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

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Why does the equality hold?

For $A \in \mathbb{R}^{n \times n}$ with $A=A^{T}$ we set: $$\lambda:= \max \{ \langle x, A x \rangle: ||x||_2=1\} \\ \mu:= \min \{ \langle x,A x \rangle: ||x||_2=1\}$$ Then for $x \in S_{||\cdot||_{...
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Equivalence relation for strings

Let R be the relation consisting of all pairs (x,y) such that x and y are strings of uppercase and lowercase English letters with the property that for every positive integer n, the nth characters in ...
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53 views

Manifolds as Homogeneous Spaces

With very little effort one can, for example, show that $S^n$ can be written as a homogeneous space as $S^n\cong G/H$, where $G$ is the group of all rotations in $\mathbb{R}^{n+1}$ about the origin ...
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Visualizing $180^\circ$ rotational symmetries of a tetrahedron

I am trying to learn about the symmetries of a regular tetrahedron. I understand the identity and all eight $120^\circ$ rotations that keep one vertex fixed, $(123),(132),(243),(234),(134),(143),(124)...
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33 views

Explaining a proof of Euler's theorem

Can someone please explain the question marked extrapolation in the following image?
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24 views

Points in complex plane invariant under inversion?

I can think of two points in the complex plane $z_1=1$ and $z_2=-1$ that are invariant under inversion, which means that $$\frac{1}{z_i}=z_i$$ Are there any other such points, or is this the whole ...
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1answer
16 views

How large is the subgroup of diffeomorphisms which preserves a vector field?

Let $M$ be a smooth manifold. Let $X \in \Gamma(TM)$ be a vector field on $M$, which vanishes at a finite number of points. (Every smooth manifold admits such a vector field). Consider the subgroup $\...
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2answers
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Is there a non abelian group that characterize a one dimensional lattice structure?

Of all groups that characterize a one dimensional lattice structure (symmetry operations including translation, $C_2$, mirror plane, inversion point), is there a non abelian one? Moreover, Can it have ...
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60 views

Show that $P$ is symmetric.

Let $P$ = $A(A^TA)^{-1}A^T$, where A is an m x n matrix with rank $n$. I feel like this is wrong, but here is my attempt: $A(A^TA)^{-1}A^T$ = $AA^{-1}(A^T)^{-1}A^T$ = $I$ And $I^T$ = $I$, so the ...
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2answers
60 views

Express rotation of the plane about a point as the product of a translation and a rotation about the origin

I need help with the following problem: a) Let $s$ be a rotation of the plane with angle $\frac{\pi}{2}$ about the point $(1,1)^t$. Write the formula for $s$ as a product $t_a \rho _\theta $ Edit: $...
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1answer
48 views

Rotations of 4-Cubes

I have recently learned the orbit stabilizer theorem, and have encountered unexpected results pertaining to the rotations of a tesseract; I am curious if there is any intuition for this. A $4$-Cube ...
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81 views

Prove symmetry of probabilities given random variables are iid and have continuous cdf

Let $Y_1, Y_2, ...$ be independent and identically distributed random variables in $(\Omega, \mathscr{F}, \mathbb{P})$ s.t. their distributions are continuous and $$F_{Y}(y) := F_{Y_1}(y) = F_{Y_2}(y) ...
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1answer
29 views

Order of affine reflections (described with complex numbers operations)

Let be the affine reflection described as an operation with complex numbers : $$s_\beta,_\nu : z \mapsto \nu + \overline{\beta z},$$  where $z, \nu \in \Bbb C$ and $\beta \in \Bbb C^1 = \{x+iy \ \...
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40 views

Symmetry and Derivative of Multivariable Functions

I think this is a fairly trivial question, but my multivariable calculus is a little rusty so I wanted to check. I have two functions $v_1(\xi,x)$ and $v_2(\xi,x)$ originating from solutions of an ODE ...
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Get the positions of a periodic system of labeled points in reference to another coordinate system

First of all, I'd like to apologize because I'm not familiar with the conventions (names and formats) used in the math community. The problem is the following, I have information about a Face ...
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50 views

Decomposing a set of complex matrices into orbits of the operation of conjugation

I need some assistance with the proof for part (b) of the following problem statement: Problem Statement: Decompose the set $\mathbb{C}^{2\times2}$ of $2\times2$ complex matrices into orbits for ...
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1answer
32 views

rotational symmetries of a polyhedron

Given is the following Polyhedron $P$ in the $\mathbb{R}^3$: I want to define the set $S$ of the rotational symmetries of the Polyhedron in $\mathbb{R}^3$. Hence an element $s \in S$ is a function ...
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88 views

Practical use of order of rotational Symmetry.

Has anyone know what is the use of finding the order of the rotational symmetry of a figure? A student of mine ask that question from me. I search it but could not find any. Plz help. https://www....
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72 views

Symmetry planes in spherical harmonic basis

Suppose I have a function $f(x):S^2\rightarrow\mathbb{C}$ in the degree four spherical harmonic basis: $$f(\theta,\varphi):=\sum_{k=-4}^4a_kY_4^k(\theta,\varphi).$$ I have two related questions: Is ...
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1answer
22 views

Undirected graph in first order logic

I'm trying to model an undirected edge in a graph. If we have an edge $(a \to b)$ given in a database, the edge $(b \to a)$ is of course implied. Given the predicate Has_Connection(a, b). It ...
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34 views

Crystallographic restriction in discrete group of isometries

I need some assistance with this proof: Problem Statement: What is the crystallographic restriction for a discrete group of isometries whose translation group $L$ has the form $\mathbb{Z}a$ with $...
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1answer
23 views

Symmetry of Predicate and Universal Quantifiers

$$ \forall x \forall y P(x,y) \implies \forall x \forall y P(x,y) \land P(y,x) $$ I guess the above statement is valid but no idea how to formally prove it, any idea?
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Solving Lie's invariant condtion for first order ODE

I write down the general equation $Y_{x}+(Y_{y}-X_{x})F-X_{y}F^{2}= XF_{x}+YF_{y}$ and assume that X=a(x) and Y=b(x)y, after that I can't see anyway to solve it for a and b. How can I get solution ...
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64 views

Isotropic left invariant Riemannian metric on $GL_n^+$?

I am trying to see if it's possible to construct a left invariant isotropic Riemannian metric on $GL_n^+$. (the group of $n \times n$ invertible real matrices with positive determinant) (When by ...
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29 views

What does a norm which is symmetric “around” an index subset look like?

I am looking at norms $\lVert\cdot\rVert$ on $\mathbb{R}^{n}$ that have the following symmetry property $$\forall \beta \in \mathbb{R}^{n} \text{: }, \lVert\beta \rVert=\lVert \beta_{J}+\beta_{J^{c}}\...
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Difference of general symmetric function that is non decreasing in its arguments

For a symmetric function $C(x,y)$ and $a,b,c,d \in [0,1]$ with $b\ge a$, $d \ge c$ and further, C(x,y) is non-decreasing in $x,y$. Then, does it hold that: $$ C(b,d) -C(a,d) -C(b,c) + C(a,c) \ge 0 $$ ...
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40 views

Is it possible to represent a set of generic things coherently without implicitly creating an order on them?

Sorry if this question seems a little incoherent, I'm not certain of the proper terminology here. I've seen unordered sets represented in various ways, but it recently occurred to me that most of ...
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2answers
56 views

Mathematics of chemistry with focus on particular symmetries

A student of mine came with a somewhat unusual request: My question about group theory and if you have a book you could recommend for some foundation: specifically chemical applications of group ...
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70 views

Is there a bi-invariant metric (not Riemannian) on $GL_n$?

Is there a bi-invariant* metric $d$ on $GL_n$ (the group of invertible marices) which generates the standard topology on it? (the subspace topology from $\mathbb{R}^{n^2}$) Note: I mean any metric (...
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Digesting a proof of Bieberbach's theorem on crystallographic groups

Here is a proof of Bieberbach's theorem: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.460.1436&rep=rep1&type=pdf I can follow the computing details, but I have no intuitive sense ...
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52 views

Is a shape formed of two tangents and radii symmetrical?

Is the kite formed by the two tangents and radii in this image symmetrical? Is there a law or reason why? I am assuming that the two tangents are of equal length, but I can't see why. Are any two ...
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Understanding Cauchy-Schwarz and Holder's inequalities.

Although these inequalities occur in various settings, and I have used them to complete a number of proofs, I can not say that I intuitively understand what their significance is. Holder's ...
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Is the follow equation representative of a Hyperbolic One dimensional conservation law?

For $y : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ and let $j : \mathbb{R} \to \mathbb{R}$ $$\frac{\partial y}{\partial x_2} + \frac{\partial (j \circ y)}{\partial x_1} = 0$$
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A norm invariant under permutations but not under signed permutations

Let $\| \cdot \|$ be a norm on $\mathbb{R}^n$. We call it axes-symmetric if $\|x\|$ does not depend on the order of the components of $x$. Equivalently if $\|x\| = \|P \cdot x \|$ for any permutation ...
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1answer
48 views

ODEs are invariant under the given Lie groups?

$\frac{dy}{dx} = \frac{x^{2}y}{x^{3}+xy+y^2}$ is invariant under $(x,y) \mapsto (\frac{x}{1+\varepsilon y},\frac{y}{1+\varepsilon y})$ I can't make both sides equal when I have a variable depends on ...
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31 views

Efficient search for equilateral triangles, squares and regular pentagons in a set of 3D points

For an algorithm to identify cubic point groups from a set of atom positions $r_i$ forming a molecule, I would need an efficient and fast algorithm to identify equilateral triangles, squares and ...
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63 views

Eigenvalues of a rotationally symmetric matrix

I have a rotationally symmetric matrix of arbitrary size, for example, \begin{equation} A = \begin{pmatrix} a & b & c & b & a \\ b & d & e & d & b \\ c & e &...
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52 views

Polyhedral symmetry in the Riemann sphere

I found an interesting set of constants while exploring the properties of nonconstant functions that are invariant under the symmetry groups of regular polyhedra in the Riemann sphere, endowed with ...
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How can I compute how many and which quartic invariants of a given group representation are linearly independent?

Given a representation $R$ of a group $G$ and the corresponding tensor product $$ R \otimes R = R_1 \oplus R_2 \oplus R_3 \oplus \ldots $$ how can I compute how many quartic terms $ \mathrm{O} (R^4) ...
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28 views

Symmetric part of A contributes to quadratic form

In my statistics note, when it talks about quadratic forms, it goes on saying: "$x^tAx=\frac12x^t(A+A^t)x$ implies that only the symmetric part of A contributes to the quadratic form." I am having ...
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457 views

Systematic solution to my soccer ball puzzle

I once received a puzzle that can be described as follows: There are $12$ black pentagonal and $20$ white hexagonal pieces. The goal is to form a soccer ball from these (aka. truncated icosahedron). ...
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668 views

Is zero the center of the numeric sequence?

The numeric sequence has symmetry on zero, with equal infinities of cancelling out + and - values on either side. Can numbers be said to have different centers of symmetry than zero? Is it possible ...
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128 views

Name of a symmetry involving complex squares

Imagine a single parameter real-valued non-zero function $f$. For any complex number $Z$, let $Z \rightarrow Z' = \Re(Z) f + i\Im(Z)/f $ Calculate $Z^2$ and $Z'^2$. Because the imaginary portion of ...
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Prove that if $A$ is a symmetric matric then $A^3$ and $A^2-2A+I$ are symmetric matrices.

I am uncertain on how to approach this proof. For most everything I've encountered concerning symmetry, it has involved taking the transpose in order to show some property. Here, I'm not certain if ...
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Symmetry of the multiplier of a l.c.s.c. abelian group

Let $G$ be a l.c.s.c. (locally compact, second countable) Abelian group, and let $\hat{G}$ be its (well-defined) dual. Consider the group $G\oplus\hat{G}$ (which is a l.c.s.c. Abelian group itself), ...
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Maximum overlap of two convex bodies

Let $\Delta_n=\big\{(x_1,x_2,\ldots,x_n): \sum_i x_i \leq 1, x_i \geq 0\big\}$ denote an n-dimensional simplex. I am trying to find an $y \in \mathbb{R}^n$ such that the $y-$shifted negative simplex ...
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54 views

Interesting $\sum_{i=0}^{m-1}(-1)^{i}(m-i)^{2} = \sum_{i=1}^{m}i$

I found that for m $\in N $ $$\sum_{i=0}^{m-1}(-1)^{i}(m-i)^{2} = \sum_{i=1}^{m}i.$$ I found it after doing an exercise. For example: $$5^{2}-4^{2}+3^{2}-2^{2}+1^{2} = 1 + 2 + 3 + 4 + 5 = 15.$$ For ...
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49 views

How can you algebraically determine if a curve is symmetric about $y=-x$?

If I have a curve implicitly defined by say $x^2+xy+y^2=1$, then it is clear that it is symmetric about $y=x$ because if I interchange x's with y's, then I have the exact same equation. However, how ...
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How to measure asymmetry of a function?

Let $f(x) = x^{2}$, so $f(x)$ is an upward symmetric parabola. It is a perfectly symmetric function since $f(x) = f(-x)$ for any value of $x$. Now, suppose $f$ is just some function. How would one ...
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Can I find the dissections of a figure based on symmetry?

Our teacher gave us a figure, and challenged us to dissect into exactly 4 shapes, that were congruent, in as many ways as possible. I won't reveal details of the specific shape. I am wondering if ...