input output stabilization of linear systems on spl zopf a formal framework is set up for the discussion of generalized autoregressive with external input models of the form ay bu where a and b are linear operators with the main emphasis being on signal spaces consisting of bounded sequences parametrized by the integers different notions of stability are explored and topological notions such as the idea of a closed system are linked with questions of stabilizability in this very general context various problems inherent in using spl zopf as the time axis are analyzed in this operatorial framework

high order asymptotic expansions for robust tests summary this paper provides high order asymptotic expansions for the m ratio and the signedroot m ratio robust test statistics which allow one to compute accurate approximations to significance or critical values using the edgeworth approximation the bartlett correction the variance correction or the saddlepoint approximation specific results are obtained for the linear regression model with the huber m estimator a monte carlo study illustrates the numerical accuracy of these approximations with respect to the usual first order approximations

energy stable model reduction of neurons by nonnegative discrete empirical interpolation the accurate and fast prediction of potential propagation in neuronal networks is of prime importance in neurosciences this work develops a novel structure preserving model reduction technique to address this problem based on galerkin projection and nonnegative operator approximation it is first shown that the corresponding reduced order model is guaranteed to be energy stable thanks to both the structure preserving approach that constructs a distinct reduced order basis for each cable in the network and the preservation of nonnegativity furthermore a posteriori error estimates are provided showing that the model reduction error can be bounded and controlled finally the application to the model reduction of a large scale neuronal network underlines the capability of the proposed approach to accurately predict the potential propagation in such networks while leading to important speedups

saddlepoint expansions in finite population problems summary saddlepoint approximations are derived for the sample mean in the case where samples are obtained randomly without replacement from a finite population asymptotic properties are investigated the great accuracy is demonstrated by numerical examples extensions to more general statistics are discussed

the orthogonal decomposition theorems for mimetic finite difference methods accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus provide reliable finite difference methods for approximating the solutions to a wide class of partial differential equations these methods mimic many fundamental properties of the underlying physical problem including conservation laws symmetries in the solution and the nondivergence of particular vector fields i e they are divergence free and should satisfy a discrete version of the orthogonal decomposition theorem this theorem plays a fundamental role in the theory of generalized solutions and in the numerical solution of physical models including the navier stokes equations and in electrodynamics we are deriving mimetic finite difference approximations of the divergence gradient and curl that satisfy discrete analogs of the integral identities satisfied by the differential operators we first define the natural discrete divergence gradient and curl operators based on coordinate invariant definitions such as gauss s theorem for the divergence next we use the formal adjoints of these natural operators to derive compatible divergence gradient and curl operators with complementary domains and ranges of values in this paper we prove that these operators satisfy discrete analogs of the orthogonal decomposition theorem and demonstrate how a discrete vector can be decomposed into two orthogonal vectors in a unique way satisfying a discrete analog of the formula vec a ggrad varphi curl vec b we also present a numerical example to illustrate the numerical procedure and calculate the convergence rate of the method for a spiral vector field

saddlepoint approximations for estimating equations summary the saddlepoint method is used to approximate to the distribution of an estimator defined by an estimating equation two different approaches are available one of which is shown to be equivalent to the technique of field hampel 1982 two recent formulae for the tail probability due to lugannani rice 1980 and to robinson 1982 respectively which are uniformly accurate over the whole range of the estimator are compared numerically with the exact results and those computed by field hampel they are found to be of comparable accuracy while avoiding the use of numerical integration the most accurate is that of lugannani rice hampel 1974 introduced a new technique for approximating to the probability density of an estimator defined by an estimating equation it is an example of what he called small sample asymptotics where high accuracy is achieved for quite small sample sizes n even down to single figures in the original version it gave an approximation to the logarithmic derivative of the density function which was integrated numerically to get the density the distribution function was obtained by a second numerical integration which could then be used to renormalize both the density and the distribution function field hampel 1982 develop the technique in detail and compare its performance with that of other approximation methods as hampel had pointed out his approach is closely related to the saddlepoint method of daniels 1954 which was applied to sample means and ratios of means following the private communication referred to by field hampel 1982 p 31 it was realized that the first numerical integration was unnecessary and that hampel s approach could be shortened to give a direct approximation to the density which is in fact a saddlepoint approximation the purpose of the present paper is to extend the use of the saddlepoint method to estimating equations there are two distinct ways of doing this which lead to different approximations of similar accuracy one appears to be more convenient for approxi mating to tail probabilities by numerical integration and was used to compute the saddlepoint approximations quoted in field hampel s tables 1 and 2 the other gives the density directly in a form equivalent to their equation 4 3 which is then integrated

saddlepoint approximations for continuous time markov processes this paper proposes saddlepoint expansions as a means to generate closed form approximations to the transition densities and cumulative distribution functions of markov processes this method is applicable to a large class of models considered in finance for which a laplace or characteristic functions but not the transition density can be found in closed form but even when such a computation is not possible explicitly we go one step further by showing how useful approximations can be obtained by replacing the laplace or characteristic functions by an expansion in small time

on discrete time linear systems with almost periodic kernels the paper considers several different stability notions for discrete time linear time varying systems specified by almost periodic kernels various input output stability notions are analysed including those in which the signal spaces consist of bounded power and asymptotically almost periodic functions further it is proved that every stable time varying discrete time system commuting with a polynomial in the shift operator is periodic an example shows that this result does not hold for unstable time varying systems

corrections and extensions of optimal control of linear systems with almost periodic input by g da prato and a ichikawa g da prato and a ichikawa consider in their 1987 paper the optimal control problem for linear systems with almost periodic inputs in order to get their results they use the space b ap 2 h the completion of the space of almost periodic functions in the quadratic mean they used that b ap 2 u inputs give almost periodic outputs by means of a scalar example we show that this does not hold in general we prove that one of their main results still holds furthermore we extend this result to all forcing terms in b ap 2 h and show that if the forcing term is almost periodic then the optimal control is almost periodic as well

on system gains for linear and nonlinear systems abstract system gains and bounds for system gains are determined for stable linear and nonlinear systems in different signal setups which include l p signal setups and certain persistent signal generalizations of the l 2 and l 1 signal setups these results show that robust h and l 1 control generalize to very versatile persistent signal settings relationships between different system gains are also derived finally an application of nonlinear system gain bounds is given by establishing induced l modelling error bounds for a class of generalized piecewise linear systems approximated with simpler linear time invariant models

mimetic discretization of two dimensional magnetic diffusion equations abstract in case of non constant resistivity cylindrical coordinates and highly distorted polygonal meshes a consistent discretization of the magnetic diffusion equations requires new discretization tools based on a discrete vector and tensor calculus we developed a new discretization method using the mimetic finite difference framework it is second order accurate on arbitrary polygonal meshes and a consistent calculation of the joule heating is intrinsic within it the second order convergence rates in l 2 and l 1 norms were verified with numerical experiments

the wilson hilferty transformation is locally saddlepoint interest in transform methods for normalising test statistics declined with the advent of computers more recently small sample asymptotic methods have been developed to approximate the distributions of complicated test statistics we propose a generalisation of a classical symmetrising transform to a similar range of statistics it is shown to behave comparably in a parametric neighbourhood to methods that exploit exponential tilting

least squares lti approximation of nonlinear systems and quasistationarity analysis least squares linear time invariant lti approximation of discrete time nonlinear systems is studied in a generalized harmonic analysis setting extending an earlier result based on quasistationary signals the least squares optimal lti model is such that the crosscorrelation between the input and the lti model output equals the crosscorrelation between the input and the output of the nonlinear system new results for limits of sample averages of signals are derived via riemann stieltjes integration theory these results are applied to crosscorrelation and quasistationarity analysis of input output signals for several important classes of nonlinear systems including stable finite memory wiener and hammerstein systems this analysis demonstrates that the assumptions used in the least squares lti approximation setup are fairly mild finally an illustrative example is provided

saddlepoint approximations for short and long memory time series a frequency domain approach abstract saddlepoint techniques provide numerically accurate small sample approximations to the distribution of estimators and test statistics except for a few simple models these approximations are not available in the framework of stationary time series we contribute to fill this gap under short or long range serial dependence for gaussian and non gaussian processes we show how to derive and implement saddlepoint approximations for two relevant classes of frequency domain statistics ratio statistics and whittle s estimator we compare our new approximations to the ones obtained by the standard asymptotic theory and by two widely applied bootstrap methods the numerical exercises for whittle s estimator show that our approximations yield accuracy s improvements while preserving analytical tractability a real data example concludes the paper

on linear models for nonlinear systems best linear time invariant lti approximations are analysed for several interesting classes of discrete nonlinear time invariant systems these include nonlinear finite impulse response systems and a class of nonsmooth systems called bi gain systems the frechet derivative of a smooth nonlinear system is studied as a potential good lti model candidate the frechet derivative is determined for nonlinear finite memory systems and for a class of wiener systems most of the concrete results are derived in an signal setting applications to linear controller design to identification of linear models and to estimation of the size of the unmodelled dynamics are discussed

on the convergence rates of energy stable finite difference schemes abstract we consider constant coefficient initial boundary value problems with a first or second derivative in time and a highest spatial derivative of order q and their semi discrete finite difference approximations with an internal truncation error of order p 1 and a boundary error of order r 0 we prove that the convergence rate is min p r q the assumptions needed for these results to hold are i the continuous problem is linear and well posed with a smooth solution ii the numerical scheme is consistent nullspace consistent nullspace invariant and energy stable these assumptions are often satisfied for summation by parts schemes

approximations for densities of sufficient estimators abstract a simple method of obtaining asymptotic expansions for the densities of sufficient estimators is described it is an extension of the one developed by o barndorff nielsen and d r cox 1979 for exponential families a series expansion in powers of n 1 is derived of which the first term has an error of order n 1 which can effectively be reduced to n 3 2 by renormalization the results obtained are similar to those given by h e daniels s 1954 saddlepoint method but the derivations are simpler a brief treatment of approximations to conditional densities is given theorems are proved which extend the validity of the multivariate edgeworth expansion to parametric families of densities of statistics which need not be standardized sums of independent and identically distributed vectors these extensions permit the treatment of problems arising in time series analysis the technique is used by j durbin 1980 to obtain approximations to the densities of partial serial correlation coefficients

saddlepoint approximations for the difference of order statistics and studentized sample quantiles summary for a sample from a given distribution the difference of two order statistics and the studentized quantile are statistics whose distribution is needed to obtain tests and confidence intervals for quantiles and quantile differences this paper gives saddlepoint approximations for densities and saddlepoint approximations of the lugannani rice form for tail probabilities of these statistics the relative errors of the approximations are n i uniformly in a neighbourhood of the parameters and this uniformity is global if the densities are log concave

on chaotic and random sequences it is demonstrated that lack of exact predictability in the sense defined in h n nazareno j a gonzalez i f costa propagating nature in chaotic systems phys rev b 57 1998 13583 13588 j a gonzalez r pino chaotic and stochastic functions physica a 276 2000 425 440 j a gonzalez l i reyes j j suarez l e guerrero g gutierez from exactly solvable chaotic maps to stochastic dynamics physica d 178 2003 26 50 does not provide a reliable characterization of chaotic and random sequences in fact there are many regular sequences xn r n 0 such that there does not exist any positive integer m and any mapping gm such that xn 1 gm xn xn 1 xn m 1 n m 1 and so such sequences are not predictable in the terminology of h n nazareno j a gonzalez i f costa propagating nature in chaotic systems phys rev b 57 1998 13583 13588 j a gonzalez r pino chaotic and stochastic functions physica a 276 2000 425 440 j a gonzalez l i reyes j j suarez l e guerrero g gutierez from exactly solvable chaotic maps to stochastic dynamics physica d 178 2003 26 50 furthermore it is shown via elementary methods that the sequence xn sin 2 zn n 0 is quite simple regular for an infinite set of nontrivially obtained irrational numbers z 1 at least for certain values thus providing counter examples to claims of high irregularity and randomness made in the series of interesting papers h n nazareno j a gonzalez i f costa propagating nature in chaotic systems phys rev b 57 1998 13583 13588 j a gonzalez r pino chaotic and stochastic functions physica a 276 2000 425 440 j a gonzalez l i reyes j j suarez l e guerrero g gutierez from exactly solvable chaotic maps to stochastic dynamics physica d 178 2003 26 50

convergence analysis of the energy and helicity preserving scheme for axisymmetric flows we give an error estimate for the energy and helicity preserving scheme ehps in second order finite difference setting on axisymmetric incompressible flows with swirling velocity this is accomplished by a weighted energy estimate along with careful and nonstandard local truncation error analysis near the geometric singularity and a far field decay estimate for the stream function a key ingredient in our a priori estimate is the permutation identities associated with the jacobians which are also a unique feature that distinguishes ehps from standard finite difference schemes

energy and helicity preserving schemes for hydro and magnetohydro dynamics flows with symmetry we propose a class of simple and efficient numerical scheme for incompressible fluid equations with coordinate symmetry by introducing a generalized vorticity stream formulation the divergence free constraints are automatically satisfied in addition with explicit treatment of the nonlinear terms and local vorticity boundary condition the navier stokes mhd respectively equation essentially decouples into 2 4 respectively scalar equation and thus the scheme is very efficient moreover with proper discretization of the nonlinear terms the scheme preserves both energy and helicity identities numerically this is achieved by recasting the nonlinear terms convection vorticity stretching geometric source lorentz force and electro motive force in terms of jacobians this conservative property is valid even in the presence of the pole singularity for axisymmetric flows the exact conservation of energy and helicity has effectively eliminated excessive numerical viscosity numerical examples have demonstrated both accuracy and efficiency of the scheme finally local mesh refinement near the boundary can also be easily incorporated into the scheme without extra cost

fully conservative higher order finite difference schemes for incompressible flow conservation properties of the mass momentum and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discrete equations existing finite difference schemes in regular and staggered grid systems are checked for violations of the conservation requirements and a few important discrepancies are pointed out in particular it is found that none of the existing higher order schemes for a staggered mesh system simultaneously conserve mass momentum and kinetic energy this deficiency is corrected through the derivation of a general family of fully conservative higher order accurate finite difference schemes for staggered grid systems finite difference schemes in a collocated grid system are also analyzed and a violation of kinetic energy conservation is revealed the predicted conservation properties are demonstrated numerically in simulations of inviscid white noise performed in a two dimensional periodic domain the proposed fourth order schemes in a staggered grid system are generalized for the case of a non uniform mesh and the resulting scheme is used to perform large eddy simulations of turbulent channel flow

analysis of finite difference schemes for unsteady navier stokes equations in vorticity formulation in this paper we provide stability and convergence analysis for a class of finite difference schemes for unsteady incompressible navier stokes equations in vorticity stream function formulation the no slip boundary condition for the velocity is converted into local vorticity boundary conditions thom s formula wilkes formula or other local formulas in the earlier literature can be used in the second order method while high order formulas such as briley s formula can be used in the fourth order compact difference scheme proposed by e and liu the stability analysis of these long stencil formulas cannot be directly derived from straightforward manipulations since more than one interior point is involved in the formula the main idea of the stability analysis is to control local terms by global quantities via discrete elliptic regularity for stream function we choose to analyze the second order scheme with wilkes formula in detail in this case we can avoid the complicated technique necessitated by the strang type high order expansions as a consequence our analysis results in almost optimal regularity assumption for the exact solution the above methodology is very general we also give a detailed analysis for the fourth order scheme using a 1 d stokes model

control variates for probability and quantile estimation in stochastic systems quantiles indicate the level of system performance that can be delivered with a specified probability while probabilities indicate the likelihood that a specified level of system performance can be achieved we present new estimators for use in simulation experiments designed to estimate such quantiles or probabilities of system performance all of the estimators exploit control variates to increase their precision which is especially important when extreme quantiles in the tails of the distribution of system performance or extreme probabilities near zero or one are of interest control variates are auxiliary random variables with known properties in this case known quantiles and a strong stochastic association with the performance measure of interest since transforming a control variate can increase its effectiveness we propose both continuous and discrete approximations to the optimal variance minimizing transformation for estimating probabilities and then invert the probability estimators to obtain corresponding quantile estimators we also propose a direct control variate quantile estimator that is not based on inverting a probability estimator an empirical study using queueing inventory and project planning examples shows that substantial reductions in mean squared error can be obtained when estimating the 0 9 0 95 and 0 99 quantiles

improved error bounds for genetic distances from dna sequences summary the genetic distance between two dna sequences may be measured by the average number of nucleotide substitutions per position that has occurred since the two sequences diverged from a common ancestor estimates of this quantity can be derived from markov models for the substitution process while the variances are estimated using the delta method and confidence intervals calculated assuming normality however when the sampling distribution of the estimator deviates from normality such intervals will not be accurate for simple one parameter models of nucleotide substitution we propose a transformation of normal confidence intervals which yields an almost exact approximation to the true confidence intervals of the distance estimators to calculate confidence intervals for more complicated models we propose the saddlepoint approximation a simulation study shows that the saddlepoint derived confidence intervals are a real improvement over existing methods

superconvergent functional output for time dependent problems using finite differences on summation by parts form finite difference operators satisfying the summation by parts sbp rules can be used to obtain high order accurate energy stable schemes for time dependent partial differential equations when the boundary conditions are imposed weakly by the simultaneous approximation term sat in general an sbp sat discretization is accurate of order p 1 with an internal accuracy of 2p and a boundary accuracy of p despite this it is shown in this paper that any linear functional computed from the time dependent solution will be accurate of order 2p when the boundary terms are imposed in a stable and dual consistent way the method does not involve the solution of the dual equations and superconvergent functionals are obtained at no extra computational cost four representative model problems are analyzed in terms of convergence and errors and it is shown in a systematic way how to derive schemes which gives superconvergent functional outputs

energy conservative sbp discretizations of the acoustic wave equation in covariant form on staggered curvilinear grids abstract we develop a numerical method for solving the acoustic wave equation in covariant form on staggered curvilinear grids in an energy conserving manner the use of a covariant basis decomposition leads to a rotationally invariant scheme that outperforms a cartesian basis decomposition on rotated grids the discretization is based on high order summation by parts sbp operators and preserves both symmetry and positive definiteness of the contravariant metric tensor to improve accuracy and decrease computational cost we also derive a modified discretization of the metric tensor that leads to a conditionally stable discretization bounds are derived that yield a point wise condition that can be evaluated to check for stability of the modified discretization this condition shows that the interpolation operators should be constructed such that their norm is close to one

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

general saddlepoint approximations of marginal densities and tail probabilities saddlepoint approximations of marginal densities and tail probabilities of general nonlinear statistics are derived these are based on the expansion of the statistic up to the second order their accuracy is shown in a variety of examples including logit and probit models and rank estimators for regression

regression to the mean for nonnormal populations summary regression to the mean is the term used to describe the effect by which individuals selected on the basis of an initial measurement which is particularly high or low compared with other members of the population have upon remeasurement values nearer to the population mean for the normal distribution simple expressions for estimates of the magnitude of this effect are available many biological variables however are not normally distributed estimates for regression to the mean for nonnormal data are given based on edgeworth series and saddlepoint approximations

robust output feedback stabilization via risk sensitive control we consider a problem of robust linear quadratic gaussian lqg control for discrete time stochastic uncertain systems with partial state measurements for a finite horizon case the problem was recently introduced by petersen et al ieee trans automat control 45 2000 398 in this paper an infinite horizon extension of the results of petersen et al ieee trans automat control 45 2000 398 is discussed we show that for a broad class of uncertain systems under consideration a controller constructed in terms of the solution to a specially parameterized risk sensitive stochastic control problem absolutely stabilizes the stochastic uncertain system

on the stabilizing effect of convection in three dimensional incompressible flows we investigate the stabilizing effect of convection in three dimensional incompressible euler and navier stokes equations the convection term is the main source of nonlinearity for these equations it is often considered destabilizing although it conserves energy due to the incompressibility condition in this paper we show that the convection term together with the incompressibility condition actually has a surprising stabilizing effect we demonstrate this by constructing a new three dimensional model that is derived for axisymmetric flows with swirl using a set of new variables this model preserves almost all the properties of the full three dimensional euler or navier stokes equations except for the convection term which is neglected in our model if we added the convection term back to our model we would recover the full navier stokes equations we will present numerical evidence that seems to support that the three dimensional model may develop a potential finite time singularity we will also analyze the mechanism that leads to these singular events in the new three dimensional model and how the convection term in the full euler and navier stokes equations destroys such a mechanism thus preventing the singularity from forming in a finite time

practical small sample asymptotics for regression problems abstract saddlepoint approximations are derived for sums of independent but not necessarily identically distributed random variables along with generalizations to estimating equations and multivariate problems these results are particularly useful for accurately approximating the distribution of regression coefficients general formulas are given for the distribution of the coefficients arising out of a generalized linear model with both canonical and noncanonical link functions we illustrate the case of logistic regression with a real data example and show how the gibbs sampler may be used to obtain confidence sets for each regression parameter based on the saddlepoint approximation

computational design for long term numerical integration of the equations of fluid motion the integral constraints on quadratic quantities of physical importance such as conservation of mean kinetic energy and mean square vorticity will not be maintained in finite difference analogues of the equation of motion for two dimensional incompressible flow unless the finite difference jacobian expression for the advection term is restricted to a form which properly represents the interaction between grid points as derived in this paper it is shown that the derived form of the finite difference jacobian prevents nonlinear computational instability and thereby permits long term numerical integrations

saddlepoint approximations for the difference of order statistics summary saddlepoint approximations for the distribution of the difference of order statistics from a continuous distribution are presented the approximations are illustrated for the case of the exponential distribution where we can find the exact probability density function of the difference of order statistics and for the case of the noncentral chi squared distribution which has application to the problem of testing a time series for linearity in both cases we show that the saddlepoint approximation is much better than the normal approximation especially for realistic sample sizes

shift invariant signal norms for fault detection and control two desirable properties of a signal norm to be used for the purpose of optimal control or fault detection are highlighted these properties are shift invariance and the permitting of persistent signals a class of norms called the window norms is analyzed among norms considered for control purposes they appear to be the only ones save the l norm that satisfy both properties the signal spaces defined by the window norms however include some interesting signals that are not in the l space the window norms have been found suitable for application in fault detection and are here also considered for optimal control it is shown that they can be taken as a support for the concept of l1 control but may also suggest a new class of optimal controllers

stable coupling of nonconforming high order finite difference methods a methodology for handling block to block coupling of nonconforming multiblock summation by parts finite difference methods is proposed the coupling is based on the construction of projection operators that move a finite difference grid solution along an interface to a space of piecewise defined functions we specifically consider discontinuous piecewise polynomial functions the constructed projection operators are compatible with the underlying summation by parts energy norm using the linear wave equation in two dimensions as a model problem energy stability of the coupled numerical method is proven for the case of curved nonconforming block to block interfaces to further demonstrate the power of the coupling procedure we show how it allows for the development of a provably energy stable coupling between curvilinear finite difference methods and a curved triangle discontinuous galerkin method the theoretical results are verified through numerical simulations on curved meshes as well as eigenval

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

general saddlepoint approximations with applications to l statistics abstract saddlepoint approximations are extended to general statistics the technique is applied to derive approximations to the density of linear combinations of order statistics including trimmed means a comparison with exact results shows the accuracy of these approximations even in very small sample sizes

on three puzzles in robust control three surprising open topics puzzles in robust control are highlighted the first and the second puzzle deal with difficulties in generalizing h sub spl infin control theory to realistic persistent signal setups it seems difficult to generalize the theory to such setups and simultaneously keep the terminology robust h sub spl infin control intact these puzzles have implications also to control oriented model validation the third puzzle due to georgiou and smith 1995 deals with difficulties in the doubly infinite time axis formulations of input output stabilization theory

a two operator approach to robust stabilization of linear systems on robust non fragile feedback stabilization is studied for causal linear time invariant lti input output systems defined in an l 2 signal space setup or equivalently in an l 2 j setup in the frequency domain two operator descriptions for the plant and the controller are used with bounded operators defined on the full space l 2 basic properties of causal lti operators on weighted l 2 spaces l 2 w which are relevant to stabilization theory are also established for different choices of the weighting function w weighted l 2 spaces allow the inclusion of persistent signals such as steps and thus provide the potential of important generalizations of the robust stabilization theory on l 2 in particular it is shown that there is a large class of weights w for which the l 2 w setup leads to h optimization this implies that robust h design can be applied to a large class of weighted l 2 spaces on problems of closability of unstable causal lti convolution operators are discussed t

symmetry preserving discretization of turbulent flow we propose to perform turbulent flow simulations in such manner that the difference operators do have the same symmetry properties as the underlying differential operators i e the convective operator is represented by a skew symmetric coefficient matrix and the diffusive operator is approximated by a symmetric positive definite matrix mimicking crucial properties of differential operators forms in itself a motivation for discretizing them in a certain manner we give it a concrete form by noting that a symmetry preserving discretization of the navier stokes equations is stable on any grid and conserves the total mass momentum and kinetic energy for the latter the physical dissipation is to be turned off of coarse being stable on any grid the choice of the grid may be based on the required accuracy solely and the main question becomes how accurate is a symmetry preserving discretization its accuracy is tested for a turbulent flow in a channel by comparing the results to those of physical experiments and previous numerical studies the comparison is carried out for a reynolds number of 5600 which is based on the channel width and the mean bulk velocity based on the channel half width and wall shear velocity the reynolds number becomes 180 the comparison shows that with a fourth order symmetry preserving method a 64 64 32 grid suffices to perform an accurate numerical simulation

a counterexample in power signals space the present author points out that in the paper by vidyasagar 1993 on nonlinear system analysis some of the results obtained are false the purpose of this paper is to give a counterexample it is shown that the set of power signals is not a vector space

new limit power function spaces to answer an open question we propose two function spaces one is banach and another is hilbert it is shown that the hilbert space is the largest one among those hilbert spaces in limit power function set whose members have associated fourier series in sense of a new basis and satisfy parseval s equality

dynamic stability of the three dimensional axisymmetric navier stokes equations with swirl in this paper we study the dynamic stability of the three dimensional axisymmetric navier stokes equations with swirl to this purpose we propose a new one dimensional model that approximates the navier stokes equations along the symmetry axis an important property of this one dimensional model is that one can construct from its solutions a family of exact solutions of the three dimensionafinal navier stokes equations the nonlinear structure of the one dimensional model has some very interesting properties on one hand it can lead to tremendous dynamic growth of the solution within a short time on the other hand it has a surprising dynamic depletion mechanism that prevents the solution from blowing up in finite time by exploiting this special nonlinear structure we prove the global regularity of the three dimensional navier stokes equations for a family of initial data whose solutions can lead to large dynamic growth but yet have global smooth solutions

on optimal lti approximation of nonlinear systems optimal linear time invariant lti approximation of discrete time nonlinear systems is studied using a quasi stationary signal description the optimality of the lti approximation is quantified as a least average squared error condition reminescent of recent studies on the performance of least squares type identification methods for lti models of nonlinear systems

empirical saddlepoint approximations for multivariate m estimators summary in this paper we investigate the use of the empirical distribution function in place of the underlying distribution function f to construct an empirical saddlepoint approximation to the density fn of a general multivariate m estimator we obtain an explicit form for the error term in the approximation investigate the effect of renormalizing the estimator carry out some numerical comparisons and discuss the regression problem the sampling distribution of a statistic is a basic tool for evaluating the properties of the statistic and for constructing inference procedures based on the statistic the density of the sampling distribution may be used to compute moments to develop approximations or to compute the distribution function and hence various tail prob abilities which can be used to construct tests or confidence intervals when the exact density is intractable it may be possible to approximate it by a normal density an edgeworth expansion or a saddlepoint approximation daniels 1954 the saddle point approximation is derived from an asymptotic expansion but is often very accurate even in quite small samples to highlight this property hampel 1973 described these and related techniques as small sample asymptotic approximations as can be seen from equation 3 later the main property of this approximation and its major advantage over edgeworth expansions is that it is always non negative and the relative error is uniformly of order n 1 for recent reviews see reid 1988 barndorff nielsen and cox 1989 and field and ronchetti 1990 the construction of these approximations typically requires knowledge of the underlying distribution however just as we can estimate the variance of a normal approxima tion or estimate the terms in an edgeworth expansion we can estimate the saddle point approximation in this paper we examine the properties of an estimate of the saddlepoint approximation our investigation is motivated by three considerations firstly the estimated approximation may be useful for comparing various estimators on a given data set if the sampling distribution is non normal at least in principle we need to compare the whole distribution and comparisons based on the estimated asymptotic variance alone may be misleading secondly the estimated approximation may be useful for evaluating the quality of simpler approximations including the normal approxima tion finally it may be useful for developing inference procedures methods like

saddlepoint approximation for the distribution of a ratio of quadratic forms in normal variables abstract in this article the saddlepoint approximations to the density and tail probability of a ratio of quadratic forms in normal variables are derived a numerical exposition via the durbin watson test statistic reveals several desirable features the approximations which involve only a limited number of computable functions provide the practitioner with an accessible and a very powerful tool

c sup algebra of strong limit power functions motivated by problems in robust control of power signal set h sub 2 e g h sub 2 is not a vector space fourier analysis cannot be carried out on h sub 2 in general we study those functions in h sub 2 which we call strong limit power we show that the set of all such functions is a sufficiently large c sup algebra fourier analysis is carried out on the functions in particular the uniqueness of the fourier expansion of a strong limit power function is established finally we point out how to analyze and reconstruct such functions

using high order methods on adaptively refined block structured meshes derivatives interpolations and filters block structured adaptive mesh refinement amr is used for efficient resolution of partial differential equations pdes solved on large computational domains by clustering mesh points only where required by large gradients previous work has indicated that fourth order convergence can be achieved on such meshes by using a suitable combination of high order derivative interpolation and filter stencils and can deliver significant computational savings over conventional second order methods at engineering error tolerances in this paper we explore the interactions between the errors introduced by the discretized operators for derivatives interpolations and filters we develop general expressions for high order derivative interpolation and filter stencils applicable in multiple dimensions using a fourier approach to facilitate high order block structured amr implementations these stencils are derived under the assumption that the fields that they are applied to are smooth for a given derivative stencil and thus a given order of accuracy the necessary interpolation order is found to be dependent on the highest spatial derivative in the pde being solved this is demonstrated empirically using one and two dimensional model equations by observing the increase in delivered accuracy as the order of accuracy of the interpolation stencil is increased we also examine the empirically observed order of convergence as the effective resolution of the mesh is increased by successively adding levels of refinement with different orders of derivative interpolation and filter stencils the procedure devised here is modular and its various pieces i e the derivative interpolation and filter stencils can be extended independently to higher orders and dimensions in the case of interpolation stencils the application of these methods often requires nontrivial logic especially near domain boundaries this logic along with the stencils used in this study are available as a freely downloadable software library

lti approximation of nonlinear systems via signal distribution theory l 2 and l 1 optimal linear time invariant lti approximation of discrete time nonlinear systems such as nonlinear finite impulse response nfir systems is studied via a signal distribution theory motivated approach the use of a signal distribution theoretic framework facilitates the formulation and analysis of many system modelling problems including system identification problems specifically a very explicit solution to the l 2 least squares lti approximation problem for nfir systems is obtained in this manner furthermore the l 1 least absolute deviations lti approximation problem for nfir systems is essentially reduced to a linear programming problem active lti modelling emphasizes model quality based on the intended use of the models in linear controller design robust stability and lti approximation concepts are studied here in a nonlinear systems context numerical examples are given illustrating the performance of the least squares ls method and the least absolute deviations lad method with lti models against nonlinear unmodelled dynamics

robust input output stabilization on z for persistent signals robustness of feedback stabilization is studied for causal linear time invariant input output systems defined on the full time axis z the signal space setting used includes also certain unbounded signals this study has been possible due to the use of two operator plant models ay bu this avoids the intrinsic difficulties of unbounded causal input output plant mappings y gu on z the full time axis robust stabilization results obtained here are analogous to those on the semi infinite time axis n in the finite dimensional case however there are certain important differences between the full time axis results and the semi infinite time axis results for infinite dimensional systems even when two operator plant models are used

an energy preserving mac yee scheme for the incompressible mhd equation we propose a simple and efficient finite difference method for the incompressible mhd equation the numerical method combines the advantage of the mac scheme for the navier stokes equation and yee s scheme for the maxwell equation in particular the semi discrete version of our scheme introduces no numerical dissipation and preserves the energy identity exactly c

summation by parts projections and stability ii we have derived stability results for high order finite difference approximations of mixed hyperbolic parabolic initial boundary value problems ibvp the results are obtained using summation by parts and a new way of representing general linear boundary conditions as an orthogonal projection by slightly rearranging the analytic equations we can prove strict stability for hyperbolic parabolic ibvp furthermore we generalize our technique so as to yield strict stability on curvilinear non smooth domains in two space dimensions finally we show how to incorporate inhomogeneous boundary data while retaining strict stability using the same procedure one can prove strict stability in higher dimensions as well

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

estimation of extremal properties in non negative networks estimation of extremal properties of complex networks of linear systems is in general a difficult task in this note such estimation problems are solved for an important class of systems namely for stable non negative networks consisting of non negative sums and convolutions of linear time invariant systems with non negative impulse responses in particular the problem of estimating the l 1 norm of a stable non negative system from rather arbitrary input output data is analysed in detail

grid stabilization of high order one sided differencing i first order hyperbolic systems we construct stable maximal order boundary closures for high order central difference methods the stability is achieved by adding a small number of additional subcell nodes near the boundaries at experimentally determined locations we find that methods up through 8th order can be stabilized by the addition of a single node up through 16th order by the addition of two nodes and up through 22nd order with three extra nodes we also consider the application of the technique to dispersion relation preserving methods and we construct and test artificial dissipation operators

squared and absolute errors in optimal approximation of nonlinear systems optimal squared error and absolute error based approximation problems for static polynomial models of nonlinear discrete time systems are studied in detail these problems have many similarities with other linear in the parameters approximation problems such as with optimal approximation problems for linear time invariant models of linear and nonlinear systems nonprobabilistic signal analysis is used close connections between the studied approximation problems and certain classical topics in approximation theory such as optimal l 2 1 1 and l 1 1 1 approximation are established by analysing conditions under which sample averages of static nonlinear functions of the input converge to appropriate riemann integrals of the static functions these results should play a significant role in the analysis of corresponding system identification and model validation problems furthermore these results demonstrate that optimal modelling based on the absolute error can offer advantages over squared error based modelling especially modelling problems in which some signals possess heavy tails can benefit from absolute value based signal and error analysis

central limit theorem in the functional approach the central limit theorem is proved within the framework of the functional approach for signal analysis in this framework a signal is modeled as a single function of time rather than a stochastic process distribution function expectation and all the familiar probabilistic parameters are built starting from this single function of time by resorting to the concept of relative measure furthermore the concept of independence among functions of time can be introduced in the paper it is shown that if a sequence of independent signals fulfills some mild regularity assumptions then the asymptotic distribution of the appropriately scaled average of such signals has a limiting normal distribution the approach is shown to be useful when only one realization of a signal is available and no ensemble of realizations is observed or exists the obtained results also allow one to rigorously justify stochastic models for signals and channels that up to now have been derived starting from a deterministic description of phenomena and for which the inferred stochastic model is built invoking a not proved ergodicity property an application to the statistical characterization of the output signal of a multipath doppler channel is presented

finite time singularities in ideal fluids with swirl three dimensional ideal incompressible fluids with swirl are studied numerically using two different methods standard finite differences and a projection method based on upwind differencing both methods give quantitatively similar results leading to the conclusion that singularities form in finite time in a manner consistent with known theoretical criteria the effect of singularities in incompressible flows on nearby compressible flows is discussed

lethargy results in lti system modelling modelling of linear time invariant lti systems is studied in an approximate modelling framework motivated by modern robust control theory as convolution type lti systems are represented by their unit impulse responses i e through a sequence of real complex numbers this paper provides also system theoretic results on modelling of sequences and their associated transfer functions the modelling results derived in this paper are mostly of lethargy type i e they establish limitations in approximate modelling of lti systems non existence of coprime factorizations is established for a number of different types of lti systems this leads to the somewhat surprising conclusion that many lti systems are not even stabilizable and that an even larger class of lti systems cannot be well approximated by rational transfer function models in addition it is discussed that often the rate of rational approximation of lti systems is not essentially better than the rate of approximation in some basis such as finite impulse response fir models furthermore different definitions of the important finite power signal setup are studied and new results given

generalized laguerre expansions of multivariate probability densities with moments we generalize the well known laguerre series approach to approximate multivariate probability density functions pdfs using multidimensional laguerre polynomials the generalized laguerre series which is defined around a gamma pdf is suited for simulating high complex natural phenomena that deviate from gaussianity combining the multivariate laguerre approximation and bayes theorem an approximation to the conditional pdfs is derived numerical results first showed the superiority of the gamma expansion over other numerical methods the ability of the gamma expansion to fit mixtures of gaussian ans super gaussian pdfs univariate and multivariate lognormal pdfs and complex geologic media is shown through different examples

brief linear quadratic control revisited infinite horizon linear quadratic control is re examined and generalized to include a class of non stationary disturbances this revision is achieved by defining a generalized infinite horizon linear quadratic control problem using a flexible functional analytic signal description specifically an instantaneous linear quadratic output feedback problem is considered in this way it is shown that the solution to the generalized linear quadratic problem is equivalent to the solution of the corresponding standard linear quadratic problem with disturbance and noise covariance matrices having in the generalized problem a different more general meaning than in the standard problem in the generalized problem these covariance matrices are majorizing matrices that have a precise meaning even in the non stationary signal case this result justifies in a nice way the wide practice of interpreting the disturbance and noise covariance matrices in linear quadratic control as design variables

high order finite difference methods for wave propagation in discontinuous media high order finite difference approximations are derived for the second order wave equation with discontinuous coefficients on rectangular geometries the discontinuity is treated by splitting the domain at the discontinuities in a multi block fashion each sub domain is discretized with compact second derivative summation by parts operators and the blocks are patched together to a global domain using the projection method this guarantees a conservative strictly stable and high order accurate scheme the analysis is verified by numerical simulations in one and two spatial dimensions

stable and accurate wave propagation in discontinuous media a time stable discretization is derived for the second order wave equation with discontinuous coefficients the discontinuity corresponds to inhomogeneity in the underlying medium and is treated by splitting the domain each homogeneous sub domain is discretized using narrow diagonal summation by parts operators and then patched to its neighbors by using a penalty method leading to fully explicit time integration this discretization yields a time stable and efficient scheme the analysis is verified by numerical simulations in one dimension using high order finite difference discretizations and in three dimensions using an unstructured finite volume discretization

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

small sample asymptotic distributions of m estimators of location summary asymptotic formulae for the distribution of m estimators i e maximum likelihood type estimators of location including the arithmetic mean are derived which numerical studies show to give relative errors for densities and tail areas of the order of magnitude of 1 down to sample sizes 3 and 4 even in the extreme tails the paper is the continuation of earlier work by the second author and is also closely related to daniels s work on the saddlepoint approximation the method consists in expanding the derivative of the logarithm of the unstandardized density of the estimator in powers of 1 n at each point using recentring by means of conjugate distributions this method yields a unified point of view for the comparison of other asymptotic methods namely saddlepoint method edgeworth expansion and large deviations approach which are also compared numerically

linear canonical stockwell transform abstract the linear canonical transform lct has long been recognized as a competent tool for optics and signal processing however the lct cannot reveal the local lct frequency contents due to its global kernel and as such makes the lct incompetent in situations demanding a joint information of time and linear canonical domain frequency in this paper we propose the linear canonical stockwell transform lcst in order to rectify the limitations of the lct and the stockwell transform st firstly we examine the resolution of the novel transform in the time and lct domains and then derive some of its basic properties such as moyal s formula inversion formula and provide a characterization of the range furthermore we derive a direct relationship between the well known wigner distribution and the proposed lcst in sequel a discrete version of the linear canonical stockwell transform is also presented finally we extend the scope of the proposed work by studying the lcst in the realm of almost periodic functions

vorticity boundary condition and related issues for finite difference schemes this paper discusses three basic issues related to the design of finite difference schemes forunsteadyviscous incompressible flows using vorticity formulations the boundary condition for vorticity an efficient time stepping procedure and the relation between these schemes and the ones based on velocity pressure formulation we show that many of the newly developed global vorticity boundary conditions can actually be written as some local formulas derived earlier we also show that if we couple a standard centered difference scheme with third or fourth order explicit runge kutta methods the resulting schemes havenocell reynolds number constraints for high reynolds number flows these schemes are stable under the cfl condition given by the convective terms finally we show that the classical mac scheme is the same as thom s formula coupled with second order centered differences in the interior in the sense that one can define discrete vorticity in a natural way for the mac scheme and get the same values as the ones computed from thom s formula we use this to derive an efficient fourth order runge kutta time discretization for the mac scheme from the one for thom s formula we present numerical results for driven cavity flow at high reynolds number 105

on the partial regularity of a 3d model of the navier stokes equations we study the partial regularity of a 3d model of the incompressible navier stokes equations which was recently introduced by the authors in 11 this model is derived for axisymmetric flows with swirl using a set of new variables it preserves almost all the properties of the full 3d euler or navier stokes equations except for the convection term which is neglected in the model if we add the convection term back to our model we would recover the full navier stokes equations in 11 we presented numerical evidence which seems to support that the 3d model develops finite time singularities while the corresponding solution of the 3d navier stokes equations remains smooth this suggests that the convection term play an essential role in stabilizing the nonlinear vortex stretching term in this paper we prove that for any suitable weak solution of the 3d model in an open set in space time the one dimensional hausdorff measure of the associated singular set is zero the partial regularity result of this paper is an analogue of the caffarelli kohn nirenberg theory for the 3d navier stokes equations

on the representation of shift invariant operators by transfer functions we give simple sufficient conditions on a function space on 0 to ensure that all shift invariant operators defined on it are represented by transfer functions this enables us to extend the result of weiss on lp 0 for 1 p but not p to more general situations

some new classes of hardy spaces abstract let b p f f sup t 1 1 2t t t f p 1 p then bp is the dual of a function algebra aq on r beurling in this paper we study the harmonic extensions of f in bp and in aq and the corresponding hardy spaces hbp haq it is shown that a parallel theory for l ll and bmo hl can be developed for the above pairs in particular we prove that for 1 a q is isomorphic to the banach space f real f p sup t 1 1 2t t t f m t f p 1 p where m t f 1 2t t t f we also prove burkholder gundy and silverstein s maximal function characterization for the new hardy space haq 1

bounded power signal spaces for robust control and modeling the nonlinear space of signals allowing wiener s generalized harmonic analysis gha the linear bounded power signal spaces of beurling marcinkiewicz and wiener and a new linear bounded power space are studied from a control and systems theory perspective specifically it is shown that the system power gain is given by the h infty norm of the system transfer function in each of these spaces for a large class of power stable finite and infinite dimensional systems the gha setup is shown to possess several limitations for the purpose of robustness analysis which motivates the use of the other more general nonstationary signal spaces the natural double sided time axis versions of bounded power signal spaces are shown to break the symmetry between hardy space h infty methods and bounded power operators e g the system transfer function being in h infty does not imply that a causal linear time invariant lti system is bounded as an operator on any of the double sided versions of the studied bounded power signal spaces