# Questions tagged [symmetry]

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

866 questions
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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 ...
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### 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 $\mu_1^j$ ...
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### Symmetry relative to the straight line. [closed]

Write the equation of symmetry relative to the line y = 2x + 1 with a slip by vector [2, -1]. I do not know how to start this task.
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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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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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### 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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### 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 (...
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### 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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### 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? ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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 $(k,d)$,...
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### 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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### 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 ...