All Questions
Tagged with discrete-geometry reference-request
174 questions
2
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Reference request: placing a set with respect to the integer grid
For $x=(x_1,...,x_n)\in \mathbb{R}^n$, let $Q_x=(x_1,x_1+1)\times ...\times (x_n,x_n+1)$ - the open cube having $x$ in its "bottom left" corner. It seems, I can prove (see a draft here) the following
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- 5,529
2
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0
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87
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A theory of (or reference for) symmetric point arrangements
I wonder where I can find something written on symmetric point arrangements (see definition below). I am interested in general references, preferably books that introduce (or papers that use) some ...
- 13.6k
1
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2
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232
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What does the extension theorem for tilings state?
I have seen several references to the so-called Extension Theorem in the context of tilings of Euclidean space.
E.g. in "The Local Theorem for Monotypic Tilings" one reads
The Extension Theorem [......
- 13.6k
1
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1
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393
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Hyperbolic embedding of a directed acyclic graph defined over strings
For integer $n$ and alphabet $\Sigma$ we construct a DAG (directed acyclic graph) $G=(V,E)$ over strings $s\in\Sigma^\star$ as follows:
$$V = \{s\in\Sigma^\star\colon |s|\le n\}$$
$$E = \{(s_1,s_2)\...
- 187
1
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2
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204
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Disks Packing Variant
Usually disk packing problems require that no two disks of the packing intersect.
Does anybody know if the problem has been studied when disks may intersect but they are not allowed to contain the ...
- 57
1
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2
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100
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Name for the weight function defined as the integer sum of coordinate entries from ${\mathbf F}_p$
In ${\mathbb F}_p^n$, $p$ prime one may define a weight function on vectors in various ways such as Hamming, or Lee weight. (These two weights correspond nicely to the respective distances from $\bar ...
- 11
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1
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75
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Given a vertex $u$ (of bounded degree $k$) and another vertex $v$ in a planar graph, what is the smallest number of "curves"?
Given a vertex $u$ (of bounded degree $k$) and another vertex $v$ in a planar graph $G$, what is the smallest number of "curves" in the plane drawn from $u$ to $v$ such that no $u$--$v$ path in $G$ ...
- 111
1
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0
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76
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Shellable non-pseudomanifolds with dimension greater than 2
Shellability of simplicial balls and spheres (simplicial complexes whose geometric realizations are homeomorphic to balls and spheres) has been studied quite extensively. There are many explicit ...
1
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0
answers
145
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Lower bound $|\sum_{x \in X} \phi(x) - \int_{\mathbb{R^2}} \phi(x) \, dx | \geq C f(\phi)$
I asked this question on math.stackexchange before, but with a bad formulation. I think the problem is quite complicated, so I decided to ask it here. Tell me if I shouldn't.
Very recently, I ...
- 183
1
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1
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379
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Bridges between geometry and combinatorics
Geometry and combinatorics are two different branches of mathematics. Does there exist any connection between them? In many cases, mathematicians solve some geometric problems by reducing them to a ...
- 613
1
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0
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69
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Can sufficiently symmetric polytopes be uniquely reconstructed from their 1-skeleton?
General convex polytopes can not be uniquely reconstructed from their 1-skeleton1, as explained here. Not even the dimension is known from the skeleton, as e.g. the complete graph $K_n,n\ge 5$ is the ...
- 13.6k
1
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0
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278
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Sphere packings with antipodal (unequal) spheres
Let $\|\cdot\|_2$ denote the Euclidean norm, let $\langle \cdot, \cdot\rangle$ denote the standard dot product, and let $\mathcal{S}^{d-1} = \{\mathbf{x} \in \mathbb{R}^d: \|\mathbf{x}\|_2 = 1\}$ ...
- 733
1
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0
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71
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Name for a Specific Planar Linear Transformation
Is there a name for linear transformations of the plane, that make $4$ points in general convex configuration co-circular, with the biggest circle through those points and, how can they be determined ...
- 13.2k
1
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0
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39
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Homology of the subcomplexes of the "diamond shaped" sphere under 1-norm in $R^n$ as a simplicial complex
The 1-norm on $\mathbb{R}^n$ is defined by $\|v\| = |v_1| + |v_2| + \cdots + |v_n|$ for a vector $v = (v_1, \ldots, v_n) \in \mathbb R^n$.
The unit sphere $S^{n-1}_1$ under the 1-norm is a simplicial ...
- 459
1
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0
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77
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Non-adjacent Pair of Edges with Minimal Weight Sum
Given an weighted, undirected Graph $G(V,E)$ without loops or parallel edges,
what is the complexity of determining a pair of non-adjacent edges, whose sum of weights is w.l.o.g. minimal?
is that ...
- 13.2k
1
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0
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70
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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$. ...
- 183
1
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0
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61
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Generalizing Concepts of Planar Euclidean Geometry to Symmetric TSP-Instances
To me it seems possible, to successfully look at symmetric TSP instances from a geometry-point of view.
Examples are:
the diagonals of the convex hull of a set of points in the euclidean plane; ...
- 13.2k
1
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0
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193
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Lattice-point enumeration question involving linear combinations of matrices
I would like to know some references to learn more about an answer to this question, if there are any references:
Let $A_1, \dots , A_m$ and $B$ be $n\times n$ symmetric matrices. Let $$S = \{(x_1, \...
- 170
1
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0
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371
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Simple development of simple curve on a cone
Let $\Lambda$ be a cone with apex $a$ and apex angle $\alpha$. Draw a simple (non-self-intersecting)
curve $C=(x,y)$ on $\Lambda$, and then develop it to a curve
$\overline{C}$ on a plane by rolling $...
- 151k
0
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3
answers
512
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The symmetry group of $\mathbb Z^d$
Let $d \ge 1$, and consider the integer lattice $\mathbb Z^d$. This is a homogeneous space, in the manner of the Erlangan Programm.
I would like to write $\mathbb Z^d = G / H$, where $G$ is the ...
- 8,512
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1
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91
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When do there exist locally regular embeddings of regular graphs?
I don't know how one can tackle the following kind of question, so any hint is welcome. I formulate a precise question in order to fix ideas, but it is to be considered as an example out of a more ...
- 2,041
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1
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173
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Which simplicial complexes are completely determined by the 1-skeleton of their dual polyhedral complexes?
Consider the following line of reasoning that shows certain simplicial complexes (of arbitrary dimension) are completely determined by corresponding graphs:
The facet complex of any simplicial ...
0
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1
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86
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Lattice-point-free body diameter
The following interesting problem was asked at Aops and I wonder if it was based on some research paper:
Let $K$ be a convex body in $\mathbb R^2$, such that the diameter of $K$ is less than $\sqrt2$....
- 3,153