Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [symmetry]

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

1619 questions
184
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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
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73
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10 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
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63
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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…
anon
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34
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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
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28
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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…
Slender Threads
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28
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5 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
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27
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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
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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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21
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3 answers

Antisymmetry in integral $I(s)=-I(1/s)=\int_0^1 \frac{\ln\left[x^s+(1-x)^{s}\right]}x dx $

I recently happened to look 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…
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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20
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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
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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
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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…
JLA
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17
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1 answer

Is there a simple maximally general generalization of Noether's theorem to arbitrary dynamical systems?

Noether's theorem informally states something like "symmetries in the dynamical law imply conserved quantities". However, the theorem is generally stated in terms of physics-specific classes of dynamical systems such as Lagrangian or Hamiltonian…
user56834
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17
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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
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