lie group symmetries and invariants of the henon heiles equations lie group symmetries and invariants of the generalized henon heiles equations are found the coupled second order equations are invariant under translations in time in general and the stretching group dilation if the linear terms in the force are absent the equivalent set of four coupled first order equations is found to be invariant under a one parameter group for three cases and the group generators are given three different approaches are reported the classical method for determining lie group symmetries a modified method for finding lie group symmetries with vector fields and the direct method for calculating the invariants for the henon heiles equations the direct method is the most efficient

configurational invariants of hamiltonian systems in this paper we explore the general conditions in order that a two dimensional natural hamiltonian system possess a second invariant which is a polynomial in the momenta and is therefore liouville integrable we examine the possibility that the invariant is preserved by the hamiltonian flow on a given energy hypersurface only weak integrability and derive the additional requirement necessary to have conservation at arbitrary energy strong integrability using null complex coordinates we show that the leading order coefficient of the polynomial is an arbitrary holomorphic function in the case of weak integrability and a polynomial in the coordinates in the strongly integrable one we review the results obtained so far with strong invariants up to degree four and provide some new examples of weakly integrable systems with linear and quadratic invariants

first integrals versus configurational invariants and a weak form of complete integrability abstract a confusion over the concept of first integrals which has been created in a recent paper by hall 13 is clarified the clear distinction between first integrals and functions which are first integrals only on a specific fixed hypersurface is discussed hall s terminology of configurational invariants is adopted for the latter case the possible relevance of knowing configurational invariants for a hamiltonian system is illustrated by results concerning a weak form of the theory on complete integrability

generalized natural mechanical systems of two degrees of freedom with quadratic integrals this paper is devoted to the problem of constructing integrable mechanical systems with two degrees of freedom whose lagrangians contain terms linear in the velocities and whose second integrals have the form of polynomials of the second degree in the velocity variables in which the coefficients depend only on the coordinates the solution of this problem is known only in the case of reversible systems for which the linear terms do not affect lagrange s equations of motion using the method developed in a previous paper the authors classify all the possible irreversible systems into three types according to a certain normal form of the line element on the configuration space the most general systems of the first two types are constructed several many parameter systems of the third type are also found some of the new cases are found to be generalizations by the introduction of some additional parameters to well known integrable problems in particle and rigid body dynamics mechanical interpretation is also given for some other cases

a direct approach to finding exact invariants for one dimensional time dependent classical hamiltonians for a classical hamiltonian h 1 2 p2 v q t with an arbitrary time dependent potential v q t exact invariants that can be expressed as series in positive powers of p i q p t n 0pnfn q t are examined the method is based on direct use of the equation di dt i t i h 0 a recursion relation for the coefficients fn q t is obtained all potentials that admit an invariant quadratic in p are found and for those potentials all invariants quadratic in p are determined the feasibility of extending the analysis to find invariants that are polynomials in p of higher degree than quadratic is discussed the systems for which invariants quadratic in p have been found are transformed to autonomous systems by a canonical transformation

a theory of exact and approximate configurational invariants abstract a theory of integrable hamiltonian systems in two dimensions is formulated and applied the four dimensional phase space problem that is the vanishing of the poisson bracket between another invariant and the hamiltonian is here reduced to the solution of a series of two dimensional configuration space equations the theory is also applicable to more than two dimensions the constraints are found which admit to integrability of the orbits for magnetic or coriolis forces as well as for forces derivable from a potential when a system admits a given invariant the invariant is found for example by quadrature a number of examples including known and apparently previously unknown invariants are given the theory of exact integrals of the motion also can be extended to the derivation of approximate invariants the orbital structure of integrable or approximately integrable systems correlates with the degree maximum power of the velocity of a standardized invariant the theory admits a variational principle among other approximation techniques for the computation of a best approximate invariant the problem of the general cubic potential with one symmetric coordinate v 1 2 ax 2 1 2 by 2 cx 2 y 1 3 dy 3 of which the well studied henon heiles potential is the special case for a b and c d is examined in detail

statistical analysis of the least squares autoregressive estimator in the presence of noise this paper analyzes the asymptotic statistical properties of the least squares autoregressive ar estimator in the presence of noise the asymptotic covariance of this estimator is derived for an ar signal with additive white or colored noise the result is shown to hold for a general gaussian or linear stationary measurement the analysis is applied in detail to first order systems a procedure for using the result to solve problems in speech recognition biomedical diagnosis and fault detection is presented results of simulations confirm the theoretical analysis

recursive solutions of estimating equations and adaptive spectral factorization a recursive spectral factorization algorithm was developed by v solo ibid vol 34 no 10 pp 1047 1051 oct 1989 to be used with recursive instrumental variables for consistent estimation of the parameters of an armax system but the convergence of the spectral factorization algorithm is questionable herein a modified adaptive spectral factorization algorithm is shown to converge and a general method for constructing convergent recursive solutions of nonlinear equations that are used to define parameter estimates is also presented

testing misspecification in vector time series models with exogenous variables summary in this paper we consider the application of hypothesis testing procedures to armax systems the score or lagrangian multiplier principle is used to obtain misspecification tests appropriate to these models the structure of the information or sensitivity matrix is investigated in order to discuss the identifiability of alternative models and is used to show the existence of a set of admissible alternatives that are locally equivalent to one another this notion is then developed to suggest why the score test is to be regarded as a pure significance test

the algebraic structure of generalized ermakov systems in three dimensions the characteristic algebra of generalized ermakov systems is sl 2 r the structure of these systems in three dimensions is obtained a subset in the form of an equation of motion with the additional requirement of so 3 symmetry is studied it includes the classical equation of the magnetic monopole the existence of three vectors of poincare type is established consideration is given to weak generalized ermakov systems in which the symmetry breaking occurs in the radial equation

integrable hamiltonian systems with vector potentials we investigate integrable two dimensional hamiltonian systems with scalar and vector potentials admitting second invariants which are linear or quadratic in the momenta in the case of a linear second invariant we provide some examples of weakly integrable systems in the case of a quadratic second invariant we recover the classical strongly integrable systems in cartesian and polar coordinates and provide some new examples of integrable systems in parabolic and elliptical coordinates

darboux integrability of 2 dimensional hamiltonian systems with homogenous potentials of degree 3 we provide a characterization of all hamiltonian systems of the form h p12 p22 2 v q1 q2 where v is a homogenous polynomial of degree 3 which are completely integrable with darboux first integrals

integrals of motion and analytic functions in autonomous dynamical systems with two degrees of freedom any integral polynomial in the velocities determines uniquely a complex analytic function conversely given any analytic function there exists a polynomial integral of a dynamical system which determines this function the analytic function provides an immediately verifiable necessary criterion which can answer the questions of whether an expression which is polynomial in the velocities can be an integral of any unknown two dimensional dynamical system the basic operation among integrals corresponds to similar operations among the corresponding analytic functions

the master integrable two dimensional system with a quartic second integral in this paper we construct a new integrable natural conservative mechanical system admitting a second invariant of the fourth degree in velocities this system is quite general and involves 21 parameters we also show that all systems with a quartic integral known up to date can be obtained from it as special cases by a relevant choice of the values of parameters the results are applied to problems of particle and rigid body dynamics new integrable cases are obtained as special versions of the new system these cases include motions in a plane lobachevsky plane sphere and surfaces of variable curvature they also include generalizations of the classical cases of kovalevskaya chaplygin and goriatchev with the addition of certain singular terms to the potential

adaptive multipath delay estimation two algorithms are proposed for the continuous tracking of multipath delay the first is an efficient method for adaptively tracking the secondary peak in the autocorrelation function caused by multipath in principle it is equivalent to the correlation method commonly used in practice the second is an adaptive inverse filter which can exactly remove the multipath transfer function the methods use an adaptive delay line interpolated by a first order filter the interpolation coefficient is explicitly estimated using the recursive gauss newton rgn algorithm often used for recursive system identification

integrability and the painleve property for low dimensional systems we examine a system described by two first order nonlinear differential equations from the point of view of integrability the singularity analysis in the complex time plane is used to investigate the painleve property which according to the ablowitz ramani segur conjecture is a prerequisite for integrability for infinite dimensional systems we show that for such low dimensional systems the painleve analysis is still a most useful guide but integrable cases also exist which do not possess the painleve property

a new class of integrable systems we present a family of dynamical systems associated with the motion of a particle in two space dimensions these systems possess a second integral of motion quadratic in velocities apart from the hamiltonian and are thus completely integrable they were found through the derivation and subsequent resolution of the integrability condition in the form of a partial differential equation pde for the potential a most important point is that the same pde was derived through considerations on the analytic structure of the singularities of the solutions weak painleve property

robust methods for recursive autoregressive moving average estimation in this paper we consider the problem of fitting recursively in time an autoregressive moving average arma model to time series data where outliers may be present although many recursive estimation procedures are available for fitting arma models they are based on the recursive least squares algorithm which is known to be badly affected by additive outliers to minimize the impact of these sort of outliers we investigate some robustified versions of the recursive maximum likelihood procedure the problem of incorporating robust scale recursion is discussed and asymptotic properties of the procedures are examined

vector autoregressive moving average identification for macroeconomic modeling a new methodology this paper develops a new methodology for identifying the structure of varma time series models the analysis proceeds by examining the echelon form and presents a fully automatic strongly consistent data driven approach to model specification a novel feature of the inferential procedures developed here is that they work in terms of a canonical representation based on the kronecker invariants in which the variables are expressed in the form of scalar dynamic structural equations derived from the varma system this feature facilitates the construction of procedures which from the perspective of macroeconomic modeling can be efficacious in that they do not rely on var approximations techniques that are applicable to both asymptotically stationary and unit root partially nonstationary cointegrated time series models are presented the inferential potential of the techniques is illustrated via simulation experiments that use data generating mechanisms based on real world examples drawn from the time series literature aspects of the kronecker invariants that impinge on practical application and that have not hitherto been discussed in the literature are explored

theory and methods abstract a self organizing filter and smoother for the general nonlinear non gaussian state space model is proposed an expanded state space model is defined by augmenting the state vector with the unknown parameters of the original state space model the state of the augmented state space model and hence the state and the parameters of the original state space model are estimated simultaneously by either a non gaussian filter smoother or a monte carlo filter smoother in contrast to maximum likelihood estimation of model parameters in ordinary state space modeling for which the recursive filter computation has to be done many times model parameter estimation in the proposed self organizing filter smoother is achieved with only two passes of the recursive filter and smoother operations examples such as automatic tuning of dispersion and the shape parameters adaptation to changes of the amplitude of a signal in seismic data state estimation for a nonlinear state space model with unknown parameters

a unified treatment of quartic invariants at fixed and arbitrary energy two dimensional hamiltonian systems admitting second invariants which are quartic in the momenta are investigated using the jacobi geometrization of the dynamics this approach allows for a unified treatment of invariants at both arbitrary and fixed energy in the differential geometric picture the quartic invariant corresponds to the existence of a fourth rank killing tensor expressing the jacobi metric in terms of a kahler potential the integrability condition for the existence of the killing tensor at fixed energy is a nonlinear equation involving the kahler potential at arbitrary energy further conditions must be imposed which lead to an overdetermined system with isolated solutions we obtain several new integrable and superintegrable systems in addition to all previously known examples

the limiting behavior of lms a realization oriented analysis is given of the gradient noise misadjustment and lag misadjustment performance of the lms least mean square algorithm new formulas are given for both of these components of excess mean square error it is shown that the traditional formula for lag misadjustment needs to be modified by adding further terms due to gradient noise and noise variance to perform the analysis it is necessary to study the convergence with probability one of the noise free fixed parameter lms algorithm convergence is found under simple conditions that improve on those previously obtained

equivalence classes of second order ordinary differential equations with only a three dimensional lie algebra of point symmetries and linearisation there are seven equivalence classes of second order ordinary differential equations possessing only three lie point symmetries and hence not linearisible by means of a point transformation we examine the representatives of these classes for linearisibility by means of other types of transformation in particular we compare the potential for linearisibility and the possession of the painleve property the complete symmetry group is realised in the standard algebra for each of the equivalence classes

an approach to testing linear times series models the purpose of this paper is to develop diagnostic tests for open loop transfer function models with autoregressive moving average stochastic disturbances using the efficient scores procedure the tests proposed are asymptotically equivalent to those based upon the likelihood ratio principle and have the advantage that they do not involve a heavy computational burden consideration is given both to the standard case of nonsingular information matrices and to the situation obtaining when there are identifiability problems and application of conventional large sample test procedures is not feasible relationships between score tests and portmanteau tests in time series analysis are also investigated some simulation evidence on the finite sample behaviour of the tests is presented and it is seen that the tests of this paper perform well

hamiltonians with high order integrals and the weak painleve concept we examine the singularity structure of the equations of motion associated to integrable two dimensional hamiltonians with second integrals of order higher than 2 we show in these specific examples that the integrability is associated to a singularity expansion of the weak painleve type new cases of integrability are discovered with still higher order integrals which are explicitly computed

lie integrable cases of the simplified multistrain two stream model for tuberculosis and dengue fever abstract we apply the techniques of lie s symmetry analysis to a caricature of the simplified multistrain model of castillo chavez and feng c castillo chavez z feng to treat or not to treat the case of tuberculosis j math biol 35 1997 629 656 for the transmission of tuberculosis and the coupled two stream vector based model of feng and velasco hernandez z feng j x velasco hernandez competitive exclusion in a vector host model for the dengue fever j math biol 35 1997 523 544 to identify the combinations of parameters which lead to the existence of nontrivial symmetries in particular we identify those combinations which lead to the possibility of the linearization of the system and provide the corresponding solutions many instances of additional symmetry are analyzed

symmetries of hamiltonian systems with two degrees of freedom we classify the lie point symmetry groups for an autonomous hamiltonian system with two degrees of freedom with the exception of the harmonic oscillator or a free particle where the dimension is 15 we obtain all dimensions between 1 and 7 for each system in the classification we examine integrability

a multi time scale generalization of recursive identification algorithm for armax systems recently chen 2010 presented a new approach to recursive identification for armax systems which is a three stage recursive scheme and assumes independent and identically distributed input signals here we observe that unless the time scale of the algorithm at one stage is reasonably faster than those at the previous stages convergence to the true value may not take place to remedy this issue this note proposes a multi time scale modification of the algorithm in 5 such that convergence is achieved in addition the new scheme handles a wider class of input signals so that the input can be designed for some purpose the advantage of the multi time scale algorithm is verified with numerical examples

adaptive algorithms for constrained arma signals in the presence of noise a family of algorithms is developed for adaptive parameter estimation of constrained autoregressive moving average arma signals in the presence of noise these algorithms utilize a priori information about the signal s properties such as its spectral type for example low pass bandpass etc or a spatial domain characteristic special applications include modeling of autoregressions ar and signals of known spectral type in the presence of noise signal deconvolution image deblurring and multipath parameter estimation selected results of simulations are included to demonstrate the performance of the algorithms

rational functions of momentum as invariants for one dimensional time dependent potentials basic theory a framework for the momentum resonance formulation of lewis and leach ann phys ny 164 47 1985 that casts new light into the nature of exact explicitly time dependent invariants for one dimensional time dependent potentials and produces additional examples of such invariants is presented the momentum resonance formulation postulates that the invariant be a rational function of momentum with simple poles which are called momentum resonances it is shown that an invariant of resonance type can be written as a functional of the potential in terms of the solution of a system of linear algebraic equations and a single necessary and sufficient condition for a potential to admit an invariant of resonance type is obtained these results are obtained by reformulating the problem in terms of a set of discrete moments that satisfy two separate recursion formulas invariants for new time dependent potentials can be obtained and previously known invariants are recovered

the he non heiles system revisited abstract the known integrable cases of the henon heiles system are shown to be closely related to the stationary flows of the known and only integrable fifth order single component and polynomial nonlinear evolution equations this is further evidence that these are the only integrable cases of the henon heiles system lax pairs are deduced for each of the integrable cases and used to construct the constants of motion a curious lax operator recently found by the painleve method is explained

rank based autoregressive order identification abstract optimal rank based procedures have been derived for testing arbitrary linear restrictions on the parameters of autoregressive moving average arma models with unspecified innovation densities the finite sample performances of these procedures are investigated here in the context of ar order identification and compared to those of classical partial correlograms and lagrange multipliers methods the results achieved by rank based methods are quite comparable in the gaussian case to those achieved by the traditional ones which under gaussian assumptions are asymptotically optimal however under non gaussian innovation densities especially heavy tailed or nonsymmetric or when outliers are present the percentages of correct order selection based on rank methods are strikingly better than those resulting from traditional approaches even in the case of very short n 25 series these empirical findings confirm the often ignored theoretical fact that the gaussian case in the arma context

polynomial integrability of the hamiltonian systems with homogeneous potential of degree 3 abstract in this paper we study the polynomial integrability of natural hamiltonian systems with two degrees of freedom having a homogeneous potential of degree k given either by a polynomial or by an inverse of a polynomial for k 2 1 3 4 their polynomial integrability has been characterized here we have two main results first we characterize the polynomial integrability of those hamiltonian systems with homogeneous potential of degree 3 second we extend a relation between the nontrivial eigenvalues of the hessian of the potential calculated at a darboux point to a family of hamiltonian systems with potentials given by an inverse of a homogeneous polynomial this relation was known for such hamiltonian systems with homogeneous polynomial potentials finally we present three open problems related with the polynomial integrability of hamiltonian systems with a rational potential

arma spectral estimation of time series with missing observations the problem of estimating the power spectral density of stationary time series when the measurements are not contiguous is considered a new autoregressive moving average arma method is proposed for this problem based on nonlinear optimization of a weighted squared error criterion the method can handle either regularly or randomly missing observations as a special case the method can handle the problem of missing sample covariances the computational complexity is modest compared to exact maximum likelihood estimation of the same parameters the performance of the algorithm is illustrated by some numerical examples and is shown to be statistically efficient in these cases

two dimensional time dependent hamiltonian systems with an exact invariant we present a direct approach to investigate the existence of an exact invariant for two dimensional hamiltonians in which the potential depends explicitly on time the method is based on an expansion of the invariant in the velocities the problem is solved completely for invariants linear and quadratic in the momenta our results contain as a particular case the results of lewis and leach on one dimensional systems

invariants at fixed and arbitrary energy a unified geometric approach invariants at arbitrary and fixed energy strongly and weakly conserved quantities for two dimensional hamiltonian systems are treated in a unified way this is achieved by utilizing the jacobi metric geometrization of the dynamics using killing tensors we obtain an integrability condition for quadratic invariants which involves an arbitrary analytic function s z for invariants at arbitrary energy the function s z is a second degree polynomial with real second derivative the integrability condition then reduces to darboux s condition for quadratic invariants at arbitrary energy the four types of classical quadratic invariants for positive definite two dimensional hamiltonians are shown to correspond to certain conformal transformations we derive the explicit relation between invariants in the physical and jacobi time gauges in this way knowledge about the invariant in the physical time gauge enables one to write down directly the components of the corresponding killing tensor for the jacobi metric we also discuss the possibility of searching for linear and quadratic invariants at fixed energy and its connection with the problem of the third integral in galactic dynamics in our approach linear and quadratic invariants at fixed energy can be found by solving a linear ordinary differential equation of the first or second degree respectively

adaptive parameter estimation for a constrained low pass system a novel algorithm is presented for adaptive parameter estimation for a constrained low pass butterworth system model the algorithm will estimate the system cutoff frequency and gain online when it is known that the true system has a low pass butterworth structure or some transfer function similar to it and its true order is used the algorithm will lead to a substantial savings in computation and more accurate results than unconstrained algorithms potential applications include filter design and adaptive decision on nyquist rate for systems

adaptive spectral factorization an on line spectral factorization algorithm is used to devise a globally convergent self tuning identifier that does not suffer from restrictions that amount to knowledge of the true system e g the positive real condition the method developed uses two ideas one idea an old one which might be called the method of split recursions is used to estimate the parameters in blocks thus one block might get the transfer function parameters while the other gets the noise parameters the other idea is to use spectral factorization to estimate moving average parameters the algorithm does have its own weaknesses e g transient behavior may not be good and it relies on a condition that is only generically true but it does not need a positive real condition to be satisfied for global convergence

a bayes procedure for the identification of univariate time series models on presente un critere d identification qui est asymptotiquement equivalent a une regle de decision bayesienne

the parsimony principle for a class of model structures a new form of the parsimony principle is presented for a class of model structures containing a common subset of parameters the principle is applied to the problem of estimating the autoregressive ar parameters of an ar process corrupted by white measurement noise

adaptive estimation in time series regression models this work develops adaptive estimators for a linear regression model with serially correlated errors we show that these results continue to hold when the order of the arma process characterizing the errors is unknown the finite sample results are promising indicating that substantial efficiency gains may be possible for samples as small as 50 observations we use these estimators to investigate the behavior of the forward foreign exchange market

comment on adaptive estimation in time series regression models by d g steigerwald abstract in this comment we discuss some serious shortcomings and errors in steigerwald 1992

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

diagnostic tests for multiple time series models this paper is concerned with the development and application of diagnostic checks for vector linear time series models a hypothesis testing procedure based upon the score or lagrangean multiplier principle is advocated and the distributions of the test statistic both under the null hypothesis and under a pitman sequence of alternatives are discussed consideration of alternative models with singular sensitivity matrices when the null hypothesis is true leads to an interpretation of the score test as a pure significance test and to a notion of an equivalence class of local alternatives portmanteau tests of model adequacy are also investigated and are seen to be equivalent to score tests

robust real time identification of linear systems with correlated noise the problem of recursive robust identification of linear discrete time single input single output dynamic systems with correlated disturbances is considered problems related to the construction of optimal robust stochastic approximation algorithms in the min max sense are demonstrated since the optimal solution cannot be achieved in practice several robustified stochastic approximation algorithms are derived on the basis of a suitable non linear transformation of normalized residuals as well as step by step optimization with respect to the weighting matrix of the algorithm the convergence of the developed algorithms is established theoretically using the ordinary differential equation approach monte carlo simulation results are presented for the quantitative performance evaluation of the proposed algorithms the results indicate the most suitable algorithms for applications in engineering practice

consistent moment selection procedures for generalized method of moments estimation this paper considers a generalized method of moments gmm estimation problem in which one has a vector of moment conditions some of which are correct and some incorrect the paper introduces several procedures for consistently selecting the correct moment conditions the procedures also can consistently determine whether there is a sufficient number of correct moment conditions to identify the unknown parameters of interest the paper specifies moment selection criteria that are gmm analogues of the widely used bic and aic model selection criteria the latter is not consistent the paper also considers downward and upward testing procedures all of the moment selection procedures discussed in this paper are based on the minimized values of the gmm criterion function for different vectors of moment conditions the procedures are applicable in time series and cross sectional contexts application of the results of the paper to instrumental variables estimation problems yields consistent procedures for selecting instrumental variables

local rank tests in a multivariate nonparametric relationship abstract consider a multivariate nonparametric model where the unknown vector of functions depends on two sets of explanatory variables for a fixed level of one set of explanatory variables we provide consistent statistical tests called local rank tests to determine whether the multivariate relationship can be explained by a smaller number of functions we also provide estimators for the smallest number of functions called local rank explaining the relationship the local rank tests and the estimators of local rank are defined in terms of the eigenvalues of a kernel based estimator of some matrix the asymptotics of the eigenvalues is established by using the so called fujikoshi expansion along with some techniques of the theory of u statistics we present a simulation study which examines the small sample properties of local rank tests we also apply the local rank tests and the local rank estimators to a demand system given by a newly constructed data set this work can be viewed as a local extension of the tests for a number of factors in a nonparametric relationship introduced by stephen donald

on certain two dimensional conservative mechanical systems with a cubic second integral in a previous paper yehia h m 1986 j mec theor appl 5 55 71 we have introduced a method for constructing integrable conservative two dimensional mechanical systems whose second integral of motion is polynomial in the velocities this method has proved successful in constructing a multitude of irreversible systems involving gyroscopic forces with a second quadratic integral yehia h m 1992 j phys a math gen 25 197 221 the objective of this paper is to apply the same method for the systematic construction of mechanical systems with a cubic integral as in our previous works the configuration space is not assumed to be a euclidean plane this widens the range of applicability of the results to diverse mechanical systems to include such problems as rigid body dynamics several new reversible and irreversible integrable systems are obtained some of these systems generalize previously known ones by introducing additional parameters which may change either or both of the configuration manifold and the potential of the forces acting on the system other systems are completely new an application is given to problems of rigid body dynamics the famous classical integrable case due to goriachev and chaplygin and all its subsequent generalizations by several authors are further generalized to include certain variable gyroscopic forces that preserve a cubic integral on the other hand the above case and another less famous case of rigid body dynamics due to goriachev and the hall toda case of particle mechanics are all obtained as special cases corresponding to different choices of certain parameters from one more general system unifying them all

first integrals for some nonlinear time dependent hamiltonian systems an observation of a simple property of canonical transformations leads to a procedure for determining first integrals for classes of hamiltonians in illustrative examples the most general result presently known from other methods is recovered a new result presented and a generalization to more than one degree of freedom discussed

new examples of integrable conservative systems on s 2 and the case of goryachev chaplygin in this paper we propose new examples of conservative systems on s 2 possessing cubic integrals in momenta we obtain a classification of a large class of such systems we show that this class contains new families of integrable systems as well as the classical case of goryachev chaplygin

the properties of the parameterization of armax systems and their relevance for structural estimation and dynamic specification armax systems in structural specification are considered topological and geometric properties of the parameter space and of the parameterization which are important for the properties of the estimators regardless of their special form are investigated and the specification of the maximum lag lengths is discussed

integrable hamiltonian systems with velocity dependent potentials the integrability of a two dimensional hamiltonian in which the potential depends explicitly on the momenta is investigated hamiltonians of this kind are encountered in the description of the motion of a particle in a magnetic field two integrable classes of potentials are identified and the second integral of motion is constructed for each of them the singularity analysis of the equations of motion is also performed confirming once more the relation between the weak painleve property and integrability

strong approximation of the recursive prediction error estimator of the parameters of an arma process abstract let n denote the recursive prediction error estimator of the parameters of an arma process and let n denote the off line prediction error estimator we show under reasonable assumptions that for any q 1 n n lq o m log n n this theorem enables us to show that most of the properties known for the off line estimator n are inherited by the on line estimator n

consistent model and moment selection procedures for gmm estimation with application to dynamic panel data models this paper develops consistent model and moment selection criteria for gmm estimation the criteria select the correct model specification and all correct moment conditions asymptotically the selection criteria resemble the widely used likelihood based selection criteria bic hqic and aic the latter is not consistent the gmm selection criteria are based on the j statistic for testing over identifying restrictions bonus terms reward the use of fewer parameters for a given number of moment conditions and the use of more moment conditions for a given number of parameters the paper also considers a consistent downward testing procedure the paper applies the model and moment selection criteria to dynamic panel data models with unobserved individual effects the paper shows how to apply the selection criteria to select the lag length for lagged dependent variables to detect the number and locations of structural breaks to determine the exogeneity of regressors and or to determine the existence of correlation between some regressors and the individual effect to illustrate the finite sample performance of the selection criteria and the testing procedures and their impact on parameter estimation the paper reports the results of a monte carlo experiment on a dynamic panel data model

the complete whittaker theorem for two dimensional integrable systems and its application the authors prove for a particle moving in a plane under the influence of a conservative force that when the motion is constrained by a second invariant quadratic in the velocities then the potential allows separability of the hamilton jacobi equation in rectangular polar elliptical cylinder or parabolic cylinder coordinates this link shows the intimate connection between quadratic invariants and the two dimensional hamilton jacobi equation they give examples of the utility of parabolic cylinder coordinates in cases of recent study

darboux s problem of quadratic integrals the problem of determining all standard classical hamiltonians in two dimensions with euclidean metric which admit constants of motion quadratic in the momenta is resolved several general results are given which make it obvious that the systems found do possess such integrals

a list of all integrable two dimensional homogeneous polynomial potentials with a polynomial integral of order at most four in the momenta we searched for integrable two dimensional homogeneous polynomial potentials with a polynomial first integral by using the so called direct method of searching for first integrals we proved that there exist no polynomial first integrals which are genuinely cubic or quartic in the momenta if the degree of homogeneous polynomial potentials is greater than four

how to construct integrable fokker planck and electromagnetic hamiltonians from ordinary integrable hamiltonians integrable hamiltonians with velocity dependent potentials including those of fokker planck type h 1 2 p2x p2y kxpx kypy are constructed from integrable hamiltonians of type h 1 2 p2x p2y v x y using certain canonical and noncanonical transformations some of the hamiltonians obtained this way are integrable only for zero energy candidates for the potential which is of interest for fokker planck models are constructed in several cases

a new necessary condition for the integrability of hamiltonian systems with a two dimensional homogeneous potential abstract recently morales ruiz and ramis obtained a strong necessary condition for the integrability of hamiltonian systems with a homogeneous potential based on their own theorem on the differential galois theory picard vessiot theory for hamiltonian systems the theorem claims that if the original hamiltonian system is integrable then the variational equation around a particular solution is solvable in the sense of the differential galois theory i e the solution is obtained only by a combination of quadratures exponential of quadratures and algebraic functions in this paper a direct and independent proof of this statement is given for hamiltonian systems with a two dimensional homogeneous potential which leads to the new necessary condition for integrability this new necessary condition well justifies the so called weak painleve conjecture of ramani et al for the first time

noether theorem for μ symmetries we give a version of noether theorem adapted to the framework of symmetries this extends to such case recent work by muriel romero and olver in the framework of symmetries and connects symmetries of a lagrangian to a suitably modified conservation law in some cases this conservation law actually reduces to a standard one we also note a relation between symmetries and conditional invariants we also consider the case where the variational principle is itself formulated as requiring vanishing variation under prolonged variation fields leading to modified euler lagrange equations in this setting symmetries of the lagrangian correspond to standard conservation laws as in the standard noether theorem we finally propose some applications and examples

symmetries of first integrals and their associated differential equations the relationship between the reduction of order through point symmetries and integration is explored with particular emphasis on the loss and gain of point contact for third order symmetries to and from nonlocal symmetries it is seen that reduction of order can even lead to the loss of all point symmetries at the third order level and their replacement at the second order level from nonlocal symmetries it is evident that nonlocal symmetries should be given more attention in applications

on the geometry of lambda symmetries and pde reduction we give a geometrical characterization of prolongations of vector fields and hence of symmetries of odes this allows an extension to the case of pdes and systems of pdes in this context the central object is a horizontal 1 form and we speak of prolongations of vector fields and symmetries of pdes we show that these are as good as standard symmetries in providing symmetry reduction of pdes and systems and explicit invariant solutions

construction of new integrable hamiltonians in two degrees of freedom a new procedure for deriving integrable hamiltonians and their constants of the motion is introduced we term this procedure the truncation program integrable hamiltonians occurring in the truncation program possess constants of the motion which are polynomials in a perturbation parameter e the relationship between this program and the whittaker program in two degrees of freedom is discussed integrable hamiltonians occurring in the whittaker program a generalization of whittaker s work possess constants of the motion which are polynomials in the momentum coordinates many previously known integrable hamiltonians are derived a new family of integrable double resonance hamiltonians and a new family of integrable hamiltonians of the form p21 p22 2 v q1 q2 are derived

unbiased parameter estimation of linear systems with colored noises we report a study of the problem of consistent parameter estimation in the general case where the measurement noise is unknown and correlated not only with the system output but also with the system input a new method is presented for eliminating the noise induced biases in the least squares ls estimates in the proposed method two filters are introduced to filter the system input and output signals respectively so that an augmented system with some known zeros and poles is obtained by shifting the space of the parameters to be estimated and using the known zeros and poles of the parameters to be estimated and using the known zeros and poles a sufficient number of linear constraints are constructed to determine the noise induced biases after eliminating the biases in the ordinary ls estimates consistent estimates of the parameters are obtained analysis shows that the results of the proposed method are independent of the noise model used

on the parsimony principle the parsimony principle is frequently used in system identification on more or less heuristical grounds recently a formal proof of this principle has been given here the assumptions used in that proof are examined to some extent it will be shown by means of some counterexamples that these assumptions cannot be relaxed unless additional restrictions are introduced a result showing how the parsimony principle should be used when estimating the parameters of regression models by the least squares method is also presented

recursive estimation of mixed autoregressive moving average order summary the order p q of an autoregressive moving average sequence y t may be estimated by minimizing a criterion log 2 p q log t t with respect to p and q where 2 is the maximum likelihood estimate of the variance of the innovations t it is suggested that instead a2 be estimated from a series of regressions of y t on y t 1 y t 2 y t p 5 t 1 t q where the t are obtained by fitting a long autoregression to the data it is shown how the sequence of regressions may for p q be economically recursively calculated by embedding them in a sequence of bivariate autoregressions asymptotic properties of the procedure are established under very general conditions

analytic integrability of hamiltonian systems with a homogeneous polynomial potential of degree 4 in the analytic case we prove the conjecture of maciejewski and przybylska j math phys 46 6 062901 2005 regarding hamiltonian systems with a homogeneous polynomial potential of degree 4 the proof of the conjecture completes the characterization of all the analytic integrable hamiltonian system with a homogeneous polynomial potential of degree 4

a self organizing state space model a self organizing filter and smoother for the general nonlinear non gaussian state space model is proposed an expanded state space model is defined by augmenting the state vector with the unknown parameters of the original state space model the state of the augmented state space model and hence the state and the parameters of the original state space model are estimated simultaneously by either a non gaussian filter smoother or a monte carlo filter smoother in contrast to maximum likelihood estimation of model parameters in ordinary state space modeling for which the recursive filter computation has to be done many times model parameter estimation in the proposed self organizing filter smoother is achieved with only two passes of the recursive filter and smoother operations examples such as automatic tuning of dispersion and the shape parameters adaptation to changes of the amplitude of a signal in seismic data state estimation for a nonlinear state space model with unknown parameters and seasonal adjustment with a nonlinear model with changing variance parameters are shown to exemplify the usefulness of the proposed method

structure identification of parameter bounding models by use of noise structure bounds the selection of a model for a given set of records is discussed for the case when the output error is required to be always within some specified bounds a new selection criterion is proposed the model should be the simplest giving sufficiently unstructured output errors the lack of structure is specified by requiring the candidate model to be capable of giving sufficiently small sample autocorrelations of the model output errors over a range of lags computational algorithms to apply the criterion are presented and illustrated by tests on ar processes

separability and lax pairs for henon heiles system the hamiltonian system corresponding to the generalized henon heiles hamiltonian h 1 2 px2 py2 1 2ax2 1 2by2 x2y ey3 is known to be integrable in the following three cases a b e 1 3 e 2 b 16a e 16 3 in the first two the system has been integrated by making use of genus one and genus two theta functions we show that in the third case the system can also be integrated by making use of elliptic functions finally using the fairbanks theorem we find lax pairs for each of the three integrable systems under consideration

inference concerning the number of factors in a multivariate nonparametric relationship this paper considers the problem of determining the number of nonparametric factors in a multivariate nonparametric relationship the definition given is broad enough to encompass a number of potential applications in econometrics including inferring the rank of demand testing whether for a given set of instruments it is possible to identify a linear model and testing arbitrage pricing theory the paper gives methods for testing hypotheses concerning the number of factors using series and kernel based nonparametric methods and also considers the problem of estimating the number of factors the methods are compared in a small simulation study and an application to determining the rank of demand systems is considered

efficient construction of canonical ladder forms for vector autoregressive processes the paper treats the problem of constructing canonical ladder realizations for vector autoregressive ar processes specified by their characteristic matrix polynomials the difficulty of this problem is rooted in the fact that the backward matrix polynomial corresponding to a given vector ar process is a nontrivial function of the forward matrix polynomial the construction calls for solving a discrete lyapunov equation in block controller form two efficient procedures for solving this equation are presented both requiring a number of operations that is proportional to at most the square of the model order applications of the new procedures to stability tests simulation of ar processes and model reduction are described

inferring the rank of a matrix abstract this paper considers methods of inference concerning the rank of matrix a based on an asymptotically normal estimate of and some identifiable specification for one such specification is 0 in which case one is interested in the rank of we first propose and examine the properties of a test of the hypothesis that the rank is of a given size against the alternative that the rank is larger we then look at the problem of estimating the rank of this matrix using model selection procedures and sequential hypothesis testing conditions for consistency of such procedures are obtained some economic applications are discussed and the various methods are compared in a monte carlo experiment

nonexistence and existence of various order integrals for two and three dimensional polynomial potentials the nonexistence of integrals of the motion which are sixth and fourth degree polynomials in the velocities was established for a range of polynomial potentials the first known systematic search for sixth order integrals was performed for cubic and quartic polynomial potentials it revealed that there exist no nondegenerate cases with such integrals for either potential similarly there exists no nondegenerate fifth or sixth degree polynomial potentials possessing quartic invariants a complete list of three dimensional cubic potentials with quadratic integrals is given all these integrable three dimensional potentials can be interpreted as orthogonal superpositions of known integrable two dimensional potentials possessing quadratic integrals they correspond to 3 of 11 possible coordinate systems in which three dimensional potentials separate