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20,843
Loose Hamilton cycles in hypergraphs
, 2008
"... We prove that any kuniform hypergraph on n vertices with minimum degree n at least + o(n) contains a loose Hamilton cycle. The proof strategy is similar to that 2(k−1) used by Kühn and Osthus for the 3uniform case. Though some additional difficulties arise in the kuniform case, our argument her ..."
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Cited by 5 (1 self)
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We prove that any kuniform hypergraph on n vertices with minimum degree n at least + o(n) contains a loose Hamilton cycle. The proof strategy is similar to that 2(k−1) used by Kühn and Osthus for the 3uniform case. Though some additional difficulties arise in the kuniform case, our argument
Loose Hamilton Cycles in Regular Hypergraphs
, 2013
"... We establish a relation between two uniform models of random kgraphs (for constant k ≥ 3) on n labeled vertices: H (k) (n, m), the random kgraph with exactly m edges, and H (k) (n, d), the random dregular kgraph. By extending to kgraphs the switching technique of McKay and Wormald, we show that ..."
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Cited by 4 (2 self)
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that, for some range of d = d(n) and a constant c> 0, if m ∼ cnd, then one can couple H (k) (n, m) and H (k) (n, d) so that the latter contains the former with probability tending to one as n → ∞. In view of known results on the existence of a loose Hamilton cycle in H (k) (n, m), we conclude that H
Loose Hamilton Cycles in Random 3Uniform Hypergraphs
, 2010
"... In the random hypergraph H = Hn,p;3 each possible triple appears independently with probability p. A loose Hamilton cycle can be described as a sequence of edges {xi,yi,xi+1} for i = 1,2,...,n/2 where x1,x2,...,xn/2,y1,y2,...,y n/2 are all distinct. We prove that there exists an absolute constant K& ..."
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Cited by 15 (11 self)
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In the random hypergraph H = Hn,p;3 each possible triple appears independently with probability p. A loose Hamilton cycle can be described as a sequence of edges {xi,yi,xi+1} for i = 1,2,...,n/2 where x1,x2,...,xn/2,y1,y2,...,y n/2 are all distinct. We prove that there exists an absolute constant K
Loose Hamilton Cycles in Random Uniform Hypergraphs
"... In the random kuniform hypergraph Hn,p;k of order n each possible ktuple appears independently with probability p. A loose Hamilton cycle is a cycle of order n in which every pair of adjacent edges intersects in a single vertex. We prove that if pnk−1 / log n tends to infinity with n then lim n→∞ ..."
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Cited by 8 (5 self)
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In the random kuniform hypergraph Hn,p;k of order n each possible ktuple appears independently with probability p. A loose Hamilton cycle is a cycle of order n in which every pair of adjacent edges intersects in a single vertex. We prove that if pnk−1 / log n tends to infinity with n then lim n
Loose Hamilton Cycles in Random kUniform Hypergraphs
, 2010
"... In the random hypergraph Hn,p;k each possible ktuple appears independently with probability p. A loose Hamilton cycle is a cycle in which every pair of adjacent edges intersects in a single vertex. We prove that if pn k−1 / log n tends to infinity with n then lim Pr(Hn,p;k contains a loose Hamilton ..."
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Cited by 2 (2 self)
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In the random hypergraph Hn,p;k each possible ktuple appears independently with probability p. A loose Hamilton cycle is a cycle in which every pair of adjacent edges intersects in a single vertex. We prove that if pn k−1 / log n tends to infinity with n then lim Pr(Hn,p;k contains a loose
Regular Hypergraphs: Asymptotic Counting and Loose Hamilton Cycles
, 2013
"... We present results from two papers by the authors on analysis of dregular kuniform hypergraphs, when k is fixed and the number n of vertices tends to infinity. The first result is approximate enumeration of such hypergraphs, provided d = d(n) = o(nκ), where κ = κ(k) = 1 for all k ≥ 4, while κ(3) ..."
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) = 1/2. The second result is that a random dregular hypergraph contains as a dense subgraph the uniform random hypergraph (a generalization of the ErdősRényi uniform graph), and, in view of known results, contains a loose Hamilton cycle with probability tending to one. 1. Regular kgraphs and k
Hamilton cycles in quasirandom hypergraphs
, 2015
"... We show that, for a natural notion of quasirandomness in kuniform hypergraphs, any quasirandom kuniform hypergraph on n vertices with constant edge density and minimum vertex degree Ω(nk−1) contains a loose Hamilton cycle. We also give a construction to show that a kuniform hypergraph satisfying ..."
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We show that, for a natural notion of quasirandomness in kuniform hypergraphs, any quasirandom kuniform hypergraph on n vertices with constant edge density and minimum vertex degree Ω(nk−1) contains a loose Hamilton cycle. We also give a construction to show that a kuniform hypergraph satisfying
Diractype results for loose Hamilton cycles in uniform hypergraphs
 JOURNAL OF COMBINATORIAL THEORY SER. B
"... A classic result of G. A. Dirac in graph theory asserts that every nvertex graph (n ≥ 3) with minimum degree at least n/2 contains a spanning (socalled Hamilton) cycle. G. Y. Katona and H. A. Kierstead suggested a possible extension of this result for kuniform hypergraphs. There a Hamilton cycl ..."
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Cited by 17 (2 self)
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for Hamilton ℓcycles if ℓ < k/2. In particular, we show that for every ℓ < k/2 every nvertex, kuniform hypergraph with minimum (k −1)degree (1/(2(k −ℓ))+o(1))n contains such a loose Hamilton ℓcycle. This degree condition is approximately tight and was conjectured by D. Kühn and D. Osthus (for ℓ = 1
Graphical models, exponential families, and variational inference
, 2008
"... The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fiel ..."
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Cited by 800 (26 self)
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The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances — including the key problems of computing marginals and modes of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide varietyof algorithms — among them sumproduct, cluster variational methods, expectationpropagation, mean field methods, maxproduct and linear programming relaxation, as well as conic programming relaxations — can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in largescale statistical models.
Modeling and simulation of genetic regulatory systems: A literature review
 JOURNAL OF COMPUTATIONAL BIOLOGY
, 2002
"... In order to understand the functioning of organisms on the molecular level, we need to know which genes are expressed, when and where in the organism, and to which extent. The regulation of gene expression is achieved through genetic regulatory systems structured by networks of interactions between ..."
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Cited by 729 (15 self)
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In order to understand the functioning of organisms on the molecular level, we need to know which genes are expressed, when and where in the organism, and to which extent. The regulation of gene expression is achieved through genetic regulatory systems structured by networks of interactions between DNA, RNA, proteins, and small molecules. As most genetic regulatory networks of interest involve many components connected through interlocking positive and negative feedback loops, an intuitive understanding of their dynamics is hard to obtain. As a consequence, formal methods and computer tools for the modeling and simulation of genetic regulatory networks will be indispensable. This paper reviews formalisms that have been employed in mathematical biology and bioinformatics to describe genetic regulatory systems, in particular directed graphs, Bayesian networks, Boolean networks and their generalizations, ordinary and partial differential equations, qualitative differential equations, stochastic equations, and rulebased formalisms. In addition, the paper discusses how these formalisms have been used in the simulation of the behavior of actual regulatory systems.
Results 1  10
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20,843