Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [invariance]

A property of an object is called invariant if, given some steps that alter the object, always remains, no matter what steps are used in what order.

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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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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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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
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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
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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
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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
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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
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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
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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
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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…
TryingHardToBecomeAGoodPrSlvr
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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
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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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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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Noether's theorem in SmoothLife?

Conway's Game of Life, being discretized in both space and time domains, has no locally conserved quantities. SmoothLife, however, is a generalization of the Game of Life to a continuous and spatially-isotropic domain. The evolution rules of…
Logan R. Kearsley
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A new graph invariant? The maximum number of non-equivalent colorings with $n$ colors.

Consider (proper) coloring of a finite graph $G$ with exactly $n$ colors and the following coloring transformation: choose an edge of the graph with the end nodes of colors $a$ and $b$ and swap the colors $a$ and $b$ in the connected component of…
DVD
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