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Results tagged with open-problems
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user 5371
If it turns out that a problem is equivalent to a known open problem, then the open-problem tag is added. After that, the question essentially becomes, "What is known about this problem? What are some possible ways to approach this problem? What are some ways that people have tried to attack it before, and with what results?"
20
votes
Is Lebesgue's "universal covering" problem still open?
The problem has been studied for various groups $G$ of isometries of $\mathbb R^n$. A set $K\subset \mathbb R^n$ is called $G$-universal cover iff every set of diameter 1 is contained in $gK$ for som …
- 22.3k
9
votes
1
answer
2k
views
The Invariant Subspace Problem: examples
Question. Is there a concrete example of a bounded linear operator on a Hilbert space for which it is not known if it has a non-trivial closed invariant subspace?
[Added 24.01.2011: According to Bern …
- 22.3k
25
votes
Polynomials having a common root with their derivatives
The strongest result in this direction that I've heard of is Sudbery's theorem (which was
originally conjectured by Popoviciu and Erdös).
Theorem. Let $P(z)$ be a polynomial of degree $n\geq 2$ a …
- 22.3k
12
votes
Nonnegative to Positive Curvature.
Yau asked in 1982 if there is any compact simply connected manifold with nonnegative curvature for which one can prove that it does not admit a metric of positive curvature. This question opens his li …
- 22.3k
16
votes
Accepted
Smallest area shape that covers all unit length curve
Whereas I don't know of any recent progress in this problem, let me mention one result for
closed curves.
Theorem. A closed plane curve of length $L$ and curvature bounded by $K$ can be contained …
- 22.3k
5
votes
Accepted
Birkhoff conjecture about integrable billiards
I haven't heard of any recent breakthroughs. The strongest result that I know is due to Misha Bialy:
Theorem. If almost every phase point of the billiard ball map in a strictly convex billiard table …
- 22.3k
14
votes
Accepted
The Ramanujan Problems
There is a survey article by Berndt, Choi, and Kang devoted to the set of 58 Ramanujan's problems. They indicate that the questions had originally appeared in the problems section of the Journal and a …
- 22.3k
56
votes
Accepted
Can we cover the unit square by these rectangles?
This problem actually goes back to Leo Moser.
The best result that I'm aware of is due to D. Jennings, who proved that all the rectangles of size $k^{-1} × (k + 1)^{-1}$, $k = 1, 2, 3 ...$, can be pac …
- 22.3k