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1answer
36 views

Eigenvectors and values of nearly identical symmetrical matrices

I am given 2 matrices which have the following traits: Let $A$ and $B$ be those matrices and $a_{i,j}$ and $b_{i,j}$ be the entries of both matrices. There are 2 disjoint subsets of the indexes, let ...
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2answers
57 views

Construct the Green s function for the equation

Construct the Green s function for the equation y^''+ 2y^'+2y=0 Which boundary conditions y(0)=0 , y(π/2)=0 Is this Green s function symmetric? What is the Green s function, if the ...
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1answer
177 views

Do Symmetric problems have Symmetric solutions?

There is a general notion of Symmetry in mathematics, that of an object being constant under some transformation. If we think of our object as being a "mathematical problem" we can see certain ...
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0answers
37 views

Constraints on eigenvalues due to the form (symmetry) of Matrix

So I have to deal with a square matrix with complex elements $A$ given by - $$A_{ij}(k) = \underbrace{\frac{1}{2}(c_{ij}+c_ji)}_{\text{symmetric under interchange of $i$ and $j$}} \exp(Ik(i-j))$$ ...
1
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1answer
186 views

Smooth diffeomorphisms preserving common symmetries

Let $A$ and $B$ be two $C^\infty$ orientation-preserving diffeomorphic, connected, bounded, open subsets of ${\mathbb R}^n$, with finitely many ends and $G$ the group of isometries of ${\mathbb R}^n$ ...
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2answers
57 views

Asymmetry in the error correction in coding theory

Does it make sense to have an error correction code which acts differently on different states (for example if we run something which runs on the binary string from $0^n \rightarrow 1^n$ involving all ...
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0answers
43 views

symmetric ranges of curve division on an oloid

taken from wikipedia , I drew an Oloid by using the functions with ; ; and with ; ; If I use any equal spaced range of t(x) on ; I will end up with an unequal range t(y), which ...
3
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1answer
93 views

Proof that a ODE system admits symmetric solutions

I have a ODE system of the form $\dot{x} = f(x)$ with $x \in \mathbb{R}^3$. Now it is claimed that if $(x_1,x_2,x_3)$ is a solution to the system that also $(-x_1,-x_2,-x_3)$ is a solution. How can I ...
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1answer
54 views

Surjectivity for permutation representation of a group action

I am having trouble proving that my function is surjective. Here is the problem statement: Problem statement: Let T be the tetrahedral rotation group. Use a suitable action of T on some set, and the ...
0
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1answer
93 views

Conjugacy classes of rotational symmetry tetrahedron

I am struggling with this question, as I don't know how to give a "short proof" as requested by our professor, as a non-graded exercise. Especially, since we have to do this without using any shapes, ...
1
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1answer
26 views

Farthest points in asymmetric 2D closed curve

Is there a mathematically proper name for the two points that are located farther away from each other in a 2D asymmetric closed curve? See the image below to get an idea of what I mean.
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3answers
223 views

center of symmetry formula

How to prove that $I(0,-1)$ is the center of symmetry of the function $$F(x)= x - \dfrac{2e^x}{(e^x -1)}$$ Is there any formula that I can directly apply?
3
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1answer
172 views

Decomposition formulas for rotational symmetries of a cube

I have a problem that I would like to check my work on. I am also stuck on the verifications for $E$ and $F$. Any help would be greatly appreciated. Thanks in advance. Problem statement: Let $G$ be ...
0
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1answer
17 views

algorithm for generating interesting normal symmetric continuous functions

Does any one know of, or can think of an algorithm which generates arbitrarily many symmetric normally distributed continuous functions? And when I say interesting I mean more complex distributions ...
0
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0answers
16 views

Types of symmetries of this frieze pattern

I would like to know if I have classified the types of symmetries of this frieze pattern correctly. Frieze pattern: The gaps between each of the three figures are equal, and the pattern extends ...
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2answers
100 views

Using counting formula to get |G| = |kernel φ||image φ|

The counting formula I am saying : Let S be a finite set on which a group G operates, and let Gs and Os be the stabilizer and orbit of an element s of S. Then |G|=|Gs||Os| or ...
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0answers
56 views

Rotation about a point

Let $s$ be the rotation of the plane with angle $\frac{\pi}{2}$ about the point $(1,1)^t$. Write the formula for $s$ as a product $t_a \rho_{\theta}$ such that $t_a$ is the translation by a vector $a$ ...
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2answers
44 views

Property on glide reflections

I need to prove that the conjugate of a glide reflection is a glide reflection. What I have tried: Let $m: X= \begin{pmatrix} x_1\\ x_2 \end{pmatrix} \mapsto \begin{pmatrix} \cos \phi & \sin \phi ...
2
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1answer
102 views

Isometries of the plane

Let $m$ be an orientation-reversing isometry. Prove algebraically the $m^2$ is a translation. What I attempted: We know that $m$ is an orientation-reversing isometry i.e. it is either a reflection or ...
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1answer
50 views

Can this polynomial transformation produce new symmetry?

I've got a polynomial transformation on $\mathbb{R}^6$, and I have a conjecture about it, but I'm having a hard time proving it. The transformation looks like this: $ u:= abcde + abc + abe + ade + ...
1
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1answer
38 views

Question on a lemma related to isometries

$\textbf{Lemma:}$ An isometry $f$ that has the form $m=t_a \rho_{\theta} $, with $\theta \neq 0$, is a rotation through the angle $\theta$ about a point in the plane. $\forall x \in ...
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0answers
11 views

Symmetry in a $p$-sphere

Let $S(p)$ be the unit sphere at the origin in the $\mathcal{l}_p$-norm. What symmetries does it have? For instance in the $\mathcal{l}_2$-norm we have unitary symmetry in the Euclidean case. In ...
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1answer
89 views

How do I proove the symmetry of Metric space?

I'm looking for the proof of that: d(x,y) = d (y,x). I know that I have to use the "non-negativity" and "triangle inequality" but I don't know how to combine them to get the result.
1
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1answer
154 views

Proving the symmetry of an equivalence relation

When proving the symmetry of an equivalence relation, must each equivalence class be closed under symmetry. for example: the relation both x and y > 10 or both x and y < 10 across all ...
2
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1answer
58 views

The intersection of two symmetry planes is a symmetry axis

I wonder if the intersection of two planes of symmetry for some three dimensional object, is a symmetry axis of that object (i.e. an axis for which there exists an angle (smaller than 360°), so that ...
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1answer
31 views

Is the graph of $X = {0,1,2,3,4}$ with the $<$ relation directed or indirected?

Let $X = {0,1,2,3,4}$ Draw the graph associated with the $<$ relation on $X$. Should this graph be directed or undirected? The answer of this question is given, but I am looking for an ...
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0answers
166 views

a conjugate of a glide reflection by any isometry of the plane is again a glide reflection

Q : Prove that a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, and show that the glide vectors have the same length. I know that: the conjugate ...
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0answers
111 views

How to prove that symmetric traceless “transverse” tensor rank s in 4 dimensions has 2s + 1 independent components?

How to prove that symmetric traceless "transverse" tensor rank $s$ in 4 dimensions has $ 2s + 1$ independent components? Let's have tensor $$ F^{\mu_{1}\dots \mu_{s}}, \quad {F^{\quad ...
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2answers
182 views

Pentagon axis of symmetry

Quick question: Can a pentagon have an axis of symmetry passing through two (one, none) of its vertices? I'm given the following definition for axis of symmetry: A figure is said to have an axis of ...
2
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2answers
131 views

Combining symbols with symmetry

So this question has probably been answered already, but I can't find a good answer through searching google or this site. Basically, if I have n symbols, how many n-length combinations of the ...
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2answers
107 views

Frieze groups, wallpaper groups

Can someone suggest a source that proves the classifications of the 7 frieze groups and 17 wallpaper groups in an elegant way?
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4answers
177 views

equilateral triangle; $3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2.$

In equilateral triangle ABC of side length d, if P is an internal point with PA = a, PB = b, and PC = c, the following pleasingly symmetrical relationship holds: $3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 ...
1
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1answer
75 views

Finding 'symmetrical range' from mean.

A machine used to make butter where its masses are normally distributed with mean m and standard deviation s.It is found that 5% from these butters are having mass more than 85g where else 10% are of ...
5
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2answers
57 views

Why this definition for “symmetry transformation”?

This question concerns section 8.5.1 in these notes: I don't understand why a symmetry transformation is defined as such. What implications is there if $\delta \mathcal L$ is a total ...
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3answers
550 views

Coloring a cube with 4 colors

There are some topics on this forum related to my question. Most of them use Burnsides Lemma. I don't know this lemma and I don't know whether it is applicable to my problem. Can someone explain the ...
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0answers
134 views

Make one cube out of 8 little cubes

As part of a puzzle, you have to stack 8 little $1\times 1\times 1$-cubes so that they form one big $2\times 2\times 2$-cube. Now I want to check all possible solution to the puzzle and therefor I'm ...
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2answers
82 views

Automorphisms of a structure as a powerful tool for studying the structure

This is just an arbitrary testimony of an often repeated slogan: "The group of automorphisms of a given structure is often a powerful tool for studying this structure." D. Lascar, On the ...
0
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1answer
108 views

When does the difference of two random variables follow a symmetric distribution?

Setup: Let $X_t$ and $Y_t$ denote two (possibly dependent) random variables with cumulative distribution functions (cdf) $F_X$ and $F_Y$. Assume the support of $F_X$ and $F_Y$ is $\mathbb{R}^+$. Let ...
1
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1answer
107 views

Minima of symmetric functions given a constraint

If $f(x,y,z,\ldots)$ is symmetric in all variables, (i.e $f$ remains the same after interchanging any two variables), and we want to find the extrema of $f$ given a symmetric constraint ...
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0answers
26 views

spheroid rotation symmetry

There is three axis and also Euler angle($\theta, \phi, \psi$), Now if we rotate the spheroid then,why is invariant with respect to rotation by $\pi$ about any axis passing through the center? I ...
1
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1answer
41 views

Is there a name for the following asymmetry property of a measure on $R$?:

Let $\mu$ be a Borel measure on $\mathbb{R}$. I am looking for a name for the following property: $\int_\mathbb{R} f d\mu \ge 0$ for all skew-symmetric Borel functions $f$ that are non-negative on ...
3
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1answer
54 views

Symmetry in reduced residue systems

This may be a stupid question, but it looks to me like the reduced residue systems modulo N are symmetrical about N/2; that is to say, that the there is the same number of integers not divisible by a ...
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0answers
193 views

Help with Hermitian Symmetry and its inverse Fourier transform in MATLAB.

I have tried to impose Hermitian symmetry on the complex number $z$ which is varies with $x$. I need to take its inverse Fourier transform. A hermitian symmetry should give a real valued inverse FT. ...
4
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1answer
37 views

Infinitesimal $SO(N)$ transformations

An infinitesimal $SO(N)$ transformation matrix can be written : $$R_{ij} = \delta_{ij}+\theta_{ij}+O(\theta^2)$$ Now it has to be shown that $\theta_{ij}$ is real and anti-symmetric. I've started ...
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1answer
44 views

Relation of Hodge dual to antisymmetric part of the

I have a question in reaction to an article by M. Born and L. Infeld (cf. [1]) concerning the relation between the hodge dual of the electromagnetic tensor and the antisymmetrization of its ...
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1answer
147 views

How many $n$-colorings up to rotation using exactly 2 of each color are there on a $2n$-polyhedron?

I'm a high-school student and I stumbled across a YouTube video explaining how Rubik's cubes work. A Rubik's cube has 6 colors, one for each side, but I started thinking about ways to $n$-color the ...
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3answers
52 views

Showing symmetry involving a matrix and its transposed matrix

I'd appreciate if someone could find a better title for this question, for I'm short of ideas right now. Given a matrix $A \in R^{n,n}$, show that $$ \frac{1}{2}(A + A^t) $$ is symmetric. I see that ...
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1answer
70 views

Standard representation of $O_h$ in $\mathbb{R}^3$

I want to give the standard representation of the complete octaedergroup $O_h$ in $\mathbb{R}^3$. To which group is $O_h$ isomorphic, and how to obtain a standard representation of the group? What ...
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3answers
133 views

Realizing groups as symmetry groups

We're supposed to think of (non-Abelian) groups as groups of symmetries of some object. Sometimes it isn't obvious what this object is. For example, the fundamental group of a topological space acts ...
1
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1answer
29 views

Finding a matrix with the following property

I have one $n \times n$ symmetric matrix $B$. Let $p$ be a scalar, I want to multiply the diagonal elements of $B$ by $p$. Let now $C$ denote the resultant matrix of the process described. Is there ...