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2
votes
2answers
142 views
On maximal regular polyhedra inscribed in a regular polyhedron
Let T, C, O, D, or I be regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively. Suppose that the outer polyhedron have edge-length 1.
For example, it's easy to prove that ...
6
votes
2answers
163 views
Do maximal polyhedra have algebraic volume?
Is it possible to prove that for every $n > 3$ the maximal possible volume of a convex polyhedron having $n$ vertices inscribed in a sphere of unit radius is an algebraic number?
Update: What ...
8
votes
2answers
110 views
Covering convex polygons with inscribed disks
The following problem came up when discussing mapping software (e.g., Google maps) with computer scientists. By $B(c,r)$ I mean the planar disk (open or closed, it doesn't matter) of radius $r$ around ...
3
votes
0answers
44 views
On understanding Discrete-Valued Stochastic Processes( time series, panel data )
It seems to me that a significant proportion of work in probability theory, statistics and machine learning are on understanding continuous-valued, relatively weakly dependent, or linear dependent ...
11
votes
1answer
297 views
Tiling the square with rectangles of small diagonals
For a given integer $k\ge3$, tile the unit square with $k$ rectangles so that the longest of the rectangles' diagonals be as short as possible. Call such a tiling optimal. The solutions is obvious in ...
2
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0answers
56 views
Criteria to decide whether a subset of the boundary complex of a polyhedron is a manifold?
Let $\mathcal{P} \subseteq \mathbb{R}^d$ be a convex polyhedron. Let $K$ be a subset of the boundary complex of $\mathcal{P}$. (Perhaps $K$ could be defined in terms of a system of linear ...
1
vote
0answers
50 views
A question on discrete numerical simulation on fluids mechanics
I read a paper "stable, circulation-preseving simplicial fuids" by Elcott,.etc. http://www.cs.jhu.edu/~misha/Fall09/Elcott07.pdf. It gives a structure preseving discretion of fluids. I have a ...
3
votes
1answer
117 views
cover and hide with squares
I am studying two numbers, related to squares, that can characterize a polygon P:
MinCoverNumber = the minimum number of axis-aligned squares required to exactly cover P (the covering squares may ...
3
votes
1answer
234 views
Interesting behaviour of Brion's formula under a degenerate change of variables
This is, probably, a question for those knowledgeable on the subject of Brion's theorem and its applications.
Lately, I've been dealing with situations of the following sort. Suppose we are given a ...
5
votes
0answers
72 views
Complexity of the union of randomly rotated unit cubes
It is a remarkable fact that the union of congrent cubes
has only at most near-quadratic combinatorial complexity,
$O^*(n^2)$ for $n$ cubes, known to be almost tight.
This contrasts with the union of ...
6
votes
2answers
110 views
Volumes of convex vs non-convex polyhedra with prescribed facets areas
It is known that given a set of Areas $A_f$ and normals $\vec{n}_f$ if $\sum_f A_f \vec{n}_f=0$ exist a unique convex polyhedron with given face areas and normals. (Minkowski theorem - See Alexandrov ...
6
votes
1answer
198 views
Pick's Theorem for rational points of bounded height
I wonder if the various lattice-point theorems, such as
Pick's Theorem or
Minkowski's Lattice Theorem,
have been generalized to the collection of points
with rational coordinates no more than height ...
5
votes
1answer
103 views
Matching on sphere to create cycle with chords
Imagine a number of chords of a sphere $S$ which nearly, but not quite, pass through
the center of $S$, in such a way that no pair of chords intersect:
I would like ...
5
votes
2answers
149 views
Solving for special rational triangles
I ran into a need for isosceles triangles that (1) have the two equal
integer side lengths $a$ (but the base $x \in \mathbb{R}$),
and (2) the apex angle $\gamma$ is a rational multiple of $\pi$.
...
3
votes
1answer
108 views
What are interesting 3-colorings of the plane without rainbow lines?
This question is about 3-colorings of the plane in which every line is bichromatic (or monochromatic), i.e., there are no three collinear points of different colors. Such colorings trivially exist, ...