AS/SC/MATH 2015 3.0 F Applied Multivariate and Vector Calculus

     2001/2002 Calendar copy: Topics covered include partial derivatives; grad, div, curl and Laplacian operators; line and surface integrals; theorems of Gauss and Stokes; double and triple integrals in various coordinate systems; extrema and Taylor series for multivariate functions.
     Other
topics covered include curves and surfaces in Cartesian, cylindrical, and spherical polar coordinates; differential vector identities; Green's theorem.
     The final grade will be based on assignments, two tests, and a final examination.
     Applied Mathematics students should, preferably, take the one-credit course MATH 2018 during the same term as this course, but in any case must take it within the first 90 credits of their programs.

Prerequisite: One of AS/SC/ MATH 1010 3.0, AS/SC/ MATH 1014 3.0, AS/SC/AK/MATH 1310 3.0, or: AS/SC/ MATH 1505 6.0 plus permission of the Course Coordinator.
Exclusions: AS/SC/MATH 2010 3.0, AS/SC/AK/MATH 2310 3.0.
Coordinator:  H.S. Freedhoff

AS/SC/MATH 2018 1.0 F Applied Mathematics Module III

     2001/2002 Calendar copy:  Designed for students in Applied Mathematics to complement and enrich material in AS/SC/MATH 2015 3.0. The module treats the theory in greater depth, and explores extended applications and modeling. One lecture hour per week. One term. One credit.
     The focus will be on problem solving. Topics will be related to vector algebra, calculus and geometry, optimization, uses of multiple integrals, curves and surfaces. Attendance will be expected. Evaluation will be based largely on assignments and participation.
     Further and updated information on the course will be available on the web site: http://www.math.yorku.ca/~muldoon/

Prerequisites: AS/SC/MATH 1014 3.0; AS/SC/MATH 1017 1.0.
Precorequisite: AS/SC/MATH 2015 3.0.
Coordinator:  Martin Muldoon (muldoon@yorku.ca

AS/SC/AK/MATH 2022 3.0 W Linear Algebra II

     2001/2002 Calendar copy: Inner product spaces, linear transformations, eigenvalues, diagonalization, least squares, quadratic forms and Markov chains. Similiar to AS/SC/AK/MATH 2222 3.0 but at a more advanced level. Required in Specialized Honours Statistics and all Applied Mathematics, Mathematics, and Mathematics for Commerce programs except the BA Program in Mathematics for Commerce.

Important! Advisors and students: Please see the note on page 12 about the new linear algebra requirements for FW 2001.

     For a general description of the subject "linear algebra'' see the minicalendar entry for MATH 1021 3.0. Additional topics: General linear transformations, matrix of a linear transformation from V to W relative to bases for V and W, effect of a change of bases.
      The text is the same as for MATH 1021 3.0.

Prerequisite: AS/SC/AK/MATH 1021 3.0 or AS/SC/ MATH 2021 3.0
or permission of the course coordinator.
Exclusion: AS/SC/AK/MATH 2222 3.0.
Coordinator:  R.G.Burns

AS/SC/AK/MATH 2030 3.0 F Elementary Probability
Formerly:
part of MATH 2030 6.0
Note that this course has moved to Fall term
from its customary Winter position.

     2001/2002 Calendar copy: Introduction to the theory of probability as preparation for further study in either mathematical or applied probability and statistics. Topics include probability spaces, conditional probability, independence, random variables, distribution functions, expectation, Chebyshev's inequality, common distributions, moment-generating functions and limit theorems.
     This introductory course in probability is designed for those students who want more than a "cookbook'' approach to the subject, for those who expect to take further courses in probability, mathematical statistics or stochastic processes, and for those majoring in Mathematics, Applied Mathematics or Statistics.
     The grading scheme and textbook have not been determined.

Prerequisite: One of AS/SC/MATH 1010 3.0,
AS/SC/MATH 1014 3.0, AS/SC/AK/MATH 1310 3.0.
Coordinator:  TBA

AS/SC/MATH 2041 3.0 F Symbolic Computation Laboratory I

     2001/2002 Calendar copy: An introduction to symbolic computing in the Maple environment. Topics from single-variable differential and integral calculus, including simple ordinary differential equations, are covered. Both mathematical understanding and applications are emphasized.
     The course provides students with an introduction to numerical computation in mathematics using Maple. Classes take place in a computer lab and students will spend most of their time working independently, under the guidance of the instructor, on assigned projects.
     The projects in MATH 2041 require a knowledge of single variable calculus, linear algebra and high school algebra. No programming knowledge is required.
     Enrolment is limited to about 30 per section.

Prerequisites: AK/AS/SC/COSC 1540 3.0, or equivalent computing experience; one of AS/SC/MATH 1010 3.0,
AS/SC/MATH 1014 3.0, AS/SC/AK/MATH 1310 3.0.
Exclusion: AS/SC/MATH 2040 6.0.
Coordinator:  TBA

AS/SC/MATH 2042 3.0 W Symbolic Computation Laboratory II

     2001/2002 Calendar copy: Advanced symbolic computing with Maple. Topics from linear algebra, differential equations, multivariate calculus, integral theorems, are covered. Both mathematical understanding and applications are emphasized.
     The general description of MATH 2042 is similar to that of MATH 2041. However, the topics covered will require greater mathematical maturity of the student. In particular, students will be required to have some knowledge of differential equations and multivariable calculus as well as the material required for MATH 2041.
     Enrolment is limited to about 30 per section.

Prerequisites: AS/SC/MATH 2041 3.0; one of AS/SC/MATH 2010 3.0, AS/SC/MATH 2015 3.0, AS/SC/AK/MATH 2310 3.0; one of AS/SC/AK/MATH 1021 3.0, AS/SC/MATH 1025 3.0, AS/SC/MATH 2021 3.0, AS/SC/AK/MATH 2221 3.0.
Corequisites: AS/SC/AK/MATH 2270 3.0;
one of AS/SC/AK/MATH 2022 3.0, AS/SC/AK/MATH 2222 3.0.
Exclusion: AS/SC/MATH 2040 6.0.
Coordinator:  TBA

AS/SC/AK/MATH 2090 3.0 FW Applications of Logic to Discrete Mathematics

     2001/2002 Calendar copy: A continuation of AS/SC/AK/MATH 1090 3.0, this course uses formal logic to study topics in discrete mathematics, including sets, relations, functions, induction, the integers.  Optional topics include program specification, sequences, recurrence relations.
Important Note: This course has Introduction to Logic for Computer Science (MATH 1090, first offered in Fall of 1998) as a strict prerequisite. Students who lack this prerequisite must check with the department before enrolling.
     Note also that this course is a program requirement in Honours COSC.
     A partial indication of the relevance of formal logic to programming is given in the course entry for COSC 3111 in a recent supplemental calendar issued by the Department of Computer Science:
     "Every program implicitly asserts a theorem to the effect that ... the program will do what its ... documentation says it will.'' Proving that theorem "is not merely a matter of luck or patient debugging ... making a correct program can be greatly aided by a logical analysis of what it is supposed to do, and for small pieces of code a proof that the code works can be produced hand-in-hand with the construction of the code itself.''
     MATH 2090 will use the mathematical logic learned in MATH 1090 to study selected topics in discrete mathematics.  Students wanting further exposure to discrete math may consider MATH 2320 3.0.
     The text will be Gries and Schneider, A Logical Approach to Discrete Math (Springer).

Prerequisite:  AS/SC/AK/MATH 1090 3.0 taken after Summer, 1998.
Coordinators:  Fall: Eli Brettler  Winter: S. Watson

AS/SC/AK/MATH 2131 3.0 W Introduction to Statistics II

     2001/2002 Calendar copy:  This course is a continuation of AS/SC/AK/MATH 2030 3.0. It provides students with an introduction to statistical methods with an emphasis on applications using continuous probability models. Note: Computer/Internet use may be required to facilitate course work.
     Topics include joint distributions, multivariate change of variables formula, conditional and marginal distributions, conditional expectation, prediction, conditional variance, covariance and correlation, moment generating functions, convergence in distribution and the continuity theorem, multivariate normal distribution and characterization, distributional theory associated with normally distributed observations, confidence intervals, tests of significance and hypotheses, likelihood and maximum likelihood estimation and the method of least squares.

Prerequisites: AS/SC/AK/MATH 2030 3.0; one of AS/SC/ MATH 2010 3.0, AS/SC/MATH 2015 3.0, AS/SC/AK/MATH 2310 3.0.
Exclusion: Not open to any student who has passed or is taking
AS/SC/AK/MATH 3131 3.0. 
Coordinator:  S. Chamberlin

AS/SC/AK/ MATH 2221 3.0 FW Linear Algebra with Applications I

     2001/2002 Calendar copy: Systems of linear equations, linear and affine subspaces of Euclidean n-space, the Gauss--Jordan algorithm, matrices and matrix algebra, determinants, vector space concepts for Euclidean n-space (linear dependence and independence, basis, dimension, etc.), various applications.

Important! Advisors and students: Please see the note on page 12 about the new linear algebra requirements for FW 2001.

     Linear algebra is a branch of mathematics which is particularly useful in other fields and in other branches of mathematics. Its frequent application in the engineering and physical sciences rivals that of calculus. Computer scientists and economists have long recognized its relevance to their discipline. Moreover, linear algebra is fundamental in the rapidly increasing quantification that is taking place in the management and social sciences. Finally, ideas of linear algebra are essential to the development of algebra, analysis, probability and statistics, and geometry.
     This course and MATH 2222 3.0 (see below) together provide a standard full-year introduction to linear algebra. While our focus will not be excessively theoretical, students will be introduced to proofs and expected to develop skills in understanding and applying concepts as well as results. Applications will be left mainly for MATH 2222 3.0.
    The text and grading scheme have not been determined as we go to press.
    Note that MATH 1540 3.0 may not be taken for credit by anyone who is taking, or anyone who has taken, MATH 2221.

Prerequisite: OAC algebra or any university mathematics course.
Exclusions: AS/SC/AK/MATH 1021 3.0,
AS/SC/MATH 1025 3.0, AS/SC/MATH 2021 3.0.
Coordinators:  Fall and Winter:  K. Bugajska

AS/SC/AK/MATH 2222 3.0 FW Linear Algebra with Applications II

     2001/2002 Calendar copy: Linear transformations and their representation by matrices, change of basis and similarity, eigenvalues and eigenvectors, diagonalization, inner product spaces, orthogonality, the Gram-Schmidt algorithm, least squares approximations, abstract vector spaces, various applications.

Important! Advisors and students: Please see the note on page 12 about the new linear algebra requirements for FW 2001.

     This course is a continuation of either MATH 1025 3.0W or MATH 2221 3.0, and requires knowledge of the topics discussed in those courses.
     The text for the {\it Fall} offering of this course will match the text for MATH 2221 from the Winter term of 1999/2000.  The text for Winter term, and the grading schemes for both terms, have not been determined as we go to press.

Prerequisite: AS/SC/AK/MATH 1021 3.0 or AS/SC/MATH 1025 3.0 or AS/SC/AK/MATH 2221 3.0.
Exclusion: AS/SC/AK/MATH 2022 3.0.
Coordinators:  Fall: TBA  Winter: K. Bugajska

AS/SC/AK/MATH 2270 3.0 W Differential Equations

     2001/2002 Calendar copy: Introduction to differential equations, including a discussion of the formation of mathematical models for real phenomena; solution by special techniques; applications; linear equations; solutions in series; other topics if time permits.
     Differential equations have played a central role in mathematics and its applications for the past three hundred years. Their importance in applications stems from the interpretation of the derivative as a rate of change, a familiar example being velocity. Many of the fundamental laws of physical science are best formulated as differential equations. In other areas, too, such as biology and economics, which involve the study of growth and change, such equations are of fundamental importance.
     In this course we will study some important types of linear differential equations and their solutions. Topics will include first-order (differential) equations; homogeneous second and higher order equations with constant coefficients; the particular solution of inhomogeneous second-order equations; series-form solutions of equations with variable coefficients; solutions by use of Laplace transforms.
    Students will use the symbolic computational computer language MAPLE to study the behaviour of differential equations. No prior experience with this language is necessary.

Prerequisites: One of AS/SC/MATH 2010 3.0, AS/SC/MATH 2015 3.0, or AS/SC/AK/MATH 2310 3.0; one of AS/SC/AK/MATH 1021 3.0, AS/SC/MATH 1025 3.0, AS/SC/MATH 2021 3.0, or AS/SC/AK/MATH 2221 3.0.
Coordinator:  TBA

AS/SC/MATH 2280 3.0 W The Mathematical Theory of Interest

     2001/2002 Calendar copy: Topics include measurement of interest, annuities, amortization of loans, bonds, sinking funds and depreciation. The course is at a level which will prepare students for the interest theory portion of the Society of Actuaries examinations.
    This is a required course for students in the Actuarial Stream, Mathematics for Commerce Honours Program. 
    The text is S.G. Kellison, The Theory of Interest, 2nd Ed. (Irwin Dorsey).

Prerequisite: AS/SC/MATH 1010 3.0 or AS/SC/MATH 1014 3.0 or AS/SC/AK/MATH 1310 3.0.
Exclusions: AS/AK/MATH 2580 6.0, AS/MATH 2581 3.0.
Coordinator:  R.L.W. Brown

AS/SC/AK/MATH 2310 3.0 FW Calculus of Several Variables with Applications

     2001/2002 Calendar copy: Vector functions, partial derivatives, gradient, multiple integrals, line integrals, optimization, applications. Offered in both terms.
    Other topics include lines, planes, curves in two and three dimensions, polar coordinates, arc length, Lagrange multipliers, change of coordinates in multiple integrals. 
    The choice of text has not been finalized yet. 
    The grading scheme will likely be based on three tests worth about 20% each and a final exam worth 40%.

Prerequisite: AS/SC/MATH 1010 3.0 or AS/SC/MATH 1014 3.0 or AS/SC/AK/MATH 1310 3.0. Students should have a knowledge of vector algebra in two and three dimensions.
Exclusions: AS/SC/MATH 2010 3.0, AS/SC/MATH 2015 3.0
Coordinators:  Fall: TBA  Winter: N. Madras

AS/SC/AK/MATH 2320 3.0 FW Discrete Mathematical Structures

     2001/2002 Calendar copy: Algebraic and combinatorial structures required in Computer Science and other disciplines. Review of sets; induction; combinatorics; graph theory, trees; big Oh-notation, complexity of algorithms; recursive definitions, recurrence relations; posets; congruence relations. This course emphasizes analysis, problem solving and proofs.
Note: This course is a program requirement in ITEC.
  AK/MATH 2442 3.0 may be viewed as a degree credit exclusion.
     Consultation with the Departments of Computer Science and of Mathematics, and with the ITEC Program, has led to the following list of topics for emphasis: "Big oh'' notation, complexity of formulae and algorithms, modular arithmetic, recursive definitions, general inductions, counting principles, recurrence relations and methods for solving them, trees and simple graph theory. The emphasis will include examples arising from algorithms and the ability to carry out analysis, problem solving, proofs and calculations which will be required in upper level courses. 
     The course does not require previous knowledge of computer science. A student of mathematics should enjoy this introduction to a variety of mathematical topics, many of which are not covered elsewhere. We will emphasize analysis, problem solving, and proofs.
     The grading scheme has not been determined. The text will be K.H. Rosen, Discrete Mathematics and its Applications, 4th Ed. (McGraw Hill).

Prerequisite: AS/SC/AK/MATH 1190 3.0,
or AS/SC/AK/MATH 1090 3.0, or AK/MATH 2441 3.0, or any 2000-level MATH course without second digit 5.  Students who have not taken AS/SC/AK/MATH 1190 3.0 or AS/SC/AK/MATH 2090 3.0 are advised to review set theory, functions, relations and induction proofs, before the course begins.
Coordinators:  Fall: S.D. Promislow  Winter: Eli Brettler

AS/SC/AK/MATH 2500 3.0 F An Introduction to the Basic Practice of Statistics

     2001/2002 Calendar copy: This course provides an introduction to the concepts of statistics with an emphasis on developing a critical attitude towards the use and misuse of statistics in business, health sciences and other areas. Note: Computer/Internet use may be required to facilitate course work.
     This course will not be offered this year by the department, though it is being offered by Atkinson College. A recommended course in the Operations Research stream of the BuSo program, it is intended for students who will require no further exposure to statistics.

Exclusions:  AS/SC/KINE 2050 3.0, AS/SC/KINE
3150 3.0, AS/SC/PSYC 2020 6.0, AS/SC/PSYC 2021 3.0, AS/SC/PSYC 2022 3.0. May not be taken by students who have taken or are taking any university course in statistics.

AS/SC/AK/MATH 2560 3.0 FW Elementary Statistics I

     2001/2002 Calendar copy: Displaying and describing distributions, normal distibution. Relationships between variables, regression and correlation. The need for design, experimental design and sampling design. Sampling distributions, bias, variability. Probability models, random variables, probability laws.
     Statistics is a collection of methods for observing and analyzing numerical data in order to make sensible decisions about them. In these courses the basic ideas of the analysis of data and of statistical inference will be introduced.
     Little mathematical background is required; high school algebra is sufficient. Mathematical proofs will be minimal; reasoning and explanations will be based mostly on intuition, verbal arguments, figures, or numerical examples. Most of the examples will be taken from our daily life; many deal with the behavioural sciences, while others come from business, the life sciences, the physical sciences, and engineering.
     Although students will be making some use of the computer to calculate statistics, to create statistical plots, and to obtain a better appreciation of statistical concepts, no previous experience in computing is required. Students will receive in class all the necessary instruction about how to use the statistical computer package Minitab.
     Students who have taken MATH2560 3.0 will normally take
MATH2570 3.0 in the second semester, where they will continue to investigate many basic statistical methods.

Prerequisite: Ontario Grade 12 Advanced Mathematics.
Exclusions: AS/SC/MATH1131 3.0,
SC/BIOL 2060 3.0, SC/BIOL3090 3.0, AS/ECON2500 3.0, AS/SC/GEOG2420 3.0, AS/SC/KINE2050 3.0, AS/SC/PHED2050 3.0, AS/POLS 3300 6.0, AS/SC/PSYC2020 6.0, AS/SC/PSYC2021 3.0, AS/SOCI3030 6.0, AK/MATH1720 6.0, AK/ECON 3470 3.0, AK/MATH2430 6.0, AK/BIOL3080 6.0, AK/BIOL3080 3.0, AK/PSYC2510 3.0.
Coordinators:  Fall and Winter:  H. Massam

AS/SC/AK/MATH 2570 3.0 W Elementary Statistics II

     2001/2002 Calendar copy: Binomial distribution, sampling distribution of sample proportions and means, central limit theorem. Confidence intervals, tests and decisions, abuse of tests. Inference for a single mean, comparing two means, and for spread. Contingency tables. Simple regression and basic analysis of variance.
     See also the description for MATH 2560 3.0.
     The text will be D.S. Moore and G.P. McCabe, Introduction to the Practice of Statistics, 3rd Ed. (W.H. Freeman and Company).
     The final grade may be based on assignments and quizzes, class tests, and a common final exam.

Prerequisite: AS/SC/AK/MATH 2560 3.0
or AS/SC/AK/MATH 1131 3.0.
Exclusions: AS/SC/MATH1132 3.0, SC/BIOL 2060
3.0, SC/BIOL3090 3.0, AS/ECON3210 3.0, AS/ECON3500 3.0, AS/SC/GEOG2420 3.0, AS/SC/KINE 3150 3.0, AS/POLS 3300 6.0, AS/SC/PSYC2020 6.0, AS/SC/PSYC2022 3.0, AS/SOCI3030 6.0, AK/MATH2430 6.0, AK/BIOL3080 6.0, AK/BIOL3090 3.0, AK/ECON 3480 3.0, AK/PSYC3010 3.0, AK/PSYC3110 3.0.
Coordinator:  Stephen Chamberlin

AS/AK/MATH 2580 6.0Mathematics of Investment and Actuarial Science

     2001/2002 Calendar copy: Theory of interest; annuities certain; amortization and sinking funds; evaluation of bonds and other investments; depreciation, depletion and capital cost; insurance, including mortality tables, principles of life annuities, premiums and reserves.
     The first four-fifths of the course deal with most of the above topics, with applications to simple and general annuities, perpetuities, loan payments, capital budgeting, and internal rates of return. In the last few weeks of the course, the theory of interest is applied to life annuities and life insurance.
     Students will use EXCEL, a spreadsheet available in the micro-computer laboratory in the Steacie building. This spreadsheet operates on both the Macintosh and IBM-compatible platforms. EXCEL will be used to simplify and illuminate equation-solving, amortization of loans and mortgages, bond schedules, depreciation tables, and mortality tables. No previous computer experience is assumed. With the help of notes and class instruction students will be introduced to the spreadsheet and to its use in mathematics of finance. Students will receive individual computer accounts on the ACADLABS server.
      Each student will also need a hand-held calculator which has power and logarithm functions. Specifically, it must be able, given values of x and y , to compute xy .
     The course should be especially interesting to students of business and economics. The emphasis will be on practical problems. Although the mathematical background required is minimal, it is preferred that students will have taken one other mathematics course at university before taking this one.
     The required text for the course will be P. Zima and R. Brown, Mathematics of Finance, 5th Ed. (McGraw-Hill Ryerson Ltd., 2001).  The manuals for EXCEL will be on reserve at the desk in Steacie Library.
     Students who wish a more advanced treatment of the material should not take this course, but enrol instead in MATH2280 3.0. In particular, this includes:
1. Honours Mathematics for Commerce students in the Actuarial Stream.
  MATH2280 3.0 is a required course for this program.
2. students contemplating a career in the actuarial
profession. They should take MATH 2280 3.0, then MATH 3280 6.0.
     In past years the final grade has been based on class testing (60%) and a final examination (40%). There will likely be some spreadsheet-based homework assignments.

Prerequisite: One full university mathematics course.
Exclusions: AS/MATH 1581 3.0, AS/SC/MATH 2280 3.0,
AS/MATH 2581 3.0.
Coordinator:  Donald Pelletier

AS/MATH 2581 3.0 W Business Mathematics II

     2001/2002 Calendar copy:  Spreadsheets and their application to business mathematics; deepening of topics in Business Mathematics I, including continuous compound interest, perpetuities, annuities where payments vary, callable bonds, bond yield rate, capital budgeting; mortality tables, life annuities, life insurance.
     This course is the sequel to MATH 1581, which must be taken as a strict prerequisite. The text will be P. Zima and R.L. Brown, Mathematics of Finance, 5th Ed. (McGraw-Hill Ryerson Limited, 2001); we expect to cover in this course all the material from this text that is skipped in MATH 1581.
     A substantial component of the course will consist of an introduction to spreadsheets, since much of the material can be particularly well treated with this tool. We will use Microsoft EXCEL in particular, but the basic concepts are common to all spreadsheets. EXCEL is available for both Macintosh and PC. There will be some in-class demonstrations of the software, but students will need individual accounts on ACADLABS in Steacie to access some of the homework problems and to develop some competence with EXCEL. 
     The grading scheme for the course has not been determined, but it will likely involve one or two tests and a final examination. The pair of half-courses 1581-2581 is equivalent to the full course 2580. Either MATH 1581 3.0 or MATH 2580 6.0 will serve to satisfy part of the Core requirements of the Business and Society Program. Roughly speaking, 1581 contains all of the easy material from 2580 while 2581 deals with the `complications'. Students who intend to take 6 rather than 3 credits in this area will find a more balanced distribution of workload by taking 2580. 

Prerequisites:  AS/MATH 1581 3.0; AK/AS/SC/COSC 1520 3.0 or
permission of the course coordinator.
Exclusions: AS/SC/MATH 2280 3.0, AS/AK/MATH 2580
6.0.
Coordinator:  Donald Pelletier