All Questions
17,339
questions
28
votes
4
answers
2k
views
If $X$ and $Y$ independent and identically distributed, then $E(|X-Y|)\leq E(|X+Y|)$. Are other proofs of this known?
I know a proof of the theorem that if $X$ and $Y$ independent and identically distributed, then $E(|X-Y|)\leq E(|X+Y|)$. The proof uses an integral representation of the absolute value,
$$\int_0^\...
- 389
28
votes
2
answers
2k
views
A 14th and 26th-power Dedekind eta function identity?
Given the Dedekind eta function $\eta(\tau)$. Define $m = (p-1)/2$ and a $24$th root of unity $\zeta = e^{2\pi i/24}$.
Let p be a prime of form $p = 12v+5$. Then for $n = 2,4,8,14$:
$$\sum_{k=0}^{p-...
- 12.1k
28
votes
2
answers
763
views
Probability of generation of ${\mathbb Z}^2$
What is the probability that three pairs $(a,b) $ , $(c,d) $ and $(e,f) $ of integers generate $\mathbb Z^2$? As usual the probability is the limit as $n\to \infty$ of the same probability for the $n\...
user6976
28
votes
5
answers
3k
views
Is Weil's bound for Kloosterman sums ever attained?
Weil's bound for Kloosterman sums states that for $(a,b)\not=(0,0)$,
$$
|K(a,b;q)|:=\left|\sum_{x\in\mathbb{F}_q^*}\chi(ax+bx^{-1})\right|\leq 2\sqrt{q},
$$
where $\chi$ is a non-trivial additive ...
- 1,008
28
votes
2
answers
5k
views
Arnold on Newton's anagram
Arnold, in his paper
The underestimated Poincaré, in Russian Math. Surveys 61 (2006), no. 1, 1–18
wrote the following:
``...Puiseux series, the theory which Newton, hundreds of years before ...
- 88.9k
28
votes
3
answers
3k
views
Sum over permutations is 1
This might be easy, but let's see.
Question 1. If $\mathfrak{S}_n$ is the group of permutations on $[n]$, then is the following true?
$$\sum_{\pi\in\mathfrak{S}_n}\prod_{j=1}^n\frac{j}{\pi(1)+\pi(...
- 41.8k
28
votes
3
answers
5k
views
Is there any pattern to the continued fraction of $\sqrt[3]{2}$? [closed]
Is there any pattern to the continued fraction of $\sqrt[3]{2}$ ? Wolfram Alpha returns for cube root of 2:
$\sqrt[3]{2}=$ [1; 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, ...
- 22.6k
28
votes
2
answers
2k
views
Contractibility of the space of Jordan curves
Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$.
If the curves are ...
- 7,047
28
votes
0
answers
2k
views
Is Feferman's unlimited category theory dead?
In 2013 Solomon Feferman in Foundations of unlimited category theory: what remains to be done (The Review of Symbolic Logic, 6 (2013) pp 6-15, link) laid out three desirable axioms for "...
- 1,436
28
votes
1
answer
1k
views
Are entire functions “essentially” determined by their maximum modulus function?
(Note: This has been asked on Math SE, but without an answer after almost two years and one offered bounty.)
For an entire function $f$ let $M(r,f)=\max_{|z|=r}|f(z)|$ be its maximum modulus function. ...
- 490
28
votes
4
answers
6k
views
Is there any way to rewrite a partial differential equation using language of differential forms, tensors, etc?
My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there any way (or any algorithm) ...
- 1,499
28
votes
2
answers
3k
views
What is the algebraic closure of the field with one element?
If doing geometry over $\mathbb F_p$ means also using its algebraic closure, it must be interesting to talk about the algebraic closure of $\mathbb F_1$ - the field with one element.
I saw that the ...
- 4,563
28
votes
6
answers
3k
views
Why is there no symplectic version of spectral geometry?
First, recall that on a Riemannian manifold $(M,g)$ the Laplace-Beltrami operator $\Delta_g:C^\infty(M)\to C^\infty(M)$ is defined as
$$
\Delta_g=\mathrm{div}_g\circ\mathrm{grad}_g,
$$
where the ...
- 1,890
28
votes
6
answers
2k
views
How fast are a ruler and compass?
This may be more of a recreational mathematics question than a research question, but I have wondered about it for a while. I hope it is not inappropriate for MO.
Consider the standard assumptions ...
- 513
28
votes
2
answers
3k
views
Is there a "finitary" solution to the Basel problem?
Gabor Toth's Glimpses of Algebra and Geometry contains the following beautiful proof (perhaps I should say "interpretation") of the formula $\displaystyle \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} ...
- 115k
28
votes
2
answers
1k
views
What are applications of commutativity theorems for rings?
Herstein's little book "Noncommutative Rings" has a chapter called Commutativity Theorems in which he proves results like Jacobson's theorem: if a ring (associative with identity, please) has the ...
- 49.6k
28
votes
1
answer
2k
views
How many polynomial Morse functions on the sphere?
Let $f$ be a homogeneous polynomial of degree $d$ in $n$ variables. Restricted to the unit sphere $S^{n-1}$, it might or might not be a Morse function.
If $f$ is a Morse function of degree $1$, you ...
- 137k
28
votes
2
answers
2k
views
morphisms representable by algebraic spaces vs morphisms representable by schemes
So I've been working with moduli stacks in algebraic geometry for a while now, with no formal training in the technicalities of the theory of algebraic stacks (ie, I've read a few articles and I learn ...
- 7,161
28
votes
4
answers
3k
views
Prove that $\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1$
Let $x>0$ and $n$ be a natural number. Prove that:
$$\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1.$$
This question is very similar to many contests problems, but ...
- 1,054
27
votes
5
answers
9k
views
Can a quotient ring R/J ever be flat over R?
If $R$ is a ring and $J\subset R$ is an ideal, can $R/J$ ever be a flat $R$-module? For algebraic geometers, the question is "can a closed immersion ever be flat?"
The answer is yes: take $J=...
- 23.7k
27
votes
3
answers
3k
views
Is “problem solving” a subject to be taught?
I am witnessing a new curriculum change in my country (Iran). It includes the change of all the mathematics textbooks at all grades. The peoples involved has sent me the textbook for seven graders (13 ...
27
votes
5
answers
2k
views
Moments of area of random triangle inscribed in a circle
The $2m$th moment of the (random) area of the triangle whose vertices are three independent, uniformly distributed random points on the unit circle appears to be $((3m)!/(m!)^3)/16^m$. Can anyone ...
- 19.4k
27
votes
1
answer
2k
views
Reconciling Lusztig's results with the Langlands philosophy
Let $\boldsymbol{G}$ be a reductive group over a finite field $\mathbb{F}_q$, $G = \boldsymbol{G}(\mathbb{F}_q)$, $W = \mathrm{W}(\mathbb{F}_q)$ the Witt vectors over $\mathbb{F}_q$, and $K = \mathrm{...
- 805
27
votes
5
answers
4k
views
Why are lacunary series so badly behaved?
Hi!
I just came across the Ostroski-Hadamard gap theorem, and while I can understand the proofs as well as the principle that the series $\sum_{n=0}^\infty z^{2^n}$ ought to have a singularity at ...
- 2,318
27
votes
1
answer
1k
views
Why are unramified maps not required to be locally of finite presentation?
I have read and heard several times that it is “important” that unramified maps are not required to be locally of finite presentation, but only locally of finite type.
Apart from this issue with ...
- 5,444
27
votes
1
answer
1k
views
Motivation for relative schemes: why should one work with schemes over a ringed topos?
Recently I've been trying to learn more about relative schemes. These were developed in M. Hakim's thesis Topos annelés et schémas relatifs under Grothendieck's guidance and appear in many of later ...
- 10.7k
27
votes
1
answer
3k
views
Fundamental group of the moduli stack of elliptic curves
I've heard that the étale fundamental group of the moduli stack of elliptic curves (over $\mathbb{Z}$) is trivial. Is there an easy proof of that? (Note that there are plenty of étale covers once one ...
- 25.3k
27
votes
1
answer
3k
views
The Galois group of a random polynomial
Intuitively, the Galois group (of a splitting field over $\mathbb Q$ of) a polynomial $f\in\mathbb Q[X]$ taken at random is most probably the full permutation group on the roots of $f$. This intuition ...
- 3,975
27
votes
10
answers
10k
views
Book recommendation for ergodic theory and/or topological dynamics?
Hello,
I'd like to hear your opinion for ergodic theory books which would suit a beginner (with background in measure theory, real analysis and topological groups). I am looking for something well ...
27
votes
3
answers
2k
views
How can classifying irreducible representations be a "wild" problem?
Let $q$ be a prime power and $U_n(\mathbb{F}_q)$ be the group of unitriangular $n\times n$-matrices. I've read and heard in several places (see e.g. this mathoverflow question) that classifying ...
- 2,450
27
votes
6
answers
5k
views
Has any open/difficult problem in ordinary mathematics been solved only/mostly by appeal to set theory?
We know that many (if not all) mathematical notions can be reduced to the talk of sets and set-membership. But it nevertheless sounds like a grueling task (if at all possible) to actually get advanced ...
27
votes
4
answers
3k
views
Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?
Apéry's proof of the irrationality of $\zeta(3)$ astounded contemporary mathematicians for its wealth of new ideas and techniques in proving the irrationality of a known constant. It is often the case ...
- 1,943
27
votes
4
answers
6k
views
A proof of the salamander lemma without Mitchell's embedding theorem?
The salamander lemma is a lemma in homological algebra from which a number of theorems quickly drop out, some of the more famous ones include the snake lemma, the five lemma, the sharp 3x3 lemma (...
27
votes
5
answers
3k
views
Is there a Morse theory proof of the Bruhat decomposition?
Let $G$ be a complex connected Lie group, $B$ a Borel subgroup and $W$ the Weyl group. The Bruhat decomposition allows us to write $G$ as a union $\bigcup_{w \in W} BwB$ of cells given by double ...
- 7,933
27
votes
2
answers
8k
views
When is the pushforward of a vector bundle still a vector bundle?
Let $X$ and $Y$ be varieties. Let $E$ be a locally free sheaf over $X$. Let $f: X \to Y$. Is there some nice criteria which ensures that $f_\ast E$ is still locally free? Sorry, if this is a very ...
- 413
27
votes
4
answers
8k
views
de Rham vs Dolbeault Cohomology
For a complex manifold $M$, one can consider (A) its de Rham cohomology, or (B) its Dolbeault cohomology. I'm looking for some motivation as to why one would bother introducing Dolbeault cohomology. ...
- 1,383
27
votes
2
answers
2k
views
Categorification of the integers
I would like to know a natural categorification of the rig of integers $\mathbb{Z}$. This should be a $2$-rig. Among the various notions of 2-rigs, we obviously have to exclude those where $+$ is a ...
- 61.5k
27
votes
2
answers
2k
views
A sum involving roots of unity
Let $n$ be a positive integer and $\zeta$ be a primitive $n$th root of unity. It is not hard to show that
\begin{align*}
\sum_{k=1}^{n-1}\frac{\zeta^k}{1-\zeta^k}=\frac{1-n}{2}.
\end{align*}
Since $\...
- 2,153
27
votes
3
answers
2k
views
Ax–Grothendieck and the Garden of Eden
It's an obvious consequence of the pigeonhole principle that any injective function over finite sets is bijective. But there are some similar results in different areas of mathematics that apply to ...
- 18.5k
27
votes
2
answers
2k
views
Are the mapping class groups of manifolds finitely presentable?
The mapping class group of a manifold is the group $\pi_0 Diff(M)$ of components of the diffeomorphism group. There are several variations: oriented manifolds and orientation preserving ...
- 27.2k
27
votes
4
answers
2k
views
Does every set of reals contain a measure-zero set of the same cardinality? Does it contain a meager set of the same cardinality?
This question arises from an issue in my post on Ashutosh's excellent question on Restrictions of the null/meager ideal.
Question 1. Does every set of reals contain a measure-zero subset
of the same ...
- 225k
27
votes
1
answer
4k
views
Intuition for Picard-Lefschetz formula
I'm trying to develop some intuition for the (local) Picard-Lefschetz formula (which I'm encountering for the first time in Deligne's paper "La Conjecture de Weil, I").
To summarize the ...
- 2,769
27
votes
1
answer
3k
views
Definitions of real reductive groups
There are several definitions of real reductive groups, sometimes subtly inequivalent. The following come to my mind:
A closed subgroup of $GL(n,\mathbb C)$ closed under conjugate transpose.
The set ...
- 971
27
votes
4
answers
3k
views
Genealogy of the Lagrange inversion theorem
A wonderful piece of classic mathematics, well-known especially to combinatorialists and to complex analysis people, and that, in my opinion, deserves more popularity even in elementary mathematics, ...
- 56.6k
27
votes
2
answers
2k
views
Are any natural examples of Gödel speed-up known?
In 1936 Gödel announced a theorem to the effect that proofs of certain theorems $T_1,T_2,\ldots$ become dramatically shorter when one passes from a formal system, such as Peano arithmetic PA, to a ...
- 12.3k
27
votes
29
answers
29k
views
Alternative undergraduate analysis texts
Other than the standard baby Rudin, Spivak, and Stein-Shakarchi, are there other alternative and comprehensive analysis texts at the undergraduate level? For example something that has general results ...
27
votes
5
answers
3k
views
Are there two non-isomorphic number fields with the same degree, class number and discriminant?
If so, do people expect certain invariants (regulator, # of complex embeddings, etc) to fully 'discriminate' between number fields?
- 2,481
27
votes
2
answers
3k
views
The Higman group
The group of Higman:
$
\langle
\
a_0, a_1, a_2, a_3 \ | \
a_0 a_1 a_0^{-1}=a_1^2,
\ a_1 a_2 a_1^{-1}=a_2^2,
\ a_2 a_3 a_2^{-1}=a_3^2,
\ a_3 a_0 a_3^{-1}=a_0^2
\ \rangle .
$
Is it simple? What is ...
- 4,695
27
votes
11
answers
4k
views
What kind of Lagrangians can we have?
In any physics book I've read the Lagrangian is introuced as as a functional whose critical points govern the dynamics of the system. It is then usually shown that a finite collection of non-...
- 2,601
27
votes
1
answer
2k
views
Does a proof of Selberg's 3.2 inequality exist?
A well-known inequality of Montgomery and Vaughan (generalizing a result of Hilbert) states that
$$ \left |\sum_{r \neq s} \frac{w_{r} \overline{w_{s}} }{\lambda_r - \lambda_s} \right| \leq \pi \...
- 11.8k