stability of explicit arkn methods for perturbed oscillators the stability of explicit arkn methods for the numerical integration of perturbed oscillators is analyzed stability regions for the second order homogeneous linear test model are obtained these stability regions represent a generalization of the stability intervals for the classical rkn methods

modified taylor approximation of functions with periodic behaviour we approximate a function with periodic behaviour by means of a small modification of its taylor polynomial this modification is based on the work of scheifele and will simplify the construction of special numerical methods for differential equations with near periodic solutions

parametric imaging with bayesian priors a validation study with 11c altropane pet abstract it has been suggested that bayesian estimation methods may be used to improve the signal to noise ratio of parametric images however there is little experience with the method and some of the underlying assumptions and performance properties of bayesian estimation remain to be investigated we used a sample population of 54 subjects studied previously with 11 c altropane to empirically evaluate the assumptions performance and some practical issues in forming parametric images by using normality tests we showed that the underpinning normality assumptions of data and parametric distribution apply to more than 80 of voxels the standard deviation of the binding potential can be reduced 30 50 using bayesian estimation without introducing substantial bias the sample size required to form the a priori information was found to be modest as little as ten subjects may be sufficient and the choice of specific subjects has little effect on bayesian estimation a realistic simulation study showed that detection of localized differences in parametric images e g by statistical parametric mapping spm could be made more reliable and or conducted with smaller sample size using bayesian estimation in conclusion bayesian estimation can improve the snr of parametric images and better detect localized changes in cohorts of subjects

exponentially fitted runge kutta methods abstract exponentially fitted runge kutta methods with s stages are constructed which exactly integrate differential initial value problems whose solutions are linear combinations of functions of the form x j exp x x j exp x r or i r j 0 1 j max where 0 j max s 2 1 the lower bound being related to explicit methods the upper bound applicable for collocation methods explicit methods with s 2 3 4 belonging to that class are constructed for these methods a study of the local truncation error is made out of which follows a simple heuristic to estimate the value error and step length control is introduced based on richardson extrapolation ideas some numerical experiments show the efficiency of the introduced methods it is shown that the same techniques can be applied to construct implicit exponentially fitted runge kutta methods

comparing methods of analyzing fmri statistical parametric maps approaches for the analysis of statistical parametric maps spms can be crudely grouped into three main categories in which different philosophies are applied to delineate activated regions these being type i error control thresholding false discovery rate fdr control thresholding and posterior probability thresholding to better understand the properties of these main approaches we carried out a simulation study to compare the approaches as they would be used on real data sets using default settings we find that posterior probability thresholding is the most powerful approach and type i error control thresholding provides the lowest levels of type i error false discovery rate control thresholding performs in between the other approaches for both these criteria although for some parameter settings this approach can approximate the performance of posterior probability thresholding based on these results we discuss the relative merits of the three approaches in an attempt to decide upon an optimal approach we conclude that viewing the problem of delineating areas of activation as a classification problem provides a highly interpretable framework for comparing the methods within this framework we highlight the role of the loss function which explicitly penalizes the types of errors that may occur in a given analysis

construction of explicit symmetric and symplectic methods of runge kutta nystrom type for solving perturbed oscillators the construction of explicit symmetric and symplectic methods of runge kutta nystrom rkn type specially adapted to the numerical solution of perturbed oscillators is analyzed based on the symmetry properties for this class of methods new explicit methods of rkn type for solving perturbed oscillators are constructed in addition we prove that this class of explicit symmetric methods are also symplectic the numerical experiments carried out show the qualitative behavior and the efficiency of the new methods when they are compared with some standard and specially adapted symmetric and symplectic methods proposed in the scientific literature for solving second order oscillatory differential systems

an unbiased parametric imaging algorithm for nonuniformly sampled biomedical system parameter estimation an unbiased algorithm of generalized linear least squares glls for parameter estimation of nonuniformly sampled biomedical systems is proposed the basic theory and detailed derivation of the algorithm are given this algorithm removes the initial values required and computational burden of nonlinear least regression and achieves a comparable estimation quality in terms of the estimates bias and standard deviation therefore this algorithm is particular useful in image wide pixel by pixel based parameter estimation e g to generate parametric images from tracer dynamic studies with positron emission tomography an example is presented to demonstrate the performance of this new technique this algorithm is also generally applicable to other continuous system parameter estimation

a robust trigonometrically fitted embedded pair for perturbed oscillators a new kind of trigonometrically fitted embedded pair of explicit arkn methods for the numerical integration of perturbed oscillators is presented in this paper this new pair is based on the trigonometrically fitted arkn method of order five derived by yang and wu in h l yang x y wu trigonometrically fitted arkn methods for perturbed oscillators appl numer math 9 2008 1375 1395 we analyze the stability properties phase lag dispersion and dissipation of the higher order method of the new pair numerical experiments carried out show that our new embedded pair is very competitive in comparison with the embedded pairs proposed in the scientific literature

a general method of bayesian estimation for parametric imaging of the brain abstract we report a general method of bayesian estimation that uses prior measurements to improve the signal to noise ratio of parametric images computed from dynamic pet scanning in our method the ordinary weighted least squares cost function is augmented by a penalty term to yield k s min k c f k t c 1 c f k s k s 0 k k t k 1 k k where c is a pet concentration history and c is its variance f is the model of the concentration history k k 1 k 2 k m t is the parameter vector k is the vector of population means for the model parameters k is its covariance k k s 0 is the conventional weighted sum of squares s 0 is chosen to control the balance between the prior and new data data from a prior population of subjects are analyzed with standard methods to provide maps of the mean parameter values and their variances as an example of this approach we used the dynamic image data of 10 normal subjects who had previously been studied with 11 c raclopride to estimate the prior distribution the dynamic data were transformed to stereotactic coordinates and analyzed by standard methods the resulting parametric maps were used to compute the voxel wise sample statistics then the cohort of priors was analyzed as a function of s using nonlinear least squares estimation and the cost function shown above as s is increased the standard error in estimating bp in single subjects was substantially reduced allowing measurement in bp in thalamus cortex brain stem etc additional studies demonstrate that a range of s values exist for which the bias is not excessive even when parameter values differ markedly from the sample mean this method can be used with any kinetic model so long as it is possible to compute a map of a priori mean parameters and their variances

a 5 3 pair of explicit arkn methods for the numerical integration of perturbed oscillators a new embedded pair of explicit rkn methods specially adapted to the numerical integration of perturbed oscillators is obtained this pair depends on a parameter v h 0 h is the integration step and is the dominant frequency and it has four stages and algebraic orders five and three the numerical experiments carried out show that the new pair is very competitive when it is compared with high quality codes proposed in the scientific literature

zero dissipative phase fitted hybrid methods for solving oscillatory second order ordinary differential equations in this paper zero dissipative phase fitted two step hybrid methods are developed for the integration of second order periodic initial value problems the phase fitted hybrid methods are constructed using similar approaches introduced by papadopoulos et al 1 this new methods are based on the existing explicit hybrid methods of order four and six numerical illustrations indicate that the new methods are much more efficient than the existing methods

a fourth algebraic order trigonometrically fitted predictor corrector scheme for ivps with oscillating solutions a scheme of trigonometrically fitted predictor corrector p c adams bashforth moulton methods is constructed in this paper our new p c method is based on the third order adams bashforth scheme as predictor and on the fourth order adams moulton scheme as corrector we tested the efficiency of our newly developed scheme against well known methods with excellent results the numerical experimentation showed that our method is considerably more efficient compared to well known methods used for the numerical solution of initial value problems with oscillating solutions

a trigonometrically fitted explicit hybrid method for the numerical integration of orbital problems abstract in this paper we present a kind of trigonometrically fitted explicit two step hybrid method which achieves algebraic order six the new method is zero dissipative phase fitted and almost p stable numerical experiments from its application to well known periodic orbital problems show that the new method is more competitive when comparing with the codes proposed by franco j m franco a class of explicit two step hybrid methods for second order ivps j comput appl math 187 2006 41 57 anastassi and simos z a anastassi t e simos a trigonometrically fitted runge kutta method for the numerical solution of orbital problems new astron 10 2005 301 309

efficient energy preserving integrators for oscillatory hamiltonian systems in this paper we focus our attention on deriving and analyzing an efficient energy preserving formula for the system of nonlinear oscillatory or highly oscillatory second order differential equations q t mq t f q t where m is a symmetric positive semi definite matrix with m 1 and f q q u q is the negative gradient of a real valued function u q this system is a hamiltonian system with the hamiltonian h p q 1 2 p t p 1 2 q t mq u q the energy preserving formula exactly preserves the hamiltonian we analyze in detail the properties of the energy preserving formula and propose new efficient energy preserving integrators in the sense of numerical implementation the convergence analysis of the fixed point iteration is presented for the implicit integrators proposed in this paper it is shown that the convergence of implicit average vector field methods is dependent on m whereas the convergence of the new energy preserving integrators is independent of m the fermi pasta ulam problem and the sine gordon equation are carried out numerically to show the competence and efficiency of the novel integrators in comparison with the well known average vector field methods in the scientific literature

scheifele two step methods for perturbed oscillators two step methods specially adapted to the numerical integration of perturbed oscillators are obtained the formulation of the methods is based on a refinement of classical taylor expansions due to scheifele g scheifele on the numerical integration of perturbed linear oscillating systems z angew math phys 22 1971 186 210 the key property is that those algorithms are able to integrate exactly harmonic oscillators with frequency w the methods depend on a parameter email protected where h is the stepsize based on the b2 series theory of coleman j p coleman order conditions for a class of two step methods for y f x y ima j numer anal 23 2003 197 220 we derive the order conditions of this new type of method the linear stability and phase properties are examined the theory is illustrated with some fourth and fifth order explicit schemes numerical results carried out on an assortment of test problems such as the integration of the orbital motion of earth satellites show the relevance of the theory

on the representation of integers by binary forms let f be a binary form with integer coefficients non zero discriminant and degree d with d at least 3 let r f z denote the number of integers of absolute value at most z which are represented by f we prove that there is a positive number c f such that r f z is asymptotic to c f z frac 2 d

zero dissipative semi implicit hybrid method for solving oscillatory or periodic problems in this paper a new semi implicit two step hybrid method with fifth algebraic order is derived for the integration of second order oscillatory initial value problems the new method possesses dispersion of order eight and dissipation of order infinity numerical experiment reveals the superiority of the new method for solving oscillatory or periodic problems over several methods of the same algebraic order in the literature

high order adaptive methods of nystrom cowell type a class of unconditionally stable multistep methods is discussed for solving initial value problems of second order differential equations which have periodic or quasiperiodic solutions this situation frequently occurs in celestial mechanics in nonlinear oscillations and various other situations the methods depend upon a parameter 0 and integrate exactly trigonometric functions along with algebraic polynomials in this paper we show a procedure for the construction of adaptive nystrom cowell formulas of arbitrarily high order of accuracy and reduce to the classical nystrom cowell methods for 0 our methods compare advantageously with other methods

counting rational points on algebraic varieties for any n 2 let z pn be a geometrically integral algebraic variety of degree d this article is concerned with the number nz b of q rational points on z which have height at most b for any e 0 we establish the estimate nz b od e n bdimz e provided that d e as indicated the implied constant depends at most on d e and n

map based kinetic analysis for voxel by voxel compartment model estimation detailed imaging of the cerebral glucose metabolism using fdg we propose a novel algorithm for voxel by voxel compartment model analysis based on a maximum a posteriori map algorithm voxel by voxel compartment model analysis can derive functional images of living tissues but it suffers from high noise statistics in voxel based pet data and extended calculation times we initially set up a feature space of the target radiopharmaceutical composed of a measured plasma time activity curve and a set of compartment model parameters and measured the noise distribution of the pet data the dynamic pet data were projected onto the feature space and then clustered using the mahalanobis distance our method was validated using simulation studies and compared with roi based ordinary kinetic analysis for fdg the parametric images exhibited an acceptable linear relation with the simulations and the roi based results and the calculation time took about 10 min we therefore concluded that our proposed map based algorithm is practical

rational points of bounded height on projective surfaces we give upper bounds for the number of rational points of bounded height on the complement of the lines on projective surfaces

fast and reliable estimation of multiple parametric images using an integrated method for dynamic spect dynamic single photon emission computed tomography spect has demonstrated the potential to quantitatively estimate physiological parameters in the brain and the heart the generalized linear least square glls method is a well established method for solving linear compartment models with fast computational speed however the high level of noise intrinsic in the spect data leads to reliability and instability problems of glls for generating parametric images an integrated method is proposed to restrict the noise in both the temporal and spatial domains to estimate multiple parametric images for dynamic spect this method comprises three steps which are optimum image sampling schedule in the projection space cluster analysis applied postreconstruction and parametric image generation with glls the simulation and experimental studies for the neuronal nicotine acetylcholine receptor tracer of 5 123i iodo a 85380 were employed to evaluate the performance of the proposed method the results of influx rate of k1 and volume of distribution of vd demonstrated that the integrated method was successful in generating low noise parametric images for high noise spect data without enhancing the partial volume effect furthermore the integrated method is computationally efficient for potential clinical applications

trigonometrically fitted nonlinear two step methods for solving second order oscillatory ivps abstract the construction of a class of nonlinear two step methods for solving second order oscillatory ivps is analyzed these methods are exact for the linear space generated by the set of functions sin t cos t trigonometrically fitted methods but they do not require a previous knowledge of the frequency or a good approximation of it some trigonometrically fitted nonlinear two step schemes with algebraic order up to four are derived and their stability and phase properties are analyzed it is shown that the new schemes can be efficiently implemented by using a special vector operation the vector product and quotient the numerical experiments carried out show that the new nonlinear two step schemes are more efficient than other standard and nonlinear methods proposed in the scientific literature for solving second order oscillatory differential systems

a new family of runge kutta type methods for the numerical integration of perturbed oscillators our task in this paper is to present a new family of methods of the runge kutta type for the numerical integration of perturbed oscillators the key property is that those algorithms are able to integrate exactly without truncation error harmonic oscillators and that for perturbed problems the local error contains the perturbation parameter as a factor some numerical examples show the excellent behaviour when they compete with runge kutta nystrom type methods

a study on statistically reliable and computationally efficient algorithms for generating local cerebral blood flow parametric images with positron emission tomography with the advent of positron emission tomography pet a variety of techniques have been developed to measure local cerebral blood flow lcbf noninvasively in humans a potential class of techniques which includes linear least squares ls linear weighted least squares wls linear generalized least squares gls and linear generalized weighted least squares gwls is proposed the statistical characteristics of these methods are examined by computer simulation the authors present a comparison of these four methods with two other rapid estimation techniques developed by huang et al 1982 and alpert 1984 and two classical methods the unweighted and weighted nonlinear least squares regression the results show that these methods can take full advantage of the contribution from the fine temporal sampling data of modern tomographs and thus provide statistically reliable estimates that are comparable to those obtained from nonlinear ls regression these methods also have high computational efficiency and the parameters can be estimated directly from operational equations in one single step therefore they can potentially be used in image wide estimation of local cerebral blood flow and distribution volume with pet

multistep numerical methods based on the scheifele g functions with application to satellite dynamics in 1971 scheifele designed a method for the integration of perturbed oscillators by refining the classical method of power series this method consists of defining a certain sequence of entire functions and finding the sought solution in the form of a linear combination of them using coefficients obtained through recurrence formulas in spite of the good behavior of the scheifele method it is only practical for use in a few particular cases in which the force function is very simple given the complexity of the preliminary calculations needed to obtain the recurrence formulas this problem is resolved in this article by converting the g functions method into a multistep scheme the key property is that those algorithms are able to integrate without truncation error harmonic oscillators involving various frequencies and that for perturbed problems the local error contains the small perturbation parameter as a factor therefore the methods are useful for finding highly accurate solutions to problems whose solutions are close to quasi periodic ones an application is performed in investigating the behavior of the proposed schemes for the integration of the orbital motion of earth satellites

an embedded pair of exponentially fitted explicit runge kutta methods an embedded pair of exponentially fitted explicit runge kutta rk methods for the numerical integration of ivps with oscillatory solutions is derived this pair is based on the exponentially fitted explicit rk method constructed in vanden berghe et al and we confirm that the methods which constitute the pair have algebraic order 4 and 3 some numerical experiments show the efficiency of our pair when it is compared with the variable step code proposed by vanden berghe et al j comput appl math 125 2000 107

improved signal to noise ratio in parametric images by cluster analysis abstract parametric images are formed by analyzing the concentration history of every voxel in pet data sets because pet concentration data at the voxel level are rather noisy noise propagation into the parametric image is often quite noticeable to address this problem a model based clustering method has been developed to generate parametric images the basic idea of the clustering method is to average over voxels whose concentration histories have the same shape we applied the method to a two parameter k 1 k 2 compartment model of local cerebral blood flow the statistic r tc t dt c t dt te k 2 t c a t dt e k 2 t c a t dt classifies curves in terms of k 2 where c t and c a t denote the tissue and blood concentration histories respectively and is the convolution operator simulation studies of noise propagation in the clustering statistic showed that 30 voxel noise yielded a 2 standard deviation in r parametric images of blood flow and partition coefficient were computed for an o 15 study with and without clustering cluster size affected bias statistical precision and computation time with clusters of 400 voxels the variance of the flow parameter was around 1 50 smaller with clustering with negligible bias and a computation time of 30 s on a 64 mhz workstation for 15 128 128 images with matlab 5 1

additive representation in short intervals ii sums of two like powers we establish that for almost all natural numbers n there is a sum of two positive integral cubes lying in the interval n n 7 18 varepsilon n here the exponent 7 18 lies half way between the trivial exponent 4 9 stemming from the greedy algorithm and the exponent 1 3 constrained by the number of integers not exceeding x that can be represented as the sum of two positive integral cubes we also provide analogous conclusions for sums of two positive integral k th powers when k ge 4

generalization of the stormer method for perturbed oscillators without explicit first derivatives abstract in this work we construct a numerical method specially designed for perturbed oscillators without explicit first derivatives the new method which generalizes the classical stormer and cowell codes for second order equations is useful when the calculation of the first derivative is not needed because this derivative is not computed some numerical examples of the artificial satellite problem show the good behaviour of the methods when they compete against classical multistep and runge kutta nystrom codes

an evaluation of the algorithms for determining local cerebral metabolic rates of glucose using positron emission tomography dynamic data measurement of the local cerebral metabolic rate of glucose lcmrglc and the individual rate constant parameters of the sup 18 f 2 fluoro 2 deoxy d glucose fdg model can provide a clearer understanding and insight to the physiological processes in the human brain and a quicker and more accurate means of diagnosis in clinical applications a systematic study using simulated and clinical tissue time activity data is presented to evaluate several existing and newly developed major algorithms used for determining lcmrglc and the individual rate constants from positron emission tomography dynamic data the computational and statistical properties of the autoradiographic approach weighted and unweighted nonlinear least squares methods patlak graphic approach weighted integration method linear least squares and generalized linear least squares methods are investigated and discussed in this paper