sufficient dimension reduction in regressions across heterogeneous subpopulations chiaromonte and co workers extended this method to regressions with qualitative predictors and developed a method partial sliced inverse regression under the assumption that the covariance matrices of the continuous predictors are constant across the levels of the qualitative predictor we extend partial sliced inverse regression by removing the restrictive homogeneous covariance condition this extension which significantly expands the applicability of the previous methodology is based on a new estimation method that makes use of a non linear least squares objective function

resolution of singularities of threefolds in positive characteristic ii abstract together with vincent cossart olivier piltant resolution of singularities of threefolds in positive characteristic i reduction to local uniformization on artin schreier and purely inseparable coverings j algebra 320 3 2008 1051 1082 this article gives a complete proof of desingularization of quasiprojective varieties of dimensional 3 on fields which are differentially finite over perfect fields

inseparable local uniformization abstract it is known since the works of zariski in the early 40s that desingularization of varieties along valuations called local uniformization of valuations can be considered as the local part of the desingularization problem it is still an open problem if local uniformization exists in positive characteristic and dimension larger than three in this paper we prove that zariski local uniformization of algebraic varieties is always possible after a purely inseparable extension of the field of rational functions and therefore any valuation can be uniformized by a purely inseparable alteration

slope inequality for families of curves over surfaces in this paper we investigate the general notion of the slope for families of curves f x rightarrow y the main result is an answer to the above question when dim y 2 and we prove a lower bound for this new slope in this case over fields of any characteristic both the notion and the slope inequality are compatible with the theory for dim y 0 1 in a very natural way and this gives a strong evidence that the slope for an n fold fibration of curves f x rightarrow y may be k x y n mathrm ch n 1 f omega x y rather than the usual stability methods the whole proof of the slope inequality here is based on a completely new method using characteristic p 0 geometry a simpler version of this method yields a new proof of the slope inequality when dim y 1

fuzzy model validation using the local statistical approach the local statistical approach for fault detection and isolation is applied to fuzzy models validation the method detects the inconsistencies between a fuzzy rule base and the modelled system it can also identify which are the faulty parameters of the fuzzy model the fisher information matrix explains the detectability of changes in the parameters of the fuzzy model simulation tests illustrate the method s credibility

generalized partially linear single index models abstract the typical generalized linear model for a regression of a response y on predictors x z has conditional mean function based on a linear combination of x z we generalize these models to have a nonparametric component replacing the linear combination t 0x t 0z by 0 t 0x t 0z where 0 is an unknown function we call these generalized partially linear single index models gplsim the models include the single index models which have 0 0 using local linear methods we propose estimates of the unknown parameters 0 0 and the unknown function 0 and obtain their asymptotic distributions examples illustrate the models and the proposed estimation methodology

adaptive estimation over anisotropic functional classes via oracle approach we address the problem of adaptive minimax estimation in white gaussian noise model under lp loss 1 p 1 on the anisotropic nikolskii classes we present the es timation procedure based on a new data driven selection scheme from the family of kernel estimators with varying bandwidths for proposed estimator we establish so called lp norm oracle inequality and use it for deriving minimax adaptive results we prove the existence of rate adaptive estimators and fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm indexes in definitions of the functional class and of the risk in particular some new asymptotics of the minimax risk are discovered including necessary and sufficient conditions for existencea uniformly consistent estimator we provide also with detailed overview of existing methods and results and formulate open problems in adaptive minimax estimation

contour regression a general approach to dimension reduction we propose a novel approach to sufficient dimension reduction in regression based on estimating contour directions of small variation in the response these directions span the orthogonal complement of the minimal space relevant for the regression and can be extracted according to two measures of variation in the response leading to simple and general contour regression scr and gcr methodology in comparison with existing sufficient dimension reduction techniques this contour based methodology guarantees exhaustive estimation of the central subspace under ellipticity of the predictor distribution and mild additional assumptions while maintaining sqrt n consistency and computational ease moreover it proves robust to departures from ellipticity we establish population properties for both scr and gcr and asymptotic properties for scr simulations to compare performance with that of standard techniques such as ordinary least squares sliced inverse regression principal hessian directions and sliced average variance estimation confirm the advantages anticipated by the theoretical analyses we demonstrate the use of contour based methods on a data set concerning soil evaporation

exploring regression structure using nonparametric functional estimation abstract average derivative functionals of regression are proposed for nonparametric model selection and diagnostics the functionals are of the integral type which under certain conditions allows their estimation at the usual parametric rate of n 1 2 we analyze asymptotic properties of the estimators of these functionals based on kernel regression these estimators can then be used for assessing the validity of various restrictions imposed on the form of regression in particular we show how they could be used to reduce the dimensionality of the model assess the relative importance of predictors measure the extent of nonlinearity and nonadditivity and under certain conditions help identify projection directions in projection pursuit models and decide on the number of these directions

svm decision boundary based discriminative subspace induction we study the problem of linear dimension reduction for classification with a focus on sufficient dimension reduction i e finding subspaces without loss of discrimination power first we formulate the concept of sufficient subspace for classification in parallel terms as for regression then we present a new method to estimate the smallest sufficient subspace based on an improvement of decision boundary analysis dba the main idea is to combine dba with support vector machines svm to overcome the inherent difficulty of dba in small sample size situations while keeping dba s estimation simplicity the compact representation of svm boundary results in a significant gain in both speed and accuracy over previous dba implementations alternatively this technique can be viewed as a way to reduce the run time complexity of svm itself comparative experiments on one simulated and four real world benchmark datasets highlight the superior performance of the proposed approach

geometric proofs of theorems of ax kochen and ersov we give an algebraic geometric proof of the theorem of ax and kochen on p adic diophantine equations in many variables unlike ax kochen s proof ours does not use any notions from mathematical logic and is based on weak toroidalization of morphisms we also show how this geometric approach yields new proofs of the ax kochen er v s ov transfer principle for local fields and of quantifier elimination theorems of basarab and pas

structural adaptation via mathbb l _p norm oracle inequalities in this paper we study the problem of adaptive estimation of a multivariate function satisfying some structural assumption we propose a novel estimation procedure that adapts simultaneously to unknown structure and smoothness of the underlying function the problem of structural adaptation is stated as the problem of selection from a given collection of estimators we develop a general selection rule and establish for it global oracle inequalities under arbitrary mathbb l p losses these results are applied for adaptive estimation in the additive multi index model

component identification and estimation in nonlinear high dimensional regression models by structural adaptation this article proposes a new method of analysis of a partially linear model whose nonlinear component is completely unknown the target of analysis is identification of the set of regressors that enter in a nonlinear way in the model function and complete estimation of the model including slope coefficients of the linear component and the link function of the nonlinear component the procedure also allows selection of the significant regression variables we also develop a test of linear hypothesis against a partially linear alternative or more generally a test that the nonlinear component is m dimensional for m 0 1 2 the approach proposed in this article is fully adaptive to the unknown model structure and applies under mild conditions on the model the only important assumption is that the dimensionality of nonlinear component is relatively small the theoretical results indicate that the procedure provides a prescribed level of the identification error and estimates the linear component with

ramification of valuations abstract in this paper we develop a very explicit theory of ramification of general valuations in algebraic function fields in characteristic zero and arbitrary dimension we obtain the strongest possible generalization of the classical ramification theory of local dedekind domains we further develop a ramification theory of algebraic functions fields of dimension two in positive characteristic we prove that local monomialization and simultaneous resolution hold under very mild assumptions and give pathological examples

counterexamples to local monomialization in positive characteristic a local monomialization of an algebraic map between varieties along a valuation is a sequence of blowups of nonsingular subvarieties along the valuation in the domain and target of the map such that the resulting map can be expressed by monomial inclusion in a suitable sense in this paper we give examples showing that a local monomialization does not always exist over fields of positive characteristic it was shown in earlier work of the author that a local monomialization always exists over fields of characteristic zero

quasi steady state reduction for compartmental systems abstract we present a method to determine an asymptotic reduction in the sense of tikhonov and fenichel for singularly perturbed compartmental systems in the presence of slow transport it turns out that the reduction can be derived from the individual interaction terms alone we apply the result to spatially discretized reaction diffusion systems and obtain based on the reduced discretized systems a heuristic to reduce reaction diffusion systems in presence of slow diffusion

on a projective resampling method for dimension reduction with multivariate responses consider the dimension reduction problem where both the response and the predictor are vectors existing estimators of this problem take one of the following routes 1 targeting the part of the dimension reduction space that is related to the conditional mean or moments of the response vector 2 pooling the estimates for the marginal dimension reduction spaces and 3 estimating the whole dimension reduction space directly by multivariate slicing however the first two approaches do not fully recover the dimension reduction space and the third is hampered by the fact that the accuracy of estimators based on multivariate slicing drops sharply as the dimension of response increases a phenomenon often called the curse of dimensionality we propose a new method that overcomes both difficulties in that it involves univariate slicing only and it is guaranteed to fully recover the dimension reduction space under reasonable conditions the method will be compared with the existing estimators by simula

a semiparametric approach to canonical analysis summary classical canonical correlation analysis is one of the fundamental tools in statistics to investigate the linear association between two sets of variables we propose a method called semiparametric canonical analysis to generalize canonical correlation analysis to incorporate the important non linear association semiparametric canonical analysis is easy to implement and interpret statistical properties are proved a consistent estimation method is developed selection of significant semiparametric canonical analysis components is discussed simulations suggest that the methods proposed have satisfactory performance in finite samples one environmental data set and one data set in social science are investigated in which non linear canonical associations are observed and interpreted

rodeo sparse greedy nonparametric regression we present a greedy method for simultaneously performing local bandwidth selection and variable selection in nonparametric regression the method starts with a local linear estimator with large bandwidths and incrementally decreases the bandwidth of variables for which the gradient of the estimator with respect to bandwidth is large the method called rodeo regularization of derivative expectation operator conducts a sequence of hypothesis tests to threshold derivatives and is easy to implement under certain assumptions on the regression function and sampling density it is shown that the rodeo applied to local linear smoothing avoids the curse of dimensionality achieving near optimal minimax rates of convergence in the number of relevant variables as if these variables were isolated in advance

uniformly attracting solutions of nonautonomous differential equations abstract understanding the structure of attractors is fundamental in nonautonomous stability and bifurcation theory by means of clarifying theorems and carefully designed examples we highlight the potential complexity of attractors for nonautonomous differential equations that are as close to autonomous equations as possible we introduce and study bounded uniform attractors and repellors for nonautonomous scalar differential equations in particular for asymptotically autonomous polynomial and periodic equations our results suggest that uniformly attracting or repelling solutions are the true analogues of attracting or repelling fixed points of autonomous systems we provide sharp conditions for the autonomous structure to break up and give way to a bewildering diversity of nonautonomous bifurcations

sequential sufficient dimension reduction for large p small n problems summary we propose a new and simple framework for dimension reduction in the large p small n setting the framework decomposes the data into pieces thereby enabling existing approaches for n p to be adapted to n p problems estimating a large covariance matrix which is a very difficult task is avoided we propose two separate paths to implement the framework our paths provide sufficient procedures for identifying informative variables via a sequential approach we illustrate the paths by using sufficient dimension reduction approaches but the paths are very general empirical evidence demonstrates the efficacy of our paths additional simulations and applications are given in an on line supplementary file

an adaptive estimation of dimension reduction space summary searching for an effective dimension reduction space is an important problem in regression especially for high dimensional data we propose an adaptive approach based on semiparametric models which we call the conditional minimum average variance estimation mave method within quite a general setting the mave method has the following advantages most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters than for that of the nonparametric function in contrast a faster consistency rate can be achieved by the mave method even without undersmoothing the nonparametric link function estimator the mave method is applicable to a wide range of models with fewer restrictions on the distribution of the covariates to the extent that even time series can be included because of the faster rate of consistency for the parameter estimators it is possible for us to estimate the dimension of the space consistently the relationship of the mave method with other methods is also investigated in particular a simple outer product gradient estimator is proposed as an initial estimator in addition to theoretical results we demonstrate the efficacy of the mave method for high dimensional data sets through simulation two real data sets are analysed by using the mave approach

a constructive approach to the estimation of dimension reduction directions in this paper we propose two new methods to estimate the dimension reduction directions of the central subspace cs by constructing a regression model such that the directions are all captured in the regression mean compared with the inverse regression estimation methods e g j amer statist assoc 86 1991 328 332 j amer statist assoc 86 1991 316 342 j amer statist assoc 87 1992 1025 1039 the new methods require no strong assumptions on the design of covariates or the functional relation between regressors and the response variable and have better performance than the inverse regression estimation methods for finite samples compared with the direct regression estimation methods e g j amer statist assoc 84 1989 986 995 ann statist 29 2001 1537 1566 j r stat soc ser b stat methodol 64 2002 363 410 which can only estimate the directions of cs in the regression mean the new methods can detect the directions of cs exhaustively consistency of the estimators and the convergence of corresponding algorithms are proved

fourier methods for estimating the central subspace and the central mean subspace in regression in regression with a high dimensional predictor vector it is important to estimate the central and central mean subspaces that preserve sufficient information about the response and the mean response using the fourier transform we have derived the candidate matrices whose column spaces recover the central and central mean subspaces exhaustively under the normality assumption of the predictors explicit estimates of the central and central mean subspaces are derived bootstrap procedures are used for determining dimensionality and choosing tuning parameters simulation results and an application to a real data are reported our methods demonstrate competitive performance compared with sir save and other existing methods the approach proposed in the article provides a novel view on sufficient dimension reduction and may lead to more powerful tools in the future

local monomialization of analytic maps abstract in this paper local monomialization theorems are proven for morphisms of complex and real analytic spaces this gives the generalization of the local monomialization theorem for morphisms of algebraic varieties over a field of characteristic zero proven in 16 and 18 to analytic spaces

examples of smash nilpotent cycles on rationally connected varieties abstract voevodsky has conjectured that numerical and smash equivalence coincide on a smooth projective variety we prove this conjecture holds for uniruled 3 folds 4 folds whose mrcc quotient has dimension 2 and for smooth complete intersections with very small degree

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

conditional moment models with data missing at random summarywe consider a general statistical model defined by moment restrictions when data are missing at random using inverse probability weighting we show that such a model is equivalent to a model for the observed variables only augmented by a moment condition defined by the missingness mechanism our framework covers parametric and semiparametric mean regressions and quantile regressions we allow for missing responses missing covariates and any combination of them the equivalence result sheds new light on various aspects of missing data and provides guidelines for building efficient estimators

sufficient dimension reduction via direct estimation of the gradients of logarithmic conditional densities sufficient dimension reduction sdr is aimed at obtaining the low rank projection matrix in the input space such that information about output data is maximally preserved among various approaches to sdr a promising method is based on the eigendecomposition of the outer product of the gradient of the conditional density of output given input in this letter we propose a novel estimator of the gradient of the logarithmic conditional density that directly fits a linear in parameter model to the true gradient under the squared loss thanks to this simple least squares formulation its solution can be computed efficiently in a closed form then we develop a new sdr method based on the proposed gradient estimator we theoretically prove that the proposed gradient estimator as well as the sdr solution obtained from it achieves the optimal parametric convergence rate finally we experimentally demonstrate that our sdr method compares favorably with existing approaches in both accuracy and computational efficiency on a variety of artificial and benchmark data sets

characteristic polyhedra of singularities without completion in hironaka j math kyoto univ 7 3 251 293 1967 defines characteristic polyhedra of a singularity embedded in spec r where r is a regular local ring unfortunately to build these polyhedra he must pick up coordinates in the completion widehat r in this article we show that in the hypersurface case whenever r is a g ring by using a different algorithm than hironaka s it is not necessary to extend r to compute the polyhedra

slow manifold for a nonlocal stochastic evolutionary system with fast and slow components abstract this work is devoted to investigating invariant manifolds for a fast slow stochastic evolutionary system with nonlocal diffusion we establish the slow reduction via a random slow manifold which captures slow dynamics of the original stochastic fast slow system a simple example is shown to illustrate this slow reduction method

direct estimation of the index coefficient in a single index model single index modeling is widely applied in for example econometric studies as a compromise between too restrictive parametric models and flexible but hardly estimable purely nonparametric models by such modeling the statistical analysis usually focuses on estimating the index coefficients the average derivative estimator ade of the index vector is based on the fact that the average gradient of a single index function f x t is proportional to the index vector unfortunately a straightforward application of this idea meets the so called curse of dimensionality problem if the dimensionality d of the model is larger than 2 however prior information about the vector can be used for improving the quality of gradient estimation by extending the weighting kernel in a direction of small directional derivative the method proposed in this paper consists of such iterative improvements of the original ade the whole procedure requires at most 2 log n iterations and the resulting estimator is n consistent under relatively mild assumptions on the model independently of the dimensionality d

topological decoupling linearization and perturbation on inhomogeneous time scales abstract we derive a linearization theorem in the framework of dynamic equations on time scales this extends a recent result from y xia j cao m han a new analytical method for the linearization of dynamic equation on measure chains j differential equations 235 2007 527 543 in various directions firstly in our setting the linear part need not to be hyperbolic and due to the existence of a center manifold this leads to a generalized global hartman grobman theorem for nonautonomous problems secondly we investigate the behavior of the topological conjugacy under parameter variation these perturbation results are tailor made for future applications in analytical discretization theory i e to study the relationship between odes and numerical schemes applied to them

murthy s conjecture on 0 cycles we show that the levine weibel chow group of 0 cycles mathrm ch d a of a reduced affine algebra a of dimension d ge 2 over an algebraically closed field is torsion free among several applications it implies an affirmative solution to an old conjecture of murthy in classical k theory

resolution of singularities and modular galois theory i shall sketch a brief history of the desingularization problem from riemann thru zariski to hironaka including the part i played in it and the work on galois theory which this led me to and how that caused me to search out many group theory gurus i shall also formulate several conjectures and suggest numerous thesis problems

local independence feature screening for nonparametric and semiparametric models by marginal empirical likelihood we consider an independence feature screening technique for identifying explanatory variables that locally contribute to the response variable in high dimensional regression analysis without requiring a specific parametric form of the underlying data model our approach accommodates a wide spectrum of nonparametric and semiparametric model families to detect the local contributions of explanatory variables our approach constructs empirical likelihood locally in conjunction with marginal nonparametric regressions since our approach actually requires no estimation it is advantageous in scenarios such as the single index models where even specification and identification of a marginal model is an issue by automatically incorporating the level of variation of the nonparametric regression and directly assessing the strength of data evidence supporting local contribution from each explanatory variable our approach provides a unique perspective for solving feature screening problems theoretical analysis shows that our approach can handle data dimensionality growing exponentially with the sample size with extensive theoretical illustrations and numerical examples we show that the local independence screening approach performs promisingly

nonparametric estimation of global functionals and a measure of the explanatory power of covariates in regression in a nonparametric regression setting with multiple random predictor variables we give the asymptotic distributions of estimators of global integral functionals of the regression surface we apply the results to the problem of obtaining reliable estimators for the nonparametric coefficient of determination this coefficient which is also called pearson s correlation ratio gives the fraction of the total variability of a response that can be explained by a given set of covariates it can be used to construct measures of nonlinearity of regression and relative importance of subsets of regressors and to assess the validity of other model restrictions in addition to providing asymptotic results we propose several data based bandwidth selection rules and carry out a monte carlo simulation study of finite sample properties of these rules and associated estimators of explanatory power we also provide two real data examples

supervised principal component analysis visualization classification and regression on subspaces and submanifolds we propose supervised principal component analysis supervised pca a generalization of pca that is uniquely effective for regression and classification problems with high dimensional input data it works by estimating a sequence of principal components that have maximal dependence on the response variable the proposed supervised pca is solvable in closed form and has a dual formulation that significantly reduces the computational complexity of problems in which the number of predictors greatly exceeds the number of observations such as dna microarray experiments furthermore we show how the algorithm can be kernelized which makes it applicable to non linear dimensionality reduction tasks experimental results on various visualization classification and regression problems show significant improvement over other supervised approaches both in accuracy and computational efficiency

gradient based kernel dimension reduction for regression this article proposes a novel approach to linear dimension reduction for regression using nonparametric estimation with positive definite kernels or reproducing kernel hilbert spaces rkhss the purpose of the dimension reduction is to find such directions in the explanatory variables that explain the response sufficiently this is called sufficient dimension reduction the proposed method is based on an estimator for the gradient of the regression function considered for the feature vectors mapped into rkhss it is proved that the method is able to estimate the directions that achieve sufficient dimension reduction in comparison with other existing methods the proposed one has wide applicability without strong assumptions on the distributions or the type of variables and needs only eigendecomposition for estimating the projection matrix the theoretical analysis shows that the estimator is consistent with certain rate under some conditions the experimental results demonstrate that the proposed metho

resolution of subgrid microscale interactions enhances the discretisation of nonautonomous partial differential equations we consider a general method for the macroscale modelling of nonlinear nonautomomous partial differential equations or difference equations our method is able to resolve the complex fluctuations which arise in nonliner nonautonomous systems these fluctuations are not accounted for by popular multiscale averaging procedures and homogenization our method is illustrated with the example of a forced burgers equation coarse grained macroscale spatial discretisations of nonlinear nonautonomous partial differential difference equations are given novel support by centre manifold theory dividing the physical domain into overlapping macroscale elements empowers the approach to resolve significant subgrid microscale structures and interactions between neighbouring elements the crucial aspect of this approach is that centre manifold theory organises the resolution of the detailed subgrid microscale structure interacting via the nonlinear dynamics within and between neighbouring elements the techniques and theory developed here may be applied to soundly discretise on a macroscale many dissipative nonautonomous partial differential difference equations such as the forced burgers equation adopted here as an illustrative example

equimultiplicity algebraic elimination and blowing up abstract given a variety x over a perfect field we study the partition defined on x by the multiplicity into equimultiple points and the effect of blowing up at smooth equimultiple centers over fields of characteristic zero we prove resolution of singularities by using the multiplicity as an invariant instead of the hilbert samuel function

principal hessian directions revisited abstract li has suggested the method of principal hessian directions for estimating plotting directions that capture curvature in regression functions this article revisits li s proposal offering a number of suggestions for improved application

local likelihood and local partial likelihood in hazard regression in survival analysis the relationship between a survival time and a covariate is conveniently modeled with the proportional hazards regression model this model usually assumes that the covariate has a log linear effect on the hazard function in this paper we consider the proportional hazards regression model with a nonparametric risk effect we discuss estimation of the risk function and its derivatives in two cases when the baseline hazard function is parametrized and when it is not parametrized in the case of a parametric baseline hazard function inference is based on a local version of the likelihood function while in the case of a nonparametric baseline hazard we use a local version of the partial likelihood this results in maximum local likelihood estimators and maximum local partial likelihood estimators respectively we establish the asymptotic normality of the estimators it turns out that both methods have the same asymptotic bias and variance in a common situation even though the local likelihood method uses information about the baseline hazard function

a new algorithm for estimating the effective dimension reduction subspace the statistical problem of estimating the effective dimension reduction edr subspace in the multi index regression model with deterministic design and additive noise is considered a new procedure for recovering the directions of the edr subspace is proposed many methods for estimating the edr subspace perform principal component analysis on a family of vectors say 1 l nearly lying in the edr subspace this is in particular the case for the structure adaptive approach proposed by hristache et al 2001a in the present work we propose to estimate the projector onto the edr subspace by the solution to the optimization problem minimize maxl 1 l lt i a l subject to a am where am is the set of all symmetric matrices with eigenvalues in 0 1 and trace less than or equal to m with m being the true structural dimension under mild assumptions n consistency of the proposed procedure is proved up to a logarithmic factor in the case when the structural dimension is not larger than 4 moreover the stochastic error of the estimator of the projector onto the edr subspace is shown to depend on l logarithmically this enables us to use a large number of vectors l for estimating the edr subspace the empirical behavior of the algorithm is studied through numerical simulations

normal forms and invariant manifolds for nonlinear non autonomous pdes viewed as odes in infinite dimensions abstract we prove that a general class of nonlinear non autonomous ode s in frechet spaces are close to ode s in a specific normal form where closeness means that solutions of the normal form ode satisfy the original ode up to a residual that vanishes up to any desired order in this normal form the centre stable and unstable coordinates of the ode are clearly separated which allows us to define invariant manifolds of such equations in a robust way the main motivation is the case where the frechet space in question is a suitable function space and the maps involved in an ode in this space are defined in terms of derivatives of the functions so that the infinite dimensional ode is a finite dimensional pde we show that our methods apply to a relevant class of nonlinear non autonomous pde s in this way

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

techniques for the study of singularities with applications to resolution of 2 dimensional schemes we give an overview of invariants of algebraic singularities over perfect fields we then show how they lead to a synthetic proof of embedded resolution of singularities of 2 dimensional schemes

kernel dimension reduction in regression we present a new methodology for sufficient dimension reduction sdr our methodology derives directly from the formulation of sdr in terms of the conditional independence of the covariate x from the response y given the projection of x on the central subspace cf j amer statist assoc 86 1991 316 342 and regression graphics 1998 wiley we show that this conditional independence assertion can be characterized in terms of conditional covariance operators on reproducing kernel hilbert spaces and we show how this characterization leads to an m estimator for the central subspace the resulting estimator is shown to be consistent under weak conditions in particular we do not have to impose linearity or ellipticity conditions of the kinds that are generally invoked for sdr methods we also present empirical results showing that the new methodology is competitive in practice

a simple algorithm for principalization of monomial ideals in this paper we give a simple constructive proof of principalization of monomial ideals and the global analog this also gives an algorithm for principalization

poincare series of resolutions of surface singularities let x spec r be a resolution of singularities of a normal surface singularity spec r with integral exceptional divisors e 1 e r we consider the poincare series g h n t n where h n r x o x n 1 e 1 n r e r we show that if r m has characteristic zero and pic 0 x is a semi abelian variety then the poincare series g is rational however we give examples to show that this series can be irrational if either of these conditions fails

a multiple index model and dimension reduction dimension reduction can be used as an initial step in statistical modeling further specification of model structure is imminent and important when the reduced dimension is still greater than 1 in this article we investigate one method of specification that involves separating the linear component from the nonlinear components leading to further dimension reduction in the unknown link function and thus better estimation and easier interpretation of the model the specified model includes the popular econometric multiple index model and the partially linear single index model as its special cases a criterion is developed to validate the model specification an algorithm is proposed to estimate the model directly asymptotic distributions for the estimators of the parameters and the nonparametric link function are derived air pollution data in chicago are used to illustrate the modeling procedure and to demonstrate its advantages over the existing dimension reduction approaches

key polynomials and minimal pairs abstract in this paper we establish the relation between key polynomials as defined in 12 and minimal pairs of definition of a valuation we also discuss truncations of valuations on a polynomial ring k x we prove that a valuation is equal to its truncation on some polynomial if and only if is valuation transcendental another important result of this paper is that if is any extension of to k x and is a complete sequence of key polynomials for without last element then for each q there exists a suitable root a q k of q such that a q q is a pseudo convergent sequence defining

effective bounds of linear series on algebraic varieties and arithmetic varieties in this paper we prove effective upper bounds for effective sections of line bundles on projective varieties and hermitian line bundles on arithmetic varieties in terms of the volumes they are effective versions of the hilbert samuel formula and the arithmetic hilbert samuel formula the treatments are high dimensional generalizations of duke math j 162 2013 1723 1770 and relative noether inequality on fibered surfaces preprint 2013 similar results are obtained independently by huayi chen majorations explicites de fonctions de hilbert samuel geometrique et arithmetique preprint 2014 with less explicit error terms

resolution of singularities of real analytic vector fields in dimension three let be an analytic vector field defined in a real analytic manifold of dimension three we prove that all the singularities of can be made elementary by a finite number of blowing ups in the ambient space

resolution of singularities for 3 folds in positive characteristic a concise complete proof of resolution of singularities of 3 folds in positive characteristic 5 is given abhyankar first proved this theorem in 1966

resolution of singularities of threefolds in positive characteristic i reduction to local uniformization on artin schreier and purely inseparable coverings abstract in dimension 3 the desingularization problem is reduced to local uniformization on artin schreier and purely inseparable coverings of regular space

on principal hessian directions for data visualization and dimension reduction another application of stein s lemma abstract modern graphical tools have enhanced our ability to learn many things from data directly with much user friendly graphical software available we are encouraged to plot a lot more often than before the benefits from direct interaction with graphics have been enormous but trailing behind these high tech advances is the issue of appropriate guidance on what to plot there are too many directions to project a high dimensional data set and unguided plotting can be time consuming and fruitless in a recent article li set up a statistical framework for study on this issue based on a notion of effective dimension reduction edr directions they are the directions to project a high dimensional input variable for the purpose of effectively viewing and studying its relationship with an output variable a methodology sliced inverse regression was introduced and shown to be useful in finding edr directions this article introduces another method for finding edr directions it begins with the observat

structure adaptive approach for dimension reduction we propose a new method of effective dimension reduction for a multiindex model which is based on iterative improvement of the family of average derivative estimates the procedure is computationally straightforward and does not require any prior information about the structure of the underlying model we show that in the case when the effective dimension m of the index space does not exceed 3 this space can be estimated with the rate n 1 2 under rather mild assumptions on the model

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

sparse non gaussian component analysis non gaussian component analysis ngca introduced in offered a method for high dimensional data analysis allowing for identifying a low dimensional non gaussian component of the whole distribution in an iterative and structure adaptive way an important step of the ngca procedure is identification of the non gaussian subspace using principle component analysis pca method this article proposes a new approach to ngca called sparse ngca which replaces the pca based procedure with a new the algorithm we refer to as convex projection