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1: 3.8 Nonlinear Equations
and the solutions are called fixed points of ϕ . …
§3.8(vii) Systems of Nonlinear Equations
§3.8(viii) Fixed-Point Iterations: Fractals
2: 26.13 Permutations: Cycle Notation
26.13.2 [ 1 2 3 4 5 6 7 8 3 5 2 4 7 8 1 6 ]
Cycles of length one are fixed points. … An element of 𝔖 n with a 1 fixed points, a 2 cycles of length 2 , , a n cycles of length n , where n = a 1 + 2 a 2 + + n a n , is said to have cycle type ( a 1 , a 2 , , a n ) . … A derangement is a permutation with no fixed points. The derangement number, d ( n ) , is the number of elements of 𝔖 n with no fixed points: …
3: Bibliography G
  • B. Gambier (1910) Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est a points critiques fixes. Acta Math. 33 (1), pp. 1–55.
  • V. V. Golubev (1960) Lectures on Integration of the Equations of Motion of a Rigid Body About a Fixed Point. Translated from the Russian by J. Shorr-Kon, Office of Technical Services, U. S. Department of Commerce, Washington, D.C..
  • 4: Bibliography K
  • S. Kowalevski (1889) Sur le problème de la rotation d’un corps solide autour d’un point fixe. Acta Math. 12 (1), pp. 177–232 (French).
  • 5: 22.19 Physical Applications
    The classical rotation of rigid bodies in free space or about a fixed point may be described in terms of elliptic, or hyperelliptic, functions if the motion is integrable (Audin (1999, Chapter 1)). …
    6: Bibliography S
  • J. Segura (2002) The zeros of special functions from a fixed point method. SIAM J. Numer. Anal. 40 (1), pp. 114–133.
  • 7: 28.33 Physical Applications
    As ω runs from 0 to + , with b and f fixed, the point ( q , a ) moves from to 0 along the ray given by the part of the line a = ( 2 b / f ) q that lies in the first quadrant of the ( q , a ) -plane. …
    8: Bibliography P
  • P. Painlevé (1906) Sur les équations différentielles du second ordre à points critiques fixès. C.R. Acad. Sc. Paris 143, pp. 1111–1117.
  • 9: 29.12 Definitions
    This result admits the following electrostatic interpretation: Given three point masses fixed at t = 0 , t = 1 , and t = k - 2 with positive charges ρ + 1 4 , σ + 1 4 , and τ + 1 4 , respectively, and n movable point masses at t 1 , t 2 , , t n arranged according to (29.12.12) with unit positive charges, the equilibrium position is attained when t j = ξ j for j = 1 , 2 , , n .
    10: 16.2 Definition and Analytic Properties
    When p q + 1 and z is fixed and not a branch point, any branch of F q p ( a ; b ; z ) is an entire function of each of the parameters a 1 , , a p , b 1 , , b q .