# Questions tagged [reference-request]

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

14,591
questions

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### Complex reflection groups: reference request

Suppose that $V$ is a finite-dimensional complex vector space, that $m\ge 2$ is an integer and that $G\subset GL(V)$ is a finite subgroup such that $V$ is an irreducible ${\mathbb{C}}[G]$-module and $...

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### Sequential definitions of continuity and related classes

It is well-known that the usual 'epsilon-delta' definition of continuity is equivalent to the sequential definition (assuming countable choice). Less well-known is the sequential definition of ...

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51
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### "Weak" partial order where only some elements are reflexive

Are there interesting examples of "almost" partial orders $\preccurlyeq$, where only some elements $x$ satisfy the reflexivity axiom $x \preccurlyeq x$, but every $x$ has at least some $y$ ...

4
votes

2
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### For every sequence of nonempty open sets there is a disjoint sequence of nonempty open sets "below" it

I am looking for any information about the following property for a compact Hausdorff space
$K$: For any sequence $\left(U_{n}\right)$ of nonempty open sets (not necessarily distinct) there is a ...

2
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0
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46
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### $\Pi^0_1$ sentences modulo "schematic entailment"

Let $\mathfrak{P}$ be the preorder of $\Delta^0_0$ (= only bounded quantifiers) formulas with one free variable in the language of arithmetic with a symbol for each primitive recursive function, under ...

0
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0
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### Diagrammatic representation of sets as irregular plane figures

I am trying to find the earliest use of plane diagrams of various shapes for representing sets. For example, this is snapshot is from the book called Some Modern Mathematics for Physicists and Other ...

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0
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96
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### Second group cohomology of a twisted fundamental group

Let $X$ be a smooth hyperbolic projective curve defined over $\mathbb{Z}[1/S]$, where $S$ is a finite set of primes, and let $\pi:=\pi_1^{\text{ét}}(X, \overline{b})$ denote its étale fundamental ...

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0
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39
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### Discrete nonabelian free subgroups of semisimple Lie groups

I understand that the following is a theorem: If $G$ is a noncompact connected semisimple Lie group, then $G$ contains a discrete nonabelian free subgroup.
I can find proofs that such a $G$ contains a ...

2
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0
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31
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### Reference request: amplification argument for hyperlinear groups

Let us define a group $G$ to be a hyperlinear group if it satisfies the conclusion of Theorem 3.6. in these notes by Vladimir Pestov. It is well-known that one can use the so-called amplification ...

1
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0
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53
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### Short path problem on Cayley graphs as language translation task (from "Permutlandski" to "Cayleylandski"(s) :). Reference/suggestion request

Context: Algorithms to find short paths on Cayley graphs of (finite) groups are of some interest - see below.
There can be several approaches to that task. One of ideas coming to my mind - in some ...

1
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0
answers

51
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### automorphisms and mellin transforms

If a real analytic function $f$ is involutive i.e. $f(f(x))=x$ and its Mellin transform can be taken on a section of the real axis, and is analytic for $x>0$, in certain cases can this imply that $\...

6
votes

2
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### Euclidean algorithm for simple closed curves

In the proof of Proposition 6.2 in Farb & Margalit, "A primer on mapping class groups", an analog of the Euclidean algorithm is used to construct a simple, closed representative (...

0
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0
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### Reference request: explicit formula for Lie derivative of matrix Lie groups

Let $M =End(n,\mathbb{C})$ be the space of complex matrices with adjoint action by $ U(n)$, i.e. acted by $B\rightarrow gBg^{-1}$ for $B\in End(n,\mathbb{C}),g\in U(n)$. Let $X_{\xi}$ be the vector ...

2
votes

2
answers

153
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### Two questions on one-dimensional dynamical systems

(1) For $\beta>1$ let $T_{\beta}:[0,1)\to [0,1)$ be given by $T_{\beta}(x)=\{\beta x\}$, where $\{\cdot\}$ denotes the non-integer part of real number. It is classical result (due to Parry I think) ...

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### Expositions of symplectic reflection groups

We will work over $\mathbb{C}$.
Remember that a finite subgroup $G$ of $\operatorname{GL}_n(\mathbb{C})$ is called a complex reflection group if it is generated by complex reflections $r$, which are ...

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0
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83
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### An interesting identity involving skew-Schur functions

Denote $\rho=(-\frac12,-\frac32,-\frac52,\dots)$. I was reading this interesting paper, where in particular, the authors claim that one can get the expression in (2.9)
\begin{align*}
\prod_{k\geq1}(1+...

4
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### Adjoining new factors for primes in UFDs

It is well-known that if we pass from a UFD to a new ring where we have factored one of the primes, it does not need to stay a UFD. The classic example is passing from $\mathbb{Z}$ to $\mathbb{Z}[\...

2
votes

1
answer

108
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### Well-posedness of PDE with $\partial_{tt}\Delta u$ - like term

I am looking for direct hints or references for the establishment of existence of suitable weak solutions admitted by a class of problems of the following type: We search $u$ satisfying
$$
\begin{...

0
votes

0
answers

48
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### Reference for packing property and König property

Can someone please suggest reference material to study about the packing property and König property of ideals and some examples?

1
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0
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### Modulus of Continuity, Heat Flow, and Derivative Estimates

Given $f : \mathbf{R}^d \to \mathbf{R}$, define $P_t f$ by
\begin{align}
(P_t f)(x) = \mathbf{E} \left[ f (x + \sqrt{t} G) \right],
\end{align}
where $G \sim \mathcal{N} (0, I_d)$ is a standard ...

3
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0
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146
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### Cardinality of the set $\#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \}$

Let $\alpha \in I$ where $I$ is some closed interval that does not contain $0$.
I am interested in upper bound for
$$
M(\alpha) = \#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \}
$$
where $N$ ...

2
votes

0
answers

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### Looking for a source on Conrad-Gabber's results about spreading out of rigid-analytic families

Brian Conrad and Ofer Gabber have some results that were announced 9 years ago here:
https://www.ihes.fr/~abbes/Gabber/OferGabber.pdf
and there's a talk by Gabber about them here:
https://www.youtube....

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0
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### MDP Average Reward independent of Initial State

Consider a Markov Decision Process where the state space $S$ and the action space $A$ are continuous and compact.
In state $s$, if action $a$ is chosen and the next state becomes $s'$, the ...

5
votes

2
answers

591
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### A version of Hilbert's Nullstellensatz for real zeros

$\newcommand\R{\Bbb R}$Let $Q(x_1,\dots,x_n)\in\R[x_1,\dots,x_n]$ be an irreducible polynomial such that the dimension of the set $Z:=\{(x_1,\dots,x_n)\in\R^n\colon Q(x_1,\dots,x_n)=0\}$ (defined, say,...

0
votes

1
answer

186
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### Atiyah sequence of a coherent sheaf

I want to show that if $X$ is a smooth complex projective variety, then analytification induces an equivalence of categories between algebraic integrable connections on $X$ and analytic integrable ...

0
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0
answers

35
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### Alexandrov's uniqueness theorem in Minkowski spacetime

Suppose $P$ is a convex polyhedron in $\mathbb{R}^{2,1}$.
Each face of $P$ comes with induced metric tensor,
if the face is space-like, then it is euclidean metric;
every time-like face is isometric ...

4
votes

0
answers

256
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### Gluing together the moduli stacks of elliptic curves over Z[1/2] and Z[1/3]?

I have a long-running desire to understand what is the "global" moduli stack of elliptic curves, as a stack over $\mathrm{Spec}(\mathbb{Z})$.
Recently I was pointed to Katz and Mazur's book, ...

3
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0
answers

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### References for orbifold curves

I am looking for a good reference (if there is any) for the theory of orbifold curves from the perspective of stacks. By an orbifold curve I mean something like a $1$-dimensional irreducible Deligne-...

6
votes

0
answers

103
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### Equation in a nilpotent group

Let $G$ be a nilpotent group of class at most $r$
(that is, $\gamma^{r+1}G=1$).
Let elements $g_1,\dotsc,g_n\in G$ be fixed.
We are interested in the set $V\subseteq\mathbb Z^n$ of solutions $x=(x_1,\...

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votes

0
answers

120
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### Friedrich Schur on the BCHD theorem (notes in English)

According to Sternberg in his book Lie Algebras,
The formula [the BCHD formula] is named after three mathematicians, Campbell, Baker, and
Hausdorff. But this is a misnomer. Substantially earlier than ...

1
vote

0
answers

86
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### Injection of $G(k)/Z(k)$ into $(G/Z)(k)$

In the first answer to the linked question it is mentioned that "the isogeny $G\to G^{ad}$ induces an injection of groups $G(k)/Z(k)\to G^{ad}(k)$". Is there a reference for this result? ...

4
votes

1
answer

109
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### Legendre's Irrationality Condition for Generalized Continued Fractions

This MathOverflow post cites that Legendre allegedly showed that given $a_{i}\in\mathbb{Z}\setminus\left\{0\right\}, b_{i}\in\mathbb{Z}$,
$$\cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cdots}...

2
votes

0
answers

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### GIT quotient and orbifolds

Let $G$ be a connected complex reductive group. Suppose $G$ acts on a smooth complex affine variety $X$. Assume the stabiliser $G_x$ of every point $x\in X$ is finite. Is it true that $X/\!/G$ is an ...

3
votes

1
answer

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### Is Toeplitz operator on the Bergman space bounded iff its symbol is bounded?

It is fairly well known that if $T_\varphi$ is a Toeplitz operator on the Hardy-Hilbert space, then $\lVert T_\varphi \rVert = \lVert \varphi \rVert _{\infty}$.
Now, if $\varphi \in L^\infty (\mathbb ...

0
votes

0
answers

32
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### Reference request: injectivity of CWT, density of dilations and translations in $L^p$

Recently, I encountered the notion of Continuous Wavelet Transform (CWT), and I find it very intriguing (for a reference, see the wiki). I believe it offers a different and more general perspective on ...

6
votes

1
answer

182
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### Attempts to define a matrix exponential over (as much as possible) general fields

Given a $n \times n$ matrix $A$ over the complex numbers, the exponential of $A$ is defined as
$$\exp(A) := \sum_{k = 1}^\infty \frac1{k!} A^k , \qquad \tag{$\star$}\label{468645_star}$$
where ...

2
votes

0
answers

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### Is $L^2([a,b]; L^2(S^2))$ the same as $L^2([a,b] \times S^2)$?

The space $L^2([a,b];L^2(S^2))$ is a Banach space with respect to the norm
$$\left\Vert f \right\Vert_1^2 = \int_{a}^b \left\Vert f(r) \right\Vert_{L^2(S^2)}^2 dr$$
The space $L^2([a,b]\times S^2)$ ...

4
votes

0
answers

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### Symmetric functions and pattern avoidance

It is known that the number of $k$-regular simple graphs with vertices labeled by $1,2,\dots,n$ can be expressed as the coefficient of $x_1^k \dots x_n^k$ in a symmetric function, which is
$$
\prod_{1\...

16
votes

3
answers

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### Conjectures in the representation theory of the symmetric group

Question: What are current open conjectures about the representation theory of the symmetric group?
I am interested mostly in the characteristic 0 case, but conjectures for the modular case can also ...

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0
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36
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### Absolute irreducibility implies free action on framed universal deformation ring

Let $\overline{\rho}: G\longrightarrow \text{GL}_n(\mathbb{F}_p)$ be a residual representation of the Galois group of a number field. Let $R_{\overline{\rho}}^{\square,\text{univ}}$ be the universal ...

6
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1
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### What is this huge generalization of the Modularity Theorem?

A friend of mine wrote:
The point is of course that the Modularity Theorem (as I stated it) is/should be really just a special case of some much bigger theorem which sets up a bijection between ...

3
votes

3
answers

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### Existence of an extension operator $E: W_0^{s,p}(\Omega)\rightarrow W^{s,p}(\mathbb{R}^d)$?

If $\Omega$ is an open Lipschitz domain with bounded boundary then their exists a continuous operator $E: W^{s,p}(\Omega) \rightarrow W^{s,p}(\mathbb{R}^d)$ where $s\in (0,1)$ such that $Eu=u$ on $\...

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0
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### Reference request: finding entries that prevent matrix from being correlation matrix

I am currently doing some research with a quantitative finance firm and my supervisor has raised an interesting question that shows up a lot with their clients: quite often, clients will want to do ...

0
votes

1
answer

98
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### Hansel's simple proof of the Skolem Mahler Lech theorem

In his paper 'A simple proof of the Skolem Mahler Lech theorem' Hansel gives a proof of the theorem in the case that the coefficients of the rational series belong to $\mathbb{Q}$. He claims that the ...

4
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0
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### Techniques to estimate PDE which are elliptic in some directions and degenerate in others

I am interested in a family of PDE which have defeated my (admittedly rather naive) attempts to prove any regularity or stability estimates. These are systems of PDE which are elliptic in some ...

2
votes

0
answers

62
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### Existence of meromorphic differential form on curve with given multiplicity of zeroes and poles

Let $m \in \mathbb{Z}^n$ be a partition of $2g-2$. Polishuk showed in his paper "Moduli spaces of curves with effective r-spin structures" (arXiv link) that if all entries of $m$ are ...

3
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0
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### Representations of a reductive Lie group vie central character and K-types

Let $G$ be a real reductive group, let $\widehat G$ denote the unitary dual and $\widehat G_{adm}\supset\widehat G$ be the admissible dual, i.e., the set of isomorphy classes of irreducible Harish-...

3
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0
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### $\mathbb{Z}_p$-points of a $\mathbb{Z}_p$-model of a reductive linear algebraic $\overline{\mathbb{Q}}_p$-group

Let $G$ be a (connected) reductive linear algebraic group over $\overline{\mathbb{Q}}_p$. By definition, this means that $G$ is a closed subgroup of some $\mathrm{GL}_n$. We can always find a ...

6
votes

1
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167
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### Strengthening of a classical set mapping theorem of Lázár

We are writing a paper in which we need to use the following theorem to prove that a certain poset satisfies c.c.c. As far as I remember I learned this theorem from András Hajnal.
Theorem 1: If $\...

2
votes

0
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84
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### Equivariant disk theorem in dimension 2

All groups I'll consider are finite.
An important part of equivariant differential topology in dimension 3 is the equivariant disk theorem, which says that for a $G$-action on a compact $3$-manifold ...