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21
A User's Guide To Discrete Morse Theory
 Proc. of the 2001 Internat. Conf. on Formal Power Series and Algebraic Combinatorics, A special volume of Advances in Applied Mathematics
, 2001
"... this paper we present an adaptation of Morse Theory that may be applied to any simplicial complex (or more general cell complex). There have been other adaptations of Morse Theory that can be applied to combinatorial spaces. For example, a Morse theory of piecewise linear functions appears in [26] a ..."
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this paper we present an adaptation of Morse Theory that may be applied to any simplicial complex (or more general cell complex). There have been other adaptations of Morse Theory that can be applied to combinatorial spaces. For example, a Morse theory of piecewise linear functions appears in [26] and the very powerful \Stratied Morse Theory" was developed by Goresky and MacPherson [19],[20]. These theories, especially the latter, have each been successfully applied to prove some very striking results
On efficient sparse integer matrix Smith normal form computations
, 2001
"... We present a new algorithm to compute the Integer Smith normal form of large sparse matrices. We reduce the computation of the Smith form to independent, and therefore parallel, computations modulo powers of wordsize primes. Consequently, the algorithm does not suffer from coefficient growth. W ..."
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Cited by 42 (20 self)
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We present a new algorithm to compute the Integer Smith normal form of large sparse matrices. We reduce the computation of the Smith form to independent, and therefore parallel, computations modulo powers of wordsize primes. Consequently, the algorithm does not suffer from coefficient growth. We have implemented several variants of this algorithm (Elimination and/or BlackBox techniques) since practical performance depends strongly on the memory available. Our method has proven useful in algebraic topology for the computation of the homology of some large simplicial complexes.
Complexes of Directed Trees
 J. Combin. Theory Ser. A
, 1998
"... To every directed graph G one can associate a complex \Delta(G) consisting of directed subforests. This construction, suggested to us by R. Stanley, is especially important in the case of a complete double directed graph Gn , where it leads to studying some interesting representations of the symm ..."
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Cited by 41 (5 self)
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To every directed graph G one can associate a complex \Delta(G) consisting of directed subforests. This construction, suggested to us by R. Stanley, is especially important in the case of a complete double directed graph Gn , where it leads to studying some interesting representations of the symmetric group and corresponds (via StanleyReisner correspondence) to an interesting quotient ring.
Combinatorial Differential Topology and Geometry
 MSRI Publications
, 1999
"... A variety of questions in combinatorics lead one to the task of analyzing a simplicial complex, or a more general cell complex. For example, a standard approach to investigating the structure of a partially ordered set is to instead study the topology of the associated ..."
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Cited by 25 (1 self)
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A variety of questions in combinatorics lead one to the task of analyzing a simplicial complex, or a more general cell complex. For example, a standard approach to investigating the structure of a partially ordered set is to instead study the topology of the associated
COMPUTING OPTIMAL MORSE MATCHINGS
, 2004
"... Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NPhard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computation ..."
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Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NPhard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computational results.
Examples of ZAcyclic and Contractible VertexHomogeneous Simplicial Complexes
 Discrete Comput. Geom
, 2001
"... It was shown in [11] that there are no (nontrivial) 2 and 3dimensional Zacyclic vertexhomogeneous simplicial complexes. In this paper we construct a 5dimensional example and further examples in higher dimensions, one of which is Oliver's example of dimension 11, the only previously kno ..."
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It was shown in [11] that there are no (nontrivial) 2 and 3dimensional Zacyclic vertexhomogeneous simplicial complexes. In this paper we construct a 5dimensional example and further examples in higher dimensions, one of which is Oliver's example of dimension 11, the only previously known example of a noncontractible Zacyclic vertexhomogeneous simplicial complex.
A note on correlations in randomly oriented graphs
, 2009
"... Abstract. Given a graph G, we consider the model where G is given a random orientation by giving each edge a random direction. It is proven that for a, b, s ∈ V (G), the events {s → a} and {s → b} are positively correlated. This correlation persists, perhaps unexpectedly, also if we first condition ..."
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Cited by 8 (5 self)
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Abstract. Given a graph G, we consider the model where G is given a random orientation by giving each edge a random direction. It is proven that for a, b, s ∈ V (G), the events {s → a} and {s → b} are positively correlated. This correlation persists, perhaps unexpectedly, also if we first condition on {s t} for any vertex t = s. With this conditioning it is also true that {s → b} and {a → t} are negatively correlated. A concept of increasing events in random orientations is defined and a general inequality corresponding to Harris inequality is given. The results are obtained by combining a very useful lemma by Colin McDiarmid which relates random orientations with edge percolation, with results by van den Berg, Häggström, Kahn on correlation inequalities for edge percolation. The results are true also for another model of randomly directed graphs.
Computing the Rank of Large Sparse Matrices over Finite Fields
"... We want to achieve efficient exact computations, such as the rank, of sparse matrices over finite fields. We therefore compare the practical behaviors, on a wide range of sparse matrices of the deterministic Gaussian elimination technique, using reordering heuristics, with the probabilistic, blackbo ..."
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We want to achieve efficient exact computations, such as the rank, of sparse matrices over finite fields. We therefore compare the practical behaviors, on a wide range of sparse matrices of the deterministic Gaussian elimination technique, using reordering heuristics, with the probabilistic, blackbox, Wiedemann algorithm. Indeed, we prove here that the latter is the fastest iterative variant of the Krylov methods to compute the minimal polynomial or the rank of a sparse matrix.
Morse Theory And Evasiveness
 Combinatorica
, 2000
"... Introduction. Consider a game played by 2 players, whom we call the hider and the seeker. Let S be a simplex of dimension n, with vertices v 0 , v 1 ; : : : ; v n , and M a subcomplex of S, known to both the hider and the seeker. Let be a face of S, known only to the hider. The seeker is permitted ..."
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Cited by 8 (1 self)
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Introduction. Consider a game played by 2 players, whom we call the hider and the seeker. Let S be a simplex of dimension n, with vertices v 0 , v 1 ; : : : ; v n , and M a subcomplex of S, known to both the hider and the seeker. Let be a face of S, known only to the hider. The seeker is permitted to ask questions of the sort \Is vertex v i in ?" The seeker's goal is to determine whether is in M , using as few questions as possible. The seeker is permitted to use the answers to the earlier questions when choosing which vertex to ask about next. We assume that the seeker chooses each question, given the answers to the previous questions, according to a deterministic algorithm, which we call a decision tree algorithm. For any decision tree algor