In this post, we will use the notion of categories and functors to make rigorous the relationship between topological spaces and their underlying fundamental and homology groups. This post follows material presented in Hatcher's "Algebraic Topology" p.162. Categories A category $~\mathcal{C}$ consists of: A collection of objects, denoted…

In the context of finite products, the box topology and product topology coincide. As the following exposition shows, this is not the case for infinite products. …

Continuing from our last primer about A* search, this post assumes knowledge of the A* algorithm. Here, we demonstrate an application of A* search to a combinatorial optimization problem and investigate the performance gains from using a more accurate heuristic. The String Character Exchange Distance Problem I'm not sure if…

This post introduces the notion of state-space search using the A* algorithm. The terminology and groundwork for general search algorithms is described. Additionally, the heuristic function as well as the properties of admissibility and consistency are defined and discussed. The search problem We will describe a minimal search framework formulated…

Mergesort is a divide and conquer sorting algorithm which can be implemented to be in place and stable. It can be shown that the number of compares in mergesort for a dataset of size N is $\approx N \log N + c N, c \in \mathbb{R}$. I couldn't sleep and…

Upcoming Posts about Diffusion Diffusion is an interesting transport process where data analysis draws from probability theory. The results are really quite intuitive. In this post, I will explain the relevant theory and results to accompany an upcoming tutorial about using ImageJ and MATLAB for analyzing particle tracking data. Theory…