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Questions tagged [reference-request]

This tag is used if a reference is needed in a paper or textbook on a specific result.

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231 votes
13 answers
40k 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 ...
Pete L. Clark's user avatar
209 votes
51 answers
80k 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/...
Benjamin Dickman's user avatar
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)\...
Emilio Pisanty's user avatar
24 votes
3 answers
12k 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 ...
Francois Ziegler's user avatar
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 ...
Tom Copeland's user avatar
  • 9,937
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 ...
M.B's user avatar
  • 2,468
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 (...
Richard Rast's user avatar
  • 1,979
69 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 ...
Joseph O'Rourke's user avatar
401 votes
84 answers
185k 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 ...
191 votes
12 answers
30k 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
16k 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 ...
Andy Putman's user avatar
  • 43.7k
50 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 ...
Thierry Zell's user avatar
  • 4,536
34 votes
4 answers
8k 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 ...
Theo Buehler's user avatar
  • 5,703
32 votes
4 answers
6k 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 ...
Mathieu Dutour Sikiric's user avatar
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 ...
Joseph O'Rourke's user avatar
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 ...
José Hdz. Stgo.'s user avatar
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 ...
Gerhard Paseman's user avatar
18 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 ...
Jesus Martinez Garcia's user avatar
10 votes
2 answers
771 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: ...
user40780's user avatar
  • 867
5 votes
1 answer
649 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 ...
user40780's user avatar
  • 867
145 votes
4 answers
66k 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 ...
Thomas Riepe's user avatar
  • 10.7k
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 $...
Joseph O'Rourke's user avatar
51 votes
7 answers
14k 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 ...
bavajee's user avatar
  • 1,157
37 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 ...
András Salamon's user avatar
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 ...
Joseph O'Rourke's user avatar
27 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. ...
kjetil b halvorsen's user avatar
23 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 ...
Justin Campbell's user avatar
16 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 ...
Alexander Burstein's user avatar
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 ...
José Hdz. Stgo.'s user avatar
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 ...
Sergei Akbarov's user avatar
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 ...
Benjamin Dickman's user avatar
11 votes
3 answers
784 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, \...
Anton Izosimov's user avatar
10 votes
2 answers
988 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 ...
Victoria Gitman's user avatar
9 votes
2 answers
644 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 $...
user 59363's user avatar
8 votes
1 answer
350 views

Any two bivariate algebraically dependent polynomials are always in the same ring generated by some bivariate polynomial?

If $f(x,y)$ and $g(x,y)$ are two algebraically dependent polynomials over some field $k$, is it true that there exists a bivariate polynomial $p(x,y)$ such that both $f(x,y)$ and $g(x,y)$ are in the ...
Adam's user avatar
  • 201
5 votes
1 answer
443 views

Hausdorff dimension of the graph of a BV function

Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function. Is the Hausdorff dimension of the graph of $u$ equal to $N$? How can we prove it? Update. In an answer to this post, it ...
Riku's user avatar
  • 819
5 votes
1 answer
237 views

Terminology for a monoid $H$ s.t. $xy \in H^\times$ only if $x, y \in H^\times$

The title has it all. Is there any consolidated terminology for referring to a (multiplicative) monoid $H$ such that $xy \in H^\times$ (if and) only if $x, y \in H^\times$? Here is a short list of ...
Salvo Tringali's user avatar
68 votes
4 answers
11k 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 ...
Andreas Thom's user avatar
  • 25.3k
60 votes
1 answer
6k views

Probability that a stick randomly broken in five places can form a tetrahedron

Edit (June 2015): Addressing this problem is a brief project report from the Illinois Geometry Lab (University of Illinois at Urbana-Champaign), dated May 2015, that appears here along with a foot-...
Benjamin Dickman's user avatar
56 votes
6 answers
6k views

Is the Mendeleev table explained in quantum mechanics?

Does anybody know if there exists a mathematical explanation of the Mendeleev table in quantum mechanics? In some textbooks (for example in "F.A.Berezin, M.A.Shubin. The Schrödinger Equation") the ...
Sergei Akbarov's user avatar
49 votes
8 answers
26k views

Roadmap for studying arithmetic geometry

I have read Hartshorne's Algebraic Geometry from chapter 1 to chapter 4, so I'd like to find some suggestions about the next step to study arithmetic geometry. I want to know how to use scheme ...
41 votes
6 answers
4k views

Measures of non-abelian-ness

Let $G$ be a finite non-abelian group of $n$ elements. I would like a measure that intuitively captures the extent to which $G$ is non-commutative. One easy measure is a count of the non-commutative ...
Joseph O'Rourke's user avatar
40 votes
11 answers
11k views

Contemporary philosophy of mathematics

Starting to write an introduction to the philosophy of mathematics, I find tons of positions that are of historical interest. Which philosophical positions are explicitly considered these days, say in ...
34 votes
6 answers
8k views

Covering a unit ball with balls half the radius

This is a direct (and obvious) generalization of the recent MO question, "Covering disks with smaller disks": How many balls of radius $\frac{1}{2}$ are needed to cover completely a ball of ...
Joseph O'Rourke's user avatar
30 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 ...
28 votes
5 answers
6k views

Summation methods for divergent series

There are many methods for assigning a value to a series that diverges, e.g. zeta function regularization, Abel summation, Cesaro summation, etc. From all of the examples I've found, two methods ...
Eric O. Korman's user avatar
26 votes
3 answers
2k views

Kasteleyn's formula for domino tilings generalized?

It seems a marvel when a bunch of irrational numbers "conspire" to become rational, even better an integer. An elementary example is $\prod_{j=1}^n4\cos^2\left(\pi j/(2n+1)\right)=1$. Kasteleyn's ...
T. Amdeberhan's user avatar
26 votes
5 answers
7k views

Proof that no differentiable space-filling curve exists

Could someone provide a reference or a sketch of a proof that no differentiable space-filling curve exists? Or piecewise differentiable? Must every continuous space-filling curve be nowhere ...
Joseph O'Rourke's user avatar

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