Poincaré conjecture

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Poincaré conjecture

[‚pwän‚kä′rā kən‚jek·chər]
(mathematics)
The question as to whether a compact, simply connected three-dimensional manifold without boundary must be homeomorphic to the three-dimensional sphere.
References in periodicals archive ?
(1) if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with [[lambda].sub.1], [[lambda].sub.2] not satisfying (3), by the Poincare's theorem, system (4) is not formally integrable;
For a discussion of this result we refer the reader to [71] where Sotomayor proves the following version of Poincare's theorem: Polynomial differential systems whose compactifications have only hyperbolic equilibria and no graphics, are generic and have at most finitely many periodic orbits (for the notion of graphic see also [28]).
Then, by Poincare's theorem [13], for each solution 1(n), n = 0, 1, 2,..., of (28), either 1(n) = 0 for all sufficiently large n [greater than or equal to] [n.sub.0], or the limit [lim.sub.n[right arrow][infinity]]l(n + 1)/l(n) exists and equals one 0] of the roots of the characteristic polynomial.
If Poincare's theorem holds, then the entropy of an individual system cannot monotonically increase since it must eventually return to its starting point.