a quadrature formula for integrals with nearby singularities the purpose of this paper is to propose a new quadrature formula for integrals with nearby singularities in the boundary element method the integrands of nearby singular boundary integrals vary drastically with the distance between the field and the source point especially field variables and their derivatives at a field point near a boundary cannot be computed accurately in the present paper a quadrature formula for l isolated singularities near the integration interval based on lagrange interpolatory polynomials is obtained the error estimation of the proposed formula is also given quadrature formulas for regular and singular integrals with conjugate poles are derived numerical examples are given and the proposed quadrature rules present the expected polynomial accuracy

exponentially small corrections in the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator is studied in a region of the coupling constant plane in which there is a sequence of exponentially small subseries beyond the standard rayleigh schrodinger perturbation theory rspt power series we give a simple algorithm for the calculation of these subseries to any desired order in terms of the rspt coefficients expressed as polynomials in the quantum number and illustrate our results with numerical borel pade summations of the expansion up to third exponentially small order

a generalization of partial autocorrelations useful in identifying arma models statistics are proposed to help identify the orders of autoregressive moving average models these are functions of the sample autocorrelations and include partial autocorrelations as special cases the large sample variances of the statistics are derived and small sample properties are explored by simulation their interpretation is illustrated using wolfer s annual sunspot numbers and wind velocity data cleveland 1972

pt symmetric pseudo perturbation recipe an imaginary cubic oscillator with spikes the pseudo perturbative shifted l expansion technique pslet is shown to be applicable in the non hermitian symmetric context the construction of bound states for several symmetric potentials is presented with special attention paid to v r ir3 ir oscillators

a genetic algorithm approach to the spectral estimation of time series with noise and missed observations this study considers the problem of estimating the autoregressive moving average arma power spectral density when measurements are corrupted by noise and by missed observations the missed observations model is based on a probabilistic structure unlike conventional cases of missed observation in parameter estimation problems the variance of noise is unavailable that is the time points of missed observations are unknown and the probability of missing data needs to be estimated in this situation spectral estimation is more difficult to solve and becomes a highly nonlinear optimization problem with many local minima in this paper we use the genetic algorithm ga method to achieve a global optimal solution with a fast convergence rate for this spectral estimation problem from the simulation results we have determined that the performance is significantly improved if the probability of data loss is considered in the spectral estimation problem

symmetrically regularized eckart poschl teller and hulthen potentials parity times time reversal symmetry of complex hamiltonians with real spectra is usually interpreted as a weaker mathematical substitute for hermiticity perhaps an equally important role is played by the related strengthened analyticity assumptions in a constructive illustration we complexify a few potentials solvable only in s wave then we continue their domain from the semi axis to the whole axis and obtain new exactly solvable models their energies turn out to be real as expected the new one dimensional spectra themselves differ quite significantly from their s wave predecessors

matching method and exact solvability of discrete symmetric square wells discrete symmetric square wells are studied their wavefunctions are found to be proportional to classical tshebyshev polynomials of complex argument the compact secular equations for energies are derived giving the real spectra in certain intervals of non hermiticity strengths z it is amusing to note that although the known square well re emerges in the usual continuum limit a twice as rich upside down symmetric spectrum is exhibited by all its present discretized predecessors

on the relationship between the s array and the box jenkins method of arma model identification abstract this paper investigates an extension of the partial autocorrelation function which we call the generalized partial autocorrelation function these generalized partial autocorrelations which are not true correlations except when p 0 are useful in examining the relationship between the s array method of gray kelley and mcintire 1978 and the box jenkins approach to arma model identification also the generalized partial autocorrelation is shown to be a useful model identification tool to be used along with the s array also discussed is a reformating of the s array into a shifted s array that the authors believe is easier to use in practice than the s array the methods of this paper are illustrated by means of examples including an analysis of the makridakis 1978 metals series data

the determination of optimum structures for the state space representation of multivariate stochastic processes when identifying a model for a multivariate stationary stochastic process an important problem is that of determining the structure of the state variable model several overlapping parameterizations can usually be fitted to a given process and the question arises as to which structure leads to the most accurate parameter estimates the accuracy of parameter estimates is often measured by the determinant of the fisher information matrix we show that all admissible structures will give asymptotically the same value to this criterion for finite data some structures may still be better than others and two heuristic structure estimation methods are analyzed some simulation results are also presented

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

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

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

complex calogero model with real energies recently the symmetry of many single particle non hermitian hamiltonians has been conjectured to be sufficient to keep their spectrum real we demonstrate this and how the similar concept of a weakened hermiticity can be extended to some exactly solvable two and three particle models

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

asymptotic solvability of an imaginary cubic oscillator with spikes for complex potentials v x ix 3 2 ix 2 2 ix 1 2 which are symmetric we show that in 1 strong coupling regime the low lying bound states almost coincide with harmonic oscillators whenever the spectrum remains real this means at all critical 1

solvability and pt symmetry in a double well model with point interactions the concept of point interactions offers one of the most suitable guides towards a quantitative analysis of properties of certain specific non hermitian so called symmetric quantum mechanical systems this is illustrated with a double well model the easy solvability of which is shown to lead to a clear picture of the mechanisms of the unavoided level crossing and of the spontaneous symmetry breaking at a certain strong non hermiticity boundary below this limit the model is shown to be suitable for an explicit illustration of technicalities related to the standard probabilistic physical interpretation of bound states in symmetric quantum mechanics

spiked and symmetrized decadic potentials supporting elementary n plets of bound states we show that the potential wells v x x10 ax8 bx6 cx4 dx2 f x2 with a central spike possess arbitrary finite multiplets of elementary exact bound states their strong asymptotic growth implies an ambiguity in their symmetrically generalized quantization via complex boundary conditions but the three eligible methods coincide at our exceptional solutions

two chi square statistics for determining the orders p and q of an arma p q process the theta lambda and eta functions have been previously proposed for use in choosing the autoregressive and moving average orders of an arma q p process visually two chi square statistics associated with these three functions are presented and used here to determine the orders of an arma process statistically it is shown that the two statistics are asymptotically equivalent to the quenouille walker s goodness of fit test statistic which is a lagrange multiplier test statistic some numerical examples are presented to illustrate the usefulness of the two chi square statistics as well as the three functions in arma modeling

coupled channel version of the pt symmetric square well a coupled pair of pt symmetric square wells is studied as a prototype of a quantum system characterized by two manifestly non hermitian commuting observables via the diagonalization of our hamiltonian h h and spin like observable we demonstrate that there exists a domain of couplings where both the respective sets of eigenvalues en n 0 1 energies and 1 spin projections remain real in such a measurable regime the model acquires a consistent probabilistic interpretation mediated by our selection of one of many available interaction dependent scalar products

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

low lying spectra in anharmonic three body oscillators with a strong short range repulsion one dimensional three body schrodinger equation is studied with binding mediated by the power law two body long range attraction v l x 2xl in superposition with a short range repulsion v k x 2 xk plus possibly further subdominant power law components such an unsolvable and non separable generalization of calogero model where one had l k 2 is shown to become separable and solvable at all integers k 0 and l 0 in the limit of the large repulsion 1

distributional borel summability of odd anharmonic oscillators it is proved that the divergent rayleigh schrodinger perturbation expansions for the eigenvalues of any odd anharmonic oscillator are borel summable in the distributional sense to the resonances naturally associated with the system

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

harmonic oscillator well with a screened coulombic core is quasi exactly solvable in the quantization scheme which weakens the hermiticity of a hamiltonian to its mere t invariance the superposition v x x2 ze2 x of the harmonic and coulomb potentials is defined at the purely imaginary effective charges ze2 if and regularized by a purely imaginary shift of x this model is quasi exactly solvable we show that at each excited n 1 th harmonic oscillator energy e 2n 3 there exists not only the well known harmonic oscillator bound state at the vanishing charge f 0 but also a normalizable n 1 plet of the further elementary sturmian eigenstates n x at eigencharges f f n 0 n 0 1 n beyond the smallest multiplicities n we recommend perturbative methods for their construction

an order recursive algorithm to solve the 3 d yule walker equations of causal 3 d ar models an order recursive algorithm is proposed to solve the 3 d yule walker equations of causal 3 d ar models it is computationally efficient and can be easily transformed into a computer program moreover it can be utilized to determine the orders of a causal 3 d ar process

script pscript t symmetric models in more dimensions and solvable square well versions of their angular schrodinger equations from the partial differential calogero s three body and smorodinsky winternitz superintegrable hamiltonians in two variables we separate the respective angular schrodinger equations and study the possibilities of their minimal symmetric complexification the simultaneous loss of the hermiticity and solvability of the respective angular potentials v is compensated by their replacement by solvable purely imaginary and piece wise constant multiple wells v0 we demonstrate that the spectrum remains real and that it exhibits a rich four series structure in the double well case