testing and estimating shape constrained nonparametric density and regression in the presence of measurement error in many applications we can expect that or are interested to know if a density function or a regression curve satisfies some specific shape constraints for example when the explanatory variable x represents the value taken by a treatment or dosage the conditional mean of the response y is often anticipated to be a monotone function of x indeed if this regression mean is not monotone in the appropriate direction then the medical or commercial value of the treatment is likely to be significantly curtailed at least for values of x that lie beyond the point at which monotonicity fails in the case of a density common shape constraints include log concavity and unimodality if we can correctly guess the shape of a curve then nonparametric estimators can be improved by taking this information into account addressing such problems requires a method for testing the hypothesis that the curve of interest satisfies a shape constraint and if the conclusion of the test is positive a technique for e

projective techniques and functional integration for gauge theories a general framework for integration over certain infinite dimensional spaces is first developed using projective limits of a projective family of compact hausdorff spaces the procedure is then applied to gauge theories to carry out integration over the non linear infinite dimensional spaces of connections modulo gauge transformations this method of evaluating functional integrals can be used either in the euclidean path integral approach or the lorentzian canonical approach a number of measures discussed are diffeomorphism invariant and therefore of interest to the connection dynamics version of quantum general relativity the account is pedagogical in particular prior knowledge of projective techniques is not assumed

an overview of parallel algorithms for the singular value and symmetric eigenvalue problems abstract solving dense symmetric eigenvalue problems and computing singular value decompositions continue to be two of the most dominating tasks in numerous scientific applications with the advent of multiprocessor computer systems the design of efficient parallel algorithms to determine these solutions becomes of paramount importance in this paper we discuss two fundamental approaches for determining the eigenvalues and eigenvectors of dense symmetric matrices on machines such as the alliant fx 8 and cray x mp one approach capitalizes upon the inherent parallelism offered by jacobi methods while the other relies upon an efficient reduction to tridiagonal form via householder s transformations followed by a multisectioning technique to obtain the eigenvalues and eigenvectors of the corresponding symmetric tridiagonal matrix for the singular value decomposition we discuss an efficient method for rectangular matrices in which the number of rows is substantially larger or smaller than the number of columns this scheme performs an initial orthogonal factorization using block householder transformation followed by a parallel one sided jacobi method to obtain the singular values and singular vectors of the resulting upper triangular matrix exceptional performance for this svd scheme is demonstrated for tall matrices of full or deficient rank having clustered or multiple singular values a hybrid method that combines one and two sided jacobi schemes is also discussed performance results for each of the above algorithms on the alliant fx 8 and cray x mp computer systems will be presented with particular emphasis given to speedups obtained over such classical eispack algorithms

on quantum gauge theories in two dimensions two dimensional quantum yang mills theory is studied from three points of view i by standard physical methods ii by relating it to the largek limit of three dimensional chern simons theory and two dimensional conformal field theory iii by relating its weak coupling limit to the theory of reidemeister ray singer torsion the results obtained from the three points of view agree and give formulas for the volumes of the moduli spaces of representations of fundamental groups of two dimensional surfaces

empirical likelihood ratio confidence intervals for a single functional summary the empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken in this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions it is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean for a class of m estimates that includes quantiles and for differentiable statistical functionals the results are nonparametric extensions of wilks s 1938 theorem for parametric likelihood ratios the intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on student s t statistic a hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced

topological models on the lattice and a remark on string theory cloning the addition of a topological model to the matter content of a conventional closed string theory leads to the appearance of many perturbatively decoupled spacetime worlds we illustrate this by classifying topological vertex models on a triangulated surface we comment on how such worlds could have been coupled in the planck era

lattice topological field theory in two dimensions the lattice definition of a two dimensional topological field theory tft is given generically and the exact solution is obtained explicitly in particular the set of all lattice topological field theories is shown to be in one to one correspondence with the set of all associative algebrasr and the physical hilbert space is identified with the centerz r of the associative algebrar perturbations of tft s are also considered in this approach showing that the form of topological perturbations is automatically determined and that all tft s are obtained from one tft by such perturbations several examples are presented including twistedn 2 minimal topological matter and the case wherer is a group ring

kernel density estimation via diffusion we present a new adaptive kernel density estimator based on linear diffusion processes the proposed estimator builds on existing ideas for adaptive smoothing by incorporating information from a pilot density estimate in addition we propose a new plug in bandwidth selection method that is free from the arbitrary normal reference rules used by existing methods we present simulation examples in which the proposed approach outperforms existing methods in terms of accuracy and reliability

nonparametric kernel regression subject to monotonicity constraints we suggest a method for monotonizing general kernel type estimators for example local linear estimators and nadaraya watson estimators attributes of our approach include the fact that it produces smooth estimates indeed with the same smoothness as the unconstrained estimate the method is applicable to a particularly wide range of estimator types it can be trivially modified to render an estimator strictly monotone and it can be employed after the smoothing step has been implemented therefore an experimenter may use his or her favorite kernel estimator and their favorite bandwidth selector to construct the basic nonparametric smoother and then use our technique to render it monotone in a smooth way implementation involves only an off the shelf programming routine the method is based on maximizing fidelity to the conventional empirical approach subject to monotonicity we adjust the unconstrained estimator by tilting the empirical distribution so as to make the least possible change in the sense of a distance measure subject to imposing the constraint of monotonicity

chern simons theory with finite gauge group we construct in detail a 2 1 dimensional gauge field theory with finite gauge group in this case the path integral reduces to a finite sum so there are no analytic problems with the quantization the theory was originally introduced by dijkgraaf and witten without details the point of working it out carefully is to focus on the algebraic structure and particularly the construction of quantum hilbert spaces on closed surfaces by cutting and pasting this includes the verlinde formula the careful development may serve as a model for dealing with similar issues in more complicated cases

lattice topological field theory on nonorientable surfaces the lattice definition of the two dimensional topological quantum field theory fukuma et al commun math phys 161 157 1994 is generalized to arbitrary not necessarily orientable compact surfaces it is shown that there is a one to one correspondence between real associative algebras and the topological state sum invariants defined on such surfaces the partition and n point functions on all two dimensional surfaces connected sums of the klein bottle or projective plane and g tori are defined and computed for arbitrary algebras in general and for the group ring a r g of discrete groups g in particular

intentionally biased bootstrap methods summary a class of weighted bootstrap techniques called biased bootstrap or b bootstrap methods is introduced it is motivated by the need to adjust empirical methods such as the uniform bootstrap in a surgical way to alter some of their features while leaving others unchanged depending on the nature of the adjustment the b bootstrap can be used to reduce bias or to reduce variance or to render some characteristic equal to a predetermined quantity examples of the last application include a b bootstrap approach to hypothesis testing in nonparametric contexts where the b bootstrap enables simulation under the null hypothesis even when the hypothesis is false and a b bootstrap competitor to tibshirani s variance stabilization method an example of the bias reduction application is adjustment of nadaraya watson kernel estimators to make them competitive with local linear smoothing other applications include density estimation under constraints outlier trimming sensitivity analysis skewness or kurtosis reduction and shrinkage

a parallel implementation of the nonsymmetric qr algorithm for distributed memory architectures one approach to solving the nonsymmetric eigenvalue problem in parallel is to parallelize the qr algorithm not long ago this was widely considered to be a hopeless task recent efforts have led to significant advances although the methods proposed up to now have suffered from scalability problems this paper discusses an approach to parallelizing the qr algorithm that greatly improves scalability a theoretical analysis indicates that the algorithm is ultimately not scalable but the nonscalability does not become evident until the matrix dimension is enormous experiments on the intel paragon system the ibm sp2 supercomputer the sgi origin 2000 and the intel asci option red supercomputer are reported

making a non parametric density estimator more attractive and more accurate by data perturbation summary motivated by both the shortcomings of high order density estimators and the increasingly large data sets in many areas of modern science we introduce new high order non parametric density estimators that are guaranteed to be positive and do not have highly oscillatory tails our approach is based on data perturbation e g by tilting or data sharpening it leads to new estimators that are more accurate than conventional kernel techniques that use positive kernels but which nevertheless enjoy the positivity property and are far less wiggly than high order kernel estimators we investigate performance by theoretical analysis and in a numerical study

on locally adaptive density estimation abstract theoretical and practical aspects of the sample point adaptive positive kernel density estimator are examined a closed form expression for the mean integrated squared error is obtained through the device of preprocessing the data by binning with this expression the exact behavior of the optimally adaptive smoothing parameter function is studied for the first time the approach differs from most earlier techniques in that bias of the adaptive estimator remains o h 2 and is not improved to the rate o h 4 a practical algorithm is constructed using a modification of least squares cross validation simulated and real examples are presented including comparisons with a fixed bandwidth estimator and a fully automatic version of abramson s adaptive estimator the results are very promising

structures and diagrammatics of four dimensional topological lattice field theories abstract crane and frenkel proposed a state sum invariant for triangulated 4 manifolds they sketched the definition of a hopf category that was to be used in their construction crane and yetter studied hopf categories and gave some examples using group cocycles that are associated to the drinfeld double of a finite group in this paper we define a state sum invariant of triangulated 4 manifolds using crane yetter cocycles as boltzmann weights our invariant is analogous to the 3 dimensional invariants defined by dijkgraaf and witten and the invariants that are defined via hopf algebras we present diagrammatic methods for the study of such invariants that illustrate connections between hopf categories and moves to triangulations

nonparametric regression to the mean available data may reflect a true but unknown random variable of interest plus an additive error which is a nuisance the problem in predicting the unknown random variable arises in many applied situations where measurements are contaminated with errors it is known as the regression to the mean problem there exists a well known solution when both the distributions of the true underlying random variable and the contaminating errors are normal this solution is given by the classical regression to the mean formula which has a data shrinkage interpretation we discuss the extension of this solution to cases where one or both of these distributions are unknown and demonstrate that the fully nonparametric case can be solved for the case of small contaminating errors the resulting nonparametric regression to the mean paradigm can be implemented by a straightforward data sharpening algorithm that is based on local sample means asymptotic justifications and practical illustrations are provided

path integration in two dimensional topological quantum field theory a positive diffeomorphism invariant generalized measure on the space of metrics of a two dimensional smooth manifold is constructed we use the term generalized measure analogously with the generalized measures of ashtekar and lewandowski and of baez a family of actions is presented which when integrated against this measure gives the two dimensional axiomatic topological quantum field theories or tqfts in terms of which durhuus and jonsson decompose every two dimensional unitary tqft as a direct sum

classification and construction of unitary topological field theories in two dimensions it is proven that unitary two dimensional topological field theories are uniquely characterized by n positive real numbers 1 n which can be regarded as the eigenvalues of a hermitian handle creation operator the number n is the dimension of the hilbert space associated with the circle and the partition functions for closed surfaces have the form zg ni 1 g 1i where g is the genus the eigenvalues can be arbitrary positive numbers it is shown how such a theory can be constructed on triangulated surfaces

a note on the usefulness of superkernels in density estimation we consider the akaike parzen rosenblatt density estimate f nh based upon any superkernel l i e an absolutely integrable function with int l 1 whose characteristic function is 1 on lbrack 1 1 rbrack and compare it with a kernel estimate g nh based upon an arbitrary kernel k we show that for a given subclass of analytic densities inf l sup k lim sup n rightarrow infty frac inf h mathbb e int f nh f inf h mathbb e int g nh f 1 where h 0 is the smoothing factor thus asymptotically the class of superkernels is as good as any other class of kernels when certain analytic densities are estimated we also obtain exact asymptotic expressions for the expected l 1 error of the kernel estimate when superkernels are used

squeezing the most out of eigenvalue solvers on high performance computers abstract this paper describes modifications to many of the standard algorithms used in computing eigenvalues and eigenvectors of matrices these modifications can dramatically increase the performance of the underlying software on high performance computers without resorting to assembler language without significantly influencing the floating point operation count and without affecting the roundoff error properties of the algorithms the techniques are applied to a wide variety of algorithms and are beneficial in various architectural settings

from 3 algebras to δ groups and symmetric cohomology we introduce groups and show how they fit in the context of lattice field theory to a topological space m we associate a group m we define the symmetric cohomology hsn g a of a group g with coefficients in a g module a the group m is determined by the action of 1 m on 2 m and an element of hs3 1 m 2 m