Markov Chains and Monte Carlo Algorithms

1. General state space Markov chains Most applications of Markov Chain Monte Carlo algorithms (MCMC) are concerned with continuous random variables, i.e the corresponding Markov chain has a continuous state space S. In this section we will give a brief overview of the theory underlying Markov chains with general state spaces. Although the basic principles…… Continue reading Markov Chains and Monte Carlo Algorithms

Transfinite Induction

\maketitle 1. Introduction This is supposed to be a primer inspired by a piece by Hilbert Levitz. The theory of transfinite ordinals is a part of set theory. While the concept is tied up with the completed infinite and high cardinalities, we’ll emphasize more constructive aspects of the theory. There have been applicatons of constructive…… Continue reading Transfinite Induction

Information Retrieval

Attention conservation notice: 680 words about Information Retrieval, and highly unoriginal. The following is very much inspired by a course by Cosma Shalizi but I felt it was worth rewriting to get to grips with the concepts. This is the first of what is hopefully a series of posts on ‘Information Retrieval’, and applications of…… Continue reading Information Retrieval

Examples of PDE

1. PDE Exercises $latex f: \mathbb{R}^n \Rightarrow \mathbb{C} $ given $latex {\tilde{M}f}&fg=000000$ is a solution of the homogeneous Cauchy problem, with $latex {\tilde{M}f(0,x) = 0}&fg=000000$, $latex {\frac{\partial \tilde{M}f}{\partial t}(0,x) = f(x)}&fg=000000$ Let $latex {g:[0,\infty) \times \mathbb{R}^n \Rightarrow \mathbb{C}}&fg=000000$ be sufficiently often continuously differentiable. Show without using any explicit formular for $latex {\tilde{M}f}&fg=000000$ that the function…… Continue reading Examples of PDE