Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
14,545
questions
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Projective limit and connected components
Let $E$ be a topological space. Let $\mathcal{K}$ be the set of the compact subsets of $E$.
$(E-K)_{K \in \mathcal{K}}$ is a projective system, because if $K,K'$ are two compacts, there are two ...
5
votes
1
answer
201
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A question about extensions of Markov semigroups
I'm cross-posting this question from MSE. It's the first time I do this so I'm unsure of etiquette regarding how to cross-post, if this irritates anyone please vote this down and I'll delete the post. ...
16
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0
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An open problem in convex geometry
Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...
5
votes
3
answers
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How do you call the problem of approximating a continuous distribution with a simple discrete distribution?
The following problem came up on the Mathematica forum as "Generating a list of integers that roughly satisfy a distribution": Given $n$, find $n$ integers (possibly with duplicates) whose ...
3
votes
1
answer
476
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Massey product in Dual Steenrod Algebra
Let $\tau_0$ be the element of dual Steenrod algebra $A_p^{*}$ at a prime $p$ which is dual to Bockstein $\beta \in A_p$. It is well known $\tau_0^2 =0$. Is it true/known that the elemnet $\xi_1$ ...
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2
answers
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Reference and best bounds of $\sum_{n\leq x}\frac{\mu(n)}{n}$
Could someone please provide information about the best possible known bounds of the sum $$A(x)=\sum_{n\leq x}\frac{\mu(n)}{n}?$$ Unconditionally, $A(x)=O(e^{-c\sqrt{\log x}})$ is known to me. Does ...
6
votes
2
answers
316
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Universal graphs on higher cardinals
The Rado graph contains every finite graph as induced subgraph, and its also holds for countable graphs. So it is an universal graph of size $\aleph_0$, which contains all graphs of size $\aleph_0$ as ...
11
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3
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On mathematical studies of the Mpemba effect
Since the days of Aristotle and Descartes, it has been known that under certain circumstances warm water freezes faster than cold water. This effect is now commonly known as the Mpemba effect, named ...
2
votes
0
answers
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Finite Heisenberg groups action on cohomology of line bundles
Let $E$ be a smooth elliptic curve over algebraically closed field $k$ of characteristic zero, $\mathcal{L}$ is a line bundle over $E$, $\operatorname{deg}(\mathcal{L})=n \geq 1$. Then I define the ...
0
votes
1
answer
2k
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Stationary distribution in general Markov Chains
This is just a reference request for a result which is very general, useful and should be well-known, but I've failed to find a good reference to cite.
The problem is to define the "most natural" ...
0
votes
1
answer
242
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Does the modified Szpiro conjecture require minimal model?
The modified Szpiro conjecture is described in
Wikipedia
and here and here.
The modified Szpiro conjecture states that: given $\varepsilon > 0$, there exists a constant $C(\varepsilon)$ such that ...
5
votes
1
answer
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On a sum involving Euler totient function
Let
$$S_a(N)=\sum_{n\le N}\frac{\varphi(an)}{n^2}.$$
The usual machinery gives an asymptotic formula
$$S_a(N)=\frac1{\zeta(2)}\cdot\frac{a^2}{\varphi_+(a)}\log N+C(a)+O(N^{-1+\varepsilon}a^{1+\...
8
votes
1
answer
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The Bialynicki-Birula Stratification of the Affine Grassmannian
Let $G$ be a connected, simply-connected complex semisimple group with affine Grassmannian $\mathcal{G}r$. Fix a maximal torus and Borel $T\subseteq B\subseteq G$. I am reading "Loop Grassmannian ...
6
votes
2
answers
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Metrics on the space of $C^{*}$ algebras
I think that there is a metric on the huge space of all $C^{*}$ algebras. What is the explicit
definition of this metric?may you introduce me a reference?
Moreover is the restriction of this ...
-3
votes
1
answer
327
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Is there a precise definition of "mathematical formula"? [closed]
In the Wikipedia article for Formula (which has no references), it is claimed that:
"The informal use of the term formula in science refers to the general construct of a relationship between given ...
2
votes
0
answers
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Reference Request: Properties of the Integer Factorization Polytope
The complexity of Integer Factorization is to my knowledge still an open problem, whereas deciding, whether a given integer is a prime number is known to be in $P$ and a proof is available online here:...
2
votes
1
answer
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A problem on the finiteness of solutions to a Diophantine equations
Given two positive integers $a,b$, and an odd prime $p$, I want to know whether the number of solutions to the following equation is finite:
$X^2=a+bp^{Y}$
where $X,Y$ are variables and are integers....
4
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0
answers
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Any references on infinite-dimensional Fourier-Plancherel theory?
Let $M$ be a measure on an infinite-dimensional topological vector space (in fact, only the measure type matters), such that $M$ is quasi-invariant under a dense subspace $S$ of shifts (let's assume ...
1
vote
0
answers
558
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Centralizer of a maximal split torus
Can you help me find a reference for the following fact?
"If $G$ is a quasi-split $p$-adic group and $T$ is a maximal split torus in $G$, then the centralizer of $T$ is a maximal torus in $G$."
Or ...
1
vote
2
answers
536
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The implicit constant in GRH
One particularity of the Generalized Riemann Hypothesis seems to deserve some clarification. In particular, what is included in the commonly accepted version of the conjecture?
GRH states that
$$\...
5
votes
1
answer
483
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Reference for the Natural Ample Line Bundle on the Affine Grassmannian
Let $G$ be a connected, simply-connected complex semisimple group. Let $$\mathcal{G}r:=G((t))/G[[t]]$$ be its affine Grassmannian. I have read that $\mathcal{G}r$ possesses a natural very ample line ...
73
votes
17
answers
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Mathematical research published in the form of poems
The article
Friedrich Wille: Galerkins Lösungsnäherungen bei monotonen Abbildungen,
Math. Z. 127 (1972), no. 1, 10-16
is written in the form of a lengthy poem, in a style similar to that
of the ...
6
votes
0
answers
261
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Stability of analytic Zariski structures
Noetherian Zariski structures are introduced by Hrushovski and Zilber(1996).
An analytic Zariski structure is a generalization of Noetherian Zariski structures, introduced by Zilber and Peatfield.
...
12
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3
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400
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(Non-)Existence of curves of low degree on affine and projective varieties
I am interested in papers that investigates the existence or non-existence of curves of low degree (relative to the degree of the ambient variety). The starting example is that of surfaces and ...
1
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0
answers
122
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Partially Observable Markov Decision Process - finding a hidden object with some positive probability
The following problem is example 5.1 from http://www.statslab.cam.ac.uk/~rrw1/oc/oc2013.pdf
A hidden object moves between two locations according to a Markov China with probability transition matrix $...
7
votes
0
answers
427
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Reference Request: Topological h-cobordism theorem in higher dimensions
I think this question on math.stackexchange is more appropriate on mathoverflow. Correct me, if you don't think so.
The h-cobordism theorem is true in the topological and in the smooth category in ...
0
votes
1
answer
201
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Reference request: on sums of the form $ax^m + by^n = h$
I know that equations of the form
$$\displaystyle ax^d + by^d = h$$
with $a,b,h \in \mathbb{Z}$ have been thoroughly investigated as a special (and interesting) case of the Thue-Mahler equation, for ...
7
votes
3
answers
2k
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Convex hulls of families of probability measures
Let $X$ be a standard Borel space, so that the space of Borel probability measures on $X$ is also a standard Borel space. We denote it by $\mathcal P(X)$.
In this paper for any family of probability ...
2
votes
0
answers
259
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Request for good research mailing list in Dynamical System & Chaos for notification of recent research results, conference, announcements [closed]
Are there some good research-level mathematics mailing list to be recommended in order to be notified of recent research results, news, announcements, conference, etc, particularly in Dynamical System ...
3
votes
1
answer
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The original proof of Wythoff's game
I am looking for the original proof of Wythoff's game. Wythoff provided the first full analysis of this game in "A modication of the game of nim, Nieuw Archief voor Wiskunde, pp. 199-202, 1907&...
18
votes
2
answers
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Reference request: Geodesic flow on a manifold with negative curvature is ergodic
I'm reading about the Mostow's rigidity theorem, and the proof uses the following (maybe well-known) result:
The geodesic flow on a manifold with negative curvature is ergodic.
The lecture note that ...
7
votes
2
answers
1k
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All mapping space between CW complexes is a CW complex?
Let $\mathrm{Map}(X,Y)$ denote the (unbased) cellular mapping space from $X$ to $Y$.
If $X$ and $Y$ are finite CW complexes, is $\mathrm{Map}(X,Y)$ a CW complex?
Can we know the cell structure of $\...
6
votes
1
answer
216
views
Finding cohesive (low exit probability) sets in a Markov process
The following is a fact about Markov chains that came up in a game theory paper. The purpose of this question is to ask if related notions or similar results are found elsewhere in probability, or are ...
9
votes
1
answer
288
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Calculations of nonabelian group cohomology of R^n
I am looking at $H^1(\mathbb{R}^n,G)$ where $G$ is a finite 2-group. I'm wondering if such things have been calculated. I'm afraid I can't say I know anything here, past the result that this ...
7
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2
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Does anyone know what is the right reference for the following simple lemma from harmonic analysis?
The lemma says that given $\lambda\geq 1$, $p\geq 1$, $a_j\geq 0$, for a collection of balls $\{B_j\}_{j\in\mathbb{N}}$ in $\mathbb{R}^n$, it holds
$$\bigg\|\sum_j a_j\chi_{\lambda B_j}\bigg\|_p\leq C(...
12
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0
answers
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Classes for which the Spectrum determines a Convex Shape
Given a planar domain $\Omega \subset \Bbb{R}^2$ bounded and open we can associate to it the spectrum of the Laplace operator with Dirichlet boundary condition. It is known that there are planar ...
3
votes
1
answer
342
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Pointwise ergodic theorem for amenable semigroups
Using tempered Følner sequences one may show a pointwise ergodic theorem for amenable groups.
(see http://www.aimsciences.org/journals/pdfsnews.jsp?paperID=2413&mode=full)
Is there a ...
13
votes
2
answers
2k
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An alternative proof of the Łojasiewicz inequality
Is there a "brute force proof" of the Łojasiewicz inequality? By "brute force" I mean a proof without introducing the machinery of semianalytic sets and so on but only using elementary results (i.e., ...
9
votes
3
answers
736
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What is the definition of picture changing operation?
What is the definition of picture changing operation?
What is a standard reference where it is defined - not just used?
9
votes
1
answer
952
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Reference request: colimits of locally presentable categories
Consider the 2-category of locally presentable categories, cocontinuous functors, and natural transformations. I believe that this 2-category is 2-cocomplete in the sense of containing all small 2-...
2
votes
0
answers
130
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Explicit local expression for Bers embedding in genus 2
Let $\mathcal T_{g,n}$ be the Teichmüller space of genus g compact Riemann surfaces with $n$ marked points. According to Riemann, this is a complex manifold of complex dimension 3g-3+n.
Bers ...
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0
answers
202
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Projective schemes with a fixed hyperplane section
Let $H$ be a hyperplane in $\mathbb P^n$, and $X \subseteq H$ be a subscheme. Let $CX \subseteq \mathbb P^n$ be the cone on $X$ from a point $p \notin H$.
Let $Hilb_{CX}$ be the Hilbert scheme whose ...
13
votes
1
answer
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When taking the fixed points commutes with taking the orbits
Let $G$ and $H$ be groups, both acting on a set $X$ on the left, in such a way that the two actions commute. (Equivalently, let $G \times H$ act on $X$.)
The set $\text{Fix}_H(X)$ of $H$-fixed ...
8
votes
4
answers
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Uniform convergence of Birkhoff averages and unique ergodicity
I am looking for a proof or a reference for the following two facts (which appear proofless in my notes from an ergodic theory course- they might be easy but i am no expert in ET):
Let $T$ be a ...
2
votes
1
answer
225
views
Is this Graph parameter known?
Let $\lambda(G)$ denote the edge-connectivity of $G$.
Consider the following parameter:
$\rho(G) = \max_{X \subset V(G)} \min(\lambda(G[X]), \lambda(G[V(G) - X]))$
Has this parameter been studied? ...
3
votes
1
answer
151
views
Definability of orderings on a formally real number field
For vector basis $b_1,..,b_n$ on a finite extension $F$ of $\mathbb{Q}$, where $-1$ is not a sum of squares, each linear order on $F$ is determined by an order on the basis. This uses information ...
5
votes
3
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542
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Who first used the cross-ratio to describe shapes in hyperbolic geometry?
I was reading this Wikipedia article today:https://en.wikipedia.org/wiki/Shape#Similarity_classes
and I realized that it strongly resembles the use of coss-ratios as "shape parameters" in hyperbolic ...
10
votes
1
answer
519
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Who first identified the universal $C^*$-algebra generated by an idempotent of norm at most $C$?
So much is known about hermitian and non-hermitian idempotents in a $C^*$-algebra, that someone must have written down the following.
Theorem The universal $C^*$-algebra generated by one element $x$...
1
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1
answer
204
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Number of solutions of a system of equation!
Let $\Theta =(\theta_1,\ldots, \theta_n)\in {\mathbb T}^n$. I want to show that the system of equations
$$
\sum_j 2\sin(\theta_i -\theta_j)+\sin(2\theta_i -2\theta_j) =0,\ \ i=1,\ldots, n,
$$
has ...
1
vote
1
answer
182
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If $u \in W^1(0,T;L^2,H^1)$ and $\varphi \in C^1([0,T]\times \Omega)$ then $\varphi u \in W^1(0,T;L^2,H^1)$?
Let $\Omega \subset \mathbb{R}^n$ be an open bounded domain.
Define $$W^1 := W^1(0,T;L^2,H^1) := \{w \in L^2(0,T;H^1(\Omega)) \mid w' \in L^2(0,T;H^{-1}(\Omega))\}$$
where $w'$ means the weak ...