# Tagged Questions

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

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### How linear map transform the unit ball?

Let $f:\mathbb{R}^n \to \mathbb{R^n}$ be a linear application, we suppose that $f$ is symmetric ($\langle f(x),y\rangle=\langle x, f(y)\rangle$), without using spectral theorem how we can see that $f$ ...
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### Can we use a symmetry argument instead of integration in BASIC probability?

Suppose $H$ is a random variable with pdf $f_H(h)$. Let $X$ and $Y$ be random variables with joint pdf $$f_{X,Y} = f_H(x) f_H(y)$$ Prove $$P(X \ge Y) = 1/2$$ Is it possible to ...
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### Truncated objects coloring

I am looking for ways to color a truncated tetrahedron allowing rotations and reflections. I know the ways to color a tetrahedron in a similar way but stumped on this. From wikipedia, both tetrahedron ...
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### How many elements does group of symmetries of this logo have?

I could only rotate but not reflect it. So is it $2$?
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### What is the name for a function that behaves symmetrically when its arguments are scaled?

In other words, is there a name for this property of a function $f$: $$f(\alpha x_1,x_2,\ldots,x_n) = f(x_1,\alpha x_2,\ldots,x_n) = \ldots = f(x_1,x_2,\ldots,\alpha x_n)$$ Edit: I appreciate the ...
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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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### 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 ...
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### 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 ...
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### 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, ...
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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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 $Q_a, Q^5_a$ ...