a generalization of the e algorithm abstract salam a on a generalization of the algorithm journal of computational and applied mathematics 46 1993 455 464 the algorithm is a quite powerful algorithm for accelerating the convergence of some classes of sequences it was generalized in two different ways some years ago in this paper a determinantal formula for its second generalization is obtained then its kernel the set of sequences which are transformed into a constant sequence is given as the solution of a difference equation

pairs of analytical eigenfunctions for the x2 λ x2 1 gx2 interaction a general method is discussed for deriving pairs of exact analytical eigenfunctions of the x2 x2 1 gx2 interaction recently discussed by blecher and leach and commented on by gallas it is shown that those pairs always consist of an odd and an even parity solution in each case and g are related by the conditions 0 and g as a result of their investigations the authors conjecture the possible existence of an infinite number of solution pairs having and g connected by several different relations

rational approximation schemes for unstable hele shaw flows abstract a rational approximation method or spectral method is developed and applied to the numerical calculation of unstable free surface flows in a hele shaw cell it is used to solve the potential problem both eulerian and lagrangian representations of the moving boundary have been used the evolution equation for the moving boundary is reduced to a system of coupled ordinary differential equations which are solved numerically by a second order implicit procedure with adaptive time steps the unstable hele shaw problem has two types of exact solution one the saffman finger remaining analytic for all time and the other leading to a cusped profile in a finite time period the efficacy of the complete numerical procedure is demonstrated by comparing the obtained numerical solutions with these exact solutions for the saffman finger we get excellent results for more than three times as long a time period as previously reported we also show that a standard linear approximation to the potential is not nearly as successful as a nonlinear approximation correct qualitative behaviour is obtained for the more difficult cusping problem

the method of lines and invariant imbedding for elliptic and parabolic free boundary problems a combination of the method of lines and invariant imbedding is suggested as a general purpose numerical algorithm for free boundary problems its effectiveness is illustrated by computing the solidification of a binary alloy in one dimension electrochemical machining and hele shaw flow in two dimensions and a stefan and ablation problem in three dimensions

quadratic fields with special class groups for every prime number p 5 it is shown that under certain hypotheses on x e q the imaginary quadratic fields q vx2p 6xp 1 have ideal class groups with noncyclic p parts several numerical examples with p 5 and 7 are presented these include the field q v 4805446 123032518648268510536 the 7 part of its class group is isomorphic to c 7 x c 7 x c 7 where c n denotes a cyclic group of order n

a unified treatment of schrodinger s equation for anharmonic and double well potentials the authors exploit some exact soluble model potentials vn alpha beta v a b lambda x6 in order to extrapolate as lambda to 0 reliable eigenvalues of the one dimensional schrodinger equations with either anharmonic or symmetric double well potentials of the form v a b 1 2ax2 bx2 b 0 their procedure which corresponds to low order rayleigh schrodinger perturbation theory is found to be competitive with both high order pade summation of conventional rspt and large scale variational calculations using harmonic oscillators basis functions

the hd and hd methods for accelerating the convergence of three center nuclear attraction and four center two electron coulomb integrals over b functions and their convergence properties abstract three center nuclear attraction and four center two electron coulomb integrals over slater type orbitals are required for ab initio and density functional theory dft molecular structure calculations they occur in many millions of terms even for small molecules and require rapid and accurate evaluation the b functions are used as a basis set of atomic orbitals these functions are well adapted to the fourier transform method that allowed analytical expressions for the integrals of interest to be developed rapid and accurate evaluation of these analytical expressions is now made possible by applying the hd and hd methods for accelerating the convergence of the semi infinite oscillatory integrals the convergence properties of the new methods are analysed the numerical results section shows the high predetermined accuracy and the substantial gain in the calculation times obtained using the new methods

numerical solutions for unstable hele shaw flows abstract a finite difference procedure using nonuniform grids is developed and applied to the unstable flow in a hele shaw cell good agreement with the analytic solutions is found for both the cusping flow and the saffman finger flow

the schrodinger equation for the x2 λx2 1 gx2 interaction certain exact odd and even wavefunctions for the potential x2 x2 1 gx2 x in are obtained the coefficients of the wavefunctions and expressions for the energies are obtained using the symbolic manipulation code reduce the chief use of these results is to provide a check for the effectiveness of numerical algorithms the results supplement those of flessas 1982 for the same potential on xe 0

hele shaw flow with a cusping free boundary abstract hele shaw flow with suction for an incompressible fluid in a porous medium is introduced as a nontrivial model problem for two dimensional transient free boundary problems in this model an initially smooth free boundary can move with unbounded velocities and develop cusps two analytic solutions are worked out for a special geometry with which numerical results can be compared it is then shown that the method of lines can reasonably well reproduce the analytic solutions although at the expense of long computer times some strategies for improving the efficiency of the method of lines including a multi grid method are described

a rodrigues formula approach to determining closed form solutions to the schrodinger equation for symmetric anharmonic oscillators a natural generalization to the rodrigues formula for the hermite polynomials leads to the definition of sets of polynomials under certain constraints they are used to explain the existence of closed form solutions to the schrodinger equation for symmetric anharmonic potential wells and to obtain further closed form solutions

lower bounds for quartic anharmonic and double well potentials rigorous and remarkably accurate lower bounds to the lower eigenvalue spectrum of the schrodinger equation with quartic anharmonic and symmetric double well potentials of the form v a b ax2 2 bx4 b 0 are presented this procedure exploits some exactly soluble model potentials and appears to be of quite general utility

applications of numerical conformal mapping abstract the application of conformal mapping methods to the solution of free surface flow problems is considered methods of numerical conformal mapping based on fourier series are extended to handle efficiently problems with time dependent boundaries they are shown to be practicable only for moderately distorted geometries extensions of the menikoff zemach method to breaking geometries are presented these latter methods are robust at quite large distortions but degrade prematurely in time dependent problems at amplitudes smaller than achieved by our recent vortex methods

a numerical study of unstable hele shaw flow abstract a numerical procedure which is based on an integral equation for the normal velocity at the interface is developed for the unstable flow with surface tension in a hele shaw cell the procedure has been validated by comparing solutions obtained by it with published results it has also been applied to a study of the effect of small changes in the initial data on the later profiles and it was found that even very small differences can lead to large differences in the profiles at later times

numerical solution of the hele shaw equations abstract an algorithm is presented for approximating the motion of the interface between two immiscible fluids in a hele shaw cell the interface is represented by a set of volume fractions we use the simple line interface calculation method along with the method of fractional steps to transport the interface the equation of continuity leads to a poisson equation for the pressure the poisson equation is discretized near the interface where the velocity field is discontinuous the discretization is based on a weak formulation of the continuity equation interpolation is used on each side of the interface to increase the accuracy of the algorithm the weak formulation as well as the interpolation are based on the computed volume fractions this treatment of the interface is new the discretized equations are solved by a modified conjugate gradient method surface tension is included and the curvature is computed through the use of osculating circles for perturbations of small amplitude a good agreement is found between the numerical results and linearized perturbation theory numerical results are presented for the finite amplitude growth of unstable fingers

exact solutions of the schrodinger equation for nonseparable anharmonic oscillator potentials in two dimensions an explicit procedure is given to construct exact closed form solutions to the time independent schrodinger equation in two dimensions 2 w x y 0 where w x y is a polynomial potential of degree greater than two not separable in cartesian coordinates several examples are discussed for which w x y is a sextic polynomial as has already been seen in studies of the corresponding one dimensional problem a complete set of eigenvalues and wave functions is not found however these closed form solutions can be used to check the accuracy and efficiency of numerical algorithms

dynamic group approach to the x2 λx2 1 gx2 potential a modified operator method based upon the so 2 1 dynamic group is applied to the one dimensional anharmonic oscillator potential v x x2 x2 1 gx2 a tilting transformation is carried out in order to improve the rate of convergence of the algebraic perturbation series very accurate results are obtained for the energy eigenvalues and for the wave functions especially in the case of small g values

a well posed numerical method to track isolated conformal map singularities in hele shaw flow we present a new numerical method for calculating an evolving 2d hele shaw interface when surface tension effects are neglected in the case where the flow is directed from the less viscous fluid into the more viscous fluid the motion of the interface is ill posed small deviations in the initial condition will produce significant changes in the ensuing motion this situation is disastrous for numerical computation as small roundoff errors can quickly lead to large inaccuracies in the computed solution our method of computation is most easily formulated using a conformal map from the fluid domain into a unit disk the method relies on analytically continuing the initial data and equations of motion into the region exterior to the disk where the evolution problem becomes well posed the equations are then numerically solved in the extended domain the presence of singularities in the conformal map outside of the disk introduces specific structures along the fluid interface our method can explicitly track the location of isolated pole and branch point singularities allowing us to draw connections between the development of interfacial patterns and the motion of singularities as they approach the unit disk in particular we are able to relate physical features such as finger shape side branch formation and competition between fingers to the nature and location of the singularities the usefulness of this method in studying the formation of topological singularities self intersections of the interface is also pointed out

an eulerian method for transient nonlinear free surface wave problems abstract an eulerian difference method is developed for the transient potential flow of an incompressible fluid with fully nonlinear free surface conditions the free surface coordinate y x t and the velocity potential x y t on the free surface are recognized as the primary unknowns to be solved as an initial value problem from the pair of nonlinear partial differential equations representing the dynamic and the kinematic conditions of the free surface the continuity relation 2 0 for the velocity potential x y t over the flow field below the free surface is recognized as a subsidiary condition to be enforced at all times the field of computation is transformed into a time invariant cartesian region with the free surface x t represented by a coordinate line or surface the iterative solution for x y t p x y t in this fixed field of computation is facilitated by the use of fast fourier transform fft the iterative process converges rapidly in terms of this converged x y t the free surface location x t and its potential x y t are advanced in time results from two planar examples are illustrated the method is equally applicable to problems in three space dimensions possibly involving interactive matching with neighboring flow fields if the initial free surface potential x y t 0 is unknown difficulties may be encountered in data specification for securing a well posed problem for solution

computation of hele shaw flows with free boundaries abstract a novel method for calculation of hele shaw flows with receding free boundaries is presented the method is applied to flows with suction from a point sink and to flow in a channel with parallel walls in each case the unknown fluid region is mapped conformally onto the unit disc the free boundary being mapped onto the unit circle this mapping which is a function of position and time is calculated numerically at points on the unit circle using a version of the boundary integral method the free boundary is thus found without explicit calculation of the pressure at internal points and the computation times are much less than those for other numerical methods for this problem numerical results are compared with explicit analytic solutions for several test problems

on some relations among numerical conformal mapping methods abstract gaier and gutknecht have shown that many numerical methods for producing the conformal map from the unit disk to a simply connected region share a common theoretical basis as solutions of nonlinear integral equations arising from the hilbert transform of a function of the boundary correspondence we give a brief presentation of this classification and extend it somewhat to include some equations for the inverse correspondence such as those of menikoff and zemach noble and schwarz christoffel the use of explicit maps and the method of bisshopp are also brought into this framework an example illustrating the use of explicit maps is given

an embedding method for the numerical solution of the cathode design problem in electrochemical machining an embedding method is considered for the numerical determination of the cathode shape for a given anode shape in a two dimensional electrolytic cell reference is made to an analytic solution to illustrate the difficulties which have to be overcome by any numerical method applied to this ill posed problem the embedding method searches amongst the direct solutions i e anodes for given cathodes since these resulting from a well posed problem are less troublesome to compute for anodes of the form g x x the transverse direction the embedding method gives good results for less than 30 60 percent of the for which the cathode first contains cusps or becomes triple valued it is shown that the embedding method performs better than some other numerical methods in overcoming the error growth difficulties associated with the illposedness of the problem

on the simultaneous eigenproblem for the x2 λ x2 1 gx2 1 interaction extension of gallas results an ansatz for the eigenfunctions of the x2 x2 1 gx2 1 interaction has been recently introduced by blecher and leach using this ansatz gallas has obtained five solution pairs for 2g 3g 2 gallas results are generalised and extended here

boundary approximation methods for potential problems associated with moving boundary problems abstract a boundary approximation method or spectral method for the numerical solution of the potential problem associated with a certain class of moving boundary problems is analyzed the questions of existence and convergence of a sequence of best approximations and the rate of convergence are studied for two dimensional problems using classical complex function theory numerical results for an example for which an exact solution is known are presented

some observations on the nature of solutions for the interaction v x x2 λx2 1 gx2 the authors study the nature of exact solutions for the nonpolynomial interaction in particular it is shown that all the solutions are supersymmetric it is also shown that this problem admits a dynamical su 2 symmetry so that an arbitrarily large part of the spectrum but not the whole spectrum can be determined exactly

a conservative finite element method for one dimensional stefan problems with appearing and disappearing phases abstract a one dimensional stefan problem involving several phases solid liquid vapor which can appear and disappear is considered a numerical method for such problems is described and numerically tested it is derived from a previous method of the same authors for one phase problems it is a front tracking method based on finite elements in space and time the treatment of the boundary conditions and the computation of the moving boundaries guarantee the exact conservation of heat energy the numerical experiments demonstrate the accuracy and efficiency of the method moreover they show that the appearing interfaces start smoothly with a vanishing initial speed to the authors knowledge this property has not been mathematically proved

a conformal map formula for difficult cases abstract a simple approximate formula is derived for the boundary map function of a conformal map which provides a remarkably good fit for mappings to regions with highly distorted boundaries the approximation follows from the reduction of a nonlocal integral equation for the map function to a local equation based on the nature of the crowding associated with distorted regions numerical examples are given

class groups of complex quadratic fields we present 75 new examples of complex quadratic fields that have 5 rank of their class groups 5 3 only one of these fields has 5 rank of its class group 3 the field q 25855935i511807 has a class group isomorphic to c 5 x c 5 x c 5 x c 5 x 7 2 x 7 11828 the fields were obtained by applying ideas of j f mestre to the 5 isogeny 11 xn 1

on quadratic fields with large 3 rank davenport and heilbronn defined a bijection between classes of binary cubic forms and classes of cubic fields which has been used to tabulate the latter we give a simpler proof of their theorem then analyze and improve the table building algorithm it computes the multiplicities of the o x general cubic discriminants real or imaginary up to x in time o x and space o x or more generally in time o x x m and space o m x for a freely chosen positive m a variant computes the 3 ranks of all quadratic fields of discriminant up to x with the same time complexity but using only m o 1 units of storage as an application we obtain the first 1618 real quadratic fields with r3 4 and prove that q 5393946914743 is the smallest imaginary quadratic field with 3 rank equal to 5

exact analytical eigenfunctions for the x2 λ x2 1 gx2 interaction the author discusses an ansatz for the eigenfunctions of the x2 x2 1 gx2 interaction introduced by blecher and leach 1987 the author shows that the ansatz provides pairs of solutions in five cases having and g connected by 6g2 4g and conjectures the possible existence of an infinite number of such solution pairs

rayleigh taylor instability and the use of conformal maps for ideal fluid flow abstract potential theory permits ideal fluid dynamics to be formulated in terms of boundary motion in two dimension the flow can then be found using conformal mapping the evolution of some rayleigh taylor instabilities is calculated well into the large amplitude nonlinear regime the rayleigh taylor calculation for atwood ratio unity is used as a prototype for a system of theoretical and numerical techniques exploiting complex variable theory and high order quadrature methods

on ranks of twists of elliptic curves and power free values of binary forms let e be an elliptic curve over q by the rank of e we shall mean the rank of the group of rational points of e mestre 31 improving on the work of neron 34 cf 13 39 and 46 has shown that there is an infinite family of elliptic curves over q with rank at least 12 however computational work see 3 4 7 and 47 suggests that a typical elliptic curve will have much smaller rank with curves of rank 0 or 1 being predominant indeed brumer 6 has proved subject to the birch swinnerton dyer conjecture the shimura taniyama weil conjecture and the generalized riemann hypothesis that the average rank of an elliptic curve ordered according to its faltings height is at most 2 3 in this article we shall study the behaviour of the rank as we run over twists of a given elliptic curve over q that is we shall restrict our attention to families of elliptic curves defined over q which are isomorphic over c there are families of quadratic cubic quartic and sextic twists see for example proposition 5 4 of chapter x of 42 let e be an elliptic curve over q with weierstrass equation y2 x ax b and for any non zero integer d let ed 2 3 denote a quadratic twist of e given by the equation dy2 x ax b let r d denote the rank of ed note that if d1 and d2 are non zero integers then ed is isomorphic to ed over q if and only if d1 d2 is the square of a rational number subject to the conjectures of birch and swinnerton dyer and of shimura taniyama and weil goldfeld 14 conjectured in 1979 that

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

schwarz christoffel methods for conformal mapping of regions with a periodic boundary abstract numerical conformal mapping methods for regions with a periodic boundary have been developed these methods are based on the generalized schwarz christoffel equation and can deal with boundary curves of arbitrary forms i e made up of one or more rectifiable jordan curves high order quadrature rules have been implemented in order to increase accuracy of the mapping this is of particular relevance to highly accurate grid generation techniques required by for example implementation of high order compact finite difference discretization schemes

schwarz christoffel mappings a general approach abstract a general method is developed for constructing equations of the schwarz christoffel type these equations define conformal transformations between regions of the complex plane each of whose boundaries may consist of straight or continuously curved line segments and corners the more familiar types of schwarz christoffel equations are rederived and some new types are obtained

computing the residue of the dedekind zeta function assuming the generalized riemann hypothesis bach has shown that one can calculate the residue of the dedekind zeta function of a number field k by a clever use of the splitting of primes p x with an error asymptotically bounded by 8 33 log d k sqrt x log x where d k is the absolute value of the discriminant of k guided by weil s explicit formula and still assuming grh we make a different use of the splitting of primes and thereby improve bach s constant to 2 33 this results in substantial speeding of one part of buchmann s class group algorithm

evaluating infinite integrals involving bessel functions of arbitrary order abstract the evaluation of intergrals of the form i n 0 x j n x d x is considered in the past the method of dividing an oscillatory integral at its zeros forming a sequence of partial sums and using extrapolation to accelerate convergence has been found to be the most efficient technique available where the oscillation is due to a trigonometric function or a bessel function of order n 0 1 here we compare various extrapolation techniques as well as choices of endpoints in dividing the integral and establish the most efficient method for evaluating infinite integrals involving bessel functions of any order n not just zero or one we also outline a simple but very effective technique for calculating bessel function zeros