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### Stochastic Covering Number of a Convex Set

Consider a convex set, say $S = [0,1]^d$. Let $X_1, X_2,\ldots,X_n, \ldots$ be i.i.d. random variables that are uniformly distributed on $S$. Denote the Euclidean ball centered at $x \in \mathbb{R}^2$ ...
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### N-balls covering n-balls

This question is a follow-on question from: Covering a unit ball with balls half the radius The questions are these: Given an arbitrary dimension d, and a unit n-ball in d-dimensional Euclidean ...
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### Can we replace 2-fold cover by n rectangles with 1-fold cover by n rectangles?

Suppose that $n$ rectangles cover every point of their union exactly twice (except for points on their boundaries). Can we partition this union into at most $n$ rectangles? I think it's pretty ...
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### Enumerating 1-Lipschitz functions on an integer grid

Let $G$ denote an integer grid consisting of $\{0,\dots,m\}\times\{0,\dots,n\}$. An integer-valued function $f:G\to\mathbb{Z}$ is said to be 1-Lipschitz if it satisfies $|f(x) - f(y)| \leq \| x-y \|$...
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### Node covering in a random graph

Given $N$ nodes randomly placed in a $D\times D$ area, i.e., the position of each node is randomly chosen. Assume that both $N$ and $D$ are sufficiantly large. An agent can move in the area at ...
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### Tiling a rectangle with weighted cells (min-max problem)

I have been struggling with a research problem. The problem can be formalized as follows: Given a $n\times m$ matrix $A$ containing cells with non-negative integer values, partition it in $J$ ...
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### Between Cover and Partition

In a cover problem, there is a complex shape (e.g. a polygon), and we have to find a set of simpler shapes (e.g. squares or rectangles), such that their union is exactly equal to the complex shape. A ...
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### Examples of Sheafification via Hypercovers

For a presheaf $F$ on a category equipped with a pretopology, one has the sheafification $F^{\sharp}$ of $F$. I know well the plus-construction of sheafification, which is presented in Artin's paper "...
If $\pi: C \rightarrow \mathbb{P}^{1}$ is a cyclic cover of $\mathbb{P}^{1}$ with Galois group $\mathbb{Z}/m \mathbb{Z}$ and thus with the (affine) formula $y^{m}= (x_{1}-a_{1})^{t_{1}}....(x_{n}-a_{... 0answers 117 views ### Besicovitch's covering theorem for ellipsoids and shadows The usual Besicovitch's covering theorem concerns closed balls in$\mathbb{R}^d$. It relies on a property called "directionally limited metric space": the principal ingredient is to say that there can'... 0answers 61 views ### Approximating Unit covering of d-dimensional points Given a$d$-dimensional disk of radius$2$in$\mathbb{R}^d$, how many disks of radius$1$suffice to cover it. Of course, it's fine if the smaller disks overlap. What matters is to specify a finite ... 0answers 95 views ### Finding good high-dimensional sphere coverings in Euclidean space Suppose we want to cover the unit sphere$\mathcal{S}^{d-1} := \{\mathbf{x} \in \mathbb{R}^d: \|\mathbf{x}\|_2 = 1\}$with spherical caps$\mathcal{C}_{\mathbf{y}} := \{\mathbf{x} \in \mathcal{S}^{d-1}...
I have come across the need to know a bound on a certain curious quantity: the covering number of the range of a continuous function $f: D \rightarrow \mathbb{R}^n$, where $D \subseteq \mathbb{R}^m$. ...