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

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Solving Lie's invariant condtion for first order ODE

I write down the general equation $Y_{x}+(Y_{y}-X_{x})F-X_{y}F^{2}= XF_{x}+YF_{y}$ and assume that X=a(x) and Y=b(x)y, after that I can't see anyway to solve it for a and b. How can I get solution ...
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2answers
63 views

Isotropic left invariant Riemannian metric on $GL_n^+$?

I am trying to see if it's possible to construct a left invariant isotropic Riemannian metric on $GL_n^+$. (the group of $n \times n$ invertible real matrices with positive determinant) (When by ...
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26 views

What does a norm which is symmetric “around” an index subset look like?

I am looking at norms $\lVert\cdot\rVert$ on $\mathbb{R}^{n}$ that have the following symmetry property $$\forall \beta \in \mathbb{R}^{n} \text{: }, \lVert\beta \rVert=\lVert ...
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23 views

Difference of general symmetric function that is non decreasing in its arguments

For a symmetric function $C(x,y)$ and $a,b,c,d \in [0,1]$ with $b\ge a$, $d \ge c$ and further, C(x,y) is non-decreasing in $x,y$. Then, does it hold that: $$ C(b,d) -C(a,d) -C(b,c) + C(a,c) \ge 0 $$ ...
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1answer
39 views

Is it possible to represent a set of generic things coherently without implicitly creating an order on them?

Sorry if this question seems a little incoherent, I'm not certain of the proper terminology here. I've seen unordered sets represented in various ways, but it recently occurred to me that most of ...
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2answers
49 views

Mathematics of chemistry with focus on particular symmetries

A student of mine came with a somewhat unusual request: My question about group theory and if you have a book you could recommend for some foundation: specifically chemical applications of group ...
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3answers
65 views

Is there a bi-invariant metric (not Riemannian) on $GL_n$?

Is there a bi-invariant* metric $d$ on $GL_n$ (the group of invertible marices) which generates the standard topology on it? (the subspace topology from $\mathbb{R}^{n^2}$) Note: I mean any metric ...
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47 views

Digesting a proof of Bieberbach's theorem on crystallographic groups

Here is a proof of Bieberbach's theorem: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.460.1436&rep=rep1&type=pdf I can follow the computing details, but I have no intuitive sense ...
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1answer
38 views

Is a shape formed of two tangents and radii symmetrical?

Is the kite formed by the two tangents and radii in this image symmetrical? Is there a law or reason why? I am assuming that the two tangents are of equal length, but I can't see why. Are any two ...
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51 views

Understanding Cauchy-Schwarz and Holder's inequalities.

Although these inequalities occur in various settings, and I have used them to complete a number of proofs, I can not say that I intuitively understand what their significance is. Holder's ...
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8 views

Is the follow equation representative of a Hyperbolic One dimensional conservation law?

For $y : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ and let $j : \mathbb{R} \to \mathbb{R}$ $$\frac{\partial y}{\partial x_2} + \frac{\partial (j \circ y)}{\partial x_1} = 0$$
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64 views

A norm invariant under permutations but not under signed permutations

Let $\| \cdot \|$ be a norm on $\mathbb{R}^n$. We call it axes-symmetric if $\|x\|$ does not depend on the order of the components of $x$. Equivalently if $\|x\| = \|P \cdot x \|$ for any permutation ...
3
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1answer
45 views

ODEs are invariant under the given Lie groups?

$\frac{dy}{dx} = \frac{x^{2}y}{x^{3}+xy+y^2}$ is invariant under $(x,y) \mapsto (\frac{x}{1+\varepsilon y},\frac{y}{1+\varepsilon y})$ I can't make both sides equal when I have a variable depends on ...
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1answer
29 views

Efficient search for equilateral triangles, squares and regular pentagons in a set of 3D points

For an algorithm to identify cubic point groups from a set of atom positions $r_i$ forming a molecule, I would need an efficient and fast algorithm to identify equilateral triangles, squares and ...
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1answer
48 views

Eigenvalues of a rotationally symmetric matrix

I have a rotationally symmetric matrix of arbitrary size, for example, \begin{equation} A = \begin{pmatrix} a & b & c & b & a \\ b & d & e & d & b \\ c & e ...
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1answer
48 views

Polyhedral symmetry in the Riemann sphere

I found an interesting set of constants while exploring the properties of nonconstant functions that are invariant under the symmetry groups of regular polyhedra in the Riemann sphere, endowed with ...
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0answers
37 views

How can I compute how many and which quartic invariants of a given group representation are linearly independent?

Given a representation $R$ of a group $G$ and the corresponding tensor product $$ R \otimes R = R_1 \oplus R_2 \oplus R_3 \oplus \ldots $$ how can I compute how many quartic terms $ \mathrm{O} (R^4) ...
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1answer
25 views

Symmetric part of A contributes to quadratic form

In my statistics note, when it talks about quadratic forms, it goes on saying: "$x^tAx=\frac12x^t(A+A^t)x$ implies that only the symmetric part of A contributes to the quadratic form." I am having ...
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1answer
445 views

Systematic solution to my soccer ball puzzle

I once received a puzzle that can be described as follows: There are $12$ black pentagonal and $20$ white hexagonal pieces. The goal is to form a soccer ball from these (aka. truncated icosahedron). ...
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2answers
663 views

Is zero the center of the numeric sequence?

The numeric sequence has symmetry on zero, with equal infinities of cancelling out + and - values on either side. Can numbers be said to have different centers of symmetry than zero? Is it possible ...
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1answer
127 views

Name of a symmetry involving complex squares

Imagine a single parameter real-valued non-zero function $f$. For any complex number $Z$, let $Z \rightarrow Z' = \Re(Z) f + i\Im(Z)/f $ Calculate $Z^2$ and $Z'^2$. Because the imaginary portion of ...
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3answers
61 views

Prove that if $A$ is a symmetric matric then $A^3$ and $A^2-2A+I$ are symmetric matrices.

I am uncertain on how to approach this proof. For most everything I've encountered concerning symmetry, it has involved taking the transpose in order to show some property. Here, I'm not certain if ...
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10 views

Symmetry of the multiplier of a l.c.s.c. abelian group

Let $G$ be a l.c.s.c. (locally compact, second countable) Abelian group, and let $\hat{G}$ be its (well-defined) dual. Consider the group $G\oplus\hat{G}$ (which is a l.c.s.c. Abelian group itself), ...
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25 views

Maximum overlap of two convex bodies

Let $\Delta_n=\big\{(x_1,x_2,\ldots,x_n): \sum_i x_i \leq 1, x_i \geq 0\big\}$ denote an n-dimensional simplex. I am trying to find an $y \in \mathbb{R}^n$ such that the $y-$shifted negative simplex ...
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1answer
54 views

Interesting $\sum_{i=0}^{m-1}(-1)^{i}(m-i)^{2} = \sum_{i=1}^{m}i$

I found that for m $\in N $ $$\sum_{i=0}^{m-1}(-1)^{i}(m-i)^{2} = \sum_{i=1}^{m}i.$$ I found it after doing an exercise. For example: $$5^{2}-4^{2}+3^{2}-2^{2}+1^{2} = 1 + 2 + 3 + 4 + 5 = 15.$$ For ...
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1answer
48 views

How can you algebraically determine if a curve is symmetric about $y=-x$?

If I have a curve implicitly defined by say $x^2+xy+y^2=1$, then it is clear that it is symmetric about $y=x$ because if I interchange x's with y's, then I have the exact same equation. However, how ...
4
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1answer
77 views

How to measure asymmetry of a function?

Let $f(x) = x^{2}$, so $f(x)$ is an upward symmetric parabola. It is a perfectly symmetric function since $f(x) = f(-x)$ for any value of $x$. Now, suppose $f$ is just some function. How would one ...
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15 views

Can I find the dissections of a figure based on symmetry?

Our teacher gave us a figure, and challenged us to dissect into exactly 4 shapes, that were congruent, in as many ways as possible. I won't reveal details of the specific shape. I am wondering if ...
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39 views

How to prove a symmetric tensor is indeed a tensor?

Our professor defined a rank (k,l) tensor as something that transforms like a tensor as follows: $$T^{\mu_1' \mu_2'...\mu_k'}{}_{\nu_1'\nu_2'...\nu_l'} ~=~ ...
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40 views

How to measure the symmetry of the curve?

In statistics, we can measure the symmetry by skewness, but if we have a curve, in other words, if we have a list of x and y values, how to measure the symmetry of its plot.
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25 views

Proofs of the multiplication or chain rule for derivatives that invoke symmetry

Introductory calculus texts sometimes include direct proofs of the multiplication and chain rules for derivatives by: Introducing a pair of differences $D_f=\frac{f(x+h)-f(x)}{h}-f'(x)$ and ...
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38 views

Determining the corresponding vector field to a group action.

Im having trouble trying to understand how to determine the corresponding vector field to a group action on a symplectic manifold. I feel this will be easier if I give two examples which are confusing ...
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18 views

Proving that rotation-inversion axis 2 and rotation axis n/2 induce rotation axis n.

How to prove that rotation axis of n/2 order and rotation-inversion axis of 2 order induce rotation-inversion axis of n order?
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2answers
46 views

How can we assume that the all parabolas in the form 'ax^2 + bx +c' are symmetrical?

So I was reading an answer to a question pertaining to the derivation of the line of symmetry. It reads as follows: The vertex occurs on the vertical line of symmetry, which is not affected by ...
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1answer
49 views

What do you call the property of a shape that looks the same for some rotations?

Specifically I mean a fragment of a sphere(e.g. for $x,y,z > 0$). It looks the same if you look at this from $(1,0,0)$ or $(0,1,0)$ or $(0,0,1)$. What do you call the property? I thought it would ...
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1answer
27 views

If $X_n$ are symmetric around $0$ then show $P(|S_n|\geq\max_{1\leq i\leq n}|X_i|)\geq\dfrac{1}{2}$

Let $\{X_k\}_{k=1}^n$ be iid random variables that are symmetric around $0$ i.e. $X=-X$ in distribution. Define $S_n=\sum_{i=1}^nX_i$. Then show $P(|S_n|\geq\max_{1\leq i\leq ...
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1answer
126 views

Space of arbitrary rotations of a cube

Suppose I have a cube $[-1,1]^3\subset\mathbb{R}^3$. I am allowed to rotate it about any angle/axis through the origin rather than just $90^\circ$ about the coordinate axes, e.g., by applying ...
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45 views

Explanation of symmetric sum in a solution

Can someone explain me why $x+y=5$ in $\text{E8}$ clearly.
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1answer
21 views

If $s$ and $t$ are symmetries of a plane such that they agree on three non collinear points then show that $s=t$

This is a problem based on "Symmetry" of the plane $\mathbb{R^2}$. Suppose $A$, $B$, $C$ are the three points in plane which are after the corresponding actions by $s$ and $t$ are in the places $D$, ...
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3answers
195 views

Why is reflection in a plane an automorphism?

I have not studied group theory, but would like to know in simple terms why reflection in a plane is an automorphism. Dr. Hermann Weyl gives the definition of automorphism in his book 'symmetry' as ...
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1answer
31 views

How is the distinction of left and right in space related to the orientation of screw?

In Dr. Hermann Weyl's book 'symmetry', he explains the difference between left and right as In space the distinction of left and right concerns the orientation of a screw. If you speak of turning ...
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70 views

A symmetric group question. [closed]

Determine the integers $n$ such that there is a Surjective homomorphism from the symmetric group $S_n$ to $S_{n-1}$. It is a question from Artin's book. Exercise 7.5.8
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1answer
58 views

Class equation question from Artin's book.

Let G be a group of order n that operates nontrivially on a set of order r. Prove that if n > r!, then G has a proper normal subgroup. Also I am not very clear of the term "operates nontrivially", ...
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1answer
550 views

Penrose's remark on impossible figures

I'd like to think that I understand symmetry groups. I know what the elements of a symmetry group are - they are transformations that preserve an object or its relevant features - and I know what the ...
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1answer
43 views

Global Optimization, symmetric solutions

Does anyone have the idea to solve the global multivariate minimization problem as below? $$\text{minimizes}\quad (x_1x_2x_3+x_1x_4x_5+x_1x_6x_7+x_2x_4x_6+x_2x_5x_7+x_3x_4x_7)-(x_1+x_2+x_4+x_7) \\ ...
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2answers
139 views

The class equation of the octahedral group

I know that the class equation of the octahedral group is this: $$1 + 8 + 6 + 6 + 3$$ I think the $8$ stands for the $8$ vertices, the $6$ could be $6$ faces and $6$ pairs of edges. Then what is the ...
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Is self-similarity a form of symmetry, is it the other way around, or is it something else?

I'm aware that self-similarity is a form of symmetry, however I'm interested in getting a more in depth explanation of the relationship. Could you consider symmetry to be a form of self-similarity? ...
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5answers
425 views

What is the intent of this problem, disguised as an eigenvalue - eigenvector problem?

Let $$ A= \begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{bmatrix} $$ $a,b,c >0$. Find eigenvalues and a basis of ...
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1answer
175 views

The Fixed Point Theorem in Artin's book

Theorem 7.3.2 Let G be a p-group, and let S be a finite set on which G operates. If the order of S is not divisible by p, there is a fixed point for the operation of G on S - an element s whose ...
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1answer
63 views

Seeking Better (Symmetry Exploiting) Solution and Generalization of An Inequality

Given positive variable $(x,y,a,b)$ where $x+y=1$, how does one "slickly" prove the following inequality? $$f(x,y) := \frac{xa+yb}{\sqrt{xa^2+yb^2}}\ge \frac{2\sqrt{ab}}{a+b}.$$ or simply $$f(x,y) := ...