A maximum total of 4 credits may be earned in Mathematics courses numbered 110 and below. A maximum total of 16 credits may be earned for calculus courses Math 112 through Math 396, and no credit can be earned for a prerequisite to a course taken after the course itself.

103. Intermediate Algebra. Only open to designated summer half-term Bridge students. (Excl).
Review of elementary algebra; rational and quadratic equations; properties of relations, functions, and their graphs; linear and quadratic functions; inequalities, logarithmic and exponential functions and equations. Equivalent to the first year of Math. 105/106.

105. Data, Functions, and Graphs. Students with credit for Math. 103 can elect Math. 105 for only 2 credits. No credit granted to those who have completed any Mathematics course numbered 110 or higher. (4). (MSA). (QR/1).
This is a course on analyzing data by means of functions and graphs. The emphasis is on mathematical modeling of real-world applications. The functions used are linear, quadratic, polynomial, logarithmic, exponential, and trigonometric. Algebra skills are assessed during the term by periodic testing.

110. Pre-Calculus (Self-Study). See Elementary Courses above. Enrollment in Math 110 is by recommendation of Math 115 instructor and override only. No credit granted to those who already have 4 credits for pre-calculus mathematics courses. (2). (Excl).
A condensed half-term version of Math 105. Offered as a self-study course through the Math Lab and directed toward students who are unable to complete a first calculus course successfully. Students study on their own and consult with tutors in the Math Lab whenever needed.

115. Calculus I. Four years of high school mathematics. See Elementary Courses above. Credit usually is granted for only one course from among Math. 112, 115, 185, and 295. No credit granted to those who have completed Math. 175. (4). (MSA). (BS). (QR/1).
This course presents calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students develop their reading, writing, and questioning skill. Topics include functions and graphs, derivatives and their application to real-life problems in various fields, and definite integrals.

116. Calculus II. Math. 115. Credit is granted for only one course from among Math. 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).
This course presents calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students develop their reading, writing, and questioning skill. Topics include the indefinite integral, techniques of integration, introduction to differential equations, infinite series.

119. Calculus II Using MAPLE. Math. 115 or score of 3 or higher on the AB or BC Advanced Placement Calculus exam. Credit is granted for only one course from among Math. 114, 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).
The material covered is approximately that of Math 116. In addition, students are taught to use the computer algebra system MAPLE as a tool to do routine calculations, to visualize and to explore. Topics include applications of the definite integral, separable differential equations, inverse functions, infinite sequences and series, conics and parametric curves.

127. Geometry and the Imagination. Three years of high school mathematics including a geometry course. No credit granted to those who have completed a 200- (or higher) level mathematics course. (4). (MSA). (BS). (QR/1).
This course introduces students to the ideas and some of the basic results in Euclidean and non-Euclidean geometry. The course begins with an historical perspective of how the ancient Greeks influenced the study of geometry prior to the 19th century. Then notions of non-Euclidean geometry and geometry in higher dimensions are introduced and studied.

128. Explorations in Number Theory. High school mathematics through at least Analytic Geometry. No credit granted to those who have completed a 200- (or higher) level mathematics course. (4). (MSA). (BS). (QR/1).
Designed for non-science concentrators and students with no intended concentration who want to learn how to think mathematically without having to take calculus first. Students are introduced to the ideas of Number Theory through lectures and experimentation by using software to investigate numerical phenomena, and to make conjectures that they try to prove.

147. Introduction to Interest Theory. Math. 112 or 115. No credit granted to those who have completed a 200- (or higher) level mathematics course. (3). (MSA). (BS).
An introduction to the mathematical concepts and techniques used by financial institutions. Topics include rates of simple and compound interest and present and accumulated values; annuity functions and applications to amortization, sinking funds and bond values; depreciation methods; introduction to life tables, life annuities and life insurance value.

156. Applied Honors Calculus II. Score of 4 or 5 on the AB or BC Advanced Placement calculus exam. Credit is granted for only one course among Math 114, 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).
Second term calculus for engineering and science concentrators. Topics include applications of integral calculus, improper integrals, sequences and series, differential equations. complex numbers, MAPLE.

175. Combinatorics and Calculus. Permission of Honors advisor. No credit granted to those who have completed a 200-level or higher Mathematics course. (4). (MSA). (BS). (QR/1).
Graph theory--connectivity, Eulerian graphs, trees, applications of trees, planarity, Euler's formula, chromatic polynomials, network flows. Enumeration--binomial theorem, recurrence relations, generating functions, inclusion-exclusion.

176. Dynamical Systems and Calculus. Credit is granted for only one course from among Math. 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).
Discrete-time and continuous dynamical systems. Topics include iterates of functions, simple differential equations, attracting and repelling fixed points and periodic orbits, chaotic motion, self-similarity, and fractals. Tools such as limits, continuity, Taylor expansions, exponentials, eigenvalues, eigenvectors, are reviewed or introduced as needed. Weekly computer workstation lab.

185. Honors Analytic Geometry and Calculus I. Permission of the Honors advisor. Credit is granted for only one course from among Math. 112, 113, 115, 185, and 295. No credit granted to those who have completed Math. 175. (4). (MSA). (BS). (QR/1).
Topics covered include functions and graphs, derivatives, differentiation of algebraic and trigonometric functions and applications, definite and indefinite integrals and applications. Other topics are included at the discretion of the instructor.

186. Honors Analytic Geometry and Calculus II. Permission of the Honors advisor. Credit is granted for only one course from among Math. 114, 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).
Topics covered include transcendental functions, techniques of integration, introduction to differential equations, conic sections, and infinite sequences and series. Other topics included at the discretion of the instructor.

203. Introduction to MAPLE and MATHEMATICA. Prior or concurrent enrollment in one term of calculus. No programming experience is assumed. No credit granted to those who have completed Math. 119. (1). (Excl). (BS). Offered mandatory credit/no credit. May be repeated for a total of two credits.
Provides students with an introduction to two powerful Computer Algebra Systems for doing algebra, calculus and statistical and graphical analysis.

215. Calculus III. Math. 116, 119, 156, 176, 186, or 296. (4). (MSA). (BS). (QR/1).
Topics include vector algebra and vector functions; analytic geometry of planes, surfaces, and solids; functions of several variables and partial differentiation; line, surface, and volume integrals and applications; vector fields and integration; Green's Theorem and Stokes' Theorem. There is a weekly computer lab using MAPLE software.

216. Introduction to Differential Equations. Math. 116, 119, 156, 176, 186, or 296. (4). (MSA). (BS).
After an introduction to ordinary differential equations, the first half of the course is devoted to topics in linear algebra, including systems of linear algebraic equations, vector spaces, linear dependence, bases, dimension, matrix algebra, determinants, eigenvalues, and eigenvectors. In the second half these tools are applied to the solution of linear systems of ordinary differential equations. Topics include: oscillating systems, the Laplace transform, initial value problems, resonance, phase portraits, and an introduction to numerical methods. This course is not intended for mathematics concentrators, who should elect the sequence 217-316.

217. Linear Algebra. Math. 215, 255, or 285. No credit granted to those who have completed or are enrolled in Math. 417, 419, or 513. (4). (MSA). (BS). (QR/1).
The topics covered include: systems of linear equations; matrix algebra; vectors, vector spaces, and subspaces; geometry of Rⁿ; linear dependence, bases, and dimension; linear transformations; Eigenvalues and Eigenvectors; diagonalization; inner products.

219. Calculus III Using MAPLE. Math. 119. (4). (MSA). (BS).
Topics include vector algebra and vector functions, introduction to Fourier series, analytic geometry of planes, surfaces, and solids; functions of several variables and partial differentiation; line, surface, and volume integrals and applications; vector fields and integration; Green's Theorem and Stokes' Theorem.

255. Applied Honors Calculus III. Math. 156. (4). (MSA). (BS).
Multivariable calculus, line, surface and volume integrals, vector fields, Green's theorem, Stokes theorem, divergence theorem, applications, MAPLE.

256. Applied Honors Calculus IV. Math. 255. (4). (MSA). (BS).
Linear algebra, matrices, systems of differential equations, initial and boundary value problems, qualitative theory of dynamical systems, nonlinear equations, numerical methods, MAPLE.

285. Honors Analytic Geometry and Calculus III. Math. 176 or 186, or permission of the Honors advisor. (4). (MSA). (BS).
Topics covered include vector algebra and vector functions; analytic geometry of planes, surfaces, and solids; function of several variables and partial differentiation; line surface and volume integrals and applications; vector fields and integration; Green's Theorem; Stokes' Theorem.

286. Honors Differential Equations. Math. 285. (3). (MSA). (BS).
Topics include first-order differential equations, higher-order linear differential equations with constant coefficients; linear systems.

288. Math Modeling Workshop. Math. 216 or 316, and Math. 217 or 417. (1). (Excl). (BS). Offered mandatory credit/no credit. May be repeated for a total of three credits.
Each weekly session describes formulation and solution of a real-world problem, using array of techniques from undergraduate mathematics.

289. Problem Seminar. (1). (Excl). (BS). May be repeated for credit with permission.
Focuses on the development of mathematical problem-solving skills. Problems are taken from classical analysis, elementary number theory, and geometry. For students with outstanding problem-solving ability.

295(195). Honors Mathematics I. Prior knowledge of first year calculus and permission of the Honors advisor. Credit is granted for only one course from among Math. 112, 113, 115, 185, and 295. (4). (MSA). (BS). (QR/1).
Introduction to mathematical analysis with emphasis on proofs and theory. Covers such topics as set theory, construction of the real number field, limits of sequences and functions, continuity, elementary functions, derivatives and integrals. Additional topics may include countability, topology of real numbers, infinite series, uniform continuity.

296(196). Honors Mathematics II. Prior knowledge of first year calculus and permission of the Honors advisor. Credit is granted for only one course from among Math. 116, 119, 156, 176, 186, and 296. (4). (Excl). (BS). (QR/1).
Topics generally include infinite series, power series, Taylor expansion, metric spaces. Other topics may include applications of analysis, Weierstrass Approximation theorem, elements of topology, introduction to linear algebra, complex numbers.

312. Applied Modern Algebra. Math. 217. Only one credit granted to those who have completed Math. 412. (3). (Excl). (BS).
Sets and functions, relations and graphs, rings and Boolean algebras, semigroups and groups, lattices. Applications chosen from such areas as switching theory, automata theory, coding theory and may include finite state machines, minimal state machines, algebraic descriptions of logic circuits, semigroup machines, binary codes, fast adders, Pólya enumeration theory, series and parallel decomposition of machines.

316. Differential Equations. Math. 215 and 217. Credit can be received for only one of Math. 216 or Math. 316, and credit can be received for only one of Math. 316 or Math. 404. (3). (Excl). (BS).
First-order equations: solutions, existence and uniqueness, and numerical techniques; linear systems: eigenvector-eigenvalue solutions of constant coefficient systems, fundamental matrix solutions, nonhomogeneous systems; higher-order equations, reduction of order, variation of parameters, series solutions; qualitative behavior of systems, equilibrium points, stability.

333. Directed Tutoring. Math. 385 and enrollment in the Elementary Program in the School of Education. (1-3). (Excl). (EXPERIENTIAL). May be repeated for a total of three credits.
An experiential mathematics course for elementary teachers. Students would tutor elementary (Math. 102) or intermediate (Math. 104) algebra in the Math. Lab. They would also participate in a weekly seminar to discuss mathematical and methodological questions.

350/Aero. 350. Aerospace Engineering Analysis. Math. 216, 256, 286, or 316. (3). (Excl). (BS).
Formulation and solution of some of the elementary initial- and boundary-value problems relevant to aerospace engineering. Application of Fourier series, separation of variables, and vector analysis to problems of forced oscillations, wave motion, diffusion, elasticity, and perfect-fluid theory.

354. Fourier Analysis and its Applications. Math. 216, 256, 286, or 316. No credit granted to those who have completed or are enrolled in Math. 454. (3). (Excl). (BS).
Fourier series; discrete Fourier transforms, and continuous Fourier transforms, with applications to subjects such as signal processing and filtering; Fourier optics; partial differential equations (Poisson, heat, and wave equations); probability theory (random walks); and Weyl's theorem on equidistribution of arithmetic sequences.

371/Engin. 371. Numerical Methods for Engineers and Scientists. Engineering 101, and Math. 216. (3). (Excl). (BS).
A survey course of the basic numerical methods which are used to solve scientific problems. In addition, concepts such as accuracy, stability and efficiency are discussed. The course provides an introduction to MATLAB, an interactive program for numerical linear algebra as well as practice in computer programming.

385. Mathematics for Elementary School Teachers. One year each of high school algebra and geometry. No credit granted to those who have completed or are enrolled in 485. (3). (Excl).
The history, development, and logical foundations of the real number system and of numeration systems including scales of notation; simple algebra, functions and graphs; and geometric relations.

395(295). Honors Analysis I. Math. 296 or permission of the Honors advisor. (4). (Excl). (BS).
Functions of several real variables. Topics are chosen from elementary linear algebra, elementary topology, differential and integral calculus of scalar- and vector-valued functions and vector valued mappings, implicit and inverse function theorems.

396(296). Honors Analysis II. Math. 395. (4). (Excl). (BS).
Differential and integral calculus of functions on Euclidean spaces. Designed primarily for mathematics Honors students who have had 295, 296 and 395. The material covered in 395, 396 and 496 is approximately that of Math. 216, 404, 451 and 551 but there is a deeper penetration into many topics.

399. Independent Reading. (1-6). (Excl). (INDEPENDENT). May be repeated for credit.
Designed especially for Honors students.

403. Mathematical Modeling Using Computer Algebra Systems. One year of calculus and junior standing. (3). (MSA). (BS). (QR/1).
This course teaches the use of Computer Algebra Systems (such as Mathematica and Maple) with emphasis on modeling real world problems in the student's area of interest

404. Intermediate Differential Equations. Math. 216. No credit granted to those who have completed Math. 256, 286, or 316. (3). (Excl). (BS).
First-order equations, second and higher-order linear equations, Wronskians, variation of parameters, mechanical vibrations, power series solutions, regular singular points, Laplace transform methods, eigenvalues and eigenvectors, nonlinear autonomous systems, critical points, stability, qualitative behavior, application to competing-species and predator-prey models, numerical methods.

412. Introduction to Modern Algebra. Math. 215, 255, or 285; and 217. No credit granted to those who have completed or are enrolled in 512. Students with credit for 312 should take 512 rather than 412. One credit granted to those who have completed 312. (3). (Excl). (BS).
Sets, functions (mapping, relations, and the common number systems (integers to complex numbers). These are then applied to the study of groups and rings. These structures are presented as abstractions from many examples. Notions such as generator, subgroup, direct product, isomorphism and homomorphism.

413. Calculus for Social Scientists. Not open to freshmen, sophomores or mathematics concentrators. (3). (Excl). (BS).
Review of algebra, functions, and graphs followed by an intuitive introduction to the rudiments of calculus. Applications of differentiation and integration to problems in the social sciences. Designed especially for seniors and graduate students with minimal mathematics background.

416. Theory of Algorithms. Math. 312 or 412 or CS 303, and CS 380. (3). (Excl). (BS).
Sorting, searching, matrix multiplication, graph problems (flows, traveling salesman), and primality and pseudoprimality testing (in connection with coding). Algorithm types as divide and conquer, backtracking, greedy and dynamic programming are analyzed using generating functions, recurrence relations, induction and recursion, graphs and trees, and permutations. A possible section on complexity theory and NP completeness.

417. Matrix Algebra I. Three courses beyond Math. 110. No credit granted to those who have completed or are enrolled in 217, 419, or 513. (3). (Excl). (BS).
Matrix operations, vector spaces, Gaussian and Gauss-Jordan algorithms for linear equations, subspaces of vector spaces, linear transformations, determinants, orthogonality, characteristic polynomials, eigenvalue problems and similarity theory. Applications include linear networks, least squares method, discrete Markov Processes, linear programming.

419/EECS 400/CS 400. Linear Spaces and Matrix Theory. Four terms of college mathematics beyond Math 110. No credit granted to those who have completed or are enrolled in 217 or 513. One credit granted to those who have completed Math. 417. (3). (Excl). (BS).
Finite dimensional linear spaces and matrix representation of linear transformations; bases, subspaces, determinants, eigenvectors, and canonical forms; and structure of solutions of systems of linear equations. Applications to differential and difference equations. The course provides more depth and content than Mathematics 417. Mathematics 513 is the proper election for students contemplating research in mathematics.

420. Matrix Algebra II. Math. 217, 417, or 419. (3). (Excl). (BS).
Similarity theory, Euclidean and unitary geometry, applications to linear and differential equations, interpolation theory, least squares and principal components, B-Splines.

422. Topics in Actuarial Mathematics I. Math. 216, 256, 286, or 316. (3). (Excl). (BS).
We explore how much insurance affects the lives of students (automobile insurance, social security, health insurance, theft insurance) as well as the lives of other family members (retirements, life insurance, group insurance). While the mathematical models are important, an ability to articulate why the insurance options exist and how they satisfy the consumer's needs are equally important. In addition, there are different options available (e.g., in social insurance programs) that offer the opportunity of discussing alternative approaches.

423. Mathematics of Finance. Math. 217 and 425; CS 183. (3). (Excl). (BS).
Introduction to mathematical models in finance and economics with particular emphasis on models for pricing derivative instruments such as options and futures. Topics include risk and return theory, portfolio theory, capital asset pricing model, random walk model, stochastic processes, Black-Scholes Analysis, numerical methods and interest rate models.

424. Compound Interest and Life Insurance. Math. 215, 255, or 285. (3). (Excl). (BS).
Rates used in compound interest theory; annuities-certain and their application to amortization, sinking funds, and bond values; and introduction to life annuities and life insurance. Both the discrete and the continuous approach are used.

425/Stat. 425. Introduction to Probability. Math. 215, 255, or 285. (3). (MSA). (BS).
Basic concepts of probability, discrete sample spaces, conditional probabilities and independence of events, random variables, expectation and variance, binomial and Poisson distribution, DeMoivre-Laplace limit theorem, and introduction to continuous probability.

427/Human Behavior 603 (Social Work). Retirement Plans and Other Employee Benefit Plans. Junior standing. (3). (Excl).
The development of employee benefit plans, both public and private. Particular emphasis is laid on modern pension plans and their relationships to current tax laws and regulations, benefits under the federal social security system, and group insurance.

431. Topics in Geometry for Teachers. Math. 215, 255, or 285. (3). (Excl). (BS).
Investigation of Euclidean geometry based on the Birkhoff or SMSG metric axiom system and reference to contemporary high school texts. Comparison with synthetic Euclid-Hilbert foundations. Historical development of absolute and hyperbolic geometry. Other non-Euclidean geometries. New directions in high school geometry, transformation groups especially isometries of the Euclidean plane as generated by reflections, similarities, and affine transformations.

433. Introduction to Differential Geometry. Math. 215, or 255 or 285, and Math. 217(3). (Excl). (BS).
Curves and surfaces in three-space, using calculus. Curvature and torsion of curves. Curvature, covariant differentiation, parallelism, isometry, geodesics, and area on surfaces. Gauss-Bonnet Theorem. Minimal surfaces.

450. Advanced Mathematics for Engineers I. Math. 216, 256, 286, or 316. (4). (Excl). (BS).
Review of curves and surfaces in implicit parametric and explicit forms; differentiability and affine approx.; implicit and inverse function theorems; chain rule for 3-space; multiple integrals, scalar and vector fields; line and surface integrals; computations of planetary motion, work, circulation and flux over surfaces; Gauss' and Stokes' Theorems; heat equations.

451. Advanced Calculus I. Math. 215 and one course beyond Math. 215; or Math. 255 or 285. Intended for concentrators; other students should elect Math. 450. (3). (Excl). (BS).
Calculus of functions of one variable, including their expansions into power series, and foundations of calculus of functions of two or more variables.

452. Advanced Calculus II. Math. 217, 417, or 419; and Math. 451. (3). (Excl). (BS).
Partial derivatives and differentiability; gradients, directional derivatives, and the chain rule; implicit function theorem; surfaces, tangent plane; max-min theory; multiple integration, change of variable; Green's and Stokes' theorems, differential forms, exterior surfaces; intro to differential geometry.

454. Boundary Value Problems for Partial Differential Equations. Math. 216, 256, 286, or 316. Students with credit for Math. 354 can elect Math. 454 for one credit. (3). (Excl). (BS).
Classical representation and convergence theorems for Fourier series; separation of variables for the 1-dim heat and wave equations in higher dim; spherical and cylindrical Bessel functions; Legendre polynomials; asymptotic integrals; discrete Fourier transform; applications to linear input-output systems, etc.

462. Mathematical Models. Math. 216, 256, 286, or 316; and 217, 417, or 419. Students with credit for 362 must have department permission to elect 462. (3). (Excl). (BS).
Construction and analysis of mathematical models involving probability, combinatorics, decision theory, optimization, games, and dynamics. Applications to some of the physical, social, life, and decision sciences.

463. Mathematical Modeling in Biology. Math. 217, 417, or 419; 286 or 316. (3). (Excl). (BS).
An introduction to the use of continuous and discrete differential equations in the biological sciences. Modeling in biology, physiology, and medicine.

464/BiomedE 464. Inverse Problems. One of Math. 217, 417, or 419; and one of Math. 216, 256, 286, or 316. (3). (Excl). (BS).
Mathematical concepts applied in the solution of inverse problems and in the analysis of related forward operators are discussed. The course content is often motivated by a particular inverse problem from a field such as medical imaging, geophysics, or non-destructive testing. Mathematical topics include ill-posedness, regularization, pseudoinverses, and transforms.

471. Introduction to Numerical Methods. Math. 216, 256, 286, or 316; and 217, 417, or 419; and a working knowledge of one high-level computer language. (3). (Excl). (BS).
Computer arithmetic; Newton's method for nonlinear equations; polynomial interpolation, numerical integration, systems of linear equations; initial value problems for ordinary differential equations; quadrature; partial pivoting, spline approximations, partial differential equations; Monte Carlo methods.

475. Elementary Number Theory. At least three terms of college mathematics are recommended. (3). (Excl). (BS).
Theory of congruences, Euler's phi-function, diophantine equations, and quadratic domains.

476. Computational Laboratory in Number Theory. Prior or concurrent enrollment in Math. 475 or 575. (1). (Excl). (BS).
Students conduct numerical explorations in IBM-PC clones, using software tailored for the purpose.

481. Introduction to Mathematical Logic. Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS).
In the first third of the course the notion of a formal language is introduced and propositional connectives, tautologies and tautological consequences are studied. The heart of the course is the study of 1st order predictive languages and their models. New elements here are quantifiers. Notions of truth, logical consequences and probability lead to completeness and compactness. Applications.

485. Mathematics for Elementary School Teachers and Supervisors. One year of high school algebra. No credit granted to those who have completed or are enrolled in 385. (3). (Excl). (BS). May not be included in a concentration plan in mathematics.
The history, development, and logical foundations of the real number system and of numeration systems including scales of notation, cardinal numbers, and the cardinal concept; and the logical structure of arithmetic (field axioms) and relations to the algorithms of elementary school instruction. Simple algebra, functions, and graphs. Geometric relationships. For persons teaching in or preparing to teach in the elementary school.

486. Concepts Basic to Secondary Mathematics. Math. 215, 255, or 285. (3). (Excl). (BS).
Foundations of algebra and geometry with implications for secondary teaching. Modern approaches to trigonometry and analytic geometry.

489. Mathematics for Elementary and Middle School Teachers. Math. 385 or 485. May not be used in any graduate program in mathematics. (3). (Excl).
The second course in a two-course sequence required of elementary teaching certificate candidates. Topics covered include: the real-number system, probability, and statistics, geometry and measurement.

490. Introduction to Topology. Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS).
The topics covered are fairly constant but the presentation and emphasis varies significantly with the instructor. Point-set topology, examples of topological spaces, orientable and non-orientable surfaces, fundamental groups, homotopy, covering spaces. Metric and Euclidean spaces are emphasized.

497. Topics in Elementary Mathematics. Math. 489. (3). (Excl). (BS). May be repeated for a total of six credits.
Selected topics in geometry, algebra, computer programming, logic, and combinatorics for prospective and in-service teachers of elementary, middle, or junior high school mathematics. Content may vary from term to term.

498. Topics in Modern Mathematics. Senior mathematics concentrators and Master Degree students in mathematical disciplines. (3). (Excl). (BS).
The course concentrates on topics in modern mathematics not represented in the standard course list. The choice of topics varies from term to term depending on faculty and student interests.

512. Algebraic Structures. Math. 451 or 513. No credit granted to those who have completed or are enrolled in 412. Math. 512 requires more mathematical maturity than Math. 412. (3). (Excl). (BS).
Description and in-depth study of the basic algebraic structure groups, rings, fields including: set theory, relations, quotient groups, permutation groups, Sylow's Theorem, quotient rings, field of fractions, extension fields, roots of polynomials, straight-edge and compass solutions, and other topics.

513. Introduction to Linear Algebra. Math. 412 or permission of Honors advisor. Two credits granted to those who have completed Math. 417; one credit granted to those who have completed Math 217 or 419. (3). (Excl). (BS).
Vector spaces, linear transformations and matrices, equivalence of matrices and forms, canonical forms, and application to linear differential equations.

520. Life Contingencies I. Math. 424 and Math. 425. (3). (Excl). (BS).
The comprehensive study of annuity and insurance functions for single lives, population theory, topics in actuarial practice.

521. Life Contingencies II. Math. 520. (3). (Excl). (BS).
The comprehensive study of annuity and insurance functions for joint lives; multiple decrement theory and applications to accidental death and disability insurances; topics in actuarial practice.

522. Actuarial Theory of Pensions and Social Security. Math. 520. (3). (Excl). (BS).
Practice and theoretical techniques of pension plan evaluation and analysis; social security projections.

523. Risk Theory. Math. 425. (3). (Excl). (BS).
Topics include utility theory, application to buying general insurance to reduce risk, compound distribution models for risk portfolios, application of stochastic processes to the ruin problem and to reinsurance.

524. Topics in Actuarial Science II. Math. 424, 425, and 520; and Stat. 426. (3). (Excl). (BS). May be repeated for a total of 9 credits.
Topics vary by term. Mathematics of Demography; Mathematics of Graduation; Survival Modes and Construction of Mortality Tables.

525/Stat. 525. Probability Theory. Math. 450 or 451. Students with credit for Math. 425/Stat. 425 can elect Math. 525/Stat. 525 for only one credit. (3). (Excl). (BS).
Axiomatic probability; combinatorics; random variables and their distributions; expectation; the mean, variance, and moment generating function; induced distributions; sums of independent random variables; the law of large numbers; the central limit theorem. Optional topics drawn from: random walks, Markov chains, and/or martingales.

526/Stat. 526. Discrete State Stochastic Processes. Math. 525 or EECS 501. (3). (Excl). (BS).
Review of discrete distributions; generating functions; compound distributions, renewal theorem; modeling of systems as Markov chains; Markov chains: first properties; Chapman-Kolmogorov equations; return and first passage times; classification of states and periodicity; absorption probabilities and the forward equation; stationary distributions and the backward equation; ergodicity; limit properties; application to branching and queueing processes; examples from engineering, biological, and social sciences; Markov chains in continuous time; embedded chains; the M/G/1 queue; Markovian decision processes; application to inventory problems; other topics at instructor's discretion.

528. Topics in Casualty Insurance. Math 217, 417, or 419. (1). (Excl).
An introduction to property and casualty insurance including policy forms, underwriting, product design and modification, rate making and claim settlement.

531. Transformation Groups in Geometry. Math. 215, 255, or 285. (3). (Excl). (BS).
Isometries and congruences in the Euclidean plane as generated by reflections, translations, and half-turns. Tilings, affine and hyperbolic geometries, and Poincaré model of the hyperbolic plane. Selected applications to ornamental design, crystallography, and regular polytopes.

535. Introduction to Algebraic Curves. Math. 513. (3). (Excl). (BS).
Plane algebraic curves, linear systems of curves, rational curves, analysis of singularities, parametrizations, formal power series, and places of a curve. Bezout's theorem, rational transformations, Luroth's theorem, dual curves, ideal of a curve, and linear series.

537. Introduction to Differentiable Manifolds. Math. 513 and 590. (3). (Excl). (BS).
Manifolds, maps, and vector bundles with applications to flows, the Frobenius theorem, Riemannian manifolds, Stokes' theorem, Sard's theorem, embedding theorems, and the tubular neighborhood theorem. This course is a prerequisite for Mathematics 635 and 693.

555. Introduction to Functions of a Complex Variable with Applications. Math. 450 or 451. (3). (Excl). (BS).
Intended primarily for students of engineering and of other cognate subjects. Doctoral students in mathematics elect Mathematics 596. Complex numbers, continuity, derivative, conformal representation, integration, Cauchy theorems, power series, singularities, and applications to engineering and mathematical physics.

556. Methods of Applied Mathematics I. Math. 217, 419, or 513; 451 and 555. (3). (Excl). (BS).
A study of some of the differential equations of mathematical physics and methods for their solution. Separation of variables for heat, wave, Laplace's and Schrödinger's equations; special functions and their integral representations and asymptotic properties; and eigenvalues as solutions of variational problems.

557. Methods of Applied Mathematics II. Math 217, 419, or 513; 451 and 555. (3). (Excl). (BS).
Elementary distributions, Green's functions and integral solutions for nonhomogeneous problems, Fourier and Hankel transforms, and Fredholm alternative and elementary methods of solution of integral equations, with additional topics as time permits.

558(658). Ordinary Differential Equations. Math. 450 or 451. (3). (Excl). (BS).
Existence and uniqueness theorems for flows, linear systems, Floquet theory, Poincaré-Bendixson theory, Poincaré maps, periodic solutions, stability theory, Hopf bifurcations, chaotic dynamics.

559. Selected Topics in Applied Mathematics. Math. 451; and 217 or 419. (3). (Excl). (BS). May be repeated for a total of six credits.
Covers a branch of mathematics which has been strongly influenced by another science. The instructor educates the student in the intuitions of that science as well as present mathematical proofs.

561/SMS 518 (Business Administration)/IOE 510. Linear Programming I. Math. 217, 417, or 419. (3). (Excl). (BS).
Formulation of problems from the private and public sectors using the mathematical model of linear programming. Development of the simplex algorithm; duality theory and economic interpretations. Postoptimality (sensitivity) analysis and applications and interpretations. Introduction to transportation and assignment problems and special purpose algorithms and advanced computational techniques. Students have an opportunity to formulate and solve models developed from more complex case studies and to use various computer programs.

562/IOE 511/Aero. 577/EECS 505/CS 505. Continuous Optimization Methods. Math. 217, 417, or 419. (3). (Excl). (BS).
Survey of continuous optimization problems. Unconstrained optimization problems: undirectional search techniques including conjugate directions and gradient and variable metric methods. Constrained optimization problems: unconstrained techniques including penalty and barrier methods and constraint techniques including methods of feasible directions, gradient projection method, and linear constraint methods. Practical applications are emphasized.

565. Combinatorics and Graph Theory. Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS).
Graph Theory: trees, k-connectivity; Eularian and Hamiltonian graphs; tournaments; graph coloring; planar graphs, Euler's formula, 5-color theorem, Kuratowski's theorem and matrix-tree theorem; Enumeration; fundamental principles, bijections, generating functions, binomial theorem, partitions and 1 series, linear recurrences, generating functions and Pólya theory.

566. Combinatorial Theory. Math. 216, 256, 286, or 316. (3). (Excl). (BS).
Permutations, combinations, generating functions, and recurrence relations. The existence and enumeration of finite, discrete configurations. Systems of representatives, Ramsey's theorem, and external problems. Construction of combinatorial designs.

567. Introduction to Coding Theory. One of Math 217, 419, 513 . (3). (Excl). (BS).
This course is an introduction to coding for error correction. Topics include: linear codes and their parameters, decoding algorithms, asymptotically good sequences of codes.

571. Numerical Methods for Scientific Computing I. Math. 217, 419, or 513; and 454. (3). (Excl). (BS).
Topics covered usually include direct and iterative methods for solving systems of linear equations, least squares methods, and computation of eigenvectors and eigenvalues.

572. Numerical Methods for Scientific Computing II. Math. 217, 419, or 513; and 454. (3). (Excl). (BS).
Topics include one step, multistep, and stiff methods for initial value problems for ODE's, stability and convergence, hyperbolic, elliptic, and parabolic PDE's, CFL condition, von Neumann stability, finite element methods, spectral methods.

575. Introduction to Theory of Numbers I. Math. 451 and 513. Students with credit for Math. 475 can elect Math. 575 for 1 credit. (3). (Excl). (BS).
Elementary theory of congruences, the quadratic reciprocity law, and properties of number theoretic functions.

582. Introduction to Set Theory. Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS).
The main topics are set algebra (union, intersection), relations and functions, orderings (partial-, linear-, well-), the natural numbers, finite and denumberable sets, the Axiom of Choice, and ordinal and cardinal numbers.

590. Introduction to Topology. Math. 451. (3). (Excl). (BS).
Topological and metric spaces,continuous functions, homeomorphism, compactness and connectedness, surface and manifolds, fundamental theorem of algebra and other topics.

591. General and Differential Topology. Math. 451. (3). (Excl). (BS).
Topological and metric spaces, continuity, subspaces, products and quotient topology, compactness and connectedness, extension theorems, topological groups, topological and differential manifolds, tangent spaces vector fields, submanifolds, inverse function theorem, immersions, submersions, partitions of unity, Sard's theorem, embedding theorems, transversality, classification of surfaces.

592. Introduction to Algebraic Topology. Math. 591. (3). (Excl). (BS).
Fundamental group, covering spaces, simplicial complexes, graphs and trees, applications to group theory, singular and simplicial homology, Eilenberg-Maclane axioms, Brouwer's and Lefschetz's fixed point theorems and other topics.

593. Algebra I. Math. 513. (3). (Excl). (BS).
Rings and modules. Euclidean rings, PIDs, classification of modules over a PID. Jordan and rational canonical forms. Structure of bilinear forms. Tensor products of modules; exterior algebras.

594. Algebra II. Math. 593. (3). (Excl). (BS).
Group theory. Permutation representations, simplicity of alternating groups for n > 4. Sylow theorems. Series in groups; solvable and nilpotent groups. Jordan-Hoelder for groups with operators. Free groups and presentations. Field extensions, norm and trace, algebraic closures. Galois theory. Transcendence degree.

596. Analysis I. Math. 451. (3). (Excl). (BS). Students with credit for Math. 555 may elect Math 596 for two credits only.
Review of analysis in R² including metric spaces, differentiable maps, Cauchy-Riemann equations, automorphisms. Analytic functions, Cauchy integral formula. Power series and Laurent expansions, fundamental theorem of algebra, harmonic functions. Functions analytic in a disk. Global properties of analytic functions. Riemann mapping theorem. Normal families.

597. Analysis II. Math. 451 and 513. (3). (Excl). (BS).
Lebesgue measure on the real line. Measurable functions and integration of R. Differentiation theory, fundamental theory of calculus. Function spaces, L-P(R), C(K), Holder and Minkowski inequalities, duality. General measure spaces, product measures, Fubini's theorem. Radon-Nikodym theorem, conditional expectation, signed measures.