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Algebraically Closed Fields Analogous to Fields of Puiseux Series. J. London Math. ,8:504-506, 1974. [15] Paulo Ribenboim. Fields: Algebraically Closed and Others. Manuscripta Mathematica, 75:115-150, 1992. [la] K. Shamseddine. New Elements of Analysis on the Levi-Ciuita Field. PhD thesis, Michigan State University, East Lansing, Michigan, USA, 1999. also MSUCL-1147. [17] K. Shamseddine and M. Berz. Exception Handling in Derivative Computation with Non-Archimedean Calculus. In M. Berz, C. Bischof, G.

For every D-admissible set U and for every E > 0, we put V(f , U,E ) = { h E A(D); Ilf-hilu < E}. Then we see that the family of finite intersections of these sets form a fundamental system of neighbourhoods o f f . Recall that by Proposition 2 [ 2 ] ,given two D-admissible sets U and V, there exists a D-admissible W which contains both U and V . , > 0, we see that V (f , W,min(a1, E ~ ) )c V(f , U,EI) nV ( f ,V, ~ 2 ) . Thus, the set of neighbourhoods of the form V(f , U ,E) where U runs the family of Dadmissible sets and E > O, defines a fundamental system of neighbourhoods o f f .

Superior limit-piercing) m+m l_ 0 and a superior limit-piercing p’ < r , it is called correctly pierced. A set D is said to be correctly pierced if every monotonous distances holes sequence of D with a non empty D-beach is correctly pierced. e. the set of diameters of the holes of D has a strictly positive lower bound. 3 37 Weighted sequences We call a weighted sequence a sequence (Tm,i,qm,i) distances holes sequence and (qm,z)I l i + ( m ) < i < ~a (monotonous ~ ) mEN a sequence of nonnegative integers.

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