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

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42
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7answers
11k 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 ...
16
votes
1answer
491 views

Why does Group Theory not come in here?

Here is a list of questions that I find quite similar, for the one and only reason that they all show much "symmetry". Which is at the same time my problem, because I don't have a very precise notion ...
126
votes
5answers
7k views

Symmetry of function defined by integral

Define a function $f(\alpha, \beta)$, $\alpha \in (-1,1)$, $\beta \in (-1,1)$ as $$ f(\alpha, \beta) = \int_0^{\infty} dx \: \frac{x^{\alpha}}{1+2 x \cos{(\pi \beta)} + x^2}$$ One can use, for ...
1
vote
1answer
77 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) ...
1
vote
3answers
2k views

Coloring a cube with 4 colors

There are some topics on this forum related to my question. Most of them use Burnsides Lemma. I don't know this lemma and I don't know whether it is applicable to my problem. Can someone explain the ...
12
votes
3answers
1k views

What's so special about the group axioms?

I've only just begun studying group theory (up to Lagrange) following on from vector spaces and I am still finding them almost frustratingly arbitrary. I'm not sure what exactly it is about the ...
11
votes
3answers
633 views

Translations in two dimensions - Group theory

I have just started learning Lie Groups and Algebra. Considering a flat 2-d plane if we want to translate a point from $(x,y)$ to $(x+a,y+b)$ then can we write it as : $$ \left( \begin{array}{ccc} ...
8
votes
4answers
279 views

equilateral triangle; $3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2.$

In equilateral triangle ABC of side length d, if P is an internal point with PA = a, PB = b, and PC = c, the following pleasingly symmetrical relationship holds: $3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 ...
7
votes
9answers
344 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 ...
6
votes
1answer
557 views

How many $n$-colorings up to rotation using exactly 2 of each color are there on a $2n$-polyhedron?

I'm a high-school student and I stumbled across a YouTube video explaining how Rubik's cubes work. A Rubik's cube has 6 colors, one for each side, but I started thinking about ways to $n$-color the ...
5
votes
1answer
297 views

Do Symmetric problems have Symmetric solutions?

There is a general notion of Symmetry in mathematics, that of an object being constant under some transformation. If we think of our object as being a "mathematical problem" we can see certain ...
0
votes
2answers
131 views

Construct the Green s function for the equation

Construct the Green s function for the equation y^''+ 2y^'+2y=0 Which boundary conditions y(0)=0 , y(π/2)=0 Is this Green s function symmetric? What is the Green s function, if the ...
5
votes
1answer
67 views

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 ...
5
votes
0answers
90 views

Symmetric Icon Fractals

I have always been fascinated by fractals. But most of all I like the Symmetric Icon fractals. There is a nice book about these fractals, written by Michael Field, called Symmetry in Chaos. I'm ...
5
votes
3answers
366 views

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 ...
1
vote
2answers
463 views

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 ...
8
votes
2answers
3k views

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. ...
6
votes
3answers
2k views

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 ...
3
votes
2answers
613 views

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 ...
2
votes
4answers
121 views

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 ...
2
votes
1answer
185 views

Question about stiffness tensor

Let's have a stiffness tensor $$ a^{ijkl}: a^{ijkl} = a^{jikl} = a^{klij} = a^{ijlk}. $$ It has a 21 independent components for anisotropic body. How does body symmetry (cubic, hexagonal etc.) ...
1
vote
1answer
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 ...
1
vote
1answer
50 views

Surfaces of constant curvature using the conformal method

I'm doing a study of surfaces with constant curvature which leads to solving the equation: $$\Delta\phi = -e^{2\phi}K_0$$ for a 2-dimensional metric with constant curvature such that rotation around ...
3
votes
2answers
542 views

Number of reflection symmetries of a basketball

Excerpt from John Horton Conway, The Symmetries of Things, pg. 12. Basketballs have two planes of reflective symmetry, as do tennis balls. I read this sentence and it immediately struck me as ...
2
votes
2answers
46 views

What is the dimension of a certain vector subspace of invariant matrices?

$\newcommand{\Sig}{\Sigma}$ Let $\Sig$ be a diagonal matrix with strictly positive entries on the diagonal. Define $V=\{B \in M_n\mid B\Sig +\Sig B^T=\Sig B +B^T \Sig \}$ (where $M_n$ is the vector ...
2
votes
1answer
64 views

Seeking Better (Symmetry Exploiting) Solution and Generalization of An Inequality

Given positive variable $(x,y,a,b)$ where $x+y=1$, how does one "slickly" prove the following inequality? $$f(x,y) := \frac{xa+yb}{\sqrt{xa^2+yb^2}}\ge \frac{2\sqrt{ab}}{a+b}.$$ or simply $$f(x,y) := ...
2
votes
2answers
152 views

Combining symbols with symmetry

So this question has probably been answered already, but I can't find a good answer through searching google or this site. Basically, if I have n symbols, how many n-length combinations of the ...
1
vote
1answer
24 views

Lie Algebra: Optimal system of one-dimensional sub-algebras of the heat equation

This is a follow up question to Invariants of a PDE by Lie Symmetries, as I tried to follow the reasoning from the book Applications of Lie Groups to Differential Equations (Peter J. Olver, Example ...
1
vote
1answer
354 views

How is $r(\theta) = \sin \frac\theta2$ symmetric about the x-axis?

I understand how it is symmetric about the $y$-axis. because $r(-\theta) = \sin \left(-\frac\theta2\right)=-\sin \left(\frac\theta2\right)=-r(\theta)$ But how is it symmetric about $x$-axis?
1
vote
1answer
39 views

Reading the sphere diagrams in point groups on wikipedia

How do you read/make sense of the sphere diagrams shown here: http://en.wikipedia.org/wiki/List_of_spherical_symmetry_groups What do the yellow shaded areas represent? What are the red ...
1
vote
5answers
117 views

What does it mean for $AA^T$ to be symmetric?

What does it mean for $AA^T$ to be symmetric? A question in my book says to show that $AA^T$ is symmetric so I took a very simple matrix to try and understand this: $A=\begin{bmatrix} 2 \\ 8 \\ ...
0
votes
0answers
46 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 ...