Intereting Posts

Show that an operation being commutative is a structural property
Intuition about the second isomorphism theorem
Prove that $N$ is normal
What was the notation for functions before Euler?
How to prove that $\sum\limits_{i=0}^p (-1)^{p-i} {p \choose i} i^j$ is $0$ for $j < p$ and $p!$ for $j = p$
What exactly IS a line integral?
Prove that all values satisfy this expression
Prove that $\lim_{\Delta x\to 0} \frac{\Delta ^{n}f(x)}{\Delta x^{n}} = f^{(n)}(x).$
Is the Formula Logically Valid?
Artinian – Noetherian rings and modules suggest study guide
Is there a formula similar to $f(x+a) = e^{a\frac{d}{dx}}f(x)$ to express $f(\alpha\cdot x)$?
0.246810121416…: Is it a algebraic number?
Number of Subspaces that contains other Space
Calculating circle radius from two points on circumference (for game movement)
From to

Let $S_{2m}$ be the group of all permutations $\pi$ of $\{1, 2, \ldots, 2m\}$. The following transition kernel $S$ generates the random transposition walk

$$

Ch(\pi, \pi’)= \begin{cases}

\frac{1}{2m} & \pi’=\pi\\[10pt]

\frac{2}{(2m)^2} & \pi’=\tau \pi\ \text{ for some transposition $\tau$}\\[10pt]

0, & \text{otherwise}

\end{cases}

$$

It is known that with symmetric probability measure $\mu$, the pair $(Ch, \mu)$ defines a reversible Markov chain.

Let $\tau=(I, J)$ be a random transposition, with $I, J$ chosen independentely and uniformly from $\{1, 2, \ldots, 2m\}$. Multiplication by $\tau$ results in taking a step in the chain defined by $Ch$.

All this structure is given in “The Concentration of Measure Phenomenon” by M. Ledoux.

- Time to reach a final state in a random dynamical system (answer known, proof unknown)
- Nice references on Markov chains/processes?
- Null-recurrence of a random walk
- Conditional return time of simple random walk
- Markov chain: join states in Transition Matrix
- Expected number of turns for a rook to move to top right-most corner?

Let $c=(c_1, c_2, \ldots, c_{2m})$ be a vector in $R^{2m}$. Define function $f:S_{2m}\longrightarrow R$ as $f(\pi):=|\sum_{k=0}^mc_{\pi(k)}-\sum_{k=m+1}^{2m}c_{\pi(k)}|$.

Question: Find the upper bound of $|f(\pi)-f(\tau \pi)|$.

Thank you for your help.

- Why is the number of possible subsequences $2^n$?
- Conjugacy classes in $A_n$.
- How does $(12\cdots n)$ and $(ab)$ generate $S_n$?
- Inequality involving rearrangement: $ \sum_{i=1}^n |x_i - y_{\sigma(i)}| \ge \sum_{i=1}^n |x_i - y_i|. $
- How many ways can four letters abcd be arranged such that a always comes before b and c always comes before d?
- Why are the periods of these permutations often 1560?
- Number of permutations which fixes a certain number of point
- Proving that any permutation in $S_n$ can be written as a product of disjoint cycles
- Composing permutations in factorial notation
- how to find the root of permutation

Consider $g(\pi)=\sum\limits_{k=0}^mc_{\pi(k)}-\sum\limits_{k=m+1}^{2m}c_{\pi(k)}$. Then $g(\tau\pi)=g(\pi)+2\cdot (c_{I}-c_{J})\cdot a_\pi(I,J)$, where $a_\pi(I,J)=+1$ if $J\leqslant m\lt I$, $a_\pi(I,J)=-1$ if $I\leqslant m\lt J$, and $a_\pi(I,J)=0$ otherwise.

Since $f=|g|$, this yields $|f(\pi)-f(\tau\pi)|\leqslant|g(\pi)-g(\tau\pi)|=2\cdot |c_{I}-c_{J}|\cdot [a_\pi(I,J)\ne0]$.

- Why should the substitution be injective when integrating by substitution?
- Determine the set $\{w:w=\exp(1/z), 0<|z|<r\}$
- Problem related polynomial ring over finite field of intergers
- A limit of a uniformly convergent sequence of smooth functions
- Harmonic Numbers series I
- A limit and a coordinate trigonometric transformation of the interior points of a square into the interior points of a triangle
- How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
- Center of the Orthogonal Group and Special Orthogonal Group
- Optimal assumptions for a theorem of differentiation under the integral sign
- How do we know that we found all solutions of a differential equation?
- Why is it called Sylvester's Law of Inertia?
- How are infinite-dimensional manifolds most commonly treated?
- Examples of famous problems resolved easily
- Why a tesselation of the plane by a convex polygon of 7 or more sides is not possible?
- How would I find a minimum weight spanning tree for W?