estimating transition probabilities from aggregate samples augmented by haphazard recaptures transition probabilities provide a convenient summary of changes in a categorical trait over time in a population the difficulties of estimating such probabilities based on only aggregate data from repeated sampling are well known we give here a method for augmenting aggregate data with haphazard recapture data which can dramatically improve the estimation precision of transition probabilities the method requires a rather high sampling fraction to provide sufficient numbers of recaptures it is based on a generalized nonlinear least squares strategy which yields transition probability estimates preserving their natural parameter space and which are asymptotically efficient the asymptotic theory is given under finite population sampling assumptions which are typical in practice

bayesian reconstructions from emission tomography data using a modified em algorithm a novel method of reconstruction from single photon emission computerized tomography data is proposed this method builds on the expectation maximization em approach to maximum likelihood reconstruction from emission tomography data but aims instead at maximum posterior probability estimation which takes account of prior belief about smoothness in the isotope concentration a novel modification to the em algorithm yields a practical method the method is illustrated by an application to data from brain scans

kahlerian twistor spaces the work of r penrose has shown the deep relationship between the conformal structure of minkowski space and the complex geometry of lines in projective three space this transformation applies to a more general class of conformal structures and in particular there is a riemannian version described in 5 which is itself of interest to mathematical physicists basically to each oriented riemannian 4manifold x one associates canonically a 6 manifold z the twistor space of x together with an almost complex structure if this structure is integrable then z becomes a complex 3 manifold with a distinguished family of projective lines on it whose geometry describes the conformal geometry of the manifold x it is natural to ask what sort of complex manifold one obtains by this process in this paper we ask the more specific question which compact 4c manifolds have ka hlerian twistor spaces 1 the answer rather surprisingly is that there are just two examples the 4 sphere s and the complex projective plane p2 c each with their standard homogeneous conformal structures the twistor space of s 4 is the 3 dimensional projective space p3 c and that of p 2 q is the manifold f2 c of flags in c 3 the essential point of the proof of this result is the remark that if z is a twistor space with a kahler metric then the canonical bundle k is negative using vanishing theorems and the riemann roch theorem we then narrow down the possibilities for z to four cases projective space the flag manifold the intersection of two quadrics in p5 and the double covering of p3 branched over a non singular quartic surface the last two candidates cannot be realized however because the value of their euler characteristic is incompatible with their being twistor spaces throughout the proof we make essential use of a real structure on z which reduces considerably the occurrence of singularities at various stages aside from this our methods are the standard ones of algebraic geometry the four examples to which the proof leads us are all fano threefolds if we drop the requirement of compactness on x then it is quite possible for an open set of a fano threefold to be a twistor space in particular we show that the family of conies lying on the intersection of two

optimal systems of nodes for lagrange interpolation on bounded intervals a survey in this brief survey special attention is paid to some recent procedures for constructing optimal interpolation processes i e with lebesgue constant having logarithmic behaviour a new result on lagrange interpolation based on the zeros of the associated jacobi polynomials and on suitable additional nodes is given

on twistor spaces of anti self dual hermitian surfaces we consider a complex surface m with anti self dual hermitian metric h and study the holomorphic properties of its twistor space z we show that the naturally defined divisor line bundle x is isomorphic to the 1 2 power of the canonical bundle of z if and only if there is a kahler metric of zero scalar curvature in the conformal class of h this has strong consequences on the geometry of m which were also found by c boyer 3 using completely different methods we also prove the existence of a very close relation between holomorphic vector fields on m and z in the case that m is compact and kahler

an iterative image space reconstruction algorthm suitable for volume ect the trend in the design of scanners for positron emission computed tomography has traditionally been to improve the transverse spatial resolution to several millimeters while maintaining relatively coarse axial resolution 1 2 cm several scanners are being built with fine sampling in the axial as well as transverse directions leading to the possibility of the true volume imaging the number of possible coincidence pairs in these scanners is quite large the usual methods of image reconstruction cannot handle these data without making approximations it is computationally most efficient to reduce the size of this large sparsely populated array by back projecting the coincidence data prior to reconstruction while analytic reconstruction techniques exist for back projected data an iterative algorithm may be necessary for those cases where the point spread function is spatially variant a modification of the maximum likelihood algorithm is proposed to reconstruct these back projected data the method the iterative image space reconstruction algorithm isra is able to reconstruct data from a scanner with a spatially variant point spread function in less time than other proposed algorithms results are presented for single slice data simulated and actual from the penn pet scanner

rigorous convergence analysis of jacobi spectral galerkin methods for volterra integral equations with noncompact kernels abstract this article deals with the implementation of jacobi spectral and pseudospectral galerkin approaches for solving weakly singular volterra integral equations with noncompact kernels the computational domain of the main problem and intervals of the involved integrals are transformed into 1 1 by some appropriate change of variables therefore one can apply the jacobi galerkin method and impose the inner product conditions on the residual of the considered equation also jacobi gauss quadrature rules are implemented to approximate the integral operator and the inner products associated to the jacobi weight function in the pseudospectral galerkin method for both cases of spectral and pseudospectral galerkin schemes rigorous convergence analysis is provided under some mild conditions on the smoothness of kernel and source functions in l and weighted l 2 norms several numerical examples are also given to confirm the theoretical prediction

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

em and beyond the basic theme of the em algorithm to repeatedly use complete data methods to solve incomplete data problems is also a theme of several more recent statistical techniques these techniques multiple imputation data augmentation stochastic relaxation and sampling importance resampling combine simulation techniques with complete data methods to attack problems that are difficult or impossible for em

on the iterative image space reconstruction algorthm for ect the isra of 1 is shown to be an iterative algorithm that aims to converge to the least squares estimates of emission densities convergence is established in the case where a unique least squares estimate exists that is elementwise strictly positive it is pointed out that in terms of asymptotic theory the resulting estimators are inferior to the maximum likelihood estimators for which the em algorithm is a computational procedure potential difficulties with the behavior of the isra are illustrated using very simple examples

maximum likelihood reconstruction for emission tomography previous models for emission tomography et do not distinguish the physics of et from that of transmission tomography we give a more accurate general mathematical model for et where an unknown emission density x y z generates and is to be reconstructed from the number of counts n d in each of d detector units d within the model we give an algorithm for determining an estimate of which maximizes the probability p n of observing the actual detector count data n over all possible densities let independent poisson variables n b with unknown means b b 1 b represent the number of unobserved emissions in each of b boxes pixels partitioning an object containing an emitter suppose each emission in box b is detected in detector unit d with probability p b d d 1 d with p b d a one step transition matrix assumed known we observe the total number n n d of emissions in each detector unit d and want to estimate the unknown b b 1 b for each the observed data n has probability or likelihood p n the em algorithm of mathematical statistics starts with an initial estimate 0 and gives the following simple iterative procedure for obtaining a new estimate new from an old estimate old to obtain k k 1 2 new b old b dd 1 n d p b d old b p b d b 1 b

compact self dual manifolds with positive scalar curvature on etudie l existence d une metrique auto duale a courbure scalaire positive sur une variete compacte simplement connexe qui n est pas conformement equivalente a ces metriques standard sur s 4 ou cp 2

on use of the em algorithm for penalized likelihood estimation summary the em algorithm is a popular approach to maximum likelihood estimation but has not been much used for penalized likelihood or maximum a posteriori estimation this paper discusses properties of the em algorithm in such contexts concentrating on rates of conver gence and presents an alternative that is usually more practical and converges at least as quickly the em algorithm is a general approach to maximum likelihood estimation rather than a specific algorithm dempster et al 1977 discussed the method and derived basic properties demonstrating that a variety of procedures previously developed rather informally could be unified the common strand to problems where the approach is applicable is a notion of incomplete data this includes the conventional sense of missing data but is much broader than that the em algorithm demon strates its strength in situations where some hypothetical experiment yields data from which estimation is particularly convenient and economical the incomplete data actually at hand are regarded as observable functions of these complete data the resulting algorithms while usually slow to converge are often extremely simple and remain practical in large problems where no other approaches may be feasible dempster et al 1977 briefly refer to the use of the same approach to the problem of finding the posterior mode maximum a posteriori estimate in a bayesian estima

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

on conformally kahler einstein manifolds we prove that any compact complex surface with c1 0 admits an einstein metric which is conformally related to a kahler metric the key new ingredient is the existence of such a metric on the blow up cp2 2cp2 of the complex projective plane at two distinct points

poon s self dual metrics and kahler geometry on montre que les metriques conformes autoduales sur des sommes connexes de cp 2 proviennent de metriques de kahler de courbure scalaire nulle sur des eclatements de c 2 en ajoutant un point a l infini et en renversant l orientation

aggregate data ecological regression and voting transitions abstract voting data typically comprise the marginal distributions of votes cast at two successive elections the fact that these obtain separately for many voting precincts or areas enables one to estimate the actual transition probabilities for movements between the options available to the voter at each election an aggregated compound multinomial model is proposed this allows log linear dependence on various covariates and specifies a simple and illuminating structure of random effects the information in such aggregated data is compared with that which would obtain had all of the transitions been observed the model is applicable to many other types of aggregated data and meets many of the difficulties inherent in ecological regression it is illustrated with an analysis of the british european election of 1984

bayesian image processing in two dimensions a bayesian image processing bip formalism which incorporates a priori amplitude and spatial probability density information was applied to two dimensional source fields for valid moderately restrictive a priori information strikingly improved results for ideal and experimental radioisotope phantom imaging data compared to a standard non bayesian formalism maximum likelihood ml were obtained the applicability of a fast fourier transform technique for convolution calculations a reduced region restriction for the initial deconvolution calculations and a relaxation parameter for accelerating convergence are considered

a theoretical study of some maximum likelihood algorithms for emission and transmission tomography this paper has the dual purpose of introducing some new algorithms for emission and transmission tomography and proving mathematically that these algorithms and related antecedent algorithms converge like the em algorithms for positron single photon and transmission tomography the algorithms provide maximum likelihood estimates of pixel concentration or linear attenuation parameters one particular innovation we discuss is a computationally practical scheme for modifying the em algorithms to include a bayesian prior the bayesian versions of the em algorithms are shown to have superior convergence properties in a vicinity of the maximum we anticipate that some of the other algorithms will also converge faster than the em algorithms

multiplicative iterative algorithms for convex programming abstract we study multiplicative iterative algorithms for the minimization of a differentiable convex function defined on the positive orthant of r n if the objective function has compact level sets and has a locally lipschitz continuous gradient then these algorithms converge to a solution of the minimization problem moreover the convergence is nearly monotone in the sense of kullback leibler divergence special cases of these algorithms have been applied in position emission tomography and are formally related to the em algorithm for position emission tomography

intrinsic autoregressions and related models on the two dimensional lattice summary stationary autoregressions on a two dimensional lattice are generalized to intrinsic models where only increments are assumed to be stationary prediction formulae and the asymptotic behaviour of the semivariogram are derived for parameter estimation we propose an approximate maximum likelihood estimator a generalization of whittle s estimator it is derived also for general intrinsic models

a multigrid expectation maximization reconstruction algorithm for positron emission tomography the problem of reconstruction in positron emission tomography pet is basically estimating the number of photon pairs emitted from the source using the concept of the maximum likelihood ml algorithm the problem of reconstruction is reduced to determining an estimate of the emitter density that maximizes the probability of observing the actual detector count data over all possible emitter density distributions a solution using this type of expectation maximization em algorithm with a fixed grid size is severely handicapped by the slow convergence rate the large computation time and the nonuniform correction efficiency of each iteration which makes the algorithm very sensitive to the image pattern an efficient knowledge based multigrid reconstruction algorithm based on the ml approach is presented to overcome these problems

numerical solutions for fredholm integral equations of the second kind with weakly singular kernel using spectral collocation method abstract we investigate the spectral collocation for fredholm integral equations of the second kind with weakly singular kernel the jacobi gauss quadrature formula is used to approximate the integral operator in the numerical implementation we obtain the convergence rates for the approximated solution of weakly singular fredholm integral equations which show that the errors of the approximate solution decay exponentially in l norm and weighted l2 norm some numerical examples are given to illustrate the theoretical results

spectral collocation method for the time fractional diffusion wave equation and convergence analysis in this paper we consider the numerical solution of the time fractional diffusion wave equation essentially the time fractional diffusion wave equation differs from the standard diffusion wave equation in the time derivative term we propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of jacobi interpolation polynomial for this equation the convergence of the method is rigorously established numerical tests are carried out to confirm the theoretical results

local likelihood estimation abstract a scatterplot smoother is applied to data of the form x 1 y 1 x 2 y 2 xn yn and uses local fitting to estimate the dependence of y on x a simple example is the running lines smoother which fits a least squares line to the y values falling in a window around each x value the value of the estimated function at x is given by the value of the least squares line at x a smoother generalizes the least squares line which assumes that the dependence of y on x is linear in this article we extend the idea of local fitting to likelihood based regression models one such application is to the class of generalized linear models nelder and wedderburn 1972 we enlarge this class by replacing the covariate form 0 x 1 with an unspecified smooth function s x this function is estimated from the data by a technique we call local likelihood estimation the method consists of maximum likelihood estimation for 0 and 1 applied in a window around each x value multiple covariates are incor

stopping rule for the mle algorithm based on statistical hypothesis testing it is known that when the maximum likelihood estimator mle algorithm passes a certain point it produces images that begin to deteriorate we propose a quantitative criterion with a simple probabilistic interpretation that allows the user to stop the algorithm just before this effect begins the mle algorithm searches for the image that has the maximum probability to generate the projection data the underlying assumption of the algorithm is a poisson distribution of the data therefore the best image according to the mle algorithm is the one that results in projection means which are as close to the data as possible it is shown that this goal conflicts with the assumption that the data are poisson distributed we test a statistical hypothesis whereby the projection data could have been generated by the image produced after each iteration the acceptance or rejection of the hypothesis is based on a parameter that decreases as the images improve and increases as they deteriorate we show that the best mle images which pass the test result in somewhat lower noise in regions of high activity than the filtered back projection results and much improved images in low activity regions the applicability of the proposed stopping rule to other iterative schemes is discussed

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

twistors kahler manifolds and bimeromorphic geometry i by considering deformations of the moishezon twistor spaces of cp2 cp2 constructed in 20 we showthat theblowupof c2 at n points in general position admits an asymptotically flat scalar flat kahler metric in each kahler class at least provided that the given points are nearly collinear department of mathematics suny at stony brook stony brook new york 11794 e mail address claude math sunysb edu this content downloaded from 157 55 39 153 on mon 19 sep 2016 04 58 00 utc all use subject to http about jstor org terms

a maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography the expectation maximization method for maximum likelihood image reconstruction in emission tomography based on the poisson distribution of the statistically independent components of the image and measurement vectors is extended to a maximum aposteriori image reconstruction using a multivariate gaussian a priori probability distribution of the image vector the approach is equivalent to a penalized maximum likelihood estimation with a special choice of the penalty function the expectation maximization method is applied to find the a posteriori probability maximizer a simple iterative formula is derived for a penalty function that is a weighted sum of the squared deviations of image vector components from their a priori mean values the method is demonstrated to be superior to pure likelihood maximization in that the penalty function prevents the occurrence of irregular high amplitude patterns in the image with a large number of iterations the so called checkerboard effect or noise artifact

heterotic string compactification and new vector bundles we propose a construction of kahler and non kahler calabi yau manifolds by branched double covers of twistor spaces in this construction we use the twistor spaces of four manifolds with self dual conformal structures with the examples of connected sum of n mathbb p 2 s we also construct k3 fibered calabi yau manifolds from the branched double covers of the blow ups of the twistor spaces these manifolds can be used in heterotic string compactifications to four dimensions we also construct stable and polystable vector bundles some classes of these vector bundles can give rise to supersymmetric grand unified models with three generations of quarks and leptons in four dimensions

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

mass in kahler geometry we prove a simple explicit formula for the mass of any asymptotically locally euclidean ale kahler manifold assuming only the sort of weak fall off conditions required for the mass to actually be well defined for ale scalar flat kahler manifolds the mass turns out to be a topological invariant depending only on the underlying smooth manifold the first chern class of the complex structure and the kahler class of the metric when the metric is actually ae asymptotically euclidean our formula not only implies a positive mass theorem for kahler metrics but also yields a penrose type inequality for the mass

equivariant connected sums of compact self dual manifolds due to a recent theorem of taubes self dual conformal structures on compact four dimensional manifolds exist in abundance 43 therefore finding examples in general is no longer an issue in this subject to deepen our understanding of self dual conformal geometry we take the path of trying to understand the possible symmetry groups of this geometry i e the groups of orientation preserving conformal transformations the first systematic approach which constructs a large collection of compact self dual manifolds with a non trivial group of symmetries is lebrun s hyperbolic ansatz 22 the condition of this ansatz is that the symmetry group contains an s l u 1 which acts on the manifold semi freely meaning that the isotropy group at any point of the manifold is either trivial or the entire group s i with this condition on the group action and some favourable topological conditions on the underlying manifold the geometrical equations of self duality on the given four dimensional manifold are reduced to the s 1 monopole equations on hyperbolic 3 space with this setup lebrun explicitly constructed a large number of compact self dual manifolds with semi free s l symmetry including the fubinistudy metric on the complex projective plane ci 2 and a one parameter family of self dual metrics on c c first found in 38 we shall call the self dual metrics on nci 2 n 1 constructed by lebrun s hyperbolic ansatz the lebrun metrics after this very successful ansatz it is natural to try to find new examples which are beyond the scope of that method

an index theorem for anti self dual orbifold cone metrics abstract recently atiyah and lebrun proved versions of the gauss bonnet and hirzebruch signature theorems for metrics with edge cone singularities in dimension four which they applied to obtain an inequality of hitchin thorpe type for einstein edge cone metrics interestingly many natural examples of edge cone metrics in dimension four are anti self dual or self dual depending upon choice of orientation on such a space there is an important elliptic complex called the anti self dual deformation complex whose index gives crucial information about the local structure of the moduli space of anti self dual metrics in this paper we compute the index of this complex in the orbifold case and give several applications

the calabi flow on kahler surfaces with bounded sobolev constant i we consider the formation of singularities along the calabi flow by assuming the uniformly bounded sobolev constants on kahler surfaces we prove that if curvature tensor is not uniformly bounded then one can form a singular model called deepest bubble such deepest bubble has to be a scalar flat ale kahler metric in certain kahler classes on toric fano surfaces the sobolev constants are a priori bounded along the calabi flow with small calabi energy we can also show in certain cases no deepest bubble can form along the flow it follows that the curvature tensor is uniformly bounded and the flow exists for all time and converges to an extremal metric subsequently to illustrate our results more clearly we focus on an example on mathbb cp 2 blown up three points at generic position our result also implies existence of constant scalar curvature metrics on mathbb cp 2 blown up three points at generic position in the kahler classes where the exceptional divisors have the same area

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

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

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

convergence analysis of the jacobi spectral collocation methods for volterra integral equations with a weakly singular kernel in this paper a jacobi collocation spectral method is developed for volterra integral equations of second kind with a weakly singular kernel we use some function transformation and variable transformations to change the equation into a new volterra integral equation deflned on the standard interval i1 1 so that the solution of the new equation possesses better regularity and the jacobi orthogonal polynomial theory can be applied conveniently in order to obtain high order accuracy for the approx imation the integral term in the resulting equation is approximated by using jacobi spectral quadrature rules the convergence analysis of this novel method is based on the lebesgue constants corresponding to the lagrange interpolation polynomials poly nomials approximation theory for orthogonal polynomials and the operator theory the spectral rate of convergence for the proposed method is established in the l 1 norm and weighted l 2 norm numerical results are presented to demonstrate the efiectiveness of the proposed method

accelerated iterative reconstruction for positron emission tomography based on the em algorithm for maximum likelihood estimation the em method that was originally developed for maximum likelihood estimation in the context of mathematical statistics may be applied to a stochastic model of positron emission tomography pet the result is an iterative algorithm for image reconstruction that is finding increasing use in pet due to its attractive theoretical and practical properties its major disadvantage is the large amount of computation that is often required due to the algorithm s slow rate of convergence this paper presents an accelerated form of the em algorithm for pet in which the changes to the image as calculated by the standard algorithm are multiplied at each iteration by an overrelaxation parameter the accelerated algorithm retains two of the important practical properties of the standard algorithm namely the selfnormalization and nonnegativity of the reconstructed images experimental results are presented using measured data obtained from a hexagonal detector system for pet the likelihood function and the norm of the data residual were monitored during the iterative process according to both of these measures the images reconstructed at iterations 7 and 11 of the accelerated algorithm are similar to those at iterations 15 and 30 of the standard algorithm for two different sets of data important theoretical properties remain to be investigated namely the convergence of the accelerated algorithm and its performance as a maximum likelihood estimator

numerical solutions for solving time fractional fokker planck equations based on spectral collocation methods abstract in this paper we consider the numerical solution of the time fractional fokker planck equation we propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of jacobi interpolation polynomial for this equation the convergence of the method is rigorously established numerical tests are carried out to confirm the theoretical results

the bootstrap to smooth or not to smooth summary the bootstrap and smoothed bootstrap are considered as alternative methods of estimating properties of unknown distributions such as the sampling error of parameter estimates criteria are developed for determining whether it is advantageous to use the smoothed bootstrap rather than the standard bootstrap key steps in the argument leading to these criteria include the study of the estimation of linear functionals of distributions and the approximation of general functionals by linear functionals consideration of an example the estimation of the standard error in the variance stabilized sample correlation coefficient elucidates previously published simulation results and also illustrates the use of computer algebraic manipulation as a useful technique in asymptotic statistics finally the various approximations used are vindicated by a simulation study