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 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 familiar with the topic, its on invariance's, so I was…
night owl
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Result of replacing $1$ to $n$ in pairs by the sum?

I have the following problem: Alice writes the numbers $1, 2, 3, 4, 5, 6, \ldots, n$ on a blackboard. Bob selects two of these numbers, erases both of them, and writes down their sum on the blackboard. For example, if Bob chose the numbers 3…
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Dodecahedron faces are buttons, vertices have counters that track the use of the buttons

As a follow up to this question, I'm trying to teach invariants by creating a game. The idea is to start with a dodecahedron where each of the 20 vertices has a counter on it and each of the 12 faces is a button. You can click a button to increase…
Brian
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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 some $2\times2$ matrix and $U^*$ is its…
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Coloring a $10×10$ grid

Given a $10×10$ grid with $9$ red blocks and $91$ white blocks, in each step we color one red block black and after that one white block red until there are $91$ black blocks and $9$ red blocks. Prove that there exits a step in which a black block…
Dojou
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Given a finite number of stones. We place every stone on an integer. Prove that, given different movements, we can only make finite number of moves

Given a finite number of stones. We place every stone on an integer (a number $x$ where $x\in Z$) and maybe multiple stones on an integer. On each move we can make one of the following movements: Remove a stone from numbers $n-1$ and $n$ and place…
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Invariant for interesting set of functions generalizing $\sin$ and $\cos$ and other properties

In an attempt to generalize $\sin(x)$ and $\cos(x)$—but just a curiosity—I found some functions with fairly interesting properties. The idea was to extent the definition of $\sin$ and $\cos$ as unique solutions to $y''+y=0$ (with $y(0)=0$ and…
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Seven vertices of a cube are labeled 0, and the remaining vertex labeled 1. Can you make all labels divisible by 3?

Seven vertices of a cube are labeled 0, and the remaining vertex labeled 1. You’re allowed to change the labels by picking an edge of the cube, and adding 1 to the labels of both of its endpoints. After repeating this multiple times, can you make…
Sunaina Pati
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Dimension of the invariant subspace

Let $\Gamma \subseteq GL_{n}(\mathbb{C})$ be a finite matrix group. Let this finite matrix group act on $f(x_1,...,x_n) \in \mathbb{C}[x_1,...,x_n]$ like so: $$\Gamma \cdot f(x_1,...,x_n) = f(\Gamma \textbf{x})$$ where $\textbf{x}$ is to be thought…
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Topological invariance of dimension

I am starting to study smooth manifolds with the book of Lee. At the beginning, he states this theorem, which is then proven later on with advanced techniques: Theorem 1.2 (Topological Invariance of Dimension). A nonempty $n$-dimensional topological…
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Rigorous proof to show that the $15$-Puzzle problem is unsolvable

So this is supposedly a very popular puzzle by Sam Loyd. (I don't want answerers to provide solutions directly from some website etc. I mean, an ingenious solution is more welcome please.) Now, as you see all numbers are arranged in ascending…
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Is a conformal transformation also a general coordinate transformation?

As far as I understand, a general coordinate transformation is induced by a diffeomorphism $f:M\rightarrow M$ where $M$ is a manifold (which can locally be described with coordinates). So if $x:M\rightarrow \mathbb{R}^m$ is a coordinate map, then…
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Rotation invariants for higher degree homogeneous polynomials (like Tr$(P^m)$ for degree 2)?

Treating rotation in $\mathbb{R}^n$ as $x\to Ox$ for orthogonal $O^T O=O O^T=1$, we can easily get complete sets of independent rotation invariants for degree 1 and 2 homogeneous polynomials: Degree 1: for $p(x)=\sum_i P_i x_i$, rotation: $P_i\to…
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Prove that the value of the expression $|a_1-b_1|+|a_2-b_2|+...+|a_n-b_n|$ does not depend on the coloring.

Now we have some $n$ of the number from the set $\{1,2,...,2n\}$ colored red and the rest of them are colored blue. Say $a_1b_2>...>b_n$ are blue. Prove that the value of the expression…
nonuser
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Intuition of invariance of power of point

Power of a point is a famous proof in elementary geometry, which states that: For a circle $\omega$ and a pont $P$ outside it, for any line through $P$ which intersects $\omega$ at $A$ and $B$, the quantity $PA.PB$ is invariant for any $A$,…
user441034
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