In [GM2], Goresky and MacPherson defined and constructed intersection complexes for topological pseudomanifolds. The complexes are defined in the derived category of sheaves of modules over a constant ring sheaf. Since analytic spaces are of this category, algebraic varieties defined over C have intersection complexes. The intersection complex of a given variety has a variation depending on a s...

We introduce complex intersection bodies and show that their properties and applications are similar to those of their real counterparts. In particular, we generalize Busemann’s theorem to the complex case by proving that complex intersection bodies of symmetric complex convex bodies are also convex. Other results include stability in the complex Busemann-Petty problem for arbitrary measures an...

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 ...

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...