knot concordance whitney towers and l 2 signatures we construct many examples of nonslice knots in 3 space that cannot be distinguished from slice knots by previously known invariants using whitney towers in place of embedded disks we define a geometric filtration of the 3 dimensional topological knot concordance group the bottom part of the filtration exhibits all classical concordance invariants including the cassongordon invariants as a first step we construct an infinite sequence of new obstructions that vanish on slice knots these take values in the l theory of skew fields associated to certain universal groups finally we use the dimension theory of von neumann algebras to define an l 2 signature and use this to detect the first unknown step in our obstruction theory

on the topological 4 genus of torus knots we prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus as an application we derive the best possible linear estimate of the topological slice genus for torus knots with non maximal signature invariant

estimating perturbative coefficients in quantum field theory and statistical physics we present a method to estimate perturbative coefficients in quantum field theory and statistical physics we show that our method works in a large number of cases and in a wide variety of areas the method is so reliable that it has enabled us to find several errors in various publications

prediction properties of aitken s iterated d 2 process of wynn s epsilon algorithm and of brezinski s iterated theta algorithm abstract the prediction properties of aitken s iterated 2 process wynn s epsilon algorithm and brezinski s iterated theta algorithm for formal power series are analyzed as a first step the defining recursive schemes of these transformations are suitably rearranged in order to permit the derivation of accuracy through order relationships on the basis of these relationships the rational approximants can be rewritten as a partial sum plus an appropriate transformation term a taylor expansion of such a transformation term which is a rational function and which can be computed recursively produces the predictions for those coefficients of the formal power series which were not used for the computation of the corresponding rational approximant

a derivation of extrapolation algorithms based on error estimates in this paper we shall emphasize the role played by error estimates and annihilation difference operators in the construction of extrapolation processes it is shown that this approach leads to a unified derivation of many extrapolation algorithms and related devices to general results about their kernels and that it opens the way to many new algorithms their convergence and acceleration properties could also be studied within this framework

characteristic boundary layers in real vanishing viscosity limits abstract in this paper we study the vanishing viscosity limit of initial boundary value problems for one dimensional mixed hyperbolic parabolic systems when the boundary is characteristic for both the viscous and the inviscid systems in particular we assume that an eigenvalue of the inviscid system vanishes uniformly we prove the stability of boundary layers expansions in small time i e before shocks for the inviscid system as long as the amplitude of the boundary layers remains sufficiently small in particular by using lagrangian coordinates we apply our result to physical systems like gasdynamics and magnetohydrodynamics with homogeneous dirichlet condition for the velocity at the boundary

method of summation of some slowly convergent series a new method of summation of slowly convergent series is proposed it may be successfully applied to the summation of generalized and basic hypergeometric series as well as some classical orthogonal polynomial series expansions in some special cases our algorithm is equivalent to wynn s epsilon algorithm weniger transformation e j weniger nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series computer physics reports 10 1989 189 371 or the technique recently introduced by cizek et al j cizek j zamastil l skala new summation technique for rapidly divergent perturbation series hydrogen atom in magnetic field journal of mathematical physics 44 3 2003 962 968 in the case of trigonometric series our method is very similar to the homeier s h transformation while in the case of orthogonal series to the k transformation two iterated methods related to the proposed method are considered some theoretical results and several illustrative numerical examples are given

the summation of the ordinary and renormalized perturbation series for the ground state energy of the quartic sextic and octic anharmonic oscillators using nonlinear sequence transformations the strongly divergent rayleigh schrodinger perturbation expansion for the ground state energy of the quartic sextic and octic anharmonic oscillator and a corresponding renormalized perturbation expansion f vinette and j ci ek j math phys 32 3392 1991 are summed by pade approximants by levin s sequence transformation d levin int j comput math b 3 371 1973 and by a closely related sequence transformation which was suggested recently e j weniger comput phys rep 10 189 1989 it is shown that the renormalized perturbation expansion can be summed much more easily than the original perturbation expansion from which it was derived and that levin s sequence transformation diverges and is not able to sum the perturbation expansions the pade summation of the renormalized perturbation expansions gives relatively good results in the quartic and sextic case in the case of the octic anharmonic oscillator even the renormalized perturbation expansion is not pade summable the best results are clearly obtained by the new sequence transformation which for instance is able to sum the renormalized perturbation expansions for the infinite coupling limits of the quartic sextic and octic anharmonic oscillator and which produces at least in the quartic and sextic case extremely accurate results

on a certain move generating link homology on the other hand the first author defined a unknot t ing operation on an oriented link diagram as in fig 0 2 11 it is proved in 11 that every knot can be deformed into a trivial knot by a finite sequence of unknotting operations it can be also proved see appendix that for two oriented and ordered n component links l k1 u k 2 u u k and l k uk u k l can be deformed into e by a finite sequence of unknotting operations if and only if ik k l ki lk k i l k i rood2 for every i 1 i n here lk denotes the linking number in s3

extended procedures for extrapolation to the limit in this paper new procedures for the extrapolation to the limit of slowly convergent sequences and functions are proposed they are based on the notions of error estimates and annihilation operators we obtain generalizations of the discrete and confluent e transformation which are the most general sequence and function transformations known so far it is shown that many transformations studied in the literature are included in our formalism particular cases of these procedures are discussed then several extensions of the e algorithm are given finally the procedure q is applied to our procedures to produce new procedures for extrapolation to the limit

two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series two new classes of nonlinear transformations the d transformation to accelerate the convergence of infinite integrals and the d transformation to accelerate the convergence of infinite series are presented in the course of the development of these transformations two interesting asymptotic expansions one for infinite integrals and the other for infinite series are derived the transformations d and d can easily be applied to infinite integrals 0f t dt whose integrands f t satisfy linear differential equations of the form f t s m k 1p k t f k t and to infinite series s r 1f r w terms f r satisfy a linear difference equation of the form f r s m k 1p k r d kf r such that in both cases the p k have asymptotic expansions in inverse powers of their arguments in order to be able to apply these transformations successfully one need not know explicitly the differential equation that the integrand satisfies or the difference equation that the terms of the series satisfy mere knowledge of the existence of such a differential or difference equation and its order m is enough this broadens the areas to which these methods can be applied the connection between the d and d transformations with some known transformations in shown the use and the remarkable efficiency of the d and d transformations are demonstrated through several numerical examples the computational aspects of these transformations are described in detail

effective characteristic polynomials and two point pade approximants as summation techniques for the strongly divergent perturbation expansions of the ground state energies of anharmonic oscillators pad e approximants are able to sum effectively the rayleigh schr odinger perturbation series for the ground state energy of the quartic anharmonic oscillator as well as the corresponding renormalized perturbation expansion e j weniger j ifmmode check c else v c fi i ifmmode check z else v z fi ek and f vinette j math phys 34 571 1993 in the sextic case pad e approximants are still able to sum these perturbation series but convergence is so slow that they are computationally useless in the octic case pad e approximants are not powerful enough and fail on the other hand the inclusion of only a few additional data from the strong coupling domain e j weniger ann phys ny to be published greatly enhances the power of summation methods the summation techniques which we consider are two point pad e approximants and effective characteristic polynomials it is shown that these summation methods give good results for the quartic and sextic anharmonic oscillators and even in the case of the octic anharmonic oscillator which represents an extremely challenging summation problem two point pad e approximants give relatively good results textcopyright 1996 the american physical society

casson towers and slice links we prove that a casson tower of height 4 contains a flat embedded disc bounded by the attaching circle and we prove disc embedding results for height 2 and 3 casson towers which are embedded into a 4 manifold with some additional fundamental group assumptions in the proofs we create a capped grope from a casson tower and use a refined height raising argument to establish the existence of a symmetric grope which has two layers of caps data which is sufficient for a topological disc to exist with the desired boundary as applications we present new slice knots and links by giving direct applications of the disc embedding theorem to produce slice discs without first constructing a complementary 4 manifold in particular we construct a family of slice knots which are potential counterexamples to the homotopy ribbon slice conjecture

on the summation of some divergent hypergeometric series and related perturbation expansions abstract divergent hypergeometric series 2 f 0 1 occur frequently in poincare type asymptotic expansions of special functions these divergent series 2 f 0 can be used for the evaluation of the corresponding special functions if suitable summation technique are applied there is considerable evidence that levin s sequence transformation 1973 and in particular also some closely related sequence transformations weniger 1989 which were derived recently sum divergent series 2 f 0 much more efficiently than pade approximants similar summation problems occur also in the case of divergent rayleigh schrodinger perturbation expansions of elementary quantum mechanical systems a comparison of the perturbation series for the quartic anharmonic oscillator with the closely related asymptotic series for the complementary error function shows that levin s sequence transformation and the recently derived new sequence transformations are again more efficient than pade approximants however the superiority of levin s sequence transformation and of the new sequence transformations is less pronounced in the case of the perturbation expansion

new summation technique for rapidly divergent perturbation series hydrogen atom in magnetic field the perturbation series for the ground state energy of the hydrogen atom in the external magnetic field is summed via the sequence transformations the formula for the large order behavior of the partial sums of the series is derived from this formula a new general sequence transformation is suggested this transformation contains free parameters that can be further optimized it is shown that if the renormalization approach is used the optimal choice of these parameters leads to the previously suggested weniger transformation

large order perturbation theory for a non hermitian pt symmetric hamiltonian a precise calculation of the ground state energy of the complex pt symmetric hamiltonian h p2 14x2 i x3 is performed using high order rayleigh schrodinger perturbation theory the energy spectrum of this hamiltonian has recently been shown to be real using numerical methods here we present convincing numerical evidence that the rayleigh schrodinger perturbation series is borel summable and show that pade summation provides excellent agreement with the real energy spectrum pade analysis provides strong numerical evidence that the once subtracted ground state energy considered as a function of 2 is a stieltjes function the analyticity properties of this stieltjes function lead to a dispersion relation that can be used to compute the imaginary part of the energy for the related real but unstable hamiltonian h p2 14 x2 ex3

joint use of the weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle point integrals ii higher order transformations the use of hyperasymptotics h and the weniger transformation wt has been proposed in a joint fashion for decoding the divergent asymptotic series generated by the steepest descent on a wide class of saddle point integrals evaluated across stokes sets r borghi phys rev e 78 026703 2008 in the present sequel the full development of the hyperasymptotic weniger transformation h wt up to the second order in h is derived numerical experiments carried out on several classes of saddle point integrals including the swallowtail diffraction catastrophe show the effectiveness of the second level h wt in particular when the integrals are evaluated beyond the asymptotic realm

development of the levin type algorithms for accelerating convergence of sequences abstract there are two aims of this paper firstly we shall introduce the determinantal representations of the new levin type algorithms and secondly we shall demonstrate further development of the levin type algorithms we consider the use of the levin type algorithms to accelerate the convergence of scalar sequence and their effectiveness for approximating the solution of a given power series is illustrated in process we shall demonstrate the convergence of each of the methods considered the approximate solution of the super enhanced levin algorithm and the efficient levin algorithm are found to be substantially more accurate than the cizek zamastil and skala transformation and the iterated aitken 2 algorithm

convergence analysis for a generalized richardson extrapolation process with an application to the d 1 transformation on convergent and divergent logarithmic sequences in an earlier work by the author the generalized richardson extrapolation process grep was introduced and some of its convergence and stability properties were discussed in a more recent work by the author a special case of grep which we now call grep 1 was considered and its properties were reviewed with emphasis on oscillatory sequences in the first part of the present work we give a detailed convergence and stability analysis of grep 1 as it applies to a large class of logarithmic sequences both convergent and divergent in particular we prove several theorems concerning the columns and the diagonals of the corresponding extrapolation table these theorems are very realistic in the sense that they explain the remarkable efficiency of grep 1 in a very precise manner in the second part we apply this analysis to the levin sidi d 1 transformation as the latter is used with a new strategy to accelerate the convergence of infinite series that converge logarithmically or to sum the divergent extensions of such series this is made possible by the observation that when the proper analogy is drawn the d 1 transformation is in fact a grep 1 we append numerical examples that demonstrate the theory

analytical and numerical studies of the convergence behavior of the j transformation abstract a new nonlinear sequence transformation the iterative j transformation was proposed recently by the author 1993 for this transformation a derivation based on the method of hierarchical consistency alternative recursive representations general properties an explicit expression for the kernel model sequences and its relation to other sequence transformations have been given the author 1994 the j transformation is of similar generality as the well known e algorithm brezinski 1980 havie 1979 in the present contribution some results on convergence acceleration properties of the j transformation are proved numerical test results are presented which show that the j transformation is a very powerful computational tool for convergence acceleration extrapolation and summation of divergent series

addition theorem of slater type orbitals a numerical evaluation of barnett coulson lowdin functions when using the one centre two range expansion method to evaluate multicentre integrals over slater type orbitals stos it may become necessary to compute numerical values of the corresponding fourier coefficients also known as barnett coulson lowdin functions bclfs bouferguene and jones 1998 j chem phys 109 5718 to carry out this task it is crucial to not only have a stable numerical procedure but also a fast algorithm in previous work bouferguene and rinaldi 1994 int j quantum chem 50 21 bclfs were represented by a double integral which led to a numerically stable algorithm but this turned out to be disappointingly time consuming the present work aims at exploring another path in which bclfs are represented either by an infinite series involving modified bessel functions or by an integral whose integrand is a smooth function both of these representations have the advantage of being symmetrical with respect to the cusp parameter a and the radial variable r as a consequence it is no longer necessary to split the integrals over r 0 into several components with a different analytical form in each of these a numerical study is also carried out to help select the most appropriate method to be used in practice

numerical computation of diffraction catastrophes with codimension eight a computational strategy aimed at evaluating diffraction catastrophes belonging to the x 9 family is presented the approach proposed is based on the use of power series expansions suitably derived for giving meaningful representation of the whole 0 x 9 subfamily jointly with a powerful sequence transformation algorithm the so called weniger transformation the convergence features of the above series expansions are investigated and several numerical experiments are carried out to assess the effectiveness of the retrieving action of the weniger transformation as well as the ease of implementation of the whole approach

an algorithm for a special case of a generalization of the richardson extrapolation process a special case of a generalization of the richardson extrapolation process is considered and its complete solution is given in closed form using this an algorithm for implementing the extrapolation is devised it is shown that this algorithm needs a very small amount of arithmetic operations and very little storage convergence and stability properties for some cases are also considered

on knots with trivial alexander polynomial we use the 2 loop term of the kontsevich integral to show that there are many knots with trivial alexander polynomial which do not have a seifert surface whose genus equals the rank of the seifert form this is one of the first applications of the kontsevich integral to intrinsically 3 dimensional questions in topology our examples contradict a lemma of mike freedman and we explain what went wrong in his argument and why the mistake is irrelevant for topological knot concordance

prediction properties of the t transformation some prediction properties of the t transformation of levin are analyzed for two kinds of limiting processes it is shown under certain circumstances that when the t transformation is applied to a power series the coefficients of the maclaurin series of the rational approximation obtained approximate those of the given series with increasing accuracy theoretical and numerical examples are appended

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

on l 1 approximation preface 1 preliminaries 2 approximation from finite dimensional subspaces of l1 3 approximation from finite dimensional subspaces in c1 k 4 unicity subspaces and property a 5 one sided l1 approximation 6 discrete lm1 approximation 7 algorithms appendices references author index subject index

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

on the melvin morton rozansky conjecture we prove a conjecture stated by melvin and morton and elucidated further by rozansky saying that the alexander conway polynomial of a knot can be read from some of the coefficients of the jones polynomials of cables of that knot i e coefficients of the colored jones polynomial we first reduce the problem to the level of weight systems using a general principle which may be of some independent interest and which sometimes allows to deduce equality of vassiliev invariants from the equality of their weight systems we then prove the conjecture combinatorially on the level of weight systems finally we prove a generalization of the melvin morton rozansky mmr conjecture to knot invariants coming from arbitrary semi simple lie algebras as side benefits we discuss a relation between the conway polynomial and immanants and a curious formula for the weight system of the colored jones polynomial

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

convergence accelerators in the computation of molecular integrals over slater type basis functions in the two range one center expansion method the so called nonlinear transformations may be of substantial help in many practical problems including the so called multicenter integrals in this work wynn s epsilon algorithm and the levin s ital u transformation have been used to improve the convergence of the series representation of three center two electron integrals obtained via the so called two range one center expansion method

joint use of the weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle point integrals when the weniger transformation is systematically used in resumming the asymptotic series arising from the application of the steepest descent method can display dramatic numerical instabilities that prevent it from improving the accuracy achievable via superasymptotics in the present paper an explanation of such pathologies through the concept of resurgence introduced by berry and howls several years ago within the context of hyperasymptotics is proposed in particular the way the topology of the whole set of the saddles influences the resummation capabilities of the weniger transformation is here investigated for the integrals defining the lowest order cuspoid diffraction catastrophes eventually a powerful and easily implementable resummation scheme based on a joint use of the weniger transformation and hyperasymptotics is proposed for taking care of the above pathologies such a joint action seems to encompass the main virtues of both approaches and the preliminary numerical results obtained from its application show that relative errors several orders of magnitude smaller than those achievable via superasymptotics can be achieved with modest implementation efforts

on a generalization of the richardson extrapolation process a convergence result for a generalized richardson extrapolation process is improved upon considerably and additional results of interest are proved an application of practical importance is also given finally some known results concerning the convergence of levin s t transformation are reconsidered in light of the results of the present work

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

an algorithm for a generalization of the richardson extrapolation process in this paper we present a recursive method designated the w m algorithm for implementing a generalization of the richardson extrapolation process that has been ntroduced in 8 compared to the direct solution of the linear systems of equations defining the extrapolation procedure this method requires a small number of arithmetic operations and very little storage the technique is also applied to solve recursively the coefficient problem associated with the rational approximations obtained by applying the d transformation of 6 13 to power series in the course of development a new recursive algorithm for implementing a very general extrapolation procedure is introduced which is similar to that given in 2 4 for solving the same problem a fortran program for the w m algorithm is also appended

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

mathematical properties of a new levin type sequence transformation introduced by cižek zamastil and skala i algebraic theory ci ek zamastil and skala j math phys 44 962 968 2003 introduced in connection with the summation of the divergent perturbation expansion of the hydrogen atom in an external magnetic field a new sequence transformation which uses as input data not only the elements of a sequence sn n 0 of partial sums but also explicit estimates n n 0 for the truncation errors the explicit incorporation of the information contained in the truncation error estimates makes this and related transformations potentially much more powerful than for instance pade approximants special cases of the new transformation are sequence transformations introduced by levin int j comput math b 3 371 388 1973 and weniger comput phys rep 10 189 371 1989 secs 7 9 numer algor 3 477 486 1992 and also a variant of richardson extrapolation philos trans r soc london ser a 226 299 349 1927 the algebraic theory of these transformations explicit expressions recurrence formulas explicit expres

error analysis of aitken s δ2 process abstract a sequence to sequence transformation called the 2 process by its developer aitken and recently analyzed by daniel shanks is successful in accelerating convergence in many convergent sequences and inducing convergence in some divergent ones it is shown here that the 2 process applied to sequences whose terms have cauchy distributions results in sequences whose terms still have the cauchy distribution and that repeated applications of the 2 process to a sequence with terms having uniform distribution simulating round off error and to sequences with terms having a normal distribution simulating measurement error yields in both cases sequences whose terms approach the cauchy distribution the result for the uniform distribution is proven that for the normal distribution is referenced

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

a new method for deriving pade approximants for some hypergeometric functions abstract in this work a unified method for obtaining the pade approximants for some hypergeometric functions is given this method is based on a set of linear equations that are obtained from the determinant expressions for pade approximants and their solution by using two simple theorems developed in the text

shape invariant potentials with pt symmetry suitable complexification of the well known hyperbolic rosen morse oscillator in one dimension is shown to give the fourth and last exactly solvable model which combines the shape and invariance

spectral analysis of the complex cubic oscillator using the exact semiclassical analysis we study the spectrum of a one parameter family of complex cubic oscillators the invariance property of the complex hamiltonians and the reality property of the spectrum are discussed analytic continuations of the spectrum in the complex parameter and their connections with the resonance problem for the real cubic oscillator are investigated the global analytic structure of the spectrum yields a branch point structure similar to the multivalued analytic structure discovered by bender and wu for the quartic oscillator

new hybrid non linear transformations of divergent perturbation series for quadratic zeeman effects the problem of hydrogen atoms in an external uniform magnetic field quadratic zeeman effect is studied by means of perturbation theory the power series for the ground state energy in terms of magnetic field strength b is divergent nevertheless it is possible to induce convergence of this divergent series by applying various non linear transformations such as those of shanks e levin t u v as well as the present omega algorithm these transformations of originally divergent perturbation series yield new sequences which then converge the induced convergence is however quite slow a new hybrid shanks levin non linear transform is devised here for accelerating these slowly converging series and sequences significant improvement in the convergence rate is obtained agreement with the exact results is excellent

stratified solutions for systems of conservation laws we study a class of weak solutions to hyperbolic systems of conservation balance laws in one space dimension called stratified solutions these solutions are bounded and regular in the direction of a linearly degenerate characteristic field of the system but not in other directions in particular they are not required to have finite total variation we prove some results of local existence and uniqueness

aitken sequences and generalized fibonacci numbers consider the sequence t generated by t i avn bv l l n 5i 2 where vi 1 t 2 a with a and b real of which the fibonacci sequence is a special case it is shown that if aitken acceleration is used on the sequence x defined by a v 1 v the resulting sequence is a subsequence of jc second if newton s method and the secant method are used with suitable starting values to solve the equation x2 ax b 0 then the sequences obtained from both of those methods are also subsequences of the original sequence

multivariate hermite interpolation by algebraic polynomials a survey abstract this is a survey of that theory of multivariate lagrange and hermite interpolation by algebraic polynomials which has been developed in the past 20 years its purpose is not to be encyclopedic but to present the basic concepts and techniques which have been developed in that period of time and to illustrate them with examples it takes classical hermite interpolation as a starting point but then successively broadens the assumptions so that finally interpolation of arbitrary functionals and the theory of singularities from algebraic geometry is discussed

on the propagation of polarization sets for systems of real principal type we define and study the polarization sets for vector valued distributions systems of real principal type are defined and we prove that for such systems of pseudodifferential operators we have a unique propagation of the polarization sets for the solutions

extrapolation methods for sommerfeld integral tails a review is presented of the extrapolation methods for accelerating the convergence of sommerfeld type integrals i e semi infinite range integrals with bessel function kernels which arise in problems involving antennas or scatterers embedded in planar multilayered media attention is limited to partition extrapolation procedures in which the sommerfeld integral is evaluated as a sum of a series of partial integrals over finite subintervals and is accelerated by an extrapolation method applied over the real axis tail segment spl alpha spl infin of the integration path where spl alpha 0 is selected to ensure that the integrand is well behaved an analytical form of the asymptotic truncation error or the remainder which characterizes the convergence properties of the sequence of partial sums and serves as a basis for some of the most efficient extrapolation methods is derived several extrapolation algorithms deemed to be the most suitable for the sommerfeld integrals are described and their performance is compared it is demonstrated that the performance of these methods is strongly affected by the horizontal displacement of the source and field points spl rho and by the choice of the subinterval break points furthermore it is found that some well known extrapolation techniques may fail for a number of values of spl rho and ways to remedy this are suggested finally the most effective extrapolation methods for accelerating sommerfeld integral tails are recommended

grope cobordism and feynman diagrams we explain how the usual algebras of feynman diagrams behave under the grope degree introduced in ct we show that the kontsevich integral rationally classifies grope cobordisms of knots in 3 space when the class is used to organize gropes this implies that the grope cobordism equivalence relations are highly nontrivial in dimension 3 we also show that the class is not a useful organizing complexity in 4 dimensions since only the arf invari ant survives in contrast measuring gropes according to height does lead to very interesting 4 dimensional information cot finally several low degree calculations are explained in particular we show that s equivalence is the same relation as grope cobordism based on the smallest tree with an internal vertex

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

a surgery sequence in dimension four the relations with knot concordance we present a systematic treatment of the classification problem for compact smooth 4 manifolds m it is modeled on the surgery exact sequence the central theorem in the classification of n manifolds n 5 the price for the extension to dimension 4 is a hole in m where a homotopy 1 skeleton should be there is no homotopy theoretic or surgical obstruction to complet ing m with a wedge of circles so that the completion has the topology of a compact smooth manifold this point set problem is all that stands between 4 manifolds and the tranquility that prevails in higher dimensions when m is simply connected only a point is missing from the model the applications of this are discussed in f and fq 1 the general theory has application to knot and link theory in particular knots with alexander polynomial 1 are characterized geometrically as knots admitting a certain

a user friendly extrapolation method for oscillatory infinite integrals developpement d une methode d extrapolation flexible pour les integrales infinies oscillatoires

determinantal representations for the j transformation the iterative j transformation homeier h h h 1993 some applications of nonlinear convergence accelerators int j quantum chem 45 545 562 is of similar generality as the well known e algorithm brezinski c 1980 a general extrapolation algorithm numer math 35 175 180 h vie t 1979 generalized neville type extrapolation schemes bit 19 204 213 the properties of the j transformation were studied recently in two companion papers homeier h h h 1994a a hierarchically consistent iterative sequence transformation numer algo 8 47 81 homeier h h h 1994b analytical and numerical studies of the convergence behavior of the j transformation j comput appl math to appear in the present contribution explicit determinantal representations for this sequence transformation are derived the relation to the brezinski walz theory brezinski c walz g 1991 sequences of transformations and triangular recursion schemes with applications in numerical analysis j comput appl math 34 361 383 is discussed overholt s process overholt k j 1965 extended aitken acceleration bit 5 122 132 is shown to be a special case of the j transformation consequently explicit determinantal representations of overholt s process are derived which do not depend on lower order transforms also families of sequences are given for which overholt s process is exact as a numerical example the euler series is summed using the j transformation the results indicate that the j transformation is a very powerful numerical tool