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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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 ...
user12806's user avatar
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5 votes
1 answer
201 views

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. ...
jkn's user avatar
  • 183
16 votes
0 answers
2k views

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? ...
alvarezpaiva's user avatar
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5 votes
3 answers
498 views

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 ...
A.G.'s user avatar
  • 151
3 votes
1 answer
476 views

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 $\xi_1$ ...
Prasit's user avatar
  • 2,013
8 votes
2 answers
793 views

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 ...
Subhajit Jana's user avatar
6 votes
2 answers
316 views

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 ...
István Kovács's user avatar
11 votes
3 answers
2k views

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 ...
UwF's user avatar
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2 votes
0 answers
285 views

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 ...
Sasha Pavlov's user avatar
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0 votes
1 answer
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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" ...
Denis's user avatar
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0 votes
1 answer
242 views

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 ...
joro's user avatar
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5 votes
1 answer
931 views

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+\...
Alexey Ustinov's user avatar
8 votes
1 answer
1k views

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 ...
Peter Crooks's user avatar
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6 votes
2 answers
429 views

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 ...
Ali Taghavi's user avatar
-3 votes
1 answer
327 views

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 ...
Brian Rushton's user avatar
2 votes
0 answers
147 views

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:...
Manfred Weis's user avatar
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2 votes
1 answer
161 views

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....
Tao Feng's user avatar
4 votes
0 answers
220 views

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 ...
Alexander Shamov's user avatar
1 vote
0 answers
558 views

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 ...
PAT's user avatar
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1 vote
2 answers
536 views

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 $$\...
Stanley Yao Xiao's user avatar
5 votes
1 answer
483 views

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 ...
Peter Crooks's user avatar
  • 4,870
73 votes
17 answers
9k views

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 views

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. ...
Mostafa Mirabi's user avatar
12 votes
3 answers
400 views

(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 ...
Stanley Yao Xiao's user avatar
1 vote
0 answers
122 views

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 $...
lost1's user avatar
  • 373
7 votes
0 answers
427 views

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 ...
Tina's user avatar
  • 71
0 votes
1 answer
201 views

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 ...
Stanley Yao Xiao's user avatar
7 votes
3 answers
2k views

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 ...
SBF's user avatar
  • 1,605
2 votes
0 answers
259 views

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 ...
topo's user avatar
  • 31
3 votes
1 answer
1k views

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&...
WythoffGame's user avatar
18 votes
2 answers
4k views

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 ...
Boyu Zhang's user avatar
7 votes
2 answers
1k views

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 $\...
Jino's user avatar
  • 699
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 ...
Ben Golub's user avatar
  • 1,058
9 votes
1 answer
288 views

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 ...
David Roberts's user avatar
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7 votes
2 answers
1k views

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(...
Changyu Guo's user avatar
  • 1,861
12 votes
0 answers
208 views

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 ...
Beni Bogosel's user avatar
  • 2,102
3 votes
1 answer
342 views

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 ...
Rob's user avatar
  • 123
13 votes
2 answers
2k views

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., ...
Italo's user avatar
  • 1,697
9 votes
3 answers
736 views

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?
Jim Stasheff's user avatar
  • 3,840
9 votes
1 answer
952 views

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-...
Theo Johnson-Freyd's user avatar
2 votes
0 answers
130 views

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 ...
Lucien's user avatar
  • 828
3 votes
0 answers
202 views

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 ...
Allen Knutson's user avatar
13 votes
1 answer
1k views

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 ...
Tom Leinster's user avatar
  • 27.2k
8 votes
4 answers
2k views

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 ...
user44316's user avatar
  • 175
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? ...
hbm's user avatar
  • 1,034
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 ...
Colin McLarty's user avatar
5 votes
3 answers
542 views

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 ...
Brian Rushton's user avatar
10 votes
1 answer
519 views

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 $x$...
Terry Loring's user avatar
  • 1,749
1 vote
1 answer
204 views

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 ...
Mohammad Khosravi's user avatar
1 vote
1 answer
182 views

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 ...
soup's user avatar
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