equations differentielles sur l espace de wiener et formules de cameron martin non lineaires proprietes algebriques des approximations sur r n une formule de perturbation des flots et convergence des approximations formule de changement de variable sur x et equation differentielle

z4 kerdock codes orthogonal spreads and extremal euclidean line sets when m is odd spreads in an orthogonal vector space of type omega 2m 2 2 are related to binary kerdock codes and extremal line sets in rr 2 m 1 with prescribed angles spreads in a 2m dimensional binary symplectic vector space are related to kerdock codes over zz 4 and extremal line sets in cc 2 m with prescribed angles these connections involve binary real and complex geometry associated with extraspecial 2 groups a geometric map from symplectic to orthogonal spreads is shown to induce the gray map from a corresponding zz 4 kerdock code to its binary image these geometric considerations lead to the construction for any odd composite m of large numbers of zz 4 kerdock codes they also produce new zz 4 linear kerdock and preparata codes 1991 mathematics subject classification primary 94b60 secondary 51m15 20c99

new bounds on the total squared correlation and optimum design of ds cdma binary signature sets the welch lower bound see welch r l ieee trans inform theory vol it 20 p 397 9 1974 on the total squared correlation tsc of signature sets is known to be tight for real valued signatures and loose for binary signatures whose number is not a multiple of four we derive new bounds on the tsc of binary signature sets for any number of signatures k and any signature length l then for almost all k l in 1 2 256 we design optimum binary signature sets that achieve the new bounds the design procedure is based on simple transformations of hadamard matrices

global flows with invariant gibbs measures for euler and navier stokes two dimensional fluids we construct a family of probability spaces p 0 associated with the euler equation for a two dimensional inviscid incompressible fluid which carries a pointwise flow t time evolution leavingp globally invariant t is obtained as the limit of galerkin approximations associated with euler equations p is also in invariant measure for a stochastic process associated with a navier stokes equation with viscosity stochastically perturbed by a white noise force

the 2d euler equations and the statistical transport equations we prove the existence of weak solutions for the forward and backward statistical transport equations associated with the 2d euler equations such solutions can be interpreted respectively as a statistical lagrangian and a statistical eulerian description of the motion of the fluid

collective coordinates and length scale competition in spatially inhomogeneous soliton bearing equations perturbed one dimensional integrable i e soliton bearing equations arise in many applied contexts when trying to improve the usually highly idealized description of problems of interest in terms of the purely integrable equations in particular when the assumption of perfect homogeneity is dropped to account for unavoidable impurities or defects perturbations depending on the spatial coordinate must be added to the original equation in this review we use the one dimensional sine gordon sg equation perturbed by a spatially periodic term as a generic paradigm to discuss the main perturbative techniques available for the study of this class of problems to place the work in context we summarize the approaches developed to date and focus on the collective coordinate approach as one of the most useful tools we introduce several versions of this perturbative method and relate them to more involved procedures we analyze in detail the application to the sg equation but the procedure is very general to illustrate this other examples of the application of collective coordinates are briefly revisited in our case study this approach helps us identify perturbative and nonperturbative regimes yielding a very simple picture of the former beyond perturbative calculations the same example of the inhomogeneous sg equation allows us to introduce a phenomenon termed length scale competition which we show to be a rather general mechanism for the appearance of complex spatiotemporal behavior in perturbed integrable systems as other instances discussed in the review show such nonperturbative results are obtained by means of numerical simulations of the full perturbed problem numerical linear stability analysis is also used to clarify the origins of the instability originated by this competition to complement our description of the techniques employed in these studies computational details of our numerical simulations are also included finally the paper closes with a discussion of the above ideas and a speculative outlook on general questions concerning the interplay of nonlinearity with disorder

the swift hohenberg equation with a nonlocal nonlinearity it is well known that aspects of the formation of localised states in a one dimensional swift hohenberg equation can be described by ginzburg landau type envelope equations this paper extends these multiple scales analyses to cases where an additional nonlinear integral term in the form of a convolution is present the presence of a kernel function introduces a new lengthscale into the problem and this results in additional complexity in both the derivation of envelope equations and in the bifurcation structure when the kernel is short range weakly nonlinear analysis results in envelope equations of standard type but whose coefficients are modified in complicated ways by the nonlinear nonlocal term nevertheless these computations can be formulated quite generally in terms of properties of the fourier transform of the kernel function when the lengthscale associated with the kernel is longer our method leads naturally to the derivation of two different novel envelope equations that describe aspects of the dynamics in these new regimes the first of these contains additional bifurcations and unexpected loops in the bifurcation diagram the second of these captures the stretched out nature of the homoclinic snaking curves that arises due to the nonlocal term

variational approximation and the use of collective coordinates we consider propagating spatially localized waves in a class of equations that contain variational and nonvariational terms the dynamics of the waves is analyzed through a collective coordinate approach motivated by the variational approximation we show that there is a natural choice of projection onto collective variables for reducing the governing nonlinear partial differential equation pde to coupled ordinary differential equations odes this projection produces odes whose solutions are exactly the stationary states of the effective lagrangian that would be considered in applying the variational approximation method we illustrate our approach by applying it to a modified fisher equation for a traveling front containing a non constant coefficient nonlinear term we present numerical results that show that our proposed projection captures both the equilibria and the dynamics of the pde much more closely than previously proposed projections

variational approximations to homoclinic snaking in continuous and discrete systems localized structures appear in a wide variety of systems arising from a pinning mechanism due to the presence of a small scale pattern or an imposed grid when there is a separation of length scales the width of the pinning region is exponentially small and beyond the reach of standard asymptotic methods we show how this behavior can be obtained using a variational method for two systems in the case of the quadratic cubic swift hohenberg equation this gives results that are in agreement with recent work using exponential asymptotics in addition the method is applied to a discrete system with cubic quintic nonlinearity giving results that agree well with numerical simulations

anticipating flows on the wiener space generated by vector fields of low regularity abstract sobolev spaces for banach space valued wiener functionals are constructed using the operator norm for the h derivatives the corresponding sobolev norms are shown to be strictly weaker than the usual malliavin sobolev norms existing results on flows on an abstract wiener space generated by a vector field are generalized to the case where the vector field is a sum of a not necessarily bounded skew adjoint operator and a non linear part whose regularity is measured by the new sobolev norms this is equivalent to a non linear perturbation of a semi group of rotations

achieving the welch bound with difference sets consider a codebook containing n unit norm complex vectors in a k dimensional space in a number of applications the codebook that minimizes the maximal cross correlation amplitude i sub max is often desirable relying on tools from combinatorial number theory we construct analytically optimal codebooks meeting in certain cases the welch lower bound when analytical constructions are not available we develop an efficient numerical search method based on a generalized lloyd algorithm which leads to considerable improvement on the achieved i sub max over existing alternatives we also derive a composite lower bound on the minimum achievable i sub max that is effective for any codebook size n

a family of alltop functions that are ea inequivalent to the cubic function sequences with optimal correlation properties are much sought after for applications in communication systems in 1980 alltop ieee trans inf theory 26 3 350 354 1980 described a set of sequences based on a cubic function and showed that these sequences were optimal with respect to the known bounds on auto and crosscorrelation subsequently these sequences were used to construct mutually unbiased bases mubs a structure of importance in quantum information theory the key feature of this cubic function is that its difference function is a planar function functions with planar difference functions have been called alltop functions this paper provides a new family of alltop functions and establishes the use of alltop functions for construction of sequence sets and mubs

on the validity of the variational approximation in discrete nonlinear schrodinger equations abstract the variational approximation is a well known tool to approximate localized states in nonlinear systems in the context of a discrete nonlinear schrodinger equation with a small coupling constant we prove error estimates for the variational approximations of site symmetric bond symmetric and twisted discrete solitons this is shown for various trial configurations which become increasingly more accurate as more parameters are taken it is also shown that the variational approximation yields the correct spectral stability result and controls the oscillatory dynamics of stable discrete solitons for long but finite time intervals

the two dimensional euler equation a statistical study we construct surface type measures on the level sets for the renormalized energy which is an invariant quantity for the two dimensional periodic euler flow and prove the existence of weak solutions living on such level sets other classes of invariant measures for the motion are also introduced

signal sets from functions with optimum nonlinearity signal sets with the best correlation property are desirable in code division multiple access cdma systems in this paper the construction of wootters and fields for mutually unbiased bases is extended into a generic construction of signal sets using planar functions then specific classes of planar functions and almost bent functions are employed to obtain q2 q q signal sets the signal sets derived from planar functions are optimal with respect to the levenstein bound and those obtained from almost bent functions nearly meet the levenstein bound the signal sets constructed in this paper could have a very small alphabet size and have applications in synchronous ds cdma systems where the number of users is greater than the signal space dimension or the spreading factor

reaction diffusion front crossing a local defect the interaction of a zeldovich frank kamenetsky reaction diffusion front with a localized defect is studied numerically and analytically for the analysis we start from conservation laws and develop simple collective variable ordinary differential equations for the front position and width their solutions are in good agreement with the solutions of the full problem finally using this reduced model we explain the pinning of the front on a large defect and obtain a quantitative criterion

variational approximations to homoclinic snaking we investigate the snaking of localized patterns seen in numerous physical applications using a variational approximation this method naturally introduces the exponentially small terms responsible for the snaking structure which are not accessible via standard multiple scales asymptotic techniques we obtain the symmetric snaking solutions and the asymmetric ladder states and also predict the stability of the localized states the resulting approximate formulas for the width of the snaking region show good agreement with numerical results

complex codebooks from combinatorial designs codebooks also called signal sets meeting the welch bounds are desirable in code division multiple access cdma systems in 2003 binary codebooks meeting welch s bounds were constructed using difference sets by ding golin and kloslashve recently a generic construction of codebooks with cyclic difference sets meeting welch s bound on the maximum cross correlation amplitude was developed by xia in this correspondence the idea of xia is extended and related constructions of optimal codebooks with both cyclic and noncyclic difference sets are presented these codebooks are optimal in the sense that they also meet this welch bound in addition complex codebooks that almost meet welch s bound on the maximum cross correlation amplitude are also constructed with almost difference sets

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

generalized welch bound equality sequences are tight frames this article shows what are called the welch 1974 bound equality wbe sequences by the signal processing community are precisely the isometric equal norm normalized uniform tight frames which are currently being investigated for a number of applications and in the real case are the spherical 2 designs of combinatorics recent applications include wavelet expansions grassmannian frames frames robust to erasures and quantum measurements this is done by giving an elementary proof of a generalization of welch s inequality to vectors which need not have equal energy and then showing that equality occurs in this exactly when the vectors form a tight frame

grassmannian beamforming for multiple input multiple output wireless systems transmit beamforming and receive combining are simple methods for exploiting the significant diversity that is available in multiple input multiple output mimo wireless systems unfortunately optimal performance requires either complete channel knowledge or knowledge of the optimal beamforming vector both are hard to realize in this article a quantized maximum signal to noise ratio snr beamforming technique is proposed where the receiver only sends the label of the best beamforming vector in a predetermined codebook to the transmitter by using the distribution of the optimal beamforming vector in independent and identically distributed rayleigh fading matrix channels the codebook design problem is solved and related to the problem of grassmannian line packing the proposed design criterion is flexible enough to allow for side constraints on the codebook vectors bounds on the codebook size are derived to guarantee full diversity order results on the density of grassmannian line packings are derived and used to develop bounds on the codebook size given a capacity or snr loss monte carlo simulations are presented that compare the probability of error for different quantization strategies

on quasi orthogonal signatures for cdma systems sum capacity optimal signatures in synchronous code division multiple access cdma systems are functions of the codebook length as well as the number of active users a new signature set must be assigned every time the number of active users changes this correspondence considers signature sets that are less sensitive to changes in the number of active users equiangular signature sequences are proven to solve a certain max min signal to interference plus noise problem which results from their interference invariance unions of orthonormal bases have subsets that come close to satisfying the welch bound bounds on the maximum number of bases with minimum maximum correlation are derived and a new construction algorithm is provided connections are made between these signature design problems grassmannian line packing frame theory and algebraic geometry