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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4
votes
3answers
507 views

Positive proportion of logarithmic gaps between consecutive primes

For $x, \lambda > 0$, define $$S_\lambda(x) := \#\{p_{n+1} \leq x : p_{n+1} - p_n \geq \lambda \log x\} ,$$ where $p_n$ is the $n$th prime number. It is known [1] that an uniform version of the ...
6
votes
1answer
243 views

Severi Formula for Intersection Multiplicities

I say in advance that I am a novice in Intersection Theory, so forgive me if my question is trivial. Let $X\subseteq\mathbb{P}^N$ be a smooth irreducible projective variety of dimension $n$ and $V, W\...
5
votes
0answers
101 views

Stable equivalence and stable Auslander algebras

Let $A$ be a representation-finite finite dimensional quiver algebra and $M$ the basic direct sum of all indecomposable $A$-modules. Recall that the Auslander algebra of $A$ is $End_A(M)$ and the ...
1
vote
0answers
242 views

Reading list in dynamical systems

So I’ve managed to gather from various sources, a plethora of books in dynamical systems. Now I’m wondering which of them to read, and in what order. So far these are the books I’ve found/been ...
6
votes
2answers
277 views

A comprehensive list of random walk inequalities?

I am interested in finding a comprehensive list of all noticeable random walk inequalities. ie. $S_n = \sum_{k\leq n} X_i$ for i.i.d symmetric $X_i$ I can only seem to find books/papers that list ...
3
votes
1answer
337 views

Quantum Field Theory: completing the “A Bridge between Mathematicians and Physicists” series

I decided to read the series "A Bridge between Mathematicians and Physicists" written by Eberhard Zeidler. But when I read the preface of the first book I realized that at first this series should be ...
2
votes
0answers
78 views

Joint drunkard walks

The drunkard walk is a game where two players have $a$ and $b$ dollars, respectively, and they play a series of fair games (both risking one dollar in each game) until one of them goes broke. My ...
4
votes
1answer
125 views

Typical and atypical modules for Lie superalgebras

There are two types highest weight representations for a Basic classical simple Lie superalgebra $\mathfrak{g}$ which are defined as typical (representation for which highest weight vector is the only ...
5
votes
1answer
186 views

Casson invariant and Euler characteristic

A slogan I frequently hear is: "the Casson invariant is the Euler characteristic of the Floer homology of flat SU(2)-connections on the integral homology sphere". Is there a single paper/reference ...
0
votes
0answers
35 views

The set that mazimizes a holomorphic mapping on the unit sphere can be made disjoint from a quarter-circle

I am hoping the below is true. If so, I can prove this: Bounding injective holomorphic mappings on $\mathbb{C}^n$ in the spirit of Andersen-Lempert for $n=2$. Mention of related ideas/topics is also ...
2
votes
0answers
72 views

Exp-decay estimate of Schrodinger equation

Consider the equation $Hu=0$ with $u\in L^2(\Omega)$, where $H=-\Delta+V$ for some bounded continuous function $V$ and $\Omega$ is an un-bounded domain(e.g. $\mathbb R^n$). If $0$ is in discrete ...
5
votes
1answer
112 views

RSK and crystal operators

Is there a good reference on how RSK (and the 3 other variants) interact with crystal operators on the semi-standard tableaux $(P,Q)$ in the image? That is, we have biwords, $W$ which are in ...
4
votes
0answers
130 views

Tight bounds for finite de Finetti's theorem

de Finetti's theorem roughly states that infinite sequence of exchangeable random variables are conditionally independent. I am looking for tight bounds for de Finetti's theorem in the following ...
15
votes
2answers
907 views

Why are Stein manifolds/spaces the analog of affine varieties/schemes in algebraic geometry?

I presume this is a GAGA-style result, but I cannot find a reference.
10
votes
1answer
268 views

What are the applications of topological quantum field theory to continuous-time dynamical systems?

From wikipedia: In dynamics, all continuous time dynamical systems, with and without noise, are Witten-type TQFTs and the phenomenon of the spontaneous breakdown of the corresponding topological ...
0
votes
1answer
53 views

Focusing and nonfocusing nonlinear terms

What is the mathematical and physical meaning of the terms focusing and nonfocusing when they refer to nonlinear terms in a dispersive equations?
4
votes
1answer
83 views

Higher cohomology for trivial module for finite groups of Lie type

Is anything known about the cohomology past $\mathrm{H}^1$ and $\mathrm{H}^2$ for the trivial module for a finite group of Lie type in cross characteristic? For the moment I just care about $\dim \...
1
vote
0answers
38 views

$W^{k,p}$ and Holder regularity for linear elliptic systems with Neumann boundary data

I'm looking for a text or paper that discusses regularity in the Sobolev and Holder sense for general linear elliptic systems of PDEs on bounded domains with Neumann boundary data. The book by ...
8
votes
1answer
238 views

History of the notion of $(G,X)$-structure

I'm currently searching for sources and historical basis on the notion of $(G,X)$-structure as it appears in Thurston's work. So far, it appears that he was the first to set it. Many mathematicans ...
5
votes
3answers
456 views

Alexandrov's generalization of Cauchy's rigidity theorem

Wikipedia states that A. D. Alexandrov generalized Cauchy's rigidity theorem for polyhedra to higher dimensions. The relevant statement in the article is not linked to any source. The sources at the ...
1
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0answers
25 views

Initial-boundary value problem for the damped wave equation with nonlinear source (in bounded domains)

It is well known that the initial-value problem for the wave equation on $\mathbb R^N$ can be studied by means of Fourier transform. What reference presents well-posedness results and qualitative ...
6
votes
1answer
383 views

What is a conic bundle and why is it called so?

I am desperately trying to understand what is a conic bundle. It seems like this is a completely standard term in algebraic geometry, there is even a page on wiki about it, but this doesn't really ...
3
votes
1answer
140 views

Spectral radius of Markov averaging operator on graphs

The definition of Markov operator which I am familiar with: For a graph $G=(V,E)$, Markov's operator upon a function $\varphi:V\rightarrow\mathbb{C}$ , $\varphi\in L^2(G,\nu)$ ($\nu(v):=\deg(v)$) ...
9
votes
1answer
316 views

Bass' conjecture implies the Parshin's conjecture

In the appendix of this paper. It is proved that Bass' conjecture for $K_n$ implies the rational Beilinson-Soulé conjecture for $K_n$. Then at the end the author claims that the same method can be ...
8
votes
1answer
246 views

What are quadric bundles?

In Mori program in dimension $3$ there is a class of Mori contractions $\phi: X\to C$ called quadric bundles, where $X$ is a three-dimensional manifold and $C$ is a curve. As far as I understand, such ...
3
votes
1answer
111 views

Most adequate software for proof checking graph theory proofs

What might be the best software for checking the validity of proofs of graph theoretical statements? Lean, HOL, ... ? One criterion would also be, what would be the easiest for a graph theorist to ...
16
votes
0answers
492 views

How is this group theoretic construct called?

Let $G$ be a finite group, $S\subset G$ a generating set, $|g| = |g|_S = $ word length with respect to $S$. Define the "defect" of $g,h$ to be $$\psi(g,h) = |g|+|h|-|gh|$$ Then $\psi:G\times G \...
6
votes
1answer
271 views

Almost complex manifold of dimension 2… locally isomorphic to ℂ?

I know that this is supposed to be standard, but I don't know how to search for it... hence the question: Let $J$ be an almost complex structure on $M:=\mathbb R^2$, i.e., a $C^\infty$ section of $\...
1
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0answers
150 views

Does the pure motive determine the Voevodsky motive?

I do not quite understand the construction of Voevodsky motives yet. Let $k$ be a field (possibly not algebraically closed), $X$ be a connected smooth projective $k$-scheme. Does the motive of $X$ in ...
1
vote
1answer
76 views

Box counting dimension of a set and Lipschitz functions

If $f$ is Lipschitz, then the following holds for the Hausdorff dimension: $$\dim_H f(A) \le \dim_H A.$$ Is the same true for the box counting dimension?
3
votes
1answer
178 views

Papers containing Ihara avoidance arguments

I am trying to understand some of the recent research in number theory. There is apparently a certain lemma, called Ihara's lemma, which can be established in some contexts and is unknown in other ...
4
votes
0answers
63 views

$\ell^\infty / ces_0$ as an ordered Banach space

Let $ces_0:=\{\xi\in\ell^\infty : \lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^{n}\xi_k=0\}$ and $q:\ell^\infty \to \ell^\infty/ces_0$ be the usual quotient map. The space $ces_0$ is closed in $\ell^\...
0
votes
0answers
67 views

On the preimage of injective holomorphic map

I am hoping the following is true. Mention of related ideas/topics are appreciated. Suppose $F:\mathbb{C}^2 \to \mathbb{C}^2$ is a injective holomorphic mapping such that $F(0)=0$ and $dF(0) = I_2$ ...
8
votes
1answer
163 views

Word length zeta function

Let $G$ be a group with a finite symmetric set $S$ of generators. Let $\ell_S(x)$ denote the word-length of a given $x\in G$. For $s\in\mathbb C$ set $$ Z(s)=\sum_{x\in G^*}\ell_S(x)^{-s}, $$ where $G^...
6
votes
0answers
276 views

Foundational Questions on Adic Spaces

There are some foundational questions on adic spaces that I can't find in the literature. It seems that these questions are pretty natural, so I guess that an answer should be known to the experts in ...
3
votes
1answer
172 views

The cone of curves of complex projective manifolds with an algebraic torus action

I would like to find a reference to the following statement: Statement. Let $X$ be a complex projective manifold with an algebraic action of a $k$-dimensional torus $(\mathbb C^*)^k$. Then the cone ...
1
vote
0answers
150 views

Lecture notes in complex algebraic geometry (which might follow Voisin's book)

I have two questions here, Whats the difference in the target audience/required-background between Voisin's two volumes and Griffiths-Harris' book on algebraic geometry which also seems highly tied ...
3
votes
1answer
107 views

Computing Deligne-Lusztig Characters in General

The goal for this question is to try to find a relatively explicit way of computing the Deligne-Lusztig characters. I understand that the $R_{T,\theta}$ can be computed if we know the values of the ...
2
votes
2answers
160 views

Hausdorff dimension of the graph of the sum of two continuous functions

How can one prove the following result on the Hausdorff dimension of the graph of the sum of two continuous functions: Let $f,g:[0,1] \to \mathbb R$ be two continuous functions. Suppose that $$\...
0
votes
1answer
73 views

Wellposedness results for the cubic Schödinger equation

Motivated by the question Relationship between the vortex filament equation and the cubic Schrödinger equation, I'd like to ask the following: Where can I find a reference on wellposedness ...
4
votes
2answers
180 views

Quasi-separatedness as a topological condition on the scheme

Let $X$ be a scheme. Is it true that the morphism $X\rightarrow \mathrm{Spec}\,\mathbb{Z}$ is quasi-separated iff the intersection of two quasi-compact open subspaces of the underlying space of $X$ is ...
0
votes
1answer
75 views

About maxima of injective holomorphic maps on $\mathbb{C}^n$

I am hoping the following is true. Mention of related ideas/topics are appreciated. Suppose $F:\mathbb{C}^n \to \mathbb{C}^n$ is a injective holomorphic mapping such that $F(0)=0$ and $dF(0) = I_n$ ...
0
votes
0answers
47 views

Relationship between the vortex filament equation and the transport equation

Let us consider the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $\chi(t,s)$ is a curve in $\mathbb R^3$. How is the Cauchy problem for the ...
0
votes
1answer
66 views

Relationship between the vortex filament equation and the cubic Schrödinger equation

How is the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $\chi(t,s)$ is a curve in $\mathbb R^3$, related to the cubic Schrödinger equation? Note 1. ...
1
vote
0answers
39 views

Derivation of the vortex filament equation from Euler equation

How can the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $\chi(t,s)$ is a curve in $\mathbb R^3$, be derived from the Euler equation $$\partial_t \...
8
votes
2answers
253 views

Hopf algebra kernels vs. algebra kernels

Let $f: H_1 \rightarrow H_2$ be a map of graded connected cocommutative Hopf algebras over a perfect field. Let $H \subset H_1$ be the Hopf algebra kernel of $f$, and let $I \subset H_1$ be the ...
2
votes
0answers
49 views

Criteria for a limit to be a proper function

This question is obviously broad; turning this broadness into something sharp is part of the problem. Given a sequence of functions defined on a Riemann Surface $R$, valued in $\Bbb C^2$, what ...
3
votes
1answer
49 views

Reference Request: $L^p(x)$/(Musielak–Orlicz space) analogue of classical $L^p$ result

Fix a non-empty open domain $\Omega\subseteq \mathbb{R}^d$ with compact closure, and a finite Borel measure $\mu$ on its closure $\overline{\Omega}$. In Halmos' book it is shown that: Classical ...
2
votes
1answer
95 views

Surveys/monographs on the vortex filament equation

Where can I find surveys on the mathematical aspects of the vortex filament equation? In particular, I'm interested in the following topics: physical motivation; notion of solutions and ...
4
votes
1answer
80 views

Volumes of double cosets $KtK$

Let $G$ be the group of rational points of a connected reductive group $\mathbb G$ defined over a non-archimedean local field $F$. For simplicity sake, I assume that $\mathbb G$ is split over $F$. Let ...