a note on the backward errors for hermite eigenvalue problems we discuss the backward error analysis of the hermite eigenproblems three kinds of problems are included generalized eigenproblem matrix polynomial eigenproblem and multiparameter eigenproblem in the case of the approximate eigenvalue being real these problems have already been solved by higham and higham siam j matrix anal 20 1998 493 tisseur lin alg appl 309 2002 339 and hachstenbach and plestenjak lin alg appl 375 2003 63 this note as a complement of their results considers the complex case

two point two time closures applied to forced isotropic turbulence abstract the direct interaction approximation and the local energy transfer theory were used to calculate freely decaying turbulence for a range of initial taylor reynolds numbers 3 r t 0 129 and forced turbulence for evolved taylor reynolds numbers r t evolved 88 and r t evolved 232 in the case of the freely decaying turbulence the new feature of the work as compared to mccomb et al j fluid mech 245 1992 279 was the ability to make comparisons with a direct numerical simulation at the same initial conditions we were also able to obtain experimental results from the simulation with an estimate of the experimental error we present a small sample of results from an extensive investigation sufficient to confirm that both closures agree with the simulation to within the experimental error in broad agreement with our earlier investigations however the main motivation of the work was to study the application of the closures to stationary turbulence and here we faced new problems such as how to force the closures or the need to compute the time history integrals of the closures over the long transient period during which the numerical simulation settles down for this reason our results should be seen as rather preliminary and tentative in nature essentially we found that agreement between the closures and the simulation was not as good as for free decay for low wavenumber behaviour but was quite good for high wavenumber spectra and quantities determined by this region such as the taylor microscale we conclude that further work is needed to establish just what is a fair comparison between forced closure calculations and the analogous direct numerical simulation

numerical simulations of solutions of a two level averaged navier stokes system abstract an averaging procedure for the navier stokes equations has been proposed in an earlier article i moise r m temam renormalization group method application to navier stokes equation discrete contin dyn syst 6 1 2000 191 210 this averaging procedure is based on a two level decomposition of the solution into low and high frequencies the aim of the present article is to investigate with the help of numerical simulations the behavior of the small scales of the corresponding system space periodic solutions with a non resonant period are considered the time evolution of the averaged and standard non averaged small scales are compared at different reynolds numbers and for different values of the cut off level used to separate large and small scales of the flow variables the numerical results illustrate the efficiency of the proposed averaging procedure for the navier stokes equations the averaged small scales provide an accurate prediction of the time averaged small scales of the navier stokes solutions as the computational cost is reduced for the averaged equations long time integrations on more than 50 eddy turnover times have been performed for cut off levels ensuring a proper resolution of the large scales in these cases development of instabilities in the averaged small scale equation is observed

asymptotic freedom non gaussian perturbation theory and the application of renormalization group theory to isotropic turbulence a model equation with an extended range of asymptotic freedom is used as the basis of a non gaussian perturbation expansion in powers of the control parameter of the conditional average re expansion in the local reynolds number allows the systematic rederivation of an earlier heuristic theory w d mccomb and a g watt phys rev lett 65 3281 1990

diffraction by a plane sector the problem of diffraction by a perfectly reflecting screen occupying an infinite sector of the equatorial plane is addressed by the random walk method the solution is represented as a superposition of the wave field completely determined by an elementary ray analysis and of the field formed by the waves diffracted by the tip of the screen the diffracted field is explicitly represented as the mathematical expectation of a specified functional on trajectories of the random motion the radial component of which runs in a complex space while the two dimensional angular component remains real valued the numerical results confirm the efficiency of the random walk approach to the analysis of diffraction by wedge shaped screens of arbitrary angles

on the universality of the kolmogorov constant in numerical simulations of turbulence motivated by a recent survey of experimental data k r sreenivasan phys fluids 2778 1995 we examine data on the kolmogorov spectrum constant in numerical simulations of isotropic turbulence using results both from previous studies and from new direct numerical simulations over a range of reynolds numbers up to 240 on the taylor scale at grid resolutions up to 5123 it is noted that in addition to k 5 3 scaling identification of a true inertial range requires spectral isotropy in the same wavenumber range we found that a plateau in the compensated three dimensional energy spectrum at k eta 0 1 0 2 commonly used to infer the kolmogorov constant from the compensated three dimensional energy spectrum actually does not represent proper inertial range behavior rather a proper if still approximate inertial range emerges at k eta 0 02 0 05 when r sub sub sub sub sub sub sub sub 0 53 for c 1 62 in excellent agreement with experiments however the one and three dimensional estimates are not fully consistent because of departures due to numerical and statistical limitations from isotropy of the computed spectra at low wavenumbers the inertial scaling of structure functions in physical space is briefly addressed since dns is still restricted to moderate reynolds numbers an accurate evaluation of the kolmogorov constant is very difficult we focus on providing new insights on the interpretation of kolmogorov 1941 similarity in the dns literature and do not consider issues pertaining to the refined similarity hypotheses of kolmogorov

new perspectives in turbulence scaling laws asymptotics and intermittency intermittency a basic property of fully developed turbulent flow decreases with growing viscosity therefore classical relationships obtained in the limit of vanishing viscosity must be corrected when the reynolds number is finite but large these corrections are the main subject of the present paper they lead to a new scaling law for wall bounded turbulence which is of key importance in engineering and to a reinterpretation of the kolmogorov obukhov scaling for the local structure of turbulence which has been of paramount interest in both theory and applications the background of these results is reviewed in similarity methods in the statistical theory of vortex motion and in intermediate asymptotics and relevant experimental data are summarized

eulerian spectral closures for isotropic turbulence using a time ordered fluctuation dissipation relation in a previous paper 1 the kraichnan wyld perturbation expansion 2 3 was used to justify the introduction of a renormalized response function connecting two point covariances at different times the resulting relationship was specialized by suitable choice of initial conditions to the form of a fluctuation dissipation relation fdr this was further developed to reconcile the time symmetry of the covariance with the causality of the response function by the introduction of time ordering along with a counter term we pointed out that this formulation provides a solution to an old problem in turbulence theory that of representing the time dependence of the covariance and response by exponential forms 4 5 we showed that the derivative with respect to difference time of the covariance with this time ordering now vanishes at the origin this allows one to study the relationships between two time spectral closures and time independent theories such as the fokker planck theory of edwards 6 or the more recent renormalization group approaches we also showed that the renormalized response function is transitive with respect to intermediate times and reported a new langevin type equation for turbulence in this paper we interpret the second order response function as a mean field propagator and show that in addition to propagating two time covariances it also links single time covariances we then make use of its newly established properties to re derive the local energy transfer let response equation 7 and show that it now contains a counter term which removes the singularity of previous propagator equations at t t we also introduce a partial propagator representation and hence reformulate the let statistical equations furthermore we specialize the two time equations to time independent form by introducing exponential time dependences and

closure approach to high order structure functions of turbulence let t p be the structure function of order p of turbulence first t2 is determined then t3 is derived from t2 and finally t p p 3 are derived from t2 and t3 this closure scheme is realized by a non gaussian statistical model we use it to study the scaling law of t p and we find that the available data on scaling exponents favor kolmogorov s 1941 theory rather than his 1962 theory we also predict the high order universal constants of inertial range scaling

novel solutions of the helmholtz equation and their application to diffraction this paper presents a series of novel representations for the solutions of the helmholtz equation in a broad class of wedge like domains including those with curvilinear non flat faces these repr

random walk approach to wave propagation in wedges and cones two and three dimensional helmholtz equations in wedge shaped and conical domains are addressed by the random walk method the solutions of the dirichlet problems in such domains are represented as mathematical expectations of specified functionals on trajectories of multidimensional random motions whose radial components run in a complex space while the angular components remain real valued this technique is applied to the sommerfeld problem of diffraction by a semi infinite screen which is explicitly solved here in the probabilistic form the numerical results confirm the efficiency of the random walk approach to the analysis of wave propagation

pseudospectra critical points and multiple eigenvalues of matrix polynomials abstract we develop a general framework for perturbation analysis of matrix polynomials more specifically we show that the normed linear space l m c n n of n by n matrix polynomials of degree at most m provides a natural framework for perturbation analysis of matrix polynomials in l m c n n we present a family of natural norms on the space l m c n n and show that the norms on the spaces c m 1 and c n n play a crucial role in the perturbation analysis of matrix polynomials we define pseudospectra of matrix polynomials in the general framework of the normed space l m c n n and show that the pseudospectra of matrix polynomials well known in the literature follow as special cases we analyze various properties of pseudospectra in the unified framework of the normed space l m c n n we analyze critical points of backward errors of approximate eigenvalues of matrix polynomials and show that each critical point is a multiple eigenvalue of an appropriately perturbed polynomial we show that common boundary points of components of pseudospectra of matrix polynomials are critical points as a consequence we show that a solution of wilkinson s problem for matrix polynomials can be read off from the pseudospectra of matrix polynomials

a formal derivation of the local energy transfer let theory of homogeneous turbulence a statistical closure of the navier stokes hierarchy which leads to equations for the two point two time covariance of the velocity field for stationary homogeneous isotropic turbulence is presented it is a generalisation of the self consistent field method due to edwards 1964 for the stationary single time velocity covariance the probability distribution functional p u t is obtained in the form of a series from the liouville equation by means of a perturbation expansion about a gaussian distribution which is chosen to give the exact two point two time covariance the triple moment is calculated in terms of an ensemble averaged infinitesimal velocity field propagator and shown to yield the edwards result as a special case the use of a gaussian zero order distribution has been found to justify the introduction of a fluctuation response relation which is in accord with modern dynamical theories in a sense this work completes the analogy drawn by edwards between turbulence and brownian motion originally edwards had shown that the noise input was determined by the correlation of the velocity field with the externally applied stirring forces but was unable to determine the system response now we find that the system response is determined by the correlation of the velocity field with internal quasi entropic forces this analysis is valid to all orders of perturbation theory and allows the recovery of the local energy transfer let theory which had previously been derived by more heuristical methods the let theory is known to be in good agreement with experimental results it is also unique among two point statistical closures in displaying an acceptable i e non markovian relationship between the transfer spectrum and the system response in accordance with experimental results as a result of the latter property it is compatible with the kolmogorov k41 spectral phenomenology

spectral analysis of structure functions and their scaling exponents in forced isotropic turbulence the pseudospectral method in conjunction with a technique for obtaining scaling exponents n from the structure functions sn r is presented as an alternative to the extended self similarity ess method and the use of generalized structure functions we propose plotting the ratio sn r s3 r against the separation r in accordance with a standard technique for analyzing experimental data this method differs from the ess technique which plots sn r against s3 r with the assumption s3 r r using our method for the particular case of s2 r we obtain the result that the exponent 2 decreases as the taylor reynolds number increases with 2 0 679 0 013 as r this supports the idea of finite viscosity corrections to the k41 prediction for s2 and is the opposite of the result obtained by ess the pseudospectral method also permits the forcing to be taken into account exactly through the calculation of the energy input in real space from the work spectrum of the stirring forces

vibration of fast trains palindromic eigenvalue problems and structure preserving doubling algorithms the vibration of fast trains is governed by a quadratic palindromic eigenvalue problem l 2a 1 email protected 0 a 1 x 0 where a 0 a email protected c n x n and a 0 t a 0 accurate and efficient solution can only be obtained using algorithms which preserve the structure of the eigenvalue problem this paper reports on the successful application of the structure preserving doubling algorithms

two dimensional problems of diffraction by finite collinear structures two closely related problems of diffraction are solved by use of the probabilistic random walk method the first concerns diffraction by a boundary of a half space with a piecewise constant boundary impedance and the second solves the problem of diffraction by a finite segment with different impedances on its sides the solutions are represented as superpositions of predefined geometric fields with several diffracted fields which are explicitly represented as mathematical expectations of certain functionals along the trajectories of specified random motions running across multisheet analytic manifolds associated with the boundary conditions numerical examples confirm the feasibility of the solutions

diffraction by a nonconvex polygon the probabilistic approach to wave propagation and diffraction is applied to a typical problem of diffraction by a nonconvex polygon the solution is obtained using a transparent technique that employs a floating coordinate system and it combines ideas from ray theory stochastic analysis and complex analysis the obtained solution is compatible with intuitive ideas about diffraction and it admits simple implementations

energy transfer and dissipation in forced isotropic turbulence a model for the reynolds number dependence of the dimensionless dissipation rate ce was derived from the dimensionless karman howarth equation resulting in ce ce c rl o 1 rl where rl is the integral scale reynolds number the coefficients c and ce arise from asymptotic expansions of the dimensionless second and third order structure functions this theoretical work was supplemented by direct numerical simulations dns of forced isotropic turbulence for integral scale reynolds numbers up to rl 5875 r 435 which were used to establish that the decay of dimensionless dissipation with increasing reynolds number took the form of a power law rln with exponent value n 1 000 0 009 and that this decay of ce was actually due to the increase in the taylor surrogate u3 l the model equation was fitted to data from the dns which resulted in the value c 18 9 1 3 and in an asymptotic value for ce in the infinite reynolds number limit of ce 0 468 0 006

a fluctuation relaxation relation for homogeneous isotropic turbulence a liouville type equation is solved perturbatively for the exact pdf of fluctuating velocities by expansion about an invariant gaussian measure which is an approximation to the symmetric part of the pdf and is constrained to give the correct covariance the antisymmetric part of the pdf is calculated to the lowest nontrivial order and leads to closed equations for the covariance in terms of a response function which arises from the liouville operator averaged against the ground state distribution as the latter is a stationary gaussian pdf we obtain an expression for the system response which is linear in the covariance

on backward errors of structured polynomial eigenproblems solved by structure preserving linearizations abstract we derive explicit computable expressions of structured backward errors of approximate eigenelements of structured matrix polynomials including symmetric skew symmetric hermitian skew hermitian even and odd polynomials we determine minimal structured perturbations for which approximate eigenelements are exact eigenelements of the perturbed polynomials we also analyze structured pseudospectra of a structured matrix polynomial and establish a partial equality between unstructured and structured pseudospectra finally we analyze the effect of structure preserving linearizations of structured matrix polynomials on the structured backward errors of approximate eigenelements and show that structure preserving linearizations which minimize structured condition numbers of eigenvalues also minimize the structured backward errors of approximate eigenelements

structured eigenvalue condition numbers and linearizations for matrix polynomials this work is concerned with eigenvalue problems for structured matrix polynomials including complex symmetric hermitian even odd palindromic and anti palindromic matrix polynomials most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size recently linearizations have been classified for which the pencil reflects the structure of the original polynomial a question of practical importance is whether this process of linearization significantly increases the eigenvalue sensitivity with respect to structured perturbations for all structures under consideration we show that this cannot happen if the matrix polynomial is well scaled there is always a structured linearization for which the structured eigenvalue condition number does not differ much this implies for example that a structure preserving algorithm applied to the linearization fully benefits from a potentially low structured eigenvalue condition number of the original matrix polynomial

more on pseudospectra for polynomial eigenvalue problems and applications in control theory definitions and characterizations of pseudospectra are given for rectangular matrix poly nomials expressed in homogeneous form p da d d 1 a d 1 da 0 it is shown that problems with infinite pseudo eigenvalues are elegantly treated in this framework for such problems stereographic projection onto the riemann sphere is shown to provide a convenient way to visualize pseudospectra lower bounds for the distance to the nearest nonregular polynomial and the nearest uncontrollable dth order system with equality for standard state space systems are obtained in terms of pseudospectra showing that pseudospectra are a fundamental tool for reasoning about matrix polynomials in areas such as control theory how and why to incorporate linear structure into pseudospectra is also discussed by example

on a class of alternating coefficient matrices quadratic eigenvalue problem in this paper we consider a class of alternating coefficient matrices quadratic eigenvalue problem aqep appearing in a wide range of applications at first the aqep is reformulated as a generalized eigenvalue problem gep by applying tisseur s linearization technique then we point out a property about skew symmetric tridiagonal matrix eigenvalue problem consequently we give that the corresponding gep can be solved by applying the cholesky ql algorithm or lanczos algorithm without complex arithmetic the structured backward error for qep would be discussed because of the special structure of the polynomial eigenvalue problems we develop structured condition number the upper bound and the lower bound are given some computational experiments and some concluding remarks are provided

norm preserving dilations and their applications to optimal error bounds the problem is given a b c to find d such that left begin array 20 c a c b d end array right leqq mu here we deal with hilbert space operators a b and c ar