257,312 research outputs found

    Long zero-free sequences in finite cyclic groups

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    A sequence in an additively written abelian group is called zero-free if each of its nonempty subsequences has sum different from the zero element of the group. The article determines the structure of the zero-free sequences with lengths greater than n/2n/2 in the additive group \Zn/ of integers modulo nn. The main result states that for each zero-free sequence (ai)i=1ℓ(a_i)_{i=1}^\ell of length ℓ>n/2\ell>n/2 in \Zn/ there is an integer gg coprime to nn such that if gaiˉ\bar{ga_i} denotes the least positive integer in the congruence class gaiga_i (modulo nn), then Σi=1ℓgaiˉ<n\Sigma_{i=1}^\ell\bar{ga_i}<n. The answers to a number of frequently asked zero-sum questions for cyclic groups follow as immediate consequences. Among other applications, best possible lower bounds are established for the maximum multiplicity of a term in a zero-free sequence with length greater than n/2n/2, as well as for the maximum multiplicity of a generator. The approach is combinatorial and does not appeal to previously known nontrivial facts.Comment: 13 page

    Long nn-zero-free sequences in finite cyclic groups

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    A sequence in the additive group Zn{\mathbb Z}_n of integers modulo nn is called nn-zero-free if it does not contain subsequences with length nn and sum zero. The article characterizes the nn-zero-free sequences in Zn{\mathbb Z}_n of length greater than 3n/2−13n/2-1. The structure of these sequences is completely determined, which generalizes a number of previously known facts. The characterization cannot be extended in the same form to shorter sequence lengths. Consequences of the main result are best possible lower bounds for the maximum multiplicity of a term in an nn-zero-free sequence of any given length greater than 3n/2−13n/2-1 in Zn{\mathbb Z}_n, and also for the combined multiplicity of the two most repeated terms. Yet another application is finding the values in a certain range of a function related to the classic theorem of Erd\H{o}s, Ginzburg and Ziv.Comment: 11 page

    Evolutionary distances in the twilight zone -- a rational kernel approach

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    Phylogenetic tree reconstruction is traditionally based on multiple sequence alignments (MSAs) and heavily depends on the validity of this information bottleneck. With increasing sequence divergence, the quality of MSAs decays quickly. Alignment-free methods, on the other hand, are based on abstract string comparisons and avoid potential alignment problems. However, in general they are not biologically motivated and ignore our knowledge about the evolution of sequences. Thus, it is still a major open question how to define an evolutionary distance metric between divergent sequences that makes use of indel information and known substitution models without the need for a multiple alignment. Here we propose a new evolutionary distance metric to close this gap. It uses finite-state transducers to create a biologically motivated similarity score which models substitutions and indels, and does not depend on a multiple sequence alignment. The sequence similarity score is defined in analogy to pairwise alignments and additionally has the positive semi-definite property. We describe its derivation and show in simulation studies and real-world examples that it is more accurate in reconstructing phylogenies than competing methods. The result is a new and accurate way of determining evolutionary distances in and beyond the twilight zone of sequence alignments that is suitable for large datasets.Comment: to appear in PLoS ON

    How Accurate Must Potentials Be for Successful Modeling of Protein Folding?

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    Protein sequences are believed to have been selected to provide the stability of, and reliable renaturation to, an encoded unique spatial fold. In recently proposed theoretical schemes, this selection is modeled as ``minimal frustration,'' or ``optimal energy'' of the desirable target conformation over all possible sequences, such that the ``design'' of the sequence is governed by the interactions between monomers. With replica mean field theory, we examine the possibility to reconstruct the renaturation, or freezing transition, of the ``designed'' heteropolymer given the inevitable errors in the determination of interaction energies, that is, the difference between sets (matrices) of interactions governing chain design and conformations, respectively. We find that the possibility of folding to the designed conformation is controlled by the correlations of the elements of the design and renaturation interaction matrices; unlike random heteropolymers, the ground state of designed heteropolymers is sufficiently stable, such that even a substantial error in the interaction energy should still yield correct renaturation.Comment: 28 pages, 3 postscript figures; tared, compressed, uuencode

    Filtering free resolutions

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    A recent result of Eisenbud-Schreyer and Boij-S\"oderberg proves that the Betti diagram of any graded module decomposes as a positive rational linear combination of pure diagrams. When does this numerical decomposition correspond to an actual filtration of the minimal free resolution? Our main result gives a sufficient condition for this to happen. We apply it to show the non-existence of free resolutions with some plausible-looking Betti diagrams and to study the semigroup of quiver representations of the simplest "wild" quiver.Comment: We correct a mistake in the proof of Corollary 4.2 in the published version of this paper. The mistake involves an incorrect definition for when two degree sequences are "sufficiently separated". The new definition weakens Theorem 1.3 somewhat, but the examples survive. We thank Amin Nematbakhsh and to Gunnar Floystad for bringing this mistake to our attention. We also correct some minor typo

    Combinatorial and Additive Number Theory Problem Sessions: '09--'19

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    These notes are a summary of the problem session discussions at various CANT (Combinatorial and Additive Number Theory Conferences). Currently they include all years from 2009 through 2019 (inclusive); the goal is to supplement this file each year. These additions will include the problem session notes from that year, and occasionally discussions on progress on previous problems. If you are interested in pursuing any of these problems and want additional information as to progress, please email the author. See http://www.theoryofnumbers.com/ for the conference homepage.Comment: Version 3.4, 58 pages, 2 figures added 2019 problems on 5/31/2019, fixed a few issues from some presenters 6/29/201
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