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

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25 views

Orbifold metric and ample line bundle

Let $X$ be a complex projective variety with only orbifold singularities, and $L$ be an ample line bundle on $X$. Is there a K\"{a}hler metric in the orbifold sense representing the first Chern class ...
2
votes
1answer
41 views

Looking for a theorem that says that the embedding $H^{1-\sigma}(M)\subset C^1(M)$ is compact for $\sigma\in (0,1)$

I am Looking for a theorem that says that the embedding $H^{1-\sigma}(M)\subset C^1(M)$ is compact for $\sigma \in (0,1)$, where $M$ is a compact manifold. Any references are appreciated. PS I am ...
5
votes
1answer
317 views

Is $\mathcal M _{g,n}$ anabelian?

Are the moduli spaces $\mathcal M _{g,n}$ expected to be anabelian? Is there anything known in that direction? Thank you very much for your help in any case!
4
votes
1answer
122 views

Centralizers of reflections in special subgroups of Coxeter groups

Let us consider a (not necessarily finite) Coxeter group $W$ generated by a finite set of involutions $S=\{s_1,...,s_n\}$ subject (as usual) to the relations $(s_is_j)^{m_{i,j}}$ with ...
4
votes
1answer
207 views

Ivanov's metaconjecture on surface homeomorphisms.

In Fifteen problems about MCG Ivanov stated the following metaconjecture: Every object naturally associated to a surface S and having a sufficiently rich structure has $Mod(S)$ as its groups of ...
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0answers
28 views

Riemannian simplicial complex and quasi-conformal complex

In this paper by Robert Young, the author defines We define a riemannian simplicial complex to be a simplicial complex with a metric which gives each simplex the structure of a riemannian ...
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1answer
117 views

Vector Bundles of small rank

I recently started the study of vector bundles on $\mathbb{P}^n$, and started to read Rao's article 'A family of vector bundles on $\mathbb{P}^3$'. There, there is a notion of spectrum of a vector ...
2
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1answer
48 views

Reference for multivariate orthogonal polynomials

I want to learn about multivariate orthogonal polynomials. Is there a good textbook/survey that you could suggest? I need to see common examples like Jack's polynomials etc .. and also general ...
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1answer
231 views
+50

Morgan Shalen compactification of $\mathbb C^2$

I'm reading the Otal's survey on the compactification of Morgan Shalen. (available here) He claims in an example (page 8) that the compactification of $\mathbb C^2$ is $S^4$, which sounds completely ...
2
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1answer
116 views

Frey's Formula and utilisation of the Hasse Invariant in “Links between Stable elliptic curves and Diophantine equations.”

In the paper "Links between Stable elliptic curves and Diophantine equations" for an elliptic curve $E$ with normal Weierstrass form $$y^2 = x^3 -g_2x -g_3$$ with $g_i \in \mathbb{Z}$ w.l.o.g. Then ...
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0answers
31 views

Tail inequality for orthomartingales/martingale difference random fields

It is known that if $(S_i= \sum_{j \leqslant i }X_i, \mathcal F_i)$ is a martingale, then for each $ \beta>1$, $\delta\in (0,\beta-1)$ and $\lambda>0$, and each integer $N \geqslant 1$, the ...
5
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1answer
131 views

Is the Poincaré metric continuous with respect to the domain?

Suppose $K \subset \mathbb{C}$ is a Cantor set and let $u:\mathbb{C} \setminus K \to \mathbb{R}$ be the maximal smooth function such that the conformal metric $e^{2u}(\mathrm{d}x^2 + \mathrm{d}y^2)$ ...
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1answer
100 views

global well posedness of cubic NLS in for initial data in $H^{s}(\mathbb R), 0<s<1$

We consider the one dimensional cubic nonlinear Shr\"odinger equation (NLS): $$i\partial_{t}\phi (x,t) +\Delta \phi (x,t)= \pm |\phi (x,t)|^{2} \phi(x,t), \ (x, t\in \mathbb R),$$ $$\phi (x,0) = ...
5
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0answers
83 views

A weak Perron-Frobenius property for sets of positive matrices

A popular form of the Perron-Frobenius theorem states the following result: if $A$ is a $d \times d$ real matrix all of whose entries are positive, then the spectral radius of $A$ is a simple ...
0
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1answer
81 views

Samuel multiplicity

Let $X$ be the hyper-surface defined by $$f:=\sum_{i=1}^k x_i^n=0$$ in $\mathbb{C}^k$. Let $Y$ be the non-reduced sub-scheme of $X$ defined by the ideal $$I=(x_1^{n-1},\dots , x_k^{n-1}) $$ What is ...
3
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1answer
226 views

'Unitary' charts on odd-dimensional spheres

Consider the odd-dimensional sphere $S^{2n+1} \subset \mathbb{C}^{n+1}$. One may talk variously about its structure as a contact, CR or Einstein-Sasaki manifold, but I'm looking for some specific ...
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1answer
77 views

Reference for higher order Campanato Lemmas, e.g. `Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}'

Whence can I reference the following fact (I have seen it quoted as `standard' in respectable places, so I hope it is so)?: Let $f : B_2(0) \to \mathbb{R}$, say $f \in L^2(B_2(0))$ . Suppose that ...
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0answers
113 views

Dynamics of electrons on a sphere

Suppose one place $n$ electrons closely surrounding the north pole of a sphere, forming a perfect planar regular $n$-gon:           Q1. What will happen if the ...
7
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2answers
589 views

Stokes theorem with corners

I've found the following version of Stokes' theorem in G. Stolzenberg's lecture notes 19: Notation: for $1 \le n \le m$ $\Lambda(m, n) = \{ \lambda: \{1,...,n\} \rightarrow \{1,...,m\} \ | \ ...
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0answers
88 views

What kind of ringed space $X$ has the property that a locally free sheaf is projective in Qcoh$(X)$?

It is well known that for an affine scheme $X$, every finitely generated locally free sheaf $\mathcal{E}$ is projective in the category Qcoh$(X)$. i.e. the functor ...
2
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1answer
144 views

Is a locally free sheaf projective in the category of $\mathcal{O}_X$-modules when $X$ is an affine scheme?

Let $X$ be an affine scheme and $\mathcal{E}$ a finitely generated locally free sheaf on $X$. It is obvious that $\mathcal{E}$ is a projective object in the category Qcoh$(X)$ since we can pass to ...
7
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0answers
126 views

Decidabilty of the Hilbert lattice and quantum logic

What is known about the decidability of (first-order formulas in) the structure $(\mathcal{L}(H),\leq)$, where $\mathcal{L}(H)$ is the collection of all closed linear subspaces of a (separable) ...
2
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0answers
76 views

Computing Voronoi poles in $\mathbb{R}^d$ (the farthest points within each cell)

Say I have a Voronoi diagram of some points $p_1,\dots,p_n\in\mathbb{R}^d$, which tesselates $\mathbb{R}^d$ into cells $V_1,\dots,V_n$. Within each cell $V_i$, the pole is defined as the vertex of ...
2
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1answer
123 views

References on the Free Loop Space

I intend to approach the paper of Wolfgang Ziller: "The Free Loop Space of Globally Symmetric Spaces", but I need the proper background on the foundations of the study of Free Loop Spaces. I obtained ...
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4answers
450 views

Rounding errors in images of Julia sets

One typically computes Julia sets by iterating a complex function, such as a polynomial or rational function. How do rounding errors affect the results? I'm looking for references on this issue, ...
3
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0answers
122 views

Group completion and algebraic K-theory

In the paper "On the group completion of a simplicial monoid", Quillen discusses the applications of group completions to the definition of algebraic K-groups. But it's before his celebrated paper on ...
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0answers
54 views

Is any affine domain J-1?

Can someone please suggest me a reference for the fact(!) that any affine domain over a field $k$ is $J-1$, i.e., its regular locus is open (I hope the result holds even if $k$ is of finite ...
3
votes
1answer
251 views

Mixed Hodge structure and cup product

I'm looking for a reference for the answer to the following questions. Let $X$ be an algebraic variety over C. When is the cup product a morphism of Mixed Hodge structures? Does $X$ have to be ...
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0answers
76 views

power laws emerging from the sandpile model

Is there a rigorous proof that the abelian sandpile model generates a power law distribution of avalanche lengths?
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1answer
123 views

Upperbounding the number of regions induced by a set of unit disks

Given a set $D$ of $n$ same radius disks, embedded in the plane, their arrangement induces a number $k$ of connected regions in $\mathbb{R}^2 \setminus \cup_{d \in D}$ . I am interested in an upper ...
4
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1answer
305 views

Inaccessible cardinal and $\Sigma_1$ reflection

A theorem of A. Levy says that, if $\kappa$ is an inaccessible cardinal, then $V_\kappa\prec_{\Sigma_1}V$ namely $V_\kappa$ is an elementary submodel when considering only $\Sigma_1$ formulas. Where ...
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4answers
671 views

Consequences of the Inverse Galois Problem

Are there any papers written about the consequences of the Inverse Galois Problem in case it is proved to be true or false? We know a lot of things that would be true if the Riemann Hypothesis holds. ...
2
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1answer
134 views

Is there a characterization of Riemannian manifolds that split off two factors?

Some Riemannian manifolds are expressed as a product manifold. Recently, I have read two articles about space-times. In both articles, the authors prove that a Riemannian manifold $\bar{M}^n$ is ...
2
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1answer
84 views

Reference request: flat surfaces

When writing a paper, I feel like to point out exact references to the following seemly easy facts concerning flat structures on a closed surface $\Sigma$ with negative Euler characteristic: The ...
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0answers
47 views

Modules over iwasawa algebra

I was reading this paper by Prof. Michael Harris. In page 107 first paragraph he states that "The estimate for $d_i^{\prime}$ then follows from the theory of the Hilbert polynomial, as in the proof of ...
0
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1answer
178 views

Reference Request: Fundamental Group Scheme

I want to learn about the Fundamental Group Scheme(First introduced by Madhav Nori). I am familiar with Basic Algebraic Geometry at the level of Eisenbud & Harris'"Geometry of Schemes" & to a ...
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0answers
104 views

Intersection Multiplicity

Let $X$ be an hyper-surface in an affine space defined by an equation $F$. We can assume that the ground field is $\mathbb{C}$ and $X$ is normal. Take functions $f_1,\dots, f_n$ on $X$ and let $B$ ...
3
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1answer
160 views

Characterising subsets of the reals as ordered spaces

There are concise and elegant characterisations of the real line as a topological space and as an ordered space in the literature. I am interested in the harder case of characterising subsets of the ...
2
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1answer
102 views

Enumerating positive fractions (reference missing)

I remember that the recursion $r(0)=0, \ \ r(n+1)=\frac{1}{2 [r(n)]+1-r(n)}$ produces a sequence of rational values $ 0 \mapsto 1 \mapsto 1/2 \mapsto 2 \mapsto 1/3 \mapsto ... $ which exausts the ...
5
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1answer
356 views

Functor generalisation

In an article I am writing, I am led to the following generalization of the notion of functor. Let $C$ and $D$ and be two categories. A generalized functor $F : C \to D$ is given by: a function $f : ...
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2answers
271 views

Choice of fibrations is like a choice of a basis of a module

In some notes on derived stacks, in describing categories of fibrant objects, the author drops this parenthetical: (Grothendieck said in his famous letter to Quillen that the choice of $\mathscr ...
0
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1answer
75 views

Norm of matrix with randomly deleted entries

Let $A$ be an $n \times n$ matrix with real entries and let $B$ be the random matrix whose $(i,j)$ entry is $$B_{i,j}=v_{i,j}A_{i,j}$$ where the $v_{i,j}$ are i.i.d Bernoulli random variables with ...
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0answers
84 views

Convergence proof for fictitious play!

In "Fictitious play property for games with identical interests" by D. Monderer and L.S. Shapley, the convergence of fictitious play to a Nash equilibrium is proved for a potential game with players ...
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2answers
491 views

Scott-Solovay unpublished paper on ``Boolean valued models of set theory''

I have read some papers from 1970$^{th}$, and in some of them, the paper of Scott and Solovay on ``Boolean valued models of set theory'' is given as a main reference, with many references to the ...
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3answers
372 views

First Explicit Irreducible Representations

Although the classification of simple Lie Algebras and their representations is fully understood, I wonder whether there is some book with exhaustive tables describing explicit irreducible ...
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2answers
184 views

The action of the center on the extended Dynkin diagram

Let $R$ be an irreducible root system with a basis $\Pi$. We obtain the Dynkin diagram $D$ and the extended Dynkin diagram ${\widetilde{D}}$ of $R$ with respect to $\Pi$. Let $Q^\vee\subset P^\vee$ ...
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0answers
84 views

If $\angle 0xy\leq \pi/2$ for every $x\in \partial K$, $y\in K$, then $K$ is a ball

Let $K$ be a bounded closed convex set in $\mathbb{R}^d$, $0$ lies in interior of $K$. Assume that for every boundary point $x$ of $K$ the plane through $x$ perpendicular to $x$ is a support plane of ...
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2answers
332 views

More recently published comprehensive reference on inequalities in the spirit of Hardy-Littlewood-Pólya

Is there a comprehensive reference book on inequalities in the spirit of the one written by G.H. Hardy, J.E. Littlewood, and G. Pólya(*), but more up-to-date (i.e., published in more recent years and ...
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1answer
1k views

Philosophical arguments in defense (or against) large cardinals

The question is essentially what is asked in the title. I split it into two parts (A) (Arguments supporting the existence of large cardinals) What are the main philosophical arguments in defense ...
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80 views

Laumon's “Sur les modules de Krichever”

I'm looking for a copy of the preprint mentioned in the title. Thank you very much in advance for any help!