By R. Gerard, J.-P. Ramis

ISBN-10: 3540126848

ISBN-13: 9783540126843

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**Sample text**

Remark 3. If ∑ω ∈u Tω 2 ≤ 1, then S(u) ≥ 0. In fact, let u = {ω1 , . . , ωr } and write, for the sake of brevity, Ti in place of Tωi . For 0 ≤ p ≤ r and for h ∈ H, set T e(v) h 2 . *
*

*Part (a) of the theorem is easy. In fact, we have T (e) = PHU(e)|H = PH |H = IH , T (s−1 ) = PHU(s−1 )|H = PHU(s)∗ |H = (PHU(s)|H)∗ = T (s)∗ , 7. P OSITIVE DEFINITE FUNCTIONS ON A GROUP 25 and ∑ ∑ (PHU(t −1 s)h(s), h(t)) = ∑ ∑ (U(t)∗U(s)h(s), h(t)) s∈G t∈G s∈G t∈G 2 = ∑ U(s)h(s) s∈G ≥0 for every finitely nonzero function h(s) from G to H. The assertion concerning continuity is obvious. Part (b). Let us consider the set H, obviously linear, of the finitely nonzero functions h(s) from G to H, and let us define on H a bilinear form2 by h, h′ = ∑ ∑(T (t −1 s)h(s), h′ (t)) s t [h = h(s), h′ = h′ (s)]. *

Xn ) is nonpositive, that is, ∂ Xi (x) ≥ 0. 12) with x(0) = x exists not only on a small neighborhood of t = 0 but on the whole t-axis. In this case τt : x → x(t) is a differentiable transformation of Rn onto itself. The functional determinant δt (x) = D(τt x) D((τt x)1 , . . , (τt x)n ) = D(x) D(x1 , . . , ∂xj ∂xj ( j = 1, . . , n) of the system of linear differential equations n ∂ Xi (τt x) dui = ∑ uk dt k=1 ∂ xk (i = 1, . . 12), and hence one derives, using Liouville’s theorem and the fact that δ0 (x) = 1, the formula δt (x) = exp − t 0 ρ (τs x)ds .

### Equations differentielles et systemes de Pfaff dans le champ complexe II. Seminaire by R. Gerard, J.-P. Ramis

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