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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34 views

Given a sequence of i.i.d. random variables, prove a result involving conditional expectation by means of symmetry argument

Let $(X_n)_{n\geq1}$ be an i.i.d. sequence with $\mathbb{E}\{|X_1|\}<\infty$. Let $S_n=X_1+\cdots+X_n$ and $\mathcal{F}_{-n}=\sigma(S_n,S_{n+1},\ldots)$. Then, one can state that $$M_{-n}=\mathbb{E}...
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Attempting to prove a conserved quantity for incompressible Navier-Stokes PDE

I am looking for feedback on this attempt to prove a conserved quantity for incompressible Navier-Stokes $$(\mathbf{u} \cdot \nabla) \mathbf{u} + \frac{\partial \mathbf{u}}{\partial t} = \nu \Delta \...
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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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59 views

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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1answer
27 views

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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1answer
22 views

If $V$ does not depend on $t$, how to show $V_{t} = V_{t+s}$, for $s\in \mathbb{R}$

I am reading a paper, and it says the following: Let $V = V(x(t))$, i.e., a function of state $x(t)$ but not depend explicitly on $t$. Now consider a mapping $$\tilde{t} = t+s, $$for any $s\in \...
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1answer
25 views

rotational symmetries of the 120-cell

I want to find the number of rotational symmetries of the 120-cell but I am not very familiar with polytopes nor counting symmetries. So, I don't know if someone can give me an idea or an example with ...
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3answers
221 views

Does every group have an object of symmetry?

I'm aware of Cayley's theorem, which says that every group is isomorphic to some subgroup of a symmetric group. But it's not clear to me whether symmetric groups themselves (apart from their name) ...
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39 views

Is $(x,y)\rightarrow (-x,-y)$ an inversion transformation?

Does anyone know whether $(x,y)\rightarrow (-x,-y)$ is an inversion transformation or not? I know that the standard inversion (parity) transformation in two dimensions should be something like $(x,y)\...
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2answers
58 views

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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1answer
36 views

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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1answer
113 views

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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2answers
25 views

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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1answer
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is the sign of symmetry distribution projected still keep symmetry?

Given a real random vector $\vec{r} \sim N(0, I^{d \times d})$ and a real matrix $B^{n\times d}$ I would obtain a new random vector $\vec{S} = sign(B \cdot \vec{r})$. Would this hold $P(\vec{S}=all+)...
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1answer
66 views

Does this inequality hold with some constant factor $c>0$?

Does there exist a real number $c>0$ such that $$ (x-1)^2+(y-1)^2-2(\sqrt{xy}-1)^2\ge c\big( (x-\sqrt{xy})^2+(y-\sqrt{xy})^2 \big) \tag{*}$$ holds for every positive real numbers $x,y$ such that $...
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20 views

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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1answer
42 views

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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1answer
16 views

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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1answer
77 views

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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2answers
150 views

Minimize $|a-1|^3+|b-1|^3$ with constant product $ab=s$

Let $0<s$, and define $$ F(s):=\min_{a,b \in \mathbb{R}^+,ab=s} \left(|a-1|^3+|b-1|^3\right). $$ I would like to find proofs for the claim $$ F(s)=\begin{cases} 1 - 3 s - 2s^{3/2}=F\big(a(s),b(s)\...
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1answer
52 views

A normal subgroup of size 4 the group of rotational symmetries of a cube

An exam question has asked me to show that there is a normal subgroup of size 4 of the group of rotational symmetries of the cube. A trick I've seen before is considering the action of a group on its ...
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Listing of point group operations

I am looking for a listing of the operations of point groups (preferrably as matrices). Nearly every textbook lists the irreducible representations and characters of the groups, but I have not found ...
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0answers
47 views

Extending a function from the positive diagonal matrices to orthogonally-invariant function on $\text{GL}_n$

I am trying to see when one can extend "partial" distances to bi-$\text{SO}_n$-invariant metrics on $\text{GL}_n^+$. On the way to there, I came to the following question: Suppose we have a ...
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29 views

Eigenvalues and Eigenvectors of Transformed Symmetric Matrix

Let $A$ be a symmetric matrix, of the following form: $$ A = \begin{bmatrix} a & b & 0 \\ b & c & 0 \\ 0 & 0 & d \end{bmatrix} $$ Furthermore, let $B$ be the transformation of ...
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142 views

Proving a complicated looking inequality in a simple way

This is again a search for alternative proofs: Let $0 <s \le 1$, and suppose that $0 <a,b $ satisfy $$ ab=s,a+b=1+\sqrt{s}. \tag{1}$$ I have a proof for the assertion $$ 2(1-\sqrt s)^3 \le |...
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1answer
38 views

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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24 views

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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76 views

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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42 views

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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1answer
16 views

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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22 views

Isometries and Symmetries of the set $\mathbb{Z}^2$ as a subset of $\mathbb{R}^2$

(a) Find all translations that are symmetries of $\mathbb{Z}^2$ (b) Find all isometries in $GO_2$ that are symmetries of $\mathbb{Z}^2$ My solutions: (a) Firstly, we define a translation $T_z(x)...
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1answer
86 views

Polya's Enumeration : Where am I going wrong?

If we write a 9 digit binary number as a $3\times3$ matrix, for example $101110101$ would be written as, $$\begin{bmatrix}1 & 0 & 1\\\ 1 & 1 & 0\\\ 1 &...
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1answer
76 views

Does $|f(\sqrt{xy})| \le |\frac{f(x)+f(y)}{2}|$ imply $f$ is a logarithm?

Let $f:\mathbb R^+ \to \mathbb R$ be a continuous function, and suppose that $$ |f(\sqrt{xy})| \le |\frac{f(x)+f(y)}{2}|, \tag{1}$$ holds for every $x,y \in \mathbb R^+$. Suppose also that $f(1)=...
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2answers
76 views

Evaluate $\int^{\pi}_0\frac{x\sin(x)}{1+\cos^2(x)}dx$ [duplicate]

I have the following task: On an interval $[0,a]$ one can use the substitution $y=a-x$ to try and exploit symmetry about the midpoint $a/2$ 1) Evaluate $\int^{\pi}_0\frac{x\sin(x)}{1+\cos^2(x)}dx$ ...
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50 views

Geometric interpretation of conjugacy classes and class equation of $D_6$ [closed]

Known that for Dihedral Group $D_6$, where $D_6=\{r,s: r^6=s^2=1, rs=sr^{-1}\}$, its conjugacy classes are given by $\{1\}, \{r,r^5\}, \{r^2,r^4\}, \{r^3\}, \{s, sr^2, sr^4\}, \{sr, sr^3, sr^5\}$, ...
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1answer
118 views

Does COVID-19 fit into the Caspar-Klug (Quasi-Equivalence) Theory for virus architecture?

The following is compiled largely from my "Applications of Group Theory to Virology" module I took at The University of York as an undergraduate back in 2012. The icosahedral group $I$ with identity $...
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Finding a function with local radial symmetry

I'm looking for a function that has local radial symmetry, for instance in 2D: $f(r,\phi)$. And with axial symmetries for many axes. I mean for instance a function that could represent this kind of ...
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2answers
39 views

Prove that if $a^2+bc \neq 0$, then the graph of $f(x)= \frac{ax+b}{cx-a}$ is symmetric about the line y=x

Prove that if $a^2+bc \ne 0$, then the graph of $f(x)= \frac{ax+b}{cx-a}$ is symmetric about the line $y=x$. Maybe this is a simple exercise, but I need help from you guys to understand how is the ...
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41 views

combinatorial block design

Suppose that $(X, B)$ is a $2-(v, k, \lambda)$ design. For $x \in X$, let $r_x$ be the number of blocks in $B$ containing $x$. Show that $r_x(k − 1) = λ(v − 1)$. Secondly, deduce that $r_x$ is ...
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2answers
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Probability of $x_1<\dots < x_n$ with symmetric multivariable density

Let $f(x_1, \dots, x_n)$ be the p.d.f. of variables $X_1, \dots, X_n$, where $f(x_1, \dots, x_n)>0$ if $X_i \in D, i=1, \dots, n$ for some set $D$, $0$ elsewhere. Suppose $f$ is symmetric, that is, ...
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1answer
20 views

Infinitesimal symmetry of a tangent distribution.

An infinitesimal symmetry of a tangent distribution $D={\rm span}(X_1,\dots,X_n)$ on $n$-dimensional manifold $M$ is a vector field $Y$ such that for every $X\in D$ a Lie bracket $[X,Y]\in D$. My ...
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1answer
43 views

Finding equivalence classes under permutation symmetry

If we write a 9 digit binary number as a $3\times3$ matrix, for example $101110101$ would be written as, $$\begin{bmatrix}1 & 0 & 1\\\ 1 & 1 & 0\\\ 1 &...
1
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0answers
47 views

Fourier transform lies on a line

A simple property of the Fourier transform is that if $f$ is real and even then its Fourier transform $\hat f$ is real, while if $f$ is real and odd then $\hat f$ is imaginary. This can be extended to ...
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2answers
45 views

Does the graph of $\tan$ have any lines of symmetry? [closed]

I think not. But if not, how do we prove this?
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1answer
25 views

Convergence as a result of symmetry

I have the following improper integral: $$\int _{-\infty }^{+\infty }\:e^{-x^2}\,{\rm d}x$$ Now, I just proved it is convergent for $[0;+\infty]$. The thing is I admit I'm a bit lazy and I don't want ...
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2answers
79 views

Pattern Recognition of Random Corks

Firstly I apologise in case the tags are inaccurate. I simply don't know where this question fits. Suppose I have a big hollow cylinder with an open top so that I can put things inside it. As a ...
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1answer
18 views

Characterising radially symmetric maps which are smooth at the origin

This is a self-answered question. I post it here since it wasn't obvious to me. It is a follow-up of this previous question of mine. Let $U \subseteq \mathbb{R}^n$ be an open set containing the ...
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0answers
12 views

Does $h^{2k}(0)=0$ imply $(h^{-1})^{2k}(0)=0$?

Let $h:[0,1] \to [0,\infty)$ be a smooth strictly increasing function satisfying $h(0)=0,h'(x)>0$ for every $x \in [0,1]$, and $h^{2k}(0)=0$ for every natural $k$. Is it true that the inverse ...
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1answer
47 views

Does Ricci flow preserve symmetries?

What does the following sentence mean: (see p.66 The Ricci Flow in Riemannian Geometry By Ben Andrews, Christopher Hopper) The Ricci flow preserves any symmetries that are present in the initial ...
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1answer
85 views

Symmetry of extended Dynkin diagrams

In a paper I was reading about Kac-Moody algebras there was remark that interested me, but I couldn't really make sense of. When discussing extended Dynkin diagrams the authors dicussed that there is ...

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