in this paper, various elementary properties of vague rings are obtained. furthermore, the concepts of vague subring, vague ideal, vague prime ideal and vague maximal ideal are introduced, and the validity of some relevant classical results in these settings are investigated.

The Fundamental Principle of Ehrenpreis asserts that any solution (in an appropiate functional space, which we will assume, to simplify, is the space of C∞ functions in R) of a system of homogeneous linear partial differential equations with constant coefficients in R, n ≥ 2, can be can be represented in terms of the exponential polynomials solutions of the system [Ehr, Pal, Bjo]. Another way t...

notions of strongly regular, regular and left(right) regular $gamma$−semigroupsare introduced. equivalent conditions are obtained through fuzzy notion for a$gamma$−semigroup to be either strongly regular or regular or left regular.

n the previous 2 chapters, we developed the fundamental thermodynamic relationships and saw how they are applied to geochemical problems. The tools now in our thermodynamic toolbox are sufficient to deal with most of the phenomena we will encounter in the second half of this book. They are not sufficient, however, to deal with all geochemical problems. In this chapter, will add a final few ther...

This paper presents Riemann test problems for ideal MHD finite volume schemes. The test problems place emphasis on the hyperbolic irregularities of the ideal MHD system, namely the occurrence of intermediate shocks and non-unique solutions. We investigate numerical solutions for the test problems obtained by several commonly used methods (Roe, HLLE, central scheme). All methods turned out to sh...

A priori estimates are derived for a class of weak solutions near a point of elliptic degeneracy. An application is given to the high-speed potential flow of an ideal fluid.

A priori estimates are derived for a class of weak solutions near a point of elliptic degeneracy. An application is given to the high-speed potential flow of an ideal fluid.

In this paper, we prove the local well-posedness for the Ideal MHD equations in the Triebel-Lizorkin spaces and obtain blow-up criterion of smooth solutions. Specially, a recovered proof of [7] for the incompressible Euler equation is given.