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Symbolic Computation of Divided Differences
, 1999
"... Divided differences are enormously useful in developing stable and accurate numerical formulas. For example, programs to compute f(x)  f(y) as might occur in integration, can be notoriously inaccurate. Such problems can be cured by approaching these computations through divided difference formulati ..."
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Divided differences are enormously useful in developing stable and accurate numerical formulas. For example, programs to compute f(x)  f(y) as might occur in integration, can be notoriously inaccurate. Such problems can be cured by approaching these computations through divided difference formulations. This paper provides a guide to divided difference theory and practice, with a special eye toward the needs of computer algebra systems that should be programmed to deal with these oftenmessy formulas.
Doing Calculus using Maple
, 1993
"... Contents 1 Introduction 2 2 First Steps 3 2.1 Logging On and Getting into Maple : : : : : : : : : : : : : : : 3 2.2 Elementary Operations in Maple : : : : : : : : : : : : : : : : 5 3 Functions 11 3.1 Defining and Evaluating Functions : : : : : : : : : : : : : : : 12 3.2 Plotting graphs of functi ..."
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Contents 1 Introduction 2 2 First Steps 3 2.1 Logging On and Getting into Maple : : : : : : : : : : : : : : : 3 2.2 Elementary Operations in Maple : : : : : : : : : : : : : : : : 5 3 Functions 11 3.1 Defining and Evaluating Functions : : : : : : : : : : : : : : : 12 3.2 Plotting graphs of functions and solving equations : : : : : : : 14 4 Limits of functions and sequences 16 5 Derivatives 17 6 Integrals 18 7 Differential Equations 21 8 Slightly advanced Maple features 22 8.1 Various Maple programs : : : : : : : : : : : : : : : : : : : : : 22 8.2 Including Maple plots and formulas in L A T E X files : : : : : : : 25 9 Where to go from here 26 10 Calculus Computer Aids in General 27 1 Introduction The use of electronic computers in learning/teaching Calculus dates back a
.4.3 Belyakov's scaling
"... to a homoclinic orbit to a saddle. The system we study is the following x = y \Gamma z y = 2:657466x +2:328733y + x 2 + xy + x 2 +0:83893461z (8.14) z = e 2:657466x \Gamma 0:83893461y 3:361277 \Gamma 0:83893461z This system was derived by augmenting a truncated unfolding of the normal ..."
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to a homoclinic orbit to a saddle. The system we study is the following x = y \Gamma z y = 2:657466x +2:328733y + x 2 + xy + x 2 +0:83893461z (8.14) z = e 2:657466x \Gamma 0:83893461y 3:361277 \Gamma 0:83893461z This system was derived by augmenting a truncated unfolding of the normal form of the Takens Bogdanov bifurcation x = y y = 2cx + (1 + c)y + x 2 + xy (8.15) which is numerically observed to have a homoclinic connection to the origin at c 1:328733,
Symbolic Computations for Random Fields Using Second Order Perturbation Method
, 2000
"... The main idea is to show the application of symbolic computations in analysis of engineering systems with random parameters. The general computational methodology is based on the stochastic second order perturbation method and its implementation in the mathematical package MAPLE 6. The entire approa ..."
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The main idea is to show the application of symbolic computations in analysis of engineering systems with random parameters. The general computational methodology is based on the stochastic second order perturbation method and its implementation in the mathematical package MAPLE 6. The entire approach is displayed on the example of simple single degree of freedom dynamical system with random spring stiffness. The results of symbolic computations are derived numerically in the form of probabilistic moments of the structural response computed for the whole analysis time domain. The general methodology can be applied to all these engineering problems where the response can be derived symbolically in deterministic case, while input parameters of the system are random variables, fields or processes characterized by the probability density function (PDF) of any type.
Polynomial oscillators as perturbations of multiple square wells
, 2008
"... We propose an extension of applicability of the RayleighSchrödinger perturbation expansions. Our innovated prescription employs an N−parametric renormalization of wavefunctions and is shown to be still able to define the corrections via an N by N matrix inversion. For the first few positive intege ..."
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We propose an extension of applicability of the RayleighSchrödinger perturbation expansions. Our innovated prescription employs an N−parametric renormalization of wavefunctions and is shown to be still able to define the corrections via an N by N matrix inversion. For the first few positive integers N, the piecewise constant forces with N discontinuities are recommended as a promising and new unperturbed model. The trigonometric form of its bound states facilitates a consequent perturbative treatement of an arbitrary polynomial potential: Constructively, we demonstrate that the necessary N 2 + N “input ” matrix elements are obtainable nonnumerically in all orders. PACS 03.65.Ge 1 1
Perturbation method
, 2008
"... with triangular propagators and anharmonicities of intermediate strength ..."
and
, 2001
"... Apparently, the energy levels merge and disappear in many PT symmetric models. This interpretation is incorrect: In squarewell model we demonstrate how the doublets of states in question continue to exist at complex conjugate energies in the strongly nonHermitian regime. PACS ..."
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Apparently, the energy levels merge and disappear in many PT symmetric models. This interpretation is incorrect: In squarewell model we demonstrate how the doublets of states in question continue to exist at complex conjugate energies in the strongly nonHermitian regime. PACS