efficient root finding algorithm with application to list decoding of algebraic geometric codes a list decoding for an error correcting code is a decoding algorithm that generates a list of codewords within a hamming distance t from the received vector where t can be greater than the error correction bound in previous work by m shokrollahi and h wasserman see ibid vol 45 p 432 7 march 1999 a list decoding procedure for reed solomon codes was generalized to algebraic geometric codes recent work by v guruswami and m sudan see ibid vol 45 p 1757 67 sept 1999 gives improved list decodings for reed solomon codes and algebraic geometric codes that work for all rates and have many applications however these list decoding algorithms are rather complicated r roth and g ruckenstein see ibid vol 46 p 246 57 jan 2000 proposed an efficient implementation of the list decoding of reed solomon codes in this correspondence extending roth and ruckenstein s fast algorithm for finding roots of univariate polynomials over polynomial rings i e the reconstruct algorithm we present an efficient algorithm for finding the roots of univariate polynomials over function fields based on the extended algorithm we give an efficient list decoding algorithm for algebraic geometric codes

a justesen construction of binary concatenated codes that asymptotically meet the zyablov bound for low rate an explicit construction of a sequence of binary codes that asymptotically meet the zyablov bound for rates lower than 0 30 is given by using justesen s construction of concatenation the outer codes are constructed from generalized hermitian curves these outer codes can be described without any algebraic geometry terminology while the proofs of some properties deeply rely on algebraic geometry

partial geometric codes we present a theory of partial geometric goppa 1981 codes which can be applied to curves for which the genus is not explicitly known that is often the case if the curve is nonsmooth or nonplanar the only algebraic geometry required is familiarity with the basic concepts yet we are able to give a full account of all one point codes including their error processors we are also able to give a completely explicit construction for the initial code in a good sequence of codes based on a set of curves introduced by garcia and stichtenoth acad sci paris vol 322 i p 1067 70 1996 for which the genus has not been determined

jordan blocks in a one dimensional markov map abstract a simple one dimensional chaotic map whose spectral decomposition of the frobenius perron operator in the space of polynomials has jordan block structure is given and analyzed the jordan block structure implies that some correlation functions have exponential decay modified by coefficients which are polynomials in t such a correlation is given for this system and a numerical simulation which agrees with the theoretical prediction is presented

on a decoding algorithm for codes on maximal curves a decoding algorithm for algebraic geometric codes that was given by a n skorobogatov and s g vladut preprint inst problems of information transmission 1988 is considered the author gives a modified algorithm with improved performance which he obtains by applying the above algorithm a number of times in parallel he proves the existence of the decoding algorithm on maximal curves by showing the existence of certain divisors however he has so far been unable to give an efficient procedure of finding these divisors

kinetic theory of area preserving maps application to the standard map in the diffusive regime the evolution of the distribution function of a dynamical system governed by a general two dimensional area preserving iterative map is studied by the methods of nonequilibrium statistical mechanics a closed non markovian master equation determines the angle averaged distribution function the density profile the complementary angle dependent part the fluctuations is expressed as a non markovian functional of the density profile whenever there exist two widely separated intrinsic time scales the master equation can be markovianized yielding an asymptotic kinetic equation the general theory is applied to the standard map in the diffusive regime i e for large stochasticity parameter and large scale length the non markovian master equation can be written and solved analytically in this approximation the two characteristic time scales are exhibited this permits the thorough study of the evolution of the density profile its tendency toward the markovian approximation and eventually toward a diffusive gaussian packet the evolution of the fluctuations is also described in detail the various relaxation processes are governed asymptotically by a single diffusion coefficient which is calculated analytically this model appears as a testing bench for the study of kinetic equations the various previous approaches to this problem are reviewed and critically discussed

fast decoding of algebraic geometric codes up to the designed minimum distance we present a decoding algorithm for algebraic geometric codes from regular plane curves in particular the hermitian curve which corrects all error patterns of weight less than d 2 with low complexity the algorithm is based on the majority scheme of feng and rao 1993 and uses a modified version of sakata s 1988 generalization of the berlekamp massey algorithm

fick s law and fractality of nonequilibrium stationary states in a reversible multibaker map nonequilibrium stationary states are studied for a multibaker map a simple reversible chaotic dynamical system the probabilistic description is extended by representing a dynamical state in terms of a measure instead of a density function the equation of motion for the cumulative function of this measure is derived and stationary solutions are constructed with the aid of derham type functional equations to select the physical states the time evolution of the distribution under a fixed boundary condition is investigated for an open multibaker chain of scattering type this system corresponds to a diffusive flow experiment through a slab of material for long times any initial distribution approaches the stationary one obeying fick s law at stationarity the intracell distribution is singular in the stable direction and expressed by the takagi function which is continuous but has no finite derivatives the result suggests that singular measures play an important role in the dynamical description of non equilibrium states

remarks on codes from hermitian curves corresp parameters and generator matrices are given for the codes obtained by applying goppa s algebraic geometric construction method to hermitian curves in pg 2 q where q 2 2s for some s in bf n automorphisms of these codes are also discussed and some results on self duality and weak self duality are given

a generalized forney formula for algebraic geometric codes this correspondence contains a straightforward generalization of decoding of bch codes to the decoding of algebraic geometric codes couched in terms of varieties ideals and grobner bases this consists of 1 a berlekamp massey type lattice shifting row reduction algorithm with majority voting similar to algorithms in the current literature 2 a realization that it produces a minimal grobner basis b for the error locator ideal i v relative to a particular weighted total degree monomial ordering 3 a factoring of that basis into several minimal plex bases that facilitates finding the variety v of error positions and 4 a direct generalization of forney s formula to calculate error magnitudes using functions spl sigma p which are by products of this factoring

carleman inequalities for the dirac and laplace operators and unique continuation on demontre les meilleures inegalites de type carleman possibles dans le cas de l operateur de dirac on obtient la propriete de prolongement unique pour d v ou v l loc r n 3n 2 2 n 3

list decoding of algebraic geometric codes we generalize sudan s see j compl vol 13 p 180 93 1997 results for reed solomon codes to the class of algebraic geometric codes designing algorithms for list decoding of algebraic geometric codes which can decode beyond the conventional error correction bound d 1 2 d being the minimum distance of the code our main algorithm is based on an interpolation scheme and factorization of polynomials over algebraic function fields for the latter problem we design a polynomial time algorithm and show that the resulting overall list decoding algorithm runs in polynomial time under some mild conditions several examples are included

codes on the klein quartic ideals and decoding corresp a sequence of codes with particular symmetries and with large rates compared to their minimal distances is constructed over the field gf 2 3 in the sequence there is for instance a code of length 21 and dimension 10 with minimal distance 9 and a code of length 21 and dimension 16 with minimal distance 3 the codes are constructed from algebraic geometry using the dictionary between coding theory and algebraic curves over finite fields established by goppa the curve used in the present work is the klein quartic this curve has the maximal number of rational points over gf 2 3 allowed by serre s improvement of the hasse weil bound which together with the low genus accounts for the good code parameters the klein quartic has the frobenius group g of order 21 acting as a group of automorphisms which accounts for the particular symmetries of the codes in fact the codes are given alternative descriptions as left ideals in the group algebra gf 2 3 g this description allows for easy decoding for instance in the case of the single error correcting code of length 21 and dimension 16 with minimal distance 3 decoding is obtained by multiplication with an idempotent in the group algebra

on the strichartz estimates for orthonormal systems of initial data with regularity abstract the classical strichartz estimates for the free schrodinger propagator have recently been substantially generalised to estimates of the form j j e i t f j 2 l p t l x q l for orthonormal systems f j j of initial data in l 2 firstly in work of frank lewin lieb seiringer and later by frank sabin the primary objective is identifying the largest possible as a function of p and q and in contrast to the classical case for such estimates the critical case turns out to be p q d 1 d d 1 d 1 we consider the case of orthonormal systems f j j in the homogeneous sobolev spaces h s for s 0 d 2 and we establish the sharp value of as a function of p q and s except possibly an endpoint in certain cases furthermore at the critical case p q d 1 d 2 s d d 1 d 1 d 2 s for general s we show the veracity of the desired estimates when p if we consider frequency localised estimates and the failure of the non localised estimates when p this exhibits the difficulty of upgrading from frequency localised estimates in this context again in contrast to the classical setting

the cauchy problem for the hartree equations under random influences abstract in this paper the cauchy problem for the hartree equations under random influences is studied in the first part the randomization of the initial conditions introduced by n burq and n tzvetkov is used to show that the hartree equations admit strong solutions for data in h for some below the critical threshold s c in the second part the global well posedness and blow up results for the hartree equations with stochastic potential are considered

on the number of correctable errors for some ag codes an algorithm that for codes from a regular plane curve corrects up to d 2 m sup 2 8 m 4 9 8 errors where d is the designed distance and m is the degree of the curve was presented in an earlier work see ibid vol 35 p 811 21 1989 it is now shown that this bound is the best possible for the algorithm considered

construction and decoding of a class of algebraic geometry codes a class of codes derived from algebraic plane curves is constructed the concepts and results from algebraic geometry that were used are explained in detail no further knowledge of algebraic geometry is needed parameters generator and parity check matrices are given the main result is a decoding algorithm which turns out to be a generalization of the peterson algorithm for decoding bch decoder codes

decoding of codes defined by a single point on a curve a decoding algorithm for certain codes from algebraic curves is presented the dual code which is used for decoding is formed by evaluating rational functions having poles at a single point a theoretical foundation is developed from which an improved bound for the minimum distance and results on decoding up to that bound are derived the decoding is done by a computationally efficient berlekamp massey type algorithm which is intrinsic to the curve

on the structure of hermitian codes and decoding for burst errors in this paper it is proved that hermitian code is a direct sum of concatenated reed solomon codes over gf q sup 2 based on this discovery first a new method for computing the dimension and tightly estimating the minimum distance of the hermitian code is derived secondly a new decoding algorithm which is especially effective in dealing with burst errors with complexity o n sup 5 3 is described finally some possible approaches for optimization of hermitian codes are discussed

unique decoding of general ag codes a unique decoding algorithm for general ag codes namely multipoint evaluation codes on algebraic curves is presented it is a natural generalization of the previous decoding algorithm which was only for one point ag codes as such it retains the same advantages of fast speed regular structure and direct message recovery upon this generalization we add a technique from the guruswami sudan list decoding that boosts the decoding speed significantly compared with other known decoding algorithms for general ag codes it has a similar decoding performance and allows streamlined practical implementation by its simple and regular structure

metaplectic representation on wiener amalgam spaces and applications to the schrodinger equation we study the action of metaplectic operators on wiener amalgam spaces giving upper bounds for their norms as an application we obtain new fixed time estimates in these spaces for schrodinger equations with general quadratic hamiltonians and strichartz estimates for the schrodinger equation with potentials v x x 2

lifespan estimates for the semi linear klein gordon equation with a quadratic potential in dimension one abstract we show for almost every m 0 the solution to the semi linear klein gordon equation with a quadratic potential in dimension one exists over a longer time interval than the one given by local existence theory using the normal form method by using an l p l q estimate for eigenfunctions of the harmonic oscillator and by carefully analysis on the nonlinearity we improve the result obtained by the author before

a systolic array implementation of the feng rao algorithm an efficient implementation of a parallel version of the feng rao algorithm on a one dimensional systolic array is presented in this paper by adopting an extended syndrome matrix syndromes of the same order lying on a slant diagonal in the extended syndrome matrix are scheduled to be examined by a series of cells simultaneously and therefore a high degree of concurrency of the feng rao algorithm can be achieved the time complexity of the proposed architecture is m g 1 by using a series of t g 1 2 1 nonhomogeneous but regular effective processors called pe cells and g trivial processors called d cells where t is designed as the half of the feng rao bound each d cell contains only delay units while each pe cell contains one finite field inverter and except the first one one or more finite field multipliers cell functions of each pe cell are basically the same and the overall control circuit of the proposed array is quite simple the proposed architecture requires in total t g 1 1 finite field inverters and t g 1 2 t g 1 2 1 2 finite field multipliers for a practical design this hardware complexity is acceptable

long time existence for the quadratic wave equation associated to the harmonic oscillator abstract we show the solution for the quadratic wave equation associated to the harmonic oscillator exists over a longer time interval than the one given by local existence theory the key point is that the nonlinearity is quadratic so that we can estimate the small divisor and perform a normal form process

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

two parameter estimates for joint spectral projections on complex spheres we prove sharp two parameter estimates for the lp l2 norm 1 p 2 of the joint spectral projectors associated to the laplace beltrami operator and to the kohn laplacian on the unit sphere s2n 1 in mathbb c n then by using the notion of contraction of lie groups we deduce the estimates recently obtained by h koch and f ricci for joint spectral projections on the reduced heisenberg group h1

a determination of the parameters of a large class of goppa codes we determine the code parameters of a large class of goppa codes constructed from certain curves described by stichtenoth see arch math vol xxiv pp 615 631 1973 in terms of the genus and the weierstrass gap sequence of a given chosen point on each tune examples are codes from hermitian artin schreier and hyperelliptic curves

carleman inequalities and the heat operator we prove a strong unique continuation theorem for functions u vanishing to infinite order in the space time variables at 0 0 and verifying the inequality 4u tu v x t u on rn 0 t when t1 n 2r 1 s v x t lt locl r x n 2r 1 s 1 n 2 r

on existence and scattering with minimal regularity for semilinear wave equations abstract we prove existence and scattering results for semilinear wave equations with low regularity data we also determine the minimal regularity that is needed to ensure local existence and well posedness and we give counterexamples to well posedness more specifically we show that equations of the type u u p with initial data u u t in r n 1 r n have a local solution if p n and we construct counterexamples if p n the existence results rely on mixed norm space time estimates of strichartz type

majority coset decoding a majority coset decoding mcd procedure that can be applied to an arbitrary geometric code is discussed in general the basic algorithm for decoding of algebraic geometric codes does not correct up to the designed minimum distance in mcd a reduction step is added to the basic algorithm in case the basic algorithm fails a majority scheme is used to obtain an additional syndrome for the error vector thus a strictly smaller cost containing the error vector is obtained in this way the basic algorithm is applied to a decreasing chain of cosets and after finitely many steps the coset will be small enough for successful application of the basic algorithm

the trace description of irreducible quasi cyclic codes the notion of a q ary irreducible quasi cyclic code of block length n and index r is introduced a trace description of such a code is provided in a fashion similar to the trace description of irreducible cyclic codes in particular it is shown that an irreducible quasi cyclic code of dimension k is completely described by an irreducible cyclic code and r elements from a field of cardinality q sup k using this fact a number of binary irreducible quasi cyclic codes of index 2 are constructed and their weight spectra obtained

δ sequences and evaluation codes defined by plane valuations at infinity we introduce the concept of sequence a sequence generates a well ordered semigroup s in z or r we show how to construct and compute parameters for the dual code of any evaluation code associated with a weight function defined by from the polynomial ring in two indeterminates to a semigroup s as above we prove that this is a simple procedure which can be understood by considering a particular class of valuations of function fields of surfaces called plane valuations at infinity we also give algorithms to construct an unlimited number of sequences of the different existing types and so this paper provides the tools to know and use a new large set of codes

dispersive estimates for principally normal pseudodifferential operators the aim of these notes is to describe some recent re sults concerning dispersive estimates for principally normal pseu dodifferential operators the main motivation for this comes from unique continuation problems such estimates can be used to prove l q carleman inequalities which in turn yield unique continuation results for various partial differential operators with rough poten tials dispersive estimates are l q estimates for nonelliptic partial differ ential operators which are a consequence of the decay properties of their fundamental solutions these decay properties follow from spa tial spreading of the singularities of the solutions since solutions prop agate in directions conormal to the characteristic set of the operator this spreading can be related to nonzero curvatures of the characteristic set dispersive estimates for constant coefficient operators are closely related to the restriction theorem in harmonic analysis various types of dispersive estimates are known to be true for op erators such as the wave operator the schrodinger operator and the linear kdv see ginibre velo 4 keel tao 11 they have proved to be useful in the study of nonlinear problems as well as of problems with unbounded potentials more recently similar estimates have been obtained for wave op erators with variable coefficients beginning with the smooth case in kapitanskii 10 mockenhaupt seeger and sogge 14 up to operators with c 2 coefficients in smith 15 and tataru 21 23 similar results were obtained for the schrodinger equation in staffilani tataru 19 c 2 coefficients and in burq gerard tzvetkov 1 smooth coeffic ients in the variable coefficient elliptic case one should also mention sogge s l q

algebraic geometry codes the theory of error correcting codes derived from curves in an algebraic geometry was initiated by the work of goppa as generalizations of bose chaudhuri hocquenghem bch reed solomon rs and goppa codes the development of the theory has received intense consideration since that time and the purpose of the paper is to review this work elements of the theory of algebraic curves at a level sufficient to understand the code constructions and decoding algorithms are introduced code constructions from particular classes of curves including the klein quartic elliptic and hyperelliptic curves and hermitian curves are presented decoding algorithms for these classes of codes and others are considered the construction of classes of asymptotically good codes using modular curves is also discussed

efficient forney functions for decoding ag codes using a forney formula to solve for the error magnitudes in decoding algebraic geometric ag codes requires producing functions spl sigma sub p which are 0 at all but one point p of the variety of the error locator ideal the best such function is produced here in a reasonably efficient way from a lex grobner basis this lex basis is in turn produced efficiently from a weighted grevlex basis by using the fglm algorithm these two steps essentially complete the efficient decoding scheme based on a forney formula started in the author s previous work see ibid vol 42 p 1263 8 1996

modular curves and codes with a polynomial construction in this paper we study q ary codes arising from modular curves x 0 11l over gf p 2 and some binary codes attached to them all these codes have a polynomial construction and have good asymptotic parameters for certain values of the parameters the q ary codes with q p 2 geq 49 are better than the varshamov gilbert bound and the binary codes are better than the ziablov bounds

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

a code decomposition approach for decoding cyclic and algebraic geometric codes an approach based on code decomposition and partial transform for constructing algorithms for decoding cyclic codes and algebraic geometric ag codes is introduced a general decoding procedure applicable to arbitrary linear codes and their subfield subcodes is then developed in particular we have developed algorithms for error and erasure decoding of cyclic codes up to their actual minimum distance error and erasure decoding of ag codes up to their designed minimum distance and decoding of subfield subcodes of ag codes geometric bch codes moreover a generalization of geometric bch codes is introduced it is shown by an example that for a simple class of ag codes the so called one point codes this generalization brings a better estimate of their minimum distance it is also shown that the algorithms developed in this paper can be applied to decode these codes up to their estimated minimum distance

algebraic decoding using special divisors the basic algorithm for decoding of algebraic geometric codes corrects up to d sub c 1 2 g 2 errors where d sub c denotes the designed minimum distance of a code and g denotes the genus of a curve the modified algorithm improves on this but applies to a restricted class of codes an extended modified algorithm that applies to all codes is formulated it will correct up to d sub c 1 2 s errors s is called the clifford defect of a curve for curves with g or 1 this defect satisfies 0

fast decoding of codes from algebraic plane curves improvement to an earlier decoding algorithm for codes from algebraic geometry is presented for codes from an arbitrary regular plane curve the authors correct up to d 2 m sup 2 8 m 4 9 8 errors where d is the designed distance of the code and m is the degree of the curve the complexity of finding the error locator is o n sup 7 3 where n is the length of the code for codes from hermitian curves the complexity of finding the error values given the error locator is o n sup 2 and the same complexity can be obtained in the general case if only d 2 m sup 2 2 errors are corrected

the extension of classical dynamics for unstable hamiltonian systems abstract classical dynamics can be formulated in terms of trajectories or in terms of statistical ensembles whose time evolution is described by the liouville equation it is shown that for the class of large nonintegrable poincare systems lps the two descriptions are not equivalent practically all dynamical systems studied in statistical mechanics belong to this class the basic step is the extension of the liouville operator l h outside the hilbert space to functions singular in their fourier transformation this function space plays an important role in statistical mechanics as functions of the hamiltonian and therefore equilibrium distribution functions belong to this class physically these functions correspond to situations characterized by persistent interactions as they are realized in macroscopic physics persistent interactions are introduced in contrast to transient interactions studied in quantum mechanics by the s matrix approach asymptotically free in and out states the eigenvalue problem for the liouville operator l h is solved in this generalized function space for lps we obtain a complex irreducible spectral representation complex means that the eigenvalues are complex numbers whose imaginary parts refer to the various irreversible processes such as relaxation times diffusion irreducible means that these representations cannot be implemented by trajectory theory as the result the dynamical group of evolution splits into two semigroups moreover the laws of classical dynamics take a new form as they have to be formulated on the statistical level they express possibilities and no more certitudes two examples of typical classical systems i e interacting particles and anharmonic lattices are studied

polynomial shift states of a chaotic map we present a piecewise linear map of the unit interval in which the resolvent of the frobenius perron operator considered in a polynomial basis has an essen tial singularity at the origin associated with the essential singularity are polynomial shift states which are obtained from creation and annihilation operators in non self dual function spaces correlation functions of general polynomial observables have decay components that vanish in a finite time

algebraic geometric codes and multidimensional cyclic codes a unified theory and algorithms for decoding using grobner bases it is proved that any algebraic geometric ag code can be expressed as a cross section of an extended multidimensional cyclic code both ag codes and multidimensional cyclic codes are described by a unified theory of linear block codes defined over point sets ag codes are defined over the points of an algebraic curve and an m dimensional cyclic code is defined over the points in m dimensional space the power of the unified theory is in its description of decoding techniques using grobner bases in order to fit an ag code into this theory a change of coordinates must be applied to the curve over which the code is defined so that the curve is in special position for curves in special position all computations can be performed with polynomials and this also makes it possible to use the theory of grobner bases next a transform is defined for ag codes which generalizes the discrete fourier transform the transform is also related to a grobner basis and is useful in setting up the decoding problem in the decoding problem a key step is finding a grobner basis for an error locator ideal for ag codes multidimensional cyclic codes and indeed any cross section of an extended multidimensional cyclic code sakata s algorithm can be used to find linear recursion relations which hold on the syndrome array in this general context the authors give a self contained and simplified presentation of sakata s algorithm and present a general framework for decoding algorithms for this family of codes in which the use of sakata s algorithm is supplemented by a procedure for extending the syndrome array

dynamical entropy for systems with stochastic perturbation dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively since for such systems the kolmogorov sinai ks entropy diverges if the diameter of the partition tends to zero we analyze the difference between the total entropy of a noisy system and the entropy of the noise itself we show that this quantity is finite and non negative and we call it the dynamical entropy of the noisy system in the weak noise limit this quantity is conjectured to tend to the ks entropy of the deterministic system in particular we consider one dimensional systems with noise described by a finite dimensional kernel for which the frobenius perron operator can be represented by a finite matrix

unique continuation and absence of positive eigenvalues for schrodinger operators on considere l operateur de laplace sur r n et une fonction v x sur un sous ensemble connexe ouvert de r n on montre que si n 3 une propriete de prolongement unique est vraie pour v l loc n 2 r n dans l espace de sobolev h loc 2 q r n avec q 2n n 2

l p eigenfunction bounds for the hermite operator author s koch herbert tataru daniel abstract we obtain l p eigenfunction bounds for the harmonic oscillator in r n and for other related operators improving earlier results of thangavelu and karadzhov we also construct suitable counterexamples which show that our estimates are sharp

the art of signaling fifty years of coding theory in 1948 shannon developed fundamental limits on the efficiency of communication over noisy channels the coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero fifty years later codes for the gaussian channel have been discovered that come close to these fundamental limits there is now a substantial algebraic theory of error correcting codes with as many connections to mathematics as to engineering practice and the last 20 years have seen the construction of algebraic geometry codes that can be encoded and decoded in polynomial time and that beat the gilbert varshamov bound given the size of coding theory as a subject this review is of necessity a personal perspective and the focus is reliable communication and not source coding or cryptography the emphasis is on connecting coding theories for hamming and euclidean space and on future challenges specifically in data networking wireless communication and quantum information theory

summability of hermite expansions ii we study the summability of n dimensional hermite expansions where n 1 we prove that the critical index for the riesz summability is n 1 2 we also prove analogues of the fejer lebesgue theorem and riemann s localisation principle when the index of the riesz means is 3n 2 6

bochner riesz profile of anharmonic oscillator l d2dx2 x we investigate spectral multipliers bochner riesz means and the convergence of eigenfunction expansion corresponding to the schrodinger operator with anharmonic potential l d2dx2 x we show that the bochner riesz profile of the operator l completely coincides with such profile of the harmonic oscillator h d2dx2 x2 it is especially surprising because the bochner riesz profile of the one dimensional standard laplace operator is known to be essentially different and the case of operators h and l resembles more the profile of multidimensional laplace operators another surprising element of the main obtained result is the fact that the proof is not based on restriction type estimates and instead an entirely new perspective has to be developed to obtain the critical exponent for bochner riesz means convergence

normal forms for semilinear superquadratic quantum oscillators on the real line we consider nonlinear hamiltonian schrodinger equations with the superquadratic oscillator d2 dx2 x2p x m where p is an integer 2 is a polynomial of degree 2p such that inf x2p x 0 and m is a multiplier i e simultaneously diagonalized with d2 dx2 x2p x a previous article grebert et al 2009 11 contains the case p 1 in rd here we deal with d 1 but we authorize any superquadratic potential under generic conditions on m related to the nonresonance of the linear part such a hamiltonian equation admits in a neighborhood of the origin a birkhoff normal form at any order consequently we deduce long time existence for solutions of the above equation with small cauchy data in the high sobolev spaces as spectral analysis spectrum and eigenfunctions of the linear part is not explicit we use helffer robert and yajima zhang s results helffer and robert 1982 12 yajima and zhang 2001 21 to understand asymptotic behavior of both spectrum and eigenfunctions

simplified understanding and efficient decoding of a class of algebraic geometric codes an efficient decoding algorithm for algebraic geometric codes is presented for codes from a large class of irreducible plane curves including hermitian curves it can correct up to d 1 2 errors where d is the designed minimum distance with it we also obtain a proof of d sub min spl ges d without directly using the riemann roch theorem the algorithm consists of gaussian elimination on a specially arranged syndrome matrix followed by a novel majority voting scheme a fast implementation incorporating block hankel matrix techniques is obtained whose worst case running time is o mn sup 2 where m is the degree of the curve applications of our techniques to decoding other algebraic geometric codes to decoding bch codes to actual minimum distance and to two dimensional shift register synthesis are also presented

regular articlegeneralized strichartz inequalities for the wave equation we make a synthetic exposition of the generalized strichartz inequalities for the wave equation obtained in 6 together with the limiting cases recently obtained in 13 with as simple proofs as possible the proofs combine stationary phase estimates dyadic decompositions the hardy littlewood inequality or the holder inequality in time and abstract duality and interpolation arguments

effective construction of algebraic geometry codes we intend to show that algebraic geometry codes may be constructed easily using blowing up theory for any projective plane algebraic curve our paper is based on a paper by le brigand and risler 1988 we try to be as explicit as possible

error locator ideals for algebraic geometric codes the error locations for an algebraic geometric code c d mp are exactly the common zeros that is a projective variety v i of a set ideal i of error locator functions the paper gives a one dimensional berlekamp massey version of the feng rao 1993 algorithm for decoding algebraic geometric codes c d mp this produces a generating set for i as an ideal of size at most spl rho the smallest positive pole order at p of any function in l mp relative to any error of weight at most e

error locating pairs for cyclic codes a general decoding method for linear codes is investigated for cyclic codes the decoding consists of solving two systems of linear equations all but four binary cyclic codes of length less than 63 can so be decoded up to their actual distance a new family of codes is given for which the decoding needs only o n sup 2 operations

spectral decomposition of tent maps using symmetry considerations the spectral decomposition of the frobenius perron operator of maps composed of many tents is determined from symmetry considerations the eigenstates involve euler as well as bernoulli polynomials

the schrodinger equation along curves and the quantum harmonic oscillator abstract we consider the schrodinger equation associated to the harmonic oscillator i t u h u where h x 2 with initial data in the sobolev space h s r d with d 2 and s 3 8 we prove almost everywhere convergence of the solution to its initial data as time tends to zero which improves a result of yajima 1990 30 to this end we consider almost everywhere convergence for the free schrodinger along curves as it turns out these problems are more or less equivalent to that of the free schodinger equation along vertical lines our results are obtained by showing the equivalence of related maximal estimates

improved geometric goppa codes i basic theory in this paper we present a construction of improved geometric goppa codes which for the case of r 2g are often more efficient than the current geometric goppa codes derived from some varieties which include algebraic curves hyperplanes surfaces and other varieties for the special case of a plane in a three dimensional projective space the improved geometric goppa codes are reduced to linear multilevel codes for these improved geometric goppa codes a designed minimum distance can be easily determined and a decoding procedure which corrects up to half the designed minimum distance is also given

on the number of correctable errors of the feng rao decoding algorithm for ag codes in this correspondence we give geometric codes on one of the curves of garcia stichtenoth 1995 with the property that the true distances are bigger than feng rao distances moreover an example for which the feng rao algorithm cannot correct more than d sub fr 1 2 1 d sub true 1 2 errors is given

decoding of hermitian codes the key equation and efficient error evaluation this paper presents a generalization of the key equation to hermitian codes the syndrome is interpreted as a power series and the product of this power series with a locator polynomial gives the error evaluator polynomial the computation of the evaluator polynomial may be done iteratively using a modified version of a previously published computationally efficient algorithm for computing locator polynomials

weighted restriction type estimates for grushin operators and application to spectral multipliers and bochner riesz summability we prove weighted restriction type estimates for grushin operators these estimates are then used to prove sharp spectral multiplier theorems as well as bochner riesz summability results with sharp exponent

decoding algebraic geometric codes up to the designed minimum distance a simple decoding procedure for algebraic geometric codes c sub omega d g is presented this decoding procedure is a generalization of peterson s decoding procedure for the bch codes it can be used to correct any d 1 2 or fewer errors with complexity o n sup 3 where d is the designed minimum distance of the algebraic geometric code and n is the codelength

intrinsic irreversibility and the validity of the kinetic description of chaotic systems irreversibility for a class of chaotic systems is seen to be an exact consequence of the dynamics through the use of a generalized spectral representation of the time evolution operator of probability densities the generalized representation is valid for one dimensional systems when the initial probability density satisfies certain physical conditions of smoothness the formalism is first applied to the one dimensional multi bernoulli map which is a simple map displaying deterministic diffusion the two dimensional invertible baker and multibaker transformations are then studied and the physical conditions determining which discrete spectral values are realized are seen to depend on the smoothness of both the density as well as the observable considered the generalized representation is constructed using a resolvent formalism the eigenstates of the diffusive systems are seen to be of a fractal nature

on the decoding of algebraic geometric codes this paper provides a survey of the existing literature on the decoding of algebraic geometric codes definitions theorems and cross references will be given we show what has been done discuss what still has to be done and pose some open problems

on automorphisms of geometric goppa codes the algebraic geometric codes which were introduced by v d goppa in 1977 7 8 we call them geometric goppa codes can be used to prove the existence of long linear codes which are better than the gilbert varshamov bound there are many good expositions of this important result ll 13 15 181 in that context one has to consider curves of high genus which admit many rational points on the other hand new interesting codes were found among the geometric goppa codes by looking at special curves e g elliptic curves hermite curves or artin schreier curves 3 5 8 13 16 171 in coding theory one is often interested in codes admitting many automorphisms e g cyclic codes goppa ls already observed that automorphisms of the underlying algebraic curve induce automorphisms of the associated geometric codes since goppa s paper is not quite correct we shall give a detailed exposition of this connexion in section 2 section 3 is the central part of our paper it is devoted to the case of geometric goppa codes associated with a curve of genus 0 so called rational goppa codes these codes are particularly interesting since many well known classes of codes e g bch codes classical goppa codes reed solomon codes cauchy codes 6 can be represented as rational goppa codes or subfield subcodes of them cl our main result is theorem 3 1 it states that in general all automorphisms of a rational goppa code are induced by automorphisms of the rational curve i e by projective transformations of the projective line

generalized reed solomon codes from algebraic geometry a few years ago tsfasman em et al using results from algebraic geometry showed that there is a sequence of codes which are generalizations of goppa codes and which exceed the gilbert varshamov bound we show that a similar sequence of codes in fact the duals of the previous codes can be found by generalizing the construction of reed solomon codes our approach has the advantage that it uses less complicated concepts from algebraic geometry

reducibility of quantum harmonic oscillator on rd with differential and quasi periodic in time potential abstract we improve the results in the following two aspects by grebert and paturel in 28 and prove that a linear schrodinger equation on r d with harmonic potential x 2 and small t quasiperiodic potential as i u t u x 2 u e v t x u 0 t x r r d reduces to an autonomous system for most values of the frequency vector r n firstly we assume the potential v is only in c t n h s r d with large enough secondly the condition on s is better than 28 as a consequence any solution of such a linear pde is almost periodic in time and remains bounded in some suitable sobolev norms

decoding geometric goppa codes using an extra place decoding geometric goppa codes can be reduced to solving the key congruence of a received word in an affine ring if the codelength is smaller than the number of rational points on the curve then this method can correct up to 1 2 d l 2 s errors where d is the designed minimum distance of the code and s is the clifford defect the affine ring with respect to a place p is the set of all rational functions which have no poles except at p and it is somehow similar to a polynomial ring for a special kind of geometric goppa code namely c sub omega d mp the decoding algorithm is reduced to solving the key equation in the affine ring which can be carried out by the subresultant sequence in the affine ring with complexity o n sup 3 where n is the length of codewords

on the long time behaviour of ensembles in a model of deterministic diffusion we describe a large model system based on the baker transformation where deterministic diffusion occurs the model is similar to one recently considered by gaspard and by hasegawa and driebe we point out the close relationship between this system and a simple random walk and analyse the evolution with time of ensembles in the system using the resolvent based version of the subdynamics formalism developed by prigogine and his collaborators we obtain an exact and rigorous description of the long time behaviour of ensembles including the irreversible approach to equilibrium for the case where the system has finite size we also consider the thermodynamic limit where the size of the system becomes infinite and derive a description of the long time behaviour in this case where correlations decay non exponentially with time

achieving the designed error capacity in decoding algebraic geometric codes a decoding algorithm for codes arising from algebraic curves explicitly constructable by goppa s construction is presented any configuration up to the greatest integer less than or equal to d 1 2 errors is corrected by the algorithm whenever d or 6g where d is the designed minimum distance of the code and g is the genus of the curve the algorithm s complexity is at most o d sup 2 n where n denotes the length of the code application to hermitian codes and connections with well known algorithms are explained

a note on hermitian codes over gf q sup 2 properties of codes defined by goppa s algebraic geometric method using hermitian curves are discussed generator and parity check matrices for such codes are derived explicitly results concerning the minimum distance and the weight distribution of the codes are provided

on the decoding of algebraic geometric codes over f sub q for q or 16 it is proved that for algebraic geometric codes on a curve over f sub q for q or 37 or on a curve of sufficiently large genus over f sub q for q or 16 there exists a polynomial decoding algorithm up to d 1 2 errors d being the designed minimum distance

on random hermite series we study integrability and continuity properties of random series of hermite functions we get optimal results which are analogues to classical results concerning fourier series like the paley zygmund or the salem zygmund theorems we also consider the case of series of radial hermite functions which are not so well behaved in this context we prove some l p bounds of radial hermite functions which are optimal when p is large