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

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69 views

A family of rank two bundles on $\mathbb CP^1$ parameterized by $H^1(O(-2n))$

Let $n$ be a positive integer. Consider the family of extensions $$0\to O(-n)\to E\to O(n)\to 0,$$ parameterized by $H^1(O(-2n))$. For each element $e\in H^1(O(-2n))$ we get a rank two bundle $E$ ...
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0answers
28 views

What are the central points of a semi-nice region in the plane?

For a convex set in Euclidean space, there is an obvious notion of its center: namely, its center of mass, which by convexity lies in the set. For a nonconvex set there is just as obviously no nice ...
3
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2answers
122 views

Hausdorff dimension of sequence space

Let $\Omega =\{0,1\}^{\mathbb{N}}$ denote the set of infinite sequences with elements $0$ or $1$. Let $d$ be the metric on $\Omega$ given by $d((x_n),(y_n))=1/2^m$, where ...
8
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2answers
295 views

$l^1$ versus $l^2$

Is there an elementary proof of this Banach space fact? If the Banach space $V$ is linearly isomorphic to $l^1$, then it does not isometrically contain euclidean spaces of arbitrarily large finite ...
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0answers
38 views

Influence of independent variables on boolean functions?

Suppose a simple connected graph $G$ where its vertices are assumed to be independent. An event with uncertainty corresponds to each vertex. My instructor guides me that even though the vertices ...
4
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1answer
202 views

Finite group action on quasi-projective varieties

Let $X$ be a smooth, quasi-projective variety, $G$ be a finite group which acts freely and properly on $X$. Denote by $\alpha:X \to X/G$ the quotient. Is $\alpha$ generically etale? Also, as I am ...
9
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0answers
250 views

A conjecture of Blakley and Dixon about odd powers of positive matrices

In a 1966 paper Blakley and Dixon conjecture the following. Let $S$ be a symmetric matrix with nonnegative entries and let $u$ be a unit vector with nonnegative entries. For integers $k\ge j$ both ...
4
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1answer
146 views

Formula relating the cup product in dimensions n and n+1

Let's will write $K_n$ for the Eilenberg-MacLane space $K(\mathbb{Z},n)$. I remind that $K_n$ is equivalent to the loop space of $K_{n+1}$. Let’s consider the map $\smallsmile:K_n\times K_m \to ...
5
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0answers
111 views

$p$-adic uniformisation of abelian varieties

In the paper $p$-adic L-functions and $p$-adic periods of modular forms of Greenberg and Stevens $\S3$ page $420$ they make the following statement: Let $A$ over $\mathbf{Q}_p$ be an abelian variety ...
3
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0answers
83 views

Reference for Superelliptic Curves

A curve is called superelliptic if $y^n = (x-\alpha_1)^{d_1}...(x-\alpha_s)^{d_s}$ where $n \ge 2$ and $d_i > 0$. From googling around, I found several papers which talk about these curves and ...
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0answers
76 views

applications of ergodic theory to periodicity of regular continued-fractions

The usual application one sees of ergodic theory to the regular continued-fractions is the Gauss-Kuzmin Theorem on the frequency of positive integers in the continued fraction expansion for almost all ...
6
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3answers
309 views

Connection between solution for Schrödinger equation and solution for heat equation

It's known, that if you write imaginary unit into a heat equation you'll get time-dependent Schrödinger equation. Recently one guy discovered a connection between solutions for these two equations ...
24
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2answers
821 views

What is the relation between sphere spectrum and supersymmetry?

In this this google+ post of Urs Schreiber, he says: "Grading over the sphere spectrum is supersymmetry" and then he redirect us to the abstract idea of superalgebra (in nLab). Are there some ...
2
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1answer
133 views

Differential equations → predicate logic mapping

I was watching Tom LaGatta's talk on category theory recently, and one of the audience brought up a statement that caught my attention (around the 56:15 mark): I was gonna say, there was a book I ...
1
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1answer
47 views

Unbounded convex domains in 2D

Let $\gamma$ be a smooth planar curve. Assume that $\gamma$ divides the plane into two domains and, it addition, that one of these domains is unbounded and convex. What can be said about the behavior ...
4
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0answers
47 views

Reference request: local cohomology in disjoint union

I have a topological space $X$ and two disjoint, closed subspaces $Y$ and $Z$ of $X$. I believe that in this situation, for any abelian sheaf $\mathcal{F}$ on $X$ and any $p \in \mathbb{N}$, there is ...
3
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1answer
167 views

Is the number of intervals you get by slicing the closed region under a nice graph continuous from below?

Let $I$ be a closed interval in $\mathbb{R}$, and let $f:I \to \mathbb{R}$ be a bounded function, smooth except at finitely many points (piece-wise smooth). There are also only finitely many critical ...
3
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1answer
113 views

How to embed $S^2\mathbb{C}^2$ into $S^2S^3\mathbb{C}^2$ and get the ideal of the twisted cubic?

Let $X:=x^3$, $Y:=x^2y$, $Z:=xy^2$ and $W:=y^3$ be the 4 independent generators of $S^3\mathbb{C}^2$, and observe that the kernel of the natural epimorphism (total symmetrisation) $$ ...
4
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1answer
117 views

how to define the injectivity radius of manifolds with boundary?

For manifolds without boundary one defines the injectivity radius as the maximal radius where the exponential map is a diffeomorphism. One can then show that the injectivity radius is the maximum ...
1
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1answer
63 views

Asymptotic for zeroes of $L(s,\chi)$ in a disk $|s|<R$

In 'Remarks on Weil's quadratic functional..' p.191 Bombieri claims any given $L$-function $L(s,\chi)$ has at least $$\big(\frac{1}{\pi}+o(1)\big)R\log R$$ zeroes in a disk $|s|<R$. Is there a ...
3
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1answer
137 views

Schubert varieties and Young diagrams

In his book Young Tableaux, Fulton asserts, in Exercise 9.4.18 on p. 152, that the Schubert variety $\Omega_{\lambda}$ is defined by the conditions $\text{dim}(V \cap F_{n+i- \lambda_{i}}) \geq i$ for ...
6
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1answer
189 views

Theorem of Bukovsky characterizing ground models

It was mentioned in a talk that Bukovsky proved the following are equivalent for inner models $M \subseteq V$: (1) There is a partial order $\mathbb P \in M$ and a $\mathbb P$-generic filter $G \in ...
3
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0answers
50 views

Transverse intersection in the $G$-orbit of paths

I know how to prove the following lemma but I assume that it is well-known. Can someone provide a reference for it? Let $d>1$ and let $M$ be a $d$-dimensional connected smooth manifold with a ...
4
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1answer
94 views

The volume around a singular isolated root when equalities are loosened

Suppose I have a system of polynomial equations in $n$ real variables $f_i(x_1,\ldots,x_n)=0$, $i=1,\ldots,m$, such that $0$ is an isolated solution. Now I replace each of the equations with a ...
3
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1answer
82 views

Symmetric matrix formula for Gaus-Legendre quadrature

While searching the web, I came across the following algorithm for the Gaus-Legendre quadrature. I wasn't able to find a reference or a proof of my own as to why it works. I'll present it, and the ...
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0answers
89 views

Why are prime spirals such a big deal? [closed]

So, I am well aware of the type of interest that goes into prime spirals, at least at the Layman level. However, I am really curious why the interest exists. See, if we look at a natural spiral, like ...
10
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1answer
579 views

Foundations of topology

I recently went to a talk of Oleg Viro where he expressed his dissatisfaction with current foundations of differential topology parallel to what has been discussed here. Also some time ago I read ...
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0answers
74 views

Reference request: "effective'' semistable reduction

I am looking for the origin of the following idea: suppose $m$ and $n$ are relatively prime integers $\geq 3$. Let $E$ be an elliptic curve over a number field $K$. Let $L/K$ be a finite extension ...
14
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0answers
375 views

Chasing a 1950s thesis from the University of Dhaka on block designs

On behalf of a friend I am searching for a thesis on block designs from the 1950s. The details are below. Author: Qazi Motahar Husein (Sometimes Husain or Hussein). Title of the Thesis: Symmetrical ...
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0answers
41 views

Literature on transformed Gaussian matrices

I am considering real $n$-by-$m$ matrices of the following type: $$ M=SM^\prime,\\ M^\prime_{ij}\sim^{iid}N(0,1). $$ Here, $S$ is a fixed $n$-by-$n$ matrix and the entries of $M^\prime$ (same size ...
5
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1answer
133 views

Reference for affine Grassmanian

Could someone please provide a precise reference in the literature where the following well-known fact is proved. Also if someone could write out the proof that would be great. ...
4
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2answers
117 views

An English version Borok's work on finite-infinite systems of ordinary differential equations

I am looking for the English translation of the paper by V. M. Borok (originally in Russian) The Cauchy problem for finite-infinite systems of linear differential equations. This work is about the ...
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2answers
401 views

Reference for Manin's idea on algebraic geometry over the symmetric monoidal model category of Motives

Reference for Y. Manin's idea of "algebraic geometry over the symmetric monoidal model category of motives." Has been sugested to me that this was made in a Manin's letter. There is an escaned copy? ...
4
votes
2answers
119 views

Classification of symmetries of tilings in surfaces?

Is there a general study of the symmetries of tilings on surfaces? Conway, Goodman-Strauss & Burgiel classified them on $\mathbb S^2, \mathbb R^2$ and $\mathbb H^2$, with their 'Magic Theorem'. ...
2
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1answer
145 views

An electronic copy of Vishik's work on $p$-adic $L$-functions for modular forms

This question is very simple. Would someone be so nice as to send me an electronic copy of M. M. Vishik, Non-Archimedean measures connected with Dirichlet series, Mat. Sb. (N.S.), 1976, Volume ...
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0answers
24 views

Fubini's Theorem for Lévy bases

Let $M$ be an infinitely divisible independently scattered and homogeneous random measure on $\mathbb R^d$ (ie a homogeneous Lévy basis). Let $\nu$ be a sigma finite measure on $\mathbb R^k$. Let ...
9
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1answer
165 views

Homotopy property of constructible sheaves on stratified spaces

Let $X$ be a stratified topological space (in my case $X$ is a compact space presented as a finite union of locally closed topological manifolds of finite dimension (strata) such that the closure of ...
4
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0answers
44 views

Looking for a Collection of Examples and Counter Examples for Assumptions about the Properties of Planar Euclidean TSP Instances?

Where can I find example and counter examples to seemingly plausible assumption about the properties of optimal solutions of planar euclidean TSP instances? The reason for asking is that the ...
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2answers
274 views

Tableaux with limited rows and complementary skew shapes

Given a partition $\mu=(\mu_1,\mu_2...,\mu_d)$, define $\bar\mu=(\mu_1-\mu_d,\mu_1-\mu_{d-1},...,\mu_1-\mu_2,0)$, the complementary shape in the $d\times \mu_1$ rectangle. Then the number of skew ...
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0answers
43 views

Multilinear Interpolation

Suppose I have a multilinear map $\Gamma(u,v)$ satisfying \begin{align} \big\| \Gamma(u,v)\big\|_{L^2} &\leq \big\| u\big\|_{L^2} \big\| v\big\|_{L^2} \\ \big\| \Gamma(u,v)\big\|_{L^\infty} ...
5
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2answers
146 views

Is $\mathbb{P}T^*M$ a sub-Riemannian manifold if $M$ is Riemannian?

(this question is about a particular aspect of a previous question, which was not duly stressed) Let $(M,g)$ a Riemannian $n$-dimensional manifold, and let $$ \widetilde{M}:=\mathbb{P}T^*M $$ be the ...
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0answers
15 views

Approximation error estimate

I would like to find a good reference for the following or a similar, probably well-known, approximation error result: Let $\Omega\subset \mathbb{R}^d$ be bounded, $p\in [1,\infty]$, $l, m\in ...
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1answer
529 views

Resolution of singularities in étale cohomology

The lack of a suitable resolution of singularities comes up often in work on étale cohomology from the 1960s and 70s, And I think even the latest version of Milne's lecture notes says "It is likely ...
2
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0answers
48 views

Action of orthogonal group on the free Lie algebra

This question is somewhat related and inspired by this post of professor Montgomery. The free Lie algebra $L(V)$ generated by an $r$-dimensional vector space $V$ is, in the language of ...
2
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1answer
146 views

Applications of Level Lowering

What are some applications/consequences of level lowering of Galois representations? I understand the application of Ribet's theorem in the proof of Fermat's last theorem but I am wondering what other ...
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2answers
223 views

Moment problem on [-1,1]: necessary and sufficient conditions

Consider a sequence of real numbers $s=(s_0,s_1,\ldots)$. When is there a Borel measure $\mu$ supported on $[-1,1]$ so that $$ s_k = \int_{[-1,1]} x^k\,\mathrm{d}\mu,\quad \forall k\in\mathbb N\;? $$ ...
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2answers
371 views

Famous results about the value of a given limit assuming it exists

Chebyshev got famous showing that if the limit $l:=\lim_{x\to\infty}\frac{\pi(x)}{x/\log x}$ exists, then necessarily $l=1$, constituting a major breakthrough towards a proof of the famous prime ...
13
votes
1answer
314 views

Homotopy fiber of a map between classifying spaces

I'm looking for a reference (and precise hypothesis if more are needed) for the following facts (or a correction, if I'm just plain wrong): Let $G$ and $H$ be topological groups and $f : G \to H$ be ...
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0answers
73 views

Reference request: Structure of $H^1({{\mathbf{Q}}_{q}},{{{{E}}_{{p}^{\infty}}}})$

I need reference on the structure of ${H}^{1}({{\mathbf{Q}}_{q}},{{E}_{{{p}^{\infty}}}})$, in particular when: (1.) $q=p$ and/or (2.) $E$ has multiplicative reduction at $q$. Here, $E$ is an ...
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0answers
112 views

Reference for supergroupoids in supersymmetry?

I would like to know some references on supergroupoids in supersymmetry. A supersymmetry is invariance under a supergroup action (nLab, Supersymmetry). It is know that groupoids provide a local ...