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

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Eigenvectors and geometrical transformation

$$A= \begin{pmatrix} 2/3 & 2/3 & -1/3 \\ 2/3 & -1/3 & 2/3 \\ -1/3 & 2/3 & 2/3 \\ \end{pmatrix}$$, I need to understand that kind of ...
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Spontaneous or not spontaneous symmetry breaking? That is the question.

I have the following system of ODEs: \begin{cases} \frac{du_{i}}{dt}=F_{i}\left(\boldsymbol{u},\boldsymbol{v}\right)+A, & i=1,\ldots M\\ \\ ...
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18 views

Show that the point group $\overline{G}$ can not be a subgroup of $G$ generated by a single glide reflection.

I have having trouble with the very last part of this problem: Q: Consider the group $G$ generated by a single glide reflection. Show that the point group is $D_1 = \{\text{id}, r\} \cong ...
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basis for symmetric square

If we have the symmetric square Sym$^{2}V$ and $V=\mathbb{C}^{2}$, why is it that $\{x^{2}, xy, y^{2}\}$ form a basis for it? So symmetric square matrices are when the main diagonal acts as a ...
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40 views

Which mathematical theory investigates distortions?

For transformations like rotations, translations or boosts Lie theory is the appropriate theory. Which theory talks in similar, systematic terms about distortions $$ M \vec{v} = m \vec{v} \quad ...
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48 views

Proving that a group is simple

I want to solve the following: Let $I$ be the group of isometries that preserve the orientation of the icosahedron. Use the class formula to show that $I$ is a simple group (i.e. It does not have ...
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16 views

Symmetry properties conserved after integration?

I have an integrand consisting of the variables a,b,w,x,y,z. Now I integrate over the variables w,x,y,z (w and x from 0 to 1 and y and z from 0 to infinity). I know that the resulting expression ...
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60 views

Simple question on group theory

Suppose we have the following system of differential equations: \begin{cases} \frac{dx_{i}}{dt}=f_{i}\left(\boldsymbol{x},\boldsymbol{y}\right), & i=1,\ldots M\\ \\ ...
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Basic questions regarding group theory and symmetry in practice

I have for some months been interested in group theory. I was very fascinated by the level of abstraction I first met when working with groups. Another aspect that has fascinated me lately is ...
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Formally correct “generator expression” for parameters of a function

I'm trying to express formally correct that a class of functions exists that have a certain property that applies to all concrete "instances" of this class. In that I try to write a "generator ...
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47 views

If $n\times n$ matrix A is symmetric, is $A^{10}$ symmetric also?

If $n\times n$ matrix A is symmetric, is $A^{10}$ symmetric also? All I can deduce is that $A^{T}=A$, but that doesn't give me much in trying to show that $A^{10}$ is symmetric. Any hints/suggestions? ...
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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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How does a Lie derivative generate a $U(1)$ isometry?

Consider a $2l$-dimensional Riemannian manifold $(M,g)$ without a boundary and let $V=V^{\mu}\frac{\partial}{\partial^{\mu}}$ be a Killing vector field, i.e. $$ \mathcal{L}_Vg_{\mu \nu} = 0 ...
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17 views

Motions that Leave a Prism Invariant

Identify the group of all motions that leave a right prism invariant. It seems like the normal dihedral group, $D_n$, with respect to the base would be a subgroup of this group. That is to say that ...
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is a line of symmetry going through the center of an octagon parallel to the ground it is on?

I came across this question recently, It wanted me to find angle EDG assuming the line crossing between the octagon/polygon is it's line of symmetry. My answer was 25 degrees, but A few friends ...
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Show that $A^TA$ has at least one positive eigenvalue if $A$ is not all-zero

I need some help on showing that $A^TA$ has at least one positive eigenvalue if $A$ is not all-zero. $A$ is rectangular and can have dependent columns in general. I can show that it cannot have ...
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31 views

Probability question in group-theoretic context

I'd like to know the probability that four randomly chosen symmetries of the cube generate the whole octahedral group $C_{2}\times S_{4}$... is there some quick way of working this out, i.e. avoiding ...
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What's so special about the group axioms?

I've only just begun studying group theory (up to Lagrange) following on from vector spaces and I am still finding them almost frustratingly arbitrary. I'm not sure what exactly it is about the ...
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43 views

Surfaces of constant curvature using the conformal method

I'm doing a study of surfaces with constant curvature which leads to solving the equation: $$\Delta\phi = -e^{2\phi}K_0$$ for a 2-dimensional metric with constant curvature such that rotation around ...
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Prove that the Gaussian quadrature points are symmetric if the interval is symmetric wrt the origin.

I need to find the proof for the symmetric Gaussian quadrature points in a symmetric interval. I thought it was a popular fact but I couldn't find it on the net. I also want to prove that the ...
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30 views

Symmetries on Hilbert spaces

Let $\mathfrak{H}$ be a Hilbert space and let $\mathcal{E}(\mathfrak{H})$ be the set of all operators $T\in B(\mathfrak{H})$ such that $0\leq T\leq 1$ (these operators are also called effects on ...
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Inverse of diagonally dominant matrix with equal off-diagonal entries

Is there an explicit expression for the inverse of strictly diagonally dominant matrix with identical off-diagonal elements? For example: $$ \begin{pmatrix} a & -b & -b \\ ...
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Doubt in Peter Olver “Applications of Lie groups to differential equations”

Book: Applications of Lie groups to differential equations. Second edition (1993). Page: 117-120. Chapter: 2. Section 2.4: Calculation of symmetry groups. Example: 2.41. The heat equation. Question ...
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Product of symmetric matrices

Let $A \in \mathbb{R}^{n \times n}$ be symmetric. I am trying to understand under which conditions on $B \in \mathbb{R}^{n \times n}$ the product $AB$ is also symmetric. It is clear that if $B$ is ...
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Symmetry of PDEs

I am trying to solve the following problem: Show that $$(u,t,x) \rightarrow (u, \frac{t}{c^2t^2 - x^2}, \frac{x}{c^2t^2 - x^2})$$ is a symmetry of the Wave equation $u_{tt} - c^2u_{xx} = 0$. Some ...
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online notes on symmetric spaces

Can anyone suggest some good online lecture notes on symmetric spaces? I am interested in reading from Helgason, which is a very tough book to read. So I am searching for some places where the ...
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Integral of symmetric function

Let $f:\mathbb{R}^n\to\mathbb{R}$ be such that $f(x_1,\dots,x_n)=f(x_{\sigma(1)},\dots,x_{\sigma(n)})$ for every $n$-permutation $\sigma$, and suppose that ...
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Solving a PDE to Yield Determining Equations

I'm going through an example in Peter Hydon's book "Symmetry Methods for Differential Equations" which finds the basis for the Lie Algebra of the point symmetry generators for Burgers' equations. ...
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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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29 views

Commutativity of matrix square root

Let $A, B \in \mathbb{R}^{n \times n}$ and let us assume that $A^{1/2}$ exists. I have often seen people write something like $$ AB = A^{1/2}\, B\; A^{1/2} $$ when both $A$ and $B$ are symmetric, in ...
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65 views

What is the smallest cubic bipartite asymmetric graph?

"According to a strengthened version of Frucht's theorem, there are infinitely many asymmetric cubic graphs." If there are any, what is the smallest cubic bipartite asymmetric graph? Kind of a ...
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Point symmetries around lines in 2D

I am trying to remember how to compute the symmetric point to an $(x=a,y=b)$ point with respect to a line, i.e. $y=mx +b$, without luck. Is there a closed form equation for this type of ...
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If $R_1$ and $R_2$ are symmetric and transitive, prove that also $R_1 \cup R_2$ is symmetric and transitive

I have to prove or confute (with a counter example) that, if $R_1$ and $R_2$ are symmetric and transitive, then $R_1 \cup R_2$ is also symmetric and transitive. Note: To prove the transitive case, ...
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Finding the Projection without using projection matrix always but to use some symmetry

I am interested in finding the t2' with the help of t1, t1' and t2. Actually I am using some projection matrix T (this will be used to project point x and y which is clear in the Image attached) on x ...
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Symmetries of Archimedean Solids

There are five platonic solids, and 13/15 (which is correct?) Archimedean Solids. The finite groups of isometries of Euclidean $3$-space are the finite subgroups of $SO(3,\mathbb{R})$ or ...
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How to find where the magnitude of the gradient of a function is maximized?

How to find where the magnitude of the gradient of ${{e}^{-({{x}^{2}}+{{y}^{2}})}}$ maximizes? I managed to calculate the magnitude - $2{{e}^{-({{x}^{2}}+{{y}^{2}})}}\sqrt{{{x}^{2}}+{{y}^{2}}}$
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Finding all operators that preserve a function

I'm not even sure what field of math this would be, and Googling "symmetry" and "functions" doesn't reveal what I'm looking for. Basically I want to find all $\{\hat{A}\}$ other than the identity ...
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3-D Realization of Symmetries of a Tetrahedron

We know that the total symmetry group of tetrahedron is $S_4$. I tried to realize these 24 symmetries, but failed to ''realize'' six of them, which correspond to the $4$-cycles $(a \,b\,c\,d)$ (being ...
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63 views

What's the exact definition of symmetry?

My question is basically the same as this, but I haven't found the answer given satisfying. The definition of symmetry I've come up with is this: Let $X \subseteq R^2$. A symmetry of $X$ is an ...
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36 views

Sketch a figure which has a group of symmetries of order 5.

I am trying to draw a shape which has only 5 symmetries I know Square has 8 Rectangle/parallelogram has 4 Triangle has 6 Circle has infinite how do i know which shape has only 5 I know that ...
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Several symmetry formulas?

I have lost the book of my math course one day before the test. I want to review several formulas, but I haven't found them anywhere online. The formulas I am looking for are: Symmetry of a line ...
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36 views

are isometries on euclidean plane translations

Let $m$ be an isometry in euclidean plane that changes orientation. Prove that $m \circ m$ is a translation.I do not have an idea how to start the proof of this exercise.
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135 views

Symmetric Properties of Roots (Quadratic Roots)

What is the proof that - $α^2$ + $β^2$ = $(α+β)^2$ - 2αβ $α^3$ + $β^3$ = $(α+β)^3$ - 3αβ(α+β) $α^4+β^4$ = ($α^3+β^3$)(α+β) - αβ($α^2+β^2$) (α+β)4 = α4 + 6α3β + $4α^2β^2$ + $6αβ^3$ + $β^4$ ...
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Is this proof show that the derivative of zeta function has no zeros in the critical strip?

suppose that :( $k \circ \zeta )(s)$= $(\zeta \circ k )(s) \neq 0 $.....$(1)$ ,and suppose that $ k(s)=\zeta(1-s)$ where : $\xi(s) = s(s-1) \pi^{s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s)$. and ...
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Fourier series Even vs. Odd and effect of integral bounds?

I understand that when you express a function in fourier series there are 3 coefficients you need to calculate ( a0, an, bn) and I have in the past made use of the symmetry of the function in my ...
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39 views

Combinatorial puzzle concerning labelled equilateral triangles

Consider equilateral triangles $\Delta$ of fixed size and in a fixed position with each side labelled by a label $l \in \{1,\dots,k\}$. Obviously there are $k^3$ such labelled triangles. Let ...
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What does it mean for vectors to be symmetrical?

Say we have three $n$-dimensional vectors $A$, $B$ and $C$. Now let's say that $A$ and $B$ are symmetrical about $C$. In two dimensions, this seems to have a fairly obvious meaning, but for higher ...
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Symmetries of a $9$ puzzle (Rubik's Slide)

Consider this Rubik's slide. With these moves (and their inverses): $$\text{Vertical shift}\: v=(147)(258)(369)$$$$\text{Rotation}\: c=(12369874)$$$$\text{Horizontal shift}\: h=(123)(456)(789)$$ Also ...
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Why is $D_5$ a subgroup of the icosahedral group

According to Wikipedia $D_5$ is a subgroup of the group of rotational symmetries of an icosahedron: http://en.wikipedia.org/wiki/Icosahedral_symmetry. I know this isn't very rigorous, but intuitively ...
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Is this claim true$(\xi \circ k)(s)=(k \circ \xi )(s)=0$ $\implies$ $k(s)=\zeta(s)=0 $ is true if and only if RH is false?

It is well known that $\xi(s)=\xi(1-s)$ is a verified functional equation for all complex $s$, where $\xi(s) = s(s-1) \pi^{s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s)$. let $k(s)=\xi(1-s)$ and ...