power series equivalent to rational functions a shifting origin kronecker type theorem and normality of pade tables letf z be a function analytic in a neighbourhood of zero for each pair of non negative integers m n form then byn toeplitz determinantd m n whose entries are the maclaurin series coefficients off namely d m n det f m j k 0 m j k j k 1 n where we definef s 0 s 0 ifs 0 a classical theorem of kronecker asserts thatf z is a rational function if and only if there existm 0 andn 0 such thatd m n 0 form m 0 andn n 0 in some important recent work such as the solution of meinardus s conjecture it has been found useful to form pade approximants not at 0 but at different points near 0 in questions regarding normality of these pade approximants with a shifting origin one considers then byn determinantd m n u which is defined by 1 but with 0 replaced byu in this spirit we prove thatf z is a rational function if and only if there exists asingle pair of positive integers m n such thatd m n u is identically zero foru in a neighbourhood of zero further we deduce that except possibly for countably many values ofu the pade table of a non rationalf z atz u is normal that isd m n u 0 for allm n 0 1 2

on the continuity of the multivariate pade operator abstract continuity of the univariate pade operator was proved in 5 6 we discuss the limitations of a multivariate generalization and prove a multivariate analogon of the continuity property

pade approximants and noise rational functions abstract we continue a study of pade approximants pa for a series perturbed by random noise this time we consider more general rational functions we begin with the simple case of a sum of two geometric series and then show how these considerations can be extended to a general rational function we do not study the most general case but rather concentrate on demonstrating how our results for geometric series extend to new situations encountered when a general rational function is considered we show that froissart doublets are a universal feature and we construct an analog of the froissart polynomial introduced in the earlier paper

the sensitivity of muscle force predictions to changes in physiologic cross sectional area abstract the mechanical effects of a muscle are related in part to the size of the muscle and to its location relative to the joint it crosses for more than a century researchers have expressed muscle size by its physiological cross sectional area pcsa researchers mathematically calculating muscle and joint forces typically use some expression of a muscle s pcsa to constrain the solution to one which is reasonable i e a solution in which small muscles may not have large forces and large muscles have large forces when expected or when there is significant electromyographic activity it is obvious that muscle mass and therefore any expression of pcsa varies significantly from person to person even in individuals of similar weight and height since it is not practical to predict the pcsa of each muscle in a living subject s limb or trunk it is important to generally understand the sensitivity of muscle force solutions to possible variations in pcsa we used nonlinear optimization techniques to predict 47 muscle forces and hip contact forces in a living subject the pcsa volume muscle fiber length of each of 47 lower limb muscle elements from two cadaver specimens and the 47 pcsa s reported by pierrynowski were input into an optimization algorithm to create three solution sets the three solutions were qualitatively similar but at times a predicted muscle force could vary as much as two to eight times in contrast the joint force solutions were within 11 of each other and therefore much less variable when using optimization techniques to predict muscle forces it must be recognized that the solution is sensitive to many assumptions and variables such as pcsa the muscle force solutions are therefore best used to determine relative values i e trends in parametric studies on the other hand the joint force solutions are less sensitive to such variations and the absolute values are more reliable

bandwidth truncation for chebyshev polynomial and ultraspherical chebyshev galerkin discretizations of differential equations the petrov galerkin ultraspherical polynomial chebyshev polynomial discretization of the highest derivative of a differential equation is a diagonal matrix the same is true for fourier galerkin discretizations nevertheless the spectral discretizations of simple problems like u x x q x u f x are usually dense matrices the villain is the multiplication matrix the galerkin representation of a term like q x u x unfortunately this part of the galerkin matrix is dense however if the ode coefficient q x has a chebyshev or fourier series that converges much more rapidly than u x then it is possible to realize great cost savings at no loss of accuracy by truncating the full n n galerkin matrix to a banded matrix where the bandwidth m n one of our themes is that when the spectral series for q x and u x have similar rates of convergence as is almost universal when a nonlinear equation is linearized for a newton krylov iteration such accuracy lossless truncation is impossible nonlinearity is but one of many causes of this sort of solution coefficient equiconvergence when bandwidth truncation is possible though our second theme is to show that a modest amount of floating point operations and memory can be saved by an unsymmetric truncation in which the number of elements retained to the left of the main diagonal is roughly double the number kept to the right our second improvement is to replace the m term spectral series for q x by its m 2 m 2 chebyshev pade rational approximation this sometimes allow one to halve the matrix bandwidth reducing the linear algebra costs by a factor of four

from stochastic partial difference equations to stochastic cellular automata through the ultra discretization we derive stochastic cellular automata with finite range interactions from stochastic partial difference equations through the ultra discrete limit a typical example for stochastic cellular automata is the domany kinzel model which is equivalent to mixed site bond oriented percolation as a special case

spurious poles in pade approximation in the theory of pade approximation locally uniform convergence has been proved only for special classes of functions for much larger classes convergence in capacity has been shown to hold true the reason for one type of convergence to hold true but the other one not can be found in poles of the approximants that may occur apparently anywhere in the complex plane because of their unwanted nature and since they do not correspond to singularities of the function f to be approximated these poles are called spurious the denominators of pade approximants satisfy orthogonality relations and consequently the location and distribution of spurious poles depend on properties of the orthogonality relations in the present paper the possibility of spurious poles is studied from the perspective of these orthogonal relations

dualities for a class of finite range probabilistic cellular automata in one dimension in this paper we study dualities for a class of one dimensional probabilistic cellular automata with finite range interactions by using a sequence of extended cellular automata

universal analytic properties of noise introducing the j matrix formalism we propose a new method in the spectral analysis of noisy time series data for damped oscillators from the jacobi three terms recursive relation for the denominators of the pade approximations built on the well known z transform of an infinite time series we build a hilbert space operator a j operator where each bound state inside the unit circle in the complex plane is simply associated with one damped oscillator while the essential spectrum of the j operator which lies on the unit circle itself is shown to represent the noise signal and noise are thus clearly separated in the complex plane for a finite time series of length 2n the j operator is replaced by a finite order j matrix jn having n eigenvalues which are time reversal covariant different classes of input noise such as blank white and uniform gaussian and pink are discussed in detail the j matrix formalism allowing us to efficiently calculate hundreds of poles of the z transform evidence of a universal behavior in the final statistical distribution of the associated poles and zeros of the z transform is shown in particular the poles and zeros tend when the length of the time series goes to infinity to a uniform angular distribution on the unit circle therefore at finite order the roots of unity in the complex plane appear to be noise attractors we show that the z transform presents the exceptional feature of allowing lossless undersampling and how to make use of this property a few basic examples are given to suggest the power of the proposed method

sensitivity of dynamic simulations of gait and dynamometer experiments to hill muscle model parameters of knee flexors and extensors we assessed and compared sensitivities of dynamic simulations to musculotendon mt parameters for gait and dynamometer experiments our aim with this comparison was to investigate whether dynamometer experiments could provide information about mt parameters that are important to reliably study mt function during gait this would mean that dynamometer experiments could be used to estimate these parameters muscle contribution to the joint torque mt torque rather than relative mt force primarily affects the resulting gait pattern and torque measured by the dynamometer in contrast to recent studies therefore we assessed the sensitivity of the mt torque rather than the sensitivity of the relative mt force based on sensitivity of the mt torque to a parameter perturbation mt parameters of the knee flexors and extensors were classified in three categories low medium and high for gait classification was based on the average sensitivity during a gait cycle for isometric and isokinetic dynamometer experiments classification was based on the highest sensitivity found in the experiments the calculated muscle contributions to the knee torque during gait and dynamometer experiments had a high sensitivity to only a limited number of mt parameters of the knee flexors and extensors suggesting that not all mt parameters need to be estimated in general the highest sensitivity was found for tendon slack length however for some muscles the sensitivity to the optimal fibre length or the maximal isometric muscle force was also high or medium the classification of the individual mt parameters for gait and dynamometer experiments was largely similar we therefore conclude that dynamometer experiments provide information about mt parameters important to reliably study mt function during gait so that subject specific estimates of mt parameters could be made based on dynamometer experiments

quantum operators in classical probability theory iv quasi duality and thinnings of interacting particle systems duality has proved to be a powerful technique in the study of interacting particle systems ips this concept can be enlarged and a quasi duality defined between various pairs of ips previously thought unrelated consequently theorems of a similar style to those involving duality can be deduced the concept of quasi duality follows naturally from our previous studies into the use ofsingle site operators an idea borrowed from quantum physics in paper ii of this series it is shown that a necessary condition for quasi duality is that the eigenvalues of the corresponding two site infinitesimal generators be the same and using this observation a number of quasi dual pairs have been found and studied it is further shown that if two different ips share a common dual then one can be considered as a thinning of the other

multivariate pade approximation multivariate pade approximation is considered in the framework of the theory developed by a cuyt 1982 the denominators are determined from determinants of matrices whose entries are homogeneous polynomials the main difference to the univariate case is a typical shift of the degree of the polynomials singularities atx 0 are analyzed since it is the rule rather than the exception that simultaneous zeros of the numerator and the denominator cannot be removed

the geometry of a parameter space of interacting particle systems we study a four parameter family of interacting particle systems containing the basic voter model and contact processes two processes in this family are related by duality or thinning if and only if their parameters belong to the same orbit of a certain one dimensional group of linear mappings this shows that many duals exist

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

on rational functions without froissart doublets in this paper we consider the problem of working with rational functions in a numeric environment a particular problem when modeling with such functions is the existence of froissart doublets where a zero is close to a pole we discuss three different parameters which allow one to monitor the absence of froissart doublets for a given general rational function these include the euclidean condition number of an underlying sylvester type matrix a parameter for determining coprimeness of two numerical polynomials and bounds on the spherical derivative we show that our parameters sharpen those found in a previous paper by two of the authors

a new perturbation analysis for signal enumeration in rotational invariance techniques the esprit algorithm is a subspace based high resolution method used in source localization and spectral analysis which provides very accurate estimates of the signal parameters however the underlying theory assumes a known model order which is usually not the case in many applications in particular it is well known that underevaluating the model order biases the estimation in this paper we analyze the perturbation induced by an erroneous model order and we present an error bound for the estimated parameters based on this theoretical framework we propose a new method for selecting an appropriate modeling order which consists in minimizing the error bound this approach is applied to both synthetic and musical signals and its performance is compared to that of existing methods such as the information theoretic criteria

spherical bessel transform via exponential sum approximation of spherical bessel function abstract a new algorithm for numerical evaluation of spherical bessel transform is proposed in this paper in this method the spherical bessel function is approximately represented as an exponential sum with complex parameters this is obtained by expressing an integral representation of spherical bessel function in complex plane and discretizing contour integrals along steepest descent paths and a contour path parallel to real axis using numerical quadrature rule with the double exponential transformation the number of terms in the expression is reduced using the modified balanced truncation method the residual part of integrand is also expanded by exponential functions using prony like method the spherical bessel transform can be evaluated analytically on arbitrary points in half open interval

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

matrix methods for pade approximation numerical calculation of poles zeros and residues abstract a representation of the pade approximation of the z transform of a signal as a resolvent of a tridiagonal matrix j n is given several formulas for the poles zeros and residues of the pade approximation in terms of the matrix j n are proposed their numerical stability is tested and compared methods for computing forward and backward errors are presented

exploiting regularities in large cellular spaces abstract we informally describe a computer algorithm for simulating deterministic cellular automata it grows smarter by selectively recording intermediate computations while operating on a compressed representation of the cellular space time no information is lost because configurations do not actually evolve instead the simulator accomodates random exploration of place times in the future of initial configurations embedded in an effectively unbounded void we use conway s life automaton as a model but the idea extends easily to other geometries dimensions and number of states

sensitivity of a hill based muscle model to perturbations in model parameters abstract musculoskeletal simulations of human movement commonly use hill muscle models to predict muscle forces but their sensitivity to model parameter values is not well understood the purpose of this study was to evaluate muscle model sensitivity to perturbations in 14 hill muscle model parameters in forward dynamic simulations of running and walking by varying each by 50 three evaluations of the muscle model were performed based on 1 calculating the sensitivity of the muscle model only 2 determining the continuous partial derivatives of the muscle equations with respect to each parameter and 3 evaluating the effects on the running and walking simulations model evaluations were found to be very sensitive percent change in outputs greater than parameter perturbation to parameters defining the series elastic component tendon force length curve of the contractile element and maximum isometric force for some parameters the range of literature values was larger than the model sensitivity model evaluations were insensitive to parameters defining the parallel elastic element force velocity curve of the contractile element and muscle activation time constants the derivative method provided similar results but also provided a generic continuous equation that can easily be applied to other motions the sensitivities of the running and walking simulations were reduced compared to the sensitivity of the muscle model alone results demonstrate the importance of evaluating sensitivity of a musculoskeletal simulation in a controlled manner and provide an indication of which parameters must be selected most carefully based on the sensitivity of a given movement

robust pade approximation via svd pade approximation is considered from the point of view of robust methods of numerical linear algebra in particular the singular value decomposition this leads to an algorithm for practical computation that bypasses most problems of solution of nearly singular systems and spurious pole zero pairs caused by rounding errors for which a matlab code is provided the success of this algorithm suggests that there might be variants of pade approximation that are pointwise convergent as the degrees of the numerator and denominator increase to infty unlike traditional pade approximants which converge only in measure or capacity

cramer rao bounds for multiple poles and coefficients of quasi polynomials in colored noise in this paper we provide analytical expressions of the cramer rao bounds for the frequencies damping factors amplitudes and phases of complex exponentials in colored noise these expressions show the explicit dependence of the bounds of each distinct parameter with respect to the amplitudes and phases leading to readily interpretable formulae which are then simplified in an asymptotic context the results are presented in the general framework of the polynomial amplitude complex exponentials pace model also referred to as the quasi polynomial model in the literature which accounts for systems involving multiple poles and represents a signal as a mixture of complex exponentials modulated by polynomials this work looks further and generalizes the studies previously undertaken on the exponential and the quasi polynomial models

on the numerical condition of a generalized hankel eigenvalue problem the generalized eigenvalue problem widetilde h y lambda h y with h a hankel matrix and widetilde h the corresponding shifted hankel matrix occurs in number of applications such as the reconstruction of the shape of a polygon from its moments the determination of abscissa of quadrature formulas of poles of pade approximants or of the unknown powers of a sparse black box polynomial in computer algebra in many of these applications the entries of the hankel matrix are only known up to a certain precision we study the sensitivity of the nonlinear application mapping the vector of hankel entries to its generalized eigenvalues a basic tool in this study is a result on the condition number of vandermonde matrices with not necessarily real abscissas which are possibly row scaled

sampling moments and reconstructing signals of finite rate of innovation shannon meets strang fix consider the problem of sampling signals which are not bandlimited but still have a finite number of degrees of freedom per unit of time such as for example nonuniform splines or piecewise polynomials and call the number of degrees of freedom per unit of time the rate of innovation classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited recently it was shown that by using an adequate sampling kernel and a sampling rate greater or equal to the rate of innovation it is possible to reconstruct such signals uniquely these sampling schemes however use kernels with infinite support and this leads to complex and potentially unstable reconstruction algorithms in this paper we show that many signals with a finite rate of innovation can be sampled and perfectly reconstructed using physically realizable kernels of compact support and a local reconstruction algorithm the class of kernels that we can use is very rich and includes functions satisfying strang fix conditions exponential splines and functions with rational fourier transform this last class of kernels is quite general and includes for instance any linear electric circuit we thus show with an example how to estimate a signal of finite rate of innovation at the output of an rc circuit the case of noisy measurements is also analyzed and we present a novel algorithm that reduces the effect of noise by oversampling

new representations of pade pade type and partial pade approximants pade approximants are rational functions with a denominator which does not vanish at zero and whose series expansion match a given series as far as possible these approximants are usually written under a rational form in this paper we will show how to write them also under two different barycentric forms and under a partial fraction form depending on free parameters according to the choice of these parameters pade type approximants can be obtained under a barycentric or a partial fraction form

froissart doublets in pade approximation in the case of polynomial noise first we study the relation between the zeros of random polynomials rn 1 and the zeros and poles of their pade approximants n n rn 1 next we consider the distribution of zeros and poles of pade approximants to the geometric series perturbed by a random polynomial noise we observe numerically interesting connections between two above problems some numerical observations on the froissart doublets have been also made

on the continuity of the pade operator let r m n be the pade approximant of order m n for a given power series f let t m n be the operator that maps f on r m n it is known that t m n satisfies a local lipschitz condition in case r m n is normal but may fail to be continuous if r m n is not normal in this paper we prove that t m n is continuous if and only if r m n has defect zero we also prove a convergence result for pade approximants in the case where the defect is positive

parameter estimation for nonincreasing exponential sums by prony like methods abstract let z j e f j with f j 0 i be distinct nodes for j 1 m with complex coefficients c j 0 we consider a nonincreasing exponential sum h x c 1 e f 1 x c m e f m x x 0 many applications in electrical engineering signal processing and mathematical physics lead to the following problem determine all parameters of h if 2 n sampled values h k k 0 2 n 1 n m are given this parameter estimation problem is a nonlinear inverse problem for noiseless sampled data we describe the close connections between prony like methods namely the classical prony method the matrix pencil method and the esprit method further we present a new efficient algorithm of matrix pencil factorization based on qr decomposition of a rectangular hankel matrix the algorithms of parameter estimation are also applied to sparse fourier approximation and nonlinear approximation

pade approximations in noise filtering resume we first recall the results on the distribution of poles and zeros of pade approximations to noisy taylor series around a point we then report the numerical experiment on noisy data coming from inelastic electron scattering cross sections on rare gases as functions of the momentum transfer at fixed energy we apply the thiele algorithm pade approximations of type ii to those data which are known to belong to an analytic function in a cut plane from to 0 in this case the poles and zeros split into three families 1 i doublet pole zero at a distance of the order of the computer roundoff 2 ii doublet pole zero at a distance of the order of the noise in the experimental data 3 iii remaining poles and zeros compatible with the analytic structure of the experimentally measured function by throwing away the first two families of poles and zeros noisy ones we were able to obtain an excellent filtering of the data compatible with the analytic structure of the analyzed function

particle systems acting on undirected graphs we study a class of interacting particle systems where the states of two neighboring sites are simultaneously changed in dependence on their previous states for two states this class is parametrized by six parameters duality and existence of stationary distributions is discussed

pade type rational and barycentric interpolation in this paper we consider the particular case of the general rational hermite interpolation problem where only the value of the function is interpolated at some points and where the function and its first derivatives agree at the origin thus the interpolants constructed in this way possess a pade type property at 0 numerical examples show the interest of the procedure the interpolation procedure can be easily modified to introduce a partial knowledge on the poles and the zeros of the function to approximated a strategy for removing the spurious poles is explained a formula for the error is proved in the real case applications are given

reconstruction of polygonal shapes from sparse fourier samples in this paper we want to reconstruct polygonal shapes in the real plane from as few fourier samples as possible that is we want to recover an original polygonal domain d with n vertices by using sparse sampling values of the fourier transform of the characteristic function of the polygonal domain we consider only simply connected polygons i e polygons with non intersecting edges for this purpose we need to reconstruct the vertices of the polygon in the case of non convex polygons we also need to reconstruct the order of the vertices to determine the correct boundary line segments the method presented here is based on the prony method

common factor detection and estimation the paper introduces a noniterative maximum likelihood ml based approach for the detection and estimation of the greatest common divisor of two given polynomials the coefficients of which are assumed to have a normal distribution with a consistently estimable covariance matrix it then goes on to show how this approach can be used to derive a noniterative ml method for estimating the orders and parameters of difference equation models with autoregressive residuals numerical examples are included to illustrate the performance of the proposed methodology

stable polefinding and rational least squares fitting via eigenvalues a common way of finding the poles of a meromorphic function f in a domain where an explicit expression of f is unknown but f can be evaluated at any given z is to interpolate f by a rational function pq such that r i f i at prescribed sample points i i 1l and then find the roots of q this is a two step process and the type of the rational interpolant needs to be specified by the user many other algorithms for polefinding and rational interpolation or least squares fitting have been proposed but their numerical stability has remained largely unexplored in this work we describe an algorithm with the following three features 1 it automatically finds an appropriate type for a rational approximant thereby allowing the user to input just the function f 2 it finds the poles via a generalized eigenvalue problem of matrices constructed directly from the sampled values f i in a one step fashion and 3 it computes rational approximants p q in a numerically stable manner in that p p q q f with small p q at the sample points making it the first rational interpolation or approximation algorithm with guaranteed numerical stability our algorithm executes an implicit change of polynomial basis by the qr factorization and allows for oversampling combined with least squares fitting through experiments we illustrate the resulting accuracy and stability which can significantly outperform existing algorithms

pade approximants for finite time ruin probabilities in this article we investigate pade approximants of hyper exponential to base on the first moments and the matrix exponential representation of pade approximated function an explicit formula is given for the laplace transform in the time to calculate finite time ruin probabilities of classical cramer lundberg model this formula generalizes the ultimate ruin probabilities formula of asmussen and rolski 1991 12 to illustrate this formula several numerical examples with different values u are given

kalman smoothing improves the estimation of joint kinematics and kinetics in marker based human gait analysis we developed a kalman smoothing algorithm to improve estimates of joint kinematics from measured marker trajectories during motion analysis kalman smoothing estimates are based on complete marker trajectories this is an improvement over other techniques such as the global optimisation method gom kalman filtering and local marker estimation lme where the estimate at each time instant is only based on part of the marker trajectories we applied gom kalman filtering lme and kalman smoothing to marker trajectories from both simulated and experimental gait motion to estimate the joint kinematics of a ten segment biomechanical model with 21 degrees of freedom three simulated marker trajectories were studied without errors with instrumental errors and with soft tissue artefacts sta two modelling errors were studied increased thigh length and hip centre dislocation we calculated estimation errors from the known joint kinematics in the simulation study compared with other techniques kalman smoothing reduced the estimation errors for the joint positions by more than 50 for the simulated marker trajectories without errors and with instrumental errors compared with gom kalman smoothing reduced the estimation errors for the joint moments by more than 35 compared with kalman filtering and lme kalman smoothing reduced the estimation errors for the joint accelerations by at least 50 our simulation results show that the use of kalman smoothing substantially improves the estimates of joint kinematics and kinetics compared with previously proposed techniques gom kalman filtering and lme for both simulated with and without modelling errors and experimentally measured gait motion

dimer automata and cellular automata abstract we define a class of discrete dynamical systems which we call dimer automata whereas in a cellular automaton the new state of one cell is a function of the states in the neighborhood in a dimer automaton the new states of two neighboring cells are functions of the states of these two cells dimer automata with synchronous dynamics seem artificial but with asynchronous dynamics such systems are very natural they are as simple as cellular automata they have some advantages in modeling spatial spread we present the definition some easy consequences a classification of one dimensional dimer automata a first approach to determine a characteristic equation and a formula for an approximate asymptotic density as well as a comparison to computer simulations finally we compare synchronous and asynchronous cellular automata

an implementation of the look ahead lanczos algorithm for non hermitian matrices the nonsymmetric lanczos method can be used to compute eigenvalues of large sparse non hermitian matrices or to solve large sparse non hermitian linear systems however the original lanczos algorithm is susceptible to possible breakdowns and potential instabilities an implementation of a look ahead version of the lanczos algorithm is presented that except for the very special situation of an incurable breakdown overcomes these problems by skipping over those steps in which a breakdown or near breakdown would occur in the standard process the proposed algorithm can handle look ahead steps of any length and requires the same number of matrix vector products and inner products as the standard lanczos process without look ahead

the second lowest extremal invariant measure of the contact process ii we continue the investigation of the behavior of the contact process on infinite connected graphs of bounded degree some questions left open by salzano and schonmann 1997 concerning the notions of complete convergence partial convergence and the criterion r s are answered the continuity properties of the survival probability and the recurrence probability are studied these order parameters are found to have a richer behavior than expected with the possibility of the survival probability being discontinuous at or above the threshold for survival a condition which guarantees the continuity of the survival probability above the survival point is introduced and exploited the recurrence probability is shown to always be left continuous above the recurrence point and a necessary and sufficient condition for its right continuity is introduced and exploited it is shown that for homogeneous graphs the survival probability can only be discontinuous at the survival point and the recurrence probability can only be discontinuous at the recurrence point for graphs which are obtained by joining a finite number of severed homogeneous trees by means of a finite number of vertices and edges the survival point the recurrence point and the discontinuity points of the survival and recurrence probabilities are located

self duality for multi state probabilistic cellular automata with finite range interactions in this paper we study a criterion of self duality for multi state probabilistic cellular automata with finite range interactions and give some models which satisfy this criterion

model reduction of large scale systems by least squares abstract in this paper we introduce an approximation method for model reduction of large scale dynamical systems this is a projection which combines aspects of the svd and krylov based reduction methods this projection can be efficiently computed using tools from numerical analysis namely the rational krylov method for the krylov side of the projection and a low rank smith type iteration to solve a lyapunov equation for the svd side of the projection for discrete time systems the proposed approach is based on the least squares fit of the r 1 th column of a hankel matrix to the preceding r columns where r is the order of the reduced system the reduced system is asymptotically stable matches the first r markov parameters of the full order model and minimizes a weighted h 2 error the method is also generalized for moment matching at arbitrary interpolation points application to continuous time systems is achieved via the bilinear transformation numerical examples prove the effectiveness of the approach the proposed method is significant because it combines guaranteed stability and moment matching together with an optimization criterion

sampling signals with finite rate of innovation the authors consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation examples of signals with a finite rate of innovation include streams of diracs e g the poisson process nonuniform splines and piecewise polynomials even though these signals are not bandlimited we show that they can be sampled uniformly at or above the rate of innovation using an appropriate kernel and then be perfectly reconstructed thus we prove sampling theorems for classes of signals and kernels that generalize the classic bandlimited and sinc kernel case in particular we show how to sample and reconstruct periodic and finite length streams of diracs nonuniform splines and piecewise polynomials using sinc and gaussian kernels for infinite length signals with finite local rate of innovation we show local sampling and reconstruction based on spline kernels the key in all constructions is to identify the innovative part of a signal e g time instants and weights of diracs using an annihilating or locator filter a device well known in spectral analysis and error correction coding this leads to standard computational procedures for solving the sampling problem which we show through experimental results applications of these new sampling results can be found in signal processing communications systems and biological systems

a physiologically based criterion of muscle force prediction in locomotion abstract the inversely nonlinear relationship of muscle contraction force and the possible contraction duration i e endurance is utilized in a method to mathematically predict individual muscle forces the method uses a nonlinear optimization scheme to solve the redundant distribution problem at the joints of interest the method is demonstrated at the elbow during isometric contraction and in the lower extremity during locomotion during gait the observed muscle activity pattern in the lower extremity as determined by emg shows substantial agreement with that activity pattern predicted when endurance is used as the optimization criterion the importance of selecting muscle prediction criteria based on sound physiological bases rather than on an arbitrary or mathematically convenient basis is emphasized

evaluating and approximating fir filters an approach based on functions of matrices most of the controllers designed for systems with timedelays involve finite impulse response fir filters in their structure in particular prediction based controllers for systems with input output delays we present theory algorithms and software to evaluate the frequency response to approximate the filters by stable linear timeinvariant systems and to compute their induced cal l 2 gain the approach is based on the theory of functions of matrices in the approach pole zero cancelations in frequency domain representations are explicitly taken into account

sensitivity of subject specific models to errors in musculo skeletal geometry subject specific musculo skeletal models of the lower extremity are an important tool for investigating various biomechanical problems for instance the results of surgery such as joint replacements and tendon transfers the aim of this study was to assess the potential effects of errors in musculo skeletal geometry on subject specific model results we performed an extensive sensitivity analysis to quantify the effect of the perturbation of origin insertion and via points of each of the 56 musculo tendon parts contained in the model we used two metrics namely a local sensitivity index lsi and an overall sensitivity index osi to distinguish the effect of the perturbation on the predicted force produced by only the perturbed musculo tendon parts and by all the remaining musculo tendon parts respectively during a simulated gait cycle results indicated that for each musculo tendon part only two points show a significant sensitivity its origin or pseudo origin point and its insertion or pseudo insertion point the most sensitive points belong to those musculo tendon parts that act as prime movers in the walking movement insertion point of the achilles tendon lsi 15 56 osi 7 17 origin points of the rectus femoris lsi 13 89 osi 2 44 and as hip stabilizers insertion points of the gluteus medius anterior lsi 17 92 osi 2 79 insertion point of the gluteus minimus lsi 21 71 osi 2 41 the proposed priority list provides quantitative information to improve the predictive accuracy of subject specific musculo skeletal models

nonlinear approximation by sums of exponentials and translates in this paper we discuss the numerical solution of two nonlinear approximation problems many applications in electrical engineering signal processing and mathematical physics lead to the following problem let h be a linear combination of exponentials with real frequencies determine all frequencies all coefficients and the number of summands if finitely many perturbed uniformly sampled data of h are given we solve this problem by an approximate prony method apm and prove the stability of the solution in the square and uniform norm further an apm for nonuniformly sampled data is proposed too the second approximation problem is related to the first one and reads as follows let varphi be a given 1 periodic window function as defined in section 4 further let f be a linear combination of translates of varphi determine all shift parameters all coefficients and the number of translates if finitely many perturbed uniformly sampled data of f are given using fourier technique this problem is transferred into the above parameter estimation problem for an exponential sum which is solved by apm the stability of the solution is discussed in the square and uniform norm too numerical experiments show the performance of our approximation methods

the rkfit algorithm for nonlinear rational approximation the rkfit algorithm outlined in m berljafa and s ga ttel generalized rational krylov decompositions with an application to rational approximation siam j matrix anal appl 2015 is a krylov based approach for solving nonlinear rational least squares problems this paper puts rkfit into a general framework allowing for its extension to nondiagonal rational approximants and a family of approximants sharing a common denominator furthermore we derive a strategy for the degree reduction of the approximants as well as methods for their conversion to partial fraction form for the efficient evaluation and root finding we also discuss commons and differences of rkfit and the popular vector fitting algorithm a matlab implementation of rkfit is provided and numerical experiments including the fitting of a mimo dynamical system and an optimization problem related to exponential integration demonstrate its applicability

stochastic models of interacting systems interacting particle systems is by now a mature area of probability theory but one that is still very active we begin this paper by explaining how models from this area arise in fields such as physics and biology we turn then to a discussion of both older and more recent results about them concentrating on contact processes voter models and exclusion processes these processes are among the most studied in the field and have the virtue of relative simplicity in their description which permits us to address the fundamental issues about their behavior without dealing with the extra complications that models from specific areas of application would require

quantum operators in classical probability theory ii the concept of duality in interacting particle systems duality has proved to be a powerful tool in the theory of interacting particle systems the approach in this paper is algebraic rather than via harris diagrams a form of duality is found which includes coalescing and annihilating duality as special cases this enables new results for the branching annihilating random walk and the biased annihilating branching process to be derived

a stable numerical method for inverting shape from moments we derive a stable technique based upon matrix pencils for the reconstruction of or approximation by polygonal shapes from moments we point out that this problem can be considered the dual of 2 d numerical quadrature over polygonal domains an analysis of the sensitivity of the problem is presented along with some numerical examples illustrating the relevant points finally an application to the problem of gravimetry is explored where the shape of a gravitationally anomalous region is to be recovered from measurements of its exterior gravitational field

signal recovery by discrete approximation and a prony like method abstract we introduce an algorithm which combines ideas of prony s approach to recover signals from given samples with approximation methods we solve two overdetermined systems of linear equations with linear programming methods and calculate the zeros of a suitable prony like polynomial we get the bandwidth m the frequencies as well as the amplitudes and some other characteristics of the signal especially it is reconstructed if sufficient many at least 2 m samples are given if we have too few samples or if they are too much noised we get an approximation of the original noiseless signal even if we have sufficient many but possible erroneous samples the obtained signal interpolates at least m of them usually the low noised or noiseless ones and of course all samples if the signal is recovered the described method behaves well to moderate sampling errors and is resistant to outliers in the samples which can be detected filtered off or corrected during the calculations to improve the quality of the computed signal

prony s method in several variables the paper gives an extension of prony s method to the multivariate case which is based on the relationship between polynomial interpolation normal forms modulo ideals and h bases though the approach is mainly of algebraic nature we also give an algorithm using techniques from numerical linear algebra to solve the problem in a fast and efficient way

an algorithm for computing a pade approximant with minimal degree denominator in this paper a new definition of a reduced pade approximant and an algorithm for its computation are proposed our approach is based on the investigation of the kernel structure of the toeplitz matrix it is shown that the reduced pade approximant always has nice properties which the classical pade approximant possesses only in the normal case the new algorithm allows us to avoid the appearance of froissart doublets induced by computer roundoff in the non normal pade table

matrix pencil method for estimating parameters of exponentially damped undamped sinusoids in noise a study of a matrix pencil method for estimating frequencies and damping factors of exponentially damped and or undamped sinusoids in noise is presented comparison of this method to a polynomial method svd prony method shows that the matrix pencil method and the polynomial method are two special cases of a matrix prediction approach and that the pencil method is more efficient in computation and less restrictive about signal probes it is found through perturbation analysis and simulation that for signals with unknown damping factors the pencil method is less sensitive to noise than the polynomial method an expression of the cramer rao bound for the exponential signals is presented

a walk on max plus algebra abstract max plus algebra is a kind of idempotent semiring over r max r with two operations max and in this paper we introduce a new model of a walk on one dimensional lattice on z as an analogue of the quantum walk over the max plus algebra and we call it max plus walk in the conventional quantum walk the summation of the l 2 norm of the states over all the positions is a conserved quantity in contrast the summation of eigenvalues of state decision matrices is a conserved quantity in the max plus walk moreover spectral analysis on the total time evolution operator is also given

parameter estimation of monomial exponential sums in one and two variables we propose a matrix pencil method to identify monomial exponential sums in the univariate case it is based on the qr factorization of hankel matrices in the bivariate case it assures the same accuracy of the univariate case in this paper we propose a matrix pencil method for the numerical identification of the parameters of monomial exponential sums in one and two variables while in the univariate case the proposed method is a variant of that developed by the authors in a preceding paper the bivariate case is treated for the first time here in the bivariate case the method we propose easily extendible to more variables reduces the problem to a pair of univariate problems and subsequently to the solution of a linear system as a result the relative errors in the univariate and in the bivariate case are almost of the same order

pade approximants and noise a case of geometric series abstract we study effects on pade approximants pa of a random noise added to coefficients of a power series numerical experiments performed during the last 30 years have shown that such noise results in the appearance of the so called froissart doublets fd pairs of zeros and poles separated by a distance of a scale of the noise for the first time we can prove that this effect actually takes place for geometric series we can also show what happens with noise induced poles or zeros that do not form fd we construct a polynomial that has as its roots mean positions of fd coefficients of this polynomial are expressed by random perturbations of coefficients of the series we study numerically this polynomial for low orders to show where in these orders fd coalesce numerical study of this polynomial to see a behaviour of fd for large orders is much easier than a study of the entire pa we also express a deviation of a pole of pas approximating the true pole of 1 1 z from 1 by perturbations of coefficients and show numerically that when order of pas increases this deviation drops down

stability of infinite clusters in supercritical percolation a recent theorem by haggstrom and peres concerning independent percolation is extended to all the quasi transitive graphs this theorem states that if 0 p1 p2 1 and percolation occurs at level p1 then every infinite cluster at level p2 contains some infinite cluster at level p1 consequences are the continuity of the percolation probability above the percolation threshold and the monotonicity of the uniqueness of the infinite cluster i e if at level p1 there is a unique infinite cluster then the same holds at level p2 these results are further generalized to graphs with a uniform percolation property the threshold for uniqueness of the infinite cluster is characterized in terms of connectivities between large balls

analytic solution of the seir epidemic model via asymptotic approximant abstract an analytic solution is obtained to the seir epidemic model the solution is created by constructing a single second order nonlinear differential equation in ln s and analytically continuing its divergent power series solution such that it matches the correct long time exponential damping of the epidemic model this is achieved through an asymptotic approximant barlow et al 2017 in the form of a modified symmetric pade approximant that incorporates this damping the utility of the analytical form is demonstrated through its application to the covid 19 pandemic

the discrete evolution model of bak and sneppen is conjugate to the classical contact process two fundamental models of critical phenomena are connected we show that the discrete bak sneppen evolution model is conjugate to the classical contact process this holds in discrete and continuous time on all graphs and for random as well as for deterministic choice of neighbors thus the extensive theory for the contact process applies to the discrete bak sneppen model too

invariant measures and decay of correlations for a class of ergodic probabilistic cellular automata we give new sufficient ergodicity conditions for two state probabilistic cellular automata pca of any dimension and any radius the proof of this result is based on an extended version of the duality concept under these assumptions in the one dimensional case we study some properties of the unique invariant measure and show that it is shift mixing also the decay of correlation is studied in detail in this sense the extended concept of duality gives exponential decay of correlation and allows to compute explicitly all the constants involved

sparse polynomial interpolation in chebyshev bases abstract we study the problem of reconstructing a sparse polynomial in a basis of chebyshev polynomials chebyshev basis in short from given samples on a chebyshev grid of 1 1 a polynomial is called m sparse in a chebyshev basis if it can be represented by a linear combination of m chebyshev polynomials for a polynomial with known and unknown chebyshev sparsity respectively we present efficient reconstruction methods where prony like methods are used the reconstruction results are mainly presented for bases of chebyshev polynomials of first and second kind respectively but similar issues can be obtained for bases of chebyshev polynomials of third and fourth kind respectively