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

# Questions tagged [symmetry]

1349 questions

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### 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

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### 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

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### 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

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### 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

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### 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…

Slender Threads

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### 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

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### 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…

blackened

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### 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

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### 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

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### 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

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### 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

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### 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

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### 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

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### 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

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### 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…

JLA

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