expressions for renyi and shannon entropies for bivariate distributions exact forms of renyi and shannon entropies are determined for 27 continuous bivariate distributions including the kotz type distribution truncated normal distribution distributions with normal and centered normal conditionals natural exponential distribution freund s exponential distribution marshall and olkin s exponential distribution exponential mixture distribution arnold and strauss s exponential distribution mckay s gamma distribution distribution with gamma conditionals gamma exponential distribution dirichlet distribution inverted beta distribution distribution with beta conditionals beta stacy distribution cuadras and auge s distribution farlie gumbel morgenstern distribution logistic distribution pearson type vii distribution pearson type ii distribution distribution with cauchy conditionals bilateral pareto distribution muliere and scarsini s pareto distribution distribution with pareto conditionals and the distribution with gumbel conditionals we believe that the results presented here will serve as an important reference for scientists and engineers in many areas

renyi divergence measures for commonly used univariate continuous distributions abstract probabilistic distances also called divergences which in some sense assess how close two probability distributions are from one another have been widely employed in probability statistics information theory and related fields of particular importance due to their generality and applicability are the renyi divergence measures this paper presents closed form expressions for the renyi and kullback leibler divergences for nineteen commonly used univariate continuous distributions as well as those for multivariate gaussian and dirichlet distributions in addition a table summarizing four of the most important information measure rates for zero mean stationary gaussian processes namely renyi entropy differential shannon entropy renyi divergence and kullback leibler divergence is presented lastly a connection between the renyi divergence and the variance of the log likelihood ratio of two distributions is established thereby extending a previous result by song j stat plan infer 93 2001 on the relation between renyi entropy and the log likelihood function a table with the corresponding variance expressions for the univariate distributions considered here is also included

point interactions in two and three dimensional riemannian manifolds we present a non perturbative renormalization of the bound state problem of n bosons interacting with finitely many dirac delta interactions on two and three dimensional riemannian manifolds using the heat kernel we formulate the problem in terms of a new operator called the principal or characteristic operator e in order to investigate the problem in more detail we then restrict the problem to one particle sector the lower bound of the ground state energy is found for a general class of manifolds e g for compact and cartan hadamard manifolds the estimate of the bound state energies in the tunneling regime is calculated by perturbation theory non degeneracy and uniqueness of the ground state is proven by the perron frobenius theorem moreover the pointwise bounds on the wave function is given and all these results are consistent with the one given in standard quantum mechanics renormalization procedure does not lead to any radical change in these cases finally renormalization group equations are derived and the function is exactly calculated this work is a natural continuation of our previous work based on a novel approach to the renormalization of point interactions developed by rajeev

hardy littlewood theory for semigroups soit t t t 0 un semi groupe symetrique sousmarkovien et soit n 2 alors on a les proprietes equivalentes a dim t t n b il existe c 2 0 s t t t f c 2 t n 2 f 1 f l 1 x t 0 en fait a b meme dans le cas n 2

estimation of the information by an adaptive partitioning of the observation space we demonstrate that it is possible to approximate the mutual information arbitrarily closely in probability by calculating the relative frequencies on appropriate partitions and achieving conditional independence on the rectangles of which the partitions are made empirical results including a comparison with maximum likelihood estimators are presented

nonlinear equations and weighted norm inequalities we study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities in particular we obtain explicit criteria for the solvability of the dirichlet problem u v u w u 0 on u 0 on on a regular domain in rn in the superlinear case q 1 the coefficients v w are arbitrary positive measurable functions or measures on we also consider more general nonlinear differential and integral equations and study the spaces of coefficients and solutions naturally associated with these problems as well as the corresponding capacities our characterizations of the existence of positive solutions take into account the interplay between v w and the corresponding green s kernel they are not only sufficient but also necessary and are established without any a priori regularity assumptions on v and w we also obtain sharp two sided estimates of solutions up to the boundary some of our results are new even if v 1 and is a ball or half space the corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so called 3g inequality by an elementary integration by parts argument this also gives a new unified proof for some classical inequalities including the carleson measure theorem for poisson integrals and trace inequalities for riesz potentials and green potentials

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

nonrelativistic lee model in three dimensional riemannian manifolds in this work we construct the nonrelativistic lee model on some class of three dimensional riemannian manifolds by following a novel approach introduced by s g rajeev e print hep th 9902025 this approach together with the help of heat kernel allows us to perform the renormalization nonperturbatively and explicitly for completeness we show that the ground state energy is bounded from below for different classes of manifolds using the upper bound estimates on the heat kernel finally we apply a kind of mean field approximation to the model for compact and noncompact manifolds separately and discover that the ground state energy grows linearly with the number of bosons n

existence of hamiltonians for some singular interactions on manifolds the existence of the hamiltonians of the renormalized point interactions in two and three dimensional riemannian manifolds and that of a relativistic extension of this model in two dimensions are proven although it is much more difficult the proof of existence of the hamiltonian for the renormalized resolvent for the non relativistic lee model can still be given to accomplish these results directly from the resolvent formula we employ some basic tools from the semigroup theory

deformations of special legendrian submanifolds in sasaki einstein manifolds in this paper we study the deformation theory of submanifolds characterized by a system of differential forms and provide a criterion for deformations of such submanifolds to be unobstructed we apply this deformation theory to special legendrian submanifolds in sasaki einstein manifolds in general special legendrian deformations have the obstruction however we show that the deformation space of special legendrian submanifolds is the intersection of two larger smooth deformation spaces of different types we also prove that any special legendrian submanifold admits smooth deformations which are not special legendrian deformations given by harmonic 1 forms

the relativistic lee model on riemannian manifolds we study the relativistic lee model on static riemannian manifolds the model is constructed nonperturbatively through its resolvent which is based on the so called principal operator and the heat kernel techniques it is shown that making the principal operator well defined dictates how to renormalize the parameters of the model the renormalization of the parameters is the same in the light front coordinates as in the instant form moreover the renormalization of the model on riemannian manifolds agrees with the flat case the asymptotic behavior of the renormalized principal operator in the large number of bosons limit implies that the ground state energy is positive in 2 1 dimensions the model requires only a mass renormalization we obtain rigorous bounds on the ground state energy for the n particle sector of the 2 1 dimensional model

estimating mutual information we present two classes of improved estimators for mutual information m x y from samples of random points distributed according to some joint probability density mu x y in contrast to conventional estimators based on binnings they are based on entropy estimates from k nearest neighbor distances this means that they are data efficient with k 1 we resolve structures down to the smallest possible scales adaptive the resolution is higher where data are more numerous and have minimal bias indeed the bias of the underlying entropy estimates is mainly due to nonuniformity of the density at the smallest resolved scale giving typically systematic errors which scale as functions of k n for n points numerically we find that both families become exact for independent distributions i e the estimator m x y vanishes up to statistical fluctuations if mu x y mu x mu y this holds for all tested marginal distributions and for all dimensions of x and y in addition we give estimators for redundancies between more than two random variables we compare our algorithms in detail with existing algorithms finally we demonstrate the usefulness of our estimators for assessing the actual independence of components obtained from independent component analysis ica for improving ica and for estimating the reliability of blind source separation

formulas for renyi information and related measures for univariate distributions in a recent paper song j stat plan infer 93 2001 51 considered renyi information of order and established its connection to the loglikelihood from this relation an intrinsic distribution measure was proposed and analytic expressions of this measure and renyi information were derived for some standard continuous distributions in this paper we derive analytical formulas for renyi and shannon entropies as well as for song s measure for 26 flexible families of univariate continuous distributions we believe that the results presented here will serve as an important reference for scientists and engineers in many areas

nondegeneracy of the ground state for nonrelativistic lee model in the present work we first briefly sketch the construction of the nonrelativistic lee model on riemannian manifolds introduced in our previous works in this approach the renormalized resolvent of the system is expressed in terms of a well defined operator called the principal operator so as to obtain a finite formulation then we show that the ground state of the nonrelativistic lee model on compact riemannian manifolds is nondegenerate using the explicit expression of the principal operator that we obtained this is achieved by combining heat kernel methods with positivity improving semi group approach and then applying these tools directly to the principal operator rather than the hamiltonian without using cut offs

analysis on graphs and noncommutative geometry abstract we study the form of the continuous time heat kernel for a second order discrete laplacian on a weighted graph the analysis is shown to be closely related to the theory of symmetric markov semigroups on noncommutative l p spaces and to the noncommutative geometry of connes the paper obtains better pointwise upper bounds on the heat kernels than those previously known by the use of a novel metric on the graph in certain cases it is shown that the new estimates are optimal of their type the metric is investigated at some length and compared with other known metrics on graphs

learning non linear functions with factor graphs we show how to use a discrete variable factor graph for learning non linear continuous functions from examples the paper proposes a scheme for embedding soft quantization in a probabilistic bayesian graph the quantized input variables are grouped into a compound variable that is mapped through a stochastic matrix into the discrete output distribution specific output values are then obtained through a process of de quantization the information flow carried by message propagation is bi directional and an algorithm for learning the factor graph parameters is explicitly derived the model that can easily merge discrete and continuous variables is demonstrated with examples and simulations

singular interactions supported by embedded curves in this work singular interactions supported by embedded curves on riemannian manifolds are discussed from a more direct and physical perspective via the heat kernel approach we show that the renormalized problem is well defined the ground state is finite and the corresponding wavefunction is positive the renormalization group invariance of the model is also discussed

the geometric complexity of special lagrangian t2 cones we prove a number of results on the geometric complexity of special lagrangian slg t2 cones in 3 every slg t2 cone has a fundamental integer invariant its spectral curve genus we prove that the spectral curve genus of an slg t2 cone gives a lower bound for its geometric complexity i e the area the stability index and the legendrian index of any slg t2 cone are all bounded below by explicit linearly growing functions of the spectral curve genus we prove that the cone on the clifford torus which has spectral curve genus zero in s5 is the unique slg t2 cone with the smallest possible legendrian index and hence that it is the unique stable slg t2 cone this leads to a classification of all rigid index 1 slg cone types in dimension three for cones with spectral curve genus two we give refined lower bounds for the area the legendrian index and the stability index one consequence of these bounds is that there exist s1 invariant slg torus cones of arbitrarily large area legendrian and stability indices we explain some consequences of our results for the programme due to joyce to understand the most common three dimensional isolated singularities of generic families of slg submanifolds in almost calabi yau manifolds

finitely many dirac delta interactions on riemannian manifolds this work is intended as an attempt to study the nonperturbative renormalization of bound state problem of finitely many dirac delta interactions on riemannian manifolds s2 h2 and h3 we formulate the problem in terms of a finite dimensional matrix called the characteristic matrix the bound state energies can be found from the characteristic equation 2 a 0 the characteristic matrix can be found after a regularization and renormalization by using a sharp cut off in the eigenvalue spectrum of the laplacian as it is done in the flat space or using the heat kernel method these two approaches are equivalent in the case of compact manifolds the heat kernel method has a general advantage to find lower bounds on the spectrum even for compact manifolds as shown in the case of s2 the heat kernels for h2 and h3 are known explicitly thus we can calculate the characteristic matrix using the result we give lower bound estimates of the discrete spectrum

entropy expressions and their estimators for multivariate distributions entropy expressions for several continuous multivariate distributions are derived point estimation of entropy for the multinormal distribution and for the distribution of order statistics from d g weinman s ph d dissertation ariz state univ tempe az 1966 exponential distribution is considered the asymptotic distribution of the uniformly minimum variance unbiased estimator for multinormal entropy is obtained simulation results on convergence of the means and variances of these estimators are provided

logarithmic sobolev inequalities for information measures for alpha ges 1 the new vajda type information measure j alpha x is a quantity generalizing fisher s information fi to which it is reduced for alpha 2 in this paper a corresponding generalized entropy power n alpha x is introduced and the inequality n alpha x j alpha x ges n is proved which is reduced to the well known inequality of stam for alpha 2 the cases of equality are also determined furthermore the blachman stam inequality for the fi of convolutions is generalized for the vajda information j alpha x and both families of results in the context of measure of information are discussed that is logarithmic sobolev inequalities lsis are written in terms of new more general entropy type information measure and therefore new information inequalities are arisen this generalization for special cases yields to the well known information measures and relative bounds

compression based methods for nonparametric prediction and estimation of some characteristics of time series we address the problem of online prediction for time series we show that any universal code or a universal data compressor can be used as a basis for constructing asymptotically optimal methods for this problem for a certain class of stationary and ergodic processes