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2
votes
2answers
37 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
votes
0answers
10 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
1answer
21 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
vote
1answer
24 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 ...
1
vote
2answers
30 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
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 ...
1
vote
2answers
42 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 ...
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, ...
6
votes
1answer
68 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
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
41 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
47 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 $G$? ...
0
votes
2answers
51 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
votes
1answer
57 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 ...
1
vote
5answers
75 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
vote
2answers
87 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
votes
1answer
47 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
votes
1answer
31 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 ...
0
votes
1answer
12 views

Difference between duality, symmetry, equivalency and invaraince?

Can someone difference in detail on these four terms? Thank you.
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 ...
4
votes
1answer
76 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
votes
2answers
21 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 $a=-\infty$ $f(x,0)=x$ $f(0,a\to\infty)\to1$ $f(1,a\to-\infty)\to0$ ...
10
votes
3answers
195 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
53 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
42 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
vote
1answer
35 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
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
1answer
37 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
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
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
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
1answer
35 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 & ...
1
vote
1answer
76 views

Number of 2n-1 equal size partitions up to symmetry

Consider the $K_{2n}$ (or just the set $\{1,\dots,2n\}$) with $S_{2n}$ acting on the vertices. Moreover consider a collection of 2n-1 partitions of the vertices into two equal sized sets (repeated ...
2
votes
0answers
46 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 ...
0
votes
1answer
44 views

What is the value of $\angle BMN$?

here $ABC$ is a isosceles triangle.$AC=BC$,$\angle C=20$, $\angle ABM=60$ and $\angle BAN=60$.What is the value of $\angle BMN$? please help me with this problem.
8
votes
1answer
238 views

Why does Group Theory not come in here?

Here is a list of questions that I find quite similar, for the one and only reason that they all show much "symmetry". Which is at the same time my problem, because I don't have a very precise notion ...
1
vote
2answers
33 views

Can this condition infer that the matrix is Hermite?

$\boldsymbol{A^H A=AA^H}$ does this imply that $\boldsymbol{A}$ is Hermite matrix? Why? $\boldsymbol{A^H}$ is the conjugate transpose of $\boldsymbol{A}$
2
votes
3answers
72 views

$A$ is a symmetric real matrix. Show that there is $B$ such that $B^3=A$

I'm having trouble with this question, I'd like someone to point me in the right direction. let $A$ be a n by n matrix with real values. show that there is another n by n real matrix $B$ such that ...
1
vote
3answers
125 views

short question regarding convention - symmetric matrices and transpose

I have a short question because wikipedia is extremly vague on this subject. Suppose I have the matrix $A=\begin{pmatrix} i & 1 \\ 1 & -i\end{pmatrix}$. Is it symmetric? I mean, in the ...
2
votes
1answer
47 views

ODE system and single PDE “equivalence”, reference request

The answers to this question Replacing large-dimensional ODE systems with one PDE suggest that, in general, one can not hope for "replacing" an ODE system with a single PDE. On the other hand, this ...
7
votes
0answers
101 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
1answer
45 views

Show the integrals are alike without explicit computation.

I saw a couple of striking integrals which are $\hspace{5em} \displaystyle \color{black}{\displaystyle \int_a^b \frac{x\,\mathrm{d}x}{\sqrt{(x^2-a^2)(b^2-x^2)}} }$$\displaystyle\ =\ $$\displaystyle ...
1
vote
1answer
35 views

What do you call a convex polyhedron whose symmetry group is transitive on the facets?

I'd like to know a name/source for the following concept: Let $P$ be a convex polyhedron in $\mathbb{R}^3$. Let $G$ be its symmetry group, and let $F$ be the collection of (top-dimensional) faces of ...
0
votes
1answer
26 views

Showing that the composite trapezoid rule is invariant to certain symmetry

I am trying to show that if $f$ holds certain symmetric properties than the composite trapezoid rule $$ T(n) = (b-a)\frac{f(a)+f(b)}{2n} + \frac{b-a}{2} \sum_{k=1}^{n-1} f(x_k) $$ is invariant over ...
0
votes
1answer
110 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 ...
1
vote
1answer
49 views

Symmetric placing of balls

Suppose I have a round domain and put my first ball in the very center (this ball alone constitutes the $0$th ring). Now around this ring (this first ball) I want to add another ring of balls in a ...
2
votes
0answers
88 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 ...
1
vote
1answer
41 views

Is exponential even?

My book presents the following simplification: $$A(w) = \frac{1}{\pi}\int_{-\infty}^{\infty} e^{-kx}cos(wx)dx = \frac{2}{\pi}\int_{0}^{\infty} e^{-kx}cos(wx)dx$$ But I can't understand why, since ...
1
vote
1answer
35 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 ...
0
votes
2answers
54 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 ...