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

on thermal stability of a reactive third grade fluid in a channel with convective cooling the walls in this paper the thermal stability of a reactive third grade liquid flowing steadily between two parallel plates with symmetrical convective cooling at the walls is investigated the system is assumed to exchange heat with the ambient following newton s cooling law and the reaction is exothermic under arrhenius kinetics neglecting the consumption of the material approximate solutions are constructed for the governing nonlinear boundary value problem using regular perturbation techniques together with a special type of hermite pade approximants and important properties of the velocity and temperature fields including bifurcations and thermal criticality conditions are discussed it is observed that a combined increase in non newtonian parameter and convective cooling enhances the thermal stability of the material

the greek banking system reanalysis of performance abstract zopounidis et al used an ordinal regression analysis to assess an additive utility model and to obtain final ranking of a representative sample of commercial greek banks c zopounidis d k despotis e stavropoulou multiattribute evaluation of greek banking performance applied stochastic models and data analysis 11 1995 97 107 in this paper we reanalyze their data by means of a new multivariate analysis method called co plot a two dimensional graphic display technique designed to analyze observations e g 16 banks and attributes e g 7 attributes simultaneously the method produces three results 1 similarity among the banks by the composite of all attributes involved five clusters are obtained 2 the structure of correlations among the attributes three clusters are obtained and 3 the mutual relationships between the banks and the attributes the banks are mapped into a partial order according to their increased performance to obtain their rating the final ranking obtained by the co plot method differed from that obtained by zopounidis et al

on the divergence of gradient expansions for kinetic energy functionals in the potential functional theory we consider the density of a fermionic system as a functional of the potential in one dimensional case where it is approximated by the thomas fermi term plus semiclassical corrections through the gradient expansion we compare this asymptotic series with the exact answer for the case of the harmonic oscillator and the morse potential it is found that the leading thomas fermi term is in agreement with the exact density but the subdominant term does not agree in terms of the asymptotic behavior because of the presence of oscillations in the exact density but their absence in the gradient expansion however after regularization of the density by convolution with a gaussian the agreement can be established even in the subdominant term moreover it is found that the expansion is always divergent and its terms grow proportionally to the factorial function of the order similar to the well known divergence of perturbation series in field theory and the quantum anharmonic oscillator pade hermite approximants allow summation of the series and one of the branches of the approximants agrees with the density

equivalence between maximum entropy principle and enforcing du tds we study here a hitherto unexplored microscopic connection between the well known thermodynamical relation du tds and jaynes maximum entropy principle maxent for determining probability distributions for the canonical ensemble

how to select and how to rank projects the promethee method abstract in this paper we present the promethee methods a new class of outranking methods in multicriteria analysis their main features are simplicity clearness and stability the notion of generalized criterion is used to construct a valued outranking relation all the parameters to be defined have an economic signification so that the decision maker can easily fix them two ways of treatment are proposed it is possible to obtain either a partial preorder promethee i or a complete one promethee ii both on a finite set of feasible actions a comparison is made with the electre iii method the stability of the results given by the two methods is analysed numerical applications are given in order to illustrate the properties of the new methods and some further problems are discussed

hypervirial perturbation theory revisited a wide range of topics in the area of hypervirial perturbation theory hvpt is discussed it is shown that with the use of a few simple procedures hvpt is capable of high accuracy for many problems results from many previous works in the literature are found to be improvable by the careful use of hvpt with appropriate choice of the unperturbed potential and of the origin at which the energy expansion is carried out two multi well problems from the literature are analysed in detail to show the value of a combination of hvpt and finite difference methods

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

high order analysis of the limit cycle of the van der pol oscillator we have applied the lindstedt poincare method to study the limit cycle of the van der pol oscillator obtaining the numerical coefficients of the series for the period and for the amplitude to order 859 hermite pade approximants have been used to extract the location of the branch cut of the series with unprecedented accuracy 100 digits both series have then been resummed using an approach based on pade approximants where the exact asymptotic behaviors of the period and the amplitude are taken into account our results improve drastically all previous results obtained on this subject

high order differential approximants we introduce a new form of differential approximant for the summation of power series the method is a special type of pade hermite approximant it consists of a high order linear differential equation with polynomial coefficients that is satisfied approximately by the partial sum of the power series this method is able to reproduce the polylogarithmic functions exactly numerical evidence suggests that this is currently one of the best methods of singularity analysis for many problems

algebraic approximants a new method of series analysis the authors propose a new method of series analysis in which the available series coefficients are fitted to an algebraic equation such a proposal restricts solutions to have algebraic singularities with rational critical exponents the advantage of the method over the method of differential approximants is that in those cases where the two methods are exact the algebraic approximant method usually though not invariably requires far fewer series coefficients for its exact specification and hence discovery the method is therefore potentially valuable in the study of two dimensional critical systems in particular

application of the quadratic approximants to simple cubic lattice trees recent work has drawn attention to the utility of algebraic approximants for series expansion analysis the special case of quadratic approximants are applied to lattice trees weakly embedded on the simple cubic lattice the resulting estimates of xc the reciprocal of the growth constant are compared with those obtained from other methods of analysis

bender wu branch points in the cubic oscillator the analytic continuation of the resonances of the cubic anharmonic oscillator to complex values of the coupling constant is studied with semiclassical and numerical methods bender wu branch points at which level crossing occurs are calculated and labelled by a process of analytic continuation the different resonances are the values that a single analytic function takes on different sheets of a riemann surface whose topology is described

non hermitian matrix description of the calpcalt symmetric anharmonic oscillators the symmetric differential schrodinger equation h e with the operator h h x p2 ax4 i x3 cx2 i x h x on l2 is studied at a 0 it is rearranged as a linear algebraic diagonalization with rigorous proof our non variational construction of bound states offers an infinite dimensional analogue to the recent finite dimensional quasi exact solution available at the less common a 0

chebyshev expansion on intervals with branch points with application to the root of kepler s equation a chebyshev hermite pade method when two or more branches of a function merge the chebyshev series of u l will converge very poorly with coefficients a n of t n l falling as o 1 n a for some small positive exponent a however as shown in j p boyd chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one dimensional bratu equation appl math comput 143 2002 189 200 it is possible to obtain approximations that converge exponentially fast in n if the roots that merge are denoted as u 1 l and u 2 l then both branches can be written without approximation as the roots of u u 1 l u u 2 l u email protected l email protected l by expanding the nonsingular coefficients of the quadratic b l and c l as chebyshev series and then applying the usual roots of a quadratic formula we can approximate both branches simultaneously with error that decreases proportional to exp email protected for some constant s 0 where n is the truncation of the chebyshev series this is dubbed the chebyshev shafer or chebyshev hermite pade method because it substitutes chebyshev series for power series in the generalized pade approximants known variously as shafer or hermite pade approximants here we extend these ideas first we explore square roots with branches that are both real valued and complex valued in the domain of interest illustrated by meteorological baroclinic instability second we illustrate triply branched functions via roots of the kepler equation f u l e email protected u email protected 0 only one of the merging roots is real valued and the root depends on two parameters l e rather than one nonetheless the chebyshev hermite pade scheme is successful over the whole two dimensional parameter plane we also discuss how to cope with poles and logarithmic singularities that arise in our examples at the extremes of the expansion domain

summation of the eigenvalue perturbation series by multi valued pade approximants application to resonance problems and double wells quadratic pade approximants are used to obtain energy levels both for the anharmonic oscillator x2 2 lambda x4 and for the double well x2 2 lambda x4 in the first case the complex valued energy of the resonances is reproduced by summation of the real terms of the perturbation series the second case is treated formally as an anharmonic oscillator with a purely imaginary frequency we use the expansion around the central maximum of the potential to obtain a complex perturbation series on the unphysical sheet of the energy function then we perform an analytical continuation of this solution to the neighbouring physical sheet taking into account the supplementary branch of quadratic approximants in this way we can reconstruct the real energy by summation of the complex series such an unusual approach eliminates the double degeneracy of states that makes ordinary perturbation theory around the minima of the double well potential incorrect

thermal stability of a reactive third grade fluid in a cylindrical pipe an exploitation of hermite pade approximation technique abstract this study is devoted to investigate the thermal stability of a reactive third grade fluid flowing steadily through a cylindrical pipe with isothermal wall it is assumed that the reaction is exothermic under arrhenius kinetics neglecting the consumption of the material approximate solutions are constructed for the governing nonlinear boundary value problem using regular perturbation techniques together with a special type of hermite pade approximants and important properties of the velocity and temperature fields including bifurcations and thermal criticality conditions are discussed

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

on the existence of compressive solitary waves in compacting media necessary conditions for the existence of compressive solitary wave solutions of a partial differential equation derived by scott and stevenson 1986 which describes the two phase fluid flow in a medium compacting under gravity are derived it is shown that for compressive solitary wave solutions to exist which satisfy certain boundary conditions it is necessary that n m 1 where n and m are the exponents in power laws relating the permeability of the medium and the viscosity of the solid matrix respectively to the voidage the effect of the value of the exponents n and m on the shape of the solitary wave is investigated by using existing analytical solutions and new numerical solutions

note a preference ranking organisation method the promethee method for multiple criteria decision making principles for a new family of outranking methods are given the main aim of the proposed promethee approach is to be as easily understood as possible by the decision maker it is based on extensions of the notion of criterion six possible extensions are considered these extensions can easily be identified by the decision maker because the parameters to be defined at most 2 have an economic significance a valued outranking graph is constructed by using a preference index two possibilities are considered to solve the ranking problem by using this valued graph promethee i provides a partial preorder and promethee ii a total preorder on the set of the possible actions

co plot a graphic display method for geometrical representations of mcdm abstract a relatively new multivariate analysis method called co plot is shown to be highly efficient for the geometrical representation of multicriteria decision problems co plot a two dimensional graphic display technique is designed for multicriteria analysis its main feature is the simultaneous analysis of actions e g observations and criteria e g variables in a simple way the method produces three results 1 similarity among actions by the composite of all criteria involved 2 the structure of correlations among the criteria and 3 the mutual relationships between the actions and the criteria the method is demonstrated on two numerical applications previously analyzed by mareschal and brans b mareschal j p brans european journal of operational research 34 1988 69 77 using the promethee methods and principal component analysis

anharmonic oscillator constructing the strong coupling expansions two novel approaches to construction of the strong coupling expansion for the anharmonic oscillator with the potential v x 1 2 x2 g 4 x4 are proposed the first one is simply a straightforward solution of the schrodinger equation via the nonlinearization technique resulting in the rapidly convergent perturbation series the second one is a version of the path integral perturbation theory but with an unconventional choice of the zeroth approximation action nine leading coefficients of the strong coupling expansion are computed they decrease rapidly the ninth one being of the order of 10 9 three leading corrections of the nonlinearization approach provide the ground state energy within a relative accuracy of 10 7 10 9 at an arbitrary coupling g the explicit formulas for corrections enable one to study the analyticity properties of the energy as a function of the coupling g in the second approach the strong coupling expansion coefficients are computed as an infinite linear series of the weak

the asymptotic behaviour of algebraic approximants we study the convergence of algebraic approximants to a function represented by a power series we consider for an arbitrary but fixed degree approximant sequences which can be generated recursively by use of sergeyev s algorithm for the exponential function a logarithmic function and a power of a binomial we find explicit formulae for the coefficients that appear in a resulting linear recurrence relation we assume that the error equation may be linearized for small errors analysis then yields the generic dominant term in the asymptotic behaviour of the error when a large number of terms of the series are used extensive numerical results confirm the behaviour finally we compare this behaviour with that for the closely related method of drazin tourigny in which the degree of the approximants grows without bound

hyperasymptotics and the linear boundary layer problem why asymptotic series diverge the simplest problem with boundary layers epsilon 2 u xx u f x is used to illustrate i why the perturbation series in powers of epsilon is asymptotic but divergent ii why the optimally truncated expansion is superasymptotic in the sense that that error is proportional to exp mbox constant epsilon and iii how to obtain an improved hyperasymptotic approximation

summation of asymptotic expansions of multiple valued functions using algebraic approximants application to anharmonic oscillators the divergent rayleigh schrodinger perturbation expansions for energy eigenvalues of cubic quartic sextic and octic oscillators are summed using algebraic approximants these approximants are generalized pade approximants that are obtained from an algebraic equation of arbitrary degree numerical results indicate that given enough terms in the asymptotic expansion the rate of convergence of the diagonal staircase approximant sequence increases with the degree different branches of the approximants converge to different branches of the function the success of the high degree approximants is attributed to their ability to model the function on multiple sheets of the riemann surface and to reproduce the correct singularity structure in the limit of large perturbation parameter an efficient recursive algorithm for computing the diagonal approximant sequence is presented

multicriteria analysis survey and new directions abstract before the advent of multicriteria analysis decision problems often took the form of the optimization of an objective function or a utility function this approach has the advantage of resulting in well defined mathematical problems but is not always representative of reality in fact the comparison of several possible decisions is rarely made according to a single point of view and the preferences on a point of view are in many cases only modelled with difficulty by a function the aim of this paper is to survey the different directions in which multicriteria analysis has developed multiattribute utility theory outranking relations interactive methods the methodology recently proposed by b roy is also presented

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

applications of hermite pade approximants to water waves and the harmonic oscillator on a lattice three types of hermite pade approximants are considered known respectively as quadratic integral and differential pade approximants the singularity structure of each type of approximant is described it is more complicated than that of the standard pade approximant and this property may often be used to estimate the types of singularity of a function from its power series expansion as well as evaluate it on its branch cuts two applications in different fields are described which illustrate these properties of the above types of hermite pade approximants the first concerns the characteristic values of mathieu s equation which are related to the energy eigenvalues of the harmonic oscillator on a lattice the second concerns the investigation of the singularity structure and values of various physical quantities associated with periodic and solitary water waves

rayleigh schrodinger perturbation theory with a strong perturbation anharmonic oscillators the bound state solutions of schrodinger s equation for the anharmonic oscillator potentials v x2 lambda x2k k 2 3 have been investigated using elementary techniques of low order variational perturbation theory for the quartic oscillator k 2 a scaled harmonic potential provides a remarkably accurate model for all lambda although this model is slightly less satisfactory for higher order anharmonicities k or 3 the perturbation procedures remain effective and can be applied successfully provided that higher order terms are calculated

a new approach to the bi dimensional representation of the dea efficient frontier with multiple inputs and outputs abstract this paper presents a new approach to the graphical presentation of dea results whatever the number of inputs and outputs are an adequate normalization of their weights is enough to generate a simple bi dimensional graph similar to that of the ccr frontier with one input and one output an advantage over other approaches to the same representation problem is that no complementary techniques are required to plot the frontier it is also proved that the distance of a dmu to the frontier is related to its efficiency the proposed approach is also valid for the bcc model

asymptotic response of observables from divergent weak coupling expansions a fractional calculus assisted pade technique appropriate constructions of pade approximants are believed to provide reasonable estimates of the asymptotic large coupling amplitude and exponent of an observable given its weak coupling expansion to some desired order in many instances however sequences of such approximants are seen to converge very poorly we outline here a strategy that exploits the idea of fractional calculus to considerably improve the convergence behavior pilot calculations on the ground state perturbative energy series of quartic sextic and octic anharmonic oscillators reveal clearly the worth of our endeavor

geometrical representations for mcda abstract in this paper geometrical representations for multicriteria decision problems are proposed this new approach provides assistance to understand the conflictual aspects of the criteria and to tackle the problem of the weights associated to them a generalized criterion including a preference function is first generated for each criterion this allows to define unicriterion preference flows for which a geometrical representation can be obtained by using the principal components analysis the actions are represented by points and criteria by axes in the pca plane a decision axis taking into account the weights associated to the criteria can be defined this technique provides the decision maker with a considerable enrichment for the understanding of his problem clusters of actions can be considered the importance of the criteria can be evaluated conflictual criteria are immediately detected incomparability between actions is emphasized and explained best compromise actions are easily selected new decision axes representing possible clusters of criteria can be considered undesirable actions can be eliminated the technique consists in a powerful new qualitative decision tool it is illustrated in the paper on some examples treated by the promethee i and ii methods a didactic and user friendly microcomputer code is available

upper and lower bounds of the ground state energy of anharmonic oscillators using renormalized inner projection renormalization is a method often used to approximate the eigenvalues of a hamiltonian that cannot be solved exactly it consists of splitting the hamiltonian into a solvable part and a remainder which is then minimized the inner projection technique first introduced by lowdin in the sixties was developed to bracket the eigenvalues between lower and upper bounds combining renormalization and lowdin s inner projection yielded the so called renormalized inner projection technique in this study this method will be applied to the quartic sextic and octic anharmonic oscillators lower and upper energy bounds are obtained for finite values of the coupling constant as well as for the infinite case the relation between the renormalized inner projection and perturbation theory will also be discussed another feature of this study is the importance of symbolic computation in allowing us to manipulate expressions with unevaluated parameters and to perform calculations in rational arithmetics or high decimal precision thus lowdin s rational approximants can be expressed explicitly as rational fractions in terms of the coupling constant and values for the limit constant can be obtained with amazing high accuracy namely 62 33 and 21 decimal places for the quartic sextic and octic oscillator respectively