Articles of symmetry

Set in $\mathbb R^²$ with an axis of symmetry in every direction

Let $A\subset \mathbb R^2$ be a set that has an axis of symmetry in every direction, that is, for any $n\in S^1$, there exists a line $D$ orthogonal to $n$, such that $A$ is invariant under the (affine) reflection of axis $D$. It is easy to show that if all of these axes intersect at […]

What is the difference between using $PAP^{-1}$ and $PAP^{T}$ to diagonalize a matrix?

What is the difference between using $PAP^{-1}$ and $PAP^{T}$ to diagonalize a matrix? Can both methods be used to diagonalize a diagonalizable matrix $A$? Also does $A$ been symmetric or not effect which method to use?

Functional equation $P(X)=P(1-X)$ for polynomials

I have encountered the following problem : Find all polynomials $P$ such as $P(X)=P(1-X)$ on $\mathbb{C}$ and then $\mathbb{R}$. I have found that on $\mathbb{C}$ such polynomials have an even degree. Because for each $a$ root, $1-a$ must be a root too. I struggle to find whether we can do something with polynomials of degree […]

Minima of symmetric functions given a constraint

If $f(x,y,z,\ldots)$ is symmetric in all variables, (i.e $f$ remains the same after interchanging any two variables), and we want to find the extrema of $f$ given a symmetric constraint $g(x,y,z,\ldots)=0$, $$\bf\text{When is it true that the extrema is achieved when }\ x=y=z=\ldots?$$ An example where this claim is true: $$ g(x,y,z) = x+y+z – […]

Why isn't an odd improper integral equal to zero

My calculus book says that the integral of $\frac1x$ cannot cross zero. Now it seems obvious that because of symmetry, there will always be an interval whose integrals are equal in magnitude and opposite insign, so cancel, even though they do not converge. Now typing into wolframalpha I got “does not converge” and only at […]

Do Symmetric Games with Nash Equilibria always have a symmetric Equilbrium?

Define a game with S players to be Symmetric if all players have the same set of options and the payoff of a player depends only on the player’s choice and the set of choices of all players. Equivalently A game is symmetric if applying a permutation to the options chosen by people induces the […]

The path to understanding Frieze Groups

What is the “Path” for understanding what Frieze Groups really are? Generally in mathematics, there is a is a path or “building blocks” approach to learning something. For example if I know how to count and I want to learn how to multiply you say the path is “Counting -> Adding (adding is fast counting) […]

Is every finite group of isometries a subgroup of a finite reflection group?

Is every finite group of isometries in $d$-dimensional Euclidean space a subgroup of a finite group generated by reflections? By “reflection” I mean reflection in a hyperplane: the isometry fixing a hyperplane and moving every other point along the orthogonal line joining it to the hyperplane to the same distance on the other side. Every […]

Is symmetry a valid option in inequalities?

Consider two questions: Q1. $$a+b+c+d+e=8$$ $$a^2+b^2+c^2+d^2+e^2=16$$ $$a,b,c,d,e\in\mathbb{I^+_0}$$ Find maximum value of ‘e’? My answer: Since when e is maximum when all other variables are equal since the equation is symmetrical in all other variables so, then $a=b=c=d=k$ (let) which gives $e=16/5$, neglecting other roots. Q2. $$\frac8x+\frac1y=1$$ Minimize $x+y+\sqrt{x^2+y^2}$. My answer: similiarly to previous problem now […]

A map which commutes with Hodge dual is conformal?

$\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\Cof}{\operatorname{Cof}}$ $\newcommand{\Det}{\operatorname{Det}}$ $\newcommand{\id}{\operatorname{Id}}$ Let $V$ and $W$ be $d$-dimensional, oriented inner-product spaces, and let $A\in\Hom(V,W)$ be an orientation-preserving map. Suppose that $A$ satisfies $$ \bigwedge^{d-k} A \circ \star_V^k= \star_W^{k} \circ \bigwedge^k A \neq 0, \tag{1} $$ for a single $1 \le k \le d-1$. Question: Is it true that $A$ is conformal? If not, […]