We give an explicit description of the Mirkovic-Vilonen cycles on the affine Grassmannian for arbitrary reductive groups. We also give a combinatorial characterization of the MV polytopes. We prove that a polytope is an MV polytope if and only if every 2-face of it is a rank 2 MV polytope.

For a given family of matrices F , we present a methodology to construct (complex) polytope Barabanov norms (and, when F is nonnegative, polytope Barabanov monotone norms and antinorms). Invariant Barabanov norms have been introduced by Barabanov and constitute an important instrument to analyze the joint spectral radius of a set of matrices. In particular, they played a key role in the disprov...

We provide an explicit combinatorial formula for the volume of the polytope of n× n doubly-stochastic matrices, also known as the Birkhoff polytope. We do this through the description of a generating function for all the lattice points of the closely related polytope of n × n real non-negative matrices with all row and column sums equal to an integer t. We can in fact recover similar formulas f...

We extend recent methods for parametric sequence alignment to the parameter space for scoring RNA folds. This involves the construction of an RNA polytope. A vertex of this polytope corresponds to RNA secondary structures with common branching. We use this polytope and its normal fan to study the effect of varying three parameters in the free energy model that are not determined experimentally....

A dissection of a convex d-polytope is a partition of the polytope into d-simplices whose vertices are among the vertices of the polytope. Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex. The size of a dissection is the number of d-simplices it contains. This paper compares triangulations of maximal size with dissecti...

The Hirsch Conjecture (1957) stated that the graph of a d-dimensional polytope with n facets cannot have (combinatorial) diameter greater than n−d. That is, any two vertices of the polytope can be connected by a path of at most n− d edges. This paper presents the first counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope wit...

Motivated by recent work in parametric sequence alignment, we study the parameter space for scoring RNA folds and construct an RNA polytope. A vertex of this polytope corresponds to RNA secondary structures with common branching. We use this polytope and its normal fan to study the effect of varying three parameters in the free energy model that are not determined experimentally. Our results in...

Let G = (V,E) be a graph with positive integral edge weights. Our problem is to find a matching of maximum weight in G. We present a simple iterative algorithm for this problem that uses a maximum cardinality matching algorithm as a subroutine. Using the current fastest maximum cardinality matching algorithms, we solve the maximum weight matching problem in O(W √ nm logn(n /m)) time, or in O(Wn...

This paper discusses ways of using a single panoramic image (captured by a rotating sensor-line camera having very-high spatial resolution) for the geometric shape recovery of a shown object. The objective is to create a sparse polyhedral model, only allowing a few interactive user inputs for a given single panoramic image. The study was motivated by the general question whether a single panora...

Automated building extraction from multi-source data has attracted great attention in recent years. This paper presents an approach to automated building extraction by fusion of image data, height data and 2D ground plans. In this approach buildings are detected using ground plans and height data. A split-and-merge process is applied to fuse image and height data and derive the parametric forms...