nonexistence of universally accelerating linear summability methods abstract let a amn b bmn be two regular toeplitz matrices suppose that for every convergent sequence sn its b transform b m where b m n 1bmnsn converges not slower than a m in the sens 0 e that b m s k a m s where s limn sn and k does not depend on m only on sn then a and b are equivalent i e they sum the same sequences even unbounded and moreover a and b essentially coincide there exists a number p such that amn bmn if m p

general approach to mathfrak su n quasi distribution functions we propose an operational form for the kernel of a mapping between an operator acting in a hilbert space of a quantum system with an symmetry group and its symbol in the corresponding classical phase space for symmetric irreps of this mapping is bijective we briefly discuss complications that will occur in the general case

acceleration of some logarithmic sequences abstract delahaye and germain bonne 1982 proved that the set of logarithmic sequences log is not accelerable nevertheless some subsets of log can be accelerated most of the algorithms are based on the asymptotic expansion of the sequence to be accelerated in this paper the point of view is quite different since an analytic property of the whole sequence is used if cn n n is a sequence of moments with an integral representation in the interval 0 1 then we w prove that the jacobi modified moments accelerate cn n n section 2 in a particular case section 3 the modified sequence can be written as a moment sequence and a second modified sequence is formed leading to further acceleration the modified moments methods are particular summation processes using orthogonal polynomials which have been extensively studied by wimp 1974 1981

extrapolation methods for improving the convergence of oligomer calculations to the infinite chain limit of quasi one dimensional stereoregular polymers abstract quasi one dimensional stereoregular polymers as for example polyacetylene are currently of considerable interest both experimentally as well as theoretically there are basically two different approaches for doing electronic structure calculations one method the so called crystal orbital method uses periodic boundary conditions and is essentially based on concepts of solid state theory the other method is essentially a quantum chemical method since it approximates the polymer by oligomers consisting of a finite number of monomer units i e by molecules of finite size in this way the highly developed technology of quantum chemical molecular programs can be used unfortunately oligomers of finite size are not necessarily able to model those features of a polymer which crucially depend on its in principle infinite extension however in such a case extrapolation techniques can be extremely helpful for example one can perform electronic structure calculations for a sequence of oligomers with an increasing number of monomer units in the next step one then can try to determine the limit of this sequence for an oligomer of infinite length with the help of suitable extrapolation methods several different extrapolation methods are discussed which are able to accomplish an extrapolation of energies and properties of oligomers to the infinite chain limit calculations for the ground state energy of polyacetylene are presented which demonstrate the practical usefulness of extrapolation methods

convergence acceleration methods the past decade abstract the aim of this paper is to present a review of the most significant results obtained the past ten years on convergence acceleration methods it is divided into four sections dealing respectively with the theory of sequence transformations the algorithms for such transformations the practical problems related to their implementation and their applications to various subjects of numerical analysis

convergence acceleration of logarithmic fixed point sequences let x n be some sequence generated by x n 1 f x n where f x email protected i email protected p ix p i p 1 a p 1 0 small it converges to zero logarithmically i e lim n x n l x n 1 thus we need algorithms for accelerating its convergence using asymptotic expansions in the analysis of the d 2 and q 2 algorithms leads to modified iterated versions of the first one and to combinations with the iterated q 2 algorithm in particular some superconvergence phenomena can be explained in this framework a similar study can be made for other nonlinear algorithms known at present moreover the above algorithms are also good accelerators for large classes of slowly convergent integrals and series

characters l 2 betti numbers and an equivariant approximation theorem let g be a group with a finite subgroup h we define the l 2 multiplicity of an irreducible representation of h in the l 2 homology of a proper g cw complex these invariants generalize the l 2 betti numbers our main results are approximation theorems for l 2 multiplicities which extend the approximation theorems for l 2 betti numbers of luck farber and elek szabo respectively the main ingredient is the theory of characters of infinite groups and a method to induce characters from finite subgroups we discuss applications to the cohomology of arithmetic groups

vector sequence transformations for the acceleration of logarithmic convergence logarithmic convergence of scalar sequences is generalized to vector sequences some sequence transformations for logarithmically convergent scalar sequences are also extended to the vector case theoretical results and numerical examples are given

wigner distribution function for finite systems we construct a wigner distribution function for finite data sets it is based on a finite optical system a linear wave guide where the finite number of discrete sensors is equal to the number of modes which the guide can carry the dynamical group for this model is su 2 and the wave functions are sets of n 2l 1 data points the wigner distribution function assigns classical c numbers to the operators of position momentum and wave guide mode

the set of logarithmically convergent sequences cannot be accelerated some theorems pennacchi germain bonne smith and ford state that methods of a certain form which are exact on geometric sequences accelerate linear convergence but no corresponding theorem is known for logarithmic convergence our study shows the reason why there is no algorithm which can accelerate all logarithmically convergent sequences we obtain this result with a generalization of remanence which is a sufficient property for a set of sequences to be unaccelerable

hyperlinearity essentially free actions and l2 invariants the sofic property we prove that connes embedding conjecture holds for the von neumann algebras of sofic groups that is sofic groups are hyperlinear hence we provide some new examples of hyperlinearity we also show that the determinant conjecture holds for sofic groups as well we introduce the notion of essentially free actions and amenable actions and study their properties

convergence acceleration by extraction of linear subsequences logarithmic sequences are usually difficult to accelerate in this paper we show how to extract a subsequence converging linearly this subsequence can be easily accelerated by classical algorithms theoretical results are given as well as an algorithmic extraction procedure numerical examples illustrate the results

a convergence acceleration method for some logarithmically convergent sequences a convergence acceleration method with one parameter for a sequence s n satisfying s n sim s n theta left c 0 frac c 1 n frac c 2 n 2 cdots right quad text as n to infty is proposed when the parameter is equal to 1 the method agrees with wynn s rho algorithm hence the method is designated as a generalized rho algorithm when s n is the nth partial sum of a series whose nth term is asymptotic to an theta 1 where theta is negative the generalized rho algorithm with parameter theta agrees essentially with the modified wynn s rho algorithm taken up by drummond error estimates and numerical examples of the generalized rho algorithm with parameter theta are given the exponent theta in the expansion above can be estimated using the generalized rho algorithm with parameter 2 in practice by estimating the exponent theta the generalized rho algorithm can be automatically applied to

accelerable subsets of logarithmic sequences abstract the aim of this paper is to study what the accelerable subsets of logsf of all logarithmically convergent sequences are three kinds of subsets of logsf and sequence transformations are given theoretical results and numerical examples are given

a su 3 wigner function for three dimensional systems we derive a wigner function with su 3 symmetry for three dimensional systems exemplified by the polarization of three dimensional electromagnetic fields

a simultaneous process for convergence acceleration and error control abstract let xn be a real sequence converging to the limit x and such that x n x n 1 x n n n v i 0 a i n i a0 0 0 for n large enough here we propose a repeated process using two transformations namely aitken s 2 process and brezinski s 2 algorithm which provide a sequence of intervals asymptotically containing x in particular we introduce for 1 a version of the modified iterated 2 process based on the convergence orders so generalizing a method proposed by sablonniere we establish that the order vp used in the pth iteration is vp 1 kp 1 1 kp 1 3 being an integer we give an estimation method for vp confirming the choice of kp 1

the epsilon algorithm and related topics the epsilon algorithm is recommended as the best all purpose acceleration method for slowly converging sequences it exploits the numerical precision of the data to extrapolate the sequence to its limit we explain its connections with pade approximation and continued fractions which underpin its theoretical base then we review the most recent extensions of these principles to treat application of the epsilon algorithm to vector valued sequences and some related topics in this paper we consider the class of methods based on using generalised inverses of vectors and the formulation specifically includes the complex case wherever possible

comparison of sequence accelerators forthe gaver method of numerical laplace transform inversion the sequence of gaver functionals is useful in the numerical inversion of laplace transforms the convergence behavior of the sequence is logarithmic therefore an acceleration scheme is required the accepted procedure utilizes salzer summation because in many cases the gaver functionals have the asymptotic behavior n t n t an 2 as n for fixed t it seems that no other acceleration schemes have been investigated in this area surely the popular nonlinear methods should be more effective however to our surprise only one nonlinear method was superior to salzer summation namely the wynn rho algorithm

a closed formula for the product of irreducible representations of su 3 we determine a closed formula in terms of p q r and s for the decomposition of the product p q r s of finite dimensional irreducible representations of su 3 we also determine in terms of p q r s m and n necessary and sufficient conditions that a term m n appears in this decomposition and its multiplicity

comparison of four nonlinear transforms on some classes of logarithmic fixed point sequences abstract let log be the set of logarithmic sequences i e of convergent sequences sn satisfying lim e n 1 e n 1 where en sn s and s limsn this paper is devoted to the comparison of four nonlinear sequence transforms on some subsets lf r of log for any positive integer r lf r denotes the set of fixed point sequences sn whose associated error sequences en sn s s limsn have an asymptotic expansion of the following type en 1 en i 1 ien1 ir where 1 c 0 r 1 2 r 1 2 1 2 is different from zero for sn lf r there holds e n a 1 n 1 r o n 1 1 r log n with a 1 r 1 1 r the four are respectively modifications of the algorithm and of the iterated 2 transform the iterated lubkin s transform and the algorithm of brezinski all of them accelerate the convergence of sequences in lf r we give accurate results concerning the asymptotic behaviour of transformed sequences we also study two algorithms combining the second and third transforms numerical examples are given to illustrate theoretical results

wigner weyl isomorphism for quantum mechanics on lie groups the wigner weyl isomorphism for quantum mechanics on a compact simple lie group g is developed in detail several features are shown to arise which have no counterparts in the familiar cartesian case notable among these is the notion of a semiquantized phase space a structure on which the weyl symbols of operators turn out to be naturally defined and figuratively speaking located midway between the classical phase space t g and the hilbert space of square integrable functions on g general expressions for the star product for weyl symbols are presented and explicitly worked out for the angle angular momentum case