# All Questions

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### Exact formula for $\sum_{n=0}^{+\infty}\frac1{n^2+1}$

Actually, as the corresponding integral ($\ln(\cos x)/(1-x)$ or something like that) cannot be expressed in closed form by Liouville theorem, this shouldn't exist, but I believe I have seen it shown ...
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### Absolute value of a polynomial as the Euclidean distance from its root

Say that you have a polynomial $f(x)$ of degree 2 in one real variable. Then, if the polynomial has only one unique root $r \in \mathbb{R}$, it factorizes as $f(x) = (x - r)^2$, which expresses the ...
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### Efficient packing in Gaussian measures on Hilbert spaces

Let $B(0,1)$ be the unit ball in a separable Hilbert space $H$ with a Gaussian measure $\mu$ on it. For a small $r > 0$, can we have $x_i \in B(0,1)$, $0 < r_i \leq r$ and a constant $K > 0$ ...
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### Math Education Paper Request [migrated]

If this is the wrong forum for this post I apologize but I'm not sure of another well-suited medium for this question (and any reference to one is appreciated). I am wondering if any research in ...
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### Uniqueness of the reduced rank QR decomposition

Let $A$ be a $n\times n$ matrix that is a real, symmetric, positive semidefinite and has rank $r$. I read about the rank reduced $QR$ decomposition (here for example) as the QR variant where $A=QR$ ...
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### A generalized theorem of Hall's marriage theorem

We all know Hall's marriage theorem as following: A bipartite graph $G$ with bipartition $\{ A,B \}$ contains a matching of $A$ if and only if $|N(S)|\geq |S|$ for all $S\subseteq A$. And I am ...
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### “Fractally self-similar” numbers

This is another question about visualization of Ford circles, the previous one being Confusion with practically implementing rational approximations. Here is an output of zooming into Ford circles at ...
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### Why standard deviation is preferred over mean deviation? [on hold]

I was doing my homework when I come across both these quantities which tells us dispersion in data. But, I am able to understand mean deviation as it tells on an average how much a value deviates from ...
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### geometric interpretation of derivation between two algebras

Given a smooth manifold $M,$ it is well known that any derivation of the algebra of smooth functions $C^{\infty}(M)$ can be seen (or it is associated to) a smooth vector field on $M.$ I am looking for ...
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### orthogonal projector onto the set of convex functions

Let $\Omega\subset \mathbb R^d$ be an open, convex domain, and consider the Hilbert space $L^2(\Omega)$. Each sum of convex functions is convex, hence the subset $Conv(\Omega)$ of all convex functions ...
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### On simple complex loops [on hold]

To apply Jordan theorem, a curve $\Gamma$ must be a simple closed continuous curve. (Its parametrization is injective) Now consider an element $[\gamma] \in H_1(\mathbb{\Omega})$ (singular homology, ...
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### General formula for composite numbers [on hold]

I propose general formula of composite numbers, except divisible by 2 and 3: Positive integers contained in two 2-dimensional arrays:$P1(i,j)=6i^2-1+(6i-1)(j-1)$ and $P2(i,j)=6i^2-1+(6i+1)(j-1)$ are ...
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### Conic sections in high dimensions

Can every $n$-dimensional ellipsoid be obtained as a (spherical) conic section? This is false for generic quadrics but seems true for ellipsoid. Does anybody have any references?
1.Is there a nontrivial pricipal bundle $P(M,G)$, with $G$ connected, such that the total space $P$ admit a foliation such that each leaf is diffeomorphic to $M$(Not necessarily via projection ...
### concentration inequality for $d$-dimensional martingale
Are any concentration inequality available for $d$-dimensional martingale. It is easy to find such inequality using the inequalities for single dimension, but that will contain the dimension $d$ in ...