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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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…
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Testing if two finite sets of points differ only by rotation (unordered, in polynomial time in size and dimension)?

Imagine we have two size $m$ sets (without order) of points $X=\{x^i\}_{i=1..m}, Y=\{y^i\}_{i=1..m} \subset \mathbb{R}^n$ and we want to answer the question if they differ only by rotation: if there exists othogonal $O^TO=OO^T=I$ such that…
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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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Invariant measures of the doubling map on the closed interval

Consider the doubling map $g\colon [0,1]\to [0,1]$ given by $g(x)=2x \, {\rm mod}\; 1$. It is clearly discontinuous at 1/2. However, its counterpart $G$ on the circle $G(e^{2\pi i \theta}) = e^{4\pi i \theta}$ ($\theta\in [0,1)$) is continuous…
Tomasz Kania
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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…
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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…
Hao S
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Examining cycles in a sequence

I am looking at a problem in Engel's problem solving strategies: Start with an $n$-tuple $S=(a_0,a_1,\ldots, a_{n-1})$ of nonnegative integers. Define the operation $T(S):=(|a_0-a_1|, |a_1-a_2|,\ldots, |a_{n-1}-a_0|)$. Now consider the sequence $S,…
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Pawn visiting every square on a chessboard exactly once and returning to its right

A pawn moves across $n\times n$ chesssboard so that in one move it can shift one square to the right, one square upward, or along a diagonal down and left. Can the pawn go through all the squares on the board, visiting each exactly once, and finish…
Gaurang Tandon
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"Invariants" of Exotic spheres

An exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. Naively, I thought that there is no algebraic topological invariant that distinguishing the exotic spheres from each…
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Invariant polynomials under an $S_4$-action

Suppose $X = \mathbb{Q}-span \langle x_1, x_2, x_3 \rangle \cong \mathbb{Q}^3$ is a $3$-dimensional $\mathbb{Q}$-vector space with some basis $(x_1, x_2,x_3)$. We let the symmetric group $S_4$ act on $X$ as the product of the standard representation…
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$(H^2-K) dA$ is globally invariant under inversions

I am reading a paper by James H. White which states that the Willmore functional is invariant under conformal mappings. Let $M$ be a smooth compact surface and $f:M\to\mathbb{R}^3$ an immersion. The Willmore functional is given…
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A question on translation invariance of functions

As I understand it, a given function, $f$ (sticking to one dimension for simplicity) is said to be translationally invariant if $$f(x+a)=f(x)$$ for any arbitrary constant $a\in\mathbb{R}$. This condition then implies that $f$ cannot have any…
user35305
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Fibonacci Loop Invariants

I've taking an Algorithms course. This is non-graded homework. The concept of loop invariants are new to me and it's taking some time to sink in. This was my first attempt at a proof of correctness today for the iterative Fibonacci algorithm. I…
Matt
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Invariant measures for stochastic processes

I have some doubts about the concept of invariant measure for a stochastic process. Let me introduce a definition. Given $(\Omega, \mathcal{E}, \mathbb{P})$ a measure space, $H$ Hilbert space, let be $Y(t)$ the unique $H-$valued stochastic process…
tomino
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Is there a name for this property of functions on groups?

Let $G$ be a group and $F:G^n \to G$ with the following property: If $x_1,…,x_n,h \in G$, then $F(hx_1,…,hx_n)=hF(x_1,…,x_n)$. Is there a name for this type of function property? It is something I’ve been investigating lately. For instance, if $G$…
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