The strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in Euclidean space. These fundamental notions occur in various branches of convex optimization (constrained approximation, convex feasibility problems, linear inequalities, for instance). It is shown that the standard constraint qualificatio...

Let A be a commutative noetherian ring of dimension n. Let P be a projective A[T ]-module. Plumstead ([P]) proved that if rankP > dimA then P splits off a free summand of rank one. It is natural to ask what happens when rankP = dimA. In this paper we investigate this question when P has trivial determinant. Let α : P I be a generic surjection (i.e. I ⊂ A[T ] is an ideal of height n). It is prov...

It is known that the standard intersection and union of type-1 fuzzy sets (i.e., the intersection and union under the minimum t-norm and maximum tconorm) are the only cutworthy operations for type1 fuzzy sets. The aim of this paper is to show that similar property holds also for type-2 fuzzy sets, with respect to some special cutting. As was already demonstrated, the intersection and union of t...

Let R be a commutative Noetherian local ring with residue class field k. In this paper, we mainly investigate direct summands of the syzygy modules of k. We prove that R is regular if and only if some syzygy module of k has a semidualizing summand. After that, we consider whether R is Gorenstein if and only if some syzygy module of k has a G-projective summand.

A geodesic metric space is δ-hyperbolic in the sense of Gromov if and only if the intersection of any two metric balls is almost a ball. In particular, R-trees can be characterised by the property that the intersection of any two metric balls is again a metric ball.

Due to the orbifold singularities, the intersection numbers on the moduli space of curves Mg,n are in general rational numbers rather than integers. We study the properties of the denominators of these intersection numbers and their relationship with the orders of automorphism groups of stable curves. We also present a conjecture about the numerical property for a general class of Hodge integrals.

A splinter is a notion of singularity that has seen numerous recent applications, especially in connection with the direct summand theorem, mixed characteristic minimal model program, Cohen–Macaulayness absolute integral closures and cohomology vanishing theorems. Nevertheless, many basic questions about these singularities remain elusive. One outstanding problem whether property spreads from p...

In a canonical way we establish an AZ{identity (see [9]) and its consequences, the LYM{inequality and the Sperner{property, for the Boolean interval lattice. Further, the Bollobas{inequality for the Boolean interval lattice turns out to be just the LYM{ inequality for the Boolean lattice. We also present an Intersection Theorem for this lattice. Perhaps more surprising is that by our approach t...

We study a prize-collecting version of the uncapacitated facility location problem and of the p-median problem. We say that the uncapacitated facility location polytope has the intersection property if adding the extra equation that fixes the number of opened facilities does not create any fractional extreme point. We characterize the graphs for which this polytope has the intersection property...

Let X be a complex K3 surface, $$\textrm{Diff}(X)$$ the group of diffeomorphisms and $$\textrm{Diff}_0(X)$$ identity component. We prove that fundamental contains free abelian countably infinite rank as direct summand. The summand is detected using families Seiberg–Witten invariants. moduli space Einstein metrics on used key ingredient in proof.