# All Questions

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### Is $2^n -1$ finitely many times the product of consecutive primes?

This question was asked at MSE but recieved no attention at all. Here it is: Are there finitely many $(n,k) \in \mathbb{N}^2$ with $2^n-1=p_1p_2\cdots p_k$ ? $p_1=3,p_2=5 , ...,p_k$ are ...
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### positions of polyhedrons with vertices on the unit sphere

Let $S^2$ be the unit $2$-sphere canonically embedded in $\mathbb{R}^3$. Let $P$ be a polyhedron whose all vertices are in $S^2$. Let $\text{Iso}(S^2)$ be the isometry group of $S^2$ and ...
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### Conditions on the hierarchy for Thurston's hyperbolization theorem

From my understanding the proof of Thurston's hyperbolization theorem for Haken $3$--manifolds consists of cutting the manifold along a hierarchy (collection of incompressible, ...
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### Polynomial differential forms on $BG$

Let $\Omega^{*}_{\text{poly}}\: : \: sSet\to dg_{\geq 0}Comm_{+}$ be the polynomial De Rahm functor on simplicial sets, where the codomain is the category of commutative differential graded algebras ...
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### Idea behind the proof of consistency of club filter of $\omega_1$ is ultrafilter + ZF + DC

I've been trying to understand Radin Forcing and some of its applications, one of which is the use of it to prove the consistency of ''Club filter of $\omega_1$ is an ultrafilter + ZF + DC''. However, ...
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### History of spectral methods to the study of real analytic $GL_2$-Eisenstein series

I'm trying to sort out the history of spectral methods in the study of real analytic $GL_2$-Eisenstein series. From what I read so far, I would say that the subject was really kicked off by the ...
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### Geodesically convex neighborhood in Finsler manifolds

It is well known that every point of a Riemannian manifold $(M,g)$ possesses a fundamental system $\{U_n\}_{n\in\mathbb N}$ of geodesically convex neighborhoods. This means that every pair of points ...
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### unwinding the definition of $H_i(KU)$ as a map of spectra $\mathbb{S}^i \to HZ \wedge KU$

I asked this on mathstackexchange but didn't get any response (or many views) so I'm asking it here, although clearly it belongs over there. In the answer to this question on mathoverflow, it says: ...
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### When does a “universal” quot scheme exist?

Suppose $M$ is a moduli space of semistable sheaves on a projective variety $X$. Let $v$ be some the discrete invariants. I would like to form a space $\mathcal Q(v) \rightarrow M$, where the fiber ...
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### Examples of beautiful theories without applications [on hold]

What are examples of beautiful theories, which have no known applications?
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### Is there any $ABCDS$ pyramid (where $ABCD$ is a rectangle) in which each 2 edges have different lengths and $|AS|+|CS|=|BS|+|DS|$? [on hold]

I had geometry quite a while ago and I wonder if anyone has any idea how to tackle this problem: Is there any $ABCDS$ pyramid (where $ABCD$ is a rectangle) in which each 2 edges (except for the base) ...
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### Vector bundles on open (affine) curves

It is well-known by Grothendieck (or earlier by Dedekind-Weber) that every vector bundle on $\mathbb{P}^1_k$ for $k$ a field decomposes into a sum of the line bundles $\mathcal{O}(k)$. As ...
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### Finding equivalent matrix combination [on hold]

I have a program I've written that is solving some problems with some matrix-vector math, but I have a feature I want to add and while I've found a work around an analytic solution would be superior. ...
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### mutual information problem [on hold]

In mutual information we have: if $x$ and $y$ are independent then $p(x,y)=p(x)p(y)$ and then $I(X;Y)=0$. Do If $I (X;Y) = 0$ when $x$ and $y$ are independent?
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### Existence and summability of cumulant [on hold]

I posted this question on math stackexchange, but no one answered. So I am seeking help here. 1) Is the statement "the $r$-th order moment exists" equivalent to "the $r$-th order cumulant exists"? ...
83 views

### Perfect centerless normal subgroups

Let $S$ be a non-trivial simple group and suppose $S \trianglelefteq G$ if $C_G(S)=1$ then $S$ is characteristic in $G$. To prove this let $\phi$ be an automorphism of $G$ and note that the ...
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### Generalized Theorem of Laguerre

There is known theorem of Laguerre, that every linear ordinary differential equation of second order $$y''+A(t)y'+B(t)y=0$$ by point transformation could be mapped into $$y'' = 0,$$ that in few words ...
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### Interpolation with double second differences [on hold]

My question is about an interpolation method used in an astronomy book that I would like to understand, and that can be found here: ...
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### mutual information entropy problem [on hold]

In mutual information we have: if $x$ and $y$ are independent then $p(x,y)=p(x)p(y)$ and then $I(X;Y)=0$. Do If $Y (X;Y) = 0$ when $x$ and $y$ are not necessarily independent?
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### Showing that a particular area is small

Note: I posted this on math.stackexchange.com earlier (original post here: http://math.stackexchange.com/questions/1471331/showing-that-a-particular-area-is-small), but it received no responses and ...
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### Is the sum of spectral projections a projection?

Let $T$ be a closed operator on a Hilbert space with discrete spectrum. Then for $\{\lambda_1,...\lambda_n\}\in\sigma(T)$ one can define the spectral projections ...
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### Which partitions of $[0,1]$ are collection of level sets of a real continuous function?

Let $f:[0,1]\to[0,1]$ be given. The level sets of $f$ (ie the collection of all sets of the form $\{x\in[0,1]:f(x)=y\}$, for each fixed $y\in[0,1]$) partition the domain of $f$. I am curious for set ...
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### Sum of Stirling numbers with exponents

I have a trouble with the following sum $\sum_{i=0}^n\binom{n}{i}S(i,m)3^i$, where $S(i,m)$ is the Stirling number of the second kind (the number of all partitions of $i$ elements into $m$ nonempty ...
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### Proposals for polymath projects

Background Polymath projects are a form of open Internet collaboration aimed towards a major mathematical goal, usually to settle a major mathematical problem. This is a concept introduced in 2009 by ...
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### What is the algebraic fundamental groups of $SO(n)$ and $Sp(2n)$?

Let $k$ be an algebraically closed field of characteristic zero. and let $$\sigma: SL_n(k)\rightarrow SL_n(k)$$ be an involution. My questions are: How could one calculate the fundamental group of ...
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### Discrete random walk with uniformly distributed transition p, set initially

I've been working on a discrete version of the "unreliable friend" distribution. It would seem that what I've come up with is equivalent to the following random walk: Choose $p$ from $U(0,1)$ Start ...
I am interested in the complexity of the following problem: Input: A list $a1\leq ⋯ \leq a_n$ of positive integers. Question: Are there two vectors $x, x'\in\{−1,0,1\}^n$ such that ...
One can study the characterization of a linear differential operator $T$ from scalar product $(f,g)=\int_{a}^{b}f(t)g(t)dt$ and the theory of adjoint operators solving $Tf=g$ by finding a right ...