Iowa State University
2005-2007 Courses and Programs
Iowa State University Catalog
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Mathematics10 |100 |200 | 300 | 400 | Graduate Courses www. math.iastate.edu Justin Peters, Chair of Department Undergraduate Study For the undergraduate curriculum in liberal arts and sciences, major in mathematics, leading to the degree bachelor of science, see Liberal Arts and Sciences, Curriculum. The program in mathematics offers training suitable for students planning to enter secondary school teaching, to work in mathematics and computation for industry or government, or to continue their studies in graduate school. The requirements for an undergraduate major in mathematics are designed so that the student may have opportunity for appropriate specialization to meet one or more of the foregoing objectives and, at the same time, obtain a thorough introduction to the mathematics underlying all of them. Graduates understand a broad range of mathematical topics and are familiar with a broad range of mathematical models. They have skills for solving problems in diverse situations. They can construct rigorous arguments to demonstrate mathematical facts. They can communicate their mathematical methods to others and can justify their assumptions. The requirements for an undergraduate major include: (a) Math 165, 166, 201, 265, 317, 301, 414, and either 266 or 267. (b) 15 additional credits in mathematics courses at the 300 level or above. (c) The courses used to satisfy a) and b) above must include one of the sequences 301, 302; 414, 415; 435, 436. (d) In addition to the credits in (b), either Math 492 or 2 credits of C I/LAS 480C. (C I/LAS 480C is available only for students seeking secondary school certification). (e) English proficiency requirement: The department requires a grade of C- or better in each of English 104 and 105 (or 105H) and an upper-level writing requirement that may be met by writing an acceptable undergraduate thesis (Math 491) or by taking at least one of Engl 302, 305, 314 or Jl MC 201. A grade of C- or better is required. The department strongly recommends that each student majoring in mathematics include in the program substantial supporting work beyond the minimum general education requirement of the college in one or more areas of application of mathematics, such as other mathematical sciences, engineering, natural science, or social science. In particular, it recommends that each student take Com S 207, 208; Phys 221, 222; and Stat 341, 342 (or Math 304). It also recommends that students contemplating graduate study in mathematics acquire a reading knowlege of French, German, or Russian. Credits earned in Math 104, 105, 140, 141, 142, 150, 151, 160, 181, 182, 195, 196, cannot be counted toward graduation by mathematics majors. The department offers a minor in mathematics which may be earned by credit in Math 165, 166, 265, 301, 307 or 317, and 266 or 267. Courses below 165 cannot be used. Graduate Study The department offers programs leading to a master of science or doctor of philosophy degree in mathematics or applied mathematics, as well as minor work for students whose major is in another department. The department also offers a program leading to the degree of master of school mathematics (M.S.M.). Students desiring to undertake graduate work leading to the M.S. or Ph.D. degree should have at least 12 semester credits of work in mathematics beyond calculus. It is desirable that these credits include advanced calculus and abstract algebra. The M.S. degree requires at least 30 credit hours and students must write a creative component or thesis and pass a comprehensive oral examination over their coursework and their creative component or thesis. See the department handbook for specific requirements. The Ph.D. degree requires a student to take 54 hours of coursework in addition to research hours, pass written qualifying examinations, pass an oral preliminary exam, and perform an original research project culminating in a dissertation which is defended by an oral exam. Ph.D. candidates must have at least one year of supervised teaching experience. See the on-line Mathematics Graduate Handbook for specific requirements. The M.S.M. degree is primarily for inservice secondary mathematics teachers. Students desiring to pursue the M.S.M degree should present some undergraduate work in mathematics beyond calculus. Candidates for the M.S.M. degree must write an approved creative component and pass a comprehensive oral examination over their course work and their creative component. Courses open for nonmajor graduate credit: 301, 302, 304, 307, 314, 317, 331, 350, 365, 373, 378, 385, 395, 414, 415, 421, 426, 435, 436, 439, 465, 471, 481, 484, 489. Courses primarily for undergraduate students Math 10. High School Algebra. (4-0) Cr. 0. F.S.SS. For students who do not have adequate facility with topics from high school algebra or do not meet the algebra admission requirement. All students should initially enroll in Math 10. The course is divided into tracks of one- and two-semester lengths. For most students a diagnostic exam will determine which track must be taken. Students will receive a grade in Math 25 or 30 respectively depending on the level of material covered. Satisfactory completion of Math 30 is recommended for students planning to take Math 140 or 151, while Math 25 is sufficient for Math 104, 105, 150, 195, Stat 101 or 105. Students must complete Math 30 to remove a deficiency in the algebra admission requirement. Topics include signed numbers, polynomials, rational and radical expressions, exponential and logarithmic expressions, and equations. Offered on a satisfactory-fail grading basis only. Math 25. High School Algebra. (4-0) Cr. 0. F.S.SS. See description of Math 10. Offered on a satisfactory-fail grading basis only. Math 30. High School Algebra. (4-0) Cr. 0. F.S.SS. See description of Math 10. Offered on a satisfactory-fail grading basis only. Math 101. Orientation in Mathematics. (1-0) Cr. R. F. For new majors. Issues to consider in planning a program of study. Sources of general information and perspectives concerning mathematics. Discussion of possible areas of study or careers. Offered on a satisfactory-fail grading basis only. Math 104. Introduction to Probability and Matrices. (3-0). Cr. 3. F.S. Prereq: Satisfactory performance on placement exam, 2 years of high school algebra, 1 year of high school geometry. Permutations, combinations, probability, binomial and multinomial theorems, matrices, expected value. Either 104 or 150 may be counted toward graduation, but not both. Math 105. Introduction to Mathematical Ideas. (3-0) Cr. 3. F.S. Prereq: Satisfactory performance on placement exam, 2 years of high school algebra, 1 year of high school geometry. Topics from mathematics and mathematical applications with emphasis on their nontechnical content. Math 140. College Algebra. (3-1) Cr. 3. F.S.SS. Prereq: Satisfactory performance on placement exam, 2 years of high school algebra; 1 year of high school geometry. Coordinate geometry, complex numbers, quadratic and polynomial equations, functions, graphing, polynomial and rational functions, exponential and logarithmic functions, systems of equations. Students in the College of Liberal Arts and Sciences may not count Math 140, 141, 142, 149, or 195 toward Group III of the General Education Requirements. Math 141. Trigonometry. (2-0) Cr. 2. F.S.SS. Prereq: Satisfactory performance on placement exam, 2 years of high school algebra; 1 year of high school geometry, or enrollment in 140. May be taken concurrently with 140. Trigonometric functions and their inverses, solving triangles, trigonometric identities and equations, graphing. Students in the College of Liberal Arts and Sciences may not count Math 140, 141, 142, 149, or 195 toward Group III of the General Education Requirements. Only one of 141, 142 may count toward graduation. Math 142. Trigonometry and Analytic Geometry. (2-1) Cr. 3. F.S.SS. Prereq: Satisfactory performance on placement exam, 2 years of high school algebra, 1 year of high school geometry, or enrollment in 140. May be taken concurrently with 140. Trigonometric functions and their inverses, solving triangles, trigonometric identities and equations, graphing, polar coordinates, complex numbers, standard equations of lines and conic sections, parametric equations. Students in the College of Liberal Arts and Sciences may not count Math 140, 141, 142, 149, or 195 toward Group III of the General Education Requirements. Only one of 141, 142 may count toward graduation. Math 150. Discrete Mathematics for Business and Social Sciences. (2-1) Cr. 3. F.S.SS. Prereq: Satisfactory performance on placement exam, 2 years of high school algebra, 1 year of high school geometry. Linear equations and inequalities, linear programming, matrix algebra, discrete probability. Either 104 or 150 may be counted toward graduation, but not both. Math 151. Calculus for Business and Social Sciences. (2-1) Cr. 3. F.S.SS. Prereq: Satisfactory performance on placement exam, 2 years of high school algebra, 1 year of high school geometry. Differential calculus, applications to max-min problems, integral calculus and applications. Will not serve as Prerequisite for 265 or 266. Only one of 151, 160, the sequence 165-166, or the sequence 181-182 may be counted towards graduation. Math 160. Survey of Calculus. (4-0) Cr. 4. F.S. Prereq: Satisfactory performance on placement exam, 2 years of high school algebra, 1 year of geometry. Analytic geometry, derivatives and integrals of elementary functions, partial derivatives, and applications. Will not serve as a Prerequisite for 265 or 266. Only one of 151, 160, the sequence 165-166, or the sequence 181-182 may be counted towards graduation. Math 165. Calculus I. (4-0) Cr. 4. F.S.SS. Prereq: Satisfactory performance on placement exam, 2 years of high school algebra, 1 year of geometry, 1 semester of trigonometry or enrollment in 141 or 142. Differential calculus, applications of the derivative, introduction to integral calculus. Only one of 151 or 160 or the sequence 165-166, or the sequence 181-182 may be counted towards graduation. Math 165H. Honors Calculus I. (4-0) Cr. 4. F. Prereq: Satisfactory performance on placement exam, 2 years of high school algebra, 1 year of geometry, 1 semester of trigonometry. High math placement scores recommended but not required. Differential calculus, applications of the derivative, introduction to integral calculus. Additional material of a theoretical, conceptual, computational, or modeling nature. Some of the work may require more ingenuity than is required in Math 165. Preference will be given to students in the University Honors Program. Only one of 151 or 160 or the sequence 165-166 or the sequence 181-182 may be counted towards graduation. Math 166. Calculus II. (4-0) Cr. 4. F.S.SS. Prereq: Grade of C- or better in 165, 165H, or high math placement scores. Integral calculus, applications of the integral, infinite series. Only one of 151, 160, the sequence 165-166, or the sequence 181-182 may be counted towards graduation. Math 166H. Honors Calculus II. (4-0) Cr. 4. F.S. Prereq: Permission of instructor and 165, 165H, or high math placement scores. Integral calculus, applications of the integral, infinite series. Additional material of a theoretical, conceptual, computational, or modeling nature. Some of the work may require more ingenuity than is required for Math 166. Preference will be given to students in the University Honors Program. Only one of 151, or 160, the sequence 165-166, or the sequence 181- 182 may be counted towards graduation. Math 181. Calculus and Difference Equations for the Life Sciences I. (4-0) Cr. 4. F.S. Prereq: Satisfactory performance on placement exam, 2 years of high school algebra, 1 year of high school geometry, 1 semester of trigonometry or enrollment in 141 or 142. Exponential and logarithm functions, derivative, first order linear difference equations and differential equations. Examples taken from laboratory experiments. Only one of 151, 160, the sequence 165- 166, or the sequence 181-182 may be counted towards graduation. Math 182. Calculus and Difference Equations for the Life Sciences II. (4-0) Cr. 4. F.S. Prereq: 181. Integral, nonlinear and second order difference equations, and differential equations. Examples taken from laboratory experiments. Only one of 151, 160, the sequence 165-166, or the sequence 181-182 may be counted towards graduation. Math 195. Mathematics for Elementary Education I.(2-2) Cr. 3. F.S. Prereq: Satisfactory performance on placement exam, 2 years high school algebra, 1 year of high school geometry, enrollment in elementary education or early childhood education. Language of sets, systems of whole numbers, topics from number theory, geometric shapes, congruence, transformations, linear measurement, problem solving. Math 196. Mathematics for Elementary Education II. (2-2) Cr. 3. F.S. Prereq: Grade of C- or better in 195. Two-and three-dimensional measurement, probability, data analysis, statistics, operations and algorithms for computing with integers, fractions, and decimals. Math 201. Introduction to Proofs. (2-0) Cr. 2. F.S. Prereq: 166 or 166H. Reading and writing simple proofs. Proofs involving the real numbers and the definitions of limit, derivative, and the definite integral. Proofs by mathematical induction. Math 265. Calculus III. (4-0) Cr. 4. F.S.SS. Prereq: Grade of C- or better in 166 or 166H. Analytic geometry and vectors, differential calculus of functions of several variables, multiple integrals, vector calculus. Math 265H. Honors Calculus III. (4-0) Cr. 4. F.S. Prereq: Permission of the instructor; and 166 or 166H. Analytic geometry and vectors, differential calculus of functions of several variables, multiple integrals, vector calculus. Additional material of a theoretical, conceptual, computational, or modeling nature. Some of the work may require more ingenuity than is required in Math 265. Preference will be given to students in the University Honors Program. Math 266. Elementary Differential Equations. (3-0) Cr. 3. F.S.SS. Prereq: Grade of C- or better in 166 or 166H. Solution methods for ordinary differential equations. First order equations, linear equations, constant coefficient equations. Eigenvalue methods for systems of first order linear equations. Introduction to stability and phase plane analysis. Math 267. Elementary Differential Equations and Laplace Transforms. (4-0) Cr. 4. F.S.SS. Prereq: Grade of C- or better in 166 or 166H. Same as 266 but also including Laplace transforms and series solutions to ordinary differential equations. Math 268. Laplace Transforms. (1-0) Cr. 1. F. Prereq: 266. Laplace transforms and series solutions to ordinary differential equations. Together, 266 and 268 are the same as 267. Math 290. Independent Study. Cr. 1 to 3 each time taken. Math 297. Intermediate Topics for School Mathematics. (2-2) Cr. 3. F.S. Prereq: Enrollment in elementary education and grade of C- or better in 196 or enrollment as mathematics major and admission to teacher education. Topics in geometry including coordinates, congruence, similarity, and transformations. Data analysis, mathematical reasoning, probability, and use of technology to learn and teach mathematics. Math 298. Cooperative Education. Cr. R. F.S.SS. Prereq: Permission of the department cooperative education coordinator; sophomore classification. Required of all cooperative education students. Students must register for this course prior to commencing each work period. Math 301. Abstract Algebra I. (3-0) Cr. 3. F.S. Prereq: 166 or 166H and 307 or 317. Theory of groups. Homomorphisms. Quotient groups. Introduction to rings. Emphasis on writing proofs. Nonmajor graduate credit. Math 302. Abstract Algebra II. (3-0) Cr. 3. S. Prereq: 301. Theory of rings and fields. Introduction to Galois theory. Emphasis on writing proofs. Nonmajor graduate credit. Math 304. Introductory Combinatorics. (3-0) Cr. 3. F. Prereq: 166 or 166H. Permutations, combinations, binomial coefficients, inclusion-exclusion principle, recurrence relations, generating functions. Additional topics selected from probability, random walks, and Markov chains. Nonmajor graduate credit. Math 307. Matrices and Linear Algebra. (3-0) Cr. 3. F.S.SS. Prereq: 2 semesters of calculus. Systems of linear equations, determinants, vector spaces, orthogonality, linear transformations, eigenvalues and eigenvectors. Emphasis on methods and techniques. Only one of 307, 317 may be counted toward graduation. Nonmajor graduate credit. Math 314. Graphs and Networks. (3-0) Cr. 3. S. Prereq: 166 or 166H. Structure and extremal properties of graphs. Topics are selected from: trees, networks, colorings, paths and cycles, connectivity, planarity, Ramsey theory, forbidden structures, enumeration, applications. Nonmajor graduate credit. Math 317. Theory of Linear Algebra. (4-0) Cr. 4. F.S. Prereq: 166. Systems of linear equations, determinants, vector spaces, inner product spaces, linear transformations, eigenvalues and eigenvectors. Emphasis on writing proofs and results. Nonmajor graduate credit. Only one of 307, 317 may be counted toward graduation. Math 331. Topology. (3-0) Cr. 3. Alt. S., offered 2007. Prereq: 307 or 317. Topological properties of metric spaces, including Rn, sequences, continuous functions, completeness, compactness. Nonmajor graduate credit. Math 341. Introduction to Theory of Probability and Statistics. (Same as Stat 341.) See Statistics. Math 342. Introduction to Theory of Probability and Statistics. (Same as Stat 342.) See Statistics. Math 350. Number Theory. (3-0) Cr. 3. Alt. S., offered 2007. Prereq: 307 or 317. Divisibility, integer representations, primes and divisors, linear diophantine equations, congruences, and multiplicative functions. Applications to cryptography. Nonmajor graduate credit. Math 365. Complex Variables with Applications. (3-0) Cr. 3. F.S. Prereq: 265. Functions of a complex variable, including differentiation, integration and series expansions, residues, evaluation of integrals, conformal mapping. Only two of 365, 385, 395 may be counted toward graduation. Nonmajor graduate credit. Math 373. Introduction to Scientific Computation. (3-0) Cr. 3. S. Prereq: 265. Vector, matrix and graphics programming in MATLAB for scientific applications. Algorithms for interpolation, systems of linear equations, least squares, nonlinear equations and optimization in one and several variables. Additional topics may include ordinary differential equations, symbolic calculation and the Fast Fourier Transform. Emphasis on effective use of mathematical software, and understanding of its strengths and limitations. Math 378. Optimization and Modeling with Evolutionary Computation. (3-0) Cr. 3. S. Prereq: One of 301, 304, Com S 330 or other discrete math; familiarity with programming. Introduction to modeling and optimization techniques known as evolutionary computation. Biological paradigms from evolution and ecology are used to solve problems in biology, engineering and areas such as combinatorial or functional optimization and modeling problems. Nonmajor graduate credit. Math 385. Introduction to Partial Differential Equations. (3-0) Cr. 3. F.S. Prereq: 265 and one of 266, 267. Separation of variables methods for elliptic, parabolic, and hyperbolic partial differential equations. Fourier series, Sturm-Liouville theory, Bessel functions, and spherical harmonics. Only two of 365, 385, 395 may be counted toward graduation. Nonmajor graduate credit. Math 395. Intermediate Engineering Mathematics. (4-0) Cr. 4. F.S. Prereq: 265 and 267. Complex variables and analytic functions, complex integration techniques, complex series, Fourier series, separation of variables in partial differential equations. Only two of 365, 385, 395 may be counted toward graduation. Nonmajor graduate credit. Math 398. Cooperative Education. Cr. R. F.S.SS. Prereq: Permission of the department cooperative education coordinator; junior classification. Required of all cooperative education students. Students must register for this course prior to commencing each work period. Math 414. Analysis I. (3-0) Cr. 3. F.S.SS. Prereq: 201; 265; and 307 or 317. A careful development of calculus of functions of a real variable: limits, continuity, differentiation, integration, series. Nonmajor graduate credit. Math 415. Analysis II. (3-0) Cr. 3. S. Prereq: 414. Sequences and series of functions of a real variable, uniform convergence, power series and Taylor series, Fourier series, topology of Rn, implicit function theorem, calculus of R2 and R3. Additional topics may include metric spaces or Stieties or Lebesgue integration. Nonmajor graduate credit. Math 421. Logic for Mathematics and Computer Science. (Same as Com S 421.) (3-0) Cr. 3. S. Prereq: 301 or 307 or 317 or Com S 330. Propositional and predicate logic, Horn logic, equational logic, resolution and unification, foundations of logic programming, reasoning about programs, program specification and verification. Nonmajor graduate credit. Math 426. Mathematical Methods for the Physical Sciences. (3-0) Cr. 3. F. Prereq: 266 or 267. A fast- paced course primarily for first-year graduate students in physics and chemistry. Emphasis on techniques needed for quantum mechanics and electrodynamics. Functions of a complex variable and contour integration, integral transforms and applications, series methods for ordinary differential equations, Green's functions, Sturm-Liouville problems and orthogonal functions, boundary-value problems for partial differential equations. Credit will not be given for both 395 and 426. Nonmajor graduate credit. Math 435. Geometry I. (3-0) Cr. 3. F. Prereq: 307 or 317. Euclidean geometry. Points, lines, circles, triangles, congruence, similarity, properties invariant under rigid motions. Synthetic, analytic, and axiomatic methods. Nonmajor graduate credit. Math 436. Geometry II. (3-0) Cr. 3. S. Prereq: 435. Continuation of Euclidean geometry with topics from elliptic, projective, or hyperbolic geometry. Emphasis on analytic methods. Nonmajor graduate credit. Math 439. Mathematics of Fractals. (3-0) Cr. 3. Alt. S., offered 2006. Prereq: 265; some knowledge of programming. Topology of metric spaces; iterated function systems; algorithms for generation of fractals; fractal dimension; Julia sets and the Mandelbrot set; applications to chaotic systems. Nonmajor graduate credit. Math 465. Advanced Calculus for Applied Mathematics. (4-0) Cr. 4. S.SS. Prereq: 265. Frequently applied concepts from multivariable calculus, presented with enough theory to promote understanding of applications. Topics may include derivative matrices, Taylor polynomials, curvilinear coordinates, Green's theorem, divergence theorem, Stokes's theorem, uniform convergence, operations on series and integrals, improper integrals. Nonmajor graduate credit. Math 471. Computational Linear Algebra and Fixed Point Iteration. (Same as Com S 471.) (3-0) Cr. 3. F.S. Prereq: 265 and either 266, or 267; knowledge of FORTRAN or C. Computational error, solutions of linear systems, least square methods, similarity methods for eigenvalues, non-linear equations, fixed point iteration in one and several variables, Newton's method in several variables. Nonmajor graduate credit. Math 481. Numerical Solution of Differential Equations and Interpolation. (Same as Com S 481.) (3-0) Cr. 3. S.SS. Prereq: 265 and either 266 or 267; knowledge of FORTRAN or C. Orthogonal polynomials, least square and spline methods, numerical differentiation and integration, Euler, Taylor, Runge-Kutta, and predictor-corrector methods for solution of systems of ordinary differential equations. Nonmajor graduate credit. Math 484. Computational Mathematics for Biologists. (Same as BCB 484.) (3-0) Cr. 3. F. A survey of graph theory, linear algebra, discrete math, and algorithms used in computational biology with examples taken from genomics, phylogenetics, and structure problems. This course provides mathematics background for BCB/Gen/Com S/Math 594. Nonmajor graduate credit. Math 489. History of Mathematics. (3-0) Cr. 3. S. Prereq: 6 credits in mathematics at the 300 level or above. History of mathematical ideas found in the undergraduate curriculum. It includes a discussion of the historical and cultural settings in which these ideas arose, and the influence of the culture on the type of mathematical ideas that developed. Some of the particular cultures and their mathematics that are studied include: Babylonian and Ancient Egyptian. Ancient Greek, Arabic, Indian, Western European and Chinese. Nonmajor graduate credit. Math 490. Independent Study. Cr. 1 to 3 each time taken. Prereq: 301 or 317; 6 credits in mathematics. No more than 9 credits of Math 490 may be counted toward graduation. Math 491. Undergraduate Thesis. Cr. 2 or 3. Writing a formal mathematics paper. Upon approval by the department, the paper will satisfy the departmental advanced English requirement. Math 492. Undergraduate Seminar. Cr. 2. S. Prereq: Consent of instructor. Introduction to mathematics research. Mathematical presentation, mathematical literature search, participating in seminar on advanced topics in mathematics. Seminar content varies. Math 497. Teaching Secondary School Mathematics. (Same as C I 497.) See Curriculum and Instruction. Math 498. Cooperative Education. Cr. R. F.S.SS. Prereq: Permission of the department cooperative education coordinator; senior classification. Required of all cooperative education students. Students must register for this course prior to commencing each work period. Courses primarily for graduate students, open to qualified undergraduate students Math 501. Introduction to Real Analysis. (3-0) Cr. 3. F. Prereq: 265 and 307 or 317. A development of the real numbers. Study of metric spaces, completeness, sequences, and continuity of functions. Differentiation and integration of real-valued functions, sequences of functions, limits and convergence, equicontinuity. Introduction to Lebesgue measure. Math 502. Numerical Analysis I. (3-0) Cr. 3. F. Prereq: 414 or 501. Numerical linear algebra including eigenvalue problems; numerical solution of nonlinear equations and optimization problems. Math 503. Numerical Analysis II. (3-0) Cr. 3. S. Prereq: 414 or 501. Does not require 502. Approximation theory, including polynomial interpolation and best approximation; numerical integration; numerical methods for ordinary differential equations. Math 504. Abstract Algebra I. (3-0) Cr. 3. F. Prereq: 302. Algebraic systems and their morphisms, including groups, rings, modules, and fields. Math 505. Abstract Algebra II. (3-0) Cr. 3. S. Prereq: 504. Continuation of 504. Math 507. Numerical Solution of Differential Equations. (Same as Com S 507.) (3-0) Cr. 3. SS. Prereq: 415 or 465. One step methods for ordinary differential equations. Finite difference methods for linear partial differential equations. Initial-boundary value problems. Math 510. Linear Algebra. (3-0) Cr. 3. F. or SS. Prereq: 302 or 307 or 317. Advanced topics in linear algebra including canonical forms, inner product spaces, bilinear forms, tensor products, and applications to other branches of mathematics. Math 511. Functions of a Single Complex Variable. (3-0) Cr. 3. S. or SS. Prereq: 414 or 465 or 501. Theory of analytic functions, integration, topology of the extended complex plane, singularities and residue theory. Math 515. Real Analysis I. (3-0) Cr. 3. F. Prereq: 414 or 501. Measure and integration. Decomposition of measures; differentiation. Metric spaces, Lp spaces, Hilbert spaces. Elementary theory of Banach spaces. Product integration, Fubini's theorem. Math 516. Real Analysis II. (3-0) Cr. 3. S. Prereq: 515. Continuation of 515. Additional topics from real analysis. Math 517. Finite Difference Methods. (3-0) Cr. 3. F. Prereq: 481 or 507. Finite difference methods for parabolic equations, with emphasis on parabolic and hyperbolic equations, hyperbolic conservation laws, and other applied PDEs. Topics include convergence, stability and implementation issues. Math 519. Methods of Applied Mathematics I. (3-0) Cr. 3. F. Prereq: 414 or 465 or 501. Techniques of classical and functional analysis with applications to partial differential equations, integral equations, and boundary value problems for ordinary differential equations. Math 520. Methods of Applied Mathematics II. (3-0) Cr. 3. S. Prereq: 519. Continuation of Math 519. Math 525. Numerical Analysis of High Performance Computing. (Same as Com S 525, Cpr E 525.) (3-0) Cr. 3. S. Prereq: Cpr E 308, or one of Math 471, 481; experience in scientific programming; knowledge of FORTRAN or C. Development, analysis, and testing of efficient numerical methods for use on current state-of-the-art high performance computers. Applications of the methods to the students' areas of research. Math 533. Cryptography. (Same as Cpr E 533, InfAs 533.) (3-0) Cr. 3. S. Prereq: Math 301 or Cpr E 310 or Com S 330. Basic concepts of secure communication, DES and AES, public-key cryptosystems, elliptic curves, hash algorithms, digital signatures, applications. Relevant material on number theory and finite fields. Math 535. Steganography and Watermarking. (Same as Cpr E 535, InfAs 535.) (3-0) Cr. 3. S. Prereq: Cpr E 531 or E E 524 or Math 533/Cpr E 533/InfAs 533. Basic principles of steganography and watermarking. Algorithms based on spatial domain approaches, transformations of data, statistical approaches. Techniques for images, video, and audio data. Communications models for data hiding. Analysis, detection and recovery of hidden data. Military, commercial and e-commerce applications. Known theoretical results. Software packages for data hiding. Social and legal issues, case studies, and digital rights management issues that affect technological development of steganography and watermarking. Current developments in the area. Math 540. Seminar in Mathematics Education. (3-0) Cr. 3. Offered on a 3-year cycle, offered SS. 2008. Prereq: Enrollment in the master of school mathematics program or professional studies in education. Research studies in mathematics learning and teaching, exemplary practices in mathematics education, and current state and national trends in the mathematics curriculum in grades K-12. Math 542. Investigating the Teaching and Learning of Secondary Mathematics. (1-0) Cr. 1. Alt. F., offered 2006. Prereq: Enrollment in master of school mathematics program, professional studies in education or by permission for secondary mathematics education majors. Research, discussion and evaluation of efforts to improve instruction in the mathematics classroom. Math 542 may be taken for credit multiple times. Math 543. Topics in Mathematics Education. (1-0) Cr. 1. F. Prereq: Teaching a mathematics course. Selected topics in collegiate mathematics education including cooperative learning, instructional use of technology, writing in mathematics, and cognitive learning theories. Research studies, exemplar practices, and trends in mathematics education. Math 545. Intermediate Calculus. (4-0) Cr. 4. Offered on a 3-year cycle, offered SS. 2007. Prereq: 3 semesters of calculus and enrollment in the master of school mathematics program. Further development of the fundamental concepts of calculus and their applications with an emphasis on a constructivist approach to learning, cooperative groups, problem solving, the use of technology. Math 546. Algorithms in Analysis and Their Computer Implementation. (2-2) Cr. 3. Offered on a 3- year cycle, offered SS. 2007. Prereq: 3 semesters in calculus or concurrent enrollment in 545 and enrollment in the master of school mathematics program. The use of technology in secondary mathematics with an emphasis on the exploration and implementation of algorithms. Math 547. Discrete Mathematics and Applications. (4-0) Cr. 4. Offered on a 3-year cycle, offered SS. 2006. Prereq: Enrollment in the master of school mathematics program. Applications of graph theory, game theory, linear programming, recursion, combinatorics and algebraic structures. Issues in integrating discrete topics into the secondary curriculum. Use of the computer to explore discrete mathematics. Math 549. Intermediate Geometry. (3-0) Cr. 3. Offered on a 3-year cycle, offered SS. 2006. Prereq: 435 or equivalent and enrollment in the master of school mathematics program. A study of geometry with emphasis on metrics, the group of isometries, the group of similarities, and the affine group. Specific spaces studied normally include the Euclidean plane, the 2-sphere, and projective 2-space. Emphasis on analytical methods. Math 554. Introduction to Stochastic Processes. (Same as Stat 554.) (3-0) Cr. 3. F. Prereq: Stat 542. Markov chains on discrete spaces in discrete and continuous time (random walks, Poisson processes, birth and death processes) and their long-term behavior. Optional topics may include branching processes, renewal theory, introduction to Brownian motion. Math 557. Ordinary Differential Equations I. (3-0) Cr. 3. F. Prereq: 415 or 465 or 501. The initial-value problem, existence and uniqueness theorems, continuous dependence on parameters, linear systems, stability and asymptotic behavior of solutions, topics from dynamical systems and two-point boundary-value problems. Math 573. Random Signal Analysis and Kalman Filtering. (Same as Aer E 573, E E 573, M E 573.) (3-0) Cr. 3. F. Prereq: E E 321 or Aer E 331 or M E 370 or M E 411 or Math 341 or 395. Elementary notions of probability. Random processes. Autocorrelation and spectral functions. Estimation of spectrum from finite data. Response of linear systems to random inputs. Discrete and continuous Kalman filter theory and applications. Smoothing and prediction. Linearization of nonlinear dynamics. Math 574. Optimal Control. (Same as Aer E 574, E E 574, M E 574.) (3-0) Cr. 3. S. Prereq: 577. The optimal control problem. Variational approach. Pontryagin's principle. Hamilton-Jacobi equation. Dynamic programming. Time-optimal, minimum fuel, minimum energy control systems. The regulator problem. Structures and properties of optimal controls. Math 575. Introduction to Robust Control. (Same as E E 575.) See Electrical Engineering. Math 576. Digital Feedback Control Systems. (Same as Aer E 576, E E 576, M E 576.) (3-0) Cr. 3. F. Prereq: 415 or Aer E 432 or E E 475 or M E 411 or M E 414; and Math 267. Sampled-data, discrete data, and the z-transform. Design of digital control systems using transform methods: root locus, frequency response and direct design methods. Design using state-space methods. Controllability, observability, pole placement, state estimators. Digital filters in control systems. Microcomputer implementation of digital filters. Finite wordlength effects. Linear quadratic optimal control in digital control systems. Simulation of digital control systems. Math 577. Linear Systems. (Same as Aer E 577, E E 577, M E 577.) (3-0) Cr. 3. F. Prereq: 415 or Aer E 331 or M E 414; and Math 307 or 317. State variable and input-output descriptions of linear continuous-time and discrete-time systems. Solution of linear dynamical equations. Controllability and observability of linear dynamical systems. Canonical descriptions of linear equations. Irreducible realizations of rational transfer function matrices. Canonical form dynamical equations. State feedback. State estimators. Decoupling by state feedback. Design of feedback systems. Stability of linear dynamical systems. Math 578. Nonlinear Systems. (Same as Aer E 578, E E 578, M E 578.) (3-0) Cr. 3. S. Prereq: 577. Classification of nonlinear control systems. Existence and uniqueness of solutions. Approximate analysis methods. Periodic orbits. Concept of stability and Lyapunove stability theory. Absolute stability of feedback systems. Input-and output stability. Passivity. Math 590. Special Topics. Cr. var. Math 594. Computational Molecular Biology. (Same as GDCB 594.) See Genetics, Development and Cell Biology. Math 597. Introductory Computational Structural Biology. (Same as BCB 597.) (3-0) Cr. 3. S. Prereq: Math 265 and some knowledge of programming. Mathematical and computational approaches to protein structure prediction and determination. Topics include molecular distance geometry, potential energy minimization, and molecular dynamics simulation. Math 599. Creative Component. Cr. var. Courses primarily for graduate students Math 601. Mathematical Logic I. (3-0) Cr. 3. Alt. F., offered 2006. Prereq: 504. First semester of full- year course. Algebraic structures in logical systems, recursive functions, consistency, undecidability and incompleteness of axiomatic theories, results of Gentzen and Gddel, theory of models, ultraproducts and ultralimits, nonstandard analysis. Math 602. Mathematical Logic II. (3-0) Cr. 3. Alt. S., offered 2007. Prereq: 601. Continuation of 601. Math 605. Design Theory and Association Schemes. (3-0) Cr. 3. Alt. F., offered 2006. Prereq: 504. Combinatorial designs and Latin squares. Construction methods including finite fields. Error- correcting codes. Adjacency matrices and algebraic combinatorics. Math 606. Enumerative Combinatorics and Ordered Sets. (3-0) Cr. 3. Alt. S., offered 2007. Prereq: 504. Ordered sets and lattices. Generating functions. Mbbius inversion and other enumeration methods. Math 607. Modern (Structural) Graph Theory. (3-0) Cr. 3. Alt. F., offered 2005. Prereq: 504. Structural and extremal theory of graphs. Topics include basic structures (trees, paths and cycles), networks, colorings, connectivity, topological graph theory, Ramsey theory, forbidden graphs and minors, introduction to random graphs, applications. Math 610. Seminar. Cr. var. Math 615. General Theory of Algebraic Structures I. (3-0) Cr. 3. Alt. F., offered 2005. Prereq: 504. First semester of full-year course. Subalgebras, homomorphisms, congruence relations, and direct products. Lattices and closure operators. Varieties and quasivarieties of algebras, free algebras, Birkhoff's theorems, clones, Mal'cev conditions. Advanced topics. Math 616. General Theory of Algebraic Structures II. (3-0) Cr. 3. Alt. S., offered 2006. Prereq: 615. Continuation of 615. Math 617. Category Theory. (3-0) Cr. 3. Alt. F., offered 2006. Prereq: 504. Categories and functors and their applications. Math 618. Boolean Algebras. (3-0) Cr. 3. Alt. S., offered 2007. Prereq: 504. Structure of Boolean algebras and their representations. Stone spaces and duality. Atomicity, completeness, distributivity, operators, extensions of homomorphisms. Examples and applications from mathematical logic and topology. Math 621. Topology. (3-0) Cr. 3. F. Prereq: Permission of instructor. Introduction to general topology. Math 622. Algebraic Topology. (3-0) Cr. 3. Alt. S., offered 2007. Prereq: 504. Foundations of algebraic topology. Simplicial complexes. Introduction to homology and cohomology. Math 624. Manifolds, Tensors and Differential Geometry. (3-0) Cr. 3. Alt. S., offered 2006. Prereq: 501 or 515. Topics selected from: Geometry of curves and surfaces. Manifolds, coordinate systems. Tensors, differential forms, Riemannian metrics. Connections, covariant differentiation, curvature tensors. Math 633. Functional Analysis I. (3-0) Cr. 3. Alt. F, offered 2005. Prereq: Permission of instructor. Fundamental theory of normed linear spaces and algebras emphasizing aspects that provide a framework for the study of boundary-value problems, eigenvalue problems, harmonic analysis, analytic function theory, and modern operator theory. Math 634. Functional Analysis II. (3-0) Cr. 3. Alt. S., offered 2006. Prereq: 633. Continuation of 633. Math 642. Advanced Probability Theory. (Same as Stat 642.) See Statistics. Math 645. Advanced Stochastic Processes. (Same as Stat 645.) (3-0) Cr. 3. S Prereq: Permission of instructor. Weak convergence. Random walks and Brownian motion. Martingales. Stochastic integration and It''s Formula. Stochastic differential equations and applications. Math 646. Mathematical Modeling of Complex Physical Systems. (3-0) Cr. 3. S. Prereq: Permission of instructor. Modeling of the dynamics of complex systems on multiple scales: Classical and dissipative molecular dynamics, stochastic modeling and Monte-Carlo simulation; macroscale non-linear dynamics and pattern formation. Math 654. Ordinary Differential Equations II. (3-0) Cr. 3. Alt. S., offered 2007. Prereq: 557. Continuation of 557. Math 655. Partial Differential Equations I. (3-0) Cr. 3. Alt. F., offered 2005. Prereq: 515 or 519. First semester of full-year course. First order equations and systems, conservation laws, general theory of linear partial differential equations of elliptic, parabolic and hyperbolic types, maximum principles, fundamental solutions, Sobolev spaces, variational and Hilbert space methods. Math 656. Partial Differential Equations II. (3-0) Cr. 3. Alt. S., offered 2006. Prereq: 655. Continuation of 655. Math 658. Dynamical Systems. (3-0) Cr. 3. Alt. S., offered 2007. Prereq: 501 or 515. Smooth mappings and flows on manifolds. Fixed points, stable, unstable and center manifolds, normal forms. Structural stability, bifurcations. Horseshoe maps, introduction to chaotic behavior. Math 666. Finite Element Methods. (3-0) Cr. 3. S. Prereq: 501 or 515. Elements of functional analysis; Sobolev spaces; variational principles and weak formulations; approximation theory in finite element spaces; analysis of finite element methods; implementation issues; applications. Math 690. Advanced Topics. Cr. var. Prereq: Permission of instructor. Math 699. Research. |
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