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+50
square cake with raisins
Alice bakes a square cake, with $n$ raisins (= points).
Bob cuts $p$ square pieces. They are axis-aligned, interior-disjoint, and each piece must contain at least $2$ raisins.
Note that a single ...
4
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103 views
Tilings of the plane
There are many possible tilings (or tesselations) of the plane:
periodic ones by a - necessarily - finite number of prototiles (e.g. regular tilings)
aperiodic ones by a finite number of prototiles ...
4
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123 views
Sphere Covering Problem
Is it possible that one can cover a sphere with 19 equal spherical caps of 30 degrees(i.e. angular radius is 30 degrees)? A table of Neil Sloane suggests it is impossible, but I want to know if anyone ...
3
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44 views
Optimal way to place a given number of points in a region?
Let $A\subset\mathbb{R}²$ and $n\in\mathbb{N}$ be a given natural number. How to find a finite subset of $A$, $P=${$p_1,...,p_n$} such that $\int_A f_P(x)$ is minimum, where $f_P(x) = ...
3
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44 views
Upper bounds on rate of q-ary codes
Among the many upper bounds for families of codes in $\mathbb F _2 ^n$, the best known bound is the one by McEliece, Rodemich, Rumsey and Welch (MRRW) which states that the rate $R(\delta)$ ...
3
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190 views
Parallel transport in discrete differential geometry - programming a game
I would like to get a better intuitive grip on how parallel transport works. I once saw a video a German guy made with a little car having a gyroscope. That car was dragged on a big beach ball and the ...
2
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52 views
Combinatorial Laplacian Spectrum
The spectrum of the combinatorial laplacian is well understood for a square lattice. What about for other lattices?
In particular: let $ f: \mathbb{Z}^2 \rightarrow \mathbb{R} $. The usual ...
2
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56 views
Twisted tori: discrete and continuous
Taking the advice of Mariano Suárez-Alvarez, I moved this question from MO to MSE:
Motivation
Let me introduce twisted (discrete) tori:
Consider the Cartesian graph product $\mathcal{C}_n = C_n ...
2
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80 views
“Round” regions on surface of convex polytope
A convex $d$-polytope $P$ is the convex hull of finitely many points. Given such a polytope with $n \gg d$ vertices, I would like to prove that its surface has to be "round" in some region.
Let me ...
2
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172 views
Accesible Area of Discrete Geometry for Undergraduate Research
This summer I will have a chance to work on a 16-week summer research project under a professor in convex/discrete geometry. I'm a first-year student with a fairly good background for my age and I've ...
2
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45 views
Polytopes-Discrete Geometry
Can someone help me solve the following question please?
Let v be a vertex of a d-polytope P such that $ 0 \in intP $ .
Prove that $ P^{*} \cap \{ y \in \mathbb{R}^d \mid\left < y, ...
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46 views
discrete harmonic extension (an exercise of Grimmett's “probability on graphs”)
I'm struggling with exercise 1.3 in Grimmett's book "probability on graphs". Take $G = (V,E)$ a finite connected graph with given positive conductances $(w_e)_{e \in E}$, and let $(x_v)_{v \in V}$ be ...
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19 views
Weighted graphs to minimise the set of distances
$c_{0}$ to $c_{3}$ are given points of the graph and the corresponding weights are $W_{1}$ to $W_{3}$. The objective is to locate $p_{1}$ and $p_{2}$ to minimise the distances $d_{0}+d_{1}$ and ...
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25 views
Fractional Helly for more than one piercing
Fractional Helly Theorem says the following:
For every $0<\alpha\leq 1$ there exists $\beta = \beta(d, \alpha)$ with the following property. Let $C_1 , C_2 , ..., C_n$ be convex sets in $R^d$, $n ...
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78 views
Lipschitz constant of the Laplace-Beltrami operator
I'm reading a paper on discrete differential geometry:
Meyer et.al.
They define the Laplace-Beltrami operator at a point $P$ by
$$\vec{K}(p) = 2k_H(P)\vec{n(P)}$$
where $\vec{n}(p)$ is the normal ...
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94 views
Complexity of Counting the number of inducing $n$-gons
Definition: A $n$-gon is simple if it has no self intersection and is in general position if no pair of its edges are parallel.
It is clear that by extending the edges of each simple $n$-gon in ...
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17 views
Simplex with edges of length at least s having smallest circumradius
Is it true that of all $k$-simplices with edge lengths greater than or equal to some parameter $s$, the regular simplex with edge lengths $s$ has the smallest circumradius?
Please supply a proof or ...
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42 views
computing centerpoint using linear programming
I was redirected from math.stackoverflow.
So I am trying to teach myself discrete geometry and have started with the centerpoint problem. Could someone please help me understand computing ...
