The symmetry tag has no wiki summary.

**21**

votes

**5**answers

1k views

### When does symmetry in an optimization problem imply that all variables are equal at optimality?

There are many optimization problems in which the variables are symmetric in the objective and the constraints; i.e., you can swap any two variables, and the problem remains the same. Let's call such ...

**16**

votes

**6**answers

3k views

### Angle Maximizing the Distance of a Projectile

It is well-known that to maximize the horizontal distance traveled by a projectile fired from the ground at a given speed, one should fire it at a $45^\circ$ angle. What's less-known, though not too ...

**12**

votes

**7**answers

1k views

### Mathematics of quasicrystals

I want to study quasicrystals from mathematical point of view, but I'm having hard time finding materials about it. If you could suggest me some books, articles or papers, I would be glad.

**12**

votes

**8**answers

3k views

### The Symmetry of a Soccer Ball

Let $P$ be a polyhedron which satisfies the following three conditions:
$P$ is built out of regular hexagons and regular pentagons.
Three faces meet at each vertex.
$P$ is topologically a sphere.
...

**11**

votes

**6**answers

989 views

### Which norms have rich isometry groups?

Let $n \ge 2$ be some positive integer. Given a norm $p : \mathbb{R}^n \to \mathbb{R}$, one can inquire about the structure and properties of its isometry group, i.e. the group of all bijections ...

**11**

votes

**2**answers

2k views

### What is the physical meaning of a Lie algebra symmetry?

The physical meaning of a Lie group symmetry is clear: for example, if you have a quantum system whose states have values in some Hilbert space $H$, then a Lie group symmetry of the system means that ...

**10**

votes

**2**answers

2k views

### Noether's Theorem in Quantum Mechanics

In classical mechanics:
If a Lagrangian L is preserved by an infinitesimal change in the state space variables qi -> qi + εKi(q) leads to only second order change in the Lagrangian:
$$ 0 = ...

**9**

votes

**2**answers

530 views

### Name this periodic tiling

Hello MO,
I've been working on a problem I'm working on in ergodic theory (finding Alpern lemmas for measure-preserving $\mathbb R^d$ actions) and have found some neat tilings, that I presume were ...

**7**

votes

**1**answer

265 views

### Is the operadic butterfly symmetric?

The operadic butterfly is a diagram in the category of operads in vector spaces. It extends the short exact sequence relating commutative, associative and Lie operads.
$$\begin{array}{ccccc}
& ...

**6**

votes

**5**answers

1k views

### Highly symmetric 6-regular graph with 20 vertices

I'm interested in (node/edge-)symmetric 6-regular graphs on 20 vertices and 60 edges, especially ones with a A5/icosahedral/dodecahedral symmetry group and especially their chromatic number. So far I ...

**6**

votes

**2**answers

834 views

### What is the relation between quantum symmetry and quantum groups?

What kind of role do quantum groups play in modern physics ?
Do quantum groups naturally arise in quantum mechanics or quantum field theories?
What should quantum symmetry refer to ?
Can we say that ...

**6**

votes

**2**answers

2k views

### Non-degeneracy of ground state in quantum mechanics

In non-relativistic quantum mechanics, what are the necessary conditions on the potential (or on the hamiltonian in general) for the ground state to be non-degenrate?

**4**

votes

**1**answer

676 views

### What is symmetry group of non-linear equation?

I am not very sure if this is a proper question, but I'm trying to investigate what the area of math can offer in researching the differential equation in polar coordinates:
$r'^2+r^2=(kt)^2$, ...

**4**

votes

**1**answer

447 views

### Differential operators preserving the space of harmonic functions (aka higher symmetries of the Laplacian)

The article http://arxiv.org/abs/hep-th/0206233 (published in Ann. of Math. (2) 161 (2005), no. 3) deals with linear differential operators $D$ for which there exists another linear differential ...

**3**

votes

**2**answers

421 views

### Does graph asymmetry imply all eigenvalues of the graph Laplacian are simple?

It is well known that
1) if there exists a non-trivial automorphism of a graph $G$ with corresponding permutation matrix $P$ then if $(v,\lambda)$ is an eigenvector-eigenvalue pair of the graph ...

**3**

votes

**1**answer

209 views

### Non-inherited symmetries of shadows of point sets

Sometimes a point set in Euclidean space may have a shadow with an unexpected symmetry. The purpose here is to ask when this happens or when it doesn't happen (in some generality).
This requires a ...

**2**

votes

**3**answers

276 views

### Symmetry Properties of Minimizers - Calculus of Variations

What methods are there to show symmetry properties of the minimizer of a problem $\inf_{u\in X}\mathcal{F}(u)$ in the calculus of variations? In general, the symmetry properties of $\mathcal{F}$ do ...

**2**

votes

**0**answers

126 views

### Diffeomorphism between open annuli preserving common symmetries

Suppose $A$ and $B$ are subsets of $\mathbb{R}^2$ homeomorphic (and thus $C^\infty$ diffeomorphic) to the open annulus (punctured $\mathbb{B}^2$) and let $G$ be the group of isometries of ${\mathbb ...

**1**

vote

**1**answer

147 views

### True or false: if a set of 2D points has valid symmetry axes, then at least one of them is equal to a principal component vector

I posted this question on math.stackexchange but got no answer, so I decided to post it here instead. Sorry about the impreciness, not professional mathematician here.
Let's assume we have a set of ...

**1**

vote

**1**answer

270 views

### Is it true that Sym[P]!=0 and Sym[Q]!=0 implies Sym[PQ]!=0 ?

Let $P,Q$ be homogenous polynomials in variables $x=x_1,\dots,x_n$ resp. $y=y_1,\dots,y_m.$
We know that $Sym_x[P]$ and $Sym_y[Q]$ are not identically zero.
Does it follow that $Sym_{x \cup y}[PQ]$ ...

**1**

vote

**0**answers

52 views

### Programmatically recognizing symmetries of a polyhedron

I asked this question on MSE a month ago, but nobody was able to answer it, so I guess the question is more difficult than I initially thought: ...

**1**

vote

**0**answers

57 views

### Semiflows and continuous symmetries

Given a differential equation on a Banach space $\mathcal{X}$ of the form $\frac{d u}{d t} = F(u)$, it is often the case that $F$ is equivariant under translations, i.e. that $T_\alpha F(u) = ...

**0**

votes

**2**answers

474 views

### Limiting set theory using symmetry

[Cross-posted from here]
If my understanding is correct, naive set theory needs to be restricted in order to avoid paradoxes including the Russell paradox. Typically, the restriction is expressed in ...

**0**

votes

**0**answers

73 views

### “Box Nodes” in Directed Graphs with Paired IO Symmetry

Consider directed graphs where all nodes have 2 inputs and 2 outputs. If we
design a box with N inputs and N outputs, what is the smallest number of
nodes it must contain to satisfy “pair symmetry” ...

**0**

votes

**0**answers

40 views

### Using symmetries of a r.v.'s distribution to boost samples and possibly do variance reduction

Suppose, for example, you are simulating samples from a (multivariate) Gaussian with mean zero and covariance $\Gamma=BB^T$. If you had generated a sample $x$, you could generate more (dependent) ...