Questions tagged [symmetry]

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

1349 questions
174
votes
6 answers

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 example, the Residue Theorem to show that $$ f(\alpha,…
Ron Gordon
  • 134,112
  • 16
  • 181
  • 296
71
votes
9 answers

Proving that $1$- and $2D$ 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 origin with probability $1$? Edit: note that while…
Isaac
  • 35,106
  • 14
  • 99
  • 136
60
votes
1 answer

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 group operation is - composition of…
whacka
  • 15,195
  • 25
  • 45
31
votes
1 answer

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 of that supposed "symmetry". Here comes: How…
Han de Bruijn
  • 16,250
  • 2
  • 41
  • 77
27
votes
3 answers

Can all groups be thought of as the symmetries of a geometrical object?

It is often said that we can think of groups as the symmetries of some mathematical object. Usual examples involve geometrical objects, for instance we can think of $\mathbb{S}_3$ as the collection of all reflections and rotation symmetries of an…
25
votes
1 answer

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 to the question, the only zero of the integrand…
Franklin Pezzuti Dyer
  • 37,332
  • 9
  • 60
  • 145
24
votes
12 answers

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 the segment $FG$. How can I approach this…
24
votes
3 answers

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 axioms that motivated them defining groups. My…
Gridley Quayle
  • 1,559
  • 1
  • 12
  • 16
20
votes
2 answers

Why is this definite integral antisymmetric in $s\mapsto s^{-1}$?

I recently happened into the following integral identity, valid for positive $s>0$: $$\int_0^1 \log\left[x^s+(1-x)^{s}\right]\frac{dx}{x}=-\frac{\pi^2}{12}\left(s-\frac{1}{s}\right).$$ The obvious question is how to show this (feel free to do so!).…
Semiclassical
  • 14,756
  • 3
  • 32
  • 84
18
votes
1 answer

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

Let $\operatorname{GL}_n^+$ be the group of real invertible matrices with positive determinant. Can we construct an explicit formula for a metric on $\operatorname{GL}_n^+$ which is left-invariant, i.e. $$d(A,B)=d(gA,gB) \, \,\forall A,B,g…
Asaf Shachar
  • 23,159
  • 5
  • 19
  • 106
18
votes
8 answers

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 reason behind the above inequality? I know this is…
Neeraj
  • 405
  • 1
  • 4
  • 8
17
votes
2 answers

Odd order moments of a symmetrical distribution

Is it true that for every symmetrical distribution all odd-order moments are equal to zero? If yes, how would I be able to prove such a thing?
nikos
  • 415
  • 2
  • 5
  • 13
17
votes
4 answers

show this inequality $ab+bc+ac+\sin{(a-1)}+\sin{(b-1)}+\sin{(c-1)}\ge 3$

let $a,b,c>0$ and such $a+b+c=3abc$, show that $$ab+bc+ac+\sin{(a-1)}+\sin{(b-1)}+\sin{(c-1)}\ge 3$$ Proposed by wang yong xi since $$\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{1}{ac}=3$$ so use Cauchy-Schwarz inequality we…
math110
  • 1
  • 15
  • 119
  • 475
15
votes
5 answers

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 group actions as follows $$\phi: G \mapsto…
Tianxiang Xiong
  • 499
  • 1
  • 3
  • 12
14
votes
1 answer

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 identity form a Lie algebra, can we conclude that…
1
2 3
89 90