# All Questions

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### Almost everywhere differentiability for a class of functions on $\mathbb{R}^2$

A while ago, I came across the following problem, which I was not able to resolve one way or the other. Let $f,g\colon\mathbb{R}^2\to\mathbb{R}$ be continuous functions such that $f(t,x)$ and ...
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### How hard is it to make a differential operator Hermitian?

Let $M$ be a closed finite-dimensional smooth manifold (over $\mathbb R$). Let $C^\infty(M) = C^\infty(M,\mathbb C)$ be the algebra of smooth complex-valued functions on $M$, with the natural complex ...
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### Cycle of length 4 in an undirected graph

Can anyone give me a hint for an algorithm to find a simple cycle of length 4 (4 edges and 4 vertices that is) in an undirected graph, given as an adjacency list? It needs to use $O(v^3)$ operations ...
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### Universal property of blowups

Can anyone help me with a proof of the following claim (see for example the book Higher algebraic geometry of Olivier Debarre, proof of Proposition 1.43, page 31): Let X be a complex manifold, and ...
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### А generalization of Gromov's theorem on polynomial growth

I was sure it is known, but it appears to be an open problem (see the answer of Terry Tao). Assume for a group $G$ there is a polynomial $P$ such that given $n\in\mathbb N$ there is set of ...
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### Entropy conjecture for distributions over $\mathbb{Z}_n$

Suppose we have two independent random variables $X$ (with distribution $p_X$) and $Y$ (with distribution $p_Y$) which take values in the cyclic group $\mathbb{Z}_n$. Let $Z = X +Y$, where the ...
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### The maximum order of finite subgroups in $GL(n,Q)$

For several years people have tried to characterize finite groups of maximal order in $\mathrm{GL}(n,\mathbb{Q})$ (or $\mathrm{GL}(n,\mathbb{Z})$) and their orders. It appear in many articles a ...
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### Non-negative integer solutions of a single Linear Diophantine Equation

Consider the following linear Diophantine Equation:: ax + by + cz = d ------------ (1) for all, a,b,c and d positive integers, and relatively prime, and ...
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### Does a compact negatively curved manfiold of dimension 4 admit a cover of finite degree?

A $3$-dimensional compact manifold of negative sectional curvature admits (by geometrisation?) a metric of curvature $-1$, and so its fundamental group has subgroups of finite index. I wonder if an ...
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### Special divisors on hyperelliptic curves

I was reading a proof that used the following result Let $C$ be a hyperelliptic of genus $\ge 3$ and $\tau \colon C \to C$ the hyperelliptic involution. If $D$ is an effective divisor of degree ...
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### Commuting Grothendieck topologies.

Let $T_1$ and $T_2$ be two Grothendieck topologies on the same small category $C$, and let $T_3 = T_1 ∪ T_2$ (by which I mean the smallest Grothendieck topology on C containing $T_1$ and $T_2$). ...
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### Open sets and Poincaré's inequality

In many references, Poincaré inequality is presented in the following way : Let $\Omega\subset \mathbb R^d$ an open bounded set. We can find a constant $C$ which depend of $\Omega$ such that for ...
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### Can isomorphisms of schemes be constructed on formal neighborhoods?

Let (A,m) be a complete local Noetherian ring and let X and Y be two schemes of finite type over A (and flat over A). Let Xn and Yn be the reductions of X and Y mod mn+1. Question: Suppose there ...
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### Different ways to construct maps and the tensor products of line bundles

Let $C$ be a curve. Then I know of two ways to create morphisms. To get morphisms from $C$, take a line bundle of any degree $L$ and use the linear system it determines to get a map into projective ...
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### Finding a metric on a tubular neighborhood of an embedded surface

Hey all. The setup for my question is an embedded surface $\Sigma \hookrightarrow M$ in a smooth, compact 4-manifold $M$. Assuming one knows the induced metric $g_{\Sigma}$ on $\Sigma$, I would like ...
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### To what extent does Spec R determine Spec of the Witt vector ring over R?

Let $R$ be a perfect $\mathbb{F}_p$-algebra and write $W(R)$ for the Witt ring [i.e., ring of Witt vectors -- PLC] on $R$. I want to know how much we can deduce about $\text{Spec } W(R)$ from ...
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### What are the fibres of a representable simplicial sheaf (in the Nisnevich topology)

Let $k$ be a field and $X$ a smooth $k$-scheme. We consider now the pointed (constant) simplical Nisnevich-sheaf $X_{+}$ that is represented by $X$. Let now $\nu\in U$ be a point of a smooth scheme, ...
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### Decompositon of the Euler class in the ideal generated by Weyl-invariant polynomials

Let $G$ be a complex reductive Lie group, $B$ be a Borel subgroup, $T\subset B$ be a maximal torus, $W$ be the Weyl group. Then the space $X:=G/B$ is a complex manifold of dimension $n$, denote by ...