rational approximation to the thomas fermi equations abstract we show that a simple and straightforward rational approximation to the thomas fermi equation provides the slope at origin with unprecedented accuracy and that pade approximants of relatively low order are far more accurate than more elaborate approaches proposed recently by other authors we consider both the thomas fermi equation for isolated atoms and for atoms in strong magnetic fields

accurate calculation of the solutions to the thomas fermi equations abstract we obtain highly accurate solutions to the thomas fermi equations for atoms and atoms in very strong magnetic fields we apply the pade hankel method numerical integration power series with pade and hermite pade approximants and chebyshev polynomials both the slope at origin and the location of the right boundary in the magnetic field case are given with unprecedented accuracy

large order perturbation expansions for the charged harmonic oscillator the charged oscillator defined by the hamiltonian h d2 dr2 r2 lambda r in the domain 0 infinity is a particular case of the family of spike oscillators which does not behave as a supersingular hamiltonian this problem is analysed around the three regions lambda to infinity lambda to 0 and lambda to infinity by using rayleigh ritz large order perturbative expansions a path is found to connect the large lambda regions with the small lambda region by means of the renormalization of the series expansions in lambda finally the riccati pade method is used to construct an implicit expansion around lambda to 0 which extends to very large values of mod lambda mod

numerical study of the skin friction on a spheroid at incidence calcul numerique du champ d ecoulement tridimensionnel incompressible laminaire les equations sont simplifiees en negligeant les termes de diffusion pour obtenir les equations de navier stokes parabolisees

on a perturbation treatment of a model for mhd viscous flow we discuss the solution of a nonlinear ordinary differential equation that appears in a model for mhd viscous flow caused by a shrinking sheet we propose an accurate numerical solution and derive simple analytical expressions our results suggest that a recent perturbation treatment of the same problem exhibits a pathological behaviour and conjecture its probable cause

topological gauge theories and group cohomology we show that three dimensional chern simons gauge theories with a compact gauge groupg not necessarily connected or simply connected can be classified by the integer cohomology grouph4 bg z in a similar way possible wess zumino interactions of such a groupg are classified byh3 g z the relation between three dimensional chern simons gauge theory and two dimensional sigma models involves a certain natural map fromh4 bg z toh3 g z we generalize this correspondence to topological spin theories which are defined on three manifolds with spin structure and are related to what might be calledz2 graded chiral algebras or chiral superalgebras in two dimensions finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group establishing links with two dimensional holomorphic orbifold models

a numerical continuation method based on pade approximants a continuation algorithm is presented with a new predictor which is based on a rational representation of the solution path this algorithm belongs to the class of asymptotic numerical methods that connect perturbation techniques with a discretization principle without the use of a correction process several examples from shell buckling and from contact mechanics are analyzed to assess the efficiency and the reliability of the method

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

on quantum gauge theories in two dimensions two dimensional quantum yang mills theory is studied from three points of view i by standard physical methods ii by relating it to the largek limit of three dimensional chern simons theory and two dimensional conformal field theory iii by relating its weak coupling limit to the theory of reidemeister ray singer torsion the results obtained from the three points of view agree and give formulas for the volumes of the moduli spaces of representations of fundamental groups of two dimensional surfaces

a quick perturbative method for schrodinger equations within the framework of perturbation theory we propose firstly an iterative method which may serve as a source of optimal unperturbed solutions in both one and more dimensions it combines the runge kutta and newton algorithm and its efficiency is illustrated on a few quartic oscillators secondly admitting also an arbitrary perturbation of potentials we generalize the existing runge kutta one dimensional rayleigh schrodinger constructions of energies and wavefunctions to more dimensions

a fractional step solution method for the unsteady incompressible navier stokes equations in generalized coordinate systems abstract a fractional step method is developed for solving the time dependent three dimensional incompressible navier stokes equations in generalized coordinate systems the primitive variable formulation uses the pressure defined at the center of the computational cell and the volume fluxes across the faces of the cells as the dependent variables instead of the cartesian components of the velocity this choice is equivalent to using the contravariant velocity components in a staggered grid multiplied by the volume of the computational cell the governing equations are discretized by finite volumes using a staggered mesh system the solution of the continuity equation is decoupled from the momentum equations by a fractional step method which enforces mass conservation by solving a poisson equation this procedure combined with a consistent approximation of the geometric quantities is done to satisfy the discretized mass conservation equation to machine accuracy as well as to gain the favorable convergence properties of the poisson solver the momentum equations are solved by an approximate factorization method and a novel zebra scheme with four color ordering is devised for the efficient solution of the poisson equation several two and three dimensional laminar test cases are computed and compared with other numerical and experimental results to validate the solution method good agreement is obtained in all cases

direct calculation of accurate siegert eigenvalues the roots of the hankel determinant built with the taylor coefficients of the regularized logarithmic derivative of the eigenfunction give both the bound state and siegert eigenvalues resonances of separable quantum mechanical systems we obtain accurate positions and widths for the resonances of some commonly chosen test examples

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

simple systematics in the energy eigenvalues of quantum anharmonic oscillators in an earlier paper we proposed bhattacharya r roy d and bhowmick s 1998 phys lett a 244 9 a remarkably simple scheme for evaluating the ground state energy of x2m quantum anharmonic oscillators in the present paper we extend the scheme to evaluate the excited state energies as well and provide a motivation for the scheme the only information needed in the calculation is the first term of the standard strong coupling expansion for each state of the oscillators we have calculated the first few terms of the standard strong coupling expansion for some excited states for the quartic sextic and octic oscillators and for a given m we provide simple expressions for the strong coupling constants as functions of n which seem to reproduce any excited state energy with a reasonable accuracy moreover for the ground state we propose a simple expression for the first term of the strong coupling expansion which is globally true i e it gives the value of the constant for any m we then calculate a large number of excited state energies for the quartic sextic and octic oscillators over a wide range of values for the coupling parameter for any of the states the problem reduces to one of solving a polynomial equation and the predicted values closely agree with those obtained by other methods

a method for solving three dimensional viscous incompressible flows over slender bodies abstract a marching iterative method for solving the three dimensional incompressible and steady reduced navier stokes equations in general orthogonal coordinate systems is described with the velocity and the pressure as dependent variables the coupled set of the linearized finite difference continuity and momentum equations are solved iteratively without any splitting or factorization errors each iteration consists of spatial marching from the upstream boundary to the downstream boundary the discrete continuity and the two linearized crossflow momentum equations are satisfied at each marching step even when the mainstream momentum equation is not converged this solution procedure is equivalent to the solution of a single poisson like equation by the successive plane over relaxation method while other available solution methods employ a jacobi type iterative scheme and therefore are less efficient several properties of the numerical method have been assessed through a series of tests performed on the laminar incompressible flow over prolate spheroids at intermediate incidence

on some polynomial potentials in d dimensions the d dimensional schrodinger s equation is analyzed with regard to the existence of exact solutions for polynomial potentials under certain conditions on the interaction parameters we show that the polynomial potentials v8 r k 18 krk 8 0 and v10 r k 110 krk 10 0 are exactly solvable by examining the polynomial solutions of certain linear differential equations with polynomial coefficients the necessary and sufficient conditions for the existence of these exact solutions are discussed finding accurate solutions for arbitrary values of the potential parameters using the asymptotic iteration method is also presented

harmonic oscillations in a quasi relativistic regime a square root anharmonic oscillator potential is analysed as a model which simulates a quasi relativistic squeezing of the harmonic oscillator spectrum several eligible namely perturbative variational hill determinant and riccati pade methods of construction of its bound states are compared

strong coupling expansions for the symmetric oscillators we study the traditional problem of convergence of perturbation expansions when the hermiticity of the hamiltonian is relaxed to a weaker symmetry an elementary and quite exceptional cubic anharmonic oscillator is chosen as an illustrative example of such models we describe its perturbative features paying particular attention to the strong coupling regime efficient numerical perturbation theory proves suitable for such a purpose

solutions of the riccati equation and their relation to the toda lattice continued fraction solutions to the riccati equation are constructed using the idea of form invariance of this equation under linear fractional transformation of the dependent variable this method has recently been applied to the scalar case and a modification and extension to the corresponding matrix equation is described it is shown that the technique is related to euler s method for generating continued fraction solutions to the corresponding second order linear differential equation several particular examples are considered and lead to some well known continued fraction representations of standard functions it is demonstrated that the elements of the continued fraction solution are solutions of the toda lattice equations the authors have therefore discovered an intriguing connection between the riccati and the above equations which deserves further investigation

the anharmonic oscillator perturbation series for cubic and quartic energy distortion twenty five terms of the perturbation series are calculated for the first five energy levels of an oscillator with lambda x4 energy distortion and a similar calculation is carried out for the first six levels for lambda x3 distortion for the positive quartic the alternating series is summed using the aitken delta 2 transformation for lambda from 0 1 to 100 an exponential integral model and a truncated binomial model are used to transform the series for negative distortion energies and the complex energy eigenvalues are calculated for lambda from 0 01 to 1

the riccati pade method for coupled channel equations we present a method for calculating the energies of bound and quasibound states of quantum mechanical problems expressed in terms of coupled channel equations the approach is based on a rational approximation to the logarithmic derivative of the eigenfunction and leads to a simple quantization condition

a family of complex potentials with real spectrum we consider a two parameter non hermitian quantum mechanical hamiltonian operator that is invariant under the combined effects of parity and time reversal transformations numerical investigation shows that for some values of the potential parameters the hamiltonian operator supports real eigenvalues and localized eigenfunctions in contrast with other parity times time reversal symmetric models which require special integration paths in the complex plane our model is integrable along a line parallel to the real axis

homotopy perturbation technique abstract the homotopy perturbation technique does not depend upon a small parameter in the equation by the homotopy technique in topology a homotopy is constructed with an imbedding parameter p 0 1 which is considered as a small parameter some examples are given the approximations obtained by the proposed method are uniformly valid not only for small parameters but also for very large parameters

a three dimensional incompressible navier stokes flow solver using primitive variables an implicit finite difference computer code has been developed to solve the incompressible navier stokes equations in a three dimensional curvilinear coordinate system the pressure field solution is based on the pseudocompressibility approach in which a time derivative pressure term is introduced into the mass conservation equation the solution procedure employs an implicit approximate factorization scheme the reynolds stresses which are uncoupled from the implicit scheme are lagged by one time step to facilitate implementing various levels of the turbulence model test problems for external and internal flows are computer and the results are compared with existing experimental data the application of this technique for general three dimensional problems is then demonstrated

an iterative process based on homotopy and perturbation techniques abstract in this paper we discuss high order iterative methods whose principle has been presented recently n damil m potier ferry a najah r chari r lahmam communications in numerical methods in engineering 15 1999 701 708 these algorithms are optimised by appropriate choices of the homotopy transformation of the perturbation technique and of the strategy of control especially the use of pade approximants yields very efficient algorithms as compared with classical first order ones a classical non linear elastic shell problem permits us to assess the efficiency of the proposed method

on an iteration method for eigenvalue problems we discuss a recently proposed asymptotic iteration method for eigenvalue problems we analyse its rate of convergence the use of adjustable parameters to improve it and the relationship with an alternative method based on the same ideas

analytic continuation of taylor series and the boundary value problems of some nonlinear ordinary differential equations abstract we compare and discuss the respective efficiency of three methods with two variants for each of them based respectively on taylor maclaurin series pade approximants and conformal mappings for solving quasi analytically a two point boundary value problem of a nonlinear ordinary differential equation ode six configurations of ode and boundary conditions are successively considered according to the increasing difficulties that they present after having indicated that the taylor series method almost always requires the recourse to analytical continuation procedures to be efficient we use the complementarity of the two remaining methods pade and conformal mapping to illustrate their respective advantages and limitations we emphasize the importance of the existence of solutions with movable singularities for the efficiency of the methods particularly for the so called pade hankel method we show that this latter method is equivalent to pushing a movable pole to infinity for each configuration we determine the singularity distribution in the complex plane of the independent variable of the solution sought and show how this distribution controls the efficiency of the two methods in general the method based on pade approximants is easy to use and robust but may be awkward in some circumstances whereas the conformal mapping method is a very fine method which should be used when high accuracy is required

a comparison between adomian decomposition method and taylor series method in the series solutions in this work we will compare the performance of the adomian decomposition method and the taylor series method applied to the solution of linear and nonlinear ordinary differential equations the comparison shows that the decomposition method is reliable efficient and easy to use from a computational viewpoint although the two methods provide the solution in an infinite series the decomposition method provides a fast convergent series of easily computable components and eliminates cumbersome computational work needed by taylor series method

the strong coupling expansion for anharmonic oscillators we accurately calculate the strong coupling expansion for anharmonic oscillators by means of a robust and stable numerical algorithm the method applies to any state and to any anharmonicity degree by means of the perturbation coefficients we estimate the location of the branch points that determine the convergence radius of the strong coupling expansion

the anharmonic oscillator complex eigenvalues for the ground state with negative quartic or cubic energy distortion the perturbed ground state eigenvalues for the harmonic oscillator with negative quartic and with cubic energy distortion are calculated by numerical integration these agree with the author s results using an asymptotic series

reply to comments on a new efficient method for calculating perturbation energies using functions which are not quadratically integrable in a previous paper sk la and c zek 1996 a new approach to calculating perturbation energies of bound states based on the use of functions which are not quadratically integrable was suggested this method has been analysed in a comment by guardiola and ros 1996 the present comment is the reply to this previous comment the aim is to give a simple and straightforward proof of the method and clarify some other points important for its application

tight bounds to the schrodinger equation eigenvalues increasing tight upper and lower bounds to the schrodinger equation eigenvalues are obtained from the riccati equation for the logarithmic derivative of the wavefunction the solution of this non linear equation is written as a pade approximant and the bounds are given by the roots of a sequence of determinants numerical results are shown for the anharmonic and quartic oscillators

a hill series method for resonances a simple algorithm is given to treat perturbed oscillator bound states and resonances the method is applied to a bound state problem for which the energy is a non analytic function of a perturbation parameter and also to two resonance problems one of which has a spectrum with an unusual dual nature

generalized zeta function representation of groups and 2 dimensional topological yang mills theory the example of gl 2 q and pgl 2 q we recall the relation between zeta function representation of groups and two dimensional topological yang mills theory through mednikh formula we prove various generalisations of mednikh formulas and define generalization of zeta function representations of groups we compute some of these functions in the case of the finite group gl 2 x1d53d q and pgl 2 x1d53d q we recall the table characters of these groups for any q compute the frobenius schur indicator of their irreducible representations and give the explicit structure of their fusion rings

on a certain boundary value problem arising in shrinking sheet flows abstract this paper presents an analysis of the boundary value problem resulting from the magnetohydrodynamic mhd viscous flow influenced by a shrinking sheet with suction for the cases of two dimensional m 1 and axisymmetric m 2 shrinking the influences of the parameter m as well as the effects of suction parameter s and hartmann number m 2 on similar entrainment velocity f and flow characteristics are studied to this purpose the resulting nonlinear ordinary differential equation is solved numerically using the 4th order runge kutta method in combination with a shooting procedure the obtained results elucidate reliability and efficiency of the technique from which interesting features between the skin friction coefficient f 0 and the entrainment velocity f as function of the mass transfer parameter s can also be obtained

large order behaviour of the strong coupling perturbation expansion for anharmonic oscillators a new formula describing the large order behaviour of the strong coupling perturbation coefficients for the anharmonic oscillators with the hamiltonian is suggested a new method for the accurate calculation of the square root branch points of the energy from the numerical values of the coefficients is also suggested the branch points and the related minimal values of the coupling constant for which the expansion converges are calculated for the ground state of the quartic sextic octic and decadic oscillators

conformal mappings versus other power series methods for solving ordinary differential equations illustration on anharmonic oscillators the simplicity and the efficiency of a quasi analytical method for solving nonlinear ordinary differential equations ode is illustrated on the study of anharmonic oscillators ao with a potential v x x2 x2m m 0 the method bervillier 2008 nucl phys b801 296 applies a priori to any ode with two point boundaries one being located at infinity the solution of which has fixed singularities in the complex plane of the independent variable x a conformal mapping of a suitably chosen angular sector of the complex plane of x upon the unit disc centered at the origin makes convergent the transformed taylor series of the generic solution so that the boundary condition at infinity can be easily imposed in principle this constraint when applied on the logarithmic derivative of the wavefunction determines the eigenvalues to an arbitrary level of accuracy in practice for 0 or slightly negative the accuracy of the results obtained is astonishingly large with regard to the modest computing power used various aspects of the method and comparisons with some seemingly similar methods based also on expressing the solution as a taylor series are shortly reviewed presented and discussed

calculation of energy eigenvalues for the quantum anharmonic oscillator with a polynomial potential we choose the squeezed vacuum state as a one parameter trial wavefunction to minimize the energy of an anharmonic oscillator with a polynomial perturbation potential using the optimal vacuum state obtained we generate the optimal generalized number state basis and set up the hamiltonian in the normal ordered form then truncating the basis we produce finite dimensional matrices in a simple manner and find their eigenvalues by standard methods we will apply our method to the morse potential whose exact eigenvalues are known as a check on the accuracy of the results that we have derived two models of the double well and the quartic potential are considered as examples

complex variable hvpt complex variable hypervirial perturbation theory is applied to the case of oscillator and coulomb potentials perturbed by a single term potential of the form vxn or vrn respectively the trial calculations reported show that this approach can produce accurate complex energies for resonant states via a simple and speedy calculation and can also be useful in studies of pt symmetry and tunnelling resonance effects

fuchsian groups coverings of riemann surfaces subgroup growth random quotients and random walks abstract fuchsian groups acting as isometries of the hyperbolic plane occur naturally in geometry combinatorial group theory and other contexts we use character theoretic and probabilistic methods to study the spaces of homomorphisms from fuchsian groups to symmetric groups we obtain a wide variety of applications ranging from counting branched coverings of riemann surfaces to subgroup growth and random finite quotients of fuchsian groups as well as random walks on symmetric groups in particular we show that in some sense almost all homomorphisms from a fuchsian group to alternating groups an are surjective and this implies higman s conjecture that every fuchsian group surjects onto all large enough alternating groups as a very special case we obtain a random hurwitz generation of an namely random generation by two elements of orders 2 and 3 whose product has order 7 we also establish the analogue of higman s conjecture for symmetric groups we apply these results to branched coverings of riemann surfaces showing that under some assumptions on the ramification types their monodromy group is almost always sn or an another application concerns subgroup growth we show that a fuchsian group has n o 1 index n subgroups where is the measure of and derive similar estimates for so called eisenstein numbers of coverings of riemann surfaces a final application concerns random walks on alternating and symmetric groups we give necessary and sufficient conditions for a collection of almost homogeneous conjugacy classes in an to have product equal to an almost uniformly pointwise our methods involve some new asymptotic results for degrees and values of irreducible characters of symmetric groups

bound state method with elementary product wavefunctions we propose a new version of the multipoint pade technique to solve schrodinger equations in their first order riccati form our compact product representation of wavefunctions interpolates or matches available approximants at several points the mechanism of matching determines the energies and eliminates spurious solutions on v x mu 2x2 lambda x2 1 gx2 its efficiency is illustrated numerically

accurate eigenvalues and eigenfunctions for quantum mechanical anharmonic oscillators the representation of the taylor expansion of the logarithmic derivative of the wavefunction by means of a pade approximant followed by an appropriate quantization condition proves a powerful way of obtaining accurate eigenvalues of the schrodinger equation in this paper we investigate in detail some of the interesting features of this approach termed riccati pade method rpm by means of its application to anharmonic oscillators we analyse the occurrence of many roots in the neighborhoods of the physical eigenvalues in the weak coupling regime and also obtain accurate coefficients of the strong coupling expansion we finally investigate the global and the local accuracy of the rpm eigenfunctions

quasi relativistic harmonic oscillator bound states the quasi relativistic harmonic oscillator bound states constructed by znojil 1996 j phys a math gen 29 2905 are investigated via a new methodical proposal compared with those obtained by an anonymous referee from a direct numerical integration method of znojil s paper our results appear to be more favourable than those obtained by znojil via quasi perturbative variational hill determinant and riccati pade methods bound states with larger angular momenta l are also constructed

comment on an application of the asymptotic iteration method to a perturbed coulomb model we discuss a recent application of the asymptotic iteration method aim to a perturbed coulomb model contrary to what was argued before we show that the aim converges and yields accurate energies for that model we also consider alternative perturbation approaches and show that one of them is equivalent to that recently proposed by another author

a new perturbative approach to nonlinear problems a recently proposed perturbative technique for quantum field theory consists of replacing nonlinear terms in the lagrangian such as 4 by 2 1 and then treating as a small parameter it is shown here that the same approach gives excellent results when applied to difficult nonlinear differential equations such as the lane emden thomas fermi blasius and duffing equations

link between the strong coupling and the weak coupling asymptotic perturbation expansions for the quartic anharmonic oscillator for eigenvalues of the hamiltonian of the quartic anharmonic oscillator we give a representation which reproduces two well known limiting cases from the theory of the anharmonic oscillator expansions the weak coupling asymptotic perturbation and the convergent strong coupling we give an estimation of the radius of convergence of the strong coupling expansion for the ground state eigenvalue

comparison of homotopy perturbation method and homotopy analysis method comparison of homotopy perturbation method hpm and homotopy analysis method is made revealing that the former is more powerful than the later furthermore the hpm is further developed in this paper by applying the modern perturbation methods

numerical solution of incompressible flows by a marching multigrid nonlinear method on decrit un schema iteratif de resolution des equations de navier stokes pour un systeme de coordonnees orthogonales curvilignes l algorithme propose est independant du nombre de reynolds et peut etre applique a la resolution des equations d euler incompressibles