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3 votes

Do lattices with small covering radius have sublattices with small covering radius?

$\DeclareMathOperator{\vol}{vol} \newcommand{\Z}{\mathbb{Z}}$ I can prove that $c_n>\sqrt{\frac{2}{\pi e}}>0.48$ for large enough $n$ [I previously thought I had a lower bound of $2$ so this is ...
Aurel's user avatar
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0 votes

Path cover with sets of nodes

The problem is NP-hard, here is a reduction to 3-dimensional matching: Given a 3DM instance of sets $X, Y, Z$ of size $n$ and $T \subseteq X \times Y \times Z$, for every $x_i \in X$, create a set of ...
SimonNW's user avatar
12 votes

Is the maximal packing density of identical circles in a circle always an algebraic number?

Yes indeed, they are all algebraic. The idea is that we can describe the critical $r$ as a first-order formula in the language of fields, something like $$\forall r_1\quad ((0 < r_1 \wedge r_1 \le ...
Aleksei Kulikov's user avatar

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