Mathematics

A review of high school mathematics topics, including basic properties of number systems, manipulation of algebraic expressions, equations and inequalities, analytic geometry, linear and quadratic functions, polynomial and rational functions, exponential and logarithm functions, trigonometric functions.  Note:  This course is designed to serve as preparation for calculus courses at the university level, such as Math 1003 and Math 1823.  It carries no credit for degree programs at UNB Fredericton.

Functions and graphs, limits, derivatives of polynomial, log, exponential and trigonometric functions. Curve sketching and extrema of functions. NOTE: Credit may be obtained for only one of MATH 1003, 1053 or 1823. 

Definition of the integral, fundamental theorem of Calculus, Techniques of integration, improper integrals. Ordinary differential equations. Taylor polynomials and series. NOTE: Credit may be obtained for only one of MATH 1013 or 1063. 

The syllabus is similar to that for MATH 1003, with more emphasis placed both on the theory of calculus and interesting applications. The course will be of special interest to students with strong mathematical backgrounds. Any interested student (with or without high school calculus) is encouraged to consult with the Mathematics Department. NOTE: Credit may be obtained for only one of MATH 1003, 1053 or 1823.

The syllabus for this course is similar to that of MATH 1013 . As with MATH 1053, more emphasis is placed on theory, mathematical rigor and interesting applications. NOTE: Credit may not be for only one of MATH 1013 or 1063 .

Lines and planes, the geometry and algebra of vectors, systems of linear equations, matrix algebra, linear independence, linear transformations, determinants, complex numbers, eigenvectors, diagonalization, rotation matrices, quadratic forms, least squares.

Polynomial, logarithmic and exponential functions. Limits and derivatives. Extreme values and related rates. Simple integration and differential equations, with stress on applications to business and economics. NOTE: Credit may be obtained for only one of MATH 1003 , 1053 or 1823.

Matrices and systems of linear equations. Linear programming concepts; graphical solution of two variable problems. Permutations and combinations. Elementary probability. Mathematics of finance. NOTE: Credit for MATH 1833 will not be given if the student has previously taken either MATH 1503 or MATH 2213.

Analytic geometry and vectors. Parametric curves. Polar, cylindrical and spherical coordinates. Functions of several variables, partial derivatives, applications to max-min. Double and triple integrals. 

Review of first order differential equations. Second order linear O.D.E.'s. Infinite series, including power series solutions to O.D.E.'s. Line and surface integrals. Theorems of Green and Stokes. Divergence Theorem.

Logic, methods of proof, mathematical induction, elementary set theory, functions and relations. NOTE: This course is designed for students desiring a good grounding in the foundations of mathematics. Theorems and proofs are an important part of the course. Credit will not be given for both MATH 2203 and CS 1303 . Students majoring in Mathematics must take MATH 2203.

This course introduces the basic concepts of linear algebra, mainly in finite dimensional real vector spaces.  Systems of linear equations, vector and matrix algebra, bases and dimension of subspaces, row and column spaces, linear transformations and matrix representations, inner products, determinants, eigenvectors and diagonalization.  Applications as time permits.

Functions of several variables, partial derivatives, multiple integrals, vector functions, Green's and Stokes' Theorems. 

An introduction to mathematical thinking. Content varies, and is focused on presenting mathematics as a living, creative discipline. A sample of topics: patterns and symmetry, tiling, non-Euclidean geometry, chaos and fractals, planetary motion, binary numerals, prime numbers, Fibonacci numbers, voting systems, the calendar. Not available for credit to students with a Major in Mathematics/Statistics.

This course is intended for students who anticipate a career as an elementary or middle school teacher. The course focuses on topics taken from the K-8 curriculum with extensions beyond classroom topics to show the 'how' and 'why' behind school mathematics. The major topics are problem solving, number concepts, number and relationship operations, patterns and relations, shape and space, as well as data management and probability. Intended for students registered in arts programs. Not available for credit to students who would have 6ch of Level 1000 mathematics in their degree programs. 

Vector spaces of functions, convergence in normed linear spaces, orthogonal polynomials, Fourier series, Fourier transform, Fast Fourier transform, introduction to wavelets, and selected applications.

Groups are the mathematical objects used to describe symmetries. This course covers the fundamentals of group theory, together with applications selected from geometry, advanced algebra and physical sciences.

First order equations, linear systems, variation of parameters, method of undetermined coefficients, Laplace transforms, power series solutions, fundamental matrix solution. Existence and uniqueness of solutions, properties of linear systems, eigenvalue problems, vector fields, phase-plane analysis. Liapunov method. 

Axiomatic systems, non-Euclidean geometry, transformations in geometries, topological properties of figures. As well as serving mathematics majors, this course will be of particular benefit to prospective mathematics teachers.

Methods of solution for first order equations. Classification of second order equations. Characteristics. Analytic and numerical methods of solution for hyperbolic, elliptic and parabolic equations. 

Primes, unique factorization, congruences, Diophantine equations, basic number theoretic functions. As well as serving mathematics majors, this course will be of particular benefit to prospective mathematics teachers.

The real number system, metric spaces, sequences and series, continuity. 

Differential calculus, integration, sequences and series of functions, completeness of basis, convergence of Fourier Series, Fourier Transforms. Additional topics may include differential forms or wavelets and wavelet transforms. 

Finite and infinite dimensional vector spaces over general fields.  Subspaces, independent and spanning sets, dimension, linear operators, determinants, inner product spaces.  As time permits, applications selected from least squares approximation, Markov chains, data compression, traffic flow, robotics, genetics, graph theory, cryptography.

Complex analytic functions, contour integrals and Cauchy's theorems; Taylor's, Laurent's and Liouville's theorems; residue calculus. 

Topics selected from: Principle of inclusion and exclusion, Mobius inversion, generating functions, systems of distinct representatives, Ramsey's Theorem, duality in external problems, duality in programming, dynamic programming, block designs, introduction to matroid theory, signal-flow graphs. (The course is also of interest to students in Computer Science and Engineering.) 

Graphs, Euler paths, tournaments, factors, spanning trees, applications; graph colourings, planar graphs, Menger's theorem, flows in networks, flow algorithms.

Topics in abstract algebra are approached from the perspective of what can be computed using such software packages as Maple, Macaulay and GAP. The topics covered will be selected from: Grobner bases, resultants, solving polynomial equations, invariant theory of finite groups, and the exact solution of differential equations. The course work will include a mixture of problem sets emphasizing theory and practical lab assignments.

Applications of algebraic and combinatorial methods to a selection of problems from coding theory, computability, information theory, formal languages, cybernetics and the social and physical sciences. 

Strategic games, n-person games in normal form, dominated strategies, Nash equilibrium, mixed strategies and mixed strategy equilibrium, games with perfect information, games with imperfect information, Bayesian games, extensive games. The course introduces basic non-cooperative game theory and analytical tools for decision makers (consumers, firms, politicians, governments). It is suitable for Mathematics, Economics, Computer Science, Management Science, Political Science, Social Science and Science students or any student with a minor in such disciplines, in particular those in the Mathematics/Statistics-Economics option. Note: this course is cross-listed as ECON 4673. Students cannot obtain credit for both MATH 3373 and ECON 4673 (or 5673).

The course introduces the basic concepts of mathematical logic, including the Axiom of Choice and its equivalents; propositional logic; languages and structures, axioms and theories, models; elements of model theory (Completeness, Compactness, Löwenheim-Skolem theorems, nonstandard models); theory of computability (ChurchTuring Thesis, recursive functions and sets, recursively enumerable sets, decision problems, the Halting Problem); Gödel's Incompleteness Theorems.

Intended for Mathematics, Science or Engineering students. Error analysis, convergence and stability. Approximation of functions by polynomials. Numerical quadrature and differentiation. The solution of linear and nonlinear equations and the solution of ordinary differential equations. This course will emphasize the understanding of numerical algorithms and stress applications in the applied sciences, as well as the influence of finite precision and arithmetic on computational results. Credit will not be given for both MATH 3413 and CS 3113 .

This course is intended to develop skills in translating a problem in the real world to a well formulated mathematical problem. The basic techniques and tools for model formulation, model analysis, numerical simulation and model interpretation will be offered. Project topics will be chosen from Biology, Physics, Chemistry, Mechanics, Engineering, Economics and elsewhere. 

Nonhomogeneous differential equations, undetermined coefficients, variation of parameters, systems of 1st and 2nd order ordinary differential equations, Laplace transforms, Fourier series.

Basic analytic geometry, spherical trigonometry, geometry of curves in space, measurements on surfaces, Gaussian surface geometry.

A non-technical survey of the development of mathematics from primitive peoples through Indian, Oriental, Babylonian, Egyptian and Greek cultures. More emphasis will be placed on Western European and post-Renaissance mathematics, and recent (post-1940) history. An attempt is made to discuss each new mathematical contribution in light of both past mathematics and social scientific forces of the day. Some background in Mathematics necessary.

A course for undergraduate students who anticipate a career as teachers. Topics build around the K-12 syllabus, with extensions beyond the classroom, to show the 'how' and 'why' behind school mathematics. Mathematical language; real numbers and other mathematical structures; Euclidean geometry; functions; mathematical connections; problem solving.

Measurement of interest, compound interest, annuities, amortization schedules and sinking funds, bonds.

A more advanced study of the topics in MATH 3803 including varying and continuous annuities and yield rates.

Survival distributions, general life insurances and life annuities, reserves. Joint annuities and last survivor annuities.

Normed spaces, the Hahn-Banach theorem, uniform boundedness theorem. The contraction mapping theorem. Existence and uniqueness for nonlinear differential equations. Further topics may include Wavelets or Banach spaces.

Prime fields and characteristic, extension fields, algebraic extensions, theory of finite fields, Galois theory, and topics which may include some of: rings, topological algebra, multilinear and exterior algebra, quadratic forms.

A deeper investigation of Euclidean and Non-Euclidean spaces of any dimension. Topics selected from: axiom systems, linear and affine transformations, conformal and linear models for Euclidean and hyperbolic spaces and their isometry groups, basic theory of convexity, combinatorial properties of polytopes.

Mathematics Honours students must complete a project under the supervision of a faculty member. The project is to include a written report and an oral presentation. Prior to being admitted into MATH 4100, the student must have been admitted to the Honours Program and have submitted an acceptable project proposal to the department. Normally students would begin preparation and research for the project during their third year of study, submit the proposal by October of their fourth (final) year of study, and complete the written and oral presentation by the end of the winter term, to graduate in May of that year. Honours students in an interdepartmental program with mathematics may choose to complete their honours project in mathematics.

Brief review of Riemann integration. Algebras of sets, outer measure, measure, measurable sets, measurable functions, the Lebesgue integral, properties of the Lebesgue integral, abstract measure spaces, integrals and derivatives, sequences of integrals, Fubini's theorem. Properties of Fourier transforms, multiresolution analysis, Daubechies wavelets. 

The theory of vector spaces and linear transformations, dual spaces, multilinear maps (including tensors and determinants); further topics chosen from canonical forms, metric vector spaces, algebras, etc.

Many of the processes studied in science, engineering and economics are described by nonlinear differential equations. This course introduces qualitative methods to find essential information about the solutions of nonlinear equations without necessarily attempting to find the solution completely. Topics include flows, stability, phase plane analysis, limit cycles, bifurcations, chaos, attractors, maps, fractals. Applications throughout.

A continuation of the topological concepts introduced in MATH 3103 . Basic results in point-set topology. 

Derivation of the Equations of Motion: Euler's equations, rotation and vorticity, Navier-Stokes equations. Potential Flow: complex potentials, harmonic functions, conformal mapping, potential flow in three dimensions. Slightly Viscous Flow: boundary layers and Prandtl boundary layer equations. Gas Flow in one dimension: characteristics and shocks.

Introduction to functionals and function spaces. Variation of a functional. Euler's equations, necessary condition for an extremum, case of several variables, invariance of Euler's equation, fixed end point problem for unknown functions, variational problems in parametric form, functionals depending on high order derivatives. 

Relativistic quantum mechanics. The negative energy problem. Classical field theory, symmetries and Noether's theorem. Free field theory and Fock space quantization. The interacting field: LSZ reduction formula, Wick's theorem, Green's functions, and Feynman diagrams. Introduction to Quantum electrodynamics and renormalization. This course is cross-listed as PHYS 4938 . Credit cannot be obtained for both Math 4443 and PHYS 4953 .

Covers in depth those functions which commonly occur in Physics and Engineering, namely, the Gamma, Beta, Bessel, Legendre, hypergeometric, Hermite and Laguerre functions. Additional or alternative special functions may be included. Applications to Physics and Engineering will be discussed.

Geometry of embedded curves and surfaces, n-dimensional manifolds, tensors, Riemannian geometry.

Along with quantum theory, general relativity is one of the central pillars of modern theoretical physics with wide-ranging implications for astrophysics and high energy physics. The essential idea is that gravitation is a manifestation of the curvature of spacetime rather than a force in the Newtonian sense.  This course will provide students with a basic working understanding of general relativity and an introduction to important applications such as black holes and cosmology.  Contents: review and geometric interpretation of special relativity, foundations of general relativity, linearized gravity and classical tests, black holes, cosmology. Note: this course is cross-listed as PHYS 4983. Credit cannot be obtained for both Math 4483 and PHYS 4983. 

The numerical solution of ordinary differential equations, and partial differential equations of elliptic, hyperbolic and parabolic type. The course is a basic introduction to finite difference methods, including the associated theory of stability, accuracy and convergence. Students will gain practical experience using state-of-the-art numerical solvers and visualization tools, while solving practical problems from the physical and biological sciences. Cross-listed as CS 4115 . 

Overview of the field of Mathematical Biology.  Development, simulation and analysis of mathematical models describing biological systems. Equal emphasis is placed on developing simple models and case studies of successful models. The principal mathematical tools are differential and difference equations, finite mathematics, probability and statistics. This course is intended for students in their third or fourth year having an interest in biological research.

A course for high school mathematics teachers. The course is built around a set of optimization problems, whose solution requires review of topics in first and second year calculus and linear algebra. Connections are made with topics in the Common Atlantic High School Mathematics Curriculum.

Brief history of structural linguistics. Introduction to mathematical methods of linguistics. Finite state automata, regular languages. Computability. Chomsky hierarchy. Phrase-structure grammars. Artificial intelligence problem. Critiques of structural linguistics. 

Basics of options, futures, and other derivative securities. Introduction to Arbitrage. Brief introduction to partial differential equations. Stochastic calculus and Ito's Lemma. Option pricing using the Black-Scholes model. Put-call parity and Hedging. Pricing of European and American call and put options. Numerical methods for the Black-Scholes model: binary trees, moving boundary problems, and linear complementarity. The barrier, and other exotic options.

Topics to be chosen jointly by student, advisor, and Department Chair. May be taken for credit more than once. Title of topic chosen will appear on transcript.