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On the frontier of polynomial computations in tropical geometry
 Journal of Symbolic Computation
"... Abstract. We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as counting the number of connected components. We cha ..."
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Cited by 13 (0 self)
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Abstract. We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as counting the number of connected components. We characterize the borderline between tractable and hard computations by proving N Phardness and #Phardness results under various strong restrictions of the input data, as well as providing polynomial time algorithms for various other restrictions. 1.
Bounds on the number of inference functions of a graphical model
 Formal Power Series and Algebraic Combinatorics (FPSAC 18
, 2006
"... Abstract. Directed and undirected graphical models, also called Bayesian networks and Markov random fields, respectively, are important statistical tools in a wide variety of fields, ranging from computational biology to probabilistic artificial intelligence. We give an upper bound on the number of ..."
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Cited by 9 (2 self)
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Abstract. Directed and undirected graphical models, also called Bayesian networks and Markov random fields, respectively, are important statistical tools in a wide variety of fields, ranging from computational biology to probabilistic artificial intelligence. We give an upper bound on the number of inference functions of any graphical model. This bound is polynomial on the size of the model, for a fixed number of parameters, thus improving the exponential upper bound given by Pachter and Sturmfels [14]. We also show that our bound is tight up to a constant factor, by constructing a family of hidden Markov models whose number of inference functions agrees asymptotically with the upper bound. Finally, we apply this bound to a model for sequence alignment that is used in computational biology.
Can Biology Lead to New Theorems?
, 2005
"... This article argues for an affirmative answer to the question in the title. In future interactions between mathematics and biology, both fields will contribute to each other, and, in particular, research in the life sciences will inspire new theorems in “pure” mathematics. This point is illustrated ..."
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This article argues for an affirmative answer to the question in the title. In future interactions between mathematics and biology, both fields will contribute to each other, and, in particular, research in the life sciences will inspire new theorems in “pure” mathematics. This point is illustrated by a snapshot of four recent contributions from biology to geometry, combinatorics and algebra.
An Introduction to Reconstructing Ancestral Genomes
, 2006
"... Abstract. Recent advances in highthroughput genomics technologies have resulted in the sequencing of large numbers of (near) complete genomes. These genome sequences are being mined for important functional elements, such as ..."
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Abstract. Recent advances in highthroughput genomics technologies have resulted in the sequencing of large numbers of (near) complete genomes. These genome sequences are being mined for important functional elements, such as
The Mystery of Two Straight Lines in Bacterial Genome Statistics
, 2007
"... In special coordinates (codon positionspecific nucleotide frequencies), bacterial genomes form two straight lines in 9dimensional space: one line for eubacterial genomes, another for archaeal genomes. All the 348 distinct bacterial genomes available in Genbank in April 2007, belong to these lines ..."
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In special coordinates (codon positionspecific nucleotide frequencies), bacterial genomes form two straight lines in 9dimensional space: one line for eubacterial genomes, another for archaeal genomes. All the 348 distinct bacterial genomes available in Genbank in April 2007, belong to these lines with high accuracy. The main challenge now is to explain the observed high accuracy. The new phenomenon of complementary symmetry for codon positionspecific nucleotide frequencies is observed. The results of analysis of several codon usage models are presented. We demonstrate that the meanfield approximation, which is also known as contextfree, or complete independence model, or Segre variety, can serve as a reasonable approximation to the real codon usage. The first two principal components of codon usage correlate strongly with genomic G+C content and the optimal growth temperature, respectively. The variation of codon usage along the third component is related to the curvature of the meanfield approximation. First three eigenvalues in codon usage PCA explain 59.1%, 7.8 % and 4.7 % of variation. The eubacterial and archaeal genomes codon usage is clearly distributed along two third order curves with genomic G+C content as a parameter.
Counting Gene Finding Functions
, 2007
"... In biology, a genefinding function is one which takes a DNA sequence and returns the most likely underlying intron/exon structure. Generalizing to graphical models, maps from observed states to their most probable hidden state are called inference functions. While the problem of counting inference ..."
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In biology, a genefinding function is one which takes a DNA sequence and returns the most likely underlying intron/exon structure. Generalizing to graphical models, maps from observed states to their most probable hidden state are called inference functions. While the problem of counting inference functions for a given model is not easy, it has been shown that the number of inference functions grows at most polynomially. In this report, we study the Neyman hidden Markov model of length n, and show that the number of inference functions is exactly 4(n − ⌈n/4⌉) + 2. 1 Genefinding in Biology In all organisms, there is a process where a genomic DNA sequence is transcribed into an RNA sequence, after which the RNA sequence can ultimately go on to become some kind of functional unit. The sequences in the genome that undergo this process are what we refer to as genes, while in between genes in the genome are intergenic sequences that do not code for any functional unit. Very often, the above process also involves the translation of the RNA sequence into an amino acid sequence via a set of rules in which three RNA bases code for
Article URL
, 2012
"... This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. A mixed integer linear programming model to reconstruct phylogenies from single nucleotide polymorphism haplotypes under the maximum parsimon ..."
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This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. A mixed integer linear programming model to reconstruct phylogenies from single nucleotide polymorphism haplotypes under the maximum parsimony criterion
Description: Algebraic combinatorics and convex geometry.
"... The field of geometric combinatorics, and combinatorial polytopes in particular, has recently received a good deal of attention. Convex polytopes whose faces correspond to combinatorial constructions have arisen in a broad spectrum of pure and applied areas. They are tied to the structures of Hopf a ..."
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The field of geometric combinatorics, and combinatorial polytopes in particular, has recently received a good deal of attention. Convex polytopes whose faces correspond to combinatorial constructions have arisen in a broad spectrum of pure and applied areas. They are tied to the structures of Hopf algebras and monoids on one side, while forming the foundation of linear