# 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

2
votes

0
answers

84
views

Any connection between extension of algebraic structure and forcing of set theory? And more, are there any approach from one of the two to other field to solve problem?

7
votes

2
answers

218
views

Let $M_g$ be the moduli space of genus g curves. In Zaal's paper ("A complete Surface in $M_6$ in Characteristic $> 2$"), the author mentioned that there is a known construction of ...

0
votes

0
answers

91
views

I've edited (ten days ago) a question on Physics Stack Exchange, this Mathematical characterization of gravitational geons, post with identifier 726281 the users of the site were kind adding in the ...

10
votes

0
answers

81
views

In an email to the categories mailing list dated 21 August 2003, Street writes:
Max reminded me of his old result (not in the LaJolla Proceedings,
but known soon after) that a monoidal V-category is ...

2
votes

1
answer

218
views

$\DeclareMathOperator\Out{Out}$Let $G=G_1\ast\cdots\ast G_k\ast F_p$ be a Grushko decomposition of a finitely generated group $G$, $\mathcal{O}$ the outer space relative to this decomposition, $[\phi]\...

6
votes

2
answers

309
views

Let $M$ be closed 3-manifold and $\alpha\in H^1(M;\mathbb Z_2)$ an arbitrary element. (In my case we know that $M$ is non-orientable and $\alpha^3=0$.) It is well known that there is a closed 2-...

4
votes

1
answer

110
views

Let $M$ be closed orientable $2n$-manifold, where $n$ is odd. It is well known that the $\mathbb Z$-module $H^\bullet(M;\mathbb Z)$ has graded-commutative multiplication and $H^{2n}(M;\mathbb Z)\simeq\...

9
votes

0
answers

219
views

I came across a reference in this MathOverflow answer to an intriguing result of Ulrich Felgner [1]: among finite non-Abelian groups, the property of being simple is first-order definable. According ...

4
votes

0
answers

107
views

To my knowledge (see, e.g., H.H. Brungs and G. Törner's Skew Power Series Rings and Derivations [J. Algebra 87 (1984), 368-379]), skew polynomial rings were first introduced by Ø. Ore in 1933: Given ...

2
votes

1
answer

144
views

I'm trying to understand whether I can use the following equality in my application -- for $u,v,w \in \mathbb{R}^d$:
$$\cos(u,w)\approx \cos(u,v)\cos(v,w)$$
Where $\cos(x,y)$ gives cosine of the angle ...

3
votes

1
answer

98
views

I would like to ask the following question.
I am searching for a reference for the following statement:
Suppose $k$ is a perfect field. Let $A$ be a (symmetric) $k$-algebra and let $M$ be a finitely ...

3
votes

1
answer

321
views

For $s \in (0,1)$, we consider the spectral fractional Laplacian
\begin{align}
(-\Delta)^{-s}u = \sum_{k=1}^{\infty}\lambda_k^{-s}(\phi_k,u)_{L^2}\phi_k
\end{align}
where
\begin{align*}
\begin{cases}
...

0
votes

0
answers

48
views

What are the current research results on the estimation of the upper bound of $D(N)$ in Chen Jingrun's theorem?
Including but not limited to Chen Jingrun's improvement 7.8342 and Wu Jie's improvement ...

7
votes

1
answer

286
views

I am wondering whether a certain instance of combinatorial reciprocity (in the sense of Stanley's classic paper "Combinatorial Reciprocity Theorems"), concerning symmetric functions, is ...

4
votes

1
answer

323
views

Let
$$
Lu=-a_{ij}(x)\partial_{ij}u+b_i(x)\partial_i u
$$
be a uniformly elliptic operator, with $A(x)=(a_{ij}(x))$ positive-definite.
Here I'm only considering smooth coefficients, and the domain $\...

2
votes

0
answers

60
views

Kostant and Kumar introduced the nil-Hecke ring for a crystallographic Coxeter group, which we will take to be $S_\infty$, which is the ring generated as a left module over the polynomial ring $\...

3
votes

0
answers

124
views

Are there any good tables of the numbers of conjugacy classes of Cartan subspaces in pseudo-Riemannian symmetric spaces? Or a good method to count them? In particular, I am interested in the ...

2
votes

2
answers

264
views

Let $\chi$ be a primitive character mod $p$ (prime) and $\rho = \beta + i \gamma$ be a non-trivial zero of $L(s, \chi)$.
I am reading a paper by Ihara and Murty where they use following estimate :
$\...

4
votes

2
answers

454
views

Question:
Did Kolmogorov develop a set of axioms for probability theory motivated by Algorithmic Information Theory in the 1960s?
Context:
In 1965, Andrey Kolmogorov considered three approaches to ...

3
votes

0
answers

90
views

From the paper
Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889.
the following result is known: Let $(E,\Sigma, \mu)$ be a ...

2
votes

1
answer

182
views

This is a question related to
Semistable curves of genus $g\geq 2$ form an Artin algebraic stack in the etale topology?
Let $\mathcal C\rightarrow \mathcal M^{ss}_g$ be the universal curve over the ...

4
votes

1
answer

178
views

I'm not sure if this is completely relevant to MO, let me know if this would be better on MSE.
I have been told today by a professor of mine that the following is a classic result of Cartan. Suppose $...

5
votes

0
answers

178
views

It happens that I stumbled on a class of infinite dimensional Lie algebras that are not Kac-Moody algebras and for which I was not really prepared for. I know some general results on infinite ...

4
votes

0
answers

63
views

For a function $f \in W^{1,p}(\mathbb R^N)$, it is well-known that there exists a constant $C_N$ (dependent on $N$) such that
$$
|f(x)-f(y)| \le C_N|x-y|(\mathcal M|\nabla f|(x) + \mathcal M|\nabla f|(...

4
votes

2
answers

543
views

Let $n$ be a positive integer and $0 \leq i < n$. Define
$$
N(i) = \# \left\{ (x_1,\dots, x_s) \in [1, n]^s: x_1^2 +\dots + x_s^2 \equiv i \mod n \right\}.
$$
I am looking for a reference for ...

0
votes

0
answers

50
views

Question: When is the operation of inversion continuous as a map between spaces of invertible functions?
Let $\mathcal{F}$ be a function space such that $f\in\mathcal{F}\implies$ $f$ is invertible and ...

5
votes

0
answers

159
views

If a smooth manifold admits a finite atlas, then for many technical purposes it is as good as compact. I was surprised to learn that every connected smooth manifold actually has a finite atlas. This ...

3
votes

0
answers

153
views

Here is a soft question I met in the book Introduction to Grothendieck Duality Theory by Altman and Kleiman.
In Chapter I the proposition 2.1 uses a term called "a character of $X$" where $X$...

1
vote

1
answer

169
views

Geometry and combinatorics are two different branches of mathematics. Does there exist any connection between them? In many cases, mathematicians solve some geometric problems by reducing them to a ...

1
vote

0
answers

37
views

I heard some results about the rigidity of higher rank action and it looks very interesting. I would like to know if there are any good survey of paper to get started in this field. Thank you in ...

0
votes

0
answers

121
views

Is there an analogy (formal or intuitive) between the notions of convergence and contractibility ?
Is the notion of convergence an instance of homotopical triviality in some formalism, or should be ?
...

3
votes

1
answer

130
views

Let $Y (N) $ be the moduli scheme of dimension two principally polarized Abelian schemes with level $N$. It is claimed in "G.Laumon - Fonctions zeta des variétés de Siegel" (Lemma 4.1) that ...

5
votes

1
answer

58
views

Let $G$ be a finite group and let $p$ be a prime number such that $p\mid |G|$.
Let $\text{IBr}(G)$ denote the set of irreducible Brauer characters of $G$ for the prime $p$.
Assume $\mathbb{F}_{q}$ is ...

2
votes

1
answer

286
views

Let $G$ be a (connected ?) algebraic group and $X$ a smooth, projective, and connected algebraic curve, both over an algebraically closed field $k$ of characteristic $0$.
My questions are then as ...

5
votes

2
answers

375
views

Let $C$ be a diagram. Consider a functor $F: C \to \mathbb{E}_{\infty}(Sp)$ from the diagram to the category of $\mathbb{E}_{\infty}$-rings in spectra. Let $R$ be the limit of this diagram.
Given the ...

1
vote

0
answers

132
views

This is a cross-post of a question I asked on StackExchange. See there for further details.
Let $L/K$ be a Galois extension of algebraic number fields of finite degree over $\mathbb{Q}$, with group $G$...

0
votes

0
answers

60
views

I recently read the following question about the Poincare-Hopf theorem in a compact submanifold. All the answers were very satisfactory to me. Is there any reference where I can look for more details ...

4
votes

0
answers

88
views

I attended a talk where the speaker said the following is due to Nielsen. I searched here and there but couldn't find the corresponding paper, if any. So, what is the earliest known proof of the ...

2
votes

1
answer

143
views

I have an explicit, monic quadratic polynomial $P(x)$ and an integer $m$. Can I bound the number of prime values in $P(0), P(1), \ldots, P(m)$? A reference would be appreciated, if available. An ...

2
votes

2
answers

441
views

Let $X$ be a Riemannian compact manifold on which acts a compact Lie group $H^+$. Let $f^+ : X \rightarrow \mathbb{R} $ be a smooth function on $X$. Consider a Lie subgroup $U$ of $H^+$ and suppose ...

4
votes

2
answers

495
views

$\newcommand{\Tors}{{\rm Tors}}
\newcommand{\tf}{{\rm\, t.f.}}
\newcommand{\Gt}{{\Gamma\!,\,\Tors}}
\newcommand{\Gtf}{{\Gamma\!,\tf}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\Z}{{\mathbb Z}}
\...

1
vote

0
answers

83
views

Let us think of the Schwartz space $\mathcal{S}(\mathbb{R}^2_+)$ on the upper half-plane $\mathbb{R}^2_+=\mathbb{R}\times(0,+\infty)$ defined as
$$
\mathcal{S}(\mathbb{R}^2_+)=\left\{f\in C^\infty(\...

5
votes

1
answer

214
views

This is a follow-up to the question of Joseph O'Rourke Which metric spaces have this superposition property?
A metric space $X$ will be called all-set-homogeneous if for any subset $A\subset X$ any ...

2
votes

0
answers

36
views

I would like to know if there is a coordinate free version of the Lagrangian of the supersymmetric sigma model on a $2$-dimensional spacetime, with target space a Kähler manifold. The action for this ...

1
vote

0
answers

77
views

I've read somewhere (but cannot locate the source) that the following classification holds: simply connected homogeneous 3-manifolds are either isometric to $S^2 \times \mathbb{R}$ or to a metric Lie ...

3
votes

0
answers

58
views

Let $w$ be a word over the alphabet $[n] := \{1, \dots, n\}$. For a fixed $S \subseteq [n]$, let $w_S$ be the word obtained from $w$ by deleting all entries not in $S$, then removing (all but one ...

6
votes

1
answer

177
views

There is a plethora of notions of randomness for Cantor space ($\phantom{}^\omega 2$) (Schnorr randomness, Martin-Löf randomness, weak 2-randomness, the various forms of higher randomness such as $\...

1
vote

1
answer

134
views

Let $0 \neq \alpha \in [0,1]$ and $q$ a positive integer.
Let $||.||$ denote the distance to the closest integer and define
$$
N_i(q) = \sum_{ \substack{ -q/2 \leq n \leq q/2 \\ \frac{i}{q} \leq || \...

0
votes

0
answers

70
views

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\const{const}$Let $[+1]:\Delta\to\Delta$ be the décalage endomorphism sending $n\mapsto n+1$
adding a new minimal element, i.e. $f:n\to m$ is sent to $...

6
votes

0
answers

222
views

When $\chi_\mathbb P(n)$ denotes the characteristic function of primes and $\mathcal H=\{h_1,h_2,\dots,h_k\}$ is some admissible $k$-tuple, the GPY sieve can be formulated as follows:
$$
S(x)=\sum_{x&...