Questions tagged [invariance]
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.
413 questions
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Rearranging Columns in a Colored Cube
An $n \times n \times n$ cube is divided into $n^3$ unit cubes, of which $n^2$ are black and the remaining ones are white.
A move consists of selecting two parallel $1 \times 1 \times n$ columns (i.e.,...
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Unique reachability of a nonnegative configuration under local updates on a cycle
Let $n \ge 3$ be an integer. Let $A_1A_2...A_n$ be a regular $n$-gon. A real number $a_i$ is assigned to each vertex $A_i$ such that
$$\sum_{i=1}^n a_i >0.$$
At each move one chooses an arbitrary ...
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Brocard inellipse area
It is well-known that the Brocard angle $\omega$ is invariant under a Poncelet rotation between the circumcircle and the Brocard inellipse. This can be shown in different ways, one of them being the ...
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Does function invariance under permutation imply existence of a stationary point invariant under permutation?
This question comes from the excellent introduction to mean-field spin glass methods by Montanari and Sen. Consider a function:
$$f\colon M_{n\times n}\longmapsto\mathbb{R}$$
which is invariant under ...
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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 (...
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Question on existence of a faithful $G$-invariant trace on $A.$
Let $(A, G, \alpha)$ be a $C^{\ast}$-dynamical system, where $G$ is a discrete group acting on $A$ by automorphisms via the action $\alpha.$ Then given a faithful trace $\tau$ on $A$ can we induce a $...
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Questions regarding the problem E5 on Engel's book Problem-Solving Strategies
Suppose not all four integers $a,b,c,d$ are equal. Start with $(a,b,c,d)$ and repeatedly replace $(a,b,c,d)$ by $(a-b,b-c,c-d,d-a)$. Then at least one number of the quadruple will become arbitrarily ...
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What's the difference between Lorentz invariance and Poincaré invariance?
I'm trying to understand null hypersurfaces.
The Poincaré Group is a superset of the Lorentz Group, but the only difference I can see is that the Poincaré Group also includes hyperspatial rotation ...
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How do solvers find counterexamples to loop invariants without iterating through values?
In formal verification, loop invariants are used to prove correctness across all iterations of a loop. A loop invariant is a property that must hold:
Before the loop starts (initialization),
After ...
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Sufficient conditions to prove invariance of a state of a $C^*$ algebra under a group G of $*$-automorphisms
This is a follow-up question of a previous question that I asked here: Doubt on invariant states and asymptotic abelianess, where it is observed that having a state $\omega$ over a $C^*$ algebra ...
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Question about scaling invariance of a PDE
The PDE
$$\frac{\partial f(x,y)}{\partial x} + \frac{\partial}{\partial y}(f(x,y)g(x,y)) = 0$$
Under the transformation
$$x'=k^bx$$
$$y'=k^ay$$
$$f'=k^{(a+b)}f$$
$$g'=k^{(a-b)}g$$
becomes
$$k^{-a} \...
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Higher-order derivative eigenfunctions
I'm looking for more information or geometric intuition on the following topic.
The exponential function is an eigenfunction of the first derivative, $d/dx=D$. That is,
$$D \exp\lambda x = \lambda \...
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Homotopy Invariance of Intersection Number
starting from this theorem:
Theorem (intersection number modulo 2): Suppose that $L^l$, $N^n$ are two
manifolds, $L$ is compact, and $𝑀^𝑚$ is a submanifold and a closed subset of $𝑁$
such that $𝑙 +...
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I reinvented the Neuberg cubic for finding "cyclic" triangle centers, but...
I "invented" a determinant $|(1,1,1),(a^2+d^2,b^2+e^2,c^2+f^2),(ad,be,cf)|$ where $a\dots f$ are the squared lengths in a quadrangle ($a,d$ opposite), resulting from a triangle plus some ...
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How should I determine if $x(t)$ is odd or even?
Homework problem:
Consider the differential equation
$$
\ddot{x}+x+\epsilon(\alpha x^2\operatorname{sgn}(x)+\beta x^3)=0,
$$
$0<\epsilon\ll1,\; \alpha>0,\; \beta>0$, where $\alpha$ and $\...
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