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

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Testing for symmetry about a curve/line

In High School Algebra , after studying how to plot a graph of $f(x)$ (rather called $y$) against $x$ in Cartesian coordinates, we studied several tests to determine the symmetry of the plotted graphs ...
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1answer
26 views

Efficient assignment of tetrahedron's chirality

Suppose we have a regular tetrahedron delimited by four points $A_{1}, A_{2}, A_{3}, A_{4}$. There are 24 permutations of vertices, but there are only two distinct terahedra that cannot be ...
2
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2answers
86 views

Functional equation $P(X)=P(1-X)$ for polynomials

I have encountered the following problem : Find all polynomials $P$ such as $P(X)=P(1-X)$ on $\mathbb{C}$ and then $\mathbb{R}$. I have found that on $\mathbb{C}$ such polynomials have an even ...
2
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2answers
55 views

Dihedral groups: Relationship between symmetries and rigid motions

In Dummit & Foote, $D_{2n},\ n \geqslant 3$ is the set of symmetries of a regular $n$-gon, where a symmetry is a rigid motion of the $n$-gon which can be effected by taking a copy of the $n$-gon, ...
3
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0answers
38 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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0answers
61 views

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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0answers
22 views

Spherical functions which are invariant under a finite rotation group

Is there a nice, clean reference which lists a basis in terms of (linear combinations of) spherical harmonics for the $L^2$ space of functions defined on the sphere $\mathbb{S}^2$ which are invariant ...
1
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1answer
30 views

General structure of a 3 by 3 persymmetric matrix with zero eigenvalues

Here is an interesting problem that might be extended to higher dimensions, I am looking for different simple ways of describing a 3 by 3 matrix with one or two zero eigenvalues. This persymmetric ...
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0answers
21 views

Difference between symmetry algebra and symmetry group

What is the difference between symmetry algebra and a symmetry group? I just wanted to know if my understanding is right. Lets say we have a system of differential equations. Then the symmetry group ...
2
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2answers
59 views

How to “rotate” a function? Or, how to write a function which has a known, rotational symmetry with respect to another function?

EDIT 2: I've posted my "real" question here: http://mathematica.stackexchange.com/questions/115766/finding-closed-form-eigenvalues-of-a-particular-matrix I have posed my question formally in LaTeX ...
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0answers
20 views

Construction of a triangle using symmetry

I need to construct a triangle $\Delta \textrm{ABC}$ knowing that $t_a = AS$, $|AS| = 6\, cm$, $|\measuredangle \textrm{BCA}| = 30°$ and $|AB| = 5.5 \,cm$. I've been told that it's possible to do it ...
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0answers
44 views

Symmetry of Rank Two Tensor with Mixed Components

I understand that a rank two tensor (t) is classified as symmetric if $t^{ij} = t^{ji}$ or $t_{ij} = t_{ji}$. Later in my reading, I came across the following quote: It is not useful to speak of ...
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0answers
13 views

Lie Algebra: Dimension of a one-parameter group (dimension of orbit)?

I am reading a book about Applications of Lie Groups to Differential Equations by Peter Olver. Lets say we have a PDE with $p$ independent variables and $q$ dependent variables In Chapter 3.1 (page ...
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0answers
20 views

Property / Feature of a symmetric or skew symmetric matrix for comparison

I have a symmetric matrix of size $n \times n$, where $n$ may range from $10$ to $50$. Let us call this matrix as $M_1$. If from $M_1$, I delete an $m$th row and an $m$th column, all other elements ...
3
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0answers
48 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 ...
0
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1answer
43 views

determine whether group is additive

I am taking a stab at group theory and in some of the questions I am working on they don't explicitly state whether a group is additive or multiplicative. $\mathbb{Z}_4$ is additive (and that makes ...
0
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1answer
29 views

Why is the orthogonal group $O_2$ generated by rotations and reflection only?

Consider the following isometries in the Euclidean group of distance preserving maps of the plane $\mbox{Iso}(\mathbb R^2)$ which is generated by the following: Rotations $\rho_\theta$ about the ...
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1answer
27 views

Lie Algebra: Optimal system of one-dimensional sub-algebras of the heat equation

This is a follow up question to Invariants of a PDE by Lie Symmetries, as I tried to follow the reasoning from the book Applications of Lie Groups to Differential Equations (Peter J. Olver, Example 3....
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0answers
17 views

Reduction of functions with Lie group symmetries

If I have a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ with a Lie group G as a symmetry, $f(Ax)=f(x),\quad A\in G$ how might I go about obtaining a reduced function $\tilde{f}$ on $\mathbb{R}^...
0
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3answers
63 views

If $A$ and $B$ are arbitrary $n \times n$ matrices, prove that $(A^TB^TBA)$ is symmetric

My attempt: $(A^TB^TBA)^T$=$(A^T)^T(B^T)^TB^TA^T$=$(AB)B^TA^T$ $\ne$ $(A^TB^TBA)$ therefore $(A^TB^TBA)$ is not symmetric.
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1answer
79 views

Find a basis of the space V of all symmetric 3x3 matricies, and thus determine the dimension of V

I need help finding the general element or matrix of $V$. Do I need to find the basis of the nullspace and basis of the image to solve this problem?
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0answers
20 views

Identifying Symmetric Matricies

If $A$ and $B$ are arbitrary $nxn$ matrices, which of the matrices must be symmetric? $something^T$ means the transpose of "something". $A^T$$A$ $A-A^T$ $A^TB^TBA$
2
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2answers
47 views

What is the dimension of a certain vector subspace of invariant matrices?

$\newcommand{\Sig}{\Sigma}$ Let $\Sig$ be a diagonal matrix with strictly positive entries on the diagonal. Define $V=\{B \in M_n\mid B\Sig +\Sig B^T=\Sig B +B^T \Sig \}$ (where $M_n$ is the vector ...
0
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1answer
22 views

Does Linear Transformation transforms orthonormal bases of symmetric matrix into orthogonal vectors?

How would you show it? I went all over the book: looked at Linear Transformation definition again, looked at orthogonality of bases for symmetric matrices. I got to the point: Since L is linear ...
8
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2answers
254 views

Reflection with respect to a parabola

I know how to find a reflection with respect to one of the axis or with respect to the origin, but let's say I want to find the reflection with respect to a parabola, how do I do it? Let's say we have ...
4
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2answers
87 views

Matrix representations of particular generators of the full octahedral group

I want to find matrix representations of the generators $a, b, c$ of the full octahedral group in the presentation $$\{a,b,c \mid a^2=1,b^3=1,(ab)^4=1,ac=ca,bc=cb\}.$$ Is there a recipe to write the ...
2
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1answer
21 views

Combination of even and odd functions

Can someone help me how to show that any function $f(x)$ defined on a symmetrically placed interval can be written as a sum of an even and a odd function? What is the special role played by "...
0
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1answer
38 views

Showing a function is odd

I have this equation: $$ f(x) = \frac{2x^2+3}{x-2} $$ and I have to prove it has half-turn symmetry around the point (2,8). I know that for a function to have half-turn symmetry, it needs to have ...
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0answers
13 views

Symmetric and reflexive closure on positive integers

Finding the symmetric and reflexive closure of the relation $R = \{(a, b) | a > b\}$ on the set of positive integers. For symmetric closure I have: $R = \{(a, b) | a > b\} U \{(b, a) | a > b\...
3
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1answer
47 views

What is the group of rotations of a volleyball(pyritohedron)?

Practice test for Abstract algebra final, very stuck on this particular question.
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0answers
11 views

Determining if Poset based on Domain and Comparison Operator?

Can someone help me with how to think about the below problem? I know that a poset is a relation which is reflexive, antisymmetric, and transitive, but unless I'm dealing with finite sets I have a lot ...
0
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1answer
34 views

Proof concerning preservation of negative eigenvalues under matrix addition of symmetric matrices

I've been working on this proof from a test review for about an hour and a half trying to figure out what to do. I have also scoured the internet in an attempt to find a similar problem to hint at ...
3
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0answers
26 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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0answers
11 views

Translational Invariance

Consider a system of objects labeled by the index $I$, each object located at position $x_{I}$. (For simplicity, we can consider one spatial dimension, or just ignore an index labeling the different ...
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0answers
17 views

Symmetry of Fourier Transform Spectrum

Given the continuous Fourier Transform $X(f)=\int_{-\infty}^{\infty}x(t)e^{-j2\pi ft}dt$ A sufficient condition for it to exist is $E_{x}=\int_{-\infty}^{\infty}|x(t)|^{2}dt < +\infty$. We can ...
0
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1answer
25 views

Symmetric Matrices and Diagonalization

Hi, I am trying to figure this problem out, but I am having difficulty. What I do know is that since A is symmetric, then S must be orthogonal. Also that S^(-1) must equal S^t (Transpose). However ...
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0answers
31 views

Metrizability of space of unitary operators on Hilbert space

Let $\mathbb{H}$ be a (complex) Hilbert space. Define $\mathbb{P}$ as a projective space over $\mathbb{H}$, i.e. $\mathbb{P}=\left(\mathbb{H}\setminus\{0\}\right)/\sim$, where $f\sim g$ iff there ...
3
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1answer
57 views

Characterizing orthogonally-invariant norms on the space of matrices

Denote by $M_n$ the space of $n \times n$ real matrices. We say a norm on $M_n$ is orthogonal invariant if: $$\|OX \|=\| XO\|=\|X \| \, \, \forall O \in O_n,X \in M_n$$ I am trying to characterize ...
4
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2answers
74 views

All-Russian Olympiad question (sum of symmetrical functions)

(All-Russian Olympiad, $1995$, $11^{th}$ Graders, Final Round) Prove that every real function, defined on all of $\mathbb R$, can be represented as a sum of two functions each of which has a vertical ...
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1answer
50 views

Commutation relation between su(N) and clifford algebra generators.

Why does the $ \gamma_5 $ matrix commute with the generators of the $su(N)$ algebra? In the case of the chiral symmetry from physics, [$Q_a$, $Q_b^5$] = $i \epsilon_{abc}Q^5_c$ where the $Q_a, Q^5_a$ ...
3
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0answers
24 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 \,: \...
2
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1answer
20 views

Can the isometry group of a metric space determine the metric?

Let $(X,d)$ be a metric space. There are always other metrics on $X$ which generates the same topology, and have the same isometry groups, for instance $\tilde d =\sqrt d$. (The same will be true for ...
2
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0answers
48 views

A brief answer for the determinant of a matrix

I am given the following matrix $A=(a_{ij})_{6 \times 6}$, where $a_{ij}=\sum_{k=1}^{10} x_k^{i+j-2}$. Remark: If $A=(a_{ij})_{10 \times 10}$ with the same $a_{ij}$ defined above, the answer is very ...
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1answer
60 views

a*b = a/b = b/a (what's this symmetry called?)

I was playing around with numbers the other day, and I found an interesting symmetry, that I would like to know if it has any specific name assigned to it. Let's assume the notation n:a to refer to ...
3
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5answers
104 views

Fastest way to show that $D_6 \to S_5$ is an injective homomorphism

I want to show that there is an injective homomorphism from $D_6 \to S_5$ where $D_6$ denotes the dihidral group of order 12 and $S_5$ the symmetric group. But I'm not sure how I can do this ...
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3answers
67 views

Does $y=(ax^{2}+bx+c)/(dx+e)$ have any lines of symmetry? [closed]

Does $y = \dfrac{ax^2 + bx + c}{dx + e}$ have any lines of symmetry. If it does, what are they, and if not, how would one prove that it doesn't have any lines of symmetry? (Please consider the ...
0
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1answer
30 views

Does $y=2/x$ have any lines of symmetry?

Lines of symmetry for $y=1/x$ are $y=x$ and $y=-x$. Does $y=2/x$ likewise have lines of symmetry?
0
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1answer
16 views

Surjective mapping of matrices under rotational and reflection symmetries

Let me preface this by saying that I'm not a mathematician and that I'm having a hard time stating my problem in the proper terms. Nevertheless, I'm faced with a problem for which I think an elegant ...
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1answer
29 views

Show if lines $L_1$ and $L_2$ are parallel then reflection $r_{L_1L_2}$ is a translation

I need to show that if lines $L_1$ and $L_2$ are parallel then the reflection $r_{L_1L_2}$ is a translation. I can draw two lines that are parallel and show how a reflection in both results in a ...
0
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2answers
22 views

necessary and sufficient conditions under which a symmetric matrix X

How to answer this question ? Provide necessary and sufficient conditions under which a symmetric matrix $X$ can be written as $X =A^T A$ for some matrix $A$.