a critical evaluation of the resolution properties of b spline and compact finite difference methods resolution properties of b spline and compact finite difference schemes are compared using fourier analysis in periodic domains and tests based on solution of the wave and heat equations in finite domains with uniform and nonuniform grids results show that compact finite difference schemes have a higher convergence rate and in some cases better resolution however b spline schemes have a more straightforward and robust formulation particularly near boundaries on nonuniform meshes

an optimized spectral difference scheme for caa problems in the implementation of spectral difference sd method the conserved variables at the flux points are calculated from the solution points using extrapolation or interpolation schemes the errors incurred in using extrapolation and interpolation would result in instability on the other hand the difference between the left and right conserved variables at the edge interface will introduce dissipation to the sd method when applying a riemann solver to compute the flux at the element interface in this paper an optimization of the extrapolation and interpolation schemes for the fourth order sd method on quadrilateral element is carried out in the wavenumber space through minimizing their dispersion error over a selected band of wavenumbers the optimized coefficients of the extrapolation and interpolation are presented and the dispersion error of the original and optimized schemes is plotted and compared an improvement of the dispersion error over the resolvable wavenumber range of sd method is obtained the stability of the optimized fourth order sd scheme is analyzed it is found that the stability of the 4th order scheme with chebyshev gauss lobatto flux points which is originally weakly unstable has been improved through the optimization the weak instability is eliminated completely if an additional second order filter is applied on selected flux points one and two dimensional linear wave propagation analyses are carried out for the optimized scheme it is found that in the resolvable wavenumber range the new sd scheme is less dispersive and less dissipative than the original scheme and the new scheme is less anisotropic for 2d wave propagation the optimized sd solver is validated with four computational aeroacoustics caa workshop benchmark problems the numerical results with optimized schemes agree much better with the analytical data than those with the original schemes

practical aspects of higher order numerical schemes for wave propagation phenomena this paper examines practical issues related to the use of compact difference based fourth and sixth order schemes for wave propagation phenomena with focus on maxwell s equations of electromagnetics an outline of the formulation and scheme optimization is followed by an assessment of the error accruing from application on stretched meshes with two approaches transformed plane method and physical space differencing in the first technique the truncation error expansion for the sixth order compact scheme confirms that the order of accuracy is preserved if a consistent mesh refinement strategy is followed and further that metrics should be evaluated numerically even if analytic expressions are available physical space differencing formulas are derived for the five point stencil by expressing the coefficients in terms of local spacing ratios the order of accuracy of the reconstruction operator is then verified with a numerical experiment on stretched meshes to ensure stability for a broad range of problems fourier analysis is employed to develop a single parameter family of up to tenth order tridiagonal based spatial filters the implementation of these filters is discussed in terms of their effect on the interior scheme as well as in a 1 d cavity where they are employed to suppress a late time instability the paper concludes after demonstrating the application of the scheme to several 3 d canonical problems utilizing cartesian as well as curvilinear meshes published in 1999 by john wiley sons ltd this article is a u s government work and is in the public domain in the united states

central limit theorems for nonlinear functionals of stationary gaussian processes let x x t t be a stationary gaussian process on p leth x be the hilbert space of variables inl2 p which are measurable with respect tox and let u s s be the associated family of time shift operators we sayy h x withe y 0 satisfies the functional central limit theorem or fclt respectively the central limit theorem of clt if in respectively where y t t equiv int limits 0 tt u s circ yds mathord left vphantom int limits 0 tt u s circ yds left var left int limits 0 t u s circ yds right right 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 right kern nulldelimiterspace left var left int limits 0 t u s circ yds right right 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 andw is a standard wiener process on 0 1 this paper provides some general sufficient conditions onx andy ensuring thaty satisfies the clt or fclt examples ofy are given which satisfy the clt but not the fclt this work extends clt s of maruyama 1976 and breuer and major 1983

noncentral convergence of multiple integrals fix 0 denote by g v 2 a gamma random variable with parameter v 2 and let n 2 be a fixed even integer consider a sequence f k of square integrable random variables belonging to the nth wiener chaos of a given gaussian process and with variance converging to 2v as k goes to infinity we prove that f k converges in distribution to 2g v 2 v if and only if e f k 4 12 e f k 3 tends to 12v 2 48v

generalized szego theorems and asymptotics of cumulants by graphical methods we obtain some general asymptotics results about a class of deterministic sums called sums with dependent indices which generalize a classical theorem fo szego the above type of sums is encountered when establishing covergence to the gaussian distribution of sums of wick products by the method of cumulants our asymptotic results reduce in this situation the proof of the central limit thoerem to the study of the connectivity of a family of associated graphs

local linear regression estimation under long range dependence strong consistency and rates we consider the nonparametric estimation of multivariate regression functions and their derivatives for a regression model with long range dependent errors we adopt a local linear fitting approach and establish strong consistency and rates for the estimators of the regression function and its derivatives the rates of convergence depend on the amount of smoothing relative to the strength of the long range dependence lrd resulting in distinct rates for small and large bandwidths moreover the conditions determining this dichotomy are different for the estimates of the regression function than for the estimates of its derivatives

regular article on a class of pade finite volume methods a class of pade finite volume methods providing an improved spectral resolution is presented and compared with well known methods the formulation is based on the sliding averages of the variables and allows the computation of derivatives of all orders using the fourier analysis these methods are examined with respect to i order of accuracy ii spectral resolution iii boundary conditions and iv stability

finite difference on grids with nearly uniform cell area and line spacing for the wave equation on complex domains a finite difference time dependent numerical method for the wave equation supported by recently derived novel elliptic grids is analyzed the method is successfully applied to single and multiple two dimensional acoustic scattering problems including soft and hard obstacles with complexly shaped boundaries the new grids have nearly uniform cell area j grids and nearly uniform grid line spacing email protected numerical experiments reveal the positive impact of these two grid properties on the scattered field convergence to its harmonic steady state the restriction imposed by stability conditions on the time step size is relaxed due to the near uniformity cell areas and grid line spacing as a consequence moderately large time steps can be used for relatively fine spatial grids resulting in greater accuracy at a lower computational cost also numerical solutions for wave problems inside annular regions of complex shapes are obtained the use of the new grids results in late time stability in contrast with other classical finite difference time dependent methods

some convergence results on quadratic forms for random fields and application to empirical covariances limit theorems are proved for quadratic forms of gaussian random fields in presence of long memory we obtain a non central limit theorem under a minimal integrability condition which allows isotropic and anisotropic models we apply our limit theorems and those of ginovian j contemp math anal 34 2 1 15 to obtain the asymptotic behavior of the empirical covariances of gaussian fields which is a particular example of quadratic forms we show that it is possible to obtain a gaussian limit when the spectral density is not in l 2 therefore the dichotomy observed in dimension d 1 between central and non central limit theorems cannot be stated so easily due to possible anisotropic strong dependence in d 1

central limit theorems for sums of wick products of stationary sequences we show by the method of cumulants that checking whether the central limit theorem for sums of wick powers of a stationary sequence holds can be reduced to the study of an associated graph problem we obtain thus central limit theorems under various integrability conditions on the cumulant spectral functions

maximum entropy and stability of a random process with a 1 f power spectrum under deterministic action abstract the principle of maximum entropy has been used to analyze the stability of the resulting process observed during the interaction of a random process with a 1 f spectrum and a deterministic action in lumped and distributed systems of nonlinear stochastic differential equations describing the coupled nonequilibrium phase transitions under the action of a harmonic force the stable resulting process is divided into two branches depending on the amplitude of the harmonic force under the action of exponential relaxation in a lumped system with an increase in the dumping coefficient the power spectrum of the resulting process becomes a spectrum of the lorentz type

a minimum distance estimator for long memory processes this paper considers a minimum distance estimator mde of the differencing para meter of the fractionally integrated white noise model the mde minimizes the differ ence between sample and population autocorrelations the paper presents calculations of asymptotic variances to examine the efficiency of the mde relative to that of the mle for values of the differencing parameter less than l 4 the mde is consistent and asymptotically normal and the asymptotic variance of the mde using the first n auto correlations approaches that of the mle as n increases however there is a substantial efficiency loss if low order autocorrelations are omitted this implies that a non parametric treatment of short run dynamics will involve a substantial loss of efficiency

quasi disjoint pentadiagonal matrix systems for the parallelization of compact finite difference schemes and filters this paper proposes a novel systematic approach for the parallelization of pentadiagonal compact finite difference schemes and filters based on domain decomposition the proposed approach allows a pentadiagonal banded matrix system to be split into quasi disjoint subsystems by using a linear algebraic transformation technique as a result the inversion of pentadiagonal matrices can be implemented within each subdomain in an independent manner subject to a conventional halo exchange process the proposed matrix transformation leads to new subdomain boundary sb compact schemes and filters that require three halo terms to exchange with neighboring subdomains the internode communication overhead in the present approach is equivalent to that of standard explicit schemes and filters based on seven point discretization stencils the new sb compact schemes and filters demand additional arithmetic operations compared to the original serial ones however it is shown that the additional cost becomes sufficiently low by choosing optimal sizes of their discretization stencils compared to earlier published results the proposed sb compact schemes and filters successfully reduce parallelization artifacts arising from subdomain boundaries to a level sufficiently negligible for sophisticated aeroacoustic simulations without degrading parallel efficiency the overall performance and parallel efficiency of the proposed approach are demonstrated by stringent benchmark tests

reml estimation asymptotic behavior and related topics the restricted maximum likelihood reml estimates of dispersion parameters variance components in a general non normal mixed model are defined as solutions of the reml equations in this paper we show the reml estimates are consistent if the model is asymptotically identifiable and infinitely informative under the location invariant class and are asymptotically normal a n if in addition the model is asymptotically nondegenerate the result does not require normality or boundedness of the rank p of design matrix of fixed effects moreover we give a necessary and sufficient condition for asymptotic normality of gaussian maximum likelihood estimates mle in non normal cases as an application we show for all unconfounded balanced mixed models of the analysis of variance the reml anova estimates are consistent and are also a n provided the models are nondegenerate the mle are consistent a n if and only if certain constraints on p are satisfied

direct numerical simulation of spatial transition to turbulence using fourth order vertical velocity second order vertical vorticity formulation a highly accurate algorithm has been developed to study the process of spatial transition to turbulence the algorithmic details of the direct numerical simulation dns of transition to turbulence in a boundary layer based on a formulation in terms of vertical velocity and vertical vorticity are presented issues concerning the boundary conditions are discussed the linear viscous terms are discretized using an implicit crank nicholson scheme and a low storage runge kutta method is used for the nonlinear terms for the spatial discretization fourth order compact finite differences have been used as these have been found to have better resolution compared to explicit differencing schemes of comparable order the number of grid points that are needed per wavelength is close to the theoretical optimum for any numerical scheme the resulting time discretized fourth order equations are split up into two second order equations resulting in helmholtz and poisson type equations the boundary conditions for the laplacian of the vertical velocity are determined using an influence matrix method a robust multigrid algorithm has been developed to solve the resulting anisotropic elliptical equations for the outflow boundary a buffer domain method which smoothly reduces the disturbances to zero in conjunction with parabolization of the navier stokes equations has been used the validation of the results for the dns solver is made both for linear and weakly nonlinear cases

the stability of numerical boundary treatments for compact high order finite difference schemes the stability characteristics of various compact fourth and sixth order spatial operators are assessed with the theory of gustafsson kreiss and sundstrom g k s for the semidiscrete initial boundary value problem these results are generalized to the fully discrete case with a recently developed theory of kreiss and wu in all cases favorable comparisons are obtained between g k s theory eigenvalue determination and numerical simulation the conventional definition of stability then is sharpened to include only those spatial discretizations that are asymptotically stable bounded left half plane eigenvalues many of the higher order schemes that are g k s stable are not asymptotically stable a series of compact fourth and sixth order schemes that are both asymptotically and g k s stable for the scalar case are then developed

optimized compact difference based finite volume schemes for linear wave phenomena high order compact difference based finite volume schemes are developed analyzed and implemented for linear wave propagation phenomena with particular emphasis on computational electromagnetics in the time domain the formulation combines the primitive function approach with five point spatially sixth and fourth order methods optimization in the semi discrete case is achieved by minimizing dispersion and isotropy error the fully discrete scheme is examined by adopting the classical fourth order runge kutta technique stability bounds are established and the coefficients for the spatial discretization are readjusted for optimum performance the scheme is then formally extended to multiple dimensions consistent boundary conditions are presented for the reconstruction operator as well as flux specification several calculations in one and three dimensions confirm the properties of the method

highly accurate compact mixed methods for two point boundary value problems finite difference method and finite element method are widely used partly for their simplicity though these methods can obtain first order or second order of accuracy in this paper we give two highly accurate fourth order and sixth order of accuracy respectively while still quite simple schemes for two point boundary value problems we call them compact mixed methods since they can obtain numerical solutions for unknown function and its first derivative simultaneously to yield best numerical solutions we combine our compact mixed schemes with a direct solver and an iterative solver then we employ fourier method to analyze differencing errors shows that our compact mixed formulae are closer to the true wavenumber numerical experiments show compact mixed schemes are very efficient and highly accurate techniques

maximum entropy formalism fractals scaling phenomena and 1 f noise a tale of tails in this report on examples of distribution functions with long tails we a show that the derivation of distributions with inverse power tails from a maximum entropy formalism would be a consequence only of an unconventional auxilliary condition that involves the specification of the average value of a complicated logarithmic function b review several models that yield log normal distributions c show that log normal distributions may mimic 1 f noise over a certain range and d present an amplification model to show how log normal personal income distributions are transformed into inverse power pareto distributions in the high income range

a cartesian based embedded geometry technique with adaptive high order finite differences for compressible flow around complex geometries an immersed boundary methodology to solve the compressible navier stokes equations around complex geometries in cartesian fluid dynamics solvers is described the objective of the new approach is to enable smooth reconstruction of pressure and viscous stresses around the embedded objects without spurious numerical artifacts a standard level set represents the boundary of the object and defines a fictitious domain into which the flow fields are smoothly extended boundary conditions on the surface are enforced by an approach inspired by analytic continuation each fluid field is extended independently constrained only by the boundary condition associated with that field unlike most existing methods no jump conditions or explicit derivation of them from the boundary conditions are required in this approach numerical stiffness that arises when the fluid solid interface is close to grid points of the mesh is addressed by preconditioning in addition the embedded geometry technique is coupled with a stable high order adaptive discretization that is enabled around the object boundary to enhance resolution the stencils used to transition the order of accuracy of the discretization are derived using the summation by parts technique that ensures stability applications to shock reflections shock ramp interactions and supersonic and low mach number flows over two and three dimensional geometries are presented

asymptotic normality of the anova estimates of components of variance in the nonnormal unbalanced hierarchal mixed model on considere le plan hierarchique non equilibre non normal et on donne des conditions douces pour une suite de tels plans telle que le vecteur d estimations anova normalisees converge vers une distribution normale multivariable

a well balanced and entropy conservative discontinuous galerkin spectral element method for the shallow water equations in this work we design an arbitrary high order accurate nodal discontinuous galerkin spectral element type method for the one dimensional shallow water equations the novel method uses a skew symmetric formulation of the continuous problem we prove that this discretisation exactly preserves the local mass and momentum furthermore we show that combined with a special numerical interface flux function the method exactly preserves the entropy which is also the total energy for the shallow water equations finally we prove that the surface fluxes the skew symmetric volume integrals and the source term are well balanced numerical tests are performed to demonstrate the theoretical findings

high order cyclo difference techniques the summation by parts energy norm is used to establish a new class of high order finite difference techniques referred to here as cyclo difference techniques these techniques are constructed cyclically from stable subelements and require no special numerical procedures near the boundaries when coupled with the simultaneous approximation term sat boundary treatment they are time asymptotically stable for an arbitrary hyperbolic system these techniques are similar to spectral multi domain techniques and are ideally suited for parallel implementation but they do not require special collocation points the principal focus of this work is on methods of sixth order formal accuracy or less however these methods could be extended in principle to any arbitrary order of accuracy

strict stability of high order compact implicit finite difference schemes temporal or strict stability of approximation to pdes is much more difficult to achieve than the classical lax stability in this paper we present a class of finite difference schemes for hyperbolic initial boundary value problems in one and two space dimensions that possess the property of strict stability the approximations are constructed so that all eigenvalues of corresponding differentiation matrix have a nonpositive real part boundary conditions are imposed by using penalty like terms fourth and sixth order compact implicit finite difference schemes are constructed and analyzed computational efficacy of the approach is corroborated by a series of numerical tests in 1 d and 2 d scalar problems

symmetrized compact scheme for receptivity study of 2d transitional channel flow in many physical problems one must maintain symmetry of numerical schemes the symmetry arising due to geometry and or physical conditions defining the flow the direct simulation of transitional plane poiseuille flow presents one such case that motivated the present study it has been shown here using the method of analyzing non periodic problems in sengupta et al 2003 that compact and other high accuracy schemes display strong directionality of the algorithm and such methods cannot be used for direct simulation of the physical flow this prevalent but unacceptable situation for dns is rectified in the present work and a new scheme is introduced that prevents this asymmetry from contaminating results the simulated transitional flow past 2d channel here using the new scheme are in agreement with experimental results and other recent views of sub critical instabilities

a staggered fourth order accurate explicit finite difference scheme for the time domain maxwell s equations abstract we consider a model explicit fourth order staggered finite difference method for the hyperbolic maxwell s equations appropriate fourth order accurate extrapolation and one sided difference operators are derived in order to complete the scheme near metal boundaries and dielectric interfaces an eigenvalue analysis of the overall scheme provides a necessary but not sufficient stability condition and indicates long time stability numerical results verify both the stability analysis and the scheme s fourth order convergence rate over complex domains that include dielectric interfaces and perfectly conducting surfaces for a fixed error level we find the fourth order scheme is computationally cheaper in comparison to the yee scheme by more than an order of magnitude some open problems encountered in the application of such high order schemes are also discussed

testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression abstract broad classes of diagnostics for serial correlation and or dynamic conditional heteroskedasticity of regression disturbances are considered the classes include statistics with good power against strongly dependent alternatives along with the usual ones that test against weak dependence and many others limiting null distributions are obtained under mild conditions on the dependence structure of the alternative against which the test is derived on moments of the disturbances and on the regressors the various test statistics have a similar overall structure and while tests against strongly dependent alternatives entail more computation than ones against weakly dependent alternatives the difference can be slight if the fast fourier transform is used

a computational stream function method for two dimensional incompressible viscous flows this work concerns the development of a numerical method based on the stream function formulation of the navier stokes equations to simulate two dimensional plane or axisymmetric viscous flows the main features of the proposed method are the use of the high order finite difference compact method for the discretization of the stream function equation the implicit pseudo transient newton krylov multigrid matrix free method for the stationary stream function equation and the fourth order runge kutta method for the integration of non stationary flows copyright 2005 john wiley sons ltd

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

high order non uniform grid schemes for numerical simulation of hypersonic boundary layer stability and transition the direct numerical simulation of receptivity instability and transition of hypersonic boundary layers requires high order accurate schemes because lower order schemes do not have an adequate accuracy level to compute the large range of time and length scales in such flow fields the main limiting factor in the application of high order schemes to practical boundary layer flow problems is the numerical instability of high order boundary closure schemes on the wall this paper presents a family of high order non uniform grid finite difference schemes with stable boundary closures for the direct numerical simulation of hypersonic boundary layer transition by using an appropriate grid stretching and clustering grid points near the boundary high order schemes with stable boundary closures can be obtained the order of the schemes ranges from first order at the lowest to the global spectral collocation method at the highest the accuracy and stability of the new high order numerical schemes is tested by numerical simulations of the linear wave equation and two dimensional incompressible flat plate boundary layer flows the high order non uniform grid schemes up to the 11th order are subsequently applied for the simulation of the receptivity of a hypersonic boundary layer to free stream disturbances over a blunt leading edge the steady and unsteady results show that the new high order schemes are stable and are able to produce high accuracy for computations of the nonlinear two dimensional navier stokes equations for the wall bounded supersonic flow

noncentral limit theorems for quadratic forms in random variables having long range dependence we study the weak convergence in d lbrack 0 1 rbrack of the quadratic form sum lbrack nt rbrack j 1 sum lbrack nt rbrack k 1 a j k h m x j h m x k adequately normalized here a s infty s infty is a symmetric sequence satisfying sum a s infty h m is the m th hermite polynomial and x j j geq 1 is a normalized gaussian sequence with covariances r k sim k d l k as k rightarrow infty where 0 d 1 and l is slowly varying we prove that for all m geq 1 the limit is brownian motion when 1 2 d 1 and it is the non gaussian rosenblatt process when 0 d 1 2

explicit and implicit multidimensional compact high resolution shock capturing methods two families of explicit and implicit compact high resolution shock capturing methods for the multidimensional compressible euler equations for fluid dynamics are constructed some of these schemes can be fourth and sixth order accurate away from discontinuities for the semi discrete case their shock capturing properties are of the total variation diminishing tvd total variation bounded tvb total variation diminishing in the mean tvdm essentially nonoscillatory eno or positive type of scheme for 1d scalar hyperbolic conservation laws and are positive schemes in more than one dimension these higher order compact schemes require the same grid stencil per spatial direction as their second order noncompact cousins the added terms over the second order noncompact cousins involve extra vector additions but no added flux evaluations due to the construction these schemes can be viewed as approximations to genuinely multidimensional schemes in the sense that they might produce less distortion in spherical type shocks and are more accurate in vortex type flows than schemes based purely on 1d extensions the extension of these families of compact schemes to coupled nonlinear systems can be accomplished using the roe approximate riemann solver the generalized steger and warming flux vector splitting or the van leer type flux vector splitting modification to existing high resolution second or third order noncompact shock capturing computer codes is minimal high resolution shock capturing properties can also be achieved via a variant of the second order lax friedrichs numerical flux without the use of riemann solvers for coupled nonlinear systems with comparable operations count to their classical shock capturing counterparts an efficient and compatible high resolution shock capturing filter for spatially fourth and sixth order classical compact and noncompact schemes is discussed the simplest extension to viscous flows can be achieved by using the standard fourth order compact or noncompact formula for the viscous terms

stability of finite difference models containing two boundaries or interfaces it is known that the stability of finite difference models of hyperbolic initial boundary value problems is connected with the propagation and reflection of parasitic waves here the waves point of view is applied to models containing two boundaries or interfaces where repeated reflection of trapped packets is a potential new source of instability our analysis accounts for various known instability phenomena in a unified way and leads to several new results three of which are as follows 1 dissipativity does not ensure stability when three or more formulas are concatenated at a boundary or internal interface 2 algebraic gks instabilities can be converted by a second boundary to exponential instabilities only when an infinite numerical reflection coefficient is present 3 gks stability and p stability can be established in certain problems by showing that the numerical reflection coefficient matrices have norm less than one

high order transverse schemes for the numerical solution of pdes many existing numerical schemes for the solution of initial boundary value problems for partial differential equations can be derived by the method of lines the pdes are converted into a system of ordinary differential equations either with initial conditions longitudinal scheme or with boundary conditions transverse scheme in particular this paper studies the performance of the transverse scheme in combination with boundary value methods moreover we do not restrict the semi discretization by the usual first or second order finite difference approximations to replace the derivative with respect to time but we use high order formulae

generalised summation by parts operators and variable coefficients abstract high order methods for conservation laws can be highly efficient if their stability is ensured a suitable means mimicking estimates of the continuous level is provided by summation by parts sbp operators and the weak enforcement of boundary conditions recently there has been an increasing interest in generalised sbp operators both in the finite difference and the discontinuous galerkin spectral element framework however if generalised sbp operators are used the treatment of the boundaries becomes more difficult since some properties of the continuous level are no longer mimicked discretely interpolating the product of two functions will in general result in a value different from the product of the interpolations thus desired properties such as conservation and stability are more difficult to obtain here new formulations are proposed allowing the creation of discretisations using general sbp operators that are both conservative and stable thus several shortcomings that might be attributed to generalised sbp operators are overcome cf nordstrom and ruggiu 2017 38 and manzanero et al 2017 39

asymptotic distributions of the sample mean autocovariances and autocorrelations of long memory time series we derive the asymptotic distributions of the sample mean autocovariances and autocorrelations for a time series whose autocovariance function has the powerlaw decay 0 o 1 as the results differ in important respects from the corresponding results for short memory processes whose autocovariance functions are absolutely summable for long memory processes the variances of the sample mean and of the sample autocovariances and autocorrelations for 0 12 are not of asymptotic order n 1 when 0 12 the asymptotic distributions of the sample autocovariances and autocorrelations are not normal

high order unconditionally stable fc ad solvers for general smooth domains i basic elements we introduce a new methodology for the numerical solution of partial differential equations in general spatial domains our algorithms are based on the use of the well known alternating direction implicit adi approach in conjunction with a certain fourier continuation fc method for the resolution of the gibbs phenomenon unlike previous alternating direction methods of order higher than one which can only deliver unconditional stability for rectangular domains the present high order algorithms possess the desirable property of unconditional stability for general domains the computational time required by our algorithms to advance a solution by one time step in turn grows in an essentially linear manner with the number of spatial discretization points used in this paper we demonstrate the fc ad methodology through a variety of examples concerning the heat and laplace equations in two and three dimensional domains with smooth boundaries applications of the fc ad methodology to hyperbolic pdes together with a theoretical discussion of the method will be put forth in a subsequent contribution the numerical examples presented in this text demonstrate the unconditional stability and high order convergence of the proposed algorithms as well the very significant improvements they can provide in one of our examples we demonstrate a one thousand improvement factor over the computing times required by some of the most efficient alternative general domain solvers

maximum entropy in a nonlinear system with a 1 f power spectrum abstract an analysis of master slave hierarchy has been made in a system of nonlinear stochastic equations describing fluctuations with a 1 f spectrum at coupled nonequilibrium phase transitions it is shown that for a system of stochastic equations there exist different probability distribution functions with power law non gaussian and gaussian tails the governing equation of a system has a probability distribution function with gaussian tails therefore distribution functions for governing equations may be used for finding the gibbs shannon entropy the local maximum of this entropy has been found it corresponds to the tuning of the parameters of the equations to criticality and points to the stability of fluctuations with a 1 f spectrum the tsallis entropy and the renyi entropy for the probability distribution functions with power law tails have been calculated the parameter q which is included in the determination of these entropies has been found from the condition that the coordinates of the maximum gibbs shannon entropy coincide with the maxima of the tsallis and renyi entropies

the theoretical accuracy of runge kutta time discretizations for the initial boundary value problem a study of the boundary error the conventional method of imposing time dependent boundary conditions for runge kutta rk time advancement reduces the formal accuracy of the space time method to first order locally and second order globally independently of the spatial operator this counter intuitive result is analyzed in this paper two methods of eliminating this problem are proposed for the linear constant coefficient case 1 impose the exact boundary condition only at the end of the complete rk cycle 2 impose consistent intermediate boundary conditions derived from the physical boundary condition and its derivatives the first method while retaining the rk accuracy in all cases results in a scheme with much reduced cfl condition rendering the rk scheme less attractive the second method retains the same allowable time step as the periodic problem however it is a general remedy only for the linear case for non linear hyperbolic equations the second method is effective only for for rk schemes of third order accuracy or less numerical studies are presented to verify the efficacy of each approach

an accurate compact treatment of pressure for colocated variables we propose a new scheme for the pressure treatment in computations of incompressible flow using a colocated grid arrangement to avoid oscillations associated with the sparse non compact stencil we introduce a compact fourth order equivalent of this stencil and study its advantages over the classical second order averaging procedure