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  2. 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.

  3. 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.

  4. 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 ...

  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. 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]

  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. Critical phenomena - Wikipedia

    en.wikipedia.org/wiki/Critical_phenomena

    The critical point is described by a conformal field theory. According to the renormalization group theory, the defining property of criticality is that the characteristic length scale of the structure of the physical system, also known as the correlation length ξ, becomes infinite. This can happen along critical lines in phase space.

  9. Structural stability - Wikipedia

    en.wikipedia.org/wiki/Structural_stability

    As a consequence of the Denjoy theorem, an orientation preserving C 2 diffeomorphism ƒ of the circle is structurally stable if and only if its rotation number is rational, ρ(ƒ) = p/q, and the periodic trajectories, which all have period q, are non-degenerate: the Jacobian of ƒ q at the periodic points is different from 1, see circle map.

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