pattern clustering based on noise modeling in wavelet space we describe an effective approach to object or feature detection in point patterns via noise modeling this is based on use of a redundant or non pyramidal wavelet transform noise modeling is based on a poisson process we illustrate this new method with a range of examples we use the close relationship between image pixelated and point representations to achieve the result of a clustering method with constant time computational cost

multiple positive solutions for a functional second order boundary value problem by using the krasnoselskii fixed point theorem on cones in banach spaces some existence results of positive solutions of a boundary value problem concerning a second order functional differential equation are given

multiscale methods for data on graphs and irregular multidimensional situations for regularly spaced one dimensional data wavelet shrinkage has proven to be a compelling method for non parametric function estimation we create three new multiscale methods that provide wavelet like transforms both for data arising on graphs and for irregularly spaced spatial data in more than one dimension the concept of scale still exists within these transforms but as a continuous quantity rather than dyadic levels further we adapt recent empirical bayesian shrinkage techniques to enable us to perform multiscale shrinkage for function estimation both on graphs and for irregular spatial data we demonstrate that our methods perform very well when compared with several other methods for spatial regression for both real and simulated data although we concentrate on multiscale shrinkage regression we present our new wavelet transforms as generic tools intended to be the basis of methods that might benefit from a multiscale representation of data either on graphs or for irregular spatial data

lipschitz stability for nonlinear volterra integrodifferential systems unlike uniform stability the linearized system inherits the property of uniform lipschitz stability from its original nonlinear system in this paper we investigate sufficient conditions for uniform lipschitz stability of nonlinear integro differential systems through their associated variational systems

detecting features in spatial point processes with clutter via model based clustering abstract we consider the problem of detecting features such as minefields or seismic faults in spatial point processes when there is substantial clutter we use model based clustering based on a mixture model for the process in which features are assumed to generate points according to highly linear multivariate normal densities and the clutter arises according to a spatial poisson process nonlinear features are represented by several densities giving a piecewise linear representation hierarchical model based clustering provides a first estimate of the features and this is then refined using the em algorithm the number of features is estimated from an approximation to its posterior distribution the method gives good results for the minefield and seismic fault problems software to implement it is available on the world wide web

model based gaussian and non gaussian clustering abstract the classification maximum likelihood approach is sufficiently general to encompass many current clustering algorithms including those based on the sum of squares criterion and on the criterion of friedman and rubin 1967 however as currently implemented it does not allow the specification of which features orientation size and shape are to be common to all clusters and which may differ between clusters also it is restricted to gaussian distributions and it does not allow for noise we propose ways of overcoming these limitations a reparameterization of the covariance matrix allows us to specify that some features but not all be the same for all clusters a practical framework for non gaussian clustering is outlined and a means of incorporating noise in the form of a poisson process is described an approximate bayesian method for choosing the number of clusters is given the performance of the proposed methods is studied by simulation with encouraging results the methods are applied to the analysis of a data set arising in the study of diabetes and the results seem better than those of previous analyses rh

error estimators for stiff differential equations abstract when solving ordinary differential equations numerically the local error is estimated at each step in the classical situation of small step sizes it is clear what is required of the error estimators stiff problems are solved with large step sizes the quality of error estimators is studied in this situatuion and it is shown how to modify unsatisfactory estimators so as to improve them greatly several formulas from the literature are treated as examples

an approximate solution for linear boundary value problems with slowly varying coefficients in this paper an approximate closed form solution for linear boundary value problems with slowly varying coefficient matrices is obtained the derivation of the approximate solution is based on the freezing technique which is commonly used in analyzing the stability of slowly varying initial value problems as well as solving them the error between the approximate and the exact solutions is given and an upper bound on the norm of the error is obtained this upper bound is proportional to the rate of change of the coefficient matrix of the boundary value problem the proposed approximate solution is obtained for a two point boundary value problem and is compared to its solution obtained numerically good agreement is observed between the approximate and the numerical solutions when the rate of change of the coefficient matrix is small

summation by parts in time we develop a new high order accurate time integration technique for initial value problems we focus on problems that originate from a space approximation using high order finite difference methods on summation by parts form with weak boundary conditions and extend that technique to the time domain the new time integration method is global high order accurate unconditionally stable and together with the approximation in space it generates optimally sharp fully discrete energy estimates in particular it is shown how stable fully discrete high order accurate approximations of the maxwells equations the elastic wave equations and the linearized euler and navier stokes equations can obtained even though we focus on finite difference approximations we stress that the methodology is completely general and suitable for all semi discrete energy stable approximations numerical experiments show that the new technique is very accurate and has limited order reduction for stiff problems

asymptotic behaviour of solutions of systems of discrete equations via liapunov type technique abstract the second liapunov method serves as a powerful tool for the investigation of the stability of the trivial solution of ordinary differential equations systems and discrete equations systems in the presented paper a liapunov type qualitative approach is used for the investigation of asymptotic behaviour of the solutions of systems of discrete equations conditions for the existence of continuum of solutions the graphs of which remain within a given prescribed set are formulated for the general systems of discrete equations u k f k u k an additional advantage of the presented approach consists of the fact that no assumption concerning the existence of the trivial solution or the existence of an equilibrium point of systems considered is assumed moreover the asymptotic behaviour of solutions of some classes of linear difference systems is given by means of concrete asymptotic formulae several illustrative examples are considered too

l0 stable split linear multistep formulas for parabolic pdes abstract new types of split linear multistep methods slmms are applied to the numerical solution of parabolic differential equations the formulas derived achieve up to fourth order in time and unlike the crank nicolson method are l0 stable

improved error bounds for freezing solutions of linear boundary value problems for the error in the freezing solutions of linear boundary value problems we obtain a bound which is sharper than that obtained recently by shahruz and schwartz appl math comput 60 1994 285 comput math appl 28 1994 75 a different freezing technique global freezing is also proposed it is shown that this new technique is easy to implement for numerical computation of the solutions moreover the corresponding solution has an error bound similar to that of the freezing method

a quantitative study on the stability of quadratic delay difference systems this paper estimates the size of the stability region around zero for quadratic delay difference systems when the initial disturbance is in the asymptotic stability region the solution of the initial value problem of the quadratic delay difference system approaches zero examples are given for a three dimensional system and three one dimensional equations to demonstrate both stability and instability examples 2 4 show that when parameters in the systems do not satisfy the stability conditions the zero solutions can be unstable three evolution features of initial disturbances are shown numerically decaying to zero being amplified but bounded and growing to infinity example 3 further shows that the stable zero solution may not be a global attractor numerical results confirm the conclusions of the main theorem in this paper and imply that our estimation of the size of the stability region are of reasonable accuracy 1998 elsevier science ltd all rights reserved

detecting mines in minefields with linear characteristics we consider the problem of detecting minefields using aerial images a first stage of image processing has reduced the image to a set of points each one representing a possible mine our task is to decide which ones are actual mines we assume that the minefield consists of approximately parallel rows of mines laid out according to a probability distribution that encourages evenly spaced linear patterns the noise points are assumed to be distributed as a poisson process we construct a markov chain monte carlo algorithm to estimate the model and obtain posterior probabilities for each point being a mine the algorithm performs well on several real minefield datasets

existence of positive solutions for second order functional differential equations we afford a existence criterion of positive solutions of a boundary value problem concerning a second order functional differential equation by using the krasnoselskii fixed point theorem on cones in banach spaces moreover we also apply our results to establish several existence theorems of multiple positive solutions for some functional differential equations

boundedness of discrete volterra equations abstract investigation of the boundedness of the solutions of volterra equations with discrete time by means of the direct liapunov method and comparison theorems is performed boundedness conditions of solutions of some particular classes of equations are derived using such conditions stability of direct quadrature methods for second kind volterra equations is analyzed

the integration of stiff initial value problems in odes using modified extended backward differentiation formulae abstract a class of modified extended backward differentiation formulae mebdf suitable for the approximate numerical integration of first order systems of stiff ordinary differential equations is introduced the computational aspects of this new class of formulae are examined in detail in particular algorithms for varying both order and stepsize are given and this leads to a variable step variable order process using highly stable formulae of order 2 8 extensive numerical results for the well known detest set are given and on this basis a comparison is made between a code incorporating mebdf a code due to hindmarsh based on conventional backward differentiation formulae and a code due to skeel and kong based on blended linear multistep methods it is shown that on this test set the mebdf code is very reliable and is competitive with the other two codes on a significant class of problems a computer listing of the mebdf code used to obtain the results presented in this paper is available from the author

adaptive linear barycentric rational finite differences method for stiff odes abstract it is our purpose to introduce a simple multistep method based on linear barycentric rational interpolation for solving ordinary differential equations also we design an adaptive version having one free parameter which is used to improve the stability properties numerical experiments of the constructed methods on some well known stiff problems indicate efficiency and capability of the methods in solving stiff problems

nonparametric maximum likelihood estimation of features in spatial point processes using voronoi tessellation abstract this article addresses the problem of estimating the support domain of a bounded point process in presence of background noise this situation occurs for example in the detection of a minefield from aerial observations a maximum likelihood estimator for a mixture of uniform point processes is derived using a natural partition of the space defined by the data themselves the voronoi tessellation the methodology is tested on simulations and compared to a model based clustering technique

boundedness of solutions of difference systems with delays we give sufficient conditions for the boundedness of all solutions of some classes of systems of difference equations with delays by comparison of their norms with the solution of certain auxiliary scalar difference equations

twin solutions to singular dirichlet problems the existence of two nonnegative solutions to dirichlet second order boundary value problems is established in this paper our nonlinearity may be singular at y 0 t 0 and or t 1

positive solutions for a floquet functional boundary value problem in this paper we prove the existence of multiple positive solutions for a boundary value problem concerning a first order functional differential equation the results are obtained by using two fixed point theorems on appropriate cones in banach spaces these theorems are based on the fixed point index theory

a unified approach to stability in integrodifferential equations via liapunov functions in this paper we investigate the problems of stability uniform stability uniform lipschitz stability asymptotic stability and uniform asymptotic stability in three classes of integrodifferential systems these are linear nonautonomous systems nonlinear systems and nonlinearly perturbed nonlinear systems a simple method for constructing liapunov functions will be used to unify the treatment of all of the above mentioned stability notions relative to the various classes of integrodifferential systems in our investigation of linear systems we benefited from the work in 4 8 123 for nonlinear systems and their perturbations we have used some of the results obtained recently in is 101 in particular we have used an extension of alekseev s l nonlinear variation of constants formula which was developed in lo the approach we use here was first formulated in 3 and utilized to study uniform lipschitz stability 2 of nonlinear differential systems the method has been extended in 12 to cover all other stability notions in nonlinear differential systems we now give a description of the contents of the paper section 1 includes all the basic definitions and preliminaries in section 2 we investigate the various stability notions of nonautonomous integrodifferential systems this extends the work in 6 81 which mainly dealt with autonomous systems we recall here from 4 that in linear systems the notions of uniform stability and uniform lipschitz stability are equivalent it was shown in 4 however that for nonlinear systems uniform

optimization of high order diagonally implicit runge kutta methods abstract this article presents constrained numerical optimization of high order linearly and algebraically stable diagonally implicit runge kutta methods after satisfying the desired order conditions undetermined coefficients are optimized with respect to objective functions which consider accuracy stability and computational cost constraints are applied during the optimization to enforce stability properties to ensure a well conditioned method and to limit the domain of the abscissa two promising third order methods are derived using this approach labelled sdirk 3 1 2 2 3 l 14 and sdirk 3 1 4 l sa 5 both optimized schemes have a good balance of properties the relative error norm of the latter the l 2 norm scaled by a function of the number of implicit stages is a factor of two smaller than comparable methods found in the literature variations on these methods are discussed relative to trade offs in their accuracy and stability properties a novel fifth order scheme sdirk 5 1 5 l 02 is derived with a significantly lower relative error norm than the comparable fifth order a stable reference method in addition the optimized scheme is l stable the accuracy and relative efficiency of the runge kutta methods are verified through numerical simulation of van der pol s equation as well as numerical simulation of vortex shedding in the laminar wake of a circular cylinder and in the turbulent wake of a naca 0012 airfoil these results demonstrate the value of numerical optimization for selecting undetermined coefficients in the construction of high order runge kutta methods with a balance between competing objectives

discrete retract principle for systems of discrete equations abstract a power tool for investigation of various problems of ordinary differential equations as well as delayed differential equations is a retraction method the development of this method is discussed here in the case of system discrete equations corresponding definition of a point of a type of strict egress for a given set with respect to the system of discrete equations u k f k u k is given in an example it is shown that this definition cannot exactly copy the analogous one known from the theory of ordinary differential equations conditions for existence of at least one solution with graph remaining in a given set are formulated the proof is based on idea of a retract principle in construction of a retract mapping the property of continuous dependence on their initial data is used illustrative applications are considered too

finding curvilinear features in spatial point patterns principal curve clustering with noise clustering about principal curves combines parametric modeling of noise with nonparametric modeling of feature shape this is useful for detecting curvilinear features in spatial point patterns with or without background noise applications include the detection of curvilinear minefields from reconnaissance images some of the points in which represent false detections and the detection of seismic faults from earthquake catalogs our algorithm for principal curve clustering is in two steps the first is hierarchical and agglomerative hpcc and the second consists of iterative relocation based on the classification em algorithm cem pcc hpcc is used to combine potential feature clusters while cem pcc refines the results and deals with background noise it is important to have a good starting point for the algorithm this can be found manually or automatically using for example nearest neighbor clutter removal or model based clustering we choose the number of features and the amount of smoothing simultaneously using approximate bayes factors

global asymptotic stability in a perturbed higher order linear difference equation abstract in this note we give a sufficient condition for the global asymptotic stability of the zero solution of the difference equation n 1 i 0 k p 1 n n i f n n n 1 n 1 n 0 1 where k and l are nonnegative integers the coefficients pi n are real numbers and the nonlinearity f satisfies the growth condition f n 0 1 l q 0 i lmax i for n 0 1 and i r 0 i where q is a constant the stability condition is formulated in terms of the fundamental solution of the unperturbed equation y n 1 i 0 k p i n y n 1

exponential fitting bdf algorithms explicit and implicit 0 stable methods we present bdf type formulas capable of the exact integration with only round off errors of differential equations the solutions of which belong to the space generated by the linear combinations of 1 eax xeax plots of their 0 stability regions in terms of are provided we will see that the explicit method is 0 stable with many parameters h plots of their regions absolute stability that include all the negative real axis are provided numerical examples show the efficiency of the proposed codes specially when we are integrating stiff problems with the explicit method

stability of finite delay difference systems the stability of finite delay difference systems has recently been discussed in l in this paper we consider the delay difference systems with infinite delay of the following form x n 1 g n x s s 1 1 1 n g n x 1 where 1 is an integer or w we assume g n 0 0 forn l l l so that 1 has the zero solution x n 0 clearly for any given integer n l 1 and a given bounded initial function 4 l 1 1 no rk there is a unique solution of l denoted by x n n 4 such that it satisfies 1 for all integers n 2 n and x n no 4 444 forn f l l no for any integer m 2 co let 2 m m 1 i and 2 z the integer set if m a for any n e z define ii411 tlnl sup l s 15 s 5 n and for any h 0 let c n 4 z rk i nl h in the sequel we always assume that the variables n r and s take integer values and the corresponding intervals are discrete intervals

a quantitative study on the stability of delay discrete singular systems for quadratic delay discrete singular systems an algebraic criterion on the stability is established and the size of the uniform stability region and asymptotic stability region around zero is estimated hence the criterion is both qualitative and quantitative with the computer techniques the criterion dependent of delay is easy test and applies to the application in the practice an illustrative simulation is given to illustrate the application of the obtained result

extrapolated stabilized explicit rungekutta methods extrapolated stabilized explicit rungekutta methods eserk are proposed to solve multi dimensional nonlinear partial differential equations pdes in such methods it is necessary to evaluate the function nt times per step but the stability region is o nt2 hence the computational cost is o nt times lower than for a traditional explicit algorithm in that way stiff problems can be integrated by the use of simple explicit evaluations in which case implicit methods usually had to be used therefore they are especially well suited for the method of lines mol discretizations of parabolic nonlinear multi dimensional pdes in this work first s stages first order methods with extended stability along the negative real axis are obtained they have slightly shorter stability regions than other traditional first order stabilized explicit rungekutta algorithms also called rungekuttachebyshev codes later they are used to derive nt stages second and fourth order schemes using richardson extrapolation the stability regions of these fourth order codes include the interval 0 01nt2 0 nt being the number of total functions evaluations which are shorter than stability regions of rock4 methods for example however the new algorithms neither suffer from propagation of errors as other rungekuttachebyshev codes as rock4 or dumka nor internal instabilities additionally many other types of higher order and also lower order methods can be obtained easily in a similar way these methods also allow adaptation of the length step with no extra cost hence the stability domain is adapted precisely to the spectrum of the problem at the current time of integration in an optimal way i e with minimal number of additional stages we compare the new techniques with other well known algorithms with good results in very stiff diffusion or reactiondiffusion multi dimensional nonlinear equations

asymptotic formulae for a particular solution of linear nonhomogeneous discrete equations abstract the aim of this contribution is to develop the asymptotic formulae which describe the asymptotic behaviour of a particular solution of linear nonhomogeneous differential equation u k a k u k g k if k it is shown that asymptotic formulae are generated by partial sums of certain formal series that satisfy the given equation a comparison with the known explicit formula for solution of the initial problem is given and moreover some illustrative examples are discussed the used method of investigation is based on results published in 1

on the stability of exponential fitting bdf algorithms we present bdf type formulas capable of the exact integration with only round off errors of differential equations whose solutions are linear combinations of an exponential with parameter l and ordinary polynomials for l 0 new formulas reduces to the classical bdf formulas plots of their 0 stability regions in terms of l are provided plots of their regions absolute stability that include all the negative real axis are provided numerical examples shows the efficiency of the proposed codes specially when we are integrating stiff oscillatory problems

extended backward differentiation methods in the numerical solution of the neutral volterra integro differential equations the numerical solution of first order volterra integro differential equations of neutral type with continuous kernel is considered preliminary results from the application of extended backward differentiation methods are given and comparisons are made with adams bashforth adams moulton predictor corrector methods and collocation methods

perturbed mebdf methods we investigate mebdf methods of cash from general linear methods point of view some perturbations of these methods are constructed which preserve the order of these formulas and improve their stability properties

two stage clustering via neural networks this paper presents a two stage approach that is effective for performing fast clustering first a competitive neural network cnn that can harmonize mean squared error and information entropy criteria is employed to exploit the substructure in the input data by identifying the local density centers a gravitation neural network gnn then takes the locations of these centers as initial weight vectors and undergoes an unsupervised update process to group the centers into clusters each node called gravi node in the gnn is associated with a finite attraction radius and would be attracted to a nearby centroid simultaneously during the update process creating the gravitation like behavior without incurring complicated computations this update process iterates until convergence and the converged centroid corresponds to a cluster compared to other clustering methods the proposed clustering scheme is free of initialization problem and does not need to pre specify the number of clusters the two stage approach is computationally efficient and has great flexibility in implementation a fully parallel hardware implementation is very possible

boundary value problems for singular second order functional differential equations abstract we consider general linear boundary value problems for equations of the form y f x y x 0 0 x x is continuous and f x y has a singularity at y 0 the results improve and extend earlier results for the case x x due to taliaferro 1979 and gatica 1989

limiting periodicity of the solutions for some volterra difference equations conditions of the limiting of periodicity of the solutions for some volterra difference equations are derived these conditions are formulated immediately in terms of the coefficients of equations and their resolvent using the second liapunov method and fixed point theorem

nearest neighbor clutter removal for estimating features in spatial point processes abstract we consider the problem of detecting features in spatial point processes in the presence of substantial clutter one example is the detection of minefields using reconnaissance aircraft images that identify many objects that are not mines our solution uses kth nearest neighbor distances of points in the process to classify them as clutter or otherwise the observed kth nearest neighbor distances are modeled as a mixture distribution the parameters of which are estimated by a simple em algorithm this method allows for detection of generally shaped features that need not be path connected in the minefield example this method yields high detection and low false positive rates another application to outlining seismic faults is considered with some success the method works well in high dimensions the method can also be used to produce very high breakdown point robust estimators of a covariance matrix

asymptotic behavior of nonlinear difference systems in this paper we study a general variational stability introduced mainly for weakly stable difference systems moreover we obtain asymptotic formulae for these systems which state new results about asymptotic behavior for perturbed systems under general hypotheses

the asymptotic stability of xn 1 axn bxn k 0 we obtain necessary and sufficient conditions for the asymptotic stability of the difference equation xn 1 axn bxn k 0 n 0 1 2 where the coefficients a and b are real numbers and the delay k is a nonnegative integer

stability analysis of delay difference systems abstract for quadratic delay difference systems a result related to stability analysis is established more specifically the zero solution is uniformly stable and uniformly asymptotically stable for suitably small delay r where r is the admissible maximum value of delay under certain conditions furthermore the estimates of the corresponding stability and asymptotic stability regions are given hence the obtained result is both qualitative and quantitative also as a by product an improved uniform asymptotic stability criterion is established

stability analysis of nonautonomous difference systems with delaying arguments abstract we present sufficient conditions for the stability of the nonautonomous difference system x k 1 j 0 m a j k x k j k z with m 1 when the n n matrices a j are slowly varying coefficients the proposed approach is based on the generalization of the freezing method for ordinary differential equations the stability conditions are formulated in terms of the corresponding cauchy s function

stability of neutral delay difference systems abstract for neutral delay difference systems of the general form two kinds of stability criteria are established in this paper one is by means of the discrete liapunov functionals while the other is with the discrete liapunov functions both make use of razumikhin techniques

exponential asymptotic stability of delay partial difference equations this paper is concerned with the linear delay partial difference equation u i j 1 a i j u i 1 j b i j u i j p i j u i j where and are two nonnegative integers a i j b i j p i j are three real sequences defined on i 0 j 0 some sufficient conditions for this equation to be exponentially asymptotically stable are derived

weak asymptotic stabilizability of discrete time systems given by set valued operators the paper deals with the weak asymptotic stabilizability property of discrete time inclusions whose right hand sides are set valued operators in banach spaces such inclusions include control systems with constrained controls questions of existence of positive eigenvalues of convex closed operators are discussed based on the method of decomposition of the state space we derive sufficient conditions for weakly asymptotic stabilizability of discrete time convex processes in terms of controllability like conditions the results are applied in weak asymptotic stabilizability of linear discrete and continuous time systems with constrained controls in banach spaces

estimate of total stability of delay difference systems abstract the concepts of total stability and total asymptotic stability for delay difference systems in the general form are defined moreover the criteria for total stability and total asymptotic stability of linear delay difference systems are established the obtained results are qualitative and quantitative in the sense that the estimates of the total stability region and the total asymptotic stability region are given

existence of positive solutions for boundary value problem of second order fde abstract we use a fixed point theorem in cones to investigate the existence of positive solutions for boundary value problem of a second order functional differential equation fde with the form y t r t y t 0 0 y t y t t r t 0 y t y t n t 1 t 1 a where y t y t for a we allow that r t has the singularity at the endpoints t 0 and t 1 of 0 1 our results include the situations that f is either superlinear or sublinear

stability criteria for a class of linear delay partial difference equations this paper is concerned with the linear delay partial difference equation a i j u i 1 j u i j 1 b i j u i j p i j u i j 0 where a i j b i j p i j i j n0 are real sequences sufficient conditions for this equation to be stable are derived some conditions for this equation to be unstable are obtained also

recent advances in methods for numerical solution of o d e initial value problems abstract this is a review paper which describes recent advances in numerical methods and computer codes for solving initial value problems of ordinary differential equations particular emphasis is placed upon stiff systems

boundedness of infinite delay difference systems x n 1 g n x s s i 1 1 n g n x 1 where i is an integer or co g is a volterra functional determined by the integer n and the values of x s for s i 1 1 n and taking values in rk the stability results of 1 have been recently established in 11 however in order to explore the existence of periodic solutions of 1 we need to know not only the stability but also the boundedness of solutions for the linear case of 1 for the relevant results we refer to 2 while for the corresponding results of infinite delay differential systems see 3 41 in this paper we are going to establish theorems on boundedness for 1 in terms of both liapunov functionals and functions with the so called razumikhin techniques before doing this we give the following notations and definitions for any integer m 2 00 let z m m 1 and z z the integer set if m a trivially for any ie e z and a given bounded initial function 4 i i 1 no rk there is a unique solution of l denoted by x n n d such that it satisfies 1 for all integers 17 2 n and x n n 4 d n forn l l l n