nsfd representations for polynomial terms appearing in the potential functions of 1 dimensional conservative systems we provide a methodology for the construction of nonstandard finite difference nsfd schemes for 1 dimensional conservative dynamical systems such systems are used to model a broad range of nonlinear oscillators to obtain the desired representations we imposed conditions following from the constraints of parity invariance and the conservation of energy a further practical computational requirement is that these schemes be explicit and linear in the dependent variable evaluated at the upper discrete time level the current work extends previous studies by mickens et al

consistent and mimetic discretizations in general relativity a discretization of a continuum theory with constraints or conserved quantities is called mimetic if it mirrors the conserved laws or constraints of the continuum theory at the discrete level such discretizations have been found useful in continuum mechanics and in electromagnetism we have recently introduced a new technique for discretizing constrained theories the technique yields discretizations that are consistent in the sense that the constraints and evolution equations can be solved simultaneously but it cannot be considered mimetic since it achieves consistency by determining the lagrange multipliers in this paper we would like to show that when applied to general relativity linearized around a minkowski background the technique yields a discretization that is mimetic in the traditional sense of the word we show this using the traditional metric variables and also the ashtekar new variables but in the latter case we restrict ourselves to the euclidean case we also argue that there appear to exist conceptual difficulties to the construction of a mimetic formulation of the full einstein equations and suggest that the new discretization scheme can provide an alternative that is nevertheless close in spirit to the traditional mimetic formulations

cluster differences scaling with a within clusters loss component and a fuzzy successive approximation strategy to avoid local minima cluster differences scaling is a method for partitioning a set of objects into classes and simultaneously finding a low dimensional spatial representation ofk cluster points to model a given square table of dissimilarities amongn stimuli or objects the least squares loss function of cluster differences scaling originally defined only on the residuals of pairs of objects that are allocated to different clusters is extended with a loss component for pairs that are allocated to the same cluster it is shown that this extension makes the method equivalent to multidimensional scaling with cluster constraints on the coordinates a decomposition of the sum of squared dissimilarities into contributions from several sources of variation is described including the appropriate degrees of freedom for each source after developing a convergent algorithm for fitting the cluster differences model it is argued that the individual objects and the cluster locations can be jointly displayed in a configuration obtained as a by product of the optimization finally the paper introduces a fuzzy version of the loss function which can be used in a successive approximation strategy for avoiding local minima a simulation study demonstrates that this strategy significantly outperforms two other well known initialization strategies and that it has a success rate of 92 out of 100 in attaining the global minimum

symplectic energy momentum preserving variational integrators the purpose of this paper is to develop variational integrators for conservative mechanical systems that are symplectic and energy and momentum conserving to do this a space time view of variational integrators is employed and time step adaptation is used to impose the constraint of conservation of energy criteria for the solvability of the time steps and some numerical examples are given

integrators for lie poisson or dynamical systems abstract we reformulate the ge marsden 1988 lie poisson integration algorithm in terms of algebra variables alone and show how to implement it for regular quadratic lie algebras to arbitrary order and for arbitrary lie algebras to first order we also describe two generalizations of ruth s method 1983 that can be used to generate very fast algorithms

on certain linear mappings between inner product and squared distance matrices abstract we obtain the spectral decomposition of four linear mappings the first is a mapping of the linear hull of all centered inner product matrices onto the linear hull of all the induced squared distance matrices it is based on the natural generalization of the cosine law of elementary euclidean geometry the other three mappings studied are 1 the adjoint and 1 extensions and applications particularly to multidimensional scaling are discussed in some detail

partial relaxation approach an eigenvalue based doa estimator framework in this paper the partial relaxation approach is introduced and applied to the direction of arrival estimation problem using spectral search unlike existing spectral based methods such as conventional beamformer capon beamformer or music that can be considered as single source approximation of multi source estimation criteria the proposed approach accounts for the existence of multiple sources at each considered direction the manifold structure of the remaining interfering signals impinging on the sensor array is relaxed which results in closed form estimates for the interference parameters thanks to this relaxation the conventional multi source optimization problem reduces to a simple spectral search following this principle we propose estimators based on the deterministic maximum likelihood weighted subspace fitting and covariance fitting methods to calculate the null spectra efficiently an iterative rooting scheme based on the rational function approximation is applied to the partial relaxation methods simulation results show that irrespective of any specific structure of the sensor array the performance of the proposed estimators is superior to the conventional methods especially in the case of low signal to noise ratio and low number of snapshots while maintaining a computational cost that is comparable to music

energy preserving schemes for nonlinear hamiltonian systems of wave equations application to the vibrating piano string the problem of the vibration of a string is well known in its linear form describing the transversal motion of a string nevertheless this description does not explain all the observations well enough nonlinear coupling between longitudinal and transversal modes seams to better model the piano string as does for instance the geometrically exact model gem this report introduces a general class of nonlinear systems nonlinear hamiltonian systems of wave equations in which fits the gem mathematical study of these systems is lead in a first part showing central properties energy preservation existence and unicity of a global smooth solution finite propagation velocity ldots space discretization is made in a classical way variational formulation and time discretization aims at numerical stability using an energy technique a definition of preserving schemes is introduced and we show that explicit schemes or partially implicit schemes which are preserving according to this definition cannot be built unless the model is linear a general energy preserving second order accurate fully implicit scheme is built for any continuous system that fits the nonlinear hamiltonian systems of wave equations class

on the effect of input signal correlation on weight misadjustment in the rls algorithm new expressions are derived for the mean weight misadjustment in the recursive least squares rls algorithm for first order markov channel estimation the expressions derived are general in that they take into account the correlation in the input it is shown that the additive system noise is amplified by a correlation amplification factor that is defined as a function of the input autocorrelation matrix eigenvalues however input correlation has almost no effect on the misadjustment due to time varying system weights these results are checked by simulations demonstrating excellent agreement with the theory

connections between the real positive semidefinite and distance matrix completion problems abstract though the real symmetric positive semidefinite psd matrices and the euclidean distance matrices are closely related exact relationships between the corresponding completion problems are not apparent we establish strong partial relationships of two types this permits the transfer of some insights from one problem to the other and allows computational tools for the psd problem to be used for the distance problem

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

a variance inequality ensuring that a pre distance matrix is euclidean we derive a variance inequality on the squares of the pre distances between n objects ensuring that the corresponding pre distance matrix is euclidean the result is discussed in relation with some transformations making a pre distance matrix euclidean

a numerical integration technique for conservative oscillators combining nonstandard finite difference methods with a hamilton s principle nonstandard finite difference schemes are constructed for several important conservative oscillators these procedures combine the nonstandard methods of mickens j sound vib 240 2001 587 with a discrete hamilton s principle

what are schur complements anyway abstract given a matrix m a b c d the schur complements of a in m are the matrices of the form s d cab where a is a generalized inverse of a we survey several recent characterizations of schur complements and discuss where they arose and how they are related

fixed point error analysis of stochastic gradient adaptive lattice filters a theoretical analysis is presented of the stochastic gradient adaptive lattice filter used as a linear one step predictor when the effects of finite precision arithmetic are taken into account only the fixed point implementation is considered both the unnormalized and normalized adaptation algorithms are analyzed expressions of the steady state mean squared values of the accumulated numerical errors in the computation of the reflection coefficients and the prediction errors of different orders have been developed the results show that the dominant term in the expressions for the mean squared values of the numerical errors is inversely proportional to the convergence parameter furthermore they indicate that the quantization errors associated with the reflection coefficients are more critical than those associated with representing the prediction error sequences signals with high correlation among samples produce larger numerical errors in the adaptive lattice filter the authors present several simulation examples that show close agreement with the theoretical results they also show that the gradient adaptive lattice filters have better numerical properties than their transversal counterparts

definiteness and semidefiniteness of quadratic forms revisited abstract it is shown how the schur complement theory can be used for the derivation of criteria for the definiteness or semidefiniteness of the restriction of a quadratic form to the null space of a matrix

fixed point roundoff error analysis of the exponentially windowed rls algorithm for time varying systems a fixed point roundoff error analysis of the exponentially windowed rls algorithm is presented it is shown that a tradeoff exists in the choice of the forgetting factor in order to reduce the sensitivity of the algorithm to additive noise must be chosen close to one on the other hand the roundoff error increases as lambda rightarrow 1 it is shown that the algorithm is stabilized with lambda rightarrow 1 to derive the theoretical results it is assumed that the input signal is a white gaussian random process finally simulations are presented which confirm the theoretical findings of the paper

on the completely positive and positive semidefinite preserving cones abstract the cone cp n q of completely positive linear transformations from m n c m n to m q is shown to be isometrically isomorphic to p nq the cone of nq by nq positive semidefinite matrices generalizations of scalar and matrix results to cp n q hp n q l m n m q where hp n q represents the hermitian preserving linear transformations are discussed relationships among the completely positives the set of positive semidefinite preservers p n and its dual m n are given left and right facial ideals of cp are characterized properties of the joint angular field of values of a finite sequence of hermitian matrices h 1 h m are studied leading to a characterization of p q p n

dykstra s algorithm for constrained least squares rectangular matrix problems abstract in a recent paper the authors applied dykstra s alternating projection algorithm to solve constrained least squares n n matrix problems we extend these results in two different directions first we make use of the singular value decomposition to solve now constrained least squares rectangular m n matrix problems that arise in several applications second we propose a new and improved implementation of the projection algorithm onto the e positive definite set of matrices this implementation does not require the computation of all eigenvalues and eigenvectors of a matrix per iteration and still guarantees convergence finally encouraging preliminary numerical results are discussed

analysis of a stabilization technique for the fixed point prewindowed rls algorithm a stable finite precision recursive least squares rls algorithm is derived for the prewindowed growing memory case forgetting factor lambda 1 the prewindowed growing memory rls algorithm diverges under fixed point implementation the random walk phenomenon due to roundoff errors in the weight update causes the divergence of the algorithm to overcome this effect these roundoff errors are modeled such that their effect is incorporated into the algorithm the steady state behavior of this new algorithm is analyzed and it is shown that the divergence phenomenon is actually eliminated and the new algorithm converges

on the probability density function of the lms adaptive filter weights the joint probability density function of the weight vector in least mean square lms adaptation is studied for gaussian data models an exact expression is derived for the characteristic function of the weight vector at time n 1 conditioned on the weight vector at time n the conditional characteristic function is expanded in a taylor series and averaged over the unknown weight density to yield a first order partial differential difference equation in the unconditioned characteristic function of the weight vector the equation is approximately solved for small values of the adaption parameter and the weights are shown to be jointly gaussian with time varying mean vector and covariance matrix given as the solution to well known difference equations for the weight vector mean and covariance matrix the theoretical results are applied to analyzing the use of the weights in detection and time delay estimation simulations that support the theoretical results are also presented

an iteration method to solve multiple constrained least squares problems abstract in this paper we propose an iteration method to solve the multiple constrained least squares matrix problem we first transform the multiple constrained least squares matrix problems into the multiple constrained matrix optimal approximation problem and then we use the idea of dykstra s algorithm to derive the basic iterative pattern we observe that we only need to solve multiple single constrained least squares matrix problems at each iteration step of the proposed algorithm we give a numerical example to illustrate the effectiveness of the proposed method to solve the original problems also we give an example to illustrate that the method proposed by escalante and li to solve the single constrained least squares matrix problem is not correct

tracking properties and steady state performance of rls adaptive filter algorithms adaptive signal processing algorithms derived from ls least squares cost functions are known to converge extremely fast and have excellent capabilities to track an unknown parameter vector this paper treats analytically and experimentally the steady state operation of rls recursive least squares adaptive filters with exponential windows for stationary and nonstationary inputs a new formula for the estimation noise has been derived involving second and fourth order statistics of the filter input as well as the exponential windowing factor and filter length furthermore it is shown that the adaptation process associated with lag effects depends solely on the exponential weighting parameter in addition the calculation of the excess mean square error due to the lag for an assumed markov channel provides the necessary information about tradeoffs between speed of adaptation and steady state error it is also the basis for comparison to the simple lms algorithm in a simple case of channel identification it is shown that the lms and rls adaptive filters have the same tracking behavior finally in the last part we present new rls restart procedures applied to transversal structures for mitigating the disastrous results of the third source of noise namely finite precision arithmetic

theory of cones abstract this survey deals with the aspects of archimedian partially ordered finite dimensional real vector spaces and order preserving linear maps which do not involve spectral theory the first section sketches some of the background of entrywise nonnegative matrices and of systems of inequalities which motivate much of the current investigations the study of inequalities resulted in the definition of a polyhedral cone k and its face lattice f k in section ii a the face lattice of a not necessarily polyhedral cone k in a vector space v is investigated in particular the interplay between the lattice properties of f k and geometric properties of k is emphasized section ii b turns to the cones k in the space of linear maps on v recall that k is the cone of all order preserving linear maps of particular interest are the algebraic structure of k as a semiring and the nature of the group aut k of nonsingular elements a k for which a 1 k as well in a short final section the cone p n of n n positive semidefinite matrices is discussed a characterization of the set of completely positive linear maps is stated the proofs will appear in a forthcoming paper

unstable periodic orbits and discretization cycles limit cycles that arise from discretizing the variable s of a nonlinear map are generally found to shadow individual unstable periodic orbits upos of the corresponding continuous map in a few cases the discretization cycles can only be explained with other mechanisms such as the near occurrence of an upo or crossover between two or more upos

low complexity variable forgetting factor mechanism for blind adaptive constrained constant modulus algorithms in this paper we propose a novel low complexity variable forgetting factor vff mechanism for blind adaptive constrained constant modulus ccm recursive least squares rls algorithms applied to linear interference suppression in direct sequence code division multiple access ds cdma systems the proposed variable forgetting factor mechanism employs an updated component related to the time average of the constant modulus cm cost function to automatically adjust the forgetting factor in order to ensure good tracking of the interference and the channel convergence and tracking analyses are carried out analytical expressions for predicting the mean squared error of the proposed adaptation technique are obtained simulation results are presented for nonstationary environments and show that the proposed variable forgetting factor mechanism achieves superior performance to previously reported methods at a reduced complexity

distance matrices and regular figures abstract a regular figure which includes all regular polygons is a set of points on a hypersphere whose center coincides with their centroid we characterize all regular figures as those whose points generate a euclidean distance matrix edm with eigenvector e the vector of all ones restricting the classical maps of schoenberg gower and critchley for all edms to the subcone of edms with eigenvector e yields new geometrical information about the generating points and a simple formula for the radius of the hypersphere

scaling radial basis functions via euclidean distance matrices a radial basis function approximation is typically a linear combination of shifts of a radially symmetric function possibly augmented by a polynomial of suitable degree that is it takes the forms x k 1nc k f x x k p x x din the mid 1980s micchelli building on pioneering work of schoenberg in the 1930s and 1940s provided simple sufficient conditions on that imply radial basis functions can interpolate scattered data however when the data density varies locally several authors such as hon and kansa 1 have suggested scaling the translates in other words it can be advantageous to replace the euclidean norm by some more general distance functional d that iss x k 1nc k f d x x k p x x dthis distance functional a need not be a metric but we shall require that d be symmetric and satisfy d g g 0 for all g d unfortunately the micchelli schoenberg theory does not obviously apply in this more general setting but some papers have observed that interpolation is well defined if the distance functional is a sufficiently small perturbation of the euclidean norm however in this study we follow a different approach which returns to the roots of schoenberg s work specifically we use schoenberg s classification of euclidean distance matrices to provide a simple technique which given a suggested distance functional d calculates a perturbed distance functional d for which the underlying interpolation matrix is invertible when the function q is strictly positive definite i e a mercer kernel or strictly conditionally positive or negative definite of order one as a simple by product of this method we can also apply the narcowich ward 2 norm estimate results easily since the minimum distance between points is now under our control via d

simultaneous classification and multidimensional scaling with external information for the exploratory analysis of a matrix of proximities or dis similarities between objects one often uses cluster analysis ca or multidimensional scaling mds solutions resulting from such analyses are sometimes interpreted using external information on the objects usually the procedures of ca mds and using external information are carried out independently and sequentially although combinations of two of the three procedures ca and mds or multidimensional scaling and using external information have been proposed in the literature the present paper offers a procedure that combines all three procedures in one analysis using a model that describes a partition of objects with cluster centroids represented in a low dimensional space which in turn is related to the information in the external variables a simulation study is carried out to demonstrate that the method works satisfactorily for data with a known underlying structure also to illustrate the method it is applied to two empirical data sets

nonlinear eigenvector algorithms for local optimization in multivariate data analysis abstract many multivariate statistical procedures such as principal component canonical correlation correspondence and discriminant analysis are based on the solution of a certain matrix approximation problem one of the most important being the reduced rank approximation its solution hinges on the singular value or the eigendecomposition of a certain matrix therefore numerical techniques to determine eigenvalues and eigenvectors play an important role in these statistical applications furthermore various modifications generalizations or refinements of the classical methods lead to some sort of nonlinear eigenproblem with no immediate unique solution however the nonlinear eigenvector approach has proved to be quite effective in solving these types of constrained optimization problems in diverse fields of multivariate data analysis since a general analysis of the properties and performance of the nonlinear eigenvector algorithm seems to be long overdue this paper collects the common features in the applications mentioned above and gives an overview from a superior perspective it unifies the treatment by contrasting the nonlinear eigenvector algorithm with the well known inverse iteration iteratively reweighted least squares and majorization algorithms extracts the mathematical tools basically from convex analysis and matrix algebra needed for the convergence proofs and discusses convergence properties

geometry and hamiltonian mechanics on discrete spaces numerical simulation is often crucial for analysing the behaviour of many complex systems which do not admit analytic solutions to this end one either converts a smooth model into a discrete in space and time model or models systems directly at a discrete level the goal of this paper is to provide a discrete analogue of differential geometry and to define on these discrete models a formal discrete hamiltonian structure in doing so we try to bring together various fundamental concepts from numerical analysis differential geometry algebraic geometry simplicial homology and classical hamiltonian mechanics for example the concept of a twisted derivation is borrowed from algebraic geometry for developing a discrete calculus the theory is applied to a nonlinear pendulum and we compare the dynamics obtained through a discrete modelling approach with the dynamics obtained via the usual discretization procedures also an example of an energy conserving algorithm on a simple harmonic oscillator is presented and its effect on the poisson structure is discussed

a roundoff error analysis of the lms adaptive algorithm the steady state output error of the least mean square lms adaptive algorithm due to the finite precision arithmetic of a digital processor is analyzed it is found to consist of three terms 1 the error due to the input data quantization 2 the error due to the rounding of the arithmetic operations in calculating the filter s output and 3 the error due to the deviation of the filter s coefficients from the values they take when infinite precision arithmetic is used the last term is of paricular interest because its mean squared value is inversely proportional to the adaptation step size both fixed and floating point arithmetics are examined and the expressions for the final mean square error are found to be similar the relation between the quantization error and the error that occurs when adaptation possibly ceases due to quantization is also investigated

distance geometry for kissing spheres abstract a kissing sphere is a sphere that is tangent to a fixed reference ball we develop in this paper a distance geometry for kissing spheres which turns out to be a generalization of the classical euclidean distance geometry

incursive discretization system bifurcation and energy conservation incursive discretization of the classical harmonic oscillator leads to system bifurcation the resulting hyperincursive representation has two alternative distinct algorithms of ordered serial noncommuting instructions and admits solutions having a discretized classical total energy that is perfectly conserved and phase space trajectories that are fully stable at all time scales hyperincursive representations can be generated for any hamiltonian system

fixed point error analysis of the normalized ladder algorithm an attempt is made to analyze the fixed point error performance of the normalized ladder algorithm for autoregressive system identification assuming rounding arithmetic a preliminary simulation study of this algorithm has shown that the bias in the estimated reflection coefficients is much more predominant than the variance of the error in the estimate the study therefore is directed to find a model for predicting the bias in the estimated reflection coefficients the analysis shows that the roundoff errors associated with the square root operations in one of the algorithm equations are mainly responsible for the bias in the estimated reflection coefficients these errors arise because of the normalization procedure that makes the quantities under the square root operations very close to one two main results are presented in the paper 1 a simplified theoretical expression for predicting the average bias in the estimated reflection coefficients at any stage is derived 2 a recursive relation for the average error arising from the finite precision arithmetic in the squared residuals is derived this relation illustrates how the errors made in one stage affect the errors in the succeeding stages simulations are performed to check the theoretical models the experimental results agree very closely with the theoretical predictions

an examination of procedures for determining the number of clusters in a data set a monte carlo evaluation of 30 procedures for determining the number of clusters was conducted on artificial data sets which contained either 2 3 4 or 5 distinct nonoverlapping clusters to provide a variety of clustering solutions the data sets were analyzed by four hierarchical clustering methods external criterion measures indicated excellent recovery of the true cluster structure by the methods at the correct hierarchy level thus the clustering present in the data was quite strong the simulation results for the stopping rules revealed a wide range in their ability to determine the correct number of clusters in the data several procedures worked fairly well whereas others performed rather poorly thus the latter group of rules would appear to have little validity particularly for data sets containing distinct clusters applied researchers are urged to select one or more of the better criteria however users are cautioned that the performance of some of the criteria may be data dependent

computing the nearest euclidean distance matrix with low embedding dimensions euclidean distance embedding appears in many high profile applications including wireless sensor network localization where not all pairwise distances among sensors are known or accurate the classical multi dimensional scaling cmds generally works well when the partial or contaminated euclidean distance matrix edm is close to the true edm but otherwise performs poorly a natural step preceding cmds would be to calculate the nearest edm to the known matrix a crucial condition on the desired nearest edm is for it to have a low embedding dimension and this makes the problem nonconvex there exists a large body of publications that deal with this problem some try to solve the problem directly and some are the type of convex relaxations of it in this paper we propose a numerical method that aims to solve this problem directly our method is strongly motivated by the majorized penalty method of gao and sun for low rank positive semi definite matrix optimization problems the basic geometric object in our study is the set of edms having a low embedding dimension we establish a zero duality gap result between the problem and its lagrangian dual problem which also motivates the majorization approach adopted numerical results show that the method works well for the euclidean embedding of network coordinate systems and for a class of problems in large scale sensor network localization and molecular conformation

distance shrinkage and euclidean embedding via regularized kernel estimation summary although recovering a euclidean distance matrix from noisy observations is a common problem in practice how well this could be done remains largely unknown to fill in this void we study a simple distance matrix estimate based on the so called regularized kernel estimate we show that such an estimate can be characterized as simply applying a constant amount of shrinkage to all observed pairwise distances this fact allows us to establish risk bounds for the estimate implying that the true distances can be estimated consistently in an average sense as the number of objects increases in addition such a characterization suggests an efficient algorithm to compute the distance matrix estimator as an alternative to the usual second order cone programming which is known not to scale well for large problems numerical experiments and an application in visualizing the diversity of vpu protein sequences from a recent study of human immunodeficiency virus type 1 further demonstrate the practical merits of the method proposed

analysis of the normalized lms algorithm with gaussian inputs the lms adaptive filter algorithm requires a priori knowledge of the input power level to select the algorithm gain parameter for stability and convergence since the input power level is usually one of the statistical unknowns it is normally estimated from the data prior to beginning the adaptation process it is then assumed that the estimate is perfect in any subsequent analysis of the lms algorithm behavior in this paper the effects of the power level estimate are incorporated in a data dependent that appears explicitly within the algorithm the transient mean and second moment behavior of the modified lms nlms algorithm are evaluated taking into account the explicit statistical dependence of upon the input data the mean behavior of the algorithm is shown to converge to the wiener weight a constant coefficient matrix difference equation is derived for the weight fluctuations about the wiener weight the equation is solved for a white data covariance matrix and for the adaptive line enhancer with a single frequency input in steady state for small expressions for the misadjustment error are also presented it is shown for the white data covariance matrix case that the averaging of about ten data samples causes negligible degradation as compared to the lms algorithm in the ale application the steady state weight fluctuations are shown to be mode dependent being largest at the frequency of the input

symplectic numerical integrators in constrained hamiltonian systems recent work reported in the literature suggests that for the long time integration of hamiltonian dynamical systems one should use methods that preserve the symplectic or canonical structure of the flow here we investigate the symplecticness of numerical integrators for constrained dynamics such as occur in molecular dynamics when bond lengths are made rigid in order to overcome stepsize limitations due to the highest frequencies this leads to a constrained hamiltonian system of smaller dimension previous work has shown that it is possible to have methods which are symplectic on the constraint manifold in phase space here it is shown that the very popular verlet method with shake type constraints is equivalent to the same method with rattle type constraints and that the latter is symplectic and time reversible this assumes that the iteration is carried to convergence we also demonstrate the global convergence of the verlet scheme in the presence of shake type and rattle type constraints we perform numerical experiments to compare these methods with the second order backward differentiation method commonly recommended for ordinary differential equations with constraints

fast recursive least squares transversal filters for adaptive filtering fast transversal filter ftf implementations of recursive least squares rls adaptive filtering algorithms are presented in this paper substantial improvements in transient behavior in comparison to stochastic gradient or lms adaptive algorithms are efficiently achieved by the presented algorithms the true not approximate solution of the rls problem is always obtained by the ftf algorithms even during the critical initialization period first n iterations of the adaptive filter this true solution is recursively calculated at a relatively modest increase in computational requirements in comparison to stochastic gradient algorithms factor of 1 6 to 3 5 depending upon application additionally the fast transversal filter algorithms are shown to offer substantial reductions in computational requirements relative to existing fast rls algorithms such as the fast kalman algorithms of morf ljung and falconer 1976 and the fast ladder lattice algorithms of morf and lee 1977 1981 they are further shown to attain steady state unnormalized or improve upon first n initialization steps the very low computational requirements of the efficient rls solutions of carayannis manolakis and kalouptsidis 1983 finally several efficient procedures are presented by which to ensure the numerical stability of the transversal filter algorithms including the incorporation of soft constraints into the performance criteria internal bounding and rescuing procedures and dynamic range increasing square root normalized variations of the transversal filters

a lagrangian dual approach to the single source localization problem the single source localization problem sslp which is nonconvex by its nature appears in several important multidisciplinary fields such as signal processing and the global positioning system in this paper we cast sslp as a euclidean distance embedding problem and study a lagrangian dual approach it is proved that the lagrangian dual problem must have an optimal solution under the generalized slater condition we provide a sufficient condition for the zero duality gap and establish the equivalence between the lagrangian dual approach and the existing generalized trust region subproblem gtrs approach studied by beck et al exact and approximate solutions of source localization problems ieee trans signal process vol 56 pp 1770 1778 2008 we also reveal new implications of the assumptions made by the gtrs approach moreover the lagrangian dual approach has a straightforward extension to the multiple source localization problem numerical simulations demonstrate that the lagrangian dual approach can produce localization of similar quality as the gtrs and can significantly outperform the well known semidefinite programming solver for the multiple source localization problem on the tested cases

the best euclidian fit to a given distance matrix in prescribed dimensions abstract in this paper we determine the euclidean distance matrix which is closest to a given not necessarily euclidian one we admit a wide class of appropriate matrix norms as measures of closeness majorization results on sums of eigenvalues of symmetric matrices serve as basic tools

properties of euclidean and non euclidean distance matrices abstract a distance matrix d of order n is symmetric with elements 1 2 d ij 2 where dii 0 d is euclidean when the 1 2 n n 1 quantities dij can be generated as the distances between a set of n points x n p in a euclidean space of dimension p the dimensionality of d is defined as the least value of p rank x of any generating x in general p 1 and p 2 are also acceptable but may include imaginary coordinates even when d is euclidean basic properties of euclidean distance matrices are established in particular when rank d it is shown that depending on whether etd e is not or is zero the generating points lie in either p 1 dimensions in which case they lie on a hypersphere or in p 2 dimensions in which case they do not the notation e is used for a vector all of whose values are one when d is non euclidean its dimensionality p r s will comprise r real and s imaginary columns of x and r s are invariant for all generating x of minimal rank higher ranking representations can arise only from p 1 r 1 s or p 1 r s 1 or p 2 r 1 s 1 so that not only are r s invariant but they are both minimal for all admissible representations x

eigenstructure of distance matrices with an equal distance subset abstract we characterize the distance matrices with an equal distance subset in terms of eigenstructure and determine edms in this class by examination of a lower dimensional matrix

gindclus generalized indclus with external information a generalized indclus model termed gindclus is presented for clustering three way two mode proximity data in order to account for the heterogeneity of the data both a partition of the subjects into homogeneous classes and a covering of the objects into groups are simultaneously determined furthermore the availability of information which is external to the three way data is exploited to better account for such heterogeneity the weights of both classifications are linearly linked to external variables allowing for the identification of meaningful classes of subjects and groups of objects the model is fitted in a least squares framework and an efficient alternating least squares algorithm is provided an extensive simulation study and an application on benchmark data are also presented

adaptive recovery of a chirped sinusoid in noise ii performance of the lms algorithm the authors study the ability of the exponentially weighted recursive least square rls algorithm to track a complex chirped exponential signal buried in additive white gaussian noise power p sub n the signal is a sinusoid whose frequency is drifting at a constant rate psi lt is recovered using an m tap adaptive predictor five principal aspects of the study are presented the methodology of the analysis proof of the quasi deterministic nature of the data covariance estimate r k a new analysis of rls for an inverse system modeling problem a new analysis of rls for a deterministic time varying model for the optimum filter and an evaluation of the residual output mean square error mse resulting from the nonoptimality of the adaptive predictor the misadjustment in terms of the forgetting rate beta of the rls algorithm it is shown that the misadjustment is dominated by a lag term of order beta sup 2 and a noise term of order beta thus a value beta sub opt exists which yields a minimum misadjustment it is proved that beta sub opt m 1 rho psi sup 2 sup 1 3 and the minimum misadjustment is equal to 3 4 p sub n m 1 beta sub opt where rho is the input signal to noise ratio snr

approximation by matrices positive semidefinite on a subspace abstract we obtain the best approximation to a matrix by matrices positive semidefinite on a subspace as a by product we present two new characterizations of euclidean distance matrices

extending multivariate distance matrix regression with an effect size measure and the asymptotic null distribution of the test statistic person centered methods are useful for studying individual differences in terms of dis similarities between response profiles on multivariate outcomes multivariate distance matrix regression mdmr tests the significance of associations of response profile dis similarities and a set of predictors using permutation tests this paper extends mdmr by deriving and empirically validating the asymptotic null distribution of its test statistic and by proposing an effect size for individual outcome variables which is shown to recover true associations these extensions alleviate the computational burden of permutation tests currently used in mdmr and render more informative results thus making mdmr accessible to new research domains

finite metric spaces of strictly negative type abstract we prove that if a finite metric space is of strictly negative type then its transfinite diameter is uniquely realized by the infinite extender load vector finite metric spaces that have this property include all spaces on two three or four points all trees and all finite subspaces of euclidean spaces we prove that if the distance matrix is both hypermetric and regular then it is of strictly negative type we show that the strictly negative type finite subspaces of spheres are precisely those which do not contain two pairs of antipodal points in connection with an open problem raised by kelly we conjecture that all finite subspaces of hyperbolic spaces are hypermetric and regular and hence of strictly negative type

momentum conserving symplectic integrators abstract in this paper we show that symplectic partitioned runge kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of hamiltonian differential equations by extended point transformation we also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of lie poisson systems on certain submanifolds of the general matrix group gl n

mechanical integrators derived from a discrete variational principle many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion in this paper we present a procedure to construct time stepping algorithms that approximate the flow of continuous odes for mechanical systems by discretizing hamilton s principle rather than the equations of motion the discrete equations share similarities to the continuous equations by preserving invariants including the symplectic form and the momentum map we first present a formulation of discrete mechanics along with a discrete variational principle we then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of noether s theorem we then use the discrete mechanics formulation to develop a procedure for constructing symplectic momentum mechanical integrators for lagrangian systems with holonomic constraints we apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators

the cone of distance matrices abstract the geometry of the cone of euclidean distance matrices edms is analyzed using a new characterization of an edm the facial structure and the angle that edms of embedding dimension one make with the center ray are found this result follows from a complete analysis of the critical points of the distance function in frobenius norm from the matrix e consisting of zero diagonal and ones elsewhere to the edms of embedding dimension one