The invariance tag has no wiki summary.
3
votes
0answers
20 views
notation for invariation
Let $\Lambda = \{T \in \operatorname{Her}_2(\mathcal{O}) ; T \ge 0\})$ and $\mathcal{O}$ the maximal order of some quadratic imaginary number field. I write $T[U] := U^* \cdot T \cdot U$ where $U$ is ...
0
votes
1answer
57 views
Prove that the existence of a bridge is an invariant
An invariant is a property $P$ that is shared by all isomorphic graphs. In other words, a property $P$ is an invariant provided that whenever $G_1$ and $G_2$ are isomorphic graphs, if $G_1$ satisfies ...
0
votes
1answer
24 views
Invariants of point sets in an affine space
A distance between a pair of points in an affine space is invariant under translation, rotation and reflection.
An angle in a triangle whose corners are tree points is also invariant under scaling.
...
1
vote
0answers
23 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\ ...
1
vote
1answer
56 views
Vertex of a pentagon-Does the algorithm always stop?
To each vertex of a pentagon,we assign an integer $x_{i}$ with sum $s=\sum x_{i}>0$.
If x,y,z are the numbers assigned to 3 successive vertices and if $y<0$,then we replace $(x,y,z)$ by ...
1
vote
1answer
65 views
Does a constant of motion always imply hamiltonian?
If a dynamical system has a constant of motion, and is not already evidently Hamiltonian, is it always possible to use a change of variables and obtain a Hamiltonian system?
Edit: the constant of ...
0
votes
2answers
45 views
Is it possible to find a 2D distribution function such that the higher order moments always exist?
Is it possible to generate a 2D distribution function $f(x,y)$ with function supports specified as $[-a,a]$ and $[-b,b]$ for $x$ and $y$ respectively, such that it always has moments which are NON ...
5
votes
0answers
240 views
Rotation invariant tensors
It is often claimed that the only tensors invariant under the orthogonal transformations (rotations) are the Kronecker delta $\delta_{ij}$, the Levi-Civita epsilon $\epsilon_{ijk}$ and various ...
-1
votes
1answer
57 views
Variance Formula
for the variance formula $\text{var}(X) = E[X^2]-(E[X])^2$
how are you suppose to work out $E[X^2]$ given the interval $[0,1]$ and $E[x]= 1/2$
1
vote
1answer
73 views
Maximal Positive Invariant Set — Some fine print
I would like to share something I noticed on the definition of Maximal Positively Invariant Sets.
Definition 1.
For a discrete-time system of the form $x_{k+1}=f(x_{k})$ (and $x_{k}\in ...
4
votes
2answers
108 views
Using the invariance principle: how to solve $n+d(n)+d(d(n))=m$?
Let $d(n)$ be the digital sum of $n$. How to solve $n+d(n)+d(d(n))=m$, where $n$ and $m$ are natural?
0
votes
1answer
99 views
endomorphism and subspace
Let $\phi: V \to V$ be an endomorphism over a $\mathbb{C}$-field. Show:
If there is a decomposition $V = V_1 \oplus V_2$ with $V_1,V_2 \neq \{0\}$ and $\phi$-invariant,
then there exist ...
2
votes
1answer
112 views
Identifying $k[x_1,x_2,y_1,y_2]^{\epsilon}$ with $k[x,y]\wedge k[x,y]$
Suppose the symmetric group $S_2$ of order 2 acts on $k^4=Spec \;k[x_1, x_2, y_1, y_2]$ by the following: for $\sigma\not=e$,
$$\sigma\circ(x_1, x_2, y_1, y_2)=(x_2,x_1,y_2,y_1).$$
That is, the ...
2
votes
2answers
54 views
Constructing the sequence: $0\rightarrow (x-y)^{S_2} \stackrel{f}{\rightarrow} k[x+y,xy]\stackrel{g}{\rightarrow} k[y]$
Let $S_2$, a group of two elements, act on $k[x,y]$ by permuting $x$ and $y$.
It is clear that
$$
0\rightarrow (x-y) \rightarrow k[x,y]\rightarrow \dfrac{k[x,y]}{(x-y)}\cong k[y] \rightarrow 0
$$
...
1
vote
0answers
60 views
$S_k$ action on $A/I$
Let $S_2$ be a finite group of order $2$ and let $S_2$ act on $k[x,y]$ by interchanging $x$ and $y$, where $k=\overline{k}$. Then since
$$
R = \left( \dfrac{k[x,y]}{(x+y)}
\right)^{S_2}
= ...
1
vote
0answers
135 views
Invariant set: sufficient condition
Given a system $\dot x = f(x)$, $x \in \mathbb{R}^n$ with a smooth $f(x)$. Let $D$ be a set in $\mathbb{R}^n$ with a smooth boundary $\partial D$ such that $\left.\langle f(x), n(x) \rangle ...
4
votes
1answer
525 views
Invariants of a matrix
I'm teaching a course in physics, and I need a simple and intuitive proof that a matrix ($3\times3$, but it doesn't matter) has exactly 1 invariant which is linear in its entries, 2 that are ...
0
votes
0answers
265 views
Time-invariant IVP
How does one know that a system (of differential equation and initial value constraints) is time-invariant (perhaps by inspection...)? What are the implications of a system with this property (esp. ...
153
votes
4answers
8k views
The Mathematics of Tetris
I am a big fan of the oldschool games and I once noticed that there is a sort parity associated to one and only one Tetris piece, the $\color{purple}{\text{T}}$ piece. This parity is found with no ...
2
votes
2answers
319 views
The Invariance Principle
I had come across a problem practicing to get better at approaching different types of problems from different field topics and this one had got me kind of stuck in what direction to go. Not so ...
2
votes
1answer
85 views
Is there any invariance under the inversion mapping?
In geometry, there is a transformation called the inversion mapping which maps nonorthogonal circles into nonorthogonal lines and vice versa.(If I make a mistake, inform me, since I am not very ...
