Degenerate Kernel Research Articles

New Design of Linear Least-Squares Fixed-Interval Smoother Using Covariance Information

This paper derives recursive linear least-squares fixed-interval smoothing algorithm using covariance information from a Wiener-Hopf integral equation. The algorithm is obtained for the white plus coloured and white Gaussian observation noises. Autocovariance functions of the signal and the coloured noise are expressed using a degenerate kernel. The degenerate kernel can represent general covariance functions of nonstationary or stationary processes by a finite sum of nonrandom functions. The efficiency of the smoother was assured by a numerical example.

New design of linear least-squares fixed-interval smoother using covariance information

This paper derives recursive linear least-squares fixed-interval smoothing algorithm using covariance information by applying an invariant imbedding method to a Wiener-Hopf integral equation. The algorithm is obtained for the white plus coloured observation noise. The signal process is nonstationary stochastic. Autocovariance functions of the signal and the coloured noise are expressed using a degenerate kernel. The degenerate kernel can represent general covariance functions of nonstationary stochastic processes by a finite sum of nonrandom functions.

Cauchy system for Fredholm resolvents with composite displacement kernels

In recent years, the invariant imbedding approach to initial-value solutions of Fredholm integral equations with degenerate or semidegenerate kernels has been discussed with emphasis. In the present paper, with the aid of invariant imbedding, the Cauchy problem for Fredholm resolvents with composite displacement kernels is reduced to that for generalized Chandrasekhar'sX-functions andY-functions. In other words, it is shown how to convert the initial-value solution of the resolvent into a system of simultaneous nonlinear integrodifferential equations ofX-functions andY-functions of a single argument. From the computational aspect, the result seems to be more tractable than the original ones.

Spectral approximation for compact integral operators by degenerate kernel methods

We establish a new improved error estimate for the solution of the integral equation eigenvalue problem by degenerate kernel methods. In [6] these estimates were proved under the assumption of normality of the original kernel as well as of the approximating degenerate kernel. Now we consider any compact integral operator and a general Banach space situation, in contrast to the Hilbert space setting in [6], This will be done by combining the techniques in [6] with the suitably transformed estimates of [5]. Our results show that degenerate kernel methods have, besides their overall property of furnishing easy approximations to eigenfunctions, for eigenvalues an order of convergence comparable to quadrature methods.

Imbedding Approach for Optimal Input Design of Linear System Identification

The design of optimal inputs for identifying parameters in MIMO continuous- time linear nonstationary systems with nonzero initial states is considered. The optimal input is shown to be a solution to a Fredholm integral equation of the second kind with a special type of symmetric semi- degenerate kernel subject to an input energy constraint . The solution methods ar e based on invariant imbedding . Initial val ue systems(I.V.S.'s) of the optimal input and the energy constr aint equation are derived for both cases where the coefficient of the energy constrained equation is pre- assigned and where the interval length of identification experiments is specified a priori . The fixed- interval and fixed- point smoothing solutions of the optimal input can be obtained by sol ving each I.V.S. of the optimal input and the total energy with respect to the interval length simultaneously . Some extensions of the results are discussed and a numerical example is shown.

Solutions of the nonlinear 3-wave equations in three spatial dimensions

Recently Ablowitz and Haberman have shown that, in three spatial dimensions, the nonlinear 3-wave evolution equation results from the compatibility condition between two well-defined first-order linear differential 3×3 systems having common solutions. We construct inversionlike integral equations (I.E.) associated with both these two linear differential systems so that the solutions of the I.E. embody their compatibility conditions. The scalar kernels of these I.E. depend upon three independent variables in such a way that there exist degenerate kernels confined in the three-dimensional coordinate space. Consequently we exhibit, for the nonlinear 3-wave evolution equations, an infinite number of solutions which, at fixed time, are confined in the three-dimensional coordinate space.

Solutions of the generalized nonlinear Schrödinger equation in two spatial dimensions

Recently, from two independent methods, the generalized nonlinear Schrödinger evolution equation in two spatial dimensions has been derived both by Ablowitz and Haberman and by Morris. Here the same extension of the Schrödinger cubic equation is obtained from a two-dimensional spatial inversionlike integral equation when a suitable time dependence is introduced. We investigate the solutions (corresponding to degenerate kernels of the inversion equation) in both cases, where the nonlinear part reduces or not to a cubic term. While in the first case, the solutions are not confined, on the contrary in the second case, we show explicitly for any finite time, that there exists an infinite number of solutions which are confined in the two-dimensional coordinate space.

Multiple scattering theory for magnetoresistance of localized impurities in metals

AbstractIt is shown that starting from a multiple scattering formalism for localized impurities in metals the linearized Boltzmann equation can be solved by the degenerate kernel technique. In this case the Boltzmann equation reduces to an algebraic set of linear equations of finite rank. An expression for the magneto‐resistance, valid for arbitrary strength of external magnetic fields is derived. In the limit of a nearly‐free‐electron‐like host band structure the well‐known results for magnetoresistance and impurity resistivity are reproduced.

Localized non‐magnetic impurities in metals: I. General theory

AbstractA general formalism is developed for localized non‐magnetic impurities in metals. In the limit of cubic symmetry and for non‐zero scattering phase shifts for l ≦ 2, it is shown, that the T‐matrix, describing the scattering properties of the impurity, is diagonal and contains all effects coming from multiple scattering and from the anisotropic band structure of the host, given only by four coefficients of the Green function of the pure host. For localized impurities the Boltzmann equation has a degenerate kernel and can be transformed in an inhomogeneous algebraic set of linear equations of finite rank. The impurity potential is calculated in the muffin‐tin approximation. Screening and recharging effects are described by a constant potential shift chosen in such a way that a Friedel sum rule for the scattering phase shifts is fulfilled.

Generalized Sobolev's phi function for the resolvent of a Milne-type integral equation with a degenerate kernel

It is well known in the field of radiative transfer that Sobolev was the first to introduce the resolvent into Milne's integral equation with a displacement kernel. Thereafter it was shown that the resolvent plays an important role in the theory of formation of spectral lines. In the theory of line-transfer problems, the kernel representation in Milne's integral equation has been used to provide an approximate solution in a manner similar to that given by the discrete ordinales method. In this paper, by means of invariant imbedding we show how to determine an exact solution of a Milne-type integral equation with a degenerate kernel, whose form is more general than the Pincherle-Gourast kernel. A Cauchy system for the resolvent is expressed in terms of generalized Sobolev's Φ- and Ψ-functions, which are computed by solving a system of differential equations for auxiliary functions. Furthermore, these functions are expressed in terms of components of the kernel representation.

The solution of ill‐conditioned linear systems arising from Fredholm equations of the first kind by steepest descents and conjugate gradients

AbstractIt is shown that the methods of steepest descents and conjugate gradients give smooth solutions of the ill‐conditioned systems of linear equations obtained by the discretization of Fredholm equations of the first kind without the use of any regularization techniques. A modification of the usual method of steepest descents is described which is equivalent to applying two cycles of the usual method with very little additional work. A number of examples are worked including one with a degenerate kernel in which the algebraic system is singular.

Method for three-body equations

A recently proposed degenerate-kernel scheme for solving Fredholm integral equations of the second kind gave very good results for Lippmann-Schwinger equations. The method is here extended to give a very simple and practical method for the three-body equations with separable two-body interactions. Numerical calculations of the three-body amplitudes are carried out at positive energies for two simple models. In spite of the more complicated singularities in the kernel as well as in the solutions of the equation, satisfactory convergence is achieved.NUCLEAR REACTIONS Faddeev equation solved. Degenerate kernel scheme used.

Reactor Lattice Calculations with a Degenerate Neutron Thermalization Kernel

In this paper an improved degenerate kernel is obtained and subsequently used instead of the exact thermalization kernel for the calculation of thermal-neutron densities in a heterogeneous reactor ...

Degenerate-kernel Boltzmann distribution functions onR p⊗U p

By means of the structural properties of the Boltzmann operator exact elementary solutions have been constructed for degenerate kernels inp-dimensional spaces. The property of the streaming operatorv·∇+vσ(v) to map a certain class of functions ϕ(x,v) defined on a subset ofR p⊗U p onto another class of functions ϕ(x) defined onR p allows one to algebraize the method of solving the transport equation and the determination of the eigenvalue spectrum.

Multigroup neutron transport with degenerate kernel

Abstract The functional analytic method of Larsen and Habetler [1] is applied to the problem of multigroup neutron transport with degenerate kernel to obtain the eigensolutions of the reduced transport operator. This particular method yields a compact and mathematically rigorous solution of the problem. Applications to the full and half-range problems are indicated.

Invariant imbedding and Fredholm integral equations with degenerate kernels

In a manner similar to that given in preceding papers by Bellman and Ueno, with the aid of the Bellman-Krein formula for the resolvent, we show how to solve Fredholm integral equations of the second kind with degenerate kernels. The standard procedure for solution is to convert it into an equivalent matrix equation, but in this paper it is transformed into a Cauchy problem which can be solved effectively by high speed digital computers.

An initial-value method for Fredholm integral equations with generalized degenerate kernels

An initial-value method is derived for integral equations with generalized degenerate kernels. Both the necessity and sufficiency of the Cauchy system are demonstrated.

Neutron Transport for a Slab with a Degenerate Scattering Kernel

Solutions for collision densities in a slab with an energy- and angle-dependent degenerate scattering kernel are developed. The slab distribution is expanded in a set of regular and singular eigenfunctions and the expansion coefficients are obtained as solutions of a matrix integral equation. From invariance principles, this integral equation depends upon the half-space generalized Milne solution. To obtain this solution one must solve a nonlinear matrix equation for a generalization of the Chandrasekhar H-function. Approximate solutions and numerical calculations for the heavy-gas scatterer are presented, including emergent distributions for the slab.

Error analysis for direct linear integral equation methods

An error analysis of projection methods for solving linear integral equations of the second kind is presented. The relationships between several direct methods for solving integral equations are examined. It is shown that the error analysis given is applicable to other methods, including a modified Nyström method and certain degenerate kernel methods.

Complete-backs cattering effects in neutron wave propagation problem

Abstract Using the Fourier transform technique, the neutron-wave-propagation problem has been studied in the presence of complete backscattering plus isotropic scattering of neutrons. The discrete and continuum solutions have been obtained in one-speed transport theory as well as in multi-velocity transport theory. A one-term degenerate kernel has been used for carrying out the analysis.