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Matroids over a ring

Alex Fink, Luca Moci (2016)

Journal of the European Mathematical Society

We introduce the notion of a matroid M over a commutative ring R , assigning to every subset of the ground set an R -module according to some axioms. When R is a field, we recover matroids. When R = , and when R is a DVR, we get (structures which contain all the data of) quasi-arithmetic matroids, and valuated matroids, i.e. tropical linear spaces, respectively. More generally, whenever R is a Dedekind domain, we extend all the usual properties and operations holding for matroids (e.g., duality), and...

Maximization of distances of regular polygons on a circle

Filip Guldan (1980)

Aplikace matematiky

This paper presents the solution of a basic problem defined by J. Černý which solves a concrete everyday problem in railway and road transport (the problem of optimization of time-tables by some criteria).

Measure and Helly's Intersection Theorem for Convex Sets

N. Stavrakas (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

Let = F α be a uniformly bounded collection of compact convex sets in ℝ ⁿ. Katchalski extended Helly’s theorem by proving for finite ℱ that dim (⋂ ℱ) ≥ d, 0 ≤ d ≤ n, if and only if the intersection of any f(n,d) elements has dimension at least d where f(n,0) = n+1 = f(n,n) and f(n,d) = maxn+1,2n-2d+2 for 1 ≤ d ≤ n-1. An equivalent statement of Katchalski’s result for finite ℱ is that there exists δ > 0 such that the intersection of any f(n,d) elements of ℱ contains a d-dimensional ball of measure...

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