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), 653-658

MSC:
Primary 12H05

DOI:
https://doi.org/10.1090/S0002-9939-1985-0792278-1

MathSciNet review:
792278

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[ASK]**Richard Askey,*Orthogonal polynomials and special functions*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR**0481145****[BAB]**A. Babakhanian,*Exponentials in differentially algebraic extension fields*, Duke Math. J.**40**(1973), 455–458. MR**0314807****[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), 1-13.**[LIR]**Leonard Lipshitz and Lee A. Rubel,*A differentially algebraic replacement theorem, and analog computability*, Preprint, Fall 1984.**[OST]**Alexander Ostrowski,*Über Dirichletsche Reihen und algebraische Differentialgleichungen*, Math. Z.**8**(1920), no. 3-4, 241–298 (German). MR**1544442**, https://doi.org/10.1007/BF01206530**[POE]**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**0347575**, https://doi.org/10.1090/S0002-9947-1974-0347575-8**[RUB]**Lee A. Rubel,*Some research problems about algebraic differential equations*, Trans. Amer. Math. Soc.**280**(1983), no. 1, 43–52. MR**712248**, https://doi.org/10.1090/S0002-9947-1983-0712248-1**[SHA]**Claude E. Shannon,*Mathematical theory of the differential analyzer*, J. Math. Phys. Mass. Inst. Tech.**20**(1941), 337–354. MR**0006251**, https://doi.org/10.1002/sapm1941201337

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
12H05

Retrieve articles in all journals with MSC: 12H05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1985-0792278-1

Article copyright:
© Copyright 1985
American Mathematical Society