A polygon splitting algorithm is a combinatorial recipe. The description and the implementation of polygon splitting should not depend on the embedding geometry. Whether a polygon is being split in Euclidean, in spherical, in oriented projective, or in hyperbolic geometry should not be part of the description of the algorithm. The algorithm should be purely combinatorial, or geometry free. The ...

As part of various obstruction theories, non-trivial Massey products have been studied in symplectic and complex geometry, commutative algebra topology for a long time. We introduce general approach to constructing the cohomology moment-angle complexes, using homotopy theoretical combinatorial methods. Our sets unifying way higher arbitrary cohomological classes generalises all existing example...

An antimatroid is a combinatorial structure abstracting the convexity in geometry. In this paper, we explore novel connections between antimatroids and matchings in a bipartite graph. In particular, we prove that a combinatorial structure induced by stable matchings or maximumweight matchings is an antimatroid. Moreover, we demonstrate that every antimatroid admits such a representation by stab...

Several combinatorial aspects of triangulations and their generalizations are studied in this thesis. A triangulation of a point con guration and a d-dimensional polyhedron whose vertices are among the points is a decomposition of the polyhedron using d-simplices with vertices among the points. The two main elds triangulations appear are combinatorial geometry in mathematics and computational g...

Spin networks are at the core of quantum gravity [1]. We have neither the space nor the competence to give an exhaustive list of the physical and philosophical interpretations of this notion (for these, see for example [2] and [3]). New (and old) approaches towards creating a finite quantum theory of general relativity would use combinatorial expressions in Feynman integrals, spin networks, spi...

A combinatorial map is a connected topological graph cellularly embeddedin a surface. This monograph concentrates on the automorphism groupof a map, which is related to the automorphism groups of a Klein surfaceand a Smarandache manifold, also applied to the enumeration of unrootedmaps on orientable and non-orientable surfaces. A number of results for theautomorphism groups ...

In analogy to valuation characterizations and kinematic formulas of convex geometry, we develop a combinatorial theory of invariant valuations and kinematic formulas for finite lattices. Combinatorial kinematic formulas are shown to have application to some probabilistic questions, leading in turn to polynomial identities for Möbius functions and Whitney numbers.

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...

human snake bitten poisoning is a serious threat in many tropical and subtropical countries such as iran. the best acceptable treatment of envenomated humans is antivenoms; however they have a series of economic and medical problems and need more improvements. in this study a combinatorial human immunoglobulin gene library against some of iranian snakes venoms was constructed. total rna prepare...

This is a personal view of some problems on minimal surfaces, Ricci flow, polyhedral geometric structures, Haken 4–manifolds, contact structures and Heegaard splittings, singular incompressible surfaces after the Hamilton–Perelman revolution. We give sets of problems based on the following themes; Minimal surfaces and hyperbolic geometry of 3–manifolds. In particular, how do minimal surfaces gi...