# 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 (...
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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,\... • 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 ... 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 ... • 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 ... 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... 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 ... 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 ... 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 ... 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 \{... 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 \... 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 ... • 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 ... • 65 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 ... • 101 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.... • 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 ... 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 ... 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 ... • 1,801 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, ... • 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)... 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 \... • 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 ... • 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\\ \... 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 ... • 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 ... • 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. ... • 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 \... 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 ... 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 ... 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, ... • 347 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 |... 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-... • 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 ... • 322 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 ... 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, ... • 149 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? • 39 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\... 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 ... 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 ... -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,... 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 ... • 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-... • 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}. ... 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 ... 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 ... • 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. ... • 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 ... • 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 ... • 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\...
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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 ...