Erik Palmgren, Constructive nonstandard analysis, Methods et analyse non standard (A. Petry, ed.), vol. 9, Cahiers du Centre de Logique, 1996, pp. 69-97.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....t i = a ih for i = 0, 1, 2 n . Then the nth polygonial approximation is the function u n : R # R defined by # # # u n (t 0 ) s u n (t n ) u n (t i 1 ) hf(t i , u n (t i 1 ) 1 # i # 2 n . 5) A constructive proof of the convergence of Euler s method can be found in [15] (i.e. convergence to a solution of (4) so that this also works as a constructive proof of the existence of such a solution) and it is here shown that the sequence of polygonial approximations u n n#N is a Cauchy sequence. The estimate of convergence is that for each k # N if m,n # k, ....

.... of Euler s method will be restricted to use only rational numbers as the endpoints of the interval (a and b) and the initial value (s) The first step is to implement the sequence of approximating functions u n n#N and this will be done by using a general polygonial function as defined in [15]: Definition 21 Given a finite sequence of pairs of real numbers (a k , b k ) n k=0 with a 0 a 1 . a n , the polygonial function P : R # R is defined as P (a k ,b k ) x) b 0 n 1 # k=0 b k 1 b k a k 1 a k (max(0, x a k ) max(0, x a k 1 ) ....

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Erik Palmgren. Constructive nonstandard analysis. In A. Petry, editor, Methodes et analyse non standard, volume 9 of Cahiers du Centre de Logique, pages 69--97, 1996.

....was, in effect, taken already in the Schmieden and Laugwitz model for infinitesimal calculus [18] from 1956. It has more recently been put forward, in extended form, as a suitable foundation for constructive nonstandard analysis by Martin Lof [8] and subsequently been developed by the author in [11, 12]. A rich class of constructive models for infinitesimals emerges from topos theory and the synthetic differential geometry of Lawvere; see Kock [4] and Moerdijk and Reyes [10] These models have in general no strong transfer principle. Nevertheless, Moerdijk [9] succeeded in giving a constructive ....

E. Palmgren. Constructive nonstandard analysis. In: A. P'etry (ed.) M'ethodes et analyse non standard, Cahiers du Centre de Logique, vol. 9, Academia, Louvain-la-Neuve 1996, 369 -- 397.

....was, in effect, taken already in the Schmieden and Laugwitz model for infinitesimal calculus [14] from 1956. It has more recently, in extended form, been put forward as a suitable foundation for constructive nonstandard analysis by Martin Lof [5] and subsequently been developed by the author in [8, 9]. A rich class of constructive models for infinitesimals emerges from topos theory and work on the synthetic differential geometry of Lawvere, see Kock [2] and Moerdijk and Reyes [7] These models have in general no strong transfer principle. Nevertheless, Moerdijk [6] succeeded in giving a ....

E. Palmgren. Constructive nonstandard analysis. In: A. P'etry (ed.) M'ethodes et analyse non standard, Cahiers du Centre de Logique, vol. 9, Academia, Louvain-la-Neuve 1996, 369 -- 397.

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Erik Palmgren, Constructive nonstandard analysis, Methods et analyse non standard (A. Petry, ed.), vol. 9, Cahiers du Centre de Logique, 1996, pp. 69-97.

No context found.

Erik Palmgren, Constructive nonstandard analysis, Methods et analyse non standard (A. Petry, ed.), vol. 9, Cahiers du Centre de Logique, 1996, pp. 69-97.

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