Some properties of generalized secant integrals: extended definition and recurrence relations

In this letter, based on Hamiltonian identity we propose a generalized expression of the second hypervirial for an arbitrary central potential wave functions in higher dimensions D and then present a useful general recurrence relation among the off-diagonal multipolar matrix elements. We show that this new proposed recurrence formula is very powerful in deriving the recurrence relations among the hydrogenic matrix elements. Some important recurrence relations can be simply obtained. It is interesting to find that the selection rule is independent of the central potential V(r) for f = rk with k = 0, 2 and that the diagonal matrix elements of V′(r) are independent of the system eigenvalues for k = 1. The application of the general recurrence relation to the isotropic harmonic oscillator case is discussed briefly.

Asymptotic behaviours and general recurrence relations

Recursiveness in matrix rational interpolation problems

CHAPTER 4 - Recurrence Relations and Algebraic Equations

The underlying stochastic nature of the requirements for the Solvency II regulations has introduced significant challenges if the required calculations are to be performed correctly, without resorting to excessive approximations, within practical timescales. It is generally acknowledged by practising actuaries within UK life offices that it is currently impossible to correctly fulfil the requirements imposed by Solvency II using existing computational techniques based on commercially available valuation packages. Our work has already shown that it is possible to perform profitability calculations at a far higher rate than is achievable using commercial packages. One of the key factors in achieving these gains is to calculate reserves using recurrence relations that scale linearly with the number of time steps. Here, we present a general vector recurrence relation which can be used for a wide range of non-unit linked policies that are covered by Solvency II; such contracts include annuities, term assurances, and endowments. Our results suggest that by using an optimised parallel implementation of this algorithm, on an affordable hardware platform, it is possible to perform the ‘brute force’ approach to demonstrating solvency in a realistic timescale (of the order of a few hours).

An algebraic approach to obtain ladder operators for hydrogenlike wave functions from the algebraic representation of the ${\mathit{L}}_{\mathit{n}}^{\mathit{k}}$ Laguerre associated polynomials directly involved is presented. The method leads to diverse ladder operators depending on the manner as the operational form of ${\mathit{L}}_{\mathit{n}}^{\mathit{k}}$ is used. As an example, this work deals with linear creation and annihilation operators of two kinds: some that influence the n energy quantum number and others acting on the l orbital number according to usual ladder properties. In addition, as a useful application of the raising and lowering operators thus obtained, generalized recurrence relations and closed formulas for the calculation of multipole matrix elements are algebraically deduced by means of a procedure that takes advantage of the relationship between the ladder operators associated with the bra and those related to the ket.

Under natural and weak hypotheses, we prove a reproducing formula for polynomials. Then we obtain a new recurrence relation between the moments of a scaling function and a new exact formula for the computation of moments of even order.

General recurrence relations between the coefficients in thenth and (n+1)th order spherical harmonic multipole expansions are derived. The particular application presented here is the derivation of the equations concerned with representing the geomagnetic field by magnetic multipoles. The equations up to the 3rd order multipole are given as an example of the method. The main advantage in using these recurrence relations rather than other methods is that the mathematics is reduced to merely a matter of successive substitutions and this allows a fast step by step generation of the required equations, in a form for which there is a simple numerical program for solution.

Solving Some General Nonhomogeneous Recurrence Relations of Order <i>r</i> by a Linearization Method and an Application to Polynomial and Factorial Polynomial Cases

The first step in any seismic hazard study is the definition of seismogenic sources and the estimation of magnitude-frequency relationships for each source. There is as yet no standard methodology for source modeling and many researchers have worked on this topic. This study is an effort to define linear and area seismic sources for Northern Iran. The linear or fault sources are developed based on tectonic features and characteristic earthquakes while the area sources are developed based on spatial distribution of small to moderate earthquakes. Time-dependent recurrence relationships are developed for fault sources using renewal approach while time-independent frequency-magnitude relationships are proposed for area sources based on Poisson process. GIS functionalities are used in this study to introduce and incorporate spatial-temporal and geostatistical indices in delineating area seismic sources. The proposed methodology is used to model seismic sources for an area of about 500 by 400 square kilometers around Tehran. Previous researches and reports are studied to compile an earthquake/fault catalog that is as complete as possible. All events are transformed to uniform magnitude scale; duplicate events and dependent shocks are removed. Completeness and time distribution of the compiled catalog is taken into account. The proposed area and linear seismic sources in conjunction with defined recurrence relationships can be used to develop time-dependent probabilistic seismic hazard analysis of Northern Iran.

Using the algebraic second quantization formalism, the general recurrence relations for some two-center harmonic oscillator integrals are obtained. The various known special cases for one-center integrals are evident in the formalism. Generating functions for these integrals are obtained and most of them are solved to find explicit expressions for polynomial, exponential, and Gaussian operator integrals. For these cases, all the generating integrals needed for utilizing the recurrence relations are also calculated in their simplest forms.

If Z and (X, Y) are independent discrete random variables with probability generating functions f(u) and g(τ, t), respectively, then the generalized random variable (V,W) has probability generating function h (τ, t) = f (g (τ, t)). This class of bivariate discrete distributions includes many of the known bivariate contagious and compound distributions. In the present paper the probabilities and factorial moments of (V,W) are expressed in terms of the probabilities and factorial moments of Z and (X ,Y) through the bipartitional polynomials; these polynomials are multivariable polynomials Ymn(fg01, fg10, fg11, ..., fgmn), fk≡fk defined by a sum over all partitions of their bipartite indices (m , n). Using properties of these polynomials, the conditional probabilities and factorial moments of W given V=m are obtained in explicit forms. Recurrence relations of the probabilities and factorial moments are obtained by using the general recurrence relation for the bipartitional polynomials. These general results are applied to the bivariate generalized Poisson, logarithmic series and general binomial distributions. Moreover, certain bivariate generalized discrete distributions with specified the generalizing random vector (X, Y) are briefly discussed.

The focusing properties of several bent crystal geometries, including the newly introduced Pestehe &amp; Askari general point-focusing system [Pestehe &amp; Askari Germi (2012),Opt. Soc. Am. A,29, 68–77; Pestehe &amp; Askari Germi (2012),J. Appl. Cryst.45, 890–901], on an arbitrarily positioned detector plane are investigated and illustrated. The properties of the focal points and the generated images are theoretically related to the local, ℓ, and directional, γ, positions of the detector plane for a given position of a point source on the Rowland circle. A general relation is derived for the detector positioning to obtain a specially focused image. This formula for the polar position of the detector plane, given by the two ℓ and γ variables, enables the exact determination of the system astigmatism and the exact calculation of the sagittal and meridional image positions for the spectrometer under study. The astigmatisms of the above-mentioned bent crystal geometries have been studied, and their sagittal and meridional focal positions and characteristics have been obtained and are illustrated. It is also shown that there is a possibility of designing a spectrometer to focus rays from a linear source onto a point on the Rowland circle.

The shapes of theoretical growth curves in atomic fluorescence spectrometry are calculated for several possible atomizer cell geometries, for several geometries of illumination measurement, for both line and continuum sources of excitation, and for several a parameter values. Factors relating the shapes of theoretical growth curves to experimental growth curves (analytical curves) are also discussed. Theoretical atomic fluorescence growth curves for hypothetical atoms having a parameters of 5, 2, 1, 0.4, and 0.1; for three different limiting cases characterizing the geometry of illumination measurement; and for both line and continuum sources are given.

Diffraction of waves by a wedge is a problem of modern acoustics whose mathematical solutions provide a theoretical basis in analyzing sound radiation and scattering phenomena. Considerable attention has previously been attracted to this class of diffraction problems, especially to the particular cases of a perfectly reflecting wedge and half-plane. The model of an impedance wedge [G. D. Malyuzhinets, 1958 Dokl. Akad. Nauk SSSR 121, 436 (1958) in Russian] is a more general and realistic one that gives new opportunities for treating acoustic diffraction by irregular surfaces [A. D. Pierce and W. J. Hadden, J. Acoust. Soc. Am. 63, 17 (1978); A. V. Osipov, Sov. Phys. Acoust. 36, 287 (1990)]. This paper investigates the wave field excited by a line source in the presence of a wedge with arbitrary values of face impedances. Based on the Malyuzhinets’ solution, various exact representations for the Green’s function are deduced to generate a system of convenient formulas for any relative positions of the source and point of observation. A new representation of the Malyuzhinets’ solution in the form of double series [A. V. Osipov, Sov. Phys. Acoust. 37, 381 (1991)] is exploited to describe the wave field of the line source placed in the vicinity of edge. In the opposite case of large distances from the edge, a uniform asymptotic expansion is presented. Applications of the obtained results to problems of sound attenuation by wedge-like barriers and sound radiation from a line source located at the edge of a corrugated surface are described.