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## Amherst College Mathematics and Statistics for 2012-13

### Mathematics

#### 105 Calculus with Algebra

MATH 105 and 106 are designed for students whose background and algebraic skills are inadequate for the fast pace of MATH 111. 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.

MATH 105 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 meetings per week, one of which is a two-hour group-work day.

Note: While MATH 105 and 106 are sufficient for any course with a MATH 111 requisite, MATH 105 alone is not. However, students who plan to take MATH 121 should consider taking MATH 105 and then MATH 111, rather than MATH 106. Students cannot register for both MATH 105 and CHEM 111 in the same semester.

Fall semester. Professor Cox.

**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

#### 106 Calculus with Elementary Functions

MATH 106 is a continuation of MATH 105. Trigonometric, logarithmic and exponential functions will be studied from the point of view of both algebra and calculus. The applications encountered in MATH 105 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 meetings per week, one of which is a two-hour group-work day.

Requisite: MATH 105. Spring semester. Professor Cox.

**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

#### 111 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 intensive section replaces one weekly class hour with a 90-to-120 minute group work day. Spring semester, Professor R. Benedetto.

**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

#### 121 Intermediate Calculus

A continuation of MATH 111. 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 MATH 111 or consent of the Department. Fall and spring semesters. Lecturer D. Benedetto.

**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

#### 130 Introduction to Statistics

(Offered as MATH 130 and ENST 240.) 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, the chi-square test, simple linear regression, and a brief introduction to analysis of variance (ANOVA). Some sections of MATH 130 have an environmental theme and are recommended for students interested in Environmental Studies. In fall 2012, the environmental section will be section 02; in spring 2013, there will NOT BE an environmental section. Four class hours per week (two will be held in the computer lab). Labs are not interchangeable between sections due to course content.

Each section limited to 20 students. Fall semester: Professors Liao and Wagaman, Visiting Professor Jeneralczuk. Spring semester: Professor Wagaman, Visiting Professor Manack and Postdoctoral Fellow Hedlin.

**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

#### 140 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.

Limited to24 students. Fall semester. Professor Leise.

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

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

#### 211 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 MATH 121 or the consent of the instructor. Fall semester: The Department. Spring semester: Assistant Professor Ching and Visiting Professor Ndangali

**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

#### 220 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

#### 250 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: MATH 121 or consent of the instructor. Fall semester. Professor Cox.

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

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

#### 260 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 2013 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. Four class hours per week, one of which will be in the computer lab.

Requisite: MATH 211 or consent of the instructor. Spring semester. Professor Leise.

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

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

#### 271 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: MATH 121 or consent of the instructor. This course and MATH 272 may not both be taken for credit. Fall semester. Visiting Professor Manack.

**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

#### 272 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: MATH 121 or consent of the instructor. This course and MATH 271 may not both be taken for credit. Spring semester. Professor Leise and Visiting Professor Ndangali.

**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

#### 330 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

#### 345 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: MATH 211. Fall semester. Professor R. Benedetto.

**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

#### 350 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: MATH 271 or 272 or consent of the instructor. Spring semester. Professor R. Benedetto.

**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

#### 355 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: MATH 211. Spring semester. Visiting Professor Manack.

**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

#### 360 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: MATH 121 or consent of the instructor. Fall semester. Professor Liao.

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

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

#### 380 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: MATH 220, 271, 272, or 355, or consent of the instructor. Spring semester. Professor Velleman.

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

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

#### 390, 490 Special Topics

Fall and spring semesters. The Department.

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

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

#### 430 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: MATH 360 or consent of the instructor. Spring semester. Professor Wagaman.

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

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

#### 455 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: MATH 355. Spring semester. Professor Ching.

**2015-16:**Not offered

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

#### 498, 499 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