Tandem duplication is a rearrangement process whereby a segment of DNA is replicated and proximally inserted. A sequence of these events is termed an evolution. Many different configurations can arise from such evolut...
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Tandem duplication is a rearrangement process whereby a segment of DNA is replicated and proximally inserted. A sequence of these events is termed an evolution. Many different configurations can arise from such evolutions, generating some interesting combinatorial properties. Firstly, new DNA connections arising in an evolution can be algebraically represented with a word producing automaton. The number of words arising from n tandem duplications can then be recursively derived. Secondly, many distinct evolutions result in the same sequence of words. With the aid of a bi-colored 2d-tree, a Hasse diagram corresponding to a partially ordered set is constructed, for which the number of linear extensions equates to the number of evolutions generating a given word sequence. Thirdly, we implement some subtree prune and graft operations on this structure to show that the total number of possible evolutions arising from n tandem duplications is Pi(n)(k=1) (4(k) - (2k + 1)). The space of structures arising from tandem duplication thus grows at a super-exponential rate with leading order term O(4(1/2n2)). Crown Copyright (C) 2015 Published by Elsevier B.V. All rights reserved.
Throughout this paper denotes a nonprincipal ultrafilter and ℐ denotes the dual ideal. ℙ(ℐ) is the poset of all partial functions p: ω → 2 such that dom(p) ∈ ℐ. In [2], Grigorieff proved that ω1 is preserved in t...
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Throughout this paper denotes a nonprincipal ultrafilter and ℐ denotes the dual ideal. ℙ(ℐ) is the poset of all partial functions p: ω → 2 such that dom(p) ∈ ℐ. In [2], Grigorieff proved that ω1 is preserved in the corresponding generic extension if and only if is a P-point. Later, when Shelah introduced the notion of a proper poset, many people observed that if is a P-point, then ℙ(ℐ) is proper. One way of proving this is to show that player II has a winning strategy in the game for ℙ(ℐ) (see [3, p. 91].)The notion of an Axiom A poset was introduced by Baumgartner [1]. If a poset satisfies Axiom A, then player II has a winning strategy in the game for ℙ, and thus, ℙ is proper. Indeed most of the naturally occurring proper posets satisfy Axiom A (e.g., Mathias's poset and Laver's poset). Thus, it is natural to ask whether or not ℙ(ℐ) satisfies Axiom A. The main result of this paper is a negative answer to this question. We will prove this by introducing another game and showing that is a P-point if and only if the corresponding game is undetermined. We will then show that if ℙ(ℐ) satisfied Axiom A, then player II would have a winning strategy in the corresponding game .We let [X]<ω = {s ⊆ X∣ ∣s∣ < ω}. We let Seq(X) denote the set of finite sequences of elements of X. If s = 〈x0, …, xn〉 ∈ Seq(X) and y ∈ X, then s * 〈y〉 = 〈x0,…,xn,y〉 ∈ Seq(X).
This paper describes a systematic approach to the enumeration of 'non-crossing' geometric configurations built an vertices of a convex n-gon in the plane. It relies on generating functions, symbolic methods, s...
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This paper describes a systematic approach to the enumeration of 'non-crossing' geometric configurations built an vertices of a convex n-gon in the plane. It relies on generating functions, symbolic methods, singularity analysis, and singularity perturbation. Consequences are both exact and asymptotic counting results for trees, forests, graphs, connected graphs, dissections, and partitions. Limit laws of the Gaussian type are also established in this framework;they concern a variety of parameters like number of leaves in trees, number of components or edges in graphs, etc. (C) 1999 Elsevier Science B.V. All rights reserved.
We prove some new results concerning the structure, the combinatorics and the arithmetics of the set PER of all the words w having two periods p and q, p less than or equal to q, which are coprimes and such that \w\=p...
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We prove some new results concerning the structure, the combinatorics and the arithmetics of the set PER of all the words w having two periods p and q, p less than or equal to q, which are coprimes and such that \w\=p+q-2. A basic theorem relating PER with the set of finite standard Sturmian words was proved in de Luca and Mignosi (1994). The main result of this paper is the following simple inductive definition of PER: the empty word belongs to PER. If w is an already constructed word of PER, then also (aw)((-)) and (bw)((-)) belong to PER, where (-) denotes the operator of palindrome left-closure, i.e. it associates to each word u the smallest palindrome word u((-)) having u as a suffix. We show that, by this result, one can construct in a simple way all finite and infinite standard Sturmian words. We prove also that, up to the automorphism which interchanges the letter a with the letter b, any element of PER can be codified by the irreducible fraction p/q. This allows us to construct for any n greater than or equal to 0 a natural bijection, that we call Farey correspondence, of the set of the Farey series of order n+1 and the set of special elements of length n of the set St of all finite Sturmian words. Finally, we introduce the concepts of Farey tree and Farey monoid. This latter is obtained by defining a suitable product operation on the developments in continued fractions of the set of all irreducible fractions p/q.
This article seeks to explain the meaning and function of Mersenne's exhaustive lists of note permutations. As is true for all of Mersenne's labors, the permutations are at their deepest level explicable in te...
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This article seeks to explain the meaning and function of Mersenne's exhaustive lists of note permutations. As is true for all of Mersenne's labors, the permutations are at their deepest level explicable in terms of his religious commitments-in this case, his faith in the ineffable plenitude of God's creation. Yet, Mersenne also intended the permutations and the combinatorial procedures that generate them to serve as the foundation of a pedagogy of compositional invention. That pedagogy, this article proposes, is tacitly informed by the rhetorical notion of copia, most comprehensively theorized by Erasmus in his influential De copia.
The spaces of coinvariants are quotient spaces of integrable s-fractur sign and l-fractur sign2 modules by subspaces generated by the actions of certain subalgebras labeled by a set of points on a complex line. When a...
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We propose a categorical setting for the study of the combinatorics of rational numbers. We find combinatorial interpretation for Bernoulli and Euler numbers and polynomials. (C) 2007 Elsevier Inc. All rights reserved.
We propose a categorical setting for the study of the combinatorics of rational numbers. We find combinatorial interpretation for Bernoulli and Euler numbers and polynomials. (C) 2007 Elsevier Inc. All rights reserved.
A subsequence is obtained from a string by deleting any number of characters;thus in contrast to a substring. a subsequence is not necessarily a contiguous part of the string. Counting subsequences under various const...
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A subsequence is obtained from a string by deleting any number of characters;thus in contrast to a substring. a subsequence is not necessarily a contiguous part of the string. Counting subsequences under various constraints has become relevant to biological sequence analysis, to machine learning, to coding theory, to the analysis of categorical time series in the social sciences, and to the theory of word complexity. We present theorems that lead to efficient dynamic programming algorithms to count (1) distinct subsequences in a string, (2) distinct common subsequences of two strings, (3) matching joint embeddings in two strings, (4) distinct subsequences with a given minimum span, and (5) sequences generated by a string allowing characters to come in runs of a length that is bounded from above. (C) 2008 Elsevier B.V. All rights reserved.
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