The covering tag has no usage guidance.

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### 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}...

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### Sheaf of relative differentials of double cover

Let $Y$ be a smooth projective $k$-variety, $D\subset Y$ a smooth (irreducible) divisor and a line bundle such that $L^2=\mathcal O_Y(D)$. Let us call $f:X\rightarrow Y$ the double cover defined by ...

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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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### induced group actions and covering maps on Eilenberg-Maclane space

Let $M$ be a finite $CW$-complex. Let $\Sigma_k$ be the symmetric group acting on $k$-letters. Suppose there is a free action of $\Sigma_k$ on $M$. Then we have a covering map
$$
f:M\to M/\Sigma_k.
...

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### How to “lift” a transitive group action on a manifold?

Let $M=G/H$ be a homogeneous manifold, with $G$ connected Lie group. Suppose that $\widetilde{M}$ is a covering of $M$.
QUESTION: is there a general prescription to obtain a Lie group $\...

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### The homology of the braid group with coefficients in the Burau representation

Let $B_n$ denote the braid group with $n$ braids. The Burau representation $B_n\to GL_n(\mathbb{Z}[t^{\pm1}])$ makes $(\mathbb{Q}[t^{\pm1}])^n$ a $B_n$-module. I am curious in knowing what $H_i(B_n, (\...

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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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### Can any $n$ dimensional (smooth, PL, topological) closed manifold be covered by $2^n$ pieces of $n$ dimensional real spaces?

For any $n$ dimensional closed manifold $M^n$, can we find an open covering $\{U_i\}_{i\in[2^n]}$ such that $M=\cup U_i$ and each $U_i\cong \mathbb R^n$? How about complex manifolds (replacing $\...

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### Maximal expansions of strongly minimal covers of hypergraphs

Let $H = (V,E)$ be a hypergraph, that is $V$ is a set and $E \subseteq {\cal P}(V)$. We assume $\bigcup E = V$. Moreover we assume that every $e\in E$ is contained in some maximal member $e'\in E$ (...

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### Is there a graph with 99 vertices in which every edge belong to a unique triangle and every nonedge to a unique quadrilateral?

99-Graph:Is there a graph with 99 vertices in which every edge(i.e. pair of joined vertices) belong to a unique triangle and every nonedge(pair of unjoined vertices) to a unique quadrilateral?

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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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### Covering number of the range of a function

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 $D \subseteq \mathbb{R}^m$. ...

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### Strongly minimal covers

Let $H=(V,E)$ be a hypergraph, that is $V$ is a set and $E\subseteq \mathcal{P}(V)$. We say that $C\subseteq E$ is a cover of $H$ if $\bigcup C = V$.
A cover $M\subseteq E$ is said to be strongly ...

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### 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'...

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### Normal Covering of a Finite Group

Suppose $G$ is a finite group and $N_1, N_2, \cdots, N_k$ are proper normal subgroups of $G$. The set $\{ N_1, \cdots, N_k\}$ is called a normal cover for $G$, if $G = \cup_{i=1}^kN_i$. I need to the ...

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### Are generically trivial finite unramified morphisms trivial

Let $S$ be a smooth affine variety over $\mathbb C$ and let $f:X\to S$ be a finite unramified morphism.
Suppose that $X(K(S))$ is non-empty. (This means that $X\to S$ has a section generically. It ...

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### Covering a finite subset of $\mathbb{N}$ with prime arithmetic progressions

Because of a problem I ran into I am trying to get a quick start in covering with arithmetic progressions.
First I want to say I am aware of this previously asked question:
Covering $\mathbb{N}$ with ...

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### Covering a set with geometric progressions

Consider the set $S_n=\{1,2,\cdots ,n\}$. What is the minimum number of distinct geometric progressions that cover $S_n$? Let us call this number $a_n$. I was wondering about this number after doing a ...

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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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### Finding a minimum covering of a polygon with interesting shapes

After reading many papers about problems of minimum polygon covering, I found out that there are four different types of units that are considered for covering polygons, in increasing order of ...

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### 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 ...

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### Unit covering of $d$-dimensional points

Given a set of points in $X$ axis, we want to cover them with minimum number of unit intervals.
For this problem we can assume that each interval in the optimal solution is starting or ending in one ...

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### Besicovitch Covering Lemma on Manifolds

The classical Besicovitch covering lemma (BCL) asserts that for any $d \geq 1$, there is a constant $N(d)$ with the following property. If $A \subset \mathbb{R}^d$ is any subset and $r : A \to (0,R]$ ...

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### Are there infinitely many natural numbers not covered by one of these 7 polynomials?

Consider the following polynomials:
$$
f_1(n_1, m_1) = 30n_1m_1 + 23n_1 + 7m_1 + 5\\
f_2(n_2, m_2) = 30n_2m_2 + 17n_2 + 13m_2 + 7\\
f_3(n_3, m_3) = 30n_3m_3 + 23n_3 + 11m_3 + 8\\
f_4(n_4, m_4) = ...

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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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### Union of linear inequalities cover whole space?

We have $n$ variables $a_0,a_1,\ldots,a_n$ such that $a_i\geq a_{i+1}$.
There are $k$ sets of linear inequality constraints on the $a_i$.
I need to check that any choice of $a_i$ satisfies at least ...

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### Image of the map induced on homology by a covering

I asked this question on math.se (http://math.stackexchange.com/questions/647930/image-of-the-map-on-homology-induced-by-a-covering), but it have not attracted much of attention.
Let $X$ and $Y$ are ...

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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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### Best and worst centrally symmetric convex covering shapes

Suppose you have a centrally symmetric convex 2D shape $C$ of area $A$, and you randomly throw
down copies of $C$ on the plane so that each $C$-center lies within a given unit square $S$,
until $S$ is ...

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### Cover of a n-simplex with balls

Consider a n-simplex. For each edge (i,j), consider a n-ball, such that vertices i and j are antipodal on this ball. Is the simplex covered by the union of these balls? Thank you.

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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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### Algorithms for covering a rectilinear polygon using the same multiple rectangles

Sorry for the crossing-posting: original post is here
All angles of the polygon (representing a room) are right. It may be convex or concave. Use rectangles of the same size (representing a sensor ...

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### what is the cyclic cover trick?

What do people mean by the "cyclic cover trick"? I have found this expression a couple of times with no complete explaination, both talking about curves and surfaces...

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### Existence of different knots in $RP^3$ having the equivalent liftings in $S^3$

I'm looking for the answer to following question. Do exist different knots in $RP^3$ which have equivalent liftings in $S^3$ under covering $p:S^3\rightarrow RP^3$?

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### holomorphic covering between points in Teichmuller space

I have the following questiom: let $X$ and $Y$ be two different points (represented by Riemann surfaces) in the Teichmuller space $T_g$ of genus $g \geq 2$ Riemann surfaces. Then of course $X$ and $Y$ ...

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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 "...

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### Can you cover the Boolean cube {0,1}^n with O(1) Hamming-balls each of radius n/2-c*sqrt(n)?

(where c>0 and the balls need not be disjoint?)
This is an embarrassingly simple question, yet somehow I couldn't find an answer (not even, "this is a well-known open problem") after spending some ...

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### Description of regular covering maps between surfaces.

This is an improved and hopefully a more precise version of the question Covering spaces of surfaces.
Question: Given a regular covering map $\pi:\Sigma_g\to\Sigma_h$, where $\Sigma_n$ denotes a ...

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### when is a locally homeo a covering map?

Let $X$ and $Y$ be locally comapct Hausdorff spaces, and $f:X\to Y$ be a surjective local homeomorphism.
When is $f$ a covering map?
It is well-known that when $f$ is proper, $f$ is a covering map.
...

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### Galois group decomposition of non-cyclic covers

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_{...

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### Equations for abelian coverings of $\mathbb{P^{1}}$

Cyclic coverings of $\mathbb{P^{1}}$ have a simple (affine) equation, namely the formula,
$y^{m}= (x_{1}-a_{1})^{t_{1}}....(x_{n}-a_{n})^{t_{n}}$. Is there such a nice equation for abelian non-cyclic ...