Two applications of Nagata rings and modules

Let [Formula: see text] be a finite type hereditary torsion theory on the category of all modules over a commutative ring. The purpose of this paper is to give two applications of Nagata rings and modules in the sense of Jara [Nagata rings, Front. Math. China 10 (2015) 91–110]. First they are used to obtain Chase’s Theorem for [Formula: see text]-coherent rings. In particular, we obtain the [Formula: see text]-version of Chase’s Theorem, where [Formula: see text] is the classical star operation in ideal theory. In the second half, we apply they to characterize [Formula: see text]-flatness in the sense of Van Oystaeyen and Verschoren [Relative Invariants of Rings-The Commutative Theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 79 (Marcel Dekker, Inc., New York, 1983)].

Charles Bouton and the Navier-Stokes Global Regularity Conjecture

The present article examines the Lie group invariants of the Navier-Stokes equation for incompressible fluids. This is accomplished by applying the invariant theory of Charles Bouton. His analyis shows that since the solutions of the NSE are relative invariants of the scaling group, they must be isobaric polynomials of x,y,z,t and thus infinitely differentiable. Then, bounded energy follows from conservation law. The total angular momentum per unit mass is a scale-invariant vector; it is analyzed and conclusions are drawn about its role in turbulence.

Structures and properties of null scroll in Minkowski 3-space

In this paper, we study structures and properties of Null scrolls. We define the (relative) invariants for Null scrolls by using a kind of standard equation. Using these (relative) invariants of Null scrolls, we give some new characterizations and classifications of Null scrolls and B-scrolls.

Characterizations of symmetric cones by means of the basic relative invariants of homogeneous cones

In this paper, we give necessary and sufficient conditions for a homogeneous cone Ω to be symmetric in two ways. One is by using the multiplier matrix of Ω, and the other is in terms of the basic relative invariants of Ω. In the latter approach, we need to show that the real parts of certain meromorphic rational functions obtained by the basic relative invariants are always positive on the tube domains over Ω. This is a generalization of a result of Ishi and Nomura [Math. Z. 259 (2008), 604–674].