ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

exactly soluble random field ising models in one dimension the authors solve exactly the one dimensional random field ising model for two classes of magnetic field distributions symmetric exponential model i and non symmetric exponential model ii for both models expressions for the free energy at all finite temperatures are presented the low temperature region is examined in more detail they obtain the zero temperature energy and entropy in closed form it is shown that the free energy of both models has an expansion in integer powers of temperature model i has a non vanishing zero point entropy for all values of the parameters as soon as randomness is diluted in model ii the zero point entropy is zero except for a discrete sequence of values of one parameter in some cases the zero temperature magnetisation is positive whereas the average magnetic field is negative the magnetisation may also change sign as a function of temperature

allison mixture and the two envelope problem in the present study we have investigated the allison mixture a variant of the parrondo s games where random mixing of two random sequences creates autocorrelation we have obtained the autocorrelation function and mutual entropy of two elements our analysis shows that the mutual information is nonzero even if two distributions have identical average values we have also considered the two envelope problem and solved for its exact probability distribution

on proof and progress in mathematics author s thurston william p abstract in response to jaffe and quinn math ho 9307227 the author discusses forms of progress in mathematics that are not captured by formal proofs of theorems especially in his own work in the theory of foliations and geometrization of 3 manifolds and dynamical systems

threshold for monotone symmetric properties through a logarithmic sobolev inequality threshold phenomena are investigated using a general approach following talagrand ann probab 22 1994 1576 1587 and friedgut and kalai proc amer math soc 12 1999 1017 10541 the general upper bound for the threshold width of symmetric monotone properties is improved this follows from a new lower bound on the maximal influence of a variable on a boolean function the method of proof is based on a well known logarithmic sobolev inequality on 0 1 n this new bound is shown to be asymptotically optimal

local tail bounds for functions of independent random variables it is shown that functions defined on 0 1 r 1 n satisfying certain conditions of bounded differences that guarantee sub gaussian tail behavior also satisfy a much stronger local sub gaussian property for self bounding and configuration functions we derive analogous locally subexponential behavior the key tool is talagrand s ann probab 22 1994 1576 1587 variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on 0 1 r 1 n for r 2

concentration of measure inequalities for markov chains and phi mixing processes we prove concentration inequalities for some classes of markov chains and f mixing processes with constants independent of the size of the sample that extend the inequalities for product measures of talagrand the method is based on information inequalities put forward by marton in case of contracting markov chains using a simple duality argument on entropy our results also include the family of logarithmic sobolev inequalities for convex functions applications to bounds on supremum of dependent empirical processes complete this work

absence of self averaging of the order parameter in the sherrington kirkpatrick model we prove that if n is the sherrington kirkpatrick sk hamiltonian and the quantity bar q n n 1 sum left langle s l right rangle h 2 converges in the variance to a nonrandom limit asn then the mean free energy of the model converges to the expression obtained by sk since this expression is known not to be correct in the low temperature region our result implies the non self averaging of the order parameter of the sk model this fact is an important ingredient of the parisi theory which is widely believed to be exact we also prove that the variance of the free energy of the sk model converges to zero asn i e the free energy has the self averaging property

the supremum of some canonical processes 1 introduction consider a random variable r v h such that eh2 l distributed like h to each t tk k 1 of 2 one can associate a r v xi ek l tkhk this defines a stochastic process xt t e2 by stochastic process we simply mean a collection of r v this process could be called the canonical process associated to h a long range and rather ambitious program would be to understand given h for which subsets t of 2 the process xt tct is bounded let us first discuss the case where h is centered gaussian the special importance of that case stems from the fact that for the question of boundedness discussed here gaussian processes of the type xt tct are the most general gaussian processes this fact is closely related to the rotational invariance of gaussian measures for a stochastic process xt tct where the set t is possibly uncountable we set

dynamic programming optimization over random data the scaling exponent for near optimal solutions a very simple example of an algorithmic problem solvable by dynamic programming is to maximize over a subseteq 1 2 ldots n the objective function a sum i xi i rm1 hspace 0 90ex 1 i in a i 1 in a for given xi i 0 this problem with random xi i provides a test example for studying the relationship between optimal and near optimal solutions of combinatorial optimization problems we show that amongst solutions differing from the optimal solution in a small proportion delta of places we can find near optimal solutions whose objective function value differs from the optimum by a factor of order delta 2 but not of smaller order we conjecture this relationship holds widely in the context of dynamic programming over random data and monte carlo simulations for the kauffman levin nk model are consistent with the conjecture this work is a technical contribution to a broad program initiated in d j aldous and a g percus proc natl acad sci usa 100 2003 pp 11211 11215 of relating such scaling exponents to the algorithmic difficulty of optimization problems

evolution of a fluctuating population in a randomly switching environment environment plays a fundamental role in the competition for resources and hence in the evolution of populations here we study a well mixed finite population consisting of two strains competing for the limited resources provided by an environment that randomly switches between states of abundance and scarcity assuming that one strain grows slightly faster than the other we consider two scenarios one of pure resource competition and one in which one strain provides a public good and investigate how environmental randomness external noise coupled to demographic internal noise determines the population s fixation properties and size distribution by analytical means and simulations we show that these coupled sources of noise can significantly enhance the fixation probability of the slower growing species we also show that the population size distribution can be unimodal bimodal or multimodal and undergoes noise induced transitions between these regimes when the rate of switching matches the population s growth rate

tight bounds on the capacity of binary input random cdma systems in this paper we consider code division multiple access cdma communication over a binary input additive white gaussian noise awgn channel using random spreading for a general class of symmetric distributions for spreading sequences in the limit of a large number of users we prove an upper bound to the capacity the bound matches the formula obtained by tanaka using the replica method we also show concentration of various relevant quantities including mutual information and free energy the mathematical methods are quite general and allow us to discuss extensions to other multiuser scenarios

lyapunov exponents one dimensional anderson localization and products of random matrices the concept of lyapunov exponent has long occupied a central place in the theory of anderson localisation its interest in this particular context is that it provides a reasonable measure of the localisation length the lyapunov exponent also features prominently in the theory of products of random matrices pioneered by furstenberg after a brief historical survey we describe some recent work that exploits the close connections between these topics we review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case finally we point out some limitations of the lyapunov exponent as a means of studying localisation properties

intrinsic noise in systems with switching environments we study individual based dynamics in finite populations subject to randomly switching environmental conditions these are inspired by models in which genes transition between on and off states regulating underlying protein dynamics similarly switches between environmental states are relevant in bacterial populations and in models of epidemic spread existing piecewise deterministic markov process approaches focus on the deterministic limit of the population dynamics while retaining the randomness of the switching here we go beyond this approximation and explicitly include effects of intrinsic stochasticity at the level of the linear noise approximation specifically we derive the stationary distributions of a number of model systems in good agreement with simulations this improves existing approaches which are limited to the regimes of fast and slow switching

the lyapunov exponent of products of random 2 2 matrices close to the identity we study products of arbitrary random real 2 2 matrices that are close to the identity matrix using the iwasawa decomposition of sl 2 we identify a continuum regime where the mean values and the covariances of the three iwasawa parameters are simultaneously small in this regime the lyapunov exponent of the product is shown to assume a scaling form in the general case the corresponding scaling function is expressed in terms of gauss hypergeometric function a number of particular cases are also considered where the scaling function of the lyapunov exponent involves other special functions airy bessel whittaker elliptic the general solution thus obtained allows us among other things to recover in a unified framework many results known previously from exactly solvable models of one dimensional disordered systems

linear smoothers and additive models we study linear smoothers and their use in building nonparametric regression models in the first part of this paper we examine certain aspects of linear smoothers for scatterplots examples of these are the running mean and running line kernel and cubic spline smoothers the eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother and several other topics such as the number of degrees of freedom of a smoother are discussed in the second part of the paper we describe how linear smoothers can be used to estimate the additive model a powerful nonparametric regression model using the backfitting algorithm we show that backfitting is the gauss seidel iterative method for solving a set of normal equations associated with the additive model we provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers

exponential inequalities for dynamical measures of expanding maps of the interval we prove an exponential inequality for the absolutely continuous invariant measure of a piecewise expanding map of the interval as an immediate corollary we obtain a concentration inequality we apply these results to the estimation of the rate of convergence of the empirical measure in various metrics and also to the efficiency of the shadowing by sets of positive measure

semianalytical solution of the random product problem of matrices and discrete time random evolution we consider the product of a large number of matrices chosen randomly with some correlation as a or b we introduce an exact functional equation whose solution gives the limiting distribution and the lyapunov exponent for the product of large number of matrices thus we suggest new semianalytical method to solve the random product problem of correlated matrices the method can be easily generalized to high order matrices as we deal with a larger systems of functional equations we have provided a known numerical algorithm for the solution of the matrix product problem we have also applied our method for a solution of discrete time random evolution our functional equation method provides exact transition point

laws of large numbers and tail inequalities for random tries and patricia trees we consider random tries and random patricia trees constructed from n independent strings of symbols drawn from any distribution on any discrete space if hn is the height of this tree we show that hn e hn tends to one in probability additional tail inequalities are given for the height depth size internal path length and profile of these trees and ordinary tries that apply without any conditions on the string distributions they need not even be identically distributed

new concentration inequalities in product spaces we introduce three new ways to measure the distance from a point to a subset of a product space and we prove corresponding concentration inequalities each of them allows to control the fluctuation of a new class of random variables

concentration for multidimensional diffusions and their boundary local times we prove that probability laws of certain multidimensional semimartingales which includes time inhomogenous diffusions under suitable assumptions satisfy quadratic transportation cost inequality under the uniform metric from this we derive concentration properties of lipschitz functions of process paths that depend on the entire history in particular we estimate concentration of boundary local time of reflected brownian motions on a polyhedral domain we work out explicit applications of consequences of measure concentration for the case of brownian motion with rank based drifts

simple and fast algorithms for linear and integer programs with two variables per inequality the authors present an o mn 2 log m algorithm for solving feasibility in linear programs with up to two variables per inequality which is derived directly from the fourier motzkin elimination method the number of variables and inequalities are denoted by n and m respectively the running time of the algorithm dominates that of the best known algorithm for the problem and is far simpler integer programming on monotone inequalities i e inequalities where the coefficients are of opposite sign is then considered this problem includes as a special case the simultaneous approximation of a rational vector with specified accuracy which is known to be np complete however it is shown that both a feasible solution and an optimal solution with respect to an arbitrary objective function can be computed in pseudo polynomial time

incremental closure for systems of two variables per inequality subclasses of linear inequalities where each inequality has at most two vari ables are popular in abstract interpretation and model checking because they strike a balance between what can be described and what can be efficiently computed this paper focuses on the tvpi class of inequalities for which each coefficient of each two variable inequality is unrestricted an implied tvpi in equality can be generated from a pair of tvpi inequalities by eliminating a given common variable echoing resolution on clauses this operation called result can be applied to derive tvpi inequalities which are entailed implied by a given tvpi system the key operation on tvpi is calculating closure satisfiability can be observed from a closed system and a closed system also simplifies the calculation of other operations a closed system can be derived by repeatedly applying the result operator the process of adding a single tvpi inequality to an already closed input tvpi system and then finding the closure of this augmented system is called incremental closure this too can be calcu lated by the repeated application of the result operator this paper studies the calculus defined by result the structure of result derivations and how deriva tions can be combined and controlled a series of lemmata on derivations are presented that collectively provide a pathway for synthesising an algorithm for incremental closure the complexity of the incremental closure algorithm is analysed and found to be o n 2 m 2 lg m where n is the number of variables and m the number of inequalities of the input tvpi system

regularity of gaussian processes on donne la validite de la conjecture de fernique 1974 sur l existence de mesures de majoration pouvant caracteriser la limitation des processus gaussiens

matrix completion from noisy entries given a matrix m of low rank we consider the problem of reconstructing it from noisy observations of a small random subset of its entries the problem arises in a variety of applications from collaborative filtering the netflix problem to structure from motion and positioning we study a low complexity algorithm introduced by keshavan montanari and oh 2010 based on a combination of spectral techniques and manifold optimization that we call here optspace we prove performance guarantees that are order optimal in a number of circumstances

su 2 wzw theory at higher genera we compute by free field techniques the scalar product of thesu 2 chern simons states on genus 1 surfaces the result is a finite dimensional integral over positions of screening charges and one complex modular parameter it uses an effective description of the cs states closely related to the one worked out by bertram 1 the scalar product formula allows to express the higher genus partition functions of the wzw conformal field theory by finite dimensional integrals it should provide the hermitian metric preserved by the knizhnik zamolodchikov bernard connection describing the variations of the cs states under the change of the complex structure of the surface

theoretical mathematics toward a cultural synthesis of mathematics and theoretical physics is speculative mathematics dangerous recent interactions between physics and mathematics pose the question with some force traditional mathematical norms discourage speculation but it is the fabric of theoretical physics in practice there can be benefits but there can also be unpleasant and destructive consequences serious caution is required and the issue should be considered before rather than after obvious damage occurs with the hazards carefully in mind we propose a framework that should allow a healthy and positive role for speculation

exact probability distribution functions for parrondo s games we study the discrete time dynamics of brownian ratchet models and parrondo s games using the fourier transform we calculate the exact probability distribution functions for both the capital dependent and history dependent parrondo s games in certain cases we find strong oscillations near the maximum of the probability distribution with two limiting distributions for odd and even number of rounds of the game indications of such oscillations first appeared in the analysis of real financial data but now we have found this phenomenon in model systems and a theoretical understanding of the phenomenon the method of our work can be applied to brownian ratchets molecular motors and portfolio optimization

additive logistic regression a statistical view of boosting boosting is one of the most important recent developments in classification methodology boosting works by sequentially applying a classification algorithm to reweighted versions of the training data and then taking a weighted majority vote of the sequence of classifiers thus produced for many classification algorithms this simple strategy results in dramatic improvements in performance we show that this seemingly mysterious phenomenon can be understood in terms of well known statistical principles namely additive modeling and maximum likelihood for the two class problem boosting can be viewed as an approximation to additive modeling on the logistic scale using maximum bernoulli likelihood as a criterion we develop more direct approximations and show that they exhibit nearly identical results to boosting direct multiclass generalizations based on multinomial likelihood are derived that exhibit performance comparable to other recently proposed multiclass generalizations of boosting in most situations and far superior in some we suggest a minor modification to boosting that can reduce computation often by factors of 10 to 50 finally we apply these insights to produce an alternative formulation of boosting decision trees this approach based on best first truncated tree induction often leads to better performance and can provide interpretable descriptions of the aggregate decision rule it is also much faster computationally making it more suitable to large scale data mining applications

stability and generalization we define notions of stability for learning algorithms and show how to use these notions to derive generalization error bounds based on the empirical error and the leave one out error the methods we use can be applied in the regression framework as well as in the classification one when the classifier is obtained by thresholding a real valued function we study the stability properties of large classes of learning algorithms such as regularization based algorithms in particular we focus on hilbert space regularization and kullback leibler regularization we demonstrate how to apply the results to svm for regression and classification

rate of convergence of the mean for sub additive ergodic sequences abstract for sub additive ergodic processes x m n with weak dependence we analyze the rate of convergence of e x 0 n n to its limit g we define an exponent given roughly by e x 0 n n g n and assuming existence of a fluctuation exponent that gives var x 0 n n 2 we provide a lower bound for of the form the main requirement is that 1 2 in the case 1 2 and under the assumption var x 0 n o n log n for some 0 we prove c for a dependent constant c these results show in particular that non diffusive fluctuations are associated to non trivial various models including first passage percolation directed polymers the minimum of a branching random walk and bin packing fall into our general framework and the results apply assuming exists in the case of first passage percolation in z d we provide a version of 1 2 without assuming existence of

majorizing measures the generic chaining majorizing measures provide bounds for the supremum of stochastic processes they represent the most general possible form of the chaining argument going back to kolmogorov majorizing measures arose from the theory of gaussian processes but they now have applications far beyond this setting the fundamental question is the construction of these measures this paper focuses on the tools that have been developed for this purpose and in particular the use of geometric ideas applications are given to several natural problems where entropy methods are powerless

the computational complexity of simultaneous diophantine approximation problems simultaneous diophantine approximation in d dimensions deals with the approximation of a vector bf alpha alpha 1 cdots alpha d of d real numbers by vectors of rational numbers all having the same denominator this paper considers the computational complexity of algorithms to find good simultaneous approximations to a given vector bf alpha of d rational numbers we measure the goodness of an approximation using the sup norm we show that a result of h w lenstra jr produces polynomial time algorithms to find sup norm best approximations to a given vector bf alpha when the dimension d is fixed we show that a recent algorithm of a k lenstra h w lenstra jr and l lovasz to find short vectors in an integral lattice can be used to find a good approximation to a given vector bf alpha in d dimensions with a denominator q satisfying 1 leqq q leqq 2 d 2 n which is within a factor sqrt 5d 2 d 1 2 of the best approximation with denominato

matrix estimation by universal singular value thresholding consider the problem of estimating the entries of a large matrix when the observed entries are noisy versions of a small random fraction of the original entries this problem has received widespread attention in recent times especially after the pioneering works of emmanuel candes and collaborators this paper introduces a simple estimation procedure called universal singular value thresholding usvt that works for any matrix that has a little bit of structure surprisingly this simple estimator achieves the minimax error rate up to a constant factor the method is applied to solve problems related to low rank matrix estimation blockmodels distance matrix completion latent space models positive definite matrix completion graphon estimation and generalized bradley terry models for pairwise comparison

discrete time ratchets the fokker planck equation and parrondo s paradox parrondo s games manifest the apparent paradox where losing strategies can be combined to win and have generated significant multidisciplinary interest in the literature here we review two recent approaches based on the fokker planck equation that rigorously establish the connection between parrondo s games and a physical model known as the flashing brownian ratchet this gives rise to a new set of parrondo s games of which the original games are a special case for the first time we perform a complete analysis of the new games via a discrete time markov chain analysis producing winning rate equations and an exploration of the parameter space where the paradoxical behaviour occurs

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

a new look at independence the concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables we attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities

numerical solution of a functional equation on a circle in this article two numerical methods are analyzed and applied to compute an invariant closed curve for a particular class of maps a second order scheme based on linear interpolation and a scheme based on interpolation by cubic splines in suitable coordinates the invariant curve is determined by the solution of a functional equation on a circle a linearized version of this functional equation is studied in detail stability of the schemes is discussed and error estimates are derived depending on the smoothness of the solution roughly speaking the analysis predicts superior behavior of the spline scheme when the solution is sufficiently smooth but a more reliable behavior of the lower order scheme in situations when a smooth solution cannot be expected this is confirmed in numerical examples the analysis of the linear functional equation does not readily generalize to nonlinear equations but the algorithms for the linear equation can be used as a building block for a treatment of nonlinear equati

a simple proof of the blowing up lemma corresp the blowing up lemma says that if the probability with respect to a product measure of a set a subseteq cal x n cal x finite n large is not exponentially small then its l n neighborhood has probability almost one for some l n o n here an information theoretic proof of the blowing up lemma generalizing it to continuous alphabets is given

critical points and supersymmetric vacua iii string m models a fundamental problem in contemporary string m theory is to count the number of inequivalent vacua satisfying constraints in a string theory model this article contains the first rigorous results on the number and distribution of supersymmetric vacua of type iib string theories compactified on a calabi yau 3 fold x with flux in particular complete proofs of the counting formulas in ashok douglas ad and denef douglas dd1 are given together with van der corput style remainder estimates

finite blocklength and moderate deviation analysis of hypothesis testing of correlated quantum states and application to classical quantum channels with memory martingale concentration inequalities constitute a powerful mathematical tool in the analysis of problems in a wide variety of fields ranging from probability and statistics to information theory and machine learning here we apply techniques borrowed from this field to quantum hypothesis testing which is the problem of discriminating quantum states belonging to two different sequences rho n n and sigma n n we obtain achievability bounds on the finite blocklength type ii stein and hoeffding errors which for i i d states are in general tighter than the corresponding bounds obtained by audenaert mosonyi and verstraete we also derive finite blocklength bounds and moderate deviation results for pairs of sequences of correlated states satisfying a non homogeneous factorization property examples of such sequences include gibbs states of spin chains with translation invariant finite range interaction as well as finitely correlated quantum states we apply our results to find bounds on the capacity of a certain class of classical quantum channels with memory which satisfy a so called channel factorization property both in the finite blocklength and moderate deviation regimes

data reduction using a discrete wavelet transform in discriminant analysis of very high dimensionality data we present a method of data reduction using a wavelet transform in discriminant analysis when the number of variables is much greater than the number of observations the method is illustrated with a prostate cancer study where the sample size is 248 and the number of variables is 48 538 generated using the proteinchip technology using a discrete wavelet transform the 48 538 data points are represented by 1271 wavelet coefficients information criteria identified 11 of the 1271 wavelet coefficients with the highest discriminatory power the linear classifier with the 11 wavelet coefficients detected prostate cancer in a separate test set with a sensitivity of 97 and specificity of 100

hilbert s 6th problem exact and approximate hydrodynamic manifolds for kinetic equations the problem of the derivation of hydrodynamics from the boltz mann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions we review a few in stances where such hydrodynamic manifolds were found analytically both as the result of summation of the chapman enskog asymptotic expansion and by the direct solution of the invariance equation these model cases comprising grad s moment systems both linear and nonlinear are studied in depth in order to gain understanding of what can be expected for the boltzmann equa tion particularly the dispersive dominance and saturation of dissipation rate of the exact hydrodynamics in the short wave limit and the viscosity modifica tion at high divergence of the flow velocity are indicated as severe obstacles to the resolution of hilbert s 6th problem furthermore we review the derivation of the approximate hydrodynamic manifold for the boltzmann equation using newton s iteration and avoiding smallness parameters and compare this to the exact solutions additionally we discuss the problem of projection of the boltzmann equation onto the approximate hydrodynamic invariant manifold using entropy concepts finally a set of hypotheses is put forward where we describe open questions and set a horizon for what can be derived exactly or proven about the hydrodynamic manifolds for the boltzmann equation in the future

large deviations of subgraph counts for sparse erdős renyi graphs abstract for any fixed simple graph h v e and any fixed u 0 we establish the leading order of the exponential rate function for the probability that the number of copies of h in the erd s renyi graph g n p exceeds its expectation by a factor 1 u assuming n h p 1 with h 1 2 where 1 is the maximum degree of h this improves on a previous result of chatterjee and the second author who obtained h c e for a constant c 0 moreover for the case of cycle counts we can take as large as 1 2 we additionally obtain the sharp upper tail for schatten norms of the adjacency matrix as well as the sharp lower tail for counts of graphs for which sidorenko s conjecture holds as a key step we establish quantitative versions of szemeredi s regularity lemma and the counting lemma suitable for the analysis of random graphs in the large deviations regime

improved algorithms for linear inequalities with two variables per inequality the authors show that a system of m linear inequalities with n variables where each inequality involves at most two variables can be solved in tilde o mn 2 time we denote tilde o f o f polylog n polylog m deterministically and in tilde o n 3 mn expected time using randomization parallel implementations of these algorithms run in tilde o n time where the deterministic algorithm uses tilde o mn processors and the randomized algorithm uses tilde o n 2 m processors the bounds significantly improve over previous algorithms the randomized algorithm is based on novel insights into the structure of the problem

interface surface tension and reentrant pinning transition in the 2d ising model we develop a new way to look at the high temperature representation of the ising model up to the critical temperature and obtain a number of interesting consequences in the two dimensional case it is possible to use these tools to prove results on phase separation lines in the whole phase coexistence regime by way of a duality transformation we illustrate the power of these techniques by studying an ising model with a boundary magnetic field in which a reentrant pinning transition takes place more precisely we show that the typical configurations of the model can be described at the macroscopic level by interfaces which are solutions of the corresponding thermodynamic variational problem this variational problem is solved explicitly there exist values of the boundary magnetic field and temperatures 0 t 2 but is pinned for t 1 t 3 it is not pinned an important property of the surface tension which is used in this paper is the sharp triangle inequality about which we report some new results the techniques used in this work are robust and can be used in a variety of different situations

spectral norm of products of random and deterministic matrices we study the spectral norm of matrices w that can be factored as w ba where a is a random matrix with independent mean zero entries and b is a fixed matrix under the 4 e th moment assumption on the entries of a we show that the spectral norm of such an m n matrix w is bounded by sqrt m sqrt n which is sharp in other words in regard to the spectral norm products of random and deterministic matrices behave similarly to random matrices with independent entries this result along with the previous work of rudelson and the author implies that the smallest singular value of a random m n matrix with i i d mean zero entries and bounded 4 e th moment is bounded below by sqrt m sqrt n 1 with high probability

exact solution of the hidden markov processes we write a master equation for the distributions related to hidden markov processes hmps and solve it using a functional equation thus the solution of hmps is mapped exactly to the solution of the functional equation for a general case the latter can be solved only numerically we derive an exact expression for the entropy of hmps our expression for the entropy is an alternative to the ones given before by the solution of integral equations the exact solution is possible because actually the model can be considered as a generalized random walk on a one dimensional strip while we give the solution for the two second order matrices our solution can be easily generalized for the l values of the markov process and m values of observables we should be able to solve a system of l functional equations in the space of dimension m 1

the statistical mechanics of random copolymers random copolymers are polymers with two or more types of monomer where the monomer sequence is determined by some random process once determined the sequence is fixed so random copolymers are an example of a system with quenched randomness we review the statistical mechanics of random copolymers focusing on self avoiding walk models where there are two types of monomers a and b which are randomly distributed along the polymer chain theoretical approximate and numerical results are reviewed for models of the random copolymer adsorption localization and collapse phase transitions we concentrate on what is known about the existence of phase transitions the morita approximation and results about self averaging we also discuss in less detail the replica trick and numerical methods including monte carlo methods exact enumeration and transfer matrix methods important open problems are identified throughout and highlighted in the conclusions

vc theory of large margin multi category classifiers in the context of discriminant analysis vapnik s statistical learning theory has mainly been developed in three directions the computation of dichotomies with binary valued functions the computation of dichotomies with real valued functions and the computation of polytomies with functions taking their values in finite sets typically the set of categories itself the case of classes of vector valued functions used to compute polytomies has seldom been considered independently which is unsatisfactory for three main reasons first this case encompasses the other ones second it cannot be treated appropriately through a naive extension of the results devoted to the computation of dichotomies third most of the classification problems met in practice involve multiple categories in this paper a vc theory of large margin multi category classifiers is introduced central in this theory are generalized vc dimensions called the dimensions first a uniform convergence bound on the risk of the classifiers of interest is derived the capacity measure involved in this bound is a covering number this covering number can be upper bounded in terms of the dimensions thanks to generalizations of sauer s lemma as is illustrated in the specific case of the scale sensitive natarajan dimension a bound on this latter dimension is then computed for the class of functions on which multi class svms are based this makes it possible to apply the structural risk minimization inductive principle to those machines

a high dimensional wilks phenomenon a theorem by wilks asserts that in smooth parametric density estimation the difference between the maximum likelihood and the likelihood of the sampling distribution converges toward a chi square distribution where the number of degrees of freedom coincides with the model dimension this observation is at the core of some goodness of fit testing procedures and of some classical model selection methods this paper describes a non asymptotic version of the wilks phenomenon in bounded contrast optimization procedures using concentration inequalities for general functions of independent random variables it proves that in bounded contrast minimization as for example in statistical learning theory the difference between the empirical risk of the minimizer of the true risk in the model and the minimum of the empirical risk the excess empirical risk satisfies a bernstein like inequality where the variance term reflects the dimension of the model and the scale term reflects the noise conditions from a mathematical statistics viewpoint the significance of this result comes from the recent observation that when using model selection via penalization the excess empirical risk represents a minimum penalty if non asymptotic guarantees concerning prediction error are to be provided from the perspective of empirical process theory this paper describes a concentration inequality for the supremum of a bounded non centered actually non positive empirical process combining the now classical analysis of m estimation building on talagrand s inequality for suprema of empirical processes and versatile moment inequalities for functions of independent random variables this paper develops a genuine bernstein like inequality that seems beyond the reach of traditional tools

on the ising model with random boundary condition the infinite volume limit behavior of the 2d ising model under possibly strong random boundary conditions is studied the model exhibits chaotic size dependence at low temperatures and we prove that the and phases are the only almost sure limit gibbs measures assuming that the limit is taken along a sparse enough sequence of squares in particular we provide an argument to show that in a sufficiently large volume a typical spin configuration under a typical boundary condition contains no interfaces in order to exclude mixtures as possible limit points a detailed multi scale contour analysis is performed

random doubly stochastic matrices the circular law hoi h nguyenabstract let x be a matrix sampled uniformly from the set of doubly stochastic matri ces of size n n we show that the empirical spectral distribution of the normalized matrixpn x ex converges almost surely to the circular law this con rms a conjecture ofchatterjee diaconis and sly

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed