twistor spaces of hypercomplex manifolds are balanced abstract a hypercomplex structure on a differentiable manifold consists of three integrable almost complex structures that satisfy quaternionic relations if in addition there exists a metric on the manifold which is hermitian with respect to the three structures and such that the corresponding hermitian forms are closed the manifold is said to be hyperkahler in the paper non hermitian yang mills connections 13 kaledin and verbitsky proved that the twistor space of a hyperkahler manifold admits a balanced metric these were first studied in the article on the existence of special metrics in complex geometry 17 by michelsohn in the present article we review the proof of this result and then generalize it and show that twistor spaces of general compact hypercomplex manifolds are balanced

local linear regression smoothers and their minimax efficiencies in this paper we introduce a smooth version of local linear regression estimators and address their advantages the mse and mise of the estimators are computed explicitly it turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency they are not only efficient in rates but also nearly efficient in constant factors in the nonparametric regression context the asymptotic minimax lower bound is developed via the heuristic of the hardest onedimensional subproblem of donoho and liu connections of the minimax risk with the modulus of continuity are made the lower bound is also applicable for estimating conditional mean regression and conditional quantiles for both fixed and random design regression problems

kahlerian twistor spaces the work of r penrose has shown the deep relationship between the conformal structure of minkowski space and the complex geometry of lines in projective three space this transformation applies to a more general class of conformal structures and in particular there is a riemannian version described in 5 which is itself of interest to mathematical physicists basically to each oriented riemannian 4manifold x one associates canonically a 6 manifold z the twistor space of x together with an almost complex structure if this structure is integrable then z becomes a complex 3 manifold with a distinguished family of projective lines on it whose geometry describes the conformal geometry of the manifold x it is natural to ask what sort of complex manifold one obtains by this process in this paper we ask the more specific question which compact 4c manifolds have ka hlerian twistor spaces 1 the answer rather surprisingly is that there are just two examples the 4 sphere s and the complex projective plane p2 c each with their standard homogeneous conformal structures the twistor space of s 4 is the 3 dimensional projective space p3 c and that of p 2 q is the manifold f2 c of flags in c 3 the essential point of the proof of this result is the remark that if z is a twistor space with a kahler metric then the canonical bundle k is negative using vanishing theorems and the riemann roch theorem we then narrow down the possibilities for z to four cases projective space the flag manifold the intersection of two quadrics in p5 and the double covering of p3 branched over a non singular quartic surface the last two candidates cannot be realized however because the value of their euler characteristic is incompatible with their being twistor spaces throughout the proof we make essential use of a real structure on z which reduces considerably the occurrence of singularities at various stages aside from this our methods are the standard ones of algebraic geometry the four examples to which the proof leads us are all fano threefolds if we drop the requirement of compactness on x then it is quite possible for an open set of a fano threefold to be a twistor space in particular we show that the family of conies lying on the intersection of two

common structure of techniques for choosing smoothing parameters in regression problems on tente de relier deux methodes de choix du parametre de lissage l une basee sur la mesure du risque et l autre sur le degre d ajustement de la regression aux donnees

on the spectral radius of weighted trees with fixed diameter and weight set abstract the spectrum of weighted graphs are often used to solve the problems in the design of networks and electronic circuits we first give some perturbational results on the spectral radius of weighted graphs when some weights of edges are modified then we derive the weighted tree with the largest spectral radius in the set of all weighted trees with fixed diameter and weight set furthermore an open problem of spectral radius on weighted paths is solved h z yang g z hu y hong bounds of spectral radii of weighted tree tsinghua science and technology 8 2003 517 520

the estimation of residual variance in nonparametric regression summary a wide class of estimators of the residual variance in nonparametric regression is considered namely those that are quadratic in the data unbiased for linear regression and always nonnegative the minimax mean squared error estimator over a natural class of regression functions is derived this optimal estimator has an interesting structure and is closely related to a minimax estimator of the regression curve itself

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

on the efficiency of selection criteria in spline regression this paper concerns the cubic smoothing spline approach to nonparametric regres sion after first deriving sharp asymptotic formulas for the eigenvalues of the smoothing ma trix the paper uses these formulas to investigate the efficiency of different selection criteria for choosing the smoothing parameter special attention is paid to the generalized maximum likelihood gml cp and extended exponential ee criteria and their marginal bayesian interpretation it is shown that a when the bayesian model that motivates gml is true using cp to estimate the smoothing parameter would result in a loss of efficiency with a factor of 10 3 proving and strengthening a conjecture proposed in stein 1990 b when the data indeed come from the cp density using gml would result in a loss of efficiency of c the loss of efficiency of the ee criterion is at most 1 543 when the data are sampled from its consistent density family the paper not only studies equally spaced observations the setting of stein 1990 but also investigates general sampling scheme of the design points and shows that the efficiency results remain the same in both cases

on nonparametric regression for iid observations in a general setting we consider the problem of sharp optimal estimation of a response function f x in a random design nonparametric regression under a general model where a pair of observations y x has a joint density p y x p y f x x we wish to estimate the response function with optimal minimax mean integrated squared error convergence as the sample size tends to traditional regularity assumptions on the conditional density p y assumed for parameter estimation are sufficient for sharp optimal nonparametric risk convergence as well as for the existence of the best constant and rate of risk convergence this best constant is a nonparametric analog of fisher information many examples are sketched including location and scale families censored data mixture models and some well known applied examples a sequential approach and some aspects of experimental design are considered as well

efficient estimation of a density in a problem of tomography the aim of tomography is to reconstruct a multidimensional function from observations of its integrals over hyperplanes we consider the model that corresponds to the case of positron emission tomography we have n i i d observations from a probability density proportional to rf where rf stands for the radon transform of the density f we assume that f is an n dimensional density such that its fourier transform is exponentially decreasing we find an estimator of f which is asymptotically efficient it achieves the optimal rate of convergence and also the best constant for the minimax risk

on the laplacian spectral radius of weighted trees with fixed matching number q and a positive weight set abstract denote by t n q w 1 w 2 w n 1 the set of n vertex weighted trees with matching number q and fixed positive weight set w n 1 w 1 w 2 w n 1 where w 1 w 2 w n 1 0 tan s w tan on the sharp upper bound of spectral radius of weighted trees j math res exposition 29 2009 293 301 determined the weighted tree in t n q w 1 w 2 w n 1 with the largest adjacent spectral radius whereas in s w tan on the laplacian spectral radius of weighted trees with a positive weight set discrete math 310 2010 1026 1036 tan determined the weighted tree in t n q w 1 w 2 w n 1 with the largest laplacian spectral radius in this paper we use a unified approach to identify the unique weighted tree in t n q w 1 w 2 w n 1 with the largest adjacent spectral radius and largest laplacian spectral radius respectively

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

adaptively local one dimensional subproblems with application to a deconvolution problem in this paper a method for finding global minimax lower bounds is introduced the idea is to adjust automatically the direction of a local one dimensional subproblem at each location to the nearly hardest one and to use locally the difficulty of the one dimensional subproblem this method has the advantages of being easily implemented and understood the lower bound is then applied to nonparametric deconvolution to obtain the optimal rates of convergence for estimating a whole function other applications are also addressed

parabolic higgs bundles and representations of the fundamental group of a punctured surface into a real group abstract we study parabolic g higgs bundles over a compact riemann surface with fixed punctures when g is a real reductive lie group and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in g with arbitrary holonomy around the punctures this generalizes simpson s results for gl n c to arbitrary complex and real reductive lie groups three interesting features are the relation between the parabolic degree and the tits geometry of the boundary at infinity of the symmetric space the treatment of the case when the logarithm of the monodromy is on the boundary of a weyl alcove and the correspondence of the orbits encoding the singularity via the kostant sekiguchi correspondence we also describe some special features of the moduli spaces when g is a split real form or a group of hermitian type

central extensions of current groups in two dimensions the paper is devoted to generalization of the theory of loop groups to the two dimensional case to every complex riemann surface we assign a central extension of the group of smooth maps from this surface to a simple complex lie group g by the jacobian of this surface this extension is topologically nontrivial as in the loop group case orbits of coadjoint representation of this extension correspond to equivalence classes of holomorphic principalg bundles over the surface when the surface is the torus elliptic curve classification of coadjoint orbits is related to linear difference equations in one variable and to classification of conjugacy classes in the loop group we study integral orbits for which the kirillov kostant form is a curvature form for some principal torus bundle the number of such orbits for a given level is finite as in the loop group case conjecturedly they correspond to analogues of integrable modules occurring in conformal field theory

semiparametric estimation of shifts on compact lie groups for image registration in this paper we focus on estimating the deformations that may exist between similar images in the presence of additive noise when a reference template is unknown the deformations are modeled as parameters lying in a finite dimensional compact lie group a general matching criterion based on the fourier transform and its well known shift property on compact lie groups is introduced m estimation and semiparametric theory are then used to study the consistency and asymptotic normality of the resulting estimators as lie groups are typically nonlinear spaces our tools rely on statistical estimation for parameters lying in a manifold and take into account the geometrical aspects of the problem some simulations are used to illustrate the usefulness of our approach and applications to various areas in image processing are discussed

the limit of the hermitian yang mills flow on reflexive sheaves abstract in this paper we study the asymptotic behavior of the hermitian yang mills flow on a reflexive sheaf we prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the harder narasimhan seshadri filtration this answers a question by bando and siu

statistical inverse estimation in hilbert scales the recovery of signals from indirect measurements blurred by random noise is considered under the assumption that prior knowledge regarding the smoothness of the signal is avialable for greater flexibility the general problem is embedded in an abstract hilbert scale in the applications sobolev scales are used for the construction of estimators we employ preconditioning along with regularized operator inversion in the appropriate inner product where the operator is bounded but not necessarily compact a lower bound to certain minimax rates is included and it is shown that in generic examples the proposed estimators attain the asymptotic minimax rate examples include errors in variables deconvolution and indirect nonparametric regression special instances of the latter are estimation of the source term in a differential equation and the estimation of the initial state in the heat equation

log density estimation in linear inverse problems we estimate a probability density function p which is related by a linear operator k to a density function q in sequences of regular exponential families based on a random sample from q in this paper deconvolution and positron emission tomography are considered the logarithm of the density function is approximated by basis functions consisting of singular functions of k while direct maximum likelihood or minimum kullback leibler density estimation in exponential families selects the parameters to match the moments of the basis functions to the sample moments in the inverse problem the moment of each singular function is related to a corresponding moment of the direct problem by a factor given by a singular value a of k thus an appropriate analogue of the maximum likelihood estimate is obtained by matching moments with respect to p to 1 a times the empirical moments associated with the sample from q bounds on the kullback leibler distance between the true density and the estimators are obtained and rates of convergence are established for log density functions having a measure of smoothness the density estimator converges to the unknown density in the kullback leibler sense and in the l2 sense at a rate determined not only by the order of smoothness of the log density and the dimension of data but also by the decay rate of the singular values of the operator a minimax lower bound for deconvolution is provided under certain conditions numerical examples using simulated data are provided to illustrate the finite sample performance of the proposed method for deconvolution and positron emission tomography

asymptotically optimal tests for conditional distributions let xl y1 xn yn be independent replicates of the random vector x y e rd m where x is rd valued and y is rm valued we assume that the conditional distribution p y e ix x qii of y given x x is a member of a parametrie family where the parameter space 0 is an open subset of rh with 0 e 0 under suitable regularity conditions we establish upper bounds for the power functions of asymptotic level a tests for the problem 0 against a sequence of contiguous alternatives as weil as asymptotically optimal tests which attain these bounds since the testing problem involves the joint density of x y as an infinite dimensional nuisance parameter its solution is not standard a monte carlo simulation exemplifies the influence of this nuisance parameter as a main tool we establish local asymptotic normality lan of certain poisson point processes which approximately describe our initial sampie

a kahler structure on the moduli space of isometric maps of a circle into euclidean space we study the spaces and and lip of smooth resp non degenerate lipschitz isometric maps of a circle into euclidean space modulo orientation preserving euclidean motions we prove that and lip are infinite dimensional kahler manifolds in particular they are complex frechet resp banach manifolds this is proved by an infinite dimensional version of the kirwan kempf ness theorem kir84 kn78 nes84 relating symplectic quotients to holomorphic quotients applied to the action ofpsl2 on the free loop space ofs2

minimax risk over hyperrectangles and implications consider estimating the mean of a standard gaussian shift when that mean is known to lie in an orthosymmetric quadratically convex set in l 2 such sets include ellipsoids hyperrectangles and l p bodies with p 2 the minimax risk among linear estimates is within 25 of the minimax risk among all estimates the minimax risk among truncated series estimates is within a factor 4 44 of the minimax risk this implies that the difficulty of estimation a statistical quantity is measured fairly precisely by the n widths a geometric quantity if the set is not quadratically convex as in the case of l p bodies with p 2 things change appreciably minimax linear estimators may be out performed arbitrarily by nonlinear estimates the ordinary kolmogorov n widths still determine the difficulty of linear estimation but the difficulty of nonlinear estimation is tied to the inner bernstein n widths which can be far smaller essential use is made of a new heuristic that the difficulty of the hardest rectangular subproblem is equal to the difficulty of the full problem

regularized policy iteration with nonparametric function spaces we study two regularization based approximate policy iteration algorithms namely reg lspi and reg brm to solve reinforcement learning and planning problems in discounted markov decision processes with large state and finite action spaces the core of these algorithms are the regularized extensions of the least squares temporal difference lstd learning and bellman residual minimization brm which are used in the algorithms policy evaluation steps regularization provides a convenient way to control the complexity of the function space to which the estimated value function belongs and as a result enables us to work with rich nonparametric function spaces we derive efficient implementations of our methods when the function space is a reproducing kernel hilbert space we analyze the statistical properties of reg lspi and provide an upper bound on the policy evaluation error and the performance loss of the policy returned by this method our bound shows the dependence of the loss on the number of samples the capacity of the function space and some intrinsic properties of the underlying markov decision process the dependence of the policy evaluation bound on the number of samples is minimax optimal this is the first work that provides such a strong guarantee for a nonparametric approximate policy iteration algorithm

adaptive piecewise polynomial estimation via trend filtering we study trend filtering a recently proposed tool of kim et al siam rev 51 2009 339 360 for nonparametric regression the trend filtering estimate is defined as the minimizer of a penalized least squares criterion in which the penalty term sums the absolute k th order discrete derivatives over the input points perhaps not surprisingly trend filtering estimates appear to have the structure of k th degree spline functions with adaptively chosen knot points we say appear here as trend filtering estimates are not really functions over continuous domains and are only defined over the discrete set of inputs this brings to mind comparisons to other nonparametric regression tools that also produce adaptive splines in particular we compare trend filtering to smoothing splines which penalize the sum of squared derivatives across input points and to locally adaptive regression splines ann statist 25 1997 387 413 which penalize the total variation of the k th derivative empirically we discover that trend filtering estimates adapt to the local level of smoothness much better than smoothing splines and further they exhibit a remarkable similarity to locally adaptive regression splines we also provide theoretical support for these empirical findings most notably we prove that with the right choice of tuning parameter the trend filtering estimate converges to the true underlying function at the minimax rate for functions whose k th derivative is of bounded variation this is done via an asymptotic pairing of trend filtering and locally adaptive regression splines which have already been shown to converge at the minimax rate ann statist 25 1997 387 413 at the core of this argument is a new result tying together the fitted values of two lasso problems that share the same outcome vector but have different predictor matrices

empirical functionals and efficient smoothing parameter selection a striking feature of curve estimation is that the smoothing parameter h 0 which minimizes the squared error of a kernel or smoothing spline estimator is very difficult to estimate this is manifest both in slow rates of convergence and in high variability of standard methods such as cross validation we quantify this difficulty by describing nonparametric information bounds and exhibit asymptotically efficient estimators of h 0 that attain the bounds the efficient estimators are substantially less variable than cross validation and other current procedures and simulations suggest that they may offer improvements at moderate sample sizes at least in terms of minimizing the squared error

a kobayashi hitchin correspondence for i holomorphic bundles abstract in this paper we introduce the notions of hermitian einstein metric and stability for i holomorphic vector bundles on bi hermitian manifolds moreover we establish a kobayashi hitchin correspondence for i holomorphic vector bundles on bi hermitian manifolds examples of such vector bundles include generalized holomorphic bundles on generalized kahler manifolds we also show that stability of a vector bundle in this sense can depend on the parameter

minimum spectral radius of a weighted graph abstract m fiedler recently introduced the following question what is the optimal distribution of nonnegative weights with total sum one among the edges of a given graph so that the spectral radius of the resulting adjacency matrix is minimum he himself has shown that the optimum solution is achieved by some decomposition of the given graph g into a collection of mutually vetex disjoint odd cycles and balanced bipartite subgraphs which maximizes a certain objective function we present a polynomial time algorithm which finds this decomposition our approach is related to matching theory

adaptive nonparametric confidence sets we construct honest confidence regions for a hilbert space valued parameter in various statistical models the confidence sets can be centered at arbitrary adaptive estimators and have diameter which adapts optimally to a given selection of models the latter adaptation is necessarily limited in scope we review the notion of adaptive confidence regions and relate the optimal rates of the diameter of adaptive confidence regions to the minimax rates for testing and estimation applications include the finite normal mean model the white noise model density estimation and regression with random design institute of mathematical statistics 2006

smoothing under diffeomorphic constraints with homeomorphic splines in this paper we introduce a new class of diffeomorphic smoothers based on general spline smoothing techniques and on the use of some tools that have been recently developed in the context of image warping to compute smooth diffeomorphisms this diffeomorphic spline is defined as the solution of an ordinary differential equation governed by an appropriate time dependent vector field this solution has a closed form expression which can be computed using classical unconstrained spline smoothing techniques this method does not require the use of quadratic or linear programming under inequality constraints and has therefore a low computational cost in a one dimensional setting incorporating diffeomorphic constraints is equivalent to imposing monotonicity thus as an illustration it is shown that such a monotone spline can be used to make monotone any unconstrained estimator of a regression function and that this monotone smoother inherits the convergence properties of the unconstrained estimator some numerical experiments are proposed to illustrate its finite sample performances and to compare them with another monotone estimator we also provide a two dimensional application on the computation of diffeomorphisms for landmark and image matching

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

on the weighted trees with given degree sequence and positive weight set abstract we determine the unique weighted tree with the largest spectral radius with respect to the adjacency and laplacian matrix in the set of all weighted trees with a given degree sequence and positive weight set moreover we also derive the weighted trees with the largest spectral radius with respect to the matrices mentioned above in the sets of all weighted trees with a given maximum degree or pendant vertex number and so on

the spectral radius of bicyclic graphs with prescribed degree sequences restricted to the bicyclic graphs with prescribed degree sequences we determine the unique graph with the largest spectral radius with respect to the adjacency matrix

recovering edges in ill posed inverse problems optimality of curvelet frames we consider a model problem of recovering a function f x 1 x 2 from noisy radon data the function f to be recovered is assumed smooth apart from a discontinuity along a c 2 curve that is an edge we use the continuum white noise model with noise level varepsilon traditional linear methods for solving such inverse problems behave poorly in the presence of edges qualitatively the reconstructions are blurred near the edges quantitatively they give in our model mean squared errors mses that tend to zero with noise level varepsilon only as o varepsilon 1 2 as varepsilon to 0 a recent innovation nonlinear shrinkage in the wavelet domain visually improves edge sharpness and improves mse convergence to o varepsilon 2 3 however as we show here this rate is not optimal in fact essentially optimal performance is obtained by deploying the recently introduced tight frames of curvelets in this setting curvelets are smooth highly anisotropic elements ideally suited for detecting and synthesizing curved edges to deploy them in the radon setting we construct a curvelet based biorthogonal decomposition of the radon operator and build curvelet shrinkage estimators based on thresholding of the noisy curvelet coefficients in effect the estimator detects edges at certain locations and orientations in the radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain we prove that the curvelet shrinkage can be tuned so that the estimator will attain within logarithmic factors the mse o varepsilon 4 5 as noise level varepsilon to 0 this rate of convergence holds uniformly over a class of functions which are c 2 except for discontinuities along c 2 curves and except for log terms is the minimax rate for that class our approach is an instance of a general strategy which should apply in other inverse problems we sketch a deconvolution example

a construction of non kahler calabi yau manifolds and new solutions to the strominger system abstract we propose a new construction of compact non kahler calabi yau manifolds with balanced metrics and study the strominger system on them in particular we obtain explicit solutions to the strominger system with degeneracies on g t 4 where g is an immersed minimal surface of genus g 3 in flat t 3

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions pointwise error global error measured in l p norms pointwise and global error in estimation of derivatives and for a wide range of smoothness classes including standard holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves finally the theory underlying the method is interesting as it exploits a correspondence between statistical questions and questions of optimal recovery and information based complexity

kernel smoothing in partial linear models on considere deux methodes d estimation l une reliee aux splines de lissage partiels l autre motivee par une analyse de residus partielle

combinatorial approach for computing the characteristic polynomial of a matrix the study of the spectrum of weighted di graphs has attracted recently much attention in the literature here we use the coates digraph as a main tool to extend in a combinatorial way some well known results from the spectral graph theory on computing the characteristic polynomial of graphs new results are related to weighted di graph and thus to any square matrix

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

asymptotic minimax risk for sup norm loss solution via optimal recovery we study the problem of estimating an unknown function on the unit interval or itsk th derivative with supremum norm loss when the function is observed in gaussian white noise and the unknown function is known only to obey lipschitz smoothness k 0 we discuss an optimization problem associated with the theory ofoptimal recovery although optimal recovery is concerned with deterministic noise chosen by a clever opponent the solution of this problem furnishes the kernel of the minimax linear estimate for gaussian white noise moreover this minimax linear estimator is asymptotically minimax among all estimates we sketch also applications to higher dimensions and to indirect measurement e g deconvolution problems

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

some results on the index of unicyclic graphs we identify in some classes of unicyclic graphs of fixed order and girth those graphs whose index i e the largest eigenvalue is maximal besides some lower and upper bounds on the indices of the graphs being considered are provided

geometrizing rates of convergence ii consider estimating a functional t f of an unknown distribution f in mathbf f from data x 1 cdots x n i i d f let omega varepsilon denote the modulus of continuity of the functional t over mathbf f computed with respect to hellinger distance for well behaved loss functions l t we show that inf t n sup mathbf f e fl t n t f is equivalent to l omega n 1 2 to within constants whenever t is linear and mathbf f is convex the same conclusion holds in three nonlinear cases estimating the rate of decay of a density estimating the mode and robust nonparametric regression we study the difficulty of testing between the composite infinite dimensional hypotheses h 0 t f leq t and h 1 t f geq t delta our results hold in the cases studied because the difficulty of the full infinite dimensional composite testing problem is comparable to the difficulty of the hardest simple two point testing subproblem

on the estimation of quadratic functionals we discuss the difficulties of estimating quadratic functionals based on observations y t from the white noise model y t int t 0 f u du sigma w t quad t in lbrack 0 1 rbrack where w t is a standard wiener process on lbrack 0 1 rbrack the optimal rates of convergence as sigma rightarrow 0 for estimating quadratic functionals under certain geometric constraints are found specifically the optimal rates of estimating int 1 0 lbrack f k x rbrack 2 dx under hyperrectangular constraints sum f x j f leq cj alpha and weighted l p body constraints sum p f sum infty 1 j r x j f p leq c are computed explicitly where x j f is the j th fourier bessel coefficient of the unknown function f we develop lower bounds based on testing two highly composite hypercubes and address their advantages the attainable lower bounds are found by applying the hardest one dimensional approach as well as the hypercube method we demonstrate that for estimating regular quadratic functionals i e the functionals which can be estimated at rate o sigma 2 the difficulties of the estimation are captured by the hardest one dimensional subproblems and for estimating nonregular quadratic functionals i e no o sigma 2 consistent estimator exists the difficulties are captured at certain finite dimensional the dimension goes to infinity as sigma rightarrow 0 hypercube subproblems

density and hazard rate estimation for right censored data by using wavelet methods this paper describes a wavelet method for the estimation of density and hazard rate functions from randomly right censored data we adopt a nonparametric approach in assuming that the density and hazard rate have no specific parametric form the method is based on dividing the time axis into a dyadic number of intervals and then counting the number of events within each interval the number of events and the survival function of the observations are then separately smoothed over time via linear wavelet smoothers and then the hazard rate function estimators are obtained by taking the ratio we prove that the estimators have pointwise and global mean square consistency obtain the best possible asymptotic mean integrated squared error convergence rate and are also asymptotically normally distributed we also describe simulation experiments that show that these estimators are reasonably reliable in practice the method is illustrated with two real examples the first uses survival time data for patients with liver metastases from a colorectal primary tumour without other distant metastases the second is concerned with times of unemployment for women and the wavelet estimate through its flexibility provides a new and interesting interpretation