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]
411 questions
284
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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 awkward.…
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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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
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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 $x^2 − x − 1?$…
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 not…
Hushus46
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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
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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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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
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2 answers
Adapted Kontsevich "Pebbling a Chessboard" game: can you win it?
In 1981, Kontsevich proposed the following problem. Imagine a chessboard that extends infinitely in only two directions: upwards and to the right. Put one pebble in each colored cell of the figure (the "L" shaped region formed by three cells in the…
hdecristo
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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…
8
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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
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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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8
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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…
Hao S
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