Texas A&M University Undergraduate Catalog 2002-2003 Edition

(MATH)

Professors G.D.Allen, G.A.Battle, G.R.Blakley, H.P.Boas, A.Boggess (Head), I.Borosh, J. H. Bramble, H.Cao, G.Chen, P. Cohen, D.C. Dobson, R. G. Douglas, K. J. Dykema, T. Erdelyi, R. E. Ewing, C. Foias, S.A.Fulling, S.C.Geller, R. Grigorchuk, D.J.Hartfiel, D.A.Hensley, A.M.Hobbs, W.B.Johnson, P. Kuchment, D.R. Larson, R. Lazarov, D.R.Lewis, F.J.Narcowich, P.Nelson,Jr., J. E. Pasciak, C.Pearcy, W.L.Perry, M.S.Pilant, G.Pisier, J.T.Pitts, K. R. Rajagopal, J. N. Reddy, W.Rundell, J F. Schielack, T.Schlumprecht, J. C. Slattery, K.C.Smith, R.R.Smith, P.F.Stiller, E.Straube, J.R.Walton, J.D.Ward, J.Zhou, J.Zinn; Associate Professors D.L.Barrow, P. Daripa, R.D.DeBlassie, R.A.Gustafson, T.R.Kiffe, P. Lima-Filho, B.D. Lowe, A. Poltoratski, M.H.Rahe, V. P.Schielack, Jr., N.Sivakumar, M.J.Stecher, S.D.Taliaferro, T.I.Vogel, C. H. Yan, P.B.Yasskin; Assistant Professors M. Aguiar, G. Berkolaiko, Y. Efendiev, P. Howard, M. Papanikolas, G. Petrova, B. Popov, J. M. Rojas, H. Schenck, Z. Sunik; Senior Lecturers A. L. Austin, A.Belmonte, L. Cardenas, M. Drost, J.Epstein, D. Manuel, M. Mogilevsky, J. Whitfield; Lecturers M. Arnold, K. Bollinger, C. Chen, L. Chen, A. Cheskidov, L. Cox, R. Eby, A. Garza, J. E. Kahlig, G. S. Klein, D. Lazarov, J. Lewis, R. Main, R. Marvit, P. Poage, S. Scarborough, N. Simpson

102. Algebra. (3-0). Credit 3. I, II, S

Sets, structure of number system; absolute values, solution sets of equations of second and higher degree, of systems of equations, and of inequalities; relations and functions, graphical representations, variation, progressions, mathematical induction, determinants.

103. Plane Trigonometry. (3-0). Credit 3.

Definitions of trigonometric functions, evaluation of functions of special angles, fundamental relations, solution of triangles, trigonometric reductions, angular measure, functions of composite angle, logarithms, inverse trigonometric functions, trigonometric equations.

131. Mathematical Concepts--Calculus. (3-0). Credit 3. I, II, S

Limits and continuity; rates of change, slope; differentiation: the derivative, maxima and minima; integration: the definite and indefinite integral techniques; curve fitting. Prerequisites: High school algebra I and II and geometry. Credit will not be given for more than one of MATH 131, 142, 151 and 171.

141. Business Mathematics I. (3-0). Credit 3. I, II, S

Linear equations and applications, linear forms and systems of linear equations, matrix algebra and applications, linear programming (graphical and simplex methods), probability and applications, statistics. Prerequisites: High school algebra I and II and geometry. Credit will not be given for more than one of MATH 141 and 166.

142. Business Mathematics II. (3-0). Credit 3. I, II, S

Derivatives, curve sketching and optimization, techniques of derivatives, logarithms and exponential functions with applications, integrals, techniques and applications of integrals, multivariate calculus. Prerequisites: High school algebra I and II and geometry or satisfactory performance on a qualifying examination. Credit will not be given for more than one of MATH 131, 142, 151 and 171.

150. Functions, Trigonometry and Linear Systems. (3-2). Credit 4. I, II, S

Graphs, functions, college algebra and trigonometry, linear systems and vectors.

151. Engineering Mathematics I. (3-2). Credit 4. I, II, S

Rectangular coordinates, vectors, analytic geometry, functions, limits, derivatives of functions, applications, integration, computer algebra (Maple). Prerequisites: High school algebra I and II, trigonometry and geometry; MATH150 or satisfactory performance on a qualifying exam. Credit will not be given for more than one of MATH 131, 142, 151 and 171.

152. Engineering Mathematics II. (3-2). Credit 4. I, II, S

Differentiation and integration techniques and their applications (area, volumes, work), improper integrals, approximate integration, analytic geometry, vectors, infinite series, power series, Taylor series, computer algebra (Maple). Prerequisite: MATH 151 or equivalent. Credit will not be given for both MATH 152 and 172.

166. Topics in Contemporary Mathematics II. (3-0). Credit 3. I, II, S

Finite mathematics, matrix theory, probability theory, game theory. Prerequisites: High school algebra I and II and geometry. Credit will not be given for more than one of MATH 141 and 166.

170. Freshman Mathematics Laboratory. (0-2). Credit 1.

Computing and problem solving laboratory; introduction to the various mathematical disciplines; development of skills in mathematical problem solving and skills in teamwork. May be taken two times for credit. Prerequisites: Concurrent enrollment in MATH 171 or 172; admission to College of Science.

171. Analytic Geometry and Calculus. (4-0). Credit 4. I, II

Vectors, functions, limits, derivatives, Mean Value Theorem, applications of derivatives, integrals, Fundamental Theorem of Calculus, computer algebra (Maple). Prerequisite: MATH 150 or satisfactory performance on a qualifying examination. Credit will not be given for more than one of MATH 131, 142, 151 and 171.

172. Calculus. (4-0). Credit 4. I, II

Techniques of integration, applications of integrals, improper integrals, sequences, infinite series, vector algebra and solid analytic geometry, computer algebra (Maple). Prerequisite: MATH 151 or 171. Credit will not be given for both MATH 152 and 172.

220. Fundamentals of Discrete Mathematics. (3-0). Credit 3.

An introduction to discrete mathematics including logic, set theory, combinatorics, graph theory. Prerequisite: MATH 172.

221. Several Variable Calculus. (4-0). Credit 4. I, II

Vector algebra and solid analytic geometry; calculus of functions of several variables; Lagrange multipliers; multiple integration, theory, methods and application; line and surface integrals, Green's and Stokes' theorems; Jacobians. Prerequisite: MATH 172 or approval of instructor. Credit will not be given for more than one of MATH 221, 251 and 253.

222. Linear Algebra. (3-0). Credit 3. I, II

Linear equations and matrices; real vector spaces, linear transformations, change of bases, determinants, eigenvalues and eigenvectors, diagonalization, inner products. Prerequisite:s MATH 152 or 172; MATH 220 or approval of instructor.

251. Engineering Mathematics III. (3-0). Credit 3. I, II, S

Vector calculus, calculus of functions of several variables, partial derivatives, directional derivatives, gradient, multiple integration, line integrals, Stokes' theorems. Prerequisite: MATH152 or equivalent. Credit will not be given for more than one of MATH 221, 251 and 253.

253. Engineering Mathematics III. (3-2). Credit 4. I, II, S

Vector calculus; calculus of functions of several variables, partial derivatives, directional derivatives, gradient, multiple integration, Green's and Stokes' theorems, computer algebra (Maple). Prerequisite: MATH152 or equivalent. Credit will not be given for more than one of MATH 221, 251 and 253.

285. Directed Studies. Credit 1 to 4.

Special problems not covered by any other lower-division course in the curriculum; intended for freshman and sophomore students. Prerequisite: Approval of department head.

289. Special Topics in... Credit 1 to 4.

Selected topics in an identified area of mathematics. May be repeated for credit. Prerequisite: Approval of instructor.

302. Discrete Mathematics. (3-0). Credit 3. I, II, S

Formal structures for describing data, algorithms and computing devices; theory and applications of sets, graphs and algebraic structures. Prerequisite: MATH 152.

304. Linear Algebra. (3-0). Credit 3. I, II, S

Introductory course in linear algebra covering abstract ideas of vector space and linear transformation as well as models and applications of these concepts, such as systems of linear equations, matrices and determinants. Prerequisite: MATH 152.

308. Differential Equations. (3-0). Credit 3. I, II, S

Linear ordinary differential equations, solutions in series, solutions using Laplace transforms, systems of differential equations. Prerequisites: MATH 251 or equivalent; knowledge of computer algebra system (Maple).

311. Topics in Applied Mathematics I. (3-0). Credit 3. I, II, S

Matrices, determinants, systems of linear equations, eigenvalues, eigenvectors, diagonalization of symmetric matrices; vector analysis, including normal derivative, gradient, divergence, curl, line and surface integrals, Gauss', Green's and Stokes' theorems. Prerequisites: MATH 221, 251 or 253; MATH 308 or concurrent enrollment therein.

325. The Mathematics of Interest. (3-0). Credit 3.

The mathematical theory associated with interest; annuities; internal rate of return; coupon bonds; valuation of noncallable bonds; yield of maturity; interest rate sensitivity; duration and convexity; reinvestment risk; total return; compound return; STRIPS; yield curve; short selling; hedge ratio; bond swaps. Prerequisites: MATH 142, 151 or 171; junior classification.

365. Structure of Mathematics I. (3-0). Credit 3. I, II, S

Informal logic, sets, relations, functions, whole numbers, numeration systems, binary operations, integers, elementary number theory, modular systems, rational numbers and the system of real numbers. Designed primarily for elementary teacher certification. Others must have consent of instructor. Prerequisite: Completion of core curriculum mathematics requirement.

366. Structure of Mathematics II. (3-0). Credit 3. I, II, S

Geometry, measurement and coordinate geometry. Designed primarily for elementary teacher certification. Others must have consent of instructor. Prerequisite: MATH 365.

367. Basic Concepts of Geometry. (3-0). Credit 3. I, II, S

Formal development of geometry: finite, non-Euclidean and Euclidean. Designed primarily for elementary mathematics teacher certification. Others must have consent of instructor. Prerequisites: MATH 131; MATH 366 or equivalent.

368. Introduction to Abstract Mathematical Structures. (3-0). Credit 3.

Mathematical proofs, sets, relations, functions, infinite cardinal numbers, algebraic structures, structure of the real line; designed primarily for elementary teacher certification. Prerequisites: MATH 131, 166, 367; approval of instructor.

375. Intermediate Real Analysis. (3-0). Credit 3.

Development of the real numbers, limits, foundations and major theorems of calculus. Designed primarily for mathematics teacher certification. Others must have consent of instructor. Prerequisites: MATH 152; MATH 220 or equivalent.

376. Intermediate Abstract Algebra. (3-0). Credit 3. II

Relations, functions, binary operators, rings, homomorphisms, integral domains and fields. Designed primarily for mathematics teacher certification. Others must have consent of instructor. Prerequisites: MATH 220 or 302; MATH 304 or equivalent.

401. Advanced Engineering Mathematics. (3-0). Credit 3. II

Engineering mathematics including Perturbation Theory, Fourier series and partial differential equations. Designed primarily for engineering majors. Others must have consent of instructor. Prerequisite: MATH 308.

403. Mathematics and Technology. (3-0). Credit 3.

Mathematical problem-solving and communication through the use of various technologies (both hardware and software). Intended primarily, but not limited to, students working toward teacher certification. Prerequisite: 400-level math class or MATH 367 or MATH 467 or approval of instructor.

407. Complex Variables. (3-0). Credit 3.

Fundamental theory of analytic functions, including residues and their applications. Prerequisite: MATH 221 or equivalent.

409. Advanced Calculus I. (3-0). Credit 3. I, II

Axioms of the real number system; point set theory of R1; compactness, completeness and connectedness; continuity and uniform continuity; sequences, series; theory of Riemann integration. Prerequisites: MATH 220 and 221.

410. Advanced Calculus II. (3-0). Credit 3. I, II

Differential and integral calculus of functions defined on Rm including inverse and implicit function theorems and change of variable formulas for integration; uniform convergence. Prerequisites: MATH 222 and 409.

411. Mathematical Probability. (3-0). Credit 3. I, II

Probability spaces, discrete and continuous random variables, special distributions, joint distributions, expectations, law of large numbers, the central limit theorem. Prerequisite: MATH 221 or equivalent.

412. Theory of Partial Differential Equations. (3-0). Credit 3. I

Formulation and solution of partial differential equations of mathematical physics; Fourier series and transform methods, complex variable methods, methods of characteristics and first order equations. Prerequisite: MATH 308 or 451 or approval of instructor.

414. Fourier Series and Wavelets. (3-0). Credit 3.

Fourier series and wavelets with applications to data compression and signal processing. Prerequisite: MATH222 or 304 or 311.

415. Modern Algebra I. (3-0). Credit 3. I

Groups, rings, fields. Prerequisite: MATH 222.

416. Modern Algebra II. (3-0) Credit 3. II

Continuation of topics introduced in MATH 415. Prerequisite: MATH 415.

417. Numerical Analysis I. (3-3). Credit 4. I, II, S

Linear systems, matrix decomposition and eigensystems, numerical integration, interpolation and numerical solution of ordinary differential equations. Prerequisites: MATH 222, 304 or 311; MATH 308 or equivalent; ability to program.

423. Linear Algebra II. (3-0). Credit 3. I, II, S

Eigenvalues, diagonal and other canonical forms for similarity and orthogonal similarity, applications to differential equations and quadratic forms. Prerequisite: MATH 222 or 304 or approval of instructor.

425. The Mathematics of Contingent Claims. (3-0). Credit 3.

The mathematical theory associated with asset price dynamics; binomial pricing models; Black-Scholes analysis; hedging; volatility smile; implied volatility trees; implied binomial trees. Prerequisite: MATH 172 or equivalent; MATH 308 or equivalent; basic probability.

431. Structures and Methods of Combinatorics. (3-0). Credit 3.

Enumerative techniques generating functions, partially ordered sets, elementary graph theory, elementary Ramsey theory. Prerequisite: MATH 220 or 302 or approval of instructor.

433. Applied Algebra. (3-0). Credit 3. II, S

Sets, functions, graphs, finite state machines, Boolean algebras, programming languages, groups and monoids, optimization and computer design. Prerequisite: MATH 220 or 302; MATH 222 or 304.

442. Mathematical Modeling. (3-0). Credit 3.

The construction of mathematical models from areas such as economics, game theory, integer programming, mathematical biology and mathematical physics. Prerequisites: MATH 304 and 308 or equivalents.

447. Topics in Analysis. (3-0). Credit 3. II

Metric spaces, compactness, completeness and connectedness; continuous functions; the theorems of Baire, Weierstrass, Arzela-Ascoli, Picard; introduction to Lebesgue integration; L2 theory of Fourier series. Prerequisite: MATH 410.

451. Theory of Ordinary Differential Equations. (3-0). Credit 3.

Linear equations of the first order; linear equations with constant coefficients; linear equations with variable coefficients; linear equations with regular singular points; existence and uniqueness of solutions to first order equations; existence and uniqueness of solutions to systems and n-th order equations. Prerequisite: MATH 221 or equivalent; MATH 222 or concurrent enrollment.

467. Modern Geometry. (3-0). Credit 3.

Rigorous development of Euclidean Geometry; Classic non-Euclidean models; Matrix representations of transformations in R3; Isometries; Transformation and symmetric groups; Similarity and Affine transformations. Prerequisite: MATH 222 or 304.

470. Communications and Cryptography. (3-0). Credit 3.

Introduction to coded communications, digital signatures, secret sharing, one-way functions, authentication, error control and data compression. Prerequisites: MATH 222 or 304 and CPSC 110 and approval of instructor.

485. Directed Studies. Credit 1 to 8. I, II, S

Special problems in mathematics not covered by any other course in the curriculum. Work may be in either theory or laboratory. Prerequisite: Approval of department head.

489. Special Topics in... Credit 1 to 4. I, II, S