My background is in the area of mathematical physics, in particular,
quantum mechanics, classical (Hamiltonian) mechanics, and dynamical
systems (including fractals and chaos).
In mathematical terms this means analysis, functional analysis, and
differential geometry (mostly ordinary and partial differential
equations). I've conducted research on nonlinear wave equations, in
particular the dynamics of soliton solutions (solitons are localized
"waves" that behave like particles and play an important role in many
applications, eg., nonlinear optics, water waves). I use both
theoretical and numerical methods (i.e., simulations). I'm also
interested in applications of partial differential equations to image
processing, and wavelets to signal processing. For more details see my Research page.
I'm involved in industrial mathematics, having worked on designing
undergraduate programs in industrial mathematics and mathematical
modelling. (Industrial mathematics is mathematics applied to problems
arising in industry, including business, engineering, and computer
science. For a brief discription see the industrial math page.) In addition I have participated in several of the PIMS/MITACS industrial mathematics workshops. I am based at the Surrey campus
which has a program in Operations Research and Applied Statistics as
part of the Industrial Mathematics undergraduate program. Information
about industrial mathematics (and applied mathematics in general) can
be found on my Applied Math resources page.
- dynamical systems (including Hamiltonian mechanics, chaos and fractals)
- nonlinear wave equations (solitons)
- variational methods in image processing ( here's a few articles to get started)
- signal processing with wavelets
- PhD 1996, University of Toronto (Mathematics).
- Ideal Denoising for Signals in Sub-Gaussian Noise, Sebastian Ferrando and Randall Pyke, to appear in Applied Computational Harmonic Analysis, 2007. Article [17 pages]
- A Characterization of Bound States for Nonlinear Wave and Schrodinger Equations, preprint 2005. Article [13 pages]
- Analyzing Network Traffic for Malicious Hacker Activity, R. Pyke, S. Kim (Random Knowledge Inc), et. al. 8th Annual PIMS MITACS Industrial Mathematics Workshop, May, 2004, http://www.pims.math.ca/industrial/2004/ipsw/
- Path Planning for an Autonomous Robot, R. Pyke, et al., Proceedings of the 7th Annual Graduate Mathematical Modelling Camp, May, 2004, http://www.pims.math.ca/Publications_and_Videos/PIMS_Proceedings/Download_GIMMC_Proceedings/
- Stability of Solitons, R. Pyke. Lecture notes, Ryerson University (2002). Article [16 pages].
- Decline Analysis,
R. Pyke , R. Forth (Alberta Energy), et. al., Proceedings of the 5th
Annual PIMS Industrial Mathematics Workshop, June, 2001, http://www.pims.math.ca/publications/proceedings/
- Nonlinear Wave Equations: Constraints on Periods and Exponential Bounds for Periodic Solutions, R.M.Pyke and I.M. Sigal. Duke Math.J., Vol. 88, No. 1, 133-180 (1997).
Abstract. PDF article. [58 pages]
- Positive Commutator Methods for Nonlinear Wave Equations, R.M.Pyke and I.M. Sigal, in Partial Differential Equations and their Applications,
proceedings of the conference held at the University of Toronto, June
1995, edited by P. Greiner, V. Ivrii, L. Seco, and C. Sulem. American
Mathematical Society (1997).
Abstract. PDF article. [15 pages]
- Virial Relations for Nonlinear Wave Equations and Nonexistence of Almost-Periodic Solutions, R. Pyke. Review of Mathematical Physics, Vol. 8, No. 7, 1001-1039 (1996).
PDF article. [53 pages]
- Notes on Symmetries, Conservation Laws, and Virial Relations, R. Pyke. Lecture Notes, University of Toronto (1996).
Abstract. PDF article. [17 pages]
- Resonances, Stability, and Effective Stability in Hamiltonian Dynamical Systems, R. Pyke. University of Toronto Mathematics Preprint (1994).
Abstract. PDF article. [31 pages]