maximal accuracy and minimal disturbance in the arthurs kelly simultaneous measurement process the accuracy of the arthurs kelly model of a simultaneous measurement of position and momentum is analysed using concepts developed by braginsky and khalili in the context of measurements of a single quantum observable a distinction is made between the errors of retrodiction and prediction it is shown that the distribution of measured values coincides with the initial state husimi function when the retrodictive accuracy is maximized this being the result first obtained by busch using somewhat different methods it is further shown that the distribution of measured values is related to the final state anti husimi function the p representation of quantum optics when the predictive accuracy is maximized the disturbance of the system by the measurement is also discussed a class of minimally disturbing measurements is characterized it is shown that the distribution of measured values then coincides with one of the smoothed wigner functions described by cartwright

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

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

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

on the moduli scheme of stable sheaves supported on cubic space curves we investigate the geometry of the simpson moduli space m p inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of stable sheaves with hilbert polynomial p m 3 m 1 it consists of two smooth rational components m 0 and m 1 of dimensions 12 and 13 intersecting each other transversally along an 11 dimensional smooth rational subvariety the component m 0 is isomorphic to the closure of the space of twisted cubics in the hilbert scheme hilb p inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i and m 1 is isomorphic to the incidence variety of the relative hilbert scheme of cubic curves contained in planes in order to obtain the result and to classify the sheaves we characterize mp inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i as geometric quotient of a certain matrix parameter space by a nonreductive group we also compute the betti numbers of the chow groups of the moduli space

a functorial construction of moduli of sheaves we show how natural functors from the category of coherent sheaves on a projective scheme to categories of kronecker modules can be used to construct moduli spaces of semistable sheaves this construction simplifies or clarifies technical aspects of existing constructions and yields new simpler definitions of theta functions about which more complete results can be proved

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 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

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

path integral approach to quantum brownian motion abstract we apply the influence functional method of feynman and vernon to the study of brownian motion at arbitrary temperature by choosing a specific model for the dissipative interaction of the system of interest with its environment we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient we show that in the limit h 0 the results obtained from the influence functional formalism reduce to the classical fokker planck equation in the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature we obtain an explicit expression for the time evolution of the complete density matrix x x t when the system starts in a particular kind of pure state we compare our results with those of other approaches to the problem of dissipation in quantum mechanics

accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1d center bundle we prove that stable ergodicity is cr open and dense among conservative partially hyperbolic diffeomorphisms with one dimensional center bundle for all r 2

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

on the bernoulli property for certain partially hyperbolic diffeomorphisms abstract we address the classical problem of equivalence between kolmogorov and bernoulli property of smooth dynamical systems in a natural class of volume preserving partially hyperbolic diffeomorphisms homotopic to anosov derived from anosov on 3 torus we prove that kolmogorov and bernoulli properties are equivalent in our approach we propose to study the conditional measures of volume along central foliation to recover fine ergodic properties for partially hyperbolic diffeomorphisms as an important consequence we obtain that there exists an almost everywhere conjugacy between any volume preserving derived from anosov diffeomorphism of 3 torus and its linearization our results also hold in higher dimensional case when central bundle is one dimensional and stable and unstable foliations are quasi isometric

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

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

partial desingularisations of quotients of nonsingular varieties and their betti numbers when a reductive group g acts linearly on a nonsingular complex projective variety x one can define a projective quotient variety x g using mumford s geometric invariant theory if the condition that every semistable point be properly stable is satisfied this quotient is the ordinary topological quotient of an open subset xss of the variety by the group in k a formula is obtained for the rational cohomology of x g under this condition the formula involves the rational cohomology of x and various linear sections of x together with the rational cohomology of the classifying spaces of g and certain reductive subgroups of g in many interesting examples the condition required in k is not satisfied thus the question arises as to what information we can obtain in general the quotient x g can now have serious singularities in contrast to the good case considered in k where the only singularities are those caused by finite isotropy groups it will be shown in this paper that there is a systematic way of blowing up x along a sequence of nonsingular subvarieties to obtain a variety x with a linear action of g such that every semistable point of x is stable the only assumption that we have to make is that there exists at least one stable point of x then x g is almost a resolution of singularities of x1 g in the sense that the most serious singularities have been resolved moreover there is a formula for the rational cohomology of x g again involving the rational cohomology of x and certain linear sections of x together with the rational cohomology of the classifying spaces of g and some reductive subgroups of g see theorem 8 14 for convenience we shall assume throughout that g is connected however the construction of x works in general and it is straightforward to modify the cohomological formulas to apply to the general case the construction of x g can also be modified to apply in some cases when x has no stable points the layout of the paper is as follows section 1 is a review of the basic facts of geometric invariant theory which will be needed and section 2 describes the relationship of geometric invariant theory with symplectic geometry and the moment map in section 3 semistability and stability in a blow up of x along a nonsingular c invariant subvariety are related to semistability and stability in x

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

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

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

measurement uncertainty relations measurement uncertainty relations are quantitative bounds on the errors in an approximate joint measurement of two observables they can be seen as a generalization of the error disturbance tradeoff first discussed heuristically by heisenberg here we prove such relations for the case of two canonically conjugate observables like position and momentum and establish a close connection with the more familiar preparation uncertainty relations constraining the sharpness of the distributions of the two observables in the same state both sets of relations are generalized to means of order rather than the usual quadratic means and we show that the optimal constants are the same for preparation and for measurement uncertainty the constants are determined numerically and compared with some bounds in the literature in both cases the near saturation of the inequalities entails that the state resp observable is uniformly close to a minimizing one

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

moduli of sheaves fourier mukai transform and partial desingularization we study birational maps among 1 the moduli space of semistable sheaves of hilbert polynomial 4m 2 on a smooth quadric surface 2 the moduli space of semistable sheaves of hilbert polynomial m 2 3m 2 on mathbb p 3 3 kontsevich s moduli space of genus zero stable maps of degree 2 to the grassmannian gr 2 4 a regular birational morphism from 1 to 2 is described in terms of fourier mukai transforms the map from 3 to 2 is kirwan s partial desingularization we also investigate several geometric properties of 1 by using the variation of moduli spaces of stable pairs

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

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

projectivity and birational geometry of bridgeland moduli spaces we construct a family of nef divisor classes on every moduli space of stable complexes in the sense of bridgeland this divisor class varies naturally with the bridgeland stability condition for a generic stability condition on a k3 surface we prove that this class is ample thereby generalizing a result of minamide yanagida and yoshioka our result also gives a systematic explanation of the relation between wall crossing for bridgeland stability and the minimal model program for the moduli space we give three applications of our method for classical moduli spaces of sheaves on a k3 surface 1 we obtain a region in the ample cone in the moduli space of gieseker stable sheaves only depending on the lattice of the k3 2 we determine the nef cone of the hilbert scheme of n points on a k3 surface of picard rank one when n is large compared to the genus 3 we verify the hassett tschinkel huybrechts sawon conjecture on the existence of a birational lagrangian fibration for the hilbert scheme in a new family of cases

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

the minimal model program for the hilbert scheme of points on p2 and bridgeland stability abstract in this paper we study the birational geometry of the hilbert scheme p 2 n of n points on p 2 we discuss the stable base locus decomposition of the effective cone and the corresponding birational models we give modular interpretations to the models in terms of moduli spaces of bridgeland semi stable objects we construct these moduli spaces as moduli spaces of quiver representations using g i t and thus show that they are projective there is a precise correspondence between wall crossings in the bridgeland stability manifold and wall crossings between mori cones for n 9 we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions

mmp for moduli of sheaves on k3s via wall crossing nef and movable cones lagrangian fibrations we use wall crossing with respect to bridgeland stability conditions to systematically study the birational geometry of a moduli space m of stable sheaves on a k3 surface x a we describe the nef cone the movable cone and the effective cone of m in terms of the mukai lattice of x b we establish a long standing conjecture that predicts the existence of a birational lagrangian fibration on m whenever m admits an integral divisor class d of square zero with respect to the beauville bogomolov form these results are proved using a natural map from the space of bridgeland stability conditions mathop mathrm stab nolimits x to the cone mathop mathrm mov nolimits x of movable divisors on m this map relates wall crossing in mathop mathrm stab nolimits x to birational transformations of m in particular every minimal model of m appears as a moduli space of bridgeland stable objects on x

a perspective on the numerical treatment of volterra equations abstract we discuss the properties and numerical treatment of various types of volterra and abel volterra integral and integro differential equations

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

hecke correspondence stable maps and the kirwan desingularization we prove that the moduli space m0 0 n 2 of stable maps of degree 2 to the moduli space n of rank 2 stable bundles of flxed odd determinant ox ix over a smooth projective curve x of genus g 3 has two irreducible components which intersect transversely one of them is kirwan s partial desingularization e mx of the moduli space mx of rank 2 semistable bundles with determinant isomorphic to ox yix for some y 2 x the other component is the partial desingularization of the git quotient phom sl 2 w pgl 2 for a vector bundle w r 1 l i2 ix of rank g over the jacobian of x we also show that the hilbert scheme h the chow scheme c of conics in n and m0 0 n 2 are related by explicit contractions