Bound states of Klein—Gordon equation for double ring-shaped oscillator scalar and vector potentials

In spherical polar coordinates, double ring-shaped oscillator potentials have supersymmetry and shape invariance for θ and r coordinates. Exact bound state solutions of Klein—Gordon equation with equal double ring-shaped oscillator scalar and vector potentials are obtained. The normalized angular wavefunction expressed in terms of Jacobi polynomials and the normalized radial wavefunction expressed in terms of the Laguerre polynomials are presented. Energy spectrum equations are obtained.     

Bound states of the SchrSdinger equation for the PSschl-Teller double-ring-shaped Coulomb potential

Poschl-Teller double-ring-shaped Coulomb （PTDRSC） potential, the Coulomb potential surrounded by PSschl- Teller and double-ring-shaped inversed square potential, is put forward. In spherical polar coordinates, PTDRSC potential has supersymmetry and shape invariance in φ,θ and τ coordinates. By using the method of supersymmetry and shape invariance, exact bound state solutions of Schr6dinger equation with PTDRSC potential are presented. The normalized φ,θ angular wave function expressed in terms of Jacobi polynomials and the normalized radial wave function expressed in terms of Laguerre polynomials are presented. Energy spectrum equations are obtained. Wave function and energy spectrum equations of the system are related to three quantum numbers and parameters of PTDRSC potential. The solutions of wave functions and corresponding eigenvalues are only suitable for the PTDRSC potential.     

Maximum likelihood polynomial regression for robust speech recognition

The linear hypothesis is the main disadvantage of maximum likelihood linear regression (MLLR).This paper applies the polynomial regression method to model adaptation and establishes a nonlinear model adaptation algorithm using maximum likelihood polynomial regression(MLPR)for robust speech recognition.In this algorithm,the nonlinear relationship between training and testing Gaussian means in every Mel channel is approximated by a set of polynomials and the polynomial coefficients are estimated from adaptation data in test environment using the expectation-maximization(EM)algorithm and maximum likelihood(ML) criterion.The experimental results show that the second-order polynomial can approximate the actual nonlinear function better and in noise compensation and speaker adaptation,the word error rates of MLPR are significantly lower than those of MLLR.The proposed MLPR algorithm overcomes the limitation of linear hypothesis well and can decrease the impact of noise,speaker and other factors simultaneously.It is especially suitable for joint adaptation of speaker and noise.     

Nucleon/nuclei polarized structure function,using Jacobi polynomials expansion

We use the constituent quark model to extract polarized parton distributions and finally polarized nucleon structure function.Due to limited experimental data which do not cover whole （x,Q 2 ） plane and to increase the reliability of the fitting,we employ the Jacobi orthogonal polynomials expansion.It will be possible to extract the polarized structure functions for Helium,using the convolution of the nucleon polarized structure functions with the light cone moment distribution.The results are in good agreement with available experimental data and some theoretical models.     

The Origin and Mathematical Characteristics of the Super-Universal Associated-Legendre Polynomials

We study the mathematical characteristics of the super-universal associated-Legendre polynomials arising from a kind of double ring-shaped potentials and obtain their polar angular wave functions with certain parity. We find that there exists the even or odd parity for the polar angular wave functions when the parameter η is taken to be positive integer, while there exist both even and odd parities when η is taken as positive non-integer real values. The relations among the super-universal associated-Legendre polynomials, the hypergeometric polynomials, and the Jacobi polynomials are also established.     

Bound states of the Schrdinger equation for the Pschl-Teller double-ring-shaped Coulomb potential

Põschl--Teller double-ring-shaped Coulomb (PTDRSC) potential, the Coulomb potential surrounded by Põschl--Teller and double-ring-shaped inversed square potential, is put forward. In spherical polar coordinates, PTDRSC potential has supersymmetry and shape invariance in φ, θ and r coordinates. By using the method of supersymmetry and shape invariance, exact bound state solutions of Schrõdinger equation with PTDRSC potential are presented. The normalized φ, θ angular wave function expressed in terms of Jacobi polynomials and the normalized radial wave function expressed in terms of Laguerre polynomials are presented. Energy spectrum equations are obtained. Wave function and energy spectrum equations of the system are related to three quantum numbers and parameters of PTDRSC potential. The solutions of wave functions and corresponding eigenvalues are only suitable for the PTDRSC potential.     

Global vector-field reconstruction of nonlinear dynamical systems from a time series with SVD method and validation with Lyapunov exponents

A method for the global vector-field reconstruction of nonlinear dynamical systems from a time series is studied in this paper. It employs a complete set of polynomials and singular value decomposition (SVD) to estimate a standard function which is certtral to the algorithm. Lyapunov exponents and dimension, calculated from the differential equations of a standard system, are used for the validation of the reconstruction. The algorithm is proven to be practical by applying it to a Roessler system.     

Representation of the quantum mechanical wavefunction by orthogonal polynomials in the energy and physical parameters

We present a formulation of quantum mechanics based on the theory of orthogonal polynomials.The wavefunction is expanded over a complete set of square integrable basis where the expansion coefficients are orthogonal polynomials in the energy and physical parameters. Information about the corresponding physical systems(both structural and dynamical) are derived from the properties of these polynomials. We demonstrate that an advantage of this formulation is that the class of exactly solvable quantum mechanical problems becomes larger than in the conventional formulation(see, for example, table 3 in the text). We limit our investigation in this work to the Askey classification scheme of hypergeometric orthogonal polynomials and focus on the Wilson polynomial and two of its limiting cases(the Meixner–Pollaczek and continuous dual Hahn polynomials). Nonetheless, the formulation is amenable to other classes of orthogonal polynomials.     

Automatic baseline correction of infrared spectra

A fast automatic algorithm is proposed for baseline correction of infrared (IR) spectral signals.It is devised based on iterative curve fitting where orthogonal polynomials are used.The algorithm can process both emission and absorption spectra automatically without human intervention.Orthogonal polynomials are used for curve fitting to reduce computation time.Both emission and absorption spectra are obtained and the results demonstrate the feasibility and practicability of this algorithm.     

Photon number cumulant expansion and generating function for photon added- and subtracted-two-mode squeezed states

For the first time, we derive the photon number cumulant for two-mode squeezed state and show that its cumulant expansion leads to normalization of two-mode photon subtracted-squeezed states and photon added- squeezed states. We show that the normalization is related to Jacobi polynomial, so the cumulant expansion in turn represents the new generating function of Jacobi polynomial.     

Photon-Subtracted Two-Mode Squeezed Thermal State and Its Photon-Number Distribution

We construct the photon-subtracted two-mode squeezed thermal state (PSTMSTS) by subtracting any number of photons from two-mode squeezed thermal state (TMSTS). It is found that the normalization factor of the density operator of PSTMSTS is a Jacobi polynomial of squeezing parameter λ and average photon number [`(n)]\bar{n} of the thermal state. We investigate the photon-number distribution (PND) of PSTMSTS and find a remarkable result that it is a quotient of two Jacobi polynomials, as well as derive a corresponding character of Jacobi polynomial.     

Classical Skew Orthogonal Polynomials and Random Matrices

Skew orthogonal polynomials arise in the calculation of the n-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials. In the case that the eigenvalue probability density function involves a classical weight function, explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials, and the structure is used to give a closed-form expression for the sum. This theory treates all classical cases on an equal footing, giving formulas applicable at once to the Hermite, Laguerre, and Jacobi cases.     

TWO NEW SOLVABLE DYNAMICAL SYSTEMS OF GOLDFISH TYPE

Two new solvable dynamical systems of goldfish type are identified, as well as their isochronous variants. The equilibrium configurations of these isochronous variants are simply related to the zeros of appropriate Laguerre and Jacobi polynomials.     

Difference schrödinger operators with linear and exponential discrete spectra

Using the factorization method, we construct finite-difference Schrödinger operators (Jacobi matrices) whose discrete spectra are composed from independent arithmetic, or geometric series. Such systems originate from the periodic, orq-periodic closure of a chain of corresponding Darboux transformations. The Charlier, Krawtchouk, Meixner orthogonal polynomials, theirq-analogs, and some other classical polynomials appear as the simplest examples forN = 1 andN = 2 (N is the period of closure). A natural generalization involves discrete versions of the Painlevé transcendents.On leave of absence from the Institute for Nuclear Research, Russian Academy of Sciences, Moscow, Russia.     

xF 3 structure function based on the associated Jacobi polynomials

We present the results of the Next-to-leading order (NLO) non-singlet QCD analysis of the experimental data of the CCFR collaboration for the xF ₃ structure function of the deep-inelastic scattering of neutrinos on the nucleon in based on the associated Jacobi polynomials expansion of the structure functions. The structure function is reconstructed from its moments by using the expansion in terms of orthogonal associated Jacobi polynomials. Our results of valence quark distributions are in good agreement with the available theoretical models.     

Orthogonal harmonic polynomials onU(2)

Electric charges and free electromagnetic waves are supposed to be described locally with the same wave differential equation. It is only the topology that is considered to be different. The calculated nonlocalU(2) individuals are characterized by polynomials that belong neither to the classical nor to the Szegö polynomials. The construction of the polynomial solution in component form, their orthogonality over singular measures, the relationships to the Jacobi polynomials, Rodriguez formulas, product decomposition, asymptotic formulas, and completeness are presented in some detail. The possibility is discussed of whether this highly nonlocal model for electric charges can have a physical significance. This work is intended to be a first step for the realization of an old idea of Einstein's (and also commented on by Dirac) to start with the electric charge, not with the Planck constant, as the primary concept for quantum theory.     

Asymptotics of Raynal-Revai coefficients at large hypermomentum

The Raynal-Revai coefficients are studied as the Wigner D functions of O(6) group generated by the kinematical rotation of two reduced Jacobi vectors in six-dimensional three-body space. These coefficients are represented as one-dimensional integrals with kernels equal to double sums of the Clebsh-Gordan coefficients and associated Legendre polynomials. Using this representation we derive the asymptotics of the Raynal-Revai coefficients at large values of the hypermomentum. The text was submitted by the author in English.     

Linear Statistics of Point Processes via Orthogonal Polynomials

For arbitrary β>0, we use the orthogonal polynomials techniques developed in (Killip and Nenciu in , 2005; Killip and Nenciu in Int. Math. Res. Not. 50: 2665–2701, 2004) to study certain linear statistics associated with the circular and Jacobi β ensembles. We identify the distribution of these statistics then prove a joint central limit theorem. In the circular case, similar statements have been proved using different methods by a number of authors. In the Jacobi case these results are new.     

Application of jacobi polynomial methods to the one-speed transport equation

The transport equations associated with radiation damage studies are often solved using expansions in Legendre polynomials. The radiation damage distribution functions which satisfy these equations may be sharply peaked in the forward direction, while the Legendre polynomials, as a set, are isotropic. This situation requires the use of many terms in the Legendre expansion in order to adequately represent the distribution functions. The Jacobi polynomials, on the other hand, can have strong peaking built into their associated weight function. To test the usefulness of the Jacobi polynomials we use them to solve the simple, one-speed, neutron transport equation. The results are then compared to the exact theory and to the results of applying Legendre methods to the same problem. This sample calculation demonstrates the advantage of the Jacobi polynomials in strongly non-isotropic situations.