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1500 questions
65
votes
14 answers
Notions of convergence not corresponding to topologies
This question concerns the ramifications of the following interesting problem that
appeared on Ed Nelson's final exam on Functional Analysis some years ago:
Exam question: Is there a metric on the measurable functions on R such that a sequence…
jon
- 801
65
votes
4 answers
Perron number distribution
A Perron number is a real algebraic integer $\lambda$ that is larger than the absolute value of any of its Galois conjugates. The Perron-Frobenius theorem says that any
non-negative integer matrix $M$ such that some power of $M$ is strictly positive…
Bill Thurston
- 24,832
65
votes
2 answers
How many cubes cover a bigger cube?
How many $n$-dimensional unit cubes
are needed to cover a cube with side
lengths $1+\epsilon$ for some
$\epsilon>0$?
For n=1, the answer is obviously two. For n=2, the drawing below shows that three unit cubes suffice, but it is impossible using…
J.C. Ottem
- 11,519
65
votes
8 answers
What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$
I would like to know if there is a known necessary and sufficient
property on an open subset of $\mathbb{R}^n$ to be diffeomorphic to $\mathbb{R}^n$ :
For example :
Are all open star-shaped subsets of $\mathbb{R}^n$ diffeomorphic to $\mathbb{R}^n$…
Oliver
- 667
65
votes
2 answers
To prove irrationality, why integrate?
I have been reading David Angell's lovely book, Irrationality and Transcendence in Number Theory, which has given me some fresh insights even with some of the easier proofs. But the book reminds me of something that I've long been puzzled by, which…
Timothy Chow
- 78,129
65
votes
2 answers
How do pure mathematicians assess whether their research ambitions can be realistically achieved?
I am an enthusiastic but ever-so-slightly naive PhD student and have been 'following my nose' a lot recently, seeing whether topics that I have studied can be generalised or translated in various ways into unfamiliar settings; exploring where the…
65
votes
6 answers
Does a referee have to check carefully the proof ?
I have always checked very carefully the papers I was refereeing when I wanted to suggest "accept". Actually I spend almost as much time checking the maths of a paper I referee than checking the maths of a paper of mine (and this is very long !).…
Hugh J
- 631
65
votes
21 answers
Situations where “naturally occurring” mathematical objects behave very differently from “typical” ones
I am looking for examples of the following situation in mathematics:
every object of type $X$ encountered in the mathematical literature, except when specifically attempting to construct counterexamples to this, satisfies a certain property $P$…
Gro-Tsen
- 29,944
- 4
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- 335
65
votes
4 answers
Tying knots with reflecting lightrays
Let a lightray bounce around inside a cube whose faces
are (internal) mirrors.
If its slopes are rational, it will eventually form a cycle.
For example, starting with a point $p_0$ in the interior
of the $-Y$ face of an axis-aligned cube, and…
Joseph O'Rourke
- 149,182
- 34
- 342
- 933
65
votes
2 answers
Does there exist a complete implementation of the Risch algorithm?
Is there a generally available (commercial or not) complete implementation of the Risch algorithm for determining whether an elementary function has an elementary antiderivative?
The Wikipedia article on symbolic integration claims that the general…
Timothy Chow
- 78,129
65
votes
3 answers
Replication crisis in mathematics
Lately, I have been learning about the replication crisis, see How Fraud, Bias, Negligence, and Hype Undermine the Search for Truth (good YouTube video) — by Michael Shermer and Stuart Ritchie. According to Wikipedia, the replication crisis (also…
John Tavers
- 301
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- 5
65
votes
22 answers
When has discrete understanding preceded continuous?
From my limited perspective, it appears that the understanding
of a mathematical phenomenon has usually been achieved,
historically, in a continuous setting
before it was fully explored in a discrete setting.
An example I have in mind is the…
Joseph O'Rourke
- 149,182
- 34
- 342
- 933
65
votes
4 answers
What is a foliation and why should I care?
The title says everything but while it is a little bit provocative let me elaborate a bit about my question. First time when I met the foliation it was just an isolated example in the differential geometry course (I was the Reeb foliation) and I…
truebaran
- 9,140
65
votes
20 answers
Do mathematical objects disappear?
I am asking this question starting from two orders of considerations.
Firstly, we can witness, considering the historical development of several sciences, that certain physical entities "disappeared": it is the case of luminiferous aether with the…
user84431
- 141
65
votes
4 answers
What are the implications of the new quasi-polynomial time solution for the Graph Isomorphism problem?
This week, news came out that Laszlo Babai has found a quasi-polynomial time algorithm to solve the Graph Isomorphism problem (that is: $O(\exp(polylog(n)))$). He is giving a series of talks this week, and the abstracts are here. I'm not an expert…
David White
- 29,779