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    K-State Undergraduate Catalog 2000-2002
     

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    Mathematics

    Louis Pigno,* Head

    Professors Burckel,* Cochrane,* Dressler,* Kapitanski,* Lee,* Miller,* Nicholls,* Peller,* Pigno,* Ramm,* Saeki,* Shult,* Smith,* Soibelman,* Strecker,* and Surowski;* Adjunct Professor Arhangel'skii; Associate Professors Bennett,* Chermak,* Crane,* Li,* Lin,* Maginnis,* Moore,* Muenzenberger,* Rosenberg,* and Yetter;* Assistant Professors Auckly,* Korten, Nagy,* Poggi-Corradini,* Vaninsky,* and Yang;* Emeriti: Professors Dixon,* Marr,* Stamey,* and Young;* Associate Professors W. Parker* and Sloat;* Instructors Chatelain, Sitz, and Woldt.

    E-mail: ugmath@math.ksu.edu
    www.ksu.edu/math

    Mathematics is the unparalleled model of an exact science, the epitome of creative art, and a language essential to understanding our modern technological world. Mathematicians design mathematical models, solve mathematical problems, and create new mathematics.

    Mathematics graduates are sought both for their specialized knowledge and for their ability to reason and think analytically and solve problems.

    Requirements
    Students may obtain either a bachelor of arts or a bachelor of science degree with a major in mathematics. For either degree, in addition to the general requirements of the university and college, mathematics majors must complete the following core courses:

    MATH 220Analytic Geometry and Calculus I 4
    MATH 221Analytic Geometry and Calculus II 4
    MATH 222Analytic Geometry and Calculus III 4
    MATH 240Elementary Differential Equations 4
    CIS 200Fundamentals of Computer
    Programming 4
    STAT 510Introductory Probability and
    Statistics I 3
    MATH 512Introduction to Modern Algebra 3
    or
    MATH 511Introduction to Algebraic Systems 3
    MATH 633Advanced Calculus I 3
    or
    MATH 520Foundations of Analysis 3
     
    For the B.A. degree, students must take 15 additional hours in mathematics numbered 400 or above; PHILO 510 may be substituted for 3 of these hours.
     
    For the B.S. degree, students must take 15 additional hours in mathematics numbered 400 and above; MATH 570 may not be used to meet this requirement.

    Students majoring in mathematics must earn a grade of 'C' or better 
     in each math course used to satisfy requirements for the major.

    All students should enroll in MATH 199 in their first fall on campus.
     
    Students may choose one of the following four programs, depending on their career interests.
     
    Applied mathematics program
    Students who intend to seek employment in business, government, or industry, should take Introduction to Modern Algebra and Advanced Calculus I (MATH 512 and 633). In addition, the following courses are recommended:

    MATH 510Discrete Mathematics 3
    MATH 540Advanced Ordinary Differential Equations 3
    MATH 551Applied Matrix Theory 3
    MATH 632Elementary Partial Differential
    Equations 3
    MATH 634Advanced Calculus II 3
    MATH 655Elementary Numerical Analysis I 3
     
    Students also should take as many additional computer science and statistics courses as possible.

    Pre-graduate program
    Students who intend to enter graduate school to work toward an advanced degree in either pure or applied mathematics should take Introduction to Modern Algebra and Advanced Calculus I (MATH 512 and 633). In addition, the following courses are recommended:

    MATH 515Introduction to Linear Algebra 3
    MATH 560Introduction to Topology 3
    MATH 634Advanced Calculus II 3
    MATH 721Analysis I 3
    MATH 722Analysis II 3
    MATH 730Abstract Algebra I 3
    MATH 731Abstract Algebra II 3
     
    Students should also take additional courses in related fields, such as computer science and statistics, and at least one foreign language, preferably French, German, or Russian.

    Actuarial mathematics program
    Students who intend to become actuaries or work in the financial sector should take as many of the following courses as possible:

    MATH 500Actuarial Mathematics 3
    MATH 510Discrete Mathematics 3
    MATH 512Introduction to Modern Algebra 3
    MATH 540Ordinary Differential Equations 3
    MATH 551Applied Matrix Theory 3
    MATH 633Advanced Calculus I 3
    MATH 655Elementary Numerical Analysis 3
    STAT 510Introductory Probability and Statistics I 3
    STAT 511Introductory Probability and Statistics II 3
    STAT 770Theory of Statistics I 3
    STAT 771Theory of Statistics II 3
     
    Students should also take courses in fields such as accounting, economics, and finance.

    Teacher preparation program
    Students who intend to become secondary school mathematics teachers may prepare for teacher certification through the College of Education while completing the requirements for a degree in mathematics. The following courses are recommended for such students:

    MATH 312Finite Applications of Mathematics 3
    MATH 506Introduction to Number Theory 3
    MATH 510Discrete Mathematics 3
    MATH 511Introduction to Algebraic Systems  3
    MATH 520Foundations of Analysis 3
    MATH 551Applied Matrix Theory 3
    MATH 570History of Mathematics 3
    MATH 572Foundations of Geometry 3
    MATH 591Topics in Mathematics for Teachers 3
     
    For specific certification requirements for secondary education, see the College of Education section of this catalog.

    Students majoring in elementary education who wish to use mathematics as an area of concentration should consider taking their 15 hours of mathematics from among the following courses:

    MATH 150Plane Trigonometry 3
    MATH 160Introduction to Contemporary Mathematics 3
    MATH 205General Calculus and Linear Algebra 3
    MATH 312Finite Applications of Mathematics 3
    MATH 313Computational Number Theory 3
    MATH 320Mathematics for Elementary
    School Teachers 3
    MATH 591Topics in Mathematics for Teachers 3
    Dual majors and dual degrees
    Students may major in mathematics and another discipline within the College of Arts and Sciences. The degree requirements of both departments must be met.

    Students may obtain a degree in mathematics and a second degree in a field in another college such as business administration or engineering. The degree requirements of both colleges must be met.

    Information for nonmajors
    Most colleges and departments require at least one mathematics course. Students should check with their advisors to determine which mathematics courses to take. Advisors are provided information that will aid them in using a student's ACT score to select the appropriate entry-level mathematics course. Advisors also have access to expanded mathematics course descriptions that will help them advise students.

    Mathematics courses
    MATH 010. Intermediate Algebra. (3) I, II, S. Preparatory course for MATH 100. Includes arithmetic (signed numbers, polynomials, algebraic fractions, exponents, and roots), solutions to equations (linear, quadratic, polynomial, root, and fractional), graphs (linear and quadratic), and geometry (area, perimeter, and the Pythagorean Theorem). Pr.: Two units of mathematics in grades 9-12 and a College Algebra PROB >= C of 43 or more on the ACT assessment; or a score of at least 7 on the mathematics placement test; or a score of at least 26 on the arithmetic placement test.
    MATH 100. College Algebra. (3) I, II, S. Fundamental concepts of algebra; algebraic equations and inequalities; functions and graphs; zeros of polynomial functions; exponential and logarithmic functions; systems of equations and inequalities. Pr.: B or better in MATH 010; or two years of high school algebra and a College Algebra PROB >= C of 60 or more on the ACT assessment; or a score of at least 18 on the mathematics placement test.
    MATH 101. The Metric System. (1) Intersession only, on sufficient demand. A systematic study of the metric system including historical background of various systems, structure of the metric system itself, and relation to existing systems; attention to competent use of metric terms in problem solving.
    MATH 150. Plane Trigonometry. (3) I, II, S. Trigonometric and inverse trigonometric functions; trigonometric identities and equations; applications involving right triangles and applications illustrating the laws of sines and cosines. Pr.: C or better in MATH 100; or two years of high school algebra and a score of 25 or more on Enhanced ACT mathematics; or a score of at least 20 on the mathematics placement exam.
    University General Education courseMATH 160. Introduction to Contemporary Mathematics. (3) I, II, S. Mathematics as used in the contemporary world. Combinatorics and probability; descriptions of data; graph theory; and various additional topics selected by the individual instructors. Pr.: MATH 100.
    MATH 199. Undergraduate Mathematics Seminar. (1) I. Topics of special interest to undergraduates in mathematics, including orientation to the mathematics curriculum, possible careers in mathematics, and cultural and professional aspects of mathematics.
    MATH 205. General Calculus and Linear Algebra. (3) I, II. Introduction to calculus and linear algebra concepts that are particularly useful to the study of economics and business administration with special emphasis on working problems. Pr.: MATH 100 with C or better grade (College Algebra in the preceding semester is recommended).
    MATH 210. Technical Calculus I. (3) I. A condensed course in analytic geometry and differential calculus with an emphasis on applications. Pr.: B or better in MATH 100 and C or better in MATH 150; or three years of college preparatory mathematics including trigonometry and a Calculus I PROB >= C of 55 or more on the ACT assessment; or a score of at least 26 on the mathematics placement test.
    MATH 211. Technical Calculus II. (3) II. A continuation of MATH 210 to include integral calculus with an emphasis on application. Pr.: C or better in MATH 210.
    MATH 220. Analytic Geometry and Calculus I. (4) I, II, S. Analytic geometry, differential and integral calculus of algebraic and trigonometric functions. Pr.: B or better in MATH 100 and C or better in MATH 150; or three years of college preparatory mathematics including trigonometry and Calculus I PROB >= C of 55 or more on the ACT assessment; or a score of at least 26 on the mathematics placement test.
    MATH 221. Analytic Geometry and Calculus II. (4) I, II, S. Continuation of MATH 220 to include transcendental functions, techniques of integration, and infinite series. Pr.: C or better in MATH 220.
    MATH 222. Analytic Geometry and Calculus III. (4) I, II, S. Continuation of MATH 221 to include functions of more than one variable. Pr.: C or better in MATH 221.
    MATH 240. Elementary Differential Equations. (4) I, II, S. Elementary techniques for solving ordinary differential equations and applications to solutions of problems in science and engineering. Pr.: C or better in MATH 222.
    MATH 312. Finite Applications of Mathematics. (3) II. Systems of equations, vector operations, linear algebra, and linear programming. Practice in setting up, solving, and interpreting mathematical models which arise in social sciences and business. Pr.: MATH 100.
    MATH 313. Computational Number Theory. (3) I, II, S. Topics in number theory selected from: divisibility, primes, modular arithmetic and special types of numbers. Emphasis is on computations. Primarily for prospective elementary school teachers of mathematics. Pr.: Sophomore standing, MATH 100.
    MATH 320. Mathematics for Elementary School Teachers I. (3) I, II. Mathematical problem solving and reasoning, development of whole number concepts and the whole number system, computation and estimation with whole numbers, number patterns and number theory, integers, fractions and rational numbers, decimals and real numbers, geometry and measurement. Pr.: MATH 100. For education majors only.
    MATH 330. Intuitive Geometry. (3) Geometric figures and patterns, properties of geometric figures, transformation and coordinate geometry, measurement. Pr.: MATH 320.
    MATH 395. Academic Excellence Workshop. (1-2) This course provides enriched supplementary instruction to selected students enrolled in selected lower-division courses. Pr.: Conc. enrollment in qualifying lower-division mathematics course and written permission of instructor.
    University General Education courseMATH 399. Honors Seminar in Mathematics. (1-3) Pr.: Membership in honors program.
    MATH 498. Senior Honors Thesis. (2) I, II, S. Open only to seniors in the arts and sciences honors program.
    MATH 499. Undergraduate Topics in Mathematics. (Var.) I, II, S. Reading courses in advanced undergraduate mathematics. Pr.: Background of courses needed for topic undertaken and consent of instructor. Repeatable for credit.
    MATH 500. Actuarial Mathematics. (3.) I. Extensive review of calculus and linear algebra including material not covered in the calculus sequence or linear algebra courses; futures and present value; annuities; amortization; yield rates; bonds and related funds; application of calculus and probability to the study of interest. Prepares students to take two of the professional examinations administered by the Society of Actuaries and the Casualty Actuanial Society. Pr.: MATH 240, MATH 551, or conc. enrollment in MATH 551.
    MATH 506. Introduction to Number Theory. (3) II. Divisibility properties of integers, prime numbers, congruences, multiplicative functions. Pr.: MATH 221.
    MATH 510. Discrete Mathematics. (3) I, II, S. Combinatorics and graph theory. Topics selected from counting principles, permutations and combinations, the inclusion/ exclusion principle, recurrence relations, trees, graph coloring, Eulerian and Hamiltonian circuits, block designs, and Ramsey Theory. Pr.: Sophomore standing and MATH 221.
    MATH 511. Introduction to Algebraic Systems. (3) I. Properties of groups, rings, domains, and fields. Examples selected from subsystems of the complex numbers, elementary number theory, and solving equations. Pr.: MATH 222.
    MATH 512. Introduction to Modern Algebra. (3) I. Introduction to the basic algebraic systems, viz., groups, rings, integral domains, fields, elementary number theory. Special emphasis will be given to methods of theorem proving. Pr.: MATH 222.
    MATH 515. Introduction to Linear Algebra. (2-3) I. Finite dimensional vector spaces; linear transformations and their matrix representations; dual spaces, invariant subspaces; Euclidean and unitary spaces; solution spaces for systems of linear equations. Pr.: MATH 512.
    MATH 520. Foundations of Analysis. (3) A study of sets and sequences, neighborhood, limit point, convergence, and open and closed set in the real line and in the plane, the concept of continuous function. Pr.: MATH 222.
    MATH 521. The Real Number System. (3) An extensive development of number systems, with emphasis upon structure. Includes systems of natural numbers, integers, rational numbers, and real numbers. Pr.: MATH 221.
    MATH 540. Advanced Ordinary Differential Equations. (3) First order scalar equations; geometry of integral curves, symmetries and exactly soluble equations; existence; uniqueness and dependence on parameters with examples. Systems of first order equations, Hamilton's equations and classical mechanics, completely integrable systems. Higher order equations. Initial value problems for second order linear equations, series solutions and special functions. Boundary value problems with applications. Introduction to perturbation theory and stability. Pr.: MATH 240.
    MATH 551. Applied Matrix Theory. (3) I, II. Matrix algebra, solutions to systems of linear equations, determinants, vector spaces, linear transformations, eigenvalues, linear programming, approximation techniques. Pr.: MATH 205 or 220.
    MATH 560. Introduction to Topology. (3) An introduction to the basic topological concepts. Topological spaces, metric spaces, closure, interior, and frontier operators, subspaces, separation and countability properties, bases, subbases, convergence, continuity, homeomorphisms, compactness, connectedness, quotients and products. The course will include a brief introduction to proof techniques and set theory. Other topics in topology also may be included. Pr.: MATH 222.
    MATH 570. History of Mathematics. (3) II. A survey of the development of mathematics from ancient to modern times. Cannot be used as part of the advanced mathematics needed for the B.S. degree in mathematics. Pr.: MATH 220.
    MATH 572. Foundations of Geometry. (3) Euclidean, non-Euclidean, and finite geometries; role of axioms; practice proving theorems in a formal system; synthetic, metric, and transformation approaches to Euclidean geometry. Pr.: MATH 221.
    MATH 591. Topics in Mathematics for Teachers. (1-3) I, II, S. Topics of importance for teachers of mathematics. May be repeated for credit. Pr.: Consent of instructor.
    MATH 615. Advanced Engineering Mathematics I. (3) I. Vector calculus; higher dimensional calculus; topics in ordinary differential equations; complex analysis. Pr.: MATH 240 and 551.
    MATH 616. Advanced Engineering Mathematics II. (3) II. Fourier series; Fourier and Laplace transforms; basic partial differential equations; basic calculus of variations. Pr.: MATH 240 and 615.
    MATH 630. Introduction to Complex Analysis. (3) I, II. Complex analytic functions and power series, complex integrals. Taylor and Laurent expansions, residues, Laplace transformation, and the inversion integral. Pr.: MATH 240.
    MATH 632. Elementary Partial Differential Equations. (3) I. Orthogonal functions, Fourier Series, boundary value problems in partial differential equations. Pr.: MATH 240.
    MATH 633. Advanced Calculus I. (3) I. Functions of one variable: limits, continuity, differentiability, Riemann-Stieltjes integral, sequences, series, power series, improper integrals. Pr.: MATH 222.
    MATH 634. Advanced Calculus II. (3) II. Functions of several variables: partial differentiation and implicit function theorems, curvilinear coordinates, differential geometry of curves and surfaces, vectors and vector fields, line and surface integrals, double and triple integrals, Green's Theorem, Stokes' Theorem, and Divergence Theorem. Pr.: MATH 633.
    MATH 655. Elementary Numerical Analysis I. (3) I. Error analysis, root finding, interpolation, approximation of functions, numerical integration and differentiation, systems of linear equations. Pr.: MATH 221, a computer language, and either MATH 515 or 551.
    MATH 656. Elementary Numerical Analysis II. (3) II. A continuation of MATH 655. Linear programming, numerical solutions of differential equations, and the use of standard packages for the solution of applied problems. Pr.: MATH 655 and 240.
    MATH 670. Mathematical Modeling. (3) Introduction of modeling procedures. Case studies in mathematical modeling projects from physical, biological, and social sciences. Pr.: Four mathematics courses numbered 500 or above.
    MATH 700. Set Theory and Logic. (3) An introduction to logic, mathematical proof, and elementary set theory; elementary logic, the basic constructions of set theory, relations, partitions, functions, cartesian products, disjoint unions, orders, and a construction of the natural numbers; also ordinal and cardinal numbers, the Axiom of Choice, and transfinite induction. Special emphasis will be given to proving theorems. Pr.: MATH 511 or 512.
    MATH 701. Elementary Topology I. (3) I. Introduction to axiomatic topology including a study of compactness, connectedness, local properties, separation axioms, and metrizability. Pr.: MATH 633.
    MATH 702. Elementary Topology II. (3) II. Path connectedness, fundamental groups, covering spaces, introduction to topological and differentiable manifolds. Pr.: MATH 701.
    MATH 704. Introduction to the Theory of Groups. (3) Introduction to abstract group theory; to include permutation groups, homomorphisms, direct products, Abelian groups. Jordan-Holder and Sylow theorem. Pr.: MATH 512.
    MATH 706. Theory of Numbers. (3) II. Divisibility, congruences, multiplicative functions, number theory from an algebraic viewpoint, quadratic reciprocity, Diophantine equations, prime numbers. Pr.: MATH 221 and either 511 or 512.
    MATH 710. Introduction to Category Theory. (3) Categories, duality, special morphism, functors, natural transformations, limits and colimits, adjoint situations, and applications. Pr.: MATH 701 and 730.
    MATH 711. Category Theory. (3) Set valued functors and concrete categories, factorization structures, algebraic and topological functors, categorical completions, Abelian categories. Pr.: MATH 710.
    MATH 713. Advanced Applied Matrix Theory. (3) A development of the concepts of eigenvalues by considering applications in differential equations and quadratic forms and estimation problems. A discussion of the Jordan canonical form, functions of matrices, vector and matrix norms, convex sets. Selected topics from the theory and application of the simplex algorithm, Markov chains, Leslie population models, Leontieff input-output model. Pr.: MATH 551 or MATH 630.
    MATH 721. Analysis I. (3) I, II, S. Metric spaces, limits, continuity, sequences and series, connectedness, compactness, Baire category, uniform convergence, theorems of Stone-Weierstrass and Arzela. Pr.: MATH 240 or graduate standing.
    MATH 722. Analysis II. (3) II. Lebesgue and Riemann-Stieltjes integration on the real line, differentiation on the real line, elementary transcendental functions. Pr.: MATH 721.
    MATH 730. Abstract Algebra I. (3) I. Groups, rings, fields, vector spaces and their homomorphisms. Elementary Galois theory and decomposition theorems for linear transformations on a finite dimensional vector space. Pr.: MATH 512 or consent of instructor.
    MATH 731. Abstract Algebra II. (3) II. Continuation of MATH 730. Pr.: MATH 730 or consent of instructor.
    MATH 740. Calculus of Variations. (3) On sufficient demand. Necessary conditions and the Euler-Lagrange equations, Hamilton-Jacobi theory, Noether's theorems, direct methods, applications to geometry and physics. Pr.: MATH 722 or equiv.
    MATH 745. Ordinary Differential Equations. (3) I. First order equations and applications, second order equations and oscillation theorems, series solutions and special functions, Sturm-Liouville problems, linear systems, autonomous systems and phase plane analysis, stability, Liapunov's method, periodic solutions, perturbation and asymptotic methods, existence and uniqueness theorems. Pr.: MATH 240.
    MATH 755. Dynamic Modeling Processes. (3) Topics to include equilibrium and stability, limit circles, reaction-diffusion, and shock phenomena, Hopf bifurcation and cusp catastrophes, chaos and strange attractors, bang-bang principle. Applications from physical and biological sciences and engineering. Pr.: MATH 240 and 551.
    MATH 757. Mathematical Control Theory. (3) Mathematical analysis of dynamical systems governed by differential equations and their optimal processes, feedback and filtering. Topics include dynamical systems with controls, axioms of control systems, input-output behaviors, stability and instability, reachability and controllability, dynamic feedback and stabilization, optimal control processes, piecewise constant control and bang-bang principle, Pontryagin maximum principle, tracking, filtering. Pr.: MATH 560, 615.
    MATH 760. Probability Theory. (3) An introduction to the mathematical theory of probability. Material covered includes combinatorial probability, random variables, independence, expectations, limit theorems, Markov chains, random walks, and martingales. Pr.: MATH 633 and STAT 510.
    MATH 772. Elementary Differential Geometry. (3) Curves and surfaces in Euclidean spaces, differential forms and exterior differentiation, differential invariants and frame fields, uniqueness theorems for curves and surfaces, geodesics, introduction to Riemannian geometry, some global theorems, minimal surfaces. Pr.: MATH 240.
    MATH 789. Combinatorial Analysis. (3) II, in alternate years. Permutations, combinations, inversion formulae, generating functions, partitions, finite geometries, difference sets, and other topics. Pr.: MATH 512.
    MATH 791. Topics in Mathematics for Secondary School Teachers. (3) Topics of importance in the preparation of secondary school teachers to teach modern mathematics. May be repeated for credit.

    Topics within Arts and Sciences:
    dMajors and Degrees dAerospace Studies dMathematics
    dDegree Requirements dAnthropology dMilitary Science
    dBachelor of Arts and Bachelor of Sciences dArt dModern Languages
    dBachelor of Fine Arts dBiochemistry dMusic
    dBachelor of Music dBiology dPhilosophy
    dBachelor of Music Education dChemistry dPhysics
    dAssociate of Arts at Fort Riley dEconomics dPolitical Science
    dAssociate of Science at Fort Riley dEnglish dPsychology
    dProgram Options dGeography dSociology, Anthropology, and Social Work
    dAdvising dGeology dSpeech Communication, Theatre, and Dance
    dUniversity Undergraduate Studies dHistory dStatistics
    dPre-Law dJournalism and Mass Communications   
    dPre-Health Professions Program dKinesiology   
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    Kansas State University
    February 7, 2001