871 research outputs found
Berezin Kernels and Analysis on Makarevich Spaces
Following ideas of van Dijk and Hille we study the link which exists between
maximal degenerate representations and Berezin kernels. We consider the
conformal group of a simple real Jordan algebra . The
maximal degenerate representations () we shall study
are induced by a character of a maximal parabolic subgroup of . These representations can be realized on a space of
smooth functions on . There is an invariant bilinear form
on the space . The problem we consider is to diagonalize this bilinear
form , with respect to the action of a symmetric subgroup
of the conformal group . This bilinear form can be written as an
integral involving the Berezin kernel , an invariant kernel on the
Riemannian symmetric space , which is a Makarevich symmetric space in the
sense of Bertram. Then we can use results by van Dijk and Pevzner who computed
the spherical Fourier transform of . From these, one deduces that the
Berezin kernel satisfies a remarkable Bernstein identity : where is an invariant differential operator on
and is a polynomial. By using this identity we compute a Hua
type integral which gives the normalizing factor for an intertwining operator
from to . Furthermore we obtain the diagonalization of the
invariant bilinear form with respect to the action of the maximal compact group
of the conformal group
On a Speculated Relation Between Chv\'atal-Sankoff Constants of Several Sequences
It is well known that, when normalized by n, the expected length of a longest
common subsequence of d sequences of length n over an alphabet of size sigma
converges to a constant gamma_{sigma,d}. We disprove a speculation by Steele
regarding a possible relation between gamma_{2,d} and gamma_{2,2}. In order to
do that we also obtain new lower bounds for gamma_{sigma,d}, when both sigma
and d are small integers.Comment: 13 pages. To appear in Combinatorics, Probability and Computin
Symmetric Spaces and Star representations II : Causal Symmetric Spaces
We construct and identify star representations canonically associated with
holonomy reducible simple symplectic symmetric spaces. This leads the a
non-commutative geometric realization of the correspondence between causal
symmetric spaces of Cayley type and Hermitian symmetric spaces of tube type.Comment: 13 page
Sum-of-squares lower bounds for planted clique
Finding cliques in random graphs and the closely related "planted" clique
variant, where a clique of size k is planted in a random G(n, 1/2) graph, have
been the focus of substantial study in algorithm design. Despite much effort,
the best known polynomial-time algorithms only solve the problem for k ~
sqrt(n).
In this paper we study the complexity of the planted clique problem under
algorithms from the Sum-of-squares hierarchy. We prove the first average case
lower bound for this model: for almost all graphs in G(n,1/2), r rounds of the
SOS hierarchy cannot find a planted k-clique unless k > n^{1/2r} (up to
logarithmic factors). Thus, for any constant number of rounds planted cliques
of size n^{o(1)} cannot be found by this powerful class of algorithms. This is
shown via an integrability gap for the natural formulation of maximum clique
problem on random graphs for SOS and Lasserre hierarchies, which in turn follow
from degree lower bounds for the Positivestellensatz proof system.
We follow the usual recipe for such proofs. First, we introduce a natural
"dual certificate" (also known as a "vector-solution" or "pseudo-expectation")
for the given system of polynomial equations representing the problem for every
fixed input graph. Then we show that the matrix associated with this dual
certificate is PSD (positive semi-definite) with high probability over the
choice of the input graph.This requires the use of certain tools. One is the
theory of association schemes, and in particular the eigenspaces and
eigenvalues of the Johnson scheme. Another is a combinatorial method we develop
to compute (via traces) norm bounds for certain random matrices whose entries
are highly dependent; we hope this method will be useful elsewhere
A Computational Method for the Rate Estimation of Evolutionary Transpositions
Genome rearrangements are evolutionary events that shuffle genomic
architectures. Most frequent genome rearrangements are reversals,
translocations, fusions, and fissions. While there are some more complex genome
rearrangements such as transpositions, they are rarely observed and believed to
constitute only a small fraction of genome rearrangements happening in the
course of evolution. The analysis of transpositions is further obfuscated by
intractability of the underlying computational problems.
We propose a computational method for estimating the rate of transpositions
in evolutionary scenarios between genomes. We applied our method to a set of
mammalian genomes and estimated the transpositions rate in mammalian evolution
to be around 0.26.Comment: Proceedings of the 3rd International Work-Conference on
Bioinformatics and Biomedical Engineering (IWBBIO), 2015. (to appear
A New Simulated Annealing Algorithm for the Multiple Sequence Alignment Problem: The approach of Polymers in a Random Media
We proposed a probabilistic algorithm to solve the Multiple Sequence
Alignment problem. The algorithm is a Simulated Annealing (SA) that exploits
the representation of the Multiple Alignment between sequences as a
directed polymer in dimensions. Within this representation we can easily
track the evolution in the configuration space of the alignment through local
moves of low computational cost. At variance with other probabilistic
algorithms proposed to solve this problem, our approach allows for the creation
and deletion of gaps without extra computational cost. The algorithm was tested
aligning proteins from the kinases family. When D=3 the results are consistent
with those obtained using a complete algorithm. For where the complete
algorithm fails, we show that our algorithm still converges to reasonable
alignments. Moreover, we study the space of solutions obtained and show that
depending on the number of sequences aligned the solutions are organized in
different ways, suggesting a possible source of errors for progressive
algorithms.Comment: 7 pages and 11 figure
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