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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

    More precisely, 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 of is negative definite. The indices of basins, passes, and peaks are 0 , 1 , {\displaystyle 0,1,} and 2 , {\displaystyle 2,} respectively.

  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. Isolated point - Wikipedia

    en.wikipedia.org/wiki/Isolated_point

    Since each I k contains only one point from F, every point of F is an isolated point. However, if p is any point in the Cantor set, then every neighborhood of p contains at least one I k, and hence at least one point of F. It follows that each point of the Cantor set lies in the closure of F, and therefore F has uncountable closure.

  6. 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.

  7. 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.

  8. Second partial derivative test - Wikipedia

    en.wikipedia.org/wiki/Second_partial_derivative_test

    At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point. (In fact, one can show that f takes both positive and negative values in small neighborhoods around (0, 0) and so this point is a saddle point of f.)

  9. 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 (,).