Articles of invariance

loop invariant fibonacci

fibonacci(int n) if n < 0 return 0; else if n = 0 return 0; else if n = 1 return 1; else x = 1; y = 0; for i from 2 to n { t = x; x = x + y; y = t; } } return x; I’m trying to find […]

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 a SINGLE eigenvector in common, the joint commutant of $A_1,A_2,…,A_k$ is trivial, how can I guarantee there exists only this single […]

Prove that if fewer than $n$ students in class are initially infected, the whole class will never be completely infected.

During 6.042, the students are sitting in an $n$ × $n$ grid. A sudden outbreak of beaver flu (a rare variant of bird flu that lasts forever; symptoms include yearning for problem sets and craving for ice cream study sessions) causes some students to get infected. Here is an example where $n = 6$ and […]

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 combinations of their tensor products. While it is easy to check that $\delta_{ij}$ and $\epsilon_{ijk}$ are indeed invariant under rotations, I would like to know if there exist any proof […]

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 $\Sigma_i Var(X_i)$ is referred to as rotational invariance and I want to know why.

How to prove that the Kronecker delta is the unique isotropic tensor of order 2?

Is there a way to prove that the Kronecker delta $\delta_{ij}$ is indeed the only isotropic second order tensor (i.e. invariant under rotation), i.e. so we can write $T_{ij} = \lambda \delta_{ij}$ for some constant $\lambda$? By rotational invariance I mean: $$ T_{ij} = T^\prime_{ij} = R_{ip} R_{jq} T_{pq}\text{,} $$ where the matrices $R_{ij}$ are […]

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 can be solved using number theory, but I am looking for a solution utilizing the concept of invariance. I […]

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 other piece in the game. Background: The Tetris playing field has width $10$. Rotation is allowed, so there are […]