# Online Mathematics Materials

In my spare time I've been developing some online educational materials for mathematics. Some of these may eventually become part of the Felynx Cougati library of multimedia mathematics materials, an ambitious project which I am peripherally involved in. Others were developed for courses I was lecturing, or as personal projects.

• Linear Partial Differential Equations and Fourier Theory (PDF; 620 pages). This is the draft manuscript for a book which was published by Cambridge University Press in 2010.

• The mathematics of voting and elections: Paradox, deception, and chaos (PDF slides for a two-hour presentation).

Contents: Various voting methods (Plurality, antiplurality, Borda count, instant runoff, etc.) and their problems and pathologies. Condorcet cycles. Arrow's impossibility theorem. Nonordinal voting systems (Approval voting, cumulative voting, relative utilitarianism) and their problems. Strategic voting and the Gibbard-Satterthwaite theorem. Representative democracy and compound-majority paradoxes (Ostrogorski, Anscombe). Voting power indices. Liberalism vs. Populism. Social choice and social welfare.

[There is also a version with a supplement, ``What is mathematics?'', which I wrote for highschool students.]

• Walrasian Equilibrium Theory: The foundation of modern mathematical microeconomics (PDF slides for a two-hour presentation).

Contents: Classical microeconomics: the supply/demand `X' graph and its shortcomings. Complements vs. substitutes; production factors; perverse supply curves; the Law of One Price.
Pure Exchange Economies: consumers (utility functions, utility maximization through trade), aggregate excess demand, Walrasian Equilibria. Arrow-Debreu existence theorem and proof.
Production Economies: firms (production technology, profit maximization), shareholders, aggregate excess demand, Walrasian Equilibria. Arrow-Debreu existence theorem and proof sketch.
Problems with Walrasian paradigm: multiple equilibria; monopolistic competition; Sonnenschein-Mantel-Debreu Theorem (`anything goes') and proof. The prevalance of disequilibrium.
Disequilibrium dynamics: Samuelson's tatonnement and shortcomings; Hahn-Negishi dynamics; Fisher price dynamics.

• Voting, Arbitration, and Fair Division: the mathematics of social choice (133 pages)

Contents: Survey of voting procedures and their shortcomings. Sen's Impossibility Theorem, Arrow's Impossibility Theorem, and the Gibbard-Satterthwaite Impossibility Theorem. Binary voting systems; May's theorem. Weighted and Vector-weighted systems. Bentham's utilitarianism; von Neumann-Morgenstern definition of cardinal utility. Bargaining Games and the Nash Arbitration Scheme. Fair division theory. [Gzipped Postscript]................ [Adobe PDF]

• Classification of Cellular Automata Some lecture notes I've prepared for an informal seminar on cellular automata I delivered at the University of Houston in spring of 2002. [Gzipped PostScript Version]...... [Adobe PDF Version]

• Visual Abstract Algebra(240 pages; working draft) An introduction to groups, rings, and fields.

• Analysis, Measure & Probability: A visual Introduction (163 pages; working draft) My attempt to develop a new pedagogical approach to real analysis, with emphasis on geometric intuition and probabilistic interpretations. [[Postscript]........[PDF]]

• Models of Philosophy (121 pages) A short monograph on how mathematical methods could be profitably employed to address certain problems in philosophy of mind, philosophy of language, philosophy of science, and political philosophy.

• Lecture notes in Linear Algebra I wrote these for Math 223, a course I taught at the University of Toronto in spring of 2000.

• Ergodic Theory, Stochastic Processes, and Information Theory (30 pages) Slides from a two hour seminar I delivered in 1999 at the University of Toronto, intended to be an introduction for nonspecialists. It is (I hope) accessible to any mathematically literate person. [Gzipped PostScript Version]...... [Adobe PDF Version]