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47
Shuffling Biological Sequences
, 1995
"... This paper considers the following sequence shuffling problem: Given a biological sequence (either DNA or protein) s, generate a random instance among all the permutations of s that exhibit the same frequencies of klets (e.g, dinucleotides, doublets of amino acids, triplets etc.). Since certain bia ..."
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Cited by 17 (0 self)
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This paper considers the following sequence shuffling problem: Given a biological sequence (either DNA or protein) s, generate a random instance among all the permutations of s that exhibit the same frequencies of klets (e.g, dinucleotides, doublets of amino acids, triplets etc.). Since certain biases in the usage of klets are fundamental to biological sequences, effective generation of such sequences is essential for the evaluation of the results of many sequence analysis tools. This paper introduces two sequence shuffling algorithms: A simple swappingbased algorithm is shown to generate a nearrandom instance and appears to work well, although its efficiency is unproven � a generation algorithm based on Euler tours is proven to produce a precisely uniform instance, and hence solve the sequence shuffling problem, in time not much more than linear in the sequence length.
On the core of a conepreserving map
 Trans. Amer. Math. Soc
, 1994
"... ABSTRACT. This is the third of a sequence of papers in an attempt to study the PerronFrobenius theory of a nonnegative matrix and its generalizations from the conetheoretic viewpoint. Our main object of interest here is the core of a conepreserving map. If A is an n x n real matrix which leaves i ..."
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Cited by 13 (3 self)
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ABSTRACT. This is the third of a sequence of papers in an attempt to study the PerronFrobenius theory of a nonnegative matrix and its generalizations from the conetheoretic viewpoint. Our main object of interest here is the core of a conepreserving map. If A is an n x n real matrix which leaves invariant a proper cone K in IR n, then by the core of A relative to K, denoted by coreK(A), we mean the convex cone. nb:1 Ai K. It is shown that when coreK(A) is polyhedral, which is the case whenever K is, then coreK(A) is generated by the distinguished eigenvectors of positive powers of A. The important concept of a distinguished Ainvariant face is introduced, which corresponds to the concept of a distinguished class in the nonnegative matrix case. We prove a significant theorem which describes a onetoone correspondence between the distinguished Ainvariant faces of K and the cycles of the permutation induced by A on the extreme rays of coreK (A), provided that the latter cone is nonzero, simplicial. By an interplay between conetheoretic and graphtheoretic ideas, the extreme rays of the core of a nonnegative matrix are fully described. Characterizations of Kirreducibility or Kprimitivity of A are also found in terms of coreK (A). Several equivalent conditions are also given on a matrix with an invariant proper cone so that its spectral radius is an eigenvalue of index one. An equivalent condition in terms of the peripheral spectrum is also found on a real matrix A with the PerronSchaefer condition for which there exists a proper invariant cone K such that coreK (A) is polyhedral, simplicial, or a single ray. A method of producing a large class of invariant proper cones for a matrix with the PerronSchaefer condition is also offered. 1.
Pólya’s permanent problem
 Electron. J. Combin
, 1996
"... A square real matrix is signnonsingular if it is forced to be nonsingular by its pattern of zero, negative, and positive entries. We give structural characterizations of signnonsingular matrices, digraphs with no even length dicycles, and square nonnegative real matrices whose permanent and determ ..."
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Cited by 10 (0 self)
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A square real matrix is signnonsingular if it is forced to be nonsingular by its pattern of zero, negative, and positive entries. We give structural characterizations of signnonsingular matrices, digraphs with no even length dicycles, and square nonnegative real matrices whose permanent and determinant are equal. The structural characterizations, which are topological in nature, imply polynomial algorithms. 1
ON GENERAL MATRICES HAVING THE PERRONFROBENIUS PROPERTY
, 2008
"... A matrix is said to have the PerronFrobenius propertyif its spectral radius is an eigenvalue with a corresponding nonnegative eigenvector. Matrices having this and similar properties are studied in this paper as generalizations of nonnegative matrices. Sets consisting of such generalized nonnegati ..."
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Cited by 9 (4 self)
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A matrix is said to have the PerronFrobenius propertyif its spectral radius is an eigenvalue with a corresponding nonnegative eigenvector. Matrices having this and similar properties are studied in this paper as generalizations of nonnegative matrices. Sets consisting of such generalized nonnegative matrices are studied and certain topological aspects such as connectedness and closure are proved. Similarity transformations leaving such sets invariant are completely described, and it is shown that a nonnilpotent matrix eventually capturing the PerronFrobenius property is in fact a matrix that already has it.
Spatially structured metapopulation models: Global and local assessment of metapopulation capacity
 Theoret. Popn Biol
, 2001
"... We model metapopulation dynamics in finite networks of discrete habitat patches with given areas and spatial locations. We define and analyze two simple and ecologically intuitive measures of the capacity of the habitat patch network to support a viable metapopulation. Metapopulation persistence cap ..."
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Cited by 9 (2 self)
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We model metapopulation dynamics in finite networks of discrete habitat patches with given areas and spatial locations. We define and analyze two simple and ecologically intuitive measures of the capacity of the habitat patch network to support a viable metapopulation. Metapopulation persistence capacity lM defines the threshold condition for longterm metapopulation persistence as lM> d, where d is defined by the extinction and colonization rate parameters of the focal species. Metapopulation invasion capacity lI sets the condition for successful invasion of an empty network from one small local population as lI> d. The metapopulation capacities lM and lI are defined as the leading eigenvalue or a comparable quantity of an appropriate ‘‘landscape’ ’ matrix. Based on these definitions, we present a classification of a very general class of deterministic, continuoustime and discretetime metapopulation models. Two specific models are analyzed in greater detail: a spatially realistic version of the continuoustime Levins model and the discretetime incidence function model with propagule sizedependent colonization rate and a rescue effect. In both models we assume that the extinction rate increases with decreasing patch area and that the colonization rate increases with patch connectivity. In the spatially realistic Levins model, the two types of metapopulation capacities coincide, whereas the incidence function model possesses a strong Allee effect
On the History of Combinatorial Optimization (till 1960)
"... Introduction As a coherent mathematical discipline, combinatorial optimization is relatively young. When studying the history of the field, one observes a number of independent lines of research, separately considering problems like optimum assignment, shortest spanning tree, transportation, and the ..."
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Cited by 9 (0 self)
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Introduction As a coherent mathematical discipline, combinatorial optimization is relatively young. When studying the history of the field, one observes a number of independent lines of research, separately considering problems like optimum assignment, shortest spanning tree, transportation, and the traveling salesman problem. Only in the 1950's, when the unifying tool of linear and integer programming became available and the area of operations research got intensive attention, these problems were put into one framework, and relations between them were laid. Indeed, linear programming forms the hinge in the history of combinatorial optimization. Its initial conception by Kantorovich and Koopmans was motivated by combinatorial applications, in particular in transportation and transshipment. After the formulation of linear programming as generic problem, and the development in 1947 by Dantzig of the simplex method as a tool, one has tried to attack about all combinatorial opti
THE GROWTH OF POWERS OF A NONNEGATIVE MATRIX
, 1980
"... Let A be a nonnegative n x n matrix. In this paper we study the growth of the powers A no, m = 1,2, 3,... when p(A) = 1. These powers occur naturally in the iteration process which is important in applications and numerical techniques. Roughly speaking, we analyze the asymptotic behavior of each en ..."
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Cited by 7 (3 self)
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Let A be a nonnegative n x n matrix. In this paper we study the growth of the powers A no, m = 1,2, 3,... when p(A) = 1. These powers occur naturally in the iteration process which is important in applications and numerical techniques. Roughly speaking, we analyze the asymptotic behavior of each entry of A on. We apply our main result to determine necessary and sufficient conditions for the convergence to the spectral radius of A of certain ratios naturally associated with the iteration above.