# Questions tagged [reference-request]

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

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

1,584
questions

223
votes

12
answers

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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 ...

204
votes

50
answers

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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

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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/...

13
votes

2
answers

2k
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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 ...

7
votes

1
answer

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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)\...

21
votes

3
answers

2k
views

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 ...

22
votes

3
answers

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views

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 ...

66
votes

4
answers

10k
views

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 ...

21
votes

5
answers

3k
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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 (...

387
votes

80
answers

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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 ...

91
votes

36
answers

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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....

60
votes

9
answers

8k
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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 ...

48
votes

15
answers

10k
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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 ...

33
votes

4
answers

7k
views

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 ...

27
votes

4
answers

6k
views

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 ...

25
votes

2
answers

3k
views

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
...

20
votes

6
answers

4k
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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
...

19
votes

2
answers

2k
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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 ...

16
votes

2
answers

7k
views

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 ...

10
votes

1
answer

634
views

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:
...

5
votes

1
answer

570
views

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 ...

63
votes

6
answers

4k
views

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 $...

35
votes

10
answers

5k
views

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 ...

33
votes

5
answers

9k
views

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 ...

23
votes

5
answers

7k
views

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 ...

14
votes

1
answer

3k
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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 ...

11
votes

3
answers

654
views

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, \...

11
votes

1
answer

1k
views

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 ...

9
votes

2
answers

884
views

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 ...

9
votes

2
answers

596
views

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 $...

8
votes

1
answer

316
views

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 ...

5
votes

1
answer

353
views

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 ...

5
votes

1
answer

227
views

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 ...

175
votes

11
answers

27k
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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 ...

139
votes

4
answers

60k
views

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 ...

68
votes

4
answers

11k
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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 ...

59
votes

1
answer

6k
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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-...

56
votes

6
answers

6k
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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 ...

47
votes

8
answers

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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 ...

45
votes

2
answers

12k
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I am now doing an article survey on the field of information geometry started by S.Amari and Barndorff-Nielson. I want to know some research situation in this field.
I have read (4) and parts of (3). ...

41
votes

11
answers

10k
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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 ...

39
votes

6
answers

4k
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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 ...

29
votes

11
answers

12k
views

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 ...

26
votes

6
answers

9k
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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. ...

26
votes

5
answers

6k
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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 ...

23
votes

3
answers

1k
views

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 ...

19
votes

3
answers

1k
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Apologies in advance if this turns out to be simple. So far I haven't found a proof or a reference.
Although I like $p$ to be a prime, I can ask the following for positive integers $n$ and $p$, ...

18
votes

2
answers

2k
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Let $S$ be a set of points in $\mathbb{R}^d$; I am especially interested in $d=2$.
Say that $S$ is an integer-distance set if every pair of points in $S$ is separated
by an integer Euclidean distance.
...

18
votes

1
answer

2k
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Wonder whether any of you know where it was that the following pearl of topology first appeared:
Prove that at any instant of time you can find three isothermal points on the surface of the Earth ...

17
votes

0
answers

841
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The (non-unique) bundle isomorphism between the bundle $J^r E$ of $r$-th order jets of sections of a vector bundle $\pi:E\rightarrow M$ and the direct sum $$\bigoplus^r_{k=0}\vee^kT^*M\otimes E\...