**05.** **Calculus with Algebra.** Mathematics 05 and 06 are designed for students whose background and algebraic skills are inadequate for the fast pace of Mathematics 11. In addition to covering the usual material of beginning calculus, these courses will have an extensive review of algebra and trigonometry. There will be a special emphasis on solving word problems.

Mathematics 05 starts with a quick review of algebraic manipulations, inequalities, absolute values and straight lines. Then the basic ideas of calculus--limits, derivatives, and integrals--are introduced, but only in the context of polynomial and rational functions. As various applications are studied, the algebraic techniques involved will be reviewed in more detail. When covering related rates and maximum-minimum problems, time will be spent learning how to approach, analyze and solve word problems. Four class hours per week. Note: While Mathematics 05 and 06 are sufficient for any course with a Mathematics 11 requisite, Mathematics 05 alone is not. However, students who plan to take Mathematics 12 should consider taking Mathematics 05 and then Mathematics 11, rather than Mathematics 06.

First semester. Visiting Professor Leise.

**06.** **Calculus with Elementary Functions.** Mathematics 06 is a continuation of Mathematics 05. Trigonometric, logarithmic and exponential functions will be studied from the point of view of both algebra and calculus. The applications encountered in Mathematics 05 will reappear in problems involving these new functions. The basic ideas and theorems of calculus will be reviewed in detail, with more attention being paid to rigor. Four class hours per week.

Second semester. Visiting Professor Leise.

**09.** **Lies, Damned Lies, and Statistics. **In 1895 H.G. Wells wrote that "Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write." Today, statistics are cited to sway our opinion on everything from which toothbrush dentists prefer to how crime rates have changed from one political administration to the next. This seminar focuses not on statistical calculations, but on the critical evaluation of statistics that are presented every day in mass media. Topics to be discussed include proper survey and study methodologies, accurate visual displays of information, fundamentals of probability, the basics of hypothesis testing and confidence intervals, as well as the true meaning of correlation and the limitations of regression models. Three class meetings per week.

Limited to 20 students. Omitted 2007-08.

**11.** **Introduction to the Calculus.** Basic concepts of limits, derivatives, anti-derivatives; applications, including max/min problems and related rates; the definite integral, simple applications; trigonometric functions; logarithms and exponential functions. Four class hours per week.

First and second semesters. Professor Benedetto.

**12.** **Intermediate Calculus.** A continuation of Mathematics 11. Inverse trigonometric and hyperbolic functions; methods of integration, both exact and approximate; applications of integration to volume and arc length; improper integrals; l’Hôpital's rule; infinite series, power series and the Taylor development; and polar coordinates. Four class hours per week.

Requisite: A grade of C or better in Mathematics 11 or consent of the Department. First and second semesters. The Department.

**13.** **Multivariable Calculus.** Elementary vector calculus; introduction to partial derivatives; multiple integrals in two and three dimensions; line integrals in the plane; Green’s theorem; the Taylor development and extrema of functions of several variables; implicit function theorems; Jacobians. Four class hours per week.

Requisite: A grade of C or better in Mathematics 12 or the consent of the instructor. First and second semesters. The Department.

**15.** **Discrete Mathematics.** This course is an introduction to some topics in mathematics that do not require the calculus. Emphasis is placed on topics that have applications in computer science, including elementary set theory, logic, mathematical induction; basic counting principles; relations and equivalence relations; graph theory; and rates of growth. Additional topics may vary from year to year. This course not only serves as an introduction to mathematical thought but it is also recommended background for advanced courses in computer science. Four class hours per week.

Second semester. Professor Cox.

**16.** **Chaotic Dynamical Systems.** Given a system such as the weather, the stock market or the population of a large city, there are many questions that can be asked about its long-term behavior. A Dynamical System is a mathematical model of such a system, and in this course, we will study dynamical systems from a mathematical point of view. In particular, we will describe the various ways in which a dynamical system can behave, and we will discover that some very simple systems can have surprisingly complex behavior. This will lead to the notion of a chaotic dynamical system. We will also discuss Newton's method, fractals, and iterations of complex functions. Three class hours per week plus a weekly one-hour computer laboratory. Offered in alternate years.

Requisite: Mathematics 13 or consent of the instructor. Omitted 2007-08.

**17. Introduction to Statistics.** This course is an introduction to applied statistical methods useful for the analysis of data from all fields. Brief coverage of data summary and graphical techniques will be followed by elementary probability, sampling distributions, the central limit theorem and statistical inference. Inference procedures include confidence intervals and hypothesis testing for both means and proportions, non-parametric alternatives to standard hypothesis tests of the mean, the chi-square test, simple linear regression, and a brief introduction to analysis of variance (ANOVA). Three class hours plus one hour of laboratory per week.

Limited to 20 students. First and second semesters. Professor Tranbarger.

**18. Regression Modeling and Design of Experiments.** This continuation of Mathematics 17 includes more detailed regression modeling using both linear and multiple regression techniques. Also covered are categorial data analysis techniques such as chi-square tests, regression modeling with indicator variables and logistic regression, followed by one and two factor analysis of variance (ANOVA). Two class hours plus two hours of laboratory per week.

Not open to students who took Math 36 in 2005-06. Requisite: Mathematics 17. Second semester. Professor Tranbarger.

**19.** **Wavelet and Fourier Analysis.** The first half of the course covers continuous and discrete Fourier transforms (including convolution and Plancherel’s formula), Fourier series (including convergence and the fast Fourier transform algorithm), and applications like heat conduction along a rod and signal processing. The second half of the course is devoted to wavelets: Haar bases, the discrete Haar transform in 1 and 2 dimensions with application to image analysis, multiresolution analysis, filters, and wavelet-based image compression like JPEG2000. Three class hours per week plus a weekly one-hour computer laboratory.

Requisite: Mathematics 13 and one of 21, 22, or 25. First semester. Visiting Professor Leise.

**20. Topics in Differential Equations.** The study of differential equations is an important part of mathematics that involves many topics, both theoretical and practical. The precise subject matter of this course will vary from year to year. For spring 2007, the topics will be nonlinear dynamics and chaos. We will study the dynamics of one- and two-dimensional flows. The focus of the course will be on bifurcation theory: how do solutions of nonlinear differential equations change qualitatively as a control parameter is varied, and how does chaos arise? To illustrate the analysis, we will consider examples from physics, biology, chemistry, and engineering. The course will also cover basic theorems concerning existence and uniqueness of solutions and continuous dependence on parameters. Three class hours per week plus a weekly one-hour computer laboratory.

Requisite: Mathematics 13 or consent of the instructor. Limited to 20 students. Omitted 2007-08.

**21.** **Linear Algebra. **The study of vector spaces over the real and complex numbers, introducing the concepts of subspace, linear independence, basis, and dimension; systems of linear equations and their solution by Gaussian elimination; matrix operations; linear transformations and their representations by matrices; eigenvalues and eigenvectors; and inner product spaces. Special attention will be paid to the theoretical development of the subject. Four class meetings per week.

Requisite: Mathematics 12 or consent of the instructor. This course and Mathematics 22 or 25 may not both be taken for credit. First semester. Professor Velleman.

**22.** **Linear Algebra with Applications. **The study of vector spaces over the real and complex numbers, introducing the concepts of subspace, linear independence, basis, and dimension; systems of linear equations and their solution by Gaussian elimination; matrix operations; linear transformations and their representations by matrices; eigenvalues and eigenvectors; and inner product spaces. Additional topics include ill-conditioned systems of equations, the LU decomposition, covariance matrices, least squares, and the singular value decomposition. Recommended for Economics majors who wish to learn linear algebra. Four class hours per week, with occasional in-class computer labs.

Requisite: Mathematics 12 or consent of the instructor. This course and Mathematics 21 or 25 may not both be taken for credit. Second semester. Visiting Professor Leise.

**24.** **Theory of Numbers.** An introduction to the theory of rational integers; divisibility, the unique factorization theorem; congruences, quadratic residues. Selections from the following topics: cryptology; Diophantine equations; asymptotic prime number estimates; continued fractions; algebraic integers. Four class hours per week. Offered in alternate years.

Requisite: Mathematics 12 or consent of the instructor. Omitted 2007-08.

**26.** **Groups, Rings and Fields.** A brief consideration of properties of sets, mappings, and the system of integers, followed by an introduction to the theory of groups and rings including the principal theorems on homomorphisms and the related quotient structures; integral domains, fields, polynomial rings. Four class hours per week.

Requisite: Mathematics 21 or 25 or both 15 and 22 or consent of the instructor. Second semester. Professor Benedetto.

**27.** **Set Theory. **Most mathematicians consider set theory to be the foundation of mathematics, because everything that is studied in mathematics can be defined in terms of the concepts of set theory, and all the theorems of mathematics can be proven from the axioms of set theory. This course will begin with the axiomatization of set theory that was developed by Ernst Zermelo and Abraham Fraenkel in the early part of the twentieth century. We will then see how all of the number systems used in mathematics are defined in set theory, and how the fundamental properties of these number systems can be proven from the Zermelo-Fraenkel axioms. Other topics will include the axiom of choice, infinite cardinal and ordinal numbers, and models of set theory. Four class hours per week.

Requisite: Mathematics 15, 21 or 25 or 28, or consent of the instructor. Second semester. Professor Velleman.

**28.** **Introduction to Analysis.** Completeness of the real numbers; topology of n-space including the Bolzano-Weierstrass and Heine-Borel theorems; sequences, properties of functions continuous on sets; infinite series, uniform convergence. The course may also study the Gamma function, Stirling’s formula, or Fourier series. Four class hours per week.

Requisite: Mathematics 13. Second semester. Professor TBA.

**29. Probability.** This course explores the nature of probability and its use in modeling real world phenomena. The course begins with the development of an intuitive feel for probabilistic thinking, based on the simple yet subtle idea of counting. It then evolves toward the rigorous study of discrete and continuous probability spaces, independence, conditional probability, expectation, and variance. Distributions covered include the Bernoulli and Binomial, Hypergeometric, Poisson, Normal, Gamma, Beta, Multinomial, and bivariate Normal. Four class hours per week.

Not open to students who have previously taken Mathematics 14. Requisite: Mathematics 12 or consent of the instructor. Omitted 2007-08.

**30. Mathematical Statistics.** This course examines the theory behind common statistical inference procedures including estimation and hypothesis testing. Beginning with exposure to Bayesian inference, the course will cover Maximum Likelihood Estimators, sufficient statistics, sampling distributions, joint distributions, confidence intervals, hypothesis testing and test selection, non-parametric procedures, and linear models. Four class hours per week.

Requisite: Probability (Mathematics 14 or 29) or consent of the instructor. Omitted 2007-08.

**31.** **Functions of a Complex Variable.** An introduction to analytic functions; complex numbers, derivatives, conformal mappings, integrals. Cauchy’s theorem; power series, singularities, Laurent series, analytic continuation; Riemann surfaces; special functions. Four class hours per week.

Requisite: Mathematics 13. First semester. Professor Starr.

**34.** **Mathematical Logic.** Mathematicians confirm their answers to mathematical questions by writing proofs. But what, exactly, is a proof? This course begins with a precise definition specifying what counts as a mathematical proof. This definition makes it possible to carry out a mathematical study of what can be accomplished by means of deductive reasoning and, perhaps more interestingly, what cannot be accomplished. Topics will include the propositional and predicate calculi, completeness, compactness, and decidability. At the end of the course we will study Gödel’s famous Incompleteness Theorem, which shows that there are statements about the positive integers that are true but impossible to prove. Four class hours per week. Offered in alternate years.

Requisite: Mathematics 15, 21 or 25 or 28, or consent of the instructor. Omitted 2007-08.

**36.** **Advanced Applied Statistics.** This continuation of Mathematics 17 includes more detailed regression modeling using both linear and multiple regression techniques. Also covered are categorical data analysis techniques such as chi-square tests, regression modeling with indicator variables and logistic regression, followed by one and two factor analysis of variance (ANOVA). Four class meetings per week.

Requisite: Mathematics 17. Omitted 2007-08.

**37.** **Topics in Mathematics.** The topics may vary from year to year. The topic for fall 2007 is Galois Theory, which is the systematic study of the roots of polynomials. The key idea, first glimpsed by Lagrange and later brought to fruition by Galois, is that there is a deep relation between group theory and the structure of the set of roots of a given polynomial. One of the most famous results of the theory is that there is no analogue of the quadratic formula for polynomials of degree five and higher; another is the impossibility of trisecting an angle with straightedge and compass. This course will develop Galois Theory from the basics of groups and fields. Four class hours per week. Offered in alternate years.

Requisite: Mathematics 26. First semester. Professor Benedetto.

**42.** **Functions of a Real Variable.** An introduction to Lebesgue measure and integration; topology of the real numbers, inner and outer measures and measurable sets; the approximation of continuous and measurable functions; the Lebesgue integral and associated convergence theorems; the Fundamental Theorem of Calculus. Four class hours per week. Offered in alternate years.

Requisite: Mathematics 28. Second semester. Professor Cox.

**44.** **Topology.** An introduction to general topology; the topology of Euclidean, metric and abstract spaces, with emphasis on such notions as continuous mappings, compactness, connectedness, completeness, separable spaces, separation axioms, and metrizable spaces. Additional topics may be selected to illustrate applications of topology in analysis or to introduce the student briefly to algebraic topology. Four class hours per week. Offered in alternate years.

Requisite: Mathematics 28. Omitted 2007-08.

**77, 78.** **Senior Departmental Honors.**

Open to seniors with the consent of the Department. First and second semesters. The Department.

**97, 98.** **Special Topics.** Independent Reading Course.

First and second semesters. The Department.

#### Related Course

**Philosophy of Mathematics. **See Philosophy 50.

Omitted 2007-08. Professors A. George and Velleman.