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NonStandard Analysis in ACL2
, 2001
"... ACL2 refers to a mathematical logic based on applicative Common Lisp, as well as to an automated theorem prover for this logic. The numeric system of ACL2 reflects that of Common Lisp, including the rational and complexrational numbers and excluding the real and complex irrationals. In conjunction ..."
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Cited by 18 (7 self)
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ACL2 refers to a mathematical logic based on applicative Common Lisp, as well as to an automated theorem prover for this logic. The numeric system of ACL2 reflects that of Common Lisp, including the rational and complexrational numbers and excluding the real and complex irrationals. In conjunction with the arithmetic completion axioms, this numeric type system makes it possible to prove the nonexistence of specific irrational numbers, such as √2. This paper describes ACL2(r), a version of ACL2 with support for the real and complex numbers. The modifications are based on nonstandard analysis, which interacts better with the discrete flavor of ACL2 than does traditional analysis.
Foundations Of Nonstandard Analysis  A Gentle Introduction to Nonstandard Extemsions
 In Nonstandard analysis (Edinburgh
"... this paper is to describe the essential features of the resulting frameworks without getting bogged down in technicalities of formal logic and without becoming dependent on an explicit construction of a specific field ..."
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Cited by 10 (2 self)
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this paper is to describe the essential features of the resulting frameworks without getting bogged down in technicalities of formal logic and without becoming dependent on an explicit construction of a specific field
Mechanical Verification of a Square Root Algorithm Using Taylor’s Theorem
 In Formal Methods in Computer Aided Design (FMCAD'02
, 2002
"... Abstract. The IBM Power4 TM processor uses series approximation to calculate square root. We formally verified the correctness of this algorithm using the ACL2(r) theorem prover. The proof requires the analysis of the approximation error on a Chebyshev series. This is done by proving Taylor’s theore ..."
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Cited by 5 (1 self)
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Abstract. The IBM Power4 TM processor uses series approximation to calculate square root. We formally verified the correctness of this algorithm using the ACL2(r) theorem prover. The proof requires the analysis of the approximation error on a Chebyshev series. This is done by proving Taylor’s theorem, and then analyzing the Chebyshev series using Taylor series. Taylor’s theorem is proved by way of nonstandard analysis, as implemented in ACL2(r). Since Taylor series of a given order have less accuracy than Chebyshev series in general, we used hundreds of Taylor series generated by ACL2(r) to evaluate the error of a Chebyshev series. 1
Taylor's Formula with Remainder
 In Proceedings of the Third International Workshop of the ACL2 Theorem Prover and its Applications
, 2002
"... In this paper, we present a proof in ACL2(r) of Taylor's formula with remainder. ..."
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Cited by 4 (4 self)
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In this paper, we present a proof in ACL2(r) of Taylor's formula with remainder.
Solutions Surstables Des équations Différentielles Complexes LentesRapides à Point Tournant
, 1999
"... : We consider singularly perturbed complex differential equations of the form "u 0 = \Psi(x; u; a; ") which have a turning point of order p and a pdimensional control parameter a. We state conditions of geometric nature, and of transversality in the parameter a, and prove that under these assumpt ..."
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Cited by 2 (0 self)
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: We consider singularly perturbed complex differential equations of the form "u 0 = \Psi(x; u; a; ") which have a turning point of order p and a pdimensional control parameter a. We state conditions of geometric nature, and of transversality in the parameter a, and prove that under these assumptions there exist solutions which are uniformly bounded with respect to the small perturbation parameter " on some macroscopic domain, and which are exponentially close to each other. The approach  based on the solution of a boundary value problem, and an application of the fixpoint theorem  generalizes a technique elaborated in a previous article [13]. These results are generalized to the case where " takes values in sectors covering a neighborhood of the origin. We derive Gevrey estimates of the formal solution concerning the parameter and show that the Borel transform of this series may be analytically continuated. Some results are obtained concerning its truncated Laplace transform...
Osittaisdifferentiaaliyhtälöiden Ratkaisukäsitykset
"... In this article we discuss general theories for nonlinear partial differential equations. We first outline some historical developments and then expose in more detail the formal theory of PDEs, based on the work of E. Cartan, Spencer and others. Then we describe the development of some nonlinear the ..."
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In this article we discuss general theories for nonlinear partial differential equations. We first outline some historical developments and then expose in more detail the formal theory of PDEs, based on the work of E. Cartan, Spencer and others. Then we describe the development of some nonlinear theories of PDEs using generalized functions. This involves the theories of Rosinger and Colombeau, in particular. The main issue is the algebraization of the theory in a certain way which leads on one hand to a very general theory but on the other hand to an essential simplification of the concept of generalized functions. Finally we study the use of field extensions and Galois theory. We follow the development of differential algebra from early ideas of Drach and Vessiot into the modern abstract theory as developed by Ritt and Kolchin and examine the difficulties of extending Galois theory to partial differential equations.
Reçu le **** * ; accepté après révision le +++++
, 2006
"... La formalisation non standard, due à P. Cartier et Y. Perrin, des oscillations rapides fournit un cadre mathématique adéquat pour de nouvelles techniques d’estimation non asymptotiques, ne nécessitant pas l’analyse statistique habituelle des bruits entachant tout capteur. On en tire diverses conséqu ..."
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La formalisation non standard, due à P. Cartier et Y. Perrin, des oscillations rapides fournit un cadre mathématique adéquat pour de nouvelles techniques d’estimation non asymptotiques, ne nécessitant pas l’analyse statistique habituelle des bruits entachant tout capteur. On en tire diverses conséquences sur les bruits multiplicatifs, la largeur des fenêtres d’estimations paramétriques et les erreurs en rafales. Pour citer cet article: M. Fliess, C. R. Acad. Sci. Paris, Ser. I 342 (2006). Noises: a nonstandard analysis. Thanks to the nonstandard formalization of fast oscillating functions, due to P. Cartier and Y. Perrin, an appropriate mathematical framework is derived for new nonasymptotic estimation techniques, which do not necessitate any statistical analysis of the noises corrupting any sensor. Various applications are deduced for multiplicative noises, for the length of the parametric estimation windows, and for burst errors. To cite this article: M. Fliess, C. R. Acad.
1 Discretisations of higher order and the theorems of Faà di Bruno and DeMoivreLaplace
"... Abstract: We study discrete functions on equidistant and nonequidistant infinitesimal grids. We consider their difference quotients of higher order and give conditions for their nearequality to the corresponding derivatives. Important tools are nonstandard notions of regularity of higher order, an ..."
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Abstract: We study discrete functions on equidistant and nonequidistant infinitesimal grids. We consider their difference quotients of higher order and give conditions for their nearequality to the corresponding derivatives. Important tools are nonstandard notions of regularity of higher order, and the formula of Faà di Bruno for higher order derivatives and a discrete version of it. As an application of such transitions from the discrete to the continuous we extend the DeMoivreLaplace Theorem to higher order: nth order difference quotients of the binomial probability distribution tend to the corresponding nth order partial differential quotients of the Gaussian distribution.