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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1answer
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Rigid motions taking a regular $n$-gon back to itself carry vertices to vertices
I have been reading Keith Conrad's expository paper Dihedral groups I and I have two questions about Theorem $2.2$, which deals with the size of $D_n$. In the first part of the proof you can read
...
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1answer
35 views
Determining Graph Automorphisms by Determining Ways of Permuting Edges
Here is an example from the book Groups, Graphs, and Trees (Ignore example 1.15; I accidently included it in my snippet of the pdf):
From my understanding, the symmetry group of a graph $\Gamma$ ...
1
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1answer
43 views
Lie symmetry methods of differential equations.
If we have a differential equation
$$
y^{(n)} = H( x, y, y', ...y^{(n-1)}) .
$$
And say we have its two infinitesimal generators admitted by this equation. What do I mean by the invariance of this ...
1
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0answers
24 views
What is the operational way of discovering scale invariance of differential equations?
Context The answer here by @Keenan Pepper gives an instance for what it means for an algebraic or trigonometric formula to be scale invariant. For quick reference, I quote his answer here but with a ...
0
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1answer
39 views
The condition for the existence of a symmetric form for the reflection formula $f(1-x)= \chi (x) f(x)$
Suppose we have a functional equation in the form
$$f(1-x)=\chi (x) f(x)$$
with given function $\chi (x)$. What is the condition on the function $\chi (x)$ so that we can write this reflection ...
0
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2answers
22 views
Partitioning evens as sum of evens
Take the set $\{a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8\}$.
We can partition according to rules.
Every member in the partition has even number of elements.
Every member in partition have to be consecutive.
...
1
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1answer
36 views
Soft: A game where players use an arbitrary structure as a guideline while trying to play identically.
The following two player game is an (inadequate) attempt to capture an idea I am very interested in exploring. If any of the learned folks here recognize the problem I'm playing with and can point me ...
3
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2answers
185 views
Solve matrix equation $XAX^*=B$ for $X$ in least squares sense
Problem
How can the following matrix equation be solved
$\underset{\mathbf{X}}{\mathrm{argmin}} \left\Vert \mathbf{X}\mathbf{A}\mathbf{X}^* - \mathbf{B} \right\Vert_F$
where $\mathbf{X} \in \...
1
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1answer
22 views
Does the orthogonal complement determine the inner product up to scaling?
Let $V$ be a real $n$-dimensional vector space, and let $g,h$ be two inner products on $V$. Fix some $1\le k\le n-1$, and denote by $\text{Gr}_k(V)$ the Grassmannian of $k$-dimensional subspaces of $V$...
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0answers
12 views
Difference between rotations, radial symmetry and central symmetry
After searching the web and not understanding the mathematical explanations of the different types of symmetry I want to ask: What is the difference between rotations, radial symmetry and central ...
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22 views
How to express a function's invariant transformations in terms of classical and exceptional groups?
Suppose I have a function $f$: $\mathbb{R}^n \rightarrow \mathbb{R}$. I'm interested in the symmetry group of $f$, i.e. transformations $T$ such that $f(T(x)) = f(x)$ for all $x$, with composition as ...
3
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1answer
28 views
If a polynomial function has an axis of symmetry at x = c, is it true that c is equal to the average of the roots?
I was trying to learn more about cool substitutions to make my life easier when solving polynomial equations.
For example, in the problem $$x(x+1)(x+2)(x+3) = 24,$$ the LHS is symmetric at $x = -3/2$...
-1
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1answer
67 views
symmetry in natural numbers
Given a finite set $E$ we can associate a group of symmetries or permutation $S_n$ where $n=|E|$ is the cardinal of $E$. My question is what if the set is infinite or more precisely countable ? Is ...
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1answer
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What function symmetric and has unique solution?
I have multiple operands, say a, b, and c. I want an operator acting on a, b, and c, but the result should be invariant of order of operands (symmetric), and there should be no other pair with same ...
1
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1answer
43 views
Is this shape a fair D-24?
I've been looking at polyhedra, incuding Platonic solids as well as Archimedean and Catalan solids. Catalan solids are face-transitive, which I believe implies that they are "fair dice", in the sense ...
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Wassertein and symmetry
So here's a scenario: I have points $(\mu_1^j,\mu_2^j)$ and I associated them the following distribution $$\rho_j=1/2\delta_{\mu_1^j}+1/2\delta_{\mu_2^j}$$
These have symmetry (exchanging $\mu_1^j$ ...
3
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symmetry of partial differential equations (Heat equation)
Morning everyone, I am doing some problem sheets for my class in Partial differential equations where we dont have an actual textbook. we are given a pack on notes. I am having an issue discerning ...
2
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1answer
34 views
Reason: non-integer powers of Hermitian matrix not Hermitian?
Let $\alpha = 2.1$.
If $A$ is symmetric real, $A^\alpha$ remains symmetric (although it could be complex-valued). If $B$ is Hermitian, $B^\alpha$ isn't Hermitian (losses the symmetry).
However, for ...
6
votes
0answers
62 views
Is there a general theory of when certain polynomials are integrable due to symmetry tricks?
Consider the functions $x^2$ and $x^4 + 2x^2y^2$ on the unit sphere $S^2$.
The surface integral of these functions over the sphere can easily be calculated by symmetry via
$$3 \iint_{S^2} x^2 \mathrm{...
6
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2answers
120 views
Proof that there exist only 17 Wallpaper Groups (Tilings of the plane)
I had a professor who once introduced us to Wallpaper Groups. There are many references that exist to understand what they are (example Wiki, Wallpaper group).
The punchline is
$$There \,\, are \,...
0
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1answer
37 views
Making the condition homogeneous
I saw it on AoPS, very great: https://artofproblemsolving.com/community/c6h1274759p6726915
JunBo-Yang used his own substitution for the condition: $a^{\,2}+ b^{\,2}+ c^{\,2}+ 3\,abc= 6$ , then:
$$a= \...
0
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1answer
24 views
example of an “almost-metric” without symmetry
It is not difficult to find an "almost-metric" $d$ that satisfies all axioms of a metric except the triangle inequality. It should also be possible to construct a function $d$ that satisfies all ...
0
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1answer
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What can I say about the form of an invariant function?
I have a general scalar function which has the properties:
\begin{align}
f(s\,a,b,c)&=s\,f(a,b,c)\\
f(s\,a,s\,b,s\,c)&=f(a,b,c)
\end{align}
where $s$ can be any real number, so the invariance ...
1
vote
0answers
25 views
Between $\ell^p$ and $\ell^\infty$ balls?
Take the $\ell^p$ ball in $\mathbb R^d$ of radius $r$ centered at the origin is
$$\{x \in \mathbb{R}^d : \sum_{i=1}^d |x_i|^p \le r^p\}.$$
Enclose it by the smallest square possible which is the $\...
7
votes
2answers
77 views
Does the action of a linear map on $k$-dimensional subspaces determine it up to scaling?
Let $V$ be a real $d$-dimensional vector space, and let $1 \le k \le d-1$ be a fixed integer. Let $A,B \in \text{Hom}(V,V)$, and suppose that $AW=BW$ for every $k$-dimensional subspace $W \le V$. Is ...
0
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0answers
44 views
What does “super-integrable” mean?
What does "super-integrable" mean?
Example from here (p. 2):
…the
Coulomb problem is super-integrable. Namely, it is not just rotation invariant,
but as well admits further integrals given by ...
1
vote
0answers
43 views
Symmetricity of binomial distribution
For each random variable, $X$, define: $Sym_X(c) = \frac{\Pr[X \geq \mu_X + c]}{\Pr[X \leq \mu_X - c]}$.
I use this definition to measure how symmetric the distribution is.
Let $X$ be a binomial ...
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12answers
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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 ...
2
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2answers
34 views
Prove that $S_3/C_3$ is a (quotient) group
Consider $S_3$ to be the symmetries of a triangle, and let $C_3$ be a subgroup that cycles the three corners, so generated by: $(1 \ 2 \ 3 )$.
Using Lagrange's theorem, compute $k = |S_3/C_3|$. ...
0
votes
1answer
37 views
commutativity of rotations and reflections
The question is as follows:
[Concerning the square embedded in the plane,] prove that the $90^{\circ}$ clockwise rotation $\sigma$ and the reflection through the north/south axis $\rho$ do not ...
1
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0answers
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How to take the determinant of a rank(1,1) tensor?
I want to find the Jacobian matrix and its determinant of the generic infinitesimal transformation:
$x'^\mu=x^\mu+\epsilon_\alpha\frac{\delta x^\mu}{\delta \epsilon_\alpha}$ where $\epsilon_\alpha$ ...
2
votes
1answer
69 views
How is this cube symmetric?
I am having a hard time with this, but I am reading in a published journal that this cube is symmetric when distributing +1 and -1 in the following corresponding points.
Positions 1-12:
$$[+,+,+,-,-,+...
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votes
1answer
38 views
Where is it used that the symmetry conserves the symplectic form in noether's theorem.
For the proof of Noether's theorem, it seems like that the only thing that's important for the symmetry map $S_g : M \to M$ is that it conserves the Hamiltonian (which will then imply that the moment ...
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0answers
39 views
What is symmetry in physics in the mathematical sense, using (Lie) groups
In physics, we sometimes say that, for example, a certain classical system has a certain symmetry, which is given by some group. I don't feel like I understand this well enough. Are there some good ...
2
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1answer
53 views
Properties of a step function
Consider the step function $\Delta: \mathbb{R}\rightarrow [0,1]$
$$
\Delta(x;\lambda,\mu)\equiv \sum_{j=1}^J \lambda_j\times 1\{\mu_j\leq x\}
$$
where
$\lambda\equiv (\lambda_1,...,\lambda_J)$ is a ...
0
votes
0answers
25 views
Flux through sphere symmetry?
Is the flux through a sphere centered at the origin of the vector field $\boldsymbol{F} = (-x,1,z)$ equal to $0$? If so, is there any simple symmetry which suggests it? I have done the calculation ...
4
votes
3answers
64 views
When does $P\text{SO}P^{-1} \subseteq \text{SO}$?
Let $P \in \text{GL}_n^{+}(\mathbb {R})$. Suppose that $P\cdot \text{SO}(n)\cdot P^{-1} \subseteq \text{SO}(n)$.
Is it true that $P \in \lambda \text{SO}(n)$ for some $\lambda \in \mathbb{R}$?
I ...
0
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1answer
59 views
Archimedean spiral - symmetry test
It is usually stated in Precalculus textbooks that in polar coordinates when a relation between $r$ and $\theta$ passes a symmetry test then the curve described by that relation has that symmetry. ...
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0answers
13 views
Representation theory and symmetry in physical systems.
I've come multiple times across statements like these
Representation theory
remains the method of choice for simplifying the physical analysis of systems
possessing (a high degree of) symmetry ...
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0answers
17 views
Symmetry of a planar system and product of inertia
I read the following sentence here:
"The products of inertia occupy the off-diagonal positions and measure the asymmetry of the mass distribution with respect to the planes of the inertia frame of ...
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1answer
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Proving symmetry and transitivity of a relation
Let R be the relation {(1,1),(1,2),(2,2),(1,3),(3,3)} on the set {1,2,3}.
I am having difficulty proving that it is symmetrical and transitive. I know for symmetry we have to prove if xRy then yRx (...
0
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1answer
34 views
Divisibility of kissing numbers
Denote by $ K(d) $ the kissing number in dimension $ d $.
I have two questions :
1) does $ d\mid K(d) $ for all $ d $?
2) does $ d\mid D $ imply $K(d)\mid K(D) $?
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vote
2answers
20 views
Symmetric relation and definition of even numbers
Let a relation $\rho$ be defined on $\mathcal{Z}$(set of integers) by '$a \rho b$ if and only if $a-b$ is even' for $ a,b \in \mathcal{Z}$.
Then, is the above relation symmetric or anti-symmetric?
...
4
votes
2answers
23 views
Does having two lines of symmetry $y=0$ and $y=-x$ imply that the shape also has lines of symmetry $x=0$ and $y=x$
I have been wondering if a shape/curve that has a line of symmetry along the lines $y=0$ and $y=-x$ is guaranteed to also have lines of symmetry along the lines $x=0$ and $y=x$.
My gut feeling tells ...
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vote
0answers
33 views
prove that a conjugate of a glide reflection is a glide reflection
This question has an answer elsewhere, a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, but they use results which were not mentioned in the book. The ...
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0answers
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Is there a theory of “almost symmetry” generalizing group theory?
Apologies for the inescapably soft question.
Does there exist a theory that aims to develop tools analogous to those of group theory, except for the study of objects that are merely almost ...
1
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1answer
54 views
Discrete Spherical Symmetry Group
Take two spheres each having a certain number (say 5) of identical dots on them. What is the approach to proving/disproving that they are equivalent under the set of spherical rotations?
One could ...
2
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1answer
34 views
In which degrees there exist non-decomposable elements in the exterior algebra?
I am trying to get a better understanding of the concept "decomposable" element in an exterior algebra. Let $V$ be a $d$-dimensional real vector space, and let $1<k<d$.
For which tuples $(k,d)$,...
8
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1answer
100 views
A characterization of the subgroup of $\text{GL}(\bigwedge^k V)$ which preserves pure tensors?
Let $V$ be a $d$-dimensional real vector space, and let $1<k<d$. Set
$$H=\{B\in\text{GL}(\bigwedge^k V) \, | \, B \,\text{ preserves pure tensors }\}$$
(i.e. $B \in H$ if it maps ...
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0answers
12 views
Orbit of a point and symmetries of a specific graph
Given is the graph:
I am interested in determining the orbit of the point 1 and also to determine the amount of symmetries that fix each of the points 1, 2 and 3?
My approach:
Notice that the ...
