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Questions tagged [symmetry]

Questions about symmetry, in group theory, geometry or elsewhere in mathematics. See https://en.wikipedia.org/wiki/Symmetry

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Is there a theory of “almost symmetry” generalizing group theory?

Apologies for the inescapably soft question. Does there exist a theory that aims to develop tools analogous to those of group theory, except for the study of objects that are merely almost ...
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215 views

Seeing symmetries

Preliminaries Let $[n] = \{0,\dots,n-1\}$ and $P([n])$ be the power set of $[n]$. Let the correlation between two subsets $x,y$ of $[n]$ be the number $\kappa(x,y) = 1 - \frac{2}{n}|x\triangle y|$ ...
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152 views

Isometries of the canonical left invariant metric on $GL_n$

Consider $GL_n(\mathbb{R})$ with the left-invariant Riemannian metric obtained by left translating the standard metric on $T_IGL_n \cong M_n \cong \mathbb{R}^{n^2}$. (i.e for $X,Y \in T_IGL_n \,: \...
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303 views

ODE system and Lie symmetries

The ODE system (see below), where $F$ is a given function together the algebraic condition (SYM) imply that $$\boxed{y(t)=k-x(t)} \tag{*}$$ for some $k$ constant. The result that $y$ is a translation ...
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Is there a general theory of when certain polynomials are integrable due to symmetry tricks?

Consider the functions $x^2$ and $x^4 + 2x^2y^2$ on the unit sphere $S^2$. The surface integral of these functions over the sphere can easily be calculated by symmetry via $$3 \iint_{S^2} x^2 \mathrm{...
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373 views

Symbols to represent each distinct symmetry of polyhedra

Is there a pictorial or symbolic way to represent each distinct symmetry of a polyhedron?
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45 views

Complex $n \times n$ matricial representation of $SO(2n,\mathbb R)$

$SO(2n,\mathbb R)$ is the special orthogonal group of $\mathbb R^{2n}$, which is the group of $2n\times2n$ real orthogonal matrices with determinant 1. If we take $U(n)$ the group of $n \times n$ ...
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118 views

All the symmetries of the Dirichlet energy are conformal

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 claim: Proposition:...
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70 views

Existence of a potential for given exact form satisfying symmetry conditions

Suppose that $\omega$ is an exact differential form satisfying $L_X \omega=0$ for $X$ in some Lie algebra of vector fields. When can I find $\eta$ such that $\omega=d \eta$ and $L_X \eta=0?$ What are ...
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148 views

Symmetric Icon Fractals

I have always been fascinated by fractals. But most of all I like the Symmetric Icon fractals. There is a nice book about these fractals, written by Michael Field, called Symmetry in Chaos. I'm ...
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155 views

Pitchfork Bifurcation vs. Period-Doubling Bifurcation

I'm trying to understand how symmetry transformations give us indication of what kind of bifurcation occurs in a particular system, and I'm currently following Ghrist et. al.'s monograph on Knots &...
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For $d >1$ is there a lattice with the following properties except hexagonal lattice and $d=2$?

The properties which (as far as I know) are unique to the hexagonal lattice: All of the cells have the same shape and there is perfect translatonal symmetry when shifting between the cells; Each cell ...
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111 views

Is the space of rigid Riemannian metrics convex?

Let $M$ be a smooth manifold. Let $g_1,g_2$ be two rigid Riemannian metrics on it. (i.e with no isometries except the identity). Is it true that every convex combination of $g_1,g_2$ is also rigid? ...
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49 views

Symmetric Fibonacci images

I was playing with the Turtle module in Python and decided to try plotting the Fibonacci series with the following scheme where $f_n$ is the $n^{th}$ Fibonacci number: Rotate the turtle $k (f_n \...
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120 views

Symmetry groups of discrete functions

I'm looking for basic information about symmetry groups of discrete functions. It is difficult to search for such information, because searching for "symmetry group" gives results that refer almost ...
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150 views

How do you call functions that fulfill $f(x)=\pm f(\pm 1/x)$?

A function $f(x)$ that fulfills $f(x)=\pm f(-x)$ is called (a)symmetric even/odd. How do you call functions that fulfill $f(x)=\color{blue}\pm f(\color{red}\pm 1/x)$? "$\color{red}{\text{Positive/...
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28 views

symmetry of partial differential equations (Heat equation)

Morning everyone, I am doing some problem sheets for my class in Partial differential equations where we dont have an actual textbook. we are given a pack on notes. I am having an issue discerning ...
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45 views

Symmetries in the equations and graphs of complex-valued polynomials

I tried to visualize the complex roots of a polynomial with real coefficients $a_i \in \mathbb{R}$: $$f(z) = z^n + a_{n-1}z^{n-1} + \dots + a_1z + a_0$$ following some obvious thoughts: For any ...
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65 views

Symmetries of a curve on the unit sphere

Let $\mathbb{S}^2 \subseteq \mathbb{R}^3$ be the unit sphere. Let $\gamma : [0, l] \to \mathbb{S}^2$ be a smooth embedded closed curve. Let $\vec{\nu}$ be one of the two continuous unit normal vector ...
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27 views

Is a map which commutes with the codifferential an isometry?

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be smooth oriented $d$-dimensional Riemannian manifolds. Let $f:\M \to \N$ be smooth, and let $\delta=d^*$ be the adjoint of ...
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Probability inequality via symmetrization

Let $X$ and $X'$ be independent and identically distributed random variables. Define the symmetrized version of $X$ as $X^s=X-X'$. If $a \geq 0$ is such that $P(X \leq -a) \leq 1-p$ and $P(X \geq a) \...
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60 views

Maple: Symmetries of strange modified heat-equation

I thought about the following PDE (the $u(x=0,t)$ is not a boundary condition! It really is a part of the PDE): $$\dfrac{\partial u(x,t)}{\partial t}=\alpha \dfrac{\partial^2 u(x,t)}{\partial x^2}+u(...
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Using Burnside's Lemma in GAP to handle special variations of the Rubik's Cube?

If you want to count the number of distinct positions of a standard 2x2x2 Rubik's Cube simple counting arguments will suffice: There are 8 corners, all distinct The 8 corners can be in any ...
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157 views

Solution of wave equation

I posted this question for the first time on Physics Stack Exchange more than one year ago. The question was closed as off topic. Even if I reworked the question no one considered the possibility ro ...
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81 views

Fixed-point subspace of $O(2)^-$, a subgroup of $O(3)$

$O(2)^-$ is generated by the $SO(2)$ of rotations about the $z$-axis and a reflection through a vertical plane. The space $V_l$ is generated by spherial harmonics, i.e., Cartan decomposition $$V_l=...
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Can a L-shaped figure be divided into 5 congruent shapes?

Given a square, remove one quarter of it. Can the resulting L-shaped figure be divided into 5 congruent shapes? If not, how can we prove that fact? I tried using circles, triangles and smaller ...
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38 views

Maximum overlap of two convex bodies

Let $\Delta_n=\big\{(x_1,x_2,\ldots,x_n): \sum_i x_i \leq 1, x_i \geq 0\big\}$ denote an n-dimensional simplex. I am trying to find an $y \in \mathbb{R}^n$ such that the $y-$shifted negative simplex ...
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98 views

Symmetry transformations in quantum mechanics.

Let $\mathcal{H}$ be the separable Hilbert space associated to some quantum system, and let $\langle\cdot,\cdot\rangle :\mathcal{H}\times\mathcal{H}\rightarrow\mathbb{C}$ denote it's inner product. ...
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140 views

Is this Fractal New?

I developed some equations relating to symmetry. When used recursively, they produce what I believe is a fractal of symmetries. The fractal is procedurally generated like a snowflake or a gasket, ...
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Joint pdf of N > 1 i.i.d. random variables isotropic if and only if they are centered gaussian?

Are centered Gaussian densities given by $$f_X(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-x^2/(2 \sigma^2)}$$ the unique densities such that the joint pdf of $N > 1$ independent and identically ...
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98 views

Idempotents and symmetries of Zn

[Soft question out of curiosity] While reading about idempotents of a ring, I used $\mathbb Z_n$ as a convenient example. In visualizing the ring's structure, I was intrigued by strange symmetries ...
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86 views

Visualisation of representations and their decomposition into irreps

A question in a Representation Theory midterm got me thinking, and made me realise I didn't really understand irreps. The question was on the subject of reps of $S_4$, and went: An obvious ...
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485 views

An isomorphism between the full tetrahedral symmetry group and the cubic rotation group?

I know that the full symmetry group of the tetrahedron and the rotation group of the cube are both isomorphic to $S_4$. But I want to show a direct, visual isomorphism. I tried looking at a ...
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66 views

Fractals vs. “neatness” / order

I've seen a lot of high level videos on fractals, etc, and how they might apply to the real world. So a tree is branches with branches with branches, and our blood vessels branch and then branch ...
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37 views

Location shifted symmetric probability distributions

Let $G: \mathbb{R}\rightarrow [0,1]$ be a cumulative distribution function (CDF) symmetric about zero, i.e., $G(x)=1-G(-x)$ at each $x\in \mathbb{R}$. Take some real numbers $\mu_1,\mu_2$. Consider ...
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What are the generators of the signed symmetric group $B_n$?

Consider the set $A_n=\{-n,-(n-1),-(n-2),\cdots,-1,1,\cdots, n-1,n\}$ where $n \in \mathbb{N}$. Let G be a subgroup of the symmetric group $S_{A_n}$ defined as follow: $G=\{\sigma \in S_{A_n}| \forall ...
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60 views

Reduce an ODE by One Dimension

I am reading Arnold's ODE but I cannot solve this problem. This is on page 79. $\mathbf{Problem.}$ Suppose a one-parameter group of symmetries of a direction field in an $n$-dimensional domain is ...
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Why contains the product of highest dimensional representation (with dim$\ne 1$) with itself the rotation around the main axis?

I observe that in the "standard finite point group" symmetries containing higher dimensional irreducible representations (over $\mathbb{R}$), the total product of the highest dimensional ...
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39 views

On sums over non-symmetric sets

Definition: We say that a subset $A\subset \mathbb{N}_0\times \mathbb{N}_0$ is symmetric if it satisfies the property: $(a,b)\in A \Rightarrow (b,a)\in A.$ We say that the subset $A$ is non-symmetric ...
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29 views

Plane shape with three axes of symmetry

If a plane shape $\phi$ has three different axes of symmetry (all belonging to the shape plane), they all interesect at the same point. This is fairly obvious but I could not find an elementary ...
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254 views

What is the definition of quarter wave symmetry of even and odd functions?

What is the definition of quarter wave symmetry? As far as I understand the following definition should be the right one (but I'm not sure): For an odd function $f(\pi-x)=f(x)$ For an even function $...
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What techniques are used to find new solutions as perturbations of known solutions?

Consider the polynomial equation $x^2 + y^2 = 1$. We know that the solutions to this equation form the unit circle, but for the moment assume that we are only given that $(0,1)$, $(0,-1)$, $(1,0)$, $(-...
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Can we characterise concircular vector fields by their flow?

Let $M$ be a Riemannian manifold. A vector field $X$ on $M$ is called concircular if $\nabla X=h \text{Id}_{TM}$ for some $h \in C^{\infty}(M)$, where $\nabla$ is the Levi-Civita connection. Every ...
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Determinant of special partitioned matrix in terms of submatrix determinants

I have a few determinant-related questions that I've been struggling with for at least a few days. I couldn't see a similar question on here. So, here it is: I wrote my own electromagnetics moment ...
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66 views

Determining size of symmetry groups with orbit-stabilizer theorem and Burnside's lemma.

The classic orbit-stabilizer theorem and Burnside lemma problems tend to have the following structure: Consider some object with a symmetry group, like a cube and its rotational symmetry group. "...
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81 views

Symmetric solutions to second order boundary value problem

Given the equation $ u''=f(u)$ with symmetric boundary conditions $u(-a)=u(a)=u_0$, is there any proof that possibly relies on dynamical systems techniques to show that the solutions are symmetric (i....
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451 views

Find the group of symmetries of the Cube

Exercise : Find the group of symmetries of the Cube. Attempt : The elements are: $3$ rotations (by $\pi/2$ or $\pi$) about the centers of $3$ pairs of opposite faces. $1$ rotation (by $\pi$) ...
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Symmetry of an object suspended inside another object

I am from a chemistry background and I haven't taken abstract algebra, so it is possible that this question may be basic. This is the problem that I have: Suppose that I have an object A, which is a ...
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Are global symmetries of smooth functionals local?

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be smooth manifolds, $\M$ oriented and equipped with a Riemannian metric. Let $L : J^1(\M,\N) \to \mathbb R$ be smooth; We ...
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81 views

Translation invariance and Zero Eigenvalues

I'm currently working on a project that relates to the work of Sandstede, School and Wulff (1997) concerning centre manifold reductions. Background: In this work we consider an Reaction Diffusion ...