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.
Questions tagged [invariance]
392 questions
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votes
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three grasshoppers jumping on a plane.
The problem is dead simple:
Three grasshoppers sit on a plane not in a line. Every second just one
of the grasshoppers hops symmetrically over one of the others. Can
they return to the initial positions after n seconds?
The very tempting…
Salvador Dali
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4
votes
2 answers
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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Prove that n is divisible by 4 in a cylic sum with variables which have only two possible values
It is known that $a_1, a_2, a_3, ... , a_n \in \left\{-1, 1 \right\}$ and $S = a_1a_2a_3a_4 + a_2a_3a_4a_5 + ... + a_na_1a_2a_3 = 0$
Prove that $n \equiv 0\space(mod\space 4)$
I know this problem can be solved using number theory, but I am looking…
Gerard
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votes
3 answers
We start with the stet $\left\{1,4,32,128,256\right\}$ Is it possible to reach the set $\left\{512,32,16,16,2\right\}$? With given rules
We start with the stet $\left\{1,4,32,128,256\right\}$. In each step we may divide one number by $2$ and multiply another number by $2$. We may repeat this step as many times as we want. Is it possible to reach the set…
HighSchool15
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finding invariant subgroups under all automorphisms
I'm trying to find group G s.t every subgroup of G is invariant under all automorphisms, or conditions for G. For example; cyclic groups and simple groups have this condition.
negar
- 143
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A question based on Invariance principle
. Suppose the positive integer n is odd. First Al writes the numbers $1, 2,..., 2n$
on the blackboard. Then he picks any two numbers a, b, erases them, and writes,
instead, $|a − b|$. Prove that an odd number will remain at the end.
I have proved it…
1
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Arthur Engel - Invariant for cyclic difference
Just started solving and came across this example:-
Consider any four integers $a,b,c,d $ where not all are equal. It is allowed to change this sequence to $a-b,b-c,c-d,d-a$. Prove that at least one of the numbers from sequence can be arbitrarily…
saket yagay
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Changing chameleons
13 green, 15 blue, and 17 red chameleons are on an island. Whenever two chameleons of different colors meet, they change to the third color. Is it possible for all of them to become the same color?
I said no. I took a look at all the chameleon…
Gerard L.
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What quantity is “invariant” in a 3-body invariant manifold?
Invariant manifolds are used to calculate low-energy trajectories for spacecraft transiting between Lagrange points. I understand that an invariant manifold is a topological manifold that is invariant under the action of the dynamical system. I…
Woody
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Show that the origin of $\mathbb{R}^3$ is not invariant under spatial translations
It is a known fact that $\mathbb{R}^3$ does not model classical physical space accurately since it includes some non-invariant structure under translations, such as the origin $(0,0,0)$. However, I am unsure as to how one would show that.
My trouble…
Promethèus
- 528
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3 answers
Prove that an algorithm cannot reach a given goal
We are given an algorithm that, in each step takes a set $\left\{a, b, c\right\}$ It takes any two variables $a, b$ at random and changes them to $0.6 + 0.8b$ and $0.8a - 0.6b$. The initial value of the algorithm is $\left\{3, 4, 12\right\}$. Prove…
Gerard
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