# The Faculty of Engineering and Applied Science Calendar2011-2012 Academic Year

Mathematics & Engineering Courses
 MTHE 217 Algebraic Structures with Applications F 3-0-.5 3.5
The purpose of the course is to provide an introduction to abstract algebraic systems, and to illustrate the concepts with applications to communication engineering. Topics covered are: symbolic logic, switching and logic circuits; set theory and mappings; equivalence relations; the integers; introduction to Boolean algebras; group theory, groups, subgroups, cyclic groups, cosets and Lagrange's theorem, quotient groups, homomorphisms and isomorphisms; applications to error-control codes, binary block codes for noisy communication channels, nearest neighbor decoding, code error detection/correction capabilities, group (linear) codes, coset decoding, generator and parity check matrices, syndrome decoding; basic properties of rings and fields. (30/0/0/12/0) PREREQUISITE: APSC 174
 MTHE 224 Applied Mathematics for Civil Engineers F 3-.5-1 4.5

The course will discuss the application of linear differential equations with constant coefficients, and systems of linear equations within the realm of civil engineering. Additionally, the course will explore relevant data analysis techniques including: graphical and statistical analysis and presentation of experimental data, random sampling, estimation using confidence intervals, linear regression, residuals and correlation. (54/0/0/0/0)

PREREQUISITES: APSC 142, APSC 172, APSC 174
EXCLUSIONS: MTHE 225 (MATH 225), MTHE 226 (MATH 226), MTHE 235 (MATH 235), MTHE 237 (MATH 237), MTHE 267 (STAT 267), MTHE 367 (STAT 367)
 MTHE 225 Ordinary Differential Equations F 3-0-.5 3.5
First order differential equations, linear differential equations with constant coefficients, and applications, Laplace transforms, systems of linear equations.
(42/0/0/0/0)
PREREQUISITES: APSC 171, APSC 172, APSC 174
 MATH 226 Ordinary Differential Equations
First order differential equations, linear differential equations with constant coefficients, and applications, Laplace transforms, systems of linear equations. (36/0/0/0/0)
~ COURSE DELETED IN 2008/09 ~
PREREQUISITES: APSC 171, APSC 172, APSC 174
 MTHE 227 Vector Analysis F 3-0-0 3
Review of multiple integrals. Differentiation and integration of vectors; line, surface and volume integrals; gradient, divergence and curl; conservative fields and potential. Spherical and cylindrical coordinates, solid angle. Green's and Stokes' theorems, the divergence theorem. (36/0/0/0/0) PREREQUISITES: APSC 171, APSC 172, APSC 174
 MTHE 228 Complex Analysis W 3-0-.5 3.5
Complex arithmetic, complex plane. Differentiation, analytic functions. Elementary functions. Elementary functions. Contour integration, Cauchy's Theorem and Integral Formula. Taylor and Laurent series, residues with applications to evaluation of integrals. (42/0/0/0/0) PREREQUISITES: APSC 171, APSC 172, APSC 174
 MTHE 232 Differential Equations W 3-0-0 3

Introduction to ordinary differential equations and their applications to the physical and social sciences. Topics may include: numerical solutions, power series and series solutions, Laplace transforms. (36/0/0/0/0)

PREREQUISITES: APSC 171, APSC 172, APSC 174
EXCLUSIONS: MATH 225, MATH 226, MATH 231, MATH 235, MATH 237
 MTHE 235 Differential Equations for Electrical and Computer Engineers F 3-0-0 3
First order differential equations, linear differential equations with constant coefficients. Laplace transforms. Systems of linear differential equations. Examples involving the use of differential equations in solving circuits will be presented. (27/0/0/9/0)
PREREQUISITES: APSC 171, APSC 172, APSC 174
 MTHE 237 Differential Equations and Computer Methods F 3-.5-0 3.5
Topics include models for dynamical systems, classification of differential equations, methods for solving differential equations, variation of parameters, systems of differential equations and connections with Linear Algebra, Laplace Transforms, numerical methods for solutions, stability of dynamical systems, non-linear equations and Lyapunov's method. Computer methods (symbolic and numerical) are introduced in the laboratory component of the course.  (18/11/0/13/0) PREREQUISITES: APSC 171, APSC 172, APSC 174
EXCLUSIONS: MATH 231, MTHE 232 (MATH 232)
 MATH 239 Applied Mathematical Modeling
A survey of important mathematical techniques used to model processes in a variety of fields. Topics include multivariable calculus and optimization, game theory, discrete-time dynamical systems, and dynamic optimization. Examples will be drawn from several areas including biology, economics, and medicine.  (18/9/5/4/0)
~ COURSE DELETED IN 2008/09 ~
PREREQUISITES: APSC 172 or MATH 120 or MATH 121 or MATH 126, APSC 174 or MATH 110 or MATH 111 recommended
 MTHE 272 Application of Numerical Methods W 3-.5-0 3.5
An introductory course on the effective use of computers in science and engineering. Topics include: solving linear and nonlinear equations, interpolation, integration, and numerical solution of ordinary differential equations. Extensive use is made of MATLAB, a high level interactive numerical package  (20/0/0/11/11) PREREQUISITES: APSC 174 or equivalent (Note:  some programming experience is important for the course)  Must be registered in BSCE
COREQUISITES: MTHE 225 or MATH 225 or MTHE 235 or MATH 235 or MTHE 232 or MATH 232 or equivalent
 MTHE 280 Advanced Calculus F 3-0-.5 3.5
Limits, Continuity, C', and linear approximations of functions of several variables. Multiple integrals and Jacobians, Line and surface integrals. The theorems of Green, Stokes, and Gauss. (42/0/0/0/0) PREREQUISITES: APSC 172, APSC 174
EXCLUSIONS: MATH 221, MTHE 227 (MATH 227)
 MTHE 281 Introduction to Real Analysis W 3-0-.5 3.5
Taylor's theorem, optimization, implicit and inverse function theorems. Elementary topology of Euclidean spaces. Sequences and series of numbers and functions. Pointwise and uniform convergence. Power series. (42/0/0/0/0) PREREQUISITES: APSC 172
 MTHE 312 Linear Algebra W 3-0-.5 3.5
Vector spaces, linear transformations and matrices. Linear equations. Determinants. Eigenvalues and eigenvectors. Normal forms. Linear functions and dual spaces. Bilinear functions, quadratic and hermitian forms. Inner product spaces, the projection theorem and applications to approximation and optimization problems. (42/0/0/0/0) PREREQUISITES: MTHE 217 (MATH 217) or permission of the instructor
 MTHE 326 Functions of a Complex Variable F 3-0-0 3
Complex numbers, analytic functions, harmonic functions. Cauchy's theorem. Taylor and Laurent series. Calculus of residues. Rouche's theorem. (36/0/0/0/0) PREREQUISITES: MTHE 280 (MATH 280), MTHE 281 (MATH 281)
 MTHE 332 Introduction to Control W 3-0-.5 3.5
Modeling control systems, linearization around an equilibrium point. Block diagrams, impulse response, transfer function, frequency response. Controllability and observability, LTI realizations. Feedback and stability, Lyapunov stability criterion, pole placement, Routh criterion. Input/output stability, design of PID controllers, Bode plots, Nyquist plots, Nyquist stability criterion, robust controllers.  (0/0/0/31/11) PREREQUISITE: MTHE 326 (MATH 326)
COREREQUISITE: MTHE 335
 MTHE 333 Control-Robotics Lab I W 0-1-0 1
This laboratory introduces the use of motion control devices such as optical encoders, pulse width amplifiers and armature controlled DC servo motors. The experiments complement the analytical and theoretical work on control taken in other third year courses. Students design and implement proportional, proportional-derivative, and proportional-integral-derivative controllers. (0/0/0/4/8)
COREQUISITE: MTHE 332
 MTHE 334 Mathematical Methods for Engineering and Physics F 3-0-.5 3.5
Motivation for signal spaces. Banach and Hilbert spaces of continuous- and discrete-time signals. Spaces of continuous and not necessarily continuous signals. The four Fourier transforms: continuous-discrete Fourier transform; continuous-continuous Fourier transform; discrete-continuous Fourier transform; discrete-discrete Fourier transform. Transform inversion using Fourier series and Fourier integrals. (28/0/0/14/0)  PREREQUISITES: MTHE 237 (MATH 237), MTHE 281 (MATH 281)
 MTHE 335 Mathematics of Engineering Systems W 3-0-.5 3.5
Linear input/output systems and their stability.  Frequency-domain and time-domain analysis.  Continuous and discrete-time modeling.  Fourier, Laplace, and Z-transforms.  Sampling and the discrete-time Fourier transform.  Applications to modulation of communications signals, filter design, and digital sampling.  (0/11/0/20/11) PREREQUISITES: MTHE 334 (MATH 334), MTHE 326 (MATH 326) or MTHE 228 (MATH 228)
 MTHE 337 Introduction to Operations Research Models W 3-0-0 3
Formulation and solution of some industrial and business problems using mathematical models. Review of probability. Markov chains and applications to inventory problems. Introduction to queuing theory and applications. Machine maintenance planning; optimal number of servers. Simulation and Monte Carlo methods. Reliability and replacement problems. Inventory and production planning problems. One, or possibly two, topics chosen from constrained optimization, network flow analysis and dynamic programming. (18/0/0/9/9)
PREREQUISITES: STAT 256 or MTHE 267 (STAT 267) or equivalent or permission of the instructor
 MTHE 338 Topics in Applied Mathematics F 3-0-.5 3.5
Methods and theory for ordinary and partial differential equations. Review of eigenvectors and eigenmodes in solutions to systems of ordinary differential equations. The principle of linear superposition and eigenfunction expansion, orthogonality, and inner product on a vector space of functions. The method of separation of variables in rectangular and cylindrical coordinate systems; sinusoidal and Bessel orthogonal functions. The wave, diffusion, and Laplace's (potential) equation. Sturm-Liouville theory: eigenvalue problems and orthogonal functions. Fourier transform and, time permitting, Laplace transform techniques. (28/0/0/14/0) PREREQUISITES: MTHE 227 (MATH 227) or MTHE 280 (MATH 280), MATH 226 or MTHE 237 (MATH 237) or MTHE 232 (MATH 232), or permission of the instructor
 MTHE 339 Evolutionary Game Theory W 3-0-0 3
This course highlights the usefulness of game theoretical approaches in solving problems in the natural sciences and economics. Basic ideas of game theory, including Nash equilibrium and mixed strategies; stability using approaches developed for the study of dynamical systems, including evolutionary stability and replicator dynamics; the emergence of co-operative behaviour; limitations of applying the theory to human behaviour. (18/9/9/0/0)
PREREQUISITES: APSC 172 or MATH 120 (or MATH 121); APSC 174 or MATH 110 (or MATH 111) recommended
EXCLUSION: MATH 239
 MTHE 406 Introduction to Coding Theory W 3-0-0 3
Construction and properties of finite fields. Polynomials, vector spaces, block codes over finite fields. Hamming distance and other code parameters. Bounds relating code parameters. Cyclic codes and their structure as ideals. Weight distribution. Special codes and their relation to designs and projective planes. Decoding algorithms. (14/0/0/12/10)
PREREQUISITES: MATH 212 or MTHE 217 (MATH 217)
 MTHE 418 Number Theory and Cryptography F 3-0-0 3
Time estimates for arithmetic and elementary number theory algorithms (division algorithm, Euclidean algorithm, congruences), modular arithmetic, finite fields, quadratic residues. Design of simple cryptographic systems, public key and RSA systems. Primality and factoring: pseudoprimes, Pollard's rho-method, index calculus. Elliptic curve cryptography. (18/0/0/9/9)
Offered in Winter Term in even-numbered years
PREREQUISITES: MATH 210 or MATH 212 or MTHE 217 (MATH 217) or MATH 211 with permission of the instructor
 MTHE 430 Modern Control Theory F 3-.5-.5 4
This course covers core topics in modern control theory: linearization of a nonlinear system around a given trajectory, existence and uniqueness theory for nonlinear and linear ordinary differential equations, the transition matrix of a linear time-varying system, controllability and observability for linear systems, weighting patterns and minimal realizations, feedback stabilization, linear state observers, free end-point and fixed end-point optimal control problems, the Riccati equation, the linear quadratic regulator. Laboratory experiments illustrate design considerations in implementing the lecture material.  (28/0/0/20/0) PREREQUISITES: MTHE 237 (MATH 237), MTHE 312 (MATH 312), MTHE 326 (MATH 326), MTHE 332 (MATH 332), or permission of the instructor
 MTHE 437 Advanced Topics in Applied Mathematics W 3-0-0 3
Subject matter to vary from year to year. (9/0/0/9/18)
~ COURSE NOT OFFERED IN 2011-2012 ~

PREREQUISITE: Permission of the instructor
 MTHE 439 Lagrangian Mechanics, Dynamics, and Control W 3-0-.5 3.5
Geometric mechanical system modelling, including configuration space, tangent bundle, kinetic energy, inertia, and force. Euler-Lagrange equations using affine connections. The last part of the course develops one of the following three applications, depending on instructor and student interest: (1) mechanical systems with nonholonomic constraints; (2) control theory for mechanical systems; (3) equilibria and stability.  (20/0/0/11/11) PREREQUISITES: MTHE 280 (MATH 280), MTHE 281 (MATH 281), MTHE 237 (MATH 237) or MATH 231, or permission of the instructor
 MTHE 472 Control of Stochastic Systems W 3-0-0 3
Optimal control of stochastic systems with applications in various fields of engineering and applied mathematics. Topics include state-space models, development of control laws by dynamic programming, controlled Markov chain models, control with complete and partial information, linear and non-linear filtering, estimation, Martingales, stochastic stability, convergence of policies, dual effect of control, adaptive control, team decisions, numerical methods, value and policy iteration algorithms. (0/0/0/18/18)
PREREQUISITES: MTHE 351 (STAT 351), MTHE 332 (MATH 332), or permission of the instructor
 MTHE 474 Information Theory F 3-0-0 3
An introduction to the fundamental principles of the theory of communication. Topics include: information measures, entropy, mutual information, divergence; modeling of information sources, discrete memoryless sources, Markov sources, entropy rate, source redundancy; fundamentals of lossless data compression, block encoding, variable-length encoding, Kraft inequality, design of Shannon-Fano and Huffman codes; fundamentals of channel coding, channel capacity, noisy channel coding theorem, channels with memory, lossless information transmission theorem; continuous-alphabet sources and channels, differential entropy, capacity of discrete-time and band-limited continuous-time Gaussian channels; rate-distortion theory, lossy data compression, rate-distortion theorem, lossy information transmission theorem. (9/0/0/17/10) PREREQUISITES: STAT 251 or MTHE 351 (STAT 351) or ELEC 326
 MTHE 477 Data Compression and Source Coding W 3-0-0 3
Fundamentals of the theoretical and practical (algorithmic) aspects of lossless and lossy data compression. Topics include: adaptive Huffman coding, arithmetic coding, the fundamental performance limits of universal lossless coding, Lempel-Ziv and related dictionary based methods, the Burrows-Wheeler transform, elements of Kolmogorov complexity theory, rate-distortion theory, scalar and vector quantization, applications to speech and image coding.  (0/0/0/21/15) PREREQUISITE: MTHE 474 (MATH 474)
 MTHE 478 Topics in Communication Theory F/W 3-0-0 3
Subject matter will vary from year to year. Possible subjects include: constrained coding and applications to magnetic and optical recording; data compression; theory and practice of error-control coding; design and performance analysis of communication networks; and other related topics. (0/0/0/18/18)
~ COURSE NOT OFFERED IN 2011-2012 ~
PREREQUISITE: Permission of the instructor
 MTHE 484 Data Networks W 3-0-0 3
This course covers performance models for data networking, delay models and loss models; analysis of multiple access systems, routing, and flow control; multiplexing; priority systems; satellite multiple access, wireless networking, wireless sensor networks. Knowledge of networking protocols is not required. (10/0/0/26/0)
~ COURSE NOT OFFERED IN 2011-2012 ~
PREREQUISITES: MTHE 455 (STAT 455) or permission of the instructor