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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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Rigid motions taking a regular $n$-gon back to itself carry vertices to vertices

I have been reading Keith Conrad's expository paper Dihedral groups I and I have two questions about Theorem $2.2$, which deals with the size of $D_n$. In the first part of the proof you can read ...
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Determining Graph Automorphisms by Determining Ways of Permuting Edges

Here is an example from the book Groups, Graphs, and Trees (Ignore example 1.15; I accidently included it in my snippet of the pdf): From my understanding, the symmetry group of a graph $\Gamma$ ...
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Lie symmetry methods of differential equations.

If we have a differential equation $$ y^{(n)} = H( x, y, y', ...y^{(n-1)}) . $$ And say we have its two infinitesimal generators admitted by this equation. What do I mean by the invariance of this ...
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center of symmetry formula

How to prove that $I(0,-1)$ is the center of symmetry of the function $$F(x)= x - \dfrac{2e^x}{(e^x -1)}$$ Is there any formula that I can directly apply?
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What is the operational way of discovering scale invariance of differential equations?

Context The answer here by @Keenan Pepper gives an instance for what it means for an algebraic or trigonometric formula to be scale invariant. For quick reference, I quote his answer here but with a ...
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Solve matrix equation $XAX^*=B$ for $X$ in least squares sense

Problem How can the following matrix equation be solved $\underset{\mathbf{X}}{\mathrm{argmin}} \left\Vert \mathbf{X}\mathbf{A}\mathbf{X}^* - \mathbf{B} \right\Vert_F$ where $\mathbf{X} \in \...
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Fitting a grid inside a circle, is the solution always symmetric?

Let us construct a grid consisting of rectangles of height $h$ and width $w$. When we place a circle of radius $r$ over this grid, there is a certain amount of rectangles that completely lie within ...
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The condition for the existence of a symmetric form for the reflection formula $f(1-x)= \chi (x) f(x)$

Suppose we have a functional equation in the form $$f(1-x)=\chi (x) f(x)$$ with given function $\chi (x)$. What is the condition on the function $\chi (x)$ so that we can write this reflection ...
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1answer
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Soft: A game where players use an arbitrary structure as a guideline while trying to play identically.

The following two player game is an (inadequate) attempt to capture an idea I am very interested in exploring. If any of the learned folks here recognize the problem I'm playing with and can point me ...
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Partitioning evens as sum of evens

Take the set $\{a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8\}$. We can partition according to rules. Every member in the partition has even number of elements. Every member in partition have to be consecutive. ...
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A map which commutes with Hodge dual is conformal?

$\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\Cof}{\operatorname{Cof}}$ $\newcommand{\Det}{\operatorname{Det}}$ $\newcommand{\id}{\operatorname{Id}}$ Let $V$ and $W$ be $d$-dimensional, ...
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Practical use of order of rotational Symmetry.

Has anyone know what is the use of finding the order of the rotational symmetry of a figure? A student of mine ask that question from me. I search it but could not find any. Plz help. https://www....
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Does the orthogonal complement determine the inner product up to scaling?

Let $V$ be a real $n$-dimensional vector space, and let $g,h$ be two inner products on $V$. Fix some $1\le k\le n-1$, and denote by $\text{Gr}_k(V)$ the Grassmannian of $k$-dimensional subspaces of $V$...
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Difference between rotations, radial symmetry and central symmetry

After searching the web and not understanding the mathematical explanations of the different types of symmetry I want to ask: What is the difference between rotations, radial symmetry and central ...
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How to express a function's invariant transformations in terms of classical and exceptional groups?

Suppose I have a function $f$: $\mathbb{R}^n \rightarrow \mathbb{R}$. I'm interested in the symmetry group of $f$, i.e. transformations $T$ such that $f(T(x)) = f(x)$ for all $x$, with composition as ...
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symmetry in natural numbers

Given a finite set $E$ we can associate a group of symmetries or permutation $S_n$ where $n=|E|$ is the cardinal of $E$. My question is what if the set is infinite or more precisely countable ? Is ...
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If a polynomial function has an axis of symmetry at x = c, is it true that c is equal to the average of the roots?

I was trying to learn more about cool substitutions to make my life easier when solving polynomial equations. For example, in the problem $$x(x+1)(x+2)(x+3) = 24,$$ the LHS is symmetric at $x = -3/2$...
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1answer
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What function symmetric and has unique solution?

I have multiple operands, say a, b, and c. I want an operator acting on a, b, and c, but the result should be invariant of order of operands (symmetric), and there should be no other pair with same ...
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Wassertein and symmetry

So here's a scenario: I have points $(\mu_1^j,\mu_2^j)$ and I associated them the following distribution $$\rho_j=1/2\delta_{\mu_1^j}+1/2\delta_{\mu_2^j}$$ These have symmetry (exchanging $\mu_1^j$ ...
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1answer
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Is this shape a fair D-24?

I've been looking at polyhedra, incuding Platonic solids as well as Archimedean and Catalan solids. Catalan solids are face-transitive, which I believe implies that they are "fair dice", in the sense ...
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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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Reason: non-integer powers of Hermitian matrix not Hermitian?

Let $\alpha = 2.1$. If $A$ is symmetric real, $A^\alpha$ remains symmetric (although it could be complex-valued). If $B$ is Hermitian, $B^\alpha$ isn't Hermitian (losses the symmetry). However, for ...
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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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Proof that there exist only 17 Wallpaper Groups (Tilings of the plane)

I had a professor who once introduced us to Wallpaper Groups. There are many references that exist to understand what they are (example Wiki, Wallpaper group). The punchline is $$There \,\, are \,...
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Manifolds as Homogeneous Spaces

With very little effort one can, for example, show that $S^n$ can be written as a homogeneous space as $S^n\cong G/H$, where $G$ is the group of all rotations in $\mathbb{R}^{n+1}$ about the origin ...
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1answer
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Making the condition homogeneous

I saw it on AoPS, very great: https://artofproblemsolving.com/community/c6h1274759p6726915 JunBo-Yang used his own substitution for the condition: $a^{\,2}+ b^{\,2}+ c^{\,2}+ 3\,abc= 6$ , then: $$a= \...
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12answers
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Stuck on a Geometry Problem

$ABCD$ is a square, $E$ is a midpoint of side $BC$, points $F$ and $G$ are on the diagonal $AC$ so that $|AF|=3\ \text{cm}$, $|GC|=4\ \text{cm}$ and $\angle{FEG}=45 ^{\circ}$. Determine the length of ...
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1answer
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example of an “almost-metric” without symmetry

It is not difficult to find an "almost-metric" $d$ that satisfies all axioms of a metric except the triangle inequality. It should also be possible to construct a function $d$ that satisfies all ...
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What can I say about the form of an invariant function?

I have a general scalar function which has the properties: \begin{align} f(s\,a,b,c)&=s\,f(a,b,c)\\ f(s\,a,s\,b,s\,c)&=f(a,b,c) \end{align} where $s$ can be any real number, so the invariance ...
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Between $\ell^p$ and $\ell^\infty$ balls?

Take the $\ell^p$ ball in $\mathbb R^d$ of radius $r$ centered at the origin is $$\{x \in \mathbb{R}^d : \sum_{i=1}^d |x_i|^p \le r^p\}.$$ Enclose it by the smallest square possible which is the $\...
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Why this definition for “symmetry transformation”?

This question concerns section 8.5.1 in these notes: I don't understand why a symmetry transformation is defined as such. What implications is there if $\delta \mathcal L$ is a total derivative? ...
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Where is it used that the symmetry conserves the symplectic form in noether's theorem.

For the proof of Noether's theorem, it seems like that the only thing that's important for the symmetry map $S_g : M \to M$ is that it conserves the Hamiltonian (which will then imply that the moment ...
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Does the action of a linear map on $k$-dimensional subspaces determine it up to scaling?

Let $V$ be a real $d$-dimensional vector space, and let $1 \le k \le d-1$ be a fixed integer. Let $A,B \in \text{Hom}(V,V)$, and suppose that $AW=BW$ for every $k$-dimensional subspace $W \le V$. Is ...
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What does “super-integrable” mean?

What does "super-integrable" mean? Example from here (p. 2): …the Coulomb problem is super-integrable. Namely, it is not just rotation invariant, but as well admits further integrals given by ...
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Symmetricity of binomial distribution

For each random variable, $X$, define: $Sym_X(c) = \frac{\Pr[X \geq \mu_X + c]}{\Pr[X \leq \mu_X - c]}$. I use this definition to measure how symmetric the distribution is. Let $X$ be a binomial ...
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commutativity of rotations and reflections

The question is as follows: [Concerning the square embedded in the plane,] prove that the $90^{\circ}$ clockwise rotation $\sigma$ and the reflection through the north/south axis $\rho$ do not ...
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2answers
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Prove that $S_3/C_3$ is a (quotient) group

Consider $S_3$ to be the symmetries of a triangle, and let $C_3$ be a subgroup that cycles the three corners, so generated by: $(1 \ 2 \ 3 )$. Using Lagrange's theorem, compute $k = |S_3/C_3|$. ...
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1answer
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How is this cube symmetric?

I am having a hard time with this, but I am reading in a published journal that this cube is symmetric when distributing +1 and -1 in the following corresponding points. Positions 1-12: $$[+,+,+,-,-,+...
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Expressing musical patterns as a mathematical model?

My goal here is to compare visual symmetry and express the temporal symmetry of music in a similar visual manner. I'm not very well versed at math and hence I'm not sure how to go about this. ...
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Hungarian Algorithm on Symmetric Matrix

I have a complete and weighted graph with an even number of vertices. I would like to separate all the vertices into pairs such that the sum of all the edge weights for each edge connecting the ...
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How to take the determinant of a rank(1,1) tensor?

I want to find the Jacobian matrix and its determinant of the generic infinitesimal transformation: $x'^\mu=x^\mu+\epsilon_\alpha\frac{\delta x^\mu}{\delta \epsilon_\alpha}$ where $\epsilon_\alpha$ ...
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What is symmetry in physics in the mathematical sense, using (Lie) groups

In physics, we sometimes say that, for example, a certain classical system has a certain symmetry, which is given by some group. I don't feel like I understand this well enough. Are there some good ...
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Properties of a step function

Consider the step function $\Delta: \mathbb{R}\rightarrow [0,1]$ $$ \Delta(x;\lambda,\mu)\equiv \sum_{j=1}^J \lambda_j\times 1\{\mu_j\leq x\} $$ where $\lambda\equiv (\lambda_1,...,\lambda_J)$ is a ...
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Flux through sphere symmetry?

Is the flux through a sphere centered at the origin of the vector field $\boldsymbol{F} = (-x,1,z)$ equal to $0$? If so, is there any simple symmetry which suggests it? I have done the calculation ...
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When does $P\text{SO}P^{-1} \subseteq \text{SO}$?

Let $P \in \text{GL}_n^{+}(\mathbb {R})$. Suppose that $P\cdot \text{SO}(n)\cdot P^{-1} \subseteq \text{SO}(n)$. Is it true that $P \in \lambda \text{SO}(n)$ for some $\lambda \in \mathbb{R}$? I ...
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1answer
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Archimedean spiral - symmetry test

It is usually stated in Precalculus textbooks that in polar coordinates when a relation between $r$ and $\theta$ passes a symmetry test then the curve described by that relation has that symmetry. ...
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Representation theory and symmetry in physical systems.

I've come multiple times across statements like these Representation theory remains the method of choice for simplifying the physical analysis of systems possessing (a high degree of) symmetry ...
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Symmetry of a planar system and product of inertia

I read the following sentence here: "The products of inertia occupy the off-diagonal positions and measure the asymmetry of the mass distribution with respect to the planes of the inertia frame of ...
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Discrete Spherical Symmetry Group

Take two spheres each having a certain number (say 5) of identical dots on them. What is the approach to proving/disproving that they are equivalent under the set of spherical rotations? One could ...
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Divisibility of kissing numbers

Denote by $ K(d) $ the kissing number in dimension $ d $. I have two questions : 1) does $ d\mid K(d) $ for all $ d $? 2) does $ d\mid D $ imply $K(d)\mid K(D) $?