polynomials for infinite domain spectral elements in this note it is found that laguerre type elements may be used in conjunction with legendre type elements to resolve more accurately regions of outflow and infinite boundaries

the zd garch model a new way to study heteroscedasticity this paper proposes a first order zero drift garch zd garch 1 1 model to study conditional heteroscedasticity and heteroscedasticity together unlike the classical garch model zd garch 1 1 model is always non stationary regardless of the sign of the lyapunov exponent gamma 0 but interestingly when gamma 0 0 it is stable with its sample path oscillating randomly between zero and infinity over time furthermore this paper studies the generalized quasi maximum likelihood estimator gqmle of zd garch 1 1 model and establishes its strong consistency and asymptotic normality based on the gqmle an estimator for gamma 0 a test for stability and a portmanteau test for model checking are all constructed simulation studies are carried out to assess the finite sample performance of the proposed estimators and tests applications demonstrate that a stable zd garch 1 1 model is more appropriate to capture heteroscedasticity than a non stationary garch 1 1 model which suffers from an inconsistent qmle of the drift term

a regression method for spatial disease rates an estimating function approach abstract epidemiologists commonly study the geographical variation of disease rates to generate and refine testable hypotheses regarding etiology poisson regression is often used to estimate the parameters that characterize the effects of risk factors on disease hierarchical models have been proposed to utilize spatial locations and neighbors as surrogates for unknown or unmeasured risk factors in the analysis of disease rates although hierarchical models are useful in modeling spatial disease rates in a scientifically meaningful way the analytic tools for them are generally computationally intensive we overcome this issue by applying a conditional spatial modeling technique using unbiased estimating functions the resulting estimator of regression coefficients is consistent and asymptotically normally distributed under mild conditions a simulation study compares the performance of the proposed scheme with the approximate inference methods for hierarchical models we apply the proposed scheme to sco

hidden integrable hierarchies of akns type a method for constructing integrable hierarchies by restricting akns flows on manifolds of finite codimension in the space of independent parameters is provided two particular types of hierachies are characterized one is given by nonlinear differential equations with coordinate dependent coefficients and the other is related to schrodinger spectral problems with energy dependent potentials of even degree

zero sets of functions and hidden hierarchies of kdv type the zero sets of kdv functions are characterized in terms of the stratification of the infinite grassmannian it is shown that these sets are related to integrable hierarchies arising from schrodinger equations with energy dependent potentials

on hamiltonian perturbations of hyperbolic systems of conservation laws ii universality of critical behaviour hamiltonian perturbations of the simplest hyperbolic equation u t a u u x 0 are studied we argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution moreover this behaviour is described by a special solution to an integrable fourth order ode

testing for in finite moments this paper proposes a test to verify whether the th moment of a random variable is finite we use the fact that under general assumptions sample moments either converge to a finite number or diverge to infinity according as the corresponding population moment is finite or not building on this we propose a test for the null that the th moment does not exist since by construction our test statistic diverges under the null and converges under the alternative we propose a randomised testing procedure to discern between the two cases we study the application of the test to raw data and to regression residuals monte carlo evidence shows that the test has the correct size and good power the results are further illustrated through an application to financial data

a spectral viscosity method based on hermite functions for nonlinear conservation laws we consider the approximation by a spectral method of the solution of the cauchy problem for a scalar conservation law in one dimension posed in the whole real line we analyze a spectral viscosity method in which the orthogonal basis considered is the one of hermite functions we prove the convergence of the approximate solution to the unique entropy solution of the problem by using compensated compactness arguments

monotonic limit theorem of bsde and nonlinear decomposition theorem of doob meyers type we have obtained the following limit theorem if a sequence of rcll supersolutions of a backward stochastic differential equations bsde converges monotonically up to y t with e sup t y t 2 then y t itself is a rcll supersolution of the same bsde theorem 2 4 and 3 6 we apply this result to the following two problems 1 nonlinear doob meyer decomposition theorem 2 the smallest supersolution of a bsde with constraints on the solution y z the constraints may be non convex with respect to y z and may be only measurable with respect to the time variable t this result may be applied to the pricing of hedging contingent claims with constrained portfolios and or wealth processes

rational legendre pseudospectral approach for solving nonlinear differential equations of lane emden type lane emden equation is a nonlinear singular equation in the astrophysics that corresponds to the polytropic models in this paper a pseudospectral technique is proposed to solve the lane emden type equations on a semi infinite domain the method is based on rational legendre functions and gauss radau integration the method reduces solving the nonlinear ordinary differential equation to solve a system of nonlinear algebraic equations the comparison of the results with the other numerical methods shows the efficiency and accuracy of this method

a complete orthonormal system of functions in l 2 infty infty space a new complete orthonormal system of functions in the l 2 infty infty space is introduced the system consists of two sequences composed of odd and even functions respectively unlike hermite and laguerre sets of functions which behave exponentially at infinity the new system exhibits asymptoticbehavior x 1 for the odd sequence and x 2 for the even one formulae representing products derivatives etc in series in the system are developed a nonlinear differential equation with a requirement for summability of the square of its solutions instead of boundary conditions is solved the example displays most of the features of the new method important for essentially nonlinear problems

hermite spectral methods for fractional pdes in unbounded domains numerical approximations of fractional pdes in unbounded domains are considered in this paper since their solutions decay slowly with power laws at infinity a domain truncation approach is not effective as no transparent boundary condition is available we develop efficient hermite collocation and hermite galerkin methods for solving a class of fractional pdes in unbounded domains directly and derive corresponding error estimates we apply these methods for solving fractional advection diffusion equations and fractional nonlinear schrodinger equations

strong invariance principles for mixing random fields given a random field z q indexed by q tuples of positive integers and satisfying a strong mixing condition we study the approximation of the partial sum field sn n z q by brownian sheet setting g alpha n 1 n q in z q n k underset raise0 3em hbox smash scriptscriptstyle geqslant mathop pi limits 1 underset raise0 3em hbox smash scriptscriptstyle leqslant i underset raise0 3em hbox smash scriptscriptstyle leqslant q i pm k n i alpha k 1 q for 0 0 we also construct an example showing that in a somewhat larger similar type domain the above approximation is generally impossible even with 0

oscillations of the zero dispersion limit of the korteweg de vries equation we attack the multiphase averaged systems for the zero dispersion limit of the kdv equation attention is paid to the most important case the single phase oscillations a scheme is developed to solve the whitham averaged system single phase averaged system this system under our scheme is transformed to a linear over determined system of euler poisson darboux type whose solution can be written down explicitly we show that for any smooth initial data which has only one hump or is a nontrivial monotone function the weak limit has single phase oscillations within a cusp in the x t plane for a short time after the breaking time for the corresponding burgers equation outside the cusp the limit satisfies the burgers equation more surprisingly we also show that the weak limit has global single phase oscillations within a cusp for any smooth nontrivial monotone initial data with only one inflection point 1993 john wiley sons inc

a uniform central limit theorem for set indexed partial sum processes with finite variance on obtient des resultats de convergence faible qui ameliorent ceux de pyke 1983 et de bass et pyke 1984

reduction of dispersionless coupled korteweg de vries equations to the euler darboux equation a quasilinear hyperbolic system of two first order equations is introduced the system is linearized by means of the hodograph transformation combined with riemann s method of characteristics in the process of linearization the main step is to explicitly express the characteristic velocities in terms of the riemann invariants the procedure is shown to be performed by quadrature only for specific combinations of the parameters in the system we then apply the method developed here to the dispersionless versions of the typical coupled korteweg de vries ckdv equations including the broer kaup ito hirota satsuma and bogoyavlenskii equations and show that these equations are transformed into the classical euler darboux equation a more general quasilinear system of equations is also considered with application to the dispersionless ckdv equations for the jaulent miodek and nutku oguz equations

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

invariance principles for self similar set indexed random fields for a stationary random field x j j in z d and some measure m on r d we consider the set indexed weighted sum process s n a sum j in z d m na cap r j frac12 x j where r j is the unit cube with lower corner j we establish a general invariance principle under a p stability assumption on the x j s and an entropy condition on the class of sets a the limit processes are self similar set indexed gaussian processes with continuous sample paths using chentsov s type representations to choose appropriate measures m and particular sets a we show that these limits can be levy fractional brownian fields or fractional brownian sheets

a maximal inequality and dependent marcinkiewicz zygmund strong laws this paper contains some extension of kolmogorov s maximal inequality to dependent sequences next we derive dependent marcinkiewicz zygmund type strong laws of large numbers from this inequality in particular for stationary strongly mixing sequences x i i z with sequence of mixing coefficients n n 0 the marcinkiewicz zygmund slln of order p holds if 0 1 1 t p 1 q p t dt where 1 denotes the inverse function of the mixing rate function t t and q denotes the quantile function of x 0 the condition is obtained by an interpolation between the condition of doukhan massart and rio implying the clt p 2 and the integrability of x 0 implying the usual slln p 1 moreover we prove that this condition cannot be improved for stationary sequences and power type rates of strong mixing

the eigenvalues of hermite and rational spectral differentiation matrices we derive expressions for the eigenvalues of spectral differentiation matrices for unbounded domains in particular we consider galerkin and collocation methods based on hermite functions as well as rational functions a fourier series combined with a cotangent mapping we show that i first derivative matrices have purely imaginary eigenvalues and second derivative matrices have real and negative eigenvalues ii for the hermite method the eigenvalues are determined by the roots of the hermite polynomials and for the rational method they are determined by the laguerre polynomials and iii the hermite method has attractive stability properties in the sense of small condition numbers and spectral radii

characterisations of set indexed brownian motion and associated conditions for finite dimensional convergence on donne deux caracterisations pour des lois de mouvement brownien en dimension finie indexees par une classe arbitraire de sous ensembles du cube unite a deux dimensions

chover type laws of the k iterated logarithm for weighted sums of strongly mixing sequences this note contains a chover type law of the k iterated logarithm for weighted sums of strong mixing sequences of random variables with a distribution in the domain of a stable law we show that the upper part of the lil is similar to other studies in the literature conversely the lower half is substantially different in particular we show that due to the failure of the classical version of the second borel cantelli lemma the upper and the lower bounds are separated with the lower bound being further and further away as the memory of the sequence increases

error estimation of hermite spectral method for nonlinear partial differential equations hermite approximation is investigated some inverse inequalities imbedding inequalities and approximation results are obtained a hermite spectral scheme is constructed for burgers equation the stability and convergence of the proposed scheme are proved strictly the techniques used in this paper are also applicable to other nonlinear problems in unbounded domains

integrable quasiclassical deformations of algebraic curves a general scheme for determining and studying integrable deformations of algebraic curves based on the use of lenard relations is presented the method is illustrated with the analysis of the hyperelliptic case an associated multi hamiltonian hierarchy of systems of hydrodynamic type is characterized

chover type laws of the k iterated logarithm for ρ mixing sequences of random variables abstract in this paper a chover type law of the k iterated logarithm is established for mixing sequences of identically distributed random variables with a distribution in the domain of an attraction of a stable distribution with an exponent in 0 2 our results generalize and improve those on chover s lil type behavior previously obtained by mikosch 1984 9 vasudeva 1984 14 and qi and cheng 1996 11 from the i i d case to mixing sequences

the τ function of the universal whitham hierarchy matrix models and topological field theories the universal witham hierarchy is considered from the point of view of topological field theories the function for this hierarchy is defined it is proved that the algebraic orbits of whitham hierarchy can be identified with various topological matter models coupled with topological gravity

algebraic varieties in the birkhoff strata of the grassmannian gr 2 harrison cohomology and integrable systems the local properties of the families of algebraic subsets wg in the birkhoff strata 2g of gr 2 containing the hyperelliptic curves of genus g are studied it is shown that the tangent spaces tg for wg are isomorphic to the linear spaces of 2 coboundaries particular subsets in wg are described by the integrable dispersionless coupled kdv systems of hydrodynamical type defining a special class of 2 cocycles and 2 coboundaries in tg it is demonstrated that the blows ups of such 2 cocycles and 2 coboundaries and gradient catastrophes for associated integrable systems are interrelated

law of the iterated logarithm for set indexed partial sum processes with finite variance is relatively compact in the uniform topology and the set of limit points is the set of functions which are absolutely continuous and whose derivatives have l 2 norm less than or equal to 1 using skorokhod embedding strassen showed that random walks also obeyed a functional lil and that the usual lil as well as many other interesting results could be obtained as a consequence the extension of strassen s results to brownian motion indexed by points in and to partial sum processes indexed by points in ir was done by pyke 7 and wichura 9 respectively for a brownian motion indexed by a large collection d of sets the functional law was proved by bass and pyke 2 and then a type of skorokhod embedding was used to extend this for partial sum processes however if h 5 is the log entropy of d defined in sect 2 it was necessary that h 6 0 moreover the sets in d had to have uniformly smooth boundaries morrow and philipp 6j were able to wea

hodograph solutions of the dispersionless coupled kdv hierarchies critical points and the euler poisson darboux equation it is shown that the hodograph solutions of the dispersionless coupled kdv dckdv hierarchies describe critical and degenerate critical points of a scalar function which obeys the euler poisson darboux equation singular sectors of each dckdv hierarchy are found to be described by solutions of higher genus dckdv hierarchies concrete solutions exhibiting shock type singularities are presented

approximation of some diffusion evolution equations in unbounded domains by hermite functions spectral and pseudospectral approximations of the heat equation are analyzed the solution is represented in a suitable basis constructed with hermite polynomials stability and convergence estimates are given and numerical tests are discussed

integrable discretization of hodograph type systems hyperelliptic integrals and whitham equations based on the well established theory of discrete conjugate nets in discrete differential geometry we propose and examine discrete analogues of important objects and notions in the theory of semi h

central limit theorems for stochastic processes under random entropy conditions necessary and sufficient conditions are found for the weak convergence of the row sums of an infinitesimal row independent triangular array nj of stochastic processes indexed by a set s to a sample continuous gaussian process when the array satisfies a random entropy condition analogous to one used by gine and zinn 1984 for empirical processes this entropy condition is satisfied when s is a class of sets or functions with the vapnik ervonenkis property and each nj f fd nj is of the form njc for some reasonable random finite signed measure vnj as a result we obtain necessary and sufficient conditions for the weak convergence of possibly non i i d partial sum processes and new sufficient conditions for empirical processes indexed by vapnik ervonenkis classes special cases include prokhorov s 1956 central limit theorem for empirical processes and shorack s 1979 theorems on weighted empirical processes

quasiclassical limit of coupled kdv equations riemann invariants and multi hamiltonian structure abstract we consider the quasiclassical limit of the first nontrivial flow in coupled kdv hierarchy written down in riemann invariants it assumes the extremely simple form r i k 1 n r k 2r i r x i i 1 n up to certain natural equivalence this turns out to be the only hydrodynamic system with n 1 compatible purely local hamiltonian structures

the durbin watson ratio under infinite variance errors abstract this paper studies the properties of the von neumann ratio for time series with infinite variance our asymptotics cover the null of iid variates and general moving average ma alternatives regression residuals are also considered in the static regression model the durbin watson statistic has the same limit distribution as the von neumann ratio in dynamic models the results are more complex in finite variance models our results specialize to those of the durbin h statistic and equivalent lm test asymptotics some monte carlo results are reported illustrating the effects of infinite variance errors and regressors in finite samples

on hamiltonian perturbations of hyperbolic systems of conservation laws i quasi triviality of bi hamiltonian perturbations we study the general structure of formal perturbative solutions to the hamiltonian perturbations of spatially one dimensional systems of hyperbolic pdes vt v x 0 under certain genericity assumptions it is proved that any bi hamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives the main tool is in constructing the so called quasimiura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bi hamiltonian structure of hydrodynamic type the quasi triviality theorem we also describe following 35 the invariants ofsuchbi hamiltonian structures with respect to the group of miura type transformations depending polynomially on the derivatives c

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

hermite pseudospectral approximations an error estimate abstract we find an error bound for the pseudospectral approximation of a function in terms of hermite functions h n x e x 2 h n x in certain weighted sobolev spaces we use properties of hermite polynomials as well as an asymptotic expression for their largest zeroes to achieve the mentioned estimate

strong law of large numbers and chover s law of the iterated logarithm under sub linear expectations abstract limit theorems for sub linear expectations are challenging field which have raised a large number of issues of interest recently the aim of this paper is to establish general strong law of large numbers and the chover s law of the iterated logarithm for a sequence of random variables under a sub linear expectation space as applications several results on strong laws of large numbers which contain marcinkiewicz strong law of large numbers and the chover s law of the iterated logarithm for extended independence and identically distributed random variables have been generalized to the sub linear expectation space context our results of strong laws of large numbers are more general than some related results previously reported obtained by zhang 2016 23 cheng 2016 5 liu et al 2015 10 and chen 2016 3 there is no report on the chover s law of the iterated logarithm under sub linear expectation and we provide a method to study this subject

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

generalized hermite spectral method matching asymptotic behaviors in this paper we propose the generalized hermite spectral method by using a family of new generalized hermite functions which are mutually orthogonal with the weight function 1 x 2 c c being an arbitrary real number we establish the basic results on the corresponding orthogonal approximation and interpolation which simulate the asymptotic behaviors of approximated functions at infinity reasonably as examples of applications the spectral schemes are provided for two model problems numerical results demonstrate their spectral accuracy in space

invariance principles for mixing sequences of random variables in this note we prove weak invariance principles for some classes of mixing sequences of l 2 integrable random variables under the condition that the variance of the sum of n random variables is asymptotic to sigma 2n where sigma 2 0 one of the results is simultaneously an extension to nonstationary case of a theorem of ibragimov and an improvement of the varphi mixing rate used by mcleish in his invariance principle for nonstationary varphi mixing sequences

heavy tail modeling and teletraffic data special invited paper huge data sets from the teletraffic industry exhibit many nonstandard characteristics such as heavy tails and long range dependence various estimation methods for heavy tailed time series with positive innovations are reviewed these include parameter estimation and model identification methods for autoregressions and moving averages parameter estimation methods include those of yule walker and the linear programming estimators of feigin and resnick as well estimators for tail heaviness such as the hill estimator and the qq estimator examples are given using call holding data and interarrivals between packet transmissions on a computer network the limit theory makes heavy use of point process techniques and random set theory

the spectral methods for parabolic volterra integro differential equations in this paper we study the numerical solutions to parabolic volterra integro differential equations in one dimensional bounded and unbounded spatial domains in a bounded domain the given parabolic volterra integro differential equation is converted to two equivalent equations then a legendre collocation method is used to solve them and finally a linear algebraic system is obtained for an unbounded case we use the algebraic mapping to transfer the problem on a bounded domain and then apply the same presented approach for the bounded domain in both cases some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method

a consistent test for nonlinear out of sample predictive accuracy abstract in this paper we draw on both the consistent specification testing and the predictive ability testing literatures and propose an integrated conditional moment type predictive accuracy test that is similar in spirit to that developed by bierens j econometr 20 1982 105 econometrica 58 1990 1443 and bierens and ploberger econometrica 65 1997 1129 the test is consistent against generic nonlinear alternatives and is designed for comparing nested models one important feature of our approach is that the same loss function is used for in sample estimation and out of sample prediction in this way we rule out the possibility that the null model can outperform the nesting generic alternative model it turns out that the limiting distribution of the icm type test statistic that we propose is a functional of a gaussian process with a covariance kernel that reflects both the time series structure of the data as well as the contribution of parameter estimation error as a consequence critical values that are data dependent and cannot be directly tabulated one approach in this case is to obtain critical value upper bounds using the approach of bierens and ploberger econometrica 65 1997 1129 here we establish the validity of a conditional p value method for constructing critical values the method is similar in spirit to that proposed by hansen econometrica 64 1996 413 and inoue econometric theory 17 2001 156 although we additionally account for parameter estimation error in a series of monte carlo experiments the finite sample properties of three variants of the predictive accuracy test are examined our findings suggest that all three variants of the test have good finite sample properties when quadratic loss is specified even for samples as small as 600 observations however non quadratic loss functions such as linex loss require larger sample sizes of 1000 observations or more in order to ensure reasonable finite sample performance

singular sector of the burgers hopf hierarchy and deformations of hyperelliptic curves we discuss the structure of shock singularities of the burgers hopf hierarchy it is shown that the set of singular solutions defines a stratification of the affine space of the flow parameters in the hierarchy the stratification is associated with the birkhoff decomposition of the grassmannian given by the set of linear spaces spanned by the hierarchy we then construct integrable hierarchy on each stratum and demonstrate that it describes a deformation of a hyperelliptic curve parametrizing the stratum the hierarchy is called the hidden burgers hopf hierarchy and we found the riemann invariant form and the hodograph solution

testing the existence of moments for garch processes abstract it is generally admitted that many financial time series have heavy tailed marginal distributions when time series models are fitted on such data the non existence of appropriate moments may invalidate standard statistical tools used for inference moreover the existence of moments can be crucial for risk management for instance when risk is measured through the expected shortfall this paper considers testing the existence of moments in the framework of garch processes while the second order stationarity condition does not depend on the distribution of the innovation higher order moment conditions involve moments of the independent innovation process we propose tests for the existence of high moments of the returns process which are based on the joint asymptotic distribution of the quasi maximum likelihood qml estimator of the volatility parameters and empirical moments of the residuals a bootstrap procedure is proposed to improve the finite sample performance of our test to achieve efficiency gains we consider non gaussian qml estimators founded on reparameterizations of the garch model and we discuss optimality issues monte carlo experiments and an empirical study illustrate the asymptotic results

exponentially accurate nonconforming least squares spectral element method for elliptic problems on unbounded domain abstract non conforming approximation methods are becoming increasingly popular because of the potential to apply to multi material and multi model analysis for both bounded and unbounded domains in this paper we present a least square approximation based method to solve the one or two dimensional elliptic problems on an unbounded domain the method gives exponential accuracy and shows superior performance when compared to other numerical methods differentiability estimates and the main stability estimate theorem using a non conforming spectral element method are also discussed the exponential convergence rate of the proposed method is also shown through rigorous error estimate and specific numerical examples

variance of set indexed sums of mixing random variables and weak convergence of set indexed processes on obtient une borne uniforme pour la variance d un processus de somme partielle indice sur un ensemble sous une condition de melange

coupled harry dym equations with multi hamiltonian structures the authors consider the isospectral flows of delta 2 sigma 1n upsilon i lambda i psi alpha psi using an unusual form of the lax approach they derive in a particularly simple manner a the second half of the lax pair b the associated hamiltonian structures c an infinite hierarchy of poisson commuting hamiltonians in this way they show that these equations possess n 1 compatible purely differential hamiltonian structures the case n 1 is just the bi hamiltonian harry dym hierarchy they thus extend the recent results on multi hamiltonian coupled kdv equations

integrable hydrodynamic chains a new approach for derivation of benney type moment chains and integrable hydrodynamic type systems is presented new integrable hydrodynamic chains are constructed all their hydrodynamical reductions are described and integrated new 2 1 integrable hydrodynamic type systems are found

critical points lauricella functions and whitham type equations a large class of semi hamiltonian systems of hydrodynamic type is interpreted as the equations governing families of critical points of functions obeying the classical linear darboux equations for conjugate nets the distinguished role of the euler poisson darboux equations and associated lauricella type functions is emphasised in particular it is shown that the classical g phase whitham equations for the kdv and nls equations are obtained via a g fold iterated darboux type transformation generated by appropriate lauricella functions