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A structure theorem for the elementary functions and its application to the identity problem (1979)

Epstein, H. I. ; Caviness, B. F.

Springer

International journal of parallel programming 8 (1979), S. 9-37

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ISSN: 1573-7640

Keywords: Symbolic mathematical computation ; algebraic simplification ; canonical form ; regular form ; structure theorem ; elementary transcendental functions ; exponential function ; logarithm function ; trigonometric functions ; hyperbolic functions ; algebraic independence ; differential algebra ; Liouville fields

Source: Springer Online Journal Archives 1860-2000

Topics: Computer Science

Notes: Abstract This paper uses elementary algebraic methods to obtain new proofs for theorems on algebraic relationships between the logarithmic and exponential functions. The main result is a multivariate version of a special case of the structure theorem due to Risch that gives, in a very explicit fashion, the possible algebraic relationships between the exponential and logarithm functions. The structure theorem has important applications to symbolic mathematical computation in that it in essence provides a canonical form for the elementary transcendental functions, and hence solves the identity problem for this class of functions. Such applications are discussed in the last section.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF00995426

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Variable basis sets in perturbation theory: Numerical finite perturbation versus analytic approach (1979)

Epstein, Saul T. ; Sadlej, Andrzej J.

New York, NY : Wiley-Blackwell

International Journal of Quantum Chemistry 15 (1979), S. 147-167

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ISSN: 0020-7608

Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics

Source: Wiley InterScience Backfile Collection 1832-2000

Topics: Chemistry and Pharmacology

Notes: We consider the problem of determining variational, external-field-dependent corrections to nonoptimal zero-field nonlinear parameters. Both a direct analytic perturbation analysis and finite perturbation methods are described in a general way and in detail for the SCF approximation. The abstract theory is illustrated by reference to the results of several explicit calculations. Also, the sensitivity of the results to the choice of zero-field values is discussed.

Additional Material: 2 Ill.