# 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.

13,375
questions

8
votes

1
answer

176
views

Here are some definitions:
A space is homotopy finite if it is homotopy equivalent to a finite CW complex.
A space finitely dominated if it is a retract of a homotopy finite space.
A space $X$ is a ...

2
votes

0
answers

65
views

In his book Algebraic Number Theory (2nd ed., Thm 2 in p.128), Lang proves the following (well-known) auxiliary result. Let $D\subset\mathbb{R}^N$ with $(N-1)$-Lipschitz parametrizable boundary. Let $...

4
votes

1
answer

207
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Let $k$ be a (complete) discretely valued field and $\ell$ a Galois extension of $k$, possibly infinite. The Galois group $\Gamma=\text{Gal}(\ell/k)$ of $\ell$ over $k$ admits a descreasing, $\mathbb ...

14
votes

1
answer

343
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In his "Théorie de chaleur" Fourier proves that the zeros of Bessel function $J_0(x)$ are all real.
I want to ask if there is a modern version of this proof exist in literature?
If someone ...

17
votes

3
answers

442
views

I believe I can prove the following result.
Let $f_n:\mathbb{R}^n\to \mathbb{R}$ be a sequence of convex functions
that converges almost everywhere to a function $f:\mathbb{R}^n\to\mathbb{R}$.
Then $...

5
votes

1
answer

284
views

I am interested in equations of the form $|\nabla d|= F(x)$, where $F(x)$ is piecewise constant and $d(x) = 0$ on $\Gamma_D$, a subset of the boundary. In particular, like in the figure, one can ...

4
votes

0
answers

123
views

Recall the constructions $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ with $[0]!_q:=1$ and the $q$-binomials (Gaussian polynomials)
$$\binom{n}k_q=\frac{[n]!_q}{[k]!_q[n-k]!_q}.$$
Given two ...

0
votes

0
answers

44
views

Is there a textbook/paper that I can reference for the following problem? I am looking for a concise proof that I can cite.
Let $G=(V,E)$ be a weighted directed acyclic graph, and consider
$s,t\in V$....

11
votes

1
answer

803
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In differential geometry it is often natural to speak of infinite-dimensional manifolds (e.g., the manifold of mappings between two smooth manifolds). Different versions of generalized smooth spaces ...

7
votes

0
answers

122
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tl;dr Is there any source that discusses the concept of a filtered Tannakian category? I'm writing a paper with this notion and want to know if it's ever been discussed.
The original book by Saavedra-...

0
votes

1
answer

115
views

Let $n$ be a positive squarefree integer, and let $h_n$ denote the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-n})$. Then, is it true that $h_n$ is odd if and only if $n$ is a ...

2
votes

1
answer

279
views

In the preprint "Level Spacing Statistics for Primes", we have found some patterns of prime spacings, which may provide new insights on the distribution of primes:
We would like to know ...

3
votes

0
answers

94
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Working in optimal control of PDEs, I came across a type of evolution problem that has instead of an initial condition a link between the initial state and the final state.
Here is a simplified ...

1
vote

0
answers

135
views

PeakFit (Systat, v. 4.12) is a software for fitting experimental peaks obtained in physics or chemical experiments. Under the miscellenous peak functions, it shows the following equations with a name, ...

2
votes

0
answers

40
views

Given a family of ring homomorphisms $ \phi_i : X \rightarrow Y_i $ where each $ Y_i $ is a topological ring and consider the initial topology on $ X $, i.e. the coarest topology such that each map is ...

1
vote

0
answers

39
views

Recently I read Rupert L. Frank's paper "Eigenvalue Bounds for the Fractional
Laplacian: A Review". For a domain $\Omega\subset\mathbf R^n$, there are two different definitions of ...

9
votes

1
answer

564
views

In IMO Shortlist 2013, there is a number theory problem:
Determine whether there exists an infinite sequence of nonzero digits $a_1,a_2,a_3,...$ and a positive integer $N$ such that for every integer $...

1
vote

0
answers

26
views

For a fixed ground set $[n]=\{1,\ldots,n\}$, and for any matroid $M$ on $[n]$, specified as a collection of bases $B_M$, the corresponding matroid basis polytope $P_M$ is defined to be the convex hull ...

3
votes

1
answer

211
views

I'm looking at the article Baker–Campbell–Hausdorff formula - Wikipedia and I have a few questions.
Under the "Special cases" section, there is a notation $\DeclareMathOperator{\ad}{ad}$
$$ \...

3
votes

0
answers

171
views

This question was asked on math.stackexchange and didn't receive any traction, so I'm turning to the MO community. Hello!
Let $C_{\bullet}$ and $D_{\bullet}$ be chain complexes of $R$-modules for some ...

7
votes

1
answer

165
views

Consider a compact metric group $G$ [A compact topological group $G$ where the topology is generated by an invariant metric]. I am particularly interested in the case where $G$ is the $n$-dimensional ...

2
votes

1
answer

82
views

Could anyone please recommend a known website where I can find a database/library that has systems of polynomial equations with $n$ variables and $m$ parameters?
I need some real examples to test my ...

3
votes

0
answers

108
views

The present quest emanates from this study by R. Stanley, including his recent MO question. Define the product (polynomials after full expansion)
$$I_n(x)=\prod_{i=1}^n(1+x^{F_{i+1}})$$
based on the ...

6
votes

1
answer

404
views

According to the Wikipedia page the center of a group ring $R[G]$ is the set:
$$
\{ p | \forall g,\, h \in G.\, p(g) = p(hgh^{-1}) \}
$$
i.e. class functions which do not distinguish elements of the ...

4
votes

0
answers

111
views

Here lattice means a poset with meets and joins. A lattice is called atomic if every element is a join of atoms. There are a few different ways to define modular for finite lattices: one is that the ...

4
votes

0
answers

81
views

$\DeclareMathOperator\Ent{Ent}$The usual logarithmic Sobolev inequality says that
$$
\Ent_\mu(f^2)\leq C\int |\nabla f|^2 d\mu
$$
where the entropy
$$
\Ent_\mu(f^2)=\int f^2 \log\left( \frac{f^2}{\int ...

8
votes

1
answer

219
views

A $\mathrm{C}^*$-algebra $\mathcal{A}\subset B(\mathsf{H})$ is a norm-closed, self-adjoint subalgebra of bounded operators on a Hilbert space. If we then take a unital self-adjoint (possibly closed) ...

1
vote

0
answers

81
views

Let me begin by a definition.
An algebraic surface $X$ over $\mathbb C$ is called a surface with normal crossing singularities if at every closed point $x\in X$, the analytic local ring
$\widehat{\...

2
votes

1
answer

329
views

Who first used the corner quotes, $\ulcorner$ and $\urcorner$, for the notion of Gödel number? They can also be written as\Godelnum with Sam Buss's macro.
They were ...

5
votes

1
answer

129
views

Consider a set theory with the following axioms:
separation: $\exists y \forall x (x \in y \leftrightarrow \phi \land x \in a)$ where $y$ is not free in $\phi$
reflection: $\phi \to \exists u \phi^u$
...

3
votes

0
answers

81
views

Let $g$ be the distribution whose Fourier coefficients are given by
$$\hat{g}(k) = \begin{cases} 0, & {k=0} \\ |k|^{s-d}, & {k\in \mathbb{Z}^d\setminus\{0\}},\end{cases} \qquad 0\leq s<d,$$
...

12
votes

4
answers

2k
views

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 ...

6
votes

1
answer

337
views

Might there be a research team that has formalised the Riemann Hypothesis? So far I have encountered two related questions:
Is there a formulation of the Riemann Hypothesis in first-order arithmetic?
...

6
votes

2
answers

189
views

It is "well-known" (e.g. stated here without proof and sketched here) that $\mathrm{Z_2}$ proves $\mathrm{Con(PA)}$ using the "usual" model-theoretic proof, that is one can build a ...

2
votes

0
answers

52
views

Consider an edge $e$ of a simplex $C$. If we increase the length of $e$ while keeping the length of other edges constant, the width of $C$ increases in some directions and decreases in other ...

9
votes

1
answer

308
views

The following is a modern, fairly general form of Bézout's theorem. (There are forms that are more general and/or more precise; bear with me.)
Define the degree of a reducible variety to be the sum of ...

1
vote

0
answers

89
views

Is there research about structures for notions of time with distributed systems of information, as with blockchains?
I am thinking of tuples $(I, T, P, A, \prec, s, \eta, u)$ where
$I$, $T$ and $P$ ...

1
vote

0
answers

78
views

Is there any result like the Bramble-Hilbert lemma for Bochner spaces?
More specifically: let $H$ a (e.g.) Hilbert space and $I$ a compact interval, $L \in L(H^k(I,H), Y)$ for $Y$ a normed space. If $...

7
votes

1
answer

254
views

What are the maximal subgroups $M < S_n$ such that $M \cap A_n$ is not maximal in $A_n$?
Maximal subgroups of $S_n$ are described by the O'Nan-Scott theorem and very extensively studied in many ...

7
votes

1
answer

155
views

What is the origin of the abacus bijection (aka the rim hook bijection, aka the Stanton-White bijection, aka James's bijection)?
Igor Pak, in his 2000 article "Ribbon tile invariants" (...

2
votes

0
answers

74
views

Recall $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ and the Gaussian polynomial $\binom{a}{b}_q=\frac{[a]!_q}{[b]!_q[a-b]!_q}$ with $[0]!_q:=1$.
Given two polynomials $U(q)=\sum_k\alpha_kq^...

1
vote

1
answer

168
views

Let $(X,\tau,d)$ be a space where $\tau$ is a topology and $d$ is a metric, where the topology $\tau$ is not necessarily compatible with $d$.
Is there a canonical name for such a structure (maybe ...

1
vote

0
answers

85
views

How can we compute the "N-wave" source-solution of the conservation law
$$u_t + (u - u^2)_x = 0, $$
that is, the entropy solution of this conservation law with the initial data $u(0,\cdot) = ...

2
votes

1
answer

509
views

I'd be applying for a Ph.D. at various grad schools in the U.S. in the coming months and while I know I'd like to pursue research in the field of Algebraic Topology, I am not knowledgeable enough yet ...

7
votes

2
answers

196
views

Let $a_n$ is a binomial sum, for example
$$
a_n := \sum_{k} \binom{n-k}{k} \quad \text{or} \quad \sum_{k=0}^n\binom{n+k}{n-k}\binom{2k}{k}
\quad \text{or} \quad \sum_{k=0}^n\sum_{\ell=0}^k\binom{n}{k}\...

3
votes

1
answer

136
views

This is a more wide-net question of Two increasingly correlated Brownian motions and Williams decomposition.
In our problem we have two correlated Brownian motions $B^{1},B^{2}$ (starting at time $t=0$...

16
votes

0
answers

300
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Some time around 1977, André Joyal constructed a topos (actually a locale, i.e., a localic topos, necessarily non-spatial) in which the closed unit interval $[0,1]$ fails to be compact. There are ...

4
votes

0
answers

96
views

For any $m,k$ define
$$ f_{m,k}(x_1,\ldots,x_n) = \sum_{1\le i_1<i_2<\cdots<i_m\le n} (x_{i_1}+\cdots+x_{i_m})^k. $$
Do these symmetric polynomials have a name and any theory?

9
votes

1
answer

300
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I have already asked this at MSE but did not get an answer.
In quantum field theory one encounters the retarded, advanced and Feynman propagators as certain solutions to a wave equation. ...

1
vote

1
answer

84
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Consider the space of non-empty, compact, and convex subsets of $\mathbb{R}^d$ equipped with the Hausdorff metric.
Are simplicial polytopes a dense subset of that space?
Probably this is just a ...