Local inversion theorems without assuming continuous differentiability

Inversion Assuming Weak Scattering

The study of weak scattering from inhomogeneous media or interface roughness has long been of interest in sonar applications. In an acoustic backscattering model of a stationary field of volume inhomogeneities, a stochastic description of the field is more useful than a deterministic description due to the complex nature of the field. A method based on linear inversion is employed to infer info...

Differentiability Theorems for Non - Linear Elliptic Equations

^ C O and Z)0(x) stand for ——^ ^ " ^ T » (bx ) . . . (o* ) y and Dz stands for all the derivatives Dz for i = \, . . . , N and 0 < |a| < mt (of course if |a | = 0 , Dz = z). Equations of the form (1 ) were discussed in my paper "Partial regularity theorems for elliptic systems" which appeared in the January 1968 issue of the Journal of Mathematics and Mechanics [17] where it was assumed that th...

Differentiability in Local Fields

Local Differentiability of Distance Functions

Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets C for which the distance function dC is continuously differentiable everywhere on an open “tube” of uniform thickness around C. Here a corresponding local theory is developed for the property of dC being continuously differentiable outside of C on some neighborhood of a point x ∈ C. This is shown to be equ...

LASERS WITHOUT INVERSION: DENSITY OPERATOR METHOD

A quantum theory of a two and three-level laser with injected atomic coherence is developed by using a density operator method, to the best of our knowledge, for the first time. The initial atomic coherence plays an essential role. At steady state, the equation of motion for the density operator yields to exhibit laser without inversion and a phase locking but no threshold for the laser fie...