The tag has no wiki summary.

learn more… | top users | synonyms

0
votes
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?
0
votes
2answers
25 views

Create a new function using symmetry w.r.t a point

Let's say I have a function $f$ which is defined on an interval $[0,1]$ . How can I create a function $g$ defined on $[0,2]$ where: $f(x)=\sqrt{x^{a} - x^b}$ ,$0<a<b$ ,$a$ and $b$ are constants ...
2
votes
1answer
51 views

Reducing size of ODE system by using symmetries: examples, references help request.

We know: A high order differential equation can be expressed as an ODE system. Knowledge of a symmetry allow one to reduce the order of a differential equation. So if we do $n$-order ODE ...
2
votes
1answer
74 views

How to show that $f(x) = \cos^2(x)\sin(x)$ is symmetric about the line $x=\frac{1}{2} \pi$?

I really have trouble with making any exercises regarding point symmetry and line symmetry. For example: Show that $f(x) = \cos^2(x)\sin(x)$ is line symmetrical in the line $x=\dfrac{1}{2} \pi$. ...
1
vote
1answer
59 views

Explanation of basic definitions in game theory.

In the article entitled Non-Cooperative Game written by Nash in 1951, he discussed about the symmetries of games. Due to my lack of basic knowledge in permutations and symmetries, I looked up some ...
1
vote
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 ...
1
vote
1answer
58 views

Prove a transformation is a variational symmetry for J

The following problem is from The Calculus of Variations by B.von Brunt (page 215, Exercise 9.2.1) Let $$ J(y)=\int_a^b xy'^2\mathrm{d}x. $$ Show that the transformation $$ X=x+\epsilon2x\ ...
0
votes
1answer
12 views

Difference between duality, symmetry, equivalency and invaraince?

Can someone difference in detail on these four terms? Thank you.
0
votes
1answer
58 views

Symmetries of a square and it's similarity to the Division Algorithm

I need help with this question: (each variable $r$ represents a rotation of the square about the axis through its centroid at $90^{\circ}$ intervals. $e$ represents nonmotion. This question is ...
0
votes
1answer
111 views

Why inner product on R^n have uniform prototype with symmetric matrix A and positive eigenvalues?

Details of the problems are given below. Assume A is a n*n symmetric matrix. Show that any inner product on R^n has this formula for some symmetric matrix A with all positive eigenvalues. The formula ...
0
votes
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 ...
0
votes
1answer
133 views

Why must a finite symmetry group be discrete?

I'm having trouble justifying why a finite symmetry group is discrete. Can someone help?
7
votes
0answers
102 views

ODE system and Lie symmetries

The ODE system (see below), where $F$ is a given function together the algebraic condition (SYM) imply that $$\boxed{y(t)=k-x(t)} \tag{*}$$ for some $k$ constant. The result that $y$ is a translation ...
3
votes
0answers
52 views

Fractals vs. “neatness” / order

I've seen a lot of high level videos on fractals, etc, and how they might apply to the real world. So a tree is branches with branches with branches, and our blood vessels branch and then branch ...
2
votes
0answers
17 views

Symmetry groups of discrete functions

I'm looking for basic information about symmetry groups of discrete functions. It is difficult to search for such information, because searching for "symmetry group" gives results that refer almost ...
2
votes
0answers
152 views

What to call this kind of symmetry in a sphere?

Geometrically, if the two hemispheres of a spherical distribution of some kind (let's say a spherical gas cloud) are similar such that the properties of the gas as seen by a person standing on a ...
2
votes
0answers
42 views

Symmetry of the pentagon and even permutation

I was doing part (iii). For the first part of that questions, $ D_{10} = \{e, \rho, \rho^2, \rho^2, \rho^4, \sigma\rho, \sigma\rho^2, \sigma\rho^3, \sigma\rho^4 \} $ where $\rho = (1 \ 2 \ 3 \ 4 \ ...
2
votes
0answers
90 views

An isomorphism between the full tetrahedral symmetry group and the cubic rotation group?

I know that the full symmetry group of the tetrahedron and the rotation group of the cube are both isomorphic to $S_4$. But I want to show a direct, visual isomorphism. I tried looking at a ...
2
votes
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 ...
1
vote
0answers
16 views

One dimensional binary string with periodic boundaries and reflection

I have a binary string $l=(l_1,l_2,\ldots,l_{2n})$ with $l\in\{0,1\}$ and the conditions $l_i \cdot l_{i+n}=0$ for all $i$ and $\sum l_i=n$. Now, I was wondering how many distinct string exist, when a ...
1
vote
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
vote
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 ...
1
vote
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. ...
1
vote
0answers
39 views

Conditions for matrix operator to preserve complex symmetry on DFT vector?

Suppose there is a DFT vector $\mathbf{X}$ (complex vector) with length N, which presents complex conjugate symmetry around its middle point, i.e., $X(1) = X(N-1)^*$, $X(2) = X(N - 2)^*$ and so forth. ...
1
vote
0answers
67 views

Dilation Invariance

Given the formula $$F(x)= \sum_{n=-\infty}^{\infty}f(x+n) $$ We know that is invariant under translations of the form $y=x+n$ for any integer $n$. However can we find a similar formula for dilations ...
0
votes
0answers
11 views

Is the autocorrelation of a function the same if one term is flipped on the y axis?

I have some questions about autocorrelation. They are very related, so I thought that one single post was appropriate for the topic. The first question is already illustrated in the subject: if I ...
0
votes
0answers
9 views

Name for symmetric irreflexive binary relation

I have an irreflexive relation $\prec$ called unpreference: if $x\prec y$ then I say $x$ is unpreferred (or not preferred) to $y$. I wish to give a name to the symmetric part of the relationship, ...
0
votes
0answers
34 views

Is it possible to split a Tetrahedal into two equal halves?

Tetrahedal has 6 sides and made up of three equilateral triangles. But, is it possible to break it evenly? I reckon it is impossible. If yes, explain how it is done. If no, why is it impossible?
0
votes
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 ...
0
votes
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 ...
0
votes
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 $a$ ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
0answers
160 views

Symmetry groups and Cayley table

Given an irregular octagon $O$ with vertices $(6,2)$, $(2,6)$, $(-1,5)$, $(-5,1)$, $(-6,-2)$, $(-2,-6)$, $(1, -5)$ and $(5,-1)$, what are the elements of the symmetry group $S(O)$ of $O$ in standard ...