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 $\alpha,s$. For $\alpha,s$ 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 $\alpha,s$ 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 $[0,1]$. 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 $\beta$, for all positive $\beta$, 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 $(G_n)\to G$ then the gas process built on $G_n$ converges in distribution to the gas process on $G$. 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 $\mathcal {T}_n$ 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 $m$ 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 $T$. The technique leads to an estimator with the risk within a logarithmic factor of $m$ 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 $m$ 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 $T$ 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 $T$ which appears in many different contexts, and account for the bias associated with this model (which is not present when $T\rightarrow\infty$). 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 $a$ 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