fast evaluation of system matrices w r t multi tree collections of tensor product refinable basis functions an algorithm is presented that for a local bilinear form evaluates in linear complexity the application of the stiffness matrix w r t a collection of tensor product multiscale basis functions assuming that this collection has a multi tree structure it generalizes an algorithm for sparse grid index sets r balder ch zenger the solution of multidimensional real helmholtz equations on sparse grids siam j sci comput 17 3 1996 631 646 and it finds its application in adaptive tensor product approximation methods

designing structured tight frames via an alternating projection method tight frames also known as general welch bound equality sequences generalize orthonormal systems numerous applications including communications coding and sparse approximation require finite dimensional tight frames that possess additional structural properties this paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems ieps which includes the frame design problem to apply this method one needs only to solve a matrix nearness problem that arises naturally from the design specifications therefore it is the fast and easy to develop versions of the algorithm that target new design problems alternating projection will often succeed even if algebraic constructions are unavailable to demonstrate that alternating projection is an effective tool for frame design the paper studies some important structural properties in detail first it addresses the most basic design problem constructing tight frames with prescribed vector norms then it discusses equiangular tight frames which are natural dictionaries for sparse approximation finally it examines tight frames whose individual vectors have low peak to average power ratio par which is a valuable property for code division multiple access cdma applications numerical experiments show that the proposed algorithm succeeds in each of these three cases the appendices investigate the convergence properties of the algorithm

adaptive wavelet schemes for elliptic problems implementation and numerical experiments recently an adaptive wavelet scheme could be proved to be asymptotically optimal for a wide class of elliptic operator equations in the sense that the error achieved by an adaptive approximate solution behaves asymptotically like the smallest possible error that can be realized by any linear combination of the corresponding number of wavelets on one hand the results are purely asymptotic on the other hand the analysis suggests new algorithmic ingredients for which no prototypes seem to exist yet it is therefore the objective of this paper to develop suitable data structures for the new algorithmic components and to obtain a quantitative validation of the theoretical results we briefly review first the main theoretical facts describe the main ingredients of the algorithm highlight the essential data structures and illustrate the results by one and two dimensional numerical examples including comparisons with an adaptive finite element scheme

an optimal adaptive wavelet method for first order system least squares in this paper it is shown that any well posed 2nd order pde can be reformulated as a well posed first order least squares system this system will be solved by an adaptive wavelet solver in optimal computational complexity the applications that are considered are second order elliptic pdes with general inhomogeneous boundary conditions and the stationary navier stokes equations

optimum sequence multisets for synchronous code division multiple access channels it is shown that the sum capacity of the symbol synchronous code division multiple access channel with equal average input energy constraints is maximized precisely by those spreading sequence multisets that meet welch s lower bound on total squared correlation it is further shown that the symmetric capacity of the channel determined by these same sequence multisets is equal to the sum capacity

adaptive solution of operator equations using wavelet frames in adaptive wavelet methods ii beyond the elliptic case of cohen dahmen and devore found comput math 2 2002 pp 203 245 an adaptive method has been developed for solving general operator equations using a riesz basis of wavelet type for the energy space the operator equation is transformed into an equivalent matrix vector system this system is solved iteratively where the application of the infinite stiffness matrix is replaced by an adaptive approximation assuming that the stiffness matrix is sufficiently compressible i e that it can be sufficiently well approximated by sparse matrices it was proved that the adaptive method has optimal computational complexity in the sense that it converges with the same rate as the best n term approximation for the solution assuming that the latter would be explicitly available the condition concerning compressibility requires that dependent on their order the wavelets have sufficiently many vanishing moments and that they be sufficiently smooth however except on tensor product domains wavelets that satisfy this smoothness requirement are not easy to construct in this paper we write the domain or manifold on which the operator equation is posed as an overlapping union of subdomains each of them being the image under a smooth parametrization of the hypercube by lifting wavelets on the hypercube to the subdomains we obtain a em frame for the energy space with this frame the operator equation is transformed into a matrix vector system after which this system is solved iteratively by an adaptive method similar to the one from the work of cohen dahmen and devore with this approach frame elements that have sufficiently many vanishing moments and are sufficiently smooth something needed for the compressibility are easily constructed by handling additional difficulties due to the fact that a frame gives rise to an underdetermined matrix vector system we prove that this adaptive method has optimal computational complexity

iterative construction of optimum signature sequence sets in synchronous cdma systems optimum signature sequence sets that maximize the capacity of single cell synchronous code division multiple access cdma systems have been identified optimum signature sequences minimize the total squared correlation tsc they form a set of orthogonal sequences if the number of users is less than or equal to the processing gain and a set of welch 1994 bound equality wbe sequences otherwise we present an algorithm where users update their transmitter signature sequences sequentially in a distributed fashion by using available receiver measurements we show that each update decreases the tsc of the set and produces better signature sequence sets progressively we prove that the algorithm converges to a set of orthogonal signature sequences when the number of users is less than or equal to the processing gain we observe and conjecture that the algorithm converges to a wbe set when the number of users is greater than the processing gain at each step the algorithm replaces one signature sequence from the set with the normalized minimum mean squared error mmse receiver corresponding to that signature sequence since the mmse filter can be obtained by a distributed algorithm for each user the proposed algorithm is amenable to distributed implementation

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

comment on tests of scaling and universality of the distributions of trade size and share volume evidence from three distinct markets in a recent publication plerou and stanley phys rev e 76 046109 2007 use the meerschaert scheffler estimator to verify the inverse half cubic law of trade size distributions we show that this procedure systematically underestimates these tail exponents

adaptive wavelet schemes for nonlinear variational problems we develop and analyze wavelet based adaptive schemes for nonlinear variational problems we derive estimates for convergence rates and corresponding work counts that turn out to be asymptotically optimal our approach is based on a new paradigm that has been put forward recently for a class of linear problems the original problem is transformed first into an equivalent one which is well posed in the euclidean metric ell 2 then conceptually one seeks iteration schemes for the infinite dimensional problem that exhibits at least a fixed error reduction per step this iteration is then realized approximately through an adaptive application of the involved operators with suitable dynamically updated accuracy tolerances the main conceptual ingredients center around nonlinear tree approximation and the sparse evaluation of nonlinear mappings of wavelet expansions we prove asymptotically optimal complexity for adaptive realizations of first order iterations and of newton s method

an optimal adaptive wavelet method without coarsening of the iterands in this paper an adaptive wavelet method for solving linear operator equations is constructed that is a modification of the method from math comp 70 2001 pp 27 75 by cohen dahmen and devore in the sense that there is no recurrent coarsening of the iterands despite this it will be shown that the method has optimal computational complexity numerical results for a simple model problem indicate that the new method is more efficient than an existing alternative adaptive wavelet method

the maximum squared correlation sum capacity and total asymptotic efficiency of minimum total squared correlation binary signature sets the total squared correlation tsc maximum squared correlation msc sum capacity c sub sum and total asymptotic efficiency tae of underloaded signature sets as well as the tsc and c sub sum of overloaded signature sets are metrics that are optimized simultaneously over the real complex field in this present work closed form expressions are derived for the msc c sub sum and tae of minimum tsc binary signature sets the expressions disprove the general equivalence of these performance metrics over the binary field and establish conditions on the number of signatures and signature length under which simultaneous optimization can or cannot be possible the sum capacity loss of the recently designed minimum tsc binary sets is found to be rather negligible in comparison with minimum tsc real complex valued welch bound equality sets

rank 2 optimal adaptive design of binary spreading codes over the real complex field the spreading code that maximizes the signal to interference plus noise ratio sinr at the output of the maximum sinr linear filter is the minimum eigenvalue eigenvector of the interference autocovariance matrix in the context of binary spreading codes the maximization problem is np hard with complexity exponential in the code length a new method for the optimization of binary spreading codes under a rank 2 approximation of the inverse interference autocovariance matrix is presented where the rank 2 optimal binary code is obtained in lower than quadratic complexity significant sinr performance improvement is demonstrated over the common binary hard limited eigenvector design which is shown to be equivalent to the rank 1 optimal solution

computation of differential operators in wavelet coordinates in found comput math 2 2002 pp 203 245 cohen dahmen and devore proposed an adaptive wavelet algorithm for solving general operator equations assuming that the operator defines a boundedly invertible mapping between a hilbert space and its dual and that a riesz basis of wavelet type for this hilbert space is available the operator equation is transformed into an equivalent well posed infinite matrix vector system this system is solved by an iterative method where each application of the infinite stiffness matrix is replaced by an adaptive approximation it was shown that if the errors of the best linear combinations from the wavelet basis with n terms are o n s for some s 0 which is determined by the besov regularity of the solution and the order of the wavelet basis then approximations yielded by the adaptive method with n terms also have errors of o n s moreover their computation takes only 0 n operations provided s s assuming that each entry of the stiffness matrix is exactly available at unit cost generally these entries have to be approximated using numerical quadrature in this paper restricting ourselves to differential operators we develop a numerical integration scheme that computes these entries giving an additional error that is consistent with the approximation error whereas in each column the average computational cost per entry is 0 1 as a consequence we can conclude that the adaptive wavelet algorithm has optimal computational complexity

cdma codeword optimization interference avoidance and convergence via class warfare interference avoidance has been shown to reduce total square correlation tsc for given ensembles of user signature waveforms codewords in a synchronous code division multiple access cdma system in all experiments we have conducted sequential application of interference avoidance produces an optimal codeword set when starting from randomly chosen initial codewords here we provide the first formal proof of convergence to optimal codeword ensembles for greedy interference avoidance algorithms augmented by a technique called class warfare whereby users which reside in more heavily loaded areas of the signal space purposely interfere with attack the reception of users in less crowded areas coordination of deliberate interference by a complete class of aggrieved user is also sometimes necessary such attacks and subsequent codeword adjustment by attacked users are shown to strictly decrease tsc along the way we also show using linear algebra and a variant of stochastic ordering equivalence between minimization of tsc and maximization of sum capacity

optimal signature design for spread spectrum steganography for any given host image or group of host images and any block transform domain of interest we find the signature vector that when used for spread spectrum ss message embedding maximizes the signal to interference plus noise ratio sinr at the output of the corresponding maximum sinr linear filter we establish that under a colored gaussian assumption on the transform domain host data the same derived signature minimizes host distortion for any target message recovery error rate and maximizes the shannon capacity of the covert steganographic link then we derive jointly optimal signature and linear processor designs for ss embedding in linearly modified transform domain host data and demonstrate orders of magnitude improvement over current ss steganographic practices optimized multisignature multimessage embedding in the same host data is studied as well

adaptive boundary element methods with convergence rates this paper presents adaptive boundary element methods for positive negative as well as zero order operator equations together with proofs that they converge at certain rates the convergence rates are quasi optimal in a certain sense under mild assumptions that are analogous to what is typically assumed in the theory of adaptive finite element methods in particular no saturation type assumption is used the main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods and an inverse type inequality involving boundary integral operators on locally refined finite element spaces

a game theoretic framework for interference avoidance various iterative algorithms for interference avoidance ia in networks with co located receivers suitable for distributed implementation have been proposed in the literature in this paper the ia problem is cast in a game theoretic framework and is formulated as a potential game this formulation accommodates previously proposed algorithms and in addition gives us a framework that enables the design of new distributed and convergent algorithms for ia including algorithms with nonidentical utility functions for the users two new convergence results for potential games are then derived the first result establishes the convergence of a class of potential games to the global solution while following best response iterations and when noise is added the second result establishes the convergence of potential games to the nash equilibria of the game while following random better response iterations the first result combined with the potential game formulation allows us to show that for a large class of network scenarios arbitrarily small noise assures the convergence of best response ia algorithms including the eigeniterations to an arbitrarily small neighborhood of the globally optimal signature sequence set the second result enables the design of reduced feedback mechanisms for ia that converge to desirable solutions

a multiplicative schwarz adaptive wavelet method for elliptic boundary value problems a multiplicative schwarz overlapping domain decomposition method is considered for solving elliptic boundary value problems by equipping the relevant sobolev spaces on the subdomains with wavelet bases adaptive wavelet methods are used for approximately solving the subdomain problems the union of the wavelet bases forms a frame for the sobolev space on the domain as a whole the resulting method is proven to be optimal in the sense that in linear complexity the iterands converge with the same rate as the sequence over n is an element of n of the best approximation from the span of the best n frame elements numerical results are given for the method applied to poisson s equation

a sparse laplacian in tensor product wavelet coordinates we construct a wavelet basis on the unit interval with respect to which both the infinite mass and stiffness matrix corresponding to the one dimensional laplacian are truly sparse and boundedly invertible as a consequence the infinite stiffness matrix corresponding to the laplacian on the n dimensional unit box with respect to the n fold tensor product wavelet basis is also sparse and boundedly invertible this greatly simplifies the implementation and improves the quantitative properties of an adaptive wavelet scheme to solve the multi dimensional poisson equation the results extend to any second order partial differential operator with constant coefficients that defines a boundedly invertible operator

optimal sequences and sum capacity of synchronous cdma systems the sum capacity of a multiuser synchronous cdma system is completely characterized in the general case of asymmetric user power constraints this solves the open problem posed by rupf and massey see ibid vol 40 p 1261 6 1994 which had solved the equal power constraint case we identify the signature sequences with real components that achieve sum capacity and indicate a simple recursive algorithm to construct them

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

reply to comment on tests of scaling and universality of the distributions of trade size and share volume evidence from three distinct markets analyzing trade by trade data for three distinct markets we showed that the cumulative distributions of trade size display power law tails pqxxq with exponents q in the levy stable domain qq 2 moreover we reported that the exponent values are consistent for all stocks irrespective of stock specific variables such as market capitalization industry sector or the specific market where the stock is traded our conclusions were based on using two distinct estimation methods racz et al now propose that one of the estimators we used has slow convergence for a pure power law particularly as tail exponents approach the boundary q 2 we examine the robustness of our results to specific estimation method by additional analysis using five distinct techniques to estimate q we find results that are fully consistent with those we had reported providing compelling evidence that our conclusions hold regardless of estimation procedure

new bounds and optimal binary signature sets part ii aperiodic total squared correlation we derive new bounds on the aperiodic total squared correlation atsc of binary antipodal signature sets for any number of signatures k and any signature length l we then present optimal designs that achieve the new bounds for several k l cases as interesting arguably side results we show that individual maximal merit factor sequences for example barker sequences are single user atsc optimal while neither the familiar gold nor the kasami set designs are atsc optimal in general the atsc optimal signature set designs provided in this work are in this sense better suited for asynchronous and or multipath code division multiplexing applications

wavelet methods for pdes some recent developments abstract this paper is concerned with recent developments of wavelet schemes for the numerical treatment of operator equations with special emphasis on two issues adaptive solution concepts and nontrivial domain geometries after describing a general multiresolution framework the key features of wavelet bases are highlighted namely locality norm equivalences and cancellation properties assuming first that wavelet bases with these properties are available on the relevant problem domains the relevance of these features for a wide class of stationary problems is explained in subsequent sections the main issues are preconditioning and the efficient adaptive application of wavelet representations of the involved operators we indicate then how these ingredients combined with concepts from nonlinear or best n term approximation culminate in an adaptive wavelet scheme for elliptic selfadjoint problems covering boundary value problems as well as boundary integral equations these schemes can be shown to exhibit convergence rates that are in a certain sense asymptotically optimal we conclude this section with some brief remarks on data structures and implementation interrelations with regularity in a certain scale of besov spaces and strategies of extending such schemes to unsymmetric or indefinite problems we address then the adaptive evaluation of nonlinear functionals of wavelet expansions as a central task arising in connection with nonlinear problems wavelet bases on nontrivial domains are discussed next the main issues are the development of fourier free construction principles and criteria for the validity of norm equivalences finally we indicate possible combinations of wavelet concepts with conventional discretizations such as finite element or finite volume schemes in connection with convection dominated and hyperbolic problems

stability of distributed power and signature sequence control for cdma systems a game theoretic framework the problem of power control and signature sequence adaptation in code division multiple access cdma systems is studied under a game theoretic framework each user tries to maximize his own utility function which may be different from others sufficient conditions for the existence of an equilibrium point in this multi objective optimization problem are identified the methodology for analyzing this class of problems is illustrated by examples

new multi parametric analytical approximations of exponential distribution with power law tails for new cars sells and other applications abstract a multi parametric family of exponential distributions with various power law tails is introduced and is shown to describe adequately the known distributions of incomes and wealth as well as the recently measured distributions of new car sales the three or four parametric families are characterized by effective temperature in the exponential part the power exponent in the power law asymptotic part the coefficient for the transition between the above two parts and the starting value if it is not equal to zero since the new car sales distributions are found to correspond to known distributions of incomes the latter may be inferred from the former

wavelet stabilization of the lagrange multiplier method we propose here a stabilization strategy for the lagrange multiplier formulation of dirichlet problems the stabilization is based on the use of equivalent scalar products for the spaces h 1 2 partial omega and h 1 2 partial omega which are realized by means of wavelet functions the resulting stabilized bilinear form is coercive with respect to the natural norm associated to the problem a uniformly coercive approximation of the stabilized bilinear form is constructed for a wide class of approximation spaces for which an optimal error estimate is provided finally a formulation is presented which is obtained by eliminating the multiplier by static condensation this formulation is closely related to the nitsche s method for solving dirichlet boundary value problems

model space of economic events abstract a method for constructing the model or virtual space of economic events when economic objects can be considered as material ones is suggested we describe change of share rates in time at stock markets as the potential difference of attracted bodies in time in this virtual space each share of each enterprise is displayed by a single particle with a unit charge it is shown that the random value of potential difference at the origin of coordinates measured at a definite time interval has the probability density coinciding with the known distribution of levy flights or levy walks a distribution of alteration in time of the standard and poor index value obtained by mantegna and stanley they shown that it is the levy walks distribution too mantegna and stanley nature 376 1995 46 is used for determination of the introduced potential dependence on coordinates in the model space a simple phenomenological model of interaction potential is introduced the potential law of each particle turns out to be closed to r 2 14 in the minimum possible three dimensional model space this model permits calculation of time of random potential correlations at a certain point of the model space these correlations could characterize the time period of making a decision by an investor at stock exchange it is shown that this time is notably shorter in unstable periods 1987 a microscopical model of interaction in the virtual space is also discussed

composite wavelet bases for operator equations this paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit n cube these bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains this covers also elliptic boundary value problems although this study is primarily motivated by our previous analysis of wavelet methods for pseudo differential equations with special emphasis on boundary integral equations in this case it is natural to model the boundary surface as a union of parametric images of the unit cube it will be shown how to construct wavelet bases on the surface which are composed of wavelet bases defined on each surface patch here the relevant properties are the validity of norm equivalences in certain ranges of sobolev scales as well as appropriate moment conditions

space time adaptive wavelet methods for parabolic evolution problems with respect to space time tensor product wavelet bases parabolic initial boundary value problems are equivalently formulated as bi infinite matrix problems adaptive wavelet methods are shown to yield sequences of approximate solutions which converge at the optimal rate in case the spatial domain is of product type the use of spatial tensor product wavelet bases is proved to overcome the so called curse of dimensionality i e the reduction of the convergence rate with increasing spatial dimension

adaptive piecewise tensor product wavelets scheme for laplace interface problems abstract a laplace type boundary value problem is considered with a generally discontinuous diffusion coefficient a domain decomposition technique is used to construct a piecewise tensor product wavelet basis that when normalized w r t the energy norm has riesz constants that are bounded uniformly in the jumps an adaptive wavelet galerkin method is applied to solve the boundary value problem with the best nonlinear approximation rate from the basis in linear computational complexity although the solutions are far from smooth numerical experiments in two dimensions show rates as for a one dimensional smooth solution the latter being possible because of the tensor product construction

ensuring convergence of the mmse iteration for interference avoidance to the global optimum viswanath and anantharam 1999 characterize the sum capacity of multiaccess vector channels for a given number of users received powers spreading gain and noise covariance matrix in a code division multiple access cdma system viswanath and anantharam present a combinatorial algorithm to generate a set of signature sequences that achieves the maximum sum capacity these sets also minimize a performance measure called generalized total square correlation tsc sub g ulukus and yates 2001 propose an iterative algorithm suitable for distributed implementation at each step one signature sequence is replaced by its linear minimum mean square error mmse filter this algorithm results in a decrease of tsc sub g at each step the mmse iteration has fixed points not only at the optimal configurations which attain the global minimum tsc sub g but also at other configurations which are suboptimal the authors of claim that simulations show that when starting with random sequences the algorithm converges to optimum sets of sequences but they give no formal proof we show that the tsc sub g function has no local minima in the sense that given any suboptimal set of sequences there exist arbitrarily close sets with lower tsc sub g therefore only the optimal sets are stable fixed points of the mmse iteration we define a noisy version of the mmse iteration as follows after replacing all the signature sequences one at a time by their linear mmse filter we add a bounded random noise to all the sequences using our observation about the tsc sub g function we can prove that if we choose the bound on the noise adequately making it decrease to zero the noisy mmse iteration converges to the set of optimal configurations with probability one for any initial set of sequences

higher order calderon zygmund estimates for the p laplace equation abstract the paper is concerned with higher order calderon zygmund estimates for the p laplace equation div a u div u p 2 u div f 1 p we are able to transfer local interior besov and triebel lizorkin regularity up to first order derivatives from the force term f to the flux a u for p 2 we show that f b q s implies a u b q s for any s 0 1 and all reasonable q 0 in the planar case the result fails for p 2 in case of higher dimensions and systems we have a smallness restriction on s the quasi banach case 0 min q 1 is included since it has important applications in the adaptive finite element analysis as an intermediate step we prove new linear decay estimates for p harmonic functions in the plane for the full range 1 p

analytical representation of stock and stock indexes returns non gaussian random walks with various jump laws abstract empirical data for stock and stock indexes returns that is available for international markets as well as for the russian stock market is introduced and discussed random walk process with a specific law of an elementary independent increment jump in some random walk space is proposed for a proper basic description an analytical representation of a random walk distribution law is derived from a general conception of stochastic elementary jumps this representation covers classic s chandrasekhar s results for gauss random walks with limited jumps and jumps having distributions with all moments as well as p levy flights with jump distributions without moments starting from the second one finally it describes truncated levy random walks for jumps having distributions with at least the second moment the natural generalization of truncated levy walks is done the random walk process is compared with known empirical data for the above discussed return fluctuations of international assets and stock indexes there is a very good agreement of the elaborated theory with empirical data

exponential distribution of financial returns at mesoscopic time lags a new stylized fact we study the probability distribution of stock returns at mesoscopic time lags return horizons ranging from about an hour to about a month while at shorter microscopic time lags the distribution has power law tails for mesoscopic times the bulk of the distribution more than 99 of the probability follows an exponential law the slope of the exponential function is determined by the variance of returns which increases proportionally to the time lag at longer times the exponential law continuously evolves into gaussian distribution the exponential to gaussian crossover is well described by the analytical solution of the heston model with stochastic volatility r 2004 elsevier b v all rights reserved

an optimal adaptive wavelet method for nonsymmetric and indefinite elliptic problems in this paper we modify the adaptive wavelet algorithm from gantumur et al an optimal adaptive wavelet method without coarsening of the iterands technical report 1325 department of mathematics utrecht university march 2005 math comp to appear so that it applies directly i e without forming the normal equation not only to self adjoint elliptic operators but also to operators of the form l a b where a is self adjoint elliptic and b is compact assuming that the resulting operator equation is well posed we show that the algorithm has optimal computational complexity

fast evaluation of nonlinear functionals of tensor product wavelet expansions for a nonlinear functional f and a function u from the span of a set of tensor product interpolets it is shown how to compute the interpolant of f u from the span of this set of tensor product interpolets in linear complexity assuming that the index set has a certain multiple tree structure applications are found in the field of adaptive tensor product solution methods for semilinear operator equations by collocation methods or after transformations between the interpolet and bi orthogonal wavelet bases by galerkin methods

signature optimization for cdma with limited feedback we study the performance of joint signature receiver optimization for direct sequence code division multiple access ds cdma with limited feedback the receiver for a particular user selects the signature from a signature codebook and relays the corresponding b index bits to the transmitter over a noiseless channel we study the performance of a random vector quantization rvq scheme in which the codebook entries are independent and isotropically distributed assuming the interfering signatures are independent and have independent and identically distributed i i d elements we evaluate the received signal to interference plus noise ratio sinr in the large system limit as the number of users processing gain and feedback bits b all tend to infinity with fixed ratios this sinr is evaluated for both the matched filter and linear minimum mean squared error mmse receivers furthermore we show that this large system sinr is the maximum that can be achieved over any sequence of codebooks numerical results show that with the mmse receiver one feedback bit per signature coefficient achieves close to single user performance we also consider a less complex and suboptimal reduced rank signature optimization scheme in which the user s signature is constrained to lie in a lower dimensional subspace the optimal subspace coefficients are scalar quantized and relayed to the transmitter the large system performance of the quantized reduced rank scheme can be approximated and numerical results show that it performs in the vicinity of the rvq bound finally we extend our analysis to the scenario in which a subset of users optimize their signatures in the presence of random interference

minimum total squared correlation quaternary signature sets new bounds and optimal designs we derive new bounds on the total squared correlation tsc of quaternary quadriphase signature sequence sets for all lengths l and set sizes k then for all k l we design minimum tsc optimal sets that meet the new bounds with equality direct numerical comparison with the tsc value of the recently obtained optimal binary sets shows under what k l realizations gains are materialized by moving from the binary to the quaternary code division multiplexing alphabet on the other hand comparison with the welch tsc value for real complex field sets shows that arguably not much is to be gained by raising the alphabet size above four for any k l the sum capacity as well as the maximum squared correlation and total asymptotic efficiency of minimum tsc quaternary sets is also evaluated in closed form and contrasted against the sum capacity of minimum tsc optimal binary and real complex sets

truncated levy distribution of sp500 stock index fluctuations distribution of one share fluctuations in a model space abstract truncated levy distributions of sp500 stock index fluctuations mantegna stanley nature 376 1995 46 are obtained in the formerly introduced model romanovsky physica a 265 1999 264 for stock market a one body random kinematics in this space corresponds to the one shares fluctuations on stock exchange the statistics of such kinematics is investigated and the asymptotic distribution of one share fluctuations was obtained all results are compared with the known experimental data mantegna stanley nature 376 1995 46 plerou et al phys rev e 60 1999 6519 and 5306 gopikrishnan et al eur j phys b 1998 138

asymptotic spectral efficiency of mimo multiple access wireless systems exploring only channel spatial correlations in this paper we use the replica method originally developed in statistical physics to investigate the asymptotic sum rate of a gaussian antenna array based multiple input multiple output mimo multiple access ma wireless channel having spatial correlations at both the transmitters and the receiver through numerical experiments the asymptotic solution can consistently produce highly accurate ergodic results for mimo ma systems with only a few antenna elements at each transmitter and receiver this offers the asymptotic solution important practical values in analyzing and designing a mimo multiple access system that makes best use of the wireless channel structure furthermore with the asymptotic solution we provide an efficient optimization algorithm to determine the asymptotic optimum transmit signal covariance matrices when only the slow varying channel spatial covariance information is available

on an adaptive multigrid solver for convection dominated problems the objective of this paper is to develop and describe an adaptive multigrid scheme for the numerical solution of convection diffusion reaction equations with dominating convection or reaction term specifically we propose a systematic construction of problem and scale dependent restriction prolongation and smoothing operators in combination with an adaptive mesh refinement strategy in particular adaptivity is not only to reduce computational complexity but also to stabilize standard galerkin discretizations so that adding artificial viscosity can be avoided the approach is suited for any choice of underlying multiresolution sequences in particular for standard finite element discretizations on nested but possibly nonuniform meshes moreover all concepts are in principle independent of the spatial dimension and avoid any special flow dependent enumeration of unknowns the performance of the scheme is illustrated by various examples in one and two spatial dimensions they are chosen so as to exhibit possibly many types of adverse or singular behavior in connection with constant and varying convection or the interplay between convective and reactive terms in the light of these tests those issues are identified that require further research

finite step algorithms for constructing optimal cdma signature sequences a description of optimal sequences for direct spread code division multiple access ds cdma is a byproduct of recent characterizations of the sum capacity this paper restates the sequence design problem as an inverse singular value problem and shows that the problem can be solved with finite step algorithms from matrix theory it proposes a new one sided algorithm that is numerically stable and faster than previous methods

stochastic volatility of financial markets as the fluctuating rate of trading an empirical study we present an empirical study of the subordination hypothesis for a stochastic time series of a stock price the fluctuating rate of trading is identified with the stochastic variance of the stock price as in the continuous time random walk ctrw framework the probability distribution of the stock price changes log returns for a given number of trades n is found to be approximately gaussian the probability distribution of n for a given time interval dt is nonpoissonian and has an exponential tail for large n and a sharp cutoff for small n combining these two distributions produces a non trivial distribution of log returns for a given time interval dt which has exponential tails and a gaussian central part in agreement with empirical observations r 2007 elsevier b v all rights reserved

adaptive wavelet algorithms for elliptic pde s on product domains with standard isotropic approximation by piecewise polynomials of fixed order in a domain d subset of r d the convergence rate in terms of the number n of degrees of freedom is inversely proportional to the space dimension d this so called curse of dimensionality can be circumvented by applying sparse tensor product approximation when certain high order mixed derivatives of the approximated function happen to be bounded in l 2 it was shown by nitsche 2006 that this regularity constraint can be dramatically reduced by considering best n term approximation from tensor product wavelet bases when the function is the solution of some well posed operator equation dimension independent approximation rates can be practically realized in linear complexity by adaptive wavelet algorithms assuming that the infinite stiffness matrix of the operator with respect to such a basis is highly compressible applying piecewise smooth wavelets we verify this compressibility for general non separable elliptic pdes in tensor domains applications of the general theory developed include adaptive galerkin discretizations of multiple scale homogenization problems and of anisotropic equations which are robust i e independent of the scale parameters resp of the size of the anisotropy

wavelets with patchwise cancellation properties we construct wavelets on general n dimensional domains or manifolds via a domain decomposition technique resulting in so called composite wavelets with this construction wavelets with supports that extend to more than one patch are only continuous over the patch interfaces normally this limited smoothness restricts the possibility for matrix compression and with that the application of these wavelets in adaptive methods for solving operator equations by modifying the scaling functions on the interval and with that on the n cube that serves as parameter domain we obtain composite wavelets that have patchwise cancellation properties of any required order meaning that the restriction of any wavelet to each patch is again a wavelet this is also true when the wavelets are required to satisfy zeroth order homogeneous dirichlet boundary conditions on part of the boundary as a result compression estimates now depend only on the patchwise smoothness of the wavelets that one may choose also taking stability into account our composite wavelets have all the properties for the application to the adaptive solution of well posed operator equations of orders 2t for t 3 2

scaling and correlation in financial time series we discuss the results of three recent phenomenological studies focussed on understanding the distinctive statistical properties of nancial time series i the probability distribution of stock price uctuations stock price uctuations occur in all magnitudes in analogy to earthquakes from tiny uctuations to very drastic events such as the crash of 19 october 1987 sometimes referred to as black monday the distribution of price uctuations decays with a power law tail well outside the levy stable regime and describes uctuations that dier by as much as 8 orders of magnitude in addition this distribution preserves its functional form for uctuations on time scales that dier by 3 orders of magnitude from 1 min up to approximately 10 days ii correlations in nancial time series while price uctuations themselves have rapidly decaying correlations the magnitude of uctuations measured by either the absolute value or the square of the price uctuations has correlations that decay as a power law persisting for several months iii volatility and trading activity we quantify the relation between trading activity measured by the number of transactions nt and the price change gt for a given stock over a time interval t t t we nd that nt displays long range power law correlations in time which leads to the interpretation that the long range correlations previously found for jgtj are connected to those of nt c 2000 elsevier science b v all rights reserved

adaptive fourier galerkin methods we study the performance of adaptive fourier galerkin methods in a periodic box in r d with dimension d 1 these methods offer unlimited approximation power only restricted by solution and data regularity they are of intrinsic interest but are also a first step towards understanding adaptivity for the hp fem we examine two nonlinear approximation classes one classical corresponding to algebraic decay of fourier coefficients and another associated with exponential decay typical of spectral approximation we investigate the natural sparsity class for the operator range and find that the exponential class is not preserved thus in contrast with the algebraic class this entails a striking different behavior of the feasible residuals that lead to practical algorithms influencing the overall optimality the sparsity degradation for the exponential class is partially compensated with coarsening we present several feasible adaptive fourier algorithms prove their contraction properties and examine the cardinality of the activated sets the galerkin approximations at the end of each iteration are quasi optimal for both classes but inner loops or intermediate approximations are sub optimal for the exponential class

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

convergence rates of multilevel and sparse tensor approximations for a random elliptic pde partial differential equations with random coefficients can be cast as parametric problems with a potentially infinite dimensional parameter domain for a class of elliptic equations we derive convergence rates of approximations based on a tensorized polynomial basis on the parameter domain and a sequence of spatial discretizations we prove that a sparse tensor product construction achieves essentially the same convergence rate as a multilevel approximation in which the spatial discretization level is chosen separately for each coefficient of the solution with respect to the polynomial basis in some cases the same rate is attained also by a single level approximation using just one spatial discretization for all coefficients we suggest an adaptive algorithm that reaches the optimal convergence rate with respect to the total number of degrees of freedom in a multilevel setting without prior knowledge of this rate numerical computations confirm theoretical results

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

sum capacity and tsc bounds in collaborative multibase wireless systems we consider a wireless system with base stations which collaborate and derive bounds on sum capacity and total squared correlation for uniform channels between users and bases the correspondence also investigates structural properties which must be satisfied by user transmit covariance matrices at the optimal sum capacity total squared correlation tsc point and shows that for multibase systems maximizing sum capacity and minimizing tsc are in general not equivalent problems

a posteriori error estimates for elliptic problems in two and three space dimensions let u in h be the exact solution of a given selfadjoint elliptic boundary value problem which is approximated by some tilde u in mathcal s mathcal s being a suitable finite element space efficient and reliable a posteriors estimates of the error u tilde u measuring the local quality of tilde u play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh a well known class of error estimates can be derived systematically by localizing the discretized defect problem by using domain decomposition techniques in this paper we provide a guideline for the theoretical analysis of such error estimates we further clarify the relation to other concepts our analysis leads to new error estimates which are specially suited to three space dimensions the theoretical results are illustrated by numerical computations

space time adaptive wavelet methods for optimal control problems constrained by parabolic evolution equations an adaptive algorithm based on wavelets is proposed for the efficient numerical solution of a control problem governed by a linear parabolic evolution equation first the constraints are represented by means of a full weak space time formulation as a linear system in ell 2 in wavelet coordinates following a recent approach by schwab and stevenson second a quadratic cost functional involving a tracking type term for the state and a regularization term for the distributed control is also formulated in terms of ell 2 sequence norms of wavelet coordinates this functional serves as a representer for a functional involving different sobolev norms with possibly nonintegral smoothness parameter standard techniques from optimization are then used to derive the first order necessary conditions as a coupled system in ell 2 coordinates for this purpose an adaptive method is proposed which can be interpreted as an inexact gradient method for the control in each iteration step the primal and adjoint systems are solved up to a prescribed accuracy by the adaptive algorithm it is shown that the adaptive algorithm converges moreover the algorithm is proved to be asymptotically optimal the convergence rate achieved for computing each of the components of the solution state adjoint state and control up to a desired target tolerance is asymptotically the same as the wavelet best n term approximation of each solution component and the total computational work is proportional to the number of computational unknowns

wavelet adaptive method for second order elliptic problems boundary conditions and domain decomposition summary wavelet methods allow to combine high order accuracy multilevel preconditioning techniques and adaptive approximation in order to solve efficiently elliptic operator equations one of the main difficulty in this context is the efficient treatment of non homogeneous boundary conditions in this paper we propose a strategy that allows to append such conditions in the setting of space refinement i e adaptive discretizations of second order problems our method is based on the use of compatible multiscale decompositions for both the domain and its boundary and on the possibility of characterizing various function spaces from the numerical properties of these decompositions in particular this allows the construction of a lifting operator which is stable for a certain range of smoothness classes and preserves the compression of the solution in the wavelet basis an explicit construction of the wavelet bases and the lifting is proposed on fairly general domains based on c 0 conforming domain decomposition techniques

adaptive wavelet schemes for parabolic problems sparse matrices and numerical results a simultaneous space time variational formulation of a parabolic evolution problem is solved with an adaptive wavelet method this method is shown to converge with the best possible rate in linear complexity thanks to the use of tensor product bases there is no penalty in complexity due to the additional time dimension special wavelets are designed such that the bi infinite system matrix is sparse this sparsity largely simplifies the implementation and improves the quantitative properties of the adaptive wavelet method numerical results for an ode and the heat equation are presented

scaling in the norwegian stock market the main objective of this paper is to investigate the validity of the much used assumptions that stock market returns follow a random walk and are normally distributed for this purpose the concepts of chaos theory and fractals are applied two independent models are used to examine price variations in the norwegian and us stock markets the first model used is the range over standard deviation or r s statistic which tests for persistence or antipersistence in the time series both the norwegian and us stock markets show significant persistence caused by long run memory components in the series in addition an average non periodic cycle of four years is found for the us stock market these results are not consistent with the random walk assumption the second model investigates the distributional scaling behaviour of the high frequency price variations in the norwegian stock market the results show a remarkable constant scaling behaviour between different time intervals this means that there is no intrinsic time scale for the dynamics of stock price variations the relationship can be expressed through a scaling exponent describing the development of the distributions as the time scale changes this description may be important when constructing or improving pricing models such that they coincide more closely with the observed market behaviour the empirical distributions of high frequency price variations for the norwegian stock market is then compared to the levy stable distribution with the relevant scaling exponent found by using the r s and distributional scaling analysis good agreement is found between the levy profile and the empirical distribution for price variations less than 6 standard deviations covering almost three orders of magnitude in the data for probabilities larger than 6 standard deviations there seem to be an exponential fall off from the levy profile in the tails which indicates that the second moment may be finite

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

fractional diffusion equation for a power law truncated levy process abstract truncated levy flights are stochastic processes which display a crossover from a heavy tailed levy behavior to a faster decaying probability distribution function pdf putting less weight on long flights overcomes the divergence of the levy distribution second moment we introduce a fractional generalization of the diffusion equation whose solution defines a process in which a levy flight of exponent is truncated by a power law of exponent 5 a closed form for the characteristic function of the process is derived the pdf of the displacement slowly converges to a gaussian in its central part showing however a power law far tail possible applications are discussed

scaling of the distribution of fluctuations of financial market indices up to more than one month we find that the distributions for dt 4d 1560 min are consistent with a power law asymptotic behavior characterized by an exponent a 3 well outside the stable levy regime 0 a 2 to test the robustness of the s p result we perform a parallel analysis on two other financial market indices database iv contains 3560 daily records of the nikkei index for the 14 year period 1984 1997 and database v contains 4649 daily records of the hang seng index for the 18 year period 1980 1997 we find estimates ofa consistent with those describing the distribution of s p 500 daily returns one possible reason for the scaling of these distributions is the long persistence of the autocorrelation function of the volatility for time scales longer than dt 3 4 d our results are consistent with a slow convergence to gaussian behavior s1063 651x 99 11211 x pacs number s 05 40 fb 05 45 tp 89 90 1n

optimality of adaptive galerkin methods for random parabolic partial differential equations galerkin discretizations of a class of parametric and random parabolic partial differential equations pdes are considered the parabolic pdes are assumed to depend on a vector y y 1 y 2 of possibly countably many parameters y j which are assumed to take values in 1 1 well posedness of weak formulations of these parametric equations in suitable bochner spaces is established adaptive galerkin discretizations of the equation based on a tensor product of a generalized polynomial chaos in the parameter domain 1 1 n and of suitable wavelet bases in the time interval i 0 t and the spatial domain d r d are proposed and their optimality is established

adaptive methods for boundary integral equations complexity and convergence estimates this paper is concerned with developing numerical techniques for the adaptive application of global operators of potential type in wavelet coordinates this is a core ingredient for a new type of adaptive solvers that has so far been explored primarily for pdes we shall show how to realize asymptotically optimal complexity in the present context of global operators asymptotically optimal means here that any target accuracy can be achieved at a computational expense that stays proportional to the number of degrees of freedom within the setting determined by an underlying wavelet basis that would ideally be necessary for realizing that target accuracy if full knowledge about the unknown solution were given the theoretical findings are supported and quantified by the first numerical experiments

sparse adaptive finite elements for radiative transfer the linear radiative transfer equation a partial differential equation for the radiation intensity u x s with independent variables email protected email protected r n in the physical domain d of dimension n 2 3 and angular variable email protected s 2 email protected r 3 y 1 is solved in the n 2 dimensional computational domain dxs 2 we propose an adaptive multilevel galerkin finite element method fem for its numerical solution our approach is based on a a stabilized variational formulation of the transport operator b on so called sparse tensor products of two hierarchic families of finite element spaces in h 1 d and in l 2 s 2 respectively and c on wavelet thresholding techniques to adapt the discretization to the underlying problem an a priori error analysis shows under strong regularity assumptions on the solution that the sparse tensor product method is clearly superior to a discrete ordinates method as it converges with essentially optimal asymptotic rates while its complexity grows essentially only as that for a linear transport problem in r n numerical experiments for n 2 on a set of example problems agree with the convergence and complexity analysis of the method and show that introducing adaptivity can improve performance in terms of accuracy vs number of degrees even further

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

composite wavelet bases with extended stability and cancellation properties the efficient solution of operator equations using wavelets requires that they generate a riesz basis for the underlying sobolev space and that they have cancellation properties of a sufficiently high order suitable biorthogonal wavelets were constructed on reference domains as the n cube via a domain decomposition approach these bases have been used as building blocks to construct biorthogonal wavelets on general domains or manifolds where in order to end up with local wavelets biorthogonality was realized with respect to a modified l 2 scalar product the use of this modified scalar product restricts the application of these so called composite wavelets to problems of orders strictly larger than 1 moreover those wavelets with supports that extend to more than one patch generally have no cancellation properties in this paper we construct local composite wavelets that are close to being biorthogonal with respect to the standard l 2 scalar product as a consequence they generate riesz bases for the sobolev spaces h s for the full range of s allowed by the continuous gluing of functions over the patch interfaces the properties of the primal and dual approximation spaces on the reference domain and in the manifold case by the regularity of the manifold moreover all these wavelets have cancellation properties of the full order induced by the approximation properties of the dual spaces on the reference domain we illustrate our findings by a concrete realization of wavelets on a perturbed sphere

adaptive wavelet methods for elliptic operator equations convergence rates this paper is concerned with the construction and analysis of wavelet based adaptive algorithms for the numerical solution of elliptic equations these algorithms approximate the solution u of the equation by a linear combination of n wavelets therefore a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of n wavelets so called n term approximation which would be obtained by keeping the n largest wavelet coefficients of the real solution which of course is unknown the main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error o n s in the energy norm whenever such a rate is possible by n term approximation the range of s 0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator moreover it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to n the adaptive algorithm applies to a wide class of elliptic problems and wavelet bases the analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization

multiuser constrained water pouring for continuous time overloaded gaussian multiple access channels in this paper signature waveforms and signal powers are jointly optimized for multiuser communications over strictly bandlimited continuous time overloaded channels corrupted by additive white gaussian noise awgn the total signal power is minimized subject to general asymmetric signal to interference plus noise ratio sinr constraints at the output of linear minimum mean squared error lmmse receivers using vectorized fourier transform vft technique the optimal solutions as well as a necessary and sufficient condition for the existence of a feasible solution are derived in the frequency domain it turns out that every optimal solution performs the same as the optimal frequency division multiple access fdma system a geometric procedure called multiuser constrained water pouring is developed to construct every possible optimal solution of which profiles of signal power and equivalent bandwidth are respectively the same as those of signal power and physical bandwidth of the optimal fdma system it is shown that orthogonal signature waveforms are assigned to oversized users and continuous time equivalents of generalized welch bound equality cte gwbe sequences are assigned to non oversized users

adaptive spectral galerkin methods with dynamic marking the convergence and optimality theory of adaptive galerkin methods is almost exclusively based on the dorfler marking this entails a fixed parameter and leads to a contraction constant bounded below away from zero for spectral galerkin methods this is a severe limitation which affects performance we present a dynamic marking strategy that allows for a superlinear relation between consecutive discretization errors and show exponential convergence with linear computational complexity whenever the solution belongs to a gevrey approximation class

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

adaptive wavelet methods for elliptic partial differential equations with random operators we apply adaptive wavelet methods to boundary value problems with random coefficients discretized by wavelets in the spatial domain and tensorized polynomials in the parameter domain greedy algorithms control the approximate application of the fully discretized random operator and the construction of sparse approximations to this operator we suggest a power iteration for estimating errors induced by sparse approximations of linear operators

levy flights from a continuous time process l evy flight dynamics can stem from simple random walks in a system whose operational time number of steps n typically grows superlinearly with physical time t thus this process is a kind of continuous time random walk ctrw dual to the typical scher montroll model in which n grows sublinearly with t models in which l evy flights emerge due to a temporal subordination allow one easily to discuss the response of a random walker to a weak outer force which is shown to be nonlinear on the other hand the relaxation of an ensemble of such walkers in a harmonic potential follows a simple exponential pattern and leads to a normal boltzmann distribution mixed models describing normal ctrw s in superlinear operational time and l evy flights under the operational time of subdiffusive ctrw s lead to a paradoxical diffusive behavior similar to the one found in transport on polymer chains the relaxation to the boltzmann distribution in such models is slow and asymptotically follows a power law

spreading code optimization and adaptation in cdma via discrete stochastic approximation the aim of this paper is to develop discrete stochastic approximation algorithms that adaptively optimize the spreading codes of users in a code division multiple access cdma system employing linear minimum mean square error mmse receivers the proposed algorithms are able to adapt to slowly time varying channel conditions one of the most important properties of the algorithms is their self learning capability they spend most of the computational effort at the global optimizer of the objective function tracking analysis of the adaptive algorithms is presented together with mean square convergence an adaptive step size algorithm is also presented for optimally adjusting the step size based on the observations numerical examples illustrating the performance of the algorithms in multipath fading channels show substantial improvement over heuristic algorithms

wavelet least squares methods for boundary value problems this paper is concerned with least squares methods for the numerical solution of operator equations our primary focus is the discussion of the following conceptual issues the selection of appropriate least squares functionals their numerical evaluation in the special light of recent developments of wavelet methods and a natural way of preconditioning the resulting systems of linear equations we describe first a general format of variational problems that are well posed in a certain natural topology in order to illustrate the scope of these problems we identify several special cases such as second order elliptic boundary value problems their formulation as a first order system transmission problems the system of stokes equations or more general saddle point problems particular emphasis is placed on the separate treatment of essential nonhomogeneous boundary conditions we propose a unified treatment based on wavelet expansions in particular we exploit the fact that weighted sequence norms of wavelet coefficients are equivalent to relevant function norms arising in the least squares context this provides access to difficult norms efficient preconditioners and in the case of first order systems optimal l2 error estimates

statistical properties of share volume traded in financial markets we quantitatively investigate the ideas behind the often expressed adage it takes volume to move stock prices and study the statistical properties of the number of shares traded q deltat for a given stock in a fixed time interval deltat we analyze transaction data for the largest 1000 stocks for the two year period 1994 95 using a database that records every transaction for all securities in three major us stock markets we find that the distribution p q deltat displays a power law decay and that the time correlations in q deltat display long range persistence further we investigate the relation between q deltat and the number of transactions n deltat in a time interval deltat and find that the long range correlations in q deltat are largely due to those of n deltat our results are consistent with the interpretation that the large equal time correlation previously found between q deltat and the absolute value of price change g deltat related to volatility are largely due to n deltat