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5
votes
3answers
68 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
54 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
72 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
23 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
41 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
72 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
29 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
40 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
45 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
104 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
1answer
81 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 ...
1
vote
1answer
53 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.
11
votes
1answer
321 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
35 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
147 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
142 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
56 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 ...
6
votes
0answers
138 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
49 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
47 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
38 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
129 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
51 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
156 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
45 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
50 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
91 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 ...
5
votes
1answer
218 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 ...
1
vote
0answers
43 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
1answer
212 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 ${\mathbb R}^n$ ...
3
votes
2answers
65 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 ...
0
votes
0answers
68 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
votes
1answer
111 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 ...
2
votes
1answer
80 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
votes
1answer
165 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
vote
1answer
41 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.
0
votes
3answers
688 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?
4
votes
1answer
247 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
votes
1answer
19 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 ...
1
vote
2answers
147 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 ...
1
vote
2answers
53 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
votes
1answer
166 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 ...
0
votes
1answer
59 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
vote
1answer
42 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 ...
0
votes
0answers
14 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
1answer
197 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
vote
1answer
522 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
votes
1answer
90 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 ...
0
votes
1answer
34 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 ...
1
vote
1answer
305 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 ...