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  2. Critical point (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Critical_point_(mathematics)

    For a function of n variables, the number of negative eigenvalues of the Hessian matrix at a critical point is called the index of the critical point. A non-degenerate critical point is a local maximum if and only if the index is n, or, equivalently, if the Hessian matrix is negative definite; it is a local minimum if the index is zero, or ...

  3. Morse theory - Wikipedia

    en.wikipedia.org/wiki/Morse_theory

    A less trivial example of a degenerate critical point is the origin of the monkey saddle. The index of a non-degenerate critical point of is the dimension of the largest subspace of the tangent space to at on which the Hessian is negative definite.

  4. Hessian matrix - Wikipedia

    en.wikipedia.org/wiki/Hessian_matrix

    Otherwise it is non-degenerate, and called a Morse critical point of . The Hessian matrix plays an important role in Morse theory and catastrophe theory, because its kernel and eigenvalues allow classification of the critical points. [2] [3] [4]

  5. Morse–Palais lemma - Wikipedia

    en.wikipedia.org/wiki/Morse–Palais_lemma

    Let (, , ) be a real Hilbert space, and let be an open neighbourhood of the origin in . Let : be a (+)-times continuously differentiable function with ; that is, + (;). Assume that () = and that is a non-degenerate critical point of ; that is, the second derivative () defines an isomorphism of with its continuous dual space by (,).

  6. Milnor number - Wikipedia

    en.wikipedia.org/wiki/Milnor_number

    If the image of f is now perturbed in a certain stable way the isolated degenerate singularity at 0 will split up into other isolated singularities which are non-degenerate. The number of such isolated non-degenerate singularities will be the number of points that have been infinitesimally glued. Precisely, another function germ g which is non ...

  7. Stationary phase approximation - Wikipedia

    en.wikipedia.org/wiki/Stationary_phase_approximation

    The second statement is that when f is a Morse function, so that the singular points of f are non-degenerate and isolated, then the question can be reduced to the case n = 1. In fact, then, a choice of g can be made to split the integral into cases with just one critical point P in each.

  8. Picard–Lefschetz theory - Wikipedia

    en.wikipedia.org/wiki/Picard–Lefschetz_theory

    The Picard–Lefschetz formula describes the monodromy at a critical point. Suppose that f is a holomorphic map from an (k+1)-dimensional projective complex manifold to the projective line P 1. Also suppose that all critical points are non-degenerate and lie in different fibers, and have images x 1,...,x n in P 1. Pick any other point x in P 1.

  9. Non-degenerate critical point - Wikipedia

    en.wikipedia.org/?title=Non-degenerate_critical...

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