The theorem presented and proved in this paper can be viewed as an extension or improvement of our recent Complete Intersection Theorem [1] and may be called the Complete Nontrivial-Intersection Theorem. It goes considerably beyond the well-known Hilton Milner Theorem [10]. We put the result into the proper perspective with a brief sketch of the key steps in its development, beginning with the ...

This paper continues our study, begun in [MS], of the relationship between the prime ideals of an algebra A and of a subalgebra R such that R ⊂A is a faithfully flat H -Galois extension, for some finite-dimensional Hopf algebra H . In that paper we defined three basic Krull relations, Incomparability (INC), t-Lying Over (t-LO), and Going Up (GU), analogous to the classical Krull relations for p...

Let H be a commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every k ∈ N, let Uk(H) denote the set of all ` ∈ N with the property that there are atoms u1, . . . , uk, v1, . . . , v` such that u1 · . . . · uk = v1 · . . . · v` (thus, Uk(H) is the union of all sets of lengths containing k). The Structure Theore...

Abstract Wefelscheid (Untersuchungen über Fastkörper und Fastbereiche, Habilitationsschrift, Hamburg, 1971) generalised the well-known Theorem of Artin/Schreier about characterization formally real fields and fundamental result Baer/Krull to near-fields. In last fifty years arose from a theory, which analyses entirety orderings field (E. Becker, L. Bröcker, M. Marshall et al.), as presented e.g...

We consider a possibility of the existence of intersection homology morphism, which would be associated to a map of analytic varieties. We assume that the map is an inclusion of codimension one. Then the existence of a morphism follows from Saito’s decomposition theorem. For varieties with conical singularities we show, that the existence of intersection homology morphism is exactly equivalent ...

Using a Riemann-Lebesgue lemma for the Fourier transform over the class of bounded variation functions that vanish at infinity, we prove the Dirichlet–Jordan theorem for functions on this class. Our proof is in the Henstock–Kurzweil integral context and is different to that of Riesz-Livingston [Amer. Math. Monthly 62 (1955), 434–437]. As consequence, we obtain the Dirichlet–Jordan theorem for f...

A quantum version of Bochner’s theorem characterising Fourier transforms of probability measures on locally compact Abelian groups gives a characterisation of the Fourier transforms of Wigner quasi-joint distributions of position and momentum. An analogous quantum Bochner theorem characterises quasi-joint distributions of components of spin. In both cases quantum states in which a true distribu...

The weighted matroid intersection problem has recently been extended to the valuated matroid intersection problem: Given a pair of valuated matroids Mi = (V,Bi, ωi) (i = 1, 2) defined on a common ground set V , find a common base B ∈ B1∩B2 that maximizes ω1(B)+ω2(B). This paper develops a Fenchel-type duality theory related to this problem with a view to establishing a convexity framework for n...