on the general structure of nonlinear evolution equations integrable by the two dimensional matrix spectral problem a generalization of the akns technique to the two dimensional arbitrary order matrix spectral problem is given the general form of the integrable equations and their backlund transformations in 1 2 dimensions are found the reduction problem is discussed

empirical processes and random projections in this note we establish some bounds on the supremum of certain empirical processes indexed by sets of functions with the same l2 norm we present several geometric applications of this result the most important of which is a sharpening of the johnson lindenstrauss embedding lemma our results apply to a large class of random matrices as we only require that the matrix entries have a subgaussian tail

the solution of the general initial value problem for the full three dimensional three wave resonant interaction abstract the general initial value problem for the full three dimensional three wave resonant interaction is solved by inverse scattering the existence and uniqueness of the solution is established when all waves have a positive energy

mean square values of hecke l series formed with r th order characters let l be a number field containing the r th roots of unity starting with the rankin selberg convolution of a metaplectic eisenstein series on the r fold cover of gl 2 with itself we construct a dirichlet series defined over l whose coefficients involve the r th order twists of a fixed hecke l function we then observe that a group of functional equations can be naturally associated with this construction combining this with the convexity theorem for holomorphic functions of several complex variables we show that this object as a function of two complex variables admits meromorphic continuation to 2 as an application we obtain asymptotic formulae for mean square values of the r th order twists of an arbitrary hecke l function defined over l

uniform bounds under increment conditions we apply a majorizing measure theorem of talagrand to obtain uniform bounds for sums of random variables satisfying increment conditions of the type considered in gal koksma theorems we give some applications

on the integrability of multidimensional nonlinear evolution equations the integrability test scheme of chen lee and liu h h chen y c lee and c s liu phys scr 20 490 1979 from one space dimension to multispace dimensions is generalized the temporal equation of the lax pair is still the linearized perturbed equation that defines the symmetries but the spectral operator in the lax pair is no longer the linear recursion operator for symmetries the absence of the linear recursion operator for symmetries in higher spatial dimensions therefore presents no direct obstacle to the chen lee liu test scheme the kadomtsev petviashvili equation is shown as an example

existence and continuity of occupation densities of stochastic integral processes let f be a square integrable function on the unit square assume that the singular numbers a i i in mathbb n of the hilbert schmidt operator associated with f admit some 0 alpha frac 1 3 such that sum infty i 1 a i alpha infty we present a purely stochastic method to investigate the occupation densities of the skorohod integral process u induced by f it allows us to show that u possesses continuous square integrable occupation densities and obviously generalizes beyond the second wiener chaos

a trace identity and its application to integrable systems of 1 2 dimensions a trace identity is proposed and it is shown that this identity can be effectively applied to establish the hamiltonian structure of the kp ds hierarchies and a new hierarchy of 1 2 dimensional systems

bi hamiltonian formulation of the kadomtsev petviashvili and benjamin ono equations it was shown recently that the kadomtsev petviashvili kp equation an integrable equation in 2 1 i e in two spatial and one temporal dimensions admits a bi hamiltonian formulation this was achieved by considering kp as a reduction of a 3 1 dimensional system in the variables x y1 y2 t it is shown here using the kp as a concrete example that equations in 2 1 possess two bi hamiltonian formulations and two recursion operators both hamiltonian operators associated with the x direction are local in contrast only one of the hamiltonian operators associated with the y direction is local furthermore using the benjamin ono equation as a concrete example it is shown that intergrodifferential equations in 1 1 admit an algebraic formulation analogous to that of equations in 2 1

sample boundedness of stochastic processes under increment conditions let t d be a compact metric space of diameter d and be an orlicz norm when is it true that all separable processes x t t t that satisfy the increment condition x t x s d t s for all s t in t are sample bounded we give optimal necessary conditions and optimal sufficient conditions in terms of the existence of a probability measure on t that satisfies an integral condition when t is a compact group and d is translation invariant we are able to compute the necessary and sufficient condition in several cases

rates of convergence for minimum contrast estimators we shall present here a general study of minimum contrast estimators in a nonparametric setting although our results are also valid in the classical parametric case for independent observations these estimators include many of the most popular estimators in various situations such as maximum likelihood estimators least squares and other estimators of the regression function estimators for mixture models or deconvolution the main theorem relates the rate of convergence of those estimators to the entropy structure of the space of parameters optimal rates depending on entropy conditions are already known at least for some of the models involved and they agree with what we get for minimum contrast estimators as long as the entropy counts are not too large but under some circumstances large entropies or changes in the entropy structure due to local perturbations the resulting the rates are only suboptimal counterexamples are constructed which show that the phenomenon is real for non parametric maximum likelihood or regression this proves that under purely metric assumptions our theorem is optimal and that minimum contrast estimators happen to be suboptimal

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

global existence and nonexistence results for a generalized davey stewartson system we consider a system of three equations which will be called generalized davey stewartson equations involving three coupled equations two for the long waves and one for the short wave propagating in an infinite elastic medium we classify the system according to the signs of the parameters conserved quantities related to mass momentum and energy are derived as well as a specific instance of the so called virial theorem using these conservation laws and the virial theorem both global existence and nonexistence results are established under different constraints on the parameters in the elliptic elliptic elliptic case

geometric phases reduction and lie poisson structure for the resonant three wave interaction hamiltonian lie poisson structures of the three wave equations associated with the lie algebras su 3 and su 2 1 are derived and shown to be compatible poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated these results can be applied to applications of nonlinear waves in for instance nonlinear optics some of the general structures presented in the latter part of this paper are implicit in the literature our purpose is to put the three wave interaction in the modern setting of geometric mechanics and to explore some new things such as explicit geometric phase formulas as well as some old things such as integrability in this context

the bernstein von mises theorem and nonregular models we study the asymptotic behaviour of the posterior distribution in a broad class of statistical models where the true solution occurs on the boundary of the parameter space we show that in this case the bayesian inference is consistent and that the posterior distribution has not only gaussian components as in the case of regular models the bernstein von mises theorem but also has gamma distribution components that depend on the behaviour of the prior distribution on the boundary and have a faster rate of convergence we also show a remarkable property of bayesian inference that for some models there appears to be no bound on efficiency of estimating the unknown parameter if it is on the boundary of the parameter space we illustrate the results on a problem from emission tomography

on the boundedness of bernoulli processes we present a positive solution to the so called bernoulli conjecture concerning the characterization of sample boundedness of bernoulli processes we also discuss some applications and related open problems

constructing weyl group multiple dirichlet series let be a reduced root system of rank r a weyl group multiple dirichlet series for is a dirichlet series in r complex variables s1 sr initially converging for re si sufficiently large that has meromorphic continuation to c and satisfies functional equations under the transformations of c corresponding to the weyl group of a heuristic definition of such series was given in bbc06 and they have been investigated in certain special cases in bbc06 bbf06 bbfh07 bbf08 bbfa cg07 cfg08 cg chi05 in this paper we generalize results in cg to construct weyl group multiple dirichlet series by a uniform method and show in all cases that they have the expected properties

a chain rule for the expected suprema of gaussian processes the expected supremum of a gaussian process indexed by the image of an index set under a function class is bounded in terms of separate properties of the index set and the function class the bound is relevant to the estimation of nonlinear transformations or the analysis of learning algorithms whenever hypotheses are chosen from composite classes as is the case for multi layer models

gaussian chaos and sample path properties of additive functionals of symmetric markov processes let x be a strongly symmetric hunt process with potential density u x y let g 2 x y 2 d x d y and let l t denote the continuous additive functional with revuz measure for a set of positive measures m g 2 subject to some additional regulariy conditions we consider families of continuous in time additive functionals l l t t r x m of x and a second order gaussian chaos h h m which is associated with l by an isomorphism theorem of dynkin a general theorem is obtained which shows that with some additional regularity conditions depending on x and m if h has a continuous version on m almost surely then so does l and furthermore that moduli of continuity for h are also moduli of continuity for l special attention is given to levy processes in r n and t n the n dimensional torus with m taken to be the set of translates of a fixed measure many concrete examples are given especially when x is brownian motion in r n and t n for n 2 and 3 for certain measures on t n and processes including brownian motion in t 3 necessary and sufficient conditions are given for the continuity of l t t r x m where m is the set of all translates of

on kubota s dirichlet series kubota 19 showed how the theory of eisenstein series on the higher metaplectic covers of sl2 which he discovered can be used to study the analytic properties of dirichlet series formed with n th order gauss sums in this paper we will prove a functional equation for such dirichlet series in the precise form required by the companion paper 2 closely related results are in eckhardt and patterson 10 the kubota dirichlet series are the entry point to a fascinating universe their residues for example are mysterious if n 3 though there is tantalizing evidence that these residues exhibit a rich structure that can only be partially glimpsed at this time when n 4 the residues are the fourier coefficients of the biquadratic theta function that were studied by suzuki 23 suzuki found that he could only determine some of the coefficients this failure to determine all the coefficients was explained in terms of the failure of uniqueness of whittaker models for the generalized theta series by deligne 9 and by kazhdan and patterson 15 on the other hand patterson 22 conjectured that the mysterious coefficients are essentially square roots of gauss sums evidence for patterson s conjecture is discussed in bump and hoffstein 6 and in eckhardt and patterson 10 where the conjecture is refined in light of numerical data partial proofs were given by suzuki in 24 and 25 another set of conjectures relevant to the mysterious coefficients of n th order theta functions were given by bump and hoffstein who considered theta functions on the n fold covers of glr for arbitrary r they are expressed as identities between rankin selberg convolutions of generalized theta series and whittaker coefficients of eisenstein series on the metaplectic group but they boil down to properties of the residues of kubota dirichlet series and their higher rank generalizations see bump and hoffstein 6 bump 4 and hoffstein 12 these conjectures are different from the patterson conjecture and there are other considerations which suggest that there may be further unproved relations beyond those described in the conjectures of patterson and bump and hoffstein

proliferation scheme for kaup boussinesq system abstract the proliferation scheme for kaup boussinesq system dispersive water wave system is discussed the recurrent procedure gives the series of integrable nonlinear equations coupled by miura maps and its lax pairs new integrable nonlinear systems are found in this series

on the structure of equations integrable by the arbitrary order linear spectral problem the general form of partial differential equations integrable by the arbitrary order linear spectral problem is found the groups of backlund transformations corresponding to these equations are constructed it is shown that partial differential equations of the class under study are hamiltonian ones some reductions of general equations are considered in particular the hamiltonian structure of the generalisations of the sine gordon equation to the groups gl n su n and so n at arbitrary n is proved

hellinger consistency of certain nonparametric maximum likelihood estimators consider a class mathscr p p theta theta in theta of probability measures on a measurable space mathscr x mathscr a dominated by a sigma finite measure mu let f theta dp theta d mu theta in theta and let theta n be a maximum likelihood estimator based on n independent observations from p theta 0 theta 0 in theta we use results from empirical process theory to obtain convergence for the hellinger distance h f hat theta n f theta 0 under certain entropy conditions on the class of densities f theta theta in theta the examples we present are a model with interval censored observations smooth densities monotone densities and convolution models in most examples the convexity of the class of densities is of special importance

reduction techniques for infinite dimensional hamiltonian systems some ideas and applications in the language of tensor analysis on differentiable manifolds we present a reduction method of integrability structures and apply it to recover some well known hierarchies of integrable nonlinear evolution equations

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

combining pac bayesian and generic chaining bounds there exist many different generalization error bounds in statistical learning theory each of these bounds contains an improvement over the others for certain situations or algorithms our goal is first to underline the links between these bounds and second to combine the different improvements into a single bound in particular we combine the pac bayes approach introduced by mcallester 1998 which is interesting for randomized predictions with the optimal union bound provided by the generic chaining technique developed by fernique and talagrand see talagrand 1996 in a way that also takes into account the variance of the combined functions we also show how this connects to rademacher based bounds

weyl group multiple dirichlet series eisenstein series and crystal bases we show that the whittaker coecients of borel eisenstein series on the metaplectic covers of glr 1 can be described as multiple dirichlet series in r complex variables whose coecients are computed by attaching a number theoretic quantity a product of gauss sums to each vertex in a crystal graph these gauss sums depend on string data previously introduced in work of lusztig berenstein and zelevinsky and littelmann these data are the lengths of segments in a path from the given vertex to the vertex of lowest weight depending on a factorization of the long weyl group element into simple reections the coecients may also be described as sums over strict gelfand tsetlin patterns the description is uniform in the degree of the metaplectic cover

weyl group multiple dirichlet series for symmetrizable kac moody root systems weyl group multiple dirichlet series introduced by brubaker bump chinta friedberg and hoffstein are expected to be whittaker coefficients of eisenstein series on metaplectic groups chinta and gunnells constructed these multiple dirichlet series for all the finite root systems using the method of averaging a weyl group action on the field of rational functions in this paper we generalize chinta and gunnells work and construct weyl group multiple dirichlet series for the root systems associated with symmetrizable kac moody algebras and establish their functional equations and meromorphic continuation introduction weyl group multiple dirichlet series were introduced in the paper 3 by brubaker bump chinta friedberg and hoffstein and have been studied in their subsequent works 4 5 7 these multiple dirichlet series are defined for a finite root system of rank r and a number field f containing the 2n th roots of unity and unify many examples in number theory that have been studied previously on a case bycase basis and are expected to be whittaker coefficients of eisenstein series on metaplectic groups this expectation is now called an eisenstein conjecture and the conjecture has been proven for the root systems of type ar 6 moreover the theory of weyl group multiple dirichlet series can be applied to the moments problem of the riemann zeta function and dirichlet l functions a nice survey on this subject can be found in 9 it is remarkable that there are two distinct constructions of these multiple dirichlet series in the work of brubaker bump and friedberg 5 6 the local coefficients are defined using the data from kashiwara s crystal graph 14 more precisely each local coefficient is given as a sum of g v over the crystals v of the same weight where g v is a product of gauss sums that are determined by the crystal graph the construction has been completed for most of the classical root systems the second construction is due to chinta and gunnells 10 11 they defined a new weyl group action on the field of rational functions in several variables the action involves gauss sums and works for all the finite root systems in a coherent way by taking an average of the weyl group action they obtained a deformed weyl character and defined the local coefficients using the deformed character the two constructions are expected to be equivalent and the equivalence in type a has been established 6 12 16 received by the editors october 10 2012 and in revised form october 12 2012 november 2 2012 april 11 2013 and april 21 2013 2010 mathematics subject classification primary 11f68 secondary 17b67 c 2014 american mathematical society reverts to public domain 28 years from publication

majorizing measures without measures we give a reformulation of majorizing measures that does not involve measures but rather special sequences of partitions this formulation is more convenient to perform chaining with different distances

on the theory of recursion operator the general structure and properties of recursion operators for hamiltonian systems with a finite number and with a continuum of degrees of freedom are considered weak and strong recursion operators are introduced the conditions which determine weak and strong recursion operators are found

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

eisenstein series on covers of odd orthogonal groups we study the whittaker coefficients of the minimal parabolic eisenstein series on the n fold cover of the split odd orthogonal group rm so 2r 1 if the degree of the cover is odd then beineke brubaker and frechette have conjectured that the p power contributions to the whittaker coefficients may be computed using the theory of crystal graphs of type c by attaching to each path component a gauss sum or a degenerate gauss sum depending on the fine structure of the path we establish their conjecture using a combination of automorphic and combinatorial representation theoretic methods surprisingly we must make use of the type a theory and the two different crystal graph descriptions of brubaker bump and friedberg available for type a based on different factorizations of the long element into simple reflections we also establish a formula for the whittaker coefficients in the even degree cover case again based on crystal graphs of type c as a further consequence we establish a lie theoretic description of the coefficients for n sufficiently large thereby confirming a conjecture of brubaker bump and friedberg

a complete characterization of coefficients of a e convergent orthogonal series and majorizing measures we characterize sequences of numbers a n such that n 1 a n n converges a e for any orthonormal system n in any l 2 space in our criteria we use the set a m n a m 2 n 1 majorizing measures on a and new descriptions of the complexity of a

exact solutions of non conservative equations for three wave and second harmonic resonance new solutions of the equations of three wave and second harmonic resonance are presented in cases where these equations are non conservative with three waves subject to both finite linear damping and complex coupling coefficients these new solutions depend on a single characteristic variable in the absence of linear damping but with complex coupling coefficients a broad class of solutions depending on three characteristic variables is found this class is a generalization of the so called one lump solutions previously known only for conservative equations with real coupling coefficients for second harmonic reasonance analogous new solutions are given for both conservative and non conservative cases

weyl group multiple dirichlet series ii the stable case to each reduced root system of rank r and each sufficiently large integer n we define a family of multiple dirichlet series in r complex variables whose group of functional equations is isomorphic to the weyl group of the coefficients in these dirichlet series exhibit a multiplicativity that reduces the specification of the coefficients to those that are powers of a single prime p for each p the number of nonzero such coefficients is equal to the order of the weyl group and each nonzero coefficient is a product of n th order gauss sums the root system plays a basic role in the combinatorics underlying the proof of the functional equations

on classification of soliton solutions of multicomponent nonlinear evolution equations we consider several ways of how one could classify the various types of soliton solutions related to multicomponent nonlinear evolution equations which are solvable by the inverse scattering method for the generalized zakharov shabat system related to a simple lie algebra g in doing so we make use of the fundamental analytic solutions the zakharov shabat dressing procedure the reduction technique and other tools characteristic for that method the multicomponent solitons are characterized by several important factors the subalgebras of g and the way these subalgebras are embedded in g the dimension of the corresponding eigensubspaces of the lax operator l as well as by additional constraints imposed by reductions

the generating operator for the n n linear system abstract the class of exactly integrable non linear evolution equations nlee related to the general first order n n linear problem is studied the set of independent scattering data j is determined and trace identities are obtained next we use a special integro differential operator and obtain the expansions of the potential and its variation in the eigenfunctions of these enable us to consider the inverse scattering method as a generalized fourier transform and to write down in a compact form the integrals of motion we prove that there exist a hierarchy of hamiltonian structures generated by at the end we calculate the symplectic form in terms of the scattering data variations

parametric estimation finite sample theory the paper aims at reconsidering the famous le cam lan theory the main features of the approach which make it different from the classical one are as follows 1 the study is nonasymptotic that is the sample size is fixed and does not tend to infinity 2 the parametric assumption is possibly misspecified and the underlying data distribution can lie beyond the given parametric family these two features enable to bridge the gap between parametric and nonparametric theory and to build a unified framework for statistical estimation the main results include large deviation bounds for the quasi maximum likelihood and the local quadratic bracketing of the log likelihood process the latter yields a number of important corollaries for statistical inference concentration confidence and risk bounds expansion of the maximum likelihood estimate etc all these corollaries are stated in a nonclassical way admitting a model misspecification and finite samples however the classical asymptotic results including the efficiency bounds can be easily derived as corollaries of the obtained nonasymptotic statements at the same time the new bracketing device works well in the situations with large or growing parameter dimension in which the classical parametric theory fails the general results are illustrated for the i i d setup as well as for generalized linear and median estimation the results apply for any dimension of the parameter space and provide a quantitative lower bound on the sample size yielding the root n accuracy

non vanishing of quadratic twists of modular l functions k have been the subject of much study both because of their intrinsicinterest and because of the prominent role they have played inkolyvagin s work on the birch and swinnerton dyer conjecture see b f h i j ko ma m m m1 o s p p waldspurger proved a fundamental theorem the ore me 1 w1 relating these central critical values to the fourier coe cients of half integral weight cusp forms for notational convenience if d is afundamental discriminant of a quadratic number eld then de ne d

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

basis adaptive sample efficient polynomial chaos base pc abstract for a large class of orthogonal basis functions there has been a recent identification of expansion methods for computing accurate stable approximations of a quantity of interest this paper presents within the context of uncertainty quantification a practical implementation using basis adaptation and coherence motivated sampling which under assumptions has satisfying guarantees this implementation is referred to as basis adaptive sample efficient polynomial chaos base pc a key component of this is the use of anisotropic polynomial order which admits evolving global bases for approximation in an efficient manner leading to consistently stable approximation for a practical class of smooth functionals this fully adaptive non intrusive method requires no a priori information of the solution and has satisfying theoretical guarantees of recovery a key contribution to stability is the use of a presented correction sampling for coherence optimal sampling in order to improve stability and accuracy within the adaptive basis scheme theoretically the method may dramatically reduce the impact of dimensionality in function approximation and numerically the method is demonstrated to perform well on problems with dimension up to 1000

bootstrap confidence sets under model misspecification a multiplier bootstrap procedure for construction of likelihood based confidence sets is considered for finite samples and a possible model misspecification theoretical results justify the bootstrap consistency for a small or moderate sample size and allow to control the impact of the parameter dimension p the bootstrap approximation works if p n is small the main result about bootstrap consistency continues to apply even if the underlying parametric model is misspecified under the so called small modeling bias condition in the case when the true model deviates significantly from the considered parametric family the bootstrap procedure is still applicable but it becomes a bit conservative the size of the constructed confidence sets is increased by the modeling bias we illustrate the results with numerical examples for misspecified constant and logistic regressions jel classification codes c13 c15

on additional motion invariants of classical hamiltonian wave systems abstract it is shown that the existence of an analytic invariant in addition to the natural ones momentum energy and in some cases number of particles leads to the existence of infinitely many such invariants nevertheless the existence of the additional motion invariant does not guarantee complete integrability complete integrability follows from the existence of an additional invariant only if the dispersion law is non degenerative with respect to decays if the dispersion law is degenerative the number of motion invariants is insufficient for complete integrability and the s matrix is factorized via decay processes one into two with real intermediate particles in this paper we present also our results concerning enumeration of degenerative dispersion laws

low rank tensor recovery via iterative hard thresholding abstract we study extensions of compressive sensing and low rank matrix recovery matrix completion to the recovery of low rank tensors of higher order from a small number of linear measurements while the theoretical understanding of low rank matrix recovery is already well developed only few contributions on the low rank tensor recovery problem are available so far in this paper we introduce versions of the iterative hard thresholding algorithm for several tensor decompositions namely the higher order singular value decomposition hosvd the tensor train format tt and the general hierarchical tucker decomposition ht we provide a partial convergence result for these algorithms which is based on a variant of the restricted isometry property of the measurement operator adapted to the tensor decomposition at hand that induces a corresponding notion of tensor rank we show that subgaussian measurement ensembles satisfy the tensor restricted isometry property with high probability under a certain almost optimal bound on the number of measurements which depends on the corresponding tensor format these bounds are extended to partial fourier maps combined with random sign flips of the tensor entries finally we illustrate the performance of iterative hard thresholding methods for tensor recovery via numerical experiments where we consider recovery from gaussian random measurements tensor completion recovery of missing entries and fourier measurements for third order tensors

integrable non isospectral flows associated with the kadomtsev petviashvili equations in 2 1 dimensions the symmetries of the kadomtsev petviashvili kp equations in 2 1 dimensions yield two hierarchies of integrable non linear evolution equations nee one is the familiar family of isospectral flows the kp hierarchy the other is non isospectral and its flows have coefficients which depend linearly on x and y the spectral methods used to solve kp can be used to solve all these nee an underlying infinite dimensional lie algebra is used to determine all the lax pairs for both families and it also determines their symplectic structures constants of the motion are constructed for the non isospectral cases

guarding against spurious discoveries in high dimensions many data mining and statistical machine learning algorithms have been developed to select a subset of covariates to associate with a response variable spurious discoveries can easily arise in high dimensional data analysis due to enormous possibilities of such selections how can we know statistically our discoveries better than those by chance in this paper we define a measure of goodness of spurious fit which shows how good a response variable can be fitted by an optimally selected subset of covariates under the null model and propose a simple and effective lamm algorithm to compute it it coincides with the maximum spurious correlation for linear models and can be regarded as a generalized maximum spurious correlation we derive the asymptotic distribution of such goodness of spurious fit for generalized linear models and l1 regression such an asymptotic distribution depends on the sample size ambient dimension the number of variables used in the fit and the covariance information it can be consistently estimated by multiplier bootstrapping and used as a benchmark to guard against spurious discoveries it can also be applied to model selection which considers only candidate models with goodness of fits better than those by spurious fits the theory and method are convincingly illustrated by simulated examples and an application to the binary outcomes from german neuroblastoma trials

nonvanishing of l series associated to cubic twists of elliptic curves has a rational solution an extensive compilation of the older history of the problem is given in dickson 6 within the past century researchers have tried to exploit either explicitly or unknowingly the fact that the curve 0 1 is in fact an elliptic curve during the nineteenth century lucas and later sylvester used a descent argument to prove that 0 1 had no solution for infinitely many d in certain congruence classes mod 9 and 18 see 6 ch xxi zagier and kramarz 19 have produced a great deal of numerical evidence about the l series of the curves based on these computations they have argued heuristically that for

cover times blanket times and majorizing measures we exhibit a strong connection between cover times of graphs gaussian processes and talagrand s theory of majorizing measures in particular we show that the cover time of any graph g is equivalent up to universal constants to the square of the expected maximum of the gaussian free field on g scaled by the number of edges in g this allows us to resolve a number of open questions we give a deterministic polynomial time algorithm that computes the cover time to within an o 1 factor for any graph answering a question of aldous and fill 1994 we also positively resolve the blanket time conjectures of winkler and zuckerman 1996 showing that for any graph the blanket and cover times are within an o 1 factor the best previous approximation factor for both these problems was o loglogn 2 for n vertex graphs due to kahn kim lovasz and vu 2000

concentration inequalities using the entropy method we investigate a new methodology worked out by ledoux and massart to prove concentration of measure inequalities the method is based on certain modified logarithmic sobolev inequalities we provide some very simple and general ready to use inequalities one of these inequalities may be considered as an exponential version of the efron stein inequality the main purpose of this paper is to point out the simplicity and the generality of the approach we show how the new method can recover many of talagrand s revolutionary inequalities and provide new applications in a variety of problems including rademacher averages rademacher chaos the number of certain small subgraphs in a random graph and the minimum of the empirical risk in some statistical estimation problems

on rank estimators in increasing dimensions abstract the family of rank estimators including han s maximum rank correlation han 1987 as a notable example has been widely exploited in studying regression problems for these estimators although the linear index is introduced for alleviating the impact of dimensionality the effect of large dimension on inference is rarely studied this paper fills this gap via studying the statistical properties of a larger family of m estimators whose objective functions are formulated as u processes and may be discontinuous in increasing dimension set up where the number of parameters p n in the model is allowed to increase with the sample size n first we find that often in estimation as p n n 0 p n n 1 2 rate of convergence is obtainable second we establish bahadur type bounds and study the validity of normal approximation which we find often requires a much stronger scaling requirement than p n 2 n 0 third we state conditions under which the numerical derivative estimator of asymptotic covariance matrix is consistent and show that the step size in implementing the covariance estimator has to be adjusted with respect to p n all theoretical results are further backed up by simulation studies

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

a theorem on majorizing measures let t d be a metric space and r 1r an increasing convex function with 0 0 we prove that if m is a probability measure m on r which is majorizing with respect to d that is supper d t 0 1 1 m b x e de then e sup x s x t 32 s t t for each separable stochastic process x t t t which satisfies e x s x t d s t 1 for all s t t s t this is a strengthening of one of the main results from talagrand ann probab 18 1990 1 491 and its proof is significantly simpler

convergence of an alternating maximization procedure we derive two convergence results for a sequential alternating maximization procedure to approximate the maximizer of random functionals such as the realized log likelihood in mle estimation we manage to show that the sequence attains the same deviation properties as shown for the profile m estimator by andresen and spokoiny 2013 that means a finite sample wilks and fisher theorem further under slightly stronger smoothness constraints on the random functional we can show nearly linear convergence to the global maximizer if the starting point for the procedure is well chosen

three dimensional three wave interactions a bilinear approach we consider the three dimensional three wave resonance equation using a bilinear approach to investigate a broad class of solutions solutions are obtained in a grammian form and their relationship to kaup s solutions examined

on the recursion operators for the gerdjikov mikhailov and valchev system we consider the recursion operator approach to the soliton equations related to an auxiliary linear system introduced recently by gerdjikov mikhailov and valchev gmv system we discuss the recursion operators obtained for the gmv system and show that they are particular cases of the generating operators related to a zakharov shabat type linear system on the algebra sl 3 c in pole gauge

double dirichlet series and the n th order twists of hecke l series let n 3 be a fixed integer and let f be a global field containing the n th roots of unity in this paper we study the collective behavior of the n th order twists of a fixed hecke l series for f to do so we introduce a double dirichlet series in two complex variables s w which is a weighted sum of the twists and obtain its meromorphic continuation we also study related sums of n th order gauss sums these objects together satisfy a nonabelian group of functional equations in s w of order 32

mean values of biquadratic zeta functions let h d be the number of primitive inequivalent binary quadratic forms of discriminant d in disquisitiones 302 and 304 8 gauss gives conjectures for the average value of h d if d 0 where in the latter case d is derived from the fundamental solution to pell s equation gauss s conjectures were first proved by lipschitz 12 for negative discriminants and siegel for positive discriminants 14 when d is a fundamental discriminant and d the primitive real dirichlet character associated to the quadratic extension q d dirichlet s class number formula relates the value of the l series of d at s 1 to the class number h d thus gauss s conjectured asymptotics are equivalent to conjectures for sums of the type

on the use of isospectral eigenvalue problems for obtaining hereditary symmetries for hamiltonian systems we present an algorithmic method for obtaining an hereditary symmetry the generalized squared eigenfunction operator from a given isospectral eigenvalue problem this method is applied to the n n eigenvalue problem considered by ablowitz and haberman and to the eigenvalue problem considered by alonso the relevant hamiltonian formulations are also determined finally an alternative method is presented in the case two evolution equations are related by a miura type transformation and their hamiltonian formulations are known

first moment of hecke l functions with quartic characters at the central point in this paper we study the first moment of central values of hecke l functions associated with quartic characters

minimax estimation of functionals of discrete distributions we propose a general methodology for the construction and analysis of essentially minimax estimators for a wide class of functionals of finite dimensional parameters and elaborate on the case of discrete distributions where the support size s is unknown and may be comparable with or even much larger than the number of observations n we treat the respective regions where the functional is nonsmooth and smooth separately in the nonsmooth regime we apply an unbiased estimator for the best polynomial approximation of the functional whereas in the smooth regime we apply a bias corrected version of the maximum likelihood estimator mle we illustrate the merit of this approach by thoroughly analyzing the performance of the resulting schemes for estimating two important information measures 1 the entropy h p sum i 1 s p i ln p i and 2 f alpha p sum i 1 s p i alpha alpha 0 we obtain the minimax l 2 rates for estimating these functionals in particular we demonstrate that our estimator achieves the optimal sample complexity n asymp s ln s for entropy estimation we also demonstrate that the sample complexity for estimating f alpha p 0 is n asymp s 1 alpha ln s which can be achieved by our estimator but not the mle for 1 we show the minimax l 2 rate for estimating f alpha p is n ln n 2 alpha 1 for infinite support size while the maximum l 2 rate for the mle is n 2 alpha 1 for all the above cases the behavior of the minimax rate optimal estimators with n samples is essentially that of the mle plug in rule with n ln n samples which we term effective sample size enlargement we highlight the practical advantages of our schemes for the estimation of entropy and mutual information we compare our performance with various existing approaches and demonstrate that our approach reduces running time and boosts the accuracy moreover we show that the minimax rate optimal mutual information estimator yielded by our framework leads to significant performance boosts over the chow liu algorithm in learning graphical models the wide use of information measure estimation suggests that the insights and estimators obtained in this paper could be broadly applicable