On complex-valued 2D eikonals. Part four: continuation past a caustic

Theories of monochromatic high-frequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by geometrical optics. These theories have recourse to eikonals that encode information on both phase and amplitude -- in other words, are complex-valued. The following mathematical principle is ultimately behind the scenes: any geometric optical eikonal, which conventional rays engender in some light region, can be consistently continued in the shadow region beyond the relevant caustic, provided an alternative eikonal, endowed with a non-zero imaginary part, comes on stage. In the present paper we explore such a principle in dimension $2.$ We investigate a partial differential system that governs the real and the imaginary parts of complex-valued two-dimensional eikonals, and an initial value problem germane to it. In physical terms, the problem in hand amounts to detecting waves that rise beside, but on the dark side of, a given caustic. In mathematical terms, such a problem shows two main peculiarities: on the one hand, degeneracy near the initial curve; on the other hand, ill-posedness in the sense of Hadamard. We benefit from using a number of technical devices: hodograph transforms, artificial viscosity, and a suitable discretization. Approximate differentiation and a parody of the quasi-reversibility method are also involved. We offer an algorithm that restrains instability and produces effective approximate solutions.Comment: 48 pages, 15 figure

Rational enhancement of enzyme performance in organic solvents. Final technical report, 1992--1996

This research focused on the following: the dependence of enzymatic activity of several model hydrolases in nonaqueous solvents; control of substrate selectivity of the protease subtilisin Carlsberg by the solvent; control of catalytic activity and enantioselectivity of this enzyme in organic solvents by immobilization support; lipase-catalyzed acylation of sugars in anhydrous hydrophobic media; the possibility of accelerating enzymatic processes in organic solvents by certain cosolvents; whether lipase catalysis in organic solvents can be enhanced by introducing interfaces in the in the reaction medium; the structure of proteins suspended in organic solvents; improving enzymatic enantioselectivity in organic solvents; analyzing the plunge in enzymatic activity upon replacing water with organic solvents; and the structural basis for the phenomenon of molecular memory of imprinted proteins in organic solvents