Topology

Closing Lemma for Diffeomorphism (Dynamical Systems) ★★★★

Author(s): Charles Pugh

Conjecture   Let $ f\in Diff^{r}(M) $ and $ p\in\omega_{f} $. Then for any neighborhood $ V_{f}\subset Diff^{r}(M) $ there is $ g\in V_{f} $ such that $ p $ is periodic point of $ g $

There is an analogous conjecture for flows ( $ C^{r} $ vector fields . In the case of diffeos this was proved by Charles Pugh for $ r = 1 $. In the case of Flows this has been solved by Sushei Hayahshy for $ r = 1 $ . But in the two cases the problem is wide open for $ r > 1 $

Keywords: Dynamics , Pertubation

Posted by Jailton Viana
updated April 24th, 2013
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Number Theory » Additive N.T.

DIS-PROOF OF BEALS CONJECTURE ★★★

Author(s):

If A(pwr)x+B(pwr)y=C(pwr)z where A,B,C,x,y,z are positive integers and x,y,z>=2,then A,B,C must have a common prime factor.

Keywords: beals conjecture

Posted by lalitha
updated April 24th, 2013
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Algebra

Sub-atomic product of funcoids is a categorical product ★★

Author(s):

Conjecture   In the category of continuous funcoids (defined similarly to the category of topological spaces) the following is a direct categorical product:
    \item Product morphism is defined similarly to the category of topological spaces. \item Product object is the sub-atomic product. \item Projections are sub-atomic projections.

See details, exact definitions, and attempted proofs here.

Keywords:

Posted by porton
updated April 21st, 2013
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Graph Theory » Coloring » Vertex coloring

Bounding the on-line choice number in terms of the choice number ★★

Author(s): Zhu

Question   Are there graphs for which $ \text{ch}^{\text{OL}}-\text{ch} $ is arbitrarily large?

Keywords: choosability; list coloring; on-line choosability

Posted by Jon Noel
updated April 11th, 2013
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