covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

generalized partially linear single index models abstract the typical generalized linear model for a regression of a response y on predictors x z has conditional mean function based on a linear combination of x z we generalize these models to have a nonparametric component replacing the linear combination t 0x t 0z by 0 t 0x t 0z where 0 is an unknown function we call these generalized partially linear single index models gplsim the models include the single index models which have 0 0 using local linear methods we propose estimates of the unknown parameters 0 0 and the unknown function 0 and obtain their asymptotic distributions examples illustrate the models and the proposed estimation methodology

statistical estimation in varying coefficient models varying coefficient models are a useful extension of classical linear models they arise naturally when one wishes to examine how regression coefficients change over different groups characterized by certain covariates such as age the appeal of these models is that the coef cient functions can easily be estimated via a simple local regression this yields a simple one step estimation procedure we show that such a one step method cannot be optimal when different coefficient functions admit different degrees of smoothness this drawback can be repaired by using our proposed two step estimation procedure the asymptotic mean squared error for the two step procedure is obtained and is shown to achieve the optimal rate of convergence a few simulation studies show that the gain by the two step procedure can be quite substantial the methodology is illustrated by an application to an environmental data set

a semiparametric threshold model for censored longitudinal data analysis motivated by an investigation of the relationship between blood pressure change and progression of microalbuminuria ma among individuals with type i diabetes we propose a new semiparametric threshold model for censored longitudinal data analysis we also study a new semiparametric bayes information criterion type criterion for identifying the parametric component of the proposed model cluster effects in the model are implemented as unknown fixed effects asymptotic properties are established for the proposed estimators a quadratic approximation used to implement the estimation procedure makes the method very easy to implement by avoiding the computation of multiple integrals and the need for iterative algorithms simulation studies show that the proposed methods work well in practice an illustration using the wisconsin diabetes registry dataset suggests some interesting findings

quantum classical correspondence via a deformed kinetic operator we propose an approach to quantum classical correspondence based on a deformation of the momentum and kinetic operators of quantum mechanics making use of the factorization method we construct classical versions of the momentum and kinetic operators which in addition to the standard quantum expressions contain terms that are functionals of the n particle density we show that this implementation of quantum classical correspondence is related to witten s deformation of the exterior derivative and laplacian introduced in the context of supersymmetric quantum mechanics the corresponding deformed action is also shown to be related to the fisher information finally we briefly consider the possible relevance of our approach to the construction of kinetic energy density functionals

fisher and jaynesian statistics compared in the description of classical fluids fisher statistics provides an information measure which like that of shannon is a functional of a distribution f the latter solves an euler equation to minimize the functional to make comparison with statistical derivations of extended thermodynamics calculations by frieden plastino and collaborators are specialized to the case where f is a phase space distribution approximating a solution of the liouville equation the euler equations are solved subject to conditions specifying values of internal energy and fluxes e g of heat and momentum the solution is compared with that of jaynes which with information theory provides an entropy model the fisher and jaynes entropies are identical only in equilibrium the distributions agree in linear approximation for a restricted set of variables this set is large enough to embrace all those needed in the classical non equilibrium description of simple fluids fisher euler solutions are not unique in non equilibrium

fisher based thermodynamics its legendre transform and concavity properties it is shown that the legendre transform structure of thermodynamics can be replicated without any change if one replaces the entropy s by fisher s information measure i also the important thermodynamic property of concavity is shown to be obeyed by i by this use of the fisher information measure we develop a thermodynamics that seems to be able to treat equilibrium and nonequilibrium situations in a manner entirely similar to the conventional one

the blume emery griffiths neural network dynamics for arbitrary temperature abstract the parallel dynamics of the fully connected blume emery griffiths neural network model is studied for arbitrary temperature by employing a probabilistic signal to noise approach a recursive scheme is found determining the time evolution of the distribution of the local fields and hence the evolution of the order parameters a comparison of this approach is made with the generating functional method allowing to calculate any physical relevant quantity as a function of time explicit analytic formula are given in both methods for the first few time steps of the dynamics up to the third time step the results are identical some arguments are presented why beyond the third time step the results differ for certain values of the model parameters furthermore fixed point equations are derived in the stationary limit numerical simulations confirm our theoretical findings

error estimates for a finite element discretization of a phase field model for mixtures we derive error estimates for finite element discretizations of phase field models that describe phase transitions in nonisothermal mixtures special attention is paid to the applicability of the result for a large class of models with nonlinear constitutive relations and to an approach that avoids an exponential dependence of the constants in the error estimate on the approximation parameter that models the thickness of the diffuse phase transition region the main assumptions on the model are a convexity condition for a function that can be interpreted as the negative local part of the entropy of the system a suitable regularity of the exact solutions and a spectrum estimate for the operator of the allen cahn equation the spectrum estimate is crucial to avoid the exponential dependence of error constants on the approximation parameters in the model this is done by a technique introduced in x feng and a prohl math comp 73 2004 pp 541 567 for phase transitions of pure materials with linear constitutive relations

high dimensional covariance matrix estimation using a factor model abstract high dimensionality comparable to sample size is common in many statistical problems we examine covariance matrix estimation in the asymptotic framework that the dimensionality p tends to as the sample size n increases motivated by the arbitrage pricing theory in finance a multi factor model is employed to reduce dimensionality and to estimate the covariance matrix the factors are observable and the number of factors k is allowed to grow with p we investigate the impact of p and k on the performance of the model based covariance matrix estimator under mild assumptions we have established convergence rates and asymptotic normality of the model based estimator its performance is compared with that of the sample covariance matrix we identify situations under which the factor approach increases performance substantially or marginally the impacts of covariance matrix estimation on optimal portfolio allocation and portfolio risk assessment are studied the asymptotic results are supported by a thorough simulation study

an adaptive composite quantile approach to dimension reduction sufficient dimension reduction li 1991 has long been a prominent issue in multivariate nonparametric regression analysis to uncover the central dimension reduction space we propose in this paper an adaptive composite quantile approach compared to existing methods 1 it requires minimal assumptions and is capable of revealing all dimension reduction directions 2 it is robust against outliers and 3 it is structure adaptive thus more efficient asymptotic results are proved and numerical examples are provided including a real data analysis

spectral comparison principles for the cahn hilliard and phase field equations and time scales for coarsening we initiate a study of the instability properties of steady solutions of the cahn hilliard ch equation and the phase field pf system in terms of the coarseness of those solutions other spectral results are obtained as well steady state solutions for the ch equation with no flux or periodic boundary conditions are also stationary states for the bistable reaction diffusion equation and the pf system with constant temperature thus it is meaningful to linearize each of these equations about a common stationary solution and discuss the stability of this state in each dynamical process converting the linearized operators to self adjoint form and expressing eigenvalues in terms of modified rayleigh quotients enables us to relate the spectra of the three operators in particular we relate the indices of instability with respect to the three processes we also discuss the change in the spectrum of the linearized ch operator in one dimension when we change the stationary solution at which it is linearized we conjecture that the eigenvalues increase with the number of oscillations of the steady state for a given mass and prove that this is so in certain cases this lends credence to the idea that motion of these dynamical systems near coarse solutions takes place on longer time scales

multigrid line smoothers for higher order upwind discretizations of convection dominated problems in this paper we present new multigrid line smoothers for the solution of higher order discretizations of convection dominated problems directly the behavior of the smoothers is analyzed theoretically with fourier smoothing and two grid analysis a parallel tri line variant is presented and evaluated the smoothers are applied to scalar convection diffusion problems discretized with limiters and systems of incompressible navier stokes and euler equations

operator norm consistent estimation of large dimensional sparse covariance matrices estimating covariance matrices is a problem of fundamental importance in multivariate statistics in practice it is increasingly frequent to work with data matrices x of dimension n x p where p and n are both large results from random matrix theory show very clearly that in this setting standard estimators like the sample covariance matrix perform in general very poorly in this large n large p setting it is sometimes the case that practitioners are willing to assume that many elements of the population covariance matrix are equal to 0 and hence this matrix is sparse we develop an estimator to handle this situation the estimator is shown to be consistent in operator norm when for instance we have p n as n oo in other words the largest singular value of the difference between the estimator and the population covariance matrix goes to zero this implies consistency of all the eigenvalues and consistency of eigenspaces associated to isolated eigenvalues we also propose a notion of sparsity for matrices that is compatible with spectral analysis and is independent of the ordering of the variables

a simple level set method for solving stefan problems a simple level set method for solving stefan problems is presented this method can be applied to problems involving dendritic solidification our method consists of an implicit finite difference scheme for solving the heat equation and a level set approach for capturing the front between solid and liquid phases of a pure substance our method is accurate with respect to some exact solutions of the stefan problem results indicate that this method can handle topology changes and complicated interfacial shapes and that it can numerically simulate many of the physical features of dendritic solidification

efficient thermal field computation in phase field models we solve the phase field equations in two dimensions to simulate crystal growth in the low undercooling regime the novelty is the use of a fast solver for the free space heat equation to compute the thermal field this solver is based on the efficient direct evaluation of the integral representation of the solution to the constant coefficient free space heat equation with a smooth source term the computational cost and memory requirements of the new solver are reasonable and no artificial boundary conditions are needed this allows one to solve for the thermal field in a computational domain whose size depends only on the size of the growing crystal and not on the extent of the thermal field which can result in significant computational savings in the low undercooling regime

thermodynamically consistent models of phase field type for the kinetics of phase transitions abstract a general framework is given for the phenomenological kinetics of phase transitions in which not only the order parameter but also the temperature may vary in time and space instead of a ginzburg landau free energy functional as used in formulating the cahn hilliard equation we use the analogous entropy functional model entropy functionals and the kinetic equations resulting from them are constructed for various cases phase transitions with and without a critical point and in the former case with or without a latent heat the class considered is general enough to include the entropy functionals for the ising model in mean field approximation the van der waals fluid and a simplified version of the density functional theory of freezing a case without critical point for which the energy is conserved but the order parameter is not provides a thermodynamically consistent derivation of the phase field equations studied by caginalp fix and others and also leads in a natural way to the lyapunov functional given by langer for these equations but the treatment also suggests that a modified version of the phase field equations might provide a more realistic model of freezing

on a nonlocal phase separation model abstract an alternative to the cahn hilliard model of phase separation for two phase systems in a simplified isothermal case is given the model is derived from a free energy with a nonlocal interacting term and allows reasonable bounds for the concentrations using the free energy as lyapunov functional the asymptotic state of the system is investigated and characterized by a variational principle

regularized estimation of large covariance matrices this paper considers estimating a covariance matrix of p variables from n observations by either banding the sample covariance matrix or estimating a banded version of the inverse of the covariance we show that these estimates are consistent in the operator norm as long as logp 2 n 0 and obtain explicit rates the results are uniform over some fairly natural well conditioned families of covariance matrices we also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and well conditioned then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix the results can be extended to smooth versions of banding and to non gaussian distributions with su ciently short tails a resampling approach is proposed for choosing the banding parameter in practice this approach is illustrated numerically on both simulated and real data

the phase field method in the sharp interface limit the phase field pf method for solidification phenomena is an open formulation based on a free energy functional two common choices for the pf potential here referred to briefly as the caginalp and kobayashi models are compared with respect to their numeric results within the classical sharp interface limit both qualitative and quantitative behavior are addressed and an assessment of the computational effort required to approximate a sharp interface problem is made it is shown that the specific form of the free energy potential does have a strong influence on the convergence of the pf results to their sharp interface limit compliance of the pf solutions with the linear kinetic model for the interface temperature is also investigated a simple one dimensional solidification problem in the presence of kinetic undercooling is solved by the pf model and also by a deforming grid method our results support the view that if care is exercised in formulating the phase temperature coupling there is a high degree of confidence in using the pf method for the numerical modeling of general solidification phenomena

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

reconstruction of two time independent coefficients in an inverse problem for a phase field system abstract in this paper we present stability results concerning the inverse problem of determining two time independent coefficients for a phase field system in a bounded domain r n for the dimension n 3 with a single observation on a subdomain and the sobolev norm of certain partial derivatives of the solutions at a fixed positive time 0 t over the whole spatial domain the proof of these results relies on an appropriate carleman estimate for the phase field system

half space analysis of the defect correction method for fromm discretization of convection a novel comprehensive discrete half space analysis for the defect correction method has been developed this analysis plays the same role for nonelliptic problem solvers as the full space fourier mode analysis plays for elliptic problem solvers numerical simulations confirm the accuracy of the half space analysis the following important findings about the defect correction method applied to the fromm discretization of the two dimensional convection equation are reported the initial convergence rate of the defect correction method is principally a function of the relative accuracy of the operators involved in the defect correction iterations the asymptotic convergence rate is about 0 5 per defect correction iteration if the driver operator is first order accurate then the initial convergence rates may be slow the number of iterations required to get into the asymptotic convergence regime or and to converge the algebraic error below the discretization error level can be proportional to h 1 3 this h dependent delay is a multidimensional phenomenon it cannot be observed in one dimensional problems and it disappears in the case of close alignment between the grid and the convection equation characteristic if the driver operator is second order accurate the defect correction solver demonstrates the asymptotic convergence rate from the very beginning only one defect correction iteration is required to converge algebraic error substantially below the discretization error level

adaptive varying coefficient linear models varying coefficient linear models arise from multivariate nonparametric regression nonlinear time series modelling and forecasting functional data analysis longitudinal data analysis and others it has been a common practice to assume that the vary coefficients are functions of a given variable which is often called an index a frequently asked question is which variable should be used as the index in this paper we explore the class of the varying coefficient linear models in which the index is unknown and is estimated as a linear combination of regression and or other variables this will enlarge the modelling capacity substantially we search for the index such that the derived varying coefficient model provides the best approximation to the underlying unknown multi dimensional regression function in the least square sense the search is implemented through the newly proposed hybrid backfitting algorithm the core of the algorithm is the alternative iteration between estimating the index through a one step scheme and estimating coefficient functions through a one dimensional local linear smoothing the generalised cross validation method for choosing bandwidth is efficiently incorporated into the algorithm the locally significant variables are selected in terms of the combined use of t statistic and akaike information criterion we further extend the algorithm for the models with two indices simulation shows that the proposed methodology has appreciable flexibility to model complex multivariate nonlinear structure and is practically feasible with average modern computers the methods are further illustrated through the canadian mink muskrat data in 1925 1994 and the pound dollar exchange rates in 1974 1983

liquid vapor transformations with surfactants phase field model and isogeometric analysis surfactants are compounds that find energetically favorable to be located at the boundaries between fluids they are able to modify the properties of those interfaces for example reducing surface tension here we propose a new model for liquid vapor flows with surfactants which captures the dynamics of the surfactant and accounts for phase transformations in the fluid the aforementioned model is derived from a free energy functional by using a coleman noll approach the proposed theory emanates from the isothermal navier stokes korteweg equations which describe single component two phase flow and naturally allow for phase transformations we believe that our model has significant potential to study the influence of surfactants in vaporization and condensation processes from a numerical point of view the proposed model poses significant challenges to existing discretization methods including stiffness in space and time internal and boundary layers as well as higher order partial differential operators to overcome these challenges we propose algorithms based on isogeometric analysis which permit an accurate and efficient discretization finally we illustrate the viability of the theoretical framework and the effectiveness of our algorithms by solving several numerical problems in two and three dimensions

analysis of an iterative scheme of fractional steps type associated to the reaction diffusion equation endowed with a general nonlinearity and cauchy neumann boundary conditions abstract the paper studies the existence uniqueness regularity and the approximation of solutions to the reaction diffusion equation endowed with a general nonlinear regular potential and cauchy neumann boundary conditions the convergence and error estimate results for an iterative scheme of fractional steps type associated to the nonlinear parabolic equation are also established the advantage of such method consists in simplifying the numerical computation necessary to be done in order to approximate the solution of nonlinear parabolic equation

quantization from an exponential distribution of infinitesimal action abstract a statistical model of quantization based on an exponential distribution of infinitesimal action is proposed a trajectory which does not extremize the action along an infinitesimal short segment of path is allowed to occur with a very small probability following an exponential law planck s constant is argued to give the average deviation from the infinitesimal stationary action

optimal smoothing in single index models single index models generalize linear regression they have applications to a variety of fields such as discrete choice analysis in econometrics and dose response models in biometrics where high dimensional regression models are often employed single index models are similar to the first step of projection pursuit regression a dimension reduction method in both cases the orientation vector can be estimated root n consistently even if the unknown univariate function or nonparametric link function is assumed to come from a large smoothness class however as we show in the present paper the similarities end there in particular the amount of smoothing necessary for root n consistent orientation estimation is very different in the two cases we suggest a simple empirical rule for selecting the bandwidth appropriate to single index models this rule is studied in a small simulation study and an application in binary response models

penalized spline estimation for partially linear single index models single index models are potentially important tools for multivariate nonparametric regression they generalize linear regression by replacing the linear combination 0tx with a nonparametric component 0 0tx where 0 is an unknown univariate link function by reducing the dimensionality from that of a general covariate vector x to a univariate index 0tx single index models avoid the so called curse of dimensionality we propose penalized spline p spline estimation of 0 in partially linear single index models where the mean function has the form 0 0tx 0tz the p spline approach offers a number of advantages over other fitting methods for single index models all parameters in the p spline single index model can be estimated simultaneously by penalized nonlinear least squares as a direct least squares fitting method our approach is rapid and computationally stable standard nonlinear least squares software can be used moreover joint inference for 0 0 and 0 is possible by

variational approach to dequantization we present a dequantization procedure based on a variational approach whereby quantum fluctuations latent in the quantum momentum are suppressed this is done by adding generic local deformations to the quantum momentum operator which give rise to a deformed kinetic term quantifying the amount of fuzziness caused by such fluctuations considered as a functional of such deformations the deformed kinetic term is shown to possess a unique minimum which is seen to be the classical kinetic energy furthermore we show that extremization of the associated deformed action functional introduces an essential nonlinearity to the resulting field equations which are seen to be the classical hamilton jacobi and continuity equations thus a variational procedure determines the particular deformation that has the effect of suppressing the quantum fluctuations resulting in dequantization of the system

a semiparametric spatial dynamic model stimulated by the boston house price data in this paper we propose a semiparametric spatial dynamic model which extends the ordinary spatial autoregressive models to accommodate the effects of some covariates associated with the house price a profile likelihood based estimation procedure is proposed the asymptotic normality of the proposed estimators are derived we also investigate how to identify the parametric nonparametric components in the proposed semiparametric model we show how many unknown parameters an unknown bivariate function amounts to and propose an aic bic of nonparametric version for model selection simulation studies are conducted to examine the performance of the proposed methods the simulation results show our methods work very well we finally apply the proposed methods to analyze the boston house price data which leads to some interesting findings

schrodinger equation from an exact uncertainty principle an exact uncertainty principle formulated as the assumption that a classical ensemble is subject to random momentum fluctuations of a strength which is determined by and scales inversely with uncertainty in position leads from the classical equations of motion to the schrodinger equation

high dimensional covariance matrix estimation in approximate factor models the variance covariance matrix plays a central role in the inferential theories of high dimensional factor models in finance and economics popular regularization methods of directly exploiting sparsity are not directly applicable to many financial problems classical methods of estimating the covariance matrices are based on the strict factor models assuming independent idiosyncratic components this assumption however is restrictive in practical applications by assuming sparse error covariance matrix we allow the presence of the cross sectional correlation even after taking out common factors and it enables us to combine the merits of both methods we estimate the sparse covariance using the adaptive thresholding technique as in cai and liu j amer statist assoc 106 2011 672 684 taking into account the fact that direct observations of the idiosyncratic components are unavailable the impact of high dimensionality on the covariance matrix estimation based on the factor structure is then studied

geometry of the schrodinger equation and stochastic mass transportation the schrodinger equation is deduced from a geometric principle lagrange hamilton jacobi and hamilton formalisms are defined almost analogously to the deterministic case which can be identified as geometric optic the form of these formalisms is identical to the deterministic formalisms furthermore it will be shown how the deterministic case is superficial to the stochastic one if planck s constant is very small an elementary proof of heisenberg s uncertainty relation finishes the paper

non extensive statistical mechanics and generalized fokker planck equation a non linear generalized fokker planck gfp equation is studied whose exact stationary solutions are the maximum entropy distributions introduced by tsallis in his generalization of statistical mechanics in the case of a constant or linearly varying drift the time dependent solutions of the gfp equation are seen to obey a suitable form of the celebrated h theorem the temporal changes of tsallis entropies are seen to be given in terms of the fisher s information measure particular time dependent solutions of the gfp equation of the maximum tsallis entropy form are obtained

an informational characterization of schrodinger s uncertainty relations heisenberg s uncertainty relations employ commutators of observables to set fundamental limits on quantum measurement the information concerning incompatibility non commutativity of observables is well included but that concerning correlation is missing schrodinger s uncertainty relations remedy this defect by supplementing the correlation in terms of anti commutators however both heisenberg s uncertainty relations and schrodinger s uncertainty relations are expressed in terms of variances which are not good measures of uncertainty in general situations e g when mixed states are involved by virtue of the wigner yanase skew information we will establish an uncertainty relation along the spirit of schrodinger from a statistical inference perspective and propose a conjecture the result may be interpreted as a quantification of certain aspect of the celebrated wigner araki yanase theorem for quantum measurement which states that observables not commuting with a conserved quantity cannot be measured exactly

dynamics of layered interfaces arising from phase boundaries the dynamics of a material in two phases is studied in the context of phase field models based on a landau ginzburg free energy functional they consist of a system of two nonlinear diffusion equat

the gradient theory of phase transitions for systems with two potential wells on generalise la theorie du gradient des transitions de phase au cas a valeur vectoriel on considere la famille des perturbations e e u w u dx e 2 u 2 dx de la fonctionnelle non convexe e 0 u w u dx ou w r n r supporte deux phases et n 1

a dynamic structure for high dimensional covariance matrices and its application in portfolio allocation abstractestimation of high dimensional covariance matrices is an interesting and important research topic in this article we propose a dynamic structure and develop an estimation procedure for high dimensional covariance matrices asymptotic properties are derived to justify the estimation procedure and simulation studies are conducted to demonstrate its performance when the sample size is finite by exploring a financial application an empirical study shows that portfolio allocation based on dynamic high dimensional covariance matrices can significantly outperform the market from 1995 to 2014 our proposed method also outperforms portfolio allocation based on the sample covariance matrix the covariance matrix based on factor models and the shrinkage estimator of covariance matrix supplementary materials for this article are available online

incompleteness of trajectory based interpretations of quantum mechanics trajectory based approaches to quantum mechanics include the de broglie bohm interpretation and nelson s stochastic interpretation it is shown that the usual route to establishing the validity of such interpretations via a decomposition of the schr dinger equation into a continuity equation and a modified hamilton jacobi equation fails for some quantum states a very simple example is provided by a quantum particle in a box described by a wavefunction that is initially uniform over the interior of the box for this example there is no corresponding continuity or modified hamilton jacobi equation and the space time dependence of the wavefunction has a known fractal structure examples with finite average energies are also constructed

sparsistency and rates of convergence in large covariance matrix estimation this paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions here sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one depending on the case of applications sparsity priori may occur on the covariance matrix its inverse or its cholesky decomposition we study these three sparsity exploration problems under a unified framework with a general penalty function we show that the rates of convergence for these problems under the frobenius norm are of order s n log p n n 1 2 where s n is the number of nonzero elements p n is the size of the covariance matrix and n is the sample size this explicitly spells out the contribution of high dimensionality is merely of a logarithmic factor the conditions on the rate with which the tuning parameter n goes to 0 have been made explicit and compared under different penalties as a result for the l 1 penalty to guarantee the sparsistency and optimal rate of convergence the number of nonzero elements should be small sn o pn at most among o pn2 parameters for estimating sparse covariance or correlation matrix sparse precision or inverse correlation matrix or sparse cholesky factor where sn is the number of the nonzero elements on the off diagonal entries on the other hand using the scad or hard thresholding penalty functions there is no such a restriction

phase field models for hypercooled solidification abstract properties of the solidification front in a hypercooled liquid so called because the temperature of the resulting solid is below the melting temperature are derived using a phase field diffuse interface model certain known properties for hypercooled fronts in specific materials are reflected within our theories such as the presence of thin thermal layers and the trend towards smoother fronts with less pronounced dendrites when the undercooling is increased within the hypercooled regime both an asymptotic analysis to derive the relevant free boundary problems and a rigorous determination of the inner profile of the diffusive interface are given of particular interest is the incorporation of anisotropy and general microscale interactions leading to higher order differential operators these features necessitate a much richer mathematical analysis than previous theories anisotropic free boundary problems are derived from our models the simplest of which involves determining the evolution of a set a solid particle whose boundary moves with velocity depending on its normal vector considerable attention is given to the identification of surface tension to comparison with previous theories and to questions of stability

symmetries of the fokker planck equation and the fisher frieden arrow of time we give here a concrete illustration of frieden s arrow of time connected with fisher s information measure it is shown that this arrow of time is related to symmetries of the fokker planck equation textcopyright 1996 the american physical society

on quantitative analysis methods for multigrid solutions limitations of the classical local mode fourier lmf analysis in application to multigrid solutions of variable coefficient problems are well known this paper introduces alternative very general quantitative analysis methods for multigrid solutions of partial differential equations the methods are applied to available nonperfect multigrid solvers that deal with practical problems the analysis methods considered in this paper focus on the main complementary parts of a multigrid cycle relaxation and coarse grid correction tasks for these parts e g error smoothing for relaxation and reduction of smooth errors for coarse grid correction are assumed to be assigned in advance idealized relaxation ir and idealized coarse grid icg iterations are introduced in these iterations one part of the cycle relaxation for ir iterations and coarse grid correction for icg iterations is replaced with an idealized imitation known to be efficient for the assigned task its complementary part is the actual part of a two grid cycle the analysis compares performances of the actual cycle and idealized iterations the ir and icg iterations are very general and can be directly applied in the most complicated simulations including highly variable or nonlinear coefficients complex geometries and unstructured grids the results of this analysis are not single number estimates they are rather convergence patterns of the iterations that may either confirm or refute expectations indicating what part of the actual solver is inefficient in carrying out the assigned task the generality of the analysis makes it a valuable tool for analyzing complicated large scale computational problems where no other analysis methods are currently available in this paper the ir and icg iterations are applied to model problems of progressive complexity the analysis proved sensitive to very delicate details of the actual multigrid cycle pointing clearly to each instance when a part of the algorithm was inefficient

a semiparametric model for cluster data in the analysis of cluster data the regression coefficients are frequently assumed to be the same across all clusters this hampers the ability to study the varying impacts of factors on each cluster in this paper a semiparametric model is introduced to account for varying impacts of factors over clusters by using cluster level covariates it achieves the parsimony of parametrization and allows the explorations of nonlinear interactions the random effect in the semiparametric model accounts also for within cluster correlation local linear based estimation procedure is proposed for estimating functional coefficients residual variance and within cluster correlation matrix the asymptotic properties of the proposed estimators are established and the method for constructing simultaneous confidence bands are proposed and studied in addition relevant hypothesis testing problems are addressed simulation studies are carried out to demonstrate the methodological power of the proposed methods in the finite sample the proposed model and methods are used to analyse the second birth interval in bangladesh leading to some interesting findings

weakly non local fluid mechanics the schrodinger equation a weakly non local extension of ideal fluid dynamics is derived from the second law of thermodynamics it is proved that in the reversible limit the additional pressure term can be derived from a potential the requirement of the additivity of the specific entropy function determines the quantum potential uniquely the relation to other known derivations of the schrodinger equation stochastic fisher information exact uncertainty is clarified

convergence analysis of the defect correction iteration for hyperbolic problems this paper explains some of the convergence behaviour of iterative implicit and defect correction schemes for the solution of the discrete steady euler equations such equations are also commonly solved by pseudo time integration the steady solution being achieved as the limit for t to infty of the solution of a time dependent problem implicit schemes are then often chosen for their favourable stability properties permitting large timesteps for efficiency an important class of implicit schemes involving first and second order accurate upwind discretisations is considered in the limit of an infinite timestep these implicit schemes approach defect correction algorithms thus our analysis is informative for both types of construction simple scalar linear model problems are introduced for the one dimensional and two dimensional cases these model problems are analyzed in detail by both fourier and matrix analyses the convergence behaviour appears to be strongly dependent on a parameter beta

multigrid and defect correction for the steady navier stokes equations theoretical and experimental convergence results are presented for nonlinear multigrid and iterative defect correction applied to finite volume discretizations of the full steady 2d compressible navier stokes equations iterative defect correction is introduced for circumventing the difficulty in solving navier stokes equations discretized with a second or higher order accurate convective part by fourier analysis applied to a model equation an optimal choice is made for the operator to be inverted in the defect correction iteration as a smoothing technique for the multigrid method collective symmetric point gauss seidel relaxation is applied with as the basic solution technique exact newton iteration applied to a continuously differentiable first order upwind discretization of the full navier stokes equations for non smooth flow problems the convergence results obtained are already competitive with those of well established navier stokes methods for smooth flow problems the present method performs better than any standard method here first order discretization error accuracy is attained in a single multigrid cycle and second order accuracy in only one defect correction cycle the method contributes to the state of the art in efficiently computing compressible viscous flows

estimation of the covariance matrix of random effects in longitudinal studies longitudinal studies are often conducted to explore the cohort and age effects in many scientific areas the within cluster correlation structure plays a very important role in longitudinal data analysis this is because not only can an estimator be improved by incorporating the within cluster correlation structure into the estimation procedure but also the within cluster correlation structure can sometimes provide valuable insights in practical problems for example it can reveal the correlation strengths among the impacts of various factors motivated by data typified by a set from bangladesh pertinent to the use of contraceptives we propose a random effect varying coefficient model and an estimation procedure for the within cluster correlation structure of the proposed model the estimation procedure is optimization free and the proposed estimators enjoy asymptotic normality under mild conditions simulations suggest that the proposed estimation is practicable for finite samples and resistent against mild forms of model m is specification finally we analyze the data mentioned above with the new random effect varying coefficient model together with the proposed estimation procedure which reveals some interesting sociological dynamics

a robust goodness of fit test for generalized autoregressive conditional heteroscedastic models summary the estimation of time series models with heavy tailed innovations has been widely discussed but corresponding goodness of fit tests have attracted less attention primarily because the autocorrelation function commonly used in constructing goodness of fit tests necessarily imposes certain moment conditions on the innovations as a bounded random variable has finite moments of all orders we address the problem by first transforming the residuals with a bounded function more specifically we consider the sample autocorrelation function of the transformed absolute residuals of a fitted generalized autoregressive conditional heteroscedastic model with the corresponding residual empirical distribution function naturally employed as the transformation a robust goodness of fit test is then constructed the asymptotic distributions of the test statistic under the null hypothesis and local alternatives are derived and monte carlo experiments are conducted to examine finite sample properties the proposed test is shown to be more powerful than existing tests when the innovations are heavy tailed

unique additive information measures boltzmann gibbs shannon fisher and beyond abstract it is proved that the only additive and isotropic information measure that can depend on the probability distribution and also on its first derivative is a linear combination of the boltzmann gibbs shannon and fisher information measures power law equilibrium distributions are found as a result of the interaction of the two terms the case of second order derivative dependence is investigated and a corresponding additive information measure is given

spreading lengths of hermite polynomials the renyi shannon and fisher spreading lengths of the classical or hypergeometric orthogonal polynomials which are quantifiers of their distribution all over the orthogonality interval are defined and investigated these information theoretic measures of the associated rakhmanov probability density which are direct measures of the polynomial spreading in the sense of having the same units as the variable share interesting properties invariance under translations and reflections linear scaling and vanishing in the limit that the variable tends towards a given definite value the expressions of the renyi and fisher lengths for the hermite polynomials are computed in terms of the polynomial degree the combinatorial multivariable bell polynomials which are shown to characterize the finite power of an arbitrary polynomial play a relevant role for the computation of these information theoretic lengths indeed these polynomials allow us to design an error free computing approach for the entropic moments weighted l q norms of hermite polynomials and subsequently for the renyi and tsallis entropies as well as for the renyi spreading lengths sharp bounds for the shannon length of these polynomials are also given by means of an information theoretic based optimization procedure moreover the existence of a linear correlation between the shannon length as well as the second order renyi length and the standard deviation is computationally proved finally the application to the most popular quantum mechanical prototype system the harmonic oscillator is discussed and some relevant asymptotical open issues related to the entropic moments mentioned previously are posed

nonparametric estimation of large covariance matrices of longitudinal data summary estimation of an unstructured covariance matrix is difficult because of its positivedefiniteness constraint this obstacle is removed by regressing each variable on its predecessors so that estimation of a covariance matrix is shown to be equivalent to that of estimating a sequence of varying coefficient and varying order regression models our framework is similar to the use of increasing order autoregressive models in approximating the covariance matrix or the spectrum of a stationary time series as an illustration we adopt fan zhang s 2000 two step estimation of functional linear models and propose nonparametric estimators of covariance matrices which are guaranteed to be positive definite for parsimony a suitable order for the sequence of auto regression models is found using penalised likelihood criteria like aic and bic some asymptotic results for the local polynomial estimators of components of a covariance matrix are established two longitudinal datasets are analysed to illustrate the methodology a simulation study reveals the advantage of the nonparametric covariance estimator over the sample covariance matrix for large covariance matrices

a new semiparametric estimation approach for large dynamic covariance matrices with multiple conditioning variables this paper studies the estimation of large dynamic covariance matrices with multiple conditioning variables we introduce an easy to implement semiparametric method to estimate each entry of the covariance matrix via model averaging marginal regression and then apply a shrinkage technique to obtain the dynamic covariance matrix estimation under some regularity conditions we derive the asymptotic properties for the proposed estimators including the uniform consistency with general convergence rates we further consider extending our methodology to deal with the scenarios i the number of conditioning variables is divergent as the sample size increases and ii the large covariance matrix is conditionally sparse relative to contemporaneous market factors we provide a simulation study that illustrates the finite sample performance of the developed methodology we also provide an application to financial portfolio choice from daily stock returns

on multigrid solution of high reynolds incompressible entering flows an approach is presented for effectively separating the solution process of the elliptic component of high reynolds incompressible steady entering flow for which classical multigrid techniques are well suited from that of the non elliptic part for which other methods are more effective it is shown by analysis and numerical calculations that such an approach is very effective in terms of asymptotic convergence as well as reduction of errors well below discretization level in a 1 fmg algorithm

nonconvex variational problems with general singular perturbations we study the effect of a general singular perturbation on a non convex variational problem with infinitely many solutions using a scaling argument and the theory of t convergence of nonlinear functionals we show that if the solutions of the perturbed problem converge in l1 as the perturba tion parameter goes to zero then the limit function satisfies a classical minimal surface problem

non isothermal phase transition models with neumann boundary conditions abstract we study the systems of nonlinear parabolic pdes for penrose fife models of non isothermal phase transitions both models with conserved and non conserved order parameters are studied with neumann boundary conditions imposed on the systems in multi dimensional spatial domain

on a nonlocal phase field system abstract we study the existence uniqueness and continuous dependence on initial data of the solution to a nonlocal phase field system on a bounded domain the system is a gradient flow for a free energy functional with nonlocal interaction also we study the asymptotic behavior of the solution and show the existence of an absorbing set in some metric space