A property of an object is called invariant if, given some steps that alter the object, it always remains, no matter what steps are used in what order.
Questions tagged [invariance]
412 questions
283
votes
5 answers
The Mathematics of Tetris
I am a big fan of the old-school games and I once noticed that there is a sort of 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…
Eric Naslund
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31
votes
4 answers
Suppose $M$ is an $m \times n$ matrix such that all rows and columns of $M$ sum to $1$. Show that $m=n$
I find this a rather awkward question, from the book "Mathematical Circles" by Fomin, Genkin and Itenberg. The question number is Question number 23 from Chapter 12 ("Invariants"). I was given a hint: use invariants, which I found even more…
Sarvesh Ravichandran Iyer
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19
votes
3 answers
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…
user54031
15
votes
3 answers
Partition 100 people, 4 from each country into 4 groups with conditions
This is a problem from the $2005$ All-Russian Olympiad. Problem is as follows:
$100$ people from $25$ countries, four from each country, sit in a circle.
Prove that one may partition them onto $4$ groups in such way that no
two countrymen, nor two…
crossvalidateme
- 391
14
votes
2 answers
Interesting Olympiad Style Problem about Invariance
Problem: The following operations are permitted with the quadratic polynomial $ax^2 +bx +c:$ (a) switch $a$ and $c$, (b) replace $x$ by $x + t$ where $t$ is any real. By repeating these operations, can you transform $x^2 − x − 2$ into $x^2 − x −…
Student
- 9,366
13
votes
2 answers
Proof that $a\nabla^2 u = bu$ is the only homogenous second order 2D PDE unchanged/invariant by rotation
Looking for feedback and maybe simpler intuition for my proof of the theorem, shown below
The statement of the theorem:
Theorem
Among all second-order homogeneous PDEs in two dimensions with constant coefficients, show that the only ones that do…
Hushus46
- 983
11
votes
1 answer
Quadratic P.S.D. differential operator that is invariant under $\textrm{SL}(2, \mathbb{R})$
Given some function $f \in L^2(\mathbb{R}^2)$, I'm interested in finding a positive semi-definite differential operator $\mathcal P: L^2(\mathbb{R}^2) \rightarrow L^2(\mathbb{R}^2)$ that is quadratic in $f$ and invariant under the the action of…
tommym
- 483
10
votes
1 answer
Matrix permutation-similarity invariants
https://en.wikipedia.org/wiki/Matrix_similarity
https://en.wikipedia.org/wiki/Permutation_matrix
The determinant and trace (and characteristic polynomial coefficients) are well-known similarity invariants of a matrix. There are more if we only allow…
mr_e_man
- 5,986
10
votes
3 answers
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…
Cherry_Developer
- 4,617
9
votes
2 answers
How can I get better at solving problems using the Invariance Principle?
I have some questions regarding the Invariance Principle commonly used in contest math. It is well known that even though invariants can make problems easier to solve, finding invariants can be really, really hard. There is this problem from Arthur…
8
votes
6 answers
We have a sequence $a_0,a_1,a_2,...,a_9$ so that each member is $1$ or $-1$. Is it possible: $a_0a_1+a_1a_2+...+a_8a_9+a_9a_0=0$
We have a sequence $a_0,a_1,a_2,...,a_9$ so that each member is $1$ or $-1$. Is it possible: $$a_0a_1+a_1a_2+...+a_8a_9+a_9a_0=0$$
This problem was given on contest, but I don't know how to solve it.
Clearly we must have $5$ terms $a_ia_{i+1}$…
nonuser
- 91,557
8
votes
1 answer
Similarity classes of matrices
Let $M_n(K)$ be the set of all $n\times n$ matrices over a field $K$. If $\mathcal{R}$ is the equivalence relation defined by matrix similarity, what does the quotient $M_n(K)/\mathcal{R}$ looks like? Is there something that characterizes it in…
Henrique Augusto Souza
- 3,130
8
votes
1 answer
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…
Hao S
- 504
7
votes
1 answer
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 quadratic, etc. When I say "invariance" I mean under…
yohBS
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7
votes
2 answers
What are some puzzles that are solved by using invariants?
This is certainly not a question with one "correct answer", but I think it's interesting and mathematical in nature. Essentially, my question is: do people know any not-well-known puzzles that are solved by means of an "interesting" invariant?
A…
Harambe
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