A differentially algebraic elimination theorem with application to analog computability in the calculus of variations
Authors:
Lee A. Rubel and Michael F. Singer
Journal:
Proc. Amer. Math. Soc. 94 (1985), 653658
MSC:
Primary 12H05
MathSciNet review:
792278
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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.
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A.
Babakhanian, Exponentials in differentially algebraic extension
fields, Duke Math. J. 40 (1973), 455–458. MR 0314807
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Richard Courant and David Hilbert, Methods of mathematical physics, Interscience, New York, 1966.
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O. Hölder, Ueber die Eigenschaft der Gamma Funktion keineralgebraischen Differentialgleichung zu genügen, Math. Ann. 28 (1887), 113.
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Leonard Lipshitz and Lee A. Rubel, A differentially algebraic replacement theorem, and analog computability, Preprint, Fall 1984.
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Alexander
Ostrowski, Über Dirichletsche Reihen und algebraische
Differentialgleichungen, Math. Z. 8 (1920),
no. 34, 241–298 (German). MR
1544442, http://dx.doi.org/10.1007/BF01206530
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Marian
Boykan Pourel, 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 0347575
(50 #78), http://dx.doi.org/10.1090/S00029947197403475758
 [RUB]
Lee
A. Rubel, Some research problems about algebraic
differential equations, Trans. Amer. Math.
Soc. 280 (1983), no. 1, 43–52. MR 712248
(84j:34005), http://dx.doi.org/10.1090/S00029947198307122481
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Claude
E. Shannon, Mathematical theory of the differential analyzer,
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MR
0006251 (3,279a)
 [ASK]
 Richard Askey, Orthogonal polynomials and special functions, SIAM, Philadelphia, Pa., 1975. MR 0481145 (58:1288)
 [BAB]
 Ararat Babakhanian, Exponentials in differentially algebraic extension fields, Duke Math. J. 40 (1973), 455460. MR 0314807 (47:3357)
 [COH]
 Richard Courant and David Hilbert, Methods of mathematical physics, Interscience, New York, 1966.
 [HOL]
 O. Hölder, Ueber die Eigenschaft der Gamma Funktion keineralgebraischen Differentialgleichung zu genügen, Math. Ann. 28 (1887), 113.
 [LIR]
 Leonard Lipshitz and Lee A. Rubel, A differentially algebraic replacement theorem, and analog computability, Preprint, Fall 1984.
 [OST]
 Alexander Ostrowski, Uber Dirichletsche Reihen und algebraische Differentialgleichungen, Math. Z. 8 (1920), 241298. MR 1544442
 [POE]
 Marian Boykan PourEl, Abstract computability and its relation to the generalpurpose analog computer, etc., Trans. Amer. Math. Soc. 199 (1974), 129. MR 0347575 (50:78)
 [RUB]
 Lee A. Rubel, Some research problems about algebraic differential equations, Trans. Amer. Math. Soc. 280 (1983), 4352. MR 712248 (84j:34005)
 [SHA]
 Claude Shannon, Mathematical theory of the differential analyzer, J. Math. Phys. 20 (1941), 337354. MR 0006251 (3:279a)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198507922781
PII:
S 00029939(1985)07922781
Article copyright:
© Copyright 1985 American Mathematical Society
