Questions tagged [covering]

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Fundamental group of a quotient by a group action

Suppose I have a quotient $X \to S$ by a finite abelian group $G$ action (I have several cases, but in all of them the group $G$ and the action could be written explicitly), where $X,S$ are surfaces (...
Manenky's user avatar
  • 21
2 votes
1 answer
185 views

Bound the probability that a point belongs to a set

Let $(a_k)_{k \geq 1}$ be random variables taking values on a finite subset $B$. Assume that $$ (1) \quad \Pr\Big (\lim_{n\rightarrow +\infty}d(\frac{1}{n}\sum_{k=1}^n 1_{[a_k = b]}, [v_\ell(b,\...
Star's user avatar
  • 88
0 votes
0 answers
23 views

Approximation of connected set by triangluation / covering by simplices

Good afternoon. I have two distinct questions: If I have connected compact in $\mathbb{R}^n$, how much $(n+1)$-simplices are needed to fill its interior such that diameter of maximal uncovered part ...
Dmitry Vilensky's user avatar
5 votes
2 answers
179 views

Can we calculate the spectral radius of the universal cover for specific graphs?

Background For a finite graph $G$, let $\tilde{G}$ denote the universal cover of $G$. For a vertex $v$, let $p_{2n}(v)$ denote the number of paths of length $2n$ that start and end at $v$. The ...
Eric Naslund's user avatar
  • 11.2k
1 vote
0 answers
129 views

Relation between projective representation and the representation of the universal cover of a Lie Group

I am reading this paper, in what says exactly: "Weare dealing with a ray representation os the conformal group AND THEREFORE with a representation of the universal covering group of the conformal ...
Gabriel Palau's user avatar
1 vote
0 answers
57 views

Admissibility of representations induced from Hecke algebra for covering groups

Assume $G$ is a semisimple algebraic group and $B$ is an Iwahori subgroup. Let $(r,E)$ be a representation of $H(G,B)$ which is an Iwahori-Hecke algebra, then Borel proved that $C_{c}(G/B)\otimes_{H}E$...
Fuutorider's user avatar
2 votes
1 answer
78 views

Minimal digraph covering with no 2-path edge sets is of size $\left( 1 + o \left( 1 \right) \right) \log_2 \chi(G)$

The last problem in 2022 IMC Day 1 strongly correlates with graph theory. In its official solution, the fundamental approach can be rephrased as follows. Give a digraph $G=(V,E)$. We call a subset of ...
Lasting Howling's user avatar
1 vote
1 answer
89 views

Hypergraphs with finite matching / covering balance

Let $H=(V,E)$ be a hypergraph such that $\emptyset\notin E$. We say that $C\subseteq V$ is a (vertex) cover if for all $e \in E$ we have $C\cap e\neq \emptyset$. The minimum size that a cover can have ...
Dominic van der Zypen's user avatar
4 votes
1 answer
188 views

Is König's Property for graphs inheritable from finite subgraphs?

Let $G = (V,E)$ be a simple, undirected graph. A set $C \subseteq V$ is said to be a (vertex) cover if $C \cap e \neq \emptyset$ for all $e\in E$. A matching is a set $M\subseteq E$ of pairwise ...
Dominic van der Zypen's user avatar
1 vote
1 answer
161 views

"Lamp-switch set-up number" of $n$ [closed]

Motivation. The following has a real-life (!) inspiration from a discussion about how to connect lamps and switches in an efficient way. Question. Let $n\in\mathbb{N}$ be a positive integer and let $\{...
Dominic van der Zypen's user avatar
0 votes
0 answers
81 views

Efficiency of covers

Let $X\neq \emptyset$ be a set. We say $C \subseteq {\cal P}(X)$ is a cover of $X$ if $\bigcup C = X$. For covers $C, D$ of $X$ we say that $C$ is more efficient than $D$ if $|C\setminus D| < |D \...
Dominic van der Zypen's user avatar
0 votes
0 answers
103 views

Optimal covering trails in 3 and 4 dimensions

A couple of years ago, I constructively solved (inside the $AABB$ $[0,3]$ X $[0,3]$ X ... X $[0,3]$) the $k$-dimensional generalization of the infamous Nine-Dot Problem by S. Loyd (see Cyclopedia of ...
Marco Ripà's user avatar
  • 1,060
1 vote
1 answer
152 views

Constant bound for the 1 dimensional Besicovitch covering theorem on real line

I recently looked through the proof of the Gagliardo–Nirenberg Interpolation Inequality, see proof and it says that for real line $R$, there exists a sequence of open intervals $\{I_k\}$, which covers ...
Xeh Deng's user avatar
0 votes
0 answers
112 views

Odd covering system without modulus 3 (mod3)

The existence of odd covering system with distinct moduli is a famous open question proposed by Erdős and Selfridge. I wonder whether a restricted condition for the problem that odd covering system ...
user1851281's user avatar
0 votes
0 answers
72 views

On finding optimal convex planar shapes to cover a given convex planar shape

Covering a specific convex shape S with n copies of another specified convex shape S' (which may be different from S) is well studied - for example, https://erich-friedman.github.io/packing/index.html....
Nandakumar R's user avatar
  • 5,371
2 votes
1 answer
54 views

Minimal vertex-covering set

If $G=(V,E)$ is a simple, undirected graph, $C\subseteq V$ is said to be a vertex cover if for every $e\in E$ we have $C\cap e \neq \emptyset.$ If $G=(V,E) $ is infinite, is there necessarily a vertex ...
Dominic van der Zypen's user avatar
4 votes
0 answers
106 views

Sufficient conditions for the Besicovitch covering theorem to hold on groups of polynomial growth

Let $G$ be a finitely generated group with symmetric generating set $S$. Then $S$ induces a distance $d$ on $G$ by letting $d(a,b) = $ the minimum $n$ such that there are generators $s_1,...,s_n$ with ...
MathidRyan's user avatar
5 votes
1 answer
176 views

Do you know explicit examples of superelliptic curves $y^{\ell} = g(x)$ (for some prime $\ell > 3$) covering some elliptic curves?

For every elliptic curve $E$ Icart in $\S 2$ of the paper explicitly constructs a superelliptic curve $S\!: y^3 = f(x)$ and a cover $\varphi\!: S \to E$. Do you know explicit examples of superelliptic ...
Dimitri Koshelev's user avatar
1 vote
1 answer
146 views

Effect of snowflaking on doubling constants

This question is related to this one. Let $(X,d)$ be a metric space, let $\epsilon\in [0,1)$ and consider the snowflake $(X,d^{1-\epsilon})$. Suppose that $(X,d)$ has a finite doubling constant, ...
ABIM's user avatar
  • 5,069
2 votes
1 answer
103 views

Smallest size of graph covered by infinite tree

Let $T$ be the universal covering tree of some finite, connected, non-tree graph, and let $n_0(T)$ be the smallest positive integer such that there exists a graph $G$ (loops and multiple edges allowed)...
Maurizio Moreschi's user avatar
4 votes
2 answers
242 views

Relationship between minimum vertex cover and matching width

Let $H$ be a 3-partite 3-uniform hypergraph with minimum vertex cover number $\tau(H)$ (i.e. $\tau(H)=\min\{|Q|: Q\subseteq V(H), e\cap Q\neq \emptyset \text{ for all } e\in E(H)\}$). Question: Is $\...
Louis D's user avatar
  • 1,666
2 votes
1 answer
188 views

On some optimal containers of a set of points on the 2D plane

Given a set of N points in general position on the plane, the problem is to give efficient algorithms to find the smallest semicircular region (semidisk) that contains the points the smallest ...
Nandakumar R's user avatar
  • 5,371
2 votes
0 answers
86 views

How many ways to cover a N×N chessboard with white and black boxes by some restrictions?

Suppose we have a N×N chessboard and the boxes ■, □. We should cover the chessboard with those boxes but there can not have the 2×2 square $\scriptstyle{\begin{array}{cc}\square&\square\\ \...
Yuhuan Lei's user avatar
2 votes
0 answers
255 views

Covering number for the unit ball in a reproducing kernel Hilbert space

I am looking for a reference for an upper bound on the covering number for the unit ball $\{ f \in \mathcal{H}: ||f||_{\mathcal{H}} || \leq 1\} $, where $\mathcal{H}$ is a reproducing kernel Hilbert ...
JohnK's user avatar
  • 121
1 vote
0 answers
39 views

Separation of balls in the torus

Let $X_1, \dots, X_N$ be $N$ balls of radius $R<<1$ in $[0,1]^d$ such that $N R^d \leqslant R^{\alpha}$ for some $\alpha > 0$ and $d(x_i,x_j)\geqslant 2R$ for any $i\not=j$. The assumption ...
JBB's user avatar
  • 11
14 votes
1 answer
894 views

Minimal good cover of the torus

Recall that an open cover $\mathfrak{U} = \{ U_\alpha \}$ of a manifold $M$ is called a good cover if all possible finite intersections $U_{\alpha_1} \cap ... \cap U_{\alpha_n}$ are contractible. ...
Dennis's user avatar
  • 253
-1 votes
1 answer
76 views

Minimal covering sets of continuous endomorphisms

For any topological space $(X,\tau)$, let $\text{End}(X)$ denote the set of continuous functions $f:X\to X$. We say that ${\cal C}\subseteq \text{End}(X)$ covers $\text{End}(X)$ if for every $f\in \...
Dominic van der Zypen's user avatar
3 votes
2 answers
121 views

Avoiding multiply covered vertices in graph edge coverings

Let $G=(V,E)$ be a simple, undirected graph with $\bigcup = E$ (that is, there are no isolated vertices). We say that $C\subseteq E$ is an edge cover of $G$ if $\bigcup C = V$. For any edge cover $C$ ...
Dominic van der Zypen's user avatar
1 vote
0 answers
49 views

Minimizing the set of multiply covered elements in a linear hypergraph

We say that a hypergraph $H=(V,E)$ is a linear hypergraph if it has the following properties: if $e_1\neq e_2\in E$ then $|e_1\cap e_2|\leq 1$, and $\bigcup E = V$. We say that $C\subseteq E$ is a ...
Dominic van der Zypen's user avatar
1 vote
1 answer
400 views

Homology of universal abelian cover of a manifold

If one define the universal abelian covering $M_0$ of a manifold $M$ as the abelian covering (i.e. normal covering with abelian group of deck transformations) that covers any other abelian covering, ...
Arnaud Maret's user avatar
1 vote
1 answer
152 views

On a combinatorial set covering property

Let $\kappa < \lambda < \mu$ be infinite cardinals. Is there a collection ${\cal U}\subseteq {\cal P}(\mu)$ of subsets of $\mu$ with the following properties? for all $U\in {\cal U}$ we have $|...
Dominic van der Zypen's user avatar
1 vote
1 answer
371 views

Growth rate of bounded Lipschitz functions on compact finite-dimensional space

Let $\mathcal X$ be a metric space of diameter $D$ and "dimension" (e.g doubling dimension) $d$. Let $L \in [0, \infty]$ and $M \in [0, \infty)$ and consider the class $\mathcal H_{M,L}$ of $L$-...
dohmatob's user avatar
  • 6,674
4 votes
1 answer
210 views

Nerve theorem for locally infinite covers by subcomplexes

Let $Y$ be a simplicial complex and let $\{Y_i\}_{i\in I}$ be a set of subcomplexes of $Y$ such that $\bigcup_{i\in I}Y_i=Y$. Let $\mathcal N$ be the nerve of this covering, and assume that for each ...
Mark Hagen's user avatar
4 votes
1 answer
72 views

Optimal pseudotransversals

A hypergraph $H=(V,E)$ consists of an non-empty set $V$ and a collection $E\subseteq {\cal P}(V)\setminus \{\emptyset\}$ of non-empty subsets of $V$. A transversal of $H$ is a set $T\subseteq V$ such ...
Dominic van der Zypen's user avatar
1 vote
1 answer
212 views

Understanding equivalent condition for covering dimension

Let dim $X$ denote the Lebesgue covering dimension for a topological space $X$. Now a result in common books concerning dimension theory states the following: If $X$ is a normal topological space, ...
user531706's user avatar
2 votes
2 answers
332 views

Image of curve along a finite etale Galois map

Let $f:X\to Y$ be a finite etale Galois morphism of varieties over $\mathbb{C}$. Let $C$ be a smooth quasi-projective connected curve in $X$. Is $f(C)$ a smooth curve?
Come Vay's user avatar
1 vote
1 answer
81 views

Choice sets in covers with small intersections

Let $X\neq \emptyset$ be a set. We say ${\cal C} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ is a cover of $X$ if $\bigcup {\cal C} = X$. A subset $S\subseteq X$ is a choice set for ${\cal C}$ if $|S\...
Dominic van der Zypen's user avatar
2 votes
1 answer
107 views

Edge covers in infinite graphs

If $G=(V,E)$ is a simple, undirected graph, then $C\subseteq V$ is an edge cover if $C\cap e \neq \emptyset$ for all $e\in E$. The "best" covers in some sense are subsets $C\subseteq V$ that meet ...
Dominic van der Zypen's user avatar
3 votes
1 answer
183 views

How to cover n sites with the smallest number of fixed radius balls?

Given $n$ "data points" in $d$ (Euclidean) space $$\mathbf{x}_j \in \mathbb{R}^d, \text{ for } j \in \{1,\dots,n\}$$ how does one find the smallest integer $m$ such that there exists $m$ "centre ...
Alec Jacobson's user avatar
-1 votes
1 answer
141 views

Covering property of complete distributive lattices

Let $(L,\land,\lor)$ be a complete distributive lattice. Given $x\neq y \in L$, is there a finite set ${\cal I}$ of closed intervals in $L$ such that no member of ${\cal I}$ contains both $x$ and $y$,...
Dominic van der Zypen's user avatar
8 votes
1 answer
239 views

The minimal number of partitions to cover all $k$ tuples

The set $N=\{1, 2, \ldots, 2k\}$ can be partitioned into pairs (e.g $(1,2),(3,4),\ldots,(2k-1,2k)$) in $\frac{(2k)!}{k!2^k}$ ways. $k$-tuple is subset of size $k$ in $N$. We say that $k$-tuple is ...
Ashot's user avatar
  • 337
7 votes
2 answers
380 views

Minimum covers of complete graphs by $4$-cycles

I am interested in coverings of the (edge set of the) complete graph $K_n$ by cycles of length $4$. It is clear that such coverings exist for each $n \ge 4$. I need to find the minimum number of $4$-...
Ashot's user avatar
  • 337
7 votes
3 answers
462 views

Minimal covering sets in families of sets intersecting in at most $1$ point

Let $X$ be an infinite set, and let ${\cal A}\subseteq{\cal P}(X)$ be a family of non-empty sets. We say $S\subseteq X$ is a cover for ${\cal A}$ if $A\cap S \neq \emptyset$ for all $A\in{\cal A}$. ...
Dominic van der Zypen's user avatar
2 votes
1 answer
214 views

Brownian motion and random walk

Let $M_{\Gamma}$ a Riemannian covering of a closed compact manifold $(M,g)$ with deck transformation $\Gamma$ (its neutral element will be denoted by $e$). If we denote by $p_t^{\Gamma}(x,y)$ the heat ...
user avatar
4 votes
2 answers
143 views

Covering all except one of the purple intersection points of $n$ red and $m$ blue lines efficiently

Consider a set of $n$ red lines and $m$ blue lines, suppose there are $nm$ distinct red-blue intersections. What is the minimum number of lines $L_1,L_2,\dots, L_n$ such that the union contains all $...
Gorka's user avatar
  • 1,825
2 votes
0 answers
143 views

Cardinality of compact doubling metric spaces with fast growing covering numbers

In this question it was established that if the growth of the number of branches of an at-most $k$-branching tree is $\Omega(k^n)$ (in the Knuth sense), then the tree has continuum many branches. ...
James Hanson's user avatar
  • 10.3k
2 votes
1 answer
100 views

Minimum number of edges to add in order to have a biclique cover

Given a bipartite graph G and a number N, what's the minimum number of edges I have to add to G in order to be able to cover the resulting graph with no more than N complete bipartite subgraphs? For ...
javic's user avatar
  • 121
4 votes
1 answer
199 views

Inscribing a "chain" into an open cover

Let $X$ be a locally connected topological space, which is covered by open sets $\{U_{\alpha},\alpha\in A\}$ and let $C$ be an arc in $X$, i.e. a homeomorphic image of an interval. Is it always ...
erz's user avatar
  • 5,285
1 vote
0 answers
64 views

When does a metric space admit finite covers by Voronoi diagrams of Delone sets?

Some preliminary definitions: For a given metric space $(X,d)$ and set $A\subset X$, the Voronoi diagram of $A$ (which I'll write $V(A)$) is the collection of sets of the form $$C_a=\{x\in X|\forall b\...
James Hanson's user avatar
  • 10.3k
5 votes
2 answers
749 views

A variant of set cover problem reformulated

Given a universe set $U$ and $n$ sets of sets $A_i$ ($i=1, \cdots, n$). Each set $A_i$ contains $k_i$ subsets of $U$, i.e., $A_i=\{B_{ij}: j=1, \cdots, k_i\}$ where $B_{ij}$ is a subset of $U$. I have ...
lchen's user avatar
  • 459