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APM Applied Mathematics Courses APM233Y1 The application of mathematical techniques to economic
analysis. Mathematical topics include linear and matrix algebra, partial differentiation,
optimization, Lagrange multipliers, differential equations. Economic applications include
consumer and producer theory, theory of markets, macroeconomic models, models of economic
growth. APM236H1 Introduction to linear programming including a rapid review
of linear algebra (row reduction, linear independence), the simplex method, the duality
theorem, complementary slackness, and the dual simplex method. A selection of the
following topics are covered: the revised simplex method, sensitivity analysis, integer
programming, the transportation algorithm. APM261H1 Formulation of problems in LP form, convexity and structure
of LP constraint sets, simplex algorithm, degeneracy, cycling and stalling, revised
method, two-phase method, duality, fundamental theorem, dual algorithm, integer
programming, sensitivity analysis, Karmarkar algorithm, network flows, transportation
algorithm, two-person zero-sum games. APM346H1 Sturm-Liouville problems, Green's functions, special
functions (Bessel, Legendre), partial differential equations of second order, separation
of variables, integral equations, Fourier transform, stationary phase method. APM351Y1 Diffusion and wave equations. Separation of variables.
Fourier series. Laplace's equation; Green's function. Schrodinger equations. Boundary
problems in plane and space. General eigenvalue problems; minimum principle for
eigenvalues. Distributions and Fourier transforms. Laplace transforms. Differential
equations of physics (electromagnetism, fluids, acoustic waves, scattering). Introduction
to nonlinear equations (shock waves, solitary waves). APM361H1 Topics selected from applied stochastic processes, queuing
theory, inventory models, scheduling theory and dynamic programming, decision methods,
simulation. A project based on a problem of current interest taken from course files or
the student's own experience is required. APM366H1 Convexity, fixed points, stable mappings optimization.
Relations orderings and utility functions; choice and decision making by individuals and
groups. Non-cooperative and cooperative games, core, Shapley value; market games. Decision
making by economic agents: consumers, producers, banks, investors, and financial
intermediaries. APM421H1 The general formulation of non-relativistic quantum mechanics
based on the theory of linear operators in a Hilbert space, self-adjoint operators,
spectral measures and the statistical interpretation of quantum mechanics; functions of
compatible observables. Schrodinger and Heisenberg pictures, complete sets of observables,
representations of the canonical commutative relations; essential self-adjointedness of
Schrodinger operators, density operators, elements of scattring theory. APM426H1 Einstein's theory of gravity. Special relativity and the
geometry of Lorentz manifolds. Gravity as a manisfestation of spacetime curvature.
Einstein's equations. Cosmological implications: big bang and inflationary universe.
Schwarzschild stars: bending of light and perihelion precession of Mercury. Topics from
black hole dynamics and gravity waves. APM436H1 Boltzmann, Euler and Navier-Stokes equations. Viscous and
non-viscous flow. Vorticity. Exact solutions. Boundary layers. Wave propragation. Analysis
of one dimensional gas flow. APM441H1 Asymptotic series. Asymptotic methods for integrals:
stationary phase and steepest descent. Regular perturbations for algebraic and
differential equations. Singular perturbation methods for ordinary differential equations:
W.K.B., strained co-ordinates, matched asymptotics, multiple scales. (Emphasizes
techniques; problems drawn from physics and engineering) APM446H1 Nonlinear partial differential equations and their physical
origin. Fourier transform; Green's function; variational methods; symmetries and
conservation laws. Special solutions (steady states, solitary waves, travelling waves,
self-similar solutions). Calculus of maps; bifurcations; stability, dynamics near
equilibrium. Propogation of nonlinear waves; dispersion, modulation, optical bistability.
Global behaviour solutions; asymptotics and blow-up. APM456H1 Differential systems with controls and reachable sets.
Non-commutativity, Lie bracket and controllability. Optimality and maximum principle.
Hamiltonian formalism and symplectic geometry. Integrability. Applications to engineering,
mechanics and geometry. APM461H1 A selection of topics from such areas as graph theory,
combinatorial algorithms, enumeration, construction of combinatorial identities. APM466H1 Introduction to the basic mathematical techniques in pricing
theory and risk management: Stochastic calculus, single-period finance, financial
derivatives (tree-approximation and Black-Scholes model for equity derivatives, American
derivatives, numerical methods, lattice models for interest-rate derivatives), value at
risk, credit risk, portfolio theory. APM496H1/497H1/498Y1/499Y1 TBA NOTE: In Prerequisites, Calc = Ontario Academic Course Calculus; A&G = Ontario Academic Course Algebra and Geometry; FM = Ontario Academic Course Finite Mathematics |
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