Questions tagged [symmetry]
Questions about symmetry, in group theory, geometry or elsewhere in mathematics. See https://en.wikipedia.org/wiki/Symmetry
3
votes
3answers
81 views
Does Young's inequality hold only for conjugate exponents?
Suppose that $ab \leq \frac{1}{p}a^p+\frac{1}{q}b^q$ holds for every real numbers $a,b\ge 0$. (where $p,q>0$ are some fixed numbers).
Is it true that $ \frac{1}{p}+\frac{1}{q}=1$?
I guess so, and ...
1
vote
2answers
30 views
A woman planning her family considers the following schemes on the assumption that boys and girls are equally likely at each delivery
A woman planning her family considers the following schemes on the assumption that boys and girls are equally likely at each delivery:
(a) Have three children.
(b) Bear children until the first girl ...
0
votes
1answer
51 views
Expressing musical patterns as a mathematical model?
I've been invited to speak on music in front of high schoolers and I decided I will speak about the repetitive patterns and symmetrical aspects of music. Now, I'm not very good at math and hence I ...
1
vote
1answer
13 views
Classify the regular polygons that fit about a common vetex
I know this can be solved using a (quasi-)symmetric groups approach from crystallography, but I wish to solve it with a more simple approach, number theory motivated.
I wish to classify the regular ...
0
votes
1answer
42 views
When studying the dihedral group of a square, do we consider only vertices or the whole points which the square covers?
When studying the dihedral group of a square, do we consider only vertices or the whole points which the square covers? Because the vertices of square also gives the same symmetries.
2
votes
0answers
30 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 ...
0
votes
0answers
14 views
Compare classical and modified algorithm of Cholesky
Let $\textbf{A}$ be a symmetric matrix with tthe modified Cholesky factorization $\textbf{A} = \textbf{R}^T\textbf{D}\textbf{R}$ and the classical factorization $\textbf{A} = \textbf{R}_c^T\textbf{R}...
1
vote
0answers
31 views
Constrained Optimization Insights
I have been experimenting with the following problem paraphrased from Khan Academy:
A manufacturer's revenue is $100h^{2/3}s^{1/3}$, where $h$ is the number of hours of labor hired, and $s$ is the ...
4
votes
1answer
104 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 ...
0
votes
2answers
125 views
How can we show that $I_{yy}=I_{zz}=I_{xx}$ and $I_{xy}=0$ using symmetry arguments?
Consider two integrals of the form $$I_{xx}=\int x^2 f(r)d^3r,~~I^\prime=\int xy ~f(r)d^3r$$ where $f(r)$ is a function of $r=|\vec{r}|$ only and has no dependence on $\theta,\phi$ in pherical polar ...
1
vote
0answers
23 views
Do cocycles “break” symmetry? [migrated]
In an article by A. Borovik, “Is mathematics special?”, he talks about the fact that carrying is a cocycle. He then says
[A student] discovered that carry is doing what cocycles frequently do: they ...
0
votes
0answers
30 views
Is there an algorithm for breaking symmetries in polycube puzzles?
Does there exist a general algorithm for dividing a polyomino/polycube packing problem into a set of subproblems which if solved will produce in aggregate all solutions to the original problem, but ...
1
vote
0answers
17 views
Products of subgroups of Euclidean Group
So, I'm reading a book and I need help with some stuff.
The book defines product of subgroups $G_1, G_2$ of $G = E(n)$ as $G_1G_2 = \{g_1g_2|g_1 \in G_1, g_2 \in G_2\}$, which is not necessarily a ...
1
vote
0answers
39 views
Finding axis of symmetry given n points and their surface normals
Given a roughly torus-shaped 3D object: it has cylindrical symmetry, one axis of rotation, and is symmetric with respect to any angle around that axis.
Imagine a donut, but with any number of ridges ...
0
votes
0answers
47 views
Set invariant under the map $x \mapsto -x$ that is a translation of symmetric sets
Let $F \subset \mathbb{R}^n$ be an open bounded (non empty) convex set and assume that it is invariant under the map $x \mapsto -x$ (which means that $F= \{ x : x \in F \} = \{ -x : x \in F\} = -F$).
...
1
vote
1answer
15 views
Find $ C_{D_{12}}(a)$=centraliser of $a$ and $ C_{D_{12}}(b)$=centraliser of $b$
Consider the Dihedral group $D_{12}=\left\langle a,b: a^6=e, \ b^2=e, \ ba=a^5b \right\rangle$ of order $12$ of symmetries of regular hexagon. Every element of $D_{12}$ can be written as $a^ib^j, \ 0 \...
1
vote
1answer
25 views
Holomorphic functions reflected through segments that aren't on the real axis
My question concerns using the Schwarz Reflection principle (or symmetry principle) to reflect regions in the domain of a holomorphic function into a symmetric (with respect to a line segment) region ...
0
votes
0answers
14 views
Prove, that $M=\{(a,b)\mid a,b \in \mathbb{N_0} \land (a-b) \text{ mod } 4 = 0\} $ is a equivalence relation
Prove, that $M=\{(a,b)\mid a,b \in \mathbb{N_0} \land (a-b) \text{ mod } 4 = 0\} $ is an equivalence-relation.
Refl.: $a-a=0 \text{ mod } 4 =0$
Sym.: $\forall x,y \in M: (x,y) \implies (y,x)$ (Not ...
2
votes
1answer
58 views
Does orthogonal-invariance of a differential imply invariance of the function?
Let $U:\text{Hom}(\mathbb{R}^d,\mathbb{R}^d) \to \mathbb{R}$ be a smooth function .
If $U$ is orthogonally-invariant, i.e. $U(QA)=U(A)$ for every $Q \in \text{SO}(n),A \in \text{Hom}(\mathbb{R}^d,\...
0
votes
2answers
36 views
Symmetry of Uniform Distribution PDF
I'm studying probability theory and came across an exercise problem that I couldn't quite understand, even with the solution, and was hoping someone could give me some insight.
Alice, Bob, and Carl ...
0
votes
1answer
16 views
Symmetry problem involving functions
I was given a definition.
A symmetry of a plane figure $F$ is an isometry that maps $F$ to itself, that is, an isometry $f:R^2 \to R^2$ such that $f(F)=F$.
I don't really understand this because ...
23
votes
1answer
363 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 ...
0
votes
0answers
12 views
Where is symmetry used in the rewriting of the following integral?
Let $F$ be a CDF that is symmetric about $t^* \in T$.
Consider the following integral:
$$\int_{t^*-\overline{\delta}}^{t^*+\overline{\delta}}cos(t)f(t)dt$$
Using change of variable and the symmetry ...
0
votes
0answers
13 views
Determine symmetry of given function
I am browsing the internet, looking for functions to determine its symmetry (because of effect of symmetry on Fourier coefficients) and I have found this function, from khan academy video:
https://...
0
votes
0answers
23 views
Example of an non-symmetrical Wilcoxon Rank Sum test statistic value
I have been given a homework in a subject called "Non-Parametric Statistics" and I'm a bit stuck with it. I would be very thankful if you could give me any advice or help, which would lead to a ...
1
vote
1answer
26 views
Symmetry group of Hopf fibration
https://en.wikipedia.org/wiki/Hopf_fibration
What is the group of transformations $\subset SO(4)$ that sends every fibre circle to another fibre circle?
I think the Lie algebra might be generated by ...
0
votes
2answers
49 views
Showing integral vanishes using symmetry
In the process of tackling a question I've ended up having to evaluate the integral $$ I = \displaystyle \int_0^1 \dfrac{4 \sin(2 \pi t) \sin (4 \pi t) + 2 \cos (2 \pi t ) \cos (4 \pi t)}{\sin^2 (2 \...
0
votes
1answer
24 views
Setting up a mesh inside a parallelepiped
This is a spatial geometry/linear algebra question with direct applications in crystallography. However, knowledge of crystallography is NOT necessary to answer the question.
I have the defining ...
2
votes
2answers
82 views
Two different color rooks are placed on a chessboard so they don't attack each other.
What is a number of ways we can do that?
Of course we have $64$ choices for the first one and then $49$ choices for the second one. So we have $64\cdot 49$ ways to put them. But some of the ...
1
vote
0answers
19 views
Symmetry in Bayesian Hypothesis testing
Let two hypothesis be:
$H_0: \rm PMF(\mu)$ with prior $(1-p)$ and
$H_1: \rm PMF(\sigma)$ with prior $p$.
Is it true that the probability of total error $ \Big( P_e= \rm type \:I \: error + type \: ...
2
votes
2answers
50 views
Prove/disprove, that the relation is reflexive, symmetric, antisymmetric and transitive
I want to prove or disprove, if the relation $R$ with
\begin{align}
iRj:\Longleftrightarrow (\forall k \in \mathbb{N} \text{ with } k \text{ is a prime number}:k \mid i \Longrightarrow k \mid j)
\end{...
1
vote
1answer
55 views
Symmetries of polynomials
Polynomials over $\mathbb{Q}$ can exhibit "geometrical symmetries" in different ways, graph-wise (as functions $P:\mathbb{R} \rightarrow \mathbb{R}$) and root-wise:
$P(x) = P (-x)$(axial symmetry ...
0
votes
0answers
29 views
How to determine basis functions partners of $x$, $x^2$, and $yz$ in $D_6$?
I have a character table for $D_6$ and I am trying to understand how to find the partners of basis functions. I am starting with $x$, $x^2$ and $yz$.
I am currently working on the $x$ function but am ...
1
vote
1answer
32 views
Rotational Symmetries of the Dodecahedron
I have been trying the build up my intuition on finding rotational symmetries of shapes and I have been looking at the dodecahedron from the platonic solids. I am convinced that I have the correct ...
0
votes
0answers
62 views
Symmetric Brick Stacking
Suppose you stack $n$ LEGO bricks ($2 \times 1$) in a plane, where
The base is contiguous
Each level is offset from the level below it by one stud.
Bricks are only stacked on top of other bricks, ...
1
vote
0answers
16 views
Can a function be uniquely determined from its symmetric components about two different points?
Can a function $f: \mathbb{R} \to \mathbb{R}$ be uniquely determined from its symmetric components about two different points? If so how?
Not sure if I'm formulating this correctly, but I've got a ...
1
vote
1answer
32 views
Determine if Relation $R$ is reflective, symmetric or transitive
Let $X = \{3, 5, 9\}, Y = \{2, 3, 6, 7\}$ and the relation, $R = \{(x, y) | y \leqslant x\}.$
For the relation, I manage to come out with $R = \{(2,3), (2,5), (2,9), (3,3), (3,5), (3,9), (6,9), (7,9)\...
3
votes
0answers
40 views
Symmetries in the equations and graphs of complex-valued polynomials
I tried to visualize the complex roots of a polynomial with real coefficients $a_i \in \mathbb{R}$:
$$f(z) = z^n + a_{n-1}z^{n-1} + \dots + a_1z + a_0$$
following some obvious thoughts:
For any ...
1
vote
1answer
21 views
Number of intersections formed by chords connecting all N evenly spaced nodes on a circle
Non-mathematician here, sorry if terminology is wrong.
For a circle with N evenly spaced nodes on the perimeter;
With chords connecting every node to every other node;
What is the total number of ...
0
votes
1answer
64 views
The integral solution lost its symmetry
I solved this integral using Mathematica:
$H(z,t)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dxdy(x\cdot y)^aE_{-t}(-ix)E_{-t}(-iy)E_{-z}(ix)E_{-z}(iy) $
Where $E_{n}(x)$ is the Exponential ...
0
votes
1answer
51 views
Symmetries and invariants.
I would like someone to give me an example of application of the following paragraph from an introductory book on Lie groups theory:
The paragraph states the following:
Symmetry transformations, if ...
0
votes
0answers
22 views
Rotational symmetry and translation symmetry on a infinitely large plane.
Given a infinitely large plane with uniform charge distribution with the charge density with the charge density being $\rho_s$ and has centrum in origin? The $z$-axis is the normal vector to the plane ...
2
votes
0answers
91 views
What are the generators of the signed symmetric group $B_n$?
Consider the set $A_n=\{-n,-(n-1),-(n-2),\cdots,-1,1,\cdots, n-1,n\}$ where $n \in \mathbb{N}$. Let G be a subgroup of the symmetric group $S_{A_n}$ defined as follow: $G=\{\sigma \in S_{A_n}| \forall ...
0
votes
0answers
38 views
Reflection operator as a product of inversion and rotation
I found an statement that says the following (Merzchbacher, Quantum Mechanics):
We can also proof that the reflection of a system with respect to a plane with normal vector $\vec{n}$, can be ...
0
votes
0answers
23 views
Metric to compare conditional probabilities
I am working on a machine translation task and there are two kind of probabilities:
$p_1 = Pr(\epsilon | \phi)$
$p_2 = Pr(\phi | \epsilon)$
Where $\epsilon$ and $\phi$ are phrases in source and ...
1
vote
1answer
50 views
Symmetry Groups [closed]
Are my answers on the right track?
How many symmetries does a square have? Describe them.
8 symmetries: 4 reflexives, 3 rotational (90, 180, and 270), 1 trivial
Working in the "symmetry group" of ...
0
votes
0answers
22 views
Reduction of appearing dilogarithms
I have a series of dilogarithms with various arguments and I was just wondering if it is possible to tell if they are all independent (that is to say, cannot be transformed into each other through e.g ...
2
votes
1answer
56 views
How to generate a symmetry group with a vector field?
I'm trying to study Lie groups applied to solution of differential equations, and I'm working with the following problem
Consider a first order homogeneous linear partial differential
equation $$...
1
vote
1answer
54 views
What does “n-fold axis” mean in symmetry groups?
Reading about the symmetries of a cube, here, they talk about (in the section "Details") "3 x rotation about a four fold axis" etc. I'm not quite sure what "four fold axis" means in the context. Can ...
3
votes
1answer
50 views
Is there a non-zero algebra homomorphism $\text{End}(V) \to \text{End}(\bigwedge^kV)$?
This question is totally out of curiosity.
Let $V$ be a real $d$-dimensional vector space. Let $1<k<d$ be fixed.
Is there a non-zero algebra homomorphism $\text{End}(V) \to \text{End}(\...
