information theoretic determination of minimax rates of convergence we present some general results determining minimax bounds on statistical risk for density estimation based on certain information theoretic considerations these bounds depend only on metric entropy conditions and are used to identify the minimax rates of convergence

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

on the rationality of knorr s virtually irreducible lattices abstract it was established by reinhard knorr that the half of brauer s height zero conjecture which assumes an abelian defect group would follow from the nonexistence of certain integral representations of abelian p groups we establish nonexistence of such representations of rank less than 14 for the elementary abelian group of order 8 and believe these to be the first results of this kind the result is made possible by imposing a simple rationality condition

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

orthogonal trajectories on riemannian manifolds and applications to plane wave type spacetimes abstract we present a result on trajectories of a lagrangian system joining two given submanifolds of a riemannian manifold under the action of an unbounded potential as an application we consider geodesics in a class of semi riemannian manifolds the plane wave type spacetimes

asymptotic equivalence for nonparametric generalized linear models we establish that a non gaussian nonparametric regression model is asymptotically equivalent to a regression model with gaussian noise the approximation is in the sense of le cam s deficiency distance the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real indexed exponential family the canonical parameter is a value f ti of a regression function f at a grid point ti nonparametric glm when f is in a holder ball with exponent we establish global asymptotic equivalence to observations of a signal f t in gaussian white noise where is related to a variance stabilizing transformation in the exponential family the result is a regression analog of the recently established gaussian approximation for the i i d model the proof is based on a functional version of the hungarian construction for the partial sum process

asymptotic equivalence of spectral density estimation and gaussian white noise we consider the statistical experiment given by a sample y 1 y n of a stationary gaussian process with an unknown smooth spectral density f asymptotic equivalence in the sense of le cam s deficiency distance to two gaussian experiments with simpler structure is established the first one is given by independent zero mean gaussians with variance approximately f i where i is a uniform grid of points in nonparametric gaussian scale regression this approximation is closely related to well known asymptotic independence results for the periodogram and corresponding inference methods the second asymptotic equivalence is to a gaussian white noise model where the drift function is the log spectral density this represents the step from a gaussian scale model to a location model and also has a counterpart in established inference methods that is log periodogram regression the problem of simple explicit equivalence maps markov kernels allowing to directly carry over inference appears in this context but is not solved here

an existence theory for relativistic brachistochrones in stationary space times we use a variational principle proven by giannoni et al j math phys 38 6367 6381 1997 for relativistic brachistochrones and techniques of global analysis to develop an existence theory for brachistochrones in stationary space times we also develop a ljusternik schnirelman theory for brachistochrones obtaining multiplicity results depending on the topology of the space time

equivalence theory for density estimation poisson processes and gaussian white noise with drift this paper establishes the global asymptotic equivalence between a poisson process with variable intensity and white noise with drift under sharp smoothness conditions on the unknown function this equivalence is also extended to density estimation models by poissonization the asymptotic equivalences are established by constructing explicit equivalence mappings the impact of such asymptotic equivalence results is that an investigation in one of these nonparametric models automatically yields asymptotically analogous results in the other models

trajectories in godel type space times under the action of a vector field abstract in this paper we consider a godel type lorentz manifold m and look for curves 0 1 m joining two fixed points in m or having prescribed period y t satisfying the system d s s 1 2 curl a s s we find timelike trajectories of these problems

asymptotic nonequivalence of nonparametric experiments when the smoothness index is an example is provided to show that the natural asymptotic equivalence does not hold between any pairs of three nonparametric experiments density problem white noise with drift and nonparametric regression when the smoothness index of the unknown nonparametric function class is 1 2

knorr lattices their characters and some subsets of fpd examples of positive height cyclic knorr lattices for elementary abelian p groups are constructed and bounds relating the height of such lattices to the size of the group are given the problem is formulated in terms of finite geometry when p 2 certain examples are shown to arise from difference sets

schur indices and abelian defect groups abstract we show that the irreducible characters lying in 2 blocks of finite solvable groups with abelian defect groups have trivial 2 local schur indices

on the energy functional on finsler manifolds and applications to stationary spacetimes in this paper we first study some global properties of the energy functional on a non reversible finsler manifold in particular we present a fully detailed proof of the palais smale condition under the completeness of the finsler metric moreover we define a finsler metric of randers type which we call fermat metric associated to a conformally standard stationary spacetime we shall study the influence of the fermat metric on the causal properties of the spacetime mainly the global hyperbolicity moreover we study the relations between the energy functional of the fermat metric and the fermat principle for the light rays in the spacetime this allows one to obtain existence and multiplicity results for light rays using the finsler theory finally the case of timelike geodesics with fixed energy is considered

lightlike rays in stationary spacetimes with critical growth abstract the aim of this paper is to extend some previous results on the existence of lightlike geodesics joining a point to a line to the case of stationary lorentzian manifolds whose metric coefficients have an optimal growth

on p convex sets and geodesics on etudie les geodesiques sur une variete contrainte m a bord irregulier on considere une classe d ensembles irreguliers m qui permet d obtenir des resultats d existence et de multiplicite pour les geodesiques

on a class of singular second order hamiltonian systems with infinitely many homoclinic solutions abstract we show existence of infinitely many homoclinic orbits at the origin for a class of singular second order hamiltonian systems u v u t u 0 t we use variational methods under the assumption that v t u satisfies the so called strong force condition

periodic solutions for dynamical systems on non complete riemannian manifolds abstract we look for periodic solutions in a dynamical system on a non complete riemannian manifold if the potential is bounded and suitable convexity assumptions hold the existence of infinitely many solutions can be proved by means of variational methods penalization arguments and classical ljusternik schnirelman theory

a complement to le cam s theorem this paper examines asymptotic equivalence in the sense of le cam between density estimation experiments and the accompanying poisson experiments the significance of asymptotic equivalence is that all asymptotically optimal statistical procedures can be carried over from one experiment to the other the equivalence given here is established under a weak assumption on the parameter space f in particular a sharp besov smoothness condition is given on f which is sufficient for poissonization namely if f is in a besov ball b p g m with ap 1 2 examples show poissonization is not possible whenever ap 0

adaptive bernstein von mises theorems in gaussian white noise we investigate bernstein von mises theorems for adaptive nonparametric bayesian procedures in the canonical gaussian white noise model we consider both a hilbert space and multiscale setting with applications in l 2 and l infty respectively this provides a theoretical justification for plug in procedures for example the use of certain credible sets for sufficiently smooth linear functionals we use this general approach to construct optimal frequentist confidence sets based on the posterior distribution we also provide simulations to numerically illustrate our approach and obtain a visual representation of the geometries involved

some cyclic virtually irreducible lattices abstract virtually irreducible lattices which satisfy a simple rationality condition and have positive height are constructed for elementary abelian 2 groups the non existence of such lattices would have implied part of brauer s height zero conjecture a characterization of rank 6 lattices for the elementary abelian group of size 16 is given

multiple critical points for indefinite functionals and applications abstract in this paper we prove a multiplicity result for critical points of an indefinite functional on a manifold by using a generalization of the ljusternik schnirelman category the abstract result is applied to prove the existence of multiple timelike trajectories for lorentzian manifolds of splitting type

on the geodesical connectedness for a class of semi riemannian manifolds we prove a variational principle for geodesics on a semi riemannian manifold m of arbitrary index k and possessing k linearly independent killing vector fields that generate a timelike distribution on m using such a principle and a suitable completeness condition for m we prove some existence and multiplicity results for geodesics joining two fixed points of m

category of loop spaces of open subsets in euclidean space has been studied in several recent papers e g l 2 5 6 91 in situations where the potential function v is singular in the sense that it is defined and c only on an open subset u of euclidean k space rk and not on the whole of rk if e wgt2 r rk is the sobolev space of t periodic functions from r to rk and e u is the subset of e of functions which map into u then the topology of e u plays a crucial role in these studies in particular the knowledge that the lusternik schnirelmann category of e u is infinite when used in conjunction with min max methods implies the existence of infinitely many critical points another subset e u which consists of those functions in e u which take the origin to a base point in u plays a dual role first of all in problems concerning the existence of geodesics in u from one subset a of u to another b of u see e g 3 the category of e u plays a role secondly the general inequality cat e u 2 cat e u see 6 when u is connected and simply connected can be applied to show that cat e u cat e u thus providing a tool for computing cat e u the topological analogue of e u is the space of free loops au c s u of continuous maps from the circle s to u and the analogue of e u is the space of based loops ru which consists of maps in au which take a base point in s to a base point in u au has the same homotopy type as e u and s2u has the same homotopy type as e u thus homotopy type invariants e g category of e u and e u may be studied through au and szu respectively the following classical theorem of serre l l is useful in this regard and is stated in terms of a general space x

normal geodesics in static spacetimes with critical asymptotic behavior abstract this paper deals with the existence of normal geodesics joining two given submanifolds in a static spacetime when the coefficient of its metric has a quadratic growth a suitable variational approach allows one to use the classical ljusternik schnirelman theory

geodesics in stationary spacetimes and classical lagrangian systems abstract we state a fundamental correspondence between geodesics on stationary spacetimes and the equations of classical particles on riemannian manifolds accelerated by a potential and a magnetic field by variational methods we prove some existence and multiplicity theorems for fixed energy solutions joining two points or periodic of the above described riemannian equation as a consequence we obtain existence and multiplicity results for geodesics with fixed energy connecting a point to a line or periodic trajectories in standard stationary spacetimes

conditional extremals in complete riemannian manifolds abstract conditional extremal curves in a complete riemannian manifold m are defined as the critical points of the squared l 2 distance between the tangent vector field of a curve and a so called prior vector field we prove that this l 2 distance satisfies the palais smale condition on the space of absolutely continuous curves joining two submanifolds of m and thus establish the existence of critical points we also prove a morse index theorem in the case where the two submanifolds are single points and use the morse inequalities to place lower bounds on the number of critical points of each index

the statistical information contained in additional observations maximum diminution of the risk functions we show that under general dimensionality conditions 4 4 dfr 1 is of order r n further the behavior of a is studied and compared for asymptotically gaussian experiments we show that the information gain increases logarithmically the gaussian and the binomial family tum out to be in some sense opposite extreme cases with the increase of information asymptotically minimal in the gaussian case and maximal in the binomial 1 introduction when considering a complicated statistical model it may be useful to construct another model which is close to the original one but statistically easier to handle the analysis of the second model may make the essential structure of the first model better understandable and help to construct suitable statistical procedures for a decision problem the usual way to get such an approximating model is to imbed the original one into a sequence of models and to expand the log likelihood function because one is more interested in approximations than in limit theorems it is necessary to estimate the closeness of the two models a natural quantity for comparing two models or in more common use of language two experiments is the deficiency distance of le cam 1964 it is based on the comparison of risk functions available in the two experiments we recall its definition

the root unroot algorithm for density estimation as implemented via wavelet block thresholding we propose and implement a density estimation procedure which begins by turning density estimation into a nonparametric regression problem this regression problem is created by binning the original observations into many small size bins and by then applying a suitable form of root transformation to the binned data counts in principle many common nonparametric regression estimators could then be applied to the transformed data we propose use of a wavelet block thresholding estimator in this paper finally the estimated regression function is un rooted by squaring and normalizing the density estimation procedure achieves simultaneously three objectives computational efficiency adaptivity and spatial adaptivity a numerical example and a practical data example are discussed to illustrate and explain the use of this procedure theoretically it is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the besov classes the estimator also automatically adapts to the local smoothness of the underlying function and attains the local adaptive minimax rate for estimating functions at a point there are three key steps in the technical argument poissonization quantile coupling and oracle risk bound for block thresholding in the non gaussian setting some of the technical results may be of independent interest

existence of variationally defined curves with higher order elliptic lagrangians abstract we present a method for proving the existence of solutions to a class of one dimensional variational problems namely those where the lagrangian is strongly elliptic the method is demonstrated by some examples of optimal interpolation problems which are motivated by applications to the mechanics and control of rigid bodies in each case the key step is to show that the variational problem satisfies the palais smale condition we do so in a general setting showing that the number of initial conditions required depends on the higher order energy bounding properties of the lagrangian