By F. Fauvet, C. Mitschi

This quantity includes 9 refereed examine papers in numerous components from combinatorics to dynamical structures, with machine algebra as an underlying and unifying topic.

Topics lined contain abnormal connections, rank aid and summability of ideas of differential structures, asymptotic behaviour of divergent sequence, integrability of Hamiltonian structures, a number of zeta values, quasi-polynomial formalism, Padé approximants concerning analytic integrability, hybrid platforms.

The interactions among computing device algebra, dynamical structures and combinatorics mentioned during this quantity could be precious for either mathematicians and theoretical physicists who're drawn to potent computation.

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Extra resources for From combinatorics to dynamical systems: journées de calcul formel, Strasbourg, March 22-23, 2002

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Vs ) of nontrivial subspaces of V is a direct sum of V if we have V = V1 ⊕ · · · ⊕ Vs . Let V = (V1 , . . , Vs ) and W = (W1 , . . , Wt ) be two direct sums of V . i) If each Vi is a subconnection, we say that V is a direct sum of subconnections of V . ii) We say that V is finer than W , and we write V ≺ W , if for any 1 exists 1 j t such that Vi ⊂ Wj . i s there iii) We say that V and W are compatible if there exists a direct sum Z, which is finer than V and W . iv) If V and W are compatible, by the intersection of V and W we mean the largest direct sum of V such that V ∧ W ≺ V and V ∧ W ≺ W .

Ki,M ( )) is a decreasing (resp. increasing) map, for the ordering of L by inclusion. Proof. Consider two lattices M ⊂ N . Fix a Smith basis (e) of for M and a Smith basis (ε) of for N . The gauge P˜ = z−K (N ) P(ε),(e) zK (M) is a gauge from N to M, hence P˜ ∈ Mn (O) and v(P˜ ) = mini,j v(Pij ) + kj, (N ) − ki, (M) 0. Since n, P ∈ GLn (O), there exists a permutation σ such that v(Pi,σ (i) ) = 0 for 1 i which implies kσ (i), (N)−ki, (M) 0. Since both sequences of elementary divisors ki, (M). The other result follows from the fact that increase, we get ki, (N) ki,M ( ) = −kn−i, (M).

Since V ≺ W , then for any i there exists ji such that Vi ⊂ Wji . If the lattice satisfies then we have = si=1 ∩ Vi ⊂ si=1 ∩ Wji . Accordingly, we get V = t s j =1 ∩ Wj ⊃ i=1 ∩ Wji = , hence W = . 10. 1 The canonical decomposition of a connection We recall the following result from [Cor2], p. 10. 1. Let (V , ∇) be a K-vector space endowed with a connection. There exists a unique regular connection ∇ r : V −→ V ⊗K such that the following holds. i) The map ϕ = ∇θ − (∇ r )θ of V is a semi-simple endomorphism of V .

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