Mathematical Perspectives on Neural Networks

By Paul Smolensky; Michael C. Mozer et al. | Go to book overview

if its real and imaginary parts are both computable. It is possible to extend this definition to functions defined over the whole real line (or over Rn) even though the Weierstrass approximation theorem does not hold.


REFERENCES

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Caldwell, J., & Pour-El, M. B. ( 1975). "On a simple definition of computable function of a real variable-- with applications to functions of a complex variable". Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 21, 1-19.

Denef, J., & Lipschitz, J. ( 1984). "Power series solutions of algebraic differential equations". Mathematische Annalen, 267, 213-238.

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Grzegorczyk, A. ( 1957a). "On the definitions of computable real continuous functions". Fundamenta Mathematicae, 44, 61-71.

Grzegorczyk, A. ( 1957b). "Some approaches to constructive analysis". In A. Heyting (Ed.). Constructivity in mathematics: 1957 Amsterdam 43-61, Studies in logic and the foundations of mathematics, Amsterdam: North Holland.

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Korn, G. A., & Korn, T. M. ( 1964). Electronic analog and hybrid computers. New York: McGraw-Hill.

Lacombe, D. ( 1955a). "Extension de la notion de fonction récursive aux fonctions d'une ou plusieurs variables réelles, I". Comptes Rendus des Séances d l'Académie des Sciences, Paris, 240, 2478-2480.

Lacombe, D. ( 1955b). "Extension de la notion de fonction récursive aux fonctions d'une ou plusieurs variables réelles II, III". Comptes Rendus des Séances d l'Académie des Sciences, Paris, 241, 13-14, 151-153.

Lacombe, D. ( 1955c). "Remarque sur les opérateurs récursifs et sur les fonctions récursives d'une variable réelle". Comptes Rendus des Séances d l'Académie des Sciences, Paris, 241, 1250-1252.

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Lacombe, D. (1958). "Sur les possibilités d'extension de la notion de fonction récursive aux fonctions d'une ou plusieurs variables reélles". Raissonement en Mathematiques et en Sciences Expérimentales: 1955Paris67-71, Colloq. Int. CNRS 70, Paris: CNRS Inst. B. Pascal.

Lipshitz, L., & Rubel, L. ( 1987). "A differentially algebraic replacement theorem and analog computability". Proceedings of the American Mathematical Society, 99, 367-372.

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