extension of finite volume compressible flow solvers to multi dimensional variable density zero mach number flows when attempting to compute unsteady variable density flows at very small or zero mach number using a standard finite volume compressible flow solver one faces at least the following difficulties i spatial pressure variations vanish as the mach numberm

testing and modeling multivariate threshold models abstract threshold autoregressive models in which the process is piecewise linear in the threshold space have received much attention in recent years in this article i use predictive residuals to construct a test statistic for detecting threshold nonlinearity in a vector time series and propose a procedure for building a multivariate threshold model the thresholds and the model are selected jointly based on the akaike information criterion the finite sample performance of the proposed test is studied by simulation the modeling procedure is then used to study arbitrage in security markets and results in a threshold cointegration between logarithms of future contracts and spot prices of a security after adjusting for the cost of carrying the contracts in this particular application thresholds are determined in part by the transaction costs i also apply the proposed procedure to u s monthly interest rates and two river flow series of iceland

detrended fluctuation analysis of intertrade durations abstract the intraday pattern long memory and multifractal nature of the intertrade durations which are defined as the waiting times between two consecutive transactions are investigated based upon the limit order book data and order flows of 23 liquid chinese stocks listed on the shenzhen stock exchange in 2003 an inverse u shaped intraday pattern in the intertrade durations with an abrupt drop in the first minute of the afternoon trading is observed based on a detrended fluctuation analysis we find a crossover of power law scaling behaviors for small box sizes trade numbers in boxes and large box sizes and strong evidence in favor of long memory in both regimes in addition the multifractal nature of intertrade durations in both regimes is confirmed by a multifractal detrended fluctuation analysis for individual stocks with a few exceptions in the small duration regime the intraday pattern has little influence on the long memory and multifractality

a family of fifth order runge kutta pairs the construction of a runge kutta pair of order 5 4 with the minimal number of stages requires the solution of a nonlinear system of 25 order conditions in 27 unknowns we define a new family of pairs which includes pairs using 6 function evaluations per integration step as well as pairs which additionally use the first function evaluation from the next step this is achieved by making use of kutta s simplifying assumption on the original system of the order conditions i e that all the internal nodes of a method contributing to the estimation of the endpoint solution provide at these nodes cost free second order approximations to the true solution of any differential equation in both cases the solution of the resulting system of nonlinear equations is completely classified and described in terms of five free parameters optimal runge kutta pairs with respect to minimized truncation error coefficients maximal phase lag order and various stability characteristics are presented these pairs were selected under the assumption that they are used in local extrapolation mode the propagated solution of a problem is the one provided by the fifth order formula of the pair numerical results obtained by testing the new pairs over a standard set of test problems suggest a significant improvement in efficiency when using a specific pair of the new family with minimized truncation error coefficients instead of some other existing pairs

a micro movement model with bayes estimation via filtering application to measuring trading noises and costs abstract we highlight a general hybrid system as the micromovement model for asset price using counting processes recently introduced with its bayes estimation via filtering we construct a new simple micromovement model and apply it to analyze trade by trade stock price data in the light of the series of works initiated by christie and schultz why do nasdaq market makers avoid odd eighth quotes finance 49 1994 1813 1840 through the new model we propose more reasonable but computationally intensive measures for trading noise including clustering noise and non clustering noise and for trading cost we employ bayes estimation via filtering to obtain parameter estimates of the new model and to provide numerical measures of trading noise and trading cost for three stocks from four chosen periods our empirical results support the important findings in christie harris schultz why did nasdaq market makers stop avoiding odd eighth quotes finance 49 1994 1841 1860 barclay christie harris kandel schultz the effects of market reform on the trading costs and depths of nasdaq stocks j finance 54 1 1999 1 34

global well posedness and scattering for the nonstandard defocusing beam equation abstract we consider the nonstandard defocusing beam equation u t t i 1 n i 4 u u u p 1 u 0 in the subcritical case we show the global well posedness and the scattering results in low dimensions by the compactness rigidity argument of kenig and merle 2006 2008 7 3 the new ingredient is that we establish the suitable strichartz estimate to overcome the difficulty caused by the complicated dispersive estimate

the large time stability of sound waves we demonstrate the existence of solutions to the full 3 3 system of compressible euler equations in one space dimension up to an arbitrary timet 0 in the case when the initial data has arbitrarily large total variation and sufficiently small supnorm the result applies to periodic solutions of the euler equations a nonlinear model for sound wave propagation in gas dynamics our analysis establishes a growth rate for the total variation that depends on a new length scaled that we identify in the problem this length scale plays no role in 2 2 systems or any system possessing a full set of riemann coordinates nor in the small total variation problem forn n systems the cases originally addressed by glimm in 1965 recent work by a number of authors has demonstrated that when the total variation is sufficiently large solutions of 3 3 systems of conservation laws can in general blow up in finite time independent of the supnorm due to amplifying instabilities created by the non trivial lie algebra of the vector fields that define the elementary waves for the large total variation problem there is an interaction between large scale effects that amplify and small scale effects that are stable and we show that the length scale on which this interaction occurs isd in the limitd we recover glimm s theorem and we observe that there exist linearly degenerate systems within the class considered for which the growth rate we obtain is sharp

h measures a new approach for studying homogenisation oscillations and concentration effects in partial differential equations new mathematical objects called h measures are introduced for studying oscillations and concentration effects in partial differential equations applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach

challenges in climate science and contemporary applied mathematics this article discusses the challenges in climate science from the emerging viewpoint of stochastic statistical properties of turbulent dynamical systems the mathematical topics discussed here include empirical information theory fluctuation dissipation theorems reduced order stochastic modeling and the development of mathematically unambiguous exactly solvable test models for climate science that capture crucial features of vastly more complex scientific problems the applied mathematics topics include the emerging development of multiscale algorithms for filtering data assimilation and superparametrization for climate science and other problems in science and engineering as well as suitable unambiguous mathematical test problems for their behavior interesting contemporary research directions and specific open problems are mentioned throughout the article the perspective here should also be useful for applications to other complex dynamical systems arising in neural science material science and environmental mechanical engineering 2011 wiley periodicals inc

singular pseudodifferential operators symmetrizers and oscillatory multidimensional shocks abstract we introduce a calculus of singular pseudodifferential operators spos depending on wavelength e and use them to solve three different types of singular quasilinear hyperbolic systems such systems arise in nonlinear geometric optics and also for example in the study of incompressible limits and of nonlinear wave equations with small nonlinear terms or small data the spos act in both slow and fast variables and are singular not only because their symbols have finite regularity and depend on 1 e but also because their derivatives fail to decay in the usual way in the dual variables there is a necessarily crude calculus with large parameter e g residual operators are just bounded on l2 but the calculus admits the proof of garding inequalities and enables us to symmetrize and sometimes even diagonalize the singular systems being considered by microlocalizing simultaneously in both slow and fast variables the paper culminates in a proof of the existence of oscillatory multidimensional shocks on a fixed time interval independent of the wavelength e as e 0 the use of spos allows us to eliminate the small divisor assumptions made in earlier work and also to construct more general oscillatory solutions in which elliptic boundary layers are present on one or both sides of the shock

multidimensional viscous shocks ii the small viscosity limit in this paper we prove the existence of curved multidimensional viscous shocks and also justify the small viscosity limit starting with a curved multidimensional inviscid shock solution to a system of hyperbolic conservation laws we show that the shock can be obtained as a small viscosity limit of solutions to an associated parabolic problem viscous shocks the two main hypotheses are a natural evans function assumption on the viscous profile together with a restriction on how much the shock can deviate from flatness the main tools are a conjugation lemma that removes xn dependence from the linearization of the parabolic problem about the viscous profile new degenerate kreiss type symmetrizers used to prove an l2 estimate for the linearized problem and a finite regularity calculus of semiclassical and mixed type classical semiclassical pseudodifferential operators 2003 wiley periodicals inc

low mach number flows in time dependent domains we perform a multiple time scale single space scale analysis of a compressible fluid in a time dependent domain when the time variations of the boundary are small with respect to the acoustic velocity we introduce an average operator with respect to the fast time the averaged leading order variables satisfy modified incompressible equations which are coupled to linear acoustic equations with respect to the fast time we discuss possible initial boundary data for the asymptotic equations inherited from the initial boundary data for the compressible equations

singular limits for the compressible euler equation in an exterior domain justification des phenomenes lies au passage des equations des fluides parfaits compressibles a celles des fluides parfaits incompressibles limites singulieres de l equation d euler des fluides compressibles dans un domaine exterieur

nonlinear geometric optics for reflecting uniformly stable pulses abstract we provide a justification with rigorous error estimates showing that the leading term in weakly nonlinear geometric optics expansions of highly oscillatory reflecting pulses is close to the uniquely determined exact solution for small wavelengths e pulses reflecting off fixed noncharacteristic boundaries are considered under the assumption that the underlying boundary problem is uniformly spectrally stable in the sense of kreiss there are two respects in which these results make rigorous the formal treatment of pulses in majda and artola 1988 16 and hunter majda and rosales 1986 10 first we give a rigorous construction of leading pulse profiles in problems where pulses traveling with many distinct group velocities are unavoidably present and second we provide a rigorous error analysis which yields a rate of convergence of approximate to exact solutions as e 0 unlike wavetrains interacting pulses do not produce resonances that affect leading order profiles however our error analysis shows the importance of estimating pulse interactions in the construction and estimation of correctors our results apply to a general class of systems that includes quasilinear problems like the compressible euler equations moreover the same methods yield a stability result for uniformly stable euler shocks perturbed by highly oscillatory pulses

nonparametric specification tests for conditional duration models this paper deals with the testing of autoregressive conditional duration acd models by gauging the distance between the parametric density and hazard rate functions implied by the duration process and their non parametric estimates we derive the asymptotic justification using the functional delta method for fixed and gamma kernels and then investigate the finite sample properties through monte carlo simulations although our tests display some size distortion bootstrapping suffices to correct the size without compromising their excellent power we show the practical usefulness of such testing procedures for the estimation of intraday volatility patterns

semiclassical and spectral analysis of oceanic waves in this work we prove that the shallow water flow subject to strong wind forcing and linearized around an adequate stationary profile develops for large times closed trajectories due to the propagation of rossby waves while poincare waves are shown to disperse the methods used in this paper involve semi classical analysis and dynamical systems for the study of rossby waves while some refined spectral analysis is required for the study of poincare waves due to the large time scale involved which is of diffractive type

a nonlinear autoregressive conditional duration model with applications to financial transaction data abstract this paper presents a new model that improves upon several inadequacies of the original autoregressive conditional duration acd model considered in engle and russell econometrica 66 5 1998 1127 1162 we propose a threshold autoregressive conditional duration tacd model to allow the expected duration to depend nonlinearly on past information variables conditions for the tacd process to be ergodic and existence of moments are established strong evidence is provided to suggest that fast transacting periods and slow transacting periods of nyse stocks have quite different dynamics based on the improved model we identify multiple structural breaks in the transaction duration data considered and those break points match nicely with real economic events

the mach stem equation and amplification in strongly nonlinear geometric optics we study highly oscillating solutions to a class of weakly well posed hyperbolic initial boundary value problems weak well posedness is associated with an amplification phenomenon of oscillating waves on the boundary in the previous works cgw14 cw14 we have rigorously justified a weakly nonlinear regime for semilinear problems in that case the forcing term on the boundary has amplitude o e 2 and oscillates at a frequency o 1 e the corresponding exact solution which has been shown to exist on a time interval that is independent of e 0 1 has amplitude o e in this paper we deal with the exact same scaling namely o e 2 forcing term on the boundary and o e solution for quasilinear problems in analogy with cgm03 this corresponds to a strongly nonlinear regime and our main result proves solvability for the corresponding wkb cascade of equations which yields existence of approximate solutions on a time interval that is independent of e 0 1 existence of exact solutions close to approximate ones is a stability issue which as shown in cgm03 highly depends on the hyperbolic system and on the boundary conditions we do not address that question here this work encompasses previous formal expansions in the case of weakly stable shock waves mr83 and two dimensional compressible vortex sheets am87 in particular we prove well posedness for the leading amplitude equation the mach stem equation of mr83 and generalize its derivation to a large class of hyperbolic boundary value problems and to periodic forcing terms the latter case is solved under a crucial nonresonant assumption and a small divisor condition

fitting a two phase threshold multiplicative error model abstract the recent literature on financial time series analysis has devoted considerable attention to nonnegative time series such as financial durations realized volatility and squared returns the class of models referred to as the multiplicative error models mem is particularly suited to model such nonnegative time series we develop a lack of fit test for fitting a two phase threshold model for the conditional mean function in an mem the proposed testing procedure can also be applied to a class of autoregressive conditional heteroscedastic threshold models we evaluate the test in a simulation study the testing procedure is illustrated by using two data examples

a variable span smoother abstract a variable span smoother based on linear fits is described local cross validation is used to estimate the optimal span as a function of abscissa value computationally efficient algorithms making use of updating formulas are presented

semi implicit extension of a godunov type scheme based on low mach number asymptotics i one dimensional flow a single scale multiple space scale asymptotic analysis provides detailed insight into the low mach number limit behavior of solutions of the compressible euler equations we use the asymptotics as a guideline for developing a low mach number extension of an explicit higher order shock capturing scheme this semi implicit scheme involves multiple pressure variables large scale differencing and averaging procedures that are discretized versions of standard operations in multiple scales asymptotic analysis advection and acoustic wave propagation are discretized explicitly and upwind and only one scalar elliptic equation is to be solved implicitly at each time step this equation is a pressure correction equation for incompressible flows when the mach number is zero in the low mach number limit the time step is restricted by a courant number based essentially on the maximum flow velocity for moderate and large mach numbers the scheme reduces to the underlying explicit higher order shock capturing algorithm

a deterministic model for the propagation of turbulent oscillations abstract the deterministic point of view on turbulent fluid motion is to consider the cauchy problem for equations of navier stokes type associated with large reynolds numbers and with singular initial data although the corresponding mathematical study has a lot progressed it remains limited by fundamental difficulties related to the presence of instabilities precisely the purpose of this article is to show on a realistic two dimensional model that up to some extent such instabilities can be managed this is achieved in the framework of a supercritical nonlinear geometric optics the aim is to provide a theory allowing to take into account the interaction of a large amplitude monophase oscillating wave with waves oscillating at smaller frequencies in the other direction the effect is that very complicated phenomena can occur in the inertial range including for instance the production of new scales

the validity of nonlinear geometric optics for weak solutions of conservation laws the method of weakly nonlinear geometric optics is one of the main formal perturbation techniques used in analyzing nonlinear wave motion for hyperbolic systems the tacit assumption in using such perturbation methods is that the corresponding solutions of the hyperbolic system remain smooth since shock waves typically form in such solutions these assumptions are rarely satisfied in practice nevertheless in a variety of applied contexts these methods give qualitatively reliable answers for discontinuous weak solutions here we give a rigorous proof for the validity of nonlinear geometric optics for general weak solutions of systems of hyperbolic conservation laws in a single space variable the methods of proof do not mimic the formal construction of weakly nonlinear asymptotics but instead rely on structural symmetries of the approximating equations stability estimates for intermediate asymptotic times and the rapid decay in variation of weak solutions for large asymptotic times

tests for parameter instability and structural change with unknown change point this paper considers tests for parameter instability and structural change with unknown change point the results apply to a wide class of parametric models that are suitable for estimation by generalized method of moments procedures the asymptotic distributions of the test statistics considered here are nonstandard because the change point parameter only appears under the alternative hypothesis and not under the null the tests considered here are shown to have nontrivial asymptotic local power against all alternatives for which the parameters are nonconstant the tests are found to perform quite well in a monte carlo experiment reported elsewhere copyright 1993 by the econometric society

neural networks with multidimensional transfer functions we present a new type of neural network nn where the data for the input layer are the value x spl epsiv r the vector y spl epsiv r sup m associated to an initial value problem ivp with y x f y x and a steplength h then the stages of a runge kutta rk method with trainable coefficients are used as hidden layers for the integration of the ivp using f as transfer function we take as output two estimations y y spl circ of ivp at the point x h training the rk method at some test problems and counting the cost of the method under the coefficients used we may achieve coefficients that help the method to perform better at a wider class of problems

compatibility conditions to allow some large amplitude wkb analysis for burger s type systems abstract in this article we discuss the problem of finding large amplitude asymptotic expansions for monophase oscillating solutions of the following multidimensional d 1 burger s type system t u v u x u 0 u r d t x r r d v c 1 r d r d more precisely we are concerned with families u e e 0 1 made of solutions to and having a development of the form u e t x h t x t x e o e where the function h t x is periodic in general due to the formation of shocks such a construction is not possible on a domain which does not depend on e 0 1 in this article we perform a detailed analysis of the restrictions to impose on the profile h and on the phase in order to remedy this among these compatibility conditions we can isolate some new interesting system of nonlinear partial differential equations we explain how to solve them and we describe how the underlying structure is propagated through the evolution equation

a family of autoregressive conditional duration models this paper develops a family of autoregressive conditional duration acd models that encompasses most specifications in the literature the nesting relies on a box cox transformation with shape parameter delta to the conditional duration process and a possibly asymmetric shocks impact curve we establish conditions for the existence of higher order moments strict stationarity geometric ergodicity and beta mixing property with exponential decay we next derive moment recursion relations and the autocovariance function of the power delta of the duration process finally we assess the practical usefulness of our family of acd models using nyse price duration data on the ibm stock while the in sample results warrant the extra flexibility provided either by the box cox transformation or by the asymmetric response to shocks we find no specification that entails satisfactory out of sample performance

optimal tests when a nuisance parameter is present only under the alternative this paper derives asymptotically optimal tests for testing problems in which a nuisance parameter exists under the alternative hypothesis but not under the null for example the results apply to tests of structural change with unknown changepoint the testing problem considered is nonstandard and the classical asymptotic optimality results for the lagrange multiplier wald and likelihood ratio do not apply a weighted average power criterion is used here to generate optimal tests this criterion is similar to that used by a wald 1943 to obtain the classical asymptotic optimality properties of wald tests in regular testing problems copyright 1994 by the econometric society

weakly nonlinear high frequency waves abstract in this paper we derive a method for finding small amplitude high frequency solutions to hyperbolic systems of quasilinear partial differential equations our solution is the sum of two parts i a superposition of small amplitude high frequency waves ii a slowly varying mean solution each high frequency wave displays nonlinear distortion of the wave profile and shocks may form shock conditions are derived for conservative systems different high frequency waves do not interact provided the frequencies and wave numbers of two waves are not linearly related to those of a third the mean solution is found by solving a linear partial differential equation this method generalizes whitham s nonlinearization technique 9 for single waves to problems where many waves are present we obtain these results by generalizing a scheme first proposed by choquet bruhat 1 which employs the method of multiple scales author

finite sample properties of the qmle for the log acd model application to australian stocks this paper concerns the properties of the quasi maximum likelihood estimator qmle of the logarithmic autoregressive conditional duration log acd model proofs of consistency and asymptotic normality of qmle for the log acd model with log normal density are presented this is an important issue as the log acd is used widely for testing various market microstructure models and effects knowledge of the distribution of the qmle is crucial for purposes of valid inference and diagnostic checking the theoretical results developed in the paper are evaluated using monte carlo experiments the experimental results also provide insights into the finite sample properties of the log acd model under different distributional assumptions finally this paper presents two extensions to the log acd model to accommodate asymmetric effects the usefulness of these novel models will be evaluated empirically using data from australian stocks

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

global entropy solutions to weakly nonlinear gas dynamics abstract entropy weak solutions with bounded periodic initial data are considered for the system of weakly nonlinear gas dynamics through a modified glimm scheme an approximate solution sequence is constructed and then a priori estimates are provided with the methods of approximate characteristics and approximate conservation laws which gives not only the existence and uniqueness but also the uniform total variation bounds for the entropy solutions

generalized morse wavelets as a superfamily of analytic wavelets the generalized morse wavelets are shown to constitute a superfamily that essentially encompasses all other commonly used analytic wavelets subsuming eight apparently distinct types of analysis filters into a single common form this superfamily of analytic wavelets provides a framework for systematically investigating wavelet suitability for various applications in addition to a parameter controlling the time domain duration or fourier domain bandwidth the wavelet shape with fixed bandwidth may be modified by varying a second parameter called for integer values of the most symmetric most nearly gaussian and generally most time frequency concentrated member of the superfamily is found to occur for 3 these wavelets known as airy wavelets capture the essential idea of popular morlet wavelet while avoiding its deficiencies they may be recommended as an ideal starting point for general purpose use

a simple justification of the singular limit for equatorial shallow water dynamics the equatorial shallow water equations at low froude number form a symmetric hyperbolic system with large variable coefficient terms although such systems are not covered by the classical klainerman majda theory of singular limits the first two authors recently proved that solutions exist uniformly and converge to the solutions of the long wave equations as the height and froude number tend to 0 their proof exploits the special structure of the equations by expanding solutions in series of parabolic cylinder functions a simpler proof of a slight generalization is presented here in the spirit of the classical theory 2008 wiley periodicals inc

a general family of explicit runge kutta pairs of orders 6 5 explicit runge kutta formula pairs of different orders of accuracy form a class of efficient algorithms for treating nonstiff ordinary differential equations so far several runge kutta pairs of order 6 5 have appeared in the literature these pairs use 8 function evaluations per step and belong to certain families of solutions of a set of 54 nonlinear algebraic equations in 44 or 45 coefficients depending on the use of the fsal first stage as last device these equations form a set of necessary and sufficient conditions that a 6 5 runge kutta pair must satisfy the solution of the latter is achieved by employing various types of simplifying assumptions in this paper we make use of the fact that all these families of pairs satisfy a common set of simplifying assumptions using only these simplifying assumptions we define a new family of 6 5 runge kutta pairs its main characteristic which is also a property that no other known family shares is that all of its nodes except the last one which eq

trilinear resonant interactions of semilinear hyperbolic waves an important question in the study of nonlinear equations is the understanding of the asymptotic behavior of families of solutions or approximate solutions one of the difficulty is known to be the nonlinear interaction of oscillations this phenomenon has been studied for special classes of solutions such as oscillatory functions with phases and amplitudes this is the domain of geometric optics in some cases general families of bounded solutions have also been considered the present paper enters the later category but is highly inspired by the results of nonlinear geometric optics geometric optics provides a precise description of the asymptotic structure of solutions to both linear e g wh and nonlinear j hk mr hmr jmr 1 jmr 2 s hyperbolic pdes whose initial data u 0 x u0 x eu1 x 0 x e have rapid oscillations with prescribed phases 0 and a single prescribed scale 1e such solutions are important in both theory and applications l kk in the lowest approximation oscillations having different spatial scales interact only when m equals zero since otherwise the amplitudes of higher frequency oscillations become negligible relative to those of lower frequencies taking m 0 requires l estimates systems that are well posed in l occur mostly in one spatial dimension for such systems arbitrary uniformly bounded sequences of initial data may be considered but comparatively little is known about the asymptotic structure of the resulting solutions since the oscillations in such solutions may be extremely complex a natural problem is to determine their non oscillatory part which may be identified with the weak limit of the sequence of solutions the highly oscillatory nature of the solutions manifests itself in the absence of strong convergence in general of course some information about those oscillations will generally be needed in order to determine the weak limit the asymptotic structure of uniformly bounded solutions depends strongly on the nature of the nonlinearity for entropy solutions of a genuinely nonlinear conservation law t 1 or of a pair of such equations in one spatial dimension dp oscillations of amplitude o 1 do not persist for positive times so a sequence of solutions converges strongly to the solution whose initial data is the weak limit of the initial data although oscillations do persist in linear systems the weak limit of a sequence of solutions to such

three dimensional instability of planar flows we study the stability of two dimensional solutions of the three dimensional navier stokes equations in the limit of small viscosity we are interested in steady flows with locally closed streamlines we consider the so called elliptic and centrifugal instabilities which correspond to the continuous spectrum of the underlying linearized euler operator through the justification of highly oscillating wentzel kramers brillouin expansions we prove the nonlinear instability of such flows the main difficulty is the control of nonoscillating and nonlocal perturbations issued from quadratic interactions

homogenization of linear and nonlinear transport equations we study the homogenization of the linear and nonlinear transport equations with oscillatory velocity fields two types of homogenized equations are derived for general n dimensional linear and nonlinear problems we derive homogenized equations by introducing additional independent variables to represent the small scales for the two dimensional linear transport equations we derive effective equations for the averaged quantities such equations take the form of either a degenerate non local diffusion equation with memory or a higher order hyperbolic equation to study the nonlinear transport equations we introduce the concept of two scale young measure and extend diperna s method to prove that it reduces to a family of dirac measures

modelling volatility recurrence intervals in the chinese commodity futures market abstract the law of extreme event occurrence attracts much research the volatility recurrence intervals of chinese commodity futures market prices are studied the results show that the probability distributions of the scaled volatility recurrence intervals have a uniform scaling curve for different thresholds q so we can deduce the probability distribution of extreme events from normal events the tail of a scaling curve can be well fitted by a weibull form which is significance tested by ks measures both short term and long term memories are present in the recurrence intervals with different thresholds q which denotes that the recurrence intervals can be predicted in addition similar to volatility volatility recurrence intervals also have clustering features through monte carlo simulation we artificially synthesise arma garch class sequences similar to the original data and find out the reason behind the clustering the larger the parameter d of the figarch model the stronger the clustering effect is finally we use the fractionally integrated autoregressive conditional duration model fiacd to analyse the recurrence interval characteristics the results indicated that the fiacd model may provide a method to analyse volatility recurrence intervals

averaging theorems for conservative systems and the weakly compressible euler equations abstract a generic averaging theorem is proven for systems of odes with two time scales that cannot be globally transformed into the usual action angle variable normal form for such systems this theorem is shown to apply to certain fourier space truncations of the non isentropic slightly compressible euler equations of fluid mechanics for the full euler equations we derive formally the generic limit equations and analyze some of their properties in the one dimensional case we prove a generic converic convergence result for the full euler equations analogous to the result for odes by making use of special properties of the one dimensional equations we prove convergence to the solution of a more complicated set of averaged equations when the genericity assumptions fail

scaling in the distribution of intertrade durations of chinese stocks abstract the distribution of intertrade durations defined as the waiting times between two consecutive transactions is investigated based upon the limit order book data of 23 liquid chinese stocks listed on the shenzhen stock exchange in the whole year 2003 a scaling pattern is observed in the distributions of intertrade durations where the empirical density functions of the normalized intertrade durations of all 23 stocks collapse onto a single curve the scaling pattern is also observed in the intertrade duration distributions for filled and partially filled trades and in the conditional distributions the ensemble distributions for all stocks are modeled by the weibull and the tsallis q exponential distributions maximum likelihood estimation shows that the weibull distribution outperforms the q exponential for not too large intertrade durations which account for more than 98 5 of the data alternatively nonlinear least squares estimation selects the q exponential as a better model in which the optimization is conducted on the distance between empirical and theoretical values of the logarithmic probability densities the distribution of intertrade durations is weibull followed by a power law tail with an asymptotic tail exponent close to 3

on the role of quadratic oscillations in nonlinear schrodinger equations ii the l 2 critical case we consider a nonlinear semi classical schrodinger equation for which quadratic oscillations lead to focusing at one point described by a nonlinear scattering operator the relevance of the nonlinearity was discussed by r carles c fermanian kammerer and i gallagher for l 2 supercritical power like nonlinearities and more general initial data the present results concern the l 2 critical case in space dimensions 1 and 2 we describe the set of non linearizable data which is larger due to the conformal invariance as an application we precise a result by f merle and l vega concerning finite time blow up for the critical schrodinger equation the proof relies on linear and nonlinear profile decompositions

testing and modeling threshold autoregressive processes abstract the threshold autoregressive model is one of the nonlinear time series models available in the literature it was first proposed by tong 1978 and discussed in detail by tong and lim 1980 and tong 1983 the major features of this class of models are limit cycles amplitude dependent frequencies and jump phenomena much of the original motivation of the model is concerned with limit cycles of a cyclical time series and indeed the model is capable of producing asymmetric limit cycles the threshold autoregressive model however has not received much attention in application this is due to a the lack of a suitable modeling procedure and b the inability to identify the threshold variable and estimate the threshold values the primary goal of this article therefore is to suggest a simple yet widely applicable model building procedure for threshold autoregressive models based on some predictive residuals a simple statistic is proposed to test for threshold nonlinearity and specify the

resonant nonlinear geometric optics for weak solutions of conservation laws abstract the o part of solutions obtained via glimm s scheme of strictly hyperbolic conservation laws with o periodic bv initial data is shown to tend as 0 to the unique entropy solution of the modulation equations of weakly nonlinear geometric optics derived by majda and rosales

explicit runge kutta nystro m methods with reduced phase errors for computing oscillating solutions we construct explicit runge kutta nystrom methods for the integration of first and second order differential equations having an oscillatory solution special attention is paid to the phase errors or dispersion of the dominant components in the numerical oscillations when these methods are applied to a linear homogeneous test model rk n methods are constructed which are dispersive of orders up to 10 whereas the algebraic order of accuracy is only 2 or 3 application of these methods to equations describing free and weakly forced oscillations and to semidiscretized hyperbolic equations reveals that the phase errors can significantly be reduced

random data cauchy theory for supercritical wave equations i local theory we study the local existence of strong solutions for the cubic nonlinear wave equation with data in hs m s 1 2 where m is a three dimensional compact riemannian manifold this problem is supercritical and can be shown to be strongly ill posed in the hadamard sense however after a suitable randomization we are able to construct local strong solution for a large set of initial data in hs m where s 1 4 in the case of a boundary less manifold and s 8 21 in the case of a manifold with boundary

multidimensional viscous shocks i degenerate symmetrizers and long time stability received by the editors september 18 2002 2000 mathematics subject classification primary 35l60 secondary 35b35 35b45 35k55 35l65 76l05

high phase lag order runge kutta and nystrom pairs we exploit the freedom in the selection of the free parameters of one family of eighth algebraic order runge kutta rk pairs and of three families of fourth sixth and eighth order rk nystrom rkn pairs with the purpose of obtaining specific pairs of the highest possible phase lag order which are also characterized by minimized principal truncation error coefficients we present a method for the analytic derivation of the dissipation order conditions for rk methods and the phase lag and dissipation order conditions for nystrom methods the rk pairs we study here are based on a one parameter generalization of some older families of pairs an algorithm and specific optimized 8 6 nystrom pairs are also provided for a class of initial value problems whose solution is known to be described by free oscillations or free oscillations of low frequency with forced oscillations of high frequency superimposed over long integration intervals these new pairs seem to offer some advantages with respect to some older pairs the latter are of the same algebraic orders as the new ones but are characterized by the minimal phase lag order according to their algebraic order and number of stages

the compressible euler equations in a bounded domain existence of solutions and the incompressible limit a short time existence theorem is proven for the euler equations for nonisentropic compressible fluid flow in a bounded domain and solutions with low mach number and almost incompressible initial data are shown to be close to corresponding solutions of the equations for incompressible flow

singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite using classical methods the authors develop a general theory of such problems this theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto fluid dynamics including new constructive local existence theorems for the time singular limit equations in particular the authors give an entirely self contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the mach number becomes small the theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures similar results are developed also for the compressible and incompressible navier stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

nonlinear geometric optics for short pulses abstract this paper studies the propagation of pulse like solutions of semilinear hyperbolic equations in the limit of short wavelength the pulses are located at a wavefront 0 where satisfies the eikonal equation and d lies on a regular sheet of the characteristic variety the approximate solutions are u e approx u t x t x e where u t x r is a smooth function with compact support in r when u satisfies a familiar nonlinear transport equation from geometric optics it is proved that there is a family of exact solutions u e exact such that u e approx has relative error o e as e 0 while the transport equation is familiar the construction of correctors and justification of the approximation are different from the analogous problems concerning the propagation of wave trains with slowly varying envelope

propagation of oscillations in real vanishing viscosity limit in this paper we study viscous perturbations of quasilinear hyperbolic systems in several space dimensions the equations involve a singular parameter which goes to zero they arise in realistic models of compressible flow the large scale motions in the atmosphere 12 they come also from theoretical considerations in non linear geometric optics 4 5 we prove that solutions u exist on a domain of space time independent on open image in new window

cheap error estimation for runge kutta methods explicit runge kutta rk methods in the form of pairs of orders p p 1 provide an attractive means for the solution of initial value problems of first order differential equations most existing rk formulas single methods as well as pairs use the minimal number of stages required for achieving a prescribed order in this article we shall study in terms of efficiency and reliability rk pairs of orders p q whenever q p 1 while in practice pairs of orders p p 1 usually require one or two more stages in addition to those already necessary for a pth order single method we show here that if p 6 7 8 or 10 then efficient pairs of orders p p 2 p p 3 or p p 4 may be easily constructed with a reduced cost in function evaluations with respect to pairs of orders p p 1 in general comparing p qt pairs used in local extrapolation mode lem the one most frequent in practice we see that while the propagated solution of a problem is in either case of the same order p pairs characterized by q p 1 use fewer function evaluations consequently they might be more efficient provided that they are accompanied by a reliable estimator and an efficient implementation could be found for their application in practical situations a new step size selection algorithm proposed here takes full advantage of the potential for increased efficiency inherited by pairs that are accompanied by no additional cost estimators this algorithm which may be also applied to nystrom pairs makes code implementation of these pairs attractive as in all cases the proportionality of the global error with respect to the requested tolerance is in practice always achieved here we shall cover the cases of pairs of orders 4 2 6 4 7 5 8 6 8 5 8 4 and 10 6 with nearly minimized truncation error coefficients which use a minimal number of stages i e in almost all cases equal to that currently known to be the minimal required for constructing a single method of order equal to that of the higher order method of the pair by studying the numerical performance of these pairs we may see that not only are these pairs as reliable as the respective pairs of the type p p 1 but in all cases they seem to be more efficient another important consequence of the numerical tests performed here is that they suggest that for a given number of stages the best rk pairs that may be attained are those for which the higher order method is of the maximal possible order

the incompressible limit of the non isentropic euler equations we study the euler equations for slightly compressible fluids that is after rescaling the limits of the euler equations of fluid dynamics as the mach number tends to zero in this paper we consider the general non isentropic equations and general data we first prove the existence of classical solutions for a time independent of the small parameter then on the whole space d we prove that the solution converges to the solution of the incompressible euler equations

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

on the nonlinear instability of euler and prandtl equations in this paper we give examples of nonlinearly unstable solutions of euler equations in the whole space 2 the half space the periodic strip the strip 1 1 and the periodic torus 2 with an application to vortex sheets using the same methods we prove an instability result for prandtl type boundary layers that appear in and 2000 john wiley sons inc

the dynamic mixed hitting time model for multiple transaction prices and times we propose a structural model for durations between events and a vector of associated marks using a multivariate brownian motion successive passage times of one latent brownian component relative to random boundaries define durations the other correlated brownian components generate the marks our model embeds the class of stochastic conditional scd and autoregressive conditional acd duration models which impose testable restrictions on the relation between the conditional expectation and conditional volatility of durations we strongly reject the scd and acd specifications for both a very liquid and less liquid nyse traded stock and characterize causality relations between volatilities and durations

fast averaging for long and short wave scaled equatorial shallow water equations with coriolis parameter deviating from linearity the equatorial shallow water equations at low froude number form a symmetric hyperbolic system with large terms containing a variable coefficient the coriolis parameter f which depends on the latitude the limiting behavior of the solutions as the froude number tends to zero was investigated rigorously a few years ago using the common approximation that the variations of f with latitude are linear in that case the large terms have a peculiar structure due to special properties of the harmonic oscillator hamiltonian which can be exploited to prove strong uniform a priori estimates in adapted functional spaces it is shown here that these estimates still hold when f deviates from linearity even though the special properties on which the proofs were based have no obvious generalization as in the linear case existence uniqueness and convergence properties of the solutions corresponding to general unbalanced data are deduced from the estimates

counter examples to concentration cancellation we study the existence and the asymptotic behavior of large amplitude high frequency oscillating waves subjected to the two dimensional burger equation this program is achieved by developing tools related to supercritical wkb analysis by selecting solutions which are divergence free we show that incompressible or compressible two dimensional euler equations are not locally closed for the weak l2 topology

resonant one dimensional nonlinear geometric optics abstract in this paper we study the existence and resonant interaction of oscillatory wave trains in one space dimension giving a rigorous proof of the validity of the corresponding expansions of weakly nonlinear optics we consider both semilinear and quasilinear systems the latter before shock formation some important features of the study are the following 1 we prove the existence of families of exact solutions which have asymptotic expansions governed by weakly nonlinear optics equations with variable coefficients nonconstant background fields and nonlinear phases are permitted our weak transversality hypotheses allow us to justify expansions where even a formal theory did not exist before 2 we make a detailed study of resonances the geometry associated with such resonances is related to the theory of planar webs 3 we study the smoothness of the profiles their regularity is ruled by a sum law analogous to that describing the propagation of singularities in one dimension 4 the expansions are justified up to the breakdown of the profiles which coincides with a suitably defined breakdown for exact solutions

runge kutta pairs of order 5 4 satisfying only the first column simplifying assumption among the most popular methods for the solution of the initial value problem are the runge kutta pairs of orders 5 and 4 these methods can be derived solving a system of nonlinear equations for its coefficients to achieve this we usually admit various simplifying assumptions the most common of them are the so called row simplifying assumptions here we neglect them and present an algorithm for the construction of runge kutta pairs of orders 5 and 4 based only in the first column simplifying assumption the result is a pair that outperforms other known pairs in the bibliography when tested to the standard set of problems of detest a cost free fourth order formula is also derived for handling dense output

long time existence of entropy solutions to the one dimensional non isentropic euler equations with periodic initial data the non isentropic euler system with periodic initial data in mathbb r 1 is studied by analyzing wave interactions in a framework of specially chosen riemann invariants generalizing glimm s functionals and applying the method of approximate conservation laws and approximate characteristics an mathcal o varepsilon 2 lower bound is established for the life span of the entropy solutions with initial data that possess varepsilon variation in each period

asymptotic single and multiple scale expansions in the low mach number limit an asymptotic analysis of the euler equations in the limit of vanishing mach number is presented which can be employed to extend the validity of numerical methods from the compressible to the low mach number regime r klein j comput phys 121 1995 pp 213 237 l hoffmann and a meister in proceedings of the fourth eccomas computational fluid dynamics conference vol 1 c hirsch et al eds athens 1998 pp 567 572 the work is based on the analysis recently proposed by klein whereby the paper is devoted to a rigorous mathematical justification of this asymptotic investigation in order to give reliable statements concerning the assumptions under which the results hold in contrast to klein a single scale as well as multiple scale expansion is used depending on the spatial domain under consideration

optimized runge kutta pairs for problems with oscillating solutions three types of methods for integrating periodic initial value problems are presented these methods are i phase fitted ii zero dissipation iii both zero dissipative and phase fitted some particular modifications of well known explicit runge kutta pairs of orders five and four are constructed numerical experiments show the efficiency of the new pairs in a wide range of oscillatory problems

stationarity of garch processes and of some nonnegative time series abstract two time series models are considered garch processes and generalized multivariate autoregressive equations x n 1 a n 1 x n b n 1 with nonnegative i i d coefficients in each case a necessary and sufficient condition ensuring the existence of a strictly stationary solution is given