametric vector optimization

In this paper, we establish new formulae for computing and/or estimating the Fréchet subdifferential of the efficient point multifunction of a parametric vector optimization problem. These formulae are presented in a broad class of conventional vector optimization problems with the presence of geometric, operator and (finite and infinite) functional constraints.     

The Pascoletti–Serafini Scalarization Scheme and Linear Vector Optimization

The Pascoletti–Serafini scalarization scheme for general vector optimization problems is studied. It is specified to linear vector optimization to give minimal representation formulae for the weakly efficient solution set and the efficient solution set. Several facts on connectedness of the solution sets of Pascoletti–Serafini’s scalar auxiliary problems, both for linear vector optimization and for nonlinear vector optimization, are established.     

An inversion formula for the Segal–Bargmann transform on a symmetric space of non-compact type

We prove analogs of the heat kernel transform inversion formulae of Golse, Leichtnam and the author [E. Leichtnam, F. Golse, M. Stenzel, Intrinsic microlocal analysis and inversion formulae for the heat equation on compact real-analytic Riemannian manifolds, Ann. Sci. École Norm. Sup. (4) 29 (6) (1996) 669–736. MR1422988 (97h:58153), Theorems 0.3, 0.4] in the setting of a Riemannian symmetric space of Helgason's non-compact type.     

Gould-Hsu反演链及其应用

In this paper,by means of Gould-Hsu inverse series relations,we establish several Gould-Hsu inversion chains.As consequence,some new transformation formulae as well as some famous hypergeometric series identities are derived.     

Quadrature formulas for the Laplace and Mellin transforms

A discrete Laplace transform and its inversion formula are obtained by using a quadrature of the integral Fourier transform which is given in terms of Hermite polynomials and its zeros. This approach yields a convergent discrete formula for the two-sided Laplace transform if the function to be transformed falls off rapidly to zero and satisfies given conditions of integrability, achieving convergence also for singular functions. The inversion formula becomes a quadrature formula for the Bromwich integral. The use of asymptotic formulae yields an algorithm to compute the discrete Laplace transform by using only exponentials.     

Implications of vector attachment and host grooming behaviour for vector population dynamics and distribution of vectors on their hosts

Towards gaining a mechanistic understanding of the co-feeding transmission dynamics of tick-borne diseases, we develop a delay differential equation model for vector-host population dynamics. In addition to the intrinsic demographic dynamics of both vector and host populations, the model has the distribution dynamics of vector individuals on hosts governed by vector attachment and host grooming behaviour. We introduce the concept of basic infestation number, derive analytic formulae for calculating it and use these formulae to characterize the distribution patterns. We also show how some of these patterns naturally lead to bi-stability and nonlinear oscillations in the vector and host populations.