sharp generalizations of the multilinear bohnenblust hille inequality we prove that the multilinear bohnenblust hille inequality is a particular case of a quite general family of optimal inequalities

global well posedness for the two dimensional nonlinear boussinesq equations with vertical dissipation abstract the present paper is devoted to the study of the global well posedness for the two dimensional nonlinear boussinesq equations with vertical dissipation in the absence of horizontal dissipation we establish a growth estimate on vertical component of velocity that is sup p 2 u 2 t l p p log p which is close to u 2 t l and is bounded via the low high decomposition technique this together with the smoothing effect in vertical direction enables us to obtain the h 1 estimate for velocity based on this we prove the existence and uniqueness of classical solution without smallness assumptions in addition we also discuss the global well posedness result for the rough initial data

global regularity for the 2d mhd equations with mixed partial dissipation and magnetic diffusion abstract whether or not classical solutions of the 2d incompressible mhd equations without full dissipation and magnetic diffusion can develop finite time singularities is a difficult issue a major result of this paper establishes the global regularity of classical solutions for the mhd equations with mixed partial dissipation and magnetic diffusion in addition the global existence conditional regularity and uniqueness of a weak solution is obtained for the 2d mhd equations with only magnetic diffusion

quantum reverse hypercontractivity its tensorization and application to strong converses in this paper we develop the theory of quantum reverse hypercontractivity inequalities and show how they can be derived from log sobolev inequalities next we prove a generalization of the stroock varopoulos inequality in the non commutative setting which allows us to derive quantum hypercontractivity and reverse hypercontractivity inequalities solely from 2 log sobolev and 1 log sobolev inequalities respectively we then prove some tensorization type results providing us with tools to prove hypercontractivity and reverse hypercontractivity not only for certain quantum superoperators but also for their tensor powers finally as an application of these results we generalize a recent technique for proving strong converse bounds in information theory via reverse hypercontractivity inequalities to the quantum setting we prove strong converse bounds for the problems of quantum hypothesis testing and classical quantum channel coding based on the quantum reverse hypercontractivity inequalities that we derive

global existence and decay of smooth solution for the 2 d mhd equations without magnetic diffusion abstract we prove the global existence and the decay estimates of small smooth solution for the 2 d mhd equations without magnetic diffusion this confirms the numerical observation that the energy of the mhd equations is dissipated at a rate independent of the ohmic resistivity

sobolev inequalities the poisson semigroup and analysis on the sphere sn abstract hypercontractive estimates are obtained for the poisson semigroup on lp sn such estimates are determined by a sharp logarithmic sobolev inequality that relates the entropy of a function on sn to its smoothness

matrix poincare φ sobolev inequalities and quantum ensembles sobolev type inequalities have been extensively studied in the frameworks of real valued functions and non commutative lp spaces and have proven useful in bounding the time evolution of classical quantum markov processes among many other applications in this paper we consider yet another fundamental setting matrix valued functions and prove new sobolev type inequalities for them our technical contributions are two fold i we establish a series of matrix poincare inequalities for separably convex functions and general functions with gaussian unitary ensembles inputs and ii we derive sobolev inequalities for matrix valued functions defined on boolean hypercubes and for those with gaussian distributions our results recover the corresponding classical inequalities i e real valued functions when the matrix has one dimension finally as an application of our technical outcomes we derive the upper bounds for a fundamental entropic quantity the holevo quantity in quantum information science since classical quantum channels are a special instance of matrix valued functions this is obtained through the equivalence between the constants in the strong data processing inequality and the sobolev inequality

lower bounds for the constants of the hardy littlewood inequalities abstract given an integer m 2 the hardy littlewood inequality for real scalars says that for all 2 m p there exists a constant c m p r 1 such that for all continuous m linear forms a l p n l p n r and all positive integers n j 1 j m 1 n a e j 1 e j m 2 m p m p p 2 m m p p 2 m 2 m p c m p r a the limiting case p is the well known bohnenblust hille inequality the behavior of the constants c m p r is an open problem in this note we provide nontrivial lower bounds for these constants

concentration of quantum states from quantum functional and transportation cost inequalities quantum functional inequalities e g the logarithmic sobolev and poincare inequalities have found widespread application in the study of the behavior of primitive quantum markov semigroups the classical counterparts of these inequalities are related to each other via a so called transportation cost inequality of order 2 tc2 the latter inequality relies on the notion of a metric on the set of probability distributions called the wasserstein distance of order 2 tc2 in turn implies a transportation cost inequality of order 1 tc1 in this paper we introduce quantum generalizations of the inequalities tc1 and tc2 making use of appropriate quantum versions of the wasserstein distances one recently defined by carlen and maas and the other defined by us we establish that these inequalities are related to each other and to the quantum modified logarithmic sobolev and poincare inequalities as in the classical case we also show that these inequalities imply certain concentration type results

on the real polynomial bohnenblust hille inequality abstract it was recently proved by f bayart and the fourth and fifth authors that the complex polynomial bohnenblust hille inequality is subexponential here we show that for real scalars this no longer holds moreover we show among other results that if d r m stands for the real bohnenblust hille constant for m homogeneous polynomials then lim sup m d r m 1 m 2 a quite surprising result having in mind that the exact value of the bohnenblust hille constants is still a mystery

the bohr radius of the n dimensional polydisk is equivalent to sqrt frac log n n we show that the bohr radius of the polydisk mathbb d n behaves asymptotically as sqrt log n n our argument is based on a new interpolative approach to the bohnenblust hille inequalities which allows us to prove that the polynomial bohnenblust hille inequality is subexponential

on the uniqueness of s functionals and m functionals under nonelliptical distributions the s functionals of multivariate location and scatter including the mve functionals are known to be uniquely defined only at unimodal elliptically symmetric distributions the goal of this paper is to establish the uniqueness of these functionals under broader classes of symmetric distributions we also discuss some implications of the uniqueness of the functionals and give examples of striclty unimodal and symmetric distributions for which the mve functional is not uniquely defined the uniqueness results for the s functionals are obtained by embedding them within a more general class of functionals which we call the m functionals with auxiliary scale the uniqueness results of this paper are then obtained for this class of multivariate functionals besides the s functionals the class of multivariate m functionals with auxiliary scale include the constrained m functionals recently introduced by kent and tyler as well as a new multivariate generalization of yohai s mm functionals

on extracting common random bits from correlated sources on large alphabets suppose alice and bob receive strings x x1 xn and y y1 yn each uniformly random in s n but so that x and y are correlated for each symbol i we have that yi xi with probability 1 e and otherwise yi is chosen independently and uniformly from s alice and bob wish to use their respective strings to extract a uniformly chosen common sequence from s k but without communicating how well can they do the trivial strategy of outputting the first k symbols yields an agreement probability of 1 e e s k in a recent work by bogdanov and mossel it was shown that in the binary case where s 2 and k k e is large enough then it is possible to extract k bits with a better agreement probability rate in particular it is possible to achieve agreement probability ke 1 2 2 ke 2 1 e 2 using a random construction based on hamming balls and this is optimal up to lower order terms in this paper we consider the same problem over larger alphabet sizes s and we show that the agreement probability rate changes dramatically as the alphabet grows in particular we show no strategy can achieve agreement probability better than 1 e k 1 s k where s 0 as s we also show that hamming ball based constructions have much lower agreement probability rate than the trivial algorithm as s our proofs and results are intimately related to subtle properties of hypercontractive inequalities

coordinatewise multiple summing operators in banach spaces we invent the new notion of coordinatewise multiple summing operators in banach spaces and use it to study various vector valued extensions of the well know bohnenblust hille inequality which originally extended littlewood s 4 3 inequality our results have application on the summability of monomial coefficients of m homogeneous polynomials p l lp as well as for the convergence theory of products of vector valued dirichlet series

robust alternatives to the f test in mixed linear models based on mm estimates mixed linear models are commonly used to analyze data in many settings these models are generally fitted by means of restricted maximum likelihood techniques relying heavily on normality the sensitivity of the resulting estimators and related tests to this underlying assumption has been identified as a weakness that can even lead to wrong interpretations very recently a highly robust estimator based on a scale estimate that is an s estimator has been proposed for general mixed linear models it has the advantage of being easy to compute and allows the computation of a robust score test however this proposal cannot be used to define a likelihood ratio type test that is certainly the most direct route to robustify an f test as the latter is usually a key tool of hypothesis testing in mixed linear models we propose two new robust estimators that allow the desired extension they also lead to resistant wald type tests useful for testing contrasts and covariate effects we study their properties theoretically and by means of simulations the analysis of a real data set illustrates the advantage of the new approach in the presence of outlying observations

asymptotic behaviour of s estimates of multivariate location parameters and dispersion matrices it is shown under appropriate conditions that rousseeuw s minimum volume estimator and other s estimators of multivariate location and dispersion parameters are consistent under certain differentiability conditions the estimates are asymptotically normally distributed with a norming factor of n 1 2

the fourth moment method higher moment analysis has typically been used to upper bound certain functions in this paper we introduce a new combinatorial method to lower bound the expectation of the absolute value of a random variable x by the expectation of a quartic in x in the special case where we are looking at the absolute value of a weighted sum of 1 1 unbiased random variables we achieve tight bounds using only a fourth moment for the total discrepancy of a set system because the fourth moment depends only on 4 wise independence our bounds will hold over polynomially sized distributions and so these bounds will be directly applicable in removing randomness to obtain nc algorithms we obtain the first nc algorithms for the problems of total discrepancy maximum acyclic subgraph tournament ranking the gale berlekamp switching game and edge discrepancy we show that for most of these applications it is truly necessary to consider a fourth moment by exhibiting a 3 wise independent distribution which does not achieve the required bounds our method is strong enough to give a new combinatorial bound on tournament ranking

robust inference with gmm estimators the local robustness properties of generalized method of moments gmm estimators and of a broad class of gmm based tests are investigated in a unified framework gmm statistics are shown to have bounded influence if and only if the function defining the orthogonality restrictions imposed on the underlying model is bounded since in many applications this function is unbounded it is useful to have procedures that modify the starting orthogonality conditions in order to obtain a robust version of a gmm estimator or test we show how this can be obtained when a reference model for the data distribution can be assumed we develop a flexible algorithm for constructing a robust gmm rgmm estimator leading to stable gmm test statistics the amount of robustness can be controlled by an appropriate tuning constant we relate by an explicit formula the choice of this constant to the maximal admissible bias on the level or and the power of a gmm test and the amount of contamination that one can reasonably assume given some information on the data finally we illustrate the rgmm methodology with some simulations of an application to rgmm testing for conditional heteroscedasticity in a simple linear autoregressive model in this example we find a significant instability of the size and the power of a classical gmm testing procedure under a non normal conditional error distribution on the other side the rgmm testing procedures can control the size and the power of the test under non standard conditions while maintaining a satisfactory power under an approximatively normal conditional error distribution

hypercontractivity for semigroups of unital qubit channels hypercontractivity is proved for products of qubit channels that belong to self adjoint semigroups the hypercontractive bound gives necessary and sufficient conditions for a product of the form e t 1 h 1 otimes cdots otimes e t n h n to be a contraction from l p to l q where l p is the algebra of 2 n dimensional matrices equipped with the normalized schatten norm and each generator h j is a self adjoint positive semidefinite operator on the algebra of 2 dimensional matrices as a particular case the result establishes the hypercontractive bound for a product of qubit depolarizing channels

on the relation between s estimators and m estimators of multivariate location and covariance we discuss the relation between s estimators and m estimators of multivariate location and covariance as in the case of the estimation of a multiple regression parameter s estimators are shown to satisfy first order conditions of m estimators we show that the influence function if x s f of s functionals exists and is the same as that of corresponding m functionals also we show that s estimators have a limiting normal distribution which is similar to the limiting normal distribution which is similar to the limiting normal distribution of m estimators finally we compare asymptotic variances and breakdown point of both types of estimators

distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding we consider the problem of ambiguous discrimination of two quantum states when we are only allowed to perform a restricted set of measurements let the bias of a povm be defined as the total variational distance between the outcome distributions for the two states to be distinguished the performance of a set of measurements can then be defined as the ratio of the bias of this povm and the largest bias achievable by any measurements we first provide lower bounds on the performance of various povms acting on a single system such as the isotropic povm and spherical 2 and 4 designs and show how these bounds can lead to certainty relations furthermore we prove lower bounds for several interesting povms acting on multipartite systems such as the set of local povms povms which can be implemented using local operations and classical communication locc separable povms and finally povms for which every bipartition results in a measurement having positive partial transpose ppt in particular our results show that a scheme of terhal et al for hiding data against local operations and classical communication 31 has the best possible dimensional dependence

factorization of absolutely continuous polynomials abstract in this paper we study the ideal of dominated p continuous polynomials that extend the nowadays well known ideal of p dominated polynomials to the more general setting of the interpolated ideals of polynomials we give the polynomial version of pietsch s factorization theorem for this new ideal although based on botelho et al in press 16 our factorization theorem requires new techniques inspired in the theory of banach lattices

new upper bounds for the constants in the bohnenblust hille inequality a classical inequality due to bohnenblust and hille states that for every positive integer m there is a constant c m 0 so that sigma n i1 im 1 vertical bar u e i1 e im vertical bar 2m m 1 m 1 2m c where c m m m 1 2m 2 m 1 2 the value of c m was improved to c m 2 m 1 2 by s kaijser and more recently h queffelec and a defant and p sevilla peris remarked that c m 2 root pi m 1 also works the bohnenblust hille inequality also holds for real banach spaces with the constants c m 2 m 1 2 in this note we show that a recent new proof of the bohnenblust hille inequality due to defant popa and schwarting provides in fact quite better estimates for c m for all values of m is an element of n in particular we will also show that for real scalars if m is even with 2 m 24 then c r m 2 1 2 c r m 2 we will mainly work on a paper by defant popa and schwarting giving some remarks about their work and explaining how to numerically improve the previously mentioned constants

improving the closed formula for subpolynomial constants in the multilinear bohnenblust hille inequalities abstract for k r or c the bohnenblust hille inequality asserts that there exists a sequence of scalars c k m m 1 such that i 1 i m 1 n u e i 1 e i m 2 m m 1 m 1 2 m c k m sup z 1 z m d n tu z 1 z m for all m linear forms u k n k n k and every positive integer n where e i i 1 n denotes the canonical basis of k n and d n represents the open unit polydisc in k n very recently 2012 it was shown that there exist constants c k m m 1 with subpolynomial growth satisfying this inequality however these constants were obtained via a complicated recursive formula we improve the best known closed non recursive approximation for these constants

robust inference for generalized linear models by starting from a natural class of robust estimators for generalized linear models based on the notion of quasi likelihood we define robust deviances that can be used for stepwise model selection as in the classical framework we derive the asymptotic distribution of tests based on robust deviances and we investigate the stability of their asymptotic level under contamination the binomial and poisson models are treated in detail two applications to real data and a sensitivity analysis show that the inference obtained by means of the new techniques is more reliable than that obtained by classical estimation and testing procedures

robust and efficient one way manova tests we propose robust tests as alternatives to the classical wilks lambda test in one way manova the robust tests use highly robust and efficient multisample multivariate s estimators or mm estimators instead of the empirical covariances the properties of several robust test statistics are compared under the null hypothesis the distribution of the test statistics is proportional to a chi square distribution as an alternative to the asymptotic distribution we develop a fast robust bootstrap method to estimate the distribution under the null hypothesis we show when it is asymptotically correct to estimate the null distribution in this way and we use simulations to verify the performance of the bootstrap based tests in finite samples we also investigate the power of the new tests as well as their robustness against outliers finally we illustrate the use of these robust test statistics on a real data example some additional results are provided as supplemental material

robust tests in regression models with omnibus alternatives and bounded influence a robust approach for testing the parametric form of a regression function versus an omnibus alternative is introduced this generalizes existing robust methods for testing subhypotheses in a regression model the new test is motivated by developments in modern smoothing based testing procedures and can be viewed as a robustification of a smoothing based conditional moment test it is asymptotically normal under both the null hypothesis and local alternatives the robustified test retains the omnibus property of the corresponding smoothing test that is it is consistent for any fixed smooth alternative in an infinite dimensional space it is shown that the bias of the asymptotic level under shrinking local contamination is bounded only if the second order hampel s influence function is bounded the test s performance is demonstrated through both monte carlo simulations and application to an agricultural dataset

a note on the polynomial bohnenblust hille inequality abstract recently in a paper published in the annals of mathematics it was shown that the bohnenblust hille inequality for complex homogeneous polynomials is hypercontractive however and to the best of our knowledge there is no result providing nontrivial lower bounds for the optimal constants for n homogeneous polynomials n 2 in this short note we provide lower bounds for these famous constants

optimal inapproximability results for max cut and other 2 variable csps in this paper we show a reduction from the unique games problem to the problem of approximating max cut to within a factor of alpha text tiny gw epsilon for all epsilon 0 here alpha text tiny gw approx 878567 denotes the approximation ratio achieved by the algorithm of goemans and williamson in j assoc comput mach 42 1995 pp 1115 1145 this implies that if the unique games conjecture of khot in proceedings of the 34th annual acm symposium on theory of computing 2002 pp 767 775 holds then the goemans williamson approximation algorithm is optimal our result indicates that the geometric nature of the goemans williamson algorithm might be intrinsic to the max cut problem our reduction relies on a theorem we call majority is stablest this was introduced as a conjecture in the original version of this paper and was subsequently confirmed in e mossel r o donnell and k oleszkiewicz proceedings of the 46th annual ieee symposium on foundations of computer science 2005 pp 21 30 a stronger version of this conjecture called plurality is stablest is still open although e mossel r o donnell and k oleszkiewicz proceedings of the 46th annual ieee symposium on foundations of computer science 2005 pp 21 30 contains a proof of an asymptotic version of it our techniques extend to several other two variable constraint satisfaction problems in particular subject to the unique games conjecture we show tight or nearly tight hardness results for max 2sat max q cut and max 2lin q for max 2sat we show approximation hardness up to a factor of roughly 943 this nearly matches the 940 approximation algorithm of lewin livnat and zwick in proceedings of the 9th annual conference on integer programming and combinatorial optimization springer verlag berlin 2002 pp 67 82 furthermore we show that our 943 factor is actually tight for a slightly restricted version of max 2sat for max q cut we show a hardness factor which asymptotically for large q matches the approximation factor achieved by frieze and jerrum improved approximation algorithms for max k cut and max bisection in integer programming and combinatorial optimization springer verlag berlin pp 1 13 namely 1 1 q 2 rm ln q q 2 for max 2lin q we show hardness of distinguishing between instances which are 1 epsilon satisfiable and those which are not even roughly q epsilon 2 satisfiable these parameters almost match those achieved by the recent algorithm of charikar makarychev and makarychev proceedings of the 38th annual acm symposium on theory of computing 2006 pp 205 214 the hardness result holds even for instances in which all equations are of the form x i x j c at a more qualitative level this result also implies that 1 epsilon vs e hardness for max 2lin q is equivalent to the unique games conjecture

the asymptotic growth of the constants in the bohnenblust hille inequality is optimal the search of sharp estimates for the constants in the bohnenblust hille inequality besides its challenging nature has quite important applications in different fields of mathematics and physics for homogeneous polynomials it was recently shown that the bohnenblust hille inequality for complex scalars is hypercontractive this result interesting by itself has found direct striking applications in the solution of several important problems for multilinear mappings precise information on the asymptotic behavior of the constants of the bohnenblust hille inequality is of particular importance for applications in quantum information theory and multipartite bell inequalities in this paper using elementary tools we prove a quite surprising result the asymptotic growth of the constants in the multilinear bohnenblust hille inequality is optimal besides its intrinsic mathematical interest and potential applications to different areas the mathematical importance of this result also lies in the fact that all previous estimates and related results for the last 80 years such as for instance the multilinear version of the famous grothendieck theorem for absolutely summing operators always present constants c m s growing at an exponential rate of certain power of m

global well posedness for the euler boussinesq system with axisymmetric data abstract in this paper we prove the global well posedness for the three dimensional euler boussinesq system with axisymmetric initial data without swirl this system couples the euler equation with a transport diffusion equation governing the temperature

absolutely summing multilinear operators via interpolation abstract we use an interpolative technique from 1 to introduce the notion of multiple n separately summing operators our approach extends and unifies some recent results for instance we recover the best known estimates of the multilinear bohnenblust hille constants due to f bayart d pellegrino and j seoane sepulveda more precisely as a consequence of our main result for 1 t 2 and m 1 we prove that i 1 i m 1 u e i 1 e i m 2 t m 2 m 1 t 2 m 1 t 2 t m j 2 m 2 2 t j t 2 t 2 t j 2 2 2 t 2 j t u for all complex m linear forms u c 0 c 0 c

on the hardness of decoding the gale berlekamp code the gale berlekamp in short gb code is the dual code of the binary product code in which the horizontal and vertical constituent codes are both the parity code it is shown that the problem of deciding whether there is a codeword of the gb code within a prescribed distance from a given received word is np complete the problem remains hard in a well defined sense even if the decoder is allowed unlimited preprocessing that depends only on the code length while the intractability of maximum likelihood decoding mld for specific codes has already been shown by bruck and naor lobstein and guruswami and vardy the result herein seems to be the first that shows hardness for a natural code in particular without any tailoring of the definition or the parameters of the code to suit the hardness proof in contrast it is also shown that with respect to any memoryless binary symmetric channel bsc with crossover probability less than 1 2 mld can be implemented in linear time for all error events except for a portion that occurs with vanishing probability

high breakdown inference for mixed linear models mixed linear models are used to analyze data in many settings these models have a multivariate normal formulation in most cases the maximum likelihood estimator mle or the residual mle reml is usually chosen to estimate the parameters however the latter are based on the strong assumption of exact multivariate normality welsh and richardson have shown that these estimators are not robust to small deviations from multivariate normality this means that in practice a small proportion of data even only one can drive the value of the estimates on their own because the model is multivariate we propose a high breakdown robust estimator for very general mixed linear models that include for example covariates this robust estimator belongs to the class of s estimators from which we can derive asymptotic properties for inference we also use it as a diagnostic tool to detect outlying subjects we discuss the advantages of this estimator compared with other robust estimators proposed previously and i

approximating the cut norm via grothendieck s inequality the cut norm a c of a real matrix a a ij i in r j in s is the maximum over all i subset r j subset s of the quantity sum i in i j in j a ij this concept plays a major role in the design of efficient approximation algorithms for dense graph and matrix problems here we show that the problem of approximating the cut norm of a given real matrix is max snp hard and we provide an efficient approximation algorithm this algorithm finds for a given matrix a a ij i in r j in s two subsets i subset r and j subset s such that sum i in i j in j a ij geq rho a c where rho 0 is an absolute constant satisfying rho 0 56 the algorithm combines semidefinite programming with a rounding technique based on grothendieck s inequality we present three known proofs of grothendieck s inequality with the necessary modifications which emphasize their algorithmic aspects these proofs contain rounding techniques which go beyond the random hyperplane rounding of goemans and williamson j acm 42 1995 pp 1115 1145 allowing us to transfer various algorithms for dense graph and matrix problems to the sparse case

quantum reverse hypercontractivity we develop reverse versions of hypercontractive inequalities for quantum channels by generalizing classical techniques we prove a reverse hypercontractive inequality for tensor products of qubit depolarizing channels we apply this to obtain a rapid mixing result for depolarizing noise applied to large subspaces and to prove bounds on a quantum generalization of non interactive correlation distillation

bootstrapping robust estimates of regression we introduce a new computer intensive method to estimate the distribution of robust regression estimates the basic idea behind our method is to bootstrap a reweighted representation of the estimates to obtain a bootstrap method that is asymptotically correct we include the auxiliary scale estimate in our reweighted representation of the estimates our method is computationally simple because for each bootstrap sample we only have to solve a linear system of equations the weights we use are decreasing functions of the absolute value of the residuals and hence outlying observations receive small weights this results in a bootstrap method that is resistant to the presence of outliers in the data the breakdown points of the quantile estimates derived with this method are higher than those obtained with the bootstrap we illustrate our method on two datasets and we report the results of a monte carlo experiment on confidence intervals for the parameters of the linear model

unbounded violation of tripartite bell inequalities we prove that there are tripartite quantum states constructed from random unitaries that can lead to arbitrarily large violations of bell inequalities for dichotomic observables as a consequence these states can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model this is in sharp contrast with the bipartite case where all violations are bounded by grothendieck s constant we will discuss the possibility of determining the hilbert space dimension from the obtained violation and comment on implications for communication complexity theory moreover we show that the violation obtained from generalized greenberger horne zeilinger ghz states is always bounded so that in contrast to many other contexts ghz states do not lead to extremal quantum correlations in this case in order to derive all these physical consequences we will have to obtain new mathematical results in the theories of operator spaces and tensor norms in particular we will prove the existence of bounded but not completely bounded trilinear forms from commutative c algebras finally we will relate the existence of diagonal states leading to unbounded violations with a long standing open problem in the context of banach algebras

sharp coincidences for absolutely summing multilinear operators abstract in this paper we prove the optimality of a family of known coincidence theorems for absolutely summing multilinear operators we connect our results with the theory of multiple summing multilinear operators and prove the sharpness of similar results obtained via the complex interpolation method

influence curves of general statistics abstract the influence curve jc of a fisher consistent functional was introduced by f hampel and plays a central role in the search for robust estimators an extension of this notion to non fisher consistent functionals is proposed in order to investigate the infinitesimal robustness of more general statistics e g those used in hypothesis testing this new definition inherits many useful properties including some on asymptotic efficiency functionals in two variables arising from two sample statistics are treated too connections with hodges lehmann shift estimators are discovered one and two sample rank statistics illustrate the theory

summing multilinear operators by blocks the isotropic and anisotropic cases abstract unifying several directions of the development of the study of summing multilinear operators between banach spaces we construct a general framework that studies under one single definition multilinear operators that are summing with respect to sums taken over any number of indices iterated and non iterated sums the isotropic and the anisotropic cases sums over arbitrary blocks and over several different sequence norms a large number of special classes of multilinear operators and of methods of generating classes of multilinear operators are recovered as particular instances ideal properties and coincidence theorems for the general classes are proved

some applications of hypercontractive inequalities in quantum information theory hypercontractive inequalities have become important tools in theoretical computer science and have recently found applications in quantum computation in this note we discuss how hypercontractive inequalities in various settings can be used to obtain fairly concise proofs of several results in quantum information theory a recent lower bound of lancien and winter on the bias achievable by local measurements which are 4 designs spectral concentration bounds for k local hamiltonians and a recent result of pellegrino and seoane sepulveda giving general lower bounds on the classical bias obtainable in multiplayer xor games

principal components analysis based on multivariate mm estimators with fast and robust bootstrap we consider robust principal components analysis pca based on multivariate mm estimators we first study the robustness and efficiency of these estimators particularly in terms of eigenvalues and eigenvectors we then focus on inference procedures based on a fast and robust bootstrap for mm estimators this method is an alternative to the approach based on the asymptotic distribution of the estimators and can also be used to assess the stability of the principal components a formal consistency proof for the bootstrap method is given and its finite sample performance is investigated through simulations we illustrate the use of the robust pca and the bootstrap inference on a real dataset

lower bounds for the complex polynomial hardy littlewood inequality abstract the hardy littlewood inequality for complex homogeneous polynomials asserts that given positive integers m 2 and n 1 if p is a complex homogeneous polynomial of degree m on l p n with m p given by p x 1 x n m a x then there exists a constant c c m p pol 1 which does not depend on n such that m a 1 c c m p pol sup z b l p n p z with p p m if m p 2 m and 2 m p m p p 2 m if 2 m p in this short note we provide nontrivial lower bounds for the constants c c m p pol for instance we prove that for m 2 and m p c c m p pol 2 m p for m even and c c m p pol 2 m 1 p for m odd estimates for the case p this is the particular case of the complex polynomial bohnenblust hille inequality were recently obtained by d nunez alarcon in 2013

the bohnenblust hille inequality for homogeneous polynomials is hypercontractive the bohnenblust hille inequality says that the 2m m 1 norm of the coefcients of an m homogeneous polynomial p on c n is bounded by kpk1 times a constant independent of n wherekk 1 denotes the supremum norm on the polydisc d n the main result of this paper is that this inequality is hypercontractive i e the constant can be taken to be c m for some c 1 combining this improved version of the bohnenblust hille inequality with other results we obtain the following the bohr radius for the polydisc d n behaves asymptotically as p logn n modulo a factor bounded away from 0 and innity

contractivity properties of a quantum diffusion semigroup we consider a quantum generalization of the classical heat equation and study contractivity properties of its associated semigroup we prove a nash inequality and a logarithmic sobolev inequality the former leads to an ultracontractivity result this in turn implies that the largest eigenvalue and the purity of a state with positive wigner function evolving under the action of the semigroup decrease at least inverse polynomially in time while its entropy increases at least logarithmically in time

two sided polynomial ideals on banach spaces abstract classes of homogeneous polynomials between banach spaces have been studied in the last three decades from the perspective of the stability with respect to the composition of a homogeneous polynomial with linear operators in an attempt to explore the underlying nonlinearity in a more consistent way and to bring the subject closer to its roots in complex analysis and also taking into account recent results in the field in this paper we propose the study of classes of homogeneous polynomials that are stable under the composition with homogeneous polynomials

on the fourier spectrum of functions on boolean cubes let f be a real valued function of degree d defined on the n dimensional boolean cube pm 1 n and f x sum s subset 1 ldots n widehat f s prod k in s x k its fourier walsh expansion the main result states that there is an absolute constant c 0 such that the ell 2d d 1 sum of the fourier coefficients of f pm 1 n longrightarrow 1 1 is bounded by c sqrt d log d it was recently proved that a similar result holds for complex valued polynomials on the n dimensional polytorus mathbb t n but that in contrast to this a replacement of the n dimensional torus mathbb t n by the n dimensional cube 1 1 n leads to a substantially weaker estimate this in the boolean case forces us to invent novel techniques which differ from the ones used in the complex or real case we indicate how our result is linked with several questions in quantum information theory

product state approximations to quantum states we show that for any many body quantum state there exists an unentangled quantum state such that most of the two body reduced density matrices are close to those of the original state this is a statement about the monogamy of entanglement which cannot be shared without limit in the same way as classical correlation our main application is to hamiltonians that are sums of two body terms for such hamiltonians we show that there exist product states with energy that is close to the ground state energy whenever the interaction graph of the hamiltonian has high degree this proves the validity of mean field theory and gives an explicitly bounded approximation error if we allow states that are entangled within small clusters of systems but product across clusters then good approximations exist when the hamiltonian satisfies one or more of the following properties 1 high degree 2 small expansion or 3 a ground state where the blocks in the partition have sublinear entanglement previously this was known only in the case of small expansion or in the regime where the entanglement was close to zero our approximations allow an extensive error in energy which is the scale considered by the quantum pcp probabilistically checkable proof and nlts no low energy trivial state conjectures thus our results put restrictions on the possible hamiltonians that could be used for a possible proof of the qpcp or nlts conjectures by contrast the classical pcp constructions are often based on constraint graphs with high degree likewise we show that the parallel repetition that is possible with classical constraint satisfaction problems cannot also be possible for quantum hamiltonians unless qpcp is false the main technical tool behind our results is a collection of new classical and quantum de finetti theorems which do not make any symmetry assumptions on the underlying states

on the global well posedness for the boussinesq system with horizontal dissipation in this paper we investigate the cauchy problem for the tridimensional boussinesq equations with horizontal dissipation under the assumption that the initial data is axisymmetric without swirl we prove the global well posedness for this system in the absence of vertical dissipation there is no smoothing effect on the vertical derivatives to make up this shortcoming we first establish a magic relationship between frac u r r and frac omega theta r by taking full advantage of the structure of the axisymmetric fluid without swirl and some tricks in harmonic analysis this together with the structure of the coupling of 1 2 entails the desired regularity

optimal hypercontractivity for fermi fields and related non commutative integration inequalities optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained these are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by nelson in the process several results of independent interest in the theory of non commutative integration are established

the 2d mhd equations with horizontal dissipation and horizontal magnetic diffusion this paper studies the global regularity of classical solutions to the 2d incompressible magnetohydrodynamic mhd equations with horizontal dissipation and horizontal magnetic diffusion it is shown here that the horizontal component of any solution admits a global in time bound in any lebesgue space l2r with 1 r and the bound grows no faster than the order of rlogr as r increases in addition we establish a conditional global regularity in terms of the lt2lx norm of the horizontal component and the global regularity of a slightly regularized version of the aforementioned mhd equations

explicit lower and upper bounds on the entangled value of multiplayer xor games the study of quantum mechanical violations of bell inequalities is motivated by the investigation and the eventual demonstration of the nonlocal properties of entanglement in recent years bell inequalities have found a fruitful re formulation using the language of multiplayer games originating from computer science this paper studies the nonlocal properties of entanglement in the context of the simplest such games called xor games when there are two players it is well known that the maximum bias the advantage over random play of players using entanglement can be at most a constant times greater than that of classical players recently perez garcia et al commun mathe phys 279 455 2008 showed that no such bound holds when there are three or more players the use of entanglement can provide an unbounded advantage and scale with the number of questions in the game their proof relies on non trivial results from operator space theory and gives a non explicit existence proof leading to a game with a very large number of questions and only a loose control over the local dimension of the players shared entanglement

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

equivalent norms in polynomial spaces and applications abstract in this paper equivalence constants between various polynomial norms are calculated as an application we also obtain sharp values of the hardy littlewood constants for 2 homogeneous polynomials on l p 2 spaces 2 p we also provide lower estimates for the hardy littlewood constants for polynomials of higher degrees

global well posedness for axisymmetric mhd system with only vertical viscosity abstract in this paper we are concerned with the global well posedness of a tri dimensional mhd system with only vertical viscosity in velocity equation for the large axisymmetric initial data by making good use of the axisymmetric structure of flow and the maximal smoothing effect of vertical diffusion we show that sup 2 p 0 t z u l p 2 p 3 4 d with this regularity for the vertical first derivative of velocity vector field we further establish losing estimates for the anisotropy tri dimensional mhd system to get the high regularity of u b which guarantees that 0 t u l d this together with the classical commutator estimate entails the global regularity of a smooth solution

two guidelines for bootstrap hypothesis testing two guidelines for nonparametric bootstrap hypothesis testing are highlighted the first recommends that resampling be done in a way that reflects the null hypothesis even when the true hypothesis is distant from the null the second guideline argues that bootstrap hypothesis tests should employ methods that are already recognized as having good features in the closely related problem of confidence interval construction violation of the first guideline can seriously reduce the power of a test sometimes this reduction is spectacular since it is most serious when the null hypothesis is grossly in error the second guideline is of some importance when the conclusion of a test is equivocal it has no direct bearing on power but improves the level accuracy of a test