Truncated Power Series Research Articles

Shock front analysis for axial shear wave propagation in a hyperelastic incompressible solid

A wavefront analysis is employed to study the propagation of axial shear waves in an incompressible hyperelastic solid, whose strain energy function is expressible as a truncated power series in terms of the basic invariants of the left Cauchy-Green tensor. Waves are generated by the application of an axial shear stress at the surface of a cylindrical cavity in an unbounded medium. Depending on the nature of the boundary condition, an acceleration front or a shock front propagates from the boundary of the cavity. For an acceleration front, the coefficients in the wavefront expansion satisfy a sequence of transport equations which can be solved analytically. For a shock front, a wavefront analysis gives approximate formulas for the wave speed, shock front and intensity of the various field variables at the front. As well, our shock front analysis is used to devise a method of estimating the breaking distance of a shock front. In order to test the validity of the results of our wavefront analysis, numerical solutions are obtained for waves initiated by a step function or by a finite duration pulse at the boundary. Our numerical solutions are found by using a recently proposed relaxation scheme for systems of conservation laws.

Identification of Transfer Function in a Feedback System and Autoregressive Analysis

Abstract Identification of open loop transfer functions in a feedback system is studied by the autoregressive analysis. Since an autoregressive model is a kind of truncated power series, bias effects are examined under an identifiable condition of the feedback system. Bias effects are mainly evaluated by zeros of innovation model equivalent to the feedback system, and disappears by a factor of minus power of the zeros, as the autoregressive model order becomes large.

Effect of diffusion and convection on the flux of depositing particles near a preadsorbed particle

A steady state convective-diffusion equation is solved using a collocation method to find the concentration profile and flux of adsorbing particles near a particle adsorbed on a line. At small values of the gravity number, NG=πd4 Δρg/6kT, the concentration profile and flux vary slowly near the preadsorbed particle, while they are highly non-uniform at large values of NG. The numerical results are compared with Brownian dynamics simulation for a range of NG values. The effect of the position of the system boundary on the collocation calculation is discussed and it is shown how the concept of flux balance may be used to improve the accuracy of the results. Finally, we develop a truncated power series that accurately fits the numerical data.

An adaptive receiver for the time- and frequency-selective fading channel

An adaptive receiver is presented in this paper for the reception of linearly modulated signals transmitted over a time- and frequency-selective fading channel. The channel is modeled as a truncated power series which represents the dispersive fading channel as a sum of three elementary flat-fading channels. The proposed receiver consists of a sequence estimator with a parallel channel estimator. The channel estimator recovers the instantaneous fading processes associated with each elementary channel and filters them to generate one-step predictions of each fading process. Some implementation difficulties and solutions are also discussed. Computer simulations using quadrature phase-shift keying (QPSK) and channels with moderate delay spreads and fade rates have been used to evaluate the performance of the receiver. The results show that our technique has potential in channels with delay spread of about 20%, signal-to-noise ratio (SNR) greater than 15 dB, and applications requiring bit-error rates (BER's) less than 10/sup -2/.

The deduction of continuously distributed activation energy populations from the moments of thermal evolution spectra

A theoretical analysis demonstrates that, if the total adatom evolution rate is measured during tempering of a substrate with an exponential tempering function, determination of the temperature moments of the rate profile leads to exact specification of the energy moments of the initial population distribution amongst varying activation energy sites. In the specific case of an energy truncated power series approximation to the initial population distribution the method allows deduction of the unknown coefficients of the power series and the truncation energy.

GITA: A REDUCE program for the normalization of polynomial Hamiltonians

For a given polynomial Hamiltonian near an equilibrium point the program GITA calculates analytically the normal Birkhoff-Gustavson form in Cartesian as well as angle-action coordinates and the formal integral of the motion. These quantities are presented in the form of a truncated power series, the highest degree of which determines the degree of the approximation. The program performs the calculation of the modified normal Birkhoff-Gustavson form for the Hamiltonian restricted by the resonance condition. The program package is written in the computer algebra system REDUCE.

A modeling via Hamilton's law with application to a model following resembling technique

As a result of direct application of Hamilton's law of varying action on the treatment of mechanical systems, solving both dynamic and control problems is done using algebraic equations only; no differential equations are needed. In these techniques both the dynamic response and control forces are assumed to be a truncated power series. The target was only to solve the concerned problem. Using the coefficients of the above-mentioned polynomials in the vector form, an input-output model for the mechanical system to follow a specified path, while it is perturbed out of that path, is introduced herein. The modeling uses only algebraic equations. In this work, a model following resembling technique is demonstrated as a useful application of this modeling way. That is to get the required input forcing function to obligate the mechanical system to behave like a reference system when it is shifted away from the specified path.

Non-symmetrical wave propagation in arteries

For a better understanding of the effects of initial stress on flow in elastic tubes, the propagation of a harmonic and non-symmetrical wave in an initially stressed thick cylindrical shell filled with an inviscid fluid is studied. Although the blood is known to be a non-Newtonian fluid, for simplicity, it is assumed to be a non-viscous, while the elastic tube is considered to be isotropic and incompressible. Utilizing the theory of small deformations superimposed on large initial static deformation, for a non-symmetrical perturbed motion the governing differential equations are obtained in cylindrical polar coordinates. Due to variability of the coefficients of the resulting differential equations of the solid body, the field equations are solved by truncated power series method. Applying the boundary conditions, the dispersion relation is obtained as a function of inner pressure, axial stretch and the thickness ratio. It is observed that the wave speed of the non-symmetrical wave is large as compared to the axially symmetrical case. Various special cases are also discussed in the paper.

Ray Method for Solving Dynamic Problems Connected With Propagation of Wave Surfaces of Strong and Weak Discontinuities

The aim of the article is to review the literature devoted to the solution of wave dynamics problems resulting in the propagation of nonstationary surfaces (volume waves) or lines (surface waves) of strong and weak discontinuities. In so doing one-term and multiple-term ray expansions, which are truncated power series with variable coefficients, are used. Jumps in all kinds of order of the time-derivatives of the functions to be desired serve as the coefficients. With their help the solutions are constructed behind the wave fronts up to the boundary of the wave motion domain. If the domains of the wave motion existence are extended, then the truncated ray series approximating the solution are not always uniformly valid over these domains. In this article, the methods for regularization of the truncated ray series are discussed, and in particular a new method which has been called by the authors as the method of “forward-area regularization”. These methods have gained recently wide application to solve the boundary-value problems connected with the wave propagation in bars, beams, plates, shells, 3D bodies, as well as with the dynamic contact interaction. In so doing, linear and nonlinear elastic, isotropic and anisotropic, thermoelastic, elastoplastic, and elastoviscoplastic media are used.

Interlaminar stress analysis of composite laminates using a sublaminate/layer model

A stress-function-based variational method developed in a previous work is extended and modified into a sublaminate/layer approach applicable to a laminated strip composed of a large number of differently orientated, anisotropic elastic plies. Lekhnitskii’s stress functions are used for two interior layers adjacent to a particular interface. The remaining layers are grouped into an upper sublaminate and a lower sublaminate. The stress functions are expanded in truncated power series of the thickness coordinate and the differential equations governing the coefficient functions are derived by using the complementary virtual work principle. The new approach limits the dimension of the eigenvalue problem to a fixed number irrespective of the number of layers in the sublaminates, so that reasonably accurate solutions of the interlaminar stresses can be computed with extreme ease. For symmetric, four-layer, angle-ply and cross-ply laminates, a comparison of the previous analysis results based on the pure layer model and new results based on two different sublaminate/layer models indicates reasonable overall agreement in the interlaminar stresses and superior agreement in the resultant peeling and shearing forces across end segments of the interface. Additional results are obtained for eight-layer, quasi-isotropic laminates under the strain loads of axial extension, bending and twisting.

Frozen orbits for satellites close to an Earth-like planet

We say that a planet is Earth-like if the coefficient of the second order zonal harmonic dominates all other coefficients in the gravity field. This paper concerns the zonal problem for satellites around an Earth-like planet, all other perturbations excluded. The potential contains all zonal coefficientsJ 2 throughJ 9. The model problem is averaged over the mean anomaly by a Lie transformation to the second order; we produce the resulting Hamiltonian as a Fourier series in the argument of perigee whose coefficients are algebraic functions of the eccentricity — not truncated power series. We then proceed to a global exploration of the equilibria in the averaged problem. These singularities which aerospace engineers know by the name of frozen orbits are located by solving the equilibria equations in two ways, (1) analytically in the neighborhood of either the zero eccentricity or the critical inclination, and (2) numerically by a Newton-Raphson iteration applied to an approximate position read from the color map of the phase flow. The analytical solutions we supply in full to assist space engineers in designing survey missions. We pay special attention to the manner in which additional zonal coefficients affect the evolution of bifurcations we had traced earlier in the main problem (J 2 only). In particular, we examine the manner in which the odd zonalJ 3 breaks the discrete symmetry inherent to the even zonal problem. In the even case, we find that Vinti's problem (J 4+J 2 2 =0) presents a degeneracy in the form of non-isolated equilibria; we surmise that the degeneracy is a reflection of the fact that Vinti's problem is separable. By numerical continuation we have discovered three families of frozen orbits in the full zonal problem under consideration; (1) a family of stable equilibria starting from the equatorial plane and tending to the critical inclination; (2) an unstable family arising from the bifurcation at the critical inclination; (3) a stable family also arising from that bifurcation and terminating with a polar orbit. Except in the neighborhood of the critical inclination, orbits in the stable families have very small eccentricities, and are thus well suited for survey missions.

Non-symmetrical waves in a prestressed elastic tube filled with an inviscid fluid

In order for a better understanding of the effect of initial stress on flow in elastic tubes, the propagation of time harmonic waves in a prestressed elastic tube filled with an inviscid fluid is studied. Although the blood is known to be a non-Newtonian fluid, for simplicity in the mathematical analysis, it is assumed to be non-viscous while the tube material is considered to be incompressible, isotropic and elastic. Utilizing the theory of small deformations superimposed on large initial static deformation, for a non-symmetrical perturbed motion, the governing differential equations are obtained in the cylindrical polar coordinates. Due to variability of the coefficients of the resulting differential equations of the solid body, the field equations are solved by a truncated power series method. Applying the boundary conditions, the dispersion relation is obtained as a function of inner pressure, axial stretch and the thickness ratio. It is observed that the wave speed of the non-symmetrical wave is large as compared to the symmetrical case. Various special cases as well as some numerical results are also discussed in the paper.

A ray method of solving problems connected with a shock interaction

Problems connected with a shock interaction of a thin elastic cylindrical bar and an elastic sphere with an Uflyand-Mindlin plate of a finite size are considered. In enumerated problems, an impact process is accompanied by material local deformations and propagation of wave surfaces of a strong or weak discontinuity in elastic bodies coming in contact. The local deformations are taken into account in terms of the Hertz's contact theory, but the dynamic deformations behind the fronts of incident and reflected waves are determined by means of truncated power series with variable coefficients (what is known as ray series). These two types of deformations are mated on the contact region boundary. As the truncated ray series, as a rule, are not uniformly applicable in the whole region of the wave solution existence, then for their improvement the method of “forward-area” regularization is used which is based on a combination of the recurrent equations of the ray method with the method of multiple scales. The method of power series (ray method) allows one to find the main characteristics of the impact theory in an analytical form.

Approximate eigenvalues, eigenvectors and inverse of a matrix with polynomial entries

LetM be a matrix with polynomial entries. This paper deals with calculating eigenvalues, eigenvectors and inverse ofM approximately in the form of truncated power series. The calculation of approximate eigenvalues is based on algorithms solving algebraic equation symbolically in terms of power series. Then, we give algorithms for calculating eigenvectors and inverse in terms of power series approximately. We show an example for each algorithm. Finally, we explain acceleration of convergence of power series and estimate of errors due to truncation of power series.

Two-level model of recording and reading of holographic gratings in photorefractive crystals: nonstationary case

A two-level model of recording and readout of holographic gratings in photorefractive crystals proposed by Rustamov has been developed for the nonstationary case. Based on the joint solution of kinetic equations and Maxwell's equations in the quasistationary approximation, a system of coupled partial differential equations for slowly varying optical wave amplitudes has been obtained. The use of a truncated power series expansion has allowed an expression to be obtained for diffraction efficiency in the nonstationary case. The dynamics of diffraction efficiency development is in satisfactory agreement with the available experimental data.

Exact solution of the dispersion equation for electromagnetic waves in sheared relativistic electron beams

The transcendental dispersion equation for electromagnetic waves propagating in the slow mode in sheared non-neutral relativistic cylindrical electron beams in strong applied magnetic fields is solved exactly. Thus, rather than truncated power series for the modified Bessel functions involved, use is made of modern algorithms able to compute such functions up to 18-digit accuracy. Consequently, new and significantly more important branches of the velocity shear instability are found. When the shear-factor and/or the geometrical parameter a/b (pipe-to-beam radius ratio) are increased, the unstable branches join, and the higher-frequency, larger-wavenumber modes are significantly enhanced. Since analytical solutions of the exact dispersion relation cannot be obtained, it is suggested that in all similar cases the methods proposed and demonstrated here should be used to carry out a rigorous stability analysis.

Power series in computer algebra

Formal power series (FPS) of the form Σ k = 0 ∞ a k ( x − x 0 ) k are important in calculus and complex analysis. In some Computer Algebra Systems (CASs) it is possible to define an FPS by direct or recursive definition of its coefficients. Since some operations cannot be directly supported within the FPS domain, some systems generally convert FPS to finite truncated power series (TPS) for operations such as addition, multiplication, division, inversion and formal substitution. This results in a substantial loss of information. Since a goal of Computer Algebra is — in contrast to numerical programming — to work with formal objects and preserve such symbolic information, CAS should be able to use FPS when possible. There is a one-to-one correspondence between FPS with positive radius of convergence and corresponding analytic functions. It should be possible to automate conversion between these forms. Among CASs only M acsyma provides a procedure powerseries to calculate FPS from analytic expressions in certain special cases, but this is rather limited. Here we give an algorithmic approach for computing an FPS for a function from a very rich family of functions including all of the most prominent ones that can be found in mathematical dictionaries except those where the general coefficient depends on the Bernoulli, Euler, or Eulerian numbers. The algorithm has been implemented by the author and A. Rennoch in the CAS M athematica , and by D. Gruntz in M aple . Moreover, the same algorithm can sometimes be reversed to calculate a function that corresponds to a given FPS, in those cases when a certain type of ordinary differential equation can be solved.

Electronic excitation transport and trapping in micellar systems: Monte Carlo simulations and density expansion approximation

A theoretical treatment of electronic energy transport and trapping in a finite volume is presented by taking, as a typical example, an ensemble of chromophores solubilized in a spherical micelle. A truncated power series expansion in the chromophore density is used to calculate the configurational average of G S( t), the probability that an initially excited donor molecule is still excited, and G D( t), the probability of finding an excitation in the sub-ensemble of donors at time t, which are directly related to experimental observables. A detailed analysis of the problem via a Monte Carlo simulation is carried out for different occupation numbers of donor and trap molecules in a micelle, and for different ratios of the micelle radius to the Förster radius. The density expansion, as checked against the numerical results, provides a quite reasonable approximation, especially at short times and low chromophore concentrations. The difference in energy transfer dynamics between a finite and infinite volume system is described. The theory developed can be useful for probing structural details of microdisperse systems via time-resolved fluorescence measurements.

Three new algorithms for multivariate polynomial GCD

Three new algorithms for multivariate polynomial GCD (greatest common divisor) are given. The first is to calculate a Gröbner basis with a certain term ordering. The second is to calculate the subresultant by treating the coefficients w.r.t. the main variable as truncated power series. The third is to calculate a PRS (polynomial remainder sequence) by treating the coefficients as truncated power series. The first algorithm is not important practically, but the second and third ones are efficient and seem to be useful practically. The third algorithm has been implemented naively and compared with the trial-division PRS algorithm and the EZGCD algorithm. Although it is too early to derive a definite conclusion, the PRS method with power series coefficients is very efficient for calculating low degree GCD of high degree non-sparse polynomials.

Software based linearisation of thermistor type nonlinearity

A simple algorithm-oriented digital technique is described that utilises a truncated power series to linearise thermistor type sensor characteristics. Early truncation of the series, made possible by the adoption of a comparison algorithm, reduces the computation time considerably. The comparison algorithm also reduces the per cent full-scale deviation in nonlinearity to well within an acceptable limit. The software developed has flexibility in the selection of sensor specification and range.