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

learn more… | top users | synonyms

1
vote
2answers
21 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}\}$ ...
5
votes
1answer
59 views
+50

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 ...
0
votes
0answers
8 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 ...
8
votes
9answers
297 views

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 ...
3
votes
1answer
18 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 ...
4
votes
1answer
49 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 ...
0
votes
1answer
671 views

What is a “unique” mirror line of symmetry?

What is a "unique" mirror line of symmetry? For example why does an equilateral triangle have three mirror lines but only one "unique"mirror line of symmetry?
0
votes
1answer
10 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 ...
0
votes
1answer
35 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|| ...
0
votes
2answers
64 views

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 ...
2
votes
0answers
68 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 ...
2
votes
1answer
67 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 ...
5
votes
1answer
41 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} $ $\textbf{T}$ ...
0
votes
1answer
22 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 ...
0
votes
1answer
425 views

How do I proove the symmetry of Metric space? [closed]

I'm looking for the proof of that: d(x,y) = d (y,x). I know that I have to use the "non-negativity" and "triangle inequality" but I don't know how to combine them to get the result.
2
votes
1answer
532 views

a conjugate of a glide reflection by any isometry of the plane is again a glide reflection

Q : Prove that a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, and show that the glide vectors have the same length. I know that: the conjugate ...
0
votes
0answers
29 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 ...
1
vote
2answers
45 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 ...
1
vote
0answers
15 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 ...
4
votes
1answer
83 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 ...
1
vote
1answer
46 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, ...
32
votes
1answer
513 views

Penrose's remark on impossible figures

I'd like to think that I understand symmetry groups. I know what the elements of a symmetry group are - they are transformations that preserve an object or its relevant features - and I know what the ...
1
vote
0answers
36 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 ...
1
vote
1answer
45 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 ...
17
votes
2answers
224 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 ...
0
votes
0answers
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 ...
1
vote
3answers
51 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 ...
5
votes
2answers
54 views

Determining the symmetry group of an infinite horizontal line.

I believe I have a satisfactory answer to the following question: Imagine we have a infinite horizontal line running through the origin, what is the associated symmetry group? Now thinking ...
0
votes
0answers
14 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 ...
0
votes
0answers
15 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 = ...
40
votes
7answers
10k views

Proving that 1- and 2-d simple symmetric random walks return to the origin with probability 1

How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the ...
0
votes
4answers
56 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 ...
0
votes
0answers
26 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 ...
2
votes
1answer
14 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 ...
0
votes
0answers
6 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 ...
2
votes
1answer
26 views

Explaining a proof of Euler's theorem

Can someone please explain the question marked extrapolation in the following image?
0
votes
0answers
30 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 ...
2
votes
1answer
42 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 ...
0
votes
0answers
90 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 ...
1
vote
3answers
46 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, ...
1
vote
1answer
28 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 \ ...
0
votes
0answers
17 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 ...
2
votes
2answers
45 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 ...
0
votes
1answer
12 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 ...
-1
votes
0answers
28 views

Is $R = \{(a,a),(b,b),(c,c)\}$ an equivalence relation on $\{a, b, c\}$?

My intuition is yes is it an eq. rel., but I'm not sure. If $a \sim a \in R$, then $a \sim a \in R$ (inverse which is just the same), and so $a \sim a \in R$ (transitivity). Is this a valid argument ...
1
vote
1answer
223 views

Symmetric random walk and Borel-Cantelli

Let $(X_n)$ be a sequence of i.i.d. r.v. such that $\mathbb{P}(X_n=1)=\mathbb{P}(X_n=-1)=\frac{1}{2}$ Also let $$S_n= \sum_{k=1}^{n} X_k$$ I am asked to show, using Borel–Cantelli lemma, that ...
3
votes
1answer
49 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 ...
1
vote
1answer
70 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) ...
2
votes
2answers
53 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: ...
2
votes
1answer
43 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 ...