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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. Critical phenomena - Wikipedia

    en.wikipedia.org/wiki/Critical_phenomena

    In systems in equilibrium, the critical point is reached only by precisely tuning a control parameter. However, in some non-equilibrium systems, the critical point is an attractor of the dynamics in a manner that is robust with respect to system parameters, a phenomenon referred to as self-organized criticality. [6]

  5. 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]

  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. Non-degenerate critical point - Wikipedia

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

    Retrieved from "https://en.wikipedia.org/w/index.php?title=Non-degenerate_critical_point&oldid=29046177"

  8. Critical point - Wikipedia

    en.wikipedia.org/wiki/Critical_point

    Critical point may refer to: Critical phenomena in physics; Critical point (mathematics), in calculus, a point where a function's derivative is either zero or nonexistent; Critical point (set theory), an elementary embedding of a transitive class into another transitive class which is the smallest ordinal which is not mapped to itself

  9. Exceptional point - Wikipedia

    en.wikipedia.org/wiki/Exceptional_point

    In quantum physics, exceptional points [1] are singularities in the parameter space where two or more eigenstates (eigenvalues and eigenvectors) coalesce. These points appear in dissipative systems , which make the Hamiltonian describing the system non-Hermitian .