A brief introduction to analytic geometry. The differential and integral calculus for algebraic, logarithmic, exponential and trigonometric functions, with applications to the natural and social sciences.

Transcendental functions; integration procedures and applications; polar coordinates; parametric equations; indeterminate forms. Introduction to partial derivatives and their geometric interpretation.

The application of non-calculus techniques in modelling "real world" problems in business, psychology, sociology, political science and ecology. The mathematical topics introduced include graphs and directed graphs, linear programming, matrices, probability, games and decisions, and difference equations. Mathematics majors may not take this course for credit.

The elements of the calculus of one variable with illustration and emphasis on its application in the biological sciences. The elementary functions, sequences and series, difference equations, differential and integral calculus.

This is a theoretical course intended primarily for mathematics, engineering and science students. Topics to be included are: inequalities and absolute values; limits and continuity using rigorous definitions the derivative and various applications, Rolle's theorem and the mean-value theorem for derivatives; the differential and antidifferentiation; the definite integral and various applications, the mean-value theorem for integrals, the fundamental theorem of calculus; logarithmic, exponential and elementary trigonometric functions.

Topics to be included are: trigonometric and hyperbolic functions; techniques of integration; polar co-ordinates; parametric equations; indeterminate forms and improper integrals; elementary geometry of surfaces; a brief introduction to partial derivatives and multiple integrals.

The Algebra of sets. Equivalence Relations, Mappings and Inverse Mappings. Review of the Real Number System. Countable and Uncountable Sets. Partially and Totally Ordered Sets. Complex Numbers and their arithmetic. Geometry and Topology of the line and the plane. Emphasis is placed on developing skills in constructing mathematical proofs.

Techniques of integration, introduction to differential equations and the elements of multivariate calculus. Illustrations and emphasis will be on biological applications. An introduction to vectors, multivariable and vector functions, difference equations, partial differentiation and multiple integration.

This course provides an overview of and practical experience in utilizing algorithms for solving numerical problems arising in applied sciences. Topics covered will include solution of a single nonlinear equation, interpolation, numerical differentiation and integration, solution of differential equations and systems of linear algebraic equations. A computer will be utilized in solving problem assignments.

Matrices and matrix operations, matrix inverses, determinants, linear equations. N-dimensional vectors: dot product, linear independence, basis and dimension. Rank of a matrix. Eigenvalues, eigenvectors and diagonalization. Applications, including least squares.

Recommended: OAC Algebra and Geometry.

Matrix Notation, Matrix Arithmetic, Matrix Inverse and Determinant, Linear Systems of Equations, and Gaussian Elimination. The Basis Theory of Vector Spaces and Linear Transformations. Matrix Representations of Linear Transformations, Change of Basis, Diagonalization. Inner Product Spaces, Quadratic Forms, and Linear Differential Operators.

First order equations, linear equations of second and higher orders, Phase Plane, Difference Equations, introduction to power series methods, Laplace transforms, formulation, solution and interpretation of differential equations of interest in science.:a

Infinite sequences and series, tests for convergence, Taylor's series. Planes and quadric surfaces. Partial differentiation, directional derivatives and gradients, maxima and minima problems, Line integrals. Multiple integrals and coordinate transformations.

Implicit differentiation and implicit function theorems for systems of equations. Mappings and coordinate transformations. Vector and scalar fields including the gradient, divergence, curl and directional derivative, and their physical interpretation. Multiple Integrals. Theorems of Green and Stokes. An introduction to the theory of Riemann integration.

Solution of differential equations which arise from problems in engineering. Linear equations of first and higher order; systems of linear equations; Laplace transforms: introduction to nonlinear equations and partial differential equations.

First order Linear Systems and their general solution by Matrix methods. Introduction to Nonlinear Systems, Stability, Limited Cycles and Chaos using numerical examples. Solution in power series of second order equations including Bessel's Equation. Introduction to Partial Differential Equations and Applications.

Symmetric Groups; the Affine Group of the Read Line; introduction to group theory; groups, subgroups, normal subgroups, factor groups, fundamental homomorphism theorem. Introduction to ring theory; rings, subrings, ideals, quotient rings, polynomial rings, Euclidean algorithm, fundamental ring homomorphism theorem. Introduction to Finite fields with Applications.

Complex vector spaces. Direct sum decompositions, Cayley-Hamilton theorem, spectral theorem for normal operators, Jordan canonical form of a matrix.

Wave equation, heat equation, Laplace equation, linearity and separation of variables; solution by Fourier series; cylindrical and spherical geometires; Bessel and Legendre functions; Fourier transforms.

Metric spaces and Normed Linear Spaces. Fixed Point Theorems and Applications. Sequences and Series of Functions. Riemann-Stieltjes integration.

Mathematical models, Introduction to linear and non-linear programming, the simplex method, convexity, Kuhn-Tucker condition. Game theory, decision analysis, and network analysis. Gueuing theory, birth and death processes.

The complex derivative and planar mappings. Analytic and harmonic functions. Conformal mappings. Elementary functions. Conformal mappings. Elementary functions. Cauchy-Goursat theorem. The Taylor and Laurent series. Calculus of residues with emphasis on applications.

Development, analysis, and interpretation of mathematical models of biological phenomena. Emphasis will be on deterministic discrete and continuous models.

A rigorous treatment of the qualitative theory of ordinary differential equations and an introduction to the modern theory of dynamical systems, existence, uniqueness, and continuity theorems. Definition and properties of dynamical systems. Linearization and local behaviour of nonlinear systems. Stable Manifold theorem. Liapunov stability. Limited cycles and Poincare-Bendixson Theorem. Introduction to bifurcations and chaotic dynamics.

Discussion of selected topics at an advanced level. Intended mainly for mathematics students in the 6th to 8th semester. Content will vary from year to year. Sample topics: probability theory, Fourier analysis, mathematical logic, operator algebras, number theory combinatorics, philosophy of mathematics, fractal geometry, chaos, Stochastic differential Equations. (Offered in odd-numbered years.)

Discussion of selected topics at an advanced level as in 63-405, but with different choice of topic.

Study of selected topics in applied mathematics at an advanced level, intended mainly for mathematical science students in the 7th or 8th semester. Sample topics are optimal control theory, nonlinear programming, and dynamical systems theory. The course will include case studies of real-world problems arising from various areas and the contribution of mathematical models to their solution. Part of the course requirement will involve the completion of a mathematical modelling project in conjunction with the departmental Mathematics and Statistics Clinic. For further information concerning the Clinic, consult the department. (Offered in even-numbered years.)

Finite symmetric groups, dihedral and cyclic groups with applications to the group of symmetries of a geometric figure in the plane. Polya-Burnside method of enumeration with applications. Galois fields with applications to combinatorial design constructions. Error correcting binary codes. (Offered in even-numbered years.)

Stone-Weierstrass approximation theorem. compactness in function spaces. Introduction to complex dynamics and the Mandelbrot set. Multivariate differential calculus.

Hilbert and Banach spaces: applications to Fourier series and numerical analysis. Hahn-Banach theorem; weak topologies. Generalized functions; application to differential equations. Completeness; uniform boundedness principle. Lebesque measure and integral; applications to probability and dynamics. Spectral theory. (Offered in even-numbered years.)

A series of modules on advanced topics in mathematical modelling techniques. Emphasis will be placed on the efficient use of computer resources and appropriate software, and the preparation of a written and an oral report. The module topics may vary from year to year, but examples include nonlinear regression, neural networks, pattern recognition, fuzzy set theory, and optimal control theory. Application areas will include the physical and biological science, engineering and environmetrics.

Theory of 1st and 2nd order partial differential equations with examples. Theory and examples of associated boundary value problems.

Classical geometry of the plane and 3-space. Non-Euclidean geometics. Elementary topology of graphs and surfaces. Topics to be selected from: Algebraic geometry; Analysis on manifolds; Reimannian geometry; Tensor analysis; Homotopy and homology groups. (Offered in odd-numbered years.)

Numerical solution of linear systems, differential equations; the algebraic eigenvalue problem, interpolation and approximation of functions, numerical quadrature.

Mathematical modelling of environmental transport systems. Linear and nonlinear compartmental models. Convective and diffusive transport. Specific models selected from Hydrology; ground-water and aquifer transport, dispersion of marine pollution, effluents in river systems; Atmospheric: pollen dispersion, plume models, dry matter suspension and deposition; Global circulation: tritium distribution. (Offered in odd-numbered years.)