Questions tagged [symmetry]
The symmetry tag has no usage guidance.
46
questions with no upvoted or accepted answers
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Symmetry of function defined by integral
(Originally posed in Math.SE in Jan 2013. Received no complete answers as of yet.)
Define a function $f(\alpha, \beta)$, $\alpha \in (-1,1)$, $\beta \in (-1,1)$ as
$$ f(\alpha, \beta) = \int_0^{\...
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91
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A characterization of root systems via their intersections with halfspaces
In a recent preprint I obtained a nice characterization of root systems as a side product.
I can imagine that this was known before, and that a source for this statement can shorten the proof of my ...
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7
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194
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Surprising symmetry in the Ramanujan bound
The condition for a connected $(q+1)$-regular graph to be Ramanujan is that every nonzero eigenvalue $\lambda$ of the graph Laplacian satisfy
$$q+1-2\sqrt{q}\le \lambda\le q+1+2\sqrt{q}.$$
With a ...
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7
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Do cocycles “break” symmetry?
In an article by A. Borovik, “Is mathematics special?”, he talks about the fact that carrying is a cocycle. He then says
[A student] discovered that carry is doing what cocycles frequently do: they ...
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84
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Can a spherical simplicial complex have more than one "central" inversion?
Let $\Delta$ be a finite connected simplicial complex. Call a simplicial map $\phi:\Delta\to\Delta$ an inversion if
$\phi$ is an involution, that is $\phi\circ\phi=\mathrm{id}$, and
$\phi$ is not ...
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5
votes
0
answers
112
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Lie groupoids as symmetries of mechanical systems?
Lie groups are well studied as symmetries of mechanical systems in symplectic/Poisson geometry. For instance, if $G$ acts freely and properly on a mechanical system modeled by a symplectic manifold $(...
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5
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0
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135
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Is there a concentration inequality depending on dimension for a symmetric function on product space?
I recently read an elegant paper of Bobkov
Bobkov, S.G., On concentration of measure on the cube, J. Math. Sci., New York 165, No. 1, 60-70 (2010); translation from Probl. Mat. Anal. 44, 55-64 (2010)....
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4
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Can a polytopal graph be "centrally symmetric" in more than one way?
Let $P,Q$ be two centrally symmetric convex polytopes, potentially of different dimensions and combinatorial type, but with the same edge-graph $G$.
The central symmetry of $P$ induces an involutory ...
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4
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0
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113
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Can we combine the symmetries of two polytopes to create a more symmetric polytope?
Suppose that there are two combinatorially equivalent (convex) polytopes $P_1,P_2\subset\Bbb R^d$, that is, both with the same face lattice $\mathcal L$.
The symmetry group $\mathrm{Aut}(P_i)\subset\...
- 12.6k
4
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0
answers
48
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Equiangular lines with symmetry requirements
Listing all possible arrangements of equiangular lines is non-trivial.
Does the problem become any easier when we additionally require that the symmetry group of that line arrangement acts ...
- 12.6k
4
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0
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156
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Cencov's "categories of figures"
In his 1982 book Statistical Decision Rules and Optimal Inference, N. N. Cencov studies statistical models (parametrized families of probability distributions) from an unconventional category-...
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4
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238
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A simple proof that all the symmetries of the Dirichlet energy are conformal
This is a cross-post.
It seems to be folklore knowledge that all the (source) symmetries of the $d$-Dirichlet energy are conformal maps.
Specifically, I have found this nice proof for the following ...
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3
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0
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107
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Are square configurations the only critical points of the energy on the circle?
$\newcommand{\S}{\mathbb{S}^1}$
$\newcommand{\la}{\lambda}$Let$$M=\{(x_1,x_2,x_3,x_4) \in (\S)^4\,\, |\,\, \text{ all the } x_i \, \text{ are distinct}\} $$
Define $E:M \to \mathbb{R}$ by
$$E(x_1,x_2,...
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3
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0
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138
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Erlangen program for "network geometry"
The subject of network geometry (Boguna et al., Network Geometry,
Nature Reviews Physics 2021) looks at "geometric aspects" of complex networks.
This is about studying a metric on the nodes, ...
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3
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0
answers
65
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Classification of maximal point groups
Have the maximal (without finite proper overgroups of the same dimensionality) finite point groups been fully classified in any dimensionality of Euclidean space greater than 4?
- 2,689
3
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Are there uniform compounds of 135 $BC_8$ polytopes?
The Coxeter group $D_8$ is an index-135 subgroup of $E_8$. One of the consequences of this is that the rectified 8-orthoplex, whose coordinates can be given as all permutations and sign changes of $\{...
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0
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103
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Are there any other regular compounds?
Ever since I first read Coxeter’s definition of a regular compound (which seems to be the most commonly used), I didn’t like it on account of it being completely different than for properly connected ...
- 2,689
3
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0
answers
105
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Where can I learn about the discrete symmetries of the complex projective plane (or space)?
I understand that $CP^1$ is the Riemann Sphere. I guess all its discrete symmetries were known for a long time and well-classified. (But suggestions or good references where this is worked out in a ...
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3
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60
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Matroids which are transitive on minimal basis exchanges
I am looking for matroids in which all minimal basis exchanges look the same, that is, the matroid is transitive on these. Let me explain what I mean by that.
Consider a finite matroid $M$. Define a ...
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3
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0
answers
313
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Isometry group of an integer
This is a cross post from MSE, as it seems the partial answer I got then was deleted, so I ask it again here.
Let $\prod_{i\in I}p_{i}^{a_{i}}$ be the prime factorization of a positive integer $n$ ...
- 6,934
3
votes
0
answers
100
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Symmetries of irregular simplices
On the wikipedia page of tetrahedron, there is a list of eight symmetry groups for a (possibly irregular) $3$-simplex (with unmarked faces). There is also a list on the page of 5-cell but doesn't ...
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3
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301
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Diffeomorphism between open annuli preserving common symmetries
Suppose $A$ and $B$ are subsets of $\mathbb{R}^2$ homeomorphic (and thus $C^\infty$ diffeomorphic) to the open annulus (punctured $\mathbb{B}^2$) and let $G$ be the group of isometries of ${\mathbb R}^...
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30
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Do digraphs with "other" symmetries have interesting properties
Question:
do digraphs $G(V,A)$, whose adjacency matrix exhibits certain symmetries, have mathematically interesting properties?
The most famous such symmetry is $(i,j)\in A\iff(j,i)\in A$ for which ...
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2
votes
0
answers
65
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Internal symmetries of partial differential relation via the nonholonomic jet bundle
On a smooth n-dimensional Riemannian manifold $M$, suppose I have the kth order partial differential relation (PDR) written in the form:
$$\mathcal{R}=\mathcal{R}\left(x^{i},u^{i},u_{j}^{i},u_{jk}^{i}\...
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2
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0
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143
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Does this geometric PDE have a solution?
Let $s(\theta), b(\theta)$ be two smooth non-constant real-valued functions on $\mathbb{S}^1$, and assume that $s$ never vanishes.
Does there exist a map $h:(0,1) \times \mathbb{S}^1 \to \mathbb{S}^1$,...
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2
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0
answers
105
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Minimal symmetry of a fibre bundle
Let $F \to E \to B$ be a topological fibre bundle with fibre $F$ and base $B$. It can be characterized by a map $B \to BAut(F)$. If it can also be characterized as a map $B \to BG$ (or say $G$ is a ...
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2
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91
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An easy way to recognize the edges of an orbit polytope?
Given a finite (orthogonal) matrix group $\Gamma\subseteq\mathrm O(\Bbb R^d)$ and a point $x\in\Bbb R^d$. The corresponding orbit polytope is
$$\mathrm{Orb}(\Gamma,x):=\mathrm{conv}\{Tx\mid T\in \...
- 12.6k
2
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0
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126
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Is the projective symmetry group of a polytope more general than its linear symmetry group?
Give a (convex) polytope $P\subset\Bbb R^d$ (the convex hull of finitely many points). Consider its linear and projective symmetry groups:
\begin{align}
\DeclareMathOperator{\Aut}{Aut}
\...
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2
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0
answers
114
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Balanced Gray codes for powers of 2
All of the binary 4-bit cyclic balanced Gray code sequences can be formed from simple reversals, bit-permutations, and circular shifts of the one Wikipedia example:
...
- 1,537
2
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0
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68
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Are spherical maps with low distortion locally expanding?
$\newcommand{\SO}[1]{\text{SO}(#1)}$
$\newcommand{\Hom}[1]{\text{Hom}(#1)}$
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\S}{\mathbb{S}}$
The question in a nutshell: Are the "best" spherical maps ...
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74
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Can we approximate this matrix field with an invertible matrix field?
Let $\mathbb{D}^2=\{ x \in \mathbb{R}^2 \, | \, |x| \le 1\}$ be the closed unit disk, and set
$$\begin{equation*}
A(x,y)=\left(
\begin{array}{cc}
x & -y \\
y & x
\end{array} \right)
\end{...
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2
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0
answers
62
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Does a map which preserve harmonic forms preserve co-closed forms (locally)?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be $d$-dimensional oriented Riemannian manifolds ($d \ge 2$). Let $f:\M \to \N$ be smooth.
Let $1 \le k \le d-1$ be fixed....
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0
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What is the symmetry group of the totally nonnegative Grassmannian $Gr_{tnn}(k,n)$?
What is the symmetry group of the totally nonnegative Grassmannian $Gr_{tnn}(k,n)$? [The latter consists of those elements of the Grassmannian that can be represented by $k \times n$-matrices all of ...
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124
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Unveiling hidden structures
One way to unveil a hidden structure of a undirected graph - given as an adjacency matrix - is to permute the rows and columns until a pattern with a maximal geometrical symmetry is found. (The ...
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102
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A PDE involving a diffeomorphism of $\mathbb{S}^1$
This question is a special case of this one.
Let $s(\theta)>0, b(\theta)$ be two smooth non-constant real-valued functions on $\mathbb{S}^1$.
Do there exist a diffeomorphism $\phi:\mathbb{S}^1 \to \...
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vote
0
answers
120
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Symmetry of points on unit sphere determined by relation between triples of points
Suppose we have $n$ points on the 3D unit sphere,
$X = (\pmb{r}_{1}, \pmb{r}_{2}, ..., \pmb{r}_{n})$.
I am interested in knowing to what extent the rotational symmetry of $X$ is determined by the ...
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Projection of cocyclic Gaussian primes on the real axis
I just stumbled upon https://math.stackexchange.com/questions/2372062/4-concylic-points-of-gaussian-primes after a quick Google search about cocyclic Gaussian primes.
As I've been investigating about ...
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1
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0
answers
44
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How dense can a transitive sets of points be?
How dense can a finite set of points on the $d$-dimensional unit sphere be if I require that the symmetry group of that arrangement is still transitive on the points?
As a measure for density I use ...
- 12.6k
1
vote
0
answers
143
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Symmetric subgraph configurations
Let $G,H$ be two simple graphs. Let's call a subgraph of $H$ that is isomorphic to $G$ a $G$-subgraph. Consider the following construction:
Construction: Let $\mathcal G=\mathcal G(G,H)$ be a graph ...
- 12.6k
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vote
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answers
104
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Is every "higher-order" harmonic morphism conformal?
$\newcommand{\TM}{\operatorname{TM}}$
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\TM}{\operatorname{T\M}}$
$\newcommand{\TN}{\operatorname{T\N}}$
$\newcommand{\TstarM}{...
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Differential equations with infinite-dimensional Lie groups
I am no expert in solving DEs by symmetry methods, but from pure interest - is it possible for a differential equation to have an infinite-dimesional Lie group as a symmetry group?
- 3,680
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0
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30
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Minimal Non-Symmetric Closed Tracks Under 45-Degree Rotations
Given an arbitrary number of clockwise and anti-clockwise 45-degree track turns, what is the smallest closed track, that is neither axisymmetric nor rotationally symmetrical? For example, AAAAAAAA ...
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162
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Isomorphic Coxeter groups
After enumerating the spherical Coxeter groups, it is easy to see that no two distinct cases are isomorphic. Does the same hold for Euclidean and hyperbolic Coxeter groups?
- 2,689
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36
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Vertex configuration to tile repeat unit
I am working with elongated triangular tiling which has a vertex configuration of 3.3.3.4.4 and noticed that the representative symmetry is not the repeat unit. Is there a general formula to convert ...
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116
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"Box Nodes" in Directed Graphs with Paired IO Symmetry
Consider directed graphs where all nodes have 2 inputs and 2 outputs. If we
design a box with N inputs and N outputs, what is the smallest number of
nodes it must contain to satisfy “pair symmetry” (...
- 1,537
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53
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Using symmetries of a r.v.'s distribution to boost samples and possibly do variance reduction
Suppose, for example, you are simulating samples from a (multivariate) Gaussian with mean zero and covariance $\Gamma=BB^T$. If you had generated a sample $x$, you could generate more (dependent) ...
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