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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 = ℤ , 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 = 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...

Metric ellipses in Minkowski planes.

Wu Senlin, Ji Donghai, Javier Alonso (2005)

Extracta Mathematicae

An ellipse in R2 can be defined as the locus of points for which the sum of the Euclidean distances from the two foci is constant. In this paper we will look at the sets that are obtained by considering in the above definition distances induced by arbitrary norms.

Metric entropy of convex hulls in Hilbert spaces

Wenbo Li, Werner Linde (2000)

Studia Mathematica

Let T be a precompact subset of a Hilbert space. We estimate the metric entropy of co(T), the convex hull of T, by quantities originating in the theory of majorizing measures. In a similar way, estimates of the Gelfand width are provided. As an application we get upper bounds for the entropy of co(T), T = t 1 , t 2 , . . . , | | t j | | ≤ a j , by functions of the a j ’s only. This partially answers a question raised by K. Ball and A. Pajor (cf. [1]). Our estimates turn out to be optimal in the case of slowly decreasing sequences ( a j ) j = 1 ∞ .

Metric Entropy of Homogeneous Spaces

Stanisław Szarek (1998)

Banach Center Publications

For a precompact subset K of a metric space and ε > 0, the covering number N(K,ε) is defined as the smallest number of balls of radius ε whose union covers K. Knowledge of the metric entropy, i.e., the asymptotic behaviour of covering numbers for (families of) metric spaces is important in many areas of mathematics (geometry, functional analysis, probability, coding theory, to name a few). In this paper we give asymptotically correct estimates for covering numbers for a large class of homogeneous...

Metrically convex functions in normed spaces

Stanisław Kryński (1993)

Studia Mathematica

Properties of metrically convex functions in normed spaces (of any dimension) are considered. The main result, Theorem 4.2, gives necessary and sufficient conditions for a function to be metrically convex, expressed in terms of the classical convexity theory.

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