Rational Function Representation Research Articles

有理関数の分数表現を用いた<i>H</i><sub>∞</sub>同定

有理関数の分数表現を用いた<i>H</i><sub>∞</sub>同定

Representation of rational functions with prefix and suffix codings

We proceed with the characterization of rational functions by means of restricted class of morphisms. Left subsequential transductions can be factored in an endmarking followed by an uniform morphism, the inverse of a prefix morphism and an alphabetic morphism. Rational functions require the inverse of a prefix morphism followed by the inverse of a suffix morphism.

Inverse-scattering view of modal structures in inhomogeneous optical waveguides

To understand the physical meaning of rational reflection coefficients in inverse-scattering theory for optical waveguide design [ J. Opt. Soc. Am. A6, 1206 ( 1989)], we studied the relationship between the poles of the transverse reflection coefficient and the modes in inhomogeneous dielectrics. By using a stratified-medium formulation we showed that these poles of the spectral reflection coefficient satisfy the same equation as the guidance condition in inhomogeneous waveguides. Therefore, in terms of wave numbers, the poles are the same as the discrete modes in the waveguide. The radiation modes have continuous real values of transverse wave numbers and are represented by the branch cut on the complex plane. Based on these results, applications of the Gel’fand–Levitan–Marchenko theory to optical waveguide synthesis with the rational function representation of the transverse reflection coefficient are discussed.

Compositional representation of rational functions

On montre que les fonctions rationnelles coincident avec les compositions de marquages terminaux, morphismes et inverses de morphismes, injectifs. Plus precisement, toute fonction rationnelle τ se represente sous la forme τ=μ m α 1 α 2 −1 α 3 ou μ m est un marquage terminal, α 1 , α 3 sont des morphismes et α 2 est un morphisme injectif

A rational function arithmetic and simplification system in common Lisp

We describe a multivariate rational function (ratio of polynomials) arithmetic simplification system written in Common Lisp. These programs form one module in a sequence of packages written with an aim to develop an efficient, freely-distributed computer algebra system. In this report, we discuss the representation of rational functions in the system and the programs for rational addition, multiplication, and exponentiation. We also describe a package that converts between expressions in Lisp prefix form and the representation of rational expressions in the system.

A method for determination of the potential from the reaction matrix at fixed energy

The inverse problem at fixed energy, in the case of a central potential with finite range, is transformed to a form that is similar to the inverse problem for the S-wave scattering. Using a rational-function representation of the input data for the reaction matrix for different partial waves and a modified Gel'fand–Levitan formulation, one can construct the potential from the partial-wave phase shift at a given energy. In this formulation of the inverse problem the properties of the Cauchy matrix have been used to perform a number of operations analytically, thus reducing the sources of numerical round-off and truncation errors. The method has been applied to construct the potential for some simple scattering problems.

Nucleon-nucleon potentials from Gel'fand-Levitan and Marchenko inversions.

Inverse scattering theory for the uncoupled channels of neutron-proton systems is developed from both the Gel'fand-Levitan and Marchenko fundamental equations. A most practical form of that theory is deduced by starting with a rational function representation of the phase shift data. By using Pad\'e approximants for the exponential function ${e}^{z}$, rational function representations for the scattering, Jost and spectral functions result. They facilitate accurate numerical solutions of both fundamental equations and from which local, energy-independent channel potentials are obtained. The Reid soft-core potential phase shifts when used as input data give potentials in excellent agreement with the original. Inversion potentials have also been generated by using as input empirical phase shifts and also those from the Paris and Bonn meson exchange interactions. Results are computed for the $^{1}\mathrm{S}_{0}$, $^{3}\mathrm{P}_{0}$, $^{3}\mathrm{P}_{1}$, $^{1}\mathrm{D}_{2}$, and $^{1}\mathrm{P}_{1}$ channels specifically and the potentials are transformed into central, tensor, spin-orbit, and quadratic spin-orbit radial form factors.

On the representation of rational functions of bounded complexity

The main result of this article is the Representation Theorem which characterizes families of rational functions of bounded complexity by appropriate mappings. This description is not only independent of the characteristic of the underlying field but also of the specific complexity measure under consideration. As an application of the Representation Theorem we derive good lower complexity bounds.

Rational function representation for accurate exchange energy functionals

A representation of the exchange energy functional as the product of a homogeneous gas functional and a rational function approximation in the density gradient is developed and shown to describe accurately both the total exchange energy and the local exchange energy density of atomic systems. For the atoms H through Kr, the error in the total exchange energy is generally <0.02 hartree except for Cr through Zn where it is <0.21 hartree.

Representations and Parallel Computations for Rational Functions

Representations of univariate rational functions over a given base of polynomials are considered, and a fast parallel algorithm for converting from one base representation to another is given. Special cases of this conversion include the following symbolic manipulation problems: Taylor expansion, partial fraction decomposition, Chinese remainder algorithm, elementary symmetric functions, Pade approximation, and various interpolation problems. If n is the input size, then all algorithms run in parallel time $O(\log ^2 n)$ and use $n^{O(1)} $ processors. They work over an arbitrary field.

Rational Function Representation of Flap Noise Spectra including Correction for Reflection Effects

A rational function is presented for the acoustic spectra generated by deflection of engine exhaust jets for under-the-wing and over-the-wing versions of externally blown flaps. The functional representation is intended to provide a means for compact storage of data and for data analysis. The expressions are based on Fourier transform functions for the Strouhal normalized pressure spectral density, and on a correction for reflection effects based on the TV-independent-source model of P. Thomas extended by use of a reflected ray transfer function. Curve fit comparisons are presented for blown flap data taken from turbofan engine tests and from largescale cold-flow model tests. Application of the rational function to scrubbing noise theory is also indicated.

MACSYMA - the fifth year

MACSYMA is a large symbolic and algebraic manipulation system which has been under development at Project MAC, M.I.T. since 1969. We first discuss some of the design decisions, such as multiple representations, that led to the current system. We then give a brief description of the facilities of MACSYMA by dividing the system into seven packages: language and interactive facilities, general representation, rational function representation and related algorithms, the integration subsystem, the power series subsystem, the MACLISP system, and miscellaneous facilities.

On the representation of rational functions as sums of squares

On the representation of rational functions as sums of squares