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

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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
777 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
46 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
16 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 ...
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0answers
31 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 ...
3
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1answer
73 views

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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1answer
40 views

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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1answer
42 views

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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3answers
67 views

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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1answer
17 views

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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0answers
28 views

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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1answer
28 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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0answers
59 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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1answer
30 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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1answer
76 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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1answer
23 views

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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1answer
34 views

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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1answer
13 views

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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0answers
179 views

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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1answer
67 views

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 - $2{{e}^{-({{x}^{2}}+{{y}^{2}})}}\sqrt{{{x}^{2}}+{{y}^{2}}}$
0
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1answer
26 views

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 ...
3
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1answer
24 views

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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1answer
65 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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1answer
56 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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0answers
21 views

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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1answer
40 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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1answer
234 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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0answers
56 views

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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1answer
27 views

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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1answer
42 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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1answer
52 views

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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1answer
27 views

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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1answer
62 views

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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209 views

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 the Riemann Hypothesis 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 ...
0
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1answer
34 views

Definition of symmetry

Let $X \subseteq R^2$. A symmetry of $X$ is isometry $f: R^2 \to R^2$ such that $f(X) = X$. For example, square has $8$ symmetries one of which is $R_{90}(a, b) = (-b, a)$. Is an element of $X$ ...
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1answer
47 views

Polygons with fixed number of axes of symmetry

Let $k$ be a positive integer. Suppose a polygon has exactly $k$ axes of symmetry. How many sides may the polygon have? A regular $n$-gon has $n$ axes of symmetry, so one answer is $k$. What are ...
2
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1answer
122 views

Maximum of a generalized Rayleigh quotient

Given two symmetric positive definite matrices $A,B\in\mathbb{R}^{n\times n}$ and $x\in\mathbb{R}^n$. How do I prove that the generalized Rayleigh quotient $R(A,B,x):=\dfrac{x\cdot A\cdot x}{x\cdot ...
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1answer
30 views

Progresive diagonalisation of a symmetric matrix

Given a symmetric real matrix $A$ there is an orthogonal transformation that brings $A$ to a block diagonal form, where $a \in \mathbb{R}$: $$ RAR^{T} = \left[ {\begin{array}{cc} a & 0 \\ 0 ...
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1answer
179 views

Symmetric matrix if symmetric linear transformation

I want to proof the following theorem: With respect to any orthonormal basis, if the 2 $\times$ 2 matrix $\bigl(\begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr)$ represents a ...
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2answers
124 views

Need help to visualise a set (reading Abelian groups)

I am reading Abstract Algebra. I cannot visualise the following example: Let $n$ be a positive integer, and consider the set $S_n$ of all permutations from the set $n = {1, 2, \ldots , n}$ to ...
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0answers
25 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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1answer
192 views

determining reflective, symmetric, transitive, anti-symmetric properties and describing equivalence classes

The question: determine if p is reflective, symmetric, transitive and/or anti-symmetric, if p is an equivalence relation, describe the equivalence classes A = Z , and $apb$ if and only if $5 | (2 a + ...
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1answer
112 views

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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1answer
133 views

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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1answer
57 views

How can I compute angles & lengths of the following tiling shape

a while back I created a tile made of arrows: I did it using a vector graphics software, without really understanding the properties of this shape. Now, let's say I want to write a program to ...
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0answers
23 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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2answers
43 views

Proving a matrix is always symmetric [duplicate]

$B$ is a square matrix of real numbers. Show that the matrix $BB^T$ is always symmetric.
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1answer
105 views

How is $r(\theta) = \sin \frac\theta2$ symmetric about the x-axis?

I understand how it is symmetric about the $y$-axis. because $r(-\theta) = \sin \left(-\frac\theta2\right)=-\sin \left(\frac\theta2\right)=-r(\theta)$ But how is it symmetric about $x$-axis?
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1answer
151 views

Why is the dihedral group closed under composition?!

I've been obsessing over this all day now. I understand associativity, presence of inverse elements and identity, but I don't get why a composition of a reflection with a rotation or other reflexions ...
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1answer
44 views

Symmetry of polar equations

In your opinion how to show symmetry in polar equations without graphing. i thought of these methods :- converting to cartesian then test . check the period of the function . please help me any ...