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1500 questions
47
votes
7 answers
Intuitive proof that the first $(n-2)$ coordinates on a sphere are uniform in a ball
It is a classical fact that if $(x_1,\ldots,x_n)$ is a random vector uniformly distributed on the sphere $S^{n-1} \subseteq \mathbb{R}^n$, then the random vector $(x_1,\ldots,x_{n-2})$ is uniformly distributed in the unit ball $B_{n-2} = \{…
Mark Meckes
- 11,286
47
votes
4 answers
How much reading do you do before you attack a problem?
When going off on a tangent from your regular area, where, presumably, you have such mastery of all cutting-edge research from your routine reading that you hardly need to do any extra (if this is false, please correct me), how much do you try to…
DoubleJay
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47
votes
4 answers
Consequences of lack of rigour
The standards of rigour in mathematics have increased several times during history. In the process some statements, previously considered correct where refuted. I wonder if these wrong statements were "applied" anywhere before (or after) refutation…
erz
- 5,385
47
votes
10 answers
Tools for Organizing Papers?
Much like a previous question on keeping research notes organized, my question is how people keep their pile of papers organized. I've got a stack of about 100 in my office, most of them classifying as "want to read", a couple "have read", and lots…
Elisha Peterson
- 590
47
votes
3 answers
Is this proof of Perron's theorem correct, and if so is it original?
A few years ago, I came up with this proof of Perron's theorem for a class presentation:
https://pi.math.cornell.edu/~web6720/Perron-Frobenius_Hannah%20Cairns.pdf
I've written an outline of it below so that you don't have to read a link.
It's close…
Hannah Cairns
- 473
47
votes
2 answers
What did Ramanujan get wrong?
Quoting his Wikipedia page (current revision):
compiled nearly 3,900 results
Nearly all his claims have now been proven correct
Which of his claims have been disproven, can any insight be gained from the mistakes of this genius?
StanOverflow
- 589
47
votes
2 answers
The two ways Feynman diagrams appear in mathematics
I've heard about two ways mathematicians describe Feynman diagrams:
They can be seen as "string diagrams" describing various type of arrows (and/or compositions operations on them) in a monoidal closed category.
They are combinatorial tools that…
Simon Henry
- 39,971
47
votes
6 answers
Intuition for Integral Transforms
It is well known that the operations of differentiation and integration are reduced to multiplication and division after being transformed by an integral transform (like e.g. Fourier or Laplace Transforms).
My question: Is there any intuition why…
vonjd
- 5,875
47
votes
5 answers
Is the determinant equal to a determinant?
Let $\det_d = \det((x_{i,j})_{1 \leq i,j\leq d})$ be the determinant of a generic $d \times d$ matrix. Suppose $k \mid d$, $1 < k < d$. Can $\det_d$ be written as the determinant of a $k \times k$ matrix of forms of degree $d/k$?
Even writing…
Zach Teitler
- 6,197
47
votes
1 answer
A three-line proof of global class field theory?
There is an idea (I think originally due to Tate) that class field theory is fundamentally a consequence of Pontrjagin duality and Hilbert Theorem 90. I'm curious whether this can phrased using modern (slightly motivic) language in something like…
Dmitry Vaintrob
- 7,912
47
votes
14 answers
Applications of the Cayley-Hamilton theorem
The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own characteristic polynomial.
Question. Does this theorem…
asv
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- 114
47
votes
9 answers
What are the reasons for considering rings without identity?
I think a major reason is because Lie algebras don't have an identity, but I'm not really sure.
teil
- 4,261
47
votes
1 answer
Transitivity on $\mathbb{N}_0$ -- a 42 problem
Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class
transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the permutation of $\mathbb{Z}$
which interchanges…
Stefan Kohl
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47
votes
1 answer
How to prove this polynomial always has integer values at all integers?
Let $m$ be any positive integer.
$$
P_m(x)=\sum_{i=0}^{m}\sum_{j=0}^{m}{x+j\choose j}{x-1\choose j}{j\choose i}{m\choose i}{i\choose m-j}\frac{3}{(2i-1)(2j+1)(2m-2i-1)}.
$$
Question: $P_m(x)$ always has integer values at all integers.
Some…
Chitsai Liu
- 2,153
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47
votes
5 answers
The resultant and the ideal generated by two polynomials in $\mathbb{Z}[x]$
I was asked the following question by a colleague and was embarrassed not to know the answer.
Let $f(x), g(x) \in \mathbb{Z}[x]$ with no root in common. Let $I = (f(x),g(x))\cap \mathbb{Z}$, that is, the elements of $\mathbb{Z}$ which are linear…
Felipe Voloch
- 30,227