Mathematics and Statistics

Chair

Stephen Anco

Professors Emeriti

Howard E. Bell, Ronald A. Kerman, Charles F. Laywine, Eric Muller

Professors

S. Ejaz Ahmed, Stephen Anco, Hichem Ben-El-Mechaiekh, Henryk Fukś, Mei Ling Huang, Omar Kihel, Yuanlin Li, Alexander Odesskii, Jan Vrbik, Thomas Wolf, Chantal Buteau, Xiojian Xu

Associate Professors

Babak Farzad, William J. Ralph

Adjunct Professors

Thomas A. Jenkyns, Peng Zhang

Instructor/Manager of Academic Support

Dorothy Levay

Director, Co-op, Career and Experiential Education

Cara Krezek

Mathematics Development Programs Co-ordinator

Neil Marshall

Online Course Developer/Administrator

Mark Willoughby

General Information

Administrative Assistant

Margaret Thomson

905-688-5550, extension 3300

Mackenzie Chown J415

brocku.ca/mathematics

The Department of Mathematics and Statistics offers several different programs of study, covering traditional and modern areas of mathematics, statistics, and mathematics education.

The BSc Honours in Mathematics and Statistics is a flexible program designed to meet a wide range of interests. There are core courses in calculus, linear algebra, differential equations, probability and statistics, as well as the use of computers to explore and solve mathematical problems. Within this program, students have the option to select a specialized concentration in one of five areas: Applied Mathematics; Pure Mathematics; Mathematics Integrated with Computers and Applications; Mathematics Education; and Statistics. Students also have the opportunity to complete an honours project or thesis under the supervision of a faculty member.

Normally the BSc Honours program requires four years to complete. A unique Accelerated Mathematics Studies stream is available for high-achieving students who want to complete the BSc Honours program in three years by following a personalized and accelerated study-plan. Students in this stream will be able to take accelerated reading courses and optional summer courses in mathematics and statistics.

For students interested in combining academic studies with real-world work experience, the Department offers Co-op programs in Mathematics or Statistics. Completion requires four and one-half years.

Minor programs in either Mathematics or Statistics, as well as a three-year pass program in Mathematics and Statistics are also offered.

The Department has a special interest in Mathematics Education and offers several programs and courses specifically for prospective teachers. These include both Concurrent and Consecutive Education programs, as well as Minors for future teachers.

Because of the diversity of course options available in all programs, students should discuss their particular interests with faculty before selecting elective courses.

Accelerated Mathematics Studies Stream

The Accelerated Mathematics Studies stream is designed for students who have exceptional mathematical abilities and who are interested in completing an accelerated BSc Honours program in three years. Strengths of each student will be assessed and a personalized study plan will be created.

Enrolment in the stream is limited due to the personalized nature of the program delivery and the individual attention given to students. In addition to the normal application procedures for admission to undergraduate degree studies, students will be assessed on the following criteria:

	required 4U subjects: Advanced Functions (MHF4U) and Calculus and Vectors (MCV4U) or equivalent or an equivalent amount of material studied in home schooling courses. Applicants with a minimum 95 percent in these two courses automatically qualify for the admission interview and the entrance examination. For applicants with a minimum 90 percent in the two 4U mathematics courses or their equivalent, permission to proceed with the admission interview and the entrance examination shall depend on the quality of the written personal statement and the portfolio of mathematical activities.
	written personal statement describing the motivations to join the accelerated stream and the personal aptitudes and experiences conducive to success in the program.
	portfolio of mathematical activities completed in addition to high school studies (e.g., participation in mathematics contests, extra-curricular mathematical training programs, independent studies, home schooling, work or related professional development, math camps, math projects at science fairs).
	passing an entrance examination (including written communication skills and mathematical proficiency).
	passing an admission interview (face-to-face or by video conferencing) aimed at assessing if the applicant has adequate knowledge and skills as well as the ability to study independently.
	strongly recommended subject: English (ENG4U) or equivalent.

Evidence of successful engagement in recognized mathematical activities, or completion of advanced mathematical training or any relevant mathematical achievements, together with scores on the entrance placement exam may qualify successful applicants for advanced standing credits. Relevant work-related experience may qualify successful applicants for challenge for credit.

Completion of a minimum of 7.0 overall credits and a minimum 85 percent major average is required for progression into year 2. A minimum of 13.0 overall credits and a minimum 85 percent major average in second year courses is required for progression into year 3.

Mathematics Co-op and Statistics Co-op

Both Co-op programs combine academic and work terms over a period of four and one-half academic years. Students spend at least two years in an academic setting studying core concepts and methodologies in Mathematics and Statistics prior to their first work placement. The study will provide the necessary academic context for the work experience.

In addition to the current fees for courses in academic study terms, the students in both co-op programs are assessed an administrative fee for each work term (see the Schedule of Fees).

Eligibility to continue in either co-op program is based on a student's major and non-major averages. A student with a minimum 70 percent major average and a minimum 60 percent non-major average may continue. A student with a major average lower than 70 percent will not be permitted to continue in either Co-op program. If a student subsequently raises his/her major average to 70 percent, the student may be readmitted only if approved by the Co-op Admissions Committee. For further information, see the Co-op Programs section of the Calendar.

All students in the Co-operative Education programs are required to read, sign and adhere to the terms of the Student Regulations Waiver and Co-op Student Manuals (brocku.ca/co-op/current-students/co-op-student-manuals) as articulated by the Co-op Programs Office. In addition, eligibility to continue in the co-op option is based on the student's major average and non-major average, and the ability to demonstrate the motivation and potential to pursue a professional career.

Each four-month co-operative education work term must be registered. Once students are registered in a co-op work term, they are expected to fulfill their commitment. If the placement accepted is for more than one four-month work term, students are committed to complete all terms. Students may not withdraw from or terminate a work term without permission from the Director, Co-op Program Office.

Certain courses are required for any degree in Mathematics (see below). Because Mathematics majors need both facility in dealing with mathematical theories and experience in the application of mathematics to real-world problems, each student should discuss his or her particular interests with faculty before selecting elective courses.

Mathematics and Computer Science Co-op Program

The Mathematics and Computer Science Co-op program combines academic and work terms over a period of four and one-half academic years. Students spend one and one-half years in an academic setting studying the fundamentals of Mathematics and Computer Science prior to their first work placement. Successful completion of courses in the core areas of Computer Science and Mathematics provides the necessary academic background for the work experience. In addition to the current fees for courses in academic study terms, Mathematics and Computer Science Co-op students are assessed an administrative fee for each work term (see the Schedule of Fees).

Eligibility to continue in the Mathematics and Computer Science Co-op program is based on the student's major and non-major averages. A student with a minimum 70 percent major average and a minimum 60 percent non-major average may continue. A student with a major average lower than 70 percent will not be permitted to continue in the Mathematics and Computer Science Co-op program. If a student subsequently raises his/her major average to 70 percent, the student may be readmitted only if approved by the Co-op Admissions Committee. For further information, see the Co-op Programs section of the Calendar.

All students in the Co-operative Education program are required to read, sign and adhere to the terms of the Student Regulations Waiver and Co-op Student Manuals (brocku.ca/co-op/current-students/co-op-student-manuals) as articulated by the Co-op Programs Office. In addition, eligibility to continue in the co-op option is based on the student's major average and non-major average, and the ability to demonstrate the motivation and potential to pursue a professional career.

Each four-month co-operative education work term must be registered. Once students are registered in a co-op work term, they are expected to fulfill their commitment. If the placement accepted is for more than one four-month work term, students are committed to complete all terms. Students may not withdraw from or terminate a work term without permission from the Director, Co-op Program Office.

The Mathematics and Computer Science Co-op program designation will be awarded to those students who have honours standing and who have successfully completed a minimum of twelve months of Co-op work experience.

Program Notes

Mathematics and Statistics Honours Program

Year 1

	MATH 1P01, 1P02 and 1P11
	MATH 1P40 or 1P66
	three elective credits (see program notes 1 and 2)

Year 2

	MATH 2P03, 2P12, 2P81 and 2P82
	MATH 2P08 or 2P40
	one-half MATH credit numbered 2(alpha)00 or above
	the Humanities context credit, Sciences context credit or Social Sciences context credit not taken in year 1
	one elective credit (see program note 3)

Year 3

	One of MATH 3P03, 3P12, 3P81, 3P82
	one of MATH 2P92, 2P94, 3P40, 3P41, 3P51, 3P52, 3P81, 3P82, 3P96
	three MATH credits numbered 2(alpha)90 or above (see program note 3)
	one elective credit (see program note 3)

Year 4

	One MATH credit numbered 2(alpha)00 or above
	three MATH credits numbered 3(alpha)90 or above (see program note 3)
	one elective credit

Accelerated Mathematics Studies Stream

Year 1

	MATH 1P01, 1P02, 1P11, 2P03 and 2P12
	MATH 1P40 or 1P66
	MATH 2P08 or 2P40
	one and one-half MATH credits numbered 1(alpha)00 or above (see program notes 1, 2 and 7)
	two elective credits (see program note 1)

Year 2

	MATH 2P81 and 2P82
	one of MATH 2P92, 3P23, 3P40, 3P41, 3P51, 3P52, 3P81, 3P82, 3P96
	one of MATH 3P03, 3P12, 3P81, 3P82
	two MATH credits numbered 2(alpha)90 or above (see program notes 3, 4, 5 and 7)
	two elective credits (see program note 1)

Year 3

	MATH 4F90
	five MATH credits numbered 3(alpha)00 or above (see program notes 3, 4, 5 and 7)
	one elective credit

Mathematics Co-op (Honours only)

Year 1

	MATH 1P01, 1P02, 1P11 and 1P40
	three elective credits (see program note 1)

Year 2

	MATH 2P03, 2P08, 2P12, 2P40, 2P81 and 2P82
	SCIE 0N90
	the Humanities context credit, Sciences context credit or Social Sciences context credit not taken in year 1
	one elective credit

Spring/Summer Sessions:

MATH 0N01 and 2C01

Year 3

Fall Term:

	MATH 3P12 and 3P81
	one MATH credit numbered 3(alpha)00 or above
	one-half elective credit

Winter Term:

MATH 0N02 and 2C02

Year 4

	MATH 3P40 and 3P82
	one credit from MATH 3P04, 3P85, 3P86, 4P06, 4P09, 4P13, 4P84, 4P92, 4P94
	one credit from MATH 4P09, 4P11, 4P84, 4P92, 4P94
	one MATH credit numbered 3(alpha)00 or above
	one elective credit

Spring/Summer Sessions:

MATH 0N03 and 2C03

Year 5

Fall Term:

	One and one-half MATH credits numbered 4(alpha)00 or above
	one elective credit

Statistics Co-op (Honours only)

Year 1

	MATH 1P01, 1P02, 1P11 and 1P40
	three elective credits (see program note 1)

Year 2

	MATH 2P03, 2P08, 2P12, 2P40, 2P81 and 2P82
	the Humanities context credit, Sciences context credit or Social Sciences context credit not taken in year 1
	one elective credit (see program note 6)

Year 3

	MATH 3P81, 3P82, 3P85 and 3P86
	one MATH credit numbered 3(alpha)00 or above (see program note 6)
	SCIE 0N90
	two elective credits (see program note 6)

Spring/Summer Sessions:

MATH 0N01 and 2C01

Year 4

Fall Term:

MATH 0N02 and 2C02

Winter Term:

	MATH 4P82 and 4P85
	one-half MATH credit (see program notes 6 and 7)
	one elective credit (see program notes 6 and 7)

Spring/Summer Sessions:

MATH 0N03 and 2C03

Year 5

Fall Term:

	MATH 4P81 and 4P84
	one and one-half MATH credits (see program notes 6 and 7)

Mathematics Pass Program

Year 1

	MATH 1P01, 1P02 and 1P11
	MATH 1P40 or 1P66
	three elective credits (see program note 1)

Year 2

	MATH 2P03
	one credit from MATH 2P08 and 2P12, MATH 2P12 and 2P40, MATH 2P12 and 2P91, MATH 2P81 and 2P82
	one-half MATH credit numbered 2(alpha)00 or above
	the Humanities context credit, Sciences context credit or Social Sciences context credit not taken in year 1
	two elective credits

Year 3

	Three MATH credits numbered 3(alpha)00 or above (see program note 3)
	two elective credits

Combined Major Program

Combined major programs have been developed by the Department of Mathematics and Statistics in co-operation with each of these departments: Biological Sciences, Chemistry, Computer Science, Economics and Physics. Program requirements are listed in the calendar sections of the co-major discipline. Students may take a combined major in Mathematics and a second discipline. For requirements in the other discipline, the student should consult the relevant department/centre. It should be noted that not all departments/centres provide a combined major option.

Mathematics and Computer Science Co-op (Honours only)

Students admitted to the Mathematics and Computer Science Co-op program must follow an approved program pattern. The most common pattern is listed below. For other approved patterns, consult the Co-op Office.

Year 1

	MATH 1P01, 1P02, 1P11, 1P40, 1P66 and 1P67
	COSC 1P02 and 1P03
	one Humanities context credit

Year 2

Fall Term:

	MATH 2P03 and 2P81
	COSC 1P50, 2P03 and 2P12
	SCIE 0N90

Winter Term:

MATH 0N01 and 2C01

Spring/Summer Sessions:

	COSC 2P13 and 3P32
	one Social Sciences context credit
	one-half elective credit

Year 3

Fall term

MATH 0N02 and 2C02

Winter Term

	MATH 2P12 or 2P92
	COSC 2P05
	one-half COSC credit (see program note 7)
	one elective credit

Spring/Summer Sessions:

MATH 0N03 and 2C03

Year 4

	MATH 2P40, 2P82, 3P60 and 4P61
	COSC 3P03 and 4F00
	one-half MATH credit
	one COSC credit (see program note 7)

Year 5

Fall Term:

	One MATH credit (see program note 7)
	one COSC credit (see program note 7)
	one-half elective credit (see program note 7)

Programs and Courses for Future Teachers

The Department of Mathematics and Statistics has identified courses that are particularly appropriate for students preparing to become teachers at either the elementary or secondary levels. Students should consult the Chair in the selection of courses.

Elementary

Three credits for a teachable subject at the Junior/Intermediate level. May include MATH 1F92, 1P05, 1P06, 1P66, 2P90, 3P23 and 3P91; MATH 1P11 or 1P12.

Secondary

For Mathematics as the first teachable subject (a minimum of five credits). An Honours degree in Mathematics is recommended.

For Mathematics as the second teachable subject, a minimum of three credits. For example: MATH 1P01, 1P02, 1P12, 2P90 and one-half MATH credit; MATH 3P23 or 4P96.

Concurrent BSc/BEd

The Department of Mathematics and Statistics and the Faculty of Education co-operate in offering two Concurrent BSc (Honours)/BEd programs. The Mathematics BSc (Honours)/BEd programs combines the BSc Honours program or BSc Integrated Studies Honours program with the teacher education program for students interested in teaching at the Intermediate/Senior level (grades 7-12) and at the Junior/Intermediate level (grades 4-10). Refer to the Education - Concurrent BSc (Honours)/BEd (Intermediate/Senior) or Education - Concurrent BSc Integrated Studies (Honours)/BEd (Junior/Intermediate) program listings for further information.

Concentration Program

Concentration in Applied Mathematics

The Concentration in Applied Mathematics program is designed for students who want a solid foundation in mathematics, statistics and computing along with exposure to modern topics like mathematics of networks, dynamical systems, cryptography, soliton theory, and mathematical physics. Graduates of this program will be able to pursue a variety of jobs in industries such as energy, aerospace and telecommunications, health/medical technology, mathematical software, and also will be well prepared to undertake graduate studies in applied mathematics or computational science or mathematical physics.

Students may earn a Concentration in Applied Mathematics by successfully completing the following courses as part of the academic work leading to a BSc (Honours) in Mathematics and Statistics:

	MATH 1P40, 1P66, 2P04, 2P08, 2P13, 3P04, 3P06, 3P08, 3P09 and 3P12
	one of MATH 2P91, 2P92, 2P94, 2P95, 2P97
	one credit from MATH 2P91, 2P97, 3P03, 3P13, 3P40, 3P60, 3P72, 3P75, 3P95, 3P97, 3P98 (see program notes 3 and 4)
	one credit from MATH 3P51, 3P52, 3P96
	three and one-half credits from MATH 3P95, 3P97, 3P98, 4F90, 4P03, 4P06, 4P09, 4P11 or 4P13, 4P71, 4P92, 4P94 (see program note 4)

Concentration in Mathematics Education

The Concentration in Mathematics Education program is for the modern teacher who wants to harness the technology that is transforming the teaching and learning of mathematics in the twenty-first century. Graduates of this program will answer the Ontario government's call for mathematics educators who are experts in teaching with a variety of technological tools. Future teachers will obtain a strong background in traditional mathematics such as calculus, algebra and geometry, and also will study the historical and contemporary contributions of mathematics to civilization. In addition, students will learn how technology can be used to create interactive learning environments, to model the real world, and to visualize information. This unique program provides several courses designed specifically for future mathematics teachers.

Students may earn a Concentration in Mathematics Education by successfully completing the following courses as part of the academic work leading to a BSc (Honours) in Mathematics and Statistics:

	MATH 1P40, 2P03, 2P12, 2P40, 2P90, 3P12, 3P23, 3P41, 3P91, 4P23 and 4P96
	MATH 2P92 or 2P94

Concentration in Mathematics Integrated with Computers and Applications (MICA)

The Concentration in Mathematics Integrated with Computers and Applications (MICA) program, focusing on technology, is ideal for students desiring careers in the application of mathematics to science, industry and finance. Graduates of this program have gone on to obtain Masters degrees in areas such as finance and computational mathematics. Students receive a solid grounding in mathematical theory and also learn how to use computer and information technology to apply and present what they have learned. The core of the MICA program consists of MATH 1P40, 2P40 and 3P40 in which students confront real world problems requiring them to create mathematical models and run computer simulations. In solving such problems, students are encouraged to develop their own strategies for using the best combination of mathematics and computing.

Students may earn a Concentration in Mathematics Integrated with Computers and Applications by successfully completing the following courses as part of the academic work leading to a BSc (Honours) in Mathematics and Statistics:

	MATH 1P40, 2P08, 2P40, 3P40, 3P52, 3P81, 3P82 and 4F90
	one and one-half credits from MATH 2P94, 3P08, 3P09, 3P51, 3P60, 3P72, 3P75, 3P85, 3P96
	MATH 3P03 or 3P12

Concentration in Pure Mathematics

The Concentration in Pure Mathematics program provides students with breadth and depth of knowledge of concepts, methodologies, techniques and aptitudes needed to become a professional mathematician. In addition to the core courses required by all concentrations, it includes advanced courses in central areas of algebra and analysis, as well as choice of electives in mathematical specialities of particular interest to the students. Suited for those who intend to pursue further studies in pure or applied mathematics as it includes foundational courses commonly required for admission to graduate programs in the mathematical sciences. Also serves those joining the work force through the development of logical reasoning, computational preferences, analytical and problem solving skills and creative thinking.

Students may earn a Concentration in Pure Mathematics by successfully completing the following courses as part of the academic work leading to a BSc (Honours) in Mathematics and Statistics:

	MATH 1P66, 2P04, 2P08, 2P13, 3P03, 3P04, 3P12 and 3P13
	one credit from MATH 2P75, 2P92, 2P97
	MATH 2P92 or 2P94
	one and one-half credits from MATH 2P97, 3P06, 3P08, 3P09, 3P60, 3P72, 3P75, 3P95, 3P96, 3P97, 3P98 (see program notes 3 and 5)
	four credits from MATH 4F90, 4P03, 4P06, 4P09, 4P11, 4P13, 4P61, 4P71, 4P92, 4P94, 4P96 (see program note 5)

Concentration in Statistics

The Concentration in Statistics program gives a student the opportunity to prepare for advanced study of statistics. Mathematical statistics theory and practical applications of statistical inferences, statistical models, stochastic models, experimental design, sampling and other methods. Development of the student's abilities in logical reasoning, creative thinking, problem solving, case studies and computation skills. Application of statistical knowledge to many areas, such as actuarial science, biological science, business, economics, education, engineering, agriculture and public health. After graduation, students can advance to graduate studies, or find jobs with employers such as Statistics Canada, hospitals, financial institutions, insurance companies and various business.

Students may earn a Concentration in Statistics by successfully completing the following courses as part of the academic work leading to a BSc (Honours) in Mathematics and Statistics:

	MATH 1P40, 2P08, 2P40, 3P81, 3P82, 3P85, 3P86, 4P81, 4P82, 4P84 and 4P85
	MATH 2P04 or 2P75

Minor Program

Minor in Mathematics

Students in other disciplines may obtain a Minor in Mathematics within their degree program by completing the following courses with a minimum 60 percent average:

	MATH 1P01 or 1P05
	MATH 1P02 or 1P06
	MATH 1P11 or 1P12
	MATH 1P40 or 1P66
	one credit from MATH 2P03, 2P08, 2P12, 2P40, 2P81, 2P82
	one MATH credit numbered 2(alpha)00 or above
	one MATH credit numbered 3(alpha)00 or above

Minor in Statistics

Students in other disciplines may obtain a Minor in Statistics within their degree program by completing the following courses with a minimum 60 percent average:

	MATH 1P01 or 1P05
	MATH 1P02 or 1P06
	MAH 1P11 or 1P12
	MATH 2P03, 2P81, 2P82, 3P81, 3P82, 3P85, 3P86, 4P81 and 4P82

Minor Programs for Teachers

Students intending to become elementary teachers, who are in another discipline, can obtain a Minor in Elementary Teaching Mathematics within their degree program by completing the following courses with a minimum 60 percent overall average:

	MATH 1P11 or 1P12
	MATH 1P66, 1P97, 1P98, 2P90 and 3P91
	MATH 2P91 or 2P92
	one-half MATH credit

Students intending to become secondary teachers, who are in another discipline, can obtain a Minor in Secondary Teaching Mathematics within their degree program by completing the following courses with a minimum 60 percent overall average:

	MATH 1P01, 1P02, 1P40 and 2P90
	MATH 1P11 or 1P12
	one of MATH 3P23, 3P41, 4P96
	two MATH credits numbered 2(alpha)00 or above

Course Descriptions

Note that not all courses are offered in every session. Refer to the applicable term timetable for details.

# Indicates a cross listed course

* Indicates primary offering of a cross listed course

Prerequisites and Restrictions

Students must check to ensure that prerequisites are met. Students may be deregistered, at the request of the instructor, from any course for which prerequisites and/or restrictions have not been met.

MATHEMATICS COURSES

MATH 1F92

Introductory Statistics

Describing and comparing data sets, linear regression analysis, basic probability theory, discrete probability distributions, binomial and normal distributions, Central Limit Theorem, confidence intervals and hypothesis tests on means and proportions, properties of t-, F- and chi-squared distributions, analysis of variance, inference on regression. Emphasis on interpretation of numerical results for all topics. Use of Minitab.

Lectures, 3 hours per week.

Prerequisite(s): one grade 11 mathematics credit.

Note: designed for non-science majors. Not open to students with credit in any university mathematics or statistics course. Major credit will not be granted to Mathematics majors.

MATH 1P01

Calculus Concepts I

Differential calculus emphasizing on concepts and the use of both theory and computers to solve problems. Precalculus topics, limits, continuity and the intermediate value theorem, derivatives and differentiability, implicit differentiation, linear approximation, mean value theorem with proof and applications, max and min, related rates, curve sketching, l'Hospital's rule, antiderivatives, Riemann sums, FTC with proof, integration by substitution. Use of Maple.

Lectures, 4 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): two grade 12 mathematics credits including MCV4U or permission of the instructor.

Note: intended for mathematics majors and/or future teachers. Students must successfully complete the Mathematics Skills Test.

Completion of this course will replace previous assigned grade and credit obtained in MATH 1P05.

MATH 1P02

Calculus Concepts II

Integral calculus emphasizing concepts, theory and computers to solve problems. Further integration techniques. Applications to areas between curves, volumes, arc length and probabilities. Multivariable calculus: partial derivatives, optimization of functions of two variables. Sequences and series: convergence tests, Taylor and Maclaurin series and applications. Differential Equations: direction fields, separable equations, growth and decay, the logistic equation, linear equations. Use of Maple.

Lectures, 4 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 1P01, 1P05 or permission of instructor.

Note: intended for mathematics majors and/or future teachers.

Completion of this course will replace previous assigned grade and credit obtained in MATH 1P06.

MATH 1P05

Applied Calculus I

Differential calculus emphasizing problem solving, calculation and applications. Precalculus topics, limits and asymptotic analysis, continuity, derivatives and differentiability, implicit differentiation, linear approximation. Applications: slope, rates of change, maximum and minimum, convexity, curve sketching, L'Hospital's rule. Antiderivatives, integrals, fundamental theorem of calculus, integration by substitution. Use of a computer algebra system.

Lectures, 4 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): two grade 12 mathematics credits including MCV4U or permission of the instructor.

Note: designed for students in the sciences, computer science, and future teachers. Students must successfully complete the Mathematics Skills Test.

Completion of this course will replace previous assigned grade and credit obtained in MATH 1P01.

MATH 1P06

Applied Calculus II

Integral calculus emphasizing problem solving, calculations and applications. Further techniques of integration. Areas between curves, volumes, arc length and probabilities. 1st order differential equations. Sequences and series: convergence tests, Taylor and Maclaurin series and applications. Use of computer algebra system.

Lectures, 4 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 1P01 or 1P05.

Note: designed for students in the sciences, computer science, and future teachers.

Completion of this course will replace previous assigned grade and credit obtained in MATH 1P02.

MATH 1P11

Linear Algebra I

Review of linear systems and matrix algebra. General determinants and applications. Complex numbers. Vector geometry in R² to R³. Dot product, norm and projections, cross product, lines and planes. Finite-dimensional vector spaces (real and complex). Linear transformations from Rⁿ to R^m; rank and kernels. Eigenvalues. Use of a computer algebra system.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): two grade 12 mathematics credits including MCV4U.

MATH 1P12

Applied Linear Algebra

Systems of linear equations with applications. Matrix algebra. Determinants. Vector geometry in R² and R³ dot product, norm and projections, cross product, lines and planes. Complex numbers. Euclidean n-space. Linear transformations from Rⁿ to R^m. Focus on applications of linear algebra to sciences and integrated use of a computer algebra system.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): one grade 12 mathematics credit or permission of instructor.

Students will not receive earned credit for MATH 1P12 if MATH 1P11 has been successfully completed.

MATH 1P20

Introduction to Mathematics

Essential mathematics skills required for university mathematics courses. Sets, real and complex numbers, solutions of inequalities and equations, functions, inverse functions, composition of functions, polynomial functions, rational functions, trigonometry, trigonometric functions, trigonometric identities, exponential functions, logarithmic functions, mathematical induction, binomial theorem, vectors and matrices.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): one grade 11 mathematics credit.

Note: not open to students with credit in any university calculus course. Cannot be used toward a Mathematics teachable subject at Brock.

MATH 1P40

Mathematics Integrated with Computers and Applications I

Exploration of ideas and problems in algebra, differential equations, and dynamical systems using computers. Topics include number theory, integers mod p, roots of equations, fractals, predator-prey models and the discrete logistic equation for population growth.

Lectures, 2 hours per week; lab, 2 hours per week.

Restriction: open to MATH (single or combined), MATH (Honours)/BEd (Intermediate/Senior) majors and minors until date specified in Registration guide.

Prerequisite(s): MATH 1P01 or 1P05.

MATH 1P66

Mathematical Reasoning

Introduction to mathematical INDGraction, logic and proofs including mathematical induction.

Lectures, 3 hours per week.

Prerequisite(s): MATH 1P20 or one grade 12 mathematics credit.

Note: MCB4U recommended.

MATH 1P67

Mathematics for Computer Science

Development and analysis of algorithms, complexity of algorithms, recursion solving recurrence relations and relations and functions.

Lectures, 3 hours per week.

Prerequisite(s): MATH 1P66.

Corequisite(s): COSC 1P03.

MATH 1P70

Mathematics in Culture

Role of mathematics in past and contemporary cultures including applications to science, society and the arts. Topics may include the stock market, social media, cryptography, history of numbers, statistics in newspapers, game theory, music, epidemics, and mathematics in movies and television.

Lectures, 3 hours per week.

Restriction: not open to Mathematics (single or combined) majors.

Note: major credit will not be granted to Mathematics majors. Cannot be used toward a Mathematics teachable subject for programs at Brock.

MATH 1P97

Calculus With Applications

Lines, polynomials, logarithms and exponential functions; two-sided limits; rates of change using derivatives; max and min of functions using derivatives; higher derivatives and concavity; area under a curve using integrals; optimization of functions of two variables using partial derivatives; growth and decay using differential equations; applications to many different disciplines; use of computer algebra systems.

Lectures, 4 hours per week.

Restriction: not open to Mathematics (single or combined) majors.

Prerequisite(s): MATH 1P20 or one grade 12 mathematics credit.

Note: may be available on site, online or blended. Designed for students in Biological Sciences, Biotechnology, Business, Earth Sciences, Economics, Environmental Geoscience, Geography and Medical Sciences. Not open to students with credit in any university calculus course. Major credit will not be granted to Mathematics majors.

MATH 1P98

Practical Statistics

Descriptive statistics; probability of events; counting rules; discrete and continuous probability distributions: binomial, Poisson and normal distributions; Central Limit Theorem; confidence intervals and hypothesis testing; analysis of variance; contingency tables; correlation and regression; emphasis on real-world applications throughout; use of statistical computer software.

Lectures, 3 hours per week.

Restriction: not open to Mathematics (single or combined) majors.

Prerequisite(s): MATH 1P20 or one grade 12 mathematics credit.

Note: designed for students in Biological Sciences, Biotechnology, Business, Earth Sciences, Economics, Environmental Geoscience and Medical Sciences. Not open to students with credit in any university statistics course. Major credit will not be granted to Mathematics majors.

MATH 1P99

Applied Statistics for Life Sciences

Visualizing and interpreting life science data using histograms, frequency tables, correlation and regression, and a variety of statistical measures; basic concepts ofprobability; applications of the binomial, Poisson and normal distributions; confidence intervals for proportions and means; hypothesis testing; contingency tables; introduction to ANOVA; emphasis on interpretation and use of Microsoft Excel software throughout.

Lectures, 3 hours per week.

Prerequisite(s): MATH 1P20 or one grade 12 mathematics credit

Note: designed for students in the Life and Applied Health Sciences. Not open to students with credit in any university statistics course.

MATH 2P03

Multivariable Calculus

Functions of two and three variables, partial derivatives, gradient, critical points, maxima and minima, Taylor expansion, inverse and implicit function theorems. Cartesian, polar, cylindrical and spherical coordinates. Curves and surfaces, parametric representation, tangent space. Two- and three-dimensional integration, line and surface integrals.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 1P02, 1P06 or permission of the instructor.

MATH 2P04

Basic Concepts of Analysis

Sets; mappings, countability; properties of the real number system; inner product, norm, and the Cauchy-Schwarz inequality; compactness and basic compactness theorems (Cantor's theorem, the Bolzano-Weierstrass theorem, the Heine-Borel theorem); connectedness; convergence of sequences; Cauchy sequences; continuous and uniformly continuous functions.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 2P03.

MATH 2P08

Ordinary Differential Equations

Linear and nonlinear differential equations. Basic existence and uniqueness theory. Analytical and numerical solution methods; asymptotic behaviour. Qualitative analysis of autonomous systems including periodic cycles and steady-states. Examples of conservative systems and dissipative systems. Modelling and applications of differential equations. Use of Maple.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 1P02, 1P06 or permission of the instructor.

MATH 2P12

Linear Algebra II

General vector spaces. Basis and dimension. Row and column spaces. Rank and nullity; the dimension theorem. Real inner product spaces. Angles and orthogonality. Orthonormal bases. Gram-Schmidt process. Eigenvalues, eigenvectors and diagonalization. Linear transformations. Applications to differential equations and least square fitting. Use of a computer algebra system.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 1P11 or 1P12.

MATH 2P13

Linear Algebra III - Advanced

Review and further study of vector spaces over arbitrary fields. General linear transformations. Kernel and range. Invertibility. Matrices of linear transformations. Similarity. Isomorphism. Complex vector spaces and inner product spaces. Unitary, normal, symmetric, skew-symmetric and Hermitian operators. Orthogonal projections and the spectral theorem. Bilinear and quadratic forms. Jordan canonical form.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 2P12.

MATH 2P40

Mathematics Integrated with Computers and Applications II

Theory and applications of mathematical modelling and simulation. Topics may include discrete dynamical systems, Monte-Carlo methods, stochastic models, the stock market, epidemics, analysis of DNA, chaotic dynamical systems, cellular automata and predator-prey.

Lectures, lab, 4 hours per week.

Prerequisite(s): MATH 1P02 or 1P06; MATH 1P40 or permission of the instructor.

Completion of this course will replace previous assigned grade and credit obtained in MATH 2F40.

MATH 2P52

Principles of Mathematics for Primary and Junior Teachers

Mathematical concepts and ideas in number systems; geometry and probability arising in the Primary and Junior school curriculum.

Lectures, seminar, 4 hours per week.

Restriction: students must have a minimum of 5.0 overall credits.

Note: designed to meet the mathematics admission requirement for the Primary/Junior Pre-service program of the Faculty of Education at Brock University. Not open to students holding credit in any grade 12 or university mathematics course.

MATH 2P75

Introductory Financial Mathematics

Mathematical models arising in finance and insurance. Compound interest, the time-value of money, annuities, mortgages, insurance, measures of risk. Introduction to stocks, bonds and options.

Lectures, lab, 4 hours per week.

Prerequisite(s): one of MATH 1P01, 1P05, 1P97 and 1P98.

MATH 2P81

Probability

Probability, events, algebra of sets, independence, conditional probability, Bayes' theorem; random variables and their univariate, multivariate, marginal and conditional distributions. Expected value of a random variable, the mean, variance and higher moments, moment generating function, Chebyshev's theorem. Some common discrete and continuous distributions: Binomial, Poisson, hypergeometric, normal, uniform and exponential. Use of SAS, Maple or other statistical packages.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 2P03 or permission of the instructor.

Note: may be taken concurrently with MATH 2P03.

MATH 2P82

Mathematical Statistics I

Random sampling, descriptive statistics, Central Limit Theorem, sampling distributions related to normality; point estimation: measurements for estimation performance, methods of moments, maximum likelihood, ordinary/weighted least squares; confidence intervals, testing procedures, and their relation for population means, difference between means, variances, ratio of variances, proportions and difference between proportions.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 2P81.

MATH 2P90

Euclidean and Non-Euclidean Geometry I

Development of Euclidean and non-Euclidean geometry from Euclid to the 19th century. Deductive nature of plane Euclidean geometry as an axiomatic system, central role of the parallel postulate and general consideration of axiomatic systems for geometry in general and non-Euclidean geometry in particular. Introduction to transformation geometry. Use of Geometer's Sketchpad.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): one MATH credit.

MATH 2P91

Discrete Optimization

Problems and methods in discrete optimization. Linear programming: problem formulation, the simplex method, software, and applications. Network models: assignment problems, max-flow problem. Directed graphs: topological sorting, dynamic programming and path problems, and the travelling salesman's problem. General graphs: Eulerian and Hamiltonian paths and circuits, and matchings.

Lectures, 3 hours per week; lab, 1 hour per week.

Prerequisite(s): MATH 1P11 or 1P12

Completion of this course will replace previous assigned grade and credit obtained in MATH 2P72.

MATH 2P92

Introduction to Combinatorics

Counting, inclusion and exclusion, pigeonhole principle, permutations and combinations, derangements, binomial expansions. Introduction to discrete probability and graph theory, Eulerian graphs, Hamilton Cycles, colouring, planarity, and trees.

Lectures, 3 hours per week; tutorial, 1 hour per week.

Prerequisite(s): two 4U mathematics credits or permission of the instructor.

Completion of this course will replace previous assigned grade and credit obtained in MATH 2P71.

MATH 2P94

Introduction to Network Analysis

Complex networks and their properties, random graphs, network formation models. Webgraph: models, search engines, page-ranking algorithms. Community clustering, community structure. Opinion formation, on-line social networks.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): two 4U mathematics credits or permission of the instructor.

Completion of this course will replace previous assigned grade and credit obtained in MATH 2P77.

MATH 2P95

Mathematics and Music

Scales and temperaments, history of the connections between mathematics and music, set theory in atonal music, group theory applied to composition and analysis, enumeration of rhythmic canons, measurement of melodic similarity using metrics, topics in mathematical music theory, applications of statistics to composition and analysis.

Lectures, 3 hours per week; lab/tutorial 1 hour per week.

Prerequisite(s): one of MATH 1P01, 1P02, 1P05, 1P06, 1P97; MATH 1P11 or 1P12 or permission of the instructor.

MATH 2P97

Problem Solving

Solving mathematical problems using insight and creative thinking. Topics may include pigeonhole principle, finite and countable sets, probability theory, congruences and divisibility, polynomials, generating functions, inequalities, limits, geometry, and mathematical games.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 1P01 or 1P05; MATH 1P11 or 1P12; MATH 2P92 (2P71) or 2P81 or permission of instructor.

Note: recommended to students wishing to participate in mathematical problem solving competitions.

MATH 2P98

Applied Statistics

Single-factor and factorial experimental design methods; nested-factorial experiments. Simple and multiple linear regression methods, correlation analysis, indicator regression; regression model building and transformations. Contingency tables, binomial tests, nonparametric rank tests. Simple random and stratified sampling techniques, estimation of sample size and related topics. Use of SAS, Maple or other statistical packages.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 1F92 or 1P98.

Note: major credit will not be granted to Mathematics majors.

MATH 3P03

Real Analysis

Approximation of functions by algebraic and trigonometric polynomials (Taylor and Fourier series); Weierstrass approximation theorem; Riemann integral of functions on Rⁿ, the Riemann-Stieltjes integral on R; improper integrals; Fourier transforms.

Lectures, 3 hours per week; tutorial, 1 hour per week.

Prerequisite(s): MATH 2P04.

MATH 3P04

Complex Analysis

Algebra and geometry of complex numbers, complex functions and their derivatives; analytic functions; harmonic functions; complex exponential and trigonometric functions and their inverses; contour integration; Cauchy's theorem and its consequences; Taylor and Laurent series; residues.

Lectures, 3 hours per week; tutorial, 1 hour per week.

Prerequisite(s): MATH 2P03.

MATH 3P06

Vector Calculus and Differential Geometry

Vector fields, vector algebra, vector calculus; gradient, curl and divergence. Polar, cylindrical and spherical coordinates. Green’s, Stokes’ and divergence theorems. Introduction to differential geometry of surfaces. Topics may include differential forms, exterior calculus, frames, Gauss-Bonnet theorem.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 2P03.

MATH 3P08

Advanced Differential Equations

Linear second-order differential equations and special functions. Introduction to Sturm-Liouville theory and series expansions by orthogonal functions. Boundary value problems for the heat equation, wave equation and Laplace equation. Green's functions. Emphasis on applications to physical sciences. Use of Maple.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 2P08.

MATH 3P09

Partial Differential Equations

Survey of linear and nonlinear partial differential equations. Analytical solution methods. Existence and uniqueness theorems, variational principles, symmetries, and conservation laws. Emphasis on applications to physical sciences. Use of Maple.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 2P08.

MATH 3P12

Applied Algebra

Group theory with applications. Topics include modular arithmetic, symmetry groups and the dihedral groups, subgroups, cyclic groups, permutation groups, group isomorphism, Burnside's theorem, cosets and Lagrange's theorem, direct products and cryptography, normal subgroups and factor groups.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 2P12 or permission of the instructor.

MATH 3P13

Abstract Algebra

Further topics in group theory: homomorphisms and isomorphism theorems, structure of finite abelian groups. Rings and ideals; polynomial rings; quotient rings. Division rings and fields; field extensions; finite fields; constructability.

Lectures, 3 hours per week; lab/tutorial 1 hour per week.

Prerequisite(s): MATH 3P12.

MATH 3P23

Great Moments in Mathematics I

Triumphs in mathematical thinking emphasizing many cultures up to 1000 AD. Analytical understanding of mathematical problems from the past, referencing the stories and times behind the people who solved them. Matching wits with great mathematicians by solving problems and developing activities related to their discoveries.

Lectures, 4 hours per week.

Prerequisite(s): one MATH credit.

Completion of this course will replace previous assigned grade and credit obtained in MATH 2P93.

MATH 3P40

Mathematics Integrated with Computers and Applications III

Concepts and programming of contemporary models and simulations used in mathematics and sciences.

Lectures, lab, 4 hours per week.

Prerequisite(s): MATH 1P11 or 1P12; MATH 2P03 and COSC 2P95.

Note: students will develop a final project in their own discipline.

Completion of this course will replace previous assigned grade and credit obtained in MATH 2F40.

MATH 3P41

Visual and Interactive Mathematics

Techniques in the visual representation of mathematical data and the interactive presentation of mathematical ideas. Topics may include modelling and simulation, visualization of real world data, interactive learning environments, and interactive websites.

Lectures, lab, 4 hours per week.

Prerequisite(s): MATH 1P02 or 1P06; MATH 1P11 or 1P12; MATH 2P40 or permission of the instructor.

Completion of this course will replace previous assigned grade and credit obtained in MATH 2F40.

MATH 3P51

Applied Mathematics with Maple

Blending mathematical concepts with computations and visualization in Maple. Modelling of physical flows, waves and vibrations. Animation of the heat equation and wave equation; applications including vibrations of rectangular and circular drums, heat flow and diffusion, sound waves. Eigenfunctions and convergence theorems for Fourier eigenfunction series. Approximations, Gibbs phenomena, and asymptotic error analysis using Maple.

Lectures, lab, 4 hours per week.

Prerequisite(s): MATH 2P03 and 2P12; MATH 2P08 or 2P40.

MATH 3P52

Partial Differential Equations in C++

Analytic solution of first order PDEs (characteristic ODE systems and their analytic solution) and the numerical solution of first and second order PDEs (discretization, derivation and comparison of different finite difference equations, stability analysis, boundary conditions), the syntax of the C++ programming language, projects in C++ solving PDEs numerically.

Lectures, lab, 4 hours per week.

Prerequisite(s): MATH 2P03 and 2P12; MATH 2P08 or 2P40.

MATH 3P60

Numerical Methods

Survey of computational methods and algorithms; basic concepts (algorithm, computational cost, convergence, stability); roots of functions; linear systems; numerical integration and differentiation; Runge-Kutta method for ordinary differential equations; finite-difference method for partial differential equations; fast Fourier transform; Monte Carlo methods. Implementation of numerical algorithms in a scientific programming language.

Lectures, 3 hours per week; lab, 1 hour per week.

Prerequisite(s): MATH 1P02 or 1P06; MATH 1P11 or 1P12; MATH 2P12 or permission of the instructor.

MATH 3P72

Continuous Optimization

Problems and methods in non-linear optimization. Classical optimization in Rⁿ: inequality constraints, Lagrangian, duality, convexity. Non-linear programming. Search methods for unconstrained optimization. Gradient methods for unconstrained optimization. Constrained optimization. Dynamic programming.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 2P03 and 2P91 (2P72).

*MATH 3P73

Game Theory

(also offered as ECON 3P73)

Representation of Games. Strategies and payoff functions. Static and dynamic games of complete or incomplete information. Equilibria concepts: Nash, Bayesian Nash and Perfect Bayesian Nash equilibria. Convexity concepts, fixed points for correspondences and minimax. Core and Shapley value of a game. Refinements and Applications.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): one of MATH 2P91 (2P72), ECON 2P30, 3P91.

MATH 3P75

Theory of Financial Mathematics

Mathematical models arising in modern investment practices. Compound interest, annuities, the time-value of money, Markowitz portfolio theory, efficient frontier, random walks, Brownian processes, future contracts, European and American options, and put-call parity. Introduction to Black-Scholes.

Lectures, lab, 4 hours per week.

Prerequisite(s): MATH 2P82.

MATH 3P81

Experimental Design

Analysis of variance; single-factor experiments; randomized block designs; Latin squares designs; factorial designs; 2^f and 3^f factorial experiments; fixed, random and mixed models; nested and nested-factorial experiments; Taguchi experiments; split-plot and confounded in blocks factorial designs; factorial replication; regression models; computational techniques and use of SAS, Maple or other statistical packages; related topics.

Lectures, 3 hours per week; lab, 1 hour per week.

Prerequisite(s): MATH 2P82.

MATH 3P82

Regression Analysis

Simple and multiple linear regression and correlation, measures of model adequacy, residual analysis, weighted least squares, polynomial regression, indicator variables, variable selection and model building, multicollinearity, time series, selected topics. Use of SAS, Maple or other statistical packages.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 1P11 or 1P12; MATH 2P82 or permission of the instructor.

Note: MATH 3P82 has been approved by VEE (Validation by Education Experience) Administration Committee of the Society of Actuaries. To receive VEE credit, candidates will need a minimum grade of 70 percent.

MATH 3P85

Mathematical Statistics II

Multivariate, marginal and conditional distributions, independence, expectation, covariance, conditional expectation. Functions of random variables, transformation techniques, order statistics. Some special and limiting distributions. Proof of central limit theorem. Point estimation: efficiency, consistency, law of large numbers, sufficiency, Rao-Blackwell theorem MVUE. Interval estimation. Hypothesis testing: Neyman-Pearson theory, likelihood ratio test, Bayesian inference.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 2P82.

MATH 3P86

Applied Multivariate Statistics

Matrix algebra and random vector, sample geometry and random sampling, multivariate normal distribution, inference about mean, comparison of several multivariate means, multivariate linear regression model, principle components, factor analysis, covariance analysis, canonical correlation analysis, discrimination and classification, cluster analysis, computational techniques and use of SAS, Maple or other statistical packages and related topics.

Lectures, 3 hours per week; lab 1 hour per week.

Prerequisite(s): MATH 1P11 or 1P12; MATH 2P82 or permission of the instructor.

MATH 3P91

Mathematics at the Junior/Intermediate/Senior Level

A treatment of mathematics and its teaching and learning at the junior, intermediate and senior levels. A major portion of the course will involve directed projects.

Lectures, seminar, 4 hours per week.

Restriction: open to MATH (Honours) BSc/BEd (Intermediate/Senior) majors, Elementary and Secondary Teaching Mathematics minors with a minimum of 9.0 overall credits.

Prerequisite(s): three MATH credits.

Note: Mathematics Integrated with Computers and Applications with a Concentration in Mathematics Education may register. Contact Department. Students in other programs will require permission of the Department.

MATH 3P94

Symmetry Groups, Matrix Representations and Applications

Symmetry groups, their invariants and matrix representations. Permutation groups, rotation groups. Representations of discrete and continuous groups by linear transformations (matrices). General properties and constructions of group representations. Representations of specific groups. Lie groups and Lie algebras. Applications in various areas of Mathematics and Theoretical Physics.

Prerequisite(s): MATH 2P12 or permission of the instructor.

*MATH 3P95

Introduction to Mathematical Physics

(also offered as PHYS 3P95)

Topics may include Calculus of variations, Lagrangian and Hamiltonian mechanics, field theory, differential forms, vector and polyvector fields, tensor fields, Lie derivative, connection, Riemannian metric, Lie groups and algebras, manifolds, and mathematical ideas of quantum mechanics. Applications to theoretical physics.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 2P03 and 2P08.

Note: MATH 2P12 is recommended.

Completion of this course will replace previous assigned grade and credit obtained in MATH (PHYS) 4P64.

MATH 3P96

Computational Ergodic Theory and Dynamical Systems

Orbits of maps, counting and invariant measures, shift transformation, ergodicity and Birkhoff ergodic theorem, central limit theorem for dynamical systems, Poincare recurrence theorem, entropy and data compression, Lyapunov exponent, invariant subspaces, construction of stable and unstable manifolds for maps, symbolic dynamics and topological entropy.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 2P03, 2P12 and 2P81.

MATH 3P97

Introductory Topology

Introduction to metric and topological spaces; connectedness, completeness, countability axioms, separation properties, covering properties, metrization of topological spaces.

Lectures, 4 hours per week.

Prerequisite(s): MATH 2P04; MATH 2P12 and 2P13 or MATH 3P12 and 3P13.

MATH 3P98

Functional Analysis

Introduction to the theory of normed linear spaces, fixed-point theorems, Stone-Weierstrass approximation on metric spaces and preliminary applications on sequence spaces.

Lectures, 4 hours per week.

Prerequisite(s): MATH 3P97.

MATH 4F90

Honours Project

Independent project in an area of pure or applied mathematics, or mathematics education.

Restriction: open to MATH (single or combined) majors with either a minimum of 14.0 credits, a minimum 70 percent major average and a minimum 60 percent non-major average or approval to year 4 (honours) and permission of the instructor.

Note: carried out under the supervision of a faculty member. The supervisor must approve the topic in advance. Presentation of the project is required.

MATH 4P03

Advanced Real Analysis

Lebesgue integration on Rⁿ; differentiation and absolute continuity; Fubini's theorem; L^p spaces, elementary theory of Banach and Hilbert spaces.

Lectures, 3 hours per week.

Prerequisite(s): MATH 3P03.

MATH 4P06

Special Topics

Advanced topics selected from ring theory, homological algebra, algebraic geometry, number theory, point-set topology, differential geometry, algebraic topology, ordinary or partial differential equations, dynamical systems or any other field of mathematics.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Restriction: permission of the instructor.

*MATH 4P09

Solitons and Nonlinear Wave Equations

(also offered as PHYS 4P09)

Linear and nonlinear travelling waves. Nonlinear evolution equations (Korteweg de Vries, nonlinear Schrodinger, sine-Gordon). Soliton solutions and their interaction properties. Lax pairs, inverse scattering, zero-curvature equations and Backlund transformations, Hamiltonian structures, and conservation laws.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): one of MATH 3P08, 3P09, 3P51, 3P52.

MATH 4P11

Topics in Groups

Advanced topics from group theory. Topics may include isomorphism theorems, Sylow theorems, finite abelian groups, free groups, nilpotent and solvable groups and some simple Lie groups.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 3P12.

MATH 4P13

Topics in Rings and Modules

Advanced topics from ring theory. Topics may include radicals, Wedderburn-Artin theorems, modules over rings and some special rings.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 3P13.

MATH 4P23

Great Moments in Mathematics II

Development of modern mathematics from medieval times to the present. Fibonacci's calculation revolution, the disputes over cubic equations, Pascal and probability, Fermat's last theorem, Newton and Calculus, Euler and infinite series, set theory and the possibilities of inconsistencies in mathematics.

Lectures, 4 hours per week.

Prerequisite(s): MATH 1P02 or 1P06; MATH 1P11 or 1P12; MATH 3P23 (2P93).

Completion of this course will replace previous assigned grade and credit obtained in MATH 3P93.

#MATH 4P61

Theory of Computation

(also offered as COSC 4P61)

Regular languages and finite state machines: deterministic and non-deterministic machines, Kleene's theorem, the pumping lemma, Myhill-Nerode Theorem and decidable questions. Context-free languages: generation by context-free grammars and acceptance by pushdown automata, pumping lemma, closure properties, decidability. Turing machines: recursively enumerable languages, universal Turing machines, halting problem and other undecidable questions.

Lectures, 3 hours per week.

Restriction: open to COSC (single or combined), BCB, CAST, CNET, GAMP and NEUR Neurocomputing stream majors.

Prerequisite(s): MATH 1P67.

Note: MATH students may take this course with permission of Department.

MATH 4P71

Combinatorics

Review of basic enumeration including distribution problems, inclusion-exclusion and generating functions. Polya theory. Finite fields. Orthogonal Latin squares, affine and projective planes. Coding theory and cryptography.

Lectures, 3 hours per week; tutorial, 1 hour per week.

Prerequisite(s): MATH 2P92 (2P71) or permission of the instructor.

MATH 4P81

Sampling Theory

Theory of finite population sampling; simple random sampling; sampling proportion; estimation of sample size; stratified random sampling; optimal allocation of sample sizes; ratio estimators; regression estimators; systematic and cluster sampling; multi-stage sampling; errors in surveys; computational techniques and use of SAS, Maple or other statistical packages and related topics.

Lectures, 3 hours per week; lab, 1 hour per week.

Prerequisite(s): MATH 3P85 or permission of the instructor.

MATH 4P82

Nonparametric Statistics

Order statistics, rank statistics, methods based on binomial distribution, contingency tables, Kolmogorov Smirnov statistics, nonparametric analysis of variance, nonparametric regression, comparisons with parametric methods. Computational techniques and use of SAS, Maple or other statistical packages.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 3P85 or permission of the instructor.

MATH 4P84

Topics in Stochastic Processes and Models

Topics may include general stochastic processes, Markov chains and processes, renewal process, branching theory, stationary processes, stochastic models, Monte Carlo simulations and related topics. Use of SAS, Maple or other statistical packages.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 3P85 or permission of the instructor.

MATH 4P85

Topics in Advanced Statistics

Topics may include advanced topics in stochastic processes and models, queueing theory, time series analysis, multivariate analysis, Bayesian statistics, advanced methods and theory in statistical inference, and related topics. Use of SAS, Maple or other statistical packages.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 3P85 or permission of the instructor.

MATH 4P90

Euclidean and Non Euclidean Geometry II

Topics in Euclidean and non-Euclidean geometry chosen from the classification of isometries in selected geometries, projective geometry, spherical geometry, finite geometries and axiomatic systems for plane Euclidean geometry.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 1P11 or 1P12; MATH 2P90.

Completion of this course will replace previous assigned grade and credit obtained in MATH 3P90.

MATH 4P92

Topics in Number Theory and Cryptography

Topics may include algebraic number theory, analytic number theory and cryptography.

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Restriction: permission of the Department.

Prerequisite(s): MATH 1P11 or 1P12; one of MATH 2P12, 2P81, 2P92 (2P71), 3P12.

*MATH 4P94

Relativity Theory and Black Holes

(also offered as PHYS 4P94)

Review of Special Relativity and Minkowski space-time. Introduction to General Relativity theory; the space-time metric, geodesics, light cones, horizons, asymptotic flatness; energy-momentum of particles and light rays. Curvature and field equations. Static black holes (Schwarzschild metric), properties of light rays and particle orbits. Rotating black holes (Kerr metric).

Lectures, 3 hours per week; lab/tutorial, 1 hour per week.

Prerequisite(s): MATH 2P03, 2P08, 3P06, PHYS 2P20 and 2P50 or permission of the instructor.

MATH 4P96

Technology and Mathematics Education

Topics may include contemporary research concerning digital technologies, such as computer algebra systems and Web 2.0, in learning and teaching mathematics, design of educational tools using VB.NET, HTML, Geometer's Sketchpad, Maple, Flash, critical appraisal of interactive learning objects in mathematics education.

Lectures, 2 hours per week; lab/tutorial, 2 hours per week.

Prerequisite(s): MATH 1P40 and two and one-half MATH credits or permission of the instructor.

CO-OP COURSES

MATH 0N01

Co-op Work Placement I

First co-op work placement (4 months) with an approved employer.

Restriction: open to MATH and MICA Co-op students.

MATH 0N02

Co-op Work Placement II

Second co-op work placement (4 months) with an approved employer.

Restriction: open to MATH and MICA Co-op students.

MATH 0N03

Co-op Work Placement III

Third co-op work placement (4 months) with an approved employer.

Restriction: open to MATH and MICA Co-op students.

MATH 0N04

Co-op Work Placement IV

Optional co-op work placement (4 months) with an approved employer.

Restriction: open to MATH and MICA Co-op students.

MATH 0N05

Co-op Work Placement V

Optional co-op work placement (4 months) with an approved employer.

Restriction: open to MATH and MICA Co-op students.

MATH 2C01

Co-op Reflective Learning and Integration I

Provides students with the opportunity to apply what they've learned in their academic studies through career-oriented work experiences at employer sites.

Restriction: open to MATH and MICA Co-op students.

Prerequisite(s): SCIE 0N90.

Corequisite(s): MATH 0N01.

Note: students will be required to prepare learning objectives, participate in a site visit, write a work term report and receive a successful work term performance evaluation.

MATH 2C02

Co-op Reflective Learning and Integration II

Provides students with the opportunity to apply what they've learned in their academic studies through career-oriented work experiences at employer sites.

Restriction: open to MATH and MICA Co-op students.

Prerequisite(s): SCIE 0N90.

Corequisite(s): MATH 0N02.

Note: students will be required to prepare learning objectives, participate in a site visit, write a work term report and receive a successful work term performance evaluation.

MATH 2C03

Co-op Reflective Learning and Integration III

Provides students with the opportunity to apply what they've learned in their academic studies through career-oriented work experiences at employer sites.

Restriction: open to MATH and MICA Co-op students.

Prerequisite(s): SCIE 0N90.

Corequisite(s): MATH 0N03.

Note: students will be required to prepare learning objectives, participate in a site visit, write a work term report and receive a successful work term performance evaluation.

MATH 2C04

Co-op Reflective Learning and Integration IV

Provides students with the opportunity to apply what they've learned in their academic studies through career-oriented work experiences at employer sites.

Restriction: open to MATH and MICA Co-op students.

Prerequisite(s): SCIE 0N90.

Corequisite(s): MATH 0N04.

Note: students will be required to prepare learning objectives, participate in a site visit, write a work term report and receive a successful work term performance evaluation.

MATH 2C05

Co-op Reflective Learning and Integration V

Provides students with the opportunity to apply what they've learned in their academic studies through career-oriented work experiences at employer sites.

Restriction: open to MATH and MICA Co-op students.

Prerequisite(s): SCIE 0N90.

Corequisite(s): MATH 0N05.

Note: students will be required to prepare learning objectives, participate in a site visit, write a work term report and receive a successful work term performance evaluation.