an implicit bidiagonal numerical method for solving the navier stokes equations in recent years much progress has been made in solving fluid dynamical problems using finite difference methods solving inviscid compressible problems in two and three dimensions has become almost routine with many suitable methods explicit or implicit available the problem of compressible viscous flows in complicated geometries remains however a major challenge here fine mesh spacing in the viscous flow regions makes the explicit methods with their simple boundary conditions extremely costly existing implicit methods can make use of large time steps but require inversions of large block tridiagonal matrices a method recently developed by maccormack eliminates this disadvantage by introducing a predictor corrector scheme requiring the inversion of only block bidiagonal matrices it is the aim of present work to extend this method to allow solution of viscous compressible problems in general coordinates for arbitrary two dimensional geometries

a law of large numbers for nearest neighbour statistics in practical data analysis methods based on proximity near neighbour relationships between sample points are important because these relations can be computed in time n log n as the number of points n associated with such methods are a class of random variables defined to be functions of a given point and its nearest neighbours in the sample if the sample points are independent and identically distributed the associated random variables will also be identically distributed but not independent despite this we show that random variables of this type satisfy a strong law of large numbers in the sense that their sample means converge to their expected values almost surely as the number of sample points n

application of the implicit maccormack scheme to the parabolized navier stokes equations application de la methode de differences finies implicite de maccormack a la resolution des equations bidimensionnelles parabolisees de navier stokes discussion des avantages et inconvenients de la methode comparaison avec le schema de beam warming en ce qui concerne la precision la stabilite le temps de calcul la place en memoire et la facilite d implementation application a un ecoulement laminaire hypersonique sur un coin de compression

a computationally efficient estimator for mutual information mutual information quantifies the determinism that exists in a relationship between random variables and thus plays an important role in exploratory data analysis we investigate a class of non parametric estimators for mutual information based on the nearest neighbour structure of observations in both the joint and marginal spaces unless both marginal spaces are one dimensional we demonstrate that a well known estimator of this type can be computationally expensive under certain conditions and propose a computationally efficient alternative that has a time complexity of order o n log n a s the number of observations n n

on the design of an optimal quantizer the problem of designing an optimal quantizer with a fixed number of levels for a wide class of error weighting functions and an arbitrary distribution function is discussed the existence of an optimal quantizer is proved and a two stage algorithm for its design is suggested in this algorithm at the first stage lloyd s iterative method i is applied for reducing the region where at the second stage the search for an optimal quantizer is performed using a hybrid of the dynamic programming algorithm and the lloyd max algorithm which achieves the absolute optimality of dynamic programming with much less computational effort for a continuous distribution with log concave density and an increasing convex weighting function of the absolute quantization error a reliable method is presented to compute the parameters of the optimal quantizer with a known precision using a generalization either of lloyd s method i or of the lloyd max algorithm

inference for exponential order statistic models based on an integrated likelihood function abstract methods of statistical inference are developed for the exponential order statistic eos model where only a subset of order statistics from a collection of n iid exponential random detection times is observable when the rate parameter for detections is unknown the maximum likelihood estimator mle of the unknown integer parameter n can be infinite with substantial probability inference for n is developed using a pseudolikelihood function obtained by integrating out the rate parameter the estimator that maximizes this function called the integrated likelihood estimator ile is shown to be finite and to have better sampling properties than the mle parameter based asymptotics are developed for the case where n is large application of the methodology is illustrated using two datasets

a hypothesis test for the end of a common source outbreak the objective of this article is to develop a hypothesis testing procedure to determine whether a common source outbreak has ended we consider the case when neither the calendar date of exposure to the pathogen nor the exact incubation period distribution is known the hypothesis testing procedure is based on the spacings between ordered calendar dates of disease onset of the cases a simulation study was performed to evaluate the robustness of the methods to various models for the incubation period of infectious diseases we investigated the impact of multiple testing on the overall outbreak wise type i error probability we derive expressions for the outbreak wise type i error probability and show that multiple testing has minimal effect on inflating that error probability the results are discussed in the context of the 2001 u s anthrax outbreak

optimal approximation and quantisation abstract a number of interesting functionals f x of a finite point set x in the euclidean space can be represented as integrals of a function that depends on the integration variable y and x restricted onto a certain set c y x that is determined by y and x and satisfies separation and uniform boundedness conditions for instance c y x can be the voronoi cell generated by x that contains y we single out the general properties of c and that ensure that the normalised infimum of f x over all sets x with cardinality n converges to a limit that can be identified using a particular form of the considered functionals include those that appear in quantisation problems for probability measures and in finding the optimal approximation of a function by splines tangent planes and triangulated surfaces for instance it is shown that the minimum l approximation error normalised by n of a sufficiently smooth bivariate convex function f using the convex triangulation converges to the l 1 1 norm of the determinant of the second derivative of f times a certain absolute constant

differentiation and regularity of semi discrete optimal transport with respect to the parameters of the discrete measure this paper aims at determining under which conditions the semi discrete optimal transport is twice dierentiable with respect to the parameters of the discrete measure and exhibits numerical applications the discussion focuses on minimal conditions on the background measure to ensure dierentiability we provide numerical illustrations in stippling and blue noise problems

computing of separated flow using the space marching conservative supra characteristics method steady hypersonic viscous flows over compression corners with streamwise separation have been computed using the space marching conservative supra characteristics method of lombard this method permits stable space marching of the parabolized navier stokes equations through large separated flow regions the present method has been used to compute surface pressure heat transfer and skin friction coefficients for two compression corner cases studied experimentally by holden and moselle the computed results compare favorably with previous navier stokes results and experimental data the present method has also been compared with the conventional beam warming scheme for solving the parabolized navier stokes equations comparisons are made for accuracy computer time and computer storage

a numerical method for solving the equations of compressible viscous flow although much progress has already been made in solving problems in aerodynamic design many new developments are still needed before the equations for unsteady compressible viscous flow can be solved routinely this paper describes one such development a new method for solving these equations has been devised that 1 is second order accurate in space and time 2 is unconditionally stable 3 preserves conservation form 4 requires no block or scalar tridiagonal inversions 5 is simple and straightforward to program estimated 10 modification for the update of many existing programs 6 is more efficient than present methods and 7 should easily adapt to current and future computer architectures computational results for laminar and turbulent flows at reynolds numbers from 3 x 10 exp 5 to 3 x 10 exp 7 and at cfl numbers as high as 10 exp 3 are compared with theory and experiment

quantization of probability distributions under norm based distortion measures ii self similar distributions for a probability measure p on rd and n n consider en inf mina v x a dp x where the infimum is taken over all subsets of rd with card n and v is a nondecreasing function under certain conditions on v we derive the precise n asymptotics of en for self similar distributions p and we find the asymptotic performance of optimal quantizers using weighted empirical measures

quantization dimension for condensation systems ii the geometric mean error abstract let f i 1 n be a family of similitudes on r 1 satisfying the strong separation condition and the self similar measure associated with f i 1 n and a probability vector t 1 t n let be the attracting measure of a condensation system associated with f i 1 n and a probability vector p 0 p 1 p n we establish a relationship between the quantization dimension of and its mass distribution on cylinder sets

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

space marching calculations about hypersonic configurations using a solution adaptive mesh algorithm a solution adaptive marching algorithm is developed and applied to a three dimensional parabolized navierstokes equation solver the resulting algorithm obtains accurate solutions by using a spatial marching adaptivegrid procedure the adaptation step redistributes grid points line by line in both crossflow directions with grid point motion controlled by forces analogous to tensional and torsional spring forces with the tensional force proportional to the error measure or weighting functions the solution adaptive marching procedure is applied to the hypersonic flow about two generic aircraft configurations the first of these is an all body type geometry with elliptical cross sections and is studied at angles of attack of 0 5 and 15 deg the second geometry is a generic blended vv ing body design results are presented that demonstrate the improvements in flow field resolution obtainable with the solution adaptive marching procedure over conventional fixed grid techniques computed pitot pressure profiles obtained using the solution adaptive algorithm show improved agreement with experimental data compared to predictions obtained using a fixed grid

adaptive reconnection based arbitrary lagrangian eulerian method ew present a new adaptive arbitrary lagrangian eulerian ale method this method is based on the reconnection based ale reale methodology of refs 35 34 6 the main elements in a standard reale method are an explicit lagrangian phase on an arbitrary polygonal in 2d mesh in which the solution and positions of grid nodes are updated a rezoning phase in which a new grid is defined by changing the connectivity using voronoi tessellation but not the number of cells and a remapping phase in which the lagrangian solution is transferred onto the new grid in the standard reale method the rezoned mesh is smoothed by using one or several steps toward centroidal voronoi tessellation but it is not adapted to the solution in any way in the current paper we present a new adaptive reale method a reale that is based on the following design principles first a monitor function or error indicator based on the hessian of some flow parameter s is utilized second an equi distribution principle for the monitor function is used as a criterion for adapting the mesh third a centroidal voronoi tessellation is used to adapt the mesh fourth we scale the monitor function to avoid very small and large cells and then smooth it to permit the use of theoretical results related to weighted centroidal voronoi tessellation in the a reale method both number of cells and their locations are allowed to change at the rezone stage on each time step the number of generators at each time step is chosen to guarantee the required spatial resolution in regions where monitor function reaches its maximum value we present all details required for implementation of new adaptive a reale method and demonstrate its performance in comparison with standard reale method on series of numerical examples

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

bounds on the geometric complexity of optimal centroidal voronoi tesselations in 3d gersho s conjecture in 3d asserts the asymptotic periodicity and structure of the optimal centroidal voronoi tessellation this relatively simple crystallization problem remains to date open we prove bounds on the geometric complexity of optimal centroidal voronoi tessellations as the number of generators tends to infinity combined with an approach of gruber in 2d these bounds reduce the resolution of the 3d gersho s conjecture to a finite albeit very large computation of an explicit convex problem in finitely many variables

the hexagon theorem we wish to int s z f z 2 s the unit square over all choices of f z which take just n values for large n it is shown that f z approaches that function which on each face of a hexagonal mesh has the value of the centroid of that face this problem in the design of quantizers for a pulse code modulation pcm system arose in which pairs of adjacent samples are quantized in an effort to capitalize on the correlation between samples and on the geometry of the plane

some remarks on absolute continuity and quantization of probability measures we introduce a notion of monotonicity of dimensions of measures we show that the upper and lower quantization dimensions are not monotone we give sufficient conditions in terms of so called vanishing rates such that implies d r d r as an application we determine the quantization dimension of a class of measures which are absolutely continuous w r t some self similar measure with the corresponding radon nikodym derivative bounded or unbounded we study the set of quantization dimensions of measures which are absolutely continuous w r t a given probability measure we prove that the infimum on this set coincides with the lower packing dimension of furthermore this infimum can be attained provided that the upper and lower packing dimensions of are equal

non parametric estimation of residual moments and covariance the aim of non parametric regression is to model the behaviour of a response vector y in terms of an explanatory vector x based only on a finite set of empirical observations this is usually perf

asymptotic uniformity of the quantization error of self similar measures let be a self similar measure on mathbb r d associated with a family of contractive similitudes s1 sn and a probability vector p1 pn let alpha n n 1 infty be a sequence of n optimal sets for of order r for each n we denote by p a alpha n a in alpha n a voronoi partition of mathbb r d with respect to n under the strong separation condition for s1 sn we show that the nth quantization error of of order r in 1 infty satisfies the following asymptotic uniformity property int limits p a alpha n rm d x a rd mu x asymp frac 1 n v n r mu quad rm for all a in alpha n

on polynomial based projection indices for exploratory projection pursuit we develop asymptotic theory for two polynomial based methods of estimating orientation in projection pursuit density approximation one of the techniques uses legendre polynomials and has been proposed and implemented by friedman 1 the other employs hermite functions issues of smoothing parameter choice and robustness are addressed it is shown that each method can be used to construct sqrt n consistent estimates of the projection which maximizes distance from normality although the former can only be employed in that manner when the underlying distribution has extremely light tails the former can be used very generally to measure low frequency departure from normality

estimating the number of faults in a system abstract we consider a reliability system with an unknown number of faults which is observed for a fixed length of time failures are assumed to follow an exponential distribution with an unknown mean and we concentrate on estimation of the number of faults v the likelihood function considered as a function of v has its maximum at one of the boundary points the observed number of faults or infinity with substantial probability likelihood interval estimates and improved point estimates are discussed the harmonic mean of the endpoints of the likelihood interval turns out to be a reasonable point estimate we comment on some extensions of this problem suggested in the software reliability literature

convergence of the lloyd algorithm for computing centroidal voronoi tessellations centroidal voronoi tessellations cvts are voronoi tessellations of a bounded geometric domain such that the generating points of the tessellations are also the centroids mass centers of the corresponding voronoi regions with respect to a given density function centroidal voronoi tessellations may also be defined in more abstract and more general settings due to the natural optimization properties enjoyed by cvts they have many applications in diverse fields the lloyd algorithm is one of the most popular iterative schemes for computing the cvts but its theoretical analysis is far from complete in this paper some new analytical results on the local and global convergence of the lloyd algorithm are presented these results are derived through careful utilization of the optimization properties shared by cvts numerical experiments are also provided to substantiate the theoretical analysis

inference and prediction for a general order statistic model with unknown population size abstract suppose that the first n order statistics from a random sample of n positive random variables are observed where n is unknown this the general order statistic model has been applied to the study of market penetration capture recapture burn in in repairable systems software reliability growth the estimation of the number of individuals exposed to radiation and the estimation of the number of unseen species inference is to be made about the unknown parameters especially n and future observations are to be predicted a bayes empirical bayes approach to inference is presented this permits the comparison of competing perhaps nonnested models for the distribution of the random variables in a natural way it also provides easily implemented inference and prediction procedures that avoid the difficulties of non bayesian methods one such difficulty is that the maximum likelihood estimator of n may be infinite results are given for the case in which vague prior information about the model

optimum quantization and its applications abstract minimum sums of moments or equivalently distortion of optimum quantizers play an important role in several branches of mathematics fejes toth s inequality for sums of moments in the plane and zador s asymptotic formula for minimum distortion in euclidean d space are the first precise pertinent results in dimension d 2 in this article these results are generalized in the form of asymptotic formulae for minimum sums of moments resp distortion of optimum quantizers on riemannian d manifolds and normed d spaces in addition we provide geometric and analytic information on the structure of optimum configurations our results are then used to obtain information on i the minimum distortion of high resolution vector quantization and optimum quantizers ii the error of best approximation of probability measures by discrete measures and support sets of best approximating discrete measures iii the minimum error of numerical integration formulae for classes of holder continuous functions and optimum sets of nodes iv best volume approximation of convex bodies by circumscribed convex polytopes and the form of best approximating polytopes and v the minimum isoperimetric quotient of convex polytopes in minkowski spaces and the form of the minimizing polytopes

nondegeneracy and weak global convergence of the lloyd algorithm in mathbb r d the lloyd algorithm originated in the context of optimal quantization and represents a fixed point iteration for computing an optimal quantizer reducing average distortion at every step it constructs a voronoi partition of the domain and replaces each generator with the centroid of the corresponding voronoi cell optimal quantization is obtained in the case of a centroidal voronoi tessellation cvt which is a special voronoi tessellation of a domain omega in mathbb r d having the property that the generators of the voronoi diagram are also the centers of mass with respect to a given density function rho geq 0 of the corresponding voronoi cells the lloyd iteration is currently the most popular and elegant algorithm for computing cvts and optimal quantizers but many questions remain about its convergence especially in d dimensional spaces d 1 in this paper we prove that any limit point of the lloyd iteration in any dimensional spaces is nondegenerate provided that omega is a convex and bounded set and rho belongs to l 1 omega and is positive almost everywhere this ensures that the fixed point map remains closed and hence the standard theory of descent methods guarantees weak global convergence of the lloyd iteration to the set of nondegenerate fixed point quantizers while previously only conjectured the convergence properties of the lloyd iteration are rigorously justified under such minimal regularity assumptions on the density functional the results presented in this paper go beyond existing convergence theories for cvts and optimal quantization related algorithms and should be of interest to both the mathematical and engineering communities

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

solution adaptive grid procedure for high speed parabolic flow solvers a solution adaptive grid procedure based bn an error equidistribution scheme is developed and applied to a parabolized navier stokes solver an improved method of selecting weighting functions is introduced that involves normalizing a combination of flowfield gradients and curvature of a number of dependent variables and then selecting the largest at each point the scheme redistributes grid points line by line with grid point motion controlled by forces analogous to tensional and torsional spring forces with the spring constants set equal to the weighting functions torsional terms are functions of the grid point positions along neighboring grid lines and provide grid smoothness and stability a grid fitting scheme is introduced for external flows in which the number of grid points in the freestream are reduced to a minimum results for several problems are presented to demonstrate the improvements obtainable with the solution adaptive grid procedure

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments