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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"Reference Request" for a lecture note by C. Skinner: Galois Representations, Iwasawa Theory, and Special Values of $L$-functions

This was originally posted on math.stackexchange as https://math.stackexchange.com/questions/4589793, where I was suggested to move it here. I'm searching a lecture note by C. Skinner named "...
2 votes
1 answer
556 views

Most important results in 2022 [closed]

Undoubtedly one of the news that attracted the most attention this year was the result of Yitang zhang on the Landau-Siegel zeros. Since it is not possible to be attentive to great results in all ...
-3 votes
0 answers
30 views

Reference Request for "Topology of Space of Holomorphic Functions" [migrated]

I want to learn "Topology of Space of Holomorphic Functions" in a rigorous way. Please advise me some references. My interest arose in this topic of Complex Analysis from the following ...
-1 votes
0 answers
37 views

Some special subgroups of nilpotent groups of nilpotency class 2 [closed]

Let $G$ be a group. Denote by $\mathrm{Z}(G)$ and $G'$ the center of $G$ and the derived subgroup of $G$, respectively. Assume $G'\subseteq\mathrm{Z}(G)$. Then, it is clear that $G$ is a nilpotent ...
2 votes
1 answer
823 views

Reference on the Collatz conjecture

I'm just looking for references in the literature for some observations I made for fun about the Collatz conjecture. The Collatz conjecture states that any positive integer $n$ can eventually be ...
4 votes
0 answers
91 views

Easiest self-contained proof of the Jewett–Krieger theorem?

Does anyone have a go-to reference for a proof of the Jewett–Krieger theorem in dynamical systems/ergodic theory? It's quite technical and I'd like to have something to show students. The best I ...
3 votes
0 answers
74 views

"Practical" references on mapping spaces as infinite-dimensional manifolds

I am studying spaces of the form $C^{k}(\mathcal{M},\mathcal{N})$ between manifolds ($k=\infty$ allowed) and I am looking for extensive references, especially analysing their topology and smooth ...
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2 votes
0 answers
42 views

Entropy of eigenvectors of a large matrix

My question pertains eigenvectors of matrices with somewhat evenly distributed entries. Let $M$ be an $N \times N$ matrix with complex entries (think of $N$ as a large integer). You can assume that $M$...
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3 votes
0 answers
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Every Polish space is the image of the Baire space by a continuous and closed map, reference

The following result was originally proven by Engelking in his 1969 paper On closed images of the space of irrationals (AMS, JSTOR, MR239571, Zbl 0177.25501) Every Polish space (i.e. every separable ...
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4 votes
0 answers
101 views

When is the topos of algebras well-pointed?

The monad version of the theorem at Topos of coalgebras over a comonad is as follows: Let $\mathcal{E}$ be an (elementary) topos. Then if a monad $T : \mathcal{E} \rightarrow \mathcal{E}$ has a right ...
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+300

explicit description of the space of global sections of a torus invariant Weil divisor over a real toric variety

Let $D$ be a Weil divisor on a normal toric variety $X$ with fan $\Sigma$ that is invariant under the action by the torus $T$. Then Proposition 4.3.2 of the textbook Toric Varieties by Cox, Little and ...
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2 votes
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A group-theoretic lemma in a paper by Ershov and He

In the proof of Lemma 2.1 in Ershov, Mikhail; He, Sue, On finiteness properties of the Johnson filtrations, ZBL06904638, the authors claim the following (without proof). Let $G$ be a finitely ...
0 votes
0 answers
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Crazy conjecture about Bernoulli umbra and reference request

For years umbral calculus have fascinated me. Bernoulli numbers (which represent powers of Bernoulli umbra) are involved in many classic power series expansions. Yet, it still remains mistery what ...
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2 votes
1 answer
136 views

Matching polynomial, but $K_2$ is replaced by $K_3$. Have these been studied?

Given a simple graph $G=(V,E)$, we can consider matchings, $M\subseteq E$, where $M$ is a matching iff no vertex is shared between different edges. The number of edges in $M$ is denoted $|M|$. The ...
5 votes
1 answer
109 views

How to classify homomorphisms from $\operatorname{PSL}(2,p)$ to $\operatorname{PGL}(n,2)$ when $2^n=p+1$?

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\GL{GL}$I am looking to classify the homomorphisms from the group $\PSL(2,p)=\Aut(\mathbb{P}^...
3 votes
0 answers
144 views

Property $(T)$ for $\mathrm{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2$

(This is in part a request for references and in part a somewhat pedagogical question.) I gave a course on expanders seven years ago, and I am giving a course on expanders again now. We will soon do ...
6 votes
0 answers
92 views

Heat Flows and spatial singularities

While working on an abstract problem, I came up with the following question: Let $\Omega_1 := \mathrm B(-1, 1)$ and $\Omega_2 := \mathrm B(1, 1)$, where $\mathrm B(x, r) \subseteq \mathbb R^2$ denotes ...
2 votes
1 answer
68 views

Reference request: “A random integral and Orlicz spaces”

I need to find the following paper: “K. Urbanik and WA Woyczynski, A random integral and Orlicz spaces, Bulletin de l'Académie Polonaise des Sciences, Série des sciences mathématiques, astronomiques ...
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4 votes
0 answers
279 views

What are some fundamental papers in derived algebraic geometry for a beginner?

If you could recommend a few papers for someone entering derived algebraic geometry outside of the classical category theory, algebraic geometry, and algebraic topology sequences? What would they be?
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6 votes
1 answer
267 views

Has anyone attempted to generalize the notion of a higher differential of $ A $ and the sheaf of differentials $ \Omega_{A/k} $?

Over a field $ k $ of characteristic zero, actions of $ \widehat{\mathbb{G}_{a}} $ on a variety $ \operatorname{Spec}(A) $ are obtained from $ A $-derivations $ \delta $ of $ A $ over $ k $ via the co-...
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Nilpotent parts of graph Laplacians

Let $W$ be the adjacency matrix of a directed graph. Let us denote by $D$ the associated in-degree matrix, whose diagonal entries are given by $D_{ii} = \sum_j W_{ij}$. The associated Laplacian $$ L =...
2 votes
0 answers
63 views

Steenbrink spectral sequence and modifications of the central fibre

If $f: X \to S$ is a proper map from a complex manifold to a disc, $Y=f^{-1}(0)$ is a divisor with strictly normal crossings and the action of monodromy on $X_t=f^{-1}(t)$ for some (hence any) $t \neq ...
4 votes
1 answer
207 views

First visit of intervals for an irrational rotation

I suppose that what I look for is known, but I can't find it. Let $\left\lbrace I_n=[a_n,b_n)\right\rbrace$ and $\left\lbrace J_n=[b_n,c]\right\rbrace$ ($n\in\mathbb{N}$) be two countable families of ...
0 votes
0 answers
20 views

Integrated risk for estimation of varying coefficient model

Consider the nonparametric varying coefficient model $$y_i = x_i'\beta(z_i)+e_i,$$ where $(x_i, z_i)$ are covariates on $[0,1]^m\times [0,1]^k$, $e_i$ are the errors, and $\beta:[0,1]^k\to [0,1]^m$ is ...
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7 votes
1 answer
514 views

Name for vector spaces with two algebra structures that satisfy the exchange law

Is there a name/reference for the following object? We have a vector space $V$ over some field with two associative bilinear operations $\circ,*:V \times V \to V$ which satisfy the interchange law, i....
3 votes
0 answers
80 views

Interpolation on Sobolev space on $[0, 1]^d$ over finite meshes

Let $\Omega = [0, 1]^d$ and suppose that $f \colon \Omega \to \mathbb{R}$ lies in order $m > d/2$ Sobolev space; i.e., $$ \|f\|_{H^m(\Omega)}^2 = \sum_{|\alpha| \leq m} \|D^\alpha f\|_{L^2(\Omega)}^...
5 votes
1 answer
208 views

Homology of spherical $3$-manifold group

I have been studying $3$-manifolds recently and I got stuck in the following situation. For lens spaces the below fact is true. Let $G$ be a finite group acting freely and orthogonally on $S^3$ so ...
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9 votes
1 answer
581 views

The $9$th tetration of $-\sqrt2$

Let $^na$ denote the $n$th tetration of $a$, so that $^0a=1$ and $$^{n+1}a=a^{^na}$$ for $n=0,1,\dots$. (For complex $x$ and $y$, here we use the definition $x^y:=e^{y\ln x}$, where $\ln$ is the ...
1 vote
0 answers
37 views

Second moment version of the multiple-sum Rogers integration formula

I know the following theorems due to Rogers. Let $X$ denote the space of $n$-dimensional unimodualar lattices in $\mathbb R^n$, equipped with the canonical Haar measure. Theorem 1(Siegel-Rogers). Let ...
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12 votes
1 answer
250 views

Can we force $\mathfrak{r}<\mathfrak{s}$?

Are there models of ZFC in which $\mathfrak{r}$ is strictly less than $\mathfrak{s}$? I've not been able to find any forcings that end up with this result. Here $\mathfrak{r}$ is the reaping number $\...
4 votes
1 answer
118 views

On partial absolute continuity

$\newcommand\B{\mathscr B}\newcommand\A{\mathscr A}\newcommand\si{\sigma}$Let $I:=[0,1]$, and let $\B$ and $\B^2$ denote the Borel $\si$-algebras over $I$ and $I^2$, respectively. Let $\A$ stand for ...
2 votes
1 answer
102 views

Results on Boolean matrices

Matrices with entries in the finite field of two elements $\mathbb{F}_2$, and with the usual operations of matrix addition and multiplication, have been intensively studied, especially due to their ...
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6 votes
2 answers
347 views

Domains that may require a good categorical background

I'm a PhD student in category theory, more specifically I study 2-dimensional category theory, that means bicategories, pseudofunctors, careful definitions of various structures you can put on this ...
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3 votes
0 answers
79 views

Baire class $1$ functions and Baire's characterization theorem

Kechris in his Classical Descriptive Set Theory book gives the following definition (Definition 24.1) and characterization (Theorem 24.15) of Baire class $1$ functions: Definition. Let $X,Y$ be ...
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9 votes
1 answer
241 views

Basic algebra of $\mathcal{O}_0(\mathfrak{sl}_n(\mathbb{C}))$ — Reference request

It is well known that the principal block $\mathcal{O}_0$ of the BGG category $\mathcal{O}$ of a semisimple Lie algebra is equivalent to the category of finitely generated modules over a certain ...
8 votes
0 answers
404 views

Reference request: infinity categories for the commutive algebraist/algebraic geometer

In a survey article Algebraic geometry in mixed characteristic, B. Bhatt writes For instance, given a commutative ring $R$ with a finitely generated ideal $I$, the assignment carrying $R$ to the $\...
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2 votes
2 answers
84 views

Concentration inequality for the spectral norm of the product of normalized Gaussian (and subgaussian) matrices in high dimensions

Let $X,Y$ be two $n\times n$ i.i.d. Gaussian matrices (entries are i.i.d N(0,1) and $X$ and $Y$ are independent). Consider their product normalized by the standard variance of entries $\frac{XY}{\sqrt ...
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3 votes
1 answer
237 views

References for applications of Young diagrams/tableaux to Quantum Mechanics

I am interested in knowing more about applications of Young diagrams and Young tableaux to Quantum Mechanics. A friend of mine suggested as a reference the following book: Wybourne, B.G.; "...
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4 votes
0 answers
59 views

Reference request: a lemma on universes and polynomial monads

I'm looking for a reference that covers things like the lemma below - it doesn't have to be the exact statement I'm going to give, anything in the general ballpark would probably be useful. Or if you ...
  • 35.1k
4 votes
2 answers
140 views

Reference request: "Tangent relation" in metric spaces

Let $X,Y$ be metric spaces. Let $f,g : X \to Y$ be two maps and $x_0 \in X$. Let us say that $f$ and $g$ are tangent at $x_0$ if the following condition is satisfied: For every $\epsilon > 0$ there ...
1 vote
0 answers
16 views

Displaceability questions of fibers on integrable hamiltonian systems

Alot is known about the (non)-displaceability of the fibers of a toric symplectic manifold. For example there is Mcduff's method of probes to prove displaceability results using the moment polytope, ...
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3 votes
0 answers
215 views

Representations of triangle groups

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PSL{PSL}$I am self-studying triangle groups and the following question comes up. Let $G$ denotes $(2,3,7)$ triangle group. It is symmetry group of $(2,...
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2 votes
1 answer
99 views

Counting numerical semigroups by largest element of minimal generating set

For a given integer $n$, I am interested in the number of different numerical semigroups one can make with a generating set consisting only of integers in $[n]$. I have done some small examples. For $...
2 votes
0 answers
147 views

Sets and their characteristic functions

There are some nice connections between properties of sets and properties of their characteristic functions. For instance: a set $C\subset \mathbb{R}$ is closed (resp. open) IFF the characteristic ...
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2 votes
1 answer
165 views

Anosov flow on the 2-sphere

Is there a simple proof that there is no Anosov flow on $S^2$? Where can I find it?
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4 votes
0 answers
70 views

Modern reference for a theorem by Bott on the Dolbeault cohomology of compact homogeneous manifolds

I am looking for a modern, maybe shorter or even easier, reference for Theorem II of Homogeneous vector bundles (R. Bott, Annals of mathematics, 1957). This is a theorem where the Dolbeault cohomology ...
4 votes
2 answers
199 views

Lower bounds for pattern complexity of aperiodic subshifts

In the setting of symbolic dynamics over $\mathbb{Z}^d$, one can define for the $n$-th pattern complexity of a given a subshift $\Omega\subseteq \mathcal{A}^{\mathbb{Z}^d}$ as $$ c_n(\Omega):= \Big\...
1 vote
1 answer
83 views

Is $L_p(X, \mu, E)$ uniformly convex for $p \in (1, \infty)$ if $E$ is a uniformly convex Banach space?

Let $(X, \Sigma, \mu)$ be a $\sigma$-finite complete measure space, $(E, |\cdot|)$ a Banach space, and $p \in (1, \infty)$. Let $L_p := L_p(X, \mu, E)$ be the Bochner space of all $\mu$-integrable ...
  • 339
5 votes
0 answers
70 views

Recognizing sections up to isotopy

Let $E$, $B$ be smooth manifolds, $\pi\colon E\to B$ be a smooth fiber bundle, and $h:B\to E$ be a smooth embedding. I would like to learn what is known about the following Question. When does there ...
5 votes
1 answer
170 views

Reference for Calderon-Zygmund $L^p$ inequalities on the sphere

The following question is motivated from Chapter 2 (Generalized Hodge Systems in 2D), particularly Section 2.3 ($L^p$ theory for Hodge systems in 2D) of Christodoulou and Klainerman's book, The global ...
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