On the plane into plane mappings of hydrodynamic type. Parabolic case

Singularities of plane into plane mappings described by parabolic two-component systems of quasi-linear partial differential equations of the first order are studied. Impediments arising in the application of the original Whitney’s approach to such a case are discussed. Hierarchy of singularities is analyzed by the double-scaling expansion method for the simplest [Formula: see text]-component Jordan system. It is shown that flex is the lowest singularity while higher singularities are given by [Formula: see text] curves which are of cusp type for [Formula: see text], [Formula: see text] Regularization of these singularities by deformation of plane into plane mappings into surface [Formula: see text] to plane is discussed. Applicability of the proposed approach to other parabolic type mappings is noted. We finally compare the results obtained for the parabolic case with non-generic gradient catastrophes for hyperbolic systems.

Properly algebraic and spectrum-finite ideals in Jordan systems

The two main results of this paper are:(i) The set of properly algebraic elements of a Jordan system (algebra, triple system or pair) over an uncountable field is an ideal.(ii) For a semiprimitive Banach Jordan system, the socle, the largest properly algebraic ideal, the largest properly spectrum-finite ideal and the largest von Neumann regular ideal all coincide.