From combinatorics to dynamical systems: journées de calcul by F. Fauvet, C. Mitschi

By F. Fauvet, C. Mitschi

This quantity includes 9 refereed learn papers in a number of parts from combinatorics to dynamical platforms, with machine algebra as an underlying and unifying subject.

Topics coated contain abnormal connections, rank relief and summability of suggestions of differential platforms, asymptotic behaviour of divergent sequence, integrability of Hamiltonian structures, a number of zeta values, quasi-polynomial formalism, Padé approximants with regards to analytic integrability, hybrid structures.

The interactions among laptop algebra, dynamical structures and combinatorics mentioned during this quantity will be necessary for either mathematicians and theoretical physicists who're drawn to powerful computation.

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

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

Example text

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 .

Let - equivalence up to order T relation ∼ as follows. T i) u ∼ u˜ if u − u˜ ∈ zT +1 T ii) ∼ ˜ if T for vectors u and u˜ in V . − ˜ ∈ zT +1 ∗ ⊗O iii) ∇ ∼ ∇˜ if ∇θ − ∇˜ θ ∈ zT +1 T ⊗O for maps ∗ and ˜ ∈ EndK V can V ⊗K V ∗ . for connections ∇ and ∇˜ on V . iv) Let (ε) and (˜ε ) be bases of V . Then (ε) ∼ (˜ε ) if εi ∼ ε˜ i for all i = 1, . . , n. 2. With the given notation, one has i) (ε) ∼ (˜ε) if and only if P − P˜ ∈ zT +1 Mn (O), where P = P(e),(ε) and P˜ = P(e),(˜ε) T for any basis (e) of .

I=1 Hence, the collection Z˜ = (Wj )1 i m,1 j ti is a direct sum of V . The direct sum Z˜ is finer than Z by construction. Let us prove that it is also finer than W . Let E = /z and E = F1 ⊕ · · · ⊕ Ft be the sum of E into the characteristic spaces of f = ∇ . For (i) (i) any 1 i m, let i /z i = Gi and ∇i i = fi . Let Gi = F1 ⊕ · · · ⊕ Fti be the (i) direct sum of the characteristic spaces of fi , and λj the corresponding eigenvalue. (i) (i) Every Fj is contained in exactly one Fk . Hence we have t i=1 Pfi (X) characteristic polynomial Pf (X) satisfies of Pf (X) = (i) Fj (i) λj =λk (i) Fj ⊂ Fk .

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