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15
votes
2answers
171 views

slick way of transforming an integral?

The function $$ (\alpha,\beta) \mapsto \int_0^\beta \frac{\sin\alpha\,d\zeta}{1+\cos\alpha\cos\zeta} $$ is a symmetric function of $\alpha$ and $\beta$. But I don't know a simpler way to see that ...
10
votes
3answers
244 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} ...
9
votes
5answers
389 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 ...
8
votes
1answer
264 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 ...
8
votes
4answers
187 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
3answers
140 views

Realizing groups as symmetry groups

We're supposed to think of (non-Abelian) groups as groups of symmetries of some object. Sometimes it isn't obvious what this object is. For example, the fundamental group of a topological space acts ...
7
votes
2answers
465 views

Do symmetric 3d solid figures always have a plane of symmetry?

Chapter 3 of Martin Gardner's The New Ambidextrous Universe begins as shown below. As you can see (highlighted), on page 13 he writes that not all solid symmetric objects have a plane of symmetry, and ...
7
votes
3answers
139 views

Breaking symmetries

Back when I was studying electromagnetism and Maxwell's equations, our teacher told us a quote. I can't recall it exactly, but the meaning was roughly the following: Symmetry in a problem is ...
7
votes
1answer
173 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 ...
6
votes
1answer
81 views

Lie algebra $\implies$ Lie group?

Lie's third theorem says that every finite-dimensional Lie algebra g over the real numbers is associated to a Lie group G. So say I have an $r-$ parameter group of symmetries whose tangents at the ...
6
votes
0answers
114 views

ODE system and Lie symmetries

The ODE system (see below), where $F$ is a given function together the algebraic condition (SYM) imply that $$\boxed{y(t)=k-x(t)} \tag{*}$$ for some $k$ constant. The result that $y$ is a translation ...
5
votes
3answers
61 views

Matrix $A=B+C$ with $B$ symmetric and $C$ antisymmetric

I am stumped on a question and am looking for some guidance on how to get it done. The problem gives you: $x_1 = \begin{bmatrix}9&-4&-2 \\-9&6&-3 \\10&-3&9\end{bmatrix}$ ...
5
votes
2answers
1k 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. ...
5
votes
1answer
191 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 ...
5
votes
2answers
62 views

Why this definition for “symmetry transformation”?

This question concerns section 8.5.1 in these notes: I don't understand why a symmetry transformation is defined as such. What implications is there if $\delta \mathcal L$ is a total ...
5
votes
1answer
172 views

Magic theorem for cylinders? Symmetry classes according to Conway's notation?

My teacher Kirsi of Mat-1.3000 in Aalto University stated 17 symmetry classes for planes and 14 for spherical things (some lecture slides here). She used Conway Thurston's notation to classify ...
5
votes
1answer
245 views

Walks of Even Length on a Bipartite Graph

Given a random walk on a simple $d$-regular bipartite graph $G$. The adjacency matrix $A'$ of $G$ may be split into blocks $$ A'=\pmatrix{ 0 &A^T\\ A&0 }, $$ The propagation operator $M=A'/d$ ...
4
votes
2answers
84 views

Automorphisms of a structure as a powerful tool for studying the structure

This is just an arbitrary testimony of an often repeated slogan: "The group of automorphisms of a given structure is often a powerful tool for studying this structure." D. Lascar, On the ...
4
votes
3answers
650 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 ...
4
votes
2answers
56 views

Intersection of two tetrahedra, point reflexion

We are given a regular tetrahedron $ABCD$ ($ABC$ is its` base and $D$ is its vertex) and we reflect it through the middle of its height (point reflexion) - and thus we obtain a congruent regular ...
4
votes
1answer
54 views

What exactly is meant by symmetry?

This is something that has troubled me for long. Artin describes 4 types of symmetry: translational, rotational, reflective and glide. I somehow understand symmetry to be a "condition" in which a ...
4
votes
1answer
45 views

Infinitesimal $SO(N)$ transformations

An infinitesimal $SO(N)$ transformation matrix can be written : $$R_{ij} = \delta_{ij}+\theta_{ij}+O(\theta^2)$$ Now it has to be shown that $\theta_{ij}$ is real and anti-symmetric. I've started ...
4
votes
1answer
62 views

Gliders, static structures in various (dynamic) systems

Structures, i.e. symmetries over time, appear in various systems: gliders in cellular automata, like Game of Life or Rule 110, unmatched string's parts in rewrite systems – unchanged in multiple ...
4
votes
1answer
89 views

Immersions of the Klein Bottle

The famous immersion of the Klein bottle lacks symmetry. (I'm talking about this one. http://en.wikipedia.org/wiki/File:Klein_bottle.svg) One can only see one plane of reflection. However, the ...
4
votes
1answer
138 views

Symmetrically splitting an octagon into quadrilaterals

I'm wondering whether it is possible to split an octagon into a finite number of quadrilaterals, such that the result is symmetric from all 8 directions (sides or points). There is one condition — any ...
4
votes
2answers
269 views

What's the intuition of the transpose of a matrix?

I know the transpose is to swap the columns and rows of a matrix. And $A^T$$A$ is a symmetric matrix which elements are the inner product of each column of $A$. But I didn't understand the intuition ...
4
votes
0answers
28 views

Symmetry groups of discrete functions

I'm looking for basic information about symmetry groups of discrete functions. It is difficult to search for such information, because searching for "symmetry group" gives results that refer almost ...
3
votes
3answers
859 views

Exponents in Odd and Even Functions

I was hoping someone could show or explain why it is that a function of the form $f(x) = ax^d + bx^e + cx^g+\cdots $ going on for some arbitrary length will be an odd function assuming $d, e, g$ and ...
3
votes
1answer
191 views

Decomposition formulas for rotational symmetries of a cube

I have a problem that I would like to check my work on. I am also stuck on the verifications for $E$ and $F$. Any help would be greatly appreciated. Thanks in advance. Problem statement: Let $G$ be ...
3
votes
1answer
64 views

Counting 0-1 matrices up to symmetry

I'm interested in counting the number of n×n 0-1 matrices with a given number of 1s up to rotation and reflection. What is the best way to do this if n is not too small? For example, consider ...
3
votes
1answer
496 views

What does cubic symmetry mean?

Is it an invariance under rotations around x-, y-, z-axis? Does this invariance separately include rotations around an arbitrary $(x$, $y$, $z)$ axis?
3
votes
3answers
54 views

Definition of Equals

DISCLAIMER: This is a first time Math.SE post from a 30-something who is only now learning math. I have read the Rules and this may not satisfy the "Questions with too many answers" criteria or ...
3
votes
1answer
45 views

Show the integrals are alike without explicit computation.

I saw a couple of striking integrals which are $\hspace{5em} \displaystyle \color{black}{\displaystyle \int_a^b \frac{x\,\mathrm{d}x}{\sqrt{(x^2-a^2)(b^2-x^2)}} }$$\displaystyle\ =\ $$\displaystyle ...
3
votes
1answer
56 views

Symmetry in reduced residue systems

This may be a stupid question, but it looks to me like the reduced residue systems modulo N are symmetrical about N/2; that is to say, that the there is the same number of integers not divisible by a ...
3
votes
2answers
61 views

Asymmetry in the error correction in coding theory

Does it make sense to have an error correction code which acts differently on different states (for example if we run something which runs on the binary string from $0^n \rightarrow 1^n$ involving all ...
3
votes
1answer
100 views

Proof that a ODE system admits symmetric solutions

I have a ODE system of the form $\dot{x} = f(x)$ with $x \in \mathbb{R}^3$. Now it is claimed that if $(x_1,x_2,x_3)$ is a solution to the system that also $(-x_1,-x_2,-x_3)$ is a solution. How can I ...
3
votes
1answer
33 views

Symmetry and singularities

The function $f(x) = x$ is point symmetric. But what's with $g(x) = (x^2 - x)/(x - 1)$? $\mathbb{D}_g = \mathbb{R} \backslash \{1\}$ $g(x) = -g(-x)$ is true for every $x \in \mathbb{R} \backslash ...
3
votes
0answers
52 views

Fractals vs. “neatness” / order

I've seen a lot of high level videos on fractals, etc, and how they might apply to the real world. So a tree is branches with branches with branches, and our blood vessels branch and then branch ...
2
votes
3answers
108 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 ...
2
votes
2answers
38 views

Is $u^TAu \geq 0$ true for all symmetric matrices $A$?

we know from the definition of inner product and norm, that $u^Tu$ is always larger than zero, except the case where $u=0$ at which case it is zero. I came across a question that infers that $u^TAu ...
2
votes
1answer
60 views

Surjectivity for permutation representation of a group action

I am having trouble proving that my function is surjective. Here is the problem statement: Problem statement: Let T be the tetrahedral rotation group. Use a suitable action of T on some set, and the ...
2
votes
1answer
108 views

Isometries of the plane

Let $m$ be an orientation-reversing isometry. Prove algebraically the $m^2$ is a translation. What I attempted: We know that $m$ is an orientation-reversing isometry i.e. it is either a reflection or ...
2
votes
2answers
133 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 ...
2
votes
1answer
46 views

If $\mathbf{X}_1$ and $\mathbf{X}_2$ are symmetries, then so is $[\mathbf{X}_1,\mathbf{X}_2]$.

Background: Suppose we have a differential equation $$ y^{(n)}=\omega(x,y,y',\ldots,y^{(n-1)}) $$ that satisfies two symmetries $\mathbf{X}_1$ and $\mathbf{X}_2$. That is, if $$ ...
2
votes
3answers
380 views

Symmetry group of a triangular lattice

The question is: Let $G$ be the group of symmetries of an equilateral triangular lattice $L$. Find the index in $G$ of the subgroup $T \cap G$. ($T$ is the group of translations) The solution ...
2
votes
1answer
128 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.) ...
2
votes
1answer
39 views

2x2 symmetric matrix is a subspace of vector space.

Can you kindly check my proof of the problem and correct if possible. The following $S=\{A\in M_{2,2} | AA^T=A^TA\}$ is a subspace of $V=M_{2,2}$ all real $2\times2$ matrices. My proof: S ...
2
votes
1answer
106 views

Help with Autonne-Takagi factorization of a complex symmetric matrix.

Let $A=A_1i+A_2$ with $A$ non singular. Now let $$B =\begin{bmatrix} A_1 & A_2\\ A_2 & -A1 \end{bmatrix}$$ With $A_1$, $A_2$ and $B$ symmetric. Is it true that: 1) $B$ is non singular 2) ...
2
votes
2answers
28 views

Symmetrical curve equation in range of [0,1]

I would like to define a function $f(x,a)$ that has the following properties: $f(0,a)=0$, excluding $a=\infty$ $f(1,a)=1$, excluding $a=-\infty$ $f(x,0)=x$ $f(0,a\to\infty)\to1$ $f(1,a\to-\infty)\to0$ ...
2
votes
1answer
52 views

ODE system and single PDE “equivalence”, reference request

The answers to this question Replacing large-dimensional ODE systems with one PDE suggest that, in general, one can not hope for "replacing" an ODE system with a single PDE. On the other hand, this ...