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1answer
19 views

Mollifying while conserving symmetries

Suppose $B = B(z,v)$ is a function in $L^1_\rm{loc}(\mathbb{R}^d \times S^{d-1})$ depending only on the values of $|z|$ and $|z \cdot v|$ (you don't make any assumptions on how $B$ depends on $|z|$ ...
2
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0answers
23 views

Does “Spherical Symmetry” as defined in General Relativity imply a Foliation of Spheres?

In Carroll's "spacetime and geometry" he defines a spherical symmetrical spacetime as a spacetime $(M,g)$ for which there exists a Lie algebra homomorphism between the Lie algebra of a subset of the ...
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1answer
49 views

Proof of axis of symmetry [duplicate]

How do you prove -b/2a the Axis of symmetry equation using the Quadratic formula?
3
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1answer
70 views

Groups of isometries

I recently read the popular scientific book "Symmetry and the Monster" and it emphasizes groups as the sets of symmetries of geometrical objects. So I was wondering, do all groups appear as symmetry ...
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2answers
49 views

Why must isometry of regular polygon fix origin?

Here is the question: Suppose $\varphi\colon\Bbb R^2\to\Bbb R^2$ is an isometry and $\varphi(\pi_n)=\pi_n$, where $\pi_n$ is the regular $n$-gon with center at origin. Why must $\varphi$ fix the ...
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0answers
36 views

Reflection Symmetry for Non-Linear Differential Equations

We are given the equations: \begin{align} \dot{x}& =\mu \, x +y+y^3 \\ \dot{y}& =2x-2y+xy^2+\gamma \, x^2y \end{align} The question at hand is to determine whether there is some sort of ...
2
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0answers
27 views

Ordering binary matrices for reflection/rotation

I have a collection of $n\times n$ binary matrices and I would like to reduce it for symmetry ($D_4$ -- reflections and rotations). The naive method of testing each pair is very slow because the ...
4
votes
2answers
125 views

Intersection of two tetrahedra, point reflexion

We are given a regular tetrahedron $ABCD$ ($ABC$ is its` base and $D$ is its vertex) and we reflect it through the middle of its height (point reflexion) - and thus we obtain a congruent regular ...
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2answers
32 views

How to prove that $n$ is prime if an $n$-Venn diagram has $n$-fold rotational symmetry

I was reading this article on "The Search for Simple Symmetric Venn Diagrams" by Frank Ruskey, Carla D. Savage, and Stan Wagon and on the first page page they prove that $n$ is prime if an $n$-Venn ...
1
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1answer
35 views

Rotation-Reflection-Symmetries

I have the following exercise: Do a rotation of $2 \cdot 72 ^\circ$. Then do a reflection of the axis $d4$. Then do a reflection of the axis $d3$. Then do a rotation of $2 \cdot 72 ^\circ$. ...
0
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1answer
66 views

Rotation and Symmetries of Equilateral Triangle

Let $\Sigma=\{ 1,2,3 \}$ be the set of the vertices of an equilateral triangle. Let $f=\sigma$ be the rotation of level with center of rotation $O$ over an angle of $\frac{2 \pi}{3}$ radians or ...
1
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1answer
25 views

Symmetries of the set of points $S$

Consider that we have a set of points $S$ in the plane $\mathbb R^2$ or in the space $\mathbb R^3$ and we also consider the one to one mappings $f:S \to S$ which have the following property: They ...
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1answer
84 views

Connection Between Fundamental Solution and Symmetries of PDE

The typical derivation of the fundamental solution of Laplace's equation is to look for a radially symmetric solution because the Laplace equation has radial symmetry, and a similar heuristic can be ...
0
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1answer
51 views

Conventional unit mesh

I'm trying to find and outline a non-primitive conventional unit mesh, I'm not sure how to go about it. I'd also like to find any mirrors of planes and rotional symmetry axes. Would this look ...
0
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2answers
77 views

Map a half sliced unit disk to upper half plane

"half sliced unit disk" Can somebody tell me how to map this conformally to the upper half plane? I think the symmetry principle should be applied here but stuck on that for hours. Pardon my hasty ...
0
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0answers
82 views

What is a “unique” mirror line of symmetry?

What is a "unique" mirror line of symmetry? For example why does an equilateral triangle have three mirror lines but only one "unique"mirror line of symmetry?
1
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2answers
27 views

Symmetry in Space

Is it possible for a non-co-planar set of points to be symmetric about a point but not symmetric about a plane? I am pretty sure this is true but I can't think of an example. Things that I think ...
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0answers
31 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
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1answer
146 views

Help with Autonne-Takagi factorization of a complex symmetric matrix.

Let $A=A_1i+A_2$ with $A$ non singular. Now let $$B =\begin{bmatrix} A_1 & A_2\\ A_2 & -A1 \end{bmatrix}$$ With $A_1$, $A_2$ and $B$ symmetric. Is it true that: 1) $B$ is non singular 2) ...
2
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2answers
45 views

Is $u^TAu \geq 0$ true for all symmetric matrices $A$?

we know from the definition of inner product and norm, that $u^Tu$ is always larger than zero, except the case where $u=0$ at which case it is zero. I came across a question that infers that $u^TAu ...
0
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0answers
15 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
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1answer
53 views

Symmetry Definition and Equation [closed]

I need some help to understand Inversian Symmetry, Conformal Symmetry, and Scale Symmetry. Could you give me some guideline?
1
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1answer
39 views

Proving that the Moment Tensor is super-symmetric

The Carathéodory theorem in the image bellow is the one about convex hull, isn't it? Would you please explain why can the tensor F be rewritten as that sum? From that representation the author ...
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2answers
39 views

How to show properties of a given relation?

I am given that R is a relation on the given set X, and I have to show if the relation is (i) reflexive, (ii) symmetric, (iii) transitive, (iv) asymmetric, and (v) give an example of an element of ...
0
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2answers
29 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 ...
1
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2answers
46 views

Understanding the shape of $\phi''(x)=F(\phi(x))$

Hello I've got a question and no idea to get a solution. Maybe someone can give me an advice. The following problem is given: There is given a function $\phi \in C^2([a,b])$. Furthermore there is a ...
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0answers
13 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, ...
6
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1answer
103 views

Lie algebra $\implies$ Lie group?

Lie's third theorem says that every finite-dimensional Lie algebra g over the real numbers is associated to a Lie group G. So say I have an $r-$ parameter group of symmetries whose tangents at the ...
2
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0answers
161 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
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0answers
72 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 \ ...
-1
votes
1answer
223 views

How to find the order of the group?

Translation: If $G$ is a finite group in which every element $g \in G$ satisfies $g^2 = e$, where $e$ is the unit element of $G$, then what are the possible values for the order $k=|G|$ of $G$? ...
2
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2answers
143 views

Number of reflection symmetries of a basketball

Excerpt from John Horton Conway, The Symmetries of Things, pg. 12. Basketballs have two planes of reflective symmetry, as do tennis balls. I read this sentence and it immediately struck me as ...
0
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1answer
118 views

Is the shortest path using Dijkstra's Algorithm symmetric?

I am writing a code for finding the shortest path using Dijkstra's algorithm for an unweighted graph. I am wondering if this shortest path is symmetric i.e. if the shortest path from say A->E is ...
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5answers
89 views

What does it mean for $AA^T$ to be symmetric?

What does it mean for $AA^T$ to be symmetric? A question in my book says to show that $AA^T$ is symmetric so I took a very simple matrix to try and understand this: $A=\begin{bmatrix} 2 \\ 8 \\ ...
1
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2answers
171 views

The path to understanding Frieze Groups

What is the "Path" for understanding what Frieze Groups really are? Generally in mathematics, there is a is a path or "building blocks" approach to learning something. For example if I know how to ...
4
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1answer
62 views

What exactly is meant by symmetry?

This is something that has troubled me for long. Artin describes 4 types of symmetry: translational, rotational, reflective and glide. I somehow understand symmetry to be a "condition" in which a ...
0
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1answer
211 views

I have a 2x2 positive-semidefinite matrix. I am trying to find the equation of its elements.

So long story short. I have a matrix $A \in S^2_+$, that is, a symmetric, positive semi-definite 2x2 matrix. Here it is: $A = \begin{bmatrix} x & y \\y & z \end{bmatrix}$. Here is what it ...
1
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1answer
91 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 ...
4
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1answer
106 views

Immersions of the Klein Bottle

The famous immersion of the Klein bottle lacks symmetry. (I'm talking about this one. http://en.wikipedia.org/wiki/File:Klein_bottle.svg) One can only see one plane of reflection. However, the ...
2
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2answers
34 views

Symmetrical curve equation in range of [0,1]

I would like to define a function $f(x,a)$ that has the following properties: $f(0,a)=0$, excluding $a=\infty$ $f(1,a)=1$, excluding $a=-\infty$ $f(x,0)=x$ $f(0,a\to\infty)\to1$ $f(1,a\to-\infty)\to0$ ...
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3answers
298 views

Translations in two dimensions - Group theory

I have just started learning Lie Groups and Algebra. Considering a flat 2-d plane if we want to translate a point from $(x,y)$ to $(x+a,y+b)$ then can we write it as : $$ \left( \begin{array}{ccc} ...
5
votes
3answers
65 views

Matrix $A=B+C$ with $B$ symmetric and $C$ antisymmetric

I am stumped on a question and am looking for some guidance on how to get it done. The problem gives you: $x_1 = \begin{bmatrix}9&-4&-2 \\-9&6&-3 \\10&-3&9\end{bmatrix}$ ...
1
vote
1answer
53 views

Plot the curve $y^2 = (x^2 +12x +36)/(x^4 -4x^3 - 12x^2 - 32x 64)$

Can you help me find the following to plot this curve, $$y^2 = \frac{x^2 +12x +36}{x^4 -4x^3 - 12x^2 - 32x 64}$$ In Explicit Form Find the Symmetry X & Y intercepts Vertical and Horizontal ...
1
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1answer
66 views

Why are all symmetry groups of regular polytopes are finite Coxeter groups.

Why are all symmetry groups of regular polytopes are finite Coxeter groups?
1
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0answers
22 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
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1answer
40 views

Spherical symmetry math

For spherical symmetry how the last four equations calculations is done? ccan you explain please? For reference see the equations 44
0
votes
1answer
70 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
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1answer
28 views

Prove the following $R \subseteq A\times B$ and $S\subseteq B\times C \rightarrow $ $ S \circ R $ is symetric

I want to prove the following $ S \circ R $ is symetric, A,B, C are sets $R \subseteq A\times B$ is Symetric $S\subseteq B\times C$ is Symetric Any Suggestions? Thanks!
0
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0answers
39 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
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1answer
44 views

Symmetric properties of eigen vectors from symmetric properties of matrix

In my physical problem I have a matrix: $\begin{bmatrix} 0 & -c_0 & -i b_1 & -c_1 \\ a_0 & 0 & a_1 & ib_1 \\ ib_1 & -c_1 & 0 & -c_0 \\ a_1 & -i b_1 & ...