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

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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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38 views

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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72 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
49 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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19 views

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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68 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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39 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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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,...
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12 views

Differential equation with ellipsoidal symmetry

I wish to solve a given PDE with ellipsoidal symmetry (I'm not sure the wording is correct). For example it could be a PDE describing the temperature field in a bottle. I am thinking of using the ...
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329 views

Why is this definite integral antisymmetric in $s\mapsto s^{-1}$?

I recently happened into the following integral identity, valid for positive $s>0$: $$\int_0^1 \log\left[x^s+(1-x)^{s}\right]\frac{dx}{x}=-\frac{\pi^2}{12}\left(s-\frac{1}{s}\right).$$ The ...
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34 views

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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130 views

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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54 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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59 views

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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34 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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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
47 views

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
62 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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51 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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87 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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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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44 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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16 views

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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53 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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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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99 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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78 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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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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35 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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24 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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17 views

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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67 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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25 views

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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58 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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72 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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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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75 views

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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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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32 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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68 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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37 views

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) ...