2003  2004
Undergraduate Catalog 



MathematicsCollege of Liberal Arts and Sciences INSTRUCTIONAL STAFF Alladi, K., Chair; Brooks, J.K., Associate Chair; Berkovich, A.; Bona, M.; Boyland, P.L.; Brechner, B.L.; Brooks, J.K.; Carter, C.; Cenzer, D.S.; Chen, Y.; Crew, R.M.; Dinculeanu, N.; Drake, D.A.; Dranishnikov, A.N.; Edwards, B.H.; Ehrlich, P.E.; Emch, G.G.; Garvan, F.G.; Glover, J.; Gopalakrishnan, J.; Groisser, D.J.; Hager, W.W.; Ho, C.Y.; Keating, K.P.; Keesling, J.E.; Khuri, R.L.; King, J.L.; Klauder, J.R.; Kutuzova, M.; Larson, J.A.; Levin, N.; Mair, B.A.; Martinez, J.; McCracken, D.L.; McCullough, S.A.; Metzler, D.; Mitchell, W.J.; Moore, T.O.; Moskow, S.; Olson, T.E.; Pilyugin, S., PopStojanovic, Z.R.; Rao, M.K.; Robinson, P.L.; Rudyak, Y.; Saxon, S.A.; Shen, L.C.; Sin, P.K.; Smith, J.Y.; Smith, R.; Summers, S.J.; Thompson, J.G.; Tiep, P.H.; Tornwall, S.B.; Townsend, M.D.; Turull, A.; Vince, A.J.; Voelklein, H.K.; Walsh, T.; White, N.L.; Wilson, D.C.; Wilson, L; Zapletal, J.; Undergraduate Coordinator: D. Groisser; Actuarial Science Advisers: Bruce Edwards, 364 Little Hall, 3920281 ext 281; Dennis Wackerly, 220 FLO, 3921941 ext 227; David Nye, 329 BUS, 3920153; Graduate Coordinator: P. Robinson MAA 4102 Introduction to Advanced Calculus for Engineers and Physical Scientists 1. Credits: 3; Prereq: grade of C or better in MAC 2313 or MAC 3474 and in MAS 4105 or MAS 3114. Theory of real numbers, functions of one variable, sequences, limits, continuity, and differentiation; continuity and differentiability of functions of several variables. MAA 4102 is not recommended for students who plan to do graduate work in mathematics; these students should take MAA 4211. (Note: credit will be given for at most one of MAA 4102, MAA 4211, and MAA 5104.) MAA 4103 Introduction to Advanced Calculus for Engineers and Physical Scientists 2. Credits: 3; Prereq: grade of C or better in MAA 4102. A continuation of MAA 4102. Theory of integration, transcendental functions, and infinite series. MAA 4102 is not recommended for students who plan to do graduate work in mathematics; these students should take MAA 4212. (Note: credit will be given for at most one of MAA 4103, MAA 4212, and MAA 5105.) MAA 4211 Advanced Calculus 1. Credits: 3; Prereq: grade of C or better in MAS 4105. An advanced treatment of limits, differentiation, integration, series; calculus of functions of several variables. (Note: credit will be given for at most one of MAA 4211, MAA 4102, and MAA 5104.) MAA 4212 Advanced Calculus 2. Credits: 3; Prereq: grade of C or better in MAA 4211, taken the previous semester. A continuation of MAA 4211. (Note: credit will be given for at most one of MAA 4212, MAA 4103, and MAA 5105.) MAA 4226 Introduction to Modern Analysis 1. Credits: 3; Prereq: grade of C or better in MAS 4105. Topology of metric spaces, numerical sequences and series, continuity, differentiation, the RiemannStielties integral, sequences and series of functions, the StoneWeierstrass, theorem, functions of several variable, Stokes’ theorem, the Lebesgue theory. (Note: credit will be given for at most one of MAA 4226 and MAA 5228.) MAA 4227 Introduction to Modern Analysis 2. Credits: 3; Prereq: grade of C of better in MAA 4226, taken the previous semester. A continuation of MAA 4226. (Note: credit will be given for at most on of MAA 4227 and MAA 5229.) MAA 4402 Elements of Complex Variables for Engineers and Physical Scientists. Credits: 3; Prereq: grade of C or better in MAC 2313 (or MAC 3474) and in MAP 2302. Complex numbers, analytic functions, CauchyRiemann equations, harmonic functions, elementary functions, integration, CauchyGoursat theorem, Cauchy integral formula, infinite series, residues and poles, conformal mapping. (Note: credit will be given for at most one of MAA 4402 and MAA 5404.) NOTE CONCERNING PRECALCULUS AND CALCULUS: A student can receive at most four hours credit for taking both MAC 1147 and MAC 1140; at most four hours credit for taking MAC 1147 and MAC 1114; and at most five hours credit for taking all three of MAC 1147, MAC 1140 and MAC 1114. If both MAC 2233 and MAC 2311 (or MAC 3472) are taken, credit will be given only for MAC 2311 (or MAC 3472). "Credit" means credit toward graduation, general education requirements, the College of Liberal Arts and Sciences basic distribution requirements, and the Gordon Rule Computation (math) requirement. MAC 1114 Trigonometry. Credits: 2. Exponential and logarithmic functions, trigonometry, analytic and additional applications of trigonometry. (M) GRM†(See note above) MAC 1140 Precalculus Algebra. Credits: 3. College algebra, functions, coordinate geometry, exponential and logarithmic functions. (M) GRM† (See note above) MAC 1147 Precalculus: Algebra and Trigonometry. Credits: 4. College algebra, functions, coordinate geometry, exponential and logarithmic functions, and trigonometry. This fastpaced course is designed as a review of algebra and trigonometry to prepare the student for calculus. This course assumes prior knowledge of intermediate algebra (Algebra II) and trigonometry. (M) GRM† (See note above) MAC 2233 Survey of Calculus 1. Credits: 3; Prereq: Strong background in Precalculus. A geometric and heuristic approach to calculus; differentiation and integration of simple algebraic and exponential functions; applications to graphing, marginal analysis, optimization, areas and volumes. (M) GRM† (See note above) MAC 2234 Survey of Calculus 2. Credits: 3; Prereq: grade of C or better in MAC 2233 or equivalent. Sequences, geometric and Taylor series; systems of linear equations, Gaussian elimination, matrices, determinants and vectors; partial differentiation, multiple integrals; applications to marginal analysis, leastsquares and Lagrange multipliers. (M) GRM† MAC 2311 Analytic Geometry and Calculus 1. Credits: 4; Prereq: Strong background in precalculus. Introduction to analytic geometry; limits; continuity; differentiation of algebraic and trigonometric functions; differentials; introduction to integration; and the fundamental theorem of calculus. (M) GRM† (Credit will be given for at most one of MAC 2233, MAC 2311, and MAC 3472.) MAC 2312 Analytic Geometry and Calculus 2. Credits: 4; Prereq: grade of C or better in MAC 2311 or in MAC 3472. Techniques of integration; applications of integration; differentiation and integration of inverse trigonometric, exponential and logarithmic functions; sequences and series. (M) GRM† (Credit will be given for at most one of MAC 2312, MAC 3512, and MAC 3473.) MAC 2313 Analytic Geometry and Calculus 3. Credits: 4; Prereq:grade of C or better in MAC 2312 or in MAC 3512 or in MAC 3473. Solid analytic geometry; vectors; partial derivatives; multiple integrals. (M) GRM† (Credit will be given for at most one of MAC 2313 and MAC 3474.) MAC 3472 Honors Calculus 1. Credits: 4; Prereq: Strong background in precalculus A first course in calculus for students who enjoy and have a sound background in mathematics. The topics covered in the sequences MAC 347234 closely parallel those covered in MAC 231123 but are treated in greater depth. (M) GRM† (Credit will be given for at most one of MAC 2311 and MAC 3472.) MAC 3473 Honors Calculus 2. Credits: 4; Prereq: grade of C or better in MAC 3472 or in MAC 2311. A continuation of MAC 3472. (M) GRM† (Credit will be given for at most one of MAC 2312, MAC 3512, and MAC 3473.) MAC 3474 Honors Calculus 3. Credits: 4; Prereq: grade of C or better in MAC 3473 or in MAC 2312 or in MAC 3512. A continuation of MAC 3473. (M) GRM† (Credit will be given for at most one of MAC 2313 and MAC 3474.) MAC 3512 Calculus 2 for Advanced Placement Students. Credits: 4; Prereq: Advancement Placement credit for MAC 2311. A calculus course for entering freshmen who have Advanced Placement credit for MAC 2311. The content of MAC 2311 and 2312 is covered: limits, continuity, differentiation, applications of differentiation, integration, logarithms and exponential functions, applications of integration, techniques to integration, sequences, series, conic sections and parametric equations. The content of the AP syllabus is reviewed during the first half of the semester, and there is a special emphasis on topics not in the AP Calculus AB syllabus, such as infinite series. (M) GRM† (Credit will be given for at most one of MAC 2312, MAC 3512, and MAC 3473.) MAD 3107 Discrete Mathematics. Credits: 3; Prereq: grade of C or better in MAC 2312, in MAC 3512, or in MAC 3473. Logic, sets, functions. Algorithms and complexity; integers and algorithms. Mathematical reasoning and induction. Counting principles; permutations and combinations; discrete probability. Advanced counting techniques, inclusionexclusion. MAD 4203 Introduction to Combinatorics 1. Credits: 3; Prereq: grade of C or better in MAC 2312 or in MAC 3512 or in MAC 3473. Permutations and combinations, binomial coefficients, inclusionexclusion, recurrence relations, Fibonacci sequences, generating functions, graph theory. (M) MAD 4204 Introduction to Combinatorics 2. Credits: 3; Prereq: grade of C or better in MAC 2312 or in MAC 3512 or in MAC 3473. Matching theory, block designs, finite projective planes and errorcorrecting codes. This course may be taken without taking MAD 4203. MAD 4401 Introduction to Numerical Analysis. Credits: 3; Prereq: experience with a scientific programming language, and a grade of C or better in MAS 4105 or in MAS 3114. Numerical integration, nonlinear equations, linear and nonlinear systems of equations, differential equations, and interpolation. MAE 3811 Mathematics for Elementary School Teachers 2. Credits: 3; Prereq: A passing score on the Arithmetic Competency Examination. Properties of and operations with rational numbers; ratio; proportion; percentages; an introduction to real numbers; elementary algebra; informal geometry and measurement; and an introduction to probability and descriptive statistics. MAP 2302 Elementary Differential Equations. Credits: 3; Prereq: grade of C or better in MAC 2312 or in MAC 3512 or in MAC 3473. Firstorder ordinary differential equations, theory of linear ordinary differential equations, solution of linear ordinary differential equations with constant coefficients, the Laplace transform and its application to solving linear ordinary differential equations. (M) GRM† MAP 3170 Introduction to Actuarial Mathematics. Credits: 3; Prereq: grade of C or better in MAC 2312 or in MAC 3512 or in MAC 3473. Amount function, simple and compound interest, present value, dollarweighted and timeweighted rates, force of interest, discrete and continuous annuities, bonds, amortization, capitalization, and applications. MAP 4102 Probability Theory and Stochastic Processes 2. Credits: 3; Prereq: grade of C or better in STA 4321. Random walks and Poisson processes, martingales, Markov chains, Brownian motion, stochastic integrals and Ito’s formula. MAP 4305 Introduction to Differential Equations for Engineers and Physical Scientists. Credits: 3; Prereq: grade of C or better in MAP 2302 and in either MAS 4105 or MAS 3114. This is a second course in differential equations. Topics covered are: systems of linear differential equations, stability theory and phase plane analysis, power series solutions of differential equations, SturmLiouville boundaryvalue problems, and special functions. (Note: credit will be given for at most one of MAP 4305 and MAP 5304.) MAP 4341 Elements of Partial Differential Equations. Credits: 3; Prereq: grade of C or better in MAP 2302 and in MAP 4305. Topics covered are: introduction to secondorder linear partial differential equations (Heat, Wave, and Laplace equations), separation of variables in PDEs, SturmLiouville eigenvalue problems, method of eigenfunction expansions (Fourier analysis), Green’s functions. Introduction to firstorder PDEs and the method of characteristics may be included at the preference of the instructor. (Note: credit will be given for at more one of MAP 4341 and MAP 5345.) MAP 4403 Mathematical Methods for Engineers. Credits: 3; Prereq: grade of C or better in MAC 2313 (or in MAC 3474),and in MAP 2302. Complex variables, including derivatives and integrals, singularities, Taylor/Laurent series and residues; Linear Algebra, including Gaussian elimination, determinants, inversion, linear independence, eigenvalues and eigenvectors, and (time permitting) QR decomposition, systems of differential equations; (time permitting) Numerical methods, including roots of equations, numerical integration, and numerical optimization. MAP 4413 Fourier Series and Transforms 1. Credits: 3; Prereq: grade of C or better in MAC 2313 (or in MAC 3474) and in MAP 2302; MAP 4305 recommended. Introduction to linear systems and transforms; Laplace, Fourier and Z transforms and their mutual relationship; convolutions. Operational calculus; computational methods including the fast Fourier transform; secondorder stationary processes and their autocorrelation functions; and problems of interpolation, extrapolation, filtering and smoothing of secondorder stationary processes. MAS 3114 Computational Linear Algebra. Credits: 3; Prereq: experience with a scientific programming language, and a grade of C or better in MAC 2312 (or in MAC 3512 or in MAC 3473) and. Linear equations, matrices and determinants. Vector spaces and linear transformations. Inner products and eigenvalues. This course emphasizes computational aspects of linear algebra. (M) GRM† MAS 3300 Numbers and Polynomials. Credits: 3; Prereq: grade of C or better in a math course taken at the University of Florida at the level of MAC 2311 or above; this requirement waived for transfer students with junior standing. This course emphasizes theorems and proofs. Topics include a algebraic and order properties of the real numbers, introduction to number theory, rational numbers and their decimal expansions, uncountability of the real numbers, complex numbers, irreducible polynomials over the integral, rational, real and complex numbers, elementary theory of equations. Taking one (but not both) of MAS 3300 and MH 3202 is very useful in preparing students for a major in mathematics. This course is also particularly useful for prospective secondaryschool mathematics teachers. (M) GRM† MAS 4105 Linear Algebra 1. Credits: 4; Prereq: grade of C or better in MAC 2313 or in MAC 3474; MAS 3300 or MHF 3202 recommended. Linear equations, matrices, vector spaces, linear transformations, determinants, eigenvalues, inner product spaces. This course includes both theory and computational skills. The student is expected to develop the ability to reason through, and coherently write up, proofs of theorems. For math majors, this course serves as a transition from a study of techniques into more conceptual math; for engineering and science majors, it serves also as a coherent foundation in linear algebra. MAS 4107 Linear Algebra 2. Credits: 4; Prereq: grade of C or better in MAS 4105. Further topics in linear algebra. GRM† MAS 4124 Introduction to Numerical Linear Algebra. Credits: 3; Prereq: experience with a scientific programming language, and a grade of C or better in MAS 4105 or in MAS 3114 . Topics in linear algebra most useful in applications with emphasis on the numerical methods involved: direct and iterative solutions to systems of linear equations; matrix norms; Householder transformations; singular value decomposition; least squares and the generalized inverse; QR method for computing eigenvalues; condition number of linear systems and eigensystems. MAS 4156 Introduction to Vector Analysis. Credits: 3; Prereq: grade of C or better in MAC 2313 or in MAC 3474. Review of vector algebra, lines, planes and space curves; vector fields, their divergence and curl; line, surface, and volume integrals, the Divergence Theorem and Stokes’ Theorem. (Note: credit will be given for at most one of MAS 4156 and MAS 5157.) MAS 4203 Introduction to Number Theory. Credits: 3; Prereq: grade of C or better in MAC 2312 or in MAC 3512 or in MAC 3473; MAS 3300 recommended. An introduction to elementary number theory and its applications to computer science and cryptology. Divisibility, primes, Euclidean Algorithm, congruences, Chinese Remainder Theorem, EulerFermat Theorem, primitive roots. Selected applications to decimal fractions, continued fractions, computer file storage and hashing functions, and publickey cryptography. (M) MAS 4301 Abstract Algebra 1. Credits: 3; Prereq: grade of C or better in MAS 4105. Sets and mappings, groups and subgroups, homomorphisms and isomorphisms, permutations, rings and domains, arithmetic properties of domains, fields. This course requires facility in writing proofs. MAS 4302 Abstract Algebra 2. Credits: 3; Prereq: grade of C or better in MAS 4301. Further topics in abstract algebra. MAT 4905 Individual Work. Credits: 1 to 3; Maximum of 10 credits; Prereq: grade of C or better in MAC 2313 (or in MAC 3474) and permission of the undergraduate coordinator. May be repeated for credit. For special topics not obtainable in the regular course offerings. MAT 4930 Special Topics in Mathematics. Credits: 1 to 3; Prereq: Permission of the undergraduate coordinator. May be repeated for credit. Maximum of 16 credits. Qualified undergraduates will take part in seminars or classes on special topics. MGF 1106 Mathematics for Liberal Arts Majors 1. Credits: 3. Students who have received credit for MGF 1202 will not receive credit for MGF 1106. This course is designed for nonscience and nonbusiness majors who need to fulfill their Gordon Rule and General Education math requirements. The course includes an introduction to set theory, logic, number theory, probability, statistics, graphing, and linear programming. (M) GRM† MGF 1107 Mathematics for Liberal Arts Majors 2. Credits: 3. A general education course that demonstrates the beauty and utility of mathematics. Topics include financial management, linear and exponential growth, mathematics in the arts, and discrete mathematics. This course may be taken without MGF 1106. (M) GRM† MHF 3202 Sets and Logic. Credits: 3; Prereq: grade of C or better in a math course taken at the University of Florida at the level of MAC 2311 or above; this requirement waived for transfer students with junior standing. Examples of sets, operations on sets, set algebra, Venn diagrams, truth tables, tautologies, applications to mathematical arguments and mathematical induction. Taking one (but not both) of MAS 3300 and MHF 3202 is also very useful in preparing students for a major in mathematics. This course can also be very useful for prospective and inserviceschool teachers and middleschool teachers. (M) GRM† MHF 4102 Elements of Set Theory. Credits: 3; Prereq: grade of C or better in MAS 4105. The basic axioms and concepts of set theory. Students present proofs. (Note: credit will be given for at most one of MHF 4102 and MHF 5107.) MHF 4203 Foundations of Mathematics. Credits: 3; Prereq: grade of C or better in MAS 4105. Models and proofs. Foundations of the real and natural numbers, algorithms, Turing machines, undecidability and independence. Examples and applications in algebra, analysis, geometry and topology. (Note: credit will be given for at most one of MHF 4203 and MHF 5207.) MHF 4404 History of Mathematics. Credits: 3; Prereq: grade of C or better in MAC 2312 or in MAC 3512 or in MAC 3473. An introduction to the history of mathematics. The course includes a historical discussion of selected mathematical topics. (M) GRM† MTG 3212 Geometry. Credits: 3; Prereq: grade of C or better in MAC 2312 or in MAC 3512 or in MAC 3473. A systematic study of the axiomatic structure of Euclidean and nonEuclidean geometry, including a comparison of the metric (Birkhoff) and synthetic (Hilbert) developments of absolute geometry; parallelism, similarity, area, lines and planes in space, solid mensuration. This course is particularly useful for prospective secondary school mathematics teachers. (M) GRM† MTG 3214 Euclidean Geometry. Credits: 3; Prereq: grade of C or better in MAC 2312 or in MAC 3512 or in MAC 3473. Axiomatic structure of Euclidean geometry: congruence, parallelism, area, similarity, circles, polygons, medians, constructions, solid geometry, spherical and hyperbolic geometry. MTG 4302 Elements of Topology 1. Credits: 3; Prereq: grade of C or better in MAS 4105. The basic concepts of general topology. (Note: credit will be given for at most one of MTG 4302 and MTG 5316.) MTG 4303 Elements of Topology 2. Credits: 3; Prereq: grade of C or better in MTG 4302. Continuation of MTG 4302. (Note: credit will be given at most one of MTG 4303 and MTG 5317.) 

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