Complexes of elliptic cylinders with a characteristic manifold of the generator element in the form of coordinate straight lines

The complex (three-parameter family) of elliptic cylinders is investigated in the three-dimensional affine space, in which the characteristic multiplicity of the forming element consists of three coordinate axes. The focal variety of the forming element of the considered variety is geometrically characterized. Geometric properties of the complex under study were obtained. It is shown that the studied manifold exists and is determined by a completely integrable system of differential equations. It is proved that the focal variety of the forming element of the complex consists of four geometrically characterized points. The center of the ray of the straight-line congruence of the axes of the cylinder, the indicatrix of the second coordinate vector, the second coordinate line and one of the coordinate planes are fixed. The indicatrix of the first coordinate vector describes a one-parameter family of lines with tangents parallel to the second coordinate vector. The end of the first coordinate vector describes a one-parameter family of lines with tangents parallel to the third coordinate vector. The indicatrix of the third coordinate vector and its end describe congruences of planes parallel to the first coordinate plane. The points of the first coordinate line and the first coordinate plane describe one-parameter families of planes parallel to the coordinate plane indicated above.

On the Morse Index of a Functional Arising in Contact Form Geometry

In this paper we study a functional at infinity associated to a contact form on a three dimensional manifold. The Morse index of this functional at infinity can be decomposed into two parts, one along the characteristic manifold and the other along the normal directions. We prove that we can redistribute the negative directions between the two subspaces through a local deformation of the contact form near a critical point at infinity.

On the Propagation of Weak Discontinuities Along Bicharacteristics in a Radiating Gas

The propagation of weak discontinuities along bicharacteristic curves in the characteristic manifold of the differential equations governing the flow of a radiating gas near the optically thin limit has been discussed. Some explicit criteria for the growth and decay of weak discontinuities along bicharacteristics are given. As a special case, when the discontinuity surface is adjacent to a region of uniform flow, the solution for the velocity gradient at the wave head is specialized to the plane, cylindrical, and spherical waves. For expandng waves, the attenuation induced by geometric factors and the radiative flux, and the growth induced by the upstream flow Mach number are discussed. It is shown that a compressive disturbance steepens into a shock only if the initial disturbance is sufficiently strong.