257,312 research outputs found
Long zero-free sequences in finite cyclic groups
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 in the additive group \Zn/ of integers modulo .
The main result states that for each zero-free sequence of
length in \Zn/ there is an integer coprime to such that if
denotes the least positive integer in the congruence class
(modulo ), then . 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 , 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 -zero-free sequences in finite cyclic groups
A sequence in the additive group of integers modulo is
called -zero-free if it does not contain subsequences with length and
sum zero. The article characterizes the -zero-free sequences in of length greater than . 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 -zero-free sequence of any given length
greater than in , 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
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?
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
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
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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