exact solutions to the schrodinger equation for the potential in two dimensions making use of an ansatz for the eigenfunctions we obtain an exact closed form solution to the non relativistic schrodinger equation with the anharmonic potential in two dimensions where the parameters of the potential a b and c satisfy some constraints

a chebyshev approximation for solving nonlinear integral equations of hammerstein type abstract a numerical method for solving fredholm volterra hammerstein integral equations is presented this method is based on replacement of the unknown function by truncated series of well known chebyshev expansion of functions the quadrature formula which we use to calculate integral terms can be estimated by fast fourier transform fft the numerical examples and the number of operations show the advantages of this method to some other usual methods

an approximation method for the solution of nonlinear integral equations a chebyshev collocation method has been presented to solve nonlinear integral equations in terms of chebyshev polynomials this method transforms the integral equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown chebyshev coefficients finally some examples are presented to illustrate the method and results discussed

perturbed factorization of the singular anharmonic oscillator eigenequation the perturbed ladder operator method is applied to the solution of the perturbed eigenequation d2 dx2 m m 1 x2 b2x2 v x ix 0 where v x b1 1 x 2 b2 1 x 4 is a singular perturbation this method which is the extension of the schrodinger infeld hull factorization method within the perturbation scheme provides closed form expressions of the perturbed eigenvalues and ladder functions by means of algebraic manipulations as an illustrative application an analytical solution of the spiked harmonic oscillator eigenequation d2 dx2 b2x2 x4 e x 0 is worked out up to the second order of the perturbation by considering specifically adapted m and dependent perturbing and unperturbed potentials in order to tentatively avoid the known difficulties of convergence of the perturbation series closed form expressions of the x4 anharmonic oscillator energies are obtained in terms of the coupling constant and the quantum number v results following from these expressions are compared wit

shrinkage to tapering estimation of large covariance matrices in this paper we introduce a shrinkage to tapering approach for estimating large covariance matrices when the number of samples is substantially fewer than the number of variables i e n p and p n the proposed estimator improves upon both shrinkage and tapering estimators by shrinking the sample covariance matrix to its tapered version we first show that under both normalized frobenius and spectral risks the minimum mean squared error mmse shrinkage to identity estimator is inconsistent and outperformed by a minimax tapering estimator for a class of high dimensional and diagonally dominant covariance matrices motivated by this observation we propose a shrinkage to tapering oracle sto estimator for efficient estimation of general large covariance matrices a closed form formula of the optimal coefficient of the proposed sto estimator is derived under the minimum frobenius risk since the true covariance matrix is to be estimated we further propose a sto approximating stoa algorithm with a data driven bandwidth selection procedure to iteratively estimate the coefficient and the covariance matrix we study the finite sample performances of different estimators and our simulation results clearly show the improved performances of the proposed sto estimators finally the proposed stoa method is applied to a real breast cancer gene expression data set

nonperturbative treatment of singular potentials a nonperturbative but absolutely convergent algorithm is applied to the determination of the eigenfunctions of hamiltonian with a singular potential of the form h d2 dx2 x2 x in the domain 0 which obey dirichlet boundary conditons the formal structure of the algorithm is identical to that of the lanczos algorithm when it is formally extended to self adjoint operators

about a numerical method of successive interpolations for functional hammerstein integral equations a new and robust numerical method for hammerstein functional integral equations is proposed the convergence and the numerical stability of the method are mathematically proved and tested on some examples

the newton method for solving the theodorsen integral equation the newton method for the solution of the theodorsen integral equation in conformal mapping is studied one step of this method consists of solving a linear integral equation the solution of which is given explicitly as the result of a riemann hilbert problem quadratic convergence of the newton method is established under certain assumptions whereas in other methods a so called condition with 1 is required to hold our method converges also for 1 we will also present a numerical implementation in which the result of one step of the newton method is approximately by a vector in r2n which can be computed with 2n log n o n multiplications in comparison one step of the newton method for the discrete theodorsen equation requires o n3 multiplications

ultraviolet divergence and scaling in a class of singular potentials for the class of singular potentials r2 r the perturbation theory in has an ultraviolet divergence for a range of which causes the perturbation series to be ordered by z where z is a fraction less than unity we use a scaling argument to find z in any dimension n

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

projection and iterated projection methods for nonliear integral equations consider the nonlinear operator equation x mathcal k x with mathcal k a completely continuous mapping of a domain in the banach space mathcal x into mathcal x and let x denote an isolated fixed point of mathcal k let mathcal k n n geqq 1 denote a sequence of finite dimensional approximating subspaces and let p n be a projection of mathcal x onto mathcal x n the projection method for solving x mathcal k x given by x n p n mathcal k x n and the iterated projection solution is defined as tilde x n mathcal k x n we analyze the convergence of x n and tilde x n to x giving a general analysis that includes both the galerkin and collocation methods a more detailed analysis is then given for a large class of urysohn integral operators in one variable showing the superconvergence of tilde x n to x

existence of conditional shape invariance symmetry for singular power potential v r ar2 br 4 cr 6 we examine the solvability conditions of the singular power potential v r ar2 br 4 cr 6 a 0 c 0 in the light of shape invariance we prove that one energy level can be obtained algebraically if a corresponding constraint condition is satisfied by the parameters we show that energies of at most two levels can be obtained algebraically for a severely restricted set of parameters resulting from the satisfaction of two such constraint conditions leading to the existence of conditional shape invariance

perturbation expansions for a class of singular potentials harrell s modified perturbation theory ann phys n y 105 379 1977 is applied and extended to obtain nonpower perturbation expansions for a class of singular hamiltonians h d2 dx2 x2 a x2 x a 0 2 known as generalized spiked harmonic oscillators the perturbation expansions developed here are valid for small values of the coupling 0 and they extend the results which harrell obtained for the spiked harmonic oscillator a 0 formulas for the excited states are also developed

unsupervised learning of gaussian mixtures based on variational component splitting in this paper we present an incremental method for model selection and learning of gaussian mixtures based on the recently proposed variational bayes approach the method adds components to the mixture using a bayesian splitting test procedure a component is split into two components and then variational update equations are applied only to the parameters of the two components as a result either both components are retained in the model or one of them is found to be redundant and is eliminated from the model in our approach the model selection problem is treated locally in a region of the data space so we can set more informative priors based on the local data distribution a modified bayesian mixture model is presented to implement this approach along with a learning algorithm that iteratively applies a splitting test on each mixture component experimental results and comparisons with two other techniques testify for the adequacy of the proposed approach

numerical solution of functional integral equations by the variational iteration method in the present article we apply the variational iteration method to obtain the numerical solution of the functional integral equations this method does not need to be dependent on linearization weak nonlinearity assumptions or perturbation theory application of this method in finding the approximate solution of some examples confirms its validity the results seem to show that the method is very effective and convenient for solving such equations

least squares approximation method for the solution of hammerstein volterra delay integral equations in this paper an efficient numerical method is developed for solving the hammerstein volterra delay integral equations by least squares ls approximation method which is based on a polynomial of degree n to compute an approximation to the solution of hammerstein volterra delay integral equations the convergence analysis of the approximation solution relative to the exact solution of the integral equation is proved and its accuracy is illustrated on two numerical examples the study of this integral equation is important because it has as a particular case the variant of a mathematical model from epidemiology

application of sinc collocation method for solving a class of nonlinear fredholm integral equations in this paper the numerical solution of nonlinear fredholm integral equations of the second kind is considered by two methods the methods are developed by means of the sinc approximation with the single exponential se and double exponential de transformations these numerical methods combine a sinc collocation method with the newton iterative process that involves solving a nonlinear system of equations we provide an error analysis for the methods so far approximate solutions with polynomial convergence have been reported for this equation these methods improve conventional results and achieve exponential convergence some numerical examples are given to confirm the accuracy and ease of implementation of the methods

averaging maximum penalized likelihood and bayesian estimation for improving gaussian mixture probability density estimates we apply the idea of averaging ensembles of estimators to probability density estimation in particular we use gaussian mixture models which are important components in many neural network applications we investigate the performance of averaging using three data sets for comparison we employ two traditional regularization approaches i e a maximum penalized likelihood approach and a bayesian approach in the maximum penalized likelihood approach we use penalty functions derived from conjugate bayesian priors such that an expectation maximization em algorithm can be used for training in all experiments the maximum penalized likelihood approach and averaging improved performance considerably if compared to a maximum likelihood approach in two of the experiments the maximum penalized likelihood approach outperformed averaging in one experiment averaging was clearly superior our conclusion is that maximum penalized likelihood gives good results if the penalty term in the cost function is appropriate for the particular problem if this is not the case averaging is superior since it shows greater robustness by not relying on any particular prior assumption the bayesian approach worked very well on a low dimensional toy problem but failed to give good performance in higher dimensional problems

regularized mixture density estimation with an analytical setting of shrinkage intensities in this paper we propose a method for p variate probability density estimation assuming a gaussian mixture model gmm our method exploits a regularization technique for improving the estimation accuracy of the gmm component covariance matrices we derive an expectation maximization algorithm for fitting our regularized gmm rgmm which exploits an analytical ledoit wolf type shrinkage estimation of the covariance matrices our method is compared with recent model based and variational bayes approximation methods using synthetic and real data sets the obtained results show that the proposed rgmm method achieves a significant improvement in the performance of multivariate probability density estimation with respect to other methods on both the synthetic and the real data sets

asymptotic error expansion of a collocation type method for hammerstein equations abstract in recent papers kumar and sloan have considered the numerical solution of the hammerstein equation y x f x a b k x t g t y t dt x a b by a method that first applied the standard collocation procedure to an equivalent equation for z t g t y t and then obtained an approximation to y by use of the equation y x f x a b k x t z t dt x a b in this paper the asymptotic error expansion of this method is obtained we show that when piecewise polynomial of gv p 1 are used the approximation solution admits an error expansion in even powers of the step size h beginning with a term h 2 p thus richardson s extrapolation can be performed on the solution and this will increase the accuracy of numerical solution greatly

study of the singular anharmonic potentials by means of the analytic continuation method here we study the singular anharmonic potentials by applying the analytic continuation method of holubec and stauffer 1985 in order to do that we have developed several approximations to the problem because this method cannot be applied when the solution has essential singularities at any point of its domain all the options here shown have the same precision giving us the eigenvalues correct to all the decimal places provided by the computer

rationalized haar wavelet bases to approximate solution of nonlinear fredholm integral equations with error analysis in this article we approximate the solutions of the nonlinear fredholm integral equations of the second kind by the method based on using the properties of rh wavelets and matrix operator also the banach fixed point theorem guarantees the convergence of the method also we get an upper bound for the error furthermore the order of convergence is analyzed the algorithm to compute the solutions and some numerical examples are also illustrated the numerical results obtained by our method have been compared with other methods

iterative method and convergence analysis for a kind of mixed nonlinear volterra fredholm integral equation in this paper the iterative method is presented for numerically solving the nonlinear volterra fredholm integral equation first considering some conditions on k i and l i i 1 2 of the integral equation then we define the equidistance collocation points and the integral part of equation is discretized by newton cotes integration formula finally the approximate solutions of integral equation are obtain by iterative method the convergence analysis of the integral equation is given some numerical examples are given to illustrate the accuracy and dependability of the method

a different approach for solving the nonlinear fredholm integral equations of the second kind in this paper we proposed a method for finding a approximate solution for the nonlinear fredholm integral equations of the second kind in this approach the nonlinearity of the kernel has no serious effect on the convergence of the solution first we convert the problem to an optimal control problem by introducing an artificial control function then by using some concepts of measure theory we make a metamorphosis in the space of the problem and finally we will get a linear programming whose solution gives rise to the approximate solution of the integral equation

singular anharmonicities and the analytic continued fractions the potential v r 2r2 2r 4 is investigated and its bound states are constructed by a generalized hill determinant method it is shown that the binding energy e and another free parameter in the underlying laurent series representation of the wave function may be obtained from a coupled pair of hill determinant equations

shrinkage algorithms for mmse covariance estimation we address covariance estimation in the sense of minimum mean squared error mmse when the samples are gaussian distributed specifically we consider shrinkage methods which are suitable for high dimensional problems with a small number of samples large p small n first we improve on the ledoit wolf lw method by conditioning on a sufficient statistic by the rao blackwell theorem this yields a new estimator called rblw whose mean squared error dominates that of lw for gaussian variables second to further reduce the estimation error we propose an iterative approach which approximates the clairvoyant shrinkage estimator convergence of this iterative method is established and a closed form expression for the limit is determined which is referred to as the oracle approximating shrinkage oas estimator both rblw and oas estimators have simple expressions and are easily implemented although the two methods are developed from different perspectives their structure is identical up to specified constants the rblw estimator provably dominates the lw method for gaussian samples numerical simulations demonstrate that the oas approach can perform even better than rblw especially when n is much less than p we also demonstrate the performance of these techniques in the context of adaptive beamforming

quantum states of a sextic potential hidden symmetry and quantum monodromy a known terminating polynomial ansatz for a finite sub set of states of the sextic central potential v a b c r ar2 br4 cr6 c 0 subject to integer related parameter constraints is confirmed to apply in two as well as three dimensions a hidden symmetry relating the eigenstates of v a b c r to those of v a b c r is used to establish the boundary between the ansatz sub set and the infinite set of higher states connections between the radial wavefunctions rn a b c r and rn a b c r also provide semiclassical insight into the origin of the parameter constraint finally the ansatz eigenvalue dispositions are related to the phenomenon of quantum monodromy

a numerical scheme for a class of nonlinear fredholm integral equations of the second kind in this paper an iterative approach for obtaining approximate solutions for a class of nonlinear fredholm integral equations of the second kind is proposed the approach contains two steps at the first one we define a discretized form of the integral equation and prove that by considering some conditions on the kernel of the integral equation solution of the discretized form converges to the exact solution of the problem following that in the next step solution of the discretized form is approximated by an iterative approach we finally on some examples show the efficiency of the proposed approach

projection pursuit mixture density estimation in this paper we seek a gaussian mixture model gmm of an n variate probability density function usually the parameters of gmms are determined in the original n dimensional space by optimizing a maximum likelihood ml criterion a practical deficiency of this method of fitting gmms is its poor performance when dealing with high dimensional data since a large sample size is needed to match the accuracy that is possible in low dimensions we propose a method for fitting the gmm based on the projection pursuit strategy this gmm is highly constrained and hence its ability to model structure in subspaces is enhanced compared to a direct ml fitting of a gmm in high dimensions our method is closely related to recently developed independent factor analysis ifa mixture models the comparisons with ml fitting of gmm in n dimensions and ifa mixtures show that the proposed method is an attractive choice for fitting gmms using small sizes of training sets

projection pursuit density estimation abstract the projection pursuit methodology is applied to the multivariate density estimation problem the resulting nonparametric procedure is often less biased than the kernel and near neighbor methods in addition graphical information is produced that can be used to help gain geometric insight into the multivariate data distribution

solution of nonlinear volterra fredholm hammerstein integral equations via rationalized haar functions abstract rationalized haar functions are developed to approximate of the nonlinear volterra fredholm hammerstein integral equations the properties of rationalized haar functions are first presented and the operational matrix of integration together with the product operational matrix are utilized to reduce the computation of integral equations into some algebraic equations the method is computationally attractive and applications are demonstrated through illustrative examples

numerical computation method in solving integral equations by using chebyshev wavelet operational matrix of integration a direct method for solving integral equations whose solutions are continuous or discontinuous by using chebyshev wavelet basis in galerkin equations is presented another method for solving volterra type integral equations which use operational matrix of integration omi for chebyshev wavelets is introduced and used to reduce this type of integral equations to a system of algebraic equations some quadrature formula for calculating inner products have presented which can be operated by fast fourier transform fft the numerical examples and the number of operations show the advantages of chebyshev wavelet galerkin method to some other usual methods and usual chebyshev basis

are large distance heegaard splittings generic in a previous paper we introduced a notion of genericity for countable sets of curves in the curve complex of a surface s based on the lebesgue measure on the space of projective measured laminations in s with this definition we prove that for each fixed g 1 the set of irreducible genus g heegaard splittings of high distance is generic in the set of all irreducible heegaard splittings our definition of genericity is different and more intrinsic then the one given via random walks

investigation on the existence of solutions for some nonlinear functional integral equations abstract we prove an existence theorem for a nonlinear integral equation which is a volterra counterpart of an integral equation arising in traffic theory using the technique of the measure of noncompactness in banach algebra we prove an existence theorem for a nonlinear functional integral equation basic fixed point theorems such as darbo s theorem are employed to obtain the aforementioned aims in banach algebra

assessing a mixture model for clustering with the integrated completed likelihood we propose an assessing method of mixture model in a cluster analysis setting with integrated completed likelihood for this purpose the observed data are assigned to unknown clusters using a maximum a posteriori operator then the integrated completed likelihood icl is approximated using the bayesian information criterion bic numerical experiments on simulated and real data of the resulting icl criterion show that it performs well both for choosing a mixture model and a relevant number of clusters in particular icl appears to be more robust than bic to violation of some of the mixture model assumptions and it can select a number of dusters leading to a sensible partitioning of the data

singular anharmonicities and the analytic continued fractions ii the potentials v r ar2 br 4 cr 6 the c 0 results of paper i j math phys 30 23 1989 are extended in spite of the presence of an additional coupling constant the laurent series solutions of the schrodinger equation that are obtained remain similar to mathieu functions indeed the recurrences for coefficients preserve their three term character their analytic continued fraction solutions still converge etc the formulas become even slightly simpler for c 0 due to a certain symmetry of the equations to be solved an acceleration of convergence is better understood and a few numerical illustrations of efficiency are also delivered

spiked harmonic oscillators a complete variational treatment is provided for a family of spiked harmonic oscillator hamiltonians h d2 dx2 bx2 x b 0 0 for arbitrary 0 a compact topological proof is presented that the set s n of known exact solutions for 2 constitutes an orthonormal basis of the hilbert space l2 0 closed form expressions are derived for the matrix elements of h with respect to s these analytical results and the inclusion of a further free parameter facilitate optimized variational estimation of the eigenvalues of h to high accuracy

on the existence of solutions of functional integral equation of urysohn type in this paper we introduce a brief historical note about the study of the existence of solutions of functional integral equations hence we prove the existence of monotonic solutions of the functional integral equation of urysohn type the technique rely on the concept of measure of noncompactness and its associated darbo fixed point theorem

eigenvectors of some large sample covariance matrix ensembles we consider sample covariance matrices s n frac 1 p sigma n 1 2 x nx n sigma n 1 2 where xn is a n p real or complex matrix with i i d entries with finite 12th moment and n is a n n positive definite matrix in addition we assume that the spectral measure of n almost surely converges to some limiting probability distribution as n and p n 0 we quantify the relationship between sample and population eigenvectors by studying the asymptotics of functionals of the type frac 1 n text tr g sigma n s n zi 1 where i is the identity matrix g is a bounded function and z is a complex number this is then used to compute the asymptotically optimal bias correction for sample eigenvalues paving the way for a new generation of improved estimators of the covariance matrix and its inverse

piecewise continuous collocation for integral equations the collocation method for integral equations of the second kind is surveyed and analyzed for the case in which the approximate solutions are only piecewise continuous difficulties with the usual function space setting of l infty d are discussed and a satisfactory sense of point evaluation is given for elements of l infty d other approaches which are discussed include i reformulation as a degenerate kernel method ii the prolongation restriction framework of noble iii other function space settings and iv reformulation as a continuous approximation problem by iterating the piecewise continuous approximate solution in the original integral equation

a simple numerical method for singular potentials a one parameter family of coordinate transformations is shown to lead to a simple finite difference method which gives highly accurate energies and expectation values for the schrodinger equation in which the potential consists of a smooth term plus a perturbing term which is singular at the origin the method is effective down to very small values of the perturbation parameter and supplements the previously reported perturbation approach which is valuable for large values

on the two dimensional stationary schrodinger equation with a singular potential abstract we study the equation u x y x y u x y 0 when the potential has the following expression r a r 2 where r are the polar coordinates in r 2 and a is a 2 periodic function of class c k with k 1 we obtain series representations for solutions in a full neighborhood of the singular point and we also give representations in terms of pseudoanalytic formal powers in sectors having the singular point as a vertex the results are obtained with the aid of the reduction of the stationary schrodinger equation to a vekua equation of a special form and by using recent developments in pseudoanalytic function theory

on the functional integral equations of mixed type and integro differential equations of fractional orders the existence of monotonic solution for the functional integral equation of the mixed typex t g t email protected 1 0k 1 t s f 1s 0 sk 2 s t f 2 t x t email protected be proved in l 1 0 1 such kind of equations plays a significant role in many branches of mathematical physics engineering and economics on the other hand our considered equation are the general form of hammerstein and volterra integral equations see 4 and 5 as an application we prove the existence of solution for the initial value problem of the fractional order integro differential equation dx t dt g t email protected 1 0k 1 t s f 1 s d bx s ds email protected 0 1

nonlinear shrinkage estimation of large dimensional covariance matrices many statistical applications require an estimate of a covariance matrix and or its inverse when the matrix dimension is large compared to the sample size which happens frequently the sample covariance matrix is known to perform poorly and may suffer from ill conditioning there already exists an extensive literature concerning improved estimators in such situations in the absence of further knowledge about the structure of the true covariance matrix the most successful approach so far arguably has been shrinkage estimation shrinking the sample covariance matrix to a multiple of the identity by taking a weighted average of the two turns out to be equivalent to linearly shrinking the sample eigenvalues to their grand mean while retaining the sample eigenvectors our paper extends this approach by considering nonlinear transformations of the sample eigenvalues we show how to construct an estimator that is asymptotically equivalent to an oracle estimator suggested in previous work as demonstrated in extensive monte carlo simulations the resulting bona fide estimator can result in sizeable improvements over the sample covariance matrix and also over linear shrinkage

on the eigenvalues of the s state radial equation of a spiked harmonic oscillator eigenvalues of the ground state of the radial schrodinger equation for a spiked harmonic oscillator potential have been evaluated employing two methods numerically via the lanczos grid technique and by means of standard pade approximants constructed from an expansion of large coupling parameter series for the energy numerical results are compared for several values of the parameters characterizing the spiked singular potential

nonsingular spiked harmonic oscillator a perturbative study of a class of nonsingular spiked harmonic oscillators defined by the hamiltonian h d2 dr 2 r 2 r in the domain 0 is carried out in the two extremes of a weak coupling and a strong coupling regimes a path has been found to connect both expansions for near 2

finite covers of random 3 manifolds a 3 manifold is haken if it contains a topologically essential surface the virtual haken conjecture posits that every irreducible 3 manifold with infinite fundamental group has a finite cover which is haken in this paper we study random 3 manifolds and their finite covers in an attempt to shed light on this difficult question in particular we consider random heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface for this model of random 3 manifold we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds our results contrast with the analogous probabilities for groups coming from random balanced presentations giving quantitative theorems to the effect that 3 manifold groups have many more finite quotients than random groups the next natural question is whether these covers have positive betti number for abelian covers of a fixed type over 3 manifolds of heegaard genus 2 we show that the probability of positive betti number is 0

energy bounds for a class of singular potentials and some related series perturbation expansions up to third order for the generalized spiked harmonic oscillator hamiltonians h d2 dx2 x2 a x2 x a 0 2 1 1 2 1 4a and small values of the coupling 0 are developed upper and lower bounds for the eigenvalues are computed by the procedure of burrows et al 1987 j phys a math gen 20 889 97 for assessing the accuracy of a truncated perturbation expansion closed form sums for some related perturbation double infinite series then immediately follow as a result of this investigation

shrinkage estimators for covariance matrices estimation of covariance matrices in small samples has been studied by many authors standard estimators like the unstructured maximum likelihood estimator ml or restricted maximum likelihood reml estimator can be very unstable with the smallest estimated eigenvalues being too small and the largest too big a standard approach to more stably estimating the matrix in small samples is to compute the ml or reml estimator under some simple structure that involves estimation of fewer parameters such as compound symmetry or independence however these estimators will not be consistent unless the hypothesized structure is correct if interest focuses on estimation of regression coefficients with correlated or longitudinal data a sandwich estimator of the covariance matrix may be used to provide standard errors for the estimated coefficients that are robust in the sense that they remain consistent under misspecification of the covariance structure with large matrices however the inefficiency of the sandwich estimator becomes worrisome we consider here two general shrinkage approaches to estimating the covariance matrix and regression coefficients the first involves shrinking the eigenvalues of the unstructured ml or reml estimator the second involves shrinking an unstructured estimator toward a structured estimator for both cases the data determine the amount of shrinkage these estimators are consistent and give consistent and asymptotically efficient estimates for regression coefficients simulations show the improved operating characteristics of the shrinkage estimators of the covariance matrix and the regression coefficients in finite samples the final estimator chosen includes a combination of both shrinkage approaches i e shrinking the eigenvalues and then shrinking toward structure we illustrate our approach on a sleep eeg study that requires estimation of a 24 x 24 covariance matrix and for which inferences on mean parameters critically depend on the covariance estimator chosen we recommend making inference using a particular shrinkage estimator that provides a reasonable compromise between structured and unstructured estimators

study of anharmonic singular potentials a simple and efficient variational method is introduced to accelerate the convergence of the eigenenergy computations for a hamiltonian h with singular potentials closed form analytic expressions in n dimensions are obtained for the matrix elements of h with respect to the eigenfunctions of a soluble singular problem with two free parameters a and b the matrix eigenvalues are then optimized with respect to a and b for a given n applications convergence rates and comparisons with earlier work are discussed in detail

entropy based incremental variational bayes learning of gaussian mixtures variational approaches to density estimation and pattern recognition using gaussian mixture models can be used to learn the model and optimize its complexity simultaneously in this brief we develop an incremental entropy based variational learning scheme that does not require any kind of initialization the key element of the proposal is to exploit the incremental learning approach to perform model selection through efficient iteration over the variational bayes optimization step in a way that the number of splits is minimized the method starts with just one component and adds new components iteratively by splitting the worst fitted kernel in terms of evaluating its entropy our experimental results on synthetic and real data sets show the effectiveness of the approach outperforming other state of the art incremental component learners

how do supersingular perturbations generate non taylor series we show that a suitable choice of the zero order problem can lead directly to the emergence of non taylor expansions in for the ground state energies of the hamiltonian family h 2 r2 r in regions 3 the discussion includes the role of dimension and the right order parameter the effect of choosing a more general potential of the form r r is also considered

exceptional and cosmetic surgeries on knots we show that the bridge distance of a knot determines a lower bound on the genera of essential surfaces and heegaard surfaces in the manifolds that result from non trivial dehn surgeries on the knot in particular knots with high bridge distance do not admit non trivial non hyperbolic surgeries or non trivial cosmetic surgeries we further show that if a knot has bridge distance at least 3 then its bridge number is bounded above by a function of seifert genus or indeed by the genus of almost any essential surface or heegaard surface in the surgered manifold

spiked oscillators exact solution a procedure to obtain the eigenenergies and eigenfunctions of a quantum spiked oscillator is presented the originality of the method lies in an adequate use of asymptotic expansions of wronskians of algebraic solutions of the schrodinger equation the procedure is applied to three familiar examples of spiked oscillators

numerical solution of nonlinear volterra fredholm hammerstein integral equations via collocation method based on radial basis functions abstract a numerical technique based on the spectral method is presented for the solution of nonlinear volterra fredholm hammerstein integral equations this method is a combination of collocation method and radial basis functions rbfs with the differentiation process drbf using zeros of the shifted legendre polynomial as the collocation points different applications of rbfs are used for this purpose the integral involved in the formulation of the problems are approximated based on legendre gauss lobatto integration rule the results of numerical experiments are compared with the analytical solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme

two cubic spline methods for solving fredholm integral equations abstract in this work we propose two methods based on the use of natural and quasi cubic spline interpolations for approximating the solution of the second kind fredholm integral equations convergence analysis is established some numerical examples are given to show the validity of the presented methods

a pointwise quasi newton method for integral equations for fully nonlinear integral equations conventional quasi newton methods such as broyden s method may not possess good convergence properties in this paper we present a new method that gives superlinear convergence for such integral equations and provides for savings over newton s method in the cost of solution of the linear equation for the quasi newton step

numerical solution of differential equations by using chebyshev wavelet operational matrix of integration in this paper we present operational matrix of integration omi of chebyshev wavelets basis and the product operation matrix pom of it some comparative examples are included to demonstrate the superiority of operational matrix of chebyshev wavelets to those of legendre wavelets

chain models and their effective hamiltonians we develop a new formalism suitable for diagonalization of hamiltonians with 2t 1 nonzero diagonals chain models in a systematic perturbation like way we get the expansions of green and or wave functions they are generated by solution of a sequence of fixed point quadratic equations for t t dimensional matrices for t 1 and t 2 this solution is feasible by elementary means so that the respective tridiagonal and pentadiagonal chain models may be considered exactly solvable in this context in an alternative first order variational perturbation formulation the method may provide the simultaneous upper and lower energy bounds for any t

variational learning for finite dirichlet mixture models and applications in this paper we focus on the variational learning of finite dirichlet mixture models compared to other algorithms that are commonly used for mixture models such as expectation maximization our approach has several advantages first the problem of over fitting is prevented furthermore the complexity of the mixture model i e the number of components can be determined automatically and simultaneously with the parameters estimation as part of the bayesian inference procedure finally since the whole inference process is analytically tractable with closed form solutions it may scale well to large applications both synthetic and real data generated from real life challenging applications namely image databases categorization and anomaly intrusion detection are experimented to verify the effectiveness of the proposed approach

collocation method for linear and nonlinear fredholm and volterra integral equations a collocation procedure is developed for the linear and nonlinear fredholm and volterra integral equations using the globally defined b spline and auxiliary basis functions the solution is collocated by cubic b spline and then the integral equation is approximated by the 5 points gauss turan quadrature formula with respect to the legendre weight function combination of these two approaches is the main idea of this paper to reduce the cost of computation and complexity the error analysis of proposed numerical method is studied theoretically numerical results are given to illustrate the efficiency of the proposed method the results are compared with the results obtained by other methods to verify that this method is accurate and efficient

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

a new collocation type method for hammerstein integral equations on presente une methode du type de collocation pour la resolution numerique des equations integrales d hammerstein on etablit la convergence de l approximation desiree vers la solution exacte

a discrete collocation type method for hammerstein equations in recent papers we have considered the numerical solution of the hammerstein equation y t f t int a b k t s g s y s ds quad t in a b by a method that first applies the standard collocation procedure to an equivalent equation for z t g t y t and then obtains an approximation to y by use of the equation y t f t int a b k t s z s ds quad t in a b we study here a discretized version of the above method this arises when numerical quadrature is used to approximate the definite integrals occurring in the said method we consider the use of interpolatory quadrature rules and we seek the discrete collocation approximation to z in certain piecewise polynomial function spaces our principal result gives the precision required of a quadrature rule to guarantee the best possible super convergence rate for the discrete approximation to y in the case where the kernel k is sufficiently smooth this rate is of the same order as the rate for an approximation to y obtain

taylor polynomial solutions of nonlinear volterra fredholm integral equations the kanwal and liu method for the solution of fredholm integral equations is applied to certain nonlinear volterra fredholm integral equations in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed

practical bayesian density estimation using mixtures of normals abstract mixtures of normals provide a flexible model for estimating densities in a bayesian framework there are some difficulties with this model however first standard reference priors yield improper posteriors second the posterior for the number of components in the mixture is not well defined if the reference prior is used third posterior simulation does not provide a direct estimate of the posterior for the number of components we present some practical methods for coping with these problems finally we give some results on the consistency of the method when the maximum number of components is allowed to grow with the sample size

numerical solution of non linear fredholm integral equations by using multiwavelets in the petrov galerkin method in this paper we use the wavelet petrov galerkin wpg method for solving one of the most important cases in non linear fredholm integral equations which is called hammerstein equations we suppose that the trial space and the test space that are used in this method to be piecewise polynomials space also we use alpert s multiwavelets as the bases for these spaces such that the wavelet basis for the test space constructed from lower degree polynomials than the trial space finally for showing efficiency of this method we use some numerical examples

sparse density estimation on the multinomial manifold a new sparse kernel density estimator is introduced based on the minimum integrated square error criterion for the finite mixture model since the constraint on the mixing coefficients of the finite mixture model is on the multinomial manifold we use the well known riemannian trust region rtr algorithm for solving this problem the first and second order riemannian geometry of the multinomial manifold are derived and utilized in the rtr algorithm numerical examples are employed to demonstrate that the proposed approach is effective in constructing sparse kernel density estimators with an accuracy competitive with those of existing kernel density estimators

unified framework to regularized covariance estimation in scaled gaussian models we consider regularized covariance estimation in scaled gaussian settings e g elliptical distributions compound gaussian processes and spherically invariant random vectors asymptotically in the number of samples the classical maximum likelihood ml estimate is optimal under different criteria and can be efficiently computed even though the optimization is nonconvex we propose a unified framework for regularizing this estimate in order to improve its finite sample performance our approach is based on the discovery of hidden convexity within the ml objective we begin by restricting the attention to diagonal covariance matrices using a simple change of variables we transform the problem into a convex optimization that can be efficiently solved we then extend this idea to nondiagonal matrices using convexity on the manifold of positive definite matrices we regularize the problem using appropriately convex penalties these allow for shrinkage towards the identity matrix shrinkage towards a diagonal matrix shrinkage towards a given positive definite matrix and regularization of the condition number we demonstrate the advantages of these estimators using numerical simulations

estimation of parameter matrices and eigenvalues in manova and canonical correlation analysis on entreprend une approche par la theorie de la decision avec l erreur quadratique comme fonction de perte

robust shrinkage estimation of high dimensional covariance matrices we address high dimensional covariance estimation for elliptical distributed samples which are also known as spherically invariant random vectors sirv or compound gaussian processes specifically we consider shrinkage methods that are suitable for high dimensional problems with a small number of samples large p small n we start from a classical robust covariance estimator tyler 1987 which is distribution free within the family of elliptical distribution but inapplicable when n p using a shrinkage coefficient we regularize tyler s fixed point iterations we prove that for all n and p the proposed fixed point iterations converge to a unique limit regardless of the initial condition next we propose a simple closed form and data dependent choice for the shrinkage coefficient which is based on a minimum mean squared error framework simulations demonstrate that the proposed method achieves low estimation error and is robust to heavy tailed samples finally as a real world application we demonstrate the performance of the proposed technique in the context of activity intrusion detection using a wireless sensor network

the generalized continued fractions and potentials of the lennard jones type for a broad class of the strongly singular potentials v r which are defined as superpositions of separate power law components the general solution of the corresponding schrodinger differential equation is constructed as an analog of mathieu functions the analogy is supported by the use of the generalized continued fractions the questions of convergence are analyzed in detail

new approach to solve two dimensional fredholm integral equations abstract in this paper we present an efficient iterative method based on quadrature formula to solve two dimensional nonlinear fredholm integral equations we investigate the convergence analysis of the proposed method the error estimation is given in terms of uniform and partial modulus of continuity in addition we show the numerical stability analysis of the method with respect to the choice of the first iteration finally some numerical experiments confirm the theoretical results and illustrate the accuracy of the method

generation of higher order pseudospectral integration matrices a new explicit expression of the higher order pseudospectral integration matrices is presented using an explicit formula for computing iterated integrals of chebyshev polynomials applications to initial value problems boundary value problems linear integral and integro differential equations are presented the present numerical results are in satisfactory agreement with the exact solutions and show the advantages of this method to some other known methods

derivative free iterative methods for nonlinear systems in this work we introduce a new operator of divided differences that preserves the convergence order when it is used for approximating the jacobian matrix in iterative method for solving nonlinear systems we obtain derivative free iterative methods with lower computational cost than the corresponding ones with different operators of divided differences we also study the global convergence of these methods by analyzing their dynamical behavior

numerical solutions of the integral equations homotopy perturbation method and adomian s decomposition method in this paper a homotopy perturbation method is proposed to solve non singular integral equations comparisons are made between adomian s decomposition method and the proposed method it is shown adomian s decomposition method is a homotopy only finally by using homotopy perturbation method a new iterative scheme like adomian s decomposition method is proposed for solving the non singular integral equations of the first kind the results reveal that the proposed method is very effective and simple