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

almost periodic higher order statistic estimation in this paper stochastic processes with higher order statistical functions decomposable into an almost periodic function plus a residual term not containing finite strength additive sinewave components are considered these processes arise in mobile communications when almost cyclostationary acs processes pass through time varying channels they include as special case the generalized almost cyclostationary processes which in turn include the acs processes in the paper the problem of estimating the fourier coefficients of the generalized fourier series expansion of the almost periodic component of higher order statistical functions is addressed estimators are proposed for cyclic temporal cross moment and cumulants they are proved to be mean square consistent and asymptotically complex normal under mild assumptions on the memory of the processes expressed in terms of summability of cross cumulants numerical results confirm the theoretical results and the derived rate of convergence to zero of bias and standard deviation of the estimators

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

numerical optimization for the purification of polluted shallow waters the optimal control theory allows us to design a wastewater treatment system in order to control marine pollution in any open area of shallow waters as was shown in previous works of the authors however there exist many closed areas for instance enclosed bays which present a serious quality problem caused by domestic industrial contaminants due to the insufficient seawater exchange in these areas it is necessary to consider a new technique in order to purify polluted waters promoting seawater exchange by the injection of clear water from the outer sea the aim of this paper is to determine the minimal quantity of injected water in order to purify the protected areas up to a fixed threshold we present the mathematical formulation of the continuous and discretized control problems and propose an algorithm for the numerical resolution finally we present numerical results obtained in the study of a real world problem

the multivariate complex normal distribution a generalization the multivariate complex normal distribution usually employed in the literature is a special case since certain restrictions have been imposed on the covariances of the real and imaginary parts of its variables a more general distribution is proposed of which the usual distribution is shown to be a special case

sampling of generalized almost cyclostationary signals in this paper the problem of sampling a continuous time generalized almost cyclostationary gacs signal is addressed the class of such nonstationary signals includes as a special case the almost cyclostationary acs signals acs signals filtered by some linear time variant channels are further examples it is shown that the discrete time signal constituted by the samples of a gacs signal is a discrete time acs signal thus discrete time acs signals can arise not only from the sampling of continuous time acs signals but from the sampling of a wider class of nonstationary signals as well namely the continuous time gacs signals in the paper relationships between generalized cyclic statistics of a continuous time gacs signal and cyclic statistics of the discrete time acs signal constituted by its samples are derived the problem of aliasing in the domain of the cycle frequencies is considered and a condition assuring that the cyclic temporal moment function of the discrete time signal can be obtained by sampling that of the continuous time signal is determined finally it is shown that starting from the sampled signal the gacs or acs nature of the continuous time signal can be conjectured provided that the analysis parameters such as the sampling period padding factor and data record length are properly chosen

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

two dimensional geodesic flows having first integrals of higher degree we present a family of riemannian metrics on two sphere having the property that the geodesic flows admit first integrals which are fiberwise homogeneous polynomials of degree greater than 2 they also have the property that all geodesics are closed

a control problem arising in the process of waste water purification abstract in this paper we state and solve an optimal control problem arisen from the management of the sewage disposal which is dumped into the sea through submarine outfalls firstly we fix oxygen and amount of organic matter as water quality indicators and we state a partial differential equations model to simulate them in a domain occupied by shallow waters constraints about water quality and economic objectives lead us to a pointwise optimal control problem with state and control constraints the theoretical analysis of the problem has been developed by the authors in martinez et al c r acad sci paris serie i 328 1999 35 martinez et al preprint dept matematica aplicada univ santiago de compostela spain 1998 we deal with the problem by using time and space discretizations and we propose two algorithms for the numerical resolution of the discretized problem finally we give numerical results obtained by applying the described techniques for a realistic problem posed in the ria of vigo spain

sampling of spectrally correlated processes the problem of sampling continuous time spectrally correlated sc processes is addressed sc processes have loeve bifrequency spectrum with spectral masses concentrated on a countable set of support curves in the bifrequency plane this class of nonstationary processes extends that of the almost cyclostationary processes and occurs in wideband mobile communications and spectral analysis with nonuniform spectral resolution the class of the discrete time sc processes is introduced and characterized it is shown that such processes can be obtained by uniformly sampling the continuous time sc processes sampling theorems are presented and sufficient conditions to avoid aliasing are provided applications are illustrated with reference to the problem of cross spectral analysis with nonuniform frequency resolution and the propagation of communications signals through mimo multipath doppler channels

canonical transformations of the extended phase space toda lattices and the stackel family of integrable systems we consider some examples of canonical transformations of the extended phase space which map a completely integrable system into another completely integrable system the proposed transformations are closed to the known maupertuis jacobi mappings and to the kepler change of the time using the kepler transformation we construct new integrable systems related to the toda lattices and the stackel family of integrable systems

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

cyclic statistic estimators with uncertain cycle frequencies for almost cyclostationary processes sufficient conditions are derived such that estimates of the second order cyclic probabilistic functions with estimated cycle frequencies are mean square consistent and asymptotically complex normal cyclic conjugate autocorrelation functions and cyclic conjugate spectra are considered under the derived conditions asymptotically as the data record length approaches infinity the estimates of cyclic statistics with estimated cycle frequencies have the same complex normal distribution as the case of exactly known cycle frequencies the results are applied to the detection of a moving cyclostationary source in the presence of strong doppler effect and for low values of snr

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

second order statistics of complex signals the second order statistical properties of complex signals are usually characterized by the covariance function however this is not sufficient for a complete second order description and it is necessary to introduce another moment called the relation function its properties and especially the conditions that it must satisfy are analyzed both for stationary and nonstationary signals this leads to a new perspective concerning the concept of complex white noise as well as the modeling of any signal as the output of a linear system driven by a white noise finally this is applied to complex autoregressive signals and it is shown that the classical prediction problem must be reformulated when the relation function is taken into consideration

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

nonparametric time series analysis for periodically correlated processes correlation functions of continuous time periodically correlated processes can be represented by a fourier series with coefficient functions it is shown that the usual estimator for stationary covariances formed from a single sample path of the process can be simply modified to provide a consistent in quadratic mean estimator for any of the coefficient functions resulting from the aforementioned representation it is shown that if the process is gaussian and b sub k tau is a fourier integral with respect to a density function g sub k lambda a two dimensional periodogram formed from a single sample function can be smoothed along a line of constant difference frequency to provide a consistent estimator for g sub k lambda this natural extension of the well known procedure for stationary processes provides a method for nonparametric spectral analysis of periodically correlated processes

on a class of integrable systems with a cubic first integral a few years ago selivanova gave an existence proof for some integrable models in fact geodesic flows on two dimensional manifolds with a cubic first integral however the explicit form of these models hinged on the solution of a nonlinear third order ordinary differential equation which could not be obtained we show that an appropriate choice of coordinates allows for integration and gives the explicit local form for the full family of integrable systems the relevant metrics are described by a finite number of parameters and lead to a large class of models mainly on the manifolds mathbb s 2 and mathbb h 2 many of these systems are globally defined and contain as special cases integrable systems due to goryachev chaplygin dullin matveev and tsiganov

a wastewater treatment problem study of the numerical convergence we deal with the numerical approximation of a problem related to the management of sewage disposal and the design of wastewater treatment systems we use a characteristics finite element method for the discretization of the state system the main difficulty is due to the existence of radon measures in the right hand side of the system we give convergence results and we present numerical experiences for a realistic problem

the cumulant theory of cyclostationary time series ii development and applications for pt i see ibid vol 42 no 12 p 3387 3408 1994 the development of the theory of nonlinear processing of cyclostationary time series that is initiated in part i is continued a new type of cumulant for complex valued variables is introduced and used to generalize the temporal and spectral moments and cumulants for cyclostationary time series from real valued to complex valued time series the relations between the temporal and spectral moments and cumulants at the inputs and outputs of several signal processing operations are determined formulas for the temporal and spectral cumulants of complex valued pulse amplitude modulated time series are derived estimators for the temporal moments and cumulants and for the cyclic polyspectra are presented and their properties are discussed the performance of these estimators is illustrated by several computer simulation examples for pulse amplitude modulated time series the theory is applied to the problems of weak signal detection and interference tolerant time delay estimation

estimation of second order cross moments of generalized almost cyclostationary processes in this paper the problem of estimating second order cross moments of generalized almost cyclostationary gacs processes is addressed gacs processes have statistical functions that are almost periodic functions of time whose generalized fourier series expansions have both frequencies and coefficients that depend on the lag shifts of the processes the class of such nonstationary processes includes the almost cyclostationary acs processes which are obtained as a special case when the frequencies do not depend on the lag shifts acs processes filtered by doppler channels and communications signals with time varying parameters are further examples it is shown that the second order cross moment of two jointly gacs processes is completely characterized by the cyclic cross correlation function moreover it is proved that the cyclic cross correlogram is an asymptotically normal mean square consistent estimator of the cyclic cross correlation function furthermore it is shown that well known consistency results for acs processes can be obtained by specializing the results of this paper

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

completely integrable 2d lagrangian systems and related integrable geodesic flows on various manifolds in this study we have formulated a theorem that generates deformations of the natural integrable conservative systems in the plane into integrable systems on riemannian and other manifolds by introducing additional parameters into their structures the relation of explicit solutions of the new and the original dynamics to the corresponding jacobi maupertuis geodesic flow is clarified for illustration we apply the result to three concrete examples of the many available integrable systems in the literature complementary integrals in those systems are polynomial in velocity with degrees 3 4 and 6 respectively as a special case of the first deformed system a new several parameter family of integrable mechanical systems and geodesic flows on s2 is constructed

improving the environmental impact of wastewater discharges with a specific simulation optimization software wastewater treatment has become one of the most important environmental problems especially for the particular case of urban areas in this work we use numerical simulation and combine optimal control theory of partial differential equations with multi objective optimization techniques to formulate and solve this type of problems specifically the main objective of this paper is to detail the mathematical tools that we can use in order to assess in decision making related to determining the sewage depuration intensities in a wastewater treatment system we show the usefulness of our techniques by solving a realistic problem posed in the estuary of vigo nw spain with the free software sos recently developed by the authors

theoretical and numerical analysis of an optimal control problem related to wastewater treatment in this work we deal with the design and management of wastewater treatment systems mainly the disposal of sea outfalls discharging polluting effluent from a sewerage system this problem is formulated as a pointwise optimal control problem with state and control constraints the main difficulties arise from the lack of regularity of the second member in the state system and from the pointwise constraints on the state variables we develop the theoretical analysis of the problem we propose an algorithm for its numerical resolution and finally we give results for a realistic problem posed in the ria of vigo spain

asymptotic moments of estimated cyclic correlation matrices the first and second moments of the cyclic cross correlogram matrix of two vector valued cyclostationary processes are derived in this paper these results generalize those of hurd 1989 to accommodate incommensurate cycle frequencies and complex vector valued discrete time nongaussian processes as an example application of the results the moments of the left singular vectors of the cyclic cross correlogram matrix are derived which are needed in the performance analysis of subspace fitting methods of direction of arrival estimation for cyclostationary signals including even those methods that do not exploit cyclostationarity but are often applied in practice to cyclostationary signals

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

stackelberg strategies for wastewater management management of wastewater has become a critical environmental problem during past decades specially for the particular case of urban areas in this work we use numerical simulation and combine optimal control theory of partial differential equations with multi objective optimization techniques to formulate and solve this type of problems from a new original hierarchical viewpoint with a leader and a follower stackelberg strategies specifically the main aim of this paper deals with applying these stackelberg techniques to the environmental problem related to determining the optimal location and the optimal purification profile when building a new treatment plant in a wastewater depuration system we analyze the control problem introduce a numerical algorithm and show the possibilities of our approach by solving a realistic problem posed in the estuary of vigo nw spain

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

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

complete involutive algebras of functions on cotangent bundles of homogeneous spaces homogeneous spaces of all compact lie groups admit riemannian metrics with completely integrable geodesic flows by means of c smooth integrals 9 10 the purpose of this paper is to give some constructions of complete involutive algebras of analytic functions polynomial in velocities on the co tangent bundles of homogeneous spaces of compact lie groups this allows us to obtain new integrable riemannian and sub riemannian geodesic flows on various homogeneous spaces such as stiefel manifolds flag manifolds and orbits of the adjoint actions of compact lie groups

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

linear time variant transformations of generalized almost cyclostationary signals i theory and method the problem of linear time variant filtering is addressed in the fraction of time fot probability framework the adopted approach which is an alternative to the classical stochastic one provides a statistical characterization of the system in terms of time averages of functions of time rather than ensemble averages of stochastic processes thus it is particularly useful when stochastic systems transform ergodic input signals into nonergodic output signals as it happens with several channel models encountered in practice the analysis is carried out with reference to the wide class of the generalized almost cyclostationary signals which includes as a special case the class of almost cyclostationary signals in this paper systems are classified as deterministic or random in the fot probability framework moreover the new concept of expectation in the fot probability framework of the impulse response function of a system is introduced for the linear time variant systems the higher order system characterization in the time domain is provided in terms of the system temporal moment function which is the kernel of the operator that transforms the additive sinewave components contained in the input lag product into the additive sinewave components contained in the output lag product moreover the higher order characterization in the frequency domain is also provided and input output relationships are derived in terms of temporal and spectral moment and cumulant functions developments and examples of application of the theory introduced here are presented in part ii of this two part paper

on a family of integrable systems on s2 with a cubic integral of motion we discuss a family of integrable systems on the sphere s2 with an additional third order integral in momenta this family contains the coryachev chaplygin top the goryachev system the system recently discovered by dullin and matveev and two new integrable systems on the non physical sphere with zero radius all these systems are isomorphic to each other

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

the higher order theory of generalized almost cyclostationary time series in this paper the class of generalized almost cyclostationary gacs time series is introduced time series belonging to this class are characterized by multivariate statistical functions that are almost periodic functions of time whose fourier series expansions can exhibit coefficients and frequencies depending on the lag shifts of the time series moreover the union over all the lag shifts of the lag dependent frequency sets is not necessarily countable almost cyclostationary acs time series turn out to be the subclass of gacs time series for which the frequencies do not depend on the lag shifts and the union of the above mentioned sets is countable the higher order characterization of gacs time series in the strict and wide sense is provided it is shown that the characterization in terms of cyclic moment and cumulant functions is inadequate for those gacs time series that are not acs then generalized cyclic moment and cumulant functions in both the time and frequency domains are introduced finally the problem of estimating the introduced generalized cyclic statistics is addressed and two examples of gacs time series are considered

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

linear time variant transformations of generalized almost cyclostationary signals ii development and applications for pt i see ibid vol 50 no 12 p 2947 61 2000 in part i the problem of the linear time variant ltv filtering is addressed in the fraction of time fot probability framework the adopted approach which is an alternative to the classical stochastic one provides a statistical characterization of the systems in terms of functions that can be estimated by a single time series the analysis is carried out with reference to the wide class of the generalized almost cyclostationary gacs signals which includes as a special case the class of the almost cyclostationary acs signals examples of applications and developments of the theory introduced in part i are presented here in part ii specifically the countability of the set of the output cycle frequencies is studied with reference to linear time variant systems for both acs and gacs not containing any acs component input signals thus the linear almost periodically time variant filtering and the product modulation are considered in detail moreover several doppler channel models are analyzed in all these examples it is shown that the fot probability approach allows one to characterize the system and its output in terms of statistical functions that can be measured by a single time series furthermore the usefulness of considering the linear filtering problem within the class of the gacs signals is clarified and several pitfalls arising from continuing to adopt for the observed time series the acs model when an increase in the data record length makes the gacs model more appropriate are pointed out

uncertainty in measurements on spectrally correlated stochastic processes in this paper the class of the spectrally correlated stochastic processes is introduced processes belonging to this class exhibit a loe spl grave ve 1963 bifrequency spectrum with spectral masses concentrated on a countable set of support curves in the bifrequency plane thus such processes have spectral components that are correlated the introduced class generalizes the almost cyclostationary acs processes that are obtained as a special case when the separation between correlated spectral components assumes values only in a countable set in such a case the support curves are lines with unit slope for the spectrally correlated processes the amount of spectral correlation existing between two separate spectral components is characterized by the bifrequency spectral correlation density function which is the density of the loe spl grave ve bifrequency spectrum on its support curves it is shown that in general when the location of the support curves is unknown the time smoothed cross periodogram can provide a reliable low bias and variance single sample path based estimate of the bifrequency spectral correlation density function in those points of the bifrequency plane where the slope of the support curves is not too far from unity moreover there exists a tradeoff between the departure of the nonstationarity from the almost cyclostationarity and the reliability of spectral correlation measurements obtainable by a single sample path furthermore in general the estimate accuracy cannot be improved as wished by increasing the data record length and the spectral resolution

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

consistent estimation of the cyclic autocorrelation the cyclic autocorrelation is often used to describe nonstationary random processes the authors investigate the conditions under which the cyclic autocorrelation can be estimated consistently in mean square for discrete time gaussian processes they extend and generalize results of hurd 1989 and refine results of boyles and gardner 1983 they derive necessary and sufficient conditions for consistency in mean square of an estimator which are in the form of a single sum of autocorrelation coefficients in the form of a double sum of autocorrelation coefficients in the bifrequency domain and in terms of the average spectrum they also discuss the rate of convergence for this estimator

second order complex random vectors and normal distributions complex random vectors are usually described by their covariance matrix this is insufficient for a complete description of second order statistics and another matrix called the relation matrix is necessary some of its properties are analyzed and used to express the probability density function of normal complex vectors various consequences are presented

variable separation theory for the null hamilton jacobi equation the theory of the separation of variables for the null hamilton jacobi equation h 0 is systematically revisited and based on levi civita separability conditions with lagrangian multipliers the separation of the null equation is shown to be equivalent to the ordinary separation of the image of the original hamiltonian under a generalized jacobi maupertuis transformation the general results are applied to the special but fundamental case of the orthogonal separation of a natural hamiltonian with a fixed value of the energy the separation is then related to conditions which extend those of stackel and kalnins and miller for the null geodesic case and it is characterized by the existence of conformal killing two tensors of special kind

the cumulant theory of cyclostationary time series i foundation the problem of characterizing the sine wave components in the output of a polynomial nonlinear system with a cyclostationary random time series input is investigated the concept of a pure nth order sine wave is introduced and it is shown that pure nth order sine wave strengths in the output time series are given by scaled fourier coefficients of the polyperiodic temporal cumulant of the input time series the higher order moments and cumulants of narrowband spectral components of time series are defined and then idealized to the case of infinitesimal bandwidth such spectral moments and cumulants are shown to be characterized by the fourier transforms of the temporal moments and cumulants of the time series it is established that the temporal and spectral cumulants have certain mathematical and practical advantages over their moment counterparts to put the contributions of the paper in perspective a uniquely comprehensive historical survey that traces the development of the ideas of temporal and spectral cumulants from their inception is provided

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

asymptotic theory of mixed time averages and kth order cyclic moment and cumulant statistics we generalize parzen s 1961 analysis of asymptotically stationary processes to mixtures of deterministic stationary nonstationary and generally complex time series under certain mixing conditions expressed in terms of joint cumulant summability we show that time averages of such mixtures converge in the mean square sense to their ensemble averages we additionally show that sample averages of arbitrary orders are jointly complex normal and provide their covariance expressions these conclusions provide us with statistical tools that treat random and deterministic signals on a common framework and are helpful in defining generalized moments and cumulants of mixed processes as an important consequence we develop consistent and asymptotically normal estimators for time varying and cyclic moments and cumulants of kth order cyclostationary processes and provide computable variance expressions some examples are considered to illustrate the salient features of the analysis

discrete time estimation of second order statistics of generalized almost cyclostationary processes in this paper the problem of estimating second order statistics of continuous time generalized almost cyclostationary gacs processes is addressed gacs processes in the wide sense have autocorrelation function almost periodic in time whose generalized fourier series expansion has both frequencies and coefficients that depend on the lag shifts almost cyclostationary acs processes are obtained as a special case when the frequencies do not depend on the lag shifts acs processes filtered by doppler channels and communications signals with time varying parameters are further examples it is shown that continuous time gacs processes do not have a discrete time counterpart the discrete time cyclic cross correlogram of the discrete time acs processes obtained by uniformly sampling gacs processes is considered as estimator of samples of the continuous time cyclic cross correlation function the asymptotic performance analysis is carried out by resorting to the hybrid cyclic cross correlogram which is partially continuous time and partially discrete time its mean square consistency and asymptotic complex normality as the number of data samples approaches infinity and the sampling period approaches zero are proved under mild conditions on the regularity of the fourier series coefficients and the finite or practically finite memory of the processes expressed in terms of summability of cumulants it is shown that the asymptotic properties of the hybrid cyclic cross correlogram are coincident with those of the continuous time cyclic cross correlogram hence discrete time estimation does not give rise to any loss in asymptotic performance with respect to continuous time estimation