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 k-lets (e.g, dinucleotides, doublets of amino acids, triplets etc.). Since certain biases in the usage of k-lets 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 swapping-based algorithm is shown to generate a near-random 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.

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A square real matrix is sign-nonsingular if it is forced to be nonsingular by its pattern of zero, negative, and positive entries. We give structural characterizations of sign-nonsingular 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

ABSTRACT. This is the third of a sequence of papers in an attempt to study the Perron-Frobenius theory of a nonnegative matrix and its generalizations from the cone-theoretic viewpoint. Our main object of interest here is the core of a cone-preserving 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 A-invariant 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 one-to-one correspondence between the distinguished A-invariant 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 cone-theoretic and graph-theoretic ideas, the extreme rays of the core of a nonnegative matrix are fully described. Characterizations of K-irreducibility or K-primitivity 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 Perron-Schaefer 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 Perron-Schaefer condition is also offered. 1.

We prove generic versions of the no-cloning and no-broadcasting theorems, applicable to essentially any non-classical finite-dimensional probabilistic model that satisfies a no-signaling criterion. This includes quantum theory as well as models supporting “super-quantum ” correlations that violate the Bell inequalities to a larger extent than quantum theory. The proof of our no-broadcasting theorem is significantly more natural and more self-contained than others we have seen: we show that a set of states is broadcastable if, and only if, it is contained in a simplex whose vertices are cloneable, and therefore distinguishable by a single measurement. This necessary and sufficient condition generalizes the quantum requirement that a broadcastable set of states commute. 1

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.

. An integer distance graph is a graph G(D) with the set of integers as vertex set and with an edge joining two vertices u and v if and only if ju \Gamma vj 2 D where D is a subset of the positive integers. We determine the chromatic number (D) of G(D) for some finite distance sets D such as sets of consecutive integers and special sets of cardinality 4. 1. Introduction For any D ` IN with IN the set of all positive integers let G(D) denote the graph with the set ZZ of integers as vertex set and with an edge joining two vertices u and v if and only if ju \Gamma vj 2 D. Such a graph G(D) is called integer distance graph or simply distance graph (of the distance set D). A coloring f : V(G) ! ff 1 ; : : : ; f k g of G is an assignment of colors to the vertices of G such that f(u) 6= f(v) for all adjacent vertices u and v. The minimum number of colors necessary to color G is the chromatic number (G). In this paper we study the chromatic number (G(D)) = (D) = (d 1 ; d 2 ; : : :) of ...

By use of the Gordan-Stiemke Theorem of the alternative we demonstrate the similarity of four theorems in combinatorial matrix theory. Each theorem contains five equivalent conditions, one of which is the existence in a given pattern of a line-sum-symmetric or constant-line-sum matrix which is semi-positive or strictly positive for the pattern. A generalization of the Gordan-Stiemke Theorem is stated in terms of complementary faces of the positive orthant and combinatorial applications are given. Many of our results are classical, but some may be new.

A matrix is said to have the Perron-Frobenius 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 Perron-Frobenius property is in fact a matrix that already has it.