A differentially algebraic elimination theorem with application to analog computability in the calculus of variations
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- by Lee A. Rubel and Michael F. Singer
- Proc. Amer. Math. Soc. 94 (1985), 653-658
- DOI: https://doi.org/10.1090/S0002-9939-1985-0792278-1
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Abstract:
An elimination theorem is proved in differential algebra, from which it follows that an analytic solution of virtually any ordinary differential equation that you can "write down" must actually solve an algebraic differential equation. As a corollary, it follows that the solutions of a large class of variational problems can be produced by an analog computer.References
- Richard Askey, Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR 0481145
- A. Babakhanian, Exponentials in differentially algebraic extension fields, Duke Math. J. 40 (1973), 455–458. MR 314807 Richard Courant and David Hilbert, Methods of mathematical physics, Interscience, New York, 1966. O. Hölder, Ueber die Eigenschaft der Gamma Funktion keineralgebraischen Differentialgleichung zu genügen, Math. Ann. 28 (1887), 1-13. Leonard Lipshitz and Lee A. Rubel, A differentially algebraic replacement theorem, and analog computability, Preprint, Fall 1984.
- Alexander Ostrowski, Über Dirichletsche Reihen und algebraische Differentialgleichungen, Math. Z. 8 (1920), no. 3-4, 241–298 (German). MR 1544442, DOI 10.1007/BF01206530
- Marian Boykan Pour-el, Abstract computability and its relation to the general purpose analog computer (some connections between logic, differential equations and analog computers), Trans. Amer. Math. Soc. 199 (1974), 1–28. MR 347575, DOI 10.1090/S0002-9947-1974-0347575-8
- Lee A. Rubel, Some research problems about algebraic differential equations, Trans. Amer. Math. Soc. 280 (1983), no. 1, 43–52. MR 712248, DOI 10.1090/S0002-9947-1983-0712248-1
- Claude E. Shannon, Mathematical theory of the differential analyzer, J. Math. Phys. Mass. Inst. Tech. 20 (1941), 337–354. MR 6251, DOI 10.1002/sapm1941201337
Bibliographic Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 94 (1985), 653-658
- MSC: Primary 12H05
- DOI: https://doi.org/10.1090/S0002-9939-1985-0792278-1
- MathSciNet review: 792278