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

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38 views

Is there a name for this kind of space?

Assume a Riemannian symmetric space $G/H$ where the decomposition of the Lie algebra of $G$ is $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$. It is a known fact that if $\mathfrak{h}$ is the Lie ...
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38 views
1
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1answer
381 views

a conjugate of a glide reflection by any isometry of the plane is again a glide reflection

Q : Prove that a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, and show that the glide vectors have the same length. I know that: the conjugate ...
1
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1answer
65 views

Group theory argument

I'm reading Group Theory in Physics by Wu-Ki Tung and on page 69 in the proof of Theorem 5.3 he makes a group theory statement that I don't get. Let me try give some notation and explanation ...
0
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0answers
47 views

Is this a spontaneous symmetry-breaking?

I have a system of equations: \begin{cases} f\left(x_{1}\right)+f\left(x_{2}\right)+P=0\\ \\ g\left(x_{1}\right)+g\left(x_{2}\right)=0 \end{cases} where $f,g$ are some functions, ...
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0answers
24 views

Classification of Differential Equations

I know that there is a theory of integrating (partial) differential equation by finding its symmetries (which form a Lie group) and making corresponding transformation of the domain. I also know ...
3
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1answer
29 views

Problem on symmetric matrices

Let $A$ be square non-singular matrix of order $n \geq 2$. If $A$ is symmetric, then $A^2$ is symmetric positive definite. If $A^2$ is symmetric positive definite, then $A$ is symmetric. I think I ...
1
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1answer
303 views

Points of symmetry of tessellation.

I was given this irregular hexagon: Then I was told to tessellate it: Now, I am being asked to find all the points on the hexagon (first picture) which are points of symmetry of my tessellation ...
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0answers
7 views

Modular arithmetic of rotation a triange

A triangle is an n-gon with 3 rotational symmetries and 3 reflection symmetries. Each rotation is 120 degrees. Suppose the triangle begins with initial angle 240 degrees and rotates through 240 ...
2
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1answer
91 views

Symmetries of a dodecahedron

Suppose we want to measure order of the group symmetries of a dodecahedron, and we know that If $G$ is a group and $S$ is a set on which $G$ acts and $s\in S$, then Order of G=(Order of stabiliser of ...
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0answers
74 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, ...
2
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0answers
32 views

Trade-off among symmetries

Take a set $X \in \mathbb{R}^2$ of nonzero measure $\mu(X) \neq 0$. I am attempting to design a set that has the following symmetries (continuous or discrete) $1.$ Scale symmetry $2.$ Rotation ...
4
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2answers
37 views

Determining the symmetry group of an infinite horizontal line.

I believe I have a satisfactory answer to the following question: Imagine we have a infinite horizontal line running through the origin, what is the associated symmetry group? Now thinking ...
1
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1answer
31 views

Are symmetric matrices necessarily positive-definite / positive semi-definite?

I am trying to prove this just to be clear about this but I don't have enough conditions to force this idea to be true, so I doubt it is. Are symmetric matrices always at least positive ...
0
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1answer
15 views

Diagonally dominant matrix for Cholesky?

I have a $10^6 \times 10^6$ dense SPD matrix, which I am called to invert, by using Cholesky factorization. However, I came across this statement: We start with the Cholesky and LU ...
0
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0answers
15 views

Proving a theorem on a rotation about a line followed by the inversion to show that it is a reflection

A theorem in my textbook is : A rotation about a line followed by the inversion about a point on that line is a reflection or a rotary reflection. I can picture this theorem in my head on a 3D space ...
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5answers
555 views

Definition of group action

I'm currently taking a class in abstract algebra, and the textbook we are using is Ted Shifrin's Abstract Algebra: A Geometric Approach. In the chapter on group actions and symmetry, he defines a ...
3
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0answers
116 views

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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0answers
11 views

Shifting a series of functions whilst maintaining symmetry

I have a function y = a*Exp[-(x - b)^2/2*c^2] + d*Exp[-Abs[-e*x]] + f Which is symmetrical when the coefficient of b is equal to 0 however it loses symmetry as ...
0
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1answer
17 views

Determine the formulas that represent the decomposition of vertices/edges/faces into orbits.

I've taken a basic stab at this problem. I feel like I am missing something big. Please help. Thanks! Q: Let $G$ be the group of rotational symmetries of a cube, let $G_v, G_e, G_f$ be the ...
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0answers
19 views

Which wallpaper patterns can coordinates be chosen so that the group $G$ operates on the lattice $L$?

For which of the seventeen wallpaper patterns can coordinates be chosen so that the group $G$ operates on the lattice $L$? FYI, this is an Artin textbook problem. Please help. Thanks!
0
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1answer
23 views

Describe the orbits of poles for the group of rotations of an octahedron.

Can anyone review my work on this problem and tell me if I'm missing anything major? Thanks! Q: Describe the orbits of poles for the group of rotations of an octahedron. There are $|G|=N=24$ ...
14
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1answer
396 views

Why does Group Theory not come in here?

Here is a list of questions that I find quite similar, for the one and only reason that they all show much "symmetry". Which is at the same time my problem, because I don't have a very precise notion ...
1
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1answer
26 views

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 ...
6
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3answers
499 views

What's the intuition of the transpose of a matrix? [duplicate]

I know the transpose is to swap the columns and rows of a matrix. And $A^T$$A$ is a symmetric matrix which elements are the inner product of each column of $A$. But I didn't understand the intuition ...
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0answers
46 views

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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0answers
32 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 ...
1
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1answer
24 views

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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1answer
74 views

What is the value of $\angle BMN$?

Here $\triangle ABC$ is a isosceles triangle with $AC=BC,\angle C=20^\circ$, $\angle ABM=60^\circ$ and $\angle BAN=50^\circ$. What is the value of $\angle BMN$? please help me with this problem.
1
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1answer
43 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 ...
0
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0answers
50 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 ...
0
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1answer
17 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 ...
2
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1answer
63 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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0answers
50 views

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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0answers
15 views

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 ...
1
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1answer
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? ...
3
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0answers
34 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 ...
1
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1answer
24 views

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 ...
1
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1answer
18 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 ...
3
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2answers
32 views

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 ...
2
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1answer
43 views

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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2answers
430 views

I have a 2x2 positive-semidefinite matrix. I am trying to find the equation of its elements.

So long story short. I have a matrix $A \in S^2_+$, that is, a symmetric, positive semi-definite 2x2 matrix. Here it is: $A = \begin{bmatrix} x & y \\y & z \end{bmatrix}$. Here is what it ...
4
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1answer
133 views

Immersions of the Klein Bottle

The famous immersion of the Klein bottle lacks symmetry. (I'm talking about this one. http://en.wikipedia.org/wiki/File:Klein_bottle.svg) One can only see one plane of reflection. However, the ...
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1answer
33 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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3answers
752 views

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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1answer
44 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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0answers
13 views

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 ...
3
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1answer
68 views

Programmatically recognizing symmetries of a polyhedron

I'm programming something, but I'm stuck at something which more math-oriented people probably can help me with. I am giving a polyhedron in the following form: for each vertex I get the cyclic order ...
2
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1answer
27 views

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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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 ...