The paper contains a combinatorial theorem (the sequence of Newton polygons of a reccurent sequence of polynomials is quasi-linear) and two applications of it in classical and quantum topology, namely in the behavior of the A-polynomial and a fixed quantum invariant (such as the Jones polynomial) under filling. Our combinatorial theorem, which complements results of Calegari-Walker [CW10] and t...

In this paper, we review the problem of matrix completion and expose its intimate relations with algebraic geometry, combinatorics and graph theory. We present the first necessary and sufficient combinatorial conditions for matrices of arbitrary rank to be identifiable from a set of matrix entries, yielding theoretical constraints and new algorithms for the problem of matrix completion. We conc...

The fundamental inverse problem in distance geometry is the one of finding positions from inter-point distances. Discretizable Molecular Distance Geometry Problem (DMDGP) a subclass (DGP) whose search space can be discretized and represented by binary tree, which explored Branch-and-Prune (BP) algorithm. It turns out that this combinatorial possesses many interesting symmetry properties were st...

Tropical geometry still is a rather young field of mathematics which nevertheless over the past couple of years has proved to be rather powerful for tackling questions in enumerative geometry (see e.g. [Mik05], [GM08], [IKS04]). It also has been used to provide alternative algorithmic means to eliminate variables (see [StY08]). People from applied mathematics such as optimisation or control the...

The matroid is called loopless if the empty subset of E is closed, and is called a combinatorial geometry if in addition all single element subsets of E are closed. A closed subset of E is called a flat of M, and every subset of E has a well-defined rank and corank in the poset of all flats of M. The notion of matroid played a fundamental role in graph theory, coding theory, combinatorial optim...

We analyze the Bianchi Identity as an instance of a basic fact of combinatorial groupoid theory, related to the Homotopy Addition Lemma. Here it becomes formulated in terms of 2-forms with values in the gauge group bundle of a groupoid, and leads in particular to the (Chern-Weil) construction of characteristic classes. The method is that of synthetic differential geometry, using “the first neig...

tournaments are a type of directed graph which have been used to study the geometry classical flag manifolds. We became interested in this graphs because combinatorial properties can be geometric [21]introduced Forman-Ricci curvature for and undirected hypergraphs obtained as particular case. In work we present basic ideas about Forman- Ricci graphs, characterize parabolic terms calculate any t...

We consider a specific class of combinatorial search or optimization problems where the search space gives rise to a closure operator and essentially the hulls are the only relevant subsets that must be checked in a brute force approach. We suggest that such a closure property can help to reduce time complexities. Moreover we propose two types of (structural) parameterization of instance classe...

We develop a fine-scale local analysis of measure entropy and measure sequence entropy based on combinatorial independence. The concepts of measure IE-tuples and measure IN-tuples are introduced and studied in analogy with their counterparts in topological dynamics. Local characterizations of the Pinsker von Neumann algebra and its sequence entropy analogue are given in terms of combinatorial i...

A pseudo-triangle is a simple polygon with exactly three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data structures in computational geometry, as planar bar-and-joint frameworks in rigidity theory and as projections of locally convex surfaces. This survey of current literature includes combinator...