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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Universal decay rate of the Fisher information along the heat flow

I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation $$ \partial_t u=\Delta u $$ ...
leo monsaingeon's user avatar
4 votes
1 answer
442 views

Generalizations of Abhyankar-Moh theorem (embeddings of the line in the plane)

Abhyankar-Moh theorem says that if $L$ is a complex line in the complex affine plane $\mathbb{C}^2$, then every embedding of $L$ into $\mathbb{C}^2$ extends to an automorphism of the plane. It seems ...
user237522's user avatar
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4 votes
1 answer
370 views

$\sum_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi, \text{and}\ \alpha \ge 1$

The Fejer-Jackson inequality as follows: $$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$ I conjecture that the inequality as follows holds: $$\sum_{...
Đào Thanh Oai's user avatar
4 votes
0 answers
733 views

Rewrite sum of radicals equation as polynomial equation

My question is about a method described in [Dr.Math forum][1] for simplifying equations involving sums of radical functions. (The following is a transcription of the example given by Dr. Vogler): --- ...
mvc's user avatar
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4 votes
2 answers
790 views

In search for a counterexample related to the Abel-Stolz theorem

Disclaimer: I posted this question seven days ago here on the Math.SE, with slightly different (however in an inessential way) comments. The question has been upvoted but no answer has been given, so ...
Daniele Tampieri's user avatar
4 votes
2 answers
235 views

Bounded density for diffusions with diffusion coefficients bounded away from $0$

Consider a diffusion given by $$X_t=\int_0^t a(s,X_s)\,dW_s$$ for $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and $a$ is a smooth enough positive function bounded away from $...
Iosif Pinelis's user avatar
4 votes
2 answers
836 views

Partitioning an orthogonal matrix into full rank square submatrices

Let $U$ be an $n \times n$ orthogonal matrix. Given an arbitrary partition ${\mathcal P}_c=\{y_1,y_2,\ldots,y_k\}$ of the columns of $U$, does there always exist a corresponding partition ${\mathcal ...
David Shuman's user avatar
3 votes
0 answers
131 views

Reference for a lemma on the asymptotic upper density of special sets with large gaps and intervals

Update. Based on Anthony Quas' comment below, the proof can be made sensibly shorter and the lemma can be slightly generalized by weakening the old assumption (iii). In a joint paper that I am ...
Salvo Tringali's user avatar
2 votes
1 answer
157 views

Boundary condition for elliptic problems and domain decomposition

This question is motivated by one that has been previously asked on this website: Elliptic problem on a domain split in two subdomains Consider an open domain $U$ split in two non-overlapping ...
user avatar
2 votes
1 answer
394 views

Tools for Removing Radicals from Equations

I am currently doing some investigations on Sylvester's 4 Point Problem Probability of 4 Points being in Convex Configuration and repeatedly face the problem of solving equations between sums of ...
Manfred Weis's user avatar
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2 votes
2 answers
301 views

Concrete example of BV function $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative

What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative? More precisely, I'd like to see an example (and a plot using Mathematica or Matlab) of a function $$...
Riku's user avatar
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2 votes
1 answer
302 views

Elliptic problem on a domain split in two subdomains

Consider the following elliptic problem in a split domain: $$ (\ast) \quad\begin{cases} -\Delta u=f_1 \quad &\text{ in } U_1\\ -\Delta u =f_2 & \text{ in } U_2\\ u=g & \text{ on } \...
user avatar
2 votes
2 answers
306 views

Weak convergence for discrete-time processes using characteristic functions

I am looking for a good reference about the analogues of the Bochner Theorem and the Lévy Continuity Theorem for probability measures on $\mathbb{R}^{\mathbb{N}}$ with the product topology. ...
Abdelmalek Abdesselam's user avatar
2 votes
1 answer
143 views

Surveys/monographs on the vortex filament equation

Where can I find surveys on the mathematical aspects of the vortex filament equation? In particular, I'm interested in the following topics: physical motivation; notion of solutions and ...
Kei's user avatar
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1 vote
1 answer
177 views

Defining homotopy via endofunctors of a simplicial category

$\newcommand\sSets{\text{sSets}}\newcommand\sing{\text{sing}}\DeclareMathOperator\Hom{Hom}\newcommand\Top{\text{Top}}$I am looking for a reference describing the notions of homotopy and ...
user420620's user avatar
0 votes
1 answer
1k views

Name of a conjecture on difference of prime numbers? [closed]

Hello Dear there is a conjecture for which I do not know how it is called. The conjecture is: Every even number can be always written as the difference between two prime numbers. Could you please ...
ali's user avatar
  • 3
0 votes
1 answer
118 views

Relationship between the vortex filament equation and the cubic Schrödinger equation

How is the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $\chi(t,s)$ is a curve in $\mathbb R^3$, related to the cubic Schrödinger equation? Note 1. ...
Kei's user avatar
  • 267
235 votes
36 answers
34k views

Conway's lesser-known results

John Horton Conway is known for many achievements: Life, the three sporadic groups in the "Conway constellation," surreal numbers, his "Look-and-Say" sequence analysis, the Conway-Schneeberger $15$-...
136 votes
15 answers
35k views

Statistics for mathematicians

I'm looking for an overview of statistics suitable for the mathematically mature reader: someone familiar with measure theoretic probability at say Billingsley level, but almost completely ignorant of ...
101 votes
30 answers
28k views

Errata for Atiyah–Macdonald

Is there a good list of errata for Atiyah–Macdonald available? A cursory Google search reveals a laughably short list here, with just a few typos. Is there any source available online which lists ...
74 votes
22 answers
17k views

Essays and thoughts on mathematics

Many distinguished mathematicians, at some point of their career, collected their thoughts on mathematics (its aesthetic, purposes, methods, etc.) and on the work of a mathematician in written ...
73 votes
17 answers
9k views

Mathematical research published in the form of poems

The article Friedrich Wille: Galerkins Lösungsnäherungen bei monotonen Abbildungen, Math. Z. 127 (1972), no. 1, 10-16 is written in the form of a lengthy poem, in a style similar to that of the ...
72 votes
8 answers
12k views

Category theory and set theory: just a different language, or different foundation of mathematics?

This is a question to research mathematicians, as well as to those concerned with the history and philosophy of mathematics. I am asking for a reference. In order to make the reference request as ...
Claus's user avatar
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71 votes
3 answers
5k views

Does iterating the derivative infinitely many times give a smooth function whenever it converges?

I am a graduate student and I've been thinking about this fun but frustrating problem for some time. Let $d = \frac{d}{dx}$, and let $f \in C^{\infty}(\mathbb{R})$ be such that for every real $x$, $$g(...
Paul Cusson's user avatar
  • 1,735
67 votes
6 answers
17k views

What is a chess piece mathematically?

Historically, the current "standard" set of chess pieces wasn't the only existing alternative or even the standard one. For instance, the famous Al-Suli's Diamond Problem (which remained ...
Morteza Azad's user avatar
59 votes
7 answers
7k views

Status of PL topology

I posted this question on math stackexchange but received no answers. Since I know there are more people knowledgeable in geometric and piecewise-linear (PL) topology here, I'm reposting the question. ...
51 votes
6 answers
9k views

Which nonlinear PDEs are of interest to algebraic geometers and why?

Motivation I have recently started thinking about the interrelations among algebraic geometry and nonlinear PDEs. It is well known that the methods and ideas of algebraic geometry have lead to a ...
mathphysicist's user avatar
48 votes
1 answer
2k views

Producing finite objects by forcing!

It is a trivial fact that forcing can not produce finite sets of ground model objects. However there are situations, where we can use forcing to prove the existence of finite objects with some ...
Mohammad Golshani's user avatar
47 votes
2 answers
5k views

Well known theorems that have not been proved

I believe that there are numerous challenging theorems in mathematics for which only a sketch of a proof exists. To meet the standards of rigor, a complete proof of these theorems has yet to be ...
45 votes
10 answers
10k views

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 ...
Pietro Majer's user avatar
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45 votes
8 answers
10k views

A down-to-earth introduction to the uses of derived categories

When I was learning about spectral sequences, one of the most helpful sources I found was Ravi Vakil's notes here. These notes are very down-to-earth and give a kind of minimum knowledge needed about ...
44 votes
6 answers
11k views

Book on mathematical "rigorous" String Theory?

I've been looking high and low for a mathematical book on String Theory. The only book I could find was "A Mathematical Introduction to String Theory" by Albeverio, Jost, Paycha and ...
42 votes
10 answers
10k views

The finite subgroups of SL(2,C)

Books can be written about the finite subgroups of $\mathrm{SL}(2,\mathbb C)$ (and their immediate family, like the polyhedral groups...) I am about to start writing notes for a short course about ...
Mariano Suárez-Álvarez's user avatar
37 votes
6 answers
3k views

Billiard dynamics under gravity

Has the dynamics of billiards in a polygon subject to gravity been studied? What I have in mind is something like this:            Still Snell's Law ...
Joseph O'Rourke's user avatar
37 votes
31 answers
27k views

A Learning Roadmap request: From high-school to mid-undergraduate studies

Dear MathOverflow community, In about a year, I think I will be starting my undergraduate studies at a Dutch university. I have decided to study mathematics. I'm not really sure why, but I'm ...
36 votes
2 answers
4k views

Kervaire invariant: Why dimension 126 especially difficult?

Is there any resource that might help non-experts gains some understanding of why the Kervaire invariant problem remains open now only in dimension $126$? ($126 =2^7-2=2^{j+1}-2$; whether $\theta_j=\...
Joseph O'Rourke's user avatar
36 votes
6 answers
12k views

The category of posets

I am trying to teach myself category theory and, as a beginner, I am looking for examples that I have a hands-on experience with. Almost every introductory text in category theory contains following ...
36 votes
1 answer
4k views

Special values of L-functions as periods

If $M$ is a pure motive over $\mathbb{Q}$, one cas define its $L$-function $L(M,s)$ which conjecturaly is a meromorphic function over $\mathbb{C}$ with finitely many poles. For example, when $M=\...
Joël's user avatar
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35 votes
2 answers
30k views

Eigenvalues of the product of two symmetric matrices

This is mostly a reference request, as this must be well-known! Let $A$ and $B$ be two real symmetric matrices, one of which is positive definite. Then it is easy to see that the product $AB$ (or $BA$...
kjetil b halvorsen's user avatar
34 votes
1 answer
5k views

Freyd-Mitchell's embedding theorem

Freyd–Mitchell's embedding theorem states that: if $A$ is a small abelian category, then there exists a ring R and a full, faithful and exact functor $F\colon A \to R\text{-}\mathrm{Mod}$. I have been ...
Bruno Stonek's user avatar
  • 2,914
34 votes
2 answers
2k views

What is the relation between the sphere spectrum and supersymmetry?

In this this google+ post of Urs Schreiber, he says: "Grading over the sphere spectrum is supersymmetry" and then he redirect us to the abstract idea of superalgebra (in nLab). Are there some ...
tttbase's user avatar
  • 1,700
34 votes
1 answer
4k views

Strong Whitney embedding theorem for non-compact manifolds

$\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds. The strong ...
Ricardo Andrade's user avatar
33 votes
1 answer
2k views

Early stabilization in the homotopy groups of spheres

Thanks to Freudenthal we know that $\pi_{n+k}(S^n)$ is independent of $n$ as soon as $n \ge k+2$. However, I was looking at the table on Wikipedia of some of the homotopy groups of spheres and noticed ...
Dylan Wilson's user avatar
  • 13.2k
32 votes
6 answers
7k views

Most important results in 2022

Undoubtedly one of the news that attracted the most attention this year was the result of Yitang Zhang on the Landau–Siegel zeros (see Consequences resulting from Yitang Zhang's latest claimed ...
31 votes
19 answers
22k views

Good books on theory of distributions

Hi all. I'm looking for english books with a good coverage of distribution theory. I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions. Thanks in advance.
30 votes
7 answers
6k views

English reference for a result of Kronecker?

Kronecker's paper Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten apparently proves the following result that I'd like to reference: Let $f$ be a monic polynomial with integer ...
Gray Taylor's user avatar
30 votes
5 answers
6k views

Verlinde's formula

"Verlinde's formula" predicts the dimension of the space of conformal blocks of a chiral CFT. Depending on... • which chiral CFT one considers (does one restrict to WZW models, or not?) &...
André Henriques's user avatar
29 votes
5 answers
6k views

Proof of the Reidemeister theorem

While preparing for my introduction to topology course, I've realized that I don't know where to find a detailed proof of the Reidemeister theorem (two link diagrams give isotopic links, iff they can ...
algori's user avatar
  • 23.2k
29 votes
6 answers
1k views

Online events during the quarantine

With many places on earth subjected to quarantine and large gathering prohibited, there are announcements of online seminars and talks open to people around the world. The talks can be conducted via ...
28 votes
2 answers
3k views

Probing a manifold with geodesics

Supposed you stand at a point $p \in M$ on a smooth 2-manifold $M$ embedded in $\mathbb{R}^3$. You do not know anything about $M$. You shoot off a geodesic $\gamma$ in some direction $u$, and learn ...
Joseph O'Rourke's user avatar

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