some limit theorems for the empirical process indexed by functions let n 1 be a sequence of classes of real valued measurable functions defined on a probability space s p under weak metric entropy conditions on n 1 and under growth conditions on we show that there are non zero numerical constantsc1 andc2 such that where n is a non decreasing function ofn related to the metric entropy of a few applications of this general result are considered we obtain a s rates of uniform convergence for the empirical process indexed by intervals as well as a s rates of uniform convergence for the empirical characteristic function over expanding intervals

free boundary regularity for a problem arising in superconductivity this paper concerns regularity properties of the mean field theory of superconductivity the problem is reminiscent of the one studied earlier by l a caffarelli l karp and h shahgholian in connection with potential theory the difficulty introduced in this paper is the existence of several patches where on each patch the solution to the problem may have different constant values however using a refined analysis we reduce the problem to the one patch case at least locally near regular free boundary points using a monotonicity formula due to georg s weiss we characterize global solutions of a related equation hence earlier regularity results apply and we conclude the c1 regularity of the free boundary

viscosity solutions of a degenerate parabolic elliptic system arising in the mean field theory of superconductivity in a type ii superconductor the magnetic field penetrates the superconducting body through the formation of vortices in an extreme type ii superconductor these vortices reduce to line singularities because the number of vortices is large it seems feasible to model their evolution by an averaged problem known as the mean field model of superconductivity we assume that the evolution law of an individual vortex which underlies the averaging process involves the current of the generated magnetic field as well as the curvature vector in the present paper we study a two dimensional reduction assuming all vortices to be perpendicular to a given direction since both the magnetic field h and the averaged vorticity are curl free we may represent them via a scalar magnetic potential q and a scalar stream function respectively we study existence uniqueness and asymptotic behaviour of solutions q of the resulting degenerate elliptic parabolic system with curvature taken into account or not by means of viscosity and weak solutions in addition we relate q to solutions h of the mean field equations without curvature finally we construct special solutions of the corresponding stationary equations with two or more superconducting phases

variational problems with two phases and their free boundaries the problem of minimizing vu 2 q2 x 2 v dx in an appropriate class of functions v is considered here q x 0 and a2 t x2 if v 0 any minimizer u is harmonic in u 0 and vu 2 has a jump a2 x 22 across the free boundary u 0 regularity and various properties are established for the minimizer u and for the free boundary introduction in this paper we consider the problem of minimizing j v f vv 2 q2 x x2 v dx vgk where q2 x 0 a22 ifu 0 and x2 u is lower semicontinuous at v 0 it is assumed that a2 0 and a a2 a22 0 the class k consists of all functions v in l loc s2 with vu g l2 i2 such that v u on a given open subset s of 3i2 and s is a domain in r the analogous problem for functions in k v k v 0 a e was studied in 1 in that paper it was proved that any local minimizer of j v in k is lipschitz continuous and if n 2 the free boundary 8 w 0 is analytic if q x is analytic the present variational problem is motivated by applications to the now of two liquids in models of jets and cavities these applications will be studied in other forthcoming papers 5 6 the present work is aimed at extending results of 1 in particular we shall establish nondegeneracy theorems the lipschitz continuity of the solution and some properties of the free boundary for n 2 the free boundary is proved to be continuously differentiable a new and rather powerful tool introduced in this paper is the monotonicity formula lemma 5 1 asserting that for a minimizer u if u x0 0 then

confidence sets for a multivariate distribution on etudie une version multivariable alternative de la bande de confiance unidimensionnelle de kolmogorov smirnov qui preserve l invariance affine

probability inequalities for empirical processes and a law of the iterated logarithm given a class f of functions on x we can view vn as a stochastic process indexed by f and consider limit theorems for this process to prove such theorems it is often helpful to have bounds on the tail of the r v sup w i n f i see for example dudley 1978 kuelbs and dudley 1980 dudley and phillip 1983 or gine and zinn 1983 our main question of interest is for what classes f can best possible bounds be obtained so first we must ask what are these best possible bounds of course

likelihood based sufficient dimension reduction we obtain the maximum likelihood estimator of the central subspace under conditional normality of the predictors given the response analytically and in simulations we found that our new estimator can preform much better than sliced inverse regression sliced average variance estimation and directional regression and that it seems quite robust to deviations from normality

concentration inequalities and asymptotic results for ratio type empirical processes let be a class of measurable functions on a measurable space s mathcal s with values in 0 1 and let pn n 1 i 1n xi be the empirical measure based on an i i d sample x1 xn from a probability distribution p on s mathcal s we study the behavior of suprema of the following type sup r n textless sigma p f leq delta n frac p n f pf phi sigma p f where pf var1 2pf and is a continuous strictly increasing function with 0 0 using talagrand s concentration inequality for empirical processes we establish concentration inequalities for such suprema and use them to derive several results about their asymptotic behavior expressing the conditions in terms of expectations of localized suprema of empirical processes we also prove new bounds for expected values of sup norms of empirical processes in terms of the largest pf and the l2 p norm of the envelope of the function class which are especially suited for estimating localized suprema with this technique we extend to function classes most of the known results on ratio type suprema of empirical processes including some of alexander s results for vc classes of sets we also consider applications of these results to several important problems in nonparametric statistics and in learning theory including general excess risk bounds in empirical risk minimization and their versions for l2 regression and classification and ratio type bounds for margin distributions in classification

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

optimal aggregation of classifiers in statistical learning classification can be considered as nonparametric estimation of sets where the risk is defined by means of a specific distance between sets associated with misclassification error it is shown that the rates of convergence of classifiers depend on two parameters the complexity of the class of candidate sets and the margin parameter the dependence is explicitly given indicating that optimal fast rates approaching o n 1 can be attained where n is the sample size and that the proposed classifiers have the property of robustness to the margin the main result of the paper concerns optimal aggregation of classifiers we suggest a classifier that automatically adapts both to the complexity and to the margin and attains the optimal fast rates up to a logarithmic factor

sample moduli for set indexed gaussian processes on etudie le comportement de la trajectoire de processus gaussiens wp indexes par des classes c de sous ensembles d un espace de probabilite x 7b a p

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

a law of the logarithm for kernel density estimators in this paper we derive a law of the logarithm for the maximal deviation between a kernel density estimator and the true underlying density function extensions to higher derivatives are included the results are applied to get optimal window widths with respect to almost sure uniform convergence

solutions of fully nonlinear elliptic equations with patches of zero gradient existence regularity and convexity of level curves in this paper we first construct viscosity solutions in the crandall lions sense of fully nonlinear elliptic equations of the form f d 2 u x g x u on u 0 in fact viscosity solutions are surprisingly weak since candidates for solutions are just continuous we only require that the test polynomials p those tangent from above or below to the graph of u at a point x 0 satisfy the correct inequality only if p x 0 0 that is we simply disregard those test polynomials for which p x 0 0 nevertheless this is enough by an appropriate use of the alexandroff bakelman technique to prove existence regularity and in two dimensions for f g cu c 0 and constant boundary conditions on a convex domains to prove that there is only one convex patch

graph connection laplacian methods can be made robust to noise recently several data analytic techniques based on graph connection laplacian gcl ideas have appeared in the literature at this point the properties of these methods are starting to be understood in the setting where the data is observed without noise we study the impact of additive noise on these methods and show that they are remarkably robust as a by product of our analysis we propose modifications of the standard algorithms that increase their robustness to noise we illustrate our results in numerical simulations

limit theorems and inequalities for the uniform empirical process indexed by intervals the uniform empirical process u is considered as a process indexed by intervals powerful new exponential bounds are established for the process indexed by both points and intervals these bounds trivialize the proof of the chibisov o reilly theorem concerning the convergence of the process with respect to 11 q 11 metrics and are used to prove an interval analogue of the chibisov o reilly theorem a strong limit theorem related to the well known holder condition for brownian bridge u is also proved connections with related work of csdki eicker jaeschke and stute are mentioned as an application we introduce a new statistic for testing uniformity which is the natural interval analogue of the classical anderson darling statistic

rates of convergence in active learning we study the rates of convergence in generalization error achievable by active learning under various types of label noise additionally we study the general problem of model selection for active learning with a nested hierarchy of hypothesis classes and propose an algorithm whose error rate provably converges to the best achievable error among classifiers in the hierarchy at a rate adaptive to both the complexity of the optimal classifier and the noise conditions in particular we state sufficient conditions for these rates to be dramatically faster than those achievable by passive learning

sudakov s typical marginals random linear functionals and a conditional central limit theorem v n sudakov sud78 proved that the one dimensional marginals of a high dimensional second order measure are close to each other in most directions extending this and a related result in the context of projection pursuit of p diaconis and d freedman dia84 we give for a probability measure p and a random a s linear functional f on a hilbert space simple sufficient conditions under which most of the one dimensional images of p under f are close to their canonical mixture which turns out to be almost a mixed normal distribution using the concept of approximate conditioning we deduce a conditional central limit theorem theorem 3 for random averages of triangular arrays of random variables which satisfy only fairly weak asymptotic orthogonality conditions

the central limit theorem for empirical processes on vapnik cervonenkis classes sufficient conditions which under weak additional assumptions are also necessary are found for the central limit theorem for empirical processes indexed by classes of functions which have the vapnik cervonenkis property this improves an earlier theorem of pollard 1982a and leads to necessary and sufficient conditions for the clt for weighted empirical processes indexed by vapnik cervonenkis classes of sets

sensitivity analysis for box cox power transformation model contrast parameters summary sensitivity analysis is an important tool for the evaluation of scientific models we consider the box cox power transformation model and study the sensitivity for the estimated slope vector when the transformation parameter is perturbed this is a key issue in the debate among bickel doksum 1981 box cox 1982 and hinkley runger 1984 on the application of the power transformation model hinkley runger 1984 conjectured that box cox s 1964 z transformation would eliminate asymptotically the sensitivity for the estimated slope vector we establish this conjecture under appropriate symmetry conditions on the joint distribution for the regressors x when the true transformation is logarithmic the conjecture holds if the distribution for x is axially symmetric for other power transformations the conjecture holds if x is multivariate normal we also consider a weaker version of hinkley runger s conjecture namely the z transformation would achieve the best possible reduction in the sensitivity we establish this weaker conjecture under less restrictive symmetry conditions which only involve the marginal distribution for xfl both conjectures might fail when those symmetry conditions are not satisfied two examples are given to illustrate the limitations of the conjectures sensitivity analysis is an important tool for the evaluation of scientific models many scientific models include some nuisance parameters which are not known precisely it is usually desirable for the model output to be stable with respect to those parameters the performance of the model can be questionable if the model output is highly sensitive to those parameters it is therefore useful to carry out a sensitivity analysis in which those parameters are perturbed the model output is derived under the perturbed parameters and compared with the model output prior to the perturbation we will study the sensitivity analysis for the box cox power transformation model with respect to the specific transformation taken let yi xi i 1 n be the observed data where yi denotes a positive scalar outcome and xi denotes a p dimensional row vector of regressors let the transformed outcome be

localization of vc classes beyond local rademacher complexities abstract in statistical learning the excess risk of empirical risk minimization erm is controlled by comp n f n where n is a size of a learning sample comp n f is a complexity term associated with a given class f and 1 2 1 interpolates between slow and fast learning rates in this paper we introduce an alternative localization approach for binary classification that leads to a novel complexity measure fixed points of the local empirical entropy we show that this complexity measure gives a tight control over comp n f in the upper bounds under bounded noise our results are accompanied by a minimax lower bound that involves the same quantity in particular we practically answer the question of optimality of erm under bounded noise for general vc classes

the central limit problem for convex bodies it is shown that every symmetric convex body which satisfies a kind of weak law of large numbers has the property that almost all its marginal distributions are approximately gaussian several quite broad classes of bodies are shown to satisfy the condition

on the inertia and range of transformations of positive definite matrices abstract results are established which relate the range and inertia of general transformations on positive definite matrices included are a bound theorem for certain eigenvalues of these transformations a characterization of positive definite preserving completely positive transformations and generalizations of the theorems of stein 8 and of stein and pfeffer 9

exploratory projection pursuit abstract a new projection pursuit algorithm for exploring multivariate data is presented that has both statistical and computational advantages over previous methods a number of practical issues concerning its application are addressed a connection to multivariate density estimation is established and its properties are investigated through simulation studies and application to real data the goal of exploratory projection pursuit is to use the data to find low one two or three dimensional projections that provide the most revealing views of the full dimensional data with these views the human gift for pattern recognition can be applied to help discover effects that may not have been anticipated in advance since linear effects are directly captured by the covariance structure of the variable pairs which are straightforward to estimate the emphasis here is on the discovery of nonlinear effects such as clustering or other general nonlinear associations among the variables although arbitrary

regularity of a free boundary with application to the pompeiu problem in the unit ball b 0 1 let u and a domain in n solve the following overdetermined problem u in b 0 1 02 u jruj 0 inb 0 1 n where denotes the characteristic function and the equation is satisfled in the sense of distributions if the complement of does not develop cusp singularities at the origin then we prove is analytic in some small neighborhood of the origin the result can be modifled to yield for more general divergence form operators as an application of this then we obtain the regularity of the boundary of a domain without the pompeiu property provided its complement has no cusp singularities

central limit theorems for stochastic processes under random entropy conditions necessary and sufficient conditions are found for the weak convergence of the row sums of an infinitesimal row independent triangular array nj of stochastic processes indexed by a set s to a sample continuous gaussian process when the array satisfies a random entropy condition analogous to one used by gine and zinn 1984 for empirical processes this entropy condition is satisfied when s is a class of sets or functions with the vapnik ervonenkis property and each nj f fd nj is of the form njc for some reasonable random finite signed measure vnj as a result we obtain necessary and sufficient conditions for the weak convergence of possibly non i i d partial sum processes and new sufficient conditions for empirical processes indexed by vapnik ervonenkis classes special cases include prokhorov s 1956 central limit theorem for empirical processes and shorack s 1979 theorems on weighted empirical processes

a model free test for reduced rank in multivariate regression we propose a test of dimension in multivariate regression this test is in the spirit of tests on the rank of the coefficient matrix in a multivariate linear model but it does not require a prespecified model the test may be particularly useful at the outset of an analysis before a multivariate model is posited because it can lead to low dimensional summary plots that are inferred to contain all of the sample information on the multivariate mean function

a multivariate distribution for sub images a new method for obtaining multivariate distributions for sub images of natural images is described the information in each sub image is summarized by a measurement vector in a measurement space the dimension of the measurement space is reduced by applying a random projection to the truncated output of the discrete cosine transforms of the sub images the measurement space is then reparametrized such that a gaussian distribution is a good model for the measurement vectors in the reparametrized space an ornstein uhlenbeck process associated with the gaussian distribution is used to model the differences between measurement vectors obtained from matching sub images the probability of a false alarm and the probability of accepting a correct match are calculated the accuracy of the resulting statistical model for matching sub images is tested using images from the middlebury stereo database with promising results in particular if the probability of accepting a correct match is relatively large then there is good agreement between the calculated and the experimental probabilities of obtaining a unique match that is also a correct match

on viscosity solutions of fully nonlinear equations with measurable ingredients we study fully nonlinear uniformly elliptic equations with measurable ingredients caffarelli s recent work on w2 p estimates for viscosity solutions has led to significant progress in this area here we present a unified treatment of this theory based on an appropriate notion of viscosity solution for instance it is shown that strong solutions are viscosity solutions that viscosity solutions are twice differentiable a e and that the pointwise derivatives satisfy the equation a e an important consequence of our approach is the possibility of passage to various kinds of limits in fully nonlinear equations this extends results of this type due to evans and krylov our work is to some extent expository the main purpose being to provide an easily accessible set of tools and techniques to study equations with measurable ingredients 1996 john wiley sons inc

on multivariate quantiles under partial orders this paper focuses on generalizing quantiles from the ordering point of view we propose the concept of partial quantiles which are based on a given partial order we establish that partial quantiles are equivariant under order preserving transformations of the data robust to outliers characterize the probability distribution if the partial order is sufficiently rich generalize the concept of efficient frontier and can measure dispersion from the partial order perspective we also study several statistical aspects of partial quantiles we provide estimators associated rates of convergence and asymptotic distributions that hold uniformly over a continuum of quantile indices furthermore we provide procedures that can restore monotonicity properties that might have been disturbed by estimation error establish computational complexity bounds and point out a concentration of measure phenomenon the latter under independence and the componentwise natural order finally we illustrate the concepts by discussing several theoretical examples and simulations empirical applications to compare intake nutrients within diets to evaluate the performance of investment funds and to study the impact of policies on tobacco awareness are also presented to illustrate the concepts and their use

on directional regression for dimension reduction we introduce directional regression dr as a method for dimension reduction like contour regression dr is derived from empirical directions but achieves higher accuracy and requires substantially less computation dr naturally synthesizes the dimension reduction estimators based on conditional moments such as sliced inverse regression and sliced average variance estimation and in doing so combines the advantages of these methods under mild conditions it provides exhaustive and consistent estimate of the dimension reduction space we develop the asymptotic distribution of the dr estimator and from that a sequential test procedure to determine the dimension of the central space we compare the performance of dr with that of existing methods by simulation and find strong evidence of its advantage over a wide range of models finally we apply dr to analyze a data set concerning the identification of hand written digits

strong limit theorems for oscillation moduli of the uniform empirical process let u n t n tfrac 1 2 gamma n t t 0 leqq t leqq 1 denote the uniform empirical process based on the first n of a sequence 1 2 of iid uniform 0 1 random variables where gamma n t n 1 sum limits i 1 n 1 0 t xi i is the empirical distribution function the oscillation modulus of u n is defined by omega n a sup u n t h u n t 0 leqq t leqq 1 h h leqq a and the lipschitz 1 2 modulus of u n is defined by tilde omega n a sup u n t h u n t h tfrac 1 2 0 leqq t leqq 1 h a leqq h leqq 1

a central limit theorem for convex sets we show that there exists a sequence varepsilon n searrow0 for which the following holds let k n be a compact convex set with a non empty interior let x be a random vector that is distributed uniformly in k then there exist a unit vector in n t0 and 0 such that sup a subset mathbb r left textit prob langle x theta rangle in a frac 1 sqrt 2 pi sigma int ae frac t t 0 2 2 sigma 2 dt right leq varepsilon n qquad ast where the supremum runs over all measurable sets a and where denotes the usual scalar product in n furthermore under the additional assumptions that the expectation of x is zero and that the covariance matrix of x is the identity matrix we may assert that most unit vectors satisfy with t0 0 and 1 corresponding principles also hold for multi dimensional marginal distributions of convex sets

rates of growth and sample moduli for weighted empirical processes indexed by sets probability inequalities are obtained for the supremum of a weighted empirical process indexed by a vapnik cervonenkis class c of sets these inequalities are particularly useful under the assumption p c c p c t 0 as t 0 they are used to obtain almost sure bounds on the rate of growth of the process as the sample size approaches infinity to find an asymptotic sample modulus for the unweighted empirical process and to study the ratio pn p of the empirical measure to the actual measure

regularity properties of a free boundary near contact points with the fixed boundary in the upper half of the unit ball b x 0 let u and omega a domain in r n x is an element of r n x 1 0 solve the following overdetermined problem deltau chi omega in b u

interior a priori estimates for solutions of fully non linear equations on etend une theorie de perturbations aux solutions d equations uniformement elliptiques d ordre 2 totalement non lineaires

decision theoretic generalizations of the pac model for neural net and other learning applications we describe a generalization of the pac learning model that is based on statistical decision theory in this model the learner receives randomly drawn examples each example consisting of an instance x in x and an outcome y in y and tries to find a hypothesis h x a where h in h that specifies the appropriate action a in a to take for each instance x in order to minimize the expectation of a loss l y a here x y and a are arbitrary sets l is a real valued function and examples are generated according to an arbitrary joint distribution on x times y special cases include the problem of learning a function from x into y the problem of learning the conditional probability distribution on y given x regression and the problem of learning a distribution on x density estimation we give theorems on the uniform convergence of empirical loss estimates to true expected loss rates for certain hypothesis spaces h and show how this implies learnability with bounded sample size disregarding computational complexity as an application we give distribution independent upper bounds on the sample size needed for learning with feedforward neural networks our theorems use a generalized notion of vc dimension that applies to classes of real valued functions adapted from pollard s work and a notion of capacity and metric dimension for classes of functions that map into a bounded metric space supersedes 89 30 and 90 52 also in information and computation vol 100 no 1 september 1992

linear dimensionality reduction survey insights and generalizations linear dimensionality reduction methods are a cornerstone of analyzing high dimensional data due to their simple geometric interpretations and typically attractive computational properties these methods capture many data features of interest such as covariance dynamical structure correlation between data sets input output relationships and margin between data classes methods have been developed with a variety of names and motivations in many fields and perhaps as a result the connections between all these methods have not been highlighted here we survey methods from this disparate literature as optimization programs over matrix manifolds we discuss principal component analysis factor analysis linear multidimensional scaling fisher s linear discriminant analysis canonical correlations analysis maximum autocorrelation factors slow feature analysis sufficient dimensionality reduction undercomplete independent component analysis linear regression distance metric learning and more this optimization framework gives insight to some rarely discussed shortcomings of well known methods such as the suboptimality of certain eigenvector solutions modern techniques for optimization over matrix manifolds enable a generic linear dimensionality reduction solver which accepts as input data and an objective to be optimized and returns as output an optimal low dimensional projection of the data this simple optimization framework further allows straightforward generalizations and novel variants of classical methods which we demonstrate here by creating an orthogonal projection canonical correlations analysis more broadly this survey and generic solver suggest that linear dimensionality reduction can move toward becoming a blackbox objective agnostic numerical technology

pattern classification using projection pursuit abstract this article discusses the adaptation of recently developed regression techniques to classifier design apart from finite sample effects projection pursuit pp regression may be used to model a desired response class as a sum of ridge functions according to a minimum expected squared error criterion this approach can be shown to furnish an optimal discriminant function which can satisfy the neyman pearson criterion over all possible thresholds basis function expansions are used instead of smoothed histograms to reduce computation since good approximation of a discriminant by a linear combination of moderate number of ridge functions may not be easy we introduce an improved method utilizing a nonlinear weighting function

c1 1 regularity in semilinear elliptic problems in this paper we give an astonishingly simple proof of c1 1 regularity in elliptic theory our technique yields both new simple proofs of old results as well as new optical results the setting we ll consider is the following let u be a solution to where b is the unit ball in n f x t is a bounded lipschitz function in x and ft is bounded from below then we prove that u c1 1 b1 2 our method is a simple corollary to a recent monotonicity argument due to caffarelli jerison and kenig 2002 wiley periodicals inc

a high dimensional wilks phenomenon a theorem by wilks asserts that in smooth parametric density estimation the difference between the maximum likelihood and the likelihood of the sampling distribution converges toward a chi square distribution where the number of degrees of freedom coincides with the model dimension this observation is at the core of some goodness of fit testing procedures and of some classical model selection methods this paper describes a non asymptotic version of the wilks phenomenon in bounded contrast optimization procedures using concentration inequalities for general functions of independent random variables it proves that in bounded contrast minimization as for example in statistical learning theory the difference between the empirical risk of the minimizer of the true risk in the model and the minimum of the empirical risk the excess empirical risk satisfies a bernstein like inequality where the variance term reflects the dimension of the model and the scale term reflects the noise conditions from a mathematical statistics viewpoint the significance of this result comes from the recent observation that when using model selection via penalization the excess empirical risk represents a minimum penalty if non asymptotic guarantees concerning prediction error are to be provided from the perspective of empirical process theory this paper describes a concentration inequality for the supremum of a bounded non centered actually non positive empirical process combining the now classical analysis of m estimation building on talagrand s inequality for suprema of empirical processes and versatile moment inequalities for functions of independent random variables this paper develops a genuine bernstein like inequality that seems beyond the reach of traditional tools

an inertial decomposition and related range results abstract with quasicommutative n square complex matrices a1 as and s square hermitian g gij relationships are given between the image s i j 1 g ij a i ha j of a linear transformation on h n being positive definite and the action of h on generalized inertial decompositions of c n