By B. Buchberger (auth.), Dr. Alfonso Miola, Dr. Marco Temperini (eds.)
New methodological elements with regards to layout and implementation of symbolic computation platforms are thought of during this quantity aiming at integrating such features right into a homogeneous software program surroundings for medical computation. The proposed technique relies on a mix of alternative recommendations: algebraic specification via modular process and of entirety algorithms, approximated and particular algebraic computing equipment, object-oriented programming paradigm, computerized theorem proving via equipment ? los angeles Hilbert and strategies of common deduction. particularly the proposed therapy of mathematical gadgets, through options for process abstraction, buildings type, and specified illustration, the programming technique which helps the layout and implementation matters, and reasoning services supported by way of the complete framework are defined.
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Additional info for Advances in the Design of Symbolic Computation Systems
Wang, D. (1990): Some examples for testing an integrated system. Tech. Rep. 4/18, P. F. Sistemi Informatici e Calcolo Parallel0, Consiglio Nazionale delle Ricerche, Rome. The uniform representation of mathematical objects by truncated power series C. Limongelli and M. Temperini 1 Introduction Systems for symbolic mathematics are based on the availability of powerful methods and techniques, which have been developed for numeric computation, symbolic and algebraic computation and automated deduction.
Ar)=OmodI, with i=l, ... ,n. Moreover let U=(Ui,j), i=l, ... ,n, j=l, ... ,r, iI, ... , in, evaluated at aI, ... , ar ). Assume that U is invertible mod I. Then, for any positive integer t, there exist a~t), ... , a~t) E R, such that (U is the Jacobian matrix of Jf"i ( and a lCf) , ... , arCf») -= 0 mod If , ajCf) = a·J - mod I ' i = 1, ... , n . 1 , ... ,r. J= Proof. The proof is given by induction on t. See Lauer (1983). o The Hensel lemma is a specialization of that theorem. The lemma is specialized by assuming n = 1, r = 2, R = Z[x] and p = I, p being a prime number.
Then, for any positive integer t there exist A~t), Ag) E Zpt [xl, such that G(t) == G mod p and H(t) == H mod p . Moreover, since n = I, we can rename fl' <1>. Thus we consider (G, H) = G· H == 0 mod p . In such case the partial derivatives are a UI,I = aG(G, H) = H , UI,2 = aH(G, H) = G a and the matrix U (1 x 2) is equal to (H, G) and is invertible mod p, if and only if G and H are relatively prime mod p. H + B . G == 0 mod p . The specialization described above can be reformulated by the following lemma.