on a non linear differential equation modelling helmholtz resonator response abstract the non linear differential equation y dot y sgn y e y y sin t proposed by hersh and walker 1 as a model for helmholtz resonator response to a sound wave in the presence of grazing mean flow is studied by matched asymptotic expansion for e 1 the waveform contains a smooth outer part and a boundary layer of rapid change approximations to two terms are found in each region and approximate value for all fourier coefficients also found and compared very favourably with numerical curve fits for the first coefficients given in reference 1 the asymptotic approximations are found to be highly accurate over the whole waveform when compared with numerical solutions

foncteurs polynomiaux stricts et modules instables sur l algebre de steenrod resume soient u f et p respectivement les categories des modules instables sur l algebre de steenrod modulo p sans bockstein des representations generiques des groupes lineaires generaux h w henn j lannes and l schwartz 1993 1 n j kuhn 1994 2 et des foncteurs polynomiaux stricts a la friedlander and suslin 1997 3 un foncteur f u f est construit dans h w henn j lannes and l schwartz 1993 1 utilisant le foncteur de lannes en degre 0 on construit dans cet article un foncteur m p u exact et commutant au produit tensoriel et a la torsion de frobenius qui constitute un relevement du foncteur oubli o p f m f o

an integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection diffusion by a steady incompressible velocity field are developed here under these conditions a rigorous stieltjes integral representation for effective diffusivity in turbulent transport is derived this representation is valid for all peclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the peclet number one consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all peclet numbers a nonlocal variational principle for effective diffusivity is developed along with applications to advection diffusion by random arrays of vortices a new class of rigorous examples is introduced these examples have an explicit stieltjes measure for the effective diffusivity furthermore the effective diffusivity behaves likek0 pe 1 2 in the limit of large peclet numbers wherek0 is the molecular diffusivity formal analogies with the theory of composite materials are exploited systematically

weyl modules and the mod 2 steenrod algebra abstract this paper continues our study of the action of the mod 2 steenrod algebra a 2 on the polynomial algebra p n f 2 t 1 t n we obtain further partial results on the hit problem of f p peterson which asks for a minimal generating set for p n as an a 2 module we also study the structure of the quotient by the hit elements as a graded representation of the finite general linear group g n gl n f 2 i e as a module over the finite group algebra f 2 g n these results were obtained in previous work of the authors for the special case of the steinberg module for g n by extending the scalars to f 2 the algebraic closure of f 2 we obtain commuting actions of a 2 and g n on p n f 2 t 1 t n while this makes no essential difference to the representation theory of g n or to the hit problem it allows us to treat the action of g n on p n as the restriction of that of the algebraic group g n gl n f 2 in particular we make use of tilting modules for g n to show that for every irreducible representation l of g n a minimal set of a 2 generators of p n must contain a copy of the corresponding dual weyl module

nonlinear asymptotic impedance model for a helmholtz resonator liner the usual nonlinear corrections for a helmholtz resonator type impedance do not seem to be based on a systematic asymptotic solution of the pertaining equations we aim to present a systematic derivation of a solution of the nonlinear helmholtz resonator equation in order to obtain analytically expressions for impedances close to resonance while including nonlinear effects the amplitude regime considered is such that when we stay away from the resonance condition the nonlinear terms are relatively small and the solution obtained is of the linear equation formed after neglecting the nonlinear terms close to the resonance frequency the nonlinear terms can no longer be neglected and algebraic equations are obtained that describe the corresponding nonlinear impedance sample results are presented including a few comparisons with measurements available in the literature the validity of the model is understood in the near resonance and nonresonance regimes

generalized random walk algorithm for the numerical modeling of complex diffusion processes a generalized form of the random walk algorithm to simulate diffusion processes is introduced unlike the usual approach at a given time all the particles from a grid node are simultaneously scattered using the bernoulli repartition this procedure saves memory and computing time and no restrictions are imposed for the maximum number of particles to be used in simulations we prove that for simple diffusion the method generalizes the finite difference scheme and gives the same precision for large enough number of particles as an example simulations of diffusion in random velocity field are performed and the main features of the stochastic mathematical model are numerically tested 2003 elsevier science b v all rights reserved

homogenization of a diffusion process in a divergence free random field on etudie le probleme d homogeneisation pour un processus de diffusion dans un champ de derive aleatoire independant de la divergence dans le cas particulier d un petit champ gaussien on derive un developpement asymptotique de la matrice de diffusion efficace en termes de sa densite spectrale

linking first occurrence polynomials over f2 by steenrod operations abstract it is proved that for every irreducible representation l of the full matrix semigroup mn f 2 the first occurrence of l as a composition factor in the polynomial algebra p f 2 x1 xn is linked by a steenrod operation to the first occurrence of l as a submodule in p this steenrod operation is given explicitly as the image of an admissible monomial in the steenrod squares sqr under the canonical anti automorphism of the mod 2 steenrod algebra a the first occurrences of both kinds are also linked to higher degree occurrences of l by elements of the milnor basis of a

closure approximations for passive scalar turbulence a comparative study on an exactly solvable model with complex features some standard closure approximations used in turbulence theory are analyzed by examining systematically the predictions these approximations produce for a passive scalar advection model consisting of a shear flow with a fluctuating cross sweep this model has a general geometric structure of a jet flow with transverse disturbances which occur in a number of contexts and it encompasses a wide variety of possible spatio temporal statistical structures for the velocity field including strong long range correlations even though the eulerian and lagrangian velocity statistics are not equal and the passive scalar statistics exhibit broader than gaussian intermittency this model is nevertheless simple enough so that many passive scalar statistics can be computed exactly and compared systematically with the predictions of the closure approximations our comparative study illustrates the strength and weaknesses of the closure approximations and points out the physical phenomena that these approximations are able or not able to describe properly in particular it is shown that the direct interaction approximation dia one of the most sophisticated closure approximations available fails to reproduce adequately the statistical features of the scalar and may even lead to absdurd predictions even though the equations it produces are rather complicated and difficult to analyze two alternative closure approximations the modified dia mdia and the renormalized lagrangian approximation rla with different levels of sophistication both are simpler to use than the dia and perform better in particular it is shown that both closure approximations always reproduce exactly the second order statistics for the scalar and that the mdia is even able to capture intermittency effects

helmholtz resonator impedance model part 1 nonlinear behavior a one dimensional preliminary design impedance prediction model is derived for helmholtz resonators constructed with circular orie ces the model addresses only the effects of resonator geometry and incident sound pressure amplitude the effects of grazing e ow are not included an experimental program was conducted to calibrate unknown empirical parameters derived in the model the model was shown to predict reasonably accurately impedance data published in the open literature for resonators constructed with single and multiple orie ces over a wide range of sound pressure amplitudes and frequencies the model provides a rapid means of calculating resonator impedance as a function of resonator geometry incident sound pressure amplitude and frequency

the morava k theories of some classifying spaces let p be a finite abelian p group with classifying space bp we compute in representation theoretic terms the morava k theories of the stable wedge summands of bp in particular we obtain a simple and purely group theoretic description of the rank of k s bg for any finite group g with an abelian p sylow subgroup a minimal amount of topology quickly reduces the problem to an algebraic one of analyzing truncated polynomial algebras as modular representations of the semigroup m 1 z p 1 main results for a fixed prime p there exists a sequence of cohomology theories k 1 k 2 with k s periodic with period 2 ps 1 these are the morava k theories 13 and generalize ordinary complex k theory in the sense that k 1 is one of the p 1 isomorphic summands of k z p recently it has been becoming clear that they play a central role in homotopy theory see for example the work of m hopkins and j smith 6 k s has further computational virtues k s x is always a free module over the coefficient ring k s and there is a kunneth isomorphism a major outstanding problem is to find good models for the spaces representing these theories related to this is the question of finding for finite groups g a group theoretic description for the rings k s bg analogous to atiyah s isomorphism 2 r g k bg the author together with j harris recently analyzed stable wedge decompositions of classifying spaces of abelian p groups 5 in this paper we compute in representation theoretic terms the morava k theories of the resulting wedge summands as a consequence we obtain a very simple and purely group theoretic description of the rank of k s bg for any finite group g with an abelian p sylow subgroup to state our results we establish some notation it is convenient to let k o be rational cohomology we let k s x always denote the reduced sth morava k theory of x if x is a space or spectrum we let k5 x equal the rank of k s x as a k s module and by abuse of notation we let ks g ks bg for a finite group g bg is the union of bg with a disjoint basepoint received bv the editors may 27 1986 1980 marhemat7es subject cl lssific ltion primary 55n22 secondary 20g05 research partially supported by the sloan foundation and the nsf t1987 american mathematical society 0002 9947 87 1 00 25 per page

subalgebras of the steenrod algebra and the action of matrices on truncated polynomial algebras let a t be the subalgebra of the steenrod algebra generated by sqi i 2t we determine which a module indecomposable summands of f2 x1 xn are free over a t by first showing this to be equivalent to the group theoretic question of determining which irreducible modular representations of mn f2 or gln f2 occur as composition factors in the truncated polynomial algebra f2 x1 xn x12 xn2 the second question is then answered

superdiffusion in nearly stratified flows in classical work matheron and the marsilly showed that superdiffusive scaling of mean square displacements occurs in transport diffusion for stratified flows with steady simple shear layers and long range spatial correlations more recently the authors have calculated a formula for the non gaussian large scale long time renormalized green function for these problems here the scaling laws and renormalized green functions for diffusion in nearly stratified flows are studied in such flows the simple shear layer with long range correlations is perturbed by incompressible flows with short range correlations here it is established that these flows belong to the same universality class as the simple shear layers with a renormalized green function with a similar structure but reflecting homogenization by the transverse displacements the tools in the analysis involve a modification of homogenization theory and also rigorous diagrammatic perturbation theory

mathematical models with exact renormalization for turbulent transport the advection diffusion of a passive scalar by incompressible velocity fields which admit a statistical description and involve a continuous range of excited spatial and or temporal scales is very important in applications ranging from fully developed turbulence to the diffusion of tracers in heterogeneous porous media a variety of renormalization theories which typically utilize partial resummation of divergent perturbation series according to various recipes have been applied to this problem in various contexts in this paper a simple model problem for the advection diffusion of a passive scalar is introduced and the complete renormalization theory is developed with full mathematical rigor explicit formulas for the anomalous time scaling in various regimes as well as the green s function for the large scale long time ensemble average are developed here formulas for the renormalized higher order statistics are also developed the simple form of the model problem is deceptive the renormalization theory for this problem exhibits a remarkable range of different renormalization phenomena as parameters in the velocity statistics are varied these phenomena include the existence of several distinct anomalous scaling regimes as the spectral parameter tilde varepsilon is varied as well as explicit regimes in tilde varepsilon where the effective equation for the ensemble average is not a simple diffusion equation but instead involves an explicit random nonlocal eddy diffusivity we use fourier analysis and the feynman kac formula as main tools in the explicit exact renormalization theory developed here

the composition factors of fp x1 x2 x3 as a gl 3 p module let gl n p denote the general linear group over the prime field fp and let sd denote the right gl n p module consisting of all homogeneous polynomials of degree d in n variables x1 x2 x over f we wish to determine the simple composition factors of sd unfortunately satisfactory models for all the simple gl n p modules themselves are not known there are p p 1 of these simple modules up to isomorphism and all of them are known to occur in the symmetric algebra s cf sd the general problem is therefore a very difficult one in our present state of knowledge the purpose of this paper is to offer a solution to this problem in the case n 3 this extends the work of david carlisle who calculated in his manchester thesis 2 the occurrences of all the l dimensional modules and developed a method for dealing with the general case let us begin with some general remarks about the problem the case n 1 is trivial as s d detd the dth power of the determinant representation the case n 2 is also easily disposed of for instance by glover 9 the simple modules are the tensor products of the sd for d 1 and d p the module s is not simple for example the pth power sf of the natural module is a proper submodule of 411 0021 8693192 3 00

on the superdiffusive behavior of passive tracer with a gaussian drift in the present article we consider a motion of a passive tracer particle whose trajectory satisfies the ito stochastic differential equation d x t v t x t dt sqrt 2 kappa d w t where w is a brownian motion v is a stationary gaussian random field with incompressible realizations independent of w and 0 we prove the superdiffusive character of the motion under certain conditions on the energy spectrum of the velocity field the result is shown both for steady time independent and time dependent and markovian velocity fields in addition we provide explicit upper and lower bounds for the hurst exponent of the trajectory all previous rigorous results concerned explicitely solvable shear flows cases

on strong anomalous diffusion superdiffusive behavior i e x2 t t2 with 1 2 is in general not completely characterized by a unique exponent we study some systems exhibiting strong anomalous diffusion i e x t q tq q where 2 1 2 and q q is not a linear function of q this feature is different from the weak superdiffusive regime i e q const 1 2 occurring in random shear flows strong anomalous diffusion can be generated by nontrivial chaotic dynamics e g lagrangian motion in 2d time dependent incompressible velocity fields 2d symplectic maps and 1d intermittent maps typically the function q q is piecewise linear this is due to two mechanisms a weak anomalous diffusion for the typical events and a ballistic transport for the rare excursions in order to have strong anomalous diffusion one needs a violation of the hypothesis of the central limit theorem this happens only in a very narrow region of the control parameters space in the presence of strong anomalous diffusion one does not have a unique exponent and therefore one has the failure of the usual scaling p x t t f x t of the probability density this implies that the effective equation at large scale and long time for p x t obeys neither the usual fick equation nor other linear equations involving temporal and or spatial fractional derivatives

numerical investigations on ergodicity of solute transport in heterogeneous aquifers 1 darcy velocities for lognormal hydraulic conductivity with small variance and finite correlation length were approximated by periodic random fields accurate simulations of two dimensional advection dispersion processes were achieved with the global random walk algorithm using 1010 particles in every transport realization reliable statistical estimations were obtained by averaging over 256 realizations the main result is a numerical evidence for the mean square convergence of the actual concentrations to the macrodispersion process predicted by a known limit theorem for small initial plumes the ergodic behavior can be expected after thousands of advection timescales when the deviation from the theoretical prediction of the cross section space averaged concentration monotonously decays and falls under 20 the increase of the transverse dimension of the initial plume slows down the approach to the quasi ergodic state and has a nonlinear effect on the variability of the actual concentrations and dispersivities