# Questions tagged [symmetry]

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

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### Prove all lines intersect at the same point [closed]

If polygon has more than 2 symmetry lines, then prove all of them will intersect at the same point.
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### Is the general solution to the Heat Equation $u_t=\gamma u_{xx}$ a homogeneous function?

Is this solution correct? The heat equation is $$\frac{\partial u(x,t)}{\partial t} = \frac{x_0^2}{t_0} \frac{\partial^2 u(x,t)}{\partial x^2}, \mbox{where } \frac{x_0^2}{t_0}=\mbox{"gamma"}$$ ...
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### Dimension of $\text{Sym}^n(\mathcal{H})$

Let $\{A_i\}$ be a set of $d-$dimensional matrices. $A_1\otimes A_2$ refers to the tensor product or Kronecker product of $A_1$ and $A_2$. We now look at symmetric tensor products i.e. linear ...
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### What does “orientation” of a platonic solid really mean?

Is there any rigorous definition of "orientation" of a platonic solid? Lots of books mention that the whole group of symmetries of platonic solids consists of rotations composed with reflections, ...
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### Formalizing “the symmetry group of a non-square rectangle”

On Wikipedia I read that the Klein four-group is "the symmetry group of a non-square rectangle". I wonder about how to formalize this. The Wikipedia article on "Symmetry group" describes the symmetry ...
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### Finding the eigenvalues of a matrix with particular symmetry

I have a matrix for which I want to get some analytical equations of the eigenvalues. The matrix is given as \begin{align} \mathbf A &= \begin{pmatrix} \epsilon_a & 0 & 0\\ 0 & \...
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### Vector equation of a line that is symmetrical to another line L with respect to plane $\Pi$

The plane $\Pi$ is defined as - $$4x - 3y + z = 1$$ The line $L$ is defined as - $$\frac{x-4}{3} = \frac{y-1}{-1} = \frac{z-5}{2}$$ I am trying to find the vector equation of a line that is ...
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### Simple symmetries for two nonlinear partial differential equations

I tried to investigate symmetry of two relatively simple partial differential equations. Firstly I define what I mean by symmetry: change of variables $t \rightarrow -t$ or $x \rightarrow -x$ or ...
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### Which automorphisms of the plane preserve the hyperbola?

Is there a reasonable characterization of all the "power-law" diffeomorphisms of finite order $f:\mathbb R^{>0} \times \mathbb R^{>0} \to \mathbb R^{>0} \times \mathbb R^{>0}$, of the form ...
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### Symmetry group of a line segment in Euclidean spaces

I am self studying the chapter of Symmetry Groups from Gallian's Abstract Algebra. There I encountered the following paragraph " It is important to realize that the symmetry group of an object ...
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### Finding the pseudo-code of an algorithm that tests if a function is antisymmetric

I am searching for a way to write a pseudocode for a algorithm that can determine if a relation $R$ (on a (finite) set $X$) is antisymmetric. Now I was thinking that I might try to make a function ...
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### Is there a simple proof for the behaviour of this solution?

Let $0 <s \le 1$, and suppose that $0 <b \le a$ satisfy $$ab=s,a+b=1+\sqrt{s}.$$ Then $a \ge 1$. I have a proof for this claim (see below), but I wonder if there are easier or alternative ...
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### A PDE is invariant under the Lie point symmetries

I am reading a paper: https://onlinelibrary.wiley.com/doi/abs/10.1002/oca.2190 For the PDE (6) in that paper: $$V_t + V_{x_1}f_1(x_1,x_2)+V_{x_2}f_2(x_1,x_2)+Q(x_1,x_2)-\frac{b^2}{4r}V^2_{x_2}=0.$$ ...
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### Under which conditions is the integral of a symmetric function symmetric?

I have an indirect utility function - $P(a,b ;\theta)=max\{\theta x, (1-\theta)b\}$ - where $a$ and $b$ are positive, deterministic parameters and $\theta$ is a random variable. I would like to study ...
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### Is there a symmetric polygon for which the centers of connected points (that are equidistant to their neighbours on its boundary) do not overlap?

I am looking for (a class of) symmetric polygon(s) that can have any number (odd and even) of points on its boundary, equidistant to their neighboring points, and when all pairs of points are ...
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### positive definite how to prove that LU decomposition is possible

Given K a symmetrical, square and positive definite matrix, how to prove that LU decomposition is possible without the need of a permutation?
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