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Questions tagged [symmetry]

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Is a symmetric, parallel (0,2)-tensor a metric?

I'm interested in affinely connected spaces, on which a metric is not necessarily defined, i.e. $(\mathcal{M},\Gamma)$. Since (as a physicist) my goal is to consider a generalized model of gravity, I ...
3
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1answer
159 views

How large can a symmetric generating set of a finite group be?

Let $G$ be a finite group of order $n$ and let $\Delta$ be its generating set. I'll say that $\Delta$ generates $G$ symmetrically if for every permutation $\pi$ of $\Delta$ there exists $f:G\...
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0answers
95 views

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-...
3
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1answer
61 views

Maximal edge length of symmetric polytopes

For me, a polytope is the convex hull of finitely many points. It is said to be vertex-transitive / edge-transitive if its symmetry group acts transitively on its vertices / edges. Let's call a ...
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65 views

Calculate the rank of a symmetric matrix

I am interested in calculate the rank of symmetric matrices. Currently I use rankMatrix of the Matrix package of R to calculate it. However it is slow. I was thinking that it might be due to not using ...
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0answers
73 views

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}{...
5
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3answers
403 views

Counting without one-to-one correspondence? [closed]

Ash and Gross in their wonderful book Fearless Symmetry found it worth mentioning (and thus suggesting) another way of counting for which "we do not even need to know how to count" (in the sense of ...
2
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0answers
43 views

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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How rich is the class of vertex- and edge-transitive polytopes?

There are only a few regular polytopes (five in 3D, six in 4D, three in any dimension above). In contrast, the class of uniform polytopes becomes very rich with higher dimensions. The class of vertex-...
2
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0answers
73 views

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 ...
3
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2answers
195 views

How to use these higher symmetries and conservation laws?

For infinite dimensional integrable systems, there are usually infinite symmetries and conservation laws. For example, the KdV equation, the KP equation. However, unlike the classical symmetries (...
3
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1answer
154 views

Symmetry in the triangular distribution

A continuous function vanishes on $(-\infty,a]$ and on $[c,\infty),$ its graph is a straight line on the interval $[a,b]$ and another straight line on $[b,c],$ and its integral is $1.$ The mean of ...
3
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0answers
149 views

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 $n$ ...
2
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1answer
50 views

Relation between symmetries and asymptotics for Painlevé equations

The first Painlevé equation $$P_I:y''=6y^2-x $$ has the symmetries $$x \mapsto \omega x \\y \mapsto \omega^3 y$$ for any fifth root of unity $\omega$. At the same time, the near-infinity asymptotics ...
5
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1answer
193 views

Odd function on the 2-sphere whose integrals over all hemispheres is zero

Let $h:\mathbb{S}^2 \to \mathbb{R} $ be a smooth function satisfying: $h(-x)=-h(x)$ For every hemisphere $A \subseteq \mathbb{S}^2$, $\int_{A}h\text{Vol}_{\mathbb{S}^2}=0$, where $\text{Vol}_\mathbb{...
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2answers
568 views

Is there a simple system that has $\text{SU}(3)$ symmetry?

The buckle at the end of a belt has $\text{SU}(2)$ symmetry, if the rotations around the three coordinate axes are taken as generators. See, for example, the paper by Hart, Francis and Kauffman, ...
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0answers
32 views

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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0answers
76 views

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?
4
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2answers
348 views

How to find the symmetry group of a differential equation

If one is given a differential equation, e. g. the KdV equation $\ u_t + u_{xxx} + uu_x = 0$, how can he find all of the symmetries of the differential equation? Is there also a method that works for ...
4
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0answers
99 views

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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0answers
85 views

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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0answers
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Symmetry-finding with SAGE?

On pp. 152-3 of Hydon's Symmetry Methods for Differential Equations (2000 ed.), he lists some computer packages for symmetry-finding. This related Mathematica StackExchange question mentions the SYM ...
1
vote
1answer
105 views

How large can the cone of $\nabla$-compatible metrics be?

Let $E$ be a smooth vector bundle over a manifold $M$, equipped with a connection $\nabla$. The set of $\nabla$-compatible metrics on $E$ forms a convex cone. This cone can be empty, however (see ...
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0answers
95 views

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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0answers
248 views

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^{\...
10
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1answer
304 views

A symmetric embedding of manifolds

Assume that $M$ is a manifold. Is there an embedding of $M$ in some $\mathbb{R}^{n}$ such that the image of $M$ in $\mathbb{R}^{n}$ is invariant under each reflection $(x_{1},x_{2},\ldots x_{i},\...
4
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1answer
199 views

Convex polyhedra, combinatorial types and Symmetry

Steinitz theorem says that combinatorial types of convex polyhedra is identified with 3-connected planar graph, called by a polyhedral graph. A symmetry of polyhedral graph means that a vertex ...
3
votes
1answer
133 views

Similarity via symmetric matrix

Let $K$ be a field extension of $F$. If two $n\times n$ matrices $A,B \in M_n(F)$ are similar via a matrix $P \in GL_n(K)$ (that is, $A=PBP^{-1}$), then there exists a matrix $Q\in GL_n(F)$ such that $...
4
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1answer
343 views

Totally geodesic subgroups in Lie groups

Let $G$ be a Lie group with a left invariant metric $g$. Let $H$ be a (closed) Lie subgroup of $G$, and assume $g$ is right-$H$-invariant. (That is $d(R_h)_e:T_eG \to T_hG$ is an isometry for every $...
2
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1answer
134 views

Heuristics for counting degrees of freedom

I have recently learned about the representation theorem for isotropic, linear operators, which says the following: Defintion: Let $M_n$ be the vector space of $n \times n$ real matrices. We say a ...
3
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0answers
661 views

Find the axis of symmetry in a point cloud

I have some dataset which describes a spherical cloud of points in 4D space. Actually, the coordinates of the points are the coefficients of unit quaternions, so you get the idea on what the data is ...
2
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1answer
92 views

Finding two structures are symmetry distinct

Would like to find if given two crystal structures whether they are symmetry distinct or symmetry equivalent in fastest (in terms of computation) possible way. The structure may have same lattice ...
8
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3answers
492 views

Parameterizing rotations of a cube

For $g\in\mathrm{SO}(3),S\subseteq \mathbb{R}^3,$ define $g\cdot S:=\{g\cdot p : p\in S\}.$ In words, if $g$ is a rotation of $\mathbb{R}^3$, $g\cdot S$ is the set of elements of $S$ rotated by $g$. ...
13
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1answer
239 views

What is the symmetry group fixing norms of elements of a unitary matrix?

Let $N\geq1$ be an integer and let us say that two matrices $U,V\in U(N)$ are related if $|U_{ij}|=|V_{ij}|$ for all indices $1\leq i,j\leq N$. When exactly are two unitary matrices related in this ...
4
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1answer
635 views

What is the current state of generalizations Noether's theorem?

The well-known Noether's theorem is a vital tool in classical physics. But it assumes some hypothesis, many of which could be removed by a detailed look. So my question is: In what directions has ...
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2answers
2k views

Who first introduced the functional definition of symmetry?

Who first introduced the definition of symmetry using functions explicitly? (That is, for instance, a symmetry of a subset $X$ of the plane is a function $F$ from the plane to the plane that preserves ...
2
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1answer
109 views

Is there always a symmetric “subset equilibrium” for an equilibrium in a symmetric game?

Consider a 2-player symmetric game given by a payoff matrix $A\in [0,1]^{n,n}$ for the row player (i.e. the column player matrix is $A^t$). Let $s=<s_1,s_2>$ be a (possibly mixed-strategies, ...
1
vote
1answer
197 views

True or false: if a set of 2D points has valid symmetry axes, then at least one of them is equal to a principal component vector

I posted this question on math.stackexchange but got no answer, so I decided to post it here instead. Sorry about the impreciness, not professional mathematician here. Let's assume we have a set of ...
3
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0answers
216 views

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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3answers
457 views

Symmetry Properties of Minimizers - Calculus of Variations

What methods are there to show symmetry properties of the minimizer of a problem $\inf_{u\in X}\mathcal{F}(u)$ in the calculus of variations? In general, the symmetry properties of $\mathcal{F}$ do ...
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0answers
87 views

“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” (...
8
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1answer
343 views

Is the operadic butterfly symmetric?

The operadic butterfly is a diagram in the category of operads in vector spaces. It extends the short exact sequence relating commutative, associative and Lie operads. $$\begin{array}{ccccc} & ...
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1answer
811 views

What is symmetry group of non-linear equation?

I am not very sure if this is a proper question, but I'm trying to investigate what the area of math can offer in researching the differential equation in polar coordinates: $r'^2+r^2=(kt)^2$, $r(t=0)=...
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0answers
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Semiflows and continuous symmetries

Given a differential equation on a Banach space $\mathcal{X}$ of the form $\frac{d u}{d t} = F(u)$, it is often the case that $F$ is equivariant under translations, i.e. that $T_\alpha F(u) = F(T_\...
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0answers
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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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7answers
2k views

Mathematics of quasicrystals

I want to study quasicrystals from mathematical point of view, but I'm having hard time finding materials about it. If you could suggest me some books, articles or papers, I would be glad.
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5answers
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Highly symmetric 6-regular graph with 20 vertices

I'm interested in (node/edge-)symmetric 6-regular graphs on 20 vertices and 60 edges, especially ones with a A5/icosahedral/dodecahedral symmetry group and especially their chromatic number. So far I ...
3
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2answers
794 views

Does graph asymmetry imply all eigenvalues of the graph Laplacian are simple?

It is well known that 1) if there exists a non-trivial automorphism of a graph $G$ with corresponding permutation matrix $P$ then if $(v,\lambda)$ is an eigenvector-eigenvalue pair of the graph ...
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2answers
611 views

Name this periodic tiling

I've been working on a problem I'm working on in ergodic theory (finding Alpern lemmas for measure-preserving $\mathbb R^d$ actions) and have found some neat tilings, that I presume were previously ...
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5answers
1k views

Which norms have rich isometry groups?

Let $n \ge 2$ be some positive integer. Given a norm $p : \mathbb{R}^n \to \mathbb{R}$, one can inquire about the structure and properties of its isometry group, i.e. the group of all bijections $F:\...