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1500 questions
45
votes
16 answers
What does the generating function $x/(1 - e^{-x})$ count?
Let $x$ be a formal (or small, since the function is analytic) variable, and consider the power series
$$ A(x) = \frac{x}{1 - e^{-x}} = \sum_{m=0}^\infty \left( -\sum_{n=1}^\infty \frac{(-x)^n}{(n+1)!} \right)^m = 1 + \frac12 x + \frac1{12}x^2 +…
Theo Johnson-Freyd
- 52,873
45
votes
2 answers
Continuous bijections vs. Homeomorphisms
This is motivated by an old question of Henno Brandsma.
Two topological spaces $X$ and $Y$ are said to be bijectively related, if there exist continuous bijections $f:X \to Y$ and $g:Y \to X$. Let´s denote by $br(X)$ the number of homeomorphism…
Ramiro de la Vega
- 11,463
45
votes
4 answers
What does "linearly disjoint" mean for abstract field extensions?
All definitions I've seen for the statement "$E,F$ are linearly disjoint extensions of $k$" are only meaningful when $E,F$ are given as subfields of a larger field, say $K$. I am happy with the equivalence of the various definitions I've seen in…
Andrew Critch
- 11,060
45
votes
8 answers
Getting nervous refereeing a paper
I am refereeing my first paper and I'm quite excited! But inexperienced and I would like to ask an advice to the Maths Community of MO. Let me tell you that I have already read Refereeing a Paper, but it seems that my question is quite different.…
Valerio Capraro
- 5,894
45
votes
0 answers
Enriched Categories: Ideals/Submodules and algebraic geometry
While working through Atiyah/MacDonald for my final exams I realized the following:
The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed with the product of ideals
$$I\cdot…
Gerrit Begher
- 3,161
45
votes
2 answers
$H^4$ of the Monster
The Monster group $M$ acts on the moonshine vertex algebra $V^\natural$.
Because $V^\natural$ is a holomorphic vertex algebra (i.e., it has a unique irreducible module), there is a corresponding cohomology class $c\in H^3(M;S^1)=H^4(M;\mathbb…
André Henriques
- 42,480
45
votes
5 answers
How to think about CM rings?
There are a few questions about CM rings and depth.
Why would one consider the concept of depth? Is there any geometric meaning associated to that? The consideration of regular sequence is okay to me. (currently I'm regarding it as a generalization…
user709
45
votes
1 answer
Example of a compact set that isn't the spectrum of an operator
This question is somewhat ill-posed (due to the word easy) and is triggered by idle curiosity:
Is there an easy example of a (separable, infinite-dimensional) Banach space $X$ and a nonempty compact set $K \subset \mathbb{C}$ such that $K$ is not…
Theo Buehler
- 5,703
45
votes
1 answer
Rolling a random walk on a sphere
A ball rolls down an inclined plane, encountering horizontal obstacles, at which it
rolls left/right with equal probability. There are regularly spaced staggered gaps that let the ball
roll down to the next, lower obstacle. The pattern resembles a…
Joseph O'Rourke
- 149,182
- 34
- 342
- 933
45
votes
56 answers
German mathematical terms like "Nullstellensatz"
There are quite a few german mathematical theorems or notions which usually are not translated into other languages. For example,
Nullstellensatz, Hauptvermutung, Freiheitssatz, Eigenvector (the "Eigen" part), Verschiebung.
For me, as a German, this…
Martin Brandenburg
- 61,443
45
votes
4 answers
Polynomial roots and convexity
A couple of years ago, I came up with the following question, to which I have no answer to this day. I have asked a few people about this, most of my teachers and some friends, but no one had ever heard of the question before, and no one knew the…
Olivier Bégassat
- 2,663
45
votes
7 answers
Are some numbers more irrational than others?
Some irrational numbers are transcendental, which makes them in some sense "more irrational" than algebraic numbers. There are also numbers, such as the golden ratio $\varphi$, which are poorly approximable by rationals. But I wonder if there is…
I. J. Kennedy
- 1,813
45
votes
12 answers
Teaching undergraduate students to write proofs
In my experience, there are roughly two approaches to teaching (North American) undergraduates to write proofs:
Students see proofs in lecture and in the textbooks, and proofs are explained when necessary, for example, the first time the instructor…
Amit Kumar Gupta
- 3,992
45
votes
10 answers
The functional equation $f(f(x))=x+f(x)^2$
I'd like to gather information and references on the following functional equation for power series $$f(f(x))=x+f(x)^2,$$$$f(x)=\sum_{k=1}^\infty c_k x^k$$
(so $c_0=0$ is imposed).
First things that can be established quickly:
it has a unique…
Pietro Majer
- 56,550
- 4
- 116
- 260
45
votes
4 answers
Does the fact that this vector space is not isomorphic to its double-dual require choice?
Let $V$ denote the vector space of sequences of real numbers that are eventually 0, and let $W$ denote the vector space of sequences of real numbers. Given $w \in W$ and $v \in V$, we can take their "dot product" $w \cdot v$, and for any give $w…
Amit Kumar Gupta
- 3,992