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## Amherst College Mathematics and Statistics for 2013-14

#### 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 151 in the same semester.

Fall semester. Professor D. Benedetto.

**2013-14:**Offered in Fall 2013

**Other years:**Offered in Fall 2007, Fall 2008, Fall 2009, Fall 2010, Fall 2011, Fall 2012, 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. Lecturer D. Benedetto.

**2013-14:**Offered in Spring 2014

**Other years:**Offered in Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, Spring 2013, 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. Fall and spring semesters. Spring semester Visiting Professor D'Ambroise.

**2013-14:**Offered in Fall 2013 and Spring 2014

**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 2014

#### 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. Professor TBA.

**2013-14:**Offered in Fall 2013 and Spring 2014

**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 2014

#### 130, 130E 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, the chi-square test, simple linear regression, and a brief introduction to analysis of variance (ANOVA). Four class hours per week (two will be held in the computer lab). Labs are not interchangeable between sections due to course content.

Limited to 20 students per section. Fall semester: Visiting Professors Jeneralczuk and Manack. Spring semester: Professors Liao and Wagaman and Lecturer Jeneralczuk.

**2013-14:**Offered in Fall 2013 and Spring 2014

**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 2014, Spring 2015

#### 135 Introduction to Statistics via Modeling

Introduction to Statistics via Modeling is an introductory statistics course that uses modeling as a unifying framework for much of statistics. The course provides a basic foundation in statistics with a major emphasis on constructing models from data. Students learn important concepts of statistics by mastering powerful and relatively advanced statistical techniques using computational tools. Topics include descriptive and inferential statistics, probability (including conditional probabilities and Bayes' rule), multiple regression and an introduction to causal inference. This is a more mathematically rigorous version of MATH 130 (students may not receive credit for both MATH 130 and MATH 135). Four class hours per week (two will be held in the computer lab).

Requisite: MATH 111. Limited to 24 students. Spring Semester. Professor Horton.

**2013-14:**Offered in Spring 2014

**Other years:**Offered in 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 to 24 students. Omitted 2013-14.

**2013-14:**Not offered

**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 and spring semester. Spring semester: Professors Dresch and Ndangali.

**2013-14:**Offered in Fall 2013 and Spring 2014

**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 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 R. Benedetto.

**2013-14:**Offered in Spring 2014

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

#### 225 Chaos and Fractals

The first half of the course will be devoted to the topic of chaos. This occurs when the long-term behavior of a system is unpredictable in predictable ways. The underlying mathematical construct is called a dynamical system, which can be discrete or continuous. We will study discrete dynamical systems in one and two dimensions and say a few words about the continuous case, which involves differential equations. The second half of the course will study fractals, which are wonderfully complicated mathematical objects that often have surprisingly simple descriptions. Some fractals will be encountered in the first half of the course, while others will be introduced in the second. A rich supply of fractals arises when dealing with complex numbers. This is where we will meet the famous Mandelbrot set.

Requesite: MATH 211 or consent of the instructor. Limited to 35 students. Fall semester. Professor Cox.

**2013-14:**Offered in Fall 2013

**Other years:**Offered in Fall 2014

#### 230 Intermediate Statistics

This course is an intermediate applied statistics course that continues the theme of hands-on data analysis begun in MATH 130. Students will learn how to evaluate an experimental study, perform appropriate statistical analysis of the data, and properly communicate their analyses. Emphasis will be placed on the use of statistical software and the interpretation of the results of data analysis. Topics covered will include basic experimental design, parametric and nonparametric methods for comparing two or more population means, analysis of variance models for multi-factor designs, multiple regression, analysis of covariance, model selection, logistic regression, and methods for analyzing various types of count data. Four class hours per week (two will be held in the computer lab).

Requisite: MATH 130 or consent of the instructor. Fall semester: Professor Horton. Spring semester: Professor Wagaman.

**2013-14:**Offered in Fall 2013 and Spring 2014

**Other years:**Offered in Spring 2008, Spring 2010, Spring 2012, 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. Spring semester. Professor Daniels.

**2013-14:**Offered in Spring 2014

**Other years:**Offered in Fall 2008, Fall 2010, Fall 2012

#### 255 Geometry

In Euclidean geometry, the parallel axiom asserts that if we have a line and a point not on the line, then there is a unique line through the point which is parallel to the given line. This seemingly obvious statement has many consequences, including the Pythagorean Theorem and the fact that the angles of a triangle sum to 180 degrees. In the nineteenth century, it was discovered that this is not the only possible geometry. The course will begin with neutral geometry, which makes no assumptions about parallel lines. We will then study non-Euclidean geometry, which uses a different parallel axiom. Familiar objects like circles and triangles behave differently in this geometry. For example, rectangles don't exist, and the angles of a triangle sum to less than 180 degrees, and the difference is proportional to the area of the triangle. This will allow us to construct an eight-sided house where every corner is a right angle. Besides proving some fun theorems, we will also study the history of non-Euclidean geometry. The final part of the course will be an introduction to differential geometry. The key concepts will be geodesics (which generalize straight lines) and curvature (which measures how the space is warped). This will allow us to make models of non-Euclidean geometry and explore how geometric ideas apply in a much wider context. Four class hours per week.

Requisite: Math 211 or consent of the instructor. Omitted 2013-14.

**2013-14:**Not offered

#### 260 Differential Equations

The study of differential equations is an important part of mathematics that involves many topics, both theoretical and practical. The course will cover first- and second-order ordinary differential equations, basic theorems concerning existence and uniqueness of solutions and continuous dependence on parameters, long-term behavior of solutions and approximate solutions. The focus of the course will be on connecting the theoretical aspects of differential equations with real-world applications from physics, biology, chemistry, and engineering. Four class hours per week.

Requisite: Math 211 or consent of the instructor. Spring semester. Professor Dresch.

**2013-14:**Offered in Spring 2014

**Other years:**Offered in Spring 2009, Spring 2011, Spring 2013, 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. Professor TBA.

**2013-14:**Offered in Fall 2013 and Spring 2014

**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 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. Professors Leise and Ndangali.

**2013-14:**Offered in Fall 2013 and Spring 2014

**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 2014, Spring 2015

#### 320 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: MATH 211 and 271 or 272. Fall semester. Professor Leise.

**2013-14:**Offered in Fall 2013

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

#### 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. Omitted 2013-14.

**2013-14:**Not offered

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

#### 335 Time Series Analysis and Applications

Many real world applications deal with a series of observations collected over time. Some familiar examples are daily stock market quotations in finance, monthly unemployment rates in economics, yearly birth rates in social science, global warming trends in environmental studies, seismic recordings in geophysics, and magnetic resonance imaging of brain waves in medicine. In this applied course, students will learn how to model the patterns in historical values of the variable(s), as well as how to use statistical methods to forecast future observations. Topics covered will include time series regression, autoregressive integrated moving average (ARIMA) models, transfer function models, state-space models and spectral analysis. If time permits, additional topics will include autoregressive conditionally heteroscedastic (ARCH) models, Kalman filtering and smoothing, and signal extraction and forecasting. Students will get practice with various applications using statistical software. Four class hours per week.

Requisite: MATH 130 or 360 or consent of the instructor. Limited to 20 students. Spring semester. Professor Liao.

**2013-14:**Not offered

#### 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. Visiting Professor Ndangali.

**2013-14:**Offered in Fall 2013

**Other years:**Offered in Fall 2007, Fall 2008, Fall 2009, Fall 2010, Fall 2011, Fall 2012, 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 Manack.

**2013-14:**Offered in Spring 2014

**Other years:**Offered in Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, Spring 2013, 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. Fall and spring semesters. Spring semester: Professor D'Ambroise.

**2013-14:**Offered in Fall 2013 and Spring 2014

**Other years:**Offered in Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, Spring 2013, 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 Wagaman.

**2013-14:**Offered in Fall 2013

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

#### 365 Stochastic Processes

A stochastic process is a collection of random variables used to model the evolution of a system over time. Unlike deterministic systems, stochastic processes involve an element of randomness or uncertainty. Examples include stock market fluctuations, audio signals, EEG recordings, and random movement such as Brownian motion and random walks. Topics will include Markov chains, martingales, Brownian motion, and stochastic integration, including Ito’s formula. Four class hours per week, with weekly in-class computer labs. \

Requisite: Math 360 or consent of instructor. Limited to 24 students. Spring semester. Professor Leise.

**2013-14:**Offered in Spring 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. Omitted 2013-14.

**2013-14:**Not offered

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

#### 385 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: MATH 220, 271, 272, or 355, or consent of the instructor. Omitted 2013-14. Professor Velleman.

**2013-14:**Not offered

#### 390, 490 Special Topics

Fall and spring semesters. The Department.

**2013-14:**Offered in Fall 2013 and Spring 2014

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

#### 410 Galois Theory

The quadratic formula shows us that the roots of a quadratic polynomial possess a certain symmetry. Galois Theory is the study of the corresponding symmetry for higher degree polynomials. We will develop this theory starting from a basic knowledge of groups, rings and fields. One of our main goals will be to prove that there is no general version of the quadratic formula for a polynomial of degree five or more. Along the way, we will also show that a circular cake can be divided into 17 (but not 7) equal slices using only a straight-edged knife.

Requisite: MATH 350 or consent of the instructor. Fall semester. Professor Ching.

**2013-14:**Offered in Fall 2013

#### 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. Omitted 2013-14.

**2013-14:**Not offered

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

#### 450 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: MATH 355. Spring semester. Professor R. Benedetto.

**2013-14:**Offered in Spring 2014

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

#### 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. Omitted 2013-14.

**2013-14:**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.

**2013-14:**Offered in Fall 2013

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