338 research outputs found
Braids inside the Birman-Wenzl-Murakami algebra
We determine the Zariski closure of the representations of the braid groups
that factorize through the Birman-Wenzl-Murakami algebra, for generic values of
the parameters . For of modulus 1 and close to 1, we prove
that these representations are unitarizable, thus deducing the topological
closure of the image when in addition are algebraically independent
Infinitesimal Hecke Algebras II
For W a finite (2-)reflection group and B its (generalized) braid group, we
determine the Zariski closure of the image of B inside the corresponding
Iwahori-Hecke algebra. The Lie algebra of this closure is reductive and
generated in the group algebra of W by the reflections of W. We determine its
decomposition in simple factors. In case W is a Coxeter group, we prove that
the representations involved are unitarizable when the parameters of the
representations have modulus 1 and are close to 1. We consequently determine
the topological closure in this case
Rates of convergence for the posterior distributions of mixtures of Betas and adaptive nonparametric estimation of the density
In this paper, we investigate the asymptotic properties of nonparametric
Bayesian mixtures of Betas for estimating a smooth density on . We
consider a parametrization of Beta distributions in terms of mean and scale
parameters and construct a mixture of these Betas in the mean parameter, while
putting a prior on this scaling parameter. We prove that such Bayesian
nonparametric models have good frequentist asymptotic properties. We determine
the posterior rate of concentration around the true density and prove that it
is the minimax rate of concentration when the true density belongs to a
H\"{o}lder class with regularity , for all positive , leading to
a minimax adaptive estimating procedure of the density. We also believe that
the approximating results obtained on these mixtures of Beta densities can be
of interest in a frequentist framework.Comment: Published in at http://dx.doi.org/10.1214/09-AOS703 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Enumeration of the Monomials of a Polynomial and Related Complexity Classes
We study the problem of generating monomials of a polynomial in the context
of enumeration complexity. In this setting, the complexity measure is the delay
between two solutions and the total time. We present two new algorithms for
restricted classes of polynomials, which have a good delay and the same global
running time as the classical ones. Moreover they are simple to describe, use
little evaluation points and one of them is parallelizable. We introduce three
new complexity classes, TotalPP, IncPP and DelayPP, which are probabilistic
counterparts of the most common classes for enumeration problems, hoping that
randomization will be a tool as strong for enumeration as it is for decision.
Our interpolation algorithms proves that a lot of interesting problems are in
these classes like the enumeration of the spanning hypertrees of a 3-uniform
hypergraph.
Finally we give a method to interpolate a degree 2 polynomials with an
acceptable (incremental) delay. We also prove that finding a specified monomial
in a degree 2 polynomial is hard unless RP = NP. It suggests that there is no
algorithm with a delay as good (polynomial) as the one we achieve for
multilinear polynomials
Some remarks on the continuity equation
We describe some relations between the properties of the Cauchy problem for
an ODE and the properties of the Cauchy problem for the associated continuity
equation in the class of measures
A note on the enumeration of directed animals via gas considerations
In the literature, most of the results about the enumeration of directed
animals on lattices via gas considerations are obtained by a formal passage to
the limit of enumeration of directed animals on cyclical versions of the
lattice. Here we provide a new point of view on this phenomenon. Using the gas
construction given in [Electron. J. Combin. (2007) 14 R71], we describe the gas
process on the cyclical versions of the lattices as a cyclical Markov chain
(roughly speaking, Markov chains conditioned to come back to their starting
point). Then we introduce a notion of convergence of graphs, such that if
then the gas process built on converges in distribution to
the gas process on . That gives a general tool to show that gas processes
related to animals enumeration are often Markovian on lines extracted from
lattices. We provide examples and computations of new generating functions for
directed animals with various sources on the triangular lattice, on the
lattices introduced in [Ann. Comb. 4 (2000) 269--284] and on a
generalization of the \mathcaligr {L}_n lattices introduced in [J. Phys. A 29
(1996) 3357--3365].Comment: Published in at http://dx.doi.org/10.1214/08-AAP580 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Nested Regular Path Queries in Description Logics
Two-way regular path queries (2RPQs) have received increased attention
recently due to their ability to relate pairs of objects by flexibly navigating
graph-structured data. They are present in property paths in SPARQL 1.1, the
new standard RDF query language, and in the XML query language XPath. In line
with XPath, we consider the extension of 2RPQs with nesting, which allows one
to require that objects along a path satisfy complex conditions, in turn
expressed through (nested) 2RPQs. We study the computational complexity of
answering nested 2RPQs and conjunctions thereof (CN2RPQs) in the presence of
domain knowledge expressed in description logics (DLs). We establish tight
complexity bounds in data and combined complexity for a variety of DLs, ranging
from lightweight DLs (DL-Lite, EL) up to highly expressive ones. Interestingly,
we are able to show that adding nesting to (C)2RPQs does not affect worst-case
data complexity of query answering for any of the considered DLs. However, in
the case of lightweight DLs, adding nesting to 2RPQs leads to a surprising jump
in combined complexity, from P-complete to Exp-complete.Comment: added Figure
A new characterization of Talagrand's transport-entropy inequalities and applications
We show that Talagrand's transport inequality is equivalent to a restricted
logarithmic Sobolev inequality. This result clarifies the links between these
two important functional inequalities. As an application, we give the first
proof of the fact that Talagrand's inequality is stable under bounded
perturbations.Comment: Published in at http://dx.doi.org/10.1214/10-AOP570 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Laplace deconvolution and its application to Dynamic Contrast Enhanced imaging
In the present paper we consider the problem of Laplace deconvolution with
noisy discrete observations. The study is motivated by Dynamic Contrast
Enhanced imaging using a bolus of contrast agent, a procedure which allows
considerable improvement in {evaluating} the quality of a vascular network and
its permeability and is widely used in medical assessment of brain flows or
cancerous tumors. Although the study is motivated by medical imaging
application, we obtain a solution of a general problem of Laplace deconvolution
based on noisy data which appears in many different contexts. We propose a new
method for Laplace deconvolution which is based on expansions of the
convolution kernel, the unknown function and the observed signal over Laguerre
functions basis. The expansion results in a small system of linear equations
with the matrix of the system being triangular and Toeplitz. The number of
the terms in the expansion of the estimator is controlled via complexity
penalty. The advantage of this methodology is that it leads to very fast
computations, does not require exact knowledge of the kernel and produces no
boundary effects due to extension at zero and cut-off at . The technique
leads to an estimator with the risk within a logarithmic factor of of the
oracle risk under no assumptions on the model and within a constant factor of
the oracle risk under mild assumptions. The methodology is illustrated by a
finite sample simulation study which includes an example of the kernel obtained
in the real life DCE experiments. Simulations confirm that the proposed
technique is fast, efficient, accurate, usable from a practical point of view
and competitive
Laplace deconvolution on the basis of time domain data and its application to Dynamic Contrast Enhanced imaging
In the present paper we consider the problem of Laplace deconvolution with
noisy discrete non-equally spaced observations on a finite time interval. We
propose a new method for Laplace deconvolution which is based on expansions of
the convolution kernel, the unknown function and the observed signal over
Laguerre functions basis (which acts as a surrogate eigenfunction basis of the
Laplace convolution operator) using regression setting. The expansion results
in a small system of linear equations with the matrix of the system being
triangular and Toeplitz. Due to this triangular structure, there is a common
number of terms in the function expansions to control, which is realized
via complexity penalty. The advantage of this methodology is that it leads to
very fast computations, produces no boundary effects due to extension at zero
and cut-off at and provides an estimator with the risk within a logarithmic
factor of the oracle risk. We emphasize that, in the present paper, we consider
the true observational model with possibly nonequispaced observations which are
available on a finite interval of length which appears in many different
contexts, and account for the bias associated with this model (which is not
present when ). The study is motivated by perfusion imaging
using a short injection of contrast agent, a procedure which is applied for
medical assessment of micro-circulation within tissues such as cancerous
tumors. Presence of a tuning parameter allows to choose the most
advantageous time units, so that both the kernel and the unknown right hand
side of the equation are well represented for the deconvolution. The
methodology is illustrated by an extensive simulation study and a real data
example which confirms that the proposed technique is fast, efficient,
accurate, usable from a practical point of view and very competitive.Comment: 36 pages, 9 figures. arXiv admin note: substantial text overlap with
arXiv:1207.223
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