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190
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4answers
12k 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 ...
10
votes
2answers
1k 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 ...
4
votes
1answer
2k 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 ...
4
votes
2answers
183 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?
3
votes
3answers
86 views

Numbers in a sequence

The number sequence $1, 9, 8, 2...$ satisfies the following rule: each element of the sequence starting from the fifth, is equal to the last digit of the sum of the previous four members. Will we ...
3
votes
1answer
61 views

Creating all strongly connected graphs with given in-degree with equal probability

I am looking for a way to sample uniformly from the space of all strongly connected directed graphs (without self-loops) of $n$ nodes and in-degree $k=(k_1,...,k_n)$ with $1 \leq k_i \leq n-1$. In ...
3
votes
2answers
111 views

Inertial Frames of Refereence

I am told that in Newtonian mechanics, no coordinate system is "superior" to any other. Also, all inertial frames are in a state of constant, rectilinear motion with respect to one another. So am I ...
3
votes
0answers
42 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 ...
2
votes
2answers
581 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
2answers
29 views

Prove that n is divisible by 4 in a cylic sum with variables which have only two possible values

It is known that $a_1, a_2, a_3, ... , a_n \in \left\{-1, 1 \right\}$ and $S = a_1a_2a_3a_4 + a_2a_3a_4a_5 + ... + a_na_1a_2a_3 = 0$ Prove that $n \equiv 0\space(mod\space 4)$ I know this problem ...
2
votes
1answer
104 views

Connection between the Tutte and characteristic polynomials?

Both the Tutte polynomial $T_G(x,y)$ and the characteristic polynomial $\phi_G(x)$ encode a great amount of structure of the input graph $G$. I've read somewhere that the Tutte polynomial has a kind ...
2
votes
2answers
60 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 $$ 0\rightarrow (x-y) \rightarrow k[x,y]\rightarrow \dfrac{k[x,y]}{(x-y)}\cong k[y] \rightarrow 0 $$ ...
2
votes
1answer
35 views

Boundedness of RHS implies existance of invariant cube

Consider a system of ODEs of the form $$\dot x_1 = f_1(x_1,x_2)-g_1(x_1,x_2)x_1 \\ \dot x_2 = f_2(x_1,x_2)-g_2(x_1,x_2)x_2,$$where $f,g$ are bounded, Lipschitz-continuous functions. (Then by the ...
2
votes
1answer
160 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
1answer
98 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 ...
2
votes
0answers
82 views

A Unique Invariant subspace for a set of matrices

Im wondering if anyone can give me a good reference or answer this question which may have already be solved. For a set of generic $n\times n$ matrices $A_1,A_2,...,A_k$, such that they share only ...
1
vote
2answers
37 views

Rotation invariance of higher than 2 dimensions

According to this $f_2(x_1,x_2) = x_1^2 + x_2^2$ is invariant under rotation. I wanted to ask if a function $f_n(x_1,x_2,...,x_n) = x_1^2 + x_2^2 + ...+ x_n^2$ is also rotation invariant. In other ...
1
vote
1answer
98 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 ...
1
vote
1answer
20 views

Prove that for every $k$ there's an invariant subspace

Let $V$, a vector space above $\mathbb{C}$ and let $T:V\to V$, a linear transformation. Show that for every $0\le k \le n$ there is an invariant subspace of $T$ with a dimension $k$. It seems ...
1
vote
1answer
29 views

Inequality involving absolute moment and variance

Suppose $X\in\Omega$ is a random variable and $f:\Omega\rightarrow [0,1]$. Is the following true: $$E[|f(X)-E[f(X)]|]^2\leq \operatorname{Var}[f(X)]?$$ This was stated without proof in a research ...
1
vote
1answer
50 views

Invariant dimension property and a ring epimorphism

In Hungerford's Algebra, p. 186, the Proposition 2.11 says Let $f:R\to S$ be a nonzero epimorphism of rings with identity. If $S$ has the invariant dimension property (IDP), then so does $R$. It is ...
1
vote
1answer
71 views

What does rotational invariance mean in statistics?

What does rotational invariance mean in statistics? The property that the normal distribution satisfies for independent normal distributed $X_i$, $\Sigma_i X_i$ is also normal with variance $\Sigma_i ...
1
vote
1answer
34 views

How to show that the cross-ratio is invariant under Moebius transformations without using generator matrices of $\text{GL}(2, \mathbb{C})$

I am given the following problem set: Show that the cross-ratio is invariant under Moebius transformation, meaning that $$D \left(L_g(z_1),L_g(z_1),L_g(z_1),L_g(z_1)\right) = D(z_1,z_2,z_3,z_4) ...
1
vote
1answer
32 views

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic.

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic. A is an integer matrix such that A has an eigenvalue which is a ...
1
vote
1answer
56 views

Expectation and Variance

I was asked to give a simple example to show that E[XY] = E[X]E[Y] is not necessarily true if and when X and Y are not independent. I can't seem to find a simple example without ending with a proof by ...
1
vote
1answer
52 views

If $M$ is invariant under $L$ is it a vector space

I'm reading Linear Algebra by Peterson, and the excercise on p.31 reads as follows: Let $L: W \to V$ be a linear operator and $V$ a vector space over $\mathbb{F}$. Show that if $M \subset V$ is ...
1
vote
1answer
114 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 ...
1
vote
1answer
103 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
726 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. ...
1
vote
1answer
32 views

Write $V=P_2(\mathbb{R})$ as a direct sum of $V=W_1\oplus W_2 \oplus W_3$

So, if I let $T:P_2(\mathbb{R}) \rightarrow P_2(\mathbb{R})$ and is a linear endomorphism given by $T(f(x))=f(x)-f(2x-1)$. Then I have to write$V=P_2(\mathbb{R})$ as a direct sum of $V=W_1\oplus W_2 ...
1
vote
0answers
69 views

Finding an invariant polynomial under a matrix action

I asked this question as a mathematica question: http://mathematica.stackexchange.com/questions/41689/finding-a-certain-invariant-polynomial-using-matrix-coordinates but maybe it will get more ...
1
vote
2answers
73 views

For general non-symmetric square matrices is there a matrix norm that is invariant under similarity transformations?

I think that there is no similarity-invariant matrix norm for general matrices. But are there similarity invariant norms for special types of matrices (e.g. for matrices whose eigevalues are different ...
1
vote
0answers
342 views

Time-invariant Differential Equation

I was wondering if it is safe to assume that $$ (1/t)y'(t) + (1/t)y(t) = 0 $$ is a time-varying differential equation. I would say these (1/t) factors cancel out and make the eq. time invariant. ...
1
vote
0answers
37 views

A set is partitioned into $k$ non-empty sets, and the difference between elements of each subset is given, reconstruct each subset

We have a set $A = \{1, 2, 3, \dots, n\}$. This set is partitioned into $k$ non-empty subsets: $$A_1 = \{a_1, a_2, \dots, a_{m_1}\}$$ $$A_2 = \{a_{m_1 + 1}, a_{{m_1} + 2}, \dots, a_{m_2}\}$$ $$.$$ ...
1
vote
0answers
46 views

finding invariant subgroups under all automorphisms

I'm trying to find group G s.t every subgroup of G is invariant under all automorphisms, or conditions for G. For example; cyclic groups and simple groups have this condition.
1
vote
0answers
128 views

How to show that the Harris corner detector is rotation invariant

I understand what it means for the Harris corner detector to be rotation invariant. But what steps do you need to take to show that it is rotation invariant?
1
vote
0answers
96 views

Calculate the variance from a function of normal random variable

I am new to the topic that I found difficulty for the question: Given the function $g(x) = e^{-X}$, $X \sim N(0,1)$, calculate the variance of $g(x)$. I know the answer is $e(e-1)$. But I don't ...
1
vote
1answer
76 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
209 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 ...
1
vote
0answers
63 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
180 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 ...
0
votes
3answers
90 views

$[$Is connected$]$ & $[$Has a simple cycle of length $m$$]$ : An Analysis on Invariance

How are the statements $$^{\dagger_1}\text{Is connected.}~~~~~^{\dagger_2}\text{Has a simple cycle of length $m$.}$$ an invariant?
0
votes
3answers
63 views

Prove that an algorithm cannot reach a given goal

We are given an algorithm that, in each step takes a set $\left\{a, b, c\right\}$ It takes any two variables $a, b$ at random and changes them to $0.6 + 0.8b$ and $0.8a - 0.6b$. The initial value of ...
0
votes
1answer
32 views

Proofs involving Subspace Invariance

Let $ V $ be a finite dimensional vector space over a Field $ F $ and let $ T \in End(V) $. i) If $ S \in End(V) $ is such that $ ST = TS $, show that $ Im(S) $ is a $T$-Invariant Subspace of $ V $ ...
0
votes
2answers
38 views

Is this proposition posible? [duplicate]

In a board, you have $13$ White round pieces, $15$ Black round pieces, and $17$ Red round pieces. In each round you can choose two different color pieces and change them with two other pieces of ...
0
votes
1answer
35 views

Proving that an ergodic and invariant map is constant a.e

I understand the first two sentences of the proof, however I cannot see how the third and final sentence holds. Why should $\mu(f \leq a)=0$ surely it should be non-zero as c is defined as the ...
0
votes
1answer
34 views

Isotropy group for the subset of Grassmannian

Consider a complex Grassmannian $Gr_{k}(C^{n})$, which is a symmetric space with symmetry group $U(n)$ (i.e. unitary group). Consider a subspace $S_{0}$ of the Grassmannian determined by the canonical ...
0
votes
1answer
90 views

Pattern for Recursive Construction

Suppose I have this recursive definition of binary strings. Let $K$ be set of binary strings. The empty string $""$ and $1$ are in $K$. If $k$ is a string in $K$, then so is $0k$, $k0$. And if $k$ is ...
0
votes
1answer
371 views

is u(x,t) is a solution of the heat equation, then so is v

Show that if $u(x,t)$ is a solution of the heat equation \begin{equation*} u_t - k u_{xx} = 0 \,,\qquad (x\in R,\ t\geq0)\,, \end{equation*} then so is \begin{equation*} v(x,t) = \frac{1}{\sqrt{kt}} ...
0
votes
1answer
131 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 ...