beyond the entropy power inequality via rearrangements a lower bound on the renyi differential entropy of a sum of independent random vectors is demonstrated in terms of rearrangements for the special case of boltzmann shannon entropy this lower bound is better than that given by the entropy power inequality several applications are discussed including a new proof of the classical entropy power inequality and an entropy inequality involving symmetrization of levy processes

on classes of non gaussian asymptotic minimizers in entropic uncertainty principles abstract in this paper we revisit the bialynicki birula and mycielski uncertainty principle and its cases of equality this shannon entropic version of the well known heisenberg uncertainty principle can be used when dealing with variables that admit no variance in this paper we extend this uncertainty principle to renyi entropies we recall that in both shannon and renyi cases and for a given dimension n the only case of equality occurs for gaussian random vectors we show that as n grows however the bound is also asymptotically attained in the cases of n dimensional student t and student r distributions a complete analytical study is performed in a special case of a student t distribution we also show numerically that this effect exists for the particular case of a n dimensional cauchy variable whatever the renyi entropy considered extending the results of abe and illustrating the analytical asymptotic study of the student t case in the student r case we show numerically that the same behavior occurs for uniformly distributed vectors these particular cases and other ones investigated in this paper are interesting since they show that this asymptotic behavior cannot be considered as a gaussianization of the vector when the dimension increases

entropy and the segal bargmann transform the segal bargmann transform a is a unitary isomorphism of the quantum mechanical configuration space l2 rn onto the segal bargmann or phase space hl2 cn of entire functions square integrable with respect to here and are gaussian measures and for z cn and measurable f rn caf z rn d t f t exp z2 2 z t if the integral converges af is defined for all f lp rn if 1 1 q 2 which implies p q and in this case a p q 1 q 2 p q 2 1 q 2 n 2q where p is the dual of p moreover a calculation with test functions shows that a p q if q 2 or p 1 q 2 cp q is given explicitly as a function of p and q and s

reverse faber krahn and mahler inequalities for the cheeger constant we prove a reverse faber krahn inequality for the cheeger constant stating that every convex body in 2 has an affine image such that the product between its cheeger constant and the square root of its area is not larger than the same quantity for the regular triangle an analogous result holds for centrally symmetric convex bodies with the regular triangle replaced by the square we also prove a mahler type inequality for the cheeger constant stating that every axisymmetric convex body in 2 has a linear image such that the product between its cheeger constant and the cheeger constant of its polar body is not larger than the same quantity for the square

the hausdorff young inequality on lie groups we prove several results about the best constants in the hausdorff young inequality for noncommutative groups in particular we establish a sharp local central version for compact lie groups and extend known results for the heisenberg group in addition we prove a universal lower bound to the best constant for general lie groups

on complex hypercontractivity abstract we give a new proof of a hypercontractivity theorem for the mehler transform with a complex parameter earlier proved by weissler 1979 j funct anal 32 102 121 and epperson 1989 j funct anal 87 1 30 the proof uses stochastic integrals and ito calculus the method also yields new proofs of some related results

rearrangement inequalities for functionals with monotone integrands the inequalities of hardy littlewood and riesz say that certain integrals involving products of two or three functions increase under symmetric decreasing rearrangement it is known that these inequalities extend to integrands of the form f u1 um where f is supermodular in particular they hold when f has nonnegative mixed second derivatives i jf for all i j this paper concerns the regularity assumptions on f and the equality cases it is shown here that extended hardy littlewood and riesz inequalities are valid for supermodular integrands that are just borel measurable under some nondegeneracy conditions all equality cases are equivalent to radially decreasing functions under transformations that leave the functionals invariant i e measure preserving maps for the hardy littlewood inequality translations for the riesz inequality the proofs rely on monotone changes of variables in the spirit of sklar s theorem

on the lp lq norm of the hankel transform and related operators abstract we investigate the l p 0 l q 0 mapping properties of the operators l f y y 0 x y f x j x y d x f c 0 0 for suitable values of the parameters and we evaluate the operator norm of l in some special and significant cases

the equality cases of the ehrhard borell inequality abstract the ehrhard borell inequality is a far reaching refinement of the classical brunn minkowski inequality that captures the sharp convexity and isoperimetric properties of gaussian measures unlike in the classical brunn minkowski theory the equality cases in this inequality are far from evident from the known proofs the equality cases are settled systematically in this paper an essential ingredient of the proofs are the geometric and probabilistic properties of certain degenerate parabolic equations the method developed here serves as a model for the investigation of equality cases in a broader class of geometric inequalities that are obtained by means of a maximum principle

existence and uniqueness of monotone measure preserving maps introduction given a pair of borel probability measures and v on rd it is natural to ask whether v can be obtained from by redistributing its mass in a canonical way in the case of the line d 1 the answer is clear as long as both measures are free from atoms x v x 0mthere is a map y x of the line to itself for which uniquely determined almost everywhere this map may be taken to be nondecreasing by a suitable choice of y x e rw o at the remaining points interpreting and v as the initial and final distribution of a one dimensional fluid the transformation y x gives a rearrangement of fluid particles yielding final state v from the initial state this rearrangement is characterized by the fact that it preserves particle ordering obviating any need for two particles to cross although the generalization of this construction to higher dimensions is the focus of this paper the one dimensional case will be pursued slightly further when the measures are absolutely continuous with respect to lebesgue d x f x dx and dr y g y dymthen formally at least neglecting regularity issues the fundamental theorem of calculus yields

some nonlinear brascamp lieb inequalities and applications to harmonic analysis abstract we use the method of induction on scales to prove certain diffeomorphism invariant nonlinear brascamp lieb inequalities we provide applications to multilinear convolution inequalities and the restriction theory for the fourier transform extending to higher dimensions recent work of bejenaru herr tataru and bennett carbery wright

redundancy analysis for qualitative variables redundancy analysis also called principal components analysis of instrumental variables is a technique for two sets of variables one set being dependent of the other its aim is maximization of the explained variance of the dependent variables by a linear combination of the explanatory variables the technique is generalized to qualitative variables it then gives implicitly a simultaneous optimal scaling of the dependent qualitative variables examples are taken from the dutch life situation survey 1977 using satisfaction with life and happiness as dependent variables the analysis leads to one well being scale defined by the explanatory variables marital status schooling income and activity

identification in restricted factor models and the evaluation of rank conditions abstract this paper presents a rank condition for the local identification in restricted factor analysis models and it provides a method for the computation of the exact rank in the condition the method can be applied to evaluate the rank of rather general jacobian matrices

new class of multiplicative algorithms for solving of entropy linear programs the general entropy linear program elp is considered two types of the new coordinate wise multiplicative algorithms with p active variables and feedback control with respect to dual variables and mixed type dual and primal variables are proposed for solving the problem study of algorithms convergence is based on a stability analysis of the auxiliary differential equations that are continuous analogues of the algorithms

improved holder and reverse holder inequalities for gaussian random vectors abstract we propose algebraic criteria that yield sharp holder types of inequalities for the product of functions of gaussian random vectors with arbitrary covariance structure while our lower inequality appears to be new we prove that the upper inequality gives an equivalent formulation for the geometric brascamp lieb inequality for gaussian measures as an application we retrieve the gaussian hypercontractivity as well as its reverse and we present a generalization of the sharp young and reverse young inequalities from the latter we recover several known inequalities in the literature including the prekopa leindler and barthe inequalities

a central limit theorem for operators abstract we prove an analogue of the central limit theorem for operators for every operator k defined on c x we construct a sequence of operators k n defined on c x 1 x n given by the n fold tensor product of k with itself modulo a proper scaling and we demonstrate that under certain orthogonality conditions this sequence converges in a weak sense to an unique operator c we show that this operator c is a member of a family of operators c that we call centered gaussian operators and which coincides with the family of operators given by a centered gaussian kernel inspired in the approximation method used by beckner in 1 to prove the sharp form of the hausdorff young inequality the present article shows that beckner s method is a special case of a general approximation method for operators in particular we characterize the hermite semi group as the family of centered gaussian operators associated with any semi group of operators

on sharp constants for dual segal bargmann lp spaces abstract we study dilated holomorphic l p space of gaussian measures over c n denoted h p n with variance scaling parameter 0 the duality relations h p n h p hold with 1 p 1 p 1 but not isometrically we identify the sharp lower constant comparing the norms on h p and h p n and provide upper and lower bounds on the sharp upper constant we prove several suggestive partial results on the sharpness of the upper constant one of these partial results leads to a sharp bound on each taylor coefficient of a function in the fock space for n 1

on the reproducing kernel of the segal bargmann space this article revolves around the properties on the lp scale of spaces of the integral kernel operator k whose kernel function is the reproducing kernel of the segal bargmann space we find sufficient conditions on p and q for k to be a hille tamarkin and hence compact operator from lp to lq with respect to the standard gaussian measure as well as with respect to a weighted measure on the codomain space we also find sufficient conditions for k to be unbounded with respect to the standard gaussian measure finally we give sufficent conditions for a toeplitz operator to be hille tamarkin on the lp scale of spaces with respect to both the standard gaussian measure and a weighted measure on the codomain space

new polya szego type inequalities and an alternative approach to comparison results for pde s abstract we prove some polya szego type inequalities which involve couples of functions and their rearrangements our inequalities reduce to the classical polya szego principle when the two functions coincide as an application we give a different proof of a comparison result for solutions to dirichlet boundary value problems for laplacian equations proved in 1

interior gradient and epsilon subgradient descent methods for constrained convex minimization we extend epsilon subgradient descent methods for unconstrained nonsmooth convex minimization to constrained problems over polyhedral sets in particular over r p this is done by replacing the usual squared quadratic regularization term used in subgradient schemes by the logarithmic quadratic distancelike function recently introduced by the authors we then obtain interior e subgradient descent methods which allow us to provide a natural extension of bundle methods and polyak s subgradient projection methods for nonsmooth convex minimization furthermore similar extensions are considered for smooth constrained minimization to produce interior gradient descent methods global convergence as well as an improved global efficiency estimate are established for these methods within a unifying principle and minimal assumptions on the problem s data

on the nonlinear brascamp lieb inequality we prove a nonlinear variant of the general brascamp lieb inequality our proof consists of running an efficient or tight induction on scales argument which uses the existence of gaussian near extremizers to the underlying linear brascamp lieb inequality lieb s theorem in a fundamental way a key ingredient is an effective version of lieb s theorem which we establish via a careful analysis of near minimizers of weighted sums of exponential functions instances of this inequality are quite prevalent in mathematics and we illustrate this with some applications in harmonic analysis

an entropy based uncertainty principle for a locally compact abelian group abstract we classify all functions on a locally compact abelian group giving equality in an entropy inequality generalizing the heisenberg uncertainty principle in particular for functions on a real line we proof a conjecture of hirschman published in 1957

duality in segal bargmann spaces abstract for 0 the bargmann projection p is the orthogonal projection from l 2 onto the holomorphic subspace l hol 2 where is the standard gaussian probability measure on c n with variance 2 n the space l hol 2 is classically known as the segal bargmann space we show that p extends to a bounded operator on l p p 2 and calculate the exact norm of this scaled l p bargmann projection we use this to show that the dual space of the l p segal bargmann space l hol p p 2 is an l p segal bargmann space but with the gaussian measure scaled differently l hol p p 2 l hol p p 2 this was shown originally by janson peetre and rochberg we show that the bargmann projection controls this dual isomorphism and gives a dimension independent estimate on one of the two constants of equivalence of the norms

volume inequalities of convex bodies from cosine transforms on grassmann manifolds abstract the l p cosine transform on grassmann manifolds naturally induces finite dimensional banach norms whose unit balls are origin symmetric convex bodies in r n reverse isoperimetric type volume inequalities for these bodies are established which extend results from the sphere to grassmann manifolds

a reverse sidorenko inequality let h be a graph allowing loops as well as vertex and edge weights we prove that for every triangle free graph g without isolated vertices the weighted number of graph homomorphisms hom g h satisfies the inequality begin aligned hom g h le prod uv in e g hom k d u d v h 1 d ud v end aligned where d u denotes the degree of vertex u in g in particular one has begin aligned hom g h 1 e g le hom k d d h 1 d 2 end aligned for every d regular triangle free g the triangle free hypothesis on g is best possible more generally we prove a graphical brascamp lieb type inequality where every edge of g is assigned some two variable function these inequalities imply tight upper bounds on the partition function of various statistical models such as the ising and potts models which includes independent sets and graph colorings for graph colorings corresponding to h k q we show that the triangle free hypothesis on g may be dropped this is also valid if some of the vertices of k q are looped a corollary is that among d regular graphs g k d d maximizes the quantity c q g 1 v g for every q and d where c q g counts proper q colorings of g finally we show that if the edge weight matrix of h is positive semidefinite then begin aligned hom g h le prod v in v g hom k d v 1 h 1 d v 1 end aligned this implies that among d regular graphs g k d 1 maximizes hom g h 1 v g for 2 spin ising models our results give a complete characterization of extremal graphs complete bipartite graphs maximize the partition function of 2 spin antiferromagnetic models and cliques maximize the partition function of ferromagnetic models these results settle a number of conjectures by galvin tetali galvin and cohen csikvari perkins tetali and provide an alternate proof to a conjecture by kahn

zur theorie der linearen und nichtlinearen integralgleichungen fredholm hat eine auflosungsformel der inhomogenen linearen integralgleichung f left s right varphi left s right lambda int limits a b k left s t right varphi left t right dt entdeckt welche das resultat enthielt das diese gleichung nach s stets aufgelost werden kann wenn nicht eine der nullstellen einer gewissen ganzen transzendenten ist fur diese werte von in hilbertscher bezeichnung die sogenannten eigenwerte des kernes k s t und nur fur diese last wie fredholm ferner zeigte die homogene gleichung 0 varphi left s right int limits a b k left s t right varphi left t right dt losungen zu die nach hilbert zum betreffenden eigenwert des kernes k s t gehorige eigenfunktionen genannt werden hilbert hat die in der theorie der partiellen und gewohnlichen differentialgleichungen so wichtigen fragen nach der existenz sogenannter normalfunktionen und der entwickelbarkeit willkurlicher funktionen nach ihnen durch einfuhrung der greenschen funktion auf das viel allgemeinere problem zuruckgefuhrt die existenz von eigenfunktionen eines in s und t symmetrischen kernes k s t zu beweisen und die gesetze der entwickelbarkeit willkurlicher funktionen nach ihnen aufzustellen die greensche funktion selbst wird durch eine integralgleichung mit unsymmetrischem kerne bestimmt wobei die fredholmschen formeln zur anwendung kommen

the contractivity of the free hamiltonian semigroup in the lp space of entire functions abstract a semigroup of operators t t t 0 generated by the number operator in the complex wave representation is given by t t z e t z for any entire function on c n the problem is to find all possible r p 1 such that t t hlr cn dgn cn dgn is contractive or bounded it turns out that t t hlr cn dgn cn dgn is contractive if and only if p 1 re2t and is unbounded if and only if p re2t furthermore all maximizers are identified they are constant functions if p 1 re2t and exponential functions if p re2t t 0

the regularity of mappings with a convex potential in this work we apply the techniques developed in cl to the problem of mappings with a convex potential between domains that is given two bounded domains q q2 of rn and two nonnegative real functions fi defined in qi that are bounded away from zero and infinity we want to study the map vf v w for a lipschitz convex v such that v maps ql onto q 2 in the a e sense and in some weak sense 1 f2 vyv det dij v f1 x in recent work y brenier showed existence and uniqueness of such a map provided that jaqil 0 under the obvious compatibility condition

symmetric rearrangements around infinity with applications to levy processes we prove a new rearrangement inequality for multiple integrals which partly generalizes a result of friedberg and luttinger arch ration mech 61 35 44 1976 and can be interpreted as involving symmetric rearrangements of domains around infty as applications we prove two comparison results for general levy processes and their symmetric rearrangements the first application concerns the survival probability of a point particle in a poisson field of moving traps following independent levy motions we show that the survival probability can only increase if the point particle does not move and the traps and the levy motions are symmetrically rearranged this essentially generalizes an isoperimetric inequality of peres and sousi geom funct anal 22 4 1000 1014 2012 for the wiener sausage in the second application we show that the q capacity of a borel measurable set for a levy process can only decrease if the set and the levy process are symmetrically rearranged this result generalizes an inequality obtained by watanabe z wahrsch verw gebiete 63 487 499 1983 for symmetric levy processes

models and methods for the analysis of correlation coefficients abstract the normal distribution plays a very prominant role in statistics and in multivariate analysis the multinormal model is even more dominant than in the classical univariate or bivariate situations in this paper we show that the assumption of continuous multinormal observations can be relaxed in various ways for continuous multivariate analysis most of the existing large sample theory can be applied without assuming multivariate normality we also develop several discrete multinormal models the first one is the classical pearson model which results from discreticizing multinormal variables the second one is inspired by correspondence analysis and the third one by loglinear analysis the models are presented in terms of correlation coefficients and applied to the spearman model of factor analyses using an example from political science

rearrangement and polarization abstract the paper has two main goals the first is to take a new approach to rearrangements on certain classes of measurable real valued functions on r n rearrangements are maps that are monotonic up to sets of measure zero and equimeasurable i e they preserve the measure of super level sets of functions all the principal known symmetrization processes for functions such as steiner and schwarz symmetrization are rearrangements and these have a multitude of applications in diverse areas of the mathematical sciences the second goal is to understand which properties of rearrangements characterize polarization a special rearrangement that has proved particularly useful in a number of contexts in order to achieve this new results are obtained on the structure of measure preserving maps on convex bodies and of rearrangements generally

multivariate methods for quantitative and qualitative data abstract in this paper several multivariate analysis methods are analyzed by means of the errors in variables model two specifications viz the principal relations and the principal factors model for the set of unobserved latent variables are introduced and shown to be complementary for particular choices of the error covariance matrix the general model reduces to several well known multivariate techniques including ordinary least squares principal components orthogonal regression and canonical correlations the same apparatus is used to arrive at a set of multivariate techniques in case the data is qualitative instead of quantitative the relationship with optimal scaling techniques for qualitative data is shown an empirical example is given of a regression involving a dependent latent variable having two ordered categorical indicator variables

gaussian optimizers for entropic inequalities in quantum information we survey the state of the art for the proof of the quantum gaussian optimizer conjectures of quantum information theory these fundamental conjectures state that quantum gaussian input states are the solution to several optimization problems involving quantum gaussian channels these problems are the quantum counterpart of three fundamental results of functional analysis and probability the entropy power inequality the sharp young s inequality for convolutions and the theorem gaussian kernels have only gaussian maximizers quantum gaussian channels play a key role in quantum communication theory they are the quantum counterpart of gaussian integral kernels and provide the mathematical model for the propagation of electromagnetic waves in the quantum regime the quantum gaussian optimizer conjectures are needed to determine the maximum communication rates over optical fibers and free space the restriction of the quantum limited gaussian attenuator to input states diagonal in the fock basis coincide

integral bounds for radar ambiguity functions and wigner distributions an upper bound is proved for the lp norm of woodward s ambiguity function in radar signal analysis and of the wigner distribution in quantum mechanics when p 2 a lower bound is proved for 1 p 2 in addition a lower bound is proved for the entropy these bounds set limits to the sharpness of the peaking of the ambiguity function or wigner distribution the bounds are best possible and equality is achieved in the lp bounds if and only if the functions f and g that enter the definition are both gaussians

shadows of convex bodies it is proved that if c is a convex body in rn then c has an affine image c of nonzero volume so that if p is any 1 codimensional orthogonal projection pca 1 cl n l n it is also shown that there is a pathological body k all of whose orthogonal projections have volume about vx times as large as iki n 1 n 0 introduction the problems discussed in this paper concern the areas of shadows orthogonal projections of convex bodies and to a lesser extent the surface areas of such bodies if c is a convex body in irn and 0 a unit vector p c will denote the orthogonal projection of c onto the 1 codimensional space perpendicular to 0 volumes and areas of convex bodies and their surfaces will be denoted with i 1 the relationship between shadows and surface areas of convex bodies is expressed in cauchy s well known formula for each n e n let vn be the volume of the n dimensional euclidean unit ball and let a an be the rotationally invariant probability on the unit sphere sn cauchy s formula states that if c is a convex body in rn then its surface area is ac i s p6ci dcr o vni jn i the classical isoperimetric inequality in rn states that any body has surface area at least as large as a euclidean ball of the same volume the first section of this paper is devoted to the proof of a local isoperimetric inequality showing that all bodies have large shadows rather than merely large surface area or average shadow the principal motivation for this result is its relationship to a conjecture of vaaler and the important problems surrounding it this theorem and its connection with vaaler s conjecture are described at the beginning of 1 an important role is played in the theory of convex bodies by the so called projection body of a convex body it is easily seen by considering polytopes received by the editors september 12 1989 1980 mathematics subject classification 1985 revision primary 52a20 52a40 the author was supported in part by nsf dms 8807243 1991 american mathematical society 0002 9947 91 1 00 25 per page

loglinear models and categorical data analysis with psychometric and econometric applications revision abstract the past decade has seen the publication of a large number of books and papers on the analysis of multi way contingency tables using loglinear and logic models the present article presents a summary of the statistical theory that underlies much of this work and provides some linkage to models and methods of special interest to psychometricians and econometricians the discussion includes a review of recent and current research on the computation of maximum likelihood estimates for loglinear and logit models especially for large multi way contingency tables

on the multilinear restriction and kakeya conjectures we prove d linear analogues of the classical restriction and kakeya conjectures in rd our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians closely related to heat flow we conclude by giving some applications to the corresponding variable coefficient problems and the so called joints problem as well as presenting some n linear analogues for n d

the perimeter inequality under steiner symmetrization cases of equality steiner symmetrization is known not to increase perimeter of sets in rn the sets whose perimeter is preserved under this symmetrization are characterized in the present paper

the borell ehrhard game a precise description of the convexity of gaussian measures is provided by sharp brunn minkowski type inequalities due to ehrhard and borell we show that these are manifestations of a game theoretic mechanism a minimax variational principle for brownian motion as an application we obtain a gaussian improvement of barthe s reverse brascamp lieb inequality

norms of quantum gaussian multi mode channels we compute the s p s p norm of a general gaussian gauge covariant multi mode channel for any 1 p where s p is a schatten space as a consequence we verify the gaussian optimizer conjecture and the multiplicativity conjecture in these cases

one parameter class of uncertainty relations based on entropy power we use the concept of entropy power to derive a one parameter class of information theoretic uncertainty relations for pairs of conjugate observables in an infinite dimensional hilbert space this class constitutes an infinite tower of higher order statistics uncertainty relations which allows one in principle to determine the shape of the underlying information distribution function by measuring the relevant entropy powers we illustrate the capability of this class by discussing two examples superpositions of vacuum and squeezed states and the cauchy type heavy tailed wave function

smoothing brascamp lieb inequalities and strong converses of coding theorems the brascamp lieb inequality in functional analysis can be viewed as a measure of the uncorrelatedness of a joint probability distribution we define the smooth brascamp lieb bl divergence as the infimum of the best constant in the brascamp lieb inequality under a perturbation of the joint probability distribution an information spectrum upper bound on the smooth bl divergence is proved using properties of the subgradient of a certain convex functional in particular in the i i d setting such an infimum converges to the best constant in a certain mutual information inequality we then derive new single shot converse bounds for the omniscient helper common randomness generation problem and the gray wyner source coding problem in terms of the smooth bl divergence where the proof relies on the functional formulation of the brascamp lieb inequality exact second order rates are thus obtained in the stationary memoryless and nonvanishing error setting these offer rare instances of strong converses second order converses for continuous sources when the rate region involves auxiliary random variables

cases of equality in certain multilinear inequalities of hardy riesz rogers brascamp lieb luttinger type abstract cases of equality in certain hardy riesz rogers brascamp lieb luttinger rearrangement inequalities are characterized

on an integral criterion for hypercontractivity of diffusion semigroups and extremal functions abstract in the line of investigation of the works by d bakry and m emery lecture notes in math vol 1123 pp 175 206 springer verlag new york berlin 1985 and o s rothaus j funct anal 42 1981 102 109 j funct anal 65 1986 358 367 we study an integral inequality behind the 2 criterion of d bakry and m emery see previous reference and its applications to hypercontractivity of diffusion semigroups with in particular a short proof of the hypercontractivity property of the ornstein uhlenbeck semigroup our exposition unifies in a simple way several previous results interpolating smoothly from the spectral gap inequalities to logarithmic sobolev inequalities and even true sobolev inequalities we examine simultaneously the extremal functions for hypercontractivity and logarithmic sobolev inequalities of the ornstein uhlenbeck semigroup and heat semigroup on spheres

a strong entropy power inequality when one of the random summands is gaussian we sharpen the entropy power inequality epi in terms of the strong data processing function for gaussian channels among other consequences this strong epi generalizes the vector extension of costa s epi to non gaussian channels in a precise sense this leads to a new reverse epi and as a corollary sharpens stam s uncertainty principle relating entropy power and fisher information or equivalently gross logarithmic sobolev inequality applications to network information theory are also given including a short self contained proof of the rate region for the two encoder quadratic gaussian source coding problem and a new outer bound for the one sided gaussian interference channel

on a reverse form of the brascamp lieb inequality we prove a reverse form of the multidimensional brascamp lieb inequality our method also gives a new way to derive the brascamp lieb inequality and is rather convenient for the study of equality cases

phase space localization of antisymmetric functions upper and lower bounds are written down for the minimum information entropy in phase space of an antisymmetric wave function in any number of dimensions similar bounds are given when the wave function is restricted to belong to any of the proper subspaces of the fourier transform operator

on lieb s conjecture for the wehrl entropy of bloch coherent states lieb s conjecture for the wehrl entropy of bloch coherent states is proved for spin 1 and spin 3 2 using a geometric representation we solve the entropy integrals for states of arbitrary spin and evaluate them explicitly in the cases of spin 1 3 2 and 2 we also give a group theoretic proof for all spin of a related inequality

extremals of functionals with competing symmetries abstract we present a new method of producing optimizing sequences for highly symmetric functionals the sequences have good convergence properties built in we apply the method in different settings to give elementary proofs of some classical inequalities such as the hardy littlewood sobolev and the logarithmic sobolev inequality in their sharp form

entropic uncertainty bound for open pointer based simultaneous measurements of conjugate observables we discuss the information entropy for a general open pointer based simultaneous measurement and show how it is bound from below this entropic uncertainty bound is a direct consequence of the structure of the entropy and can be obtained from the formal solution of the measurement dynamics furthermore the structural properties of the entropy allow us to give an intuitive interpretation of the noisy influence of the pointers and the environmental heat bath on the measurement results

quantitative analysis of qualitative data this paper presents an overview of an approach to the quantitative analysis of qualitative data with theoretical and methodological explanations of the two cornerstones of the approach alternating least squares and optimal scaling using these two principles my colleagues and i have extended a variety of analysis procedures originally proposed for quantitative interval or ratio data to qualitative nominal or ordinal data including additivity analysis and analysis of variance multiple and canonical regression principal components common factor and three mode factor analysis and multidimensional scaling the approach has two advantages a if a least squares procedure is known for analyzing quantitative data it can be extended to qualitative data and b the resulting algorithm will be convergent three completely worked through examples of the additivity analysis procedure and the steps involved in the regression procedures are presented

sum rate capacity for symmetric gaussian multiple access channels with feedback the feedback sum rate capacity is established for the symmetric j user gaussian multiple access channel gmac the main contribution is a converse bound that combines the dependence balance argument of hekstra and willems 1989 with a variant of the factorization of a convex envelope of geng and nair 2014 the converse bound matches the achievable sum rate of the fourier modulated estimate correction strategy of kramer 2002

gaussian kernels have only gaussian maximizers a gaussian integral kernel g x y on rn x rn is the exponential of a quadratic form in x and y the fourier transform kernel is an example the problem addressed here is to find the sharp bound of g as an operator from l p rp to l q rn and to prove that the l p rn functions that saturate the bound are necessarily gaussians this is accomplished generally for 1 q in some special cases besides greatly extending previous results in this area the proof technique is also essentially different from earlier ones a corollary of these results is a fully multidimensional multilinear generalization of young s inequality

compactness via symmetrization consider two types of translation invariant functionals i and j on rm and a sequence of functions fn whose corresponding symmetric rearrangements f n are convergent we show that fn themselves converge up to translations if either limn i fn limn i fn or limn j fn limn j fn these compactness results lead to applications in variational problems and stability problems in stellar dynamics

the weakly nonlinear large box limit of the 2d cubic nonlinear schrodinger equation we consider the cubic nonlinear schrodinger nls equation set on a two dimensional box of size l with periodic boundary conditions by taking the large box limit l to infty in the weakly nonlinear regime characterized by smallness in the critical space we derive a new equation set on r 2 that approximates the dynamics of the frequency modes this nonlinear equation turns out to be hamiltonian and enjoys interesting symmetries such as its invariance under fourier transform as well as several families of explicit solutions a large part of this work is devoted to a rigorous approximation result that allows to project the long time dynamics of the limit equation into that of the cubic nls equation on a box of finite size

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

optimal heat kernel estimates for schrodinger operators with magnetic fields in two dimensions sharp smoothing estimates are proven for magnetic schrodinger semigroups in two dimensions under the assumption that the magnetic field is bounded below by some positive constantb0 as a consequence thel norm of the associated integral kernel is bounded by thel norm of the mehler kernel of the schrodinger semigroup with the constant magnetic fieldb0

multiplicativity properties of entrywise positive maps abstract multiplicativity of certain maximal p q norms of a tensor product of linear maps on matrix algebras is proved in situations in which the condition of complete positivity cp is either augmented by or replaced by the requirement that the entries of a matrix representative of the map are non negative ep in particular for integer t multiplicativity holds for the maximal 2 2t norm of a product of two maps whenever one of the pair is ep for the maximal 1 t norm for pairs of cp maps when one of them is also ep and for the maximal 1 2t norm for the product of an ep and a 2 positive map similar results are shown in the infinite dimensional setting of convolution operators on l 2 r with the pointwise positivity of an integral kernel replacing entrywise positivity of a matrix these results apply in particular to gaussian bosonic channels

dimension dependent hypercontractivity for gaussian kernels we derive sharp local and dimension dependent hypercontractive bounds on the markov kernel of a large class of diffusion semigroups unlike the dimension free ones they capture refined properties of markov kernels such as trace estimates they imply classical bounds on the ornstein uhlenbeck semigroup and a dimensional and refined transportation talagrand inequality when applied to the hamilton jacobi equation hypercontractive bounds on the ornstein uhlenbeck semigroup driven by a non diffusive levy semigroup are also investigated curvature dimension criteria are the main tool in the analysis

remarks on the renyi entropy of a sum of iid random variables in this note we study a conjecture of madiman and wang which predicted that the generalized gaussian distribution minimizes the renyi entropy of the sum of independent random variables through a variational analysis we show that the generalized gaussian fails to be a minimizer for the problem

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 hypercontractive approach to exactly bounding an operator with complex gaussian kernel abstract we study operators g l q r n l p r n given by a nondegenerate complex gaussian kernel in many cases such an operator attains its bound at a maximizing gaussian function g l q r n to show this we prove a complexified version of edward nelson s hypercontractive theorem

multiplicative interior gradient methods for minimization over thenonnegative orthant we introduce a new class of multiplicative iterative methods for solving minimization problems over the nonnegative orthant the algorithm is akin to a natural extension of gradient methods for unconstrained minimization problems to the case of nonnegativity constraints with the special feature that it generates a sequence of iterates which remain in the interior of the nonnegative orthant we prove that the algorithm combined with an appropriate line search is weakly convergent to a saddle point of the minimization problem when the minimand is a differentiable function with bounded level sets if the function is convex then weak convergence to an optimal solution is obtained moreover by using an appropriate regularized line search we prove that the level set boundeness hypothesis can be removed and full convergence of the iterates to an optimal solution is established in the convex case

simultaneous equation systems as moment structure models with an introduction to latent variable models abstract an introduction to latent variable covariance structure models is provided estimation and testing of such models with normal and non normal variables is reviewed and a relatively inexpensive asymptotically distribution free estimator is introduced the traditional econometric simultaneous equation system is reconceptualized as a random vector structural equation model extensions of the model to deal with latent variables and a wider variety of structural phenomena are made via the joreskog keesling wiley lisrel approach and a simpler approach due to bentler and weeks these representation systems are extended via first and second moments about the origin of measured variables explained in terms of the structural coefficients and the moments of explanatory variables citations to further literature are provided

latent variable structural equation modeling with categorical data abstract structural equation modeling with latent variables is overviewed for situations involving a mixture of dichotomous ordered polytomous and continuous indicators of latent variables special emphasis is placed on categorical variables models in psychometrics econometrics and biometrics are interrelated via a general model due to muthen limited information least squares estimators and full information estimation are discussed an example is estimated with a model for a four wave longitudinal data set where dichotomous responses are related to each other and a set of independent variables via latent variables with a variance component structure

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem