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Questions tagged [symmetry]

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

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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
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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
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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
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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
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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
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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}...
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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
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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
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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 ...
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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 ...
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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
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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 ...
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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 ...
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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
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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
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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
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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
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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
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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
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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
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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 ...
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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 ...
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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://...
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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
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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
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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 \...
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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
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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 ...
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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
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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
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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
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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 ...
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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
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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, ...
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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
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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)\...
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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 ...
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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 ...
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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
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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 ...
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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}(\...