asymptotic equivalence of nonparametric regression and white noise the principal result is that under conditions to any nonparametric regression problem there corresponds an asymptotically equivalent sequence of white noise with drift problems and conversely this asymptotic equivalence is in a global and uniform sense any normalized risk function attainable in one problem is asymptotically attainable in the other with the difference in normalized risks converging to zero uniformly over the entire parameter space the results are constructive a recipe is provided for producing these asymptotically equivalent procedures some implications and generalizations of the principal result are also discussed

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

waldhausen s classification theorem for finitely uniformizable 3 orbifolds we define a map between two orbifolds with respect to this map we generalize 3 manifold theory to 3 orbifolds as the main goal we generalize the waldhausen s classification theorem of haken 3 manifolds to finitely uniformizable 3 orbifolds for applications of the developed theory we introduce an invariant for links and tangles by using the orbifold fundamental group with the invariant we classify a class of links and show the untangling theorem

a new kernel based approach for linear system identification this paper describes a new kernel based approach for linear system identification of stable systems we model the impulse response as the realization of a gaussian process whose statistics differently from previously adopted priors include information not only on smoothness but also on bibo stability the associated autocovariance defines what we call a stable spline kernel the corresponding minimum variance estimate belongs to a reproducing kernel hilbert space which is spectrally characterized compared to parametric identification techniques the impulse response of the system is searched for within an infinite dimensional space dense in the space of continuous functions overparametrization is avoided by tuning few hyperparameters via marginal likelihood maximization the proposed approach may prove particularly useful in the context of robust identification in order to obtain reduced order models by exploiting a two step procedure that projects the nonparametric estimate onto the space of nominal models the continuous time derivation immediately extends to the discrete time case on several continuous and discrete time benchmarks taken from the literature the proposed approach compares very favorably with the existing parametric and nonparametric techniques

a bernstein von mises theorem for smooth functionals in semiparametric models a bernstein von mises theorem is derived for general semiparametric functionals the result is applied to a variety of semiparametric problems in i i d and non i i d situations in particular new tools are developed to handle semiparametric bias in particular for nonlinear functionals and in cases where regularity is possibly low examples include the squared l 2 norm in gaussian white noise nonlinear functionals in density estimation as well as functionals in autoregressive models for density estimation a systematic study of bvm results for two important classes of priors is provided namely random histograms and gaussian process priors

sobolev bounds on functions with scattered zeros with applications to radial basis function surface fitting in this paper we discuss sobolev bounds on functions that vanish at scattered points in a bounded lipschitz domain that satisfies a uniform interior cone condition the sobolev spaces involved may have fractional as well as integer order we then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions rbfs these estimates include situations in which the target function does not belong to the native space of the rbf

bayesian aspects of some nonparametric problems we study the bayesian approach to nonparametric function estimation problems such as nonparametric regression and signal estimation we consider the asymptotic properties of bayes procedures for conjugate gaussian priors we show that so long as the prior puts nonzero measure on the very large parameter set of interest then the bayes estimators are not satisfactory more specifically we show that these estimators do not achieve the correct minimax rate over norm bounded sets in the parameter space thus all bayes estimators for proper gaussian priors have zero asymptotic efficiency in this minimax sense we then present a class of priors whose bayes procedures attain the optimal minimax rate of convergence these priors may be viewed as compound or hierarchical mixtures of suitable gaussian distributions

asymptotics of karhunen loeve eigenvalues and tight constants for probability distributions of passive scalar transport in this paper we study the asymptotics of the probability distribution function for a certain model of freely decaying passive scalar transport in particular we prove rigorous large n or semiclassical asymptotics for the eigenvalues of the covariance of a fractional brownian motion using these asymptotics along with some standard large deviations results we are able to derive tight asymptotics for the rate of decay of the tails of the probability density for a generalization of the majda model of scalar intermittency originally due to vanden eijnden we are also able to derive asymptotically tight estimates for the closely related problem of small l2 ball probabilities for a fractional brownian motion

bayesian inverse problems with gaussian priors the posterior distribution in a nonparametric inverse problem is shown to contract to the true parameter at a rate that depends on the smoothness of the parameter and the smoothness and scale of the prior correct combinations of these characteristics lead to the minimax rate the frequentist coverage of credible sets is shown to depend on the combination of prior and true parameter with smoother priors leading to zero coverage and rougher priors to conservative coverage in the latter case credible sets are of the correct order of magnitude the results are numerically illustrated by the problem of recovering a function from observation of a noisy version of its primitive

a versatile technique for the optimal approximation of random processes by functional quantization this paper presents the mathematical foundation of a novel and versatile technique for the approximation of random processes using the functional quantization concept in this approach random processes are represented by optimal quantizers that is finite collections of deterministic functions and corresponding probability masses that are carefully constructed to attain certain optimal properties although the computational cost of obtaining such optimal quantizers is not negligible their use in applications is simple and relatively inexpensive the paper has two objectives first we present an accessible overview of functional quantization theory second we introduce a novel methodology to compute optimal quantizers which is based on the classical lloyd s method and monte carlo simulation for validation purposes the proposed methodology is tested using two fundamental gaussian random processes brownian motion and fractional brownian motion for which optimal quantizers are known the results obtained compare very well with previous results reported in the literature then to demonstrate the unique versatility of the methodology the application to a non gaussian process is presented

asymptotic equivalence of density estimation and gaussian white noise signal recovery in gaussian white noise with variance tending to zero has served for some time as a representative model for nonparametric curve estimation having all the essential traits in a pure form the equivalence has mostly been stated informally but an approximation in the sense of le cam s deficiency distance would make it precise the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss in nonparametrics a first result of this kind has recently been established for gaussian regression we consider the analogous problem for the experiment given by n i i d observations having density f on the unit interval our basic result concerns the parameter space of densities which are in a holder ball with exponent 1 2 and which are uniformly bounded away from zero we show that an i i d sample of size n with density f is globally asymptotically equivalent to a white noise experiment with drift f l 2 and variance 1 4n l this represents a nonparametric analog of le cam s heteroscedastic gaussian approximation in the finite dimensional case the proof utilizes empirical process techniques related to the hungarian construction white noise models on f and log f are also considered allowing for various automatic asymptotic risk bounds in the i i d model from white noise

coxeter groups and meridional rank of links we prove the meridional rank conjecture for twisted links and arborescent links associated to bipartite trees with even weights these links are substantial generalizations of pretzels and two bridge links respectively lower bounds on meridional rank are obtained via coxeter quotients of the groups of link complements matching upper bounds on bridge number are found using the wirtinger numbers of link diagrams a combinatorial tool developed by the authors

on coverage and local radial rates of credible sets in the mildly ill posed inverse signal in white noise model we construct confidence sets as credible balls with respect to the empirical bayes posterior resulting from a certain two level hierarchical prior the quality of the posterior is characterized by the contraction rate which we allow to be local that is depending on the parameter the issue of optimality of the constructed confidence sets is addressed via a trade off between its size the local radial rate and its coverage probability we introduce excessive bias restriction ebr more general than self similarity and polished tail condition recently studied in the literature under ebr we establish the confidence optimality of our credible set with some local oracle radial rate we also derive the oracle estimation inequality and the oracle posterior contraction rate the obtained local results are more powerful than global adaptive minimax results for a number of smoothness scales follow as consequence in particular the ones considered by szabo et al ann statist 43 2015 1391 1428

dynamics of bimeromorphic maps of surfaces we classify bimeromorphic self maps f x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of compact kahler surfaces x in terms of their actions f h 1 1 x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i on cohomology we observe that the growth rate of f n is invariant under bimeromorphic conjugacy and that by conjugating one can always arrange that f n f n we show that the sequence f n can be bounded grow linearly grow quadratically or grow exponentially in the first three cases we show that after conjugating f is an automorphism virtually isotopic to the identity f preserves a rational fibration or f preserves an elliptic fibration respectively in the last case we show that there is a unique up to scaling expanding eigenvector for f that is nef and that f is bimeromorphically conjugate to an automorphism if and only if 2 0 we go on in this case to construct a dynamically natural positive current representing and we study the growth rate of periodic orbits of f we conclude by illustrating our results with a particular family of examples

the semiparametric bernstein von mises theorem in a smooth semiparametric estimation problem the marginal posterior for the parameter of interest is expected to be asymptotically normal and satisfy frequentist criteria of optimality if the model is endowed with a suitable prior it is shown that under certain straightforward and interpretable conditions the assertion of le cam s acclaimed but strictly parametric bernstein von mises theorem univ california publ statist 1 1953 277 329 holds in the semiparametric situation as well as a consequence bayesian point estimators achieve efficiency for example in the sense of hajek s convolution theorem z wahrsch verw gebiete 14 1970 323 330 the model is required to satisfy differentiability and metric entropy conditions while the nuisance prior must assign nonzero mass to certain kullback leibler neighborhoods ghosal ghosh and van der vaart ann statist 28 2000 500 531 in addition the marginal posterior is required to converge at parametric rate which appears to be the most stringent condition in examples the results are applied to estimation of the linear coefficient in partial linear regression with a gaussian prior on a smoothness class for the nuisance

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 hyperelliptic involutions of hyperbolic 3 manifolds by classical results a compact riemann surface or equivalently an orientable closed hyperbolic 2 manifold has at most one hyperelliptic involution i e the quotient by the involution is the 2 sphere a hyperelliptic involution is central in the automorphism or isometry group of such a hyperelliptic surface its fixed points are exactly the weierstrass points of the surface see e g fk in the present paper we consider hyperelliptic involutions of hyperbolic 3manifolds by definition these are isometric involutions whose quotient is the 3 sphere the situation is more complicated in dimension three in particular a hyperbolic 3 manifold can have more than one hyperelliptic involution our main result states that there is a universal bound on the number of non conjugate hyperelliptic involutions of closed hyperbolic 3 manifolds this bound is quite small and we compute it for various situations our methods also determine how different hyperelliptic involutions are related there remains a small gap between these upper bounds and the number of different involutionswhichwe can actually construct see r1 rz we remark that the question on the number of different hyperelliptic involutions of hyperbolic 3 manifolds has explicitly been asked in mv we obtain also results for hyperelliptic isometries of orders greater than two in particular for prime numbers p 2 we give a complete solution to the problem on the number of different hyperelliptic isometries of order p i e the upper bound we obtain is sharp let isom m denote the orientation preserving isometry group of a hyperbolic 3 manifoldm

mixed fractional brownian motion a spectral take abstract this paper provides yet another look at the mixed fractional brownian motion fbm this time from the spectral perspective the main result is an asymptotic approximation for the eigenvalues of its covariance operator using this approximation we derive the exact asymptotics of the l 2 small ball probabilities which was previously known only with logarithmic accuracy the obtained expression exhibits an interesting stratification of scales which occurs at certain values of the hurst parameter of the fractional component some of them have been previously encountered in other problems involving such mixtures

superefficiency in nonparametric function estimation fixed parameter asymptotic statements are often used in the context of nonparametric curve estimation problems e g nonparametric density or regression estimation in this context several forms of superefficiency can occur in contrast to what can happen in regular parametric problems here every parameter point e g unknown density or regression function can be a point of superefficiency we begin with an example which shows how fixed parameter asymptotic statements have often appeared in the study of adaptive kernel estimators and how superefficiency can occur in this context we then carry out a more systematic study of such fixed parameter statements it is shown in four general settings how the degree of superefficiency attainable depends on the structural assumptions in each case

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

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

a space quantization method for numerical integration abstract we propose a new method sqm for numerical integration of c functions 0 2 defined on a convex subset c of rd with respect to a continuous distribution it relies on a space quantization of c by a n tuplex x1 xn cn fd is approximated by a weighted sum of the f xi s the integration error bound depends on the distortion en x of the voronoi tessellation of x this notion comes from information theoretists its main properties existence of a minimizing n tuple in cn asymptotics of mincn en as n are presented for a wide class of measures a simple stochastic optimization procedure is proposed to compute in any dimension d x and the characteristics of its voronoi tessellation some new results on the competitive learning vector quantization algorithm when 2 are obtained as a by product some tests simulations and provisional remarks are proposed as a conclusion

adaptive bayesian inference on the mean of an infinite dimensional normal distribution we consider the problem of estimating the mean of an infinite break dimensional normal distribution from the bayesian perspective under the assumption that the unknown true mean satisfies a smoothness condition we first derive the convergence rate of the posterior distribution for a prior that is the infinite product of certain normal distributions and compare with the minimax rate of convergence for point estimators although the posterior distribution can achieve the optimal rate of convergence the required prior depends on a smoothness parameter q when this parameter q is unknown besides the estimation of the mean we encounter the problem of selecting a model in a bayesian approach this uncertainty in the model selection can be handled simply by further putting a prior on the index of the model we show that if q takes values only in a discrete set the resulting hierarchical prior leads to the same convergence rate of the posterior as if we had a single model a slightly weaker result is presented when q is unrestricted an adaptive point estimator based on the posterior distribution is also constructed primary subjects 62g20 secondary subjects 62c10 62g05 keywords adaptive bayes procedure convergence rate minimax risk posterior distribution model selection

a bayesian approach to non parametric monotone function estimation the paper proposes two bayesian approaches to non parametric monotone function estimation the first approach uses a hierarchical bayes framework and a characterization of smooth monotone functions given by ramsay that allows unconstrained estimation the second approach uses a bayesian regression spline model of smith and kohn with a mixture distribution of constrained normal distributions as the prior for the regression coefficients to ensure the monotonicity of the resulting function estimate the small sample properties of the two function estimators across a range of functions are provided via simulation and compared with existing methods asymptotic results are also given that show that bayesian methods provide consistent function estimators for a large class of smooth functions an example is provided involving economic demand functions that illustrates the application of the constrained regression spline estimator in the context of a multiple regression model where two functions are constrained to be monotone

on the bernstein von mises theorem with infinite dimensional parameters if there are many independent identically distributed observations governed by a smooth finite dimensional statistical model the bayes estimate and the maximum likelihood estimate will be close furthermore the posterior distribution of the parameter vector around the posterior mean will be close to the distribution of the maximum likelihood estimate around truth thus bayesian confidence sets have good frequentist coverage properties and conversely however even for the simplest infinite dimensional models such results do not hold the object here is to give some examples

convergence rates for bayesian density estimation of infinite dimensional exponential families we study the rate of convergence of posterior distributions in density estimation problems for log densities in periodic sobolev classes characterized by a smoothness parameter p the posterior expected density provides a nonparametric estimation procedure attaining the optimal minimax rate of convergence under hellinger loss if the posterior distribution achieves the optimal rate over certain uniformity classes a prior on the density class of interest is induced by a prior on the coefficients of the trigonometric series expansion of the log density we show that when p is known the posterior distribution of a gaussian prior achieves the optimal rate provided the prior variances die off sufficiently rapidly for a mixture of normal distributions the mixing weights on the dimension of the exponential family are assumed to be bounded below by an exponentially decreasing sequence to avoid the use of infinite bases we develop priors that cut off the series at a sample size dependent truncation point when the degree of smoothness is unknown a finite mixture of normal priors indexed by the smoothness parameter which is also assigned a prior produces the best rate a rate adaptive estimator is derived

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

statistics of parameter estimates a concrete example most mathematical models include parameters that need to be determined from measurements the estimated values of these parameters and their uncertainties depend on assumptions made about noise levels models or prior knowledge but what can we say about the validity of such estimates and the influence of these assumptions this paper is concerned with methods to address these questions and for didactic purposes it is written in the context of a concrete nonlinear parameter estimation problem we will use the results of a physical experiment conducted by allmaras et al at texas a m university m allmaras et al siam rev 55 2013 pp 149 167 to illustrate the importance of validation procedures for statistical parameter estimation we describe statistical methods and data analysis tools to check the choices of likelihood and prior distributions and provide examples of how to compare bayesian results with those obtained by non bayesian methods based on different types of assumptions we explain

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

asymptotic normality of semiparametric and nonparametric posterior distributions in this article it is shown that the marginal semiparametric and nonparametric posterior distributions for a parameter of interest behave like an ordinary parametric posterior distribution this in practice provides support of the utility of marginal semiparametric and nonparametric posterior distributions in particular the marginal semiparametric and nonparametric posterior distributions are asymptotically normal and centered at the corresponding maximum likelihood estimates mles or posterior means with covariance matrix the inverse of the fisher information additionally the semiparametric and nonparametric mles for the parameter of interest and the marginal posterior means are asymptotically normal and centered at the true parameter with the same covariance matrix the results are a semiparametric version and a nonparametric version of the parametric bayesian central limit theorem that establish a connection between the semiparametric and the nonparametric bayesian inference and their frequentis

a geometric proof of the fintushel stern formula abstract the fintushel stern formula asserts that the casson invariant of a brieskorn homology sphere p q r equals 1 8 the signature of its milnor fiber we give a geometric proof of this formula as opposite to computational methods used in the original proof the formula is also refined to relate equivariant casson invariants to equivariant signatures

asymptotics and the theory of inference asymptotic analysis has always been very useful for deriving distributions in statistics in cases where the exact distribution is unavailable more importantly asymptotic analysis can also provide insight into the inference process itself suggesting what information is available and how this information may be extracted the development of likelihood inference over the past twenty some years provides an illustration of the interplay between techniques of approximation and statistical theory

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

non standard components of the character variety for a family of montesinos knots we show that there are montesinos knots with n 1 geq 4 tangles whose character varieties contain arbitrarily many irreducible components of dimension d for any 1 le d le n 2 moreover these irreducible components can be chosen so that the trace of the meridian is non constant

finite group actions on 3 manifolds if g is a finite group acting smoothly on a closed surface f it is well known that g leaves invariant some riemannian metric of constant curvature on f thus any action of g on the 2 sphere s 2 is conjugate in diff s 2 to an orthogonal action if g acts on the torus sx s 1 there is a g invariant flat metric on s a s 1 and if g acts on a surface f with negative euler number then f admits a g invariant hyperbolic metric recently thurston th 1 th 2 th 3 has described the eight 3 dimensional geometries which provide geometric structures for closed 3 manifolds in the same way that the 2 sphere s 2 the euclidean plane e 2 and the hyperbolic plane h 2 provide geometric structures for surfaces see also the survey article by scott sc4 thurston also conjectured that if m is a closed 3 manifold with a geometric structure modelled on one of these eight geometries say x then any smooth action of a finite group g on m should leave invariant some metric on m inducing the geometry x we will say that g preserves the geometric structure on m in this case it should be noted that the restriction to smooth actions of g on m is essential for bing bi showed that there are involutions of s 3 whose fixed set is a wild 2 sphere however in dimension two it was proved by eilenberg ei that any action of a finite group on a surface is conjugate to a smooth action in this paper our main result asserts that thurston s conjecture holds for five of the eight geometries the result is the following

studying discrete dynamical systems through differential equations abstract in this paper we consider dynamical systems generated by a diffeomorphism f defined on u an open subset of r n and give conditions over f which imply that their dynamics can be understood by studying the flow of an associated differential equation x x x also defined on u in particular the case where f has n 1 functionally independent first integrals is considered in this case x is constructed by imposing that it shares with f the same set of first integrals and that the functional equation f x det d f x x x u has some non zero solution several examples for n 2 3 are presented most of them coming from several well known difference equations

nonexistence of informative unbiased estimators in singular problems in many nonparametric problems such as density estimation nonparametric regression and so on all the existing informative estimators are biased asymptotic or finite sample there has long been a suspicion that either informative unbiased estimators do not exist for such problems or they must be quite complicated in this paper we clarify the nonexistence of informative unbiased estimators in all singular problems both for fixed sample size and asymptotically this includes most problems with optimal rate of convergence slower than n1 2 we also discuss situations in regular problems where such nonexistences can occur

principal points and self consistent points of elliptical distributions the k principal points of a p variate random vector x are those points 1 k r p which approximate the distribution of x by minimizing the expected squared distance of x from the nearest of the j any set of k points y 1 y k partitions r p into domains of attraction d 1 d k according to minimal distance following hastie and stuetzle we call y 1 y k self consistent if e xx d j y j for j 1 k principal points are a special case of self consistent points in this paper we study principal points and self consistent points of p variate elliptical distributions the main results are the following 1 if k self consistent points of x span a subspace of dimension q p then this subspace is also spanned by q principal components that is self consistent points of elliptical distributions exist only in principal component subspaces 2 the subspace spanned by k principal points of x is identical with the subspace spanned by the principal components associated with the largest roots this proves a conjecture of flury we also discuss implications of our results for the computation and estimation of principal points

k integrals and k lie symmetries in discrete dynamical systems abstract we generalize the concept of symplectic maps to that of k symplectic maps maps whose kth iterates are symplectic similarly k symmetries and k integrals are symmetries resp integrals of the kth iterate of the map it is shown that k symmetries and k integrals are related by the k symplectic structure as in the k 1 continuous case noether s theorem examples are given of k integrals and their related k symmetries for k 1 4

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

hyperbolic structures on 3 manifolds i deformation of acylindrical manifolds this is the first in a series of papers showing that haken manifolds have hyperbolic structures this first was published the second two have existed only in preprint form and later preprints were never completed this eprint is only an approximation to the published version which is the definitive form for part i and is provided for convenience only all references and quotations should be taken from the published version since the theorem numbering is different and not all corrections have been incorporated into the present version parts ii and iii will be made available as eprints shortly

simultaneous confidence bands for penalized spline estimators in this article we construct simultaneous confidence bands for a smooth curve using penalized spline estimators we consider three types of estimation methods a as a standard fixed effect nonparametric model b using the mixed model framework with the spline coefficients as random effects and c a full bayesian approach the volume of tube formula is applied for the first two methods and compared with bayesian simultaneous confidence bands from a frequentist perspective we show that the mixed model formulation of penalized splines can help obtain at least approximately confidence bands with either bayesian or frequentist properties simulations and data analysis support the proposed methods the r package confbands accompanies the article

on some algebraic difference equations un 2un ψ un 1 in r related to families of conics or cubics generalization of the lyness sequences abstract in this paper and in a forthcoming one we study difference equations in r of the types 2 u n 2 u n a b u n 1 u n 1 2 4 u n 2 u n a b u n 1 c u n 1 2 c u n 1 6 u n 2 u n a b u n 1 c u n 1 2 c d u n 1 u n 1 2 which are linked to families of conics cubics and quartics respectively these equations generalize lyness one u n 2 u n a u n 1 studied in several papers whose behavior was completely elucidated in g bastien m rogalski in press through methods which are transposed in the present paper for the study of 1 and 2 and in the forthcoming one for 3 in particular we prove in the present paper a form of chaotic behavior for solutions of difference equations 1 and 2 and find all the possible periods for these solutions

on the convergence of the laplace approximation and noise level robustness of laplace based monte carlo methods for bayesian inverse problems the bayesian approach to inverse problems provides a rigorous framework for the incorporation and quantification of uncertainties in measurements parameters and models we are interested in designing numerical methods which are robust w r t the size of the observational noise i e methods which behave well in case of concentrated posterior measures the concentration of the posterior is a highly desirable situation in practice since it relates to informative or large data however it can pose a computational challenge for numerical methods based on the prior measure we propose to employ the laplace approximation of the posterior as the base measure for numerical integration in this context the laplace approximation is a gaussian measure centered at the maximum a posteriori estimate and with covariance matrix depending on the logposterior density we discuss convergence results of the laplace approximation in terms of the hellinger distance and analyze the efficiency of monte carlo methods based on it in particular we show that laplace based importance sampling and laplace based quasi monte carlo methods are robust w r t the concentration of the posterior for large classes of posterior distributions and integrands whereas prior based importance sampling and plain quasi monte carlo are not numerical experiments are presented to illustrate the theoretical findings

bayesian nonparametric inference for random distributions and related functions in recent years bayesian nonparametric inference both theoretical and computational has witnessed considerable advances however these advances have not received a full critical and comparative analysis of their scope impact and limitations in statistical modelling many aspects of the theory and methods remain a mystery to practitioners and many open guestions remain in this paper we discuss and illustrate the rich modelling and analytic possibilities that are available to the statistician within the bayesian nonparametric and or semiparametric framework

an algebraic geometric approach to integrable maps of the plane we show that the dynamics of a birational map on an elliptic curve over a field is typically conjugate to addition by a point under the associated group law when the field is taken to be the function field of rational complex functions of one variable this amounts to an algebraic geometric version of the arnold liouville integrability theorem for planar integrable maps by products of this approach are that birational maps preserving foliations are necessarily the composition of two involutions and that relationships between birational maps preserving the same foliation can be described in terms of the respective points they add on the corresponding weierstrass curves when the result is applied to finite fields it helps explain some universal features of the periodic orbit distribution function for the reductions of integrable maps

frequentist coverage of adaptive nonparametric bayesian credible sets we investigate the frequentist coverage of bayesian credible sets in a nonparametric setting we consider a scale of priors of varying regularity and choose the regularity by an empirical bayes method next we consider a central set of prescribed posterior probability in the posterior distribution of the chosen regularity we show that such an adaptive bayes credible set gives correct uncertainty quantification of polished tail parameters in the sense of high probability of coverage of such parameters on the negative side we show by theory and example that adaptation of the prior necessarily leads to gross and haphazard uncertainty quantification for some true parameters that are still within the hyperrectangle regularity scale

lie symmetries of birational maps preserving genus 0 fibrations we prove that any planar birational integrable map which preserves a fibration given by genus 0 curves has a lie symmetry and some associated invariant measures the obtained results allow to study in a systematic way the global dynamics of these maps using this approach the dynamics of several maps is described in particular we are able to give for particular examples the explicit expression of the rotation number function and the set of periods of the considered maps

from multiple gaussian sequences to functional data and beyond a stein estimation approach summary we expand the notion of gaussian sequence models to n experiments and propose a stein estimation strategy which relies on pooling information across experiments an oracle inequality is established to assess conditional risks given the underlying effects based on which we can quantify the size of relative error and obtain a tuning free recovery strategy that is easy to compute produces model parsimony and extends to unknown variance we show that the simultaneous recovery is adaptive to an oracle strategy which also enjoys a robustness guarantee in a minimax sense a connection to functional data is established via le cam theory for fixed and random designs under general regularity settings we further extend the model projection to general bases with mild conditions on correlation structure and conclude with potential application to other statistical problems simulated and real data examples are provided to lend empirical support to the methodology proposed and to illustrate the potential for substantial computational savings

bernstein von mises theorem for linear functionals of the density in this paper we study the asymptotic posterior distribution of linear functionals of the density by deriving general conditions to obtain a semi parametric version of the bernstein von mises theorem the special case of the cumulative distributive function evaluated at a specific point is widely considered in particular we show that for infinite dimensional exponential families under quite general assumptions the asymptotic posterior distribution of the functional can be either gaussian or a mixture of gaussian distributions with different centering points this illustrates the positive but also the negative phenomena that can occur in the study of bernstein von mises results

minimal index torsion free subgroups of kleinian groups a kleinian group is a discrete subgroup of psl 2 c the full group of orientation preserving isometries of 3 dimensional hyperbolic space in the language of t1 q h3 is a hyperbolic 3 orbifold that is a metric 3 orbifold in which all sectional curvatures are 1 and for which is the orbifold fundamental group see t1 for further details a fuchsian group is a discrete subgroup of psl 2 r and as such acts discontinuously on the hyperbolic plane we define to be of finite co volume resp co area if the volume resp area of the quotient orbifold q is finite by selberg s lemma if is a kleinian resp fuchsian group of finite covolume resp co area it contains a torsion free subgroup of finite index by definition any torsion free subgroup cannot contain any finite subgroups of so the index must be a multiple of the lowest common multiple of all orders of finite subgroups of see cfjr for a proof being of finite co volume implies there are only finitely many conjugacy classes of finite subgroups thus we make the following

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

the π orbifold group of a link on associe a un enlacement l dans s 3 un groupe o l appele le groupe de variete d orbite de l on etudie certaines proprietes du groupe o l reliees a la geometrie de l enlacement l

on pi hyperbolic knots with the same 2 fold branched coverings we consider the following problem from the kirby s list problem 3 25 let k be a knot in s 3 and m k its 2 fold branched covering space describe the equivalence class k of k in the set of knots under the equivalence relation k 1 approx k 2 if m k 1 is homeomorphic to m k 2 it is known that there exist arbitrarily many different hyperbolic knots with the same 2 fold branched coverings due to mutation along conway spheres thus the most basic class of knots to investigate are knots which do not admit conway spheres in this paper we solve the above problem for knots which do not admit conway spheres in the following sense we give upper bounds for the number of knots in the equivalence class k of a knot k and we describe how the different knots in the equivalence class of k are related

optimal partitioning for linear mixed effects models applications to identifying placebo responders a longstanding problem in clinical research is distinguishing drug treated subjects that respond due to specific effects of the drug from those that respond to nonspecific or placebo effects of the treatment linear mixed effect models are commonly used to model longitudinal clinical trial data in this paper we present a solution to the problem of identifying placebo responders using an optimal partitioning methodology for linear mixed effects models since individual outcomes in a longitudinal study correspond to curves the optimal partitioning methodology produces a set of prototypical outcome profiles the optimal partitioning methodology can accommodate both continuous and discrete covariates the proposed partitioning strategy is compared and contrasted with the growth mixture modeling approach the methodology is applied to a two phase depression clinical trial where subjects in a first phase were treated openly for 12 weeks with fluoxetine followed by a double blind discontinuation phase where

the strong law of large numbers for k means and best possible nets of banach valued random variables let b be a uniformly convex banach space x a b valued random variable and k a given positive integer number a random sample of x is substituted by the set of k elements which minimizes a criterion we found conditions to assure that this set converges a s as the sample size increases to the set of k elements which minimizes the same criterion for x

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

on recurrence equations associated with invariant varieties of periodic points a recurrence equation is a discrete integrable equation whose solutions are all periodic and the period is fixed we show that infinitely many recurrence equations can be derived from the information about invariant varieties of periodic points of higher dimensional integrable maps

an analysis of bayesian inference for nonparametric regression abstract the observation model y sub i beta i n epsilon sub i 1 or n is considered where the epsilon s are i i d mean zero and variance sigma sq and beta is an unknown smooth function a gaussian prior distribution is specified by assuming beta is the solution of a high order stochastic differential equation the estimation error delta beta beta average is analyzed where beta average is the posterior expectation of beta asymptotic posterior and sampling distributional approximations are given for abs val del square when abs val square is one of a family of norms natural to the problem it is shown that the frequentist coverage probability of a variety of 1 alpha posterior probability regions tends to be larger than 1 alpha but will be infinitely often less than any epsilon 0 as n approaches infinity with prior probability 1 a related continuous time signal estimation problem is also studied keywords bayesian inference nonparametric regression confidence regions signal extraction smoothing splices

mixed gaussian processes a filtering approach this paper presents a new approach to the analysis of mixed processes xt bt gt t 0 t xt bt gt t 0 t where btbt is a brownian motion and gtgt is an independent centered gaussian process we obtain a new canonical innovation representation of xx using linear filtering theory when the kernel k s t 2 s tegtgs s t k s t 2 s tegtgs s t has a weak singularity on the diagonal our results generalize the classical innovation formulas beyond the square integrable setting for kernels with stronger singularity our approach is applicable to processes with additional fractional structure including the mixed fractional brownian motion from mathematical finance we show how previously known measure equivalence relations and semimartingale properties follow from our canonical representation in a unified way and complement them with new formulas for radon nikodym densities

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

rates of convergence of posterior distributions we compute the rate at which the posterior distribution concentrates around the true parameter value the spaces we work in are quite general and include infinite dimensional cases the rates are driven by two quantities the size of the space as measured by bracketing entropy and the degree to which the prior concentrates in a small ball around the true parameter we consider two examples