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The courses MATH 100 (algebra and trigonometry) and MATH 103 (precalculus) are available for students who need to improve basic skills that are required in other courses such as calculus, chemistry, psychology, economics, and statistics. However, neither MATH 100 nor MATH 103 may be counted toward the area requirement in natural science and/or mathematics. Courses equivalent to MATH 100 may not be transferred for College credit.

Students planning to major in the social sciences, arts, or humanities who wish to take a mathematics course but omit the study of calculus may choose from MATH 108 (Modes of Mathematical Thinking), MATH 111 (Elementary Probability Theory), or MATH 112 (Elementary Statistics). Even though it is not a prerequisite for MATH 112, MATH 111 is frequently taken prior to MATH 112. Note that both MATH 111 and MATH 112 may be counted toward the area requirement in natural science and/or mathematics.

The study of calculus is the foundation of college mathematics for students planning to major in mathematics or the physical sciences or anticipating a career or graduate study in any of the natural sciences, engineering, or applied social sciences (such as economics). There are essentially three programs of study available in the calculus:

- MATH 121, 122 is a terminal one-year sequence intended for business and social science majors;
- MATH 131, 132, 221 is the traditional calculus sequence intended for students of mathematics and the natural sciences, as well as for students intending to pursue graduate work in the applied social sciences.

Students with no previous calculus may elect MATH 121, MATH 131. An alternative to MATH 121 is MATH 121S, which places greater emphasis on problem solving and giving more individual attention to the student.

Students who have previously passed a calculus course in high school may elect MATH 122, 131, 132, or 221 as their first course, depending upon placement, preparation, and interest. A strong high school calculus course is generally adequate preparation for MATH 132 as a first calculus course, even if advanced placement credit has not been awarded for MATH 131. Students planning to take any advanced course in mathematics should not take MATH 122, because credit in MATH 122 must be forfeited if one takes MATH 132 (or its equivalent).

Exceptionally well prepared students (who place out of both MATH 131 and 132) may choose either MATH 221 or MATH 225 (Differential Equations) as their first course.

Advanced placement credit in the calculus sequence is granted on the basis of the College Entrance Examination Board Advanced Placement Test (either AB or BC). A score of 4 or 5 on the AB test or a score of 3 on the BC test gives the student credit for MATH 131. A score of 4 or 5 on the BC test gives the students credit for both MATH 131 and 132. Students who wish to enter the calculus sequence but who have not received advanced placement credit should consult the First-Year Handbook for placement guidelines based on grades and achievement test scores. The Department of Mathematics offers short advisory placement tests during fall orientation.

Pre-commerce students are required to take a statistics course, usually MATH 112, and one other mathematics course, usually MATH 111, 121, 122, or MATH 131.

**Warning** There are numerous instances of equivalent courses offered by
the Department of Mathematics as well as by the Department of Applied
Mathematics in the School of Engineering and Applied Science. A student
may not offer for degree credit two equivalent courses, e.g., MATH 131
and APMA 101, or MATH 131 and MATH 121.

**MATH 100 - (3) (Y)
Algebra and Trigonometry**

Study of computational and algebraic skills, patterns of quantitative problem solving, and mathematical thought. Includes computations, linear and quadratic equations, functions and graphs, and the trigonometry of triangles. (Credit/No Credit; chargeable against allowable non-College credits.)

**MATH 103 - (4) (S)
Precalculus**

Prerequisite: high school algebra II and geometry or MATH 100

Study of computational skills, patterns of quantitative problem solving, and mathematical thought. Includes linear and quadratic equations, polynomials, inverse functions, logarithms, arithmetic and geometric sequences, trigonometric functions, and linear systems. (Does not satisfy College science requirement.)

**MATH 108 - (3) (Y)
Modes of Mathematical Thinking**

Study of logic, number systems, functions, analytic geometry, equations, matrices, enumeration, computer algebra systems. Intended for liberal arts students and emphasizes the connection between analytic-algebraic and geometric reasoning in the understanding of mathematics. Facilitated by the use of a modern computer algebra system, such as Maple.

**MATH 110 - (3) (Y)
Mathematics for Elementary Teachers**

Prerequisite or corequisite: EDHS 201

A study of numbers, operations, and their properties; geometric figures and their properties; and introductory probability and statistics.

**MATH 111 - (3) (S)
Probability/Finite Mathematics**

Study of finite probability theory including combinatorics, equiprobable models, conditional probability and Bayes' theorem, expectation and variance, and Markow chains.

**MATH 112 - (3) (S)
Introduction to Statistics**

Study of probability distributions, tests of hypotheses, chi-square tests, sampling, and regression and correlation.

**MATH 121 - (4) (S)
Applied Calculus I**

Topics include limits and continuity; differentiation and integration of algebraic and elementary transcendental functions; and applications to maximum-minimum problems, curve sketching and exponential growth. Credit is not given for both MATH 121 and MATH 131.

**MATH 121S - (4) (Y)
Introduction to Calculus**

Prerequisite: Permission of instructor

Topics include limits and continuity; differentiation and integration of algebraic and elementary transcendental functions; and applications to maximum-minimum problems, curve sketching and exponential growth.

**MATH 122 - (3) (Y)
Applied Calculus II**

Prerequisite: MATH 121 or equivalent

A second calculus course for business, biology, and social science students. Analyzes functions of several variables, their graphs, partial derivatives and optimization; multiple integrals. Includes a review of basic single variable calculus (MATH 121 or equivalent) and an introduction to differential equations and infinite series. Credit is not given for both MATH 122 and MATH 132.

**MATH 131 - (4) (S)
Calculus I**

Prerequisite: Background in algebra, trigonometry, exponentials, logarithms, and analytic geometry

Introductory calculus with emphasis on techniques and applications. Recommended for natural science majors and students planning additional work in mathematics. The differential and integral calculus for functions of a single variable is developed through the fundamental theorem of calculus. Credit is not given for both MATH 121 and MATH 131.

**MATH 132 - (4) (S)
Calculus II**

Prerequisite: MATH 131 or permission of instructor

Continuation of 131. Applications of the integral, techniques of integration, infinite series, vectors. Credit is not given for both MATH 122 and MATH 132.

**MATH 132A - (5) (Y)
Calculus II**

Prerequisite: MATH 121 or permission of instructor

Continuation of MATH 121 for students who wish to cover the material of MATH 132.

**MATH 221 - (4) (S)
Calculus III **

Prerequisite: MATH 132 or its equivalent

A study of functions of several variables including lines and planes in space, differentiation of functions of several variables, maxima and minima, multiple integration, line integrals, and volume.

**MATH 225 - (4) (S)
Ordinary Differential Equations**

Prerequisite: MATH 132 or its equivalent

Introduction to the methods, theory and applications of differential equations. Topics include first-order, second and higher-order linear equations, series solutions, linear systems of first-order differential equations, j1 and the associated matrix theory. Additional topics may include numerical methods, non-linear systems, boundary value problems, and additional applications. Section 1 of MATH 225 in the spring covers, in addition to the basic topics, an introduction to Sturm-Liouville theory, Fourier series and boundary value problems and their connection with partial differential equations. Physics majors should enroll in section 1. No knowledge of physics is assumed in MATH 225, section 1.

**MATH 300 - (3) (Y)
Foundations of Analysis**

Prerequisite: MATH 132 or equivalent

Selection of topics from logic and the construction of mathematical proofs, basic set theory, number systems, continuity of functions and foundations of analysis. Introduces at an intermediate level the standards of mathematical rigor and abstraction that are encountered in advanced mathematics, based on the material of the calculus and other basic mathematics.

**MATH 306 - (3) (Y)
Algorithms**

Prerequisite: MATH 132 and computer proficiency

The study of abstract algorithms to solve mathematical problems and their implementation in a high-level language. Topics from sorting problems, recursive algorithms, and dynamic data structures.

**MATH 311 - (3) (Y)
Introduction to Mathematical Probability**

Prerequisite: MATH 132. A knowledge of double integrals is recommended

Sample spaces, combinatorial analysis, discrete and continuous random variables, classical distributions, expectation, Chebyshev theorem, independence, central limit theorem, conditional probability, generating functions.

**MATH 312 - (3) (Y)
Introduction to Mathematical Statistics**

Prerequisite: MATH 311

Sampling theory, point estimation, interval estimation, testing hypothesis (including the Neyman-Pearson lemma and likelihood ratio tests), regression and correlation.

**MATH 331 - (3) (S)
Basic Real Analysis**

Prerequisite: MATH 132

Concentrates on proving the basic theorems of calculus, with due attention to the beginner with little or no experience in the techniques of proof. Topics include limits, continuity, differentiability, the Bolzano-Weierstrass theorem, Taylor's theorem, integrability of continuous functions, and uniform convergence.

**MATH 332 - (3) (Y)
Complex Variables With Applications**

Prerequisite: MATH 221

Topics include analytic functions, Cauchy formulas, power series, residue theorem, conformal mapping, and Laplace transforms.

**MATH 351 - (3) (S)
Elementary Linear Algebra**

Prerequisite: MATH 132

Topics include matrices, elementary row operations, inverses, vector spaces and bases, inner products and Gram-Schmidt orthogonalization, orthogonal matrices, linear transformations and change of basis, eigenvalues, eigenvectors, and symmetric matrices.

**MATH 354 - (3) (Y)
Survey of Algebra**

Prerequisite: MATH 132 or equivalent

An introductory survey of the major topics of modern algebra: groups, rings, and fields. Applications to other areas of mathematics, such as geometry and number theory are presented. Explores the rational, real, and complex number systems, and the algebra of polynomials.

**MATH 355 - (3) (IR)
Algebraic Automata Theory**

Prerequisite: MATH 351

An introduction to the theory of sequential machines, including an introduction to the theory of finite permutation groups and transformation semigroups. Includes examples from biological and electronic systems as well as computer science, the Krohn-Rhodes decomposition of a state machine, and Mealy machines.

**MATH 404 - (3) (Y)
Discrete Mathematics**

Prerequisite: MATH 354 or permission of instructor

Topics include combinatorial principles, the binomial and multinomial theorems, partitions, discrete probability, algebraic structures, trees, graphs, symmetry groups, Polya's enumeration formula, linear recursions, and generating functions.

**MATH 408 - (3) (Y)
Operations Research**

Prerequisite: MATH 132 and MATH 351

Development of mathematical models and their solutions, including linear programming, the simplex algorithm, dual programming, parametric programming, integer programming, transportation models, assignment models, and network analysis.

**MATH 414/514 - (3) (Y)
Mathematics of Derivative Securities**

Prerequisite: MATH 221 or MATH 122 or its equivalent, and a knowledge of probability and statistics. MATH 311 or its equivalent is recommended.

Topics include arbitrage arguments, valuation of futures, forwards and swaps, hedging, option-pricing theory, and sensitivity analysis.

**MATH 430 - (3) (S)
Elementary Numerical Analysis**

Prerequisite: MATH 225 and computer proficiency

Topics include Taylor's theorem, solution of nonlinear equations, interpolation and approximation by polynomials, numerical quadrature. Further topics may be selected from numerical solutions of ordinary differential equations, Fourier series, and least-square approximation.

**MATH 452 - (3) (IR)
Algebraic Coding Theory**

Prerequisite: MATH 351 and MATH 354 or permission of instructor

An introduction to the use of algebraic techniques for communicating information in the presence of noise. Topics include the basic concepts of coding theory: linear codes, bounds for codes, BCH codes and their decoding algorithms. Other topics may include quadratic residue codes, Reed-Muller codes, algebraic geometry codes, and connections with groups, designs, and lattices.

**MATH 453 - (3) (IR)
Number Theory**

Prerequisite: MATH 354 or permission of instructor

Topics include congruences, quadratic reciprocity, Diophantine equations, and number-theoretic functions, among others.

**MATH 493 - (3) (IR)
Independent Study**

Reading and study programs in areas of interest to the individual student. This course is primarily for juniors and seniors who have developed an interest in a branch of mathematics not covered in a regular course. It is the responsibility of the student to obtain a faculty advisor to approve and direct the program.

**MATH 501 - (3) (Y)
The History of the Calculus**

Prerequisite: MATH 221 and MATH 351 or permission of instructor

Study of the evolution of the various mathematical ideas leading up to the development of the calculus in the seventeenth century, and how those ideas were perfected and extended by succeeding generations of mathematicians. Emphasizes primary source materials when possible.

**MATH 503 - (3) (Y)
The History of Mathematics**

Prerequisite: MATH 221 and MATH 351 or permission of instructor

Study of the development of mathematics from classical antiquity through the end of the nineteenth century, focusing on the critical periods in the evolution of such areas as geometry, number theory, algebra, probability and set theory. Emphasizes primary source materials when possible.

**MATH 509 - (3) (Y)
Mathematical Probability**

Prerequisite: Three semesters of calculus, and graduate standing. Students who have received credit for MATH 311 may not take MATH 509 for credit.

Study of the development and analysis of probability models through the basic concepts of sample spaces, random variables, probability distributions, expectations, and conditional probability. Additional topics include distributions of transformed variables, moment generating functions, and the central limit theorem.

**MATH 510 - (3) (Y)
Mathematical Statistics**

Prerequisite: MATH 509 and graduate standing. Students who have received credit for MATH 312 may not take MATH 510 for credit.

Topics include methods of estimation, general concepts of hypothesis testing, linear models and estimation by least squares, categorical data, and nonparametric statistics.

**MATH 511 - (3) (Y)
Stochastic Processes**

Prerequisite: MATH 311 or permission of instructor

Topics in probability theory selected from Random walks, Markov processes, Brownian motion, Poisson processes, branching processes, stationary time series, linear filtering and prediction, queuing process, and renewal theory.

**MATH 521 - (3) (Y)
Advanced Calculus with Applied Mathematics**

Prerequisite: MATH 221, MATH 225

Topics include vector analysis, Green's, Stokes', divergence theorems, conservation of energy, potential energy functions. Emphasis on physical interpretation. Also includes Sturm-Liouville problems, Fourier series, special functions, orthogonal polynomials, and Green's functions.

**MATH 522 - (3) (Y)
Partial Differential Equations and Applied Mathematics**

Prerequisites: MATH 521 (MATH 351 recommended)

Introduction to complex variables and partial differential equations. Topics include analytic functions, complex integration, power series, residues, conformal mapping; separation of variables, boundary value problems, Laplace's equation, wave equation, and heat equation.

**MATH 525 - (3) (IR)
Advanced Ordinary Differential Equations**

Prerequisites: MATH 221, MATH 225, MATH 351 or permission of instructor

Emphasis on the qualitative geometrical theory of ordinary differential equations. Topics include all or most of the following: Picard's method and basic existence and uniqueness theorems; linear systems; the phase plane and Sturm's theorems; the Poincaré-Bendixon theorem; and Lyapunov's method and stability. Other topics presented as time permits.

**MATH 526 - (3) (IR)
Partial Differential Equations**

Prerequisite: MATH 221, MATH 225 and MATH 351 or permission of instructor

A theoretical introduction from a classical viewpoint. Topics include harmonic and subharmonic functions, wave and heat equations, Cauchy-Kowalewski and Holmgren theorems, characteristics and Hamilton- Jacobi theory.

**MATH 530 - (3) (IR)
Computer Methods in Numerical Analysis **

Prerequisites: MATH 351, MATH 430 and computer proficiency

A study of the underlying mathematical principles and the use of sophisticated software for numerical problems such as spline interpolation, ordinary differential equations, nonlinear equations, optimization, and singular-value decomposition of a matrix.

**MATH 531, 532 - (3) (Y)
Introduction to Real Analysis I, II**

Prerequisite: MATH 221, MATH 351

Topics include the basic topology of Euclidean spaces, continuity and differentiation of functions on Euclidean spaces; Riemann-Stieltjes integration, convergence of sequences and series of functions; and equicontinuous families of functions, Weierstrass theorem, inverse function theorem and implicit function theorem, integration of differential forms, and Stokes' Theorem.

**MATH 551, 552 - (3) (Y)
Introduction to Abstract Algebra I, II**

Prerequisite: MATH 351 or permission of instructor

Introduction to algebraic systems: groups, rings, fields, vector spaces and their general properties; subsystems, quotient systems, homomorphisms. Includes basic examples, such as permutation groups, polynomial rings, and groups and rings of matrices. Additional topics may include applications to linear algebra and number theory.

**MATH 570 - (3) (Y)
Introduction to Geometry**

Prerequisite: MATH 221 and MATH 351 or permission of instructor

Topics selected from analytic geometry, affine geometry, projective geometry, hyperbolic and non-Euclidean geometry.

**MATH 572 - (3) (Y)
Introduction to Differential Geometry**

Prerequisite: MATH 221

Topics selected by the instructor from the theory of curves and surfaces in Euclidean space and the theory of manifolds.

**MATH 577- (3) (Y)
General Topology**

Prerequisite: MATH 221; corequisite: MATH 551 or equivalent

Topics include topological spaces and continuous functions; product and quotient topologies; compactness and connectedness; separation and metrication; and the fundamental group and covering spaces.

**MATH 583 - (3) (IR)
Seminar**

Prerequisite: Permission of instructor

Presentation of selected topics in mathematics.

**MATH 596 - (3) (S)
Supervised Study in Mathematics**

Prerequisite: Permission of instructor and graduate standing

In exceptional circumstances, a student may undertake a rigorous program of supervised study designed to expose the student to a particular area of mathematics. Regular homework assignments and scheduled examinations are required.

**Courses in Statistics** Courses at the 300-500 levels offered by the
Division of Statistics may be found later in this chapter.

Continue to: Program in Medieval Studies

Return to: Chapter 6 Index