The Hairy Ball Theorem via Sperner's Lemma

@article{Jarvis2004TheHB,
  title={The Hairy Ball Theorem via Sperner's Lemma},
  author={Tyler Jarvis and James Tanton},
  journal={The American Mathematical Monthly},
  year={2004},
  volume={111},
  pages={599 - 603}
}
(2004). The Hairy Ball Theorem via Sperner's Lemma. The American Mathematical Monthly: Vol. 111, No. 7, pp. 599-603. 

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References

SHOWING 1-10 OF 14 REFERENCES
A Simple Special Case of Sharkovskii's Theorem
In this note I c R is a bounded closed interval and f: I -> I is a continuous map; fn denotes the n-fold composition of f with itself. A point x E I is a periodic point for f with period p if fP(x) =
COEXISTENCE OF CYCLES OF A CONTINUOUS MAP OF THE LINE INTO ITSELF
The basic result of this investigation may be formulated as follows. Consider the set of natural numbers in which the following relationship is introduced: n1 precedes n2 (n1 ≼ n2) if for any
A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line
Two theorems are proved—the first and the more important of them due to Šarkovskii—providing complete and surprisingly simple answers to the following two questions: (i) given that a continuous mapT
Dynamics in One Dimension
Periodic orbits.- Turbulence.- Unstable manifolds and homoclinic points.- Topological dynamics.- Topological dynamics (continued).- Chaotic and non-chaotic maps.- Types of periodic orbits.-
Periodic Points of Continuous Functions
Analysis of the cyclic behavior of points under repeated application of a function yields insights into population patterns.
On Fixed Points
char ticky=0x22;char *first="main(){printf(";char *second="char ti cky=0x22;char *first=%c%s%c;char *second=%c%s%c;char *third=%c%s%c ;%s%c%s%c%s";char
A note about Sharkovskii's theorem
  • 2003
Fixed Points (trans
  • Mathematical World
  • 1991
Fixed points, (Translated from Russian by V. Minachin)
  • Mathematical World 2, American Math. Soc., Providence,
  • 1991
Mathematics and logic
...
1
2
...