trace expansions for elliptic cone operators with stationary domains we analyze the behavior of the trace of the resolvent of an ellip tic cone differential operator as the spectral parameter tends to infinity the resolvent splits into two components one associated with the minimal exten sion of the operator and another of finite rank depending on the particular choice of domain we give a full asymptotic expansion of the first compo nent and expand the component of finite rank in the case where the domain is stationary the results make use and develop further our previous investi gations on the analytic and geometric structure of the resolvent the analysis of nonstationary domains considerably more intricate is pursued elsewhere

filtering and frequency interpretations of singular spectrum analysis new filtering and spectral interpretations of singular spectrum analysis ssa are provided it is shown that the variables reconstructed from diagonal averaging of reduced rank approximations to the trajectory matrix can be obtained from a noncausal convolution filter with zero phase characteristics the reconstructed variables are readily constructed using a two pass filtering algorithm that is well known in the signal processing literature when the number of rows in the trajectory matrix is much larger than number of columns many results reported in the signal processing literature can be used to derive the properties of the resulting filters and their spectra new features of the reconstructed series are revealed using these results two examples are used to illustrate the results derived in this paper

optimal shadowing and noise reduction abstract the shadowing problem is that of finding a deterministic orbit as close as possible to a given noisy orbit we present an optimal solution to this problem in the sense of least mean squares which also provides an effective and convenient numerical method for noise reduction for data generated by a dynamical system given a noisy orbit y and a dynamical system we derive a set of nonlinear equations whose solution x is the deterministic orbit with the smallest possible euclidean distance to y we present a numerical method for solving these equations the quality of the solution depends on the initial noise level when is known exactly the noise can be reduced to machine precision over long trajectory segments with higher noise levels there are regions where the algorithm has difficulty but significant overall noise reductions are still achieved if must be learned from the data the noise reduction is limited by the accuracy of the learning algorithm and the number of available data points but large reductions are still possible in some cases

angle dependent ultrasonic detection and imaging of brachytherapy seeds using singular spectrum analysis transrectal ultrasound guided brachytherapy uses small titanium shelled radioactive seeds to locally treat prostate cancer during the implantation procedure needles inserted transperitoneally cause gland movement resulting in seed misplacement and suboptimal dosimetry in a previous study an algorithm based on singular spectrum analysis ssa applied to envelope detected ultrasound signals was proposed to determine seed locations j mamou and e j feleppa j acoust soc am 121 1790 1801 2007 successful implementation of the ssa algorithm could allow correcting dosimetry errors during the implantation procedure the algorithm demonstrated promise when the seed orientation was parallel to the needle and normal to the ultrasound beam in this present study the algorithm was tested when the seed orientation deviated up to 22 degrees from normality experimental data from a seed in an ideal environment and in beef were collected with a single element spherically focused 5 mhz transducer simulations were designed and evaluated with the algorithm finally objective quantitative scoring metrics were developed to evaluate the algorithm performance and for comparison with b mode images the results quantitatively established that the ssa algorithm always outperformed b mode images and that seeds could be detected accurately up to a deviation of approximately 10 degrees

boundary value problems for first order elliptic wedge operators we develop an elliptic theory based in l 2 of boundary value problems for general wedge differential operators of first order under only mild assumptions on the boundary spectrum in particular we do not require the indicial roots to be constant along the base of the boundary fibration our theory includes as a special case the classical theory of elliptic boundary value problems for first order operators with and without the shapiro lopatinskii condition and can be thought of as a natural extension of that theory to the geometrically and analytically relevant class of wedge operators wedge operators arise in the global analysis on manifolds with incomplete edge singularities our theory settles in the first order case the long standing open problem to develop a robust elliptic theory of boundary value problems for such operators

bounded imaginary powers of cone differential operators on higher order mellin sobolev spaces and applications to the cahn hilliard equation abstract extending earlier results on the existence of bounded imaginary powers for cone differential operators on weighted l p spaces h p 0 b over a manifold with conical singularities we show how the same assumptions also yield the existence of bounded imaginary powers on higher order mellin sobolev spaces h p s b s 0 as an application we consider the cahn hilliard equation on a manifold with possibly warped conical singularities relying on our work for the case of straight cones we first establish r sectoriality and thus maximal regularity for the linearized equation and then deduce the existence of a short time solution with the help of a theorem by clement and li we also obtain the short time asymptotics of the solution near the conical point

a nonlinear filtering technique for multi oscillator systems abstract the recent application of methods from nonlinear systems theory to the analysis of time series can be used to provide a novel technique for filtering such series when they represent the coupled output of several distinct oscillators the method is a nonlinear analogue of fourier spectral analysis but is potentially more powerful in that it allows for local in time cycle to cycle variability in oscillator amplitude

integration by parts homogeneous chaos expansions and smooth densities by iterating a martingale representation result a homogeneous chaos expansion is obtained using the martingale representation the integration by parts formula of the malliavin calculus is derived using properties of stochastic flows the infinite dimensional calculus of variations is not required

operator theoretic framework for forecasting nonlinear time series with kernel analog techniques abstract kernel analog forecasting kaf alternatively known as kernel principal component regression is a kernel method used for nonparametric statistical forecasting of dynamically generated time series data this paper synthesizes descriptions of kernel methods and koopman operator theory in order to provide a single consistent account of kaf the framework presented here illuminates the property of the kaf method that under measure preserving and ergodic dynamics it consistently approximates the conditional expectation of observables that are acted upon by the koopman operator of the dynamical system and are conditioned on the observed data at forecast initialization more precisely kaf yields optimal predictions in the sense of minimal root mean square error with respect to the invariant measure in the asymptotic limit of large data the presented framework facilitates moreover the analysis of generalization error and quantification of uncertainty extensions of kaf to the construction of conditional variance and conditional probability functions as well as to non symmetric kernels are also shown illustrations of various aspects of kaf are provided with applications to simple examples namely a periodic flow on the circle and the chaotic lorenz 63 system

the swift hohenberg equation on conic manifolds abstract we consider the swift hohenberg equation on manifolds with conical singularities and show existence uniqueness and maximal regularity of the short time solution in terms of mellin sobolev spaces moreover we give a necessary and sufficient condition so that the above solution exists for all times space asymptotic expansion of the solution near the singularity is also provided and its relation to the local geometry is shown the same problem is considered on closed manifolds and similar results are obtained by using the above singular analysis theory

modeling daily realized futures volatility with singular spectrum analysis abstract using singular spectrum analysis ssa we model the realized volatility and logarithmic standard deviations of two important futures return series the realized volatility and logarithmic standard deviations are constructed following the methodology of andersen et al j am stat ass 96 2001 42 55 using intra day transaction data we find that ssa decomposes the volatility series quite well and effectively captures both the market trend accounting for about 34 38 of the total variance in the series and more importantly a number of underlying market periodicities reliable identification of any periodicities is extremely important for options pricing and risk management and we believe that ssa can be a useful addition to the financial practitioners toolbox

properties of solutions of stochastic differential equations with continuous state dependent switching abstract this work is concerned with several properties of solutions of stochastic differential equations arising from hybrid switching diffusions the word hybrid highlights the coexistence of continuous dynamics and discrete events the underlying process has two components one component describes the continuous dynamics whereas the other is a switching process representing discrete events one of the main features is the switching component depending on the continuous dynamics in this paper weak continuity is proved first then continuous and smooth dependence on initial data are demonstrated in addition it is shown that certain functions of the solutions verify a system of kolmogorov s backward differential equations moreover rates of convergence of numerical approximation algorithms are dealt with

interpretation of biomechanical simulations of normal and chaotic vocal fold oscillations with empirical eigenfunctions empirical orthogonal eigenfunctions are extracted from biomechanical simulations of normal and chaotic vocal fold oscillations for normal phonation two dominant empirical eigenfunctions capture the vibration patterns of the folds and exhibit a 1 1 entrainment the eigenfunctions show some correspondence to theoretical low order normal modes of a simplified three dimensional elastic continuum and to the normal modes of a linearized two mass model the eigenfunctions also facilitate a physical interpretation of energy transfer mechanisms in vocal fold dynamics subharmonic regimes and chaotic oscillations are observed during simulations of a lax cover in which case at least three empirical eigenfunctions are necessary to capture the resulting vocal fold oscillations these chaotic oscillations might be understood in terms of a desynchronization of a few of the low order modes and may be related to mechanisms of creaky voice or vocal fry furthermore some of the empirical eigenfunctions captured during complex oscillations correspond to higher order normal modes described in earlier theoretical work the empirical eigenfunctions may also be useful in the design of lower order models valid over the range for which the empirical eigenfunctions remain more or less constant and may help facilitate bifurcation analyses of the biomechanical simulation

zeta regularized determinants for conic manifolds we study relative zeta regularized determinants of laplace type operators on compact conic manifolds we establish gluing formulae for relative zeta regularized determinants for arbitrary self adjoint extensions of the laplace beltrami operator we express the relative determinants for these as a ratio of the determinants of certain finite matrices for the self adjoint extensions corresponding to dirichlet and neumann conditions the formula is particularly simple and elegant

singular spectrum analysis in nonlinear dynamics with applications to paleoclimatic time series we distinguish between two dimensions of a dynamical system given by experimental time series statistical dimension gives a theoretical upper bound for the minimal number of degrees of freedom required to describe tje attractor up to the accuracy of the data taking into account sampling and noise problems the dynamical dimension is the intrinsic dimension of the attractor and does not depend on the quality of the data singular spectrum analysis ssa provides estimates of the statistical dimension ssa also describes the main physical phenomena reflected by the data it gives adaptive spectral filters associated with the dominant oscillations of the system and clarifies the noise characteristics of the data we apply ssa to four paleoclimatic records the principal climatic oscillations and the regime changes in their amplitude are detected about 10 degrees of freedom are statistically significant in the data large noise and insufficient sample length do not allow reliable estimates of the dynamical dimension

characterization of a two parameter mixed mode electrochemical behavior regime using neural networks abstract we use nonlinear signal processing techniques based on artificial neural networks to construct a continuous time model set of ordinary differential equations odes from experimental observations of mixed mode oscillations during the galvanostatic oxidation of hydrogen on platinum in the presence of bismuth and chloride ions the data was in the form of time series of the potential for different values of the applied current and chloride ion concentration we use the model to reconstruct the experimental dynamics and to explore the associated bifurcation structures in phase space using numerical bifurcation techniques we locate stable and unstable periodic solutions calculate eigenvalues and identify bifurcation points this approach constitutes a promising data post processing procedure for investigating phase space and parameter space of real experimental systems it allows us to infer phase space structures which the experiments can only probe with limited measurement precision and only at a discrete number of operating parameter settings for example the fitted model suggests the existence of a sub critical hopf bifurcation near the range of parameters probed in the experiments this might explain the experimental difficulty in locating small amplitude simple oscillations

dimension measurement on high dimensional systems abstract using experimental data from a variety of laboratory electrical systems the upper limits on dimension measurement are examined and it is shown that contrary to standard scaling estimates it can be possible to reliably measure correlation dimensions larger than 10 the practical implications of the constraints on such a measurement are examined including the role of statistical errors geometrical artifacts dimension algorithm metric choice and the process power spectrum these ideas are applied to electrical 1 f noise from both macroscopic and microscopic sources to show that it cannot be explained by a deterministic model with fewer than roughly 20 degrees of freedom

nonlinear laplacian spectral analysis of rayleigh benard convection the analysis of physical datasets using modern methods developed in machine learning presents unique challenges and opportunities these datasets typically feature many degrees of freedom which tends to increase the computational cost of statistical methods and complicate interpretation in addition physical systems frequently exhibit a high degree of symmetry that should be exploited by any data analysis technique the classic problem of rayleigh benard convection in a periodic domain is an example of such a physical system with trivial symmetries this article presents a technique for analyzing the time variability of numerical simulations of two dimensional rayleigh benard convection at large aspect ratio and intermediate rayleigh number the simulated dynamics are highly unsteady and consist of several convective rolls that are distributed across the domain and oscillate with a preferred frequency intermittent extreme events in the net heat transfer as quantified by the time weighted probability distribution function of the nusselt number are a hallmark of these simulations nonlinear laplacian spectral analysis nlsa is a data driven method which is ideally suited for the study of such highly nonlinear and intermittent dynamics but the trivial symmetries of the rayleigh benard problem such as horizontal shift invariance can mask the interesting dynamics to overcome this issue the vertical velocity is averaged over parcels of similar temperature and height which substantially compresses the size of the dataset and removes trivial horizontal symmetries this isothermally averaged dataset which is shown to preserve the net convective heat flux across horizontal surfaces is then used as an input to nlsa the analysis generates a small number of orthogonal modes which describe the spatiotemporal variability of the heat transfer a regression analysis shows that the extreme events of the net heat transfer are primarily associated with a family of modes with fat tailed probability distributions and low frequency temporal power spectra on the other hand the regular oscillation of the heat transfer is associated with a pair of modes with nearly uniform probability distributions physical mechanisms for the regular oscillation and the extreme heat transfer events are hypothesized based on an analysis of the spatio temporal structure of these modes finally proposals are made for this approach to be applied to the study of other problems in turbulent convection including three dimensional rayleigh benard convection and moist atmospheric convection

convergence of the k 2 entropy for random noises with power law spectra abstract we discuss a counter example to the traditional expectation that stochastic processes lead to a non convergence of the k 2 entropy in computed or measured time series we consider a simple class of colored random noises with power law spectra which have already been shown to generate finite and predictable values of the correlation dimension we show that the k 2 entropy of these random noises converges as well these results imply that the observation of a finite or null value of the k 2 entropy in the analysis of data is not enough to infer that the system is dominated by a deterministic process such as low dimensional chaos

the very unusual properties of the resolvent heat kernel and zeta function for the operator d2 dr2 1 4r2 in this paper we analyze the resolvent the heat kernel and the spectral zeta function of the operator d2 dr2 1 4r2 over the finite interval the structural properties of these spectral functions depend strongly on the chosen self adjoint realization of the operator a choice being made necessary because of the singular potential present only for the friedrichs realization standard properties are reproduced for all other realizations highly nonstandard properties are observed in particular for k n we find terms like logt k in the small t asymptotic expansion of the heat kernel furthermore the zeta function has s 0 as a logarithmic branch point

reverse time diffusion processes corresp given a diffusion process on bf r n described by a stochastic differential equation forward in time we develop a corresponding stochastic differential equation in reverse time which yields the same sample paths this stochastic differential equation can be used in problems of estimation and smoothing

bounded imaginary powers of differential operators on manifolds with conical singularities we study the minimal and maximal closed extension of a differential operator a on a manifold b with conical singularities when a acts as an unbounded operator on weighted l p spaces over b 1 p under suitable ellipticity assumptions we can define a family of complex powers a z z we also obtain sufficient information on the resolvent of a to show the boundedness of the purely imaginary powers examples concern unique solvability and maximal regularity for the solution of the cauchy problem for the laplacian on conical manifolds as well as certain quasilinear diffusion equations

topology from time series we describe methods for the study of topological properties of the invariant manifolds of experimental dynamical systems we explain how to compute such invariants as the euler characteristic and betti numbers using time series data and suggest a number of potential applications

state space reconstruction parameters in the analysis of chaotic time series the role of the time window length abstract the most common state space reconstruction method in the analysis of chaotic time series is the method of delays mod many techniques have been suggested to estimate the parameters of mod i e the time delay and the embedding dimension m we discuss the applicability of these techniques with a critical view as to their validity and point out the necessity of determining the overall time window length w for successful embedding emphasis is put on the relation between w and the dynamics of the underlying chaotic system and we suggest to set w p the mean orbital period p is approximated from the oscillations of the time series the procedure is assessed using the correlation dimension for both synthetic and real data for clean synthetic data values of w larger than p always give good results given enough data and thus p can be considered as a lower limit w p for noisy synthetic data and real data an upper limit is reached for w which approaches p for increasing noise amplitude

resolvents of cone pseudodifferential operators asymptotic expansions and applications we study the structure and asymptotic behavior of the resolvent of elliptic cone pseudodifferential operators acting on weighted sobolev spaces over a compact manifold with boundary we obtain an asymptotic expansion of the resolvent as the spectral parameter tends to infinity and use it to derive corresponding heat trace and zeta function expansions as well as an analytic index formula

detecting switch dynamics in chaotic time waveform using a parametrized family of nonlinear predictors abstract an algorithm is presented for detecting switch dynamics in chaotic time waveform by the switch dynamics we mean that the chaotic time waveform is measured from a dynamical system whose bifurcation parameters are occasionally switched among a set of slightly different parameter values first the switched chaotic time waveform is divided into windows of short term time waveforms from the set of windowed time waveforms qualitatively similar parametrized family of nonlinear predictors is constructed qualitatively similar parametrized family means that the family of nonlinear predictors exhibits qualitatively similar bifurcation phenomena as the original by characterizing the windows of short term chaotic time waveforms in terms of the qualitative parameters of nonlinear predictors switch dynamics of their associated bifurcation parameters are detected for the lorenz equations the rossler equations and the mackey glass equations efficiency of the algorithm is demonstrated in the experiment chaotic time waveforms contaminated with observational noise is considered

self consistent asset pricing models we discuss the foundations of factor or regression models in the light of the self consistency condition that the market portfolio and more generally the risk factors is are constituted of the assets whose returns it is they are supposed to explain as already reported in several articles self consistency implies correlations between the return disturbances as a consequence the alphas and betas of the factor model are unobservable self consistency leads to renormalized betas with zero effective alphas which are observable with standard ols regressions when the conditions derived from internal consistency are not met the model is necessarily incomplete which means that some sources of risk cannot be replicated or hedged by a portfolio of stocks traded on the market even for infinite economies analytical derivations and numerical simulations show that for arbitrary choices of the proxy which are different from the true market portfolio a modified linear regression holds with a non zero value ai at the origin between an asset i s return and the proxy s return self consistency also introduces orthogonality and normality conditions linking the betas alphas as well as the residuals and the weights of the proxy portfolio two diagnostics based on these orthogonality and normality conditions are implemented on a basket of 323 assets which have been components of the s p500 in the period from january 1990 to february 2005 these two diagnostics show interesting departures from dynamical self consistency starting about 2 years before the end of the internet bubble assuming that the capm holds with the self consistency condition the ols method automatically obeys the resulting orthogonality and normality conditions and therefore provides a simple way to selfconsistently assess the parameters of the model by using proxy portfolios made only of the assets which are used in the capm regressions finally the factor decomposition with the self consistency condition derives a risk factor decomposition in the multi factor case which is identical to the principal component analysis pca thus providing a direct link between model driven and data driven constructions of risk factors this correspondence shows that pca will therefore suffer from the same limitations as the capm and its multi factor generalization namely lack of out of sample explanatory power and predictability in the multi period context the self consistency conditions force the betas to be time dependent with specific constraints r 2007 elsevier b v all rights reserved

adjoints of elliptic cone operators we study the adjointness problem for the closed extensions of a general b elliptic operator a x diff m b m e 0 initially defined as an unbounded operator a c c m e x l 2 b m e x l 2 b m e inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i the case where a is a symmetric semibounded operator is of particular interest and we give a complete description of the domain of the friedrichs extension of such an operator

minute ahead stock price forecasting based on singular spectrum analysis and support vector regression abstract time series modeling and forecasting is an essential and hard task in financial engineering and optimization various models have been proposed in the literature and tested on daily data however a limited attention has been given to intraday data in this regard the current work presents a model for intraday stock price prediction that uses singular spectrum analysis ssa and support vector regression svr coupled with particle swarm optimization pso in particular the ssa decomposes stock price time series into a small number of independent components used as predictors the svr is applied to the task of forecasting and pso is employed to optimize svr parameters the performance of our proposed model is compared to the performance of four models widely used in financial prediction the wavelet transform wt coupled with feedforward neural network ffnn autoregressive moving average arma process polynomial regression polyreg and naive model finally the mean absolute error mae mean absolute percentage error mape and the root mean of squared errors rmse are used as main performance metrics by applying all models to six intraday stock price time series the forecasting results from simulations show that the presented ssa pso svr largely outperforms the conventional wt ffnn arma polynomial regression and naive model in terms of mae mape and rmse therefore our proposed predictive system ssa pso svr shows evident potential for noisy financial time series analysis and forecasting

recognizing chaotic time waveforms in terms of a parametrized family of nonlinear predictors abstract consider a chaotic dynamical system which exhibits a variety of chaotic time waveforms with a change in the bifurcation parameters this paper presents an algorithm for estimating the underlying bifurcation parameters of the chaotic time waveforms in experimental situation in which no a priori analytical knowledge of the dynamical system is available first we construct qualitatively similar parametrized family of nonlinear predictors only from several sets of chaotic time waveforms qualitatively similar parametrized family means that the family of nonlinear predictors exhibits qualitatively similar bifurcation phenomena as the original chaotic time waveforms are then characterized in terms of the qualitatively similar bifurcation parameters of the nonlinear predictors we call the characterization of chaotic time waveforms in terms of the underlying bifurcation parameters chaotic time waveform recognition several numerical experiments using the rossler equations show the efficiency of the algorithm the effect of observational noise included in chaotic time waveforms is also considered

on the detection of determinism in a time series the nonlinearly scaled distributions of the strengths of the orthogonal modes in the data of a time series are compared with that derived from its surrogate counterpart to assess its chaoticity or the stochastic nature chaoticity manifests in the relatively decreasing strengths of the weaker modes with increasing dimension of the orthogonal spaces mapping the process the distributions of the strengths of the modes for the data and the surrogate are compared statistically singular value decomposition is used for the implementation the ability of the proposed method to distinguish between chaotic and stochastic dynamics is demonstrated through a number of simulated series including linear and nonlinear stochastic noise and processes described by lorenz rossler and mackey glass equations henon map and ecological map and some real life examples

on the non equivalence of observables in phase space reconstructions from recorded time series in practical problems of phase space reconstruction it is usually the case that the reconstruction is much easier using a particular recorded scalar variable this seems to contradict the general belief that all variables of a dynamical system are equivalent in phase space reconstruction problems this paper will argue that in many cases the choice of a particular scalar time series from which to reconstruct the original dynamics could be critical it is argued that different dynamical variables do not provide the same level of information observability of the underlying dynamics and as a consequence the quality of a global reconstruction critically depends on the recorded variable examples in which the choice of observables is critical are discussed and the level of information contained in a given variable is quantified in the case where the original system is known a clear example of such a situation arises in the r ossler system for which the performance of a global vector field reconstruction technique is investigated using time series of variables x y or z taken one at a time

a note on regime switching kolmogorov s forward and backward equations using stochastic flows abstract using stochastic flows probabilistic solutions of markovian regime switching forward and backward kolmogorov s equations are discussed these equations may be related to some pricing equations in mathematical finance their solutions are derived by differentiating a family of conditional expectations

resolvents of elliptic cone operators we prove the existence of sectors of minimal growth for general closed extensions of elliptic cone operators under natural ellipticity conditions this is achieved by the construction of a suitable parametrix and reduction to the boundary special attention is devoted to the clarification of the analytic structure of the resolvent

dimensions and entropies of strange attractors from a fluctuating dynamics approach abstract it is shown that the fluctuations in the divergence of near by trajectories on strictly deterministic strange attractors can be modelled by stochastic concepts in particular we propose kramers moyal type equations for correlation functions between points on the attractor the drift terms are the lyapunov exponents the diffusion terms depend on the above fluctuations from this we obtain bounds on generalized dimensions and entropies numerical results show that in nearly all studied cases henon map zaslavskii map mackey glass eq the attractors are fractal measures in the sense of farmer information dimension hausdorff dimension metric entropy topological entropy

direct solutions of kolmogorov s equations by stochastic flows abstract solutions of kolmogorov s forward and backward equations are obtained by considering a family of conditional expectations and the use of stochastic flows to justify differentiation in the time variable

symplectic geometry spectrum analysis of nonlinear time series various time series decomposition techniques including wavelet transform singular spectrum analysis empirical mode decomposition and independent component analysis have been developed for non linear dynamic system analysis in this paper we describe a symplectic geometry spectrum analysis sgsa method to decompose a time series into a set of independent additive components sgsa is performed in four steps embedding symplectic qr decomposition grouping and diagonal averaging the obtained components can be used for de noising prediction control and synchronization we demonstrate the effectiveness of sgsa in reconstructing and predicting two noisy benchmark nonlinear dynamic systems the lorenz and mackey glass attractors examples of prediction of a decadal average sunspot number time series and a mechanomyographic signal recorded from human skeletal muscle further demonstrate the applicability of the sgsa method in real life applications

some extensions of radial basis functions and their applications in artificial intelligence abstract radial basis functions have recently found interesting applications in artificial intelligence and in particular in the problem of learning to perform a particular task from a set of examples however in many practical cases the radial basis functions method cannot be applied in a straight forward manner because it does not take into account some features that are typical of the problem of learning from examples in this paper we show some extensions of the standard theory introduced in order to deal with a large number of examples and with problems in which different variables play very different roles we present some examples and also point out some open problems

an experimental observation of a new type of intermittency abstract a novel hysteretic form of regular intermittency has been observed experimentally in a variant of the taylor couette problem where the fluid flow is confined between a stationary outer cylinder and a rotating inner cylinder and end plates the results cannot all be explained within the standard pomeau manneville classification of intermittency and an extension to this theory is proposed to account for the observed phenomena a satisfactory agreement is found between experiment and theory both for the reinjection burst frequency scaling law and the influence of noise on the regularity of the bursts

application of singular spectrum analysis to the smoothing of raw kinematic signals motion capture systems currently used in biomechanical analysis introduce systematic measurement errors that appear in the form of noise in recorded displacement signals the noise is unacceptably amplified when differentiating displacements to obtain velocities and accelerations to avoid this phenomenon it is necessary to smooth the displacement signal prior to differentiation in order to eliminate the noise introduced by the experimental system the use of singular spectrum analysis ssa is presented in this paper as an alternative to traditional digital filtering methods ssa is a novel non parametric technique based on principles of multivariate statistics the original time series is decomposed into a number of additive time series each of which can be easily identified as being part of the modulated signal or as being part of the random noise several examples that demonstrate the superiority of this technique over other methods used in biomechanical analysis are presented in this paper

green s formulas for cone differential operators green s formulas for elliptic cone differential operators are established this is achieved by an accurate description of the maximal domain of an elliptic cone differential operator and its formal adjoint thereby utilizing the concept of a discrete asymptotic type from this description the singular coefficients replacing the boundary traces in classical green s formulas are deduced

statistical analysis of geopotential height gh timeseries based on tsallis non extensive statistical mechanics abstract in this paper we perform statistical analysis of time series deriving from earth s climate the time series are concerned with geopotential height gh and correspond to temporal and spatial components of the global distribution of month average values during the period 1948 2012 the analysis is based on tsallis non extensive statistical mechanics and in particular on the estimation of tsallis q triplet namely q stat q sens q rel the reconstructed phase space and the estimation of correlation dimension and the hurst exponent of rescaled range analysis r s the deviation of tsallis q triplet from unity indicates non gaussian tsallis q gaussian non extensive character with heavy tails probability density functions pdfs multifractal behavior and long range dependences for all timeseries considered also noticeable differences of the q triplet estimation found in the timeseries at distinct local or temporal regions moreover in the reconstructive phase space revealed a lower dimensional fractal set in the gh dynamical phase space strong self organization and the estimation of hurst exponent indicated multifractality non gaussianity and persistence the analysis is giving significant information identifying and characterizing the dynamical characteristics of the earth s climate

singular sturm liouville theory on manifolds in this paper we investigate schrodinger operators l g a x on a compact riemannian manifold m g where the potential function a x is assumed to be continuous but not necessarily bounded outside of some closed set m of measure zero under certain geometric hypotheses on and growth conditions on a x as x we prove that the dirichlet extension of l is bounded from below with discrete spectrum in many cases a x is allowed to approach as x we also consider conditions on and a x under which the sturm liouville theory of l is singular in that no boundary conditions are needed to specify the eigenvalues and eigenfunctions of l in particular this occurs when the domain of l does not depend on boundary conditions for example when l is essentially self adjoint or more generally essentially dirichlet a new property that we define the behavior of l on weighted sobolev spaces is also discussed in most of the paper we assume that is a k dimensional submanifold without boundary but in the last few sections we generalize our results to stratified sets

reverse time markov processes corresp the problem of representing a markov process as a reverse time markov process is discussed diffusions and finite state markov processes are two important classes of markov processes for which this problem has been considered a gap in a paper of castanon is pointed out which discusses diffusions reverse time finite and discrete state markov processes are then considered using martingale methods

phase portrait reconstruction using multivariate singular systems analysis abstract the singular systems analysis ssa method of broomhead and king is applied to the reconstruction of phase portraits from experimental time series of temperature measurements taken simultaneously at up to 16 probes within a rotating thermal convection experiment and used to characterize a number of different kinds of flow including the determination of their correlation dimension results are compared with equivalent reconstructions using more conventional method of delays embedding and ssa applied in the time domain only to data obtained from a single thermocouple probe multivariate ssa m ssa is shown to offer significantly improved reconstructions in terms of signal noise ratio and structural uniformity of the attractor over earlier methods this is due mainly to the ability of m ssa to combine simultaneously information sampling the complete range of the spatial cross correlation function in signals with complex spatio temporal structure

heat kernel coefficients for laplace operators on the spherical suspension in this paper we compute the coefficients of the heat kernel asymptotic expansion for laplace operators acting on scalar functions defined on the so called spherical suspension or riemann cap subjected to dirichlet boundary conditions by utilizing a contour integral representation of the spectral zeta function for the laplacian on the spherical suspension we find its analytic continuation in the complex plane and its associated meromorphic structure thanks to the well known relation between the zeta function and the heat kernel obtainable via mellin transform we compute the coefficients of the asymptotic expansion in arbitrary dimensions the particular case of a d dimensional sphere as the base manifold is studied as well and the first few heat kernel coefficients are given explicitly

analytic torsion on manifolds with edges abstract let m g be an odd dimensional incomplete compact riemannian singular space with a simple edge singularity we study the analytic torsion on m and in particular consider how it depends on the metric g if g is an admissible edge metric we prove that the torsion zeta function is holomorphic near s 0 hence the torsion is well defined but possibly depends on g in general dimensions we prove that the torsion depends only on the asymptotic structure of g near the singular stratum of m when the dimension of the edge is odd we prove that the analytic torsion is independent of the choice of even admissible edge metrics the main tool is the construction via the methodology of geometric microlocal analysis of the heat kernel for the friedrichs extension of the hodge laplacian in all degrees in this way we obtain detailed asymptotics of this heat kernel and its trace

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

singular spectrum analysis a toolkit for short noisy chaotic signals abstract singular spectrum analysis ssa is developed further based on experience with applications to geophysical time series it is shown that ssa provides a crude but robust approximation of strange attractors by tori in the presence of noise the method works well for short noisy time series the lagged covariance matrix of the processes studied is the basis of ssa we select subsets of eigenelements and associated principal components pcs in order to provide i a noise reduction algorithm ii a detrending algorithm and iii an algorithm for the identification of oscillatory components reconstructed components rcs are developed to provide optimal reconstruction of a dynamic process at precise epochs rather than averaged over the window length of the analysis ssa is combined with advanced spectral analysis methods the maximum entropy method mem and the multi taper method mtm to refine the interpretation of oscillatory behavior a combined ssa mem method is also used for the prediction of selected subsets of rcs the entire toolkit is validated against a set of four prescribed time series generated by known processes quasi periodic or chaotic it is also applied to a time series of global surface air temperatures 130 years long which has attracted considerable attention in the context of the global warming issue and provides a severe test for noise reduction and prediction

predicting monthly streamflow using data driven models coupled with data preprocessing techniques 1 in this paper the accuracy performance of monthly streamflow forecasts is discussed when using data driven modeling techniques on the streamflow series a crisp distributed support vectors regression cdsvr model was proposed for monthly streamflow prediction in comparison with four other models autoregressive moving average arma k nearest neighbors knn artificial neural networks anns and crisp distributed artificial neural networks cdann with respect to distributed models of cdsvr and cdann the fuzzy c means fcm clustering technique first split the flow data into three subsets low medium and high levels according to the magnitudes of the data and then three single svrs or anns were fitted to three subsets this paper gives a detailed analysis on reconstruction of dynamics that was used to identify the configuration of all models except for arma to improve the model performance the data preprocessing techniques of singular spectrum analysis ssa and or moving average ma were coupled with all five models some discussions were presented 1 on the number of neighbors in knn 2 on the configuration of ann and 3 on the investigation of effects of ma and ssa two streamflow series from different locations in china xiangjiaba and danjiangkou were applied for the analysis of forecasting forecasts were conducted at four different horizons 1 3 6 and 1 2 month ahead forecasts the results showed that models fed by preprocessed data performed better than models fed by original data and cdsvr outperformed other models except for at a 6 month ahead horizon for danjiangkou for the perspective of streamflow series the ssa exhibited better effects on danjingkou data because its raw discharge series was more complex than the discharge of xiangjiaba the ma considerably improved the performance of ann cdann and cdsvr by adjusting the correlation relationship between input components and output of models it was also found that the performance of cdsvr deteriorated with the increase of the forecast horizon

construction and analysis of differential equations from experimental time series of oscillatory systems abstract we propose a novel method to derive an autonomous system of ordinary differential equations from an experimental time series this model describe quantitatively transient behaviour of a system with limit cycle dynamics and is simplified to an evolution equation for the amplitude and frequency the landau equation in addition the model is used for a qualitative prediction of the influence of periodic excitation on the experimental system using the van der pol transformation and floquet theory in particular the instability of the locking solution including the hopf bifurcation the intermittency scenario and the period doubling cascade is considered the method of constructing a differential equation from an experimental time signal is successfully applied to a velocity fluctuation of the laminar periodic wake behind a circular cylinder

data adaptive wavelets and multi scale singular spectrum analysis using multi scale ideas from wavelet analysis we extend singular spectrum analysis ssa to the study of nonstationary time series including the case where intermittency gives rise to the divergence of their variance the wavelet transform resembles a local fourier transform within a finite moving window whose width w proportional to the major period of interest is varied to explore a broad range of such periods ssa on the other hand relies on the construction of the lag correlation matrix c onm lagged copies of the time series over a fixed window width w to detect the regular part of the variability in that window in terms of the minimal number of oscillatory components herew dm1t with1t as the time step the proposed multi scale ssa is a local ssa analysis within a moving window of widthm w n wheren is the length of the time series multi scale ssa variesw while keeping a fixed w m ratio and uses the eigenvectors of the corresponding lag correlation matrix c m as data adaptive wavelets successive eigenvectors of c m correspond approximately to successive derivatives of the first mother wavelet in standard wavelet analysis multi scale ssa thus solves objectively the delicate problem of optimizing the analyzing wavelet in the time frequency domain by a suitable localization of the signal s correlation matrix we present several examples of application to synthetic signals with fractal or power law behavior which mimic selected features of certain climatic or geophysical time series the method is applied next to the monthly values of the southern oscillation index soi for 1933 1996 the soi time series is widely believed to capture major features of the el nino southern oscillation enso in the tropical pacific our methodology highlights an abrupt periodicity shift in the soi near 1960 this abrupt shift between 5 and 3 years supports the devil s staircase scenario for the enso phenomenon preliminary results of this study were presented at the xxii general assembly of the european geophysical society vienna may 1997 and at the fall meeting of the american geophysical union san francisco december 1997 2000 elsevier science b v all rights reserved

an approximate analytic solution of the lorenz equations multiple time perturbation theory is applied to the lorenz equations to obtain approximate solutions these solutions are used to obtain the form of the mapping relating two successive maxima of one of the variables of the equations good agreement is obtained with the numerical results presented by lorenz

faint phase synchronization detection through structured orthomax rotations in singular spectrum analysis multivariate singular spectrum analysis m ssa with a structured varimax rotation is a method that allows a deep characterization of phase synchronization ps phenomena in an almost automatic fashion it has been increasingly used in the study of ps in networks of nonlinear real world and numeric systems this paper investigates the impact of the other recently developed structured orthomax rotations on the m ssa ability to characterize ps the results show that by using the structured quartimax rotation a very faint and intermittent ps regime can be detected in contrast with the structured varimax which demands a stronger more consolidated ps regime this is due to the fact that the different rotations do not have the same efficiency in achieving a simple structure of the m ssa eigenvectors nevertheless for well established ps regimes the same robustness of the original m ssa approach against high levels of additive gaussian noise was found for the structured quartimax and biquartimax rotations however for all approaches we found an overshoot of the qualitative range for the ps onset due to noise

distinguishing between low dimensional dynamics and randomness in measured time series the success of current attempts to distinguish between low dimensional chaos and random behavior in a time series of observations is considered first we discuss stationary stochastic processes which produce finite numerical estimates of the correlation dimension and k2 entropy under naive application of correlation integral methods we then consider several straightforward tests to evaluate whether correlation integral methods reflect the global geometry or the local fractal structure of the trajectory this determines whether the methods are applicable to a given series if they are we evaluate the significance of a particular result for example by considering the results of the analysis of stochastic signals with statistical properties similar to those of observed series from the examples considered it is clear that the correlation integral should not be used in isolation but as one of a collection of tools to distinguish chaos from stochasticity

using renyi parameter to improve the predictive power of singular value decomposition entropy on stock market abstract this paper generalizes the method of traditional singular value decomposition entropy by incorporating orders q of renyi entropy we analyze the predictive power of the entropy based on trajectory matrix using shanghai composite index and dow jones index data in both static test and dynamic test in the static test on sci results of global granger causality tests all turn out to be significant regardless of orders selected but this entropy fails to show much predictability in american stock market in the dynamic test we find that the predictive power can be significantly improved in sci by our generalized method but not in dji this suggests that noises and errors affect sci more frequently than dji in the end results obtained using different length of sliding window also corroborate this finding

reconstruction expansion as a geometry based framework for choosing proper delay times abstract the quality of attractor reconstruction using the method of delays is known to be sensitive to the delay parameter here we develop a new computationally efficient approach to choosing that quantifies reconstruction expansion from the identity line of the embedding space we show that reconstruction expansion is related to the concept of reconstruction signal strength and that increased expansion corresponds to diminished effects of measurement error thus reconstruction expansion represents a simple geometrical framework for choosing furthermore we describe the role of dynamical error in attractor expansion and argue that algorithms for determining should be considered as attempts at estimating an upper bound to the optimal delay

on the full calculus of pseudo differential operators on boundary groupoids with polynomial growth abstract in this paper we enlarge the space of uniformly supported pseudo differential operators on some groupoids by considering kernels satisfying certain asymptotic estimates we show that such enlarged space contains the compact parametrix and the generalized inverse of uniformly supported operators with fredholm vector representation

flow transitions resembling bifurcations of the logistic map in simulations of the baroclinic rotating annulus we present evidence for a sequence of bifurcations in simulations of the differentially heated baroclinic rotating annulus similar to bifurcations of the logistic map the met office oxford rotating annulus laboratory simulation morals code is used to construct a detailed numerical regime diagram for the annulus and the distribution of regimes in parameter space is discussed the bifurcations are observed in a sequence of runs at high temperature forcing identified by poincare sections of the dominant temperature mode amplitude time series higher order return maps and predictions using quadratic fits to the data are used to verify this result and lyapunov exponents are calculated to identify and quantify the chaotic parts of the sequence

approximating stable and unstable manifolds in experiments we introduce a procedure to reveal invariant stable and unstable manifolds given only experimental data we assume a model is not available and show how coordinate delay embedding coupled with invariant phase space regions can be used to construct stable and unstable manifolds of an embedded saddle we show that the method is able to capture the fine structure of the manifold is independent of dimension and is efficient relative to previous techniques

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

correlation of local stretchings as a way of characterising chaotic dynamics amid noise abstract a positive lyapunov exponent can result from either chaotic dynamics or noise as a possible means of assessing the degree to which the exponent is noise affected and as a possible index in its own right the localised divergence of two neighbouring trajectories from a fiducial trajectory is compared in the expectation of finding strong correlation when the source of stretching is deterministic chaos and little or no correlation when the source is coloured or uncoloured noise the method is applied to various equation systems and to a periodic time series from the self excited oscillation of flexible tube collapsed by external pressure and conveying a flow

observation of a strange attractor abstract phase space portraits have been constructed and analyzed for noisy nonperiodic data obtained in an experiment on a nonequilibrium homogeneous chemical reaction the phase space trajectories define a limit set that is an attractor following a perturbation the trajectory quickly returns to the attracting set this attracting set is shown to be strange nearby trajectories separate exponentially on the average moreover the poincare sections exhibit the stretching and folding that is characteristic of strange attractors

reconstructing bifurcation diagrams only from time waveforms abstract a simple algorithm is proposed for reconstructing bifurcation diagrams only from time waveforms of a dynamical system without knowing an explicit form of the dynamics and its dependence on the parameters reconstruction means that the obtained bifurcation diagram is qualitatively similar to that of the original system the algorithm consists of two steps first a nonlinear predictor is sought using a particular class of functions with many parameters second the karhunen loeve transform is used to find only significant parameters contributing the bifurcations the algorithm is tested against two families of dynamical system the henon family and the coupled logistic delayed logistic family

the partially observed stochastic minimum principle using stochastic flows and the generalized differentiation formula of bismut and kunita the change in cost due to a strong variation of an optimal control is explicitly calculated differentiating this expression gives a minimum principle in both the partially observed and stochastic open loop situations in the latter case the equation satisfied by the adjoint process is obtained by applying a martingale representation result

infinitesimal rigidity of cone manifolds and the stoker problem for hyperbolic and euclidean polyhedra the deformation theory of hyperbolic and euclidean cone manifolds with all cone angles less than 2 plays an important role in many problems in low dimensional topology and in the geometrization of 3 manifolds furthermore various old conjectures dating back to stoker about the moduli space of convex hyperbolic and euclidean polyhedra can be reduced to the study of deformations of cone manifolds by doubling a polyhedron across its faces this deformation theory has been understood by hodgson and kerckhoff 7 when the singular set has no vertices and by weis 32 when the cone angles are less than we prove here an infinitesimal rigidity result valid for cone angles less than 2 stating that infinitesimal deformations which leave the dihedral angles fixed are trivial in the hyperbolic case and reduce to some simple deformations in the euclidean case the method is to treat this as a problem concerning the deformation theory of singular einstein metrics and to apply analytic methods about elliptic operators on stratified spaces this work is an important ingredient in the local deformation theory of cone manifolds by the second author 22 see also the concurrent work by weis 33

reduced models of atmospheric low frequency variability parameter estimation and comparative performance abstract low frequency variability lfv of the atmosphere refers to its behavior on time scales of 10 100 days longer than the life cycle of a mid latitude cyclone but shorter than a season this behavior is still poorly understood and hard to predict the present study compares various model reduction strategies that help in deriving simplified models of lfv three distinct strategies are applied here to reduce a fairly realistic high dimensional quasi geostrophic 3 level qg3 atmospheric model to lower dimensions i an empirical dynamical method which retains only a few components in the projection of the full qg3 model equations onto a specified basis and finds the linear deterministic and the stochastic corrections empirically as in selten 1995 5 ii a purely dynamics based technique employing the stochastic mode reduction strategy of majda et al 2001 62 and iii a purely empirical multi level regression procedure which specifies the functional form of the reduced model and finds the model coefficients by multiple polynomial regression as in kravtsov et al 2005 3 the empirical dynamical and dynamical reduced models were further improved by sequential parameter estimation and benchmarked against multi level regression models the extended kalman filter was used for the parameter estimation overall the reduced models perform better when more statistical information is used in the model construction thus the purely empirical stochastic models with quadratic nonlinearity and additive noise reproduce very well the linear properties of the full qg3 model s lfv i e its autocorrelations and spectra as well as the nonlinear properties i e the persistent flow regimes that induce non gaussian features in the model s probability density function the empirical dynamical models capture the basic statistical properties of the full model s lfv such as the variance and integral correlation time scales of the leading lfv modes as well as some of the regime behavior features but fail to reproduce the detailed structure of autocorrelations and distort the statistics of the regimes dynamical models that use data assimilation corrections do capture the linear statistics to a degree comparable with that of empirical dynamical models but do much less well on the full qg3 model s nonlinear dynamics these results are discussed in terms of their implications for a better understanding and prediction of lfv

reconstructing attractors from filtered time series abstract in this paper we investigate the criteria necessary to reconstruct an attractor from a time series that has been passed through an iir filter to guarantee that the whole of the attractor is reconstructed requires uniform bounds on the contraction rate of the dynamics however we show that if we use a condition that is only based on the average contraction rates the lyapunov exponents then a set of high probability can still be reconstructed we argue that in practice this is all that is necessary

the ubiquitous ζ function and some of its usual and unusual meromorphic properties in this contribution we announce a complete classification and new exotic phenomena of the meromorphic structure of functions associated with conic manifolds proved in 37 in particular we show that the meromorphic extensions of these functions have in general countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multiplicity moreover we give a precise algebraic combinatorial formula to compute the coefficients of the leading order terms of the singularities