{"id":68,"date":"2013-10-06T04:44:41","date_gmt":"2013-10-06T04:44:41","guid":{"rendered":"http:\/\/sites.ulethbridge.ca\/roussel\/?p=68"},"modified":"2019-08-20T11:50:59","modified_gmt":"2019-08-20T17:50:59","slug":"the-big-five-oh-part-1","status":"publish","type":"post","link":"https:\/\/sites.ulethbridge.ca\/roussel\/2013\/10\/06\/the-big-five-oh-part-1\/","title":{"rendered":"The big five-oh, Part 1"},"content":{"rendered":"

No, I haven’t turned 50 yet. However, my 50th refereed paper<\/a> has now appeared in print. This therefore seems like an appropriate time to look back on some of the research I have done since I started out as a graduate student at the University of Toronto. (I had two prior papers from my undergraduate work, but these were both in areas of science I didn’t pursue.) My intention here isn’t to write a scholarly review paper, so you won’t find a detailed set of citations here. My full list of publications<\/a> is, in any event, available on my web site.<\/p>\n

My M.Sc. and Ph.D. theses were both on the application of invariant manifold theory to steady-state kinetics. I was introduced to these problems by my supervisor at the University of Toronto, Simon J. Fraser. Simon is a great person to work for. He is supportive, and full of ideas, but he also lets you pursue your own ideas. I had a great time working for him, and learned an awful lot of nonlinear dynamics from him.<\/p>\n

One way to think of the evolution of a chemical system is as a motion in a phase space, typically a space whose axes are the concentrations of the various chemical species involved, but sometimes including other relevant variables like temperature. The \u00a0phase space of a chemical system is typically very high-dimensional. The reactions that transform one species into another occur on many different time scales. The net result is that we can picture the motion in phase space as involving a hierarchy of collapse processes onto surfaces of lower and lower dimension, the fastest processes being responsible for the first collapse events, followed by slower and slower processes.1<\/sup> These surfaces are invariant manifolds of the differential equations, and we developed methods to compute them. Given the equation of a low-dimensional manifold, we obtain a reduced model of the motion in phase space, i.e. one involving the few variables necessary to describe motion on this manifold.<\/p>\n

Invariant manifold theory has been a fertile area of research for me over the years. I continue to publish in this area from time to time. In fact, one of my current M.Sc. students, Blessing Okeke, is working on a set of problems in this area. Expect more work on these problems in the future!<\/p>\n

Toward the end of my time in Toronto, my supervisor, Simon J. Fraser, allowed me to spend some time working with Carmay Lim<\/a> who, at the time, was cross-appointed to several departments at the University of Toronto, and worked out of the Medical Sciences Building. This was a very productive time, and I learned a lot from Carmay, particularly about doing research efficiently.<\/p>\n

We worked on a set of applied problems on the lignification of wood using an interesting piece of hardware called a cellular automata machine<\/a>. This was a special-purpose computer built to efficiently simulate two-dimensional cellular automata. The machine was programmed in Forth<\/a>, a programming language most of you have probably never heard of, with some bits written in assembly language for extra efficiency. For a geek like me, programming this machine was great fun. I think we did some useful work, too, as our work on lignification kinetics still gets cited from time to time.<\/p>\n

I had been to the 1992 SIAM Conference on Applications of Dynamical Systems in Snowbird which, I think, was just the second of what would become a long-lived series of conferences. There, I had discovered that there was a lot of interest in delay-differential equations (DDEs), as the tools necessary to analyze these equations were being sharpened. I had thought about the possibility of applying DDEs to chemical modelling, and decided to apply to work with Michael Mackey<\/a> at McGill University, who was an expert on the application of DDEs in biological modelling. McGill was a great environment, and I learned a lot from Michael and his students. The most significant outcome of my time in Montreal was a paper published in the Journal of Physical Chemistry<\/em> on the use of DDEs in chemical modelling.2<\/sup><\/p>\n

I pursued this style of modelling in a handful of papers. Eventually, I got interested in the use of delays to simplify models that can’t be described by differential equations, namely stochastic systems.3<\/sup> This is another one of those ideas that I have kept following down through the years.<\/p>\n

In my next blog post, I will reflect on some of the work I have done since arriving in Lethbridge.<\/p>\n

1<\/sup>Marc R. Roussel and Simon J. Fraser (1991) On the geometry of transient relaxation. J. Chem. Phys.<\/em> 94<\/b>, 7106\u20137113.
\n2<\/sup>Marc R. Roussel (1996) The use of delay-differential equations in chemical kinetics. J. Phys. Chem.<\/em> 100<\/b>, 8323\u20138330.
\n3<\/sup>Marc R. Roussel and Rui Zhu (2006) Validation of an algorithm for delay stochastic simulation of transcription and translation in prokaryotic gene expression. Phys. Biol.<\/em> 3<\/b>, 274\u2013284.<\/p>\n","protected":false},"excerpt":{"rendered":"

No, I haven’t turned 50 yet. However, my 50th refereed paper has now appeared in print. This therefore seems like an appropriate time to look back on some of the research I have done since I started out as a graduate student at the University of Toronto. (I had two prior papers from my undergraduate […]<\/p>\n","protected":false},"author":66,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"categories":[7],"tags":[],"class_list":["post-68","post","type-post","status-publish","format-standard","hentry","category-scientific-miscellanea"],"_links":{"self":[{"href":"https:\/\/sites.ulethbridge.ca\/roussel\/wp-json\/wp\/v2\/posts\/68","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.ulethbridge.ca\/roussel\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.ulethbridge.ca\/roussel\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.ulethbridge.ca\/roussel\/wp-json\/wp\/v2\/users\/66"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.ulethbridge.ca\/roussel\/wp-json\/wp\/v2\/comments?post=68"}],"version-history":[{"count":4,"href":"https:\/\/sites.ulethbridge.ca\/roussel\/wp-json\/wp\/v2\/posts\/68\/revisions"}],"predecessor-version":[{"id":72,"href":"https:\/\/sites.ulethbridge.ca\/roussel\/wp-json\/wp\/v2\/posts\/68\/revisions\/72"}],"wp:attachment":[{"href":"https:\/\/sites.ulethbridge.ca\/roussel\/wp-json\/wp\/v2\/media?parent=68"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.ulethbridge.ca\/roussel\/wp-json\/wp\/v2\/categories?post=68"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.ulethbridge.ca\/roussel\/wp-json\/wp\/v2\/tags?post=68"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}