a relativistic hypergeometric function we survey our work on a function generalizing 2f1 this function is a joint eigenfunction of four askey wilson type hyperbolic difference operators reducing to the askey wilson polynomials for certain discrete values of the variables it is defined by a contour integral generalizing the barnes representation of 2f1 it has various symmetries including a hidden d4 symmetry in the parameters by means of the associated hilbert space transform the difference operators can be promoted to self adjoint operators provided the parameters vary over a certain polytope in the parameter space for a dense subset of parameter shifts give rise to an explicit evaluation in terms of rational functions of exponentials hyperbolic functions and plane waves

the optimal symmetrical points for polynomial interpolation of real functions in the tetrahedron abstract the main result of this paper is the computation of the mean optimal symmetrical interpolation points in the tetrahedron up to degree 9 this interpoation set has the smallest lebesgue constant known today

some orthogonal very well poised functions recently ismail masson and suslov established a continuous orthogonality relation and some other properties of a bessel function on a q quadratic grid askey suggested that the bessel type orthogonality found in the above paper at the level really has a general character and can be extended up to the level very well poised fuctions are known as a nonterminating version of the classical askey wilson polynomials in this paper we prove askey s conjecture and discuss some properties of the orthogonal functions another type of the orthogonality relation for a very well poised function was recently found by askey rahman and suslov

efficient algorithms for universal portfolios a constant rebalanced portfolio is an investment strategy that keeps the same distribution of wealth among a set of stocks from day to day there has been much work on cover s universal algorithm which is competitive with the best constant rebalanced portfolio determined in hindsight cover 1991 helmbold et al 1998 blum and kalai 1999 foster and vohra 1999 vovk 1998 cover and ordentlich 1996a cover 1996c while this algorithm has good performance guarantees all known implementations are exponential in the number of stocks restricting the number of stocks used in experiments helmbold et al 1998 cover and ordentlich 1996a ordentlich and cover 1996b cover 1996c blum and kalai 1999 we present an efficient implementation of the universal algorithm that is based on non uniform random walks that are rapidly mixing applegate and kannan 1991 lovasz and simonovits 1992 frieze and kannan 1999 this same implementation also works for non financial applications of the universal algorithm such as data compression cover 1996c and language modeling chen et al 1999

properties of generalized univariate hypergeometric functions based on spiridonov s analysis of elliptic generalizations of the gauss hypergeometric function we develop a common framework for 7 parameter families of generalized elliptic hyperbolic and trigonometric univariate hypergeometric functions in each case we derive the symmetries of the generalized hypergeometric function under the weyl group of type e 7 elliptic hyperbolic and of type e 6 trigonometric using the appropriate versions of the nassrallah rahman beta integral and we derive contiguous relations using fundamental addition formulas for theta and sine functions the top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with ruijsenaars relativistic hypergeometric function and the askey wilson function respectively we show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions we also describe an intimate connection between the hyperbolic and trigonometric theory which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions

a lower bound on compression of unknown alphabets many applications call for universal compression of strings over large possibly infinite alphabets however it has long been known that the resulting redundancy is infinite even for i i d distributions it was recently shown that the redundancy of the strings patterns which abstract the values of the symbols retaining only their relative precedence is subliner in the blocklength n hence the per symbol redundancy diminishers to zero in this paper we show that pattern redundancy is at least 1 5log2e n1 3 bits to do so we construct a generating function whose coefficients lower bound the redundancy and use hayman s saddle point approximation technique to determine the coefficients asymptotic behavior

the performance of universal encoding universal coding theory is surveyed from the viewpoint of the interplay between delay and redundancy the price for universality turns out to be acceptably small

integration by table look up for p version finite elements on curved tetrahedra abstract we develop a method of evaluating element level matrices and vectors associated with p version finite element computations on tetrahedral meshes with curved boundaries the procedure uses optimal interpolation formula q chen i babuska approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle comput methods appl mech engrg 128 1995 405 417 q chen i babuska approximate optimal polynomial interpolation points in the tetrahedron comput methods appl mech engrg 137 1997 89 94 for non polynomial portions of integrands followed by exact integration exact integrals of product combinations of polynomial interpolants and shape functions are stored so that different integrals may be efficiently evaluated by a table look up we present error analysis for our scheme applied to curvilinear tetrahedral meshes with exact geometric representation of the boundary of the domain as defined in a geometric modelling system numerical results are presented for problems involving the helmholtz equation

a generalized hypergeometric function satisfying four analytic difference equations of askey wilson type the hypergeometric function 2 f 1 can be written in terms of a contour integral involving gamma functions we generalize this barnes representation by using a certain generalized gamma function as a building block in this way we obtain a new 2 f 1 generalization with various symmetry features we determine the analyticity properties of the r function in all of its eight arguments and show that it is a joint eigenfunction of four distinct askey wilson type difference operators two acting on v and two on the askey wilson polynomials can be obtained by a suitable discretization of v or

steady state mse performance analysis of mixture approaches to adaptive filtering in this paper we consider mixture approaches that adaptively combine outputs of several parallel running adaptive algorithms these parallel units can be considered as diversity branches that can be exploited to improve the overall performance we study various mixture structures where the final output is constructed as the weighted linear combination of the outputs of several constituent filters although the mixture structure is linear the combination weights can be updated in a highly nonlinear manner to minimize the final estimation error such as in singer and feder 1999 arenas garcia figueiras vidal and sayed 2006 lopes satorius and sayed 2006 bershad bermudez and tourneret 2008 and silva and nascimento 2008 we distinguish mixture approaches that are convex combinations where the linear mixture weights are constrained to be nonnegative and sum up to one singer and feder 1999 arenas garcia figueiras vidal and sayed 2006 affine combinations where the linear mixture weights are constrained to sum up to one bershad bermudez and tourneret 2008 and finally unconstrained linear combinations of constituent filters kozat and singer 2000 we investigate mixture structures with respect to their final mean square error mse and tracking performance in the steady state for stationary and certain nonstationary data respectively we demonstrate that these mixture approaches can greatly improve over the performance of the constituent filters our analysis is also generic such that it can be applied to inhomogeneous mixtures of constituent adaptive branches with possibly different structures adaptation methods or having different filter lengths

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

a best points interpolation method for efficient approximation of parametrized functions we present an interpolation method for efficient approximation of parametrized functions the method recognizes and exploits the low dimensional manifold structure of the parametrized functions to provide good approximation basic ingredients include a specific problem dependent basis set defining a low dimensional representation of the parametrized functions and a set of best interpolation points capturing the spatial parameter variation of the parametrized functions the best interpolation points are defined as solution of a least squares minimization problem which can be solved efficiently using standard optimization algorithms the approximation is then determined from the basis set and the best interpolation points through an inexpensive and stable interpolation procedure in addition an a posteriori error estimator is introduced to quantify the approximation error and requires little additional cost numerical results are presented to demonstrate the accuracy and efficiency of the method copyright 2007 john wiley sons ltd

fast universalization of investment strategies a universalization of a parameterized investment strategy is an online algorithm whose average daily performance approaches that of the strategy operating with the optimal parameters determined offline in hindsight we present a general framework for universalizing investment strategies and discuss conditions under which investment strategies are universalizable we present examples of common investment strategies that fit into our framework the examples include both trading strategies that decide positions in individual stocks and portfolio strategies that allocate wealth among multiple stocks this work extends in a natural way cover s universal portfolio work we also discuss the runtime efficiency of universalization algorithms while a straightforward implementation of our algorithms runs in time exponential in the number of parameters we show that the efficient universal portfolio computation technique of kalai and vempala proceedings of the 41st annual ieee symposium on foundations of computer science redondo beach ca 2000 pp 486 491 involving the sampling of log concave functions can be generalized to other classes of investment strategies thus yielding provably good approximation algorithms in our framework

gain from the two envelope problem via information asymmetry on the suboptimality of randomized switching the two envelope problem or exchange problem is one of maximizing the payoff in choosing between two values given an observation of only one this paradigm is of interest in a range of fields from engineering to mathematical finance as it is now known that the payoff can be increased by exploiting a form of information asymmetry here we consider a version of the two envelope game where the envelopes contents are governed by a continuous positive random variable while the optimal switching strategy is known and deterministic once an envelope has been opened it is not necessarily optimal when the content s distribution is unknown a useful alternative in this case may be to use a switching strategy that depends randomly on the observed value in the opened envelope this approach can lead to a gain when compared with never switching here we quantify the gain owing to such conditional randomized switching when the random variable has a generalized negative exponential distribution and compare this to the optimal switching strategy we also show that a randomized strategy may be advantageous when the distribution of the envelope s contents is unknown since it can always lead to a gain

a generalized hypergeometric function ii asymptotics and d 4 symmetry in previous work we introduced and studied a function r a a bf c v hat v that generalizes the hypergeometric function in this paper we focus on a similarity transformed function mathcal e a a gamma v hat v with parameters 4 related to the couplings c 4 by a shift depending on a a we show that the function is invariant under all maps w with w in the weyl group of type d 4 choosing a a positive and bf gamma hat v real we obtain detailed information on the re v asymptotics of the function in particular we explicitly determine the leading asymptotics in terms of plane waves and the c function that implements the similarity r

iterative nonparametric estimation of a log optimal portfolio selection function let stock market vectors form a stationary ergodic sequence for fixed d spl isin n a log optimal portfolio selection function of the past d observed vectors is iteratively estimated on the basis of a training sequence by use of gradients and nonparametric regression strong consistency is obtained under a boundedness and spl alpha mixing condition without further assumptions on the distribution

approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle the main results of this paper are the analysis of the quality of approximation of polynomial interpolation and the computation of the approximate optimal interpolation points in the triangle we introduce various norms for the interpolation operator computational results indicate that for a given polynomial degree the set that minimizes the mean l2 norm of the interpolation operator is close to the smallest lebesgue constant interpolation set in particular for the triangle this set gives the smallest lebesgue constant currently known

pq adaptive solid finite elements for three dimensional plates and shells this article presents a pq adaptive finite element procedure for three dimensional computation of thin walled structures like plates and shells the approach is based on a hexahedral element with high order shape functions the polynomial degree of the shape functions in the three spatial directions of the element is adjusted by means of an adaptive procedure to this end a hierarchical error indicator is applied to automatically control the polynomial degree q in thickness and p in the in plane direction of plate and shell like structures the efficiency and reliability of the proposed adaptive method is demonstrated using two numerical examples

a comparison of high order interpolation nodes for the pyramid the use of pyramid elements is crucial to the construction of efficient hex dominant meshes m bergot g cohen and m durufle j sci comput 42 2010 pp 345 381 for conforming nodal finite element methods with mixed element types it is advantageous for nodal distributions on the faces of the pyramid to match those on the faces and edges of hexahedra and tetrahedra we adapt existing procedures for constructing optimized tetrahedral nodal sets for high order interpolation to the pyramid with constrained face nodes including two generalizations of the explicit warp and blend construction of nodes on the tetrahedron t warburton j engrg math 56 2006 pp 247 262 comparisons between nodal sets show that the lowest lebesgue constants are given by warp and blend nodes for order n leq 7 and fekete nodes for n 7 though numerical experiments show little variation in the conditioning and accuracy of all surveyed nonequidistant points

the finite cell method for three dimensional problems of solid mechanics this article presents a generalization of the recently proposed finite cell method to three dimensional problems of linear elasticity the finite cell method combines ideas from embedding or fictitious domain methods with the p version of the finite element method besides supporting a fast simple generation of meshes it also provides high convergence rates mesh generation for a boundary representation of solids and for voxel based data obtained from ct scans is addressed in detail in addition the implementation of non homogeneous neumann boundary conditions and the computation of cell matrices based on a composed integration is presented the performance of the proposed method is demonstrated by three numerical examples including the elastostatic computation of a human bone biopsy

on the condition number of high order finite element methods influence of p refinement and mesh distortion abstract in this article the condition number of the stiffness matrix k is compared for three high order finite element methods fems i e the p version of the fem the spectral element method sem and the nurbs based isogeometric analysis iga note that only problems in linear elasticity are considered in the analysis it is well known that the condition number is one factor strongly influencing the number of significant digits for direct solvers or the required iteration count for iterative solution schemes therefore it is important to investigate the effect of the choice of the shape functions and the element distortion on k based on numerous one and two and three dimensional examples these influences are comprehensively studied and the numerical results are compared with condition number estimates extracted from the literature overall a good agreement is observed for p fem sem and two dimensional iga while discrepancies are noted for three dimensional isogeometric elements these are important findings as theoretical results may be only available for very restricted scenarios where one element geometries constant jacobi matrices of the element maps etc are considered

universal prediction this paper consists of an overview on universal prediction from an information theoretic perspective special attention is given to the notion of probability assignment under the self information loss function which is directly related to the theory of universal data compression both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings

hyperbolic beta integrals abstract hyperbolic beta integrals are analogues of euler s beta integral in which the role of euler s gamma function is taken over by ruijsenaars hyperbolic gamma function they may be viewed as q q bibasic analogues of the beta integral in which the two bases q and q are interrelated by modular inversion and they entail q analogues of the beta integral for q 1 the integrals under consideration are the hyperbolic analogues of the ramanujan integral the askey wilson integral and the nassrallah rahman integral we show that the hyperbolic nassrallah rahman integral is a formal limit case of spiridonov s elliptic nassrallah rahman integral

randomized switching in the two envelope problem the two envelope problem is a conundrum in decision theory that is subject to longstanding debate it is a counterintuitive problem of decidability between two different states in the presence of uncertainty where a player s payoff must be maximized in some fashion the problem is a significant one as it impacts on our understanding of probability theory decision theory and optimization it is timely to revisit this problem as a number of related two state switching phenomena are emerging in physics engineering and economics literature in this paper we discuss this wider significance and offer a new approach to the problem for the first time we analyse the problem by adopting cover s switching strategy this is where we randomly switch states with a probability that is a smoothly decreasing function of the observed value of one state surprisingly we show that the player s payoff can be increased by this strategy we also extend the problem to show that a deterministic switching strategy based on a thresholded decision once the amount in an envelope is observed is also workable

speaking of infinity i i d strings we study the redundancy of three approaches to compression of independent and identically distributed i i d strings over large possibly infinite alphabets standard compression of the string itself and compression of the string s shape and pattern which describe its symbols relative magnitude and precedence respectively we determine the rate at which per symbol standard redundancy increases to infinity as the alphabet size increases show that the maximum per symbol shape redundancy is between 0 027 and 1 and compare these to results showing that per symbol pattern redundancy diminishes to zero for all alphabet sizes we relate these concepts to ordered and unordered partitions of integers and sets and use this framework to explore relations between several combinatorial quantities including the bell fubini and second type stirling numbers

hierarchical basis for stabilized finite element methods for compressible flows the quest for more accurate numerical simulations of the navier stokes equations has motivated the study of high order piecewise polynomial basis functions also known as k version finite element methods as a means to attain this accuracy in the most cost effective manner stabilized finite element methods e g streamline upwind petrov galerkin have been proven to attain optimal rates of convergence for any polynomial order basis as well as perform well on a variety of flows including turbulence applications we present a stabilized finite element formulation for fluid dynamics using mesh entity based hierarchical basis functions the implementation is shown to attain theoretical convergence results for the linear model problem as well as the navier stokes equations

universal compression of markov and related sources over arbitrary alphabets recent work has considered encoding a string by separately conveying its symbols and its pattern the order in which the symbols appear it was shown that the patterns of independent and identically distributed i i d strings can be losslessly compressed with diminishing per symbol redundancy in this correspondence the pattern redundancy of distributions with memory is considered close lower and upper bounds are established on the pattern redundancy of strings generated by hidden markov models hmms with a small number of states showing in particular that their per symbol pattern redundancy diminishes with increasing string length the upper bounds are obtained by analyzing the growth rate of the number of multidimensional integer partitions and the lower bounds using hayman s theorem

first order analytic difference equations and integrable quantum systems we present a new solution method for a class of first order analytic difference equations the method yields explicit minimal solutions that are essentially unique special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic trigonometric and elliptic type euler s gamma function being of rational type we study these generalized gamma functions in considerable detail the scattering and weight functions u and w functions associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions we obtain detailed information on these u and w functions exploiting the difference equations they satisfy

on discretely entropy stable weight adjusted discontinuous galerkin methods curvilinear meshes abstract we construct entropy conservative and entropy stable discontinuous galerkin dg discretizations for time dependent nonlinear hyperbolic conservation laws on curvilinear meshes the resulting schemes preserve a semi discrete quadrature approximation of a continuous global entropy inequality the proof requires the satisfaction of a discrete geometric conservation law which we enforce through an appropriate polynomial approximation we extend the construction of entropy conservative and entropy stable dg schemes to the case when high order curvilinear mass matrices are approximated using low storage weight adjusted approximations and describe how to retain global conservation properties under such an approximation for certain types of curvilinear meshes these weight adjusted approximations deliver optimal rates of convergence finally the high order accuracy local conservation and discrete conservation or dissipation of entropy for under resolved solutions are verified through numerical experiments for the compressible euler equations on triangular and tetrahedral meshes

spectral collocation on triangular elements we consider the poisson problem on a segment of the unit disc and on triangles on the segment we transform the poisson problem by means of polar coordinates in these new coordinates we have a problem in a rectangle which can easily be mapped onto the square here standard chebyshev collocation techniques can be applied then the segment is mapped onto a triangle where the same spectral scheme may be used by numerical tests we observed the expected high spectral accuracy due to the corner singularity a singular behaviour of the solution can be expected here we improved the accuracy by auxiliary mapping techniques further it is shown that finite difference preconditioning can be successfully applied in order to construct an efficient iterative solver finally a domain decomposition technique is applied to the patching of a rectangular and a triangular element

quantum parrondo s games parrondo s paradox arises when two losing games are combined to produce a winning one a history dependent quantum parrondo game is studied where the rotation operators that represent the toss of a classical biased coin are replaced by general su 2 operators to transform the game into the quantum domain in the initial state a superposition of qubits can be used to couple the games and produce interference leading to quite different payoffs to those in the classical case

the associated askey wilson polynomials we derive some contiguous relations for very well poised 8 7 series and use them to construct two linearly independent solutions of the three term recurrence relation of the associated askey wilson polynomials we then use these solutions to find explicit representations of two families of associated askey wilson polynomials we identify the corresponding continued fractions as quotients of two very well poised 807 series and find the weight functions

a pseudo spectral scheme for the incompressible navier stokes equations using unstructured nodal elements a pseudo spectral scheme for solving the incompressible navier stokes equations using unstructured nodal triangles is proposed efficient algorithms are developed with numerical evidence that indicates optimal rates of convergence can be achieved navier stokes simulations of kovasznay shear layer roll up and row past a cylinder are included to show comparisons between the different nodal sets considered and an alternative modal approach c 2000 academic press

shell finite cell method a high order fictitious domain approach for thin walled structures abstract this article presents a generalization of the recently proposed finite cell method to thin walled structures this approach uses a combination of well known fictitious domain methods with high order hierarchical ansatz spaces known from the p version of the finite element method whereas the original concept embeds a three dimensional structure in a simple domain being meshed into a grid of cube shaped cells the extension presented in this paper applies the fictitious domain approach to a two dimensional master domain defined in the parameter plane of the geometric model implementation details are discussed and numerical benchmark problems show the high accuracy and computational efficiency of the new approach it is also remarked that the present approach can easily be carried over to isogeometric analysis opening an attractive possibility to simulate trimmed nurbs surfaces

a generalized hypergeometric function iii associated hilbert space transform for generic parameters a a c 0 2 4 we associate a hilbert space transform to the relativistic hypergeometric function r a a bf c v hat v studied in previous papers restricting the couplings c to a certain polytope we show that the renormalized r function kernel gives rise to an isometry from the even subspace of l 2 mathbb r hat w hat v d hat v to the even subspace of l2 w v dv where hat w hat v and w v are positive and even weight functions we prove that the orthogonal complement of the range of this isometry is spanned by n pairwise orthogonal functions the latter are in essence askey wilson polynomials arising from the r function by choosing hat v i kappa n with kappa 0 ldots kappa n 1 distinct negative numbers the two commuting analytic difference operators acting on the variable v for which r is a joint eigenfunction give rise to two commuting self adjoint hamiltonians on the even subspace of l2 w v dv we explicitly determine the relation of the time dependent scattering theory for these dynamics to their joint spectral transform

sequential prediction over hierarchical structures we study sequential compound decision problems in the context of sequential prediction of real valued sequences in particular we consider finite state fs predictors that are constructed based on a hierarchical structure such as the order preserving patterns of the sequence history we define hierarchical equivalence classes by tying certain models at a hierarchy level in a recursive manner in order to mitigate undertraining problems these equivalence classes defined on a hierarchical structure are then used to construct a super exponential number of sequential fs predictors based on their combinations and permutations we then introduce truly sequential algorithms with computational complexity only linear in the pattern length that 1 asymptotically achieve the performance of the best fs predictor or the best linear combination of all the fs predictors in an individual sequence manner without any stochastic assumptions over any data length n under a wide range of loss functions 2 achieve the mean square error of the best linear combination of all fs filters or predictors in the steady state for certain nonstationary models we illustrate the superior convergence and tracking capabilities of our algorithm with respect to several state of the art methods in the literature through simulations over synthetic and real benchmark data

universal modeling and coding the problems arising in the modeling and coding of strings for compression purposes are discussed the notion of an information source that simplifies and sharpens the traditional one is axiomatized and adaptive and nonadaptive models are defined with a measure of complexity assigned to the models a fundamental theorem is proved which states that models that use any kind of alphabet extension are inferior to the best models using no alphabet extensions at all a general class of so called first in first out fifo arithmetic codes is described which require no alphabet extension devices and which therefore can be used in conjunction with the best models because the coding parameters are the probabilities that define the model their design is easy and the application of the code is straightforward even with adaptively changing source models

higher order conformal decomposition fem cdfem abstract a higher order accurate finite element method is proposed which uses automatically generated meshes based on implicit level set data for the description of boundaries and interfaces in two and three dimensions the method is an alternative for fictitious domain and extended finite element methods the domain of interest is immersed in a background mesh composed by higher order elements the zero level sets are identified and meshed followed by a decomposition of the cut background elements into conforming sub elements adaptivity is a crucial ingredient of the method to guarantee the success of the mesh generation it ensures the successful decomposition of cut elements and enables improved geometry descriptions and approximations it is confirmed that higher order accurate results with optimal convergence rates are achieved with the proposed conformal decomposition finite element method cdfem

ruijsenaars hypergeometric function and the modular double of u q sl 2 c abstract simultaneous eigenfunctions of two askey wilson second order difference operators are constructed as formal matrix coefficients of the principal series representation of the modular double of the quantized universal enveloping algebra u q sl 2 c these eigenfunctions are shown to be equal to ruijsenaars hypergeometric function under a proper parameter correspondence

a numerical comparison of seven grids for polynomial interpolation on the interval abstract seven types of chebyshev like grids in one dimension are compared according to four different criteria for accuracy the grid which minimizes the lebesgue constant is the best because it performs fairly well by all four criteria for the same reason minimizing the lebesgue constant seems to be the most useful measure of optimality because grids that are good according to this criterion are good when measured by other criteria too cfm optimality which is the property that all cardinal functions lagrange fundamental polynomials have maxima at the interpolation points seems to be the least discriminating criterion because all seven grids generate cardinal functions that have maxima at or very near the interpolation points the difference between these grids on all four criteria always less than a factor of two and usually much smaller are sufficiently modest so that the final choice between grids should probably not be made because of accuracy but rather based on other criteria such as ease of programming analytical simplicity conformality with other approximations and timestep in applications to partial differential equations to explore such nonaccuracy issues we also compared six of the grids on the basis of maximum allowed timestep when the grid is used to discretize the spatial coordinate and an explicit scheme is used for time marching the lebesgue optimal grid is also nearly optimal in the sense of allowing the longest timestep with a pseudospectral collocation algorithm the lebesgue grid allows a timestep three halves as long as that of the legendre lobatto spectral element method

overlapping schwarz methods for unstructured spectral elements a parallel and scalable domain decomposition method for unstructured and hybrid spectral element discretizations of elliptic problems is introduced and studied the spectral elements are affine images of the reference triangle or square in two dimensions and of the reference tetrahedron pyramid prism or cube in three dimensions the method is based on overlapping schwarz techniques applied to the schur complement of the discrete system and is implemented as a preconditioner for a krylov space method numerical results in two and three dimensions show that the iteration counts of our method are bounded by a constant independent of the spectral degree and the number of subdomains the resulting elliptic solver can be used in navier stokes simulations using the spectral element code nektar

asymptotic optimality and asymptotic equipartition properties of log optimum investment on montre comment un investisseur peut repartir ses fonds entre plusieurs opportunites d investissement pour maximiser le taux de croissance de son capital

a universal data compression system a universal data compression algorithm is described which is capable of compressing long strings generated by a finitely generated source with a near optimum per symbol length without prior knowledge of the source this class of sources may be viewed as a generalization of markov sources to random fields moreover the algorithm does not require a working storage much larger than that needed to describe the source generating parameters

a comparative study of finite element and spectral element methods in seismic wavefield modeling the computational accuracy and efficiency of finite element method and spectral element method sem are investigated thoroughly in time domain elastic wavefield modeling the diagonal mass matrices of the fem and sem free from matrix inversion are compared comprehensively by making full use of the mass lumped technique and quadrature rules we investigate the fem and sem based respectively on quadrilateral with the polynomials of degrees one and two and on triangular grids with the polynomials of degrees one and three generally the numerical solutions based on quadrilateral grids have a higher precision than those computed on triangular grids when the same order of polynomials is used the fem has a comparable accuracy to the sem with the same number of interpolant points in view of the triangular and quadrilateral sems the former suffers from larger computational costs and relatively lower accuracy compared with the latter furthermore the convergence study proves that the triangular sem produces consistently larger errors than the quadrilateral sem for any order and element sizes however the triangular sem can adapt to arbitrary complex geometries effectively in terms of efficiency the fem has an efficiency comparable with the sem on condition that the order of interpolation polynomials is identical in addition a perfectly matched layer pml boundary condition in variational form is deduced by introducing four intermediate variables in frequency domain the pml avoids convolution calculation and obtains an exact solution through inverse fourier transform in time domain the numerical examples verify the validity and effectiveness of the pml

nurbs enhanced finite element method nefem the development of nurbs enhanced finite element method nefem is revisited this technique allows a seamless integration of the cad boundary representation of the domain and the finite element method fem the importance of the geometrical model in finite element simulations is addressed and the benefits and potential of nefem are discussed and compared with respect to other curved finite element techniques

universal portfolios with side information we present a sequential investment algorithm the spl mu weighted universal portfolio with side information which achieves to first order in the exponent the same wealth as the best side information dependent investment strategy the best state constant rebalanced portfolio determined in hindsight from observed market and side information outcomes this is an individual sequence result which shows the difference between the exponential growth wealth of the best state constant rebalanced portfolio and the universal portfolio with side information is uniformly less than d 2n log n 1 k n log 2 for every stock market and side information sequence and for all time n here d k m 1 is the number of degrees of freedom in the state constant rebalanced portfolio with k states of side information and m stocks the proof of this result establishes a close connection between universal investment and universal data compression

recursive parameter free explicitly defined interpolation nodes for simplices a rule for constructing interpolation nodes for n th degree polynomials on the simplex is presented these nodes are simple to define recursively from families of 1d node sets such as the lobatto

stable spectral methods for conservation laws on triangles with unstructured grids this paper presents an asymptotically stable scheme for the spectral approximation of linear conservation laws defined on a triangle lagrange interpolation on a general two dimensional nodal set is employed and by imposing the boundary conditions weakly through a penalty term the scheme is proven stable in l 2 this result is established for a general unstructured grid in the triangle a special case for which the nodes along the edges of the triangle are chosen as the legendre gauss lobatto quadrature points is discussed in detail the eigenvalue spectrum of the approximation to the advective operator is computed and is shown to result in an o n 2 restriction on the time step when considering explicit time stepping c 1999 elsevier science s a all rights reserved