# Mathematics

Dr. Janet McShane

Chair of the Department

The mathematics curriculum allows students to prepare for careers in business, industry, actuarial science and teaching, or for further graduate study. Year long course sequences in Linear Algebra, Abstract Algebra, and Analysis prepare students for further work in pure or applied mathematics. A selection of courses from Linear Algebra, Probability and Statistics, and Operations Research form the basis for work in finance, engineering, and all of the sciences. Students may select from a variety of advanced topics offered by faculty with expertise in an area of pure or applied mathematics. Among these courses are Machine Learning, Non-linear Optimization, and Applied Network Theory. The courses introduce mathematics that is in use in today's technological workforce. The algorithm employed by Google to rank pages is an example how mathematics is used in our everyday world.

Classes are small, giving students the opportunity to build strong relationships with faculty. Students are invited to participate in national competitions such as the Putnam Exam and the Mathematical Modeling Contest. Each year students who wish to participate in the nationwide Putnam exam are coached by faculty to prepare for this exam.

Many of our students present projects at the Spuyten Duyvil Undergraduate Mathematics Conference, which was founded by the Manhattan College Mathematics and Computer Science Department. Funds are available to support undergraduate research projects during the summer and studentshave also attended summer undergraduate research and intensive study programs.

Any student wishing to participate in the Study Abroad program will find the Department makes every effort to provide the needed support to allow them to finish their required course work.

The Department is a member of the national honor society, Pu Mu Epsilon. And students who make presentations at conferences may be nominated for membership in Sigma Xi,

**General Requirements: **Major courses should be taken in accordance with the PLAN OF STUDY listed below. The order in which core courses are taken is flexible. A minimum grade of C in each of the major courses is required. Before taking any major course, the student must obtain a grade of C or better in the prerequisite courses.

## Major in Mathematics

A major program in mathematics is available in the School of Science leading to either a Bachelor of Science or a Bachelor of Arts degree. The requirements for these degrees are shown below.

**BS in Mathematics**

MATH 185 | Calculus I ^{*} | 3 |

MATH 186 | Calculus II ^{*} | 3 |

MATH 243 | Discrete Foundations | 3 |

MATH 272 | Linear Algebra I | 3 |

MATH 285 | Calculus III ^{*} | 3 |

MATH 286 | Differential Equations | 3 |

MATH 331 | Probability | 3 |

MATH 377 | Algebra I | 3 |

MATH 387 | Analysis I | 3 |

MATH 471 | Linear Algebra II | 3 |

MATH 478 | Algebra II | 3 |

MATH 489 | Problem Seminar | 3 |

MATH 490 | Complex Analysis | 3 |

Math Electives | 6 | |

CMPT 101 | Computer Science I | 3 |

CMPT 102 | Computer Science II | 3 |

PHYS 101 | Physics I | 4 |

PHYS 102 | Physics II | 4 |

Natural Sciences | 8 | |

Total Credits | 67 |

* | Students who major in mathematics and are selected for the honors sequence will be enrolled in the Honors sections of MATH 185 and MATH 186. MATH 287 replaces MATH 285. |

**BA in Mathematics**

MATH 185 | Calculus I ^{*} | 3 |

MATH 186 | Calculus II ^{*} | 3 |

MATH 243 | Discrete Foundations | 3 |

MATH 272 | Linear Algebra I | 3 |

MATH 285 | Calculus III ^{*} | 3 |

MATH 286 | Differential Equations | 3 |

MATH 331 | Probability | 3 |

MATH 377 | Algebra I | 3 |

MATH 387 | Analysis I | 3 |

MATH 471 | Linear Algebra II | 3 |

MATH 478 | Algebra II | 3 |

MATH 489 | Problem Seminar | 3 |

MATH 490 | Complex Analysis | 3 |

Math Electives | 6 | |

CMPT 101 | Computer Science I | 3 |

CMPT 102 | Computer Science II | 3 |

3 SCI xxx courses | 9 | |

Total Credits | 60 |

* | Students who major in mathematics and are selected for the honors sequence will be enrolled in the Honors sections of MATH 185 Calculus I and MATH 186 Calculus II. MATH 287 Honors Calculus III replaces MATH 285 Calculus III. |

**Requirements for a Second Major in Mathematics**:

Students from the schools of Arts, Business, Engineering and Science who are interested in declaring a *second* major in Mathematics should see the department chair for the required form. The requirements are 36 credits of courses conferring credit for the major in the school of science, and include Math 185-186 (or Math 187-188),Math 243, Math 272, Math 377, Math 387. At least 2 courses must be chosen from at the 400 level.

### Requirements for Students in the School of Education & Health

Students in Adolescence Education Mathematics earn a "second major" in mathematics by completing the following sequence, as required by their degree program.

#### Adolescence Education

Mathematics Requirements ^{*} | ||

MATH 185 | Calculus I | 3 |

MATH 186 | Calculus II | 3 |

MATH 243 | Discrete Foundations | 3 |

MATH 272 | Linear Algebra I | 3 |

MATH 285 | Calculus III | 3 |

MATH 361 | Introduction to Higher Geometry | 3 |

MATH 331 | Probability | 3 |

MATH 377 | Algebra I | 3 |

MATH 387 | Analysis I | 3 |

MATH 432 | Statistical Inference | 3 |

MATH 489 | Problem Seminar | 3 |

MATH 422 | Seminar for Mathematics Education | 3 |

Computer Science Requirements | ||

CMPT 101 | Computer Science I | 3 |

CMPT 102 | Computer Science II | 3 |

CMPT 214 | Teaching and Learning with Technology | 3 |

* | Sequencing of courses is very important, so that only one course is required during the semester when the student is doing student teaching. MATH 331 Probability (Fall) and MATH 432 Statistical Inference (Spring) are a one-year sequence and should be taken in the junior year. |

**Childhood Education**

All Students majoring in Childhood Education take the following 6 credit Core sequence.

MATH 221 | Mathematics for Elementary School Teachers I | 3 |

MATH 222 | Mathematics for Elementary School Teachers II | 3 |

Childhood Education majors may also choose to concentrate in Mathematics: These students must take Math 185, Math 186 Math 186, Math 243, Math 272, Math 285, Math 361, Math 221-222 plus two additional electives approved by the Mathematics Department.

Students in Childhood Education who choose a General Studies concentration with Mathematics as one area, must take Math 185, Math 186, Math 243, Math 361, Math 230, in addition to the core requirements of Math 221-222. These students may chose to declare a minor in mathematics

**Minor in Mathematics**

A minimum of 15 credits, or five approved courses, including MATH 185 Calculus I (formerly MATH 103), MATH 186 Calculus II (formerly MATH 104), MATH 285 Calculus III (formerly Math 201) or MATH 287 Honors Calculus III((formerly Math 209) . A grade of C or above is required for all courses meeting the requirements for a minor in mathematics. A student may use at most one course transferred from another institution toward the minor. At least four of the courses used toward the minor must be taken at Manhattan College. If a student has a grade point average of 3.6 on three courses beyond Calculus I and II, taken at Manhattan College, then the Department Chair may approve a minor.

Note: MATH 100, MATH 111, MATH 151, MATH 153, MATH 221, MATH 222, MATH 230, MATH 321, and MATH 422 may not be used toward the Math minor. MATH 361 is not suitable for engineering students. In the old numbering system, these courses correspond to , MATH 102, MATH 105, MATH 211, MATH 221, MATH 222, MATH 307, and MATH 466.

Exceptions must be approved by the Department Chair.

**Application: **When the required courses are completed, a student must get a Minor Form from the department secretary, fill it out and have it signed by the Chair of the Department. An approved form will be forwarded to the appropriate dean. If a form is not approved, the student will be notified.

### Math Minor Requirements for Students in the School of Engineering

MATH 185 | Calculus I | 3 |

MATH 186 | Calculus II | 3 |

MATH 285 | Calculus III | 3 |

MATH 286 | Differential Equations | 3 |

One of the following: | 3 | |

Discrete Foundations | ||

Linear Algebra I | ||

Probability (not approved for EECE or CE) ^{*} | ||

Vector Calculus (not approved for EECE) ^{*} | ||

Partial Differential Equations | ||

Statistical Inference (not approved for EECE or CE) ^{*} | ||

Operations Research | ||

Complex Analysis (recommended for EECE) ^{*} | ||

Selected 300-400 level courses, including Topics courses, may be approved. |

MATH 361 will not approved for Engineering majors.

* | Course duplication with courses in the major department |

Courses taught in the School of Engineering with similar content to those in the Department of Mathematics are not eligible to count toward a minor in mathematics. CEEN 308, CIVL 305, EECE 307, EECE 315, and MECH 314 will not count toward the minor.

### Math Minor Requirements for Students in the School of Arts and the School of Business

MATH 154 | Calculus Business Decisions | 3 |

or MATH 155 | Calculus for the Life Sciences I | |

or MATH 185 | Calculus I | |

or MATH 187 | Honors Calculus I | |

MATH 156 | Calculus for the Life Sciences II | 3 |

or MATH 186 | Calculus II | |

or MATH 188 | Honors Calculus II | |

Three of the following: | 9 | |

Discrete Foundations | ||

Linear Algebra I | ||

Differential Equations | ||

Probability | ||

Vector Calculus | ||

Partial Differential Equations | ||

Statistical Inference | ||

Operations Research | ||

Complex Analysis | ||

Selected 300-400 level courses, including Topics courses, may be approved. |

## PLAN OF STUDY

## Bachelor of Science in Mathematics

First Year | |||
---|---|---|---|

Fall | Credits | Spring | Credits |

MATH 185 | 3 | MATH 186 | 3 |

CMPT 101 | 3 | CMPT 102 | 3 |

MLL^{*} | 3 | MLL^{*} | 3 |

ENGL 110 | 3 | RELS 110 | 3 |

LLRN 102 | 3 | Social Science | 3 |

SCI 100 | 1 | ||

16 | 15 | ||

Second Year | |||

Fall | Credits | Spring | Credits |

MATH 285 or 287 | 3 | MATH 286 | 3 |

MATH 243 | 3 | MATH 272 | 3 |

PHYS 101 | 4 | PHYS 102 | 4 |

PHIL 150 | 3 | ENGL 150 | 3 |

Social Science | 3 | Free Elective | 3 |

16 | 16 | ||

Third Year | |||

Fall | Credits | Spring | Credits |

MATH 331 | 3 | MATH 387 | 3 |

MATH 377 | 3 | MATH 478 | 3 |

MATH 471 | 3 | RELS Catholic Studies | 3 |

Natural Science^{**} | 4 | Natural Science^{**} | 4 |

HIST 150 | 3 | MUSC 150 or ART 150 | 3 |

16 | 16 | ||

Fourth Year | |||

Fall | Credits | Spring | Credits |

MATH 489 | 3 | MATH 490 | 3 |

MATH Elective | 3 | MATH Elective | 3 |

Free Electives | 9 | Free Electives | 6 |

RELS Global/Contemporary | 3 | ||

15 | 15 | ||

Total Credits: 125 |

* | |

** | |

## Bachelor of Arts in Mathematics

First Year | |||
---|---|---|---|

Fall | Credits | Spring | Credits |

MATH 185 | 3 | MATH 186 | 3 |

CMPT 101 | 3 | CMPT 102 | 3 |

MLL^{*} | 3 | MLL^{*} | 3 |

ENGL 110 | 3 | RELS 110 | 3 |

LLRN 102 | 3 | Social Science | 3 |

SCI 100 | 1 | ||

16 | 15 | ||

Second Year | |||

Fall | Credits | Spring | Credits |

MATH 285 or 287 | 3 | MATH 286 | 3 |

MATH 243 | 3 | MATH 272 | 3 |

SCI XXX^{**} | 3 | SCI XXX | 3 |

PHIL 150 | 3 | SCI XXX | 3 |

Social Science | 3 | ENGL 150 | 3 |

15 | 15 | ||

Third Year | |||

Fall | Credits | Spring | Credits |

MATH 331^{***} | 3 | MATH 387 | 3 |

MATH 377 | 3 | MATH 478 | 3 |

MATH 471 | 3 | RELS Catholic Studies | 3 |

HIST 150 | 3 | MUSC 150 or ART 150 | 3 |

Free Electives | 3 | Free Electives | 3 |

15 | 15 | ||

Fourth Year | |||

Fall | Credits | Spring | Credits |

MATH 489 | 3 | MATH 490 | 3 |

MATH Elective | 3 | RELS Global/Contemporary | 3 |

Social Sciences | 3 | MATH Elective | 3 |

Free Electives | 6 | Free Electives | 6 |

15 | 15 | ||

Total Credits: 121 |

* | One year sequence of a Modern Foreign Language. |

** | Students may opt for one (1) full year of a lab science (8 credits). In this case, an additional course must be taken to achieve a total of 120 credits. Student may also opt to replace SCIXX with 9 credits of courses from within a single discipline in the School of Science. |

*** | If MATH 432 Statistical Inferenceis taken as a Math elective, it is recommended it be taken immediately following MATH 331 Probability. |

### Courses

**MATH 096. Bridge Course for Business. 0 Credits.**

This online course is intended for incoming Business students identified by the Mathematics placement exam. (Linear Analysis for Business). It includes a review of the fundamentals of algebra, functions and their graphs, logarithmic and exponential functions, Excel spreadsheets.

**MATH 100. Pre-Calculus Mathematics. 3 Credits.**

Basic set theory, functions, and their graphs. Topics from algebra, theory of equations, trigonometry and analytic geometry. Intended to prepare students for a course in calculus. (Meets four hours per week.).

**MATH 111. Precalculus for Business Students. 3 Credits.**

Review of elementary algebra, introduction to analytic geometry, functions and their graphs, logarithmic and exponential functions, polynomial functions. (Meets four hours per week).

**MATH 151. Topics in Modern MathModern Mathematics. 3 Credits.**

(Formerly MATH 102) An introduction to a mathematical topic of particular interest to students in humanities or social sciences. Possibilities include but not limited to the mathematics of social choice, mathematics and art, mathematics and music, modeling for environmental issues, network theory, consumer mathematics. Descriptions of particular topic chosen will be available at the time of offering.

**MATH 153. Linear Math Analysis. 3 Credits.**

(Formerly Math 105) Survey of business applications. Topics include financial mathematics: simple and compound interest, annuities and amortization of loans; linear optimization: solving systems of linear equations and matrix algebra; probability: elementary counting techniques, odds, and expected value. Prerequisite: A satisfactory score on the mathematics placement exam or a C or better in MATH 111 or an equivalent course is required to enroll in MATH 153.

**MATH 154. Calculus Business Decisions. 3 Credits.**

(Formerly Math 106) A one-semester course in the calculus of functions of one variable, intended for students in Business. Polynomial, rational, and exponential and logarithm functions. Limits, derivatives, techniques and applications of differentiation. Indefinite and definite integrals, applications of the integral. Prerequisite: A satisfactory score on the mathematics placement exam or a C or better in MATH 111 or an equivalent course is required to enroll in MATH 154.

**MATH 155. Calculus for the Life Sciences I. 3 Credits.**

(Formerly MATH 121) Limits, continuity, exponential/logarithmic functions, differentiation/antidifferentiation. An introduction to the definite integral. Meets four hours per week. Prerequisite: A satisfactory score on the mathematics placement exam or a C or better in MATH 100 or an equivalent course is required to enroll in MATH 155.

**MATH 156. Calculus for the Life Sciences II. 3 Credits.**

(Formerly MATH 122) Applications chosen from the life sciences, including population, decay, growth models, stability, and matrix methods. Volumes of solids, integration techniques, difference/differential equations. Meets four hours per week. Pre-requisite: A grade of C or better in MATH 155 or MATH 185 or MATH 121.

**MATH 185. Calculus I. 3 Credits.**

(Formerly Math 103) Limits, derivatives, continuity, differentiation and an introduction to the definite integral, and area between curves.(Meets four hours per week). Students are reminded that a grade of C- or lower in MATH 185 (formerly MATH 103) may indicate inadequate preparation for MATH 186 (formerly MATH 104). Prerequisite: A satisfactory score on the mathematics placement exam or a C or better in MATH 100 or an equivalent course is required to enroll in MATH 185.

**MATH 186. Calculus II. 3 Credits.**

(Formerly Math 104) Applications of the definite integral, transcendental functions, integration techniques and infinite series. (Meets four hours per week). Students are reminded that a grade of C- or lower in MATH 185 (formerly MATH 103) may indicate inadequate preparation for MATH 186. Prerequisite: A C or better in MATH 185, MATH 103, MATH 155, MATH 121, MATH 106 or an equivalent course is required to enroll in MATH 186.

**MATH 187. Honors Calculus I. 3 Credits.**

(Formerly MATH 109) Rigorous development of differential and integral calculus. Restricted to select students who will take this courses in lieu of Mathematics 185 (Meets four hours per week). Pre-requisite: Students will be selected by the professor.

**MATH 188. Honors Calculus II. 3 Credits.**

(Formerly MATH 110) Rigorous development of differential and integral calculus. Restricted to select students who will take this courses in lieu of Mathematics 186 (Meets four hours per week). Pre-requisite: Students will be selected by the professor.

**MATH 221. Mathematics for Elementary School Teachers I. 3 Credits.**

Courses for prospective teachers in elementary school who are not majoring in mathematics. The content and method will follow the current standards of the National Council of Teachers of Mathematics for the elementary level. Topics include tools for problem solving, numeration systems, and number theory.

**MATH 222. Mathematics for Elementary School Teachers II. 3 Credits.**

Courses for prospective teachers in elementary school who are not majoring in mathematics. The content and method will follow the current standards of the National Council of Teachers of Mathematics for the elementary level. Topics include tools for problem solving, geometry, and trigonometry.

**MATH 230. Elementary Statistics. 3 Credits.**

(Formerly MATH 211) An introduction to statistical methods: descriptive statistics, association between two variables, basic probability, discrete random variables, binomial and normal random variables, sampling distribution, confidence intervals, tests of significance.

**MATH 243. Discrete Foundations. 3 Credits.**

(Formerly MATH 213) An introduction to the mathematical concepts and techniques of discrete mathematics: Topics include principles of logical argument, modular arithmetic and congrence classes, induction, sets, functions, relations, summations, the binomial theorem cardinality of sets. Topics form the mathematical basis of computer science and are pivotal to the study of advanced mathematics. Pre-requisite: MATH 216.

**MATH 244. Discrete Mathematics II. 3 Credits.**

A continuation of MATH 243. Principles of counting, permutations and combinations, discrete probability, inclusion-exclusion, combinatorial arguments, graphs, matrices, isomorphism, markov chains, graph properties, tress, search trees, weighted trees, introduction to analysis of algorithms. Pre-requisite: MATH 243.

**MATH 272. Linear Algebra I. 3 Credits.**

(Formerly MATH 215)Linear equations and matrices, vector spaces, sub spaces, linear independence, bases, dimension, inner product spaces, linear transformations, eigenvalues and eigenvectors, orthogonal matrices and diagonalization. Prerequisite: MATH 285.

**MATH 285. Calculus III. 3 Credits.**

(Formerly Math 201) Algebraic and geometric aspects of vectors, functions of several variables, partial derivatives and multiple integrals. Prerequisite: A grade of C or better in MATH 186 or MATH 156 or MATH 104 or MATH 122.

**MATH 286. Differential Equations. 3 Credits.**

(Formerly MATH 203) This course focuses on techniques of solving first and second order ordinary differential equations. Methods include separation of variables, variation of parameters, and the Laplace transform. Applications include linear and nonlinear models. Prerequisite: MATH 201 or MATH 209 or MATH 285 or MATH 287 or permission of the instructor.

**MATH 287. Honors Calculus III. 3 Credits.**

(Formerly 209) Honors equivalent to Calculus III. Algebraic and geometric aspects of vectors, functions of several variables, partial derivatives and multiple integrals. Intended to be for students who have completed the honors section of MATH 104 or MATH 186. Prerequisite: A grade of C or better in the Honors section of MATH 186 or MATH 104 or permission of the instructor.

**MATH 320. Fundamental Concepts: Probability and Data Analysis. 3 Credits.**

A course for prospective teachers of K-8 mathematics. The course will explore data analysis as a four-step investigative process involving question development, data production, data analysis and contextual conclusions. Topics may include describing and comparing data distributions for both categorical and numerical data, exploring bivariate relationships, exploring elementary probability, and using random sampling as a basis for informal inference. Course includes use of appropriate software. (Education only).

**MATH 321. Fundamental Concepts: Algebra and Number Theory. 3 Credits.**

For prospective teachers of K-8 mathematics. Topics chosen from expressions and equations, ratio, proportional relationships and inversely proportional relationships, arithmetic and geometric sequences, factors and multiples (including greatest common factor and least common multiple), prime numbers and the Fundamental Theorem of Arithmetic, divisibility tests, rational versus irrational numbers, the division algorithm, modular arithmetic, functions (linear, quadratic, and exponential). Restricted to students in EDUC.

**MATH 322. Fundamental Concepts: Geometry and Measurement. 3 Credits.**

For prospective teachers for K-8 mathematics. Perimeter, area, surface area, volume, and angle; geometric shapes, geometric solids, transformations, dilations, symmetry, congruence, similarity; modeling with similar triangles, and the Pythagorean Theorem and its converse. The use of appropriate software is an important component of the course. (Educ. only).

**MATH 331. Probability. 3 Credits.**

(Formerly MATH 420) Basic theorems in probability, random variables, distribution functions, expected values; binomial, Poisson and normal distributions. Fall. Prerequisite: MATH 104 or MATH 186.

**MATH 336. Applied Statistics. 3 Credits.**

(Formerly Math 333) A calculus based survey of probability and statistics with applications in social, natural sciences and engineering. Topics include probability, discrete and continuous random variables, point and interval estimation, hypothesis testing, linear models (encompassing regression and ANOVA). Prerequisite: MATH 186 or MATH 156 or equivalent.

**MATH 361. Introduction to Higher Geometry. 3 Credits.**

(Formerly MATH 311) A survey of axiomatic and modern geometry intended for future middle and high school teachers. Topics covered will include incidence axioms, congruence theorems for triangles, the circle theorems, implications of the fifth postulate, congruence theorems for quadrilaterals, parallelism, similarity, transformational geometry, matrix transformations, and an introduction to spherical and hyperbolic geometry. The course will incorporate the use of Geometers Sketchpad or equivalent software as a tool for verification of conjectures. Offered every other year. Spring. Prerequisites: MATH 213, MATH 215 or MATH 243, MATH 272.

**MATH 375. Internship for Juniors. 3 Credits.**

Students participate in an off-campus training experience closely related to one of the areas of mathematics. Frequent meetings with the advisor plus a paper are required. Prerequisites: Junior status, 3.0 GPA, and permission of the student's advisor or the Chair.

**MATH 377. Algebra I. 3 Credits.**

(Formerly Math 315) The first part of a two-semester sequence. An introduction to algebraic structures with an emphasis on groups, covering normal subgroups, cosets. Lagrange's theorem and the fundamental homomorphism theorems. Fall.
Prerequisites: MATH 213, MATH 215 or MATH 243, MATH 272.

**MATH 385. Vector Calculus. 3 Credits.**

(Formerly Math 305)Review of vector algebra. Vector-valued functions. Divergence and curl. Multiple integrals; different coordinate systems. Line integrals, Greens Theorem, independence of path, conservative force fields. Surface integrals, Divergence Theorem, Stokes Theorem, Applications. Prerequisite: MATH 201 or MATH 209 or MATH 285 or MATH 287.

**MATH 386. Partial Differential Equations. 3 Credits.**

(Formerly MATH 308) Classification of partial differential equations. Characteristics. Derivation of the classical linear second order equations. Fourier series. Separation of variables. Initial and boundary value problems. Cauchy, Dirichlet, and Neumann problems. Prerequisite: MATH 203 or MATH 286.

**MATH 387. Analysis I. 3 Credits.**

(Formerly Math 313) A rigorous treatment of differential calculus of one variable: sequences, limits, continuity, the derivative. Fall. Prerequisites: MATH 201 and MATH 213 or MATH 285 or MATH 287 and MATH 243.

**MATH 422. Seminar for Mathematics Education. 3 Credits.**

(Formerly Math 466) This course is intended for prospective mathematics teachers. Topics in high school mathematics are examined from an advanced perspective. Topics include the real and complex numbers, functions, equations, and trigonometry. (Enrollment restricted to students in the School of Education and Health.) Spring. Prerequisites: MATH 213, MATH 215 or MATH 243 and MATH 272.

**MATH 432. Statistical Inference. 3 Credits.**

Sampling distributions, point estimation, interval estimation, testing statistical hypotheses, regression and correlation. Spring. Prerequisite: MATH 420 or MATH 331.

**MATH 433. Advanced Mathematical Statistics I. 3 Credits.**

(Formerly MATH 423) Analysis of variance, regression analysis, non-parametric and sequential tests of hypotheses. Prerequisite: MATH 421 or MATH 432.

**MATH 448. Combinatorics and Graph Theory. 3 Credits.**

Fundamental concepts in combinatorics include binomial coefficients, inclusion-exclusion, and generating functions. Topics in graph theory include connectivity, planarity, colorings and chromatic polynomials, and max-flow-min-cut in networks, and other applications. Not open to students with credit for CMPT 335. Pre-requisite: MATH 272.

**MATH 455. Operations Research. 3 Credits.**

(Formerly MATH 425) Optimization, linear programming, simplex method, duality theory. Transportation problems, scheduling problems, queuing theory.
Prerequisite: MATH 215 or MATH 272 or permission of instructor.

**MATH 457. Machine Learning. 3 Credits.**

An introduction to the field of Machine Learning and its real-world applications. Topics include Supervised & Unsupervised Learning, Bayesian Decision Theory, Nonparametric Methods, Linear Discriminant Functions, Multilayer Neural Networks, Stochastic Methods and Cluster Analysis. Pre-requisite: MATH 285.

**MATH 464. Topology. 3 Credits.**

(Formerly Math 417) Beginning with a review of set theory and basic topological definitions, topological spaces are studied with metric spaces considered as examples. Compactness, connectedness, metrization theorems. An introduction to homotopy theory. Prerequisite: MATH 213 or MATH 243 or permission of instructor.

**MATH 471. Linear Algebra II. 3 Credits.**

(Formerly Math 325) A continuation of the topics introduced in MATH 241( formerly MATH 215) with emphasis on orthogonality, inner product spaces, eigenvalues and eigenvectors, diagonalization, quadratic forms and numerical linear algebra. Fall. Prerequisite: MATH 215 or MATH 272.

**MATH 475. Internship for Seniors. 3 Credits.**

Students participate in an off-campus training experience closely related to their area of mathematics. Frequent meetings with the advisor plus a paper are required. Prerequisites: Senior status, 3.0 GPA, and permission of the student's advisor or the Chair.

**MATH 478. Algebra II. 3 Credits.**

(Formerly Math 316)The second part of a two-semester sequence. Further study of algebraic structures, such as rings, fields and integral domains. The homomorphism theorems and applications. Spring. Prerequisite: MATH 315 or MATH 377.

**MATH 488. Analysis II. 3 Credits.**

(Formerly Math 314) A continuation of MATH 387 (or MATH 313). Topology of the real numbers, uniform convergence, Riemann integral, infinite series, Taylor and Fourier series, metric spaces. Prerequisite: MATH 203 and MATH 213 or MATH 286 or MATH 243.

**MATH 489. Problem Seminar. 3 Credits.**

(Formerly Math 460) A capstone course for senior mathematics majors. Problems will be chosen to integrate the themes of the major. Oral presentations and mathematical writing and proof will be emphasized. Spring. Prerequisites: MATH 313 or MATH 315 or MATH 387, MATH 377 or permission of instructor.

**MATH 490. Complex Analysis. 3 Credits.**

(Formerly MATH 407) The complex plane, functions, limits and continuity. Analytic functions, Cauchy- Riemann equations. Cauchy integral theorem and consequences. Additional topics may include: Power series, Taylor and Laurent series, classification of singularities, the Residue Theorem and its applications, conformal mapping, selected applications. Spring. Prerequisite: MATH 286 or MATH 203, MATH 243 recommended or permission of instructor.

**MATH 491. Topics in Mathematics. 3 Credits.**

(Formerly MATH 461-462) Admission only by permission of the Chair of the Department. This course is offered when demand warrants. Prerequisite: Permission of the chair.

**MATH 492. Topics in Mathematics. 3 Credits.**

(Formerly Math 461-462) Admission only by permission of the Chair of the Department. This course is offered when demand warrants. Prerequisite: Permission of the chair.

**MATH 497. Mathematics Seminar. 3 Credits.**

(Formerly MATH 467) A course limited to students of superior ability who wish to study some advanced topic mutually agreed upon by them, the instructor and the Department Chair. Prerequisite: Permission of the instructor and chair.

**MATH 499. Independent Study. 3 Credits.**

(Formerly Math 469) Individual study or research under faculty supervision. Prerequisite: Permission of the instructor and chair.