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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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Introduction to quaternion's geometrical intuitions starting from a set of three complex numbers

Introduction Recently I made a question here about the binary-tetrahedral-group and I started by analyzing quaternion-projections in the complex plane and their associated symmetries. Introduction to ...
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Introduction to the Binary Tetrahedral group and the 24-cell

Context and introduction I was playing with complex number sequences $Z_n=r_n\omega^n=u_n+iv_n$ represented in space and realized that it's always possible to associate up to 48 naturally symmetric ...
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Difficulty Understanding Notation for the Symmetries in Tiling Patterns

I am starting to learn group theory, and am reading about the symmetries in tiling patterns here. I have a few questions about the examples provided: In figure 9, a vertical line through the red ...
Chris Daniel's user avatar
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General form of symmetric function of 3 dimensional vectors

In the paper Deep Sets by Zaheer et al, they prove a theorem (eq 18 in appendix) that states that any general scalar symmetric function of $M$ variables can be written as $$f(x_1, x_2, \dots, x_M) = \...
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What do people call an equation that describes a relationship between a functional and one or more of its functional derivatives?

"Functional differential equation" seems to already be used for something different (https://en.wikipedia.org/wiki/Functional_differential_equation). I'm talking about something like the ...
William Wright's user avatar
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Scaling and Helmholtz equation

Solutions to Laplace equation and powers thereof have some convenient invariance properties that solutions to Helmholtz equation $\Delta u-u=f$ apparently lacks, especially in regards to scaling, ...
undefined's user avatar
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Writing symmetry operations as composition of group action and stabilizer

In the attached image let $K$ be the union of all colored-in edges. The symmetry group of the cube is restricted by imposing that K should be mapped onto itself. I want to find the restricted symmetry ...
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Explicit example of a set of coset representatives of $U(n)$ within $O(2n)$

I understand how to identify a unitary group $U(n)$ with the elements of the orthogonal group $O(2n)$ which commute with a linear complex structure $J$. I am also aware of the "two-out-of-three&...
Andrius Kulikauskas's user avatar
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Symmetry Intuition for Dice Probability Question

My question is a follow-up to this question: Is this a fair game (three players throw dice)? Player A tosses three six-sided fair dice, and B tosses a fair D-20. Whoever has the larger number/sum of ...
Abhay Agarwal's user avatar
3 votes
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Conditions for two linear transforms of iid random variables to have a symmetric joint distribution

Suppose $Z$ is a $T \times 1$ vector with i.i.d. $Z_t, t=1,2,\ldots,T$, and $E(Z_{t})=0$. Now given two constant $T \times 1$ vectors $a$ and $b$, define two random variables: $X=a^{\prime}Z$ and $Y=b^...
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How best to find orientation of a mirror plane

As part of a much larger 3D-symmetry-group-finding algorithm, I have this problem about searching for mirror planes: Let's say I have a circumscribed polygon (i.e. all of its vertices are the same ...
Sebastian Mostek's user avatar
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Showing a subset is dense in the symmetric tensor product of a Hilbert space

Let $\mathcal{H}$ be a Hilbert space and $\mathcal{H}^{\odot p}$ denote the $p$-fold symmetric tensor product. I want to show that the set $$U=\text{Span}\{u^{\otimes p} : u\in \mathcal{H}\}$$ is ...
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Dihedral group $D_3$(also $D_4$) - reflection compose rotation

I’ve read in a book(image link)that for dihedral group $D_3, a \circ r_1 = b $where a means reflection about $AO$ ($O$ is the centroid of an equilateral triangle $ABC$), $r_1$ means rotation about O ...
A Ghosh 's user avatar
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The symmetry of multiplications with the number 9

Yesterday while I was watching my 8 year old son doing his math-exercises which was multiplications with 9, I noticed this symmetry in the results of multiplying 9 with the range 1 to 10, which I ...
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Higher order Laplacian-like operators

We have the second order differential operator Laplacian for sphere $\Delta=\frac{1}{\sin\theta}\partial_\theta(\sin\theta\partial_\theta) + \cot^{\;2}\theta {\partial_\phi}^2$ and for plane $\Delta={\...
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Dirac Points: Triangular vs. Honeycomb Lattices

I'm reading the paper 'Honeycomb Lattice Potentials and Dirac Points' by Fefferman&Weinstein. To my understanding they claim that the existence of Dirac Cones at K/K' points is entirely determined ...
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Back to basics: Why do we care about symmetry of functions?

I am teaching a high school class on basic properties of functions, and like to motivate each of the properties with an example of why we even care to look at these properties. E.g. monotonicity ...
arridadiyaat's user avatar
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Why does $R_{a,\theta}S_L(a)=R_\theta S_L(a)+(1-R_\theta)a$?

I'm working on a problem, Show that $R_{a,\theta}S_L(x)=T_c$, where $c=(1-R_\theta)(a-S_L(a))$. Here $R_{a,\theta}$ is rotation by angle $\theta$ about point $a$; $S_L$ is reflection in line $L$; $...
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Prove symmetry of probabilities given random variables are iid and have (not absolutely) continuous cdf

Let $Y_1, Y_2, \ldots$ be independent and identically distributed random variables in $(\Omega, \mathscr{F}, \mathbb{P})$ s.t. their distributions are continuous. Denote common distribution as $$F(y) :...
BCLC's user avatar
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Is the $k$th covariant derivative of a function $f\colon M\to\mathbb R$ symmetric?

In the flat case ($\mathbb R^N$ with the Euclidean metric) it is true that the $k$th covariant derivative $\nabla^kf$ is symmetric because its coordinates are: $$(\nabla^kf)_{i_1\ldots i_k}(x)=\dfrac{\...
Raoní Cabral Ponciano's user avatar
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Matrices generate finite subgroup of $SL_2(\mathbb{Z})$

Show that $A:= \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} $ and $ B:=\begin{bmatrix} 0 & -1 \\ 1 & 1 \end{bmatrix} $ generate a finite subgroup of $SL_2(\mathbb{Z})$ (set of ...
Xaver Wallenstein's user avatar
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Differences in symmetry group of a triangle in $\mathbb{R}^3$ vs. $\mathbb{R}^2$?

Given a regular triangle in the xy-plane with corners in $(1,0,0); (-\frac{1}{2},\frac{\sqrt{3}}{2},0);(-\frac{1}{2},-\frac{\sqrt{3}}{2},0)$, show that the symmetry group $G \subset O(3)$ of this ...
Merkel_Bot's user avatar
2 votes
2 answers
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Why is my approach wrong for this probability question?

I came across the following question: You roll a fair $40$-sided and fair $60$-sided dice. What is the probability that the number on the $60$-sided dice is strictly larger than that on the $40$-...
Abhay Agarwal's user avatar
1 vote
1 answer
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How to exploit symmetries in vector fields

Doing electrostatics, I've found that everyone makes assumptions that nobody proves, and it's related to symmetries. If we have a system that we want to calculate the E-field, one starts with the ...
Tito Diego's user avatar
7 votes
1 answer
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Irrational numbers as "periods" in discrete sequences based on complex exponentials

Main string description To keep the core of the question short, I leave the context introduction at the end of this text. Here I describe a construction that is related to that introduction, but to ...
phionez's user avatar
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Intuition for Symmetry while calculating Mean of Hypergeometric Distribution

I'm studying discrete probability, and one of the example problems in my text is: Let $X \sim \ Hypergeometric(b, r, k)$. Determine $E(X)$. Definition of Hypergeometric Distribution: The random ...
Abhay Agarwal's user avatar
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2 answers
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Each symmetry of a triangle corresponds to exactly one permutation

I am trying to understand a statement about symmetries of a triangle. The set of lecture notes I am working through assert that (1) every permutation corresponds to a symmetry and (2) every symmetry ...
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Symmetric hyperbolic 3-manifold

Suppose one has a complete closed hyperbolic 3-manifold upon which there is an anti-conformal involution (a symmetry). Can one represent the manifold as a quotient of $\mathbb{H}^3$ by some discrete ...
Alex's user avatar
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Using Symmetry in calculating expectations

Shuffle an ordinary deck of 52 playing cards containing four aces. Then turn up the cards from the top until the first ace appears. On the average how many cards are required to be turned before ...
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Symmetries in regular "floral" circle patterns (seed of life etc.)

Consider $n$ equally sized circles $c_i$ in the plane with their centers $x_i$ in the vertices $v_i$ of a regular $n$-gon and all meeting in the center of that $n$-gon. Examples: Fig. 1: circle ...
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Defining Schwars rearrangement for other kind of symmetric domains

Given a Lebesgue measurable subset $E \subset \mathbb{R}^N$ we denote its $N-$dimensional Lebesgue measure by $|E|$. Let $\Omega \subset \mathbb{R}^N$ a bounded measurable set and $u : \Omega \to \...
Lucas Linhares's user avatar
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Symmetry of the partial derivative of a diffeomorphic function

I am wondering if there is symmetry among the single partial derivative of a multivariate diffeomorphic function. Let $f: \mathcal{X} \rightarrow \mathcal{X}$ be a diffeomorphism defined over $\...
ajl123's user avatar
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Relations Symmetry and Transitivity

Given the following Relations over the set $M := \{α, β, γ\}$ $R1 := \{(α, α), (α, β), (β, α), (β, β), (γ, γ)\}$ How is $R1$ transitive? The condition for transitivity is $(a,y)\in R1 \text{ and }(...
robsmayer's user avatar
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1 answer
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System of 3 non-linear equations in 3 unknowns - sum and product of solutions without needing to actually solve?

Consider three, rather generic, equations in three variables $x,y,z$: $$ \begin{cases} y z+x=a\\ z x+y=b\\ x y+z=c \end{cases} $$ Now, assuming that there is a solution to this system, is it possible ...
Red Five's user avatar
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On the distribution of odd/even length symmetric continued fractions.

In this question it is shown that if a continued fraction of some rational $\frac{p}{q}=[a_0; a_1, \dots, a_n]$ is symmetrical (i.e. $a_k = a_{n-k}$) then $q^2 \equiv(-1)^n \pmod{p} $. A converse is ...
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Compact asymmetric form in the plane?

Not being a mathematician, I may be imprecise in asking my question. For experiments on the visual perception of symmetries in the plane, I'm looking for (a) a closed curve in the plane, that is (b) ...
mk9y's user avatar
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1 answer
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Curves vs Lines: A Symmetry Question

For every line in the 2d plane, we can construct a shape with an "inside and outside" (often a circle) such that the shape is cut by the line into two symmetrical parts. Does this property ...
MrMustache's user avatar
2 votes
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50 views

Putting a group together around a circular table

A group of ten students includes triplets Alex, Alison and Alice wearing black, white and yellow t-shirts respectively. They are to be seated in groups of five around two tables: one green and one red....
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Grimmett and Stirzaker P57 (Letter Matching) updated

https://math.stackexchange.com/posts/2854522/edit I was stuck on step 4 of the derivation below. A secretary types n different letters together with matching envelopes, she then drops the pile down ...
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Riemannian geometry and Symmetries, Hyperbolic Spaces

I am currently trying to connect Hyperbolic geometry through several models with Riemannian geometry. At first I have transformed the metric tensor from sphere in $R^3$ and the pseudo-sphere in $R^{2,...
S_d_pap's user avatar
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1 answer
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A proof of Pólya-Szegő inequality

Denote by $|A|$ the $N-$dimensional Lebesgue measure of a Borel set $A \subset \mathbb{R}^N$ and define $$ A^\ast := B_{R}(0), \quad R = \left(\frac{N}{\omega_N}|A| \right)^{\frac{1}{N}}, $$ where $\...
Lucas Linhares's user avatar
1 vote
1 answer
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How we define the Lie symmetry of a stochastic differential equation?

In the literature of symmetry groups, Sophus Lie define the symmetry of a pde or an ode by a vector field defined in the tangent space of the submanifold (defined by solutions of the pde or ode) and ...
Anas Cobain's user avatar
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Geometrically determining the symmetry of a tetrahedron (using solid geometry, not group theory)

It is well established that a regular tetrahedron has 12 orientation preserving symmetries, the group $A_4$. To better understand these symmetries, I set out to identify them geometrically: Prove that ...
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Why or how to prove that the infinitesimal symmetries of a differential equation form a Lie algebra?

Given a partial differential equation, after computing the Lie point symmetries of this PDE, means a one-parameter group (Lie group) of transformation that leaves the set of solutions of the PDE ...
Anas Cobain's user avatar
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Is the tetrahedron necessarily mirror-asymmetric?

I was shocked to watch Anton Petrov's latest video, "Wow, Incredible Evidence That Universe Is Not Symmetric After All", where he says that the Tetrahedron is the simplest object that is not ...
Miss Understands's user avatar
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1 answer
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What is the correct sign for the four-vector potential gauge transform; $A_\mu\to A_\mu\pm\partial_\mu\lambda$ and where does this gauge originate? [closed]

I have three questions regarding the following extract(s), I have marked red the parts for which I do not understand for later reference. The convention followed for the Minkowski metric in these ...
Sirius Black's user avatar
1 vote
1 answer
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Non-zero expectation of $p(z)^kz$ for non-even univariate polynomial $p$ with Gaussian variable $z$.

Suppose that $p:\mathbb{R} \rightarrow \mathbb{R}$ is a polynomial and let $z$ be a standard Gaussian variable $(z \sim \mathcal{N}(0,1))$. I am interested in the behavior of the expectation $\mathbb{...
Yatin Dandi's user avatar
1 vote
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Claims about solution of nonlinear differential equation with symmetries

I have a nonlinear ordinary differential equation (of sixth order). If $y(x)$ satisfies the given nonlinear differential equation and its associated boundary conditions, both $y(1-x)$ and $-y(1-x)$ ...
akr's user avatar
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Symmetry about half of the domain in nonlinear ODE solutions

Is there a way to predict in advance whether the solution to a sixth-order nonlinear ODE will exhibit symmetry about half of its domain? The ODE is expressed as $f(x, y', y'', y''', y'''', y''''', y'''...
akr's user avatar
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Geometric centroid and symmetry

I have some intuition that for shapes with some rotational symmetry, the symmetry is typically about the geometric centroid. For example, for a cuboid and cube this is true. What about in general, for ...
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