Solution of the distributional equation and Green’s functions for scattering problems

A general method is described to obtain Green’s functions involved in nuclear scattering theory problems. By imposing specified boundary conditions on the general solution to the distribution equation, one can readily obtain the Green’s function for the radial Schrödinger equation.     

Plancherel Inversion as Unified Approach to Wavelet Transforms and Wigner Functions

We demonstrate that the Plancherel transform for Type-I groups provides one with a natural, unified perspective for the generalized continuous wavelet transform, on the one hand, and for a class of Wigner functions, on the other. We first prove that a Plancherel inversion formula, well known for Bruhat functions on the group, holds for a much larger class of functions. This result allows us to view the wavelet transform as essentially the inverse Plancherel transform. The wavelet transform of a signal is an L²-function on an appropriately chosen group while the Wigner function is defined on a coadjoint orbit of the group and serves as an alternative characterization of the signal, which is often used in practical applications. The Plancherel transform maps L²-functions on a group unitarily to fields of Hilbert-Schmidt operators, indexed by unitary irreducible representations of the group. The wavelet transform can essentially be looked upon as a restricted inverse Plancherel transform, while Wigner functions are modified Fourier transforms of inverse Plancherel transforms, usually restricted to a subset of the unitary dual of the group. Some known results on both Wigner functions and wavelet transforms, appearing in the literature from very different perspectives, are naturally unified within our approach. Explicit computations on a number of groups illustrate the theory. Communicated by Gian Michele Graf submitted 05/06/01, accepted: 19/09/02     

Numerical calculation of elliptic integrals and elliptic functions. III

Summary This paper is a continuation of [2, 3]. It contains anALGOL program for the incomplete elliptic integral of the third kind based on a theory described in [4]. This program replaces the inadequate one based on the Gauß-transformation which was published in [2]. In addition, anAlgol program for a general complete elliptic integral is presented. Editor's note. In this fascicle, prepublication of algorithms from the Special Functions Series of the Handbook for Automatic Computation is continued. Algorithms are published inAlgol 60 reference language as approved by the IFIP. Contributions in this series should be styled after the most recently published ones.This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by the U. S. Army Research Office —Durham under Contract DA-31-124-ARO-D-257.     

Applications of Wigner’s theorem to positive maps preserving norm of operator products

We apply Wigner’s theorem to positive maps on standard operator algebras that preserve norm of operator products or sum of singular values of operator products. It follows that such preservers are of the form ϕ(A) = U AU* with U either a unitary or antiunitary operator. Selected from Journal of Mathematical Study, 2004, 37(1): 11–16     

Exponential functions of Kronecker products and trace calculation

An identity for the trace of an exponential function of Kronecker products of matrices is derived. This identity is important for the calculation of the grand potential for Fermi systems.     

Chord-length distribution functions and Rice formulae. Application to random media

We consider a stationary and isotropic bi-phasic (pore and solid) medium, draw many lines through it, and see each line as a one-dimensional level-cut process with value 0 or 1 according to whether a regular stationary process X is less or greater than a given level. The intervals corresponding to the points at which X is in a given phase are named chords. We are interested in obtaining information on the chord-length distribution functions. Working with the Palm probability measure and using level crossings techniques, in particular Rice methods, we can obtain not only the exact analytical formula of the chord-length distribution function but also the joint distribution function of the lengths of two successive chords. Finally, we indicate some concrete applications for the computation of usual stereological parameters.     

Hilbert spaces of analytic functions,inverse scattering and operator models.I

This paper develops a method for obtaining linear fractional representations of a givenn×n matrix valued function which is analytic and contractive in either the unit disc or the open upper half plane. The method depends upon the theory of reproducing kernel Hilbert spaces of vector valued functions developed by de Branges. A self-contained account of the relevant aspects of these spaces to this study is included. In addition, the methods alluded to above are used in conjunction with some ideas of Krein, to develop models for simple, closed symmetric [resp. isometric] operators with equal deficiency indices. A number of related issues and applications are also discussed.     

Hilbert spaces of analytic functions,inverse scattering and operator models.II

This is the second and final part of a paper which appeared in a preceding issue of this journal. Herein the methods developed in the earlier sections of this paper are used first, in conjunction with some ideas of Krein, to develop models for simple, closed symmetric [resp. isometric]operators with finite and equal deficiency indices. A number of other related issues and applications are then discussed briefly. These include entropy inequalities, interpolation, parametrization ofJ inner matrices, the Schur algorithm and canonical equations. Finally, a list of misprints for Part I is incorporated at the end.     

Application of variational principle to scattering problems

The validity of the variational principle for scattering problems is examined in the case of ionization of atomic hydrogen by electron impact. The effective charge seen by the scattered electron is determined mathematically in a rigorous way excluding any empirical assumptions. It is shown that the elaborated approach gives effective charge values that are reasonable and have clear physical meaning.     

Incomparable integrals and approximate calculation of monotone Boolean functions

The number of incomparable k-dimensional intervals in the Boolean n-cube is estimated. The result is used to estimate the complexity of approximate computation of an arbitrary monotone Boolean function of n variables.     

Representation of the space of polyanalytic functions as a direct sum of orthogonal subspaces. Application to rational approximations

Suppose thatD={z:|z|<1}, L ₂ (D) is the space of functions square-integrable over area inD,A _k (D) is the set of allk-analytic functions inD, (A ₁ (D)=A(D) is the set of all analytic functions inD),A _k L ₂ (D)=L ₂ (D)∩A _k (D),A ₁ L ₂ (D)=AL ₂ (D),