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Kansas State University

Mathematics

Louis Pigno, Head

Professors Bennett, Burckel, Cochrane, Lin, Miller, Nagy, Pigno, Ramm, Rosenberg, Saeki, Smith, Soibelman, and Strecker; Associate Professors Auckly, Chermak, Crane, Li, Maginnis, Moore, Muenzenberger, Pinner, Poggi-Corradini, Yang, and Yetter; Assistant Professors Alfonseca, Fang, Hansen, Hoehn, Korten, Roitman, and Ryabogin; Emeriti: Professors Dixon, Dressler, Lee, Marr, Shult, Stamey, Surowski, and Young; Associate Professors W. Parker and Sloat; Instructors Sitz and Woldt.

785-532-6750 Fax: 785-532-0546

E-mail: ugmath@math.ksu.edu

www.math.ksu.edu/main

Our modern technological world has many fields that require mathematical expertise. The Department of Mathematics at K-State offers courses for those pursuing a career in mathematics as well as those needing quantitative and problem-solving skills for use in other fields.

Mathematics graduates are sought both for their specialized knowledge and for their ability to reason and think analytically and solve problems. Well into the future, there will be a demand for mathematically trained people in technologically related positions dealing with applications and in education at all levels.

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 I4
MATH 221Analytic Geometry and Calculus II4
MATH 222Analytic Geometry and Calculus III4
MATH 240Elementary Differential Equations4
CIS 111Fundamentals of Computer Programming3
or
CIS 200Fundamentals of Software Design and Implementation4
or both of the following
CIS 105Introduction to Computer Programming1
and
CIS 209C Programming for Engineers3
STAT 510Introductory Probability and
Statistics I3
MATH 512Introduction to Modern Algebra3
or
MATH 511Introduction to Algebraic Systems3
MATH 633Advanced Calculus I3
or
MATH 520Foundations of Analysis3
 
For the BA 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 BS degree, students must take 15 additional hours in mathematics numbered 400 and above.
 
Students majoring in mathematics must earn a grade of C or better in each math and statistics 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 apply mathematics in business, government, or industry should take the following courses:

CIS 200Fundamentals of Software Design and Implementation4
CIS208C Language Laboratory1
MATH 510Discrete Mathematics3
MATH 512Introduction to Modern Algebra3
MATH 540Advanced Ordinary Differential Equations3
MATH 551Applied Matrix Theory3
MATH 632Elementary Partial Differential
Equations3
MATH 633Advanced Calculus I3
MATH 634Advanced Calculus II3
MATH 655Elementary Numerical Analysis I3
STAT 510Introductory Probability and Statistics I3
 

Students should also 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 the following courses:

CIS 111Fundamentals of Computer Programming3
MATH 506Introduction to Number Theory3
MATH 512Introduction to Modern Algebra3
MATH 515Introduction to Linear Algebra3
MATH 560Introduction to Topology3
MATH 630Introduction to Complex Analysis3
MATH 633Advanced Calculus I3
MATH 634Advanced Calculus II3
STAT 510Introductory Probability and Statistics I3
 

Students should also take additional courses in related fields such as computer science and statistics.

Actuarial mathematics program

Students who intend to become actuaries or work in the financial sector should take the following courses:

CIS 111Fundamentals of Computer Programming3
CIS208C Language Laboratory1
MATH 500Mathematical Theory of Interest3
MATH 501Mathematical Foundations of Actuarial Science3
MATH 510Discrete Mathematics3
MATH 512Introduction to Modern Algebra3
MATH 551Applied Matrix Theory3
MATH 633Advanced Calculus I3
MATH 655Elementary Numerical Analysis3
STAT 510Introductory Probability and Statistics I3
STAT 511Introductory Probability and Statistics II3
 

For Verification of Educational Experience (VEE) credit and/or preparation for the first four actuarial exams and/or VEE exams, students should take courses such as:

ECON 110Principles of Macroeconomics3
ECON 120Principles of Microeconomics3
FINAN 450Principles of Finance3
FINAN 520Equity Securities and Markets3
STAT 705Regression and Correlation Analyses2
STAT 770Theory of Statistics I3
STAT 771Theory of Statistics II3
 

Teacher preparation program

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

CIS 111Fundamentals of Computer Programming3
MATH 312Finite Applications of Mathematics3
MATH 506Introduction to Number Theory3
MATH 510Discrete Mathematics3
MATH 511Introduction to Algebraic Systems 3
MATH 520Foundations of Analysis3
MATH 551Applied Matrix Theory3
MATH 570History of Mathematics3
MATH 572Foundations of Geometry3
MATH 591Topics in Mathematics for Teachers3
STAT 510Introductory Probability and Statistics I3
 

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 Trigonometry3
MATH 160Introduction to Contemporary Mathematics3
MATH 205General Calculus and Linear Algebra3
MATH 312Finite Applications of Mathematics3
MATH 313Computational Number Theory3
MATH 320Mathematics for Elementary
School Teachers3
MATH 591Topics in Mathematics for Teachers3

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, education, or engineering. The degree requirements of both colleges and majors 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 scores 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 by K-State; 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 by K-State; 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.

MATH 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 by K-State; 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 by K-State; 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. Repeatable for credit for different qualifying courses.

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. Mathematical Theory of Interest. (3) I. Extensive introduction to the mathematical theory of interest with applications to the field of actuarial science. Prepares students to take the second professional examination administered by the Society of Actuaries and the Casualty Actuarial Society. Pr.: MATH 221.

MATH 501. Mathematical Foundations of Actuarial Science. (3) II. Introduction to calculus-based applications and models with an emphasis on applications to insurance. Prepares students to take the first professional examination administered by the Society of Actuaries and Casualty Actuarial Society. Pr.: MATH 222 and STAT 510 or conc. enrollment in STAT 510.

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. (3) II. 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) II. 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) I. 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) II. 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, S. 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) II. 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. Pr.: MATH 220.

MATH 572. Foundations of Geometry. (3) I. 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 599. Topics in Mathematics. (1-3) Courses on special topics to be taught on sufficient demand.

MATH 630. Introduction to Complex Analysis. (3) I. 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) II. 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, II, S. 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) I. 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 630.

MATH 715. Applied Mathematics I. (3) I. Topics from vector calculus, higher-dimensional calculus, ordinary differential equations, matrix theory, linear algebra, and complex analysis. Pr.: MATH 222.

MATH 716. Applied Mathematics II. (3) II. Topics from Fourier series, Fourier and Laplace transforms, partial differential equations, calculus of variations, and linear algebra. Pr.: MATH 715.

MATH 721. Introduction to Real Analysis. (3) I. Limits, continuity, uniform convergence, completeness, differentiation, Riemann integration. Pr.: MATH 634 or graduate standing.

MATH 722. Introduction to Functions of Several Variables. (3) II. Analysis of functions of several variables, including differentiability, partial differentiability, maxima and minima, inverse function theorem, implicit function theorem, integration, Fubini's theorem. Pr.: MATH 634 or graduate standing.

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, 715.

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.

MATH 799. Topics in Mathematics. (1-3) Courses on special topics to be taught on sufficient demand.