analysis parameter estimation and optimal control of non linear systems via general orthogonal polynomials general orthogonal polynomials are introduced to approximate the solution of a class of non linear systems using the integration operational matrix the product operational matrix and the derived non linear operational matrix the dynamical equation of a non linear system can be reduced to a set of simultaneous non linear algebraic equations thus greatly simplifying the solution the parameter identification problem for non linear systems is also dealt with an approximate solution for a non linear optimal control problem with quadratic performance measure is also considered three examples are given to demonstrate the validity and applicability of the orthogonal polynomial approximations

an intermediate distribution between gaussian and cauchy distributions abstract in this paper we construct an intermediate distribution linking the gaussian and the cauchy distribution we provide the probability density function and the corresponding characteristic function of the intermediate distribution because many kinds of distributions have no moment we introduce weighted moments specifically we consider weighted moments under two types of weighted functions the cut off function and the exponential function through these two types of weighted functions we can obtain weighted moments for almost all distributions we consider an application of the probability density function of the intermediate distribution on the spectral line broadening in laser theory moreover we utilize the intermediate distribution to the problem of the stock market return in quantitative finance

a nonlinear schrodinger equation with two symmetric point interactions in one dimension we consider a time dependent one dimensional nonlinear schrodinger equation with a symmetric double well potential represented by two dirac s among our results we give an explicit formula for the integral kernel of the unitary semigroup associated with the linear part of the hamiltonian then we establish the corresponding strichartz type estimate and we prove local existence and uniqueness of the solution to the original nonlinear problem

levy flights in inhomogeneous environments and 1 f noise abstract complex dynamical systems which are governed by anomalous diffusion often can be described by langevin equations driven by levy stable noise in this article we generalize nonlinear stochastic differential equations driven by gaussian noise and generating signals with 1 f power spectral density by replacing the gaussian noise with a more general levy stable noise the equations with the gaussian noise arise as a special case when the index of stability 2 we expect that this generalization may be useful for describing 1 f fluctuations in the systems subjected to levy stable noise

the point like limit for a nls equation with concentrated nonlinearity in dimension three abstract we consider a scaling limit of a nonlinear schrodinger equation nls with a nonlocal nonlinearity showing that it reproduces in the limit of cutoff removal a nls equation with nonlinearity concentrated at a point the regularized dynamics is described by the equation i t e t e t g e e e t 2 e e t e where e 0 weakly and the function g embodies the nonlinearity and the scaling and has to be fine tuned in order to have a nontrivial limit dynamics the limit dynamics is a nonlinear version of point interaction in dimension three and it has been previously studied in several papers as regards the well posedness blow up and asymptotic properties of solutions our result is the first justification of the model as the point limit of a regularized dynamics

strichartz inequalities and the nonlinear schrodinger equation on compact manifolds kklwe prove strichartz estimates with fractional loss of derivatives for the schrodinger equation on any riemannian compact manifold as a consequence we infer low regularity local well posedness results in any dimension as well as global existence results for the cauchy problem of nonlinear schrodinger equations on surfaces in the case of defocusing polynomial nonlinearities and on three manifolds in the case of cubic defocusing nonlinearities we also discuss the optimality of these strichartz estimates on spheres

l p boundedness of wave operators for two dimensional schrodinger operators let be the two dimensional schrodinger operator with the real valued potential v which satisfies the decay condition at infinity for we show that the wave operators are bounded in for any 1 p under the condition that h has no zero bound states or zero resonance extending the corresponding results for higher dimensions as w intertwine h 0 and the absolutely continuous part h p ac of h f h p ac w f h 0 w for any borel function f on 1 this reduces the various l p mapping properties of f h p ac to those of f h 0 the convolution operator by the fourier transform of the function f 2

analysis and optimal control of time varying linear systems via block pulse functions succinct and clear sighted operational properties of block pulse functions are fully applied to the analysis and optimal control of time varying linear systems with quadratic performance index piecewise constant solutions equally distributed which are simple in form and convenient for use or implementation are consequently obtained another advantage of this method is that any positive integer can be chosen as the number of sub intervals whereas in the case of walsh function approximation the choice can only be made from 2 4 8 16 32 and so on the algorithms developed in the paper are illustrated by appropriate examples

block pulse series analysis of linear systems incorporating observers the paper considers the analysis of linear systems incorporating observers via block pulse functions and presents recursive algorithms convenient for computers compared with walsh function approaches the results are much simpler in form an example is given to illustrate the use of the method

non linear systems identification and optimal control we consider analytical control affine non linear continuous systems by the introduction of a notation based on the tensor product we can use the matrix calculus to solve the problems of model identification by orthogonal functions and the construction of a non linear state feedback for the optimal control problem

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

self modulation processes and resulting generic 1 f fluctuations we analyze high precision data of transaction intervals in a foreign exchange market and show that it is nicely approximated by a non stationary poisson process whose expectation value is given by a moving average of its own trace generalizing this result we introduce novel stochastic processes called the self modulation processes by the self modulation effect clustering occurs automatically resulting in fat tailed interval distributions including the zipf s law in an extreme case we prove rigorously that the corresponding power spectrum follows the 1 f spectrum

trading activity as driven poisson process comparison with empirical data we propose the point process model as the poissonian like stochastic sequence with slowly diffusing mean rate and adjust the parameters of the model to the empirical data of trading activity for 26 stocks traded on nyse the proposed scaled stochastic differential equation provides the universal description of the trading activities with the same parameters applicable for all stocks

additive multiplicative stochastic models of financial mean reverting processes we investigate a generalized stochastic model with the property known as mean reversion that is the tendency to relax towards a historical reference level besides this property the dynamics is driven by multiplicative and additive wiener processes while the former is modulated by the internal behavior of the system the latter is purely exogenous we focus on the stochastic dynamics of volatilities but our model may also be suitable for other financial random variables exhibiting the mean reversion property the generalized model contains as particular cases many early approaches in the literature of volatilities or more generally of mean reverting financial processes we analyze the long time probability density function associated to the model defined through an it o langevin equation we obtain a rich spectrum of shapes for the probability function according to the model parameters we show that additive multiplicative processes provide realistic models to describe empirical distributions for the whole range of data

on the design of discrete time optimal dynamic controllers using the partial state observer abstract this paper considers the problem of designing optimal dynamic controllers based on a partial state observer for discrete time linear time invariant systems with inaccessible state it is assumed that the initial system state is an unknown random vector with known mean and covariance the performance index is taken to be the expectation with respect to the initial state of the standard quadratic one for the discrete time regulator necessary and sufficient conditions for optimality are derived provided that the subspace which ensures the existence of a partial state observer is given a design algorithm is also obtained which can be directly solved for the parameters of the optimal observer and requires the solution of a matrix riccati equation of the observer order as in the continuous time case the separation property in the restricted sense is seen to hold finally an example is presented to illustrate the procedure

superstatistics thermodynamics and fluctuations a thermodynamiclike formalism is developed for superstatistical systems based on conditional entropies this theory takes into account large scale variations of intensive variables of systems in nonequilibrium stationary states ordinary thermodynamics is recovered as a special case of the present theory and corrections to it can systematically be evaluated a generalization of einstein s relation for fluctuations is presented using a maximum entropy condition

application of chebyshev polynomials to the optimal control of time varying linear systems the operational matrix of backward integration for the shifted chebyshev polynomials is introduced in this study the general expression of the shifted chebyshev polynomial approximation for any two arbitrary functions is also presented a linear time varying optimal control system with a quadratic performance measure is solved by using the shifted chebyshev polynomials only a small number of chebyshev polynomials is needed to produce an excellent result and the outcome is much better than the solution obtained by using the block pulse function so computer memory capacity and computing time can be saved considerably

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

shifted chebyshev series analysis and identification of time varying bilinear systems shifted chebyshev series are developed in this study to analyse and identify time varying bilinear systems effective computational formulations which are very straightforward and convenient for digital computations are achieved consequently the computational difficulty can be remarkably reduced two illustrative examples are given to demonstrate the superiority of the presently proposed method

fourier series direct method for variational problems a direct method for solving variational problems using fourier series is presented an operational matrix of integration is first introduced and is utilized to reduce a variational problem to the solution of algebraic equations illustrative examples are also given

autoregressive conditional duration a new model for irregularly spaced transaction data this paper proposes a new statistical model for the analysis of data which arrive at irregular intervals the model treats the time between events as a stochastic process and proposes a new class of point processes with dependent arrival rates the conditional intensity is developed and compared with other self exciting processes because the model focuses on the expected duration between events it is called the autoregressive conditional duration acd model asymptotic properties of the quasi maximum likelihood estimator are developed as a corollary to arch model results strong evidence is provided for duration clustering for the financial transaction data analyzed both deterministic time of day effects and stochastic effects are important the model is applied to the arrival times of trades and therefore is a model of transaction volume and also to the arrival of other events such as price changes models for the volatility of prices are estimated with price based durations and examined from a market microstructure point of view

regular articlegeneralized strichartz inequalities for the wave equation we make a synthetic exposition of the generalized strichartz inequalities for the wave equation obtained in 6 together with the limiting cases recently obtained in 13 with as simple proofs as possible the proofs combine stationary phase estimates dyadic decompositions the hardy littlewood inequality or the holder inequality in time and abstract duality and interpolation arguments

laguerre polynomial analysis in optimal control systems incorporating observers this paper considers a method of analysis using the laguerre polynomial approach for finding the optimal control of linear systems incorporating an observer a recursive algorithm convenient for computers is also derived in which small matrices are always used to find a continuous solution the unique property of laguerre polynomials leads to excellent results which are demonstrated with an example

the operational matrices of integration and differentiation for the fourier sine cosine and exponential series for the fourier sine cosine series basis vector varphi t and the fourier exponential series basis vector psi t a linear nonsingular transfor marion t is determined such that psi t t varphi t this result is then used to show that the operational matrices of integration p and q for varphi t and psi t respectively are related by the expression tp qt analogous results are derived for the corresponding operational matrices of differentiation d and r general expressions are derived for t p q d and r

the valuation of equity warrants under the fractional vasicek process of the short term interest rate abstract motivated by the empirical evidence of long range dependence in short term interest rates and considering the long maturities of equity warrants we propose the fractional vasicek model to describe the dynamics of the short rate in the pricing environment of equity warrants using the partial differential equation approach we present a valuation model for equity warrants under the assumption that the short rate follows the fractional vasicek process after identifying the pricing model for equity warrants we provide the parameter estimation procedure for the proposed pricing model since obtaining the values of equity warrants from the proposed model needs to solve a nonlinear equation we employ a hybrid intelligent algorithm to get around this optimization problem furthermore to illustrate the practicality of our proposed model we conduct an empirical study to ascertain the performance of our proposed model using the data from china s warrant market and the china foreign exchange trade system the comparison of traditional models such as the black scholes model the noreen wolfson model the lauterbach schultz model and the ukhov model with our proposed model is also presented the empirical results show that the mean absolute percentage error of our pricing model is 10 30 by contrast the black scholes model the noreen wolfson model the lauterbach schultz model and the ukhov model applied to the same warrant produce mean absolute errors of 35 26 37 67 33 40 32 81 respectively thus the long memory property in stochastic interest rates cannot be ignored in determining the valuation of equity warrants

dispersive estimates for schrodinger equations with threshold resonance and eigenvalue let h v x be a three dimensional schrodinger operator we study the time decay in lp spaces of scattering solutions e ithpcu where pc is the orthogonal projection onto the continuous spectral subspace of l2 r3 for h under suitable decay assumptions on v x it is shown that they satisfy the so called lp lq estimates e ithpcu p 4 t 3 1 2 1 p u q for all 1 q 2 p with 1 p 1 q 1 if h has no threshold resonance and eigenvalue and for all 3 2 q 2 p 3 if otherwise

common scaling patterns in intertrade times of u s stocks we analyze the sequence of time intervals between consecutive stock trades of thirty companies representing eight sectors of the u s economy over a period of 4 yrs for all companies we find that i the probability density function of intertrade times may be fit by a weibull distribution ii when appropriately rescaled the probability densities of all companies collapse onto a single curve implying a universal functional form iii the intertrade times exhibit power law correlated behavior within a trading day and a consistently greater degree of correlation over larger time scales in agreement with the correlation behavior of the absolute price returns for the corresponding company and iv the magnitude series of intertrade time increments is characterized by long range power law correlations suggesting the presence of nonlinear features in the trading dynamics while the sign series is anticorrelated at small scales our results suggest that independent of industry sector market capitalization and average level of trading activity the series of intertrade times exhibit possibly universal scaling patterns which may relate to a common mechanism underlying the trading dynamics of diverse companies further our observation of long range power law correlations and a parallel with the crossover in the scaling of absolute price returns for each individual stock support the hypothesis that the dynamics of transaction times may play a role in the process of price formation

the computation of ck spline functions abstract we present in this paper an algorithm for computing ck spline functions i e spline functions which generate k time continuous and derivable approximations these functions seem to be very efficient in several applied nonlinear differential problems like nonlinear control or nonlinear simulation

taylor series analysis of linear optimal control systems incorporating observers in this paper the taylor series method is successfully applied to the analysis of linear optimal control systems incorporating observers the basic idea is that the state and error variables are expressed in terms of taylor series a very powerful computational algorithm is proposed to study the problem for any length of time taking only two or three terms of expansion coefficients is enough to calculate the state and error functions one illustrative example is given and a very satisfactory result is obtained

nonlinear garch model and 1 f noise abstract auto regressive conditionally heteroskedastic arch family models are still used by practitioners in business and economic policy making as a conditional volatility forecasting models furthermore arch models still are attracting an interest of the researchers in this contribution we consider the well known garch 1 1 process and its nonlinear modifications reminiscent of ngarch model we investigate the possibility to reproduce power law statistics probability density function and power spectral density using arch family models for this purpose we derive stochastic differential equations from the garch processes in consideration we find the obtained equations to be similar to a general class of stochastic differential equations known to reproduce power law statistics we show that linear garch 1 1 process has power law distribution but its power spectral density is brownian noise like however the nonlinear modifications exhibit both power law distribution and power spectral density of the 1 f form including 1 f noise

point process model of 1 f noise vs a sum of lorentzians we present a simple point process model of 1 f noise covering different values of the exponent the signal of the model consists of pulses or events the interpulse interevent interarrival recurrence or waiting times of the signal are described by the general langevin equation with the multiplicative noise and stochastically diffuse in some interval resulting in a power law distribution our model is free from the requirement of a wide distribution of relaxation times and from the power law forms of the pulses it contains only one relaxation rate and yields 1 f spectra in a wide range of frequencies we obtain explicit expressions for the power spectra and present numerical illustrations of the model further we analyze the relation of the point process model of 1 f noise with the bernamont surdin mcwhorter model representing the signals as a sum of the uncorrelated components we show that the point process model is complementary to the model based on the sum of signals with a wide range distribution of the relaxation times in contrast to the gaussian distribution of the signal intensity of the sum of the uncorrelated components the point process exhibits asymptotically a power law distribution of the signal intensity the developed multiplicative point process model of 1 f noise may be used for modeling and analysis of stochastic processes in different systems with the power law distribution of the intensity of pulsing signals

tsallis distributions and 1 f noise from nonlinear stochastic differential equations probability distributions that emerge from the formalism of nonextensive statistical mechanics have been applied to a variety of problems in this article we unite modeling of such distributions with the model of widespread 1 f noise we propose a class of nonlinear stochastic differential equations giving both the q exponential or q gaussian distributions of signal intensity revealing long range correlations and 1 f behavior of the power spectral density the superstatistical framework to get 1 f noise with q exponential and q gaussian distributions of the signal intensity is proposed as well

dispersion and strichartz estimates for some finite rank perturbations of the laplace operator we consider the dispersion properties in lp spaces of schrodinger hamiltonians with a large number of obstacles modelled by rank one perturbations we obtain both for the dispersion an strichartz estimates nonperturbative results with respect to the coupling constants

the econometrics of ultra high frequency data ultra high frequency data is defined to be a full record of transactions and their associated characteristics the transaction arrival times and accompanying measures can be analyzed as marked point processes the acd point process developed by engle and russell 1998 is applied to ibm transactions arrival times to develop semiparametric hazard estimates and conditional intensities combining these intensities with a garch model of prices produces ultra high frequency measures of volatility both returns and variances are found to be negatively influenced by long durations as suggested by asymmetric information models of market micro structure

a long range memory stochastic model of the return in financial markets abstract we present a nonlinear stochastic differential equation sde which mimics the probability density function pdf of the return and the power spectrum of the absolute return in financial markets absolute return as a measure of market volatility is considered in the proposed model as a long range memory stochastic variable the sde is obtained from the analogy with an earlier proposed model of trading activity in the financial markets and generalized within the nonextensive statistical mechanics framework the proposed stochastic model generates time series of the return with two power law statistics i e the pdf and the power spectral density reproducing the empirical data for the one minute trading return in the nyse

time dependent propagator with point interaction we compute the time dependent schrodinger propagator with a point interaction in dimension n or 3 including the new cases of n 2 and the most general interaction supported by a point for n 1 we also give the small time asymptotics for n or 3 the case n 2 has the peculiarity of involving logarithmic terms in the expansion

dispersive analysis of charge transfer models we prove dispersive estimates for the time dependent schrodinger equation with a charge transfer hamiltonian as a by product we also obtain another proof of asymptotic completeness of the wave operators for a charge transfer model established earlier by k yajima and j m graf we also consider a more general matrix non self adjoint charge transfer problem this model appears naturally in the study of nonlinear multisoliton systems and is specifically motivated by the problem of asymptotic stability of multisoliton states of a nonlinear schrodinger equation 2004 wiley periodicals inc

multiplicative point process as a model of trading activity signals consisting of a sequence of pulses show that inherent origin of the 1 f noise is a brownian 1uctuation of the average interevent time between subsequent pulses of the pulse sequence in this paper we generalize the model of interevent time to reproduce a variety of self a5ne time series exhibiting power spectral density s f scaling as a power of the frequency f furthermore we analyze the relation between the power law correlations andthe origin of the power law probability distribution of the signal intensity we introduce a stochastic multiplicative model for the time intervals between point events and analyze the statistical properties of the signal analytically and numerically such model system exhibits power law spectral den

time decay for solutions of schrodinger equations with rough and time dependent potentials in this paper we establish dispersive estimates for solutions to the linear schrodinger equation in three dimensions frac 1 i partial t psi triangle psi v psi 0 qquad psi s f 0 1 where v t x is a time dependent potential that satisfies the conditions sup t v t cdot l frac 3 2 mathbb r 3 sup x in mathbb r 3 int mathbb r 3 int infty infty frac v hat tau x x y d tau dy c 0 here c0 is some small constant and v hat tau x denotes the fourier transform with respect to the first variable we show that under these conditions 0 1 admits solutions l t l2x 3 l2t l6x 3 for any f l2 3 satisfying the dispersive inequality psi t infty le c t s frac32 f 1 text for all times t s 0 2 for the case of time independent potentials v x 0 2 remains true if int mathbb r 6 frac v x v y x y 2 dxdy 4 pi 2 text and v mathcal k sup x in mathbb r 3 int mathbb r 3 frac v y x y dy 4 pi we also establish the dispersive estimate with an e loss for large energies provided v mathcal k v 2 infty

nonlinear stochastic models of 1 f noise and power law distributions starting from the developed generalized point process model of 1 f noise b kaulakys et al phys rev e 71 2005 051105 we derive the nonlinear stochastic differential equations for the signal exhibiting 1 f b noise and 1 x l distribution density of the signal intensity with different values of b and l the processes with 1 f b are demonstrated by the numerical solution of the derived equations with the appropriate restriction of the diffusion of the signal in some finite interval the proposed consideration may be used for modeling and analysis of stochastic processes in different systems with the powerlaw distributions long range memory or with the elements of self organization r 2006 elsevier b v all rights reserved

shifted chebyshev series analysis of linear optimal control systems incorporating observers the analysis of optimal control systems incorporating observers has been approached by the shifted chebyshev series in this study the method simplifies the system of equations into the successive solution of a set of linear algebraic equations one example is illustrated and only a small number of the chebyshev polynomials is needed to obtain a very attractive and excellent result

explanation of power law behavior of autoregressive conditional duration processes based on the random multiplicative process autoregressive conditional duration acd processes which have the potential to be applied to power law distributions of complex systems found in natural science life science and social science are analyzed both numerically and theoretically an acd 1 process exhibits the singular second order moment which suggests that its probability density function pdf has a power law tail it is verified that the pdf of the acd 1 has a power law tail with an arbitrary exponent depending on a model parameter on the basis of theory of the random multiplicative process a relation between the model parameter and the power law exponent is theoretically derived it is confirmed that the relation is valid from numerical simulations an application of the acd 1 to intervals between two successive transactions in a foreign currency market is shown

modeling non stationarities in high frequency financial time series abstract we study tick by tick financial returns for the ftse mib index of the italian stock exchange borsa italiana we confirm previously detected non stationarities scaling properties reported before for other high frequency financial data are only approximately valid as a consequence of our empirical analyses we propose a simple model for non stationary returns based on a non homogeneous normal compound poisson process it turns out that our model can approximately reproduce several stylized facts of high frequency financial time series moreover using monte carlo simulations we analyze order selection for this class of models using three information criteria akaike s information criterion aic the bayesian information criterion bic and the hannan quinn information criterion hq for comparison we perform a similar monte carlo experiment for the acd autoregressive conditional duration model our results show that the information criteria work best for small parameter numbers for the compound poisson type models whereas for the acd model the model selection procedure does not work well in certain cases