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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Let $A$ be an $n \times n$ matrix with real entries such that $A^2 + I = 0$ then $n$ is even.

Let $A$ be an $n \times n$ matrix with real entries such that $A^2 + I = 0$ then $n$ is even. And if $n = 2k$, then $A$ is similar over the field of real numbers to a matrix of the block form $$\...
4
votes
1answer
470 views

(Average) Number of cycles of length m in permutations on N with k cycles

Suppose we have permutations on $[1,2,...,n]$ that have exactly $k$ cycles (which there are $|s(n,k)|$ of where $s(n,k)$ is the Stirling number of the first kind). What is the average number of ...
0
votes
1answer
94 views

Prove $\sum \sqrt{{\frac {2{a}^{2}b}{a+c}}} \leqq a+b+c$ for $a,b,c>0$

For $a,b,c>0$. Prove that $$\sum \sqrt{{\frac {2{a}^{2}b}{a+c}}} \leqq a+b+c \,\,-----(1)$$ My solution$:$ By C-S, we need to prove: $$(\sum ab) \cdot (\sum \frac{2a}{a+c}) \leqq (a+b+c)^2\, (\...
0
votes
1answer
59 views

Prove $2\left(x^2+y^2+z^2+1)(x^3y+y^3z+z^3x+xyz\right) \le \left(x^2+y^2+z^2+3xyz\right)^2.$

For $x,y,z\geqq 0$ and $x+y+z=1.$ Prove that$:$ $$2\left(x^2+y^2+z^2+1)(x^3y+y^3z+z^3x+xyz\right) \leqq \left(x^2+y^2+z^2+3xyz\right)^2.$$ Let $y=\hbox{mid} \{x,y,z\}$ and $\text{P}= \left[\left(x^2+...