Form Of Power Series Expansion Research Articles

Analytic solutions of systems of functional equations

The analytic properties of integral manifolds of systems of difference equations are studied. Analytic solutions of systems of functional equations are determined in the form of power series expansions.

Resonances and Power Series Solutions of the Forced van der Pol Oscillator

To the first approximation, the nonresonant steady-state motion of the forced van der Pol oscillator is a combination of a free and a forced response. When the forcing amplitude increases beyond a certain critical value, which depends on the forcing frequency, the free response will decay and the motion is represented by just a particular solution that has the same frequency as the forcing term. In the first part of this paper, a general expression is determined for all of the higher-order resonant frequencies associated with a formal multiple timescale perturbation expansion of the solution. These frequencies are shown to be dense on the real axis. Then a formal power series expansion is developed for a particular solution in terms of the small damping parameter $\varepsilon $, which is shown to be valid when the forcing frequency is not close to a certain subset of the resonant frequencies. Using Padé approximants, information is obtained about the location and nature of the singularities in the complex $\varepsilon $-plane that limit the convergence of the series. Three pairs of singularities whose locations change as the forcing amplitude varies are discovered. In particular, when the forcing amplitude exceeds a certain value, a pair of singularities along the imaginary axis becomes dominant. The emergence of these singularities as the dominant ones is discussed in relation to the quenching of the free oscillation in the total response.

Quadratic regulatory theory for analytic non-linear systems with additive controls

Quadratic regulator problems over finite and infinite time intervals are solved for non-linear systems which are analytic in the state and linear in the control. The solution approach is based on using a formal power series expansion method for solving a non-linear Hamiltonian system. The technique of Carleman linearization facilitates the approach. The unique solution to the non-linear optimal regulator problem is obtained in terms of a convergent power series which satisfies the Hamilton-Jacobi-Bellman equation of dynamic programming. The realization of the optimal regulator is investigated and sufficient conditions are derived from the optimal regulator using the technique of Carleman linearization. Several examples are given to illustrate the theory.

Concentration fluctuation of stiff polymers. I. Static structure factor

The static structure factor g(k) of the concentrated solution of stiff polymers is calculated based on the generalized random phase approximation (RPA) which accounts for the nematic interaction (i.e., the interaction causing the nematic phase of stiff polymers) between the segments. The structure factor is obtained in the form of power series expansion with respect to k. The result is: (i) the correlation length ξ is independent of the nematic interaction, and is given by the conventional RPA; (ii) the sign of the coefficient of k4 changes from positive to negative as the nematic interaction is increased; and (iii) the structure factor of rigid rod, which can be calculated exactly, shows no anomaly below the critical concentration c*, but suddenly shows anomaly above c*.

On the use of generalized Euler transformation in handling divergent series

A parametrized version of the Euler transformation, introduced fairly recently, is employed to study the behaviour of functions, given their formal power-series (alternating) expansions in λ with finite radii of convergence, in the limit λ»∞. The strategy requires only the first few low-order data. Results are tested with quite a few known cases and found remarkably satisfactory. The role of some other methods in this context are briefly discussed.

On the formal development of elastic foundation models

In developing foundation models, the majority of researchers proceeded in two ways: (1) Start with the Winkler model and, in order to bring it closer to reality, assume some kind of interaction between the spring elements, or (2) start with the continuum and then introduce simplifying assumptions with respect to displacements and/or stresses. A third approach, developed mainly in recent decades, is a procedure which makes use of formal power series expansions. The purpose of present paper is to discuss this third approach for generating foundation models. It is shown that the a priori assumption that the contact pressure is related to the surface deflections and its derivatives, p(x)=f(w, w′, w″, ...), is too restrictive and that the proper form is p(x)=F(w, w′, w″, ..., p′, p″, ...). The paper concludes with a comparison of the obtained models and a discussion of the sequencing of foundation models of higher order beyond the Winkler foundation.

On forward Delbrück scattering

Alternative very transparent expressions for the forward complex amplitude for Delbruck scattering, in the form of a single integral representation and in the form of power series expansions in the photon energy are derived. In particular, the series expansions, obtained by using the method of the Mellin transform, are rapidly convergent and are found to be very convenient for numerical calculations. Tables of numerical results are also given.

Two-point Padé approximants and the quotient difference algorithm

In a recent paper Sidi considered the two-point Padé approximants to a function for which formal power series expansions at the origin and at infinity are given, and provided determinant representations for them. In comparing his approach with the quotient-difference algorithm, used by McCabe and Murphy on the same problem, certain properties of the coefficients generated by the quotient-difference algorithm are overlooked. It is the purpose of this short note to describe these properties.

Some aspects of two-point Padé approximants

Recently McCabe and Murphy have considered the two-point Padé approximants to a function for which (formal) power series expansions at the origin and at infinity are given. In this paper these approximations are slightly modified and determinant representations for them are given. The existence of various three-term recursion relations for the numerators and denominators of these approximants is shown. Based on these, a new continued fraction representation for these approximants is obtained and also an efficient recursive method is proposed for the determination of the coefficients of all the approximants that obtain from a given number of terms of the power series.

A proposal for a nonperturbative analytical approach in plasmas

The interaction between a highly dense plasma and the strong electromagnetic field of a powerful laser, as i t is required for induced fusion, is quite difficult to be t reated theoret ical ly because of the various physical phenomena accompanying the evolution of the two fluids (ions and electrons) and the nonlineari ty of the equations of motion. Furthermore the strong coupling between the pump field (laser) and the mat ter is such to suggest the introduction of new analyt ical approximate methods instead of the perturbat ive approaches valid in the weak-signal cases. To this end we are going to focus here our at tent ion on the Pad6 approximants and related approaches such as the socalled s-algorythm ah'eady applied successfully in other branches of Physics (1). If we are given a formal power series expansion (divergent, asymptot ic or weakly convergent) in one variable (2)

About the relation [X, P]=i

By means of an example it is illustrated that the usual canonical commutation relation is not equivalent to the canonical commutation relation in the Weyl form. The example is also used to show that several well-known operator identities, in the author's opinion, are false because they were derived by formal power series expansions.

Solutions to a Problem in Power Series Reversion

This paper presents the general solution of the following problem in two forms. Let $f(x,y)$ be defined by the formal power series $f(x,y) = \sum _{m = 0}^\infty \sum _{n = 0}^\infty f_{mn} x^m y^n $ with $f_{00} \ne 0$. If v satisfies $v(x,y) = f(xv^a ,yv^b )$, where a and b are constants, then find the formal power series expansion of $v^c(x,y)$, where c is also a constant. A special case.of this problem, which occurs in a paper by R. A. Handelsman and J. S. Lew [1], has been proposed as a problem to be solved by computer using a symbolic algebra system [2].

Nonlinear Diffusion in Critical Binary Mixture

In the present note we consider the diffusion current in a critical binary mixture from the viewpoint of the nonlinear transport. As was shown by Kawasaki and ·Gunton1> in their study of the nonlinear shear viscosity a rem~rkable feature arises in the ·nonlinear transport; the nonlinear transport coefficient is :ilonanalytic with respect to the thermal ·driving force. This peculiar effect is due to.· the existence of the ·diffusion mode which brings the socalled long time tail into the time correlation function of transverse velocity. Thus, the formal power series expansion of the nonlinear transport coefficient with respect to the thermal driving force breaks down. The resummation of this series expansion Dl.ay give the convergent result and at the same time the nonanalytic transport c::~ef­ ficient. Physically this nonanalyticity originates from the fact that in small wave numbers the decay of the tr'ansverse velocity interferes with the velocity gradient.U In the present reference system the similar effect can be expected since the decay of the local concentration and that of the transverse velocity are of diffusion type. The present study will be made in the framework of the linearized hydrodynamic' equations which can be derived from the mode-coupling theory.2> The steady concentration gradient is present in the xdirection and its magnitude is denoted as d; Then, the linearized hydrodynamic equations are easily obtained as

Robust Regression: Asymptotics, Conjectures and Monte Carlo

Maximum likelihood type robust estimates of regression are defined and their asymptotic properties are investigated both theoretically and empirically. Perhaps the most important new feature is that the number $p$ of parameters is allowed to increase with the number $n$ of observations. The initial terms of a formal power series expansion (essentially in powers of $p/n$) show an excellent agreement with Monte Carlo results, in most cases down to 4 observations per parameter.

An analogue of l’Hospital’s rule

Formal power series expansions have proved as useful in differential algebra as in other fields of mathematics. The present work magnifies one small aspect of their theory (leading terms) to get a simple result that is independent of the expansions themselves and still of considerable utility.

Evaluation of relativistic and radiative corrections to the energy of two-electron atomic states

Hylleraas-Scherr-Knight variational perturbation wave functions, known through 20th perturbation order, have been used to evaluate- in the form of power series expansions in Z −1-the lowest-order relativistic and radiative correction s to the energy of the two-electron atomic states (1s) 21S (2p) 23P. The mass polarization correctionm, which contributes to the same order of magnitude, has been determined as well. The total theoretical energies are tabulated for the first 10 elements of the helium isoelectronic sequence, and the values are compared with available experimental results.

On the steady rotation of an axisymmetric solid in a conducting fluid at high Hartmann numbers

In an earlier paper [Shail, 1] one of the present authors considered the effect of a magnetic field on the frictional couple experienced by an axisymmetric solid insulator which rotates slowly in a bounded viscous conducting fluid, the applied magnetic field being parallel to the axis of rotation. Results for the couple in various geometrical configurations were obtained in the form of power series expansions in the Hartmann number M , and hence are valid only for M ≪ 1. However, because of the increased rigidity given to the fluid by the applied field, it is physically evident that the magnetic field will have a more dramatic effect on the flow pattern for large values of M . Thus one object of this paper is to investigate the frictional couple on a solid insulator rotating in an unbounded fluid for values of M ≫ 1.

On some formal power series expansions

On some formal power series expansions

Self-Consistent Potential for Individual Particle Motions in the Nuclei with Closed Shells

The principle of mass renormalization in the quantum electrodynamics is applied to derive the equations which the potential for the individual particle motions satisfies self-consistently in the nuclei with the doubly closed shells. The potential is to be such that the excitation energy of each single particle (hole) state is invariant when the residual interactions are taken into account. Since this condition is concerned only with the diagonal matrix elements of the potential, the other matrix elements are so determined that the wave functions of the unperturbed ground and single particle (hole) states can best simulate the corresponding perturbed ones. The perturbed ground state is supposed not to have the components with one (particle and hole) pair excitation. Each perturbed single particle (hole) state should have the components of the different unperturbed single particle (hole) states as small as possible. The equations for the potential are given in a form of power series expansion with respect to the two-body interactions and are explicitly written up to the second order. The result turns out to be practically identical with that for the real part of the optical potential. The relation with Sawada's theory in an infinite system is pointed out.

Resistivity and Hall Effect in Liquid Metals

We assume a liquid metal in which the unscreened electron-ion interaction is the sum of spherically symmetric potentials. Using an exact expression for the conductivity tensor in a magnetic field H, we obtain expressions for the Hall constant $R$ and for the zero-magnetic-field frequency-dependent conductivity tensor ${{\ensuremath{\sigma}}_{\ensuremath{\mu}\ensuremath{\mu}}}^{0}(\ensuremath{\omega})$ in the form of power-series expansions in the potential. Effects due to the electron spin are ignored. Calculations are done for Zn, Li, and Na using the pseudopotentials calculated by Harrison. For Zn it is shown that the series for ${{\ensuremath{\sigma}}_{\ensuremath{\mu}\ensuremath{\mu}}}^{0}(\ensuremath{\omega})$ converges slowly if it converges at all. It is concluded that the small-pseudopotential approximation is not convenient for, and possibly not valid for, dealing with electrical transport in liquid metals.