a study on the dimension of global sections of adjoint bundles for polarized manifolds abstract let x be a smooth complex projective variety of dimension n and let l be an ample line bundle on x in this paper in order to investigate the dimension of h 0 k x t l more systematically we introduce the invariant a i x l for every integer i with 0 i n furthermore we study this invariant for the case where l is ample and spanned by global sections as applications we get a lower bound resp an upper bound for the dimension of h 0 k x t l if l is ample and spanned by global sections resp very ample

numerical analysis for a system coupling curve evolution to reaction diffusion on the curve we consider a finite element approximation for a system consisting of the evolution of a closed planar curve by forced curve shortening flow coupled to a reaction diffusion equation on the evolving curve the scheme for the curve evolution is based on a parametric description allowing for tangential motion whereas the discretization for the pde on the curve uses an idea from g dziuk and c m elliott ima j numer anal 27 2007 pp 262 292 we prove optimal error bounds for the resulting fully discrete approximation and present numerical experiments these confirm our estimates and also illustrate the advantage of the tangential motion of the mesh points in practice

l 2 estimates for the evolving surface finite element method in this paper we consider the evolving surface nite element method for the advection and di usion of a conserved scalar quantity on a moving surface in an earlier paper using a suitable variational formulation in time dependent sobolev space we proposed and analysed a nite element method using surface nite elements on evolving triangulated surfaces an optimal order h error bound was proved for linear nite elements in this work we prove the optimal error bound in l t uniformly in time

higher order finite element methods and pointwise error estimates for elliptic problems on surfaces we define higher order analogues to the piecewise linear surface finite element method studied in g dziuk finite elements for the beltrami operator on arbitrary surfaces in partial differential equations and calculus of variations springer verlag berlin 1988 pp 142 155 and prove error estimates in both pointwise and l 2 based norms using the laplace beltrami problem on an implicitly defined surface gamma as a model pde we define lagrange finite element methods of arbitrary degree on polynomial approximations to gamma which likewise are of arbitrary degree then we prove a priori error estimates in the l 2 h 1 and corresponding pointwise norms that demonstrate the interaction between the pde error that arises from employing a finite dimensional finite element space and the geometric error that results from approximating gamma we also consider parametric finite element approximations that are defined on gamma and thus induce no geometric error computational examples confirm the sharpness of our error estimates

assessing the performance of an allocation rule the problem of estimating the error rates of a sample based rule on the basis of the same sample used in its construction is considered the apparent error rate is an obvious nonparametric estimate of the conditional error rate of a sample rule but unfortunately it provides too optimistic an assessment attention is focussed on the formation of improved estimates mainly through appropriate bias correction of the apparent error rate in this respect the role of the bootstrap a computer based methodology is highlighted

estimation of error rate for linear discriminant functions by resampling non gaussian populations abstract this article presents simulation results comparing various resampling estimators of classification error rate for linear discriminant type classification algorithms three non gaussian multivariate populations are studied namely exponential cauchy and uniform simulations are conducted for small sample sizes two class and three class problems and 2 d 3 d and 5 d distributions estimation procedures and sample sizes are the same as in our previous study of gaussian populations again 200 bootstrap replications are used for each simulation trial for exponential and uniform distributions the 0 632 estimator generally performs best however for cauchy distributions the convex bootstrap and the e 0 often outperform the 0 632 estimator

error rate estimation by mixture decomposition abstract we show how error rate estimation may be viewed as a problem of mixture decomposition we apply the idea to the average conditional error rate approach to estimation and explore the effectiveness of one particular decomposition method by simulation

a fully discrete evolving surface finite element method in this paper we consider a time discrete evolving surface finite element method for the advection and diffusion of a conserved scalar quantity on a moving surface in earlier papers using a suitable variational formulation in time dependent sobolev space we proposed and analyzed a finite element method using surface finite elements on evolving triangulated surfaces ima j numer anal 25 2007 pp 385 407 math comp to appear optimal order l 2 gamma t and h 1 gamma t error bounds were proved for linear finite elements in this work we prove optimal order error bounds for a backward euler time discretization

on the asymptotic properties of smoothed estimators of the classifiction error rate abstract a smoothed correction of the classification error rate counting estimate is examined the consistency of the estimate under very general conditions is established for normal distributions asymptotic theory reveals that the relative deficiency of the counting estimate with respect to the smoothed one can tend to infinity as the sample size increases

fully implicit linearly implicit and implicit explicit backward difference formulae for quasi linear parabolic equations quasi linear parabolic equations are discretised in time by fully implicit backward difference formulae bdf as well as by implicit explicit and linearly implicit bdf methods up to order five under appropriate stability conditions for the various methods considered we establish optimal order a priori error bounds by energy estimates which become applicable via the nevanlinna odeh multiplier technique

a convergent evolving finite element algorithm for mean curvature flow of closed surfaces a proof of convergence is given for semi and full discretizations of mean curvature flow of closed two dimensional surfaces the numerical method proposed and studied here combines evolving finite elements whose nodes determine the discrete surface like in dziuk s method and linearly implicit backward difference formulae for time integration the proposed method differs from dziuk s approach in that it discretizes huisken s evolution equations for the normal vector and mean curvature and uses these evolving geometric quantities in the velocity law projected to the finite element space this numerical method admits a convergence analysis in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five the error analysis combines stability estimates and consistency estimates to yield optimal order h 1 norm error bounds for the computed surface position velocity normal vector and mean curvature the stability analysis is based on the matrix vector formulation of the finite element method and does not use geometric arguments the geometry enters only into the consistency estimates numerical experiments illustrate and complement the theoretical results

estimating the error rate of a prediction rule improvement on cross validation abstract we construct a prediction rule on the basis of some data and then wish to estimate the error rate of this rule in classifying future observations cross validation provides a nearly unbiased estimate using only the original data cross validation turns out to be related closely to the bootstrap estimate of the error rate this article has two purposes to understand better the theoretical basis of the prediction problem and to investigate some related estimators which seem to offer considerably improved estimation in small samples

linearly implicit full discretization of surface evolution stability and convergence of full discretizations of various surface evolution equations are studied in this paper the proposed discretization combines a higher order evolving surface finite element method for space discretization with higher order linearly implicit backward difference formulae for time discretization the stability of the full discretization is studied in the matrix vector formulation of the numerical method the geometry of the problem enters into the bounds of the consistency errors but does not enter into the proof of stability numerical examples illustrate the convergence behaviour of the full discretization

convergence of finite elements on an evolving surface driven by diffusion on the surface for a parabolic surface partial differential equation coupled to surface evolution convergence of the spatial semidiscretization is studied in this paper the velocity of the evolving surface is not given explicitly but depends on the solution of the parabolic equation on the surface various velocity laws are considered elliptic regularization of a direct pointwise coupling a regularized mean curvature flow and a dynamic velocity law a novel stability and convergence analysis for evolving surface finite elements for the coupled problem of surface diffusion and surface evolution is developed the stability analysis works with the matrix vector formulation of the method and does not use geometric arguments the geometry enters only into the consistency estimates numerical experiments complement the theoretical results

bias associated with the discriminant analysis approach to the estimation of mixing proportions abstract the estimation of the mixing proportion 1 in which the first of two multivariate normal groups occur is considered on the basis of a sample drawn from a mixture of them under the model studied there are also data of known origin available from each of the groups for the formation of a discriminant rule r the bias of the estimator of 1 based on the proportion of the mixture data assigned to the first group by r is investigated also the case where the data of known origin are taken to be correlated is considered

nonparametric confidence and tolerance regions in canonical variate analysis canonical variate analysis is a much used descriptive multivariate technique for grouped data in which the group means are represented by points in a low typically two dimensional space if inferential use is to be made of the technique some knowledge of the variability of these points under repeated sampling is necessary the traditional approach is to construct confidence or tolerance circles around each point by using approximate asymptotic results based on normal assumptions for the data jackknife and bootstrap resampling schemes are proposed in this paper as a basis for constructing nonparametric regions these regions are compared with the normal based circles on the well known fisher iris data and also by means of a small monte carlo study incorporating nonnormal as well as normal data it is suggested that the normal based circles may often be inappropriate and areas of further investigation are identified

estimation of allocation rates in a cluster analysis context abstract a sample of multivariate observations is assumed to be drawn from a mixture of a given number of underlying populations the mixture likelihood approach to clustering is used to allocate each individual in the sample to its population of origin on the basis of the estimated posterior probabilities of population membership estimation of the correct allocation rate is considered for individual populations as well as for the overall mixture by averaging functions of the maximum of these posterior probabilities the estimates of the correct allocation rates provide a means of assessing the performance of the mixture approach to clustering the bootstrap technique is investigated for its effectiveness in reducing the bias of the estimates so obtained results are reported for three real data sets and a simulation study it is demonstrated that the proposed estimates generally provide useful information on the unobservable allocation rates of the mixture approach encouraging results are obtained for

on the adjunction mapping of very ample vector bundles of corank one let e be a very ample vector bundle of rank n 1 over a smooth complex projective variety x of dimension n 3 the structure of x e being known when k x det e 0 we investigate the structure of the adjunction mapping when 0 k x det e n