This tag is used if a reference is needed in a paper or textbook on a specific result.

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7
votes
2answers
78 views

Reference for a strong intermediate value theorem for measures

Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states ...
1
vote
1answer
95 views

Nuclearity noncommutative torus

I read that the Noncommutative torus (rotation algebra) is nuclear when $\theta\in\mathbb{R}\setminus\mathbb{Q}$. Unfortunately, I haven't found a proof. Could someone give me a reference and/or an ...
0
votes
1answer
67 views

Stability conditions of coherent sheaves on abelian 3-folds

My work for now consists on understanding stability conditions of coherent sheaves on abelian 3-folds. I have the book by D. Huybrechts (the geometry of moduli spaces of sheaves), But I would like to ...
0
votes
0answers
99 views

A question about a subset in R^n homeomorphic to an open subset [on hold]

Let A be a subset of n-dimensional Eucliean space R^n, A is homeomorphic to an open subset of R^n. Then whether A is also an open subset of R^n? Is it a theorem in somewhere? Thank you very much.
3
votes
1answer
136 views

Determinant of the oriented adjacency matrix of a tree

Let $(V,E)$ be a finite oriented directed graph, with vertices and edges ordered, and $M$ the $|V|\times |E|$ matrix with entries $$ m_{ve} = \begin{cases} 1 &\text{if $e$ points at $v$}\\ -1 ...
2
votes
3answers
520 views

Koebe–Andreev–Thurston theorem - where can I find a proof?

Koebe–Andreev–Thurston theorem (known also as the circle packing theorem) says that any planar graph can be realized by a set of (interior-) disjoint disks corresponding to vertices, such that two ...
2
votes
0answers
57 views

What is the computational complexity to compute the integral numerically?

Given $$\int_{\Delta}\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}$$ where $P_i$ is polynomial(that is $P_1(x_1,x_2,\dots,x_n), P_2(x_1,x_2,\dots,x_n)$ are polynomial) whose coefficients are ...
2
votes
1answer
111 views

A perfect $(n,k)$ shuffle function

Suppose you have a deck of $n$ cards; e.g., $n{=}12$: $$ (1,2,3,4,5,6,7,8,9,10,11,12) \;. $$ Cut the deck into $k$ equal-sized pieces, where $k|n$; e.g., for $k{=}4$, the $12$ cards are partitioned ...
4
votes
0answers
72 views

$A_\infty$ structure on sum of twists of structure sheaf

Fix $n$ and let $P^n$ be projective $n$-space. Let $S = k[x_0, \dots, x_n]$. Set $A^0 = \bigoplus_{d \ge 0} H^0(P^n, \mathcal{O}(d))$ and $A^n = \bigoplus_{d < -n} H^n(P^n, \mathcal{O}(d))$. I ...
4
votes
2answers
177 views

Powers of finite simple groups [duplicate]

I have heard about the following result: for each finite simple non-abelian group $S$ and each natural number $r\ge 2$ there exists a number $n=n(r,S)$ such that the power $S^n$ is $r$-generator but ...
1
vote
2answers
245 views

Notion of manifold curvature?

Consider a particular embedding of a $C^2$ manifold $\mathcal{M}\subseteq\mathbb{R}^m$. Given $p\in\mathcal{M}$, suppose $\epsilon>0$ is small enough that the portion $U$ of $\mathcal{M}$ which is ...
2
votes
1answer
55 views

Calculating the “Belvedere Hull” of a Simple Planar Polygon

As an informal motivation the problem, imagine a tower with polygonal footprint, that is located in a beautiful landscape, the "Belvedere Hull" is then related to the directions, in which one would ...
3
votes
1answer
107 views

Systems of equations in Boolean Algebra

I have to study systems of equations in a Boolean algebra, the matrix is $m\times n$ with $m\neq n$. The Boolean algebra is actually the simplest one, it contains only $0$ and $1$, let us denote it by ...
10
votes
1answer
661 views

Von Neumann's consistency proof

In the paper Zur Hilbertschen Beweistheorie, John Von Neumann has proposed a consistency proof for a fragment of first-order arithmetic (the fragment without induction and with the successor axioms ...
1
vote
1answer
81 views

Piercing of subspaces in a projective space?

The "piercing subspace" problem may be stated as follows: There are given several subspaces in a projective space, rather non-intersecting. Find an additional subspace of a prescribed dimension that ...
4
votes
1answer
199 views

Picard of the product of two curves

Can anyone point to me where I can find the proof that the Picard group of the product of two curves is isomorphic to the product of the Picard groups times the hom among the Jacobians? Does the ...
8
votes
1answer
166 views

amenable + without $BS(m,n)$+finite $K(G,1)$implies virtually cyclic?

I heard from someone that the following problem is an open question. (Open Problem 1)For a countable discrete group $G$, suppose it does not contain any Baumslag-Solitar subgroups $BS(m,n):=\langle ...
2
votes
1answer
265 views

fixed point and homotopy fixed points

Let $G$ be a group and $X$ be a $G$-space (finite G-CW-complexe when needed). Let $p$ a prime number and $G= \mathbf{Z}/p\mathbf{Z}$, If I'm not wrong Miller-Lannes,... theory provides tools and ...
5
votes
1answer
164 views

What does an endomorphism in a triangulated category give rise to?

Let $D\xrightarrow[]\varphi D\xrightarrow[]kE\xrightarrow[]j\Sigma D$ be an exact triangle in a triangulated category. I am trying to figure out what structure emerges from this on the base of the ...
7
votes
2answers
244 views

Closure of the orbits of the $SL(2,\mathbb{Z})$-action on $\mathbb{R}^2$

I'm coming with a very basic question for which I can't find an answer. Please forgive me if I didn't search efficiently enough. What can the closure of an orbit of an element $X$ of $\mathbb{R}^2$ ...
5
votes
1answer
288 views

Representation theory of the general linear group over a finite prime field

I am re-posting a question I asked on math.se here because I am unsatisfied with the answers I obtained. The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely ...
11
votes
0answers
236 views

References for $K_{4k}(\mathbb{Z})$

Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be ...
7
votes
2answers
555 views

Every free abelian group is slender, why?

Wikipedia states that every free abelian group is slender. Where can I find a proof? If this is not trivial, then I will also need a reference to use in my paper.
1
vote
1answer
125 views

Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite area such that

Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite areas such that for each simply finite area there exist a diffeomorphic map that ...
0
votes
0answers
48 views

Convex optimization solver taking oracles as objective function [closed]

Is there any solver for convex optimization in C++ (or some dedicated scheme while no solver is yet available) that could solve a convex optimization problem with objective function value given by an ...
9
votes
1answer
336 views

Between compact and locally uniform: What is the name of this convergence?

Let $X$ be a topological space, $(Y,d)$ a metric space, $f\in Y^X$, and $(f_n)$ a sequence in $Y^X$ with the following property: For every $x_0\in X$ and every $\varepsilon>0$, there exist a ...
8
votes
2answers
258 views

What is the BRST-anti-BRST formalism?

What is the BRST-anti-BRST formalism? Is the Sp(2) doublet the ghost, antighost pair? Introductory accounts of this subject seem to be hard to find. I would appreciate a reference for someone who ...
2
votes
0answers
161 views

Diophantine equations over cyclotomic fields

Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of ...
7
votes
7answers
536 views

What is the best reference for Spectral theory?

I'm studying Bernard Aupetit: A Primer on Spectral Theory but the textbook we are using is a little bit heavy going for me. Is there a best book to learn about these things? Thank you.
3
votes
0answers
76 views

Oscillatory integral moments of $L(\frac{1}{2} + it, f \times f)$

Understanding moments and subconvexity bounds for $L$-functions is a big topic with a lot of activity. I'm currently looking at a related problem, bounding $$ \int_0^T L\left(\tfrac{1}{2} + it, f ...
10
votes
1answer
200 views

Subquotients in the Verma filtration on Verma modules

Let $\lambda$ be a dominant integral weight of $\mathfrak g$, a finite-dimensional reductive Lie algebra over $\mathbb C$. Let $M(w\cdot \lambda)$ denote the Verma module with high weight $w\cdot ...
1
vote
0answers
42 views

Name for generalization of bivariate weighted-homogeneous polynomials

A polynomial $f = \sum_j c_j X^{\alpha_j}Y^{\beta_j}\in\mathbb K[X,Y]$ is said weighted-homogeneous if there exist $p$, $q$ and $d$ (where $p$ and $q$ are not both $0$) such that ...
2
votes
1answer
100 views

A good reference for uniformization theorem for compact and non-compact Riemann surface

I am looking for a good reference for the uniformization theorem for Riemann surfaces, which states that each simply connected Riemannian surface is conformally equivalent to the complex plane ...
6
votes
1answer
275 views

Applications of the Small and Great Theorems of Picard

I just presented the two famous theorems of Picard (Small and Great) in a graduate course, but I have not managed to discover a good number of interesting applications. List of applications (rather ...
4
votes
1answer
77 views

Orthogonal polynomial under linear transformation

Let $M_n(x) = x^n$ be the standard monomials. The binomial formula allows one to expand $M_n(ax+b)$ as a linear combination of $M_k(x)$, for $k \leq n$, giving $$ M_n(ax+b) = (ax+b)^n = \sum_{k=0}^n ...
1
vote
0answers
42 views

request for any expository works in pointwise convergence of double Fourier series and especially a paper by Hardy

Quart. J. Math. Volume 37, Issue 1, Pages 53-79 On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters. Hardy, G.H. I am not ...
5
votes
1answer
179 views

Book about the history of mathematics for weather prediction

Can someone recommend a book about the history of mathematics being used for weather prediction, preferable one which covers recent developments?
1
vote
2answers
127 views

Speed and absence of non-constant bounded harmonic functions

For a (symmetric) random walks on countable groups generated by $\mu$, there is a "brute-force computation" argument of Avez (1974) that shows that if the entropy $h_\mu$ is trivial then there are no ...
5
votes
1answer
64 views

Is nfcp equivalent to stable + eliminates $\exists^\infty$?

Let $T$ be a complete first-order theory. Recall that a formula $\phi(\overline{x},\overline{y})$ has the finite cover property (fcp) if for all $n$, there exist $\overline{a}_1,\dots,\overline{a}_n$ ...
3
votes
0answers
95 views

torsion free for the 2nd cohomology group?

Let $G$ denotes an infinite coutable discrete group with Kazhdan's property (T), My question is: is it known that the 2nd cohomology group $H^2(G,\mathbb{Z}G)$ is torsion free? Thanks in advance! ...
2
votes
1answer
116 views

For compact complex surfaces $h^{1,0}$ is either $h^{0,1}$ or $h^{0,1} - 1$. Do we need to use the Enriques-Kodaira classification?

In the Wikipedia article on the Enriques-Kodaira classification, before the classification itself, the following sentence appears: For compact complex surfaces $h^{1,0}$ is either $h^{0,1}$ or ...
9
votes
3answers
183 views

Reference request: Systems of linear PDES with constant coefficients

I am looking for a reference for the following statement: Assume that $P_1, \dots, P_k \in \mathbb R[x_1, \dots, x_m]$ and consider a system of PDEs \begin{align} P_i(\partial / \partial x_1, \dots, ...
1
vote
0answers
107 views

Noncommutativization of fixed point theory

What papers or references have been devoted for a noncommutativization of "Fixed point theory". Here the terminology Noncommutativiztion, as usual, indicates to that famous table with 2 columns: ...
5
votes
1answer
160 views

Uniformization of a plane minus cantor set

Let $\mathbb{D}$ be the unit disk endowed with the Poincaré metric and $G$ be a Fuchsian group such that the hyperbolic surface $\mathbb{D}/G$ is homeomorphic to the plane minus a Cantor set. ...
4
votes
1answer
115 views

Dependence of solutions on parameters in partial differential equations

In the standard homogenization problem $$-\nabla.\left(A\left(x,\frac{x}{\epsilon}\right)\nabla u^{\epsilon}(x)\right)=f\ \mbox{in } \Omega,$$ the homogenized matrix $A_0$ is given in terms of ...
5
votes
1answer
350 views

Harmonic Crystal using Random Walk

Me and my advisor were looking for a specific proof of the disorder in $2d$ harmonic crystals. We could not find a paper or a textbook with it, so I thought trying my luck here. Basically, it is a ...
2
votes
0answers
112 views

Weyl group of a symmetric space

Let $G/K$ be a symmetric space of a non-compact type, i.e. $G$ is a semi-simple connected Lie group, and $K$ is its maximal compact subgroup. Helgason in his book "Differential geometry and symmetric ...
1
vote
0answers
98 views

How does perturbation method guarantee its solution for the perturbed pde $\Delta u + \epsilon u^2 =0$

[This may not be a research-level question; if it violates any term of this website, I will delete it right away] My question is quite simple: Suppose we are given a PDE of with a boundary condition ...
3
votes
1answer
88 views

Proper Model Category

Let R be a commuative ring. Consider the category of simplicial R-modules with the projective model stucture. Can someone give me a precise reference which proves that this model category is proper? ...
0
votes
1answer
25 views

Sufficient and necessary condition for BIBO stability

I am looking for a reference for the proof of the next claim: "BIBO—bounded input bounded output—stability. We claim that a necessary and sufficient condition for a system described by a linear, ...