University of Toronto at Scarborough 2001/2002 Calendar
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(B.Sc.)
Discipline Representative: Until June 30, 2001
J. Scherk (416-287-7269)
July 1, 2001 to June 30, 2002 - T.B.A.
Our Mathematics began in the ancient Mesopotamian
civilizations. The Babylonians already knew much of the mathematics
taught traditionally in our schools. Their algebra and geometry
was phrased in terms of crops and fields and money. Since the
Renaissance, much of mathematics has come from problems in physics
and astronomy; for example, calculus arose from problems in mechanics.
In turn mathematics has provided the theoretical framework and
tools in the Physical Sciences. In the 19th century some parts
of mathematics appeared to develop away from their origins in
the physical world. To the great surprise of many scientists and
mathematicians, some of the "pure" mathematics has turned
out to be essential in many aspects of 20th century science. Differential
geometry provides the language for general relativity and cosmology,
and Hilbert space theory and group representations are the tools
for quantum mechanics. Similarly, graph theory, combinatorics
and number theory play a major role in computer science.
Please refer to the Physical
Sciences Scarborough preamble on page 148 for a list of the Programs
offered. Descriptions of these Programs will be found on subsequent
pages of this section.
Supervisor: E. Moore (416-287-7267)
The Specialist Program in Mathematics is designed
to give students a thorough grounding in the main areas of Mathematics,
together with an understanding of the close relationship between
Mathematics and other Sciences. It is aimed at students who may
be interested in teaching, law, government or industry, or who
may decide to pursue a career in research.
[This Program is comparable to the Specialist Program
in Mathematics and Applications on the St. George Campus.]
CSCB70H Fundamental Data Structures and Techniques
MATB24H Linear Algebra II
MATC01H Groups and Symmetry
MATB41H Techniques of the Calculus of Several Variables I
MATB42H Techniques of the Calculus of Several Variables II
MATB43H Introduction to Analysis
STAB22H Statistics
MATB44H Ordinary Differential Equations
MATC46H Differential Equations
MATC34H Complex Variables
STAB47H Introduction to Probability Theory and Mathematical
Statistics
two of:
MATB61H Linear Programming and Optimization
MATC35H Chaos, Fractals and Dynamics
MATC38H Introduction to Real Analysis
MATC61H Stochastic Differential Equations
MATC65H Complex Variables II
one of:
MATC44H Introduction to Combinatorics
MATC32H Graph Theory and Algorithms for Its Applications
CSCC50H Numerical Algebra and Optimization
CSCC51H Numerical Approximation, Integration and
Ordinary Differential Equations
1.0 F.C.E. from MAT at B-, C-, or D-level.
PSCD02H Current Questions in Mathematics and Science
or
PSCD03H Computers in Contemporary Society
NOTE: PSCD01H
is a required course for ETP students
Recommended course: PHYB21H
Supervisor: M. Evans (416-287-7274)
The Specialist Program in Mathematics has been withdrawn. Students currently registered in it will be allowed to complete it. Please consult with the Supervisor of Studies.
Interested students should consider the Statistics Stream of the Specialist Program in Mathematics and Its Applications (below).
Supervisor: E. Moore (416-287-7267)
CORE FOR ALL PROGRAM STREAMS:
STAB22H Statistics
MATB24H Linear Algebra II
MATB41H Techniques of the Calculus of Several Variables I
MATB42H Techniques of the Calculus of Several Variables II
MATB43H Introduction to Analysis
MATB44H Ordinary Differential Equations
STAB47H Introduction to Probability Theory and Mathematical
Statistics**
MATC01H Groups and Symmetry
MATC34H Complex Variables
* CSCA57H is required for the Computational Physical Sciences stream
** STAB47H must be taken in second year for the Statistics
stream
MATC02H Fields and Groups
MATC15H Introduction to Number Theory
MATC25H Classical Plane Geometries and their Transformations
[MATC32H Graph Theory and Algorithms for its Applications
or
MATC44H Introduction to Combinatorics]
Three of:
MATB61H Linear Programming and Optimization
MATC46H Differential Equations
MATC35H Chaos, Fractals and Dynamics
MATC38H Introduction to Real Analysis
MATC63H Differential Geometry I
MATC65H Complex Variables II
Four C- or D-level CSC, MAT or STA half-courses
[PSCD02H Current Questions in Mathematics and Science
or
PSCD03H Computers in Contemporary Society]
NOTE:
Students following this stream are encouraged to apply for acceptance
into the Early Teacher Project. ETP students are required to take
PSCD01H.
[PSCD02H Current Questions in Mathematics and Science
or
PSCD03H Computers in Contemporary Society]
One of:
CSCC54H Computer-Based Simulation Models
CSCD18H Computer Graphics
MATC61H Stochastic Differential Equations
MATC65H Complex Variables II
MATD01H Readings in Mathematics
[PSCD02H Current Questions in Mathematics and Science
or
PSCD03H Computers in Contemporary Society]
See Joint Mathematics stream in Computer Science
Specialist Program
12 half-courses chosen with the approval of the program
supervisor for Mathematics and Its Applications
[PSCD02H Current Questions in Mathematics and Science
or
PSCD03H Computers in Contemporary Society]
Students must choose one of the following options:
Supervisor: H.S. Rosenthal
(416-287-7268)
MATA26Y Calculus
MATA23H Linear Algebra I
MATB24H Linear Algebra II
CSCA58H Introduction to Computer Science
CSCB70H Fundamental Data Structures and Techniques
or
CSCB38H Discrete Mathematics for Computer Science
MATC01H Groups and Symmetry
MATB41H Techniques of the Calculus of Several Variables I
MATB42H Techniques of the Calculus of Several Variables II
STAB22H Statistics
a) 1.5 F.C.E. from:
MATB43H, MATB44H (C51H), MATC46H (C56H), MATC35H,
MATC38H (C57H), MATC34H (C60H), MATC65H
b) 0.5 F.C.E. from:
MATC25H (B30H), MATB61H, MATC02H (B32H), MATC15H
(B70H), MATC30H, MATC44H (C31H), MATC32H, MATC61H, MATC63H (C54H)
c) 1.0 F.C.E. from:
CSCB28H, CSCB38H, CSCB58H, CSCB70H, CSCC24H, CSCC50H, CSCC51H, CSCC54H, CSCC64H, CSCC78H, CSCC85H, CSC300 / PSCD03, CSC318, CSC340, CSC
D-level, CSC 400-level, MATC44H (C31H), MATC32H,
MATD01H, MATD02H, STAB47H, STAC42H, STAC62H, STAC67H, STA300-level,
STA400-level
Supervisor: M. Evans (416-287-7274)
CSCA58H Introduction to Computer Science
MATA23H Linear Algebra I
MATB24H Linear Algebra II
MATA26Y Calculus
CSCB70H Fundamental Data Structures and Techniques
MATB41H Techniques of the Calculus of Several Variables I
MATB42H Techniques of the Calculus of Several Variables II
STAB22H Statistics
STAB47H Introduction to Probability Theory and Mathematical
Statistics
a) 2.0 F.C.E. from:
STAC42H, STAC52H, STAC57H, STAC62H, STAC67H, any
300- and 400-level STA courses on St. George campus.
b) 1.0 F.C.E. from:
MATB43H, MATB44H, MATC35H, MATC38H, MATC46H, MATC34H,
C65H, MATC25H, MATC91H, MATC15H, C30H, C32H, C43H, MATC44H, MATC32H,
MATC63H (C54H), STAC42H, STAC52H, STAC57H, STAC62H, STAC67H, any
300- and 400-level STA course on St. George campus. CSCB28H, CSCB38H,
CSCB58H, CSCC24H, CSCC50H, CSCC51H, CSCC54H, CSCC64H, CSCC78H,
CSCC85H, CSC300/PSCD03, CSC318, CSC340, MATC44H CSCC32H, any D-level
or 400-level CSC course.
Matrices, linear systems, elementary matrices and the inverse of a matrix. Vector spaces over R, subspaces, basis and dimension. Real inner product spaces, geometry in Rn, lines and hyperplanes. Linear transformation, kernel, range, matrix representation, isomorphisms. The determinant, Cramer's rule, the adjoint matrix. Eigenvalues, eigenvectors, similarity, diagonalization. Projections, Gram-Schmidt process, orthogonal transformations and orthogonal diagonalization, isometries, quadratic forms, conics, quadric surfaces.
Two one-hour lectures per week and one two-hour tutorial per week.
Exclusion: (MATA04), MAT223
Prerequisite: OAC Calculus & OAC Algebra and
Geometry
First term of MATA26Y. Students in academic difficulty in MATA26Y may withdraw from MATA26Y and enrol in MATA24S in the Spring term.
These students are informed of this option by the end of the Fall term. Classes begin in the first week of the Spring term; late enrolment is not permitted. MATA24S together with MATA25F is equivalent for program and prerequisite purposes to MATA26Y. Students not enrolled in MATA26Y in the Fall term will not be allowed to enrol in MATA24S, with the following exception. A student who has successfully completed PHYA10F and wishes to enrol in PHYA21S, but is not enrolled in MATA26Y, may enrol in MATA24S with the consent of the instructor.
Two one-hour lectures per week and a one hour tutorial per week.
Exclusion: MATA26Y, (MATA28Y), MATA27H; MATA29Y, MAT130, 133, 134, 135, 137, 139, 149
Prerequisites: Enrolment in MATA26Y, and withdrawal
from MATA26Y after midterm, with a mark of at least 20% in the
midterm.
Second term content of MATA26Y; the final examination includes topics covered in MATA24H. Offered in the Summer Session only; students who have not previously enrolled in MATA24H will NOT be allowed to enrol in MATA25H. MATA24H together with MATA25F is equivalent for program and MAT prerequisite purposes to MATA26Y.
Two one-hour lectures per week and a one hour tutorial per week.
Exclusion: MATA26Y, (MATA28Y), MATA27H MATA29Y, MAT123, 124, 125, 126, 133, 135, 137
Prerequisites: MATA24S successfully completed.
This course includes: limits and continuity, derivatives, related rates, extremum problems, graph sketching, Newton's method, indefinite and definite integrals, numerical integration, Taylor approximation and differential equations.
MATA26Y introduces the basic techniques of calculus with a strong emphasis on methods of approximation. The course will develop these ideas by the investigation of specific examples. MATA26Y is a demanding course which will equip the student for most sciences and for further work in mathematics.
Two one-hour lectures per week and a one hour tutorial per week.
Exclusion: (MATA28Y), MATA27H, MATA29Y, MAT123, 124, 125, 126, 133, 135, 137,
Prerequisite: [OAC Calculus] & [one of OAC Algebra
and Geometry or OAC Finite Mathematics]
Areas to be covered include: R^n as a model space in economics, contour maps; exponential functions and logarithms, with applications to finance; review of differential calculus in one variable, marginal analysis; vectors, differential calculus of several variables and optimization, with applications to economics.
Two one-hour lectures per week and a one hour tutorial per week.
Exclusions: MATA26Y (MATA28Y), MATA29Y, MATA24H, MATA25H, MAT123, 124, 125, 126, 133, 135, 137
Prerequisite: OAC Calculus
NOTE:
This course is intended for students in Management Programs. It
does not satisfy the requirements for any Program in the Physical
or Life Sciences, and it may not be used as a prerequisite for
any further Mathematics course.
An introductory course in Mathematics built around applications in Life Science and Environmental Science. Discrete probability theory, applications to genetics, exponential and logarithm functions, plotting data; basic calculus: limits and derivatives, maxima and minima, integration, growth rates, diffusion rates; differential equations, population dynamics; vectors and matrices in two and three dimensions; basic statistics: hypothesis testing, distributional analysis.
Two one-hour lectures per week and a one hour tutorial per week.
Exclusion: MATA26Y, MATA28Y, MATA27H, MATA24H, MATA25H, MATB41H, MAT123, 124, 125, 126, 133, 135, 137; JMB170
Prerequisites: OAC Calculus and [OAC Algebra and Geometry or OAC Finite Mathematics]
NOTE: This
course will not satisfy the Mathematics requirements for most
Programs in Physical Sciences, nor will it normally serve as a
pre-requisite for further courses in Mathematics. Students who
are not sure which MAT A-level course they should choose are encouraged
to consult with the supervisor(s) of Programs in their area(s)
of interest.
Fields. Vector spaces over a field. Linear transformations, dual spaces. Diagonalizability, direct sums. Invariant subspaces, Cayley-Hamilton theorem. Complex inner product, orthogonality, the adjoint of a linear operator, the projection matrix and the method of least squares. Normal,
self-adjoint and unitary operators. Spectral theorem. Conditioning and Rayleigh quotient. Jordan canonical form.
Two one-hour lectures per week and a two hour tutorial per week.
Exclusion: MATA04Y, MAT224
Prerequisite: MATA23H, MAT223
A study of Vector algebra in Rn, lines and planes in R3, complex numbers, matrices, determinants and linear equations, functions of several variables, partial derivatives, gradient, tangent plane, Jacobian matrix and chain rule, Taylor series, extremal problems, extremal problems with constraints and Lagrange multipliers, multiple integrals, spherical and cylindrical coordinates, law of transformation of variables.
Two one-hour lectures per week and a one hour tutorial per week.
Exclusion: MATA28Y, MATA29Y, MAT230, 234, 235, 237, 239, 257
Prerequisite: MATA23H, MATA26Y
Fourier series. Vector fields in Rn, Divergence and curl, curves, parametric representation of curves, path and line integrals, surfaces, parametric representations of surfaces, surface integrals. Green's, Gauss', and Stokes' theorems will also be covered. An introduction to differential forms, total derivative.
Two one-hour lectures per week and a one hour tutorial per week.
Exclusion: MAT230, 234, 235, 237, 239, 257
Prerequisite: MATB41H
This course is designed for students whose interest
in mathematics has been stimulated by their experience in first
year mathematics courses, and who wish to acquire the analytic
techniques which are essential for more advanced work. There will
be a fundamental emphasis on rigorous analytic proofs. Students
will study the least upper bound principle for R, limits in R
and R2, continuous functions in one and two variables, space filling
curves and nowhere differentiable functions, existence of extrema
on closed and bounded sets, mean value theorems and the fundamental
theorems
of the calculus, the Riemann integral.
Two one-hour lectures per week and a one hour tutorial per week.
Exclusion: (MATA27Y); MAT246
Prerequisite: MATA26Y & [MATA23H & MATB24H (MATA04Y)]
Corequisite: MATB42H
(Formerly MATC51H)
Ordinary differential equations of the first and second order, existence and uniqueness; solutions by series and integrals; linear systems of first order; non-linear equations; difference equations.
Two one-hour lectures per week and a one hour tutorial per week.
Exclusion: (MATC51), MAT244, 267
Prerequisite: MATA26Y & MATA23H
Co-requisite: MATB41H & MATB24H
Linear programming, simplex algorithm, duality theory, interior point method; quadratic and convex optimization, stochastic programming; applications to portfolio optimization and operations research.
Two one-hour lectures per week and a one hour tutorial per week.
Exclusion: APM261H
Prerequisite: MATA23H
Co-requisite: MATB42H
Congruences and fields. Permutations and permutation groups. Linear groups. Abstract groups, homomorphisms, subgroups. Symmetry groups of regular polygons and Platonic solids, wallpaper groups. Group actions, class formula. Cosets, Lagrange's theorem. Normal subgroups, quotient groups. Emphasis on examples and calulations.
Two one-hour lectures per week and a one hour tutorial per week.
Exclusion: (MATC91H, MATB31), MAT301, (MAT347)
Prerequisite: MATB24H
(Formerly MATB32H)
Introduction to abstract group theory: Sylow theorems, groups of small order. Simple groups. Classification of finite abelian groups. Fields and Galois theory: Polynomials over a field. Field extensions, adjunction of roots of a polynomial.
Constructibility, trisection of angles, construction of regular polygons. Galois groups of polynomials, in particular cubics, quartics. Solvable groups. Insolvability of quintics by radicals.
Two one-hour lectures per week and a one hour tutorial per week.
Exclusion: (MATB32H), MAT302 (MAT347)
Prerequisite: MATC01H
(Formerly MATB70)
Elementary topics in number theory; arithmetic functions; polynomials over the residue classes modulo m, characters on the residue classes modulo m; quadratic reciprocity law, representation of numbers as sums of squares.
Two one-hour lectures per week and a one hour tutorial per week.
Exclusion: (MATB70), MAT315
Prerequisite: MATA26Y & MATB24H, (MATA04Y)
An introduction to geometry with a selection of topics from the following: symmetry and symmetry groups, finite geometries and applications, non-Euclidean geometry.
Two one-hour lectures per week and a one hour tutorial per week.
Exclusion: (MATB30), MAT365 (325)
Prerequisite: MATA23H
Co-requisite: MATC01H
(Formerly MATC60H)
Theory of functions of one complex variable, analytic and meromorphic functions. Cauchy's theorem, residue calculus, conformal mappings, introduction to analytic continuation and harmonic functions.
Two one-hour lectures per week and a one hour tutorial per week.
Exclusion: (MATC60H), MAT334
Prerequisite: MATB42
Areas covered include: metric spaces, dynamics on
the real line, fixed points, periodic points, attractors, repellors,
Sarkovskii's theoren parametrized families of functions and bifurcations,
period doubling, dynamics of the logistic
map, symbolic dynamics, chaos, topological equivalence of the logistic map and the shift map, Newton's method; dynamics on the complex line, iterations of rational functions, Julia sets, Mandelbrot set.
Exclusion: MAT335
Prerequisite: MATB43H
Metric spaces, completeness, uniform convergence. Topics in measure theory: the Lebesgue integral, Riemann-Stieltjes integral, Lp spaces, Fourier series.
Exclusion: MAT337 (338)
Prerequisite: MATB42H & MATB43H
Basic counting principles, generating functions, permutations with restrictions. Fundamentals of graph theory with algorithms; applications (including network flows). Combinatorial structures including block designs and finite geometries.
Exclusion: (MATC31), MAT344
Prerequisite: MATB24H
(Formerly MATC56)
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.
Two one-hour lectures per week and a one hour tutorial per week.
Exclusion: (MATC56), MAT267
Prerequisite: MATB44H & MATB24H
Co-requisite: MATB42
Brownian motions, Fokker-Planck equation, stopping times, reflection principle, Girsanov theorem; Stochastic calculus, Ito's lemma, martingales; stochastic optimization, Black-Scholes equation. The course provides an introduction to methods of interest in financial mathematics.
Two one-hour lectures per week and a one hour tutorial per week.
Prerequisite: MATB42H, STAB47H
Corequisites: MATB61H & MATC46H
Recommended: STAC62H
Independent study under the direction of a faculty member.
Prerequisite: A G.P.A. of 2.5 or more and consent
of the instructor.
Independent study under the direction of a faculty member.
Prerequisite: A GPA of 2.5 or more and consent of
the instructor.
MATC30H3 Geometry II
Exclusion: (MATB25Y)
Prerequisite: MATC25H
MATC31H3 Combinatorics
Prerequisites: MATB44H or (CSCB38) & at
least one other B-level course in mathematics or computer science
MATC32H3 Graph Theory
Exclusion: (JMCC32H)
Prerequisites: [MATB31H or CSCB38H] & at
least one other B-level course in Mathematics or Computer Science
(expected to be offered in 2002/2003)
MATC53Y3 Real Analysis
Exclusion: (MATC50H), (MATC55H), MAT338, 350
Prerequisite: MATB32H (MATB49H) & MATB41H &
MATB42H & MATB43H
MATC63H3 Differential Geometry
Exclusions: MATC54, MAT363
Prerequisites: MATB43
MATC65H3 Complex Variables II
Exclusion: MAT354
Prerequisites: MATB34
(Expected to be offered in 2002/2003)
University of Toronto at Scarborough 2001/2002 Calendar
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