Questions tagged [reference-request]

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

Filter by
Sorted by
Tagged with
3 votes
0 answers
33 views

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 $...
inkspot's user avatar
  • 3,082
0 votes
0 answers
42 views

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 ...
Sam Sanders's user avatar
  • 3,977
-1 votes
0 answers
51 views

"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 $y$ ...
Jannik Pitt's user avatar
  • 1,179
4 votes
2 answers
246 views

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 ...
erz's user avatar
  • 5,425
2 votes
0 answers
46 views

$\Pi^0_1$ 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 ...
Noah Schweber's user avatar
0 votes
0 answers
37 views

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 ...
AChem's user avatar
  • 803
1 vote
0 answers
96 views

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 ...
kindasorta's user avatar
  • 1,701
1 vote
0 answers
39 views

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 ...
Iian Smythe's user avatar
  • 3,011
2 votes
0 answers
31 views

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 ...
Keivan Karai's user avatar
  • 6,104
1 vote
0 answers
53 views

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 ...
Alexander Chervov's user avatar
1 vote
0 answers
51 views

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 $\...
geocalc33's user avatar
  • 101
6 votes
2 answers
194 views

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 (...
MRJ's user avatar
  • 73
0 votes
0 answers
51 views

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 ...
0207's user avatar
  • 123
2 votes
2 answers
153 views

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) ...
Jörg Neunhäuserer's user avatar
5 votes
0 answers
30 views

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 ...
jg1896's user avatar
  • 2,733
1 vote
0 answers
83 views

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+...
T. Amdeberhan's user avatar
4 votes
0 answers
83 views

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}[\...
Pace Nielsen's user avatar
  • 18.2k
2 votes
1 answer
108 views

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{...
l'étudiant's user avatar
0 votes
0 answers
48 views

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?
Sowbarnika R's user avatar
1 vote
0 answers
55 views

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 ...
πr8's user avatar
  • 706
3 votes
0 answers
146 views

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 $N$ ...
Johnny T.'s user avatar
  • 3,595
2 votes
0 answers
79 views

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....
Kim's user avatar
  • 4,124
0 votes
0 answers
33 views

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 ...
Euclid's user avatar
  • 3
5 votes
2 answers
591 views

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,...
Iosif Pinelis's user avatar
0 votes
1 answer
186 views

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 ...
Tanny Sieben's user avatar
0 votes
0 answers
35 views

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 ...
Anton Petrunin's user avatar
4 votes
0 answers
256 views

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, ...
David Roberts's user avatar
  • 33.9k
3 votes
0 answers
78 views

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-...
Alekos Robotis's user avatar
6 votes
0 answers
103 views

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,\...
Semen Podkorytov's user avatar
-3 votes
0 answers
120 views

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 ...
Tom Copeland's user avatar
  • 9,931
1 vote
0 answers
86 views

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? ...
Μάρκος Καραμέρης's user avatar
4 votes
1 answer
109 views

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}...
Vessel's user avatar
  • 145
2 votes
0 answers
126 views

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 ...
Dr. Evil's user avatar
  • 2,681
3 votes
1 answer
212 views

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 ...
ashK's user avatar
  • 117
0 votes
0 answers
32 views

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 ...
Zhang Yuhan's user avatar
6 votes
1 answer
182 views

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 ...
rosan98's user avatar
  • 361
2 votes
0 answers
115 views

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 $L^2([a,b]\times S^2)$ ...
Laithy's user avatar
  • 885
4 votes
0 answers
86 views

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\...
minhtoan's user avatar
  • 1,454
16 votes
3 answers
913 views

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 ...
Mare's user avatar
  • 26.2k
1 vote
0 answers
36 views

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 ...
kindasorta's user avatar
  • 1,701
6 votes
1 answer
869 views

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 ...
John Baez's user avatar
  • 21.6k
3 votes
3 answers
204 views

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 $\...
Perelman's user avatar
  • 143
1 vote
0 answers
63 views

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 ...
Martin Skilleter's user avatar
0 votes
1 answer
98 views

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 ...
trinket34's user avatar
4 votes
0 answers
63 views

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 ...
Aidan Backus's user avatar
2 votes
0 answers
62 views

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 ...
Matthias's user avatar
  • 203
3 votes
0 answers
96 views

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-...
Nandor's user avatar
  • 119
3 votes
0 answers
85 views

$\mathbb{Z}_p$-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 ...
Otto's user avatar
  • 31
6 votes
1 answer
167 views

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 $\...
Lajos Soukup's user avatar
  • 1,477
2 votes
0 answers
84 views

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 ...
Evan Scott's user avatar

1
2 3 4 5
292