analysis of multiple scattering iterations for high frequency scattering problems i the two dimensional case we present an analysis of a recently proposed integral equation method for the solution of high frequency electromagnetic and acoustic scattering problems that delivers error controllable solutions in frequency independent computational times within single scattering configurations the method is based on the use of an appropriate ansatz for the unknown surface densities and on suitable extensions of the method of stationary phase the extension to multiple scattering configurations in turn is attained through consideration of an iterative neumann series that successively accounts for further geometrical wave reflections as we show for a collection of two dimensional cylindrical convex obstacles this series can be rearranged into a sum of periodic orbits of increasing period each corresponding to contributions arising from waves that reflect off a fixed subset of scatterers when these are transversed sequentially in a periodic manner here we analyze the properties of these periodic orbits in the high frequency regime by deriving precise asymptotic expansions for the currents i e the normal derivative of the fields that they induce on the surface of the obstacles as we demonstrate these expansions can be used to provide accurate estimates of the rate at which their magnitude decreases with increasing number of reflections which defines the overall rate of convergence of the multiple scattering series moreover we show that the detailed asymptotic knowledge of these currents can be used to accelerate this convergence and thus to reduce the number of iterations necessary to attain a prescribed accuracy

a theorem on divergence in the general sense for continued fractions if the odd and even parts of a continued fraction converge to different values the continued fraction may or may not converge in the general sense we prove a theorem which settles the question of general convergence for a wide class of such continued fractions we apply this theorem to two general classes of q continued fraction to show that if g q is one of these continued fractions and q 1 then either g q converges or does not converge in the general sense we also show that if the odd and even parts of the continued fraction kn 1 an 1 converge to different values then limn an

the rogers ramanujan continual fraction abstract a survey of many theorems on the rogers ramanujan continued fraction is provided emphasis is given to results from ramanujan s lost notebook that have only recently been proved

applications of the heine and bauer muir transformations to rogers ramanujan type continued fractions abstract in this paper we show that various continued fractions for the quotient of general ramanujan functions g a q b q g a b may be derived from each other via bauer muir transformations the separate convergence of numerators and denominators play a key part in showing that the continued fractions and their bauer muir transformations converge to the same limit we also show that these continued fractions may be derived from either heine s continued fraction for a ratio of 1 2 functions or other similar continued fraction expansions of ratios of 1 2 functions further by employing essentially the same methods a new continued fraction for g a q b q g a b is derived finally we derive a number of new versions of some beautiful continued fraction expansions of ramanujan for certain combinations of infinite products with the following being an example a b q a b q a b q a b q a b 1 a b 1 a 2 1 b 2 q 1 a b q 2 a b q 2 b a q 2 q 1 a b q 4 1 a 2 q 2 1 b 2 q 2 q 3 1 a b q 6 a b q 4 b a q 4 q 3 1 a b q 8

the dirichlet problem for viscous compressible isothermal navier stokes equations in two dimensions we consider the navier stokes equations for compressible isothermal flow in the steady two dimensional case and show the existence of a weak solution for homogeneous dirichlet boundary conditions

exact and quasiexact solvability of second order superintegrable quantum systems i euclidean space preliminaries we show that second order superintegrable systems in two dimensional and three dimensional euclidean space generate both exactly solvable es and quasiexactly solvable qes problems in quantum mechanics via separation of variables and demonstrate the increased insight into the structure of such problems provided by superintegrability a principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable most past separation of variables treatments of qes problems via partial differential equations have only incorporated separability not multiseparability also we propose another definition of es and qes the quantum mechanical problem is called es if the solution of schrodinger equation can be expressed in terms of hypergeometric functions fnm and is qes if the schrodinger equation admits polynomial solutions with coefficients necessarily satisfying a three term or higher order of recurrence relations in three dimensions we give an example of a system tha

orthogonal laurent polynomials and strong moment theory a survey these topics are found in many parts of 20th century mathematics and its applications in mathematical physics chemistry statistics and engineering historically the analytic theory of continued fractions has played a central role in both the origin and the development of the other topics continued fractions are intimately related to pad e approximants and special functions emphasis is given to the development of strong moment theory and orthogonal laurent polynomials and to the related continued fractions quadrature formulas integral transforms and linear functionals by a strong moment problem we mean the following for a given bisequence n n of real numbers does there exist a distribution function such that

on 2d euler equations i on the energy casimir stabilities and the spectra for linearized 2d euler equations in this paper we study a linearized two dimensional euler equation this equation decouples into infinitely many invariant subsystems each invariant subsystem is shown to be a linear hamiltonian system of infinite dimensions another important invariant besides the hamiltonian for each invariant subsystem is found and is utilized to prove an unstable disk theorem through a simple energy casimir argument holm et al phys rep 123 1 116 1985 the eigenvalues of the linear hamiltonian system are of four types real pairs c c purely imaginary pairs id id quadruples c id and zero eigenvalues the eigenvalues are computed through continued fractions the spectral equation for each invariant subsystem is a poincare type difference equation i e it can be represented as the spectral equation of an infinite matrix operator and the infinite matrix operator is a sum of a constant coefficient infinite matrix operator and a compact infinite matrix operator we have obtained a complete spectral theory

a sleszyňski pringsheim theorem for vector valued continued fractions and its optimal error bounds the first purpose of this work is to prove a convergence theorem for vector valued continued fractions which is an exact extension of a well known theorem of sleszynski pringsheim and our proof is based on the truncation error analysis for such continued fractions the second purpose is to give refined error bounds for such continued fractions these bounds are also improvements of the scalar results and the numerical example given explains the effect of our results

mathematica evidence that ramanujan kills baker gammel wills a 1961 conjecture of baker gammel and wills asserts that if a function f is meromorphic in the unit ball and analytic at zero then a subsequence of its diagonal pade approximants converges uniformly in compact subsets omitting poles inasmuch as the denominators of the pade approximants are complex orthogonal polynomials and the convergence of sequences of pade approximants is determined largely by the behaviour of their poles the conjecture deals with distribution of zeros of complex orthogonal polynomials in this paper we present numerical evidence derived using the mathematica package that ramanujan s continued fractionh q z 1 qz 1 q 2z 1 q 3z 1 provides a counterexample provided q is appropriately chosen on the unit circle

collision statistics for random flights with anisotropic scattering and absorption for a broad class of random walks with anisotropic scattering kernels and absorption we derive explicit formulas that allow expressing the moments of the collision number n v performed in a volume v as a function of the particle equilibrium distribution our results apply to arbitrary domains v and boundary conditions and allow assessing the hitting statistics for systems where the typical displacements are comparable to the domain size so that the diffusion limit is possibly not attained an example is discussed for one dimensional random flights with exponential displacements where analytical calculations can be carried out

stieltjes continued fractions for polygamma functions speed of convergence it is well known that the polygamma functions k z dk 1 dzk 1 log z k 0 1 2 can be represented in the hall plane region arg z 0 in the present paper it is shown that the coefficients am k have the asymptotic behavior am k m2 16 m from this it is deduced that the nth approximant gn k z of g k z converges at the rate g k z gn k z a nb n 1 where the positive constants a and b depend upon k and z but are independent of n

some theorems on the rogers ramanujan continued fraction in ramanujan s lost notebook in his first two letters to g h hardy and in his notebooks ramanujan recorded many theorems about the rogers ramanujan continued fraction in his lost notebook he offered several further assertions the purpose of this paper is to provide proofs for many of the claims about the rogers ramanujan and generalized rogers ramanujan continued fractions found in the lost notebook these theorems involve among other things modular equations transformations zeros and class invariants

recurrence times in quasi periodic motion statistical properties role of cell size parameter dependence we investigate the probabilistic properties of recurrence times for the simplest form of aperiodic deterministic dynamics quasi periodic motion previous results using number theory techniques predict two fundamental recurrence times for uniform quasi periodic motion on a two dimensional torus while no analogous analytic result seems to exist for higher dimensional tori the two dimensional uniform case is reanalyzed from a more geometric point of view and new workable expressions are derived that enable us fully to understand and predict the recurrence phenomenon and to analyze its parameter dependence emphasis is placed on the statistical properties and in particular on the variability of recurrence times around their mean in relation to local farey tree structure higher dimensional tori are considered and seen to also display a high variability in their finite time recurrence behavior the results are finally extended to the non uniform quasi periodic case

algebraic description of the finite stieltjes moment problem abstract the stieltjes problem of moments seeks for a nondecreasing positive distribution function on the semi axis 0 so that its moments match a given infinite sequence of positive real numbers m 0 m 1 in his seminal paper investigations on continued fractions published in 1894 stieltjes gave a complete solution including the conditions for the existence and uniqueness in relation to his main goal the convergence theory of continued fractions one can also reformulate the stieltjes problem of moments as looking for a sequence of positive distribution functions 1 2 where the nth distribution function has n points of increase and m 0 m 1 m 2 n 1 represent its first 2n moments i e as the sequence of the finite stieltjes moment problems this view can be linked to iterative solution of large linear algebraic systems providing that m 0 m 1 are moments of some linear self adjoint and coercive operator a on a hilbert space with respect to a given vector f the finite stieltjes moment problems determine the iterations of the conjugate gradient method applied for solving a u f and vice versa here the existence and uniqueness is guaranteed by the properties of the operator a reformulation for finite sequences matrices and finite vectors is obvious this fundamental link raises a question on how the solution of the finite stieltjes moment problem can be described purely algebraically this has motivated the presented exposition built upon ideas published previously by several authors since the description uses matrices of moments it is not intended for numerical computations

vector continued fractions using a generalized inverse a real vector space combined with an inverse involution for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale the choice of sign for different components of the vector inverse permits construction of vector analogues of the jacobi continued fraction these vector jacobi fractions are related to vector and scalar valued polynomial functions of the vectors which satisfy recurrence relations similar to those of orthogonal polynomials the vector jacobi fraction has strong convergence properties which are demonstrated analytically and illustrated numerically

a note on bounds for the derivatives of continued fractions although it is difficult to differentiate analytic functions defined by continued fractions it is relatively easy in some cases to determine uniform bounds on such derivatives by perceiving the continued fraction as an infinite composition of linear fractional transformations and applying an infinite chain rule for differentiation

continued fractions with three limit points abstract on page 45 in his lost notebook ramanujan asserts that a certain q continued fraction has three limit points more precisely if a n b n denotes its n th partial quotient and n tends to in each of three residue classes modulo 3 then each of the three limits of a n b n exists and is explicitly given by ramanujan ramanujan s assertion is proved in this paper moreover general classes of continued fractions with three limit points are established

orthogonal laurent polynomials and two point pade approximants associated with dawson s integral starting from the coefficients of the power expansions for the dawson s integral both around the origin and infinity a linear functional acting on the space of laurent polynomials is defined in this paper properties of this functional are studied in connection with certain sequences of orthogonal laurent polynomials and two point pade approximants

rogers ramanujan and the baker gammel wills pade conjecture in 1961 baker gammel and wills conjectured that for functions f meromorphic in the unit ball a subsequence of its diagonal pade approximants converges uniformly in compact subsets of the ball omitting poles of f there is also apparently a cruder version of the conjecture due to pade himself going back to the early twentieth century we show here that for carefully chosen q on the unit circle the rogers ramanujan continued fraction

counting statistics a feynman kac perspective by building upon a feynman kac formalism we assess the distribution of the number of collisions in a given region for a broad class of discrete time random walks in absorbing and nonabsorbing media we derive the evolution equation for the generating function of the number of collisions and we complete our analysis by examining the moments of the distribution and their relation to the walker equilibrium density some significant applications are discussed in detail in particular we revisit the gambler s ruin problem and generalize to random walks with absorption the arcsine law for the number of collisions on the half line

approximants of sleszynski pringsheim continued fractions abstract we characterize the sequences zn of complex numbers which are sequences of approximants of continued fractions k an bn with an 1 bn and study some of their properties in particular we give truncation error bounds for such continued fractions

continued fractions with applications i introductory examples ii more basics iii convergence criteria iv continued fractions and three term recurrence relations v correspondence of continued fractions vi hypergeometric functions vii moments and orthogonality viii pade approximants ix some applications in number theory x zero free regions xi digital filters and continued fractions xii applications to some differential equations appendix some continued fraction expansions references index

control of a heat diffusion system through a fractional order nonlinear algorithm the application of the fc concepts has increased significantly in different fields of science and engineering because fc captures properties that classical integer order models neglect this paper studies a heat diffusion system that is described through the fractional operator s 0 5 under the control of a fractional nonlinear algorithm the tuning of the algorithm follows the optimization of performance control indices the results demonstrate the good performance of the proposed controller

on the divergence of the rogers ramanujan continued fraction on the unit circle this paper studies ordinary and general convergence of the rogers ramanujan continued fraction let the continued fraction expansion of any irrational number t 0 1 be denoted by 0 e 1 t e 2 t and let the i th convergent of this continued fraction expansion be denoted by c i t d i t let s t 0 1 e i 1 t d i t infinitely often where 5 1 2 let y s exp 2 it t e s it is shown that if y e y s then the rogers ramanujan continued fraction r y diverges at y s is an uncountable set of measure zero it is also shown that there is an uncountable set of points g c y s such that if y g then r y does not converge generally it is further shown that r y does not converge generally for y 1 however we show that r y does converge generally if y is a primitive 5m th root of unity for some m e n combining this result with a theorem of i schur then gives that the continued fraction converges generally at all roots of unity

orthogonality and recurrence for ordered laurent polynomial sequences we consider orderings of nested subspaces of the space of laurent polynomials on the real line more general than the balanced orderings associated with the ordered bases 1 z 1 z z 2 z 2 and 1 z z 1 z 2 z 2 we show that with such orderings the sequence of orthonormal laurent polynomials determined by a positive linear functional satisfies a three term recurrence relation reciprocally we show that with such orderings a sequence of laurent polynomials which satisfies a recurrence relation of this form is orthonormal with respect to a certain positive functional

structured matrices continued fractions and root localization of polynomials we give a detailed account of various connections between several classes of objects hankel hurwitz toeplitz vandermonde and other structured matrices stietjes and jacobi type continued fractions cauchy indices moment problems total positivity and root localization of univariate polynomials along with a survey of many classical facts we provide a number of new results

scalar levin type sequence transformations sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series the basic idea is to construct from a given sequence sn a new sequence s t sn where each sdepends on a finite number of elements sn1 snm often the sn are the partial sums of an infinite series the aim is to find transformations such that s converges faster than or sums sn transformations t sn n that depend not only on the sequence elements or partial sums sn but also on an auxiliary sequence of so called remainder estimates n are of levin type if they are linear in the sn and nonlinear in the n such remainder estimates provide an easy to use possibility to use asymptotic information on the problem sequence for the construction of highly efficient sequence transformations as shown first by levin it is possible to obtain such asymptotic information easily for large classes of sequences in such a way that the n are simple functions of a few sequence elements sn then nonlinear sequence transformations are obtained special cases of such levin type transformations belong to the most powerful currently known extrapolation methods for scalar sequences and series here we review known levin type sequence transformations and put them in a com mon theoretical framework it is discussed how such transformations may be constructed by either a model sequence approach or by iteration of simple transformations as illus tration two new sequence transformations are derived common properties and results on convergence acceleration and stability are given for important special cases extensions of the general results are presented also guidelines for the application of levin type sequence transformations are discussed and a few numerical examples are given keywords convergence acceleration extrapolation summation of divergent se ries stability analysis hierarchical consistency iterative sequence transformation levin type transformations algorithm linear convergence logarithmic con vergence fourier series power series rational approximation subject classifications ams mos 65b05 65b10 65b15 40a05 40a25 42c15

orthogonal rational functions on the real half line with poles in 0 the main objective is to generalize previous results obtained for orthogonal laurent polynomials and their application in the context of stieltjes moment problems to the multipoint case the measure of orthogonality is supposed to have support on 0 while the orthogonal rational functions will have poles that are assumed to be in the neighborhood of 0 and in this way orthogonal laurent polynomials will be a special case obtained when all the poles are at 0 and we shall introduce the restrictions on the measure and the locations of the poles gradually and derive recurrence relations christoffel darboux relations and the solution of the rational stieltjes moment problem under appropriate conditions

surprising variants of cauchy s formula for mean chord length we examine isotropic and anisotropic random walks which begin on the surface of linear n square n n or cubic n n n lattices and end upon encountering the surface again the mean length of walks is equal to n and the distribution of lengths n generally scales as n 1 5 for large n our results are interesting in the context of an old formula due to cauchy that the mean length of a chord through a convex body of volume v and surface s is proportional to v s it has been realized in recent years that cauchy s formula holds surprisingly even if chords are replaced by irregular insect paths or trajectories of colliding gas molecules the random walk on a lattice offers a simple and transparent understanding of this result in comparison to other formulations based on boltzmann s transport equation in continuum

a note on the relation between two convergence acceleration methods for ordinary continued fractions in this note we relate two methods of convergence acceleration for ordinary continued fractions the first one is due to lorentzen and waadeland jacobsen and waadeland 1980 1998 the second one to waadeland 1986 1987 1987 1987 1988

numerical evidence that the perturbation expansion for a non hermitian pt symmetric hamiltonian is stieltjes recently several studies of non hermitian hamiltonians having pt symmetry have been conducted most striking about these complex hamiltonians is how closely their properties resemble those of conventional hermitian hamiltonians this paper presents further evidence of the similarity of these hamiltonians to hermitian hamiltonians by examining the summation of the divergent weak coupling perturbation series for the ground state energy of the pt symmetric hamiltonian h p2 14x2 i x3 recently studied by bender and dunne for this purpose the first 193 nonzero coefficients of the rayleigh schrodinger perturbation series in powers of 2 for the ground state energy were calculated pade summation and pade prediction techniques recently described by weniger are applied to this perturbation series the qualitative features of the results obtained in this way are indistinguishable from those obtained in the case of the perturbation series for the quartic anharmonic oscillator which is known to be a stieltjes se

green s matrix from jacobi matrix hamiltonian we propose two ways for determining the green s matrix for problems admitting hamiltonians that have infinite symmetric tridiagonal i e jacobi matrix form on some basis representation in addition to the recurrence relation coming from the jacobi matrix the first approach also requires the matrix elements of the green s operator between the first elements of the basis in the second approach the recurrence relation is solved directly by continued fractions and the solution is continued analytically to the whole complex plane both approaches are illustrated with the non trivial but calculable example of the d dimensional coulomb green s matrix we give the corresponding formulas for the d dimensional harmonic oscillator as well

effective laguerre asymptotics it is known that the generalized laguerre polynomials can enjoy subexponential growth for large primary index in particular for certain fixed parameter pairs a z one has the large n asymptotic behavior l n a z sim c a z n a 2 1 4 e 2 sqrt nz we introduce a computationally motivated contour integral that allows efficient numerical laguerre evaluations yet also leads to the complete asymptotic series over the full parameter domain of subexponential behavior we present a fast algorithm for symbolic generation of the rather formidable expansion coefficients along the way we address the difficult problem of establishing effective i e rigorous and explicit error bounds on the general expansion a primary tool for these developments is an exp arc method giving a natural bridge between converging series and effective asymptotics

solving composite sum of powers via pade approximation and orthogonal polynomials with application to optimal pwm problem abstract this paper presents methods for solving the polynomial system j 1 k x j i j k 1 n x j i p i i 1 2 n which is called the composite sum of powers it is shown that these polynomial equation can be reduced to a single variable polynomial equations by exploiting the modified newton s identities in this paper we generalize this identity and solve it via pade approximation theory and the related theory of formal orthogonal polynomials fops because the solution forms the roots of fops we present several interesting computational procedures such as the use of three term reccurence formulas determinantal formulations and the computation of the eigenvalues of tridiagonal matrices the computation of this special polynomial system arise in practical engineering task of solving optimal odd symmetry single phase pulse width modulated pwm problem

optimal designs for rational models and weighted polynomial regression sion model of degree p with efficiency function 1 x are presented interest in these designs stems from the fact that they are equivalent to locally d optimal designs for inverse quadratic polynomial models for the unrestricted design space and p n the d optimal designs put equal masses on p 1 points which coincide with the zeros of an ultraspherical polynomial while for p n they are equivalent to d optimal designs for certain trigonometric regression models and exhibit all the curious and interesting features of those designs for the restricted design space 1 1 sufficient but not necessary conditions for the d optimal designs to be based on p 1 points are developed in this case the problem of constructing p 1 point d optimal designs is equivalent to an eigenvalue problem and the designs can be found numerically for n 1 and 2 the problem is solved analytically and specifically the d optimal designs put equal masses at the points 1 and at the p 1 zeros of a sum of n 1 ultraspherical polynomials a conjecture which extends these analytical results to cases with n an integer greater than 2 is given and is examined empirically 1 introduction weighted polynomial regression models with variance functions which depend on the explanatory variable have played and continue to play an important role in the development of classical optimal design theory the reasons for this are essentially twofold first there is a wealth of elegant mathematics associated with the construction of d optimal designs for many of these models in particular for certain classes of variance functions it is possible to show that the d optimal designs put equal masses on p 1 points of support where p is the degree of the polynomial embedded in the model and furthermore it is possible to use tools from the theory of differential equations and of canonical moments in order to establish that these support points coincide with the zeros of classical orthogonal polynomi

a worpitzky theorem for vector valued continued fractions generalizations of worpitzky s convergence theorem for ordinary continued fractions are proved for vector valued continued fractions defined by the samelson inverse for vectors our convergence proof is constructive and thus yields more refined truncation error estimates this work shows that some of our results are exact generalizations or improvements of the scalar ones

the lanczos algorithm for extensive many body systems in the thermodynamic limit we establish rigorously the scaling properties of the lanczos process applied to an arbitrary extensive many body system which is carried to convergence n and the thermodynamic limit n taken in this limit the solution for the limiting lanczos coefficients are found exactly and generally through two equivalent sets of equations given initial knowledge of the exact cumulant generating function the measure and the orthogonal polynomial system associated with the lanczos process in this regime are also given explicitly some important representations of these lanczos functions are given including taylor series expansions and theorems controling their general properties are proven

g continued fractions for basic hypergeometric functions ii in this paper we apply a modification of a generalized pringsheim s theorem to obtain a gcontinued fraction expansion for the quotient of two contiguous basic hypergeometric functions in arbitrarily many variables as an application we obtain a g continued fraction extension of the rogers ramanujan continued fraction 2003 elsevier inc all rights reserved

continued fractions with circular twin value sets we prove that if the continued fraction k a n 1 has circular twin value sets v 0 v 1 then k a n 1 converges except in some very special cases the results generalize previous work by jones and thron

complex jacobi matrices complex jacobi matrices play an important role in the study of asymptotics and zero distribution of formal orthogonal polynomials fops the latter are essential tools in several fields of numerical analysis for instance in the context of iterative methods for solving large systems of linear equations or in the study of pade approximation and jacobi continued fractions in this paper we present some known and some new results on fops in terms of spectral properties of the underlying infinite jacobi matrix with a special emphasis to unbounded recurrence coefficients here we recover several classical results for real jacobi matrices the inverse problem of characterizing properties of the jacobi operator in terms of fops and other solutions of a given three term recurrence is also investigated this enables us to give results on the approximation of the resolvent by inverses of finite sections with applications to the convergence of pade approximants

cauchy s formulas for random walks in bounded domains cauchy s formula was originally established for random straight paths crossing a body b rn and basically relates the average chord length through b to the ratio between the volume and the surface of the body itself the original statement was later extended in the context of transport theory so as to cover the stochastic paths of pearson random walks with exponentially distributed flight lengths traversing a bounded domain some heuristic arguments suggest that cauchy s formula may also hold true for pearson random walks with arbitrarily distributed flight lengths for such a broad class of stochastic processes we rigorously derive a generalized cauchy s formula for the average length travelled by the walkers in the body and show that this quantity depends indeed only on the ratio between the volume and the surface provided that some constraints are imposed on the entrance step of the walker in b similar results are also obtained for the average number of collisions performed by the walker in b

lyapounov exponents and quenched large deviations for multidimensional random walk in random environment assign to the lattice sizes z z d i i d random 2d dimensional vectors z z e e 1 whose entries take values in the open unit interval and add up to one given a realization of this environment let x n n 0 be a markov chain on z d which when at z moves one step to its neighbor z e with transition probability z z e we derive a large deviation principle for x n n by means of a result similar to the shape theorem of first passage percolation and related models this result produces certain constants that are the analogue of the lyapounov exponents known from brownian motion in poissonian potential or random walk in random potential we follow a strategy similar to sznitman

numerical analysis 2000 vol ii interpolation and extrapolation multivariate polynomial interpolation is a basic and fundamental subject in approximation theory and numerical analysis which has received and continues receiving not deep but constant attention in this short survey we review its development in the rst 75 years of this century including a pioneering paper by kronecker in the 19th century c 2000 elsevier science b v all rights reserved