deformed shape invariance and exactly solvable hamiltonians with position dependent effective mass known shape invariant potentials for the constant mass schrodinger equation are taken as effective potentials in a position dependent effective mass pdem one the corresponding shape invariance condition turns out to be deformed its solvability imposes the form of both the deformed superpotential and the pdem a lot of new exactly solvable potentials associated with a pdem background are generated in this way a novel and important condition restricting the existence of bound states whenever the pdem vanishes at an end point of the interval is identified in some cases the bound state spectrum results from a smooth deformation of that of the conventional shape invariant potential used in the construction in others one observes a generation or suppression of bound states depending on the mass parameter values the corresponding wavefunctions are given in terms of some deformed classical orthogonal polynomials

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

pseudo hermiticity versus pt symmetry iii equivalence of pseudo hermiticity and the presence of antilinear symmetries we show that a diagonalizable non hermitian hamiltonian h is pseudo hermitian if and only if it has an antilinear symmetry i e a symmetry generated by an invertible antilinear operator this implies that the eigenvalues of h are real or come in complex conjugate pairs if and only if h possesses such a symmetry in particular the reality of the spectrum of h implies the presence of an antilinear symmetry we further show that the spectrum of h is real if and only if there is a positive definite inner product on the hilbert space with respect to which h is hermitian or alternatively there is a pseudo canonical transformation of the hilbert space that maps h into a hermitian operator

discretization of nonlinear models by sinc collocation interpolation methods abstract this paper deals with the solution of initial boundary value problems for nonlinear evolution equations in one and two space dimensions the solution technique is based on collocation interpolation methods which use sinc functions the paper aims to provide a guideline towards a large number of nonlinear problems of interest in applied sciences

the joint graphical lasso for inverse covariance estimation across multiple classes we consider the problem of estimating multiple related gaussian graphical models from a high dimensional data set with observations belonging to distinct classes we propose the joint graphical lasso which borrows strength across the classes in order to estimate multiple graphical models that share certain characteristics such as the locations or weights of nonzero edges our approach is based upon maximizing a penalized log likelihood we employ generalized fused lasso or group lasso penalties and implement a fast admm algorithm to solve the corresponding convex optimization problems the performance of the proposed method is illustrated through simulated and real data examples

pt symmetric pseudo perturbation recipe an imaginary cubic oscillator with spikes the pseudo perturbative shifted l expansion technique pslet is shown to be applicable in the non hermitian symmetric context the construction of bound states for several symmetric potentials is presented with special attention paid to v r ir3 ir oscillators

doubly functional graphical models in high dimensions we consider estimating a functional graphical model from multivariate functional observations in functional data analysis the classical assumption is that each function has been measured over a densely sampled grid however in practice the functions have often been observed with measurement error at a relatively small number of points we propose a class of doubly functional graphical models to capture the evolving conditional dependence relationship among a large number of sparsely or densely sampled functions our approach first implements a nonparametric smoother to perform functional principal components analysis for each curve then estimates a functional covariance matrix and finally computes sparse precision matrices which in turn provide the doubly functional graphical model we derive some novel concentration bounds uniform convergence rates and model selection properties of our estimator for both sparsely and densely sampled functional data in the high dimensional large p small n regime we demonstrate via simulations that the proposed method significantly outperforms possible competitors our proposed method is applied to a brain imaging dataset

on the distribution of dominance in populations of social organisms bumble bees may possess a scalar character called dominance which changes according to certain rules as a result of encounters between pairs of organisms an equation for the distribution of dominance in a population is derived based on a set of plausible axioms the resulting boltzmann like integrodifferential equation is analyzed analytically and or numerically for certain important special cases

harmonic wavelets towards the solution of nonlinear pde in this paper harmonic wavelets which are analytically defined and band limited are studied together with their differentiable properties harmonic wavelets were recently applied to the solution of evolution problems and more generally to describe evolution operators in order to consider the evolution of a solitary profile and to focus on the localization property of wavelets it seems to be more expedient to make use of functions with limited compact support either in space or in frequency the connection coefficients of harmonic wavelets are explicitly computed in the following at any order and characterized by some recursive formulas in particular they are functionally and finitely defined by a simple formula for any order of the basis derivatives

mapping of shape invariant potentials under point canonical transformations the authors give explicit point canonical transformations which map twelve types of shape invariant potentials which are known to be exactly solvable into two potential classes the eigenfunctions in these two classes are given by hypergeometric and confluent hypergeometric functions respectively

remarks on the local time rescaling in path integration the local time transformation used in path integration is shown to be equivalent to standard procedures applied to schrodinger s equation

variational analysis for a generalized spiked harmonic oscillator a variational analysis is presented for the generalized spiked harmonic oscillator hamiltonian operator d2 dx 2 bx 2 a x 2 x where is a real positive parameter the formalism makes use of a basis provided by exact solutions of schrodinger s equation for the gol dman and krivchenkov hamiltonian and the corresponding matrix elements that were previously found for all the discrete eigenvalues the method provides bounds which improve as the dimension d of the basis set is increased extension to the n dimensional case in arbitrary angular momentum subspaces is also presented by minimizing over the free parameter a we are able to reduce substantially the number of basis functions needed for a given accuracy

functional data analysis for sparse longitudinal data we propose a nonparametric method to perform functional principal components analysis for the case of sparse longitudinal data the method aims at irregularly spaced longitudinal data where the number of repeated measurements available per subject is small in contrast classical functional data analysis requires a large number of regularly spaced measurements per subject we assume that the repeated measurements are located randomly with a random number of repetitions for each subject and are determined by an underlying smooth random subject specific trajectory plus measurement errors basic elements of our approach are the parsimonious estimation of the covariance structure and mean function of the trajectories and the estimation of the variance of the measurement errors the eigenfunction basis is estimated from the data and functional principal components score estimates are obtained by a conditioning step this conditional estimation method is conceptually simple and straightforward to implement

a basis for variational calculations in d dimensions in this paper we derive expressions for matrix elements i hj for the hamiltonian h qa q rq in d 2 dimensions the basis functions in each angular momentum subspace are of the form the matrix elements are given in terms of the gamma function for all d the significance of the parameters t and p and scale s are discussed applications to a variety of potentials are presented including potentials with singular repulsive terms of the form r 0 perturbed coulomb potentials d r br ar2 and potentials with weak repulsive terms such as r2 r4 0

joint estimation of multiple graphical models from high dimensional time series in this manuscript we consider the problem of jointly estimating multiple graphical models in high dimensions we assume that the data are collected from n subjects each of which consists of t possibly dependent observations the graphical models of subjects vary but are assumed to change smoothly corresponding to a measure of closeness between subjects we propose a kernel based method for jointly estimating all graphical models theoretically under a double asymptotic framework where both t n and the dimension d can increase we provide the explicit rate of convergence in parameter estimation it characterizes the strength one can borrow across different individuals and the impact of data dependence on parameter estimation empirically experiments on both synthetic and real resting state functional magnetic resonance imaging rs fmri data illustrate the effectiveness of the proposed method

from sparse to dense functional data and beyond nonparametric estimation of mean and covariance functions is important in functional data analysis we investigate the performance of local linear smoothers for both mean and covariance functions with a general weighing scheme which includes two commonly used schemes equal weight per observation obs and equal weight per subject subj as two special cases we provide a comprehensive analysis of their asymptotic properties on a unified platform for all types of sampling plan be it dense sparse or neither three types of asymptotic properties are investigated in this paper asymptotic normality l 2 convergence and uniform convergence the asymptotic theories are unified on two aspects 1 the weighing scheme is very general 2 the magnitude of the number n i of measurements for the i th subject relative to the sample size n can vary freely based on the relative order of n i to n functional data are partitioned into three types non dense dense and ultra dense functional data for the obs and subj schemes these two weighing schemes are compared both theoretically and numerically we also propose a new class of weighing schemes in terms of a mixture of the obs and subj weights of which theoretical and numerical performances are examined and compared

solution of nonlinear initial boundary value problems by sinc collocation interpolation methods abstract this paper deals with the solution of initial boundary value problems for nonlinear evolution equations in one and two space dimensions the solution technique is based on collocation interpolation methods which use sinc functions the application refers to some direct and inverse type problems

an exactly soluble schrodinger equation with smooth position dependent mass the one dimensional generalized schrodinger equation for a system with smooth potential and mass step is resolved exactly the wave function depends on the heun s function which is a solution of a second order fuchsian equation with four singularities the behavior of the transmission coefficient as a function of energy is compared to that of the case of an abrupt potential and mass step two limiting cases are also studied when the width of the mass step is vanishing and when the smooth potential and mass step tend to an abrupt potential and mass step

on multisymplecticity of sinc gauss legendre collocation discretizations for some hamiltonian pdes abstract in this paper a new multisymplectic scheme based on sinc collocation discretization in space and gauss legendre collocation discretization in time for some hamiltonian pdes is developed the scheme preserves the multisymplectic geometry structure exactly by satisfying the discrete multisymplectic conservation law moreover the discrete energy and momentum conservative properties of the multisymplectic integrator are also discussed in order to testify the superiority of the multisymplectic method it is applied to nonlinear dirac equation numerical experiments are given to illustrate the accuracy of the approximation and the conserved quantities

part of the d dimensional spiked harmonic oscillator spectra the pseudoperturbative shifted l expansion technique pslet is generalized for states with an arbitrary number of nodal zeros interdimensional degeneracies emerging from the isomorphism between the angular momentum and the dimensionality of the central force schrodinger equation are used to construct part of the d dimensional spiked harmonic oscillator bound state spectra pslet results are found to compare excellently with those from direct numerical integration and generalized variational methods

asymptotic normality and optimalities in estimation of large gaussian graphical models the gaussian graphical model a popular paradigm for studying relationship among variables in a wide range of applications has attracted great attention in recent years this paper considers a fundamental question when is it possible to estimate low dimensional parameters at parametric square root rate in a large gaussian graphical model a novel regression approach is proposed to obtain asymptotically efficient estimation of each entry of a precision matrix under a sparseness condition relative to the sample size when the precision matrix is not sufficiently sparse or equivalently the sample size is not sufficiently large a lower bound is established to show that it is no longer possible to achieve the parametric rate in the estimation of each entry this lower bound result which provides an answer to the delicate sample size question is established with a novel construction of a subset of sparse precision matrices in an application of le cam s lemma moreover the proposed estimator is proven to have optimal convergence rate when the parametric rate cannot be achieved under a minimal sample requirement the proposed estimator is applied to test the presence of an edge in the gaussian graphical model or to recover the support of the entire model to obtain adaptive rate optimal estimation of the entire precision matrix as measured by the matrix ell q operator norm and to make inference in latent variables in the graphical model all of this is achieved under a sparsity condition on the precision matrix and a side condition on the range of its spectrum this significantly relaxes the commonly imposed uniform signal strength condition on the precision matrix irrepresentability condition on the hessian tensor operator of the covariance matrix or the ell 1 constraint on the precision matrix numerical results confirm our theoretical findings the roc curve of the proposed algorithm asymptotic normal thresholding ant for support recovery significantly outperforms that of the popular glasso algorithm

domain decomposition and discretization of continuous mathematical models abstract this paper deals with the development of decomposition of domains methods related to the discretization by collocation interpolation methods of continuous models described by nonlinear partial differential equations the objective of this paper is to show how generalized collocation and domain decomposition methods can be applied for problems where different models are used within the same domain

model selection and estimation in the gaussian graphical model summary we propose penalized likelihood methods for estimating the concentration matrix in the gaussian graphical model the methods lead to a sparse and shrinkage estimator of the concentration matrix that is positive definite and thus conduct model selection and estimation simultaneously the implementation of the methods is nontrivial because of the positive definite constraint on the concentration matrix but we show that the computation can be done effectively by taking advantage of the efficient maxdet algorithm developed in convex optimization we propose a bic type criterion for the selection of the tuning parameter in the penalized likelihood methods the connection between our methods and existing methods is illustrated simulations and real examples demonstrate the competitive performance of the new methods

a constrained l1 minimization approach to sparse precision matrix estimation this article proposes a constrained l1 minimization method for estimating a sparse inverse covariance matrix based on a sample of n iid p variate random variables the resulting estimator is shown to have a number of desirable properties in particular the rate of convergence between the estimator and the true s sparse precision matrix under the spectral norm is when the population distribution has either exponential type tails or polynomial type tails we present convergence rates under the elementwise l norm and frobenius norm in addition we consider graphical model selection the procedure is easily implemented by linear programming numerical performance of the estimator is investigated using both simulated and real data in particular the procedure is applied to analyze a breast cancer dataset and is found to perform favorably compared with existing methods

solution of nonlinear evolution problems by parallelized collocation interpolation methods abstract this paper deals with the solution of initial boundary value problems for nonlinear evolution equations the solution technique is based on collocation interpolation methods which are improved in order to reduce computational errors at fixed discretizations of the independent variables the method consists in parallelizing the approximation of the space derivatives so that the same approximation is reached by a lower number of collocation points and hence by a lower computation time the analysis includes a theoretical and computational estimate both of the approximation error and of the computational time

proof of dynamical scaling in smoluchowski s coagulation equation with constant kernel smoluchowski s coagulation equation for irreversible aggregation with constant kernel is considered in its discrete version open image in new window wherect c1 t is the concentration ofl particle clusters at timet we prove that for initial data satisfyingc1 0 0 and the condition 0 cl 0 0 the solutions behave asymptotically likec1 t t 2 c lt 1 ast withlt 1 kept fixed the scaling function c is 1 gr where rho sum l lc l 0 a conserved quantity is the initial number of particles per unit volume an analous result is obtained for the continuous version of smoluchowski s coagulation equation frac partial partial t c v text t int 0 v du text c v u text t text c u text t 2c v text t int 0 infty du text c u text t wherec v t is the oncentration of clusters of sizev

functional graphical models graphical models have attracted increasing attention in recent years especially in settings involving high dimensional data in particular gaussian graphical models are used to model the conditio

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

comprehensive analysis of conditionally exactly solvable models we study a quantum mechanical potential introduced previously as a conditionally exactly solvable ces model besides an analysis following its original introduction in terms of the point canonical transformation we also present an alternative supersymmetric construction of it we demonstrate that from the three roots of the implicit cubic equation defining the bound state energy eigenvalues there is always only one that leads to a meaningful physical state finally we demonstrate that the present ces interaction is in fact an exactly solvable natanzon class potential

dynamic covariance models abstractan important problem in contemporary statistics is to understand the relationship among a large number of variables based on a dataset usually with p the number of the variables much larger than n the sample size recent efforts have focused on modeling static covariance matrices where pairwise covariances are considered invariant in many real systems however these pairwise relations often change to characterize the changing correlations in a high dimensional system we study a class of dynamic covariance models dcms assumed to be sparse and investigate for the first time a unified theory for understanding their nonasymptotic error rates and model selection properties in particular in the challenging high dimensional regime we highlight a new uniform consistency theory in which the sample size can be seen as n4 5 when the bandwidth parameter is chosen as h n 1 5 for accounting for the dynamics we show that this result holds uniformly over a range of the variable used for modeling t

numerical methods based on whittaker cardinal or sinc functions this paper summarizes the results known to date for using sine functions composed with other functions as bases for approximations in numerical analysis described in this paper are methods of interpolation and approximation of functions and their derivatives quadrature the approximate evaluation of transforms hilbert fourier laplace hankel and mellin and the approximate solution of differential and integral equations the methods have many advantages over classical methods which use polynomials as bases in addition all of the methods converge at an optimal rate if singularities on the boundary of approximation are ignored

a nonparametric graphical model for functional data with application to brain networks based on fmri abstractwe introduce a nonparametric graphical model whose observations on vertices are functions many modern applications such as electroencephalogram and functional magnetic resonance imaging fmri produce data are of this type the model is based on additive conditional independence aci a statistical relation that captures the spirit of conditional independence without resorting to multi dimensional kernels the random functions are assumed to reside in a hilbert space no distributional assumption is imposed on the random functions instead their statistical relations are characterized nonparametrically by a second hilbert space which is a reproducing kernel hilbert space whose kernel is determined by the inner product of the first hilbert space a precision operator is then constructed based on the second space which characterizes aci and hence also the graph the resulting estimator is relatively easy to compute requiring no iterative optimization or inversion of large matrices we estab

quasi relativistic harmonic oscillator bound states the quasi relativistic harmonic oscillator bound states constructed by znojil 1996 j phys a math gen 29 2905 are investigated via a new methodical proposal compared with those obtained by an anonymous referee from a direct numerical integration method of znojil s paper our results appear to be more favourable than those obtained by znojil via quasi perturbative variational hill determinant and riccati pade methods bound states with larger angular momenta l are also constructed

d dimensional generalization of the point canonical transformation for a quantum particle with position dependent mass the d dimensional generalization of the point canonical transformation for a quantum particle endowed with a position dependent mass in the schrodinger equation is described illustrative examples including the harmonic oscillator coulomb spiked harmonic kratzer morse oscillator p schl teller and hulthen potentials are used as reference potentials to obtain exact energy eigenvalues and eigenfunctions for target potentials at different position dependent mass settings

the nonparanormal semiparametric estimation of high dimensional undirected graphs recent methods for estimating sparse undirected graphs for real valued data in high dimensional problems rely heavily on the assumption of normality we show how to use a semiparametric gaussian copula or nonparanormal for high dimensional inference just as additive models extend linear models by replacing linear functions with a set of one dimensional smooth functions the nonparanormal extends the normal by transforming the variables by smooth functions we derive a method for estimating the nonparanormal study the method s theoretical properties and show that it works well in many examples

gaussian graphical model estimation with false discovery rate control this paper studies the estimation of a high dimensional gaussian graphical model ggm typically the existing methods depend on regularization techniques as a result it is necessary to choose the regularized parameter however the precise relationship between the regularized parameter and the number of false edges in ggm estimation is unclear in this paper we propose an alternative method by a multiple testing procedure based on our new test statistics for conditional dependence we propose a simultaneous testing procedure for conditional dependence in ggm our method can control the false discovery rate fdr asymptotically the numerical performance of the proposed method shows that our method works quite well

graph estimation with joint additive models in recent years there has been considerable interest in estimating conditional independence graphs in high dimensions most previous work has assumed that the variables are multivariate gaussian or that the conditional means of the variables are linear in fact these two assumptions are nearly equivalent unfortunately if these assumptions are violated the resulting conditional independence estimates can be inaccurate we propose a semi parametric method graph estimation with joint additive models which allows the conditional means of the features to take on an arbitrary additive form we present an efficient algorithm for our estimator s computation and prove that it is consistent we extend our method to estimation of directed graphs with known causal ordering using simulated data we show that our method performs better than existing methods when there are non linear relationships among the features and is comparable to methods that assume multivariate normality when the conditional means are linear we illustrate our method on a cell signaling data set

nonparametric mixed effects models for unequally sampled noisy curves we propose a method of analyzing collections of related curves in which the individual curves are modeled as spline functions with random coefficients the method is applicable when the individual curves are sampled at variable and irregularly spaced points this produces a low rank low frequency approximation to the covariance structure which can be estimated naturally by the em algorithm smooth curves for individual trajectories are constructed as best linear unbiased predictor blup estimates combining data from that individual and the entire collection this framework leads naturally to methods for examining the effects of covariates on the shapes of the curves we use model selection techniques akaike information criterion aic bayesian information criterion bic and cross validation to select the number of breakpoints for the spline approximation we believe that the methodology we propose provides a simple flexible and computationally efficient means of functional data analysis