# Tag Info

Accepted

### Why do bees create hexagonal cells ? (Mathematical reasons)

There are two principles at play here: a mathematical principle that favors hexagonal networks, and a physical principle that favors a network with straight walls. The mathematical principle that ...
• 172k
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### Understanding sphere packing in higher dimensions

There are two things you need to understand. The first is how to prove sphere packing bounds via harmonic analysis ("linear programming bounds"). My lecture notes from PCMI 2014 give an exposition ...
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### Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?

There are many ways to interpret this question depending on how you "randomly" choose your vectors. Here's one example. Take the set of vectors $v\in\{-1,1\}^N$, where the coordinates of $v$ are ...
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### Is there a contractible hyperbolic 3-orbifold of finite volume?

Yes. For example, let $M$ be the figure-eight knot complement. So $M$ is a hyperbolic manifold with volume a bit more than 2. The manifold $M$ has a two-fold symmetry $\tau$ that fixes, pointwise, a ...
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### Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?

Pick two random unit vectors. After picking the first vector, switch to a coordinate system in which this is the first basis vector. The probability distribution for the second vector is assumed to be ...
Accepted

### Is it possible to completely embed complete Heyting Algebras into upsets of a poset?

No, not in general: for instance, the real interval $([0,1],{\le})$, or any non-atomic complete Boolean algebra, do not have such an embedding. This follows from the following characterization: ...
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### Is it still not known whether the construction of shortest nonzero vector of a lattice w.r.t. $l^2$-norm is NP-hard?

The NP-hardness of the shortest vector problem in $L_2$ norm is discussed in this 2015 lecture by Vinod Vaikuntanathan. An algorithm for this problem would give a randomised algorithm for any problem ...
• 172k

### Kissing number lower bound vs. upper bound - precise meanings?

Lots of questions here, I'll see how many I can address. To be clear, $K_L$ and $K_U$ are summaries of our current knowledge. There is some true kissing number in each dimension, and hopefully we'll ...
• 149k
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### Is this obfuscation scheme unbreakable?

"Is this obfuscation scheme unbreakable?" "Well.. no." said people a couple of years later. On GGHRSW13 specifically: Cryptanalyses of Candidate Branching Program Obfuscators See also (concurrent, ...
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### Is the image of an $S$-arithmetic subgroup under a surjective $k$-morphism $S$-arithmetic?

Let $k$ be a global field, and $S$ a non-empty finite set of places of $k$ containing the archimedean places. It is certain that you meant to assume $G$ and $H$ are smooth and affine (hence the ...

### in search of a transformation between determinants

This doesn't answer the original question but answers the later SNF question for the matrix $B_n$. Let $C_n$ be the $n\times n$ matrix whose $(i,j)$-entry ($1\leq i,j\leq n$) is $\binom{x+1}{2j-i}$. ...
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There is such a transformation, of the form predicted in Linear transformation that preserves the determinant. Denoting $R$ the involution matrix $e_i\mapsto e_{n+1-i}$, it turns out that the matrix ...