Mathematics

Mathematics is a discipline which has been said to be the Queen of the Sciences, and is the foundation of most modern quantitative and qualitative studies. The permanence and universality of mathematics throughout the ages is a consequence of its very nature. Mathematics is cumulative, developing from the earliest use of numbers by prehistoric civilizations to the highly deductive nature of geometry as developed by the Greeks, from the practical applications of calculus developed in the seventeenth century to the modern use of number theory in computer cryptography. Mathematics has many faces, from practical uses of its statistical tools to theoretical studies of abstract relationships. Our goal is to introduce students to all facets of the discipline, and to give them an appreciation of the historical, theoretical and applied nature of the discipline, as well as a full understanding of the beauty of the subject.

The Department offers a broad variety of courses and programs in Mathematics. Beginning courses may introduce students to the applications to which Calculus may be applied or the practical uses of statistics; more advanced courses deal with topics ranging from geometry to game theory. All courses in the Mathematics curriculum offer a blend of theory and practical applications. Many of the courses offered include a substantial computational component, and students are encouraged to use the mathematical software tools available. Courses are designed to address the needs of a wide variety of users, from the casual to the professional. Some students may enrol in a course to familiarize themselves with university level mathematics, while others will take a series of courses related directly to their chosen study area. Those choosing to pursue a minor or major in mathematics will be exposed to more advanced courses which blend Mathematical theory and practice.

Mount Allison has been very successful in placing many of its students in graduate programs in Mathematics, while many others have found employment after graduation in one of many fields for which mathematical understanding is an asset. Teaching, actuarial work, law and medicine are all areas requiring the ability to think and reason logically and for which a mathematical background can prove beneficial.

Disciplinary B.A. and B.Sc. Programs

B.A. or B.Sc. MINOR in Mathematics is 24 credits earned as follows:

6	from MATH 1111, 1121
12	from MATH 2111, 2121, 2211, 2221, 2311, 2321.
6	from Mathematics at the 3/4000 level.

B.A. MAJOR in Mathematics is 60 credits earned as follows:

15	from MATH 1111, 1121, 2111, 2211, 2221
3	from MATH 2121, 2311
3	from MATH 3111, 3141, 4131
3	from MATH 3011, 3211, 3221, 3231
3	from MATH 3151, 3311, 3411, 3511
15	from Mathematics at the 3/4000 level
3	from COMP 1631
15	credits from complementary disciplines chosen in consultation with the Program Advisor

B.A. HONOURS in Mathematics is 72 credits earned as follows:

18	from MATH 1111, 1121, 2111, 2121, 2211, 2221
3	from COMP 1631
21	from MATH 3011, 3111, 3121, 3211, 3221, 3311, 3411, 4131, 4221
15	from Mathematics at the 3/4000 levels
9	from Mathematics or Computer Science at the 3/4000 levels
6	from Computer Science, Economics, or Mathematics at the 2000 level or above, or from COMM 3411, LING 2001, 3001, PHIL 2611, PHIL 3631

B.A. or B.Sc. HONOURS in Computer Science and Mathematics is 75 or 87 credits earned as follows:

18	from COMP 1631, 1731, 2211*, 2611, 2631, 2931
15	from MATH 1111, 1121, 2111, 2121, 2221
9	from MATH 3111, 3211, 3311
3	from MATH 3011, 3221, 3231, 3251, 4221
3	from Mathematics at the 3/4000 levels
3	from COMP 3361, 3971
15	from COMP 3411, 3611, 3721, 3911, 4721
9	from Computer Science or Mathematics at the 3/4000 level
12	from CHEM 1001, 1021; PHYS 1051, 1551 (only for B.Sc.)

Note: * COMP 2211 is cross-listed as MATH 2211

B.Sc. MAJOR in Mathematics is 60 credits earned as follows:

18	from MATH 1111, 1121, 2111, 2121, 2211, 2221
3	from MATH 3111, 3141, 4131
3	from MATH 3011, 3211, 3221, 3231
3	from MATH 3151, 3311, 3411, 3511
15	from Mathematics at the 3/4000 level
6	from COMP 1631, 1731
6	from CHEM 1001, 1021
6	PHYS 1051, 1551

B.Sc. HONOURS in Mathematics is 78 credits earned as follows:

18	from MATH 1111, 1121, 2111, 2121, 2211, 2221
6	from COMP 1631, 1731
21	from MATH 3011, 3111, 3121, 3211, 3221, 3311, 3411, 4131, 4221
15	from Mathematics at the 3/4000 level
6	from Mathematics or Computer Science at the 3/4000 level
6	from CHEM 1001, 1021
6	from PHYS 1051, 1551

Interdisciplinary B.A. Program

B.A. HONOURS in Economics and Mathematics is 81 credits earned as follows:

21	from ECON 1001, 1011, 2001, 2011, 2101, 2111, 2701
21	from MATH 1111, 1121, 2111, 2121, 2211, 2221, 2311
3	from COMP 1631
12	from MATH 3011, 3111, 3211, 3311
12	from ECON 4700, 4801, 4811, 4821
6	from ECON at the 3/4000 levels which may include ECON 4990
6	from MATH 3121, 3131, 3151, 3221, 3251, 3321, 3411, 3431, 3511, 3531, 4111, 4131, 4141, 4221. Note that MATH 4951 may also be included if the Program Co-ordinators approve its content.

Interdisciplinary B.Sc. Program

B.Sc. HONOURS in Mathematics and Physics is 90 credits earned as follows:

18	from MATH 1111, 1121, 2111, 2121, 2211, 2221
3	from COMP 1631
6	from CHEM 1001, 1021
12	from PHYS 1051, 1551, 2251, 2801
3	from MATH/PHYS 3451
9	from MATH 3111, 3211, 3311
6	from MATH 3141, 4131
6	from MATH 3121, 3131, 3221, 3151, 3231, 3321, 3411, 3431, 3531, 4111, 4151, 4211
18	from PHYS 3101, 3201, 3701, 3811, 3821, 4411
3	from Physics at the 4000 level
6	from PHYS 4990

PLACEMENT IN MATHEMATICS

Students wishing to take the introductory calculus course (Mathematics 1111) are required to write a Mathematics Placement Test to determine their level of mathematical preparation. Based on their test scores and the University regulations, students will be placed in Mathematics 1011 or Mathematics 1111. The Mathematics Placement Test will be administered prior to the beginning of classes. Students will be allowed to re-write the test during the first week of classes.

MATHEMATICS COURSES

Note: The listing of a course in the Calendar is not a guarantee that the course is offered every year.

Note: Students must obtain a grade of at least C- in all courses used to fulfill prerequisite requirements. Otherwise, written permission of the appropriate Department Head or Program Co-ordinator must be obtained.

MATH 1011 (3CR)
SETS, FUNCTIONS AND RELATIONS
Format: lecture 3 hours, laboratory 1.5 hours
This course will focus on the real number system, inequalities, plane analytic geometry (lines and conics), functions, inverse functions, polynomials, rational functions, trigonometric functions, exponential and logarithmic functions. Fundamental methods of graphing functions, using non-calculus based techniques, will be emphasized. This course is primarily intended for non-science students or as a prerequisite for MATH 1111 for those students who have not passed the Mathematics Placement Test. Science students who have passed the Mathematics Placement Test require the permission of the Department of Mathematics and Computer Science to enrol in this course. This course cannot be used to satisfy the Bachelor of Science degree requirement of a course in MATH/COMP (7.3.3). Credit will not be given for this course if credit has already been granted for MATH 1111.

MATH 1111 (3CR)
INTRODUCTION TO CALCULUS I
Format: lecture 3 hours, laboratory 1.5 hours
Prereq: A passing score on the Mathematics Placement Test, or MATH 1011; or permission of the Department. Students enrolling in Mathematics 1111 should normally have completed a university preparatory course in Mathematics designed to prepare them for University calculus
Derivatives of the algebraic and exponential functions are developed. Applications include curve sketching, related rates, and optimization problems.

MATH 1121 (3CR)
INTRODUCTION TO CALCULUS II
Format: lecture 3 hours, laboratory 1.5 hours
Prereq: MATH 1111; or permission of the Department
The derivatives of trigonometric functions are introduced, various techniques of integration studied and some applications presented. Among these applications are: area between curves, volume work and elementary differential equations.

MATH 1131 (3CR)
MATHEMATICS FOR LIFE AND ENVIRONMENTAL SCIENCE
Format: lecture 3 hours, laboratory 1.5 hours
Prereq: MATH 1111; or permission of the Department
Exclusion: MATH 1121, 1251
An application-oriented continuation of the study of calculus and an introduction to other topics of interest to students in the life and environmental sciences.

MATH 1251 (3CR)
FINITE MATHEMATICS
Format: lecture 3 hours
Exclusion: MATH 1131
This course introduces students at all levels to the most applicable branches of finite mathematics and is particularly suitable for students in the social and behavioural sciences and commerce. Topics discussed include Markov chains, linear programming and game theory.

MATH 2111 (3CR)
MULTIVARIABLE CALCULUS
Format: lecture 3 hours
Prereq: MATH 1121; or permission of the Department
Exclusion: MATH 2111 (Vector Calculus)
Topics include: Sequences and series, power series, Taylor and MacLaurin series; conic sections, quadric surfaces, cylindrical and spherical co-ordinates in three space; functions of several variables: continuity, partial derivatives, tangent planes, chain rule, maximum and minimum values, Lagrange multipliers, double and triple integrals.

MATH 2121 (3CR)
ELEMENTARY DIFFERENTIAL EQUATIONS
Format: lecture 3 hours
Prereq: MATH 2111; or permission of the Department
This is an introduction to the techniques and applications of first and second order differential equations. Included will be: applications of first order equations to areas such as growth and decay, cooling and diffusion, mixture problems, chemical reactions, the logistic equation, orthogonal trajectories, higher order differential equations and applications, and power series solutions of differential equations.

MATH/COMP 2211 (3CR)
DISCRETE STRUCTURES
Format: lecture 3 hours
Prereq: MATH 1111; or permission of the Department
Note: This course is cross-listed as COMP 2211 and may therefore count as three credits in either discipline.
Exclusion: MATH 2211 Discrete Mathematics
An introduction to the terminology and concepts of discrete mathematics, covering such topics as: logical arguments, proofs and algorithm verification, sets, relations, functions and cardinality of sets, induction and recursion, enumeration, algorithms and complexity.

MATH 2221 (3CR)
LINEAR ALGEBRA
Format: lecture 3 hours
Prereq: MATH 1111; or permission of the Department
An introductory course in linear algebra covering such topics as linear equations, matrices, determinants, vector spaces, linear transformations, inner products, eigenvalues, and eigenvectors. Whenever possible, concepts are given a geometric interpretation in two and three-dimensional space.

MATH 2311 (3CR)
STATISTICS I
Format: lecture 3 hours
Prereq: University preparatory level Mathematics or MATH 1011 and either MATH 1111, or registration in second year or higher; or permission of the Department
Note: Students may count for credit towards a degree a maximum of 6 credits from BIOL 2701, GENS 2431, MATH 2311, and PSYC 2001
This course is an introduction to some of the concepts and techniques of probability and statistics. Topics include descriptive statistics, elementary probability, probability distributions, statistical estimation, hypothesis testing, and the use of a statistical software package in analyzing data. Examples come from a wide variety of disciplines.

MATH 2321 (3CR)
STATISTICS II
Format: lecture 3 hours, laboratory 1 hour
Prereq: MATH 2311 or 3311; or permission of the Department
Exclusion: ECON 2701
This is a second course in the concepts and techniques of probability and statistics. The course covers a selection of topics from analysis of variance, linear and nonlinear regression, correlation estimation and prediction, independence, Wilcoxon and goodness-of-fit tests and includes data analysis using statistical software. Examples come from a wide variety of sources and disciplines.

MATH 3011 (3CR)
SET THEORY AND MATHEMATICAL LOGIC
Format: lecture 3 hours
Prereq: MATH 2211; or permission of the Department
This course provides a mathematical introduction to the basic ideas of set theory and logic. Topics covered may include: axiom of choice, cardinal and ordinal numbers, Boolean algebras and their applications, completeness, decidability, philosophies of mathematics.

MATH 3031 (3CR)
HISTORY OF MATHEMATICS
Format: lecture 3 hours
Prereq: MATH 1121 (or 1131) and 6 credits from MATH 2111, 2121, 2211 and 2221
A survey of the history of Mathematics. Topics include: the achievements of early civilizations, the developments in Europe leading to the calculus and its consequences, the growth of rigor in the 18th and 19th centuries, the axiomatic method in the 20th century.

MATH 3111 (3CR)
REAL ANALYSIS I
Format: lecture 3 hours
Prereq: MATH 2111, 2121, 2211; or permission of the Department
Exclusion: MATH 3110
A systematic and rigorous study of the real numbers and functions of a real variable, emphasizing limits and continuity.

MATH 3121 (3CR)
REAL ANALYSIS II
Format: lecture 3 hours
Prereq: MATH 3111; or permission of the Department
Exclusion: MATH 3110
A continuation of Mathematics 3111 including the study of concepts from the Calculus, including differentiation and integration.

MATH 3131 (3CR)
MATHEMATICAL METHODS FOR DIFFERENTIAL EQUATIONS
Format: lecture 3 hours
Prereq: MATH 2121 and 2221; or permission of the Department
This course is designed primarily for students in mathematics, physics, or engineering. It covers systems of 1st and 2nd order ordinary differential equations, Laplace and Fourier transforms, power series solutions for equations with singular points, and Fourier series.

MATH 3141 (3CR)
VECTOR CALCULUS
Format: lecture 3 hours
Prereq: MATH 2111; or permission of the Department
Topics covered include vectors in the plane and in three space, vector functions, curves, tangent and normal vectors, velocity and acceleration; curvature and arc length, directional derivatives and the gradient, vector fields, line integrals, the Fundamental Theorem of line integrals, divergence and curl, Green's Theorem, parametrized surfaces, surface area and surface integrals, flux, Stokes' Theorem, and the Divergence Theorem.

MATH 3151 (3CR)
AN INTRODUCTION TO MATHEMATICAL MODELLING
Format: lecture 3 hours
Prereq: MATH 2121 and 2221; or permission of the Department
This course provides an introduction to the nature of theoretical mathematical modelling illustrated by examples drawn from the physical and engineering sciences, pursuit and conflict problems, population dynamics (mathematical ecology), traffic flow, sociological problems (voting, kinship, cultural stability) and other areas depending on the interests of the class.

MATH 3211 (3CR)
MODERN ALGEBRA I
Format: lecture 3 hours
Prereq: MATH 2211 and MATH 2221; or permission of the Department
An introduction to the theory of groups and rings.

MATH 3221 (3CR)
ADVANCED LINEAR ALGEBRA
Format: lecture 3 hours
Prereq: MATH 2221; (MATH 2211 is recommended); or permission of the Department
An advanced course in linear algebra, covering selected topics from: change of basis and similarity of matrices; multilinear forms and determinants; canonical forms, Primary Decomposition Theorem, Jordan form; semisimple and normal operators; spectral theory; quadratic forms; applications to geography, electrical networks, linear programming, differential equations, or the geometry of conic sections.

MATH 3231 (3CR)
NUMBER THEORY
Format: lecture 3 hours
Prereq: MATH 2211; or permission of the Department
An introductory half-course in the theory of numbers covering such topics as: Euclidean algorithm, Fundamental Theorem of Arithmetic, congruences, diophantine equations, Fermat and Wilson Theorems, quadratic residues, continued fractions, Prime number theorem.

MATH 3251 (3CR)
INTRODUCTION TO COMBINATORICS AND GRAPH THEORY
Format: lecture 3 hours
Prereq: MATH 2211, 2221; or permission of the Department
Topics covered include enumeration (permutations and combinations, inclusion-exclusion and pigeonhole principles, recurrence relations and generating functions), algorithmic graph theory (minimum-weight spanning trees and minimum-weight paths) and combinatorial design theory (latin squares and finite geometries, balanced incomplete block designs, triple systems).

MATH/ECON 3301 (3CR)
INTRODUCTION TO GAME THEORY
Format: lecture 3 hours, laboratory 1 hour
Prereq: ECON 1001 and ECON 1011; or MATH 1111; or permission of the Department
Note: This course is cross-listed as ECON 3301 and therefore may count as 3 credits in either discipline
This course introduces the basic tools and methods of Game Theory. Game Theory is a mathematically oriented approach to understanding the strategic interaction of self-interested agents. Emphasis is on non-cooperative games. Topics include backwards induction, iterative deletion of dominated strategies, Nash equilibrium, repeated games, some equilibrium refinements, evolutionary game theory, and Bayesian Nash equilibria.

MATH 3311 (3CR)
PROBABILITY AND STATISTICS I
Format: lecture 3 hours
Prereq: MATH 2111; or permission of the Department
Exclusion: MATH 3310
An introduction to the mathematical theory of probability. Topics covered include: sample space, events, axioms, conditional probability, Bayes Theorem, random variables, combinatorial probability, moment generating functions, transformations of random variables, univariate and jint distributions with reference to the binomial, hypergeometric, normal, Gamma, Poisson, and others; convergence of sequences of variables, central Limit Theorem.

MATH 3321 (3CR)
PROBABILITY AND STATISTICS II
Format: lecture 3 hours
Prereq: MATH 3311; or permission of the Department
Exclusion: MATH 3310
An introduction to mathematical statistics. Topics covered include: Estimation, unbiasedness, efficiency, Cramer-Rao lower bound, consistency, sufficiency, maximum likelihood estimators, hypothesis testing, power of tests, likelihood ration, regression analysis and analysis of variance.

MATH/COMP 3411 (3CR)
NUMERICAL ANALYSIS
Format: lecture 3 hours
Prereq: MATH 1121, 2221, and COMP 1711 or 1731 or 1751; or permission of the Department
Note: This course is cross listed as COMP 3411 and may therefore count as three credits in either discipline.
This course is an introduction to numerical methods for solving a variety of problems in mathematics, the natural sciences, and engineering. Topics to be studied include numerical solution of linear and nonlinear systems of equations, Gauss elimination, pivoting strategies, numerical stability, PLU factorization, tridiagonal matrices, polynomial and cubic spline approximation and interpolation.

MATH 3431 (3CR)
ORDINARY DIFFERENTIAL EQUATIONS
Format: lecture 3 hours
Prereq: MATH 2121 and MATH/COMP 3411; or permission of the Department
This course utilizes both numerical and theoretical techniques to study ordinary differential equations. Topics include numerical, integration, Runge-Kutta and multistep methods, stability, introduction to qualitative methods, phase-plane analysis, stability of non-linear systems, Lyapunov's method, chaos theory.

MATH/PHYS 3451 (3CR)
METHODS OF MATHEMATICAL PHYSICS
Format: lecture 3 hours, laboratory 3 hours
Prereq: MATH 2111, MATH 2221, PHYS 2251
Note: This course is cross listed as PHYS 3451 and may therefore count as three credits in either discipline.
This course provides students with a selection of mathematical skills needed in more advanced physics courses. Frequently utilized mathematical methods in theoretical physics are introduced in close connection to physics applications. The assumptions behind the relevant theorems are mentioned in order to discuss their limitations, however, more rigourous mathematical proofs are not generally covered. Topics include vector and tensor analysis, use of special functions, operators and eigenvalue problems. Fourier analysis, and complex variable techniques in physics. The lab component of the course will use symbolic algebra and numerical software, such as Maple, to solve associated physics problems.

MATH/COMP 3511 (3CR)
LINEAR PROGRAMMING
Format: lecture 3 hours
Prereq: MATH 2221, 3 credits in Computer Science; or permission of the Department
Note: This course is cross listed as COMP 3511 and may therefore count as three credits in either discipline.
Among the topics covered are linear and integer programming, the simplex and revised simplex methods, duality theory and sensitivity analysis, and various applications.

MATH/COMP 3531 (3CR)
SIMULATION AND MODELLING
Format: lecture 3 hours
Prereq: MATH 1111; one of MATH 2311, 3311, PSYC 2001 and 2011; three credits in Computer Science; or permission of the Department
Note: This course is cross listed as COMP 3531 and may therefore count as three credits in either discipline.
An introduction to the simulation technique for studying mathematical models. Specific titles include: systems theory and system models, continuous system simulation, discrete system simulation, Monte Carlo methods, random number generators, and simulation languages. Emphasis will be placed upon computer implementation of the methods studied.

MATH 4111 (3CR)
TOPOLOGY
Format: lecture 3 hours
Prereq: MATH 3111; or permission of the Department
Exclusion: MATH 4110
An introduction to the study of metric and topological spaces, convergence, and continuous functions.

MATH 4121 (3CR)
TOPICS IN ANALYSIS
Format: lecture 3 hours
Prereq: MATH 3111; or permission of the Department
Exclusion: MATH 4110
This course covers selected topics in Analysis, depending on the background and interests of the students involved.

MATH 4131 (3CR)
COMPLEX VARIABLES WITH APPLICATIONS
Format: lecture 3 hours
Prereq: MATH 2111; or permission of the Department
This course is designed primarily for students in mathematics and physics. It covers analytic functions, Cauchy-Riemann equations, conformal mapping, complex integrals, Cauchy's integral theorem, Taylor and Laurent Series, residues, evaluation of real integrals, and inverse transforms.

MATH 4141 (3CR)
MEASURE AND INTEGRATION
Format: lecture 3 hours
Prereq: MATH 3110 or 3121; or permission of the Department
Topology of Rn, Lebesque Measure, Measurable Functions, the Lebesque integral, the convergence theorems and products measures. As time permits, other topics such as abstract measure theory, Lp-spaces and absolute continuity will be covered.

MATH 4151 (3CR)
BOUNDARY AND EIGENVALUE PROBLEMS
Format: lecture 3 hours
Prereq: MATH 3131; or permission of the Department
This course is designed primarily for students in mathematics, physics, or engineering. It extends the material studied in Mathematics 3131. Topics include separation of variables (product method), generalized Fourier series, Sturm-Liouville theory, Legendre polynomials, Bessel Functions, Green's functions, and calculus of variations.

MATH 4221 (3CR)
MODERN ALGEBRA II
Format: lecture 3 hours
Prereq: MATH 3211; or permission of the Department
The classical theory of fields and rings and their applications.

MATH/COMP 4631 (3CR)
THEORY OF COMPUTATION
Format: lecture 3 hours
Prereq: COMP/MATH 2211, COMP 1721 or 1731; or permission of the Department
Note: This course is cross listed as COMP 4631 and may therefore count as three credits in either discipline.
This course is an introduction to theoretical aspects of Computer Science such as formal language and automata theory and complexity theory.

MATH/COMP 4651 (3CR)
CRYPTOGRAPHY
Format: lecture 3 hours
Prereq: COMP 1631 or 1711, 1721 or 1731, COMP/MATH 2211; or permission of the Department
Note: This course is cross listed as COMP 4651 and may therefore count as three credits in either discipline.
This course is an introduction to cryptographic algorithms and to the cryptanalysis of these algorithms, with an emphasis on the fundamental principles of information security. Topics include: classical cryptosystems, modern block and stream ciphers, public-key ciphers, digital signatures, hash functions, key distribution and agreement.

MATH 4950/4951 (6/3CR)
INDEPENDENT STUDY IN MATHEMATICS
Format: Independent Study
Prereq: Permission of the Department/Program Advisor. Students must obtain consent of an instructor who is willing to be a supervisor and must register for the course prior to the last day for change of registration in the term during which the course is being taken.
Note: A program on Independent Study cannot duplicate subject matter covered through regular course offerings.
Note: Students may register for MATH 4950/51 more than once, provided the subject matter differs.
This course permits senior students, under the direction faculty members, to pursue their interest in areas not covered, or not covered in depth, by other courses through a program of independent study.