We recall that the calculation of homology with integer coecients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which in general are sparse. We provide a review of several algorithms for the calculation of Smith Normal Form of sparse matrices and compare their running times for actual boundary matrices. Then we describe alternative approaches to the calculation of simplicial homology. The last section then describes motivating examples and actual experiments with the GAP package that was implemented by the authors. These examples also include as an example of other homology theories some calculations of Lie algebra homology.

Abstract—In this paper, we investigate the exact conditions under which the ℓ 1 and ℓ 0 minimizations arising in the context of sparse error correction or sparse signal reconstruction are equivalent. We present a much simplified condition for verifying equivalence, which leads to a provably correct algorithm that computes the exact sparsity of the error or the signal needed to ensure equivalence. In the case when the encoding matrix is imbalanced, we show how an optimal diagonal rescaling matrix can be computed via linear programming, so that the rescaled system enjoys the widest possible equivalence. I.

In this paper we apply new geometric and combinatorial methods to the study of phylogenetic mixtures. The focus of the geometric approach is to describe the geometry of phylogenetic mixture distributions for the two state random cluster model, which is a generalization of the two state symmetric (CFN) model. In particular, we show that the set of mixture distributions forms a convex polytope and we calculate its dimension; corollaries include a simple criterion for when a mixture of branch lengths on the star tree can mimic the site pattern frequency vector of a resolved quartet tree. Furthermore, by computing volumes of polytopes we can clarify how “common ” non-identifiable mixtures are under the CFN model. We also present a new combinatorial result which extends any identifiability result for a specific pair of trees of size six to arbitrary pairs of trees. Next we present a positive result showing identifiability of ratesacross-sites models. Finally, we answer a question raised in a previous paper concerning “mixed branch repulsion ” on trees larger than quartet trees under the CFN model.

This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem CompletenessC, which in turn can be solved by a simplicial homology computation. Like other convex hull algorithms, our algorithm is polynomial (in the size of input plus output) for simplicial or simple input. We show that the “no”-case of CompletenessC has a certificate that can be checked in polynomial time (if integrity of the input is guaranteed).

Contrary to popular belief, we show that the optimal parameters for IBM Model 1 are not unique. We demonstrate that, for a large class of words, IBM Model 1 is indifferent among a continuum of ways to allocate probability mass to their translations. We study the magnitude of the variance in optimal model parameters using a linear programming approach as well as multiple random trials, and demonstrate that it results in variance in test set log-likelihood and alignment error rate. 1

One fundamental problem in the field of network coding is to determine the network coding capacity of networks under various network coding schemes. In this thesis, we address the problem with two approaches: matroidal networks and capacity regions. In our matroidal approach, we prove the converse of the theorem which states that, if a network is scalar-linearly solvable then it is a matroidal network associated with a representable matroid over a finite field. As a consequence, we obtain a correspondence between scalar-linearly solvable networks and representable matroids over finite fields in the framework of matroidal networks. We prove a theorem about the scalar-linear solvability of networks and field characteristics. We provide a method for generating scalar-linearly solvable networks that are potentially different from

Abstract In this paper, we apply new geometric and combinatorial methods to the study of phylogenetic mixtures. The focus of the geometric approach is to describe the geometry of phylogenetic mixture distributions for the two state random cluster model, which is a generalization of the two state symmetric (CFN) model. In particular, we show that the set of mixture distributions forms a convex polytope and we calculate its dimension; corollaries include a simple criterion for when a mixture of branch lengths on the star tree can mimic the site pattern frequency vector of a resolved quartet tree. Furthermore, by computing volumes of polytopes we can clarify how “common ” non-identifiable mixtures are under the CFN model. We also present a new combinatorial result which extends any identifiability result for a specific pair of trees of size six to arbitrary pairs of trees. Next we present a positive result showing identifiability of rates-across-sites models. Finally, we answer a question raised in a previous paper concerning “mixed branch repulsion ” on trees larger than quartet trees under the CFN model.

This paper presents a novel source code optimization technique called advanced code hoisting. It aims at moving portions of code from inner loops to outer ones. In contrast to existing code motion techniques, this is done under consideration of control flow aspects. Depending on the conditions of if-statements, moving an expression can lead to an increased number of executions of this expression. This paper contains formal descriptions of the polyhedral models used for control flow analysis so as to suppress a code motion in such a situation. Due to the inherent portability of source code transformations, a very detailed benchmarking using 8 different processors was performed. The application of our implemented techniques to real-life multimedia benchmarks leads to average speed-ups of 25.5%–52 % and energy savings of 33.4%–74.5%. Furthermore, advanced code hoisting leads to improved pipeline and cache behavior and smaller code sizes. 1.

In this paper we explore, from an algorithmic point of view, the extent to which the facial angles and combinatorial structure of a convex polyhedron determine the polyhedron—in particular the edge lengths and dihedral angles of the polyhedron. Cauchy’s rigidity theorem of 1813 states that the dihedral angles are uniquely determined. Finding them is a significant algorithmic problem which we express as a spherical graph drawing problem. Our main result is that the edge lengths, although not uniquely determined, can be found via linear programming. We make use of significant mathematics on convex polyhedra by Stoker, Van Heijenoort, Gale, and Shepherd.