choice of kernel order in density estimation the selection of the order i e number of vanishing moments of the kernel in a kernel density estimator is considered from two points of view first theoretical properties are investigated by a mean integrated squared error analysis of the problem second and perhaps more importantly cross validation is proposed as a practical method of choice and theoretical backing for this is provided through an asymptotic optimality result

simulating time harmonic flows with the lattice boltzmann method we simulate time harmonic flows by the lattice boltzmann method we propose a general scheme to choose simulation parameters under the constraints of fixed reynolds and womersley numbers and with a specified simulation error under these constraints parameters are chosen to minimize the execution time numerical stability is studied in a range of reynolds and womersley numbers as an example of time harmonic flow simulations results of blood flow in a human abdominal aorta are presented

analysis of the lattice boltzmann treatment of hydrodynamics it is shown that the lattice boltzmann model can be made to obey the three dimensional thermal navier stokes equations by fixing 26 moments of the equilibrium distribution function this can be achieved on a cubic lattice with particle velocities connecting each lattice site to its first second and third nearest neighbors however this minimal model is not linearly stable in the limit of low transport coefficients additional particle velocities must be selected to achieve linear stability

smoothing splines regression derivatives and deconvolution abstract the statistical properties of a cubic smoothing spline and its derivative are analyzed it is shown that unless unnatural boundary conditions hold the integrated squared bias is dominated by local effects near the boundary similar effects are shown to occur in the regularized solution of a translation kernel integral equation these results are derived by developing a fourier representation for a smoothing spline author

on a problem concerning spacings let un n 1 2 be a sequence of independent uniformly distributed random variables on 0 1 and si n i 1 n be the spacings induced by the order statistics of u1 un 1 denote begin array l m l n rm max min s i j n rm 1 mathbin lower 3ex hbox buildrel over smash scriptstyle vphantom x i mathbin lower 3ex hbox buildrel over smash scriptstyle vphantom x n l 1 rm 0 mathbin lower 3ex hbox buildrel over smash scriptstyle vphantom x j l end array the exact distribution of m2 n is determined and based on analysis of something called fibonacci distribution the weak and almost sure convergence of the sequence m2 n are discussed furthermore the limiting distribution of ml n is determined for any fixed l and the equation p mathop lim limits n to infty rm nl n m l n n log n 1 1 is shown for some sequence ln

grid refinement for lattice bgk models lattice boltzmann models proposed at the end of the 1980s as the noise free version of lattice gas models are based on gas kinetic representation of fluid flow their recent modifications the lattice bgk models provide especially simple effective and stable algorithms for the solution of hydrodynamical problems a local second order grid refinement scheme for the lattice bgk model is proposed in this work the refinement scheme and a boundary fitting scheme for complicated geometries are applied to simulate a benchmark problem of flow past a cylinder in a channel with small and moderate reynolds numbers

a semiparametric approach to density estimation abstract one method of fitting a parametric density function f x is first to estimate by maximum likelihood say and then to estimate f x by on the other hand when the parametric model does not hold the true density f x may be estimated nonparametrically as in the case of a kernel estimate the key idea proposed is to fit a combination of the parametric and nonparametric estimates namely where 0 1 is unknown the parameter is estimated from the data and its estimate is then used in as the proposed density estimate the main point is that we expect to be close to unity when the parametric model prevails and close to zero when it does not hold we show that under certain conditions converges to the parametric density when the parametric model holds and approaches the true f x when the parametric model does not hold the procedure was applied to a number of actual data sets in each case the maximum likelihood estimate was readily obtained and the semiparametric density es

the flat model structure on given a cotorsion pair a b in an abelian category c with enough a objects and enough b objects we define two cotorsion pairs in the category ch c of unbounded chain complexes we see that these two cotorsion pairs are related in a nice way when a b is hereditary we then show that both of these induced cotorsion pairs are complete when a b is the flat cotorsion pair of r modules this proves the flat cover conjecture for possibly unbounded chain complexes and also gives us a new flat model category structure on ch r in the last section we use the theory of model categories to show that we can define ext n r m n using a flat resolution of m and a cotorsion coresolution of n

classification of probability laws by tail behavior abstract in this article i refine parzen s density quantile tail exponent classification of probability laws by tail behavior by subdividing his medium tailed class into medium short medium medium and medium long i give a physical interpretation of short medium and long tailed distributions in terms of the limiting size of the extreme spacings in a random sample from the distribution i show that this classification by the limiting size of extreme spacings fits nicely within the framework of my refinement of parzen s density quantile classification

the consistency of automatic kernel density estimates we consider the parzen rosenblatt kernel density estimate on rd with data dependent smoothing factor sufficient conditions on the asymptotic behavior of the smoothing factor are given under which the estimate is pointwise consistent almost everywhere for all densities f to be estimated when the smoothing factor is a function only of the sample size n it is shown that these conditions are also necessary a generalization of results by deheuvels the consistency of various automatic kernel density estimates is a simple consequence of these theorems

some criticisms of stochastic shrinkage and ridge regression with counterexamples the existence theorems for reduced mean squared error ridge regression and shrunken estimators are commonly used to justify the use of completely stochastic versions of these same estimators but there are no guarantees for improvement over the original unbiased estimators in particular this is demonstrated for a case where all possible values of the stochastic factor are known to satisfy the relevant inequality of the corresponding existence theorem examples of estimators with guaranteed improvement are given

a data dependent approach to density estimation a data dependent approach to density estimation is proposed here the proposed method requires boundedness and some weak integrability condition on the unknown density but not any assumption of smoothness applications to histogram kernel spline and orthogonal series methods are discussed

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

on bandwidth choice for spatial data density estimation bandwidth choice is crucial in spatial kernel estimation in exploring non gaussian complex spatial data the paper investigates the choice of adaptive and non adaptive bandwidths for density estimation given data on a spatial lattice an adaptive bandwidth depends on local data and hence adaptively conforms with local features of the spatial data we propose a spatial cross validation scv choice of a global bandwidth this is done first with a pilot density involved in the expression for the adaptive bandwidth the optimality of the procedure is established and it is shown that a non adaptive bandwidth choice comes out as a special case although the cross validation idea has been popular for choosing a non adaptive bandwidth in data driven smoothing of independent and time series data its theory and application have not been much investigated for spatial data for the adaptive case there is little theory even for independent data conditions that ensure asymptotic optimality of the scv selected bandwidth are derived actually also extending time series and independent data optimality results further for the adaptive bandwidth with an estimated pilot density oracle properties of the resultant density estimator are obtained asymptotically as if the true pilot were known numerical simulations show that finite sample performance of the scv adaptive bandwidth choice works quite well it outperforms the existing r routines such as the rule of thumb and the so called second generation sheather jones bandwidths for moderate and big data sets an empirical application to a set of spatial soil data is further implemented with non gaussian features significantly identified

recent developments in nonparametric density estimation advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where restrictions are imposed upon the class of densities or on the form of the likelihood function nonparametric density estimators that are data adaptive and lead to locally smoothed estimators are also discussed these include variable partition histograms estimators based on statistically equivalent blocks nearest neighbor estimators variable kernel estimators and adaptive kernel estimators for the multivariate case extensions of methods of univariate density estimation are usually straightforward but can be computationally expensive a method of multivariate density estimation that did not spring from a univariate generalization is described namely projection pursuit density estimation in which both dimensionality reduction and density estimation can be pursued at the same time finally some areas of related research are mentioned such as nonparametric estimation of functionals of a density robust parametric estimation semiparametric models and density estimation for censored and incomplete data directional and spherical data and density estimation for dependent sequences of observations

nonparametric policy analysis abstract this article considers the problem of predicting the mean effect of a change in the distribution of certain policy related variables on a dependent variable y this is conventionally done using a parametric model if however the conditional expectation of y given policy and nonpolicy variables x is unaltered by the policy intervention and if the support of x after the policy intervention lies within the support of x before the intervention then this analysis can be performed nonparametrically the proposed nonparametric estimator is developed for the model yj g xj a dj uj where g is a continuous unknown function of the continuous variables x dj is an m vector of dummy variables a is an m vector of unknown parameters representing fixed cell specific effects and uj is an error term with e uj xj dj 0 the estimand is b eg xj eg xj where x and x respectively denote the values of x before and after the policy intervention a nonparametric estimator bn is propo

lattice boltzmann model for free surface flow and its application to filling process in casting a generalized lattice boltzmann model to simulate free surface is constructed in both two and three dimensions the proposed model satisfies the interfacial boundary conditions accurately a distinctive feature of the model is that the collision processes is carried out only on the points occupied partially or fully by the fluid to maintain a sharp interfacial front the method includes an anti diffusion algorithm the unknown distribution functions at the interfacial region are constructed according to the first order chapman enskog analysis the interfacial boundary conditions are satisfied exactly by the coefficients in the chapman enskog expansion the distribution functions are naturally expressed in the local interfacial coordinates the macroscopic quantities al the interface are extracted from the least square solutions of a locally linearized system obtained from the known distribution functions the proposed method does not require any geometric front construction and is robust for any interfacial topology simulation results of realistic filling process are presented rectangular cavity in two dimensions and hammer box campbell box sheffield box and motorblock in three dimensions to enhance the stability at high reynolds numbers various upwind type schemes are developed free slip and no slip boundary conditions are also discussed

cross validation in density estimation considerable interest has been shown recently in the use of cross validatory procedures much of this attention is due to the efforts of stone 1974a b 1977a b who has provided a general discussion of the method of cross validatory choice dawid 1974 suggested that in certain circumstances cross validation might lead to inconsistency and some of these fears were confirmed by stone 1977b in the present paper we examine a popular cross validatory argument for selecting the smoothing parameter or window size in the nonparametric estimation of a density we derive results describing the order of magnitude of the cross validatory window and show that the resulting estimator will perform suboptimally our results are confirmed by monte carlo trials alternative techniques such as that proposed by silverman 1978a seem preferable habbema hermans van den broek 1974 suggested a method for discriminant analysis based on nonparametric density estimation they considered kernel type estimates and in the univariate case these usually have the form

cellular automaton fluids a model for flow in porous media numerical models of fluid flow through porous media can be developed from either microscopic or macroscopic properties the large scale viewpoint is perhaps the most prevalent darcy s law relates the chief macroscopic parameters of interest flow rate permeability viscosity and pressure gradient and may be invoked to solve for any of these parameters when the others are known in practical situations however this solution may not be possible attention is then typically focused on the estimation of permeability and numerous numerical methods based on knowledge of the microscopic pore space geometry have been proposed because the intrinsic inhomogeneity of porous media makes the application of proper boundary conditions difficult microscopic flow calculations have typically been achieved with idealized arrays of geometrically simple pores throats and cracks i propose here an attractive alternative which can freely and accurately model fluid flow in grossly irregular geometries this new method s

on the relationship between stability of extreme order statistics and convergence of the maximum likelihood kernel density estimate by michel broniatowski paul deheuvels and luc devroyeiuniversite pierre et marie curie universite pierre et marie curie andmcgill universitylet f be a density on the real line and let f be the kernel estimate of fin which the smoothing factor is obtained by maximizing the cross validatedlikelihood product according to the method of duin and habbema hermansand vandenbroek under mild regularity conditions on the kernel and f weshow among other things that f jf f 0 almost surely if and only if thesample extremes of f are strongly stable

totally acyclic complexes over noetherian schemes abstract we define a notion of total acyclicity for complexes of flat quasi coherent sheaves on a semi separated noetherian scheme and study these complexes using the pure derived category of flat quasi coherent sheaves we prove that a scheme is gorenstein if and only if every acyclic complex of flat quasi coherent sheaves is totally acyclic our formalism also removes the need for a dualising complex in several known results for rings including jorgensen s proof of the existence of gorenstein projective precovers

acceleration of lattice bgk schemes with grid refinement abstract the lbgk method with local grid refinement has been shown to be an efficient and accurate tool for the simulation of incompressible viscous flows over complex geometries in the present study further improvement of this concept is proposed enabling the use of smaller amount of time steps on refined grids without impairing the spatial or temporal accuracy this extension of the lbgk method has been proved by analytical and numerical investigations the gain in computational time was found to be significant

data based optimal smoothing of orthogonal series density estimates let f be a density possessing some smoothness properties and let x 1 cdots x n be independent observations from f some desirable properties of orthogonal series density estimates f n m lambda of f of the form f n m lambda t sum n nu 1 frac hat f nu 1 lambda nu 2m phi nu t where phi nu is an orthonormal sequence and hat f nu 1 n sum n j 1 phi nu x j is an estimate of f nu int phi nu t f t dt are discussed the parameter lambda plays the role of a bandwidth or smoothing parameter and m controls a shape factor the major novel result of this note is a simple method for estimating lambda and m from the data in an objective manner to minimize integrated mean square error the results extend to multivariate estimates

weak convergence of k nn density and regression estimators with varying k and applications on considere le probleme sous une condition de lissite du second ordre dans le cas unidimensionnel

on overfitting generalization and randomly expanded training sets an algorithmic procedure is developed for the random expansion of a given training set to combat overfitting and improve the generalization ability of backpropagation trained multilayer perceptrons mlps the training set is k means clustered and locally most entropic colored gaussian joint input output probability density function estimates are formed per cluster the number of clusters is chosen such that the resulting overall colored gaussian mixture exhibits minimum differential entropy upon global cross validated shaping numerical studies on real data and synthetic data examples drawn from the literature illustrate and support these theoretical developments

implementation aspects of 3d lattice bgk in many realistic fluid dynamical simulations the specification of the boundary conditions the error sources and the number of time steps to reach a steady state are important practical considerations in this paper we study these issues in the case of the lattice bgk model the objective is to present a comprehensive overview of some pitfalls and shortcomings of the lattice bgk method and to introduce some new ideas useful in practical simulations we begin with an evaluation of the widely used bounce back boundary condition in staircase geometries by simulating flow in an inclined tube it is shown that the bounce back scheme is first order accurate in space when the location of the non slip wall is assumed to be at the boundary nodes moreover for a specific inclination angle of 45 degrees the scheme is found to be second order accurate when the location of the non slip velocity is fitted halfway between the last fluid nodes and the first solid nodes the error as a function of the relaxation parameter is in that case qualitatively similar to that of flat walls next a comparison of simulations of fluid flow by means of pressure boundaries and by means of body force is presented a good agreement between these two boundary conditions has been found in the creeping flow regime for higher reynolds numbers differences have been found that are probably caused by problems associated with the pressure boundaries furthermore two widely used 3d models namelyd3q15andd3q19 are analysed it is shown that thed3q15model may induce artificial checkerboard invariants due to the connectivity of the lattice finally a new iterative method which significantly reduces the saturation time is presented and validated on different benchmark problems

heat transfer in lattice bgk modeled fluid the thermal lattice bgk model is a recently suggested numerical tool aiming at solving problems of thermohydrodynamics the quality of the lattice bgk simulation is checked in this paper by calculating temperature profiles in the couette flow under different eckert and mach numbers a revised lower order model is proposed to improve the accuracy and the higher order model is proved to be advantageous in this respect especially in the flow regime with a higher mach number

cosupports and minimal pure injective resolutions of affine rings abstract we prove that any affine ring r over a field k has full cosupport i e the cosupport of r is equal to spec r using this fact we give a complete description of all terms in a minimal pure injective resolution of r provided that k 1 and dim r 2 or k 1 and dim r 2 as a corollary we obtain a partial answer to a conjecture by gruson

uniform consistency of automatic and location adaptive delta sequence estimators the class of delta sequence estimators for a probability density includes the kernel histogram and orthogonal series types because each can be characterized as a collection of averages of some function that is indexed by a smoothing parameter there are two important extensions of this class the first allows a random smoothing parameter for example that specified by a cross validation method the second allows the smoothing parameter to be a function of location for example an estimator based on nearest neighbor distance in this paper a general method is presented which establishes uniform consistency for all of these estimators

an euler solver based on locally adaptive discrete velocities a new discrete velocity model is presented to solve the three dimensional euler equations the velocities in the model are of an adaptive nature both the origin of the discrete velocity space and the magnitudes of the discrete velocities are dependent on the local flow and are used in a finite volume context the numerical implementation of the model follows the near equilibrium flow method of nadiga and pullin and results in a scheme which is second order in space in the smooth regions and between first and second order at discontinuities and second order in time the three dimensional code is included for one choice of the scaling between the magnitude of the discrete velocities and the local internal energy of the flow the method reduces to a flux splitting scheme based on characteristics as a preliminary exercise the result of the sod shock tube simulation is compared to the exact solution

linear lattice boltzmann schemes for acoustic parameter choices and isotropy properties in this paper we investigate the numerous possible parameter choices for linear lattice boltzmann schemes according to the definition of the order of isotropy given in augier et al in press 4 this property written in a general framework that includes all of the d dq q schemes can be understood through a group operation the resulting relationships between the parameters of the scheme defining the equilibrium states and relaxation times yield a rigorous methodology that should be followed if one is to ensure isotropy at a given order for acoustic applications in two spacial dimensions namely d 2q 9 and d 2q 1 3 schemes this methodology is used to propose a full description of the sets of parameters that are isotropic at order m email protected 1 2 3 4 5 for d 2q 9 and email protected 1 2 for d 2q 1 3 we include numerical illustrations for the d 2q 9 scheme obtained with the code lbmpy developed in the laboratory of mathematics of the university paris sud

homological dimensions and regular rings abstract a question of avramov and foxby concerning injective dimension of complexes is settled in the affirmative for the class of noetherian rings a key step in the proof is to recast the problem on hand into one about the homotopy category of complexes of injective modules analogous results for flat dimension and projective dimension are also established

heuristic optimality criterion algorithm for shape design of fluid flow this paper presents a heuristic optimality criterion algorithm for shape design of fluid flow in this algorithm the lattice boltzmann method lbm is utilized to calculate the flow field of a fluid domain which is divided into elemental cells a heuristic optimality criterion is applied for cells at the solid fluid interface i e the dynamic pressure for fluid cells and the viscous stress on their neighboring solid cells an automatic program is processed step by step to exchange the positions of solid and fluid cells identified by the optimality criterion with the objective of decreasing the flow resistance at the constraint of constant fluid volume to illustrate the procedure of this algorithm for shape design of fluid flow two simple examples are presented one with fluid flowing through a right angle elbow and the other through a converging t junction numerical results show that this algorithm can successfully reduce the total pressure drop of the system demonstrating its potential applications in engineering optimal design

multifractal analysis of flow velocity simulated with the lattice model approach in idealized three dimensional porous media 1 the strange attractor formalism has been used to check whether the flow velocity in idealized three dimensional porous media could be described by the multifractal approach the flow velocity magnitude was simulated with a lattice model in six media with different porosities the results obtained from the analysis of the mass exponent functions and the multifractal and generalized dimensions spectra show that the velocity of the simulated flows has a multifractal type of scaling that becomes stronger as the porosity decreases a correlation analysis has been performed showing the relationships between the statistical properties and the multifractal parameters of the velocity magnitude distribution

a novel lattice bgk approach for low mach number combustion an extended lattice boltzmann bgk model is presented for the simulation of low mach number flows with significant density changes for applications to reactive flows this new model is coupled with a finite difference scheme for solving the transport equations of energy and species with a boundary fitting formulation and local grid refinement the scheme enables accurate and efficient computations of low mach number reactive flows in complex geometry on the simplest cartesian grids examples of reactive flows around porous burners are presented

error analysis and correction for lattice boltzmann simulated flow conductance in capillaries of different shapes and alignments we study the relative error in conductance calculations for simulated flow of a single component single phase fluid through a capillary in three dimensions by the lattice boltzmann lb method with bounce back boundary conditions the relative error with respect to analytical results for capillary cross sections of circular triangular and square shapes are calculated as a function of the cross section diameter a and for different alignment of the cross section relative to the underlying lattice grid it is shown when the shapes are not aligned perfectly to the lattice that the relative error decreases systematically with the size a as 1 a when a is evaluated by mapping the computed cross sectional area in terms of the enclosed number of grid points to the respective geometrical shapes concerned for perfectly aligned geometries viz the square capillary aligned to the lb lattice grid or rotated with its side along the diagonal of the lb grid the relative error decreases as 1 a 2 a simple method is suggested to locate the boundary wall depending on its orientation relative to the grid such that the exact conductance of the new shape matches the lb computed conductance

review papers recent developments in nonparametric density estimation abstract advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where re

absolute relative and tate cohomology of modules of finite gorenstein dimension we study finitely generated modules m over a ring r noetherian on both sides if m has finite gorenstein dimension mbox g dim rm in the sense of auslander and bridger then it determines two other cohomology theories besides the one given by the absolute cohomology functors rm ext n r m relative cohomology functors rm ext n mathcal g m are defined for all non negative integers n they treat the modules of gorenstein dimension 0 as projectives and vanish for n mbox g dim rm tate cohomology functors widehat rm ext n r m are defined for all integers n all groups widehat rm ext n r m n vanish if m or n has finite projective dimension comparison morphisms varepsilon mathcal g n colon rm ext n mathcal g m to rm ext n r m and varepsilon r n colon rm ext n r m to widehat rm ext n r m link these functors we give a self contained treatment of modules of finite g dimension establish basic properties of relative and tate cohomology and embed the comparison morphisms into a canonical long exact sequence 0 to rm ext 1 mathcal g m to cdots to rm ext n mathcal g m to rm ext n r m to widehat rm ext n r m to rm ext n 1 mathcal g m to cdots we show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings 2000 mathematical subject classification 16e05 13h10 18g25

extent to which least squares cross validation minimises integrated square error in nonparametric density estimation let ho o and c be the windows which minimise mean integrated square error integrated square error and the least squares cross validatory criterion respectively for kernel density estimates it is argued that o not ho should be the benchmark for comparing different data driven approaches to the determination of window size asymptotic properties of ho o and c o and of differences between integrated square errors evaluated at these windows are derived it is shown that in comparison to the benchmark o the observable window c performs as well as the so called optimal but unattainable window ho to both first and second order

empirical functionals and efficient smoothing parameter selection a striking feature of curve estimation is that the smoothing parameter h 0 which minimizes the squared error of a kernel or smoothing spline estimator is very difficult to estimate this is manifest both in slow rates of convergence and in high variability of standard methods such as cross validation we quantify this difficulty by describing nonparametric information bounds and exhibit asymptotically efficient estimators of h 0 that attain the bounds the efficient estimators are substantially less variable than cross validation and other current procedures and simulations suggest that they may offer improvements at moderate sample sizes at least in terms of minimizing the squared error

mean squared error properties of generalized ridge estimators abstract mean squared errors are derived and examined for a number of generalized ridge gr regression estimators and the estimators are compared in a number of specific regression models the potential of gr estimators is discussed

using additive noise in back propagation training the possibility of improving the generalization capability of a neural network by introducing additive noise to the training samples is discussed the network considered is a feedforward layered neural network trained with the back propagation algorithm back propagation training is viewed as nonlinear least squares regression and the additive noise is interpreted as generating a kernel estimate of the probability density that describes the training vector distribution two specific application types are considered pattern classifier networks and estimation of a nonstochastic mapping from data corrupted by measurement errors it is not proved that the introduction of additive noise to the training vectors always improves network generalization however the analysis suggests mathematically justified rules for choosing the characteristics of noise if additive noise is used in training results of mathematical statistics are used to establish various asymptotic consistency results for the proposed method numerical simulations support the applicability of the training method

initial and boundary conditions for the lattice boltzmann method an alternative approach of implementing initial and boundary conditions for the lattice boltzmann method is presented the basic idea is to calculate the lattice boltzmann populations at a boundary node from the fluid variables that are specified at this node using the gradients of the fluid velocity the numerical performance of the lattice boltzmann method is tested on several problems with exact solutions and is also compared to an explicit finite difference projection method the discretization error of the lattice boltzmann method decreases quadratically with finer resolution both in space and in time the roundoff error of the lattice boltzmann method creates problems unless double precision arithmetic is used

cosupport computations for finitely generated modules over commutative noetherian rings abstract we show that the cosupport of a commutative noetherian ring is precisely the set of primes appearing in a minimal pure injective resolution of the ring as an application of this we prove that every countable commutative noetherian ring has full cosupport we also settle the comparison of cosupport and support of finitely generated modules over any commutative noetherian ring of finite krull dimension finally we give an example showing that the cosupport of a finitely generated module need not be a closed subset of spec r providing a negative answer to a question of sather wagstaff and wicklein 29

complete modules and torsion modules suppose that r is a ring and that a is a chain complex over r inside the derived category of differential graded a modules there are naturally defined subcategories of a torsion objects and of a complete objects under a finiteness condition on a we develop a morita theory for these subcat egories find conceptual interpretations for some associated algebraic functors and in appropriate commutative situations identify the associated functors as local homology or local cohomology some of the results are suprising even in the case r z and a z p 1 introduction let r be a ring and r mod the derived category of chain complexes of left modules see section 1 2 we choose a fixed complex a which is perfect in other words isomorphic in mod to a complex of finite length in which the entries are finitely generated projective modules we de clare another complex n to be a trivial if hom a a0 0 where hom denotes the chain complex of morphisms in r mod going further we say that x is a torsion if hom x at 0 for all a trivial n9 and that x is a complete if homr n9x 0 for all a trivial n we then study the category ators of a torsion complexes and the category acomp of a complete complexes both of these are triangulated full subcategories of r mod it turns out that these categories are equivalent to one another and also equivalent to the derived category of dif ferential graded modules over the endomorphism complex of a we construct approximation functors celu r mod at0rs and a r mod acomp for an object m of r mod9 the complex celu m is a kind of a cellular approxi mation to m in the sense that it is the best approximation to m which can be cobbled together from a and its suspensions the complex ma is the bousfield localization of m with respect to a homology theory on r mod derived from a we provide algebraic formulas for the functors and find that the functors are related in interesting ways one of which involves an arithmetic square we also show that if cciia r has certain finite dimensionality properties then an object m of r mod is a torsion or a complete if and only if the homology groups 77 m individually satisfy appropriate torsion or completeness conditions suppose now that r is a commutative ring that 7 c r is a finitely generated ideal that a r i9 and that k is the associated koszul complex the complex a

the amount of noise inherent in bandwidth selection for a kernel density estimator abstract any practical method of constructing a bandwidth must depend only on a statistical sample and should produce some sort of estimate of this bandwidth the purpose of this paper is to show that there a well defined limits to the accuracy of all data driven bandwidth estimates put another way there is an unbridgeable gap between the minimum integrated square error attained using a optimal bandwidth and the minimum achievable integrated square error using a data driven bandwidth estimate additional keywords stochastic processes cross validation and random variables

minimal pure injective resolutions of flat modules abstract every module has a minimal pure injective resolution for a flat modul over a noetherian ring the structure of the pure injective modules appearing in such a resolution is now known and can be used to give information about minimal pure injective resolutions of flat modules this information can in turn be used to give new proofs of results about the projective dimension of flat modules and at the same time sharpen these results a change of ring theorem gives the perhaps surprising result that the pure injective dimensions of the coordinate rings of affine algebraic varieties over some fixed ground field are the same for all varieties of a given dimension n

lattice gas automata for flow through porous media abstract lattice gas hydrodynamic models for flows through porous media in two and three dimensions are described the computational method easily handles arbitrary boundaries and a large range of reynolds numbers darcy s law is confirmed for poiseuille flow and for complicated boundary flows multiply connected pore structures similar to actual sandstone with fixed fractal dimension and porosity are generated permeability as a function of fractal dimension and porosity is calculated and compared with results of other methods and experiments

monte carlo study of three data based nonparametric probability density estimators abstract although the theoretical properties of modern nonparametric probability density estimators have been studied for 25 years there remains the practical problem of how to specify the amount of bias or smoothing in a density estimate based on a random sample in this paper we review and evaluate three recently developed data based algorithms that completely specify a density estimate from a random sample using monte carlo techniques we compare the statistical accuracy of these algorithms as measured by the integrated mean squared error in addition we examine the sensitivity of these algorithms to outliers and estimate computer time requirements one conclusion we draw is that the statistical accuracy of these data based algorithms seems comparable to levels predicted by theoretical models

accurate method for including solid fluid boundary interactions in mesoscopic model fluids particle models are attractive methods for simulating the dynamics of complex mesoscopic fluids many practical applications of this methodology involve flow through a solid geometry as the system is modeled using particles whose positions move continuously in space one might expect that implementing the correct stick boundary condition exactly at the solid fluid interface is straightforward after all unlike discrete methods there is no mapping onto a grid to contend with in this article we describe a method that for axisymmetric flows imposes both the no slip condition and continuity of stress at the interface we show that the new method then accurately reproduces correct hydrodynamic behavior right up to the location of the interface as such computed flow profiles are correct even using a relatively small number of particles to model the fluid

the lattice boltzmann equation a new tool for computational fluid dynamics abstract we present a series of applications which demonstrate that the lattice boltzmann equation is an adequate computational tool to address problems spanning a wide spectrum of fluid regimes ranging from laminar to fully turbulent flows in two and three dimensions

an asymptotically efficient solution to the bandwidth problem of kernel density estimation a data driven method of choosing the bandwidth h of a kernel density estimator is heuristically motivated by considering modifications of the kullback leibler or pseudo likelihood cross validation function it is seen that this means of choosing h is asymptotically equivalent to taking the h that minimizes some compelling error criteria such as the average squared error and the integrated squared error thus for a given kernel function the bandwidth can be chosen optimally without making precise smoothness assumptions on the underlying density

simulation of flows around an impulsively started circular cylinder by taylor series expansion and least squares based lattice boltzmann method the two dimensional incompressible viscous flow past an impulsively started circular cylinder for a wide range of reynolds numbers re 20 9500 is studied computationally by using an explicit taylor series expansion and least squares based lattice boltzmann method the final equation for distribution function in our method is in an explicit form and essentially has no limitation on choice of mesh structure and lattice model it can be easily applied to simulation of flows with curved boundaries such as the problem considered in this work for the flow past an impulsively started circular cylinder numerical results obtained by present method agree very well with experimental data and computational results of navier stokes equations available in the literature

multiscale lattice boltzmann schemes with turbulence modeling abstract the viability of multiscale lattice boltzmann schemes for the numerical simulation of turbulent flows is discussed and numerically demonstrated for turboaxial machine applications the extension of boundary fitting formulas based on wall functions is proposed which enables the efficient computation of turbulent flows in complex curvilinear geometry using a simple cartesian grid examples of two dimensional turbulent flows in an axial compressor cascade are presented

a comparison of cross validation techniques in density estimation in the setting of nonparametric multivariate density estimation theorems are established which allow a comparison of the kullback leibler and the least squares cross validation methods of smoothing parameter selection the family of delta sequence estimators including kemel orthogonal series histogram and histospline estimators is considered these theorems also show that either type of cross validation can be used to compare different estimators e g kernel versus orthogonal series 1 introduction consider the problem of trying to estimate a d dimensional probability density function f x using a random sample xl xn from f most proposed estimators of f depend on a smoothing parameter say x e dr whose selection is crucial to the performance of the estimator in this paper for the large class of delta sequence estimators theorems are obtained which allow comparison of two smoothing parameter selectors which are known to be asymptotically optimal an important consequence of these results is that either smoothing parameter selector may be used for a data based comparison of two density estimators for example kernel versus orthogonal series another attractive feature of these results is that they are set in a quite general framework special cases of which provide simpler proofs of several recent asymptotic optimality results in sections 2 and 3 the family of delta sequence estimators and the smoothing parameter selectors are given the theorems are stated in section 4 with some remarks in section 5 the rest of the paper consists of proofs

a comparison study of multi component lattice boltzmann models for flow in porous media applications a comparison study of three different multi component lattice boltzmann models is carried out to explore their capability of describing binary immiscible fluid systems the shan chen pseudo potential model the oxford free energy model and the colour gradient model are investigated using the multi relaxation time scheme mrt algorithm to study the flow of binary immiscible fluids we investigate poiseuille flow of layered immiscible binary fluids and capillary fingering phenomena and evaluate the results against analytical solutions in addition we examine the capability of the various models to simulate fluids with significant viscosity and density contrast and suitable interface thickness this is of great importance for large scale flow in porous media applications where it is important to minimise the interfacial thickness from a computational point of view we find that the shan chen model can simulate high density ratios up to 800 for binary fluids with the same viscosity imposing a viscosity contrast will lead to highly diffusive interfaces in the shan chen model and therefore this will affect significantly the numerical stability the free energy model and the colour gradient model have similar capabilities on this point they can simulate binary fluids having the same density but with significant viscosity contrast this is of great importance to study the flow of water supercritical co 2 and oil in porous media for co 2 storage and enhanced oil recovery eor operations

flat covers of complexes abstract in this article we define and study flat complexes over any ring also we prove that any complex over a commutative noetherian ring with finite krull dimension has a flat cover and a dg flat cover

minimal semi flat cotorsion replacements and cosupport abstract over a commutative noetherian ring r of finite krull dimension we show that every complex of flat cotorsion r modules decomposes as a direct sum of a minimal complex and a contractible complex moreover we define the notion of a semi flat cotorsion complex as a special type of semi flat complex and provide functorial ways to construct a quasi isomorphism from a semi flat complex to a semi flat cotorsion complex consequently every r complex can be replaced by a minimal semi flat cotorsion complex in the derived category over r furthermore we describe structure of semi flat cotorsion replacements by which we recover classic theorems for finitistic dimensions in addition we improve some results on cosupport and give a cautionary example we also explain that semi flat cotorsion replacements always exist and can be used to describe the derived category over any associative ring

investigation of stability and hydrodynamics of different lattice boltzmann models stability and hydrodynamic behaviors of different lattice boltzmann models including the lattice boltzmann equation lbe the differential lattice boltzmann equation dlbe the interpolation supplemented lattice boltzmann method islbm and the taylor series expansion and least square based lattice boltzmann method tllbm are studied in detail our work is based on the von neumann linearized stability analysis under a uniform flow condition the local stability and hydrodynamic dissipation behaviors are studied by solving the evolution operator of the linearized lattice boltzmann equations numerically our investigation shows that the lbe schemes with interpolations such as dlbe islbm and tllbm improve the numerical stability by increasing hyper viscosities at large wave numbers small scales it was found that these interpolated lbe schemes with the upwind interpolations are more stable than those with central interpolations because of much larger hyper viscosities

on gorenstein projective injective and flat dimensions a functorial description with applications gorenstein homological dimensions are refinements of the classical homological dimensions and finiteness singles out modules with amenable properties reflecting those of modules over gorenstein rings as opposed to their classical counterparts these dimensions do not immediately come with practical and robust criteria for finiteness not even over commutative noetherian local rings in this paper we enlarge the class of rings known to admit good criteria for finiteness of gorenstein dimensions

laws of the iterated logarithm for order statistics of uniform spacings let x 1 x 2 cdots be a sequence of independent uniformly distributed random variables on lbrack 0 1 rbrack and let k n be the k th largest spacing induced by the order statistics of x 1 cdots x n 1 we show that lim sup nk n log n 2 log 2n 1 k quad text almost surely and lim inf nk n log n log 3n c quad text almost surely where log 2 leq c leq 0 and log j is the j times iterated logarithm

nonparametric maximum likelihood estimation by the method of sieves maximum likelihood estimation often fails when the parameter takes values in an infinite dimensional space for example the maximum likelihood method cannot be applied to the completely nonparametric estimation of a density function from an operatorname iid sample the maximum of the likelihood is not attained by any density in this example as in many other examples the parameter space positive functions with area one is too big but the likelihood method can often be salvaged if we first maximize over a constrained subspace of the parameter space and then relax the constraint as the sample size grows this is grenander s method of sieves application of the method sometimes leads to new estimators for familiar problems or to a new motivation for an already well studied technique we will establish some general consistency results for the method and then we will focus on three applications

the homotopy category of flat modules and grothendieck duality let r be a ring we prove that the homotopy category k r proj is always aleph 1 compactly generated and depending on the ring r it may or may not be compactly generated we use this to give a description of k r proj as a quotient of k r flat the remarkable fact is that this new description of k r proj generalizes to non affine schemes this will appear in murfet s thesis

lattice boltzmann computational fluid dynamics in three dimensions the recent development of the lattice gas method and its extension to the lattice boltzmann method have provided new computational schemes for fluid dynamics both methods are fully paralleled and can easily model many different physical problems including flows with complicated boundary conditions in this paper basic principles of a lattice boltzmann computational method are described and applied to several three dimensional benchmark problems in most previous lattice gas and lattice boltzmann methods a face centered hyper cubic lattice in four dimensional space was used to obtain an isotropic stress tensor to conserve computer memory we develop a model which requires 14 moving directions instead of the usual 24 directions lattice boltzmann models describing two phase fluid flows and magnetohydrodynamics can be developed based on this simpler 14 directional lattice comparisons between three dimensional spectral code results and results using our method are given for simple periodic geometries an important property of the lattice boltzmann method is that simulations for flow in simple and complex geometries have the same speed and efficiency while all other methods including the spectral method are unable to model complicated geometries efficiently

cellular automaton fluids 1 basic theory continuum equations are derived for the large scale behavior of a class of cellular automaton models for fluids the cellular automata are discrete analogues of molecular dynamics in which particles with discrete velocities populate the links of a fixed array of sites kinetic equations for microscopic particle distributions are constructed hydrodynamic equations are then derived using the chapman enskog expansion slightly modified navier stokes equations are obtained in two and three dimensions with certain lattices viscosities and other transport coefficients are calculated using the boltzmann transport equation approximation some corrections to the equations of motion for cellular automaton fluids beyond the navier stokes order are given

akaike s information criterion and kullback leibler loss for histogram density estimation we take a detailed look at akaike s information criterion aic and kullback leibler cross validation klcv in the problem of histogram density estimation two different definitions of number of unknown parameters inaic are considered a careful description is given of the influence of density tail properties on performance of both types ofaic and onklcv a number of practical conclusions emerge in particular we find thataic will often give problems when used with heavy tailed unbounded densities but can perform quite well with compactly supported densities in the latter case both types ofaic produce similar results and those results will sometimes be asymptotically equivalent to the ones obtained fromklcv however depending on the shape of the true density theklcv method can fail to balance bias and variance components of loss with the result thatklcv andaic may produce very different results

laws of the iterated logarithm for nonparametric density estimators we establish a law of the iterated logarithm for a triangular array of independent random variables and apply it to obtain laws for a large class of nonparametric density estimators we consider the case of rosenblatt parzen kernel estimators trigonometric series estimators and orthogonal polynomial estimators in detail and point out that our technique has wider application

kernel density estimation and intrinsic alignment for shape priors in level set segmentation in this paper we make two contributions to the field of level set based image segmentation firstly we propose shape dissimilarity measures on the space of level set functions which are analytically invariant under the action of certain transformation groups the invariance is obtained by an intrinsic registration of the evolving level set function in contrast to existing approaches to invariance in the level set framework this closed form solution removes the need to iteratively optimize explicit pose parameters the resulting shape gradient is more accurate in that it takes into account the effect of boundary variation on the object s pose secondly based on these invariant shape dissimilarity measures we propose a statistical shape prior which allows to accurately encode multiple fairly distinct training shapes this prior constitutes an extension of kernel density estimators to the level set domain in contrast to the commonly employed gaussian distribution such nonparametric density estimators are suited to model aribtrary distributions we demonstrate the advantages of this multi modal shape prior applied to the segmentation and tracking of a partially occluded walking person in a video sequence and on the segmentation of the left ventricle in cardiac ultrasound images we give quantitative results on segmentation accuracy and on the dependency of segmentation results on the number of training shapes