Convex analysis” is developed for functions defined on integer lattice points. We investigate the class of functions which enjoy a variant of Steinitz’s exchange property. It includes linear functions on matroids, valuations on matroids (in the sense of Dress and Wenzel), and separable concave functions on the integral base polytope of submodular systems. It is shown that a function ω has the S...

A weight function ! : 2 → R¿0 from the set of all subsets of [n]={1; : : : ; n} to the nonnegative real numbers is called shift-monotone in {m+1; : : : ; n} if !({a1; : : : ; aj})¿!({b1; : : : ; bj}) holds for all {a1; : : : ; aj}; {b1; : : : ; bj}⊆ [n] with ai6bi; i = 1; : : : ; j, and if !(A)¿!(B) holds for all A; B⊆ [n] with A⊆B and B\A⊆{m + 1; : : : ; n}. A family F⊆ 2 is called intersectin...

The idea of probabilistic metric space was introduced by Menger and he showed that probabilistic metric spaces are generalizations of metric spaces. Thus, in this paper, we prove some of the important features and theorems and conclusions that are found in metric spaces. At the beginning of this paper, the distance distribution functions are proposed. These functions are essential in defining p...

We prove that if X,X are closed subschemes of a torus T over a non-Archimedean field K, of complementary codimension and with finite intersection, then the stable tropical intersection along a (possibly positive-dimensional, possibly unbounded) connected component C of Trop(X) ∩ Trop(X) lifts to algebraic intersection points, with multiplicities. This theorem requires potentially passing to a s...

Let A be a regular local ring containing 1/2, which is either equicharacteristic, or is smooth over a d.v.r. of mixed characteristic. We prove that the product maps on derived Grothendieck-Witt groups of A satisfy the following property: given two elements with supports which do not intersect properly, their product vanishes. This gives an analogue for “oriented intersection multiplicities” of ...

Let A be a regular local ring containing 1/2, which is either equicharacteristic, or is smooth over a d.v.r. of mixed characteristic. We prove that the product maps on derived Grothendieck-Witt groups of A satisfy the following property: given two elements with supports which do not intersect properly, their product vanishes. This gives an analogue for “oriented intersection multiplicites” of S...

Fix integers n, r ≥ 4 and let F denote a family of r-sets of an n-element set. Suppose that for every four distinct A,B,C,D ∈ F with |A∪B ∪C ∪D| ≤ 2r, we have A∩B ∩C ∩D 6= ∅. We prove that for n sufficiently large, |F| ≤ ( n−1 r−1 ) , with equality only if ⋂ F∈F F 6= ∅. This is closely related to a problem of Katona and a result of Frankl and Füredi [10], who proved a similar statement for thre...

Let D be a domain and let S torsion-free monoid such that has characteristic 0 or the quotient group of satisfies ascending chain condition on cyclic subgroups. We give characterization when algebra [ ] is weakly Krull. As corollaries, we reobtain results Krull resp. factorial, due to Chouinard Chang. Furthermore, deduce generalized algebras analogous our main result characterize domains among ...

In this paper we consider families of compact convex sets in R such that any subfamily of size at most d has a nonempty intersection. We prove some analogues of the central point theorem and Tverberg’s theorem for such families.

We introduce a notion of weak intersection number of a collection of sets, modifying the notion of intersection number due to J.L. Kelley, and obtain an analogue of Kelley’s characterization of Boolean algebras which support a finitely additive strictly positive measure. We also consider graph-theoretic reformulations of the notions of intersection number and weak intersection number. Kelley [4...