small ball probabilities for the slepian gaussian fields the d dimensional slepian gaussian random field s t t r d is a mean zero gaussian process with covariance function es s s t i d i i max 0 a i s i t i for a i 0 and t t i t d r d small ball probabilities for s t are obtained under the l 2 norm on 0 1 d and under the sup norm on 0 1 2 which implies talagrand s result for the brownian sheet the method of proof for the sup norm case is purely probabilistic and analytic and thus avoids ingenious combinatoric arguments of using decreasing mathematical induction in particular riesz product techniques are new ingredients in our arguments

adaptive nonparametric bayesian inference using location scale mixture priors we study location scale mixture priors for nonparametric statistical problems including multivariate regression density estimation and classification we show that a rate adaptive procedure can be obtained if the prior is properly constructed in particular we show that adaptation is achieved if a kernel mixture prior on a regression function is constructed using a gaussian kernel an inverse gamma bandwidth and gaussian mixing weights

on the gaussian measure of the intersection the gaussian correlation conjecture states that for any two symmetric convex sets in n dimensional space and for any centered gaussian measure on that space the measure of the intersection is greater than or equal to the product of the measures in this paper we obtain several results which substantiate this conjecture for example in the standard gaussian case we show there is a positive constant c such that the conjecture is true if the two sets are in the euclidean ball of radius c n further we show that if for every n the conjecture is true when the sets are in the euclidean ball of radius n then it is true in general our most concrete result is that the conjecture is true if the two sets are arbitrary centered ellipsoids

tail behaviour for suprema of empirical processes abstract this document considers multi variate empirical processes with an empirical distribution function based on i i d variables with certain distribution functions

small deviations of riemann liouville processes in l_ lowercase q spaces with respect to fractal measures we investigate riemann liouville processes r h with h 0 and fractional brownian motions b h for 0 h 1 and study their small deviation properties in the spaces l q 0 1 mu of special interest here are thin fractal measures mu that is those that are singular with respect to the lebesgue measure we describe the behavior of small deviation probabilities by numerical quantities of mu called mixed entropy numbers characterizing size and regularity of the underlying measure for the particularly interesting case of self similar measures the asymptotic behavior of the mixed entropy is evaluated explicitly we also provide two sided estimates for this quantity in the case of random measures generated by subordinators while the upper asymptotic bound for the small deviation probability is proved by purely probabilistic methods the lower bound is verified by analytic tools concerning entropy and kolmogorov numbers of riemann liouville operators

rates of clustering for some gaussian self similar processes the analogue of strassen s functional law of the iterated logarithm in known for many gaussian processes which have suitable scaling properties and here we establish rates at which this convergence takes place we provide a new proof of the best upper bound for the convergence tok by suitably normalized brownian motion and then continue with this method to get similar bounds for the brownian sheet and other self similar gaussian processes the previous method which produced these results for brownian motion in 1 was highly dependent on many special properties unavailable when dealing with other gaussian processes

bounds on eigenvalues of finite element systems bounds on eigenvalues of various finite element systems are analysed by using an element eigenvalue theorem together with the global eigenvalue theorem both two dimensional continuum dynamics and heat conduction problems are considered these bounds provide stable time steps for explict time integration schemes a reduced eigenproblem at element quadrature point level with all zero eigenvalues suppressed is also presented in this paper the simplified eigenproblem results in simple formulas for calculating the eigenvalues

projection schemes for one point quadrature shell elements abstract a new projection scheme to improve the performance of one point quadrature shell elements is presented the projection is performed elementwise and enforces invariance of the internal power to rigid body motion and drill rotation the projection is especially suitable for 6 degree of freedom per node explicit programs and can be easily implemented as an integral part of the subroutine that calculates the internal forces on the element level problems that could not be solved accurately with one point quadrature elements such as the twisted beam problem show excellent results even for very coarse meshes when the projection is used

extrapolation estimates for entropy numbers we establish abstract extrapolation results for entropy numbers of operators in banach spaces the results apply to extrapolation in the source spaces and also in the target spaces as an illustration of the abstract results applications to limiting sobolev embeddings are given

nonlinear dynamic analysis of shells with the triangular element tric abstract tric is a facet triangular shell element which is based on the natural mode method it has been shown that the tric shell element satisfies the individual element test and in the framework of the nonconsistent formulation the convergence requirements are fulfilled while it has been proved to be very efficient in linear and nonlinear static problems moreover another major advantage in the formulation of this element is the incorporation of the transverse shear deformations in a way that defies the shear locking phenomenon in this work the derivation of the consistent and lumped mass matrices of the tric element is presented so that it can be used in linear and nonlinear dynamic problems both translational and rotational inertia are included in the consistent mass matrix which is conceived using kinematical and geometrical arguments consistent with the assumed natural rigid body and straining modes of the element all the kinematical and geometrical arguments that are invoked for the derivation of the consistent mass matrix are briefly presented moreover two formulations of the lumped mass matrix of tric are derived the first formulation is based entirely on geometrical considerations whereas the second is based on lumping the consistent mass matrix of tric finally the element s robustness and accuracy will be shown by applying it to properly selected benchmark examples of nonlinear shell dynamics while its computational efficiency will be demonstrated by comparing the cpu performance of the element with the other available shell elements

sharp asymptotics of the functional quantization problem for gaussian processes the sharp asymptotics for the l 2 quantization errors of gaussian measures on a hilbert space and in particular for gaussian processes is derived the condition imposed is regular variation of the eigenvalues

on the lower tail probabilities of some random series the behavior of tail probabilities p s r r 0 is investigated where s is a series s j z j generated by some sequence of positive numbers j and by a sequence z j of independent copies of a positive random variable z we present the exact asymptotic expression for p s r by means of laplace transform eexp s under weak assumptions on the behavior of the tail probabilities of z in the vicinity of zero the bounds of accuracy are also given and under weak supplementary smoothness conditions the asymptotic properties of the density of s are investigated

the first exit time of a brownian motion from an unbounded convex domain consider the first exit time d of a d 1 dimensional brownian motion from an unbounded open domain d set x y rd 1 dvtxy f x x rd starting at vspace 0 5pt x0 f x0 1 rd 1 for some x0 rd where the function f x on rd is convex and f x as the euclidean norm x very general estimates for the asymptotics of log pr d t are given by using gaussian techniques in particular for f x exp x p p 0 limt t 1 logt 2 plog pr d t j2 2 where d 2 2 and j is the smallest positive zero of the bessel function j

asymptotic analysis via mellin transforms for small deviations in l2 norm of integrated brownian sheets we use mellin transforms to compute a full asymptotic expansion for the tail of the laplace transform of the squared l2 norm of any multiply integrated brownian sheet through reversion we obtain corresponding strong small deviation estimates

integrated brownian motions and exact l2 small balls we will introduce a class of m times integrated brownian motions the exact asymptotic expansions for the l2 small ball probabilities will be discussed for members of this class of which the usual m times integrated brownian motion is an example another example will be what we call an euler integrated brownian motion we will also find very sharp estimates for the asymptotics of the eigenvalues of the covariance operator of integrated brownian motions and will therefore obtain exact not just logarithmic asymptotics

a unified stress update algorithm for explicit transient shell dynamics with combined isotropic kinematic hardening in eulerian rate type phenomenological finite elasto plasticity models abstract this paper discusses applications of the objective stress rates to a unified stress update algorithm for transient shell dynamic analysis within the context of explicit time integration we propose a unified stress update algorithm via establishing the eulerian rate type constitutive equations of hypoelastic plastic materials the constitutive equations are often given on the additive decomposition of the rate of deformation tensor under the assumption that the elastic part of the rate of deformation tensor may be characterized to be hypoelastic with grade zero i e the fourth rank constant isotropic tensor several objective stress rates are derived through the lie derivative of the cauchy or kirchhoff stress tensor those stress rates cover for instance the jaumann green naghdi truesdell oldroyd and cotter rivlin stress rates further the unified stress update algorithm is implemented along with the so called radial return method and the phenomenological plasticity theory with combined isotropic kinematic hardenings among the objective stress rates employed here we emphasize especially the stress update procedure of the green naghdi stress rate for transient shell dynamics because this algorithm developed in this work is not shown even in the representative commercial codes in the implementation of the unified stress update algorithm we applied our algorithms to the belytschko lin tsay shell theory to validate its accuracy and effectiveness moreover a slightly modified secant algorithm for the zero transverse normal stress of the shell is also developed the ultimate objectivity of this work is to provide a guideline for the best choice of an appropriate objective stress rate for phenomenological elastoplastic models several numerical examples are demonstrated including non contact transient shell dynamics examples and contact impact examples such as a ball plate impact problem by which the accuracy and effectiveness of the unified stress update algorithm developed in this work are addressed

optimal bayes classifiers for functional data and density ratios bayes classifiers for functional data pose a challenge this is because probability density functions do not exist for functional data as a consequence the classical bayes classifier using density quotients needs to be modified we propose to use density ratios of projections on a sequence of eigenfunctions that are common to the groups to be classified the density ratios can then be factored into density ratios of individual functional principal components whence the classification problem is reduced to a sequence of nonparametric one dimensional density estimates this is an extension to functional data of some of the very earliest nonparametric bayes classifiers that were based on simple density ratios in the one dimensional case by means of the factorization of the density quotients the curse of dimensionality that would otherwise severely affect bayes classifiers for functional data can be avoided we demonstrate that in the case of gaussian functional data the proposed functional bayes classifier reduces to a functional version of the classical quadratic discriminant a study of the asymptotic behavior of the proposed classifiers in the large sample limit shows that under certain conditions the misclassification rate converges to zero a phenomenon that has been referred to as perfect classification the proposed classifiers also perform favorably in finite sample applications as we demonstrate in comparisons with other functional classifiers in simulations and various data applications including wine spectral data functional magnetic resonance imaging fmri data for attention deficit hyperactivity disorder adhd patients and yeast gene expression data

contact impact by the pinball algorithm with penalty and lagrangian methods contact impact algorithms which are sometimes called slideline algorithms are a computationally time consuming part of many explicit simulations of non linear problems because they involve many branches so they are not amenable to vectorization which is essential for speed on supercomputers the pinball algorithm is a simplified slideline algorithm which is readily vectorized its major idea is to embed pinballs in surface elements and to enforce the impenetrability condition only to pinballs it can be implemented in either a lagrange multiplier or penalty method it is shown that in any lagrange multiplier method no iterations are needed to define the contact surface examples of solutions and running times are given

levy area of wiener processes in banach spaces the goal of this paper is to construct canonical levy area processes for banach space valued brownian motions via dyadic approximations the significance of the existence of canonical levy area processes is that a stochastic integration theory can be established for such brownian motions in banach spaces existence of flows for stochastic differential equations with infinite dimensional noise then follows via the results of lyons and lyons and qian see e g system control and rough paths 2000 oxford univ press this investigation involves a careful analysis on the choice of tensor norms motivated by the applications to infinite dimensional stochastic differential equations

strong deviation theorems for functionals of the nonnegative continuous random variables abstract by using the approach of laplace transform we establish the strong deviation theorems represented by inequalities and described by asymptotic average log likelihood ratio for functionals of the nonnegative continuous random variables as corollaries we obtain several strong laws of large numbers for the random variable functionals the results of this paper extend and improve the corresponding results of some current literatures

adaptive bayesian estimation using a gaussian random field with inverse gamma bandwidth we consider nonparametric bayesian estimation inference using a rescaled smooth gaussian field as a prior for a multidimensional function the rescaling is achieved using a gamma variable and the procedure can be viewed as choosing an inverse gamma bandwidth the procedure is studied from a frequentist perspective in three statistical settings involving replicated observations density estimation regression and classification we prove that the resulting posterior distribution shrinks to the distribution that generates the data at a speed which is minimax optimal up to a logarithmic factor whatever the regularity level of the data generating distribution thus the hierachical bayesian procedure with a fixed prior is shown to be fully adaptive

large deviations for local times and intersection local times of fractional brownian motions and riemann liouville processes in this paper we prove exact forms of large deviations for local times and intersection local times of fractional brownian motions and riemann liouville processes we also show that a fractional brownian motion and the related riemann liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times as a consequence of our large deviation estimates we derive laws of iterated logarithm for the corresponding local times the key points of our methods 1 logarithmic superadditivity of a normalized sequence of moments of exponentially randomized local time of a fractional brownian motion 2 logarithmic subadditivity of a normalized sequence of moments of exponentially randomized intersection local time of riemann liouville processes 3 comparison of local and intersection local times based on embedding of a part of a fractional brownian motion into the reproducing kernel hilbert space of the riemann liouville process

physical stabilization of the 4 node shell element with one point quadrature abstract a new four node shell element for nonlinear analysis which is useful for explicit time integration with single point quadrature is presented an assumed strain method is used to stabilize the zero energy modes of the element the stabilization procedure is based on the hu washizu variational principle and does not require the input of any stabilization parameters example problems show good accuracy and rate of convergence compared with fully integrated 4 node elements however the physically stabilized element is substantially faster than the fully integrated element

rates of contraction of posterior distributions based on gaussian process priors we derive rates of contraction of posterior distributions on nonparametric or semiparametric models based on gaussian processes the rate of contraction is shown to depend on the position of the true parameter relative to the reproducing kernel hilbert space of the gaussian process and the small ball probabilities of the gaussian process we determine these quantities for a range of examples of gaussian priors and in several statistical settings for instance we consider the rate of contraction of the posterior distribution based on sampling from a smooth density model when the prior models the log density as a fractionally integrated brownian motion we also consider regression with gaussian errors and smooth classification under a logistic or probit link function combined with various priors

eigenvalues and stable time steps for the bilinear mindlin plate element eigenvalues are obtained for the 4 node mindlin plate element with one point quadrature both isotropic and some anisotropic matenals are considered but for the latter only bounds are obtained on the eigenvalues these eigenvalues provide stable time steps for explicit time integration algorithms

functional quantization of gaussian processes quantization consists in studying the lr error induced by the approximation of a random vector x by a vector quantized version taking a finite number n of values for rm valued random vectors the theory and practice is quite well established and in particular the asymptotics as n of the resulting minimal quantization error for nonsingular distributions is well known it behaves like c x r m n 1 m this paper is a transposition of this problem to random vectors in an infinite dimensional hilbert space and in particular to stochastic processes xt t 0 1 viewed as l2 0 1 dt valued random vectors for gaussian vectors and the l2 error we present detailed results for stationary and optimal quantizers we further establish a precise link between the rate problem and shannon kolmogorov s entropy of x this allows us to compute the exact rate of convergence to zero of the minimal l2 quantization error under rather general conditions on the eigenvalues of the covariance operator typical rates are o logn a a 0 they are obtained for instance for the fractional brownian motion and the fractional ornstein uhlenbeck process the exponent a is closely related with the l2 regularity of the process

a stabilization procedure for the quadrilateral plate element with one point quadrature a stabilization procedure is developed for controlling the kinematic modes of the four node bilinear quadrilateral element when single point quadrature is used these kinematic modes manifest themselves by spatial oscillations or singularity of the total stiffness in this stabilization procedure additional generalized strains are defined which are activated by the kinematic modes these generalized modes are furthermore not activated by rigid body motions regardless of the shape of the quadrilateral by using a scaling law developed in an earlier paper the stabilization parameters are defined so they do not adversely affect the element s performance several problems which are subject to kinematic modes are presented to illustrate the performance of this stabilization procedure for linear problems

the small ball problem for the brownian sheet we show that the logarithm of the probability that the brownian sheet has a supremum at most e over 0 1 2 is of order e 2 log 1 e 3

probabilities of randomly centered small balls and quantization in banach spaces we investigate the gaussian small ball probabilities with random centers find their deterministic a s equivalents and establish a relation to infinite dimensional high resolution quantization

probability estimates for multiparameter brownian processes soit f une fonction de repartition sur 0 1 d et w f le processus gaussien qui est la limite faible du processus empirique determine par f si g est une fonction sur 0 1 d on obtient des bornes superieures et inferieures pour p sup t 0 1 d w f t g t e

physical shape functions in finite element analysis of moderately thick plates a new rectangular finite element for moderately thick plates is presented the element is based on the concept of physical shape functions i e functions which contain in themselves physical and geometrical properties of the element the analytic formulae of stiffness geometric stiffness and mass matrices are presented for isotropic material the element is free from locking and zero energy modes not corresponding to rigid body motions several examples are presented for static initial stability and free vibration problems

a reduced integration solid shell finite element based on the eas and the ans concept geometrically linear problems in this paper a new reduced integration eight node solid shell finite element is presented the enhanced assumed strain eas concept based on the hu washizu variational principle requires only one eas degree of freedom to cure volumetric and poisson thickness locking one key point of the derivation is the taylor expansion of the inverse jacobian with respect to the element center which closely approximates the element shape and allows us to implement the assumed natural strain ans concept to eliminate the curvature thickness and the transverse shear locking the second crucial point is a combined taylor expansion of the compatible strain with respect to the center of the element and the normal through the element center leading to an efficient and locking free hourglass stabilization without rank deficiency hence the element requires only a single integration point in the shell plane and at least two integration points in thickness direction the formulation fulfills both the membrane and the bending patch test exactly which has to the authors knowledge not yet been achieved for reduced integration eight node solid shell elements in the literature owing to the three dimensional modeling of the structure fully three dimensional matenal models can be implemented without additional assumptions

a semiparametric bernstein von mises theorem for gaussian process priors this paper is a contribution to the bayesian theory of semiparametric estimation we are interested in the so called bernstein von mises theorem in a semiparametric framework where the unknown quantity is f with the parameter of interest and f an infinite dimensional nuisance parameter two theorems are established one in the case with no loss of information and one in the information loss case with gaussian process priors the general theory is applied to three specific models the estimation of the center of symmetry of a symmetric function in gaussian white noise a time discrete functional data analysis model and cox s proportional hazards model in all cases the range of application of the theorems is investigated by using a family of gaussian priors parametrized by a continuous parameter

on the small ball inequality in all dimensions abstract let h r denote an l normalized haar function adapted to a dyadic rectangle r 0 1 d we show that for choices of coefficients r we have the following lower bound on the l norms of the sums of such functions where the sum is over rectangles of a fixed volume n d 1 2 r 2 n r h r x l 0 1 d 2 n r 2 n r for some 0 1 2 the point of interest is the dependence upon the logarithm of the volume of the rectangles with n d 1 2 on the left above the inequality is trivial while it is conjectured that the inequality holds with n d 2 2 this is known in the case of d 2 michel talagrand the small ball problem for the brownian sheet ann probab 22 3 1994 1331 1354 mr 95k 60049 and a recent paper of two of the authors dmitriy bilyk michael t lacey on the small ball inequality in three dimensions duke math j 2006 in press arxiv math ca 0609815 proves a partial result towards the conjecture in three dimensions in this paper we show that the argument of dmitriy bilyk michael t lacey on the small ball inequality in three dimensions duke math j 2006 in press arxiv math ca 0609815 can be extended to arbitrary dimension we also prove related results in the subjects of the irregularity of distribution and approximation theory the authors are unaware of any prior results on these questions in any dimension d 4

limit laws for local times of the brownian sheet letb s t s t 0 be a brownian sheet then for alls 0 the processbs t b s t t 0 is a scaled brownian motion which admits a local timels x t which is jointly continuous inx r ands t 0 s1 2ls is a standard brownian local time we prove that mathop lim limits lambda to infty lambda 2 log p mathop sup limits 1 lambda 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2

a low order hexahedral finite element for modelling shells a thin eight node tri linear displacement hexahedral finite element is the starting point for the derivation of a constant membrane stress resultant constant bending stress resultant shell finite element the derivation begins by introducing a taylor series expansion for the stress distribution in the isoparametric co ordinates of the element the effect of the taylor series expansion for the stress distribution is to explicitly identify those strain modes of the element that are conjugate to the mean or average stress and the linear variation in stress the constant membrane stress resultants are identified with the mean stress components and the constant bending stress resultants are identified with the linear variation in stress through the thickness along with in plane linear variations of selected components of the transverse shear stress further a plane stress constitutive assumption is introduced and an explicit treatment of the finite element s thickness is introduced a number of elastic simulations show the useful results that can be obtained tip loaded twisted beam point loaded hemisphere point loaded sphere tip loaded raasch hook and a beam bent into a ring all of the gradient divergence operators are evaluated in closed form providing unequivocal evaluations of membrane and bending strain rates along with the appropriate divergence calculations involving the membrane stress and bending stress resultants the fact that a hexahedral shell finite element has two distinct surfaces aids sliding interface algorithms when a shell folds back on itself when subjected to large deformations

strong limit theorems for arbitrary stochastic sequences in this paper we establish two strong limit theorems for arbitrary stochastic sequences as corollaries we generalize some known results

nonlinear thin shell analysis using a curved triangular element abstract this paper describes a simple but robust curved triangular thin shell element the element has twelve degrees of freedom namely the displacement components in the corner nodes and the rotations about the element sides the element is based upon the well known facet element with constant membrane strains and constant stress couples and is augmented to take into account the influence of the changes of curvature on the membrane deformations due to the quadratic character of this influence curved geometries can be modelled in order to permit arbitrarily large rotational increments reference vectors are attached to each element side these reference vectors are updated by means of orthogonal transformations the latter are determined uniquely by the subsequent orientations of the associated element sides the changes of curvature are evaluated using these reference vectors and a corotated reference configuration to describe time independent elastoplasticity a layered shell model is applied while besseling s fraction model is adopted for each layer a number of numerical examples illustrate the performance of the element

exact l2 small ball behavior of integrated gaussian processes and spectral asymptotics of boundary value problems we find the exact small deviation asymptotics for the l 2 norm of various m times integrated gaussian processes closely connected with the wiener process and the ornstein uhlenbeck process using a general approach from the spectral theory of linear differential operators we obtain the two term spectral asymptotics of eigenvalues in corresponding boundary value problems this enables us to improve the recent results from 15 on the small ball asymptotics for a class of m times integrated wiener processes moreover the exact small ball asymptotics for the m times integrated brownian bridge the m times integrated ornstein uhlenbeck process and similar processes appear as relatively simple examples illustrating the developed general theory

orlicz spaces bounds for special classes of hyperbolic sums abstract let r r 1 r d denote a dyadic rectangle in the unit cube 0 1 d d 3 let h r be the l normalized haar function supported on r in 10 the conjectured signed small ball inequality r 2 n r h r n d 2 where r 1 was proven under the additional assumption that the coefficients also satisfy the splitting property r r 1 r 2 r d with r 1 r 2 r d 1 we give another proof of this result using a duality argument based on this approach we also show r 2 n r h r exp l a n d 2 1 a 2 a for any integer n 1 and any choice of coefficients r 1 1 which satisfy the splitting property the above inequality has been conjectured for general coefficients r 1 1 in d 3 these bounds are investigated further for more general coefficients r 1 1 the proof of the sharpness of the l lower bound of hyperbolic sums with coefficients satisfying the splitting property is also provided

the small time chung wichura law for levy processes with non vanishing brownian component we give a small time functional version of chung s other law of the iterated logarithm for levy processes with non vanishing brownian component this is an analogue of the other law of the iterated logarithm at large times for levy processes and random walks with finite variance as extended to a functional version by wichura as one of many possible applications we mention a functional law for a two sided passage time process

a uniform strain hexahedron and quadrilateral with orthogonal hourglass control the treatment of zero energy modes which arise due to one point integration of first order isoparametric finite elements is addressed a method for precisely isolating these modes for arbitrary geometry is developed two hourglass control schemes viscous and elastic are presented and examined in addition a convenient one point integration scheme which analytically integrates the element volume and uniform strain modes is presented

weighted covering numbers of convex sets abstract in this paper we define the notions of weighted covering number and weighted separation number for convex sets and compare them to the classical covering and separation numbers this sheds new light on the equivalence of classical covering and separation we also provide a formula for computing these numbers via a limit of classical covering numbers in higher dimensions

multiple quadrature underintegrated finite elements new multiple quadrature point underintegrated finite elements with hourglass control are developed the elements are selectively underintegrated to avoid volumetric and shear locking and save computational time an approach for hourglass control is proposed such that the stabilization operators are obtained simply by taking the partial derivatives of the generalized strain rate vector with respect to the natural co ordinates so that the elements require no stabilization parameter to improve accuracy over the traditional one point quadrature elements several quadrature points are used to integrate the internal forces especially for tracing the plastic fronts in the mesh during loading and unloading in elastic plastic analysis two and four point quadrature elements are proposed for use in the two and three dimensional elements respectively other multiple quadrature points can also be employed several numerical examples such as thin beam plate and shell problems are presented to demonstrate the applicability of the proposed elements

approximation metric entropy and small ball estimates for gaussian measures a precise link proved by kuelbs and li relates the small ball behavior of a gaussian measure on a banach space e with the metric entropy behavior of k the unit ball of the reproducing kernel hilbert space of in e we remove the main regularity assumption imposed on the unknown function in the link this enables the application of tools and results from functional analysis to small ball problems and leads to small ball estimates of general algebraic type as well as to new estimates for concrete gaussian processes moreover we show that the small ball behavior of a gaussian process is also tightly connected with the speed of approximation by finite rank processes

the effects of element shape on the critical time step in explicit time integrators for elasto dynamics summary in this paper the effects of element shape on the critical time step are investigated the common rule of thumb used in practice is that the critical time step is set by the shortest distance within an element divided by the dilatational compressive wave speed with a modest safety factor for regularly shaped elements many analytical solutions for the critical time step are available but this paper focusses on distorted element shapes the main purpose is to verify whether element distortion adversely affects the critical time step or not two types of element distortion will be considered namely aspect ratio distortion and angular distortion and two particular elements will be studied four noded bilinear quadrilaterals and three noded linear triangles the maximum eigenfrequencies of the distorted elements are determined and compared to those of the corresponding undistorted elements the critical time steps obtained from single element calculations are also compared to those from calculations based on finite element patches with multiple elements copyright 2014 john wiley sons ltd

thomas bayes walk on manifolds convergence of the bayes posterior measure is considered in canonical statistical settings where observations sit on a geometrical object such as a compact manifold or more generally on a compact metric space verifying some conditions a natural geometric prior based on randomly rescaled solutions of the heat equation is considered upper and lower bound posterior contraction rates are derived

new rectangular plate elements based on twist kirchhoff theory abstract we introduce a new framework for the development of thin plate finite elements the twist kirchhoff theory a family of rectangular plate elements is derived that takes advantage of the special structure of this new theory particular attention is focused on the lowest order member of the family an eight degree of freedom four node element with mid side rotations whose stiffness matrix is exactly computed with one point gaussian quadrature we prove a convergence theorem for it and various error estimates these are also generalized to the higher order elements in the family numerical tests corroborate the theoretical results

direct and sparse construction of consistent inverse mass matrices general variational formulation and application to selective mass scaling summary classical explicit finite element formulations rely on lumped mass matrices a diagonalized mass matrix enables a trivial computation of the acceleration vector from the force vector recently non diagonal mass matrices for explicit finite element analysis fea have received attention due to the selective mass scaling sms technique sms allows larger time step sizes without substantial loss of accuracy however an expensive solution for accelerations is required at each time step in the present study this problem is solved by directly constructing the inverse mass matrix first a consistent and sparse inverse mass matrix is built from the modified hamiltons principle with independent displacement and momentum variables usage of biorthogonal bases for momentum allows elimination of momentum unknowns without matrix inversions and directly yields the inverse mass matrix denoted here as reciprocal mass matrix rmm secondly a variational mass scaling technique is applied to the rmm it is based on the penalized hamiltons principle with an additional velocity variable and a free parameter using element wise bases for velocity and a local elimination yields variationally scaled rmm thirdly examples illustrating the efficiency of the proposed method for simplex elements are presented and discussed copyright 2014 john wiley sons ltd

metric entropy and the small ball problem for gaussian measures abstract we establish a precise link between the small ball problem for a gaussian measure on a separable banach space and the metric entropy of the unit ball of the hubert space h generating this link allows us to compute small ball probabilities from metric entropy results and vice versa

strong convergence of pairwise nqd random sequences abstract strong limit theory is one of the most important problems in probability theory some results on the convergence of pairwise nqd random sequences have been presented this paper further analyzes the strong convergence of pairwise nqd sequences and generalizes partial results of wu q y wu convergence properties of pairwise nqd random sequences acta math sinica 45 3 2002 617 624 in chinese since no general moment inequalities are given as so far we avoid this problem and obtain a class of strong limit theorem for nqd sequences and some corresponding conclusions by use of truncation methods and generalized three series theorem which are the supplements to the previous fruits

small deviations of weighted fractional processes and average non linear approximation we investigate the small deviation problem for weighted fractional brownian motions in l q norm 1 q let b h be a fractional brownian motion with hurst index 0 h 1 if 1 r h 1 q then our main result asserts formula math provided the weight function p satisfies a condition slightly stronger than the r integrability thus we extend earlier results for brownian motion i e h 1 2 to the fractional case our basic tools are entropy estimates for fractional integration operators a non linear approximation technique for gaussian processes as well as sharp entropy estimates for l q sums of linear operators defined on a hilbert space

information rates of nonparametric gaussian process methods we consider the quality of learning a response function by a nonparametric bayesian approach using a gaussian process gp prior on the response function we upper bound the quadratic risk of the learning procedure which in turn is an upper bound on the kullback leibler information between the predictive and true data distribution the upper bound is expressed in small ball probabilities and concentration measures of the gp prior we illustrate the computation of the upper bound for the matern and squared exponential kernels for these priors the risk and hence the information criterion tends to zero for all continuous response functions however the rate at which this happens depends on the combination of true response function and gaussian prior and is expressible in a certain concentration function in particular the results show that for good performance the regularity of the gp prior should match the regularity of the unknown response function

a robust non linear mixed hybrid quadrilateral shell element in the paper a non linear quadrilateral shell element for the analysis of thin structures is presented the variational formulation is based on a hu washizu functional with independent displacement stress and strain fields the interpolation matrices for the mid surface displacements and rotations as well as for the stress resultants and strains are specified restrictions on the interpolation functions concerning fulfillment of the patch test and stability are derived the developed mixed hybrid shell element possesses the correct rank and fulfills the in plane and bending patch test using newton s method the finite element approximation of the stationary condition is iteratively solved our formulation can accommodate arbitrary non linear material models for finite deformations in the examples we present results for isotropic plasticity at finite rotations and small strains as well as bifurcation problems and post buckling response the essential feature of the new element is the robustness in the equilibrium iterations it allows very large load steps in comparison to other element formulations copyright 2005 john wiley sons ltd

mixture inner product spaces and their application to functional data analysis we introduce the concept of mixture inner product spaces associated with a given separable hilbert space which feature an infinite dimensional mixture of finite dimensional vector spaces and are dense in the underlying hilbert space any hilbert valued random element can be arbitrarily closely approximated by mixture inner product space valued random elements while this concept can be applied to data in any infinite dimensional hilbert space the case of functional data that are random elements in the l 2 space of square integrable functions is of special interest for functional data mixture inner product spaces provide a new perspective where each realization of the underlying stochastic process falls into one of the component spaces and is represented by a finite number of basis functions the number of which corresponds to the dimension of the component space in the mixture representation of functional data the number of included mixture components used to represent a given random element in l 2 is specifically adapted to each random trajectory and may be arbitrarily large key benefits of this novel approach are first that it provides a new perspective on the construction of a probability density in function space under mild regularity conditions and second that individual trajectories possess a trajectory specific dimension that corresponds to a latent random variable making it possible to use a larger number of components for less smooth and a smaller number for smoother trajectories this enables flexible and parsimonious modeling of heterogeneous trajectory shapes we establish estimation consistency of the functional mixture density and introduce an algorithm for fitting the functional mixture model based on a modified expectation maximization algorithm simulations confirm that in comparison to traditional functional principal component analysis the proposed method achieves similar or better data recovery while using fewer components on average its practical merits are also demonstrated in an analysis of egg laying trajectories for medflies

quadratic functionals and small ball probabilities for the m fold integrated brownian motion let the gaussian process xm t be the m fold integrated brownian motion for positive integer m the laplace transform of the quadratic functional of xm t is found by using an appropriate self adjoint integral operator the result is then used to show the power of a general connection between small ball probabilities for the gaussian process the connection is discovered by introducing an independent random shift the interplay between our results and the principal eigenvalues for nonuniform elliptic generators on an unbounded domain is also discussed

small ball probabilities for gaussian random fields and tensor products of compact operators we find the logarithmic l 2 small ball asymptotics for a large class of zero mean gaussian fields with covariances having the structure of tensor product the main condition imposed on marginal covariances is the regular behavior of their eigenvalues at infinity that is valid for a multitude of gaussian random functions including the fractional brownian sheet ornstein uhlenbeck sheet etc so we get the far reaching generalizations of well known results by csaki 1982 and by li 1992 another class of gaussian fields considered is the class of additive fields studied under the supremum norm by chen and li 2003 our theorems are based on new results on spectral asymptotics for the tensor products of compact self adjoint operators in hilbert space which are of independent interest

small ball probabilities of gaussian fields lower bounds on the small ball probability are given for brownian sheet type gaussian fields as well as for general gaussian fields with stationary increments in d in particular a sharp bound is found for the fractional levy brownian fields

explicit algorithms for the nonlinear dynamics of shells abstract a finite element formulation and algorithm for the nonlinear analysis of the large deflection materially nonlinear response of impulsively loaded shells is presented a unique feature of this algorithm is the use of a bilinear four node quadrilateral element with single point quadrature and a simple hourglass control which is orthogonal to rigid body modes on an element level and does not compromise the consistency of the equations the geometric nonlinearities are treated by using a corotational description wherein a coordinate system that rotates with the material is embedded at the integration point thus the algorithm is directly applicable to anisotropic materials without any corrections for frame invariance of material property tensors this algorithm can treat about 200 element time steps per cpu second on a cyber 170 730 computer in the explicit time integration mode numerous results are presented for both elastic and elastic plastic problems with large strains that show that the method in most cases is comparable in accuracy with an earlier version of this algorithm employing a cubic triangular plate shell element but substantially faster

duality of metric entropy abstractfor two convex bodies kand t in r n the covering number of kbyt denoted n k t is de ned as the minimal number of translates oft needed to cover k let us denote by k the polar body of k andby d the euclidean unit ball in r n we prove that the two functionsof t n k td and n d tk are equivalent in the appropriate sense uniformly over symmetric convex bodies k r n and over n n inparticular this veri es the duality conjecture for entropy numbers of linearoperators posed by pietsch in 1972 in the central case when either thedomain or the range of the operator is a hilbert space 1 introduction for two convex bodies kand t in r n the covering number of kby t denoted n k t is de ned as the minimal number of translates of tneeded to cover kn k t min n x 1 x n r n k i n x i t we denote by dthe euclidean unit ball in r n in this paper we prove thefollowing duality result for covering numbers theorem 1 main theorem there exist two universal constants

sharp asymptotics of the kolmogorov entropy for gaussian measures abstract let u denote the unit ball of the cameron martin space of a gaussian measure on a hilbert space the sharp asymptotics for the kolmogorov metric entropy numbers of u is derived the condition imposed is regular variation of the eigenvalues of the covariance operator a consequence is a precise link including constants to the functional quantization problem

krein s spectral theory and the paley wiener expansion for fractional brownian motion in this paper we develop the spectral theory of the fractional brownian motion fbm using the ideas of krein s work on continuous analogous of orthogonal polynomials on the unit circle we exhibit the functions which are orthogonal with respect to the spectral measure of the fbm and obtain an explicit reproducing kernel in the frequency domain we use these results to derive an extension of the classical paley wiener expansion of the ordinary brownian motion to the fractional case

an improved treatment of transverse shear in the mindlin type four node quadrilateral element abstract an improved displacement methodology based on mindlin theory is developed and applied to a four node twelve degrees of freedom quadrilateral element special attention is directed toward enhancement of the transverse shear energy representation particularly a non uniform order kinematic interpolation scheme is accompanied by what is termed element appropriate shear correction factors the element matrices are consistently derived from hamilton s variational principle using exact numerical integration throughout the element stiffness matrix possesses correct rank and is well conditioned over the entire mindlin theory range of thickness to length ratios exceptionally good element performance is demonstrated by solutions to a set of static and vibrational problems

a new one point quadrature quadrilateral shell element with drilling degrees of freedom abstract in this paper we discuss the development of a new four node general shell element with single point quadrature used for the analysis of non linear geometrical and material problems one of the main features of the present development is the implementation of a rotation component around the shell normal i e drilling rotation to belytschko s family of shell elements thus at each node six degrees of freedom i e three translations and three rotations make the element easy to connect to space beams stiffeners or intersecting shells a projection scheme for warping correction is proposed so that the element is accurate for both flat and warped configurations all locking phenomena such as transverse shear in plane shear and membrane locking are controlled by an assumed strain method also a physical stabilization approach for control of spurious zero energy modes is proposed without requiring any artificial stabilization parameters this approach is inexpensive and accurate because it updates and stores hourglass stresses only at the mid surface rather than at all the integration points through the shell thickness to demonstrate the features of the new shell element it was applied to the analysis of large scale non linear static dynamic and impact problems using elastic isotropic anisotropic elastoplastic and composite damage models the element performed well for the problems that sometimes cause difficulties for other shell element techniques

a linear quadrilateral shell element with fast stiffness computation abstract a new quadrilateral shell element with 5 6 nodal degrees of freedom is presented assuming linear isotropic elasticity a hellinger reissner functional with independent displacements rotations and stress resultants is used within the mixed formulation the stress resultants are interpolated using five parameters for the membrane forces as well as for the bending moments and four parameters for the shear forces the hybrid element stiffness matrix resulting from the stationary condition is integrated analytically this leads to a part obtained by one point integration and a stabilization matrix the element possesses the correct rank is free of locking and is applicable within the whole range of thin and thick shells the in plane and bending patch tests are fulfilled and the computed numerical examples show that the convergence behaviour of the stress resultants is very good in comparison to comparable existing elements the essential advantage is the fast stiffness computation due to the analytically integrated matrices

a note on chung s strong law of large numbers abstract in this paper we generalize the classical chung s strong law of large numbers we also prove some results about almost sure a summability for sequences of independent random variables where a is a suitable real infinite matrix

small values of gaussian processes and functional laws of the iterated logarithm we estimate small ball probabilities for locally nondeterministic gaussian processes with stationary increments a class of processes that includes the fractional brownian motions these estimates are used to prove chung type laws of the iterated logarithm

identifying nonlinear covariate effects in semimartingale regression models letxt be a semimartingale which is either continuous or of counting process type and which satisfies the stochastic differential equationdxt yt t zt dt dmt wherey andz are predictable covariate processes m is a martingale and is an unknown nonrandom function we study inference for by introducing an estimator for a t z int limits 0 z int limits 0 t alpha s x dsdx and deriving a functional central limit theorem for the estimator the asymptotic distribution turns out to be given by a gaussian random field that admits a representation as a stochastic integral with respect to a multiparameter wiener process this result is used to develop a test for independence ofx from the covariatez a test for time homogeneity of and a goodness of fit test for the proportional hazards model t z 1 t a2 z used in survival analysis

dynamics of the bilinear mindlin plate element plate and shell finite elements based upon mindlin plate theory and evaluated by one point quadrature are attractive because of their effectiveness for both thick and thin shells at minimal computing cost however these elements are rank deficient and must be stabilized to obtain reliable behaviour methods for achieving full rank of the stiffness and for stabilizing element behaviour in static analysis and in explicit dynamic calculations exist and are quite effective this paper addresses the remaining issue of controlling spurious modes of response in vibration analysis and implicit dynamic solutions several alternatives for the element mass formulation are examined in detail we show that non physical dynamic modes present a potential problem with most mass matrix formulations and that spurious modes other than the familiar hourglassing motion are possible a combination of projection methods and reduced quadrature is suggested which eliminates these deficiencies and produces accurate numerical results the remaining techniques investigated give rise to anomalous behaviour which make them unsuitable for general use