the jackknife estimate of variance tukey s jackknife estimate of variance for a statistic s x 1 x 2 cdots x n which is a symmetric function of i i d random variables x i is investigated using an anova like decomposition of s it is shown that the jackknife variance estimate tends always to be biased upwards a theorem to this effect being proved for the natural jackknife estimate of operatorname var s x 1 x 2 cdots x n 1 based on x 1 x 2 cdots x n

the properties of the cross match estimate and split sampling by noting the connection with k sample u statistics we find a simple decomposition of the variance of the cross match estimate which can be regarded as a generalization of efron and stein we apply the decomposition in assessing efficiencies of several plans of using the weighted samples from an importance scheme the applications of the formula to multiple imputations lead to a method of crossing jointly imputed data to gain more accuracy

classification of homogeneous data with large alphabets given training sequences generated by two distinct but unknown distributions on a common alphabet we study the problem of determining whether a third sequence was generated according to the first or second distribution to model sources such as natural language for which the underlying distributions are difficult to learn from realistic amounts of data we allow the alphabet size to grow and therefore the probability distributions to change with the block length our primary focus is the situation in which the underlying probabilities are all of the same order and in this regime we show that consistent classification is possible if and only if the alphabet grows subquadratically with the block length we also show that some commonly used statistical tests are suboptimal in that they are consistent only if the alphabet grows sublinearly

dispersive estimates for schrodinger operators in dimension two we prove l1 2 l 2 for the two dimensional schrodinger operator v with the decay rate t 1 we assume that zero energy is neither an eigenvalue nor a resonance this condition is formulated as in the recent paper by jensen and nenciu on threshold expansions for the two dimensional resolvent

a weighted dispersive estimate for schrodinger operators in dimension two let h v where v is a real valued potential on mathbb r 2 satisfying v x lesssim langle x rangle 3 we prove that if zero is a regular point of the spectrum of h v then w 1 e ith p ac f l infty mathbb r 2 lesssim frac 1 t log 2 t w f l 1 mathbb r 2 t geq 2 with w x log 2 x 2 this decay rate was obtained by murata in the setting of weighted l 2 spaces with polynomially growing weights

lp boundedness of the wave operator for the one dimensional schrodinger operator given a one dimensional perturbed schrodinger operator h d2 dx2 v x we consider the associated wave operators w defined as the strong l2 limits lim s to pm infty e ish e ish 0 we prove that w are bounded operators on lp for all 1 p provided 1 x 2 v x in l 1 or else 1 x v x in l 1 and 0 is not a resonance for p we obtain an estimate in terms of the hilbert transform some applications to dispersive estimates for equations with variable rough coefficients are given

l p boundedness of wave operators for two dimensional schrodinger operators let be the two dimensional schrodinger operator with the real valued potential v which satisfies the decay condition at infinity for we show that the wave operators are bounded in for any 1 p under the condition that h has no zero bound states or zero resonance extending the corresponding results for higher dimensions as w intertwine h 0 and the absolutely continuous part h p ac of h f h p ac w f h 0 w for any borel function f on 1 this reduces the various l p mapping properties of f h p ac to those of f h 0 the convolution operator by the fourier transform of the function f 2

probabilistic and worst case analyses of classical problems of combinatorial optimization in euclidean space the classical problems reviewed are the traveling salesman problem minimal spanning tree minimal matching greedy matching minimal triangulation and others each optimization problem is considered for finite sets of points in d and the feature of principal interest is the value of the associated objective function special attention is given to the asymptotic behavior of this value under probabilistic assumptions but both probabilistic and worst case analyses are surveyed

a central limit theorem for generalized quadratic forms random variables of the form w n mathop sum limits 1 leqq i leqq n mathop sum limits text 1 leqq j leqq n w ijn x i x j are considered with x i independent not necessarily identically distributed and w ijn borel functions such that w ijn x i x j is square integrable and has vanishing conditional expectations e w ijn x i x j x i e w ijn x i x j x j 0 text a text s text a central limit theorem is proved under the condition that the normed fourth moment tends to 3 under some restrictions the condition is also necessary finally conditions on the individual tails of w ijn x i x j and an eigenvalue condition are given that ensure asymptotic normality of w n

explicit formulae for the wave operators of perturbed self adjoint operators abstract we give explicit formulae for the wave operators of the position operator h 0 and the operator h obtained from h 0 by a rank one perturbation then by the spectral theorem we deduce a general formula for wave operators of unbounded self adjoint operators with absolute continuous spectrum

a berry esseen bound for symmetric statistics the rate of convergence of the distribution function of a symmetric function of n independent and identically distributed random variables to its normal limit is investigated under appropriate moment conditions the rate is shown to be o left n frac 1 2 right this theorem generalizes many known results for special cases and two examples are given possible further extensions are indicated

on the wave equation with a large rough potential we prove an optimal dispersive l decay estimate for a three dimensional wave equation perturbed with a large nonsmooth potential belonging to a particular kato class the proof is based on a spectral representation of the solution and suitable resolvent estimates for the perturbed operator

dispersive and strichartz estimates for the three dimensional wave equation with a scaling critical class of potentials we consider the three dimensional linear wave equation perturbed by a potential v belonging to a scaling critical class which is the closure of bounded compactly supported functions with respect to the global kato norm without any further smallness assumptions we first prove the dispersive estimate for the solution of the wave equation as a byproduct we prove strichartz estimates under a slightly stronger condition

tensor analysis of anova decomposition abstract the analogy between anova for general n factor crossed layouts and the anova type decomposition of square integrable statistics is demonstrated using the notions and notations of tensor analysis and multilinear algebra a theory of tensors is developed in such a way that a it can be immediately applied in computer programs and b it can be easily generalized to l 2 spaces

the lp lp estimate for the schrodinger equation on the half line abstract in this paper we prove the lp lp estimate for the schrodinger equation on the half line and with homogeneous dirichlet boundary condition at the origin

asymptotic expansions in time for solutions of schrodinger type equations let ia i p d v be a dissipative operator in l2 rn where p d is an elliptic differential operator of order m with real constant coefficients and v is a compact operator from the weighted sobolev space hm s rn to h0 p s rn s r for some m 1 let r z be the resolvent of a then an asymptotic expansion of r z as z approaches a critical value of the polynomial p is given the coefficient operators in the expansion are computed explicitly by using the resolvent expansion and the results of m murata 9 an asymptotic expansion of e ita as t is given

the w k p continuity of the schrodinger wave operators on the line we prove that the wave operators for the schrodinger equation on the line are continuous on the sobolev spaces w k p 1 p moreover if the potential is exceptional and where f 1 x 0 is a jost solution at zero energy the wave operators are continuous on w k 1 and on w k

generalized entropy power inequalities and monotonicity properties of information new families of fisher information and entropy power inequalities for sums of independent random variables are presented these inequalities relate the information in the sum of n independent random variables to the information contained in sums over subsets of the random variables for an arbitrary collection of subsets as a consequence a simple proof of the monotonicity of information in central limit theorems is obtained both in the setting of independent and identically distributed i i d summands as well as in the more general setting of independent summands with variance standardized sums

time decay of scaling critical electromagnetic schrodinger flows we obtain a representation formula for solutions to schrodinger equations with a class of homogeneous scaling critical electromagnetic potentials as a consequence we prove the sharp l 1 to l infty time decay estimate for the 3d inverse square and the 2d aharonov bohm potentials

on the scattering problem for the nonlinear schrodinger equation with a potential in 2d abstract we consider the scattering problem for the nonlinear schrodinger equation with a potential in two space dimensions appropriate resolvent estimates are proved and applied to estimate the operator a s appearing in commutator relations the equivalence between the operators v s 2 and s 2 in l 2 norm sense for 0 s 1 is investigated by using free resolvent estimates and gaussian estimates for the heat kernel of the schrodinger operator v our main result guarantees the global existence of solutions and time decay of the solutions assuming initial data have small weighted sobolev norms moreover the global solutions obtained in the main result scatter

a batch means methodology for estimation of a nonlinear function of a steady state mean we study the estimation of steady state performance measures from an d valued stochastic process y y t t 0 representing the output of a simulation in many applications we may be interested in the estimation of a steady state performance measure that cannot be expressed as a steady state mean r e g the variance of the steady state distribution the ratio of steady state means and steady state conditional expectations these examples are particular cases of a more general problem the estimation of a nonlinear function f r of r we propose a batch means based methodology that allows us to use jackknifing to reduce the bias of the point estimator asymptotically valid confidence intervals for f r are obtained by combining three different point estimators classical batch means and jackknife with two different variability estimators classical and jackknife the performances of the point estimators are discussed by considering asymptotic expansions for their biases and mean squared error

dispersive estimates for schrodinger equations with threshold resonance and eigenvalue let h v x be a three dimensional schrodinger operator we study the time decay in lp spaces of scattering solutions e ithpcu where pc is the orthogonal projection onto the continuous spectral subspace of l2 r3 for h under suitable decay assumptions on v x it is shown that they satisfy the so called lp lq estimates e ithpcu p 4 t 3 1 2 1 p u q for all 1 q 2 p with 1 p 1 q 1 if h has no threshold resonance and eigenvalue and for all 3 2 q 2 p 3 if otherwise

on the flows associated to selfadjoint operators on metric measure spaces let x be a metric space with a doubling measure satisfying mu b gtrsim r b n for any ball b with any radius r b 0 let l be a non negative selfadjoint operator on l 2 x we assume that e tl satisfies a gaussian upper bound and that the flow e itl satisfies a typical l 1 l infty dispersive estimate of the form begin aligned vert e itl vert l 1 rightarrow l infty lesssim t frac n 2 end aligned then we prove a similar l 1 l infty dispersive estimate for a general class of flows e it phi l with phi r of power type near 0 and near infty in the case of fractional powers phi l l nu nu in 0 1 we deduce dispersive estimates for e itl nu with data in sobolev besov or hardy spaces h p l with p in 0 1 associated to the operator l

structure of wave operators for a scaling critical class of potentials we prove a structure formula for the wave operators in bbb r 3 w pm mathop hbox s lim limits t to pm infty e it delta v p c e it delta and their adjoints for a scaling invariant class of scalar potentials v in b b left v mid sum k in bbb z 2 k 2 big chi x in 2 k 2 k 1 x v x big l 2 infty right under the assumption that zero is neither an eigenvalue nor a resonance for delta v the formula implies the boundedness of wave operators on l p spaces 1 leq p leq infty on weighted l p spaces and on sobolev spaces as well as multilinear estimates for e ith p c when v decreases rapidly at infinity we obtain an asymptotic expansion of the wave operators the first term of the expansion is of order langle y rangle 4 commutes with the laplacian and exists when v in langle x rangle 3 2 epsilon l 2 we also prove that the scattering operator s w w is an integrable combination of isometries the proof is based on an abstract version of wiener s theorem applied in a new function space

on the lp boundedness of wave operators for two dimensional schrodinger operators with threshold obstructions abstract let h v be a schrodinger operator on l 2 r 2 with real valued potential v and let h 0 if v has sufficient pointwise decay the wave operators w s lim t e i t h e i t h 0 are known to be bounded on l p r 2 for all 1 p if zero is not an eigenvalue or resonance we show that if there is an s wave resonance or an eigenvalue only at zero then the wave operators are bounded on l p r 2 for 1 p this result stands in contrast to results in higher dimensions where the presence of zero energy obstructions is known to shrink the range of valid exponents p

dispersive estimate for the 1d schrodinger equation with a steplike potential abstract we prove a sharp dispersive estimate p a c u t x c t 1 2 u 0 l 1 r for the one dimensional schrodinger equation i u t u x x v x u v 0 x u 0 where 1 x 2 v l 1 r and v 0 is a step function real valued and constant on the positive and negative real axes

a remark on linearly unstable standing wave solutions to nls abstract we obtain an optimal growth estimate of a semigroup generated by a linearized operator around a standing wave solution nonlinear schrodinger equations in two dimension using the growth estimate of the semigroup we prove that a linearly unstable standing wave solution is orbitally unstable and that instability of the standing wave solution is mainly caused by a mode of an eigenfunction associated with the rightmost or the leftmost eigenvalues of the linearized operator our result is obtained by using the method of yajima and cuccagna that proved l p boundedness of the wave operator

hoeffding decompositions and urn sequences let x x1 x2 be a nondeterministic infinite exchangeable sequence with values in 0 1 we show that x is hoeffding decomposable if and only if x is either an i i d sequence or a polya sequence this completes the results established in peccati ann probab 32 2004 1796 1829 the proof uses several combinatorial implications of the correspondence between hoeffding decomposability and weak independence our results must be compared with previous characterizations of i i d and polya sequences given by hill lane and sudderth ann probab 15 1987 1586 1592 and diaconis and ylvisaker ann statist 7 1979 269 281 the final section contains a partial characterization of hoeffding decomposable sequences with values in a set with more than two elements

dispersion and strichartz estimates for some finite rank perturbations of the laplace operator we consider the dispersion properties in lp spaces of schrodinger hamiltonians with a large number of obstacles modelled by rank one perturbations we obtain both for the dispersion an strichartz estimates nonperturbative results with respect to the coupling constants

dispersive estimates for schrodinger operators in dimensions one and three we consider l 1 l estimates for the time evolution of hamiltonians h v in dimensions d 1 and d 3 with bound t 2 we require decay of the potentials but no regularity in d 1 the decay assumption is 1 x v x dx whereas in d 3 it is v x c 1 x 3

a remark on lp boundedness of wave operators for two dimensional schrodinger operators let h v be a two dimensional schrodinger operator with a real potential v x satisfying the decay condition 6 let h0 we show that the wave operators are bounded in lp r2 under the condition that h has no zero resonances or bound states in this paper the condition imposed in a previous paper k yajima commun math phys 208 125 152 1999 is removed

dispersive analysis of charge transfer models we prove dispersive estimates for the time dependent schrodinger equation with a charge transfer hamiltonian as a by product we also obtain another proof of asymptotic completeness of the wave operators for a charge transfer model established earlier by k yajima and j m graf we also consider a more general matrix non self adjoint charge transfer problem this model appears naturally in the study of nonlinear multisoliton systems and is specifically motivated by the problem of asymptotic stability of multisoliton states of a nonlinear schrodinger equation 2004 wiley periodicals inc

scattering in the energy space for the nls with variable coefficients we consider the nls with variable coefficients in dimension n ge 3 begin aligned i partial t u lu f u 0 qquad lv nabla b cdot a x nabla b v c x v qquad nabla b nabla ib x end aligned on mathbb r n or more generally on an exterior domain with dirichlet boundary conditions for a gauge invariant defocusing nonlinearity of power type f u simeq u gamma 1 u we assume that l is a small long range perturbation of delta plus a potential with a large positive part the first main result of the paper is a bilinear smoothing interaction morawetz estimate for the solution as an application under the conditional assumption that strichartz estimates are valid for the linear flow e itl we prove global well posedness in the energy space for subcritical powers gamma 1 frac 4 n when the domain is mathbb r n by extending the strichartz estimates due to tataru am j math 130 3 571 634 2008 we prove that the conditional assumption is satisfied and deduce well posedness and scattering in the energy space

multidimensional inverse scattering for the nonlinear klein gordon equation with a potential in this paper we solve the multidimensional inverse scattering problem for the nonlinear klein gordon equation on rn n 2 2 t2u x t u x t u x t v0 x u x t j 1 vj x u 2 j0 j u x t 0 we prove that the small amplitude limit of the scattering operator determines uniquely all the vj j 0 1 our proof gives as well a method for the reconstruction of the vj j 0 1

on precise center stable manifold theorems for certain reaction diffusion and klein gordon equations abstract we consider positive radial and exponentially decaying steady state solutions of the general reaction diffusion and klein gordon type equations and present an explicit construction of infinite dimensional invariant manifolds in the vicinity of these solutions the result is a precise stable manifold theorem for the reaction diffusion equation and a precise center stable manifold theorem for the klein gordon equation which include the co dimension of the manifolds and the decay rates for even perturbations

time decay for solutions of schrodinger equations with rough and time dependent potentials in this paper we establish dispersive estimates for solutions to the linear schrodinger equation in three dimensions frac 1 i partial t psi triangle psi v psi 0 qquad psi s f 0 1 where v t x is a time dependent potential that satisfies the conditions sup t v t cdot l frac 3 2 mathbb r 3 sup x in mathbb r 3 int mathbb r 3 int infty infty frac v hat tau x x y d tau dy c 0 here c0 is some small constant and v hat tau x denotes the fourier transform with respect to the first variable we show that under these conditions 0 1 admits solutions l t l2x 3 l2t l6x 3 for any f l2 3 satisfying the dispersive inequality psi t infty le c t s frac32 f 1 text for all times t s 0 2 for the case of time independent potentials v x 0 2 remains true if int mathbb r 6 frac v x v y x y 2 dxdy 4 pi 2 text and v mathcal k sup x in mathbb r 3 int mathbb r 3 frac v y x y dy 4 pi we also establish the dispersive estimate with an e loss for large energies provided v mathcal k v 2 infty

orthogonal decomposition of finite population statistics and its applications to distributional asymptotics we study orthogonal decomposition of symmetric statistics based on samples drawn without replacement from finite populations several applications to finite population statistics are given we establish one term edge worth expansions for general asymptotically normal symmetric statistics prove an efron stein inequality and the consistency of the jackknife estimator of variance our expansions provide second order a s approximations to wu s jackknife histogram

elementary symmetric polynomials of increasing order the asymptotic behaviour of elementary symmetric polynomials sn k of order k based on n independent and identically distributed random variables x1 xn is investigated for the case that both k and n get large if k o n frac 1 2 then the distribution function of a suitably normalised sn k is shown to converge to a standard normal limit the speed of this convergence to normality is of order mathcal o kn frac 1 2 provided k mathcal o log 1 nlog 2 1 nn frac 1 2 and certain natural moment assumptions are imposed this order bound is sharp and cannot be inferred from one of the existing berry esseen bounds for u statistics if k at the rate n1 2 then a non normal weak limit appears provided the xi s are positive and sn k is standardised appropriately on the other hand if k at a rate faster than n1 2 then it is shown that for positive xj sthere exists no linear norming which causes sn k to converge weakly to a nondegenerate weak limit

moduli spaces for principal bundles in arbitrary characteristic in this article we solve the problem of constructing moduli spaces of semistable principal bundles and singular versions of them over smooth projective varieties over algebraically closed ground fields of positive characteristic

on normal approximations to u statistics let x1 xn be i i d random observations taking their values in a measurable space consider a u statistic of order k says s x1 xn based on the sample x1 xn assume that s can be represented in the form s l t with el et 0 where l is a linear statistic and t is a stochastically smaller statistic assume further that the linear statistic l is approximately normal and that we have a bound for error of such approximation our objective is to show that a similar bound holds for the distribution ofs provided that we correct it by adding vart i ln 2 3 1 vart where vart et 2 is the variance of t it occurs that one has to correct the normal approximation as well replacing it by some asymptotically as n 1 equivalent distribution furthermore we show that the class of u statistics up to negligible error terms is closed under studentization or self normalization for example using the jackknife estimators of variances this leads to error bounds of normal approximations for non deterministically normalized u statistics our results extend reflne and yield a number of related known results the bounds have a simple and convenient form for applications and analysis since they involve only the variance of the noise termt in the case k 3 of higher order statistics it remains open whether one can improve the bounds removing the logarithmic factor ln 2 3 1 vart we provide as well an optimal bound under a lower moment assumption ejtj fi 1 fi 2