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Linear programming duality and morphisms

Winfried Hochstättler, Jaroslav Nešetřil (1999)

Commentationes Mathematicae Universitatis Carolinae

In this paper we investigate a class of problems permitting a good characterisation from the point of view of morphisms of oriented matroids. We prove several morphism-duality theorems for oriented matroids. These generalize LP-duality (in form of Farkas' Lemma) and Minty's Painting Lemma. Moreover, we characterize all morphism duality theorems, thus proving the essential unicity of Farkas' Lemma. This research helped to isolate perhaps the most natural definition of strong maps for oriented matroids....

Local cohomology of logarithmic forms

G. Denham, H. Schenck, M. Schulze, M. Wakefield, U. Walther (2013)

Annales de l’institut Fourier

Let Y be a divisor on a smooth algebraic variety X . We investigate the geometry of the Jacobian scheme of Y , homological invariants derived from logarithmic differential forms along Y , and their relationship with the property that Y be a free divisor. We consider arrangements of hyperplanes as a source of examples and counterexamples. In particular, we make a complete calculation of the local cohomology of logarithmic forms of generic hyperplane arrangements.

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...

Minimality of toric arrangements

Giacomo d'Antonio, Emanuele Delucchi (2015)

Journal of the European Mathematical Society

We prove that the complement of a toric arrangement has the homotopy type of a minimal CW-complex. As a corollary we deduce that the integer cohomology of these spaces is torsionfree. We apply discrete Morse theory to the toric Salvetti complex, providing a sequence of cellular collapses that leads to a minimal complex.

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