All Questions
2,597 questions
231
votes
13
answers
42k
views
Is there an introduction to probability theory from a structuralist/categorical perspective?
The title really is the question, but allow me to explain.
I am a pure mathematician working outside of probability theory, but the concepts and techniques of probability theory (in the sense of ...
- 65.4k
212
votes
52
answers
82k
views
Ways to prove the fundamental theorem of algebra
This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?
Please give a new way in each answer, and if possible give reference. I start by giving two:
...
32
votes
2
answers
4k
views
Similarities between Post's Problem and Cohen's Forcing
Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated 4/...
- 7,831
20
votes
2
answers
7k
views
Question about functional derivatives
This page on Wikipedia defines the so-called functional derivative as follows: "Given a manifold $M$ representing (continuous/smooth) functions $\rho$ (with certain boundary conditions, etc.) and a ...
- 3,515
7
votes
1
answer
2k
views
Pochhammer symbol of a differential, and hypergeometric polynomials
I have a minor result which I'm sure has come up somewhere before but I can't seem to find it.
Consider a confluent hypergeometric function of the form
$$\newcommand{\ff}{{}_1F_1}
\ff(b+k;b;z)\...
- 2,358
25
votes
3
answers
13k
views
Fourier transform of the unit sphere
The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula
$$
\int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu ...
- 31.5k
13
votes
2
answers
2k
views
Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus
I was exploring some raising and lowering operators related to an infinitesimal generator for fractional integro-derivatives and found an Appell sequence of polynomials, i.e., an infinite sequence of ...
- 10.5k
41
votes
4
answers
16k
views
Product of Borel sigma algebras
If $X$ and $Y$ are separable metric spaces, then the Borel $\sigma$-algebra $B(X \times Y)$ of the product is the $\sigma$-algebra generated by $B(X)\times B(Y)$. I am embarrassed to admit that I ...
- 31.5k
23
votes
5
answers
4k
views
Morse-Kelley set theory consistency strength
I've come across several references to MK (Morse-Kelley set theory), which includes the idea of a proper class, a limitation of size, includes the axiom schema of comprehension across class variables (...
- 1,979
22
votes
3
answers
2k
views
Hecke equidistribution
For a prime $p\equiv 1\pmod{4}$, we can write $p=a^2+b^2=N(a+bi)$. Therefore
$$
a+bi=p^{1/2}e^{i\varphi}
$$
where $\varphi\in [0,2\pi]$. I know that Hecke proved that $\varphi$ is equidistributed. I ...
- 2,508
5
votes
1
answer
630
views
Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution
Examples of infinite dimensional involutions
Edit 2/25/23, as suggested by YCOR below: (Start)
The first return on a Google search on involution--from late Latin 'a rolling up'--gives the Oxford ...
- 10.5k
406
votes
85
answers
189k
views
Proofs without words
Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results?
(One could ask if this is of interest to mathematicians, and I ...
70
votes
4
answers
11k
views
$C^1$ isometric embedding of flat torus into $\mathbb{R}^3$
I read (in a paper by Emil Saucan) that the flat torus may be isometrically embedded
in $\mathbb{R}^3$ with a $C^1$ map by the Kuiper extension of the Nash Embedding Theorem,
a claim repeated in this ...
- 151k
45
votes
7
answers
9k
views
What's an example of a space that needs the Hahn-Banach Theorem?
The Hahn-Banach theorem is rightly seen as one of the Big Theorems in functional analysis. Indeed, it can be said to be where functional analysis really starts. But as it's one of those "there ...
- 26.8k
8
votes
4
answers
530
views
Inside-out polygonal dissections
A dissection of a polygon $P$
is a partition of $P$ into a finite number of pieces, which can then be rearranged
(via planar translations and rotations) and joined (without overlap) to form a new ...
- 151k
8
votes
3
answers
1k
views
Ramanujan's Master Formula: A proof and relation to umbral calculus
The Ramanujan's master theorem states that:
$$
\int_0^{\infty}x^{s-1}\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}a_nx^ndx=\Gamma(s)a_{-s}
$$
I found a really strange proof recently on a personal blog:
Define
$...
- 302
196
votes
12
answers
31k
views
Do you know important theorems that remain unknown?
Do you know of any very important theorems that remain unknown? I mean results that could easily make into textbooks or research monographs, but almost
nobody knows about them. If you provide an ...
96
votes
36
answers
17k
views
The concept of duality
I have been thinking for sometime about asking this question, but because I did not want to have two "big-list" questions open at the same time, I did not ask this one. Now its time has come....
62
votes
9
answers
9k
views
Fundamental groups of noncompact surfaces
I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact about surface topology ...
- 44.8k
52
votes
15
answers
11k
views
Explicit computations using the Haar measure
This question is somewhat related to my previous one on Grassmanians. The few times I've encountered the Haar measure in the course of my mathematical education, it's always been used in a very ...
- 4,586
40
votes
5
answers
10k
views
Is there a natural measures on the space of measurable functions?
Given a set Ω and a σ-algebra F of subsets, is there some natural way to assign something like a "uniform" measure on the space of all measurable functions on this space? (I suppose first ...
- 1,475
34
votes
4
answers
9k
views
Why are the integers with the cofinite topology not path-connected?
An apparently elementary question that bugs me for quite some time:
(1) Why are the integers with the cofinite topology not path-connected?
Recall that the open sets in the cofinite topology on a ...
- 5,743
33
votes
4
answers
7k
views
Classification of finite groups of isometries
Consider the problem of classifying the finite groups of isometries of $\mathbb{R}^n$.
For $n=2$ it is cyclic and dihedral groups.
For $n=3$ they are well known, probably from Kepler and are related ...
32
votes
6
answers
3k
views
Can distribution theory be developed Riemann-free?
I imagine most people who frequent MO have been indoctrinated into the point of view that the Riemann integral can be safely discarded once one has taken the time to develop the Lebesgue integral. ...
- 29.2k
27
votes
5
answers
3k
views
Nice applications for Schwartz distributions
I am to teach a second year grad course in analysis with focus on Schwartz distributions. Among the core topics I intend to cover are:
Some multilinear algebra including the Kernel Theorem and ...
26
votes
2
answers
4k
views
3D models of the unfoldings of the hypercube?
There are (apparently) 261 distinct unfoldings of the 4D hypercube, a.k.a., the
tesseract, into 3D.1
These unfoldings (or "nets") are analogous to the 11 unfoldings of
the 3D cube into the plane.2
...
- 151k
20
votes
6
answers
4k
views
Erik Westzynthius's cool upper bound argument: update?
Version 2 of this writeup is
available, and includes a newer and simple upper bound thanks to
MathOverflow 88777 as
well as indirect references to future writeups. Details of further work
...
- 13k
20
votes
2
answers
2k
views
On a result attributed to W. Ljunggren and T. Nagell
I've read in a number of places that, building on previous work of T. Nagell, W. Ljunggren proved in 1 that the Diophantine equation
$$\frac{x^{n}-1}{x-1} = y^{2}$$
doesn't admit solutions in ...
- 8,842
19
votes
2
answers
8k
views
The canonical line bundle of a normal variety
I have heard that the canonical divisor can be defined on a normal variety X since the smooth locus has codimension 2. Then, I have heard as well that for ANY algebraic variety such that the canonical ...
- 2,215
10
votes
2
answers
803
views
Reference for Wang tile
I am working on projects in solving ground state of generalized Ising models. One recent work involves tiling with basic tiles that filled the whole lattice. For example, we could obtain results:
...
- 867
5
votes
1
answer
672
views
coloring in lattice
This is a mathematical question raised from engineering and physics:
Is there some established mathematical approach in filling a physical lattice with some colored basis (black and white here)? For ...
- 867
148
votes
4
answers
69k
views
What are "perfectoid spaces"?
This talk is about a theory of "perfectoid spaces", which "compares objects in characteristic p with objects in characteristic 0". What are those spaces, where can one read about them?
Edit: A bit ...
- 10.8k
68
votes
4
answers
12k
views
Nelson's program to show inconsistency of ZF
At the end of the paper Division by three by Peter G. Doyle and John H. Conway, the authors say:
Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel ...
- 25.5k
64
votes
6
answers
5k
views
Shortest closed curve to inspect a sphere
Let $S$ be a sphere in $\mathbb{R}^3$. Let $C$ be a closed curve in $\mathbb{R}^3$ disjoint from and
exterior to $S$
which has the property that every point $x$ on $S$ is visible to some point $y$ of $...
- 151k
54
votes
7
answers
15k
views
Why are local systems and representations of the fundamental group equivalent
My question: Let X be a sufficiently 'nice' topological space. Then there is an equivalence between representations of the fundamental group of X and local systems on X, i.e. sheaves on X locally ...
- 1,197
38
votes
5
answers
10k
views
Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?
The Diophantine equation
$$x^3+y^3+z^3=3$$
has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions ...
- 2,350
36
votes
10
answers
6k
views
Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature
A curve in the plane is determined, up to orientation-preserving
Euclidean
motions, by its curvature function, $\kappa(s)$.
Here is one of my favorite examples, from
Alfred Gray's book,
Modern ...
- 151k
31
votes
11
answers
13k
views
Uniformization theorem for Riemann surfaces
How does one prove that every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere, and these are not conformally equivalent to ...
29
votes
6
answers
10k
views
how to find/define eigenvectors as a continuous function of matrix?
I asked this (with background) here
https://stats.stackexchange.com/questions/38494/principal-component-analysis-bootstrap-and-probability-of-eigenvalue-collision
but did not really get any answers. ...
- 2,600
28
votes
6
answers
12k
views
Almost orthogonal vectors
This is to do with high dimensional geometry, which I'm always useless with. Suppose we have some large integer $n$ and some small $\epsilon>0$. Working in the unit sphere of $\mathbb R^n$ or $\...
- 18.7k
26
votes
6
answers
8k
views
prime ideals in C([0,1])
It is clear that each maximal ideal in ring of continuous functions over $[0,1]\subset \mathbb R$ corresponds to a point and vice-versa.
So, for each ideal $I$ define $Z(I) =\{x\in [0,1]\,|\,f(x)=0, ...
- 5,063
24
votes
5
answers
8k
views
totally disconnected and zero-dimensional spaces
When do the notions of totally disconnected space and zero-dimensional space coincide? From what I gather, there are at least three common notions of topological dimension: covering dimension, small ...
- 3,605
23
votes
3
answers
6k
views
Density of smooth functions under "Hölder metric"
This question came up when I was doing some reading into convolution squares of singular measures. Recall a function $f$ on the torus $T = [-1/2,1/2]$ is said to be $\alpha$-Hölder (for $0 < \alpha ...
- 505
18
votes
2
answers
1k
views
A combinatorial interpretation for $n$-ary trees for negative $n$
The ordinary generating function $T_n=T_n(x)$ for the $n$-ary trees satisfies the functional equation
$$
T_n=1+xT_n^n.
$$
This is usually defined for $n\ge 0$, but the functional equation can be ...
- 1,190
14
votes
1
answer
4k
views
What is the source of this E̶r̶d̶ő̶s̶ quote?
Namely, the following one
"All problems appeared once in the [American Mathematical] Monthly."
I remember reading it several years ago... When I first posed the question, I believed that I had ...
- 8,842
14
votes
4
answers
2k
views
What can be preserved in mathematics if all constructions are carried out in ZF?
This is inspired by this discussion. I see that the debates about the necessity of the axiom of choice in this or that statement are still ongoing. In this regard, I became interested in whether there ...
- 7,426
11
votes
3
answers
882
views
Reference request: Systems of linear PDES with constant coefficients
I am looking for a reference for the following statement:
Assume that $P_1, \dots, P_k \in \mathbb R[x_1, \dots, x_m]$ and consider a system of PDEs
\begin{align}
P_i(\partial / \partial x_1, \dots, \...
- 241
11
votes
1
answer
1k
views
Extending an assignment property from Q to R (or C)
Property of any odd number of nonnegative integers:
Given $x_1 \leq \cdots \leq x_{2n + 1}$ with each $x_i \in \mathbb{Z}_{\geq 0}$, suppose that for any $x_i$ we remove, the remaining numbers can be ...
- 7,831
9
votes
2
answers
674
views
Powers of finite simple groups
I have heard about the following result: for each finite simple non-abelian group $S$ and each natural number $r\ge 2$ there exists a number $n=n(r,S)$ such that the power $S^n$ is $r$-generator but $...
- 331
9
votes
2
answers
1k
views
Relationship between fragments of the axiom of choice and the dependent choice principles
The dependent choice principle ${\rm DC}_\kappa$ states that if $S$ is a nonempty set and $R$ is a binary relation such that for every $s\in S^{\lt\kappa}$, there is $x\in S$ with $sRx$, then there ...
- 2,240