My most recent talk was on the use of a pair of unwinding numbers in order to obtain continuous expressions of symbolic integrals of rational functions of a real variable. This works by unwinding paths of integration that though they explicitly traverse the real line, they may implicitly cross branch cuts or go to infinity.
Consider an integral of the form
\[ \int_a^x\frac{p(t)}{q(t)}dt=g(x), \]
where \(p(x)\) and \(q(x)\) are real ordinary or trigonometric polynomials and \(g(x)\) is some rational (ordinary or trigonometric) function of \(x\) plus a sum of logarithms. Provided that \(q(x)\) has no real zeros, the integrand is continuous and bounded on the entire real line and so \(g(x)\) is also. In fairly generic cases, however, the expression for \(g(x)\) returned by some computer algebra systems, such as Maple, is discontinuous because the argument becomes singular.
We show that with an unwinding number \(\mathcal{K}_{\theta}\) that counts logarithmic branch crossings and another unwinding number \(\mathcal{K}_{r}\) that counts odd order pole passes through infinity on the Riemann sphere, the expression can be made continuous in a conceptually clear and computationally inexpensive way.
The following is a list of the talks that I have given in both mathematics and philosophy.
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Conference Presentations:
- (2013) "Feasible Epistemology: Contributions of Applied Mathematics to Scientific Inference. PGSA Colloquium Series, Department of Philosophy, The University of Western Ontario, 16 October.
- (2013) “Rational Discovery of the Natural World: An Algebro-Geometric Response to Steiner” MathFest 2013/Canadian Society for the History and Philosophy of Mathematics (MathFest 2013/CSHPM) Conference, Hartford CT, 1-3 August.
- (2011) with Corless, RM. “Computation for Confirmation.” Conference on The Plurality of Numerical Methods and their Philosophical Analysis. Université Paris—Panthéon-Sorbonne, Paris, 3-4 November.
- (2010) with Fillion, N. “Computation in Scientific Explanation,” in the course Contemporary Philosophy of Science, Department of Philosophy, University of Guelph, 17 November.
- (2010) with Fillion, N. “Explanation and Abstraction: The Case of Backward Error Analysis” Philosophy of Science Association Biennial Meeting, Montreal, Quebec, 4-6 November.
- (2010) with Fillion, N. “Modeling and Explanation: Lessons from Modern Error Theory.” Canadian Society for the History and Philosophy of Science (CSHPS) Conference, Concordia University, Montreal, Quebec, 28-30 May.
- (2010) with Fillion, N. “A Step Forward with Backward Error,” PGSA Colloquium Series, Department of Philosophy, The University of Western Ontario, 12 March.
- (2009) “The Conversion of Phenomena to Theory: Lessons on Applicability from the Development of Electromagnetism.” Canadian Mathematical Society/Canadian Society for the History and Philosophy of Mathematics (CMS/CSHPM) Conference, Memorial University, St. John’s, Newfoundland, 6-8 June.
- (2009) “From the World to Mathematics and Back Again: What We Can Understand About Applicability from the Development of Electromagnetism.” PGSA Collo- quium Series, Department of Philosophy, The University of Western Ontario, 25 March.
- (2008) “Theories, Models and Representation: Lessons from Solid State Physics.” Canadian Society for the History and Philosophy of Science (CSHPS) Conference, University of British Columbia, Vancouver, British Columbia, 3-5 June.
- (2008) “Theories, Models and Representation: Lessons from Solid State Physics.” PGSA Colloquium Series, Department of Philosophy, University of Western Ontario, 12 March.
- (2005) “Interpretations of Probability in Quantum Mechanics.” PGSA Conference, Department of Philosophy, University of Waterloo, June.
Posters
(2008) "Theories, Models and Representation: Lessons from Solid State Physics.” Western Research Day, 28 March, and Arts and Humanities Research Day, 2 April.
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Conference Presentations:
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(2014) with Corless, RM, Jeffrey, DJ. "Unwinding Paths on the Riemann Sphere for Continuous Integrals of Rational Functions." 20th Conference on Applications of Computer Algebra, Fordham University, New York City, 9-12 July.
- (2014) with Corless, RM, Jeffrey, DJ. "Unwinding Paths on the Riemann Sphere for Continuous Integrals of Rational Functions." Encuentro de Álgebra Computacional y Aplicaciones XIV (EACA--Meeting on Computer Algebra and Applications), Institut d'Estudis Catalans, Barcelona, 18-20 June.
Posters
- (2011) "Dynamics Backward: Backward Error Analysis for Ordinary Differential Equations." Epistemology of Modeling & Simulation National Conference, Pittsburgh, 1-3 April.
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