Introduction Logarithmic concavity is a property of a sequence of real numbers, occurring throughout algebraic geometry, convex geometry, and combinatorics. A sequence of positive numbers a0,... ,ad is log-concave if a2 i ≥ ai−1ai+1 for all i. This means that the logarithms, log(ai), form a concave sequence. The condition implies unimodality of the sequence (ai), a property easier to visualize:...

The Generalized Discretizable Molecular Distance Geometry Problem is a distance geometry problems that can be solved by a combinatorial algorithm called “Branch-and-Prune”. It was observed empirically that the number of solutions of YES instances is always a power of two. We give a proof that this event happens with probability one.

51 participants attended this meeting. As in previous Dagstuhl seminars about Computational Geometry, the exchange of ideas and information about the latest developments in the fields of computational and combinatorial geometry was the main purpose of this meeting. In addition, there was a focus on a few topics on individual days of the meeting, featuring longer keynote talks as well as more te...

We investigate the combinatorial geometry obtained from algebraic closure over a fixed subfield in an algebraically closed field. The main result classifies the subgeometries which are isomorphic to projective planes. This is applied to give new examples of algebraic characteristic sets of matroids. The main technique used, which is motivated by classical projective geometry, is that a particul...

Graph theory based methods are powerful means for representing structural systems so that their geometry and topology can be understood clearly. The combination of graph theory based methods and some metaheuristics can offer effective solutions for complex engineering optimization problems. This paper presents a Charged System Search (CSS) algorithm for the free shape optimizations of thin-wall...

On a geometrical view, the conception of map geometries is introduced , which is a nice model of the Smarandache geometries, also new kind of and more general intrinsic geometry of surfaces. Some open problems related combinatorial maps with the Riemann geometry and Smarandache geometries are presented.

In Section 1 we overview combinatorial results on normal polytopes, old and new. These polytopes represent central objects of study in the contemporary discrete convex geometry, on the crossroads of combinatorics, commutative algebra, and algebraic geometry. In Sections 2 and 3 we describe two very different possible ways of advancing the theory of normal polytopes to next essential level, invo...