All Questions
4,828 questions
3
votes
3
answers
390
views
Can we uniquely define a graph to have the topology of a polytope via proper edge length selection?
I'll ask you to consider a situation wherein one has a series of edges for a graph, $(e_1, e_2, ..., e_N) \in E$, each with a specifiable length $(l_1, l_2, ..., l_N) \in L$, and the goal is to insure ...
- 17.9k
2
votes
1
answer
384
views
Feasible space of SDP
Typically the non-empty feasible space of a SDP has some curved boundary which is why the feasible space has infinitely many extreme points. Is it ever possible to have a SDP whose non-empty feasible ...
- 1,243
12
votes
3
answers
1k
views
F→E→B bundle with B,E,F hyperbolic: possible?
It would be interesting to me obtain an answer to the following easy to state question:
Does there exist a (smooth) fibre bundle $\pi\colon E\rightarrow B$ with typical fibre $F$ such that $E$, $B$ ...
- 10.6k
9
votes
2
answers
477
views
An extension of Gaussian Isoperimetry
The Gaussian isoperimetric inequality (Tsirelson,Sudakov, Borell) states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces have the minimal Gaussian ...
- 6,694
3
votes
1
answer
316
views
Hausdorff dimension and Mertens function
Hello,
when one plots the Mertens function, it really looks like a fractal. So does anyone know the (approximate) value of the Hausdorff dimension of the set $\{(x,y),y=M(x),x\in\mathbb{R}^+\}$?
...
- 41.1k
1
vote
1
answer
234
views
Model for shipping widgets in an optimal way
I am a programmer and have the following requirement.
We are trying to figure out the optimal way to ship widgets. Below is the scenario:
We need to ship 1,000,000 widgets
We have two different size ...
- 3,934
7
votes
1
answer
360
views
Standard reference for equivalence of PU(2) action on $\mathbb{C}\mathbb{P}^1$ and SO(3) action on $S^2$
The equivalence I describe below is well-known, but I'd like a simple standard reference for it.
Consider $\mathbb{C}\mathbb{P}^1$, the set of one-dimensional subspaces of $\mathbb{C}^2$, which has a ...
CommunityBot
- 1
1
vote
0
answers
785
views
A curious property of the Gergonne point
Ha, finally no knot theory :-)
First of all, let's define the "power line" of three circles.
(Very probably, someone had the idea before me, but no math forum
ever came up with something .)
Call the ...
- 41.1k
10
votes
1
answer
688
views
$G$-structures of finite type.
A $G$-structure $\pi : B_G \rightarrow M$ is said to be of $finite$ $type$ if $\mathfrak{g}^{(k)} = 0$ for some $k \in \mathbb{N}$, where $\mathfrak{g}^{(k)}$ denotes the $k$th prolongation of the Lie ...
- 108k
30
votes
2
answers
2k
views
Maneuvering with limited moves on $S^2$
This question comes to me via a friend, and apparently has something to do with quantum physics. However, stripped of all physics, it seems interesting enough on its own. I assume someone has asked ...
- 301
2
votes
0
answers
147
views
System dynamic of space euclidean and hyperbolic tilings
Theorem 2.9. (Rudolph [Rud89]) Suppose $X_{T}$ is a finite local complexity (FLC)
tiling space. Then $X_{T}$ is compact in the tiling metric d. Moreover, the action $T$ of
$R^{d}$ by translation is on ...
- 21
2
votes
1
answer
304
views
existence of l1 embedding using LP feasibility
hello
Let (A, d) be an n-point metric space
for $t \geq 1$,the task it to find an integer $m$ and an embedding $f : A \rightarrow R^m$ s.t.
$\forall x,y \in A$ : $d(x,y) \leq d_1(f(x), f(y)) \leq t*...
- 16.8k
1
vote
1
answer
531
views
Split sum into equal terms
Given a sum of $l$ integers $r_1+...+r_k+...+r_l$ and an integer $t$.
Find indices
$1 < p_1 <...< p_h <...< p_{t-1} < l$
such that in sum
$(r_1+...+r_{p_1})+...+(r_{p_{h-1}+1}+......
- 18.6k
0
votes
1
answer
219
views
Find elements $\rho_i$ such that $H < B : [\langle H, \rho_i \rangle : H ] = 2$
Let $\Gamma (G;(G_i)_{i \in I})$ be a coset geometry (in the sense of Buekenhout) firm, residually connected and flag transitive with Borel subgroup $B$. Consider $H$ any subgroup of $B$. I want to ...
- 499
7
votes
1
answer
362
views
Nonexpansive multi-valued maps in $\ell^2$
Let $C$ be a nonempty bounded closed convex subset, say the unit ball, of $\ell^2(\mathbb{N})$. Let $T: C\to 2^C$ be a map such that $T(x)$ is nonempty closed for each $x$, and that $$D(Tx,Ty)\le \|x-...
CommunityBot
- 1
0
votes
1
answer
2k
views
Quaternion between two quaternions [closed]
Hello,
I have an orientation P1 in a 3D space, represented as a quaternion [w x y z].
Then P1 is rotated using another quaternion (q1) with the formula
P2=q1*P1*q1'...
- 96.4k
4
votes
1
answer
626
views
Embed the intersection of an n-dimensional unit $L_1$ sphere and a hyperplane into an (n-1)-dimensional unit $L_1$ sphere.
In $\mathbb{R}^n$, given an unit $L_1$ sphere $\mathcal{B}_n: |x_1|+|x_2|+\ldots+|x_n|\leq 1$ and a hyperplane $\mathcal{P}: a_1x_1+a_2x_2+\ldots+a_nx_n=0$. Does there always exist a rotation such ...
- 59
5
votes
2
answers
1k
views
Critical Radius for Infinite Dimensional Sphere Packing
Hello. I'd like to consider the open unit ball in an infinite dimensional Hilbert space and ask when can we fit infinitely many open balls of radius $r<1$ inside.
For example, when $r=1/(1+\sqrt2)$...
- 60.5k
2
votes
1
answer
404
views
Alexandrov curvature of a compact length space
I've found lots of (more or less precise) definitions of the Alexandrov curvature, but I'm mainly interested in that of "Alexandrov curvature bounded below". Could anyone give me that or give me a ...
- 151k
0
votes
2
answers
333
views
Explicit example of a smooth - but not analytic- closed curve without self-intersections
There exist smooth - but not analytic - closed curves without self-intersections. I just would like to see a simple example of such a curve.
- 1,883
2
votes
1
answer
1k
views
Lebesgue covering dimension
Roughly from wikipedia: The covering dimension of a topological space $X$ is defined to be the minimum value of $n$ such that every finite open cover of $X$ has a finite open refinement in which no ...
- 45k
10
votes
2
answers
1k
views
Dense sphere packings which are not lattice packings
This question is about dense sphere packings in euclidean space $\mathbb R^n$. By a sphere packing I understand any arrangement of mutually disjoint solid open spheres in $\mathbb R^n$, all of the ...
- 4,015
5
votes
1
answer
817
views
Generalization of Moise's theorem
I am looking for a generalization of Moise's theorem, which the few professors that I asked treat as a "known geometric fact" but none could find a reference to an article proving it.
The claim is ...
- 56.6k
4
votes
2
answers
682
views
Capacity of Balls in Hyperbolic Space
Given $M$ a Riemannian manifold and $\Omega\subset M$ the capacity of $\Omega$ is defined as
$$
\mathrm{cap}(\Omega)=\inf \int_{M\setminus\Omega}{|\mathrm{grad} \varphi|^2 dV}
$$
where $\varphi$ ...
- 45k
16
votes
0
answers
763
views
Lipschitz Homeomorphisms Between Spheres of N-dimensional Spaces
Let $B_p^N$ be the unit ball of $\mathbb{R}^N$ under the $\ell_p^N$ norm.
Question: Let $C_N$ be the infimum of all $C$ for which there is a homeomorphism $f_N$ from $B_\infty^N$ onto $B_2^N$ so ...
- 31.5k
5
votes
2
answers
629
views
Approximate search space on a 5x5x5 cube with 3 different possible classes?
Hey all,
I read the meta, and I realize this question might be pretty elementary for this site, but I'm having trouble computing this, and I know it won't take too much insight for someone to give me ...
- 153
0
votes
1
answer
327
views
Minimum cardinality of the intersection of 2D rectangles
Let $S$ be a set of 2D points $(x,y)$ with positive real coordinates, i.e. $x,y>0$. An 2D rectangle $R$ is called an ${Origin-Rectangle}$ if it is decided by the origin $(0,0)$ and another point $(...
- 13k
1
vote
0
answers
246
views
How to derive an energy measure of metric deforming
The problem is an abstract from applied science.
Given an $n$ dimensional Riemann manifold with metric $\langle M, g\rangle$, we could define deformation of the metric $g(t)$ where $t\in [0,1]$, for ...
- 135
0
votes
1
answer
250
views
Hyperbolic isometries in cocompact Hadamard (i.e. cat(0) proper simply connected) spaces
Swenson proved in "A cut point theorem for ${\rm CAT}(0)$ groups" that a locally compact Hadamard space with a geometric action by a group $G$ admits a hyperbolic isometry (that lie in $G$).
Is it ...
- 281
1
vote
1
answer
248
views
Is there a good approximating polygon for every smooth set?
Suppose we have an open set S whose boundary is a closed Jordan curve that has a unique tangent at each point. Is it true that for every epsilon there is a P polygon contained in S such that there is ...
- 3,270
2
votes
3
answers
2k
views
Are there non-measurable sets with smooth boundary?
I learned analysis a while ago, so let me define what I want. Suppose we have a set whose boundary is a closed Jordan curve that has a unique tangent at each point. Is it true that this set is (...
- 13k
7
votes
0
answers
208
views
How do metrics behave under joining along a manifold embedded in the boundary?
How do metrics behave under joining along a manifold embedded in the boundary?
This is, more-or-less, part of Problem 4.66 in Kirby's List:
Problem 4.66 How do metrics (e.g. Riemannian, Lorentz, ...
- 1,897
0
votes
1
answer
526
views
How the distance between sets is called?
Hello,
I've recently write down some measure for sets and now I wonder how it is called or where it is described?
The measure itself is the following:
Let $A$ & $B$ -- two sets of values from a ...
- 25.8k
2
votes
1
answer
370
views
Large subgroups of the Hamming cube
Let's consider the abelian group $\mathbb{Z}^N_2$ equipped with the Hamming metric (the hypercube).
Suppose I have a subgroup of this hypercube (not necessarily a subcube) which is generated by a set ...
- 6,342
18
votes
2
answers
3k
views
A question about the proof of Mostow rigidity
I have recently been studying a proof of Mostow rigidity (along the lines of Mostow's original argument), and I'm left a little confused about something. We start with an isomorphism $\alpha: \Gamma \...
- 68.9k
1
vote
1
answer
342
views
Half-space comparison of perimeter
Claim: suppose that $E$ is a set of finite perimeter, and $H$ is a half space. Then $P(F\cap H)\le P(F)$. In words: restricting a Caccioppoli set to a half-space will not increase the perimeter.
My ...
- 320
13
votes
1
answer
329
views
Spectral properties of finite metric sets
Given a finite metric set $S=\{P_1,\dots,P_n\}$, one gets a real symmetric matrix $M=M(S)$
with rows and columns indexed by elements of $S$ by setting
$M_{i,j}=d(P_i,P_j)$.
It is easy to see that $M$...
- 11.4k
12
votes
3
answers
707
views
A "round" lattice with low kissing number?
Historically, the lattices with high density were studied intensively, e.g. E_8 lattice or Leech Lattice. However, there are situations that lattices with low kissing number are required. Specifically,...
CommunityBot
- 1
8
votes
1
answer
1k
views
Is there an elementary way to show the triangular inequality for this expression ?
Consider the space $X$ of all scalar products on $\mathbb{R}^n$. For a scalar product $s$ and a base $B:=b_1\ldots,b_n$ let $M_{s,B}$ denote the matrix, whose $(i,j)$-th entry is $(s(b_i,b_j))$ . ...
- 28.6k
8
votes
2
answers
991
views
Higher-order axiomatisations of Euclidean Geometry?
I am currently thinking about the possibility to axiomatise Euclidean Geometry using higher-order axioms. The idea is that all objects are points, and that we only have two primitive notions: A three ...
- 11
8
votes
0
answers
358
views
Coloring toroidal polyhedra with convex faces?
Consider a toroidal polyhedron, which is a topological torus, in which all faces are planar, two faces meet in at most an edge, and adjacent faces are not coplanar. The Szilassi polyhedron has 7 non-...
13
votes
0
answers
751
views
$\epsilon$-nets with respect to the cut norm
The cut norm $||A||\_C$ of a real matrix $A = (a_{i,j}) \in \mathcal{R}^{n\times n}$ is the maximum over all $I \subseteq [n], J \subseteq [n]$ of the quantity $\left|\sum_{i \in I, j \in J}a_{i,j}\...
CommunityBot
- 1
3
votes
0
answers
233
views
How many set partitions on a big cube’s boundary arise from cubomino decompositions of the solid cube?
Introduction. This is a counting question about configurations that can appear on the outside of assembled Soma cube-like puzzles. More specifically, it’s about the ways in which the pieces of an ...
- 588
4
votes
1
answer
488
views
Are there non-tiling polyhedra that pack arbitrarily well?
The fact that an upper bound on the packing density $< 1$ has only recently been exhibited for regular tetrahedra in $\mathbb{R}^3$ (see this question) suggests that proving concrete bounds of ...
CommunityBot
- 1
2
votes
1
answer
802
views
Points on circumsphere of n-simplex
Project an n-simplex of side length $a$ on it's ($n-1$)-dimensional circumsphere by a ray starting at the center. Denote the images of the $n+1$ faces of dimension $n-1$ of the simplex by $A_1,\dots ...
- 25.1k
8
votes
1
answer
673
views
Estimating the Volume of the Metric Polytope
A metric on $n$ points $N$ can be represented as a vector $x \in \mathbb{R}_+^{n \choose 2}$.
For each pair of distinct $i, j \in N$, we have $d(i,j) = d(j,i) = x_{i,j}$. The set of all metrics is ...
CommunityBot
- 1
18
votes
2
answers
4k
views
Turning pants inside-out (or backwards) while tied together
An entertaining topological party trick that I have seen performed is to turn your pants inside-out while having your feet tied together by a piece of string. For a demonstration, check out this ...
- 11
5
votes
1
answer
636
views
analogues of Cayley plane as homogenous spaces
The Cayley projective plane $\mathbb{OP}^2$ can be defined as a homogenous space $\mathrm{F_4/Spin(9)}$, where $\mathrm{F_4}$ is the compact exceptional simple Lie group. The other possible approach ...
- 8,597
7
votes
2
answers
611
views
Escher, Conway, Kali, etc.
One can express the symmetry types of, say, Escher's "Circle Limit" prints using
Conway's orbifold notation, best known in the context of symmetries of Euclidean
plane patterns.
For example, Circle ...
- 96.4k
9
votes
3
answers
752
views
Non-Kahler manifolds where the different Laplacians are compatible
On a Kahler manifold, the different Laplacians are compatible: $\Delta_d=2\Delta_{\bar{\partial}}=2\Delta_{\partial}$.
Are there non-Kahler Hermitian manifolds where the above identity holds?
- 7,902