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Questions tagged [symmetry]

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

54
votes
9answers
24k 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 ...
4
votes
3answers
2k views

Show that if $c_1, c_2, \ldots, c_{\phi(m)}$ is a reduced residue system modulo m, $m \neq 2$ then $c_1 + \cdots+ c_{\phi(m)} \equiv 0 \pmod{m}$

Show that if $c_1, c_2,\ldots, c_{\phi(m)}$ is a reduced residue system modulo $m$, $m \neq 2$, and $m$ is a positive integer, then $c_1 +\cdots+ c_{\phi(m)} \equiv 0 \pmod{m}$ From the problem ...
10
votes
9answers
4k 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 ...
26
votes
1answer
962 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 ...
7
votes
2answers
269 views

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, ...
1
vote
1answer
366 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) ...
6
votes
1answer
1k 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 ...
4
votes
1answer
233 views

Visualizing tuples $(a,b,x,y)$ of the extended Euclidean algorithm in a four-dimensional tesseract. Are there hidden symmetries?

I am trying to visualize the possible symmetries in the Euclidean four-dimensional space of the $4$-tuples of points $(a,b,x,y)$ generated by the extended Euclidean algorithm, where $ax+by=gcd(a,b)$. ...
161
votes
5answers
8k 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 ...
46
votes
1answer
1k 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 ...
8
votes
1answer
606 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 ...
1
vote
3answers
5k 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 ...
0
votes
1answer
78 views

How to determine the size of the complete game tree for basic [M]?

You can read the rules of the game here, or actually play it free on the mobile mbrane app, but it's not required to address the question. Essentially: players take turns placing integers onto an ...
20
votes
3answers
2k 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 ...
12
votes
3answers
2k 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} ...
23
votes
1answer
389 views

Why does $\int_1^\sqrt2 \frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$ equal to $0$?

In this question, the OP poses the following definite integral, which just happens to vanish: $$\int_1^\sqrt2 \frac{1}{x}\ln\bigg(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\bigg)dx=0$$ As noticed by one commenter ...
8
votes
2answers
799 views

Reflection with respect to a parabola

I know how to find a reflection with respect to one of the axis or with respect to the origin, but let's say I want to find the reflection with respect to a parabola, how do I do it? Let's say we have ...
8
votes
4answers
360 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 ...
2
votes
1answer
1k views

Understanding the symmetries of the Riemann tensor

I know that the Riemann tensor $R_{\alpha \beta \gamma \lambda}$ for $n$ dimensions has $$\dfrac{n^2(n^2-1)}{12} \tag{1}$$ independent components because of its properties: Symmetry: $R_{\alpha\beta\...
7
votes
1answer
722 views

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 ...
8
votes
1answer
216 views

A volume form on the sphere which gives equal areas to all hemispheres is invariant under the antipodal map?

Let $\omega$ be a volume form on $\mathbb{S}^2$ with the property that the induced area (w.r.t $\omega$) of all the hemispheres is the same. Is it true that $\omega$ is invariant under the antipodal ...
4
votes
1answer
140 views

Which matrices commute with $\operatorname{SO}_n$?

$\newcommand{\GLp}{\operatorname{GL}_n^+}$ $\newcommand{\SO}{\operatorname{SO}_n}$ Let $n>2$, and Let $A \in \GLp$ be an invertible real $n \times n$ matrix, which commutes with $\SO$. Is it true ...
2
votes
0answers
49 views

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 (...
0
votes
2answers
202 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 ...
2
votes
2answers
64 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
108 views

Can't understand integration-through-symmetry argument

In a previous question of mine, I asked how the volume of the region $$xy>zw \quad\wedge \quad x>-y$$ bounded by the unit $4$-ball (so $x^2+y^2+z^2+w^2<1$) could be calculated. The answer ...
2
votes
1answer
128 views

Does pushforward map preserve integral curves?

Let $X$ be a vector field on the manifold $M$ and let $Y$ be a vector field on $N$. Denote with $e^{tX}$, $e^{tY}$ the flows they generate, that is $$ u(t)=e^{tX}p\quad \text{is the unique solution ...
23
votes
12answers
3k views

Stuck on a Geometry Problem

$ABCD$ is a square, $E$ is a midpoint of side $BC$, points $F$ and $G$ are on the diagonal $AC$ so that $|AF|=3\ \text{cm}$, $|GC|=4\ \text{cm}$ and $\angle{FEG}=45 ^{\circ}$. Determine the length of ...
16
votes
1answer
644 views

Is there an explicit left invariant metric on the general linear group?

Consider $GL_n^+$, the group of (real) invertible matrices with positive determinant. Is it possible to find an explicit formula for a metric on $GL_n^+$ which is left-invariant, i.e $$d(A,B)=d(gA,...
6
votes
1answer
2k views

Inverse of diagonally dominant matrix with equal off-diagonal entries

Is there an explicit expression for the inverse of strictly diagonally dominant matrix with identical off-diagonal elements? For example: $$ \begin{pmatrix} a & -b & -b \\ ...
14
votes
5answers
1k views

Definition of group action

I'm currently taking a class in abstract algebra, and the textbook we are using is Ted Shifrin's Abstract Algebra: A Geometric Approach. In the chapter on group actions and symmetry, he defines a ...
7
votes
2answers
77 views

Does the action of a linear map on $k$-dimensional subspaces determine it up to scaling?

Let $V$ be a real $d$-dimensional vector space, and let $1 \le k \le d-1$ be a fixed integer. Let $A,B \in \text{Hom}(V,V)$, and suppose that $AW=BW$ for every $k$-dimensional subspace $W \le V$. Is ...
6
votes
1answer
149 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 ...
5
votes
0answers
148 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
1answer
99 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 ...
2
votes
2answers
1k 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 ...
5
votes
3answers
601 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 ...
4
votes
1answer
186 views

Can the isometry group of a metric space determine the metric?

Let $(X,d)$ be a metric space. There are always other metrics on $X$ which generates the same topology, and have the same isometry groups, for instance $\tilde d =\sqrt d$. (The same will be true for ...
3
votes
3answers
2k views

$A$ is a symmetric real matrix. Show that there is $B$ such that $B^3=A$

I'm having trouble with this question, I'd like someone to point me in the right direction. let $A$ be a n by n matrix with real values. show that there is another n by n real matrix $B$ such that $...
3
votes
3answers
95 views

Every $k$ dimensional subspace is $S$-invariant implies $S$ is a multiple of the identity

Let $V$ be a $d$-dimensional real vector space. Let $S:V \to V$ be an invertible linear map, and let $2 \le k \le d-1$ be fixed. Suppose that every $k$-dimensional subspace is $S$-invariant. Then, I ...
9
votes
2answers
1k 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 $g(x,y,z,\...
9
votes
2answers
6k 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. ...
7
votes
3answers
4k 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 ...
7
votes
1answer
90 views

Visualizing and understanding the roots of $f(z) = z^2 - e^{i\varphi}$

[I've added another animated picture below, showing how the actions of the groups $Q_\alpha$, $R_\beta$ coincide every now and then.] The roots of $f(z) = z^2 - e^{i\varphi}$ are simply $\pm e^{i\...
3
votes
1answer
58 views

A closed manifold of negative Ricci curvature has no conformal vector fields

I am trying to find a reference for the following claim, which is probably well-known: Let $M$ be a closed manifold of dimension $\ge 3$, with negative Ricci curvature. Then every conformal vector ...
3
votes
2answers
1k 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 of a ...
2
votes
4answers
506 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
0answers
89 views

Why contains the product of highest dimensional representation (with dim$\ne 1$) with itself the rotation around the main axis?

I observe that in the "standard finite point group" symmetries containing higher dimensional irreducible representations (over $\mathbb{R}$), the total product of the highest dimensional ...
2
votes
2answers
796 views

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 ...
2
votes
0answers
37 views

Location shifted symmetric probability distributions

Let $G: \mathbb{R}\rightarrow [0,1]$ be a cumulative distribution function (CDF) symmetric about zero, i.e., $G(x)=1-G(-x)$ at each $x\in \mathbb{R}$. Take some real numbers $\mu_1,\mu_2$. Consider ...