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

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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 $A$.
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1answer
36 views

“Minimising” a linear combination of orthonormal vectors

I am working with two $24\times1$ orthonormal vectors, and I wish to find the linear combination of these for which the maximum possible number of entries in the resultant vector are equal to (or ...
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1answer
46 views

Analysis of coefficients of $x^k+\dfrac1{x^k}$ polynomials

Given $x+\dfrac1x=n$, I derived several expressions in terms of $n$ to solve for $x^k+\dfrac1{x^k}$ and put them in a chart as shown below. My questions is how are the coefficients of these ...
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1answer
52 views

Show that $(x'Ay)x=(xx'A)y$

Suppose $x,y$ are vectors and $A$ is a symmetric invertible matrix. Show that $(x'Ay)x=(xx'A)y$. How can one prove the above? I am aware that matrix multiplication is not necessarily associative. ...
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41 views

How to use symmetry in triple integrals/ in $\mathbb{R}^3$

Hi I think I still get confused on one aspect of calculating triple integrals. I am used to utilizing symmetry in $\mathbb{R}^{2}$ , but I think there is something major I am missing in $\mathbb{R}^{3}...
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1answer
34 views

Why, given an object with rotational symmetry, is the axis of symmetry a principal axis?

When consulting textbooks and notes online about principle axes of inertia, I couldn't find a source which directly addressed the reasoning/proof behind following statement: "Given an object with a ...
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1answer
62 views

When does a linear combination of trigonometric functions have an axis of symmetry?

I am trying to find out when a linear combination of $\sin(ax)$ and $\cos(bx)$ has an axis of symmetry. Clearly, $\sin(x)+\cos(x)$ has an axis of symmetry at $\pi/4$. It seems as if $\sin(3 x)+\cos(...
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80 views

Can a L-shaped figure be divided into 5 congruent shapes?

Given a square, remove one quarter of it. Can the resulting L-shaped figure be divided into 5 congruent shapes? If not, how can we prove that fact? I tried using circles, triangles and smaller ...
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61 views

Resources for solving Euclidean geometry problems using symmetries

I know a number of books that treat geometry from the viewpoint of transformations/symmetries. However, very few of them actually teach someone to solve Euclidean geometry problems using said ...
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1answer
41 views

Use only Symmetry in question related to triangle to prove $\angle KPQ = \angle PKL$

In figure $\triangle APB$ , $\triangle QBC$ & $\triangle ARC$ are isosceles. $PK$ $\perp AC$, also $Ql$ $\perp AC$ $P,B,L$ are co linear. Also $\angle APB =\angle BQC = \angle ARC = 120^o$. How ...
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Show that $H = \bigcup_{n > 0} S_n$ is not equal to Sym$(X)$ for $X = \{1,2,3,4,…\}$.

Let $X = \{1,2,3...\}$ be the set of positive natural numbers, $S_n$ the permutation group, and Sym$(X)$ the set of all bijections from X to X with operation composition. I have the following ...
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31 views

Easiest way to prove a congruency

Let's take ABCD a convex quadrilateral, such that the diagonals AC and BD are perpendicular, and the angles ABC and CDA are equal. What is the easiest way to prove that the triangles ABC and CDA are ...
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32 views

Symmetry in graphs of polynomials

I am learning graph skectching. Its well known that quadratic polynomials over reals are symmetric about their minima/maxima. But today I discovered an interesting result that Cubic polynomials are ...
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1answer
43 views

Group generated by two reflections order 2

I have two reflections $r$, $s$ with $r^{2}=s^{2}=e$. Also $rs \neq sr$ and $(rs)^4 = (sr)^4= e$. I know $r$ and $s$ generate a group of order $16$ but that was by writing each matrix multiplication ...
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21 views

Anti-symmetric ways

A car dealer lines up his best objects for sale. He has 'n' Porsche and 'n' Ferrari. How many anti-symmetric ways are there to arrange these cars? (Anti-symmetric means that if ith from left is a ...
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1answer
76 views

finding irrational positive eigenvalues of a real symmetric $3 \times 3$ matrix

I have a real symmetric $3 \times 3$ matrix and I know all its eigenvalues are positive and irrational. All I care about are the eigenvalues (don't need the eigenvectors). What is the most efficient ...
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1answer
38 views

Symmetries of octahedron with $2$-faces action

Want to do the $2$-faces action. We use the Orbit stabilizer theorem. Let $X$ be the set of faces (any face can go to any face), $X=\{1,2,3,4,5,6,7,8 \}$. Where $1,2,3,4$ are the front faces of the ...
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Why is $J_n$ not symmetric, for $n\notin\mathbb Z$, while Bessel's equation is still symmetric?

Bessel's equation, $$x^2y''+xy'+(x^2-n^2)y=0,$$ has even parity, regardless of the value of $n$. So a solution of this equation must be even or odd. However, the Bessel functions $J_n$, which are ...
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27 views

Multivariable Calculus - Symmetry in integrals, cancelling out

So, I've got the integral $\iiint_R$$(xy+z^2)dV$ over the set $0\le z \le 1-|x|-|y|$. So, I've gotten that this makes out a pyramid in $\mathbb{R}^3$ with corners in $(0,0,1)$, $(0,1,0)$ and $(1,0,0)$,...
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23 views

Method of determining symmetries in an irregular polygon (2D or 3D)?

Thank you in advance for helping. Given a polygon with $n$ vertices, $$P = \begin{bmatrix} x_{1} & x_{2} & ... & x_{n} \\ y_{1} & y_{2} & ... & y_{n} \end{bmatrix}$$ how does ...
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Is there any significance in such Heegner numbers (or class number 1) representation symmetry?

$$\mathrm{A003173}(n) = 1+\frac{(1 + \sqrt{3})^{n-1} - (1 - \sqrt{3})^{n-1}}{2\sqrt{3}}$$ for $n = 1,2,3,4$ . $$\mathrm{A003173}(n) = 19+24\frac{(1 + \sqrt{3})^{n-6} - (1 - \sqrt{3})^{n-6}}{2\sqrt{3}...
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Is there an explicit left invariant metric on the general linear group?

Consider $GL_n^+$, the group of (real) invertible matrices with positive determinant. Is it possible to find an explicit formula for a metric on $GL_n^+$ which is left-invariant, i.e $$d(A,B)=d(gA,...
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10 views

Mathematical theory for equally distributed dipole structures with inner equilibration

I'm looking for a mathematical theory for equally distributed dipole structures with inner equilibration. I know, that there exist two magnetic clusters, where the north and the south poles equally ...
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Is there a classical analog of Bloch's theorem?

In quantum mechanics, having a spatially periodic Hamiltonian imposes a lot of structure on solutions of Schrodinger's equation (e.g. band structure), primarily due to Bloch's theorem. In perfect ...
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27 views

How to show if the following subset $W$ is a subspace of a vector space $V$?

$1.$ $V=P_n(\mathbb{R}), $and $ W=\{p(x)\in P_n(\mathbb{R})\mid p(1)+p(2)+p(3)=0 \}$ $2.$ $V=M_{n\times n}(\mathbb{R}), $and $ W=\{A\in M_{n\times n}(\mathbb{R}) \mid A \text{ is not symmetric}\}$ $...
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25 views

Generating space representation of all elements of a point group using generators

A well known set of groups are the 32 three-dimensional crystallographic point groups which elements represent the transformation matrices of the symmetry elements (rotation, reflection etc.). If one ...
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1answer
45 views

Any bi-invariant distance on a group is inverse-invariant?

$\newcommand{\inv}{\text{inv}}$ Let $G$ be a Lie group. Assume $d$ is a metric on $G$ (in the sense of metric spaces) which is bi-invariant. Is it true that the inverse automorphism must be an ...
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1answer
86 views

Finding a matrix given eigenvalues and eigenvectors.

I am asked to construct a $4 \times 4$ symmetric matrix, with given eigenvalues and eigenvectors. I understand how to actually get $A$ as a product of $P^T, D$ and $P$, when $D$ is the diagonal matrix,...
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13 views

How to prove that the pointreflection at the midpoint of two several points out of a regular pointlattice fix the lattice?

How to prove that the pointreflection at the midpoint of two several points $A,B\in\mathfrak{L}$ in a regular pointlattice $\mathfrak{L}$ fix the lattice $\mathfrak{L}$? We call $\mathfrak{L}\...
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36 views

How do we get the inequality?

Proposition: If $A \in \mathbb{R}^{n \times n}$ a symmetric matrix then $||A||= \sup \{ ||Ax||_2: ||x||_2=1\}= \sup \{ |\langle x, Ax \rangle|: ||x||_2=1\}$. Proof: It suffices to show that $||A|| ...
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1answer
45 views

Is this 2-tensor symmetric? It satisfies these conditions

I have some scalar field $p$ and a 2nd order tensor $\textbf{T}$ such that $div( \ \textbf{T} \ \textbf{grad}(p) \ )=0$ $\textbf{curl}(\ \textbf{T} \ \textbf{grad}(p) \ )=\textbf{0} $ $\textbf{T}$ ...
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90 views

Nodal Lines of the Eigenvalue problem $\Delta u=\lambda u$

I have really enjoyed performing the method of separation of variables to identify the eigenfunctions and nodal lines (the set of points for which each eigenfunctions vanishes) of the 2-D wave ...
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69 views

What are the non-vanishing directions of $A_{a(b} B_{cd)} - B_{a(b} A_{cd)}$? ($A$ and $B$ symmetric tensors)

Question Consider two symmetric tensors $A_{ab}=A_{ba},\, B_{ab}=B_{ba}$ which are generally of full rank (non-degenerate, all eigenvalues non-zero but of general sign) and their tensor product under ...
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29 views

How to show the space of inverse-invariant metrics on a Lie group is infinite dimensional?

Let $G$ be a Lie group. I am trying to convinve myself there are 'many' Riemannian metrics on $G$ for which the inverse automorphism is an isometry. Denote the iverse by $i$. For any metric $g$ on $G$...
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Why $e^x$ is always greater than $x^e$?

I find it very strange that $$ e^x \geq x^e \, \quad \forall x \in \mathbb{R}^+.$$ I have scratched my head for a long time, but could not find any logical reason. Can anybody explain what is the ...
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2answers
54 views

There exists an orthogonal antisymmetric real $n\times n$-matrix iff $n$ is even

How do I prove that there exists an orthogonal antisymmetric $n\times n$-matrix with real coefficients iff $n$ is even? I know that orthogonal and antisymmetric means $AA^T=I$ and $A=-A^T$, thus $A^2=...
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0answers
20 views

Weyl Transformations and Group actions

I have the following question. Let $(M,g_{ab})$ be a Riemannian manifold $M$ with metric $g$, and with an action of a Lie group $G$. Moreover, the Riemannian metric $g_{ab}$ is taken to be invariant ...
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47 views

Axis of approximate symmetry in irregular polygon

I'm searching for an axis of approximate reflection symmetry in irregular convex polygons with straight boundaries. Considering the polygons are irregular, the axis of approximate symmetry (defined as ...
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1answer
68 views

Extend isometry on some cube vertices to the entire cube

Let $K\subset V=\{-1,1\}^n$ be a set of vertices of the $n$-dimensional hypercube $D=[-1,+1]^n$ and let $f:K\to V$ be an isometry with respect to the Euclidean metric inherited from $\mathbb R^n.$ We ...
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Problems in Wikipedia about symmetry group of the non-equilateral isosceles triangle.

I think this wikipedia article - https://en.wikipedia.org/wiki/Symmetry_group#Two_dimensions - is wrong when it states that "$D_2$, which is isomorphic to the Klein four-group, is the symmetry group ...
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84 views

Is every linear operator which is $SO(n)$-invariant necessarily isotropic?

Let $M_n$ be the vector space of $n \times n$ real matrices. We say a linear operator $\alpha:M_n \to M_n$ is hemitropic* if: $(*) \, \, \alpha(S^TXS)=S^T\alpha(X)S \, , \, \forall S \in SO(n)$ and ...
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81 views

Rotational Symmetry for Rangoli Designs

I am viewing these Rangoli Patterns And it says Which one has no line symmetry but matches four times as it turns around? But to me, they all have line symmetry, i.e. you fold it in half, ...
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Nontrivial Symmetry of an Integral Depending on a Parameter

We have defined the following function \begin{equation} \chi(\gamma)=\frac{1}{2\pi}\int{\rm d}^2{\bf x}_2\frac{{\bf x}_{10}^2}{{\bf x}_{20}^2{\bf x}_{21}^2}\left[\left(\frac{{\bf x}_{21}^2}{{\bf x}_{...
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47 views

$O(d,d)$ vector

The $O(d,d)$ matrix $M$ ($M\in O(d,d)$) is defined as the matrix satisfying $$ M^T \eta M=\eta,\quad \eta=\left(\begin{matrix}0 & 1\\1 & 0 \end{matrix}\right).$$ So if we are given some ...
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70 views

Killing fields and symmetries

In my answer to Why are Killing fields relevant in physics? I wrote: The flow of a Killing field can drag the whole manifold. Since Killing fields are divergence free and thus their flows have no ...
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1answer
48 views

Invariants of a PDE by Lie Symmetries

I have a little Problem in understanding how to derive invariants form Lie Symmetries (or theire infinitesimals). As one can show the heat equation $u_t=u_{xx}$ has the following symmetries (...
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symetrical coordinates of algebraic variety

Let $$M = \{ \mathbf{x} \in \mathbb{R}^n : x_i \geq 0 \}$$ and $c \in \mathbb{R}$, $\alpha_i \in \mathbb{Z} \setminus \{0\} $ for $i \in \{1,\dots,n\}$ $$ \mathcal{S} = \{ \mathbf{x} \in M : c = \...
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67 views

How to show that an odd function always goes through zero?

I have the standard definition of an odd-function from wikipedia: Again, let f(x) be a real-valued function of a real variable. Then f is odd if the following equation holds for all x and -x in ...
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38 views

Group of rotational symmetries of regular tetrahedron is isomorphic to $A_4$

Let $G$ be the group of rotational symmetries of a regular tetrahedron. I'm trying to think of an argument proving that since $G\cong H$, where $H\leq S_4$, $H=A_4$. There are 12 rotational ...
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1answer
16 views

Finding Equality around Axis of Symmetry

I have a particular function that for even numbers $m$ obeys the following equation: $$f_{m,n}\left(\frac{2}{m}-x\right)=(-1)^nf_{m,n}(x)$$ Now when I put in odd values for $m$ and plot the function,...