Algebraic aspects of nonlinear differential equations by Manin Yu.I.

By Manin Yu.I.

Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. eleven, pp. 5–152, 1978.

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Extra resources for Algebraic aspects of nonlinear differential equations (J.Sov.Math. 11, 1979, 1-122)

Example text

Recall that a critical point x ∈ U is called nondegenerate if the Hessian matrix D2 f (x) is nonsingular. Morse’s lemma, due originally to Morse [202], states that after a local, possibly nonlinear, change of coordinates, the function f is identical to its quadratic form q(x) := f (x0 ) + D2 f (x0 )(x − x0 ), x − x0 . Thus, the quadratic function q(x) determines the behavior of the function f around x0 . Morse’s original proof uses the Gram–Schmidt process. A modern version of the proof can be found in Milnor [197].

Thus, f is bounded from below on K. Suppose that f does not have a global minimizer on K. Define Fn := {x ∈ K : f (x) > f ∗ + 1/n}. Then Fn is an open subset of K and K = ∪∞ n=1 Fn . As above, we have K = Fn for some n > 1, that is, f (x) > f ∗ + 1/n for all x ∈ K, a contradiction to the definition of f ∗ . We remark that the second proof is more general, since it is valid verbatim on all compact topological spaces, not only compact metric spaces. The compactness assumption can be relaxed somewhat.

Define the square root of B, C := U diag{ d1 , . . , dn } U T . Note that C −1 BC −1 = I. Now A := C −1 AC −1 has the spectral decomposition V T AV = Λ = diag{λ1 , . . , λn }. Setting X = C −1 V , we see that X T AX = Λ, X T BX = V T C −1 BC −1 V = V T V = I, completing the proof. 1 Counting Roots of Polynomials in Intervals The number of positive (and negative) eigenvalues can be counted by a simple rule dating back to Descartes in seventeenth century. 22. Let a0 , a1 , . . , an be a sequence of real numbers.

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