# Questions tagged [cyclic-decomposition]

Use this tag for questions about (1) in group theory, expressing a permutation in terms of its constituent cycles, (2) in commutative algebra and linear algebra, writing a finitely generated module over a principal ideal domain as the direct sum of cyclic modules and one free module, or (3) in graph theory, partitioning the vertices of a graph into subsets such that the vertices in each subset lie on a cycle.

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### Minimum number of swaps required to such that all pairs are together

This is from a programming question that seems to have concepts of cyclic decomposition involved which I find difficult to grasp. Consider an arrangement [2,5,6,1,3,4,0,7], find minimum swaps to so ...
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### Proving $\frac{a}{b^3}+\frac{b}{c^3}+\frac{c}{a^3}\geqslant \frac{a+b}{b^3+c^3}+\frac{b+c}{c^3+a^3}+\frac{c+a}{a^3+b^3}$

For $a,b,c>0.$ Prove$:$ $$\dfrac{a}{b^3}+\dfrac{b}{c^3}+\dfrac{c}{a^3}\geqslant \dfrac{a+b}{b^3+c^3}+\dfrac{b+c}{c^3+a^3}+\dfrac{c+a}{a^3+b^3}\quad (\text{Tran Quoc Thinh})$$ It's easy with ...
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### Proving $\sum_{\text{cyc}}\, (23a-5b-c)(a-b)^2(a+b-3c)^2 \geqslant 0$

Problem (KaiRain's problem). For $a,b,c\geqslant 0.$ Prove $$\displaystyle \sum_{\text{cyc}}\, (23a-5b-c)(a-b)^2(a+b-3c)^2 \geqslant 0$$ I only found a proof by $pqr.$ (Note that from pqr's proof we ...
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### Why is a hamming (7,4) coding scheme symmetric?

I'm trying to understand the explanation on pages 2/5 of the file and 18 in the book http://www.inference.org.uk/mackay/itprnn/ps/17.21.pdf on why the (7,4) Hamming code is symmetric. The explanation ...
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### Time Series Decomposition and Stationary

I am currently learning about time series. Understand that in order to apply ARMA model, the time series must be stationary, which implies no trend, no cyclic pattern. Also I learnt that time series ...
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### How to find the cyclic vectors when finding the Rational form of a given matrix?

For a given matrix, say $A$, of dimension $n$, in $\mathbb{F}^{n\times n}$, the characteristic polynomial of $A$ is of the form $$c_A = p_1^{d_1}*...*p_k^{d_k},$$ and the minimal polynomial of the ...
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### Calculate Smith normal form, cyclic group decomposition

Can someone please check my working for the following problem? Let $A$ be the abelian group generated by elements $x,y,z$ with relations $7x+5y+2z=0, 3x+3y=0, 13x+11y+2z=0$. Decompose $A$ as a ...
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### How to maximize 3-cycles in this graph type inspired by block design?

I've come up with this question and I've been playing with it for a couple weeks by now without any definite breakthrough; it seems there should be a better approach than straight naive brute force, ...
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### On the intuitive idea behind decomposition into cyclic modules

Let $R$ be a ring and $M$ be a free $R$-module. If $M$ has a basis of size $n$ then we have $M \cong R^n$. I find this rather intuitive. With that as a building block, I also find rather intuitive ...
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### Existence of unique maximal proper T-invariant subspace implies indecomposability

Let $V$ be a finite-dimensional vector space over a field $\mathbb{F}$ and let $T$ be a linear operator on $V$. Show that $T$ is indecomposable if and only if there is a unique maximal proper $T$-...
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### Prove or disprove: The minimum number of transpositions needed to decompose $\sigma$ is $n-S$.

Assume that $\sigma \in S_n$ and $\sigma = \alpha_1 \dots\alpha_s$ is the decomposition of $\sigma$ into disjoint cycles, such that all of the members of $\{1,2,\dots,n\}$ are appeared in the members ...
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### Partitioning a graph in cycles of four

I have the following question: Suppose that in a simple undirected graph with $4n$ vertices, each vertex has degree at least $2n$. Is it true that we can always partition the set of vertices in $n$ ...
### How could I describe homomorphisms from $C_m \rightarrow C_n$?
E.g.: If $m = 21$ and $n = 15$. I realise I could enumerate the number of bijective mappings but I suspect I would be excluding other homomorphisms. How would I best approach this problem?