Harry Gingold

Globally analytic triangularization of a matrix function

Abstract A well-known fundamental theorem of Schur guarantees that every constant matrix A is unitarily similar to a triangular matrix. If A(t) is a matrix-valued function analytic on [ a, b ] with real eigenvalues, it is shown in this paper via a constructive algorithm that there exists a unitary matrix U(t) analytic on [ a, b ] such that U∗(t)A(t)U(t) is triangular. In the process of doing so, we also show that the Gram-Schmidt orthogonalization process preserves global analyticity. A similar algorithm which preserves periodicity works also for a periodic analytic matrix.

An Asymptotic Decomposition Method Applied to Multi-Turning Point Problems

An asymptotic decomposition technique is developed. It is designed and used for 2 by 2 first order singularly perturbed linear differential systems. A new set of decoupled linear integral equations is introduced in the process of the asymptotic analysis. Its usefulness is demonstrated with multi-turning point problems. An adiabatic theorem in quantum mechanics is proved in a general case of degenerate energy levels.

In general, the less degeneracy the less transition. A principle for time-dependent Hamiltonian systems in quantum mechanics

A principle in quantum mechanics is proposed: ‘‘In general, the less degeneracy the less transition.’’ Mathematical support of this principle is given in a setting of a slowly varying time‐dependent Schrodinger equation via a theorem of asymptotic decomposition. Formulas that quantitatively relate transition and degeneracy are developed. Ramifications of those formulas are discussed.

Local principles of wave propagation in inhomogeneous media

Four local principles are proven for waves propagating in a layered medium with a variable wave speed. These principles are (1) that inhomogeneities increase the amplitude of waves generated by a source of fixed strength, (2) that inhomogeneities reduce spatial oscillation, or increase the wavelength, (3) that inhomogeneities decrease transmission, or increase reflection, and (4) that transmission increases monotonically with frequency. Definitions of inhomogeneity, local wave function, and local reflection and transmission coefficients are made as a basis for stating these principles.

Transcendentally small reflection of waves for problems with/without turning points near infinity : a new uniform approach

In this paper the generalized Liouville–Green approximation is used to study the wave reflection with a turning point at infinity. The method provided here unifies the work by many authors in finding the nontrivial behavior of the reflection coefficient for high‐energy particles above barrier in the semiclassical limit.

On the asymptotic summation of difference systems

A new method for asymptotic summation of linear systems of difference equations is proposed and studied. It is based on the introduction of a certain summation equation that pinpoints sufficient conditions for asymptotic summation. These conditions serve as a framework from which new and old theorems follow. In particular the analogues of the fundamental theorems of Levinson and Hartman and Wintner are shown to follow from one and the same framework. Examples are given that are not amenable to other techniques.

Reflection of sound waves by sound‐speed inhomogeneities

An analysis is given of the influence of the rate of change of sound speed on reflection and transmission in a perfect gas. Asymptotic formulas, valid for both low and high frequencies, are developed to compute the reflection and transmission coefficients for one‐dimensional waves propagating through variable‐speed layers. The sound speed may have a discontinuous first derivative. It is shown that local reflection effects are proportional to the square of the derivative of the logarithm of the sound speed. The method predicts reflections for gradually varying sound‐speed profiles having continuous derivatives of all orders. In the special case where the sound speed is a piecewise linear function, the general method produces an exact solution in addition to the asymptotic formulas. This exact solution is valid for arbitrarily large sound‐speed gradient and reduces to the classical result for discontinuous sound speed in the limit of an infinite gradient over an infinitesimal distance. The exact solution pr...

Dichotomies and moving singularities

We consider a linear differential system εσ Φ (t,ε)Y′ =A(t, ε)Y, with ε a small parameter and Φ(t, ε) a function which may vanish in the domain of definition. Under some conditions imposed on the eigenvalues of the matrixA(t, ε), there exists an invertible matrixH(t, ε) which is continuous on ([0,a] × [0, ε0]). The transformationY=H(t, ε)Z takes then dimensional linear system into two differential systems of orderk andn−k respectively, withk<n. Thus the investigaton ofn dimensional systems encountered in singular perturbation as well as in stability theory is considerably simplified.

Differential Equations with Moving Singularities

A moving singularity of a differential equation is a singular point whose location depends on a parameter. The solutions of initial-value problems for such equations are investigated in this paper, with particular reference to the convergence behavior of these solutions as the parameter tends to a singular limit.

Complexity analysis of a top down algorithm motivated by power product expansions

Abstract This paper introduces the Binomial Expansion Cryptosystem, a prototype for a crpytosytem which could also be of interest to small elite organizations. Let be a set of positive integers. The Binomial Expansion Cryptosystem exploits the one to one correspondence between a finite integer power product expansion, , and its associated power series representation , by taking the product representation, converting it into series format, and transmitting a set of N coefficients, namely . The crux of our decryption amounts to solving a finite number of Diophantine equations. It is accomplished through a so called Generalized Top-Down Algorithm and transforms into . Algebraic and complexity properties of the Generalized Top-Down Algorithm are analyzed. Preliminary security issues are discussed. Numerous detailed examples are provided.