Pade Approximation Method Research Articles

An Asymptotic Method for Effective Properties of Visco‐Elastic Composite Materials

AbstractWe consider visco‐elastic two‐components composite materials, consisting of an infinite regular arrays of fibers inclusions embedded in a matrix. Analytical formulas for Laplace transforms of the effective shear relaxation and creep functions are derived. For this aim an asymptotic technique, based on the homogenization method, the three‐phase model, the method of boundary perturbation, Pade and quasifractional approximants, is used Obtained results also allow us to evaluate the effective complex shear modulus and loss tangent for stationary harmonic oscillations of considered composite material. Derived solutions are valid for all values of the components concentrations and properties. Our method may be effectively used for determining various effective properties of visco‐elastic composite materials with different inclusions geometries.

Magnetic Properties and Critical Behaviour of the B-Spinel CdCr2xIn2?2xS4 (0.9 ? x ? 1)

We study the magnetic properties and the critical behaviour of the B-spinel CdCr2xIn2—2xS4 systems in the range of concentration 0.9 ≤ x ≤ 1. Using the mean-field theory combined with the probability distribution of interaction, the exchange integrals were calculated up to the second nearest neighbours. The critical temperature and the critical exponents for the magnetic sysceptibility and the correlation length γ and ν, respectively, are deduced by applying the high-temperature series expansion (HTS) extrapolated with the Pade approximants (PA) method. The obtained values of Tc are in good agreement with those predicted by the magnetic measurements. The values of γ and ν are sensitive to the dilution ratio x. For the compounds situated in the region 0.98 ≤ x ≤ 1, they are close to those of the 3D Heisenberg model. For the concentration x = 0.9, γ and ν deviate slowly from the values predicted by this model and approach the values found in re-entrant systems.

Asymptotic Padé-approximant methods and QCD current correlation functions

Asymptotic Pade-approximant methods are utilized to estimate the leading-order unknown (i.e., not-yet-calculated) contributions to the perturbative expansions of two-current QCD correlation functions obtained from scalar-channel fermion and gluon currents, as well as from vector-channel fermion currents. Such contributions to the imaginary part of each correlator are polynomials of logarithms whose coefficients (other than the constant term within the polynomial) may be extracted from prior-order contributions by use of the renormalization-group (RG) equation appropriate for each correlator. We find surprisingly good agreement between asymptotic Pade-approximant predictions and RG-determinations of such coefficients for each correlation function considered, although such agreement is seen to diminish with increasing numbers of quark flavours. The RG-determined coefficients we obtain are then utilized in conjunction with asymptotic Pade-approximant methods to predict the RG-inaccessible constant terms of the leading-order unknown contributions for all three correlators. The vector channel predictions lead to estimates for the order $(\alpha_s^4)$ contribution to $R(s) \equiv[\sigma(e^+ e^- \to hadrons) / \sigma(e^+ e^- \to \mu^+ \mu^-)]$ for three, four, and five flavours.

Model Conversions of Uncertain Linear System via the Interval Tuning Bilinear Approximation Method.

This paper presents a new tuning bilinear approximation method together with the interval arithmetic operation to convert a continuous-time (descrete-time) uncertain system to an equivalent discrete-time (continuous-time) uncertain model.Based on the bilinear approximation method, we induct one parameter for tuning the error bound between the exact and approximate uncertain system matrices.The bounds of interval system matrices via the proposed interval tuning bilinear approximation method are narrower than those of the existing interval bilinear and the interval pade approximation methods(1).The resulting interval models (the enclosing interval models) are able to tightly enclose the exact uncertian models.Also the approximate discrete-time interval solution is able to tightly enclose the exact interval solution of the continuous-time uncertain state-space equation.The proposed method will be helpful for analysis and synthesis of uncertain systems.

Fractional-rational regularization of a system of linear equations over the skew-field of quaternions

We generalize the Pade approximation method of transforming a power series into a rational function for systems of linear equations over the skew-field of quaternions. We carry out a regularization of the solution for the case when the operator equation is ill-posed.

Renormalization-scale-invariant continuation of truncated QCD (QED) series — an analysis beyond large- β0 approximation

An approximation algorithm is proposed to transform truncated QCD (or QED) series for observables. The approximation is a modification of the Baker-Gammel approximants, and is independent of the renormalization scale (RSci) μ — the coupling parameter α(μ) in the series and in the resulting approximants can evolve according to the perturbative renormalization group equation (RGE) to any chosen loop order. The proposed algorithm is a natural generalization of the recently proposed method of diagonal Pade approximants, the latter making the result RScl invariant in large- β 0 approximation for α(μ). The algorithm described below can extract a large amount of information from a calculated available truncated perturbative series for an observable, by implicitly resumming large classes of diagrams.

Asymptotic Padé Approximants and the SQCD β-function

We present a prediction for the four loop $\beta$-function for SQCD based on the method of Asymptotic Pad\'e Approximants.

Explicit Lower Bounds for Rational Approximation to Algebraic Numbers

In this paper, we apply Pade approximation methods to derive completely explicit measures of irrationality for certain classes of algebraic numbers. Our approach is similar to that taken previously by G.V. Chudnovsky but has some fundamental advantages with regards to determining implicit constants. Our general results may be applied to produce specific bounds of the flavour of . Further examples are tabulated and applications to Diophantine equations are briefly discussed as are other topics of related interest. 1991 Mathematics Subject Classification: primary 11J68, 11J82; secondary 11D41.

Model conversions of uncertain linear systems using interval multipoint Pade approximation

Many dynamic systems and industrial control processes can be represented by a multirate sampled-data uncertain system, which consists of a continuous-time uncertain subsystem and a multirate discrete-time uncertain subsystem. The uncertainties in these systems arise from unmodeled dynamics, parameter variations, sensor noises, actuator constraints, etc. As is the common practice the sampled-data uncertain system needs to be converted to a purely continuous-time or discrete-time uncertain model, so that the well-established analysis and design methods in the continuous-time or discrete-time domain can be directly applied to the equivalent model. This paper presents a new interval multipoint Pade approximation method for converting a continuous-time (discrete-time) uncertain linear system to an equivalent discrete-time (continuous-time) uncertain model. The system matrices characterizing the state-space descriptions of the original uncertain systems are represented by interval matrices. Using the approximate uncertain models obtained based on interval analysis and multipoint Pade approximation the dynamic states of the resulting models have been shown to be able to closely match those of the original uncertain systems for a relatively longer sampling period.

Series analysis of tricritical behaviour: mean-field model and partial differential approximants

A partial differential approximant (PDA) analysis of series for the tricritical point of the mean-field model of Glasser, Privman, and Schulman has been developed. All features of the tricritical point are exactly reproduced by an overwhelming majority of the trial PDAs in contrast to the mixed success of the slicewise Pade approximant method. The effect of noise on the approximant convergence is also studied.

Digital interval modeling and hybrid control of uncertain systems

This paper addresses two issues: (1) converting a continuous-time uncertain system to an equivalent discrete-time interval model; and (2) constructing a robust digital control law from a robust analogue control law for hybrid control of sampled-data uncertain systems. The system matrices characterizing the state-space representation of the original continuous-time uncertain systems are assumed to be interval matrices. The Pade approximation method together with a geometric-series approximation method is employed to obtain the generalized enclosing discrete-time interval models. The generalized enclosing interval models are able to tightly enclose the exact discrete-time uncertain model, and can be utilized for digital simulation and digital design of continuous-time uncertain systems. A new family of digitally redesigned interval controllers is constructed from a continuous-time robust controller for robust digital control of continuous-time uncertain systems. Using the newly digitally redesigned interval controllers, the dynamic states of the digitally controlled sampled-data uncertain systems are able to closely match those of the original analogously controlled continuous-time uncertain systems for a relatively longer sampling period.

Vibration Analysis of Viscoelastically Damped Sandwich Shells

The simplified governing equations and corresponding boundary conditions of vibration of viscoelastically damped unsymmetrical sandwich shells are given. The asymptotic solution to the equations is then discussed. If only the first terms of the asymptotic solution of all variables are taken as an approximate solution, the result is identical with that obtained from the modal strain energy method. By taking more terms of the asymptotic solution with successive calculations and use of the Pade approximants method, accuracy of natural frequencies and modal loss factors of sandwich shells can be improved. The lowest three or four natural frequencies and modal loss factors of simply supported cylindrical sandwich shells are calculated.

Model conversions of uncertain linear systems via the interval Pade approximation method

This paper presents a new interval Pade approximation method to convert a continuous-time (discrete-time) uncertain linear system to an equivalent discrete-time (continuous-time) uncertain model via interval arithmetic operations. Based on the inclusion theorem related to the interval arithmetic, the interval Pade's approximants and their associated interval error matrices with interval arguments are obtained via the Pade's approximants and their associated error matrices with degenerate (real) arguments, respectively. Tighter error bounds of various approximate uncertain models with respect to their exact uncertain models are determined and used to modify the obtained Pade's approximants, so that the resulting approximate uncertain models are able to tightly enclose the original uncertain systems. Thus, the analysis and design of the original uncertain systems can be indirectly carried out using the converted uncertain models in either the continuous-time or the discrete-time domain.

Acoustic scattering in three‐dimensional fluid media

The scattering of acoustic waves from three‐dimensional compressible fluid scatterers is considered. Particular attention is paid to cases where the scatterers have moderate magnitude in compressibility contrast and nondimensional wave number. The perturbation method based on Pade approximants developed by Chandra and Thompson [J. Acoust. Soc. Am. 92, 1047–1055 (1992)] is extended to allow the solution of problems in three dimensions. It is shown that the functional form afforded by the Pade approximant model allows one to represent and evaluate the characteristic resonances and mode shapes of the scattered pressure field. These modes are a function of the compressibility contrast and the frequency of the incident pressure wave. Numerical results are shown to compare favorably with analytical solutions for scattering from a sphere. The Pade approximant method is shown to be feasible for calculating the pressure scattered from an inhomogeneous distribution of scatterers as well as for determining internal ...

Study of the strongly interacting Higgs physics model with heavy fermions

We study the effect of a possible fourth heavy generation of fermions on the Higgs sector of the standard model. We show, from the violation of elastic unitarity, that the scale of strong interactions is well below 1 TeV even with a Higgs mass as low as 500 GeV provided the fourth generation fermion mass is equal or larger than the Higgs mass. The diagonal Pade approximant method is then used to unitarize the partial wave amplitudes. It is found that, for the fourth generation fermion masses which are comparable to or larger than the Higgs mass, the Landau ghosts in the I = 0 and I = 2 channels of the reconstructed amplitudes move too close to the physical region to be accepted.

Digital redesign of a continuous-time controller using the pade and inverse-pade approximation method

This paper presents a mixed method consisting of the Pade approximation method and the inverse-Pade approximation method for digital redesign of a continuous-time state-feedback controller. Using this digitally redesigned controller, the states of digitally controlled sampled-data system closely match those of the original continuous-time controlled system for a relatively longer sampling period. It is shown that the digitally redesigned controller based on the bilinear and inverse-bilinear transform method is a specific class of the proposed controller.

Series analysis of tricritical behaviour: mean-field model and slicewise Pade approximants

A mean-field model is proposed as a test case for tricritical series analyses methods. Derivation of the 50th order series for the magnetization is reported. As the first application this series is analyzed by the traditional slicewise Pade approximant method popular in earlier studies of tricriticality.

Pade approximations for detectability in K-clutter and noise

The Pade approximation (PA) method is used to analyze the detection performance of single and multiple pulse radar systems operating in K-distributed clutter and thermal noise. Simple approximations for false-alarm and detection probabilities are obtained, using lower order moments for the detection decision statistic. Both envelope and squaring detector laws are considered, with noncoherent integration, for independent and correlated K clutter. The target is assumed to be pulse-to-pulse Rayleigh fading. The methods are a substantial application of the PA methods we have previously published. >

Accurate calculation of Coulomb sums: Efficacy of Padé-like methods.

The adequacy of numerical sequence accelerative transforms in providing accurate estimates of Coulomb sums is considered, referring particularly to distorted lattices. Performance of diagonal Pade approximants (DPA) in this context is critically assessed. Failure in the case of lattice vacancies is also demonstrated. The method of multiple-point Pade approximants (MPA) has been introduced for slowly convergent sequences and is shown to work well for both regular and distorted lattices, the latter being due either to impurities or vacancies. Viability of the two methods is also compared. In divergent situations with distortions owing to vacancies, a strategy of obtaining reliable results by separate applications of both DPA and MPA at appropriate places is also sketched. Representative calculations involve two basic cubic-lattice sums, one slowly convergent and the other divergent, from which very good quality estimates of Madelung constants for a number of common lattices follow.

A transient finite element analysis of linear viscoelastic material model

Since the attenulation of propagating waves through soil/rock is related to the localized material properties as well as the strain developed, the commonly used Rayleigh-type damping model and its variations are not suitable for dynamic finite element analysis of such materials. A linear viscoelastic material model based on the concept of the relaxation spectrum is manipualted in place of the damping model in this paper. The method proposed by Day and Minster11 to transform the convolutional form of the stress–strain relationship to a set of differential operators using the Pade approximant method is generalized to non-scalar waves and implemented for transient finite element analyses. A time-marching scheme is also proposed to incorporate the resultant differential operators into the governing equation of motion. The accuracy related to the Pade approximant method and the time-marching scheme is investigated by critically analysing some scalar wave propagation problems. The proposed technique is further verified using two one-dimensional stress wave propagation problems and a two-dimensional transient propagating wave through an unbounded linear viscoelastic medium. Some encouraging results have been obtained using the proposed technique and guidelines for using this technique are also presented. Comparisons of analytical solutions obtained by Fourier synthesis and numerical results have been provided.