symbolic model checking for real time systems abstract we describe finite state programs over real numbered time in a guarded command language with real valued clocks or equivalently as finite automata with real valued clocks model checking answers the question which states of a real time program satisfy a branching time specification given in an extension of ctl with clock variables we develop an algorithm that computes this set of states symbolically as a fixpoint of a functional on state predicates without constructing the state space for this purpose we introduce a calculus on computation trees over real numbered time unfortunately many standard program properties such as response for all nonzeno execution sequences during which time diverges cannot be characterized by fixpoints we show that the expressiveness of the timed calculus is incomparable to the expressiveness of timed ctl fortunately this result does not impair the symbolic verification of implementable real time programs those whose safety constraints are machine closed with respect to diverging time and whose fairness constraints are restricted to finite upper bounds on clock values all timed ctl properties of such programs are shown to be computable as finitely approximable fixpoints in a simple decidable theory

calculating the hilbert transform on spaces with energy concentration convergence and divergence regions in many different applications it is important to determine the hilbert transform of a given function however it is generally impossible to calculate it in closed form therefore hilbert transform approximations are used this paper studies the convergence and divergence behavior of general classes of such approximation methods these classes are characterized by two very natural axioms and they include basically all known traditional numerical algorithms the convergence of these methods is investigated on a family of signal spaces of continuous functions with finite energy these spaces are parametrized by a number which measures the energy concentration in the low frequency components of the signal it is shown that stable methods only exist on signal spaces with a sufficient energy concentration and this paper gives some explicit examples of convergent methods on all other spaces in the family of signal spaces every sampling based hilbert transform approximation shows a blowup behavior of its peak value i e on these spaces every sampling based hilbert transform approximation diverges

computing the hilbert transform on the real line we introduce a new method for computing the hilbert transform on the real line it is a collocation method based on an expansion in rational eigenfunctions of the hilbert transform operator and implemented through the fast fourier transform an error analysis is given and convergence rates for some simple classes of functions are established numerical tests indicate that the method compares favorably with existing methods

an automatic quadrature for cauchy principal value integrals an automatic quadrature is presented for computing cauchy principal value integrals q f c faf t t c dt a c b for smooth functions f t after subtracting out the singularity we approximate the function f t by a sum of chebyshev polynomials whose coefficients are computed using the ftt the evaluations of q f c for a set of values of c in a b are efficiently accomplished with the same number of function evaluations numerical examples are also given

a method to evaluate the hilbert transform on 0 the author proposes a method to approximate the hilbert transform on the real positive semiaxis by a suitable lagrange interpolating polynomial the method employs truncated gaussian rules and uses the interlacing properties of the zeros of generalized laguerre polynomials the error estimate in a weighted uniform norm is proved and some numerical tests show the efficacy of the proposed procedure

a novel rotationally invariant region based hidden markov model for efficient 3 d image segmentation we present a novel 3 d region based hidden markov model rbhmm for efficient unsupervised 3 d image segmentation our contribution is twofold first rbhmm employs a more efficient representation of the image data than current state of the art hmm based approaches that are based on either voxels or rectangular lattices grids thus resulting in a faster optimization process second our proposed novel tree structured parameter estimation algorithm for the rbhmm provides a locally optimal data labeling that is invariant to object rotation which is a highly valuable property in segmentation tasks especially in medical imaging where the segmentation results need to be independent of patient positioning in scanners in order to minimize methodological variability in data analysis we demonstrate the advantages of our proposed technique over grid based hmms by validating on synthetic images of geometric shapes as well as both simulated and clinical brain mri scans for the geometric shapes data our method produced consistently accurate segmentation results that were also invariant to object rotation for the brain mri data our white matter and gray matter segmentation resulted in substantially higher robustness and accuracy levels with improved dice similarity indices of 4 60 p 0 0022 and 7 71 p 0 0001 respectively

gaussian quadrature formulae of the third kind for cauchy principal value integrals basic properties and error estimates abstract let n 1 f be the polynomial of degree n 1 interpolating the function f at the points x1 x2 xn with pn xi 0 i e at the nodes of the classical gaussian quadrature formula for the numerical approximation of the cauchy principal value integral 1 1 x x 1 d x with 1 1 and f c1 1 1 we present the quadrature formula qn 1g3 given by q n 1 g3 1 n 1 x n 1 x d x 1 n 1 1 we show that this quadrature formula does not have the disadvantages of the other two well known quadrature formulae based on the same set of nodes in particular we prove that the sequence based on the same set of nodes in particular we prove that the sequence qn 1g3 f converges to the true value of the integral uniformly for all 1 1 we give estimates for the error term furthermore we state some relations connecting the present quadrature formula to the previously introduced formulae

approximation of semigroups and related operator functions by resolvent series we consider the approximation of semigroups e tau a and of the functions varphi j tau a that appear in exponential integrators by resolvent series the interesting fact is that the resolvent series expresses the operator functions e tau a and varphi j tau a respectively in efficiently computable terms this is important for semigroups where the new approximation is different from well known approximations by rational functions as well as for the application of exponential integrators which are currently of high interest and which are usually studied in a semigroup setting on banach spaces the approximation of the operator functions varphi j tau a in a general strongly continuous semigroup setting has not been discussed in the literature so far but this is crucial for an application of these integrators with unbounded operators or bounded operators like discretization matrices with large norm and eigenvalues somewhere in the left half plane

evaluating of dawson s integral by solving its differential equation using orthogonal rational chebyshev functions abstract dawson s integral is u y exp y 2 0 y exp z 2 d z we show that by solving the differential equation d u d y 2 yu 1 using the orthogonal rational chebyshev functions of the second kind sb 2 n y l which generates a pentadiagonal petrov galerkin matrix one can obtain an accuracy of roughly 3 8 n digits where n is the number of terms in the spectral series the sb series is not as efficient as previously known approximations for low to moderate accuracy however because the n term approximation can be found in only o n operations the new algorithm is the best arbitrary precision strategy for computing dawson s integral

uniform approximations to finite hilbert transform and its derivative interpolatory integration rules of numerical stability are presented for approximating cauchy principal value p v integrals 11 f t t c dt and hadamard finite part f p integrals 11f t t c 2 dt 1 c 1 respectively for a given smooth function f t present quadrature rules consist of interpolating f t at abscissae in the interval of integration 1 1 except for the pole c where neither the function value f c nor its derivative f c is required followed by subtracting out the singularities we demonstrate that the use of both endpoints 1 as abscissae in interpolating f t is essential for uniformly approximating the integrals namely for bounding the approximation errors independently of the values of c in fact for the f p integrals the use of double abscissae at both endpoints 1 as well as simple abscissae in 1 1 enables the uniform approximations while the use of simple abscissae at both endpoints 1 and those in 1 1 is sufficient for the p v integrals these facts suggest that finite hilbert transforms p v integrals and their derivatives f p integrals with varied values of c could be approximated efficiently with the same number of abscissae respectively some numerical examples are given

a theory of stochastic systems part ii process algebra this paper introduces pronounce spades a stochastic process algebra for discrete event systems that extends traditional process algebra with timed actions whose delay is governed by general a o continuous probability distributions the operational semantics is defined in terms of stochastic automata a model that uses clocks like in timed automata to symbolically represent randomly timed systems cf the accompanying paper p r d argenio j p katoen a theory of stochastic systems part i stochastic automata inf comput 2005 to appear we show that stochastic automata and are equally expressive and prove that the operational semantics of a term up to conversion of clocks is unique modulo symbolic bisimulation open probabilistic and structural bisimulation are proven to be congruences for and are equipped with an equational theory the equational theory is shown to be complete for structural bisimulation and allows to derive an expansion law

computing the hilbert transform and its inverse we construct a new method for approximating hilbert transforms and their inverse throughout the complex plane both problems can be formulated as riemann hilbert problems via plemelj s lemma using this framework we re derive existing approaches for computing hilbert transforms over the real line and unit interval with the added benefit that we can compute the hilbert transform in the complex plane we then demonstrate the power of this approach by generalizing to the half line combining two half lines we can compute the hilbert transform of a more general class of functions on the real line than is possible with existing methods

a high order solver for the heat equation in 1d domains with moving boundaries we describe a fast high order accurate method for the solution of the heat equation in domains with moving dirichlet or neumann boundaries and distributed forces we assume that the motion of the boundary is prescribed our method extends the work of greengard and strain comm pure appl math xliii 1990 pp 949 963 our scheme is based on a time space chebyshev pseudo spectral collocation discretization which is combined with a recursive product quadrature rule to accurately and efficiently approximate convolutions with green s function for the heat equation we present numerical results that exhibit up to eighth order convergence rates assuming n time steps and m spatial discretization points the evaluation of the solution of the heat equation at the same number of points in space time requires mathcal o n m log m work thus our scheme can be characterized as fast that is it is work optimal up to a logarithmic factor

uniform approximation to cauchy principal value integrals with logarithmic singularity abstract an approximation of clenshaw curtis type is given for cauchy principal value integrals of logarithmically singular functions i f c 1 1 f x log x c x c d x c 1 1 with a given function f using a polynomial p n of degree n interpolating f at the chebyshev nodes we obtain an approximation i p n c i f c we expand p n in terms of chebyshev polynomials with o n log n computations by using the fast fourier transform our method is efficient for smooth functions f for which p n converges to f fast as n grows and so simple to implement this is achieved by exploiting three term inhomogeneous recurrence relations in three stages to evaluate i p n c for f z analytic on the interval 1 1 in the complex plane z the error of the approximation i p n c is shown to be bounded uniformly using numerical examples we demonstrate the performance of the present method

elementary solutions for a model boltzmann equation in one dimension and the connection to grossly determined solutions abstract the fourier transformed version of the time dependent slip flow model boltzmann equation associated with the linearized bgk model is solved in order to determine the solution s asymptotics the ultimate goal of this paper is to demonstrate that there exists a robust set of solutions to this model boltzmann equation that possess a special property that was conjectured by truesdell and muncaster that solutions decay to a subclass of the solution set uniquely determined by the initial mass density of the gas called the grossly determined solutions first we determine the spectrum and eigendistributions of the associated homogeneous equation then using case s method of elementary solutions we find analytic time dependent solutions to the model boltzmann equation for initial data with a specialized compact support condition under the fourier transform in doing so we show that the spectrum separates the solutions into two distinct parts one that behaves as a set of transient solutions and the other limiting to a stable subclass of solutions thus we demonstrate that for gas flows with this specialized initial density condition in time all gas flows for the one dimensional model boltzmann equation act as grossly determined solutions

a complete orthonormal system of functions in l 2 infty infty space a new complete orthonormal system of functions in the l 2 infty infty space is introduced the system consists of two sequences composed of odd and even functions respectively unlike hermite and laguerre sets of functions which behave exponentially at infinity the new system exhibits asymptoticbehavior x 1 for the odd sequence and x 2 for the even one formulae representing products derivatives etc in series in the system are developed a nonlinear differential equation with a requirement for summability of the square of its solutions instead of boundary conditions is solved the example displays most of the features of the new method important for essentially nonlinear problems

on uniform approximations to hypersingular finite part integrals abstract in this paper new uniform approximation schemes for computation of hypersingular finite part integrals are studied the methods are verified to be supremely qualified for oscillatory integrands in addition based on the results on chebyshev approximations the uniform convergence and computational error bounds are considered preliminary numerical results show the stability and efficiency of the schemes

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

computations of the complex error function rational expansions for computing the complex error function w z e z 2 text erfc iz are presented these expansions have the following attractive properties 1 they can be evaluated using a polynomial evaluation routine such as homer s method 2 the polynomial coefficients can be computed once and for all by a single fast fourier transform fft and 3 high accuracy is achieved uniformly in the complex plane with only a small number of terms comparisons reveal that in some parts of the complex plane certain competitors may be more efficient however the difference in efficiency is never great and the new algorithms are simpler than existing ones a complete program takes eight lines of matlab code

bisimulation for probabilistic transition systems a coalgebraic approach the notion of bisimulation as proposed by larsen and skou for discrete probabilistic transition systems is shown to coincide with a coalgebraic definition in the sense of aczel and mendier in terms of a set functor this coalgebraic formulation makes it possible to generalize the concepts to a continuous setting involving borel probability measures under reasonable conditions generalized probabilistic bisimilarity can be characterized categorically application of the final coalgebra paradigm then yields an internally fully abstract semantical domain with respect to probabilistic bisimulation

self similar slip instability on interfaces with rate and state dependent friction we examine the development of a frictional instability with diverging sliding rate at the interface of elastic bodies in contact evolution of friction is determined by a slip rate and state depe

a generalization of the wiener rational basis functions on infinite intervals part ii numerical investigation in part i we introduced the generalized wiener rational basis functions and here in part ii we continue our investigation with numerical experiments wiener s generalized basis can utilize the fast fourier transform for integer values of the decay parameter s we outline two algorithms for doing so in addition the issue of galerkin representations for polynomial nonlinearities of expansions is addressed the wiener basis set is compared against domain truncation methods fourier and chebyshev polynomials hermite functions sinc interpolations and mapped chebyshev expansions and we show that for both exponentially and algebraically decaying functions the wiener approximation is as good as or superior to these alternatives in addition we carry out preliminary investigations regarding tuning of the decay parameter s numerical simulations of korteweg de vries type equations show the effectiveness of the wiener expansion we also explore the practical use of the wiener basis functions on the semi infinite interval which is compared against laguerre function methods and other jacobi polynomial mappings

an algorithm based on the fft for a generalized chebyshev interpolation an algorithm for a generalized chebyshev interpolation procedure increasing the number of sample points more moderately than doubling is pre sented the fft for a real sequence is incorporated into the algorithm to enhance its efficiency numerical comparison with other existing algorithms is given

numerical integration in the presence of an interior singularity abstract the convergence of sequences of integration rules applied to an improperly integrable function with an interior singularity is investigated the rules studied include the gauss jacobi rules and the interpolatory integration rules based on the zeros of 1 x 2 m p n 2m x m 0 1 for certain values of and the results are then applied to the study of the convergence of hunter s method for cauchy principal value integrals

reactive generative and stratified models of probabilistic processes abstract we introduce three models of probabilistic processes namely reactive generative and stratified these models are investigated within the context of pccs an extension of milner s sccs in which each summand of a process summation expression is guarded by a probability and the sum of these probabilities is 1 for each model we present a structural operational semantics of pccs and a notion of bisimulation equivalence which we prove to be a congruence we also show that the models form a hierarchy the reactive model is derivable from the generative model by abstraction from the relative probabilities of different actions and the generative model is derivable from the stratified model by abstraction from the purely probabilistic branching structure moreover the classical nonprobabilistic model is derivable from each of these models by abstraction from all probabilities

the theory of interactive generalized semi markov processes in this paper we introduce the calculus of interactive generalized semi markov processes igsmps a stochastic process algebra which can express probabilistic timed delays with general distributions and synchronizable actions with zero duration and where choices may be probabilistic non deterministic and prioritized igsmp is equipped with a structural operational semantics which generates semantic models in the form of generalized semi markov processes gsmps i e probabilistic systems with generally distributed time extended with action transitions representing interaction among system components this is obtained by expressing the concurrent execution of delays through a variant of st semantics which is based on dynamic names the fact that names for delays are generated dynamically by the semantics makes it possible to define a notion of observational congruence for igsmp that abstracts from internal actions with zero duration simply as a combination of standard observational congruence and probabilistic bisimulation we also present a complete axiomatization for observational congruence over igsmp finally we show how to derive a gsmp from a given igsmp specification in order to evaluate the system performance and we present a case study

gauss kronrod integration rules for cauchy principal value integrals kronrod extensions to two classes of gauss and lobatto integration rules for the evaluation of cauchy principal value integrals are derived since in one frequently occurring case the kronrod extension involves evaluating the derivative of the integrand a new extension is introduced using n 2 points which requires only values of the integrand however this new rule does not exist for all n and when it does several significant figures are lost in its use 1 introduction in this paper we shall consider kronrod extensions ke to integration rules based on gauss and lobatto points for the evaluation of cauchy principal value cpv integrals of the form fix vyx 1 since the existence theory of ke s for regular integrals insures such extensions for only certain classes of weight functions w x we shall restrict our attention here to the most important class namely w x 1 x2y x 2 where 0 p 2 in the gauss case and p 1 in the lobatto case 9 the only other relevant weight functions are the jacobi weight functions w x 1 x a l x with a i s or s y a f and w x x2 i rx2 oo r 1 5 for our case w x 1 x2 m 1 2 the corresponding orthogonal polynomials are the gegenbauer polynomials c x usually written c x which have the following normalization 11 p 174 2 w x c x cmil x dx 8 mhnil where 3 hn i ft n 2p r p n u n r p r 2p

gaussian mrf rotation invariant features for image classification features based on markov random field mrf models are sensitive to texture rotation this paper develops an anisotropic circular gaussian mrf acgmrf model for retrieving rotation invariant texture features to overcome the singularity problem of the least squares estimate method an approximate least squares estimate method is designed and implemented rotation invariant features are obtained from the acgmrf model parameters using the discrete fourier transform the acgmrf model is demonstrated to be a statistical improvement over three published methods the three methods include a laplacian pyramid an isotropic circular gmrf icgmrf and gray level cooccurrence probability features

a novel method for computing the hilbert transform with haar multiresolution approximation in this paper an algorithm for computing the hilbert transform based on the haar multiresolution approximation is proposed and the l 2 error is estimated experimental results show that it outperforms the library function hilbert in matlab the mathworks inc 1994 2007 finally it is applied to compute the instantaneous phase of signals approximately and is compared with three existing methods

corrigendum generalized chebyshev interpolation and its application to automatic quadrature math comp 41 1983 no 164 537 553 mr0717701 84m 65037 by hasegawa t torii and i ninomiya

segmentation of brain mr images through a hidden markov random field model and the expectation maximization algorithm the finite mixture fm model is the most commonly used model for statistical segmentation of brain magnetic resonance mr images because of its simple mathematical form and the piecewise constant nature of ideal brain mr images however being a histogram based model the fm has an intrinsic limitation no spatial information is taken into account this causes the fm model to work only on well defined images with low levels of noise unfortunately this is often not the the case due to artifacts such as partial volume effect and bias field distortion under these conditions fm model based methods produce unreliable results here the authors propose a novel hidden markov random field hmrf model which is a stochastic process generated by a mrf whose state sequence cannot be observed directly but which can be indirectly estimated through observations mathematically it can be shown that the fm model is a degenerate version of the hmrf model the advantage of the hmrf model derives from the way in which the spatial information is encoded through the mutual influences of neighboring sites although mrf modeling has been employed in mr image segmentation by other researchers most reported methods are limited to using mrf as a general prior in an fm model based approach to fit the hmrf model an em algorithm is used the authors show that by incorporating both the hmrf model and the em algorithm into a hmrf em framework an accurate and robust segmentation can be achieved more importantly the hmrf em framework can easily be combined with other techniques as an example the authors show how the bias field correction algorithm of guillemaud and brady 1997 can be incorporated into this framework to achieve a three dimensional fully automated approach for brain mr image segmentation

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 the uniform convergence of gaussian quadrature rules for cauchy principal value integrals and their derivatives the convergence of the aforementioned quadrature rules for integrands possessing h6lder continuous derivatives of an appropriate order is proved to be uniform and not only pointwise the rate of convergence is also established and an application to the numerical solution of singular integral equations is made

approximating cauchy type singular integral by an automatic quadrature scheme an automatic quadrature scheme is developed for the approximate evaluation of the product type indefinite integral q f x y c x y r t k t c f t dt 1 x y 1 1

the eigenvalues of hermite and rational spectral differentiation matrices we derive expressions for the eigenvalues of spectral differentiation matrices for unbounded domains in particular we consider galerkin and collocation methods based on hermite functions as well as rational functions a fourier series combined with a cotangent mapping we show that i first derivative matrices have purely imaginary eigenvalues and second derivative matrices have real and negative eigenvalues ii for the hermite method the eigenvalues are determined by the roots of the hermite polynomials and for the rational method they are determined by the laguerre polynomials and iii the hermite method has attractive stability properties in the sense of small condition numbers and spectral radii

an efficient method for approximate solution of a singular integral equation with cauchy kernel abstract in this paper the least squares method and the reproducing kernel space are employed to obtain approximate solution of a general form of singular integral equations with cauchy kernel this approach is based on the construction of the subspace which is dense in the reproducing kernel space the numerical examples illustrate the high accuracy of the method

clenshaw curtis type quadrature rule for hypersingular integrals with highly oscillatory kernels abstract the clenshaw curtis type quadrature rule is proposed for the numerical evaluation of the hypersingular integrals with highly oscillatory kernels and weak singularities at the end points for any smooth functions g x based on the fast hermite interpolation this paper provides a stable recurrence relation for these modified moments convergence rates with respect to the frequency k and the number of interpolation points n are considered these theoretical results and high accuracy of the presented algorithm are illustrated by some numerical examples

uniform convergence of optimal order quadrature rules for cauchy principal value integrals abstract for the numerical evaluation of cauchy principal value integrals of the form e 1 1 f e c s 1 1 we consider a quadrature method based on spline interpolation of odd degree 2 k 1 k n 0 we show that these rules converge uniformly for 1 1 in particular we calculate the exact order of magnitude of the error and show that it is equal to the order of the optimal remainder in the class of functions with bounded s th derivative if s e s 2 k 1 2 k 2 finally we compare the rule to the well known quadrature rule of elliott and paget which only converges pointwise

interlacing properties of the zeros of the orthogonal polynomials and approximation of the hilbert transform abstract in this paper starting from interlacing properties of the zeros of the orthogonal polynomials the authors propose a new method to approximate the finite hilbert transform for this method they give error estimates in uniform norm

numerical and perturbative computations of solitary waves of the benjamin ono equation with higher order nonlinearity using christov rational basis functions computation of solitons of the cubically nonlinear benjamin ono equation is challenging first the equation contains the hilbert transform a nonlocal integral operator second its solitary waves decay only as o 1 x 2 to solve the integro differential equation for waves traveling at a phase speed c we introduced the artificial homotopy h u x x c u 1 d u 2 du 3 0 d 0 1 and solved it in two ways the first was continuation in the homotopy parameter d marching from the known benjamin ono soliton for d 0 to the cubically nonlinear soliton at d 1 the second strategy was to bypass continuation by numerically computing perturbation series in d and forming pade approximants to obtain a very accurate approximation at d 1 to further minimize computations we derived an elementary theorem to reduce the two parameter soliton family to a parameter free function the soliton symmetric about the origin with unit phase speed solitons for higher order benjamin ono equations are also computed and compared to their korteweg devries counterparts all computations applied the pseudospectral method with a basis of rational orthogonal functions invented by christov which are eigenfunctions of the hilbert transform

convergence of product formulas for the numerical evaluation of certain two dimensional cauchy principal value integrals we consider product rules of interpolatory type for the numerical approximation of certain two dimensional cauchy principal value integrals we present convergence results which generalize those known in the one dimensional case

peano kernels and bounds for the error constants of gaussian and related quadrature rules for cauchy principal value integrals we show that if f in c k 1 1 k ge 2 the error term of every modified positive interpolatory quadrature rule for cauchy principal value integrals of the type mathop int nolimits 1 1 w x f x over x lambda dx lambda in 1 1 fulfills r n f lambda o n k ln n uniformly for all lambda in 1 1 and hence it is of optimal order of magnitude in the classes c k 1 1 k 2 3 4 ldots here w is a weight function with the property 0 le w x sqrt 1 x 2 le c we give explicit upper bounds for the peano type error constants of such rules this improves and completes earlier results by criscuolo and mastroianni calcolo 22 1985 391 441 and numer math 54 1989 445 461 and ioakimidis math comp 44 1985 191 198 for the special case of the gaussian rule we show that the restriction k ge 2 can be dropped the results are based on a new representation of the peano kernels of these formulae via the peano kernels of the underlying classical quadrature formulae this representation may also be useful in connection with some different problems

the rate of convergence of fourier coefficients for entire functions of infinite order with application to the weideman cloot sinh mapping for pseudospectral computations on an infinite interval we analytically compute the asymptotic fourier coefficients for several classes of functions to answer two questions the numerical question is to explain the success of the weideman cloot algorithm for solving differential equations on an infinite interval their method combines fourier expansion with a change of coordinate using the hyperbolic sine function the sinh mapping transforms a simple function like exp z2 into an entire function of infinite order this raises the second analytical question what is the fourier rate of convergence for entire functions of an infinite order the answer is sometimes even slower than a geometric series in this case the fourier series converge only on the real axis even when the function u z being expanded is free of singularities except at infinity earlier analysis ignored stationary point contributions to the asymptotic fourier coefficients when u z had singularities off the real z axis but we show that sometimes these stationary point terms are more important than residues at the poles of u z

error estimates for interpolatory quadrature formulae in this paper we study the remainder of interpolatory quadrature formulae for this purpose we develop a simple but quite general comparison technique for linear functionals applied to quadrature formulae it allows to eliminate one of the nodes and to estimate the remainder of the old formula in terms of the new one by repeated application we may compare with quadrature formulae having only a few nodes left or even no nodes at all with the help of this method we obtain asymptotically best possible error bounds for the clenshaw curtis quadrature and other polya type formulae our comparison technique can also be applied to the problem of definiteness i e the question whether the remainderr f of a formula of orderm can be represented asc f m by successive elimination of nodes we obtain a sequence of sufficient criteria for definiteness including all the criteria known to us as special cases finally we ask for good and worst quadrature formulae within certain classes we shall see that amongst all quadrature formulae with positive coefficients and fixed orderm the gauss type formulae are worst interpreted in terms of peano kernels our theorem yields results on monosplines which may be of interest in themselves

uniform convergence of derivatives of lagrange interpolation abstract uniform convergence of derivatives of lagrange interpolation at the union of zeros of jacobi polynomials and some additional points is investigated

quadrature rule for indefinite integral of algebraic logarithmic singular integrands an automatic quadrature method is presented for approximating the indefinite integral of functions having algebraic logarithmic singularities q x y c f email protected x yf t t c alog t c dt 1 1 within a finite range 1 1 for some smooth function f t that is approximated by a finite sum of chebyshev polynomials we expand the given indefinite integral in terms of chebyshev polynomials by using auxiliary algebraic logarithmic functions present scheme approximates the indefinite integral q x y c f uniformly namely bounds the approximation error independently of the value c as well x and y this fact enables us to evaluate the integral transform q x y c f with varied values of x y and c efficiently some numerical examples illustrate the performance of the present quadrature scheme

a practical algorithm for computing cauchy principal value integrals of oscillatory functions abstract a new automatic quadrature scheme is proposed for evaluating cauchy principal value integrals of oscillatory functions 1 1 f x exp i x x 1 dx 1 1 r the desired approximation is obtained by expanding the function f in the series of chebyshev polynomials of the first kind and then by constructing the indefinite integral for a properly modified integrand to overcome the singularity the method is proved to converge uniformly with respect to both and for any function f satisfying max 1 x 1 f x

piecewise polynomial quadratures for cauchy singular integrals in this paper we propose piecewise polynomial methods for the approximation of cauchy principal value integrals and develop a simple efficient and numerically stable algorithm for the evaluation of the weights of the resulting piecewise polynomial quadratures we present two examples to illustrate the advantages of these quadratures versus the gauss jacobi quadratures

error bounds for gaussian quadrature of analytic functions for gaussian quadrature rules over a finite interval applied to analytic or meromorphic functions we develop error bounds from contour integral representations of the remainder term as in previous work on the subject we consider both circular and elliptic contours in contrast with earlier work however we attempt to determine exactly where on the contour the kernel of the error functional attains its maximum modulus we succeed in answering this question for a large class of weight distributions including all jacobi weights when the contour is a circle in the more difficult case of elliptic contours we can settle the question for certain special jacobi weight distributions with parameters pm frac 1 2 and we provide empirical results for more general jacobi weights we further point out that the kernel of the error functional at any complex point outside the interval of integration can be evaluated accurately and efficiently by a recursive procedure the same procedure is useful also t

a sinc quadrature subroutine for cauchy principal value integrals a subroutine is presented for the evaluation of cauchy principal value integrals based on a truncated sum of a sinc function series with the function evaluated at shifted sinc points the subroutine returns an approximation to the integral together with estimates of the relative and the absolute errors numerical results demonstrate that the new method works well on analytic functions with end point singularities

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

functions of the second kind for freud weights and series expansions of hilbert transforms abstract let w e q where q r r is even continuous and of smooth polynomial growth at infinity then we call w 2 e 2 q a freud weight the most typical examples being w 2 x exp x 1 corresponding to the weight w 2 we can form the sequence of orthonormal polynomials p j w 2 x j 0 the functions of the second kind are q j w 2 x h p j w 2 x j 0 where h denotes the hilbert transform that is for g l r h g x p v g t t x dt here p v denotes principal value for a large class of freud weights we obtain bounds on q j j 0 in the l and l p norms we also estimate the generalized function of the second kind q j w 2 v x h p j w v x for a fixed function v we then apply these estimates to investigate the convergence of series of the second kind which form the basis of henrici s method of approximating cauchy principal value integrals

bisimulation through probabilistic testing abstract we propose a language for testing concurrent processes and examine its strength in terms of the processes that are distinguished by a test by using probabilistic transition systems as the underlying semantic model we show how a testing algorithm can distinguish with a probability arbitrarily close to one between processes that are not bisimulation equivalent we also show a similar result in a slightly stronger form for a new process relation called 2 3 bisimulation which lies strictly between that of simulation and bisimulation finally the ultimately strength of the testing language is shown to identify a new process relation called probabilistic bisimulation which is strictly stronger than bisimulation

generalized chebyshev interpolation and its application to automatic quadrature a generalized chebyshev interpolation procedure increasing a fixed number of sample points at a time is developed and analvzed it is incorporated into an efficient automatic quadrature scheme of clenshaw curtis tvpe numerical examples indicate that the present method is efficient not onlv for well behaved functions but for those w ith discontinuous low order derivatives by virtue of adequate error estimation as wvell as saving of sample points

a definiteness criterion for linear functionals and its application to cauchy principle value quadrature abstract let g be a linear functional on a k 1 1 the class of all functions with an absolutely continuous k 1 th derivative let r and let g be given by g g in this situation a theo bras and schmeiser 1981 states that g is definite if g is definite in this paper we give conditions on g which ensure the definiteness of g an application of our result yields error bounds for a quadrature rule for cauchy principal value integrals

on the convergence of interpolatory type quadrature rules for evaluating cauchy integrals the aim of this work is to analyse the stability and the convergence for the quadrature rule of interpolatory type based on the trigonometric approximation for the discretization of the cauchy principal value integrals 11 f t d we prove that the quadrature rule has almost optimal stability property behaving in the form o logn 1 sin2x x cos t using this result we show that the rule has an exponential convergence rate when the function f is differentiable enough when f possesses continuous derivatives up to order p 0 and the derivative f p t satisfies holder continuity of order we can also prove that the rule has the convergence rate of the form o a b log n n2 np p where is as small as we like a and b are constants depending only on x

a method for the practical evaluation of the hilbert transform on the real line abstract we investigate a method for the numerical evaluation of the weighted hilbert transform over the entire real line exp x 2 f x x t 1 d x where the integral is taken in the cauchy principal value sense the method is based on a combination of two polynomial approximation processes in this way we obtain a procedure that is numerically stable and efficient the algorithm can be implemented in a fast way and existing well known software packages may be used from the point of view of theoretical error bounds the method is competitive generalizations to other approximation methods are also possible

image classification by a two dimensional hidden markov model for block based classification an image is divided into blocks and a feature vector is formed for each block by grouping statistics extracted from the block conventional block based classification algorithms decide the class of a block by examining only the feature vector of this block and ignoring context information in order to improve classification by context an algorithm is proposed that models images by two dimensional 2 d hidden markov models hmms the hmm considers feature vectors statistically dependent through an underlying state process assumed to be a markov mesh which has transition probabilities conditioned on the states of neighboring blocks from both horizontal and vertical directions thus the dependency in two dimensions is reflected simultaneously the hmm parameters are estimated by the em algorithm to classify an image the classes with maximum a posteriori probability are searched jointly for all the blocks applications of the hmm algorithm to document and aerial image segmentation show that the algorithm outperforms cart sup tm lvq and bayes vq

exit criteria and monotonicity in compound quadrature of gaussian type in this short note for compound quadrature rules of gaussian type we prove stopping rules and monotonicity results based on peano kernel methods

uniform approximations to cauchy principal value integrals of oscillatory functions this paper presents an interpolatory type integration rule for the numerical evaluation of cauchy principal value integrals of oscillatory integrands 1 1e i w xf x email protected where 1 t 1 for a given smooth function f x the proposed method is constructed by interpolating f x at practical chebyshev points and subtracting out the singularity a numerically stable procedure is obtained and the corresponding algorithm can be implemented by fast fourier transform the validity of the method has been demonstrated by several numerical experiments

interlacing of zeros of shifted sequences of one parameter orthogonal polynomials we study the interlacing property of zeros of laguerre polynomials of adjacent degree where the free parameters differ by an integer and of the same degree where the free parameter is shifted continuously similar interlacing results are proven for the positive zeros of gegenbauer polynomials

new error bounds for modified quadrature formulas for cauchy principal value integrals abstract for the numerical approximation of cauchy principal value integrals we consider the so called modified quadrature formulas i e formulas obtained by first subtracting out the singularity and then applying a classical quadrature formula we are interested in error bounds holding uniformly for all possible positions of the singular point the standard error bounds are based on suprema of derivatives but they often overestimate the true errors by a factor that grows with the number of nodes of the quadrature formula we give new bounds involving the total variation var s and l p norms t s t p of some derivative of the integrand function these bounds give additional possibilities for sharper estimations of the error

convergence results for piecewise linear quadratures for cauchy principal value integrals etablissement des conditions de convergence simple et uniforme des quadratures lineaires par morceaux pour les integrales a valeur principale de cauchy

a theory of stochastic systems part i stochastic automata this paper presents the theoretical underpinning of a model for symbolically representing probabilistic transition systems an extension of labelled transition systems for the modelling of general discrete as well as continuous or singular probability spaces these transition systems are particularly suited for modelling softly timed systems real time systems in which the time constraints are of random nature for continuous probability spaces these transition systems are infinite by nature stochastic automata represent their behaviour in a finite way this paper presents the model of stochastic automata their semantics in terms of probabilistic transition systems and studies several notions of bisimulation furthermore the relationship of stochastic automata to generalised semi markov processes is established

some numerical algorithms to evaluate hadamard finite part integrals abstract some algorithms are described for the numerical evaluation of hadamard finite part integrals of type 1 1 f x x t 2 v dx t where v is a jacobi weight convergence results and some numerical examples are given

numerical integration of functions with poles near the interval of integration abstract an automatic quadrature scheme is presented for approximating integrals of functions that are analytic in the interval of integration but contain pole or poles of order 2 i e a double pole on the real axis or a complex conjugate pair of double poles near the interval of integration the present scheme is based on product integration rules of interpolatory type using function values of the abscissae only in the interval of integration the integral is approximated and evaluated by using recurrence relations and some extrapolation method after the smooth part of the integrand is expanded in terms of the chebyshev polynomials the fast fourier transform fft technique is used to generate efficiently the sequence of the finite chebyshev series expansions until an approximation of the integral satisfying the required tolerance is obtained with an adequate estimate of the error numerical examples are included to illustrate the performance of the method

on the uniform convergence of gaussian quadrature rules for cauchy principal value integrals in a previous paper the authors proposed a modified gaussian rule m wf t to compute the integral wf t in the cauchy principal value sense associated with the weightw and they proved the convergence in closed sets contained in the integration interval the main purpose of the present work is to prove uniform convergence of the sequence m wf t on the whole integration interval and to give estimates for the remainder term the same results are shown for particular subsequences of the gaussian rules m wf t for the evaluation of cauchy principal value integrals a result on the uniform convergence of the product rules is also discussed and an application to the numerical solution of singular integral equations is made