Here is a really quick derivation of the Cauchy-Riemann equations of complex analysis. Consider a function of a complex variable, , where , such that: , where and are real-valued functions. An analytic function is one that is expressible as a power series in . That is, . Then, . We formally differentiate this equation […]

# Category: Physics

I’ve been fascinated by the triangle offense for a long time. I think it is a beautiful way to play basketball, and the right way to play basketball, in the half-court, a “system-based” way to play. For those of you that are interested, I highly recommend Tex Winter’s classic book on the topic. There is […]

My new paper has now been published in Annalen der Physik, which is a great honour, because 100 years ago, Einstein’s General Theory of Relativity was also published in the same journal. This paper describes a method by which one is able to determine whether a given spatially flat cosmological model produces finite-time singularities, and […]

What goes into making a cosmological model? Here is a presentation (that was part of my Ph.D. dissertation) that I have reproduced and embedded here to describe what actually goes into the making of a cosmological model. After describing some general properties, I describe specifically a early-universe model that contains a viscous fluid and a […]

A recent lecture and a series of interviews has been posted online where cosmologist George F.R. Ellis discusses the issue of fine-tuning in biology at considerable length and in considerable detail. Of course, the larger theme here is that to discuss and understand things like Darwinian evolution properly, one needs to have an understanding of […]

A big part of my research involves dynamical systems theory. A lot of people don’t know what this is, at least, they don’t have a very good idea. It has not helped that the vast majority of Canadian university physics programs have deemphasized classical mechanics and differential equations, but that is an another story! Anyways, […]

Since I expect the concept of gravitational waves to once become very popular in the next few days, I wrote some quick notes on them, I.e., where they come from. They are handwritten, as I didn’t have time to LaTeX them, but, hopefully, they’ll be useful to interested readers! Also note that, gravitational waves are […]

One of my earliest works was deriving equations which themselves were forms of Einstein’s field equations that described the state of the early universe, which may have had dominant viscous effects. I was delighted to learn that these equations were published in Springer’s Handbook of Spacetime Cosmology textbook. Here is a snapshot of the textbook […]

Our new paper was accepted for publication in Physical Review D. The goal of the paper was to calculate the probability that a multiverse could emerge from a more general background spacetime, in this case, Bianchi Type I coupled to a chaotic inflaton potential. Basically, we found that a multiverse being generated from such a […]