a 2 adic approach to the analysis of cyclic codes this paper describes how 2 adic numbers can be used to analyze the structure of binary cyclic codes and of cyclic codes defined over z sub 2 a a spl ges 2 the ring of integers modulo 2 sup a it provides a 2 adic proof of a theorem of mceliece that characterizes the possible hamming weights that can appear in a binary cyclic code a generalization of this theorem is derived that applies to cyclic codes over z sub 2 a that are obtained from binary cyclic codes by a sequence of hensel lifts this generalization characterizes the number of times a residue modulo 2 sup a appears as a component of an arbitrary codeword in the cyclic code the limit of the sequence of hensel lifts is a universal code defined over the 2 adic integers this code was first introduced by calderbank and sloane 1995 and is the main subject of this paper binary cyclic codes and cyclic codes over z sub 2 a are obtained from these universal codes by reduction modulo some power of 2 a special case of particular interest is cyclic codes over z sub 4 that are obtained from binary cyclic codes by means of a single hensel lift the binary images of such codes under the gray isometry include the kerdock preparata and delsart goethals codes these are nonlinear binary codes that contain more codewords than any linear code presently known fundamental understanding of the composition of codewords in cyclic codes over z sub 4 is central to the search for more families of optimal codes this paper also constructs even unimodular lattices from the hensel lift of extended binary cyclic codes that are self dual with all hamming weights divisible by 4 the leech lattice arises in this way as do extremal lattices in dimensions 32 through 48

binary kloosterman sums modulo 256 and coefficients of the characteristic polynomial kloosterman sums are exponential sums on finite fields that have important applications in cryptography and coding theory we use stickelberger s theorem and the gross koblitz formula to determine the value of the binary kloosterman sum at a modulo 64 modulo 128 and modulo 256 in terms of coefficients of the characteristic polynomial of a

on the analytical summation of fourier series and its relation to the asymptotic behaviour of fourier transforms forms of the poisson summation formula psf appropriate for the summation of semi infinite and infinite fourier series are derived application of these results to the acceleration of convergence of various types of series with monotonically decreasing coefficient functions yields transformed series with terms that decay either exponentially or with the inverse first or second power of the index variable these two very different convergence properties are explained in terms of the asymptotic properties of the relevant fourier transforms which are in turn related to the power series expansions of the summand functions in the original fourier series the result is that the poisson summation formula works best for fourier cosine series in which the summand functions are expansible in even powers and for fourier sine series in which the summand functions have power series with odd powers here application of the psf produces series of terms that decay exponentially with increasing argument x in contrast application of the semi infinite version of the psf to fourier cosine series of terms with odd power expansions or to fourier sine series of terms with even power expansions yields transformed series involving functions of the form which decay approximately as if the summand function in the fourier series has a power series with both even and odd powers the transformed series involves sine and cosine integral functions which decay approximately as fourier series of these last three types in general require additional acceleration for example by application of the kummer transformation

a performance comparison of associated legendre projections the associated legendre projection provides a means for accelerating the dynamical core of a global weather or climate model therefore the goal is to determine the fastest possible projection algorithm of which this paper compares six the standard method which computes the projection using a forward and backward associated legendre transform the direct method which uses a single projection matrix when this approach results in fewer operations the fast multipole method the weighted orthogonal complement method the seminaive method and a projection based on transforms proposed by mohlenkamp timing results indicate that all the projections behave like o n3 algorithms up to at least n 200 spectral truncation for this range of resolutions the weighted orthogonal complement has the lowest operation count best cache utilization and best overall timings

relevance of memory in minority games by considering diffusion on de bruijn graphs we study in detail the dynamics of the histories in the minority game a model of competition between adaptative agents such graphs describe the structure of the temporal evolution of m bit strings each node standing for a given string i e a history in the minority game we show that the frequency of visit of each history is not given by 1 2 m in the limit of large m when the transition probabilities are biased consequently all quantities of the model do significantly depend on whether the histories are real or uniformly and randomly sampled we expose a self consistent theory of the case of real histories which turns out to be in very good agreement with numerical simulations

continuous time dynamics of the thermal minority game we study the continuous time dynamics of the thermal minority game we find that the dynamical equations of the model reduce to a set of stochastic differential equations for an interacting disordered system with nontrivial random diffusion this is the simplest microscopic description which accounts for all the features of the system within this framework we study the phase structure of the model and find that its macroscopic properties strongly depend on the initial conditions

on the minority game analytical and numerical studies abstract we investigate further several properties of the minority game we have recently introduced we explain the origin of the phase transition and give an analytical expression of 2 n in the n 2 m region the ability of the players to learn a given payoff is also analyzed and we show that the darwinian evolution process tends to a self organized state in particular the lifetime distribution is a power law with exponent 2 furthermore we study the influence of identical players on their gain and on the systems performance finally we show that large brains always take advantage of small brains

a test of a modified algorithm for computing spherical harmonic coefficients using an fft this note describes the results of testing a modified method based on an algorithm by dilts 1 for computing the spherical harmonic expansion coefficients for a function on a sphere using a two dimensional fft the new version of the dilts program eliminates problems of overflow and large storage requirements encountered when harmonic degree values of l 16 are computed thus allowing the determination of coefficients for high l values l 300 however results from timing tests indicate that the new version of the dilts program is not practical for computing spherical harmonic expansion coefficients for large l due to the very long cpu times required for execution of the program the most effective technique remains the one pioneered by brown 2

a united approach to accelerating trigonometric expansions abstract the fourier series of a smooth function on a compact interval usually has slow convergence due to the fact that the periodic extension of the function has jumps at the interval endpoints for various symmetry conditions polynomial interpolation methods have been developed for performing a boundary correction the resulting variants of krylov approximants are a sum of a correction polynomial and a fourier sum of the corrected function 1 8 in this paper we review these methods and derive estimates in the maximum norm we further show that derivatives of the krylov approximants are again krylov approximants of derivatives of the considered function this enables us to give a unified treatment of the problem of simultaneous approximation

irrelevance of memory in the minority game by means of extensive numerical simulations we show that all the distinctive features of the minority game introduced by challet and zhang physica a 256 514 1998 are completely independent of the memory of the agents the only crucial requirement is that all the individuals must possess the same information irrespective of whether this information is true or false

an integral transform approach for a mixed boundary problem involving a flowing partially penetrating well with infinitesimal well skin 1 a flowing partially penetrating well with infinitesimal well skin is a mixed boundary because a cauchy condition is prescribed along the screen length and a neumann condition of no flux is stipulated over the remaining unscreened part an analytical approach based on the integral transform technique is developed to determine the laplace domain solution for such a mixed boundary problem in a confined aquifer of finite thickness first the mixed boundary is changed into a homogeneous neumann boundary by substituting the cauchy condition with a neumann condition in terms of well bore flux that varies along the screen length and is time dependent despite the well bore flux being unknown a priori the modified model containing this homogeneous neumann boundary can be solved with the laplace and the finite fourier cosine transforms to determine well bore flux screen length is discretized into a finite number of segments to which the cauchy condition is reinstated this reinstatement also restores the relation between the original model and the solutions obtained for a given time the numerical inversion of the laplace domain solution yields the drawdown distributions well bore flux and the well discharge this analytical approach provides an alternative for dealing with the mixed boundary problems especially when aquifer thickness is assumed to be finite

efficient calculation of the free space periodic green s function electromagnetic scattering from periodic structures can be formulated in terms of an integral equation that has as its kernel a periodic green s function the periodic green s function can be derived as a response to an array of line point sources spatial domain or as a response to series of current sheets spectral domain these responses are a fourier transform pair and are slowly convergent summations the convergence problems in each domain arise from unavoidable singularities in the reciprocal domain a method is discussed to overcome the slow convergence by using the poisson summation formula and summing in a combination of spectral and spatial domains a parameter study is performed to determine an optimum way to weigh the combination of domains simple examples of scattering from a one dimensional array of strips and two dimensional array of plates are used to illustrate the concepts

mesoscopic approach to minority games in herd regime abstract we study minority games in efficient regime by incorporating the utility function and aggregating agents with similar strategies we develop an effective mesoscale notion of the state of the game using this approach the game can be represented as a markov process with substantially reduced number of states with explicitly computable probabilities for any payoff the finiteness of the number of states is proved interesting features of an extensive random variable called aggregated demand viz its strong inhomogeneity and presence of patterns in time can be easily interpreted using markov theory and quenched disorder approach we can explain important macroscopic characteristics of the game behavior of variance per capita and predictability of the aggregated demand we prove that in the case of linear payoff many attractors in the state space are possible

crowd anticrowd theory of the minority game abstract the minority game is a simple yet highly non trivial agent based model for a complex adaptive system here we provide an explanation of the game s fluctuations which is both intuitive and quantitative and which applies over the entire parameter range of interest the physical idea behind our theory is to describe the interplay between crowds of like minded agents and their anticorrelated partners anticrowds

continuum time limit and stationary states of the minority game we discuss in detail the derivation of stochastic differential equations for the continuum time limit of the minority game we show that all properties of the minority game can be understood by a careful theoretical analysis of such equations in particular i we confirm that the stationary state properties are given by the ground state configurations of a disordered soft spin system ii we derive the full stationary state distribution iii we characterize the dependence on initial conditions in the symmetric phase and iv we clarify the behavior of the system as a function of the learning rate this leaves us with a complete and coherent picture of the collective behavior of the minority game strikingly we find that the temperaturelike parameter which is introduced in the choice behavior of individual agents turns out to play the role at the collective level of the inverse of a thermodynamic temperature

convergence acceleration of modified fourier series in one or more dimensions modified fourier series have recently been introduced as an adjustment of classical fourier series for the approximation of nonperiodic functions defined on d variate cubes such approximations offer a number of advantages including uniform convergence however like fourier series the rate of convergence is typically slow in this paper we extend eckhoff s method to the convergence acceleration of multivariate modified fourier series by suitable augmentation of the approximation basis we demonstrate how to increase the convergence rate to an arbitrary algebraic order moreover we illustrate how numerical stability of the method can be improved by utilising appropriate auxiliary functions in the univariate setting it is known that eckhoff s method exhibits an auto correction phenomenon we extend this result to the multivariate case finally we demonstrate how a significant reduction in the number of approximation coefficients can be achieved by using a hyperbolic cross index set

associated legendre functions on the cut abstract algorithms and a code are described for the computation of the associated legendre functions pvu cos pv u cos qvu cos and the normalized legendre polynomial p v u cos in the ranges 0 1 2 0 1 2 1 2 v the algorithms are based on power series expansions and recurrence relations they are executed in extended range arithmetic thereby admitting very extensive ranges of and v without causing overflow or underflow

acceleration of algebraically converging fourier series when the coefficients have series in powers of 1 n if the coefficients in a fourier cosine series f x f email protected n 0 a ncos nx decrease as a small negative power of n then one may need millions of terms to sum the series to high accuracy we show that if the a n are known analytically and have a power series in 1 n then it is straightforward to approximate f x as a series of what we shall the lanczos krylov lk functions we describe the similar methodology for sine series general fourier series are merely the sum of a cosine series with a sine series and thus are implicitly handled too for cosine coefficients that involve only even powers of n and sine coefficients that are functions of odd powers of n the lk functions may be expressed in terms of bernoulli polynomials the lk functions for cosine coefficients involving odd powers of n and for sine coefficients in even powers of n are not known explicitly these are also known as clausen functions we provide rapidly convergent series to compute these clausen functions to high accuracy our method includes the endpoint subtraction ideas of lanczos and krylov but is more general the sum n 1 1 n 1 1 email protected cos nx where l 0 is a constant arises in phase transitions in absorbed monolayers on metal surfaces it is easily summed by our method which correctly incorporates the logarithmic singularities at x email protected

generating functional analysis of the dynamics of the batch minority game with random external information we study the dynamics of the batch minority game with random external information using generating functional techniques introduced by de dominicis the relevant control parameter in this model is the ratio a5 p n of the number p of possible values for the external information over the number n of trading agents in the limit n we calculate the location a c of the phase transition signaling the onset of anomalous response and solve the statics for a a c exactly the temporal correlations in global market fluctuations turn out not to decay to zero for infinitely widely separated times for a a c the stationary state is shown to be nonunique for a 0 we analyze our equations in leading order in a and find asymptotic solutions with diverging volatility s5o a 21 2 as regularly observed in simulations but also asymptotic solutions with vanishing volatility s5o a 1 2 the former however are shown to emerge only if the agents initial strategy valuations are below a specific critical value

numerical comparisons of nonlinear convergence accelerators as part of a continuing program of numerical tests of convergence accelerators we have compared the iterated aitken s a2 method wynn s algorithm brezinski s 0 algorithm and levin s u transform on a broad range of test problems linearly convergence alternating monotone and irregular sign series logarithmically convergent series power method and bernoulli method sequences alternating and monotone asymptotic series and some perturba tion series arising in applications in each category either the algorithm or the u transform gives the best results of the four methods tested in some cases differences among methods are slight and in others they are quite striking 1 introduction we reported in 19 on a numerical test program that compared eleven different scalar convergence accelerators on a broad range of infinite series problems in an attempt to find an across the board accelerator that would handle both linear and logarithmic convergence and both alternating and monotone series or sequences the best candidates to emerge from that study were levin s u transform 14 and brezinski s 0 algorithm 6 we also provided some theoretical support for that conclusion in terms of the wide range of series summed exactly by these methods and by proving that 02 and the levin transforms all accelerate linear convergence the other levin transforms t and v produced essentially equivalent numerical results to u except that t fails on logarithmic convergence we have continued the test program started with 19 to include monotone linearly convergent series series with irregular sign patterns specifically fourier series and sequences whose terms do not depend explicitly on the previous term and or the index e g bernoulli s method since convergence accelerators are often known to transform divergent series to convergent ones we have also included in our test program divergent power series asymptotic series and a few perturbation series the methods considered in this paper are the u and 0 algorithms noted above and

diophantine equations with binomial coefficients and perturbations of symmetric boolean functions this paper presents a study of perturbations of symmetric boolean functions in particular it establishes a connection between exponential sums of these perturbations and diophantine equations of the form sum l 0 n binom n l x l 0 where x j belongs to some fixed bounded subset gamma of mathbb z the concepts of trivially balanced symmetric boolean function and sporadic balanced boolean function are extended to this type of perturbations an observation made by canteaut and videau for symmetric boolean functions of fixed degree is extended to be specific it is proved that excluding the trivial cases balanced perturbations of fixed degree do not exist when the number of variables grows some sporadic balanced perturbations are presented finally a beautiful but unexpected identity between exponential sums for perturbations of two different symmetric boolean functions is also included in this work

statistical properties of the attendance time series in the minority game abstract we study the statistical properties of the attendance time series corresponding to the number of agents making a particular decision in the minority game mg we focus on the analysis of the probability distribution and the autocorrelation function of the attendance over a time interval in the efficient phase of the game in this regime both the probability distribution and the autocorrelation function are shown to have similar behaviour for time differences corresponding to multiples of 2 2m which is twice the number of possible history bit strings in a mg with agents making decisions based on the most recent m outcomes of the game

crowd effects and volatility in markets with competing agents abstract we present analytic and numerical results for two models namely the minority model and the bar attendance model which offer simple paradigms for a competitive marketplace both models feature heterogeneous agents with bounded rationality who act using inductive reasoning we find that the effects of crowding are crucial to the understanding of the macroscopic fluctuations or volatility in the resulting dynamics of these systems

on the efficient parallel computation of legendre transforms in this article we discuss a parallel implementation of efficient algorithms for computation of legendre polynomial transforms and other orthogonal polynomial transforms we develop an approach to the driscoll healy algorithm using polynomial arithmetic and present experimental results on the accuracy efficiency and scalability of our implementation the algorithms were implemented in ansi c using the bsplib communications library we also present a new algorithm for computing the cosine transform of two vectors at the same time

computational harmonic analysis for tensor fields on the two sphere in this paper we describe algorithms for the numerical computation of fourier transforms of tensor fields on the two sphere s2 these algorithms reduce the computation of an expansion on tensor spherical harmonics to expansions in scalar spherical harmonics and hence can take advantage of recent improvements in the efficiency of computation of scalar spherical harmonic transforms

computation of spherical harmonic expansion coefficients via fft s abstract a method for numerically expanding an arbitrary function on the sphere in a series of spherical harmonics which makes use of the speed of a fast fourier transform is described discussions of the operation count storage requirements accuracy and an algebraic and a numerical example are included a comparison with straightforward integration is made throughout also a new method for evaluating the spherical harmonics is discussed

a novel sampling theorem on the sphere we develop a novel sampling theorem on the sphere and corresponding fast algorithms by associating the sphere with the torus through a periodic extension the fundamental property of any sampling theorem is the number of samples required to represent a band limited signal to represent exactly a signal on the sphere band limited at l all sampling theorems on the sphere require o l2 samples however our sampling theorem requires less than half the number of samples of other equiangular sampling theorems on the sphere and an asymptotically identical but smaller number of samples than the gauss legendre sampling theorem the complexity of our algorithms scale as o l3 however the continual use of fast fourier transforms reduces the constant prefactor associated with the asymptotic scaling considerably resulting in algorithms that are fast furthermore we do not require any precomputation and our algorithms apply to both scalar and spin functions on the sphere without any change in computational complexity or computation time we make our implementation of these algorithms available publicly and perform numerical experiments demonstrating their speed and accuracy up to very high band limits finally we highlight the advantages of our sampling theorem in the context of potential applications notably in the field of compressive sampling

convergence acceleration of fourier series by analytical and numerical application of poisson s formula a form of the poisson summation formula appropriate for the transformation of infinite fourier series is derived and shown to be applicable to slowly converging series arising in the solution of laplace s equation in a rectangular electrochemical cell the result is a rapidly convergent fourier series which is particularly useful in giving the potential close to the electrode surface results obtained from the analytical expression are found to be in close agreement with those resulting from numerical evaluation of the fourier integrals using simpson s rule and the euler transformation and direct summation by using goertzel s 1958 algorithm

tight bounds for chevalley warning ax katz type estimates with improved applications following the work of chevalley and warning ax obtained a bound on the p divisibility of exponential sums involving multi variable polynomials of fixed degree d over a finite field of characteristic p this bound was subsequently improved by katz more recently moreno and moreno and adolphson and sperber derived bounds that in many instances improved upon the ax katz result here we derive a tight bound on the p divisibility of the exponential sums while exact computation of this bound requires the solution of a system of modular equations approximations are provided which in several classes of examples improve on the results of chevalley and warning ax and katz adolphson and sperber and moreno and moreno all of the above results readily translate into bounds on the p divisibility of the number of zeros of multi variable polynomials an important consequence of one of our main results is a method to find classes of examples for which bounds on divisibility of the number of solutions of a system of polynomial equations over finite fields are tight in particular we give classes of examples for which the moreno moreno bound is tight it is important to note that we have also found applications of our results to coding theory computation of the covering radius of certain codes and to waring s problem over finite fields these will be described elsewhere

statistical mechanics of systems with heterogeneous agents minority games we study analytically a simple game theoretical model of heterogeneous interacting agents we show that the stationary state of the system is described by the ground state of a disordered spin model which is exactly solvable within the simple replica symmetric ansatz such a stationary state differs from the nash equilibrium where each agent maximizes her own utility the latter turns out to be characterized by a replica symmetry broken structure numerical results fully agree with our analytical findings

a unified approach for predicting sound radiation from baffled rectangular plates with arbitrary boundary conditions this paper discusses sound radiation from a baffled rectangular plate with each of its edges arbitrarily supported in the form of elastic restraints the plate displacement function is universally expressed as a 2 d fourier cosine series supplemented by several 1 d series the unknown fourier expansion coefficients are then determined by using the rayleigh ritz procedure once the vibration field is solved the displacement function is further simplified to a single standard 2 d fourier cosine series in the subsequent acoustic analysis thus the sound radiation from a rectangular plate can always be obtained from the radiation resistance matrix for an invariant set of cosine functions regardless of its actual dimensions and boundary conditions further this radiation resistance matrix unlike the traditional ones for modal functions only needs to be calculated once for all plates with the same aspect ratio in order to determine the radiation resistance matrix effectively an analytical formula is derived in the form of a power series of the non dimensional acoustic wavenumber the formula is mathematically valid and accurate for any wavenumber several numerical examples are presented to validate the formulations and show the effect of the boundary conditions on the radiation behavior of planar sources

deterministic dynamics in the minority game the minority game mg behaves as a stochastically disturbed deterministic system due to the coin toss invoked to resolve tied strategies averaging over this stochasticity yields a description of the mg s deterministic dynamics via mapping equations for the strategy score and global information the strategy score map contains both restoring force and bias terms whose magnitudes depend on the game s quenched disorder approximate analytical expressions are obtained and the effect of market impact is discussed the global information map represents a trajectory on a de bruijn graph for small quenched disorder a eulerian trail represents a stable attractor it is shown analytically how antipersistence arises the response to perturbations and different initial conditions is also discussed