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## Amherst College Mathematics and Statistics for 2009-10

### Mathematics

#### 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. Students cannot register for both Mathematics 05 and Chemistry 11 in the same semester.

Fall semester. Visiting Professor Condon.

**2015-16:**Offered in Fall 2015

**Other years:**Offered in Fall 2007, Fall 2008, Fall 2009, Fall 2010, Fall 2011, Fall 2012, Fall 2013, Fall 2014

#### 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.

Requisite: Mathematics 05. Spring semester. Visiting Professor Condon.

**2015-16:**Offered in Spring 2016

**Other years:**Offered in Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, Spring 2013, Spring 2014, Spring 2015

#### 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 2009-10.

**2015-16:**Not offered

#### 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.

Fall and spring semesters. In the Fall semester, the intensive section (Section 01) is open only to students listed as eligible on the Mathematics placement list. The Department.

**2015-16:**Offered in Fall 2015 and Spring 2016

**Other years:**Offered in Fall 2007, Spring 2008, Fall 2008, Spring 2009, Fall 2009, Spring 2010, Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015

#### 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. Fall and spring semesters. The Department.

**2015-16:**Offered in Fall 2015 and Spring 2016

**Other years:**Offered in Fall 2007, Spring 2008, Fall 2008, Spring 2009, Fall 2009, Spring 2010, Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015

#### 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. Fall semester: Professor Leise and Visiting Professor Hutz. Spring semester: Visiting Professor Hutz.

**2015-16:**Offered in Fall 2015 and Spring 2016

**Other years:**Offered in Fall 2007, Spring 2008, Fall 2008, Spring 2009, Fall 2009, Spring 2010, Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015

#### 14 Mathematical Modeling

Mathematical modeling is the process of translating a real world problem into a mathematical expression, analyzing it using mathematical tools and numerical simulations, and then interpreting the results in the context of the original problem. Discussion of basic modeling principles and case studies will be followed by several projects from areas such as environmental studies and biology (e.g., air pollution, ground water flow, populations of interacting species, social networks). This course has no requisites; projects will be tailored to each student’s level of mathematical preparation. Four class hours per week, with occasional in-class computer labs.

Omitted 2009-10.

**2015-16:**Offered in Spring 2016

**Other years:**Offered in Fall 2008, Fall 2012, Fall 2014

#### 15 Discrete Mathematics

This course is an introduction to some topics in mathematics that do not require the calculus. The topics covered include logic, elementary set theory, functions, relations and equivalence relations, mathematical induction, counting principles, and graph theory. Additional topics may vary from year to year. This course serves as an introduction to mathematical thought and pays particular attention to helping students learn how to write proofs. Four class hours per week.

Spring semester. Professor Velleman.

**2015-16:**Offered in Fall 2015 and Spring 2016

**Other years:**Offered in Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, Spring 2013, Spring 2014, Spring 2015

#### 17 Introduction to Statistics

(Offered as Mathematics 17 and Environmental Studies 24.) 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). In a semester when two sections of Math 17 are offered, Section 02 is recommended for students interested in Environmental Studies. Two class hours plus two hours of laboratory per week.

Each section limited to 20 students. Fall and spring semesters. Professors Liao and Wagaman.

**2015-16:**Offered in Fall 2015 and Spring 2016

**Other years:**Offered in Fall 2007, Spring 2008, Fall 2008, Spring 2009, Fall 2009, Spring 2010, Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015

#### 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 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). Two class hours plus two hours of laboratory per week.

Requisite: Mathematics 17 or consent of the instructor. Spring semester. Professor Liao.

**2015-16:**Not offered

**Other years:**Offered in Spring 2008, Spring 2010, Spring 2012, Fall 2013, Spring 2014

#### 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. In spring 2007 the topics were nonlinear dynamics and chaos. We studied the dynamics of one- and two-dimensional flows. The focus was 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 considered examples from physics, biology, chemistry, and engineering. The course also covered 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 2009-10.

**2015-16:**Offered in Spring 2016

**Other years:**Offered in Spring 2009, Spring 2011, Spring 2013, Spring 2014, Spring 2015

#### 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 may not both be taken for credit. Fall semester. Professor Cox.

**2015-16:**Offered in Fall 2015 and Spring 2016

**Other years:**Offered in Fall 2007, Spring 2008, Fall 2008, Spring 2009, Fall 2009, Fall 2010, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015

#### 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 may not both be taken for credit. Spring semester. Professor Leise.

**2015-16:**Offered in Fall 2015 and Spring 2016

**Other years:**Offered in Fall 2007, Spring 2008, Fall 2008, Spring 2009, Spring 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015

#### 23 Multivariate Data Analysis

Real world experiments often provide data that consist of many variables. When confronted with a large number of variables, there may be many different directions to proceed, but the direction chosen is ultimately based on the question(s) being asked. In biology, one could ask which observed characteristics distinguish females from males in a given species. In archeology, one could examine how the observed characteristics of pottery relate to their location on the site, look for clusters of similar pottery types, and gain valuable information about the location of markets or religious centers in relation to residential housing. This course will explore how to visualize large data sets and study a variety of methods to analyze them. Methods covered include principal components analysis, factor analysis, classification techniques (discriminant analysis and classification trees) and clustering techniques. This course will feature hands-on data analysis in weekly computer labs, emphasizing application over theory. Four class hours per week.

Limited to 20 students. Fall semester. Professor Wagaman.

**2015-16:**Not offered

**Other years:**Offered in Fall 2009, Fall 2012, Fall 2014

#### 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 2009-10.

**2015-16:**Offered in Spring 2016

**Other years:**Offered in Fall 2008, Fall 2010, Fall 2012, Spring 2014, Spring 2015

#### 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 22 or consent of the instructor. Spring semester. Visiting Professor Condon.

**2015-16:**Offered in Fall 2015 and Spring 2016

**Other years:**Offered in Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, Spring 2013, Spring 2014, Fall 2014, Spring 2015

#### 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, 22, or 28, or consent of the instructor. Fall semester. Professor Velleman.

**2015-16:**Offered in Fall 2015

**Other years:**Offered in Spring 2008, Fall 2009, Spring 2013

#### 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. Spring semester. Visiting Professor Hutz.

**2015-16:**Offered in Fall 2015 and Spring 2016

**Other years:**Offered in Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015

#### 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.

Requisite: Mathematics 12 or consent of the instructor. Omitted 2009-10.

**2015-16:**Offered in Fall 2015

**Other years:**Offered in Fall 2008, Fall 2010, Fall 2011, Fall 2012, Fall 2013, Fall 2014

#### 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: Mathematics 29 or consent of the instructor. Omitted 2009-10.

**2015-16:**Offered in Spring 2016

**Other years:**Offered in Spring 2009, Spring 2011, Spring 2013, Spring 2015

#### 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. Fall semester. Professor Condon.

**2015-16:**Offered in Fall 2015

**Other years:**Offered in Fall 2007, Fall 2008, Fall 2009, Fall 2010, Fall 2011, Fall 2012, Fall 2013, Fall 2014

#### 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, 22, or 28, or consent of the instructor. Omitted 2009-10.

**2015-16:**Not offered

**Other years:**Offered in Fall 2008, Fall 2010, Spring 2015

#### 37 Topics in Mathematics

The topics may vary from year to year. Four class hours per week.

Omitted 2009-10.

**2015-16:**Not offered

**Other years:**Offered in Fall 2007

#### 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. Spring semester. Professor Leise.

**2015-16:**Offered in Spring 2016

**Other years:**Offered in Spring 2008, Spring 2010, Spring 2012, Spring 2014

#### 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 2009-10.

**2015-16:**Not offered

**Other years:**Offered in Spring 2009, Spring 2011, Spring 2013, Spring 2015

#### 77, 78 Senior Departmental Honors

Open to seniors with the consent of the Department. Fall semester. The Department.

**2015-16:**Offered in Fall 2015

**Other years:**Offered in Fall 2009, Fall 2010, Fall 2011, Fall 2012, Fall 2013, Fall 2014

#### 97, 98 Special Topics

Fall semester. The Department.

**2015-16:**Offered in Fall 2015 and Spring 2016

**Other years:**Offered in Fall 2009, Fall 2010, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015

### Statistics

#### 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 21 or 22. Fall semester. Professor Leise.

**2015-16:**Offered in Fall 2015

**Other years:**Offered in Fall 2007, Fall 2009, Fall 2011, Fall 2013

#### Related Courses

PHIL-50 Philosophy of Mathematics (Course not offered this year.)