optimal rates of convergence for deconvolving a density abstract suppose that the sum of two independent random variables x and z is observed where z denotes measurement error and has a known distribution and where the unknown density f of x is to be estimated one application is the estimation of a prior density for a sequence of location parameters a second application arises in the errors in variables problem for nonlinear and generalized linear models when one attempts to model the distribution of the true but unobservable covariates this article shows that if z is normally distributed and f has k bounded derivatives then the fastest attainable convergence rate of any nonparametric estimator of f is only log n k 2 therefore deconvolution with normal errors may not be a practical proposition other error distributions are also treated stefanski carroll 1987a estimators achieve the optimal rates the results given have versions for multiplicative errors where they imply that even optimal rates are exceptionally slow

mean and variance of implicitly defined biased estimators such as penalized maximum likelihood applications to tomography many estimators in signal processing problems are defined implicitly as the maximum of some objective function examples of implicitly defined estimators include maximum likelihood penalized likelihood maximum a posteriori and nonlinear least squares estimation for such estimators exact analytical expressions for the mean and variance are usually unavailable therefore investigators usually resort to numerical simulations to examine the properties of the mean and variance of such estimators this paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters we derive the approximations using the implicit function theorem the taylor expansion and the chain rule the expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation as illustrations we demonstrate that the approximations work well in two tomographic imaging applications with poisson statistics we also describe a plug in approximation that provides a remarkably accurate estimate of variability even from a single noisy poisson sinogram measurement the approximations should be useful in a wide range of estimation problems

convergence rates for regularized solutions of integral equations from discrete noisy data given data yi kg u ei where the e s are random errors the u s are known g is an unknown function in a reproducing kernel space with kernel r and k is a known integral operator it is shown how to calculate convergence rates for the regularized solution of the equation as the evaluation points u become dense in the interval of interest these rates are shown to depend on the eigenvalue asymptotics of krk where r is the integral operator with kernel r the theory is applied to abel s equation and the estimation of particle size densities in stereology rates of convergence of regularized histogram estimates of the particle size density are given

imposing various boundary conditions on positive definite kernels abstract this work is motivated by the frequent occurrence of boundary value problems with various boundary conditions in the modeling of some problems in engineering and physical science here we propose a new technique to force the positive definite kernels such as some radial basis functions to satisfy the boundary conditions exactly it can improve the applications of existing methods based on positive definite kernels and radial basis functions especially the kernel based pseudospectral method for handling the differential equations with more complicated boundary conditions in the proposed technique some new kernels are constructed using the positive definite kernels in a manner that they satisfy the required conditions in addition we prove the positive definiteness of the newly constructed kernel and also the non singularity of the collocation matrix is proved under some conditions the proposed method is verified through the numerical solution of some benchmark problems such as a singularly perturbed steady state convection diffusion problem two and three dimensional poissons equations with various boundary conditions

an iterative multistep kernel based method for nonlinear volterra integral and integro differential equations of fractional order abstract an iterative multistep kernel based method is proposed for the nonlinear volterra integral equations and nonlinear volterra integro differential equations of fractional order which can produce reliable globally smooth numerical solutions an error estimate of the positive definite kernel interpolation and the convergence and error analysis of the proposed iterative scheme are investigated here we focused on positive definite radial basis kernels and further a new and applicable shape parameter selection strategy is proposed the proposed multi step method set up and solve several small local problems instead of a single large problem which makes it suitable for problems with long time simulations in order to show the efficiency and versatility of the proposed method some numerical experiments are reported the comparison of the numerical results with the analytical solutions and the best reported results in the literature confirm the good accuracy and applicability of the proposed method

deconvolution of microfluorometric histograms with b splines abstract we consider the problem of estimating a probability density from observations from that density which are further contaminated by random errors we propose a method of estimation using spline functions discuss the numerical implementation of the method and prove its consistency the problem is motivated by the analysis of dna content obtained by microfluorometry and an example of such an analysis is included

synchronization of oscillators with random nonlocal connectivity in this paper we study the existing observation in literature about synchronization of a large number of coupled maps with random nonlocal connectivity chate and manneville chaos 2 307 1992 these connectivities which lack any spatial significance can be realized in neural nets and electrical circuits it is quite interesting and of practical importance to note that a huge number of maps can be synchronized with this connectivity we show that this synchronization stems from the fact that the connectivity matrix has a finite gap in the eigenvalue spectrum in the macroscopic limit we give a quantitative explanation for the gap we compare the analytic results with the ones quoted in the above reference we also study the departures from this highly collective behavior in the low connectivity limit and show that the behavior is almost statistical for very low connectivity of late there has been considerable attention paid to the study of coupled map lattices cml in various contexts they have been used as a computationally simple and analytically tractable model for spatiotemporal systems 1 the studies on cml s have been either in one and two dimensional lattices or with global coupling in which case there is no notion of lattice geometry 2 the higher dimensional connectivities 3 or hierarchial connectivities 4 5 are studied very little one more system that has been studied is the cml with random nonlocal couplings 6 the motivation is twofold first this is an effectively high dimensional system the phenomenology in cml in higher dimensions has not been studied much and needs further investigation second there are systems like neural nets in which the local connections do not have any spatial significance there also exist systems like electrical circuits 7 in which connectivity is at one s will and such a coupling can be easily realized thus the studies of different connectivities and their effects will be useful in designing well controlled systems in this system of random nonlocal connectivity chate and manneville have presented preliminary results 6 which show that synchronization of a large number of oscillators is easily achieved with this connectivity synchronization of oscillators in spatially extended systems such as coupled oscillators is important from various points of view by achieving synchronization one can try to build huge but more controllable and better behaved systems which are effectively low dimensional 7 sometimes synchronization may serve other purposes such as sending codes that are difficult to break 8 in various contexts this problem has been subjected to several studies in the past few years 9 we would like to show that the phenomenon of synchronization can be generally understood by investigating the eigenvalue spectrum of the connectivity matrix and can help us to understand the existing observations in this paper we will explicitly illustrate how one can separate the mode leading to spatial homogeneity from the rest we will show that there exists a finite gap between the growth rates the spatially homogeneous mode and the rest in the model studied in ref 6 we will also study the departures from this behavior for lower connectivities for the linear stability analysis of the synchronized state we will study the eigenvalue spectrum of the connectivity matrix with random nonlocal connectivity we will also study the eigenspectrum of the product of such matrices we would note that a similar model of random connectivity matrix of size n3n with k nonzero elements in each row has been investigated by cook and derrida in ref 10 in connection with the random energy model the generalized random energy model and directed polymers in random media they have obtained exact analytic results for products of such matrices in the case where the matrices are sparse and the distribution function of nonzero elements is not a d function however in the model studied by chate and manneville 6 all connections have the same weight i e the distribution function of nonzero elements is a d function

a shooting reproducing kernel hilbert space method for multiple solutions of nonlinear boundary value problems in this work an iterative method is proposed to predict and demonstrate the existence and multiplicity of solutions for nonlinear boundary value problems in addition the proposed method is capable of calculating analytical approximations for all branches of solutions this method is a combination of reproducing kernel hilbert space method and a shooting like technique which takes advantage of two powerful methods for solving nonlinear boundary value problems the formulation and implementation of this iterative method is discussed for nonlinear second order with two and three point boundary value problems also the convergence of the proposed method is proved to demonstrate the computational efficiency the mentioned method is implemented for some nonlinear exactly solvable differential equations including strongly nonlinear bratu equation and nonlinear reaction diffusion equation it is also applied successfully to two nonlinear three point boundary value problems with unknown exact solutions in the last example a new branch of solutions is found which shows the power of the method to search for multiple solutions and indicates that the method may be successful in cases where purely analytic methods are not

cross validated spline methods for the estimation of three dimensional tumor size distributions from observations on two dimensional cross sections abstract we study the problem of estimating the distribution of the three dimensional radiuses of a collection of spheres given measurements of the two dimensional radiuses of a sample of planar cross sections this problem arises in the estimation of the tumor size distribution of spherical microtumors induced in mouse livers following injection of a carcinogen we first convert this problem to a form suitable for the application of cross validated spline methods for the solution of ill posed integral equations given noisy data then we develop special numerical techniques that will allow the spline methods to be accurately applied to integral equations like those associated with the present problem we apply the resulting method to some mouse liver data the subject mouse liver has been completely dissected allowing a rare comparison of the estimate with the truth the statistical properties of the estimate are explored via monte carlo methods the interplay between statistical and numerical analyti

an iterative reproducing kernel method in hilbert space for the multi point boundary value problems abstract in this paper an iterative method is proposed to solve the nonlinear bitsadze samarskii boundary value problems with multi point boundary conditions the algorithm is based on the reproducing kernel hilbert space method we use an iterative scheme to overcome the nonlinearity of the problem the convergence and error estimate of the iterative scheme are established the reproducing kernel hilbert space method is used to generate an approximation of the linearized problem in fact the reproducing kernel hilbert space method is combined with an iterative scheme to approximate the solution and an error estimate of the approximate solution is derived in order to show the efficiency and versatility of the proposed method some numerical results are reported the comparison of numerical results with the analytical solution and the best results reported in the literature confirms the good accuracy and applicability of the proposed method

homotopy perturbation reproducing kernel method for nonlinear systems of second order boundary value problems in this paper based on homotopy perturbation method hpm and reproducing kernel method rkm a new method is presented for solving nonlinear systems of second order boundary value problems bvps hpm is based on the use of traditional perturbation method and homotopy technique the hpm can reduce a nonlinear problem to a sequence of linear problems and generate a rapid convergent series solution in most cases rkm is also an analytical technique which can solve powerfully linear bvps homotopy perturbation reproducing kernel method hp rkm combines advantages of these two methods and therefore can be used to solve efficiently systems of nonlinear bvps three numerical examples are presented to illustrate the strength of the method

solving the generalized regularized long wave equation on the basis of a reproducing kernel space on the basis of a reproducing kernel space an iterative algorithm for solving the generalized regularized long wave equation is presented the analytical solution in the reproducing kernel space is shown in a series form and the approximate solution u n is constructed by truncating the series to n terms the convergence of u n to the analytical solution is also proved results obtained by the proposed method imply that it can be considered as a simple and accurate method for solving such evolution equations

global synchronization of complex networks perturbed by the poisson noise in this paper the problem of stochastic synchronization analysis is investigated for complex networks perturbed by the poisson noise by using the key tool such as the infinitesimal operator for stochastic differential equations driven by the poisson process this paper proposes a globally exponentially synchronization criterion in mean square for complex networks perturbed by the poisson noise finally numerical examples are provided to demonstrate the effectiveness of the proposed approach

theoretical study of lesion detectability of map reconstruction using computer observers the low signal to noise ratio snr in emission data has stimulated the development of statistical image reconstruction methods based on the maximum a posteriori map principle experimental examples have shown that statistical methods improve image quality compared to the conventional filtered backprojection fbp method however these results depend on isolated data sets here the authors study the lesion detectability of map reconstruction theoretically using computer observers these theoretical results can be applied to different object structures they show that for a quadratic smoothing prior the lesion detectability using the prewhitening observer is independent of the smoothing parameter and the neighborhood of the prior while the nonprewhitening observer exhibits an optimum smoothing point the authors also compare the results to those of fbp reconstruction the comparison shows that for ideal positron emission tomography pet systems where data are true line integrals of the tracer distribution the map reconstruction has a higher snr for lesion detection than fbp reconstruction due to the modeling of the poisson noise for realistic systems map reconstruction further benefits from accurately modeling the physical photon detection process in pet

synchronization of fractional order complex dynamical networks abstract in this letter the synchronization of complex dynamical networks with fractional order chaotic nodes is studied a fractional order controller for synchronization of complex network is presented some new sufficient synchronization criteria are proposed based on the lyapunov stability theory and the lasalle invariance principle these synchronization criteria can apply to an arbitrary fractional order complex network in which the coupling configuration matrix and the inner coupling matrix are not assumed to be symmetric or irreducible it means that this method is more general and effective numerical simulations of two fractional order complex networks demonstrate the universality and the effectiveness of the proposed method

a study of least squares and maximum likelihood for image reconstruction in positron emission tomography positron emission tomography pet is a radiologic tool offering a unique capability for measuring tissue metabolism in vivo a number of biological and physical factors limit the resolution of pet so often the statistical aspects of image reconstruction have an appreciable effect on the quality of information obtained from a study to a first approximation reconstruction involves the solution of a linear inverse problem with a line integral radon type transform standard filtered back projection reconstruction is based on the method of least squares although computationally efficient the method does not enforce positivity constraints leading to undesirable negative artifacts in the results maximum likelihood based approaches to reconstruction do not suffer from this problem but their computational complexity has limited the ability to determine quantitatively the improvements in image quality in this paper asymptotic approximations and numerical simulations are used to examine the least squares and maximum likelihood methods in some detail the studies are carried out for idealized representations of conventional and time of flight tomographs the asymptotic analysis indicates that for a range of sobolev norms the rates of estimation of least squares and maximum likelihood reconstructions are of the same order this is borne out by numerical studies however in these studies maximum likelihood is found to be more efficient than least squares on a conventional distance angle tomograph the root mean square error is on the order of 10 20 smaller for maximum likelihood reconstructions the corresponding reduction in the root mean square error on a time of flight tomograph is in the 5 range similar results are found using more complex region of interest based measures of reconstruction accuracy in addition it is found that much of the effect of maximum likelihood can apparently be realized by postprocessing least squares solutions in an ad hoc manner to reduce negativity artifacts since the postprocessing scheme is computationally fast this suggests that in pet it may be possible to realize the benefits of maximum likelihood without the substantial computational overhead of the em algorithm

global synchronization criteria of linearly coupled neural network systems with time varying coupling in this paper global synchronization of linearly coupled neural network nn systems with time varying coupling is investigated the dynamical behavior of the uncoupled system at each node is general which can be chaotic or others the coupling configuration is time varying i e the coupling matrix is not a constant matrix based on lyapunov function method and the specific property of householder transform some criteria for the global synchronization are obtained by these criteria one can verify whether the coupled system with time varying coupling is globally synchronized which is important and useful for both understanding and interpreting synchronization phenomena and designing coupling configuration finally two simulations are given to demonstrate the effectiveness of the theoretical results

error estimation for the reproducing kernel method to solve linear boundary value problems in the previous works the authors presented the reproducing kernel method rkm for solving various boundary value problems however an effective error estimation for this method has not yet been discussed the aim of this paper is to fill this gap in this paper we shall give the error estimation for the reproducing kernel method to solve linear boundary value problems

some error estimates for the reproducing kernel hilbert spaces method in this paper we derive some effective error estimates for the reproducing kernel hilbert space method applied to a general class of linear initial or boundary value problems the first error estimate is computable and yields a worst case bound in the form of a percentage of the norm of the true solution which has not yet been discussed according to the knowledge of the authors the second error estimate is a residual based error estimate which is expressed in terms of the fill distance so that convergence is studied for the fill distance tends to zero this is a generalization and improvement of the existing error estimates some numerical results are presented to demonstrate the applicability of the estimates

reproducing kernel method for solving singularly perturbed differential difference equations with boundary layer behavior in hilbert space abstract we consider singularly perturbed differential difference equation which contains negative shift in the differentiated term with boundary layer behavior rkhsm reproducing kernel hilbert space method without gram schmidt orthogonalization process is considered in the present paper we decompose the domain of the problem into two subintervals one of them has not the boundary layer and the other one has the side of the interval in which the boundary layer exists is important if the boundary layer of this problem exists on the left side of interval the rkhsm will provide a proper approximation of solution otherwise for the implement of rkhsm we need to change the variable of the singularly perturbed problem to shift the boundary layer region

synchronization criteria for a generalized complex delayed dynamical network model abstract it has been demonstrated that most complex networks display synchronization phenomena and this problem has attracted great attention of various fields including science and engineering in this paper a generalized complex dynamical network model with time varying delays was presented kronecker product was adopted to investigate this model moreover several synchronization criteria were derived for both delay independent and delay dependent asymptotical stability especially it has been shown that synchronization of such dynamical network is determined by the linear matrix inequality consisting of coupling configuration matrices inner coupling matrices and isolated cells at last illustrative examples were given to validate the above acquired

stabilizing weighted complex networks real networks often consist of local units which interact with each other via asymmetric and heterogeneous connections in this paper the v stability problem is investigated for a class of asymmetric weighted coupled networks with nonidentical node dynamics which includes the unweighted network as a special case pinning control is suggested to stabilize such a coupled network the complicated stabilization problem is reduced to measuring the semi negative property of the characteristic matrix which embodies not only the network topology but also the node self dynamics and the control gains it is found that network stabilizability depends critically on the second largest eigenvalue of the characteristic matrix the smaller the second largest eigenvalue is the more the network is pinning controllable numerical simulations of two representative networks composed of non chaotic systems and chaotic systems respectively are shown for illustration and verification

adaptive pinning cluster synchronization of fractional order complex dynamical networks the problem about cluster synchronization of fractional order cdns is studied via a pinning adaptive approach in this paper based on the stability theory of fractional differential equations some sufficient criteria for local and global cluster synchronization of fractional order cdns are derived in this paper the coupling configuration matrix can be asymmetric as well as reducible and the inner coupling matrix can also be asymmetric moreover the number of pinning nodes in each cluster can be evaluated especially when the coupling strength is large enough and the coupling configuration matrix is symmetric cluster synchronization can be achieved via pinning a single node in each cluster finally some typical examples are given to illustrate the correctness and effectiveness of our results a surprising finding is that the synchronization performance will become better as the fractional order decreases in this simulation

a smoothed em approach to indirect estimation problems with particular reference to stereology and emission tomography there are many practical problems where the observed data are not drawn directly from the density g of real interest but rather from another distribution derived from g by the application of an integral operator the estimation of g then entails both statistical and numerical difficulties a natural statistical approach is by maximum likelihood conveniently implemented using the em algorithm but this provides unsatisfactory reconstructions of g in this paper we modify the maximum likelihood em approach by introducing a simple smoothing step at each em iteration in our experience this algorithm converges in relatively few iterations to good estimates of g that do not depend on the choice of starting configuration some theoretical background is given that relates this smoothed em algorithm to a maximum penalized likelihood approach two applications are considered in detail the first is the classical stereology problem of determining particle size distributions from data collected on a plane section through a composite medium the second concerns the recovery of the structure of a section of the human body from external observations obtained by positron emission tomography for this problem we also suggest several technical improvements on existing methodology

clustering massive datasets with application in software metrics and tomography clustering datasets is not an easy problem in general and the difficulty is compounded for massive datasets this article develops under gaussian assumptions a multistage algorithm that clusters an initial sample filters out observations that can be reasonably classified by these clusters and iterates the preceding procedure on the remainder a final step uses the estimated class probabilities and dispersions to classify each observation in the dataset results on test experiments indicate good performance application to datasets from software metrics and positron emission tomography required no more than five stages each suggesting that the procedure is practical to implement

chaos synchronization of general complex dynamical networks recently it has been demonstrated that many large scale complex dynamical networks display a collective synchronization motion here we introduce a time varying complex dynamical network model and further investigate its synchronization phenomenon based on this new complex network model two network chaos synchronization theorems are proved we show that the chaos synchronization of a time varying complex network is determined by means of the inner coupled link matrix the eigenvalues and the corresponding eigenvectors of the coupled configuration matrix rather than the conventional eigenvalues of the coupled configuration matrix for a uniform network especially we do not assume that the coupled configuration matrix is symmetric and its off diagonal elements are nonnegative which in a way generalizes the related results existing in the literature

from image deblurring to optimal investments maximum likelihood solutions for positive linear inverse problems the problem of recovering an input signal from a blurred output in an input output system with linear distortion is ubiquitous in science and technology when the blurred output is not degraded by statistical noise the problem is entirely deterministic and amounts to a mathematical inversion of a linear system with positive parameters subject to positivity constraints on the solution we show that all such linear inverse problems with positivity restrictions lininpos problems for short can be interpreted as statistical estimation problems from incomplete data based on infinitely large samples and that maximum likelihood ml estimation and the em algorithm provide a straightforward method of solution for such problems

some error estimates for solving volterra integral equations by using the reproducing kernel method in this paper some error estimates for the reproducing kernel method rkm in solving a class of volterra integral equations are established applicability of the estimates are demonstrated by testing some various examples

positron emission tomography by markov chain monte carlo with auxiliary variables we propose a new algorithm for positron emission tomography pet image reconstruction the algorithm belongs to the family of markov chain monte carlo methods with auxiliary variables the idea is to iteratively generate hidden variables at one step and use them for image restoration at another step the well known model of vardi et al j amer statist assoc 80 1985 8 for pet is combined with the bayesian model of lasota and niemiro pattern recognition 36 2003 931 for the underlying images this latter model takes advantage of the fact that medical images often consist of relatively few grey levels of unknown intensity the algorithm of lasota and niemiro pattern recognition 36 2003 931 is used in the image restoration part of the pet algorithm essentially as a noise filtering and smoothing device it is now equipped with an additional data reconstruction step we include simulation results which suggest that the method is truly reliable we also describe a version of the basic algorithm in which a random simulation step is replaced by computation of expected value similarly as in the em algorithm

on nonparametric estimation of intercept and slope distributions in random coefficient regression an experiment records stimulus and response for a random sample of cases the relationship between response and stimulus is thought to be linear the values of the slope and intercept varying by case from such data we construct a consistent asymptotically normal nonparametric estimator for the joint density of the slope and intercept our methodology incorporates the radial projection slice theorem for the radon transform a technique for locally linear nonparametric regression and a tapered fourier inversion computationally the new density estimator is more feasible than competing nonparametric estimators one of which is based on moments and the other on minimum distance considerations

a time varying complex dynamical network model and its controlled synchronization criteria today complex networks have attracted increasing attention from various fields of science and engineering it has been demonstrated that many complex networks display various synchronization phenomena in this note we introduce a time varying complex dynamical network model we then further investigate its synchronization phenomenon and prove several network synchronization theorems especially we show that synchronization of such a time varying dynamical network is completely determined by the inner coupling matrix and by the eigenvalues and the corresponding eigenvectors of the coupling configuration matrix of the network

variability assessment in positron emission tomography and related generalized deconvolution models abstract the problem of variance assessment for positron emission tomography pet image reconstructions is considered in the context of generalized deconvolution a refinement of an approximate technique proposed by carson and colleagues is examined computational implications of representing the reconstruction kernel in terms of a weighted sum of gaussian densities are developed bias and variance characteristics of the resulting variance estimators are examined by numerical simulation for typical regions the error in estimated standard deviations is found to be on the order of 10 the use of smoothing to obtain more reliable pointwise variance estimators is described and some theoretical analysis of this technique is carried out for the pet application simulations suggest that the percent improvement in the root mean squared error accuracy of pointwise variance estimators obtained by smoothing can be on the order of 30 a practical application of the methodology to a pet study is presented

nonparametric residue analysis of dynamic pet data with application to cerebral fdg studies in normals kinetic analysis is used to extract metabolic information from dynamic positron emission tomography pet uptake data the theory of indicator dilutions developed in the seminal work of meier and zierler 1954 provides a probabilistic framework for representation of pet tracer uptake data in terms of a convolution between an arterial input function and a tissue residue the residue is a scaled survival function associated with tracer residence in the tissue nonparametric inference for the residue a deconvolution problem provides a novel approach to kinetic analysis critically one that is not reliant on specific compartmental modeling assumptions a practical computational technique based on regularized cubic b spline approximation of the residence time distribution is proposed nonparametric residue analysis allows formal statistical evaluation of specific parametric models to be considered this analysis needs to properly account for the increased flexibility of the nonparametric estimator the met

solving a nonlinear system of second order boundary value problems in this paper a method is presented to obtain the analytical and approximate solutions of linear and nonlinear systems of second order boundary value problems the analytical solution is represented in the form of series in the reproducing kernel space in the mean time the approximate solution un x is obtained by the n term intercept of the analytical solution and is proved to converge to the analytical solution some numerical examples are studied to demonstrate the accuracy of the present method results obtained by the method indicate the method is simple and effective

speed of estimation in positron emission tomography and related inverse problems we study a continuous idealization of the image reconstruction problem in positron emission tomography considered as an example of bivariate density estimation based on indirect observations given a large sample of indirect observations we consider the size of the equivalent sample of observations whose original exact positions would allow equally accurate estimation of the image of interest both for indirect and for direct observations we establish exact minimax rates of convergence of estimation for all possible estimators over suitable smoothness classes of functions the approach of the paper is applicable to a wide class of linear inverse problems

bandwidth selection for indirect density estimation based on corrupted histogram data abstract motivated by a number of practical applications we consider a class of indirect nonparametric density estimation problems in which the observed data consist of a histogram of binned empirically corrected counts due to variability in the process of correction the histogram cannot be modeled in terms of simple scaled poisson statistics this departure necessitates the development of a new methodology for bandwidth selection a variant of the method of unbiased risk estimation is proposed the methodology is studied using numerical simulations and asymptotic analysis tools in the context of a class of idealized density deconvolution problems the methodology is adapted for application to the practical reconstruction problem of positron emission tomography pet realistic numerical simulations and physical phantom data are presented to validate the approach in this setting some illustrations with cerebral glucose utilization and myocardial blood flow studies from some actual patient data sets

a statistical model for positron emission tomography abstract positron emission tomography pet still in its research stages is a technique that promises to open new medical frontiers by enabling physicians to study the metabolic activity of the body in a pictorial manner much as in x ray transmission tomography and other modes of computerized tomography the quality of the reconstructed image in pet is very sensitive to the mathematical algorithm to be used for reconstruction in this article we tailor a mathematical model to the physics of positron emissions and we use the model to describe the basic image reconstruction problem of pet as a standard problem in statistical estimation from incomplete data we describe various estimation procedures such as the maximum likelihood ml method using the em algorithm the method of moments and the least squares method a computer simulation of a pet experiment is then used to demonstrate the ml and the least squares reconstructions the main purposes of this article are to report on what we believe is an

computational method based on reproducing kernel for solving singularly perturbed differential difference equations with a delay abstract in this paper reproducing kernel hilbert space method rkhsm based on collocation scheme is used to solve singularly perturbed second order differential difference equations we implement rkhsm without gram schmidt orthogonalization process for singularly perturbed differential difference equation with boundary layer behavior and also oscillatory behavior with small delay rkhsm in this study is based on the division of the problem domain into two subintervals one with boundary layer and the other one without such a boundary layer several numerical examples are studied to demonstrate the accuracy of the present method results of the present scheme indicate that new algorithm has the following advantages small computational work fast convergence and high precision