on a new class of completely integrable nonlinear wave equations i infinitely many conservation laws we point out a class of nonlinear wave equations which admit infinitely many conserved quantities these equations are characterized by a pair of exact one forms the implication that they are closed gives rise to equations the characteristics and riemann invariants of which are readily obtained the construction of the conservation laws requires the solution of a linear second order equation which can be reduced to canonical form using the riemann invariants the hodograph transformation results in a similar linear equation we discuss also the symplectic structure and backlund transformations associated with these equations

structural modeling and psychometrika an historical perspective on growth and achievements the field of linear structural equation modeling with continuous variables is reviewed trends in psychometric theory and data analysis across the five decades of publication ofpsychometrika are discussed especially the clarification of concepts of population and sample explication of the parametric structure of models delineation of concepts of exploratory and confirmatory data analysis expansion of statistical theory in psychometrics estimation via optimization of an explicit objective function and implementation of general function minimization methods developments in the ideas of factor analysis latent variables as well as structural and causal modeling are noted some major conceptual achievements involving general covariance structure representations multiple population models and moment structures are reviewed the major statistical achievements of normal theory generalized least squares estimation elliptical and distribution free estimation and higher moment estimation are discussed computer programs that implement some of the theoretical developments are described

the foundations of factor analysis summary a new approach to factor analysis and related latent variable methods is proposed which is based on data reduction using the idea of bayesian sufficiency considerations of symmetry invariance and independence are used to determine an appropriate family of models the results are expressed in terms of linear functions of the manifest variables after the manner of principal components analysis the approach justifies some of the practices based on the normal theory factor model and lays a foundation for the treatment of nonnormal including categorical variables 1 background factor analysis is a widely used statistical technique but its theoretical foundations are somewhat obscure and subject to dispute it is one of a family of multivariate methods which also includes latent structure and latent trait analysis the common feature of the models underlying these methods is that the observed random variables are assumed to depend on latent that is unobserved random variables there is sometimes debate about whether these latent variables are real in any sense but they can be viewed simply as constructs designed to simplify and summarize the complex web of interrelated variables with which nature confronts us by expressing these relationships in terms of a small number of latent variables or factors the models effect a reduction in dimensionality which aids comprehension an early theoretical account of the subject is by anderson rubin 1956 and more recent and comprehensive treatments are provided by lawley maxwell 1971 and harman 1968 the present paper is an attempt to provide a coherent framework within which existing methodology can be evaluated and a base from which new methods can be developed our approach is to start from a very general statement of the problem in terms of the distributions of the random variables involved and then to invoke ideas of symmetry invariance and conditional independence to determine the class of models that it is reasonable to consider this not only exhibits the unity of the various models already in existence but resolves many of the ambiguities and obscurities with which the subject has been bedevilled an early approach on these lines is given by anderson 1959 and was recognized by birnbaum in his contribution to lord novick 1968 the same line was followed by bartholomew 1980 in spite of this the practical implications do not seem to have been made fully explicit or generally recognized

discrete lagrange s equations and canonical equations based on the principle of least action by assuming that time is discrete discrete lagrange s equations and discrete canonical equations are derived on the basis of the principle of least action the discrete mechanics contains no differentiation and integration but contains difference quotients and summations the time interval is not required to be infinitesimal the discrete canonical equations conserve the hamiltonian exactly

high order schemes for conservative or dissipative systems we propose a new method for designing high order finite difference schemes that inherit conservation or dissipation properties from conservative or dissipative systems such as hamiltonian systems with without damping terms the proposed method has a feature that the computational costs of the resulting schemes do not increase in practice even when the order of accuracy is increased

parametrization of factor analysis models abstract for general factor analysis or equivalently for error in variables models the so called identifiability problem is apparently still unsolved although raised and discussed by k pearson already in 1901 in this paper we discuss the identifiability or what we rather prefer to name model selection problem for factor analysis models representing two blocks of variables we show that there is a continuum of minimal models which connect together two extreme representations of the pure regression type this continuum of models can be parametrized in terms of a projection matrix describing the part of the modelled vector which is represented exactly i e with no random modelling error by the factor analysis model any procedure of model selection is just a procedure for choosing the exact part of the modelled variables any choice results in certain modelling error for the remaining variables whose variance can be computed explicitly the analysis is done in the static case but carries over almost without changes to dynamic factor analysis models see picci and pinzoni 1986 by use of spectral representation

a symplectic integration algorithm for separable hamiltonian functions abstract we derive an algorithm to numerically integrate differential equations derivable from a separable hamiltonian function this symplectic algorithm is accurate to fourth order in the time step and preserves exactly the poincare cartan integral invariants associated with the topology of the phase flow we compare the efficiency and accuracy of this method to that of existing integrators both symplectic and non symplectic by integrating the equations of motion corresponding to a nonlinear pendulum a particle in the field of a standing wave and a harmonic oscillator perturbed by a plane wave

bivariate survival models induced by frailties abstract when two observed survival times depend via a proportional hazards model on the same unobserved variable called a frailty this common dependence induces an association between the observed times this article considers the class of bivariate survival distributions that can arise in this way extends it to allow negative association and shows that the observable bivariate distribution determines the unobserved frailty distribution up to a scale parameter a cross ratio function easily estimated even from censored data is the key to both the characterization results and inferential procedures and diagnostic plots that are introduced and illustrated by real examples the models considered are the natural counterpart for survival data of the latent variable models that have long been used in factor analysis for continuous data or in latent structure analysis for binary data the main distinguishing feature of survival data is the possibility of censoring which makes the standard methods difficu

two classes of linearly implicit local energy preserving approach for general multi symplectic hamiltonian pdes abstract two classes of efficient and robust schemes are proposed for the general multi symplectic hamiltonian systems using the invariant energy quadratization ieq approach the schemes are linear second order accurate local energy preserving and preserve the global energy they are not restricted to specific forms of the nonlinear part of the state function and only require solving linear equations at each time step we applied the new schemes to various multi symplectic hamiltonian pdes to demonstrate their effectiveness computational efficiency and accuracy

weakly split spherical tits systems in quasi reductive groups we will prove that any weakly split spherical tits system b n in g bf g k bf g a quasi reductive k group such as a connected reductive k group satisfying some natural conditions is standard in particular if bf g is anisotropic over k then such a tits system is trivial i e b g n

an extended discrete gradient formula for oscillatory hamiltonian systems in this paper incorporating the idea of the discrete gradient method into the extended runge kutta nystrom integrator we derive and analyze an extended discrete gradient formula for the oscillatory hamiltonian system with the hamiltonian where represents generalized positions represents generalized momenta and is a symmetric and positive semi definite matrix the solution of this system is a nonlinear oscillator basically many nonlinear oscillatory mechanical systems with a partitioned hamiltonian function lend themselves to this approach the extended discrete gradient formula presented in this paper exactly preserves the energy h p q we derive some properties of the new formula the convergence is analyzed for the implicit schemes based on the discrete gradient formula and it turns out that the convergence of the implicit schemes based on the extended discrete gradient formula is independent of m which is a significant property for the oscillatory hamiltonian system thus it transpires that a larger step size can be chosen for the new energy preserving schemes than that for the traditional discrete gradient methods when applied to the oscillatory hamiltonian system illustrative examples show the competence and efficiency of the new schemes in comparison with the traditional discrete gradient methods in the scientific literature

mechanical systems subject to holonomic constraints differential algebraic formulations and conservative integration abstract the numerical integration in time of the equations of motion for mechanical systems subject to holonomic constraints is considered schemes are introduced for the direct treatment of a differential algebraic form of the equations that preserve the constraints the total energy and other integrals such as linear and angular momentum arising from affine symmetries moreover the schemes can be shown to preserve the property of time reversibility in an appropriate sense an example is given to illustrate various aspects of the proposed methods

new energy preserving algorithms for nonlinear hamiltonian wave equation equipped with neumann boundary conditions abstract in this paper using the blend of spatial discretization by second order or fourth order finite difference methods fdm and time integration by the generalized average vector field gavf method or the generalized adapted average vector field gaavf method we propose and analyze novel energy preserving algorithms for solving the nonlinear hamiltonian wave equation equipped with homogeneous neumann boundary conditions firstly two kinds of finite difference methods are considered to discretize the spatial derivative which can be of order two and order four respectively in all the spatial grid points the conservation laws of the discrete energy are established after the semi discretization a hamiltonian system of odes is derived whose hamiltonian can be regarded as the approximate energy of the original continuous system then the gavf formula and the gaavf formula are developed and applied to the derived hamiltonian odes to yield some novel and efficient algorithms which can exactly preserve the discrete energy the numerical simulation is implemented and the numerical results demonstrate the spatial and temporal accuracy and the remarkable energy preserving property of the new algorithms presented in this paper

hamiltonian systems discrete time approximation losslessness passivity and composability abstract in this paper a passive integrator dedicated to input output hamiltonian systems approximation is presented in a first step a discrete hamiltonian framework endowed with a lie derivative like formula is introduced it is shown that the discrete dynamics encodes energy conservation and passivity additionally the characterization of the discrete dynamics in terms of dirac structure is shown to be invariant by interconnection the class is thus composable networked systems belong to the class in a second step the discrete dynamics is considered as a one step integration method the method is shown to be convergent and provides a discrete time approximation of an input output hamiltonian system accordingly the discrete dynamics inherits intrinsic energetic characteristics storage function and dissipation rate from the original system the method is thus tagged as passive integrator as an illustration the closed loop behavior of interconnected subsystems and the stabilization of a rigid body spinning around its center of mass are presented

explicit volume preserving and symplectic integrators for trigonometric polynomial flows we introduce explicit volume preserving and symplectic integrators for the case of generalized trigonometric polynomial flows the method is demonstrated using the arter flow and computational trials are conducted using a 4 dimensional vector field

integral preserving integrators ordinary differential equations having a first integral may be solved numerically using one of several methods with the integral preserved to machine accuracy one such method is the discrete gradient method it is shown here that the order of the method can be bootstrapped repeatedly to higher orders of accuracy the method is illustrated using the henon heiles system

describing the elephant structure and function in multivariate data there is a unity underlying the diversity of models for the analysis of multivariate data essentially they constitute a family models most generally nonlinear for structural functional relations between variables drawn from a behavior domain

hamiltonian conserving discrete canonical equations based on variational difference quotients abstract new discrete mechanics based on the assumption of the discrete time is proposed the discrete mechanics does not contain any continuous differentiation but contains only difference quotients resulting discrete hamiltonian s canonical equations are single time step difference equations and exactly conserve the hamiltonian the canonical equations give the numerical results more accurately than the heun scheme and the 4th order runge kutta scheme

ordinal consistency and ordinal true scores this paper argues that test data are ordinal that latent trait scores are only determined ordinally and that test data are used largely for ordinal purposes therefore it is desirable to develop a test theory based only on ordinal assumptions a set of ordinal assumptions is presented including an ordinal version of local independence from these assumptions it is first shown that the gamma correlation between two tests is the product of their gamma correlations with the true latent order the theory is generalized to allow for heterogeneous tests by defining a weighted average local independence the tau correlations between total score and the latent order can be found in both homogeneous and heterogeneous cases and a system of differential item weighting to maximize the tau correlation between weighted items and the latent order is provided thus a purely ordinal test theory seems possible

efficient solution of optimal control problems using hybrid systems we consider the synthesis of optimal controls for continuous feedback systems by recasting the problem to a hybrid optimal control problem synthesize optimal enabling conditions for switching between locations in which the control is constant an algorithmic solution is obtained by translating the hybrid automaton to a finite automaton using a bisimulation and formulating a dynamic programming problem with extra conditions to ensure non zenoness of trajectories we show that the discrete value function converges to the viscosity solution of the hamilton jacobi bellman equation as a discretization parameter tends to zero

modeling complex systems by generalized factor analysis we propose a new modeling paradigm for large dimensional aggregates of stochastic systems by generalized factor analysis gfa models these models describe the data as the sum of a flocking plus an uncorrelated idiosyncratic component the flocking component describes a sort of collective orderly motion which admits a much simpler mathematical description than the whole ensemble while the idiosyncratic component describes weakly correlated noise we first discuss static gfa representations and characterize in a rigorous way the properties of the two components the extraction of the dynamic flocking component is discussed for time stationary linear systems and for a simple classes of separable random fields

numerical solution of isospectral flows in this paper we are concerned with the problem of solving numerically isospectral flows these flows are characterized by the differential equation l b l l l 0 l 0 where l 0 is a d x d symmetric matrix b l is a skew symmetric matrix function of l and b l is the lie bracket operator we show that standard runge kutta schemes fail in recovering the main qualitative feature of these flows that is isospectrality since they cannot recover arbitrary cubic conservation laws this failure motivates us to introduce an alternative approach and establish a framework for generation of isospectral methods of arbitrarily high order

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

moufang polygons and irreducible spherical bn pairs of rank 2 i let g be a group with an irreducible spherical bn pair of rank 2 satisfying the additional condition there exists a normal nilpotent subgroup u of b with b tu where t b n and jw ja16 forthe weyl gr oup w n b n we show that g corresponds to a moufang polygon and hence is essentially known r 2003 elsevier science usa all rights reserved

energy preserving numerical schemes of high accuracy for one dimensional hamiltonian systems we present a class of non standard numerical schemes which are modifications of the discrete gradient method they preserve the energy integral exactly up to the round off error the considered class contains locally exact discrete gradient schemes and integrators of arbitrary high order in numerical experiments we compare our integrators with some other numerical schemes including the standard discrete gradient method the leap frog scheme and a symplectic scheme of fourth order we study the error accumulation for a very long time and the conservation of the energy integral

on the equivalence between sor type methods for linear systems and the discrete gradient methods for gradient systems abstract inspired by the iterative nature of many discretization methods for continuous dynamical systems connections between iterative numerical methods in numerical linear algebra and continuous dynamical systems have been studied since 1970s for stationary iterative methods solving linear systems chu 1988 2008 discussed a connection to continuous dynamical systems by using the explicit euler method however further understanding of stationary iterative methods might be limited due to the use of the explicit euler method this paper presents a new connection based on the so called discrete gradient methods between sor type methods and gradient systems there the key of the discussion is the equivalence between sor type methods and the discrete gradient methods applied to gradient systems the discussion leads to new interpretations for sor type methods for example a new derivation of sor type methods is found these methods monotonically decrease a certain quadratic function and a new interpretation of the relaxation parameter is obtained besides while studying the new connection a new discrete gradient is also obtained

latent variable models for ordered categorical data abstract the multivariate analysis of ordered categorical data is an important growth point in statistics especially as applied to the social sciences this paper is concerned with the situation in which the associations between a set of categorical variables are assumed to arise from their dependence on a small number of latent variables the problem of modelling this relationship is first posed in very general terms and then specialized to particular cases for which statistical methods are available or in course of development such models can be viewed as tools for data reduction or for scaling and both aspects are reviewed the possible relevance of the method in economics and their relationship to other methods for categorical variables is briefly discussed

estimation of generalized linear latent variable models generalized linear latent variable models gllvms as defined by bartholomew and knott enable modelling of relationships between manifest and latent variables they extend structural equation modelling techniques which are powerful tools in the social sciences however because of the complexity of the log likelihood function of a gllvm an approximation such as numerical integration must be used for inference this can limit drastically the number of variables in the model and can lead to biased estimators we propose a new estimator for the parameters of a gllvm based on a laplace approximation to the likelihood function and which can be computed even for models with a large number of variables the new estimator can be viewed as an m estimator leading to readily available asymptotic properties and correct inference a simulation study shows its excellent finite sample properties in particular when compared with a well established approach such as lisrel a real data example on the measurement of wealth for the computation of multidimensional inequality is analysed to highlight the importance of the methodology

linearization of universal field equations the universal field equations recently constructed as examples of higher dimensional dynamical systems which admit an infinity of inequivalent lagrangians are shown to be linearised by a legendre transformation this establishes the conjecture that these equations describe integrable systems while this construction is implicit in general there exists a large class of solutions for which an explicit form may be written comment 11pp dtp 92 47 ni 92 011

conditional association and unidimensionality in monotone latent variable models latent variable models represent the joint distribution of observable variables in terms of a simple structure involving unobserved or latent variables usually assuming the conditional independence of the observable variables given the latent variables these models play an important role in educational measurement and psychometrics in sociology and in population genetics and are implicit in some work on systems reliability we study a broad class of latent variable models namely the monotone unidimensional models in which the latent variable is a scalar the observable variables are conditionally independent given the latent variable and the conditional distribution of the observables given the latent variable is stochastically increasing in the latent variable all models in this class imply a new strong form of positive dependence among the observable variables namely conditional positive association this positive dependence condition may be used to test whether any model in this class can provide an adequate fit to observed data various applications generalizations and a numerical example are discussed

robustness of statistical inference in factor analysis and related models summary a class of latent variable models which includes the unrestricted factor analysis model is considered it is shown that minimum discrepancy test statistics and estimators derived under normality assumptions retain their asymptotic properties when the common factors are not normally distributed but the unique factors do have a multivariate normal distribution the minimum discrepancy test statistics and estimators considered include the usual likelihood ratio test statistic and maximum likelihood estimators

collocation and relaxed collocation for the fer and the magnus expansions we consider the fer and the magnus expansions for the numerical solution of the nonlinear matrix lie group ode y gamma t y y y 0 y 0 where y evolves in a matrix lie group g and gamma t y is in the lie algebra hbox fr g ghe departing from a geometrical approach that distinguishes between those operations performed in the group and those performed in the tangent space we construct lie group invariant methods based on collocation we prove that as long as the two expansions are correctly truncated the collocation nodes c 1 c 2 ldots c nu yield numerical methods whose order is the same as in the classical setting we also relax the collocation conditions thereby devising explicit methods of order 3 to conclude we discuss the proposed methods in a numerical experiment that arises in hamiltonian mechanics comparing the results with projection techniques

locally exact modifications of numerical schemes we present a new class of exponential integrators for ordinary differential equations locally exact modifications of known numerical schemes local exactness means that they preserve the linearization of the original system at every point in particular locally exact integrators preserve all fixed points and are a stable we apply this approach to popular schemes including euler schemes the implicit midpoint rule and the trapezoidal rule we found locally exact modifications of discrete gradient schemes for symmetric discrete gradients and coordinate increment discrete gradients preserving their main geometric property exact conservation of the energy integral for arbitrary multidimensional hamiltonian systems in canonical coordinates numerical experiments for a two dimensional anharmonic oscillator show that locally exact schemes have very good accuracy in the neighbourhood of stable equilibrium much higher than suggested by the order of new schemes locally exact modification sometimes increases the order but in many cases leaves it unchanged

preserving multiple first integrals by discrete gradients we consider systems of ordinary differential equations with known first integrals the notion of a discrete tangent space is introduced as the orthogonal complement of an arbitrary set of discrete gradients integrators which exactly conserve all the first integrals simultaneously are then defined in both cases we start from an arbitrary method of a prescribed order say a runge kutta scheme and modify it using two approaches one is based on projection and the other on local coordinates the methods are tested on the kepler problem

products of conjugacy classes in finite and algebraic simple groups abstract we prove the arad herzog conjecture for various families of finite simple groups if a and b are nontrivial conjugacy classes then a b is not a conjugacy class we also prove that if g is a finite simple group of lie type and a and b are nontrivial conjugacy classes either both semisimple or both unipotent then a b is not a conjugacy class we also prove a strong version of the arad herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes as a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product a special case of this has been used by prasad to prove a uniqueness result for tits systems in quasi reductive groups our final result is a generalization of the baer suzuki theorem for p elements with p 5

transformation theory how normal is a family of distributions this paper concerns the following question if x is a real valued random variate having a one parameter family of distributions mathscr f to what extent can mathscr f be normalized by a monotone transformation in other words does there exist a single transformation y g x such that y has nearly a normal distribution for every distribution of x in mathscr f the theory developed answers the question without considering the form of g at all in those cases where the answer is positive simple formulas for calculating g are given the paper also considers the relationship between normalization and variance stabilization

on a class of discretizations of hamiltonian nonlinear partial differential equations abstract we present a new class of discretizations of partial differential equations pdes that preserve a momentum like integral of the motion this results in an effective translational invariance for the dynamical problem and the absence of a peierls nabarro barrier that is usually present in discretizations of hamiltonian pdes a general method to construct such discretizations for any nonlinearity is given and the properties of the resulting differential difference equations are analyzed in a number of different cases for nonlinear klein gordon as well for nonlinear schrodinger type systems while for the former nonintegrability of the dynamical problem is evident in the latter case numerical evidence suggests that the behavior is close to the one of integrable systems we also show how static solutions of the equations can be constructed for these discretizations and discuss the similarities and differences of the method with previously reported ones

the lack of continuity and the role of infinite and infinitesimal in numerical methods for odes the case of symplecticity when numerically integrating canonical hamiltonian systems the long term conservation of some of its invariants for example the hamiltonian function itself assumes a central role the classical approach to this problem has led to the definition of symplectic methods among which we mention gauss legendre collocation formulae indeed in the continuous setting energy conservation is derived from symplecticity via an infinite number of infinitesimal contact transformations however this infinite process cannot be directly transferred to the discrete setting by following a different approach in this paper we describe a sequence of methods sharing the same essential spectrum and then the same essential properties which are energy preserving starting from a certain element of the sequence on i e after a finite number of steps

integrable generalizations of the two dimensional born infeld equation the born infeld equation in two dimensions is generalized to higher dimensions lorentz invariance is retained and the resulting system is completely integrable via linearization by legendre transformation this generalization retains homogeneity in second derivatives of the field

decoupled local global energy preserving schemes for the n coupled nonlinear schrodinger equations abstract we develop two local energy preserving integrators and a global energy preserving integrator for the general multisymplectic hamiltonian system when applied to the 1d and multi dimensional n coupled nonlinear schrodinger equations the given schemes have the exact preservation of the local global conservation law and are decoupled in the components n n 1 2 n i e each of the components can be solved independently the decoupled feature is significant and helpful for overcoming the computational difficulty of the n coupled n 3 nonlinear schrodinger equations especially of the multi dimensional case the composition method is employed to improve the accuracy of the schemes in time and the discrete fast fourier transform is used to reduce the computational complexity several numerical experiments are carried out to exhibit the behaviors of the wave solutions numerical results confirm the theoretical results

unified approach to hamiltonian systems poisson systems gradient systems and systems with lyapunov functions or first integrals systems with a first integral i e constant of motion or a lyapunov function can be written as linear gradient systems dot x l x nabla v x for an appropriate matrix function l with a generalization to several integrals or lyapunov functions the discrete time analogue delta x delta t l bar nabla v where bar nabla is a discrete gradient preserves v as an integral or lyapunov function respectively

new simple groups with a bn pair abstract we show that there are simple groups with a spherical bn pair of rank 2 which are non moufang and hence not of algebraic origin

moufang sets of finite morley rank of odd type abstract we show that for a wide class of groups of finite morley rank the presence of a split bn pair of tits rank 1 forces the group to be of the form psl 2 and the bn pair to be standard our approach is via the theory of moufang sets specifically we investigate infinite and so called hereditarily proper moufang sets of finite morley rank in the case where the little projective group has no infinite elementary abelian 2 subgroups and show that all such moufang sets are standard and thus associated to psl 2 f for f an algebraically closed field of characteristic not 2 provided the hua subgroups are nilpotent further we prove that the same conclusion can be reached whenever the hua subgroups are l groups and the root groups are not simple

products and commutators of classes in algebraic groups we classify pairs of conjugacy classes in almost simple algebraic groups whose product consists of finitely many classes this leads to several interesting families of examples which are related to a generalization of the baer suzuki theorem for finite groups we also answer a question of pavel shumyatsky on commutators of pairs of conjugacy classes in simple algebraic groups it turns out that the resulting examples are exactly those for which the product also consists of only finitely many classes

on the energy theorem for the lagrange equations in the discrete case a method for discretizing lagrange equations is proposed in the conservative case the solutions of the system obtained this way verify the energy integral exactly

on the preservation of invariants by explicit runge kutta methods a new strategy to preserve invariants in the numerical integration of initial value problems with explicit runge kutta methods is presented it is proved that this technique retains the order of the original method has an easy and cheap implementation and can be used in adaptive runge kutta codes some numerical experiments with the classical code of dormand and prince dopri5 4 based on a pair of embedded methods with orders 5 and 4 are presented to show the behavior of the new method for several problems which possess invariants

improving the accuracy of the discrete gradient method in the one dimensional case we present two numerical schemes of high accuracy for one dimensional dynamical systems they are modifications of the discrete gradient method and keep its advantages including stability and conservation of the energy integral however their accuracy is higher by several orders of magnitude

discrete gradient methods for solving odes numerically while preserving a first integral we show that all direct methods for preserving a first integral during the numerical integration of an ordinary differential equation fit into the unified framework of discrete gradient methods using this framework we construct several new integral preserving schemes

preserving algebraic invariants with runge kutta methods abstract we study runge kutta methods for the integration of ordinary differential equations and the retention of algebraic invariants as a general rule we derive two conditions for the retention of such invariants the first is a condition on the coefficients of the methods the second is a pair of partial differential equations that otherwise must be obeyed by the invariant this paper extends previous work on multistep methods in iserles technical report na1997 13 damtp university of cambridge 1997 the cases related to the retention of quadratic and cubic invariants perhaps of greatest relevance in applications are thoroughly discussed we conclude recommending a generalized class of runge kutta schemes namely lie group type runge kutta methods these are schemes for the solution of odes on lie groups but can be employed together with group actions to preserve a larger class of algebraic invariants without restrictions on the coefficients

a new class of energy preserving numerical integration methods the first ever energy preserving b series numerical integration method for ordinary differential equations is presented and applied to several hamiltonian systems related novel lie algebraic results are also discussed

explicit estimating equations for semiparametric generalized linear latent variable models summary we study generalized linear latent variable models without requiring a distributional assumption of the latent variables using a geometric approach we derive consistent semiparametric estimators we demonstrate that these models have a property which is similar to that of a sufficient complete statistic which enables us to simplify the estimating procedure and explicitly to formulate the semiparametric estimating equations we further show that the explicit estimators have the usual root n consistency and asymptotic normality we explain the computational implementation of our method and illustrate the numerical performance of the estimators in finite sample situations via extensive simulation studies the advantage of our estimators over the existing likelihood approach is also shown via numerical comparison we employ the method to analyse a real data example from economics

linear models based on noisy data and the frisch scheme we address the problem of identifying linear relations among variables based on noisy measurements this is a central question in the search for structure in large data sets often a key assumption is that measurement errors in each variable are independent this basic formulation has its roots in the work of charles spearman in 1904 and of ragnar frisch in the 1930s various topics such as errors in variables factor analysis and instrumental variables all refer to alternative viewpoints on this problem and on ways to account for the anticipated way that noise enters the data in the present paper we begin by describing certain fundamental contributions by the founders of the field and provide alternative modern proofs to certain key results we then go on to consider a modern viewpoint and novel numerical techniques to the problem the central theme is expressed by the frisch kalman dictum which calls for identifying a noise contribution that allows a maximal number of simultaneous linear relations a

a note on the efficient implementation of hamiltonian bvms we discuss the efficient implementation of hamiltonian bvms hbvms a recently introduced class of energy preserving methods for canonical hamiltonian systems see brugnano et al 8 and references therein also sketching their blended formulation we also discuss the case of separable problems for which the structure of the problem can be exploited to gain efficiency

split bn pairs of rank 2 the octagons abstract let g be a group with an irreducible spherical bn pair of rank 2 where b contains a normal nilpotent subgroup u with b u b n then g is essentially a group of lie type this completes the classification of split bn pairs of rank 2 generalizing the corresponding result for finite groups due to fong and seitz

goodness of fit for generalized linear latent variables models generalized linear latent variables models gllvm enable the modeling of relationships between manifest and latent variables where the manifest variables are distributed according to a distribution of the exponential family e g binomial or normal and to the multinomial distribution for ordinal manifest variables these models are widely used in social sciences to test the appropriateness of a particular model one needs to define a goodness of fit test statistic gfi in the normal case one can use a likelihood ratio test or a modified version proposed by satorra and bentler 2001 s b gfi that compares the sample covariance matrix to the estimated covariance matrix induced by the model in the binary case pearson type test statistics can be used if the number of observations is sufficiently large in the other cases including the case of mixed types of manifest variables there exists gfi based on a comparison between a pseudo sample covariance and the model covariance of the manifest vari

a numerical integrator for the two fixed centres problem conserving all constants of motion the two fixed centres problem describes the motion of a particle influenced by the gravitational pull of two fixed particles this is a well known integrable hamiltonian system having three constants of motion this paper presents a discrete two dimensional two fixed centres problem which conserves two of these constants of motion moreover this discrete system preserves the discrete analogue of the third constant of motion a canonical transformation is introduced to separate the variables and remove singularities of the system a combination of this transformation and energy preserving method plays an important role in deriving the discrete system with a variable time step

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

numerical simulation for a nonlinear partial differential equation with variable coefficients by means of the discrete variational derivative method partial differential equations with possibly discontinuous coefficients play an important part in engineering physics and ecology in this paper we will study nonlinear partial differential equations with variable coefficients arising from population models generally speaking it is difficult to analyze the behavior of nonlinear partial differential equations therefore we usually rely on the numerical approximation currently there is an increasing interest in designing numerical schemes that preserve energy properties for differential equations we will design the numerical schemes that preserve discrete energy property and show numerical experiments for a nonlinear partial differential equation with variable coefficients

long time behaviour of discretizations of the simple pendulum equation we compare several discretizations of the simple pendulum equation in a series of numerical experiments the stress is put on the long time behaviour the chosen numerical schemes are either symplectic maps or integrable energy preserving maps or both therefore they preserve qualitative features of solutions such as periodicity we describe characteristic periodic time dependences of numerical estimates of the period and the amplitude and explain them as systematic numerical by effects produced by any method finally we propose a new numerical scheme which is a modification of the discrete gradient method this modified discrete gradient method preserves almost exactly the period of small oscillations for any time step