Could anyone suggest me some list of references to understand the convergence rate of Markov chain on Metric space( Preferably on Complete and Separable) which moves as follows: $$ X_{n+1}= f_{\omega_n}(X_n)$$ where $f_1,\dots, f_s$ are measurable functions on $\mathcal X$, a metric space with ...

This is a problem out of the text Introduction to the Theory of Numbers by Niven, Zuckerman, and Montogmery and I am having quite a bit of trouble with it. I tried to prove it directly, but that didn't work. The question is Show that $\binom{p^{\alpha}-1}k \equiv (-1)^k \pmod p$ for $ 0 \leq k ...

Suppose we have a system of equations Ax = b. Let $K_n$ be a kylov matrix i.e. $K_n= [\array{ b & Ab & A^2b &... &A^nb ] }$. Applying QR decomposition to $K_n$, we obtain $K_n=Q_n R_n$ and $ H_n = Q_n^* AQ_n$ where Hn is the projection of A to Kn. $H_n$ is a Hessenberg matrix. The book I a...