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
4 votes
0 answers
29 views

Where can I learn more about the topology on $\mathbb{R}$ induced by the map $\mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$?

Consider the (continuous, injective, abelian group homomorphism) map $\Phi \colon \mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$ (where the target is given the product topology) taking $x\in \...
6 votes
1 answer
379 views

How hard is it to find the first layer of this basic $\mathbb{Z}_p$-extension?

$\DeclareMathOperator\Gal{Gal}$Let $p$ be a prime number and $\zeta_{p^n}$ be a primitive $p^n$-th root of unity. We know that there is a unique subfield $\mathbb{Q}_1$ of $\mathbb{Q}(\zeta_{p^2})$ ...
3 votes
2 answers
217 views

General version of $d$-separation

I find the $d$-separation criterion (see, e.g., Theorem 2 here; note however the preceding definition, which basically means we are treating discrete random variables) a really useful sufficient ...
1 vote
1 answer
127 views

Feynman-Kac formula with non-zero boundary condition

Let $D \subseteq \mathbb{R}^m$ be a bounded domain. The Feynman-Kac formula for the heat equation with initial condition $u(t, x) = f(x)$ and boundary condition $u(t, x)|_{\partial D} = 0$ is given by ...
3 votes
1 answer
2k views

Does this hexagon theorem have a name?

Question : Do you know this property of a hexagon? Consider the configuration: Six points $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$ in a plane and let six points $B_i \in A_iA_{i+1}$ for $i=1, 2,\dots, ...
2 votes
1 answer
828 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 ...
79 votes
18 answers
24k views

What programming language should a professional mathematician know? [closed]

More and more I am becoming convinced that one should know at least one programming language very well as a mathematician of this century. Is my conviction justified, or not applicable? If I am right,...
2 votes
1 answer
559 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
66 views

"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 "...
18 votes
5 answers
2k views

Is there a relation between 4-dimensional general relativity and exotic smooth structures on $\mathbb{R}^4$?

Let's say General Relativity is the study of the Einstein equation on smooth Lorentzian manifolds, i.e. pseudo-Riemannian manifolds of signature $(n-1,1)$. I've heard more than once people say that ...
3 votes
0 answers
106 views

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 ...
0 votes
0 answers
60 views
+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 ...
4 votes
2 answers
166 views

Counting integers with k large prime divisors

If $x \ge y \ge 1$ are real numbers and if $k$ is a positive integer, take $\Phi_k(x, y)$ to be the number of integers $\le x$ with exactly $k$ prime factors and no prime factor $\le y$. If $y$ is ...
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 ...
-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 ...
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}^...
-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 ...
1 vote
1 answer
45 views

Multidimensional intersection property

Consider the multidimensional annulus $\{(p,\theta)\} = \mathbb R^n\times\mathbb T^n$ endowed by the $1$-form $\omega=p\,d\theta$. A diffeomorphism $A$ of this annulus onlo itself is said to be exact ...
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 ...
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 ...
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 ...
2 votes
0 answers
366 views

Test function with Steklov average for Caccioppoli-type inequality for porous medium equation

Let $m>1$ and consider locally bounded weak solutions $u(x, t)$ of the parabolic porous medium equation, meaning that $$u\in C_{loc}\left([0, T); L_{loc}^{2}(U)\right)\cap \left\{u^{\frac{m+1}{2}}\...
4 votes
1 answer
231 views

Some questions on a paper of Wilking

I am currently trying to understand Wilking's paper "A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities". If I may, I wish to clarify a few points where I ...
1 vote
1 answer
88 views

Number of distinct near-squares primes dividing an odd perfect number

I'm curious about if the following question is in the literature or what work can be done about it. Denote the number of distinct primes dividing an odd perfect number $N$ with the arithmetic function ...
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 $M$...
0 votes
1 answer
68 views

Ito-Levy decomposition for $\alpha$-stable processes?

The Ito-Levy decomposition is well-known as a characterization of Levy processes. What does it give for the specific case of $\alpha$-stable Levy processes?
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 ...
2 votes
0 answers
99 views

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
55 views

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 ...
2 votes
1 answer
218 views

Growth rate of an outer automorphism of a free product

$\DeclareMathOperator\Out{Out}$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]\...
2 votes
1 answer
56 views

Results in Computational Geometry Utilizing Doubling Dimension of a Metric Space

According to Wikipedia, However, many results from classical harmonic analysis and computational geometry extend to the setting of metric spaces with doubling measures. My question is: what are some ...
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 ...
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 ...
13 votes
1 answer
889 views

Do Baumslag-Solitar Group von Neumann algebras have Property $\Gamma$?

A type $II_{1}$ factor $M$ with trace $\tau$ has Property $\Gamma$ if for every finite subset $\{ x_{1}, x_{2},..., x_{n} \} \subseteq M$ and each $\epsilon >0$, there is a unitary element $u$ in $...
25 votes
6 answers
5k views

Proof of Krylov-Bogoliubov theorem

Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel ...
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 ...
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-...
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 ...
4 votes
2 answers
439 views

Mathematical analysis of Lewisian concepts, esp. natural properties

David Lewis was one of the great philosophers of our time. He was a genuine philosopher, his focus was on theoretical metaphysics. And he had something to say about mathematics. His last book - he ...
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)}^...
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?
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 ...
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.; "...
0 votes
0 answers
20 views

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 ...
21 votes
2 answers
1k views

Forbidden mirror sequences

Let $\cal{M}$ be a finite collection of two-sided mirrors, each an open unit-length segment in $\mathbb{R^2}$, and such that the segments when closed are disjoint. A ray of light that reflects off the ...
9 votes
1 answer
139 views

Bibliography request: Entropy for continued fractions

Given a strictly positive real number $x$ we set $e(x)=\log(1+x)$ if $x$ is an integer and $$e(x)=\log(1+x)+\frac{1+\lbrace x\rbrace}{1+x}\left(e(1/\lbrace x\rbrace)-\log(1+\lbrace x\rbrace)\right)$$ ...
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 ...
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....

1
2 3 4 5
268