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MathematicsNatural Sciences Division FacultyNuh Aydin Robert M. Fesq Jr. Bradley A. Hartlaub Judy A. Holdener Keith E. Howard Brian D. Jones Robert M. McLeod Robert S. Milnikel Jr. Michael D. Radmacher Carol S. Schumacher Stephen P. Slack Assistant Professor For well over two thousand years, mathematics has been a part of the human search for understanding. Mathematical discoveries have come both from the attempt to describe the natural world and from the desire to arrive at a form of inescapable truth from careful reasoning that begins with a small set of self-evident assumptions. These remain fruitful and important motivations for mathematical thinking, but in the last century mathematics has been successfully applied to many other aspects of the human world: voting trends in politics, the dating of ancient artifacts, the analysis of automobile traffic patterns, and long-term strategies for the sustainable harvest of deciduous forests, to mention a few. Today, mathematics as a mode of thought and expression is more valuable than ever before. Learning to think in mathematical terms is an essential part of becoming a liberally educated person. Kenyon’s program in mathematics endeavors to blend interrelated but distinguishable facets of mathematics: theoretical ideas and methods, modeling real-world situations, the statistical analysis of data, and scientific computing. The curriculum is designed to develop competence in each of these aspects of mathematics in a way that responds to the interests and need of individual students. New StudentsFor those students who want only an introduction to mathematics, or perhaps a course to satisfy a distribution requirement, selection from MATH 105, 106, 108, 110, 111, 117, and 118 is appropriate. Students who think they might want to continue the study of mathematics beyond one year, either by pursuing a major or minor in mathematics or as a foundation for courses in other disciplines, usually begin with the calculus sequence (MATH 111, 112, and 213). Students who have already had calculus or who want to take more than one math course may choose to begin with the Elements of Statistics (MATH 106) and Data Analysis (MATH 206) or Introduction to Computer Science (MATH 118). A few especially well-prepared students take Linear Algebra (MATH 224) or Foundations (MATH 222) in their first year. (Please see Associate Professor Brad Hartlaub for further information.) Pre-calculus (MATH 110) is a course for students with a weak background in mathematics who wish to prepare for calculus. MATH 111 is an introductory course in calculus. Students who have completed a substantial course in calculus might qualify for one of the successor courses, MATH 112 or 213. MATH 106 is an introduction to statistics, which focuses on quantitative reasoning skills. MATH 118 introduces students to computer programming. Please read the course descriptions for further information concerning these courses, and look for the The ready availability of powerful computers has made the computer one of the primary tools of the mathematician. Students will be expected to use appropriate computer software in many of the mathematics courses. However, no prior experience with the software packages or programming is expected, except in those courses that have MATH 118 as a prerequisite. Course Requirements for the MajorThere are two concentrations within the mathematics major: classical mathematics and statistics. The coursework required for completion of the major in each concentration is given below. Classical Mathematics A student must have credit for the following core courses: In addition, majors must have credit for at least three other courses selected with the consent of the department. However, MATH 110 may not be used to satisfy the requirements for the major. Statistics A student must have credit for the following core courses: Three semesters of calculus (MATH 111, 112, 213 or the equivalent) In addition to the core courses, majors must also have credit for two of the following: Additional Requirements for the MajorMajors should present to the department, through their advisor and prior to the start of the senior year, a written statement on how their major program will meet expectations that go beyond the accumulation of units of credit, as follows: 1. Mathematics is a vital component in the methods used by other disciplines. Therefore, majors are expected to present a program of study that includes courses (at least 1 unit) that use mathematics in significant ways. While many such courses may be found in the natural sciences, suitable courses may also be found in other disciplines, such as economics. 2. Majors are expected to attain a depth of study within mathematics, as well as breadth. Therefore majors are expected to present a program of study that will fulfill these expectations. Ordinarily, depth of study results from election of a two-course sequence that systematically progresses into one of the branches of mathematics, algebra, analysis, or geometry, broadly conceived. A concentration in statistics within the mathematics major will automatically meet the expectation of depth of study. Senior ExerciseThe Senior Exercise begins promptly in the fall of the senior year with independent study on a topic of interest to the student and approved by the department. The independent study culminates in the writing of a paper which is due in November. (Juniors are encouraged to begin thinking about possible topics before they leave for the summer.) Students are also required to take the Major Field Test in Mathematics produced by the Educational Testing Service. Evaluation of the Senior Exercise is based on the student’s performance on the paper and the standardized exam. A detailed guide on the Senior Exercise is available from the department office or on the math department website, accessible via Kenyon’s site, www.kenyon.edu. Suggestions for Majoring in MathematicsStudents wishing to keep open the option of a major in mathematics typically begin with the study of calculus in their first year and normally complete the calculus sequence, MATH 222 (Foundations of Analysis) and either MATH 118 or MATH 106 by the end of the sophomore year. A major is usually declared no later than the second semester of the sophomore year. Those considering a mathematics major should consult with a member of the mathematics department in order to plan their course of study. The requirements for the major are minimal. Anyone who is planning a career in the mathematical sciences, or who intends to read for honors, is encouraged to consult with one or more members of the department concerning further studies that would be appropriate. Similarly, any student who wishes to propose a variation of the major program is encouraged to discuss the plan with a member of the department prior to submitting a written proposal for a decision by the department. Students who are interested in teaching mathematics at the high-school level should take MATH 230 (Geometry), since this course is required for certification in most states, including Ohio. Honors in MathematicsThe Honors Program in mathematics requires three semesters of honors work: the Junior Honors Seminar in the spring of the junior year and two semesters of Senior Honors. The purpose of the Junior Honors Seminar is to allow honors students to explore widely so as to broaden their mathematical horizons and at the same time decide on a topic (or topics) on which to concentrate during their senior year. Students must have the consent of the department to undertake honors work. To be considered for the Honors Program, students must have an excellent academic record both in their mathematics courses and overall, and they must show promise for continued in-depth study of mathematics. Requirements for the MinorsThere are two minors in mathematics. Each minor deals with core material of a part of the discipline, and each reflects the logically structured nature of mathematics through a pattern of prerequisites. A minor consists of satisfactory completion of the courses indicated. MathematicsThe calculus sequence MATH 111, 112, 213, and four courses from the following: MATH 105, 106, 107, 108, 118, 222, 224, 227, 230, 232, 324, 333, 335, 336, 237, 341, 347, 352, 435, 460, 461. (Students may count at most one of the following: MATH 105, 106, 107, 108, and 118.) StatisticsFive courses in statistics from the following: MATH 106, 206 216, 226, 316, 336, 436. (Students may count at most one statistics course from another department. For example, ECON 375or PSYC 200 may be substituted for one of the courses listed above.) Our goal is to provide a solid introduction to basic statistical methods, including data analysis, design and analysis of experiments, statistical inference, and statistical models, using professional software such as Minitab, SAS, and Maple. Deviations from the list of approved courses must be ratified by the mathematics department. Students considering a minor in mathematics are urged to speak with a member of the department about the selection of courses and the ways in which the major discipline and a proposed mathematics minor are related. First-Semester CoursesElements of Statistics This is a basic course in statistics. The topics to be covered are the nature of statistical reasoning, graphical and descriptive statistical methods, design of experiments, sampling methods, probability, probability distributions, sampling distributions, estimation, and statistical inference. Confidence intervals and hypothesis tests for means and proportions will be studied in the one- and two-sample settings. Minitab, a statistical software package, will be used, and students will be engaged in a wide variety of hands-on projects. Enrollment limited. Pre-calculus This course prepares students for the study of calculus. It is particularly directed to those planning to enter the calculus sequence that begins with MATH 111. Primary emphasis is placed on the study of real valued functions, particularly polynomial, rational, logarithmic, exponential, trigonometric, and inverse trigonometric functions. Conceptual understanding will be emphasized. Computer labs that use graphing programs and a computer algebra system will be employed. Students with 1/2 unit of credit for calculus may not receive credit for MATH 110. Enrollment limited. Calculus A The first in a three-semester calculus sequence, this course covers the basic ideas of differential calculus. Differential calculus is concerned primarily with the fundamental problem of determining instantaneous rates of change. In this course we will study instantaneous rates of change from both a qualitative geometric and a quantitative analytic perspective. We will cover in detail the underlying theory, techniques, and applications of the derivative. The problem of anti-differentiation, identifying quantities given their rates of change, will also be introduced. The course will conclude by relating the process of anti-differentiation to the problem of finding the area beneath curves, thus providing an intuitive link between differential calculus and integral calculus. Those who have had a year of high-school calculus but do not have advanced placement credit for MATH 111 should take the Calculus Placement Exam to determine whether they are ready for MATH 112. Students who have 1/2 unit of credit for calculus may not receive credit for MATH 111. Prerequisites: solid grounding in algebra, trigonometry, and elementary functions. Enrollment limited. Calculus B The second in a three-semester calculus sequence, this course is concerned primarily with the basic ideas of integral calculus and the Riemann sums that serve as its foundation. We will cover in detail the ideas of integral calculus, including integration and the fundamental theorem, techniques of integration, numerical methods, and applications of integration. Analysis of differential equations by separation of variables, Euler’s method, and slope fields will be a part of the course, as will the ideas of convergence related to improper integrals, sequences, series and Taylor Series. Prerequisite: MATH 111 or permission of the instructor. Enrollment limited. Special Topic: Elementary Modeling of “Real-World” Processes The purpose of this course is to provide a significant modeling experience to serve as a bridge between the study of various fields and the mathematical relations that govern them. We will investigate meaningful problems that arise in a number of academic disciplines including the life sciences, physical sciences, social sciences, economics, and management. This is not a course in solving word problems with a particular path to the correct solution. Instead, it is course in which you will be given world problems in which the ideas of solutions are relative to assumptions and are particularly not unique. Indeed, the arguments that serve to define the problem will be just as important as the methods used to “solve” the problem. We wish to pay special attention to the many steps involved in problem solving, including problem identification, model identification or selection, identification and collection of data, model validation, formulation of solutions, and finally implementation and maintenance. Prerequisite: MATH 110 or permission of the instructor. Enrollment limited. An Introduction to Computer Science This course presents an introduction to computer science intended for those planning to take additional courses in computing, for those with a strong foundation in mathematics, and for those intending to major in science or mathematics or one of the social sciences where a strong background in computation is desirable. This course will expose the student to a variety of applications where an algorithmic approach is natural and will include both numerical and non-numerical computation. The principles of structured programming will be emphasized. Enrollment limited. Calculus C The third in a three-semester calculus sequence, this course examines differentiation and integration in three dimensions. Topics of study include functions of more than one variable, vectors and vector algebra, partial derivatives, optimization, and multiple integrals. Some of the following topics from vector calculus will also be covered as time permits: vector fields, line integrals, flux integrals, curl, and divergence. Prerequisite: MATH 112 or permission of the instructor. Linear Algebra I Linear algebra grew out of the study of the problem of organizing and solving systems of equations. Today, ideas from linear algebra are highly useful in most areas of higher-level mathematics. Moreover, there are numerous uses of linear algebra in other disciplines, including computer science, physics, chemistry, biology, and economics. This course involves the study of vector spaces, an appealing geometric way of formulating many of the most important ideas in the subject. Two familiar vector spaces from calculus are the plane and 3-space. In addition, students in MATH 224 examine matrices, which may be thought of as functions between vector spaces. In the past, linear algebra involved tedious calculations. Now we have computers to do this work for us, allowing us to spend more time on concepts and intuition. A computer algebra system such as Maple will likely be used. Prerequisite: MATH 112 (12) or permission of instructor. Design and Analysis of
Experiments This course will focus on standard methods of designing and analyzing experiments. Simple comparative designs, factorial designs, block designs, and appropriate post-hoc comparisons will be discussed. These techniques are commonly used by statisticians and experimental scientists in a wide variety of fields. Statistical software will be introduced and heavily used throughout the course. No prior experience with the software is necessary. Each student will be asked to design an experiment, conduct the experiment, and collect and analyze the appropriate data. Prerequisite: MATH 106 (6) or permission of instructor. Enrollment limited. Special Topic: Introduction to Bayesian Statistics Bayesian methods provide a framework for incorporating prior information in statistical inferences, together with information derived through experimentation or observation. The field is so named because its methodologies rely on Bayes’ Rule, a basic law of probability studied in many introductory probability courses. The course will begin with a review of the basic concepts of probability, which are critical to understanding the foundations of the subject. Some issues concerning experimental design, sampling, and the scientific method will be discussed. We will move on to study inferential methods, including proportion models for one and two populations and models for population means. As time permits, we may cover some of following topics: models for two population means, regression methods, and various density models. These problems will be presented within a Bayesian framework. We will also compare and contrast Bayesian methods with classical methods. Prerequisite: MATH106 or permission of the instructor. Enrollment limited. Special Topic: Number Theory Seminar Patterns within the set of natural numbers have enticed mathematicians for well over two millennia, making number theory one of the oldest branches of mathematics. Still, numerous number theoretic problems remain open to this day, and many of these problems continue to entice the mathematical masses. In this course we will explore some of the classical problems in number theory, focusing primarily on open problems and partial results related to these problems. Our approach will be different from that of most upper-level math courses. Rather than working through a textbook, our study will be driven by the unknown. We will work through a set of (carefully chosen) mathematical papers. Students in this seminar will be responsible for reading and presenting these papers to one another. Along the way, we will cover much of the material typically included in a standard number theory course: divisibility, primes and their distribution, congruences, number theoretic functions, and more. Prerequisite: MATH 222 or permission of the instructor. Enrollment limited. Probability This course provides a mathematical introduction to probability. Topics include basic probability theory, random variables, discrete and continuous distributions, mathematical expectation, functions of random variables, and asymptotic theory. Prerequisite: MATH 213. Real Analysis I This course is a first introduction to Real Analysis. “Real” refers to the real numbers. Much of our work will revolve around the real number system. We will start by carefully considering the axioms that describe it. Students will be asked to consider many functions that take on real values—that is, each object in our domain will be associated with a real number. For instance, every point in the plane can be associated with its distance from the origin. Two points in the plane give rise to a real number: the distance between them. The concept of distance will be a major theme of the course. “Analysis” is one of the principle branches of mathematics. One often hears that analysis is the theoretical underpinnings of the calculus, but though this has a kernel of truth, it is an answer that misleads by oversimplifying. Certainly, analysis had its inception in the attempt to give a careful, mathematically sound explanation of the ideas of the calculus. But over the last century, analysis has grown out of its original packaging and is now much more than simply the theory of the calculus. Analysis is the mathematics of “closeness”—the mathematics of limiting processes. The idea of continuity can be phrased in terms of limits. Both derivatives and integrals are the end results of taking a limit. Compactness is a property of sets that underlies many of the most important theorems encountered in calculus. These and related ideas will be the subject of the course. Prerequisites: MATH 213 and MATH 222. Topology Topology is a relatively new branch of geometry that studies very general properties of geometric objects, how these objects can be modified, and the relations between them. Three key concepts in topology are compactness, connectedness, and continuity, and the mathematics associated with these concepts is the focus of the course. Compactness is a general idea helping us to more fully understand the concept of limit, whether of numbers, functions, or even geometric objects. For example, the fact that a closed interval (or square, or cube, or n-dimensional ball) is compact is required for basic theorems of calculus. Connectedness is a concept generalizing the intuitive idea that an object is in one piece: the most famous of all the fractals, the Mandelbrot Set, is connected, even though its best computer- graphics representation might make this seem doubtful. Continuous functions are studied in calculus, and the general concept can be thought of as a way by which functions permit us to compare properties of different spaces or as a way of modifying one space so that it has the shape or properties of another. Economics, chemistry, and physics are among the subjects that find topology useful. The course will touch on selected topics that are used in applications. Prerequisite: permission of instructor. Individual Study This course enables students to study a topic of special interest under the direction of a member of the mathematics department. Prerequisites: permission of instructor and department chair. Senior Honors The content of this course is variable and adapted to the needs of senior candidates for honors in mathematics. Prerequisite: permission of department. Second-Semester CoursesElements of Statistics See first-semester course description. Calculus A See first-semester course description. Calculus B See first-semester course description. An Introduction to Computer Science See first-semester course description. Data Analysis This course follows MATH 106 and focuses on (1) additional topics in statistics, including linear regression, nonparametric methods, discrete data analysis, and analysis of variance; (2) efficient use of statistical software in data analysis and statistical inference; and (3) writing and presenting statistical reports, including graphics. The MATH 106-206 sequence provides a foundation for statistical work in applied fields such as econometrics, psychology, and biology. It also serves as preparation for study of theoretical probability and statistics. Prerequisite: MATH 106. Calculus C See first-semester course description. Data Structures and Program Design This course is intended as a second course in programming, as well as an introduction to the concept of computational complexity and the major abstract data structures (such as arrays, stacks, queues, link lists, graphs, and trees), their implementation and application, and the role they play in the design of efficient algorithms. Students will be required to write several programs using a high-level language. Prerequisite: MATH 118. Foundations This course introduces students to mathematical reasoning and rigor in the context of set-theoretic questions. The course will cover basic logic and set theory, relations—including orderings, functions, and equivalence relations—and the fundamental aspects of cardinality. Emphasis will be placed on helping students in reading, writing, and understanding mathematical reasoning. Students will be actively engaged in creative work in mathematics. The course should be taken no later than the spring semester of the sophomore year. Advanced first-year students interested in mathematics are encouraged to consider taking this course in their first year. (Please see a member of the mathematics faculty if you think you might want to do this.) Prerequisite: MATH 213 or permission of instructor. Linear Regression Models This course will focus on linear regression models. Simple linear regression with one predictor variable will serve as the starting point. Models, inferences, diagnostics, and remedial measures for dealing with invalid assumptions will be examined. The matrix approach to simple linear regression will be presented and used to develop more general multiple regression models. Building and evaluating models for real data will be the ultimate goal of the course. Time series models, nonlinear regression models, and logistic regression models may also be studied if time permits. Prerequisites: MATH 106 and MATH 224 or permission of instructor. Differential Equations Differential equations arise naturally to model dynamical systems such as occur in physics, biology, and economics, and have given major impetus to other fields in mathematics, such as topology and the theory of chaos. This course covers basic analytic, numerical, and qualitative methods for the solution and understanding of ordinary differential equations. Computer-based technology will be used. Prerequisite or co-requisite: MATH 213. Abstract Algebra I The phrase “abstract algebra” correctly suggests some sort of a generalization of a topic most of us learned in high school, though it goes very much beyond that, of course. Three of the most important structures in abstract algebra are groups, rings, and fields; all three are, in fact, abstractions of familiar objects—the integers form a group or ring, while the real numbers give us an example of a field. Each of these structures has the property that any two of the subjects in the system may be “combined” in some way to produce a new object in the system. In the system of integers, for example, this “combining” might be addition or multiplication. Groups and rings are fundamental tools for any mathematician and many scientists, but these concepts are beautiful and worthy of study in their own right—group theory and ring theory currently are both very active areas of mathematical research. In this course, the student examines the basics of groups and rings, with emphasis on the many examples of these algebraic structures. A possible example might be a study of symmetry with the aid of group theory. Prerequisite: MATH 222 or permission of the instructor. Mathematical Models This course introduces students to the concepts, techniques, and power of mathematical modeling. Both deterministic and probabilistic models will be explored, with examples taken from the social, physical, and life sciences. Students engage cooperatively and individually in the formulation of mathematical models and in learning mathematical techniques used to investigate those models. Prerequisites: MATH 106 and MATH 112 or permission of instructor. Junior Honors The goal of the Junior Honors Seminar is twofold: to develop a greater understanding of a broad selection of mathematical topics, and to gain the experience of independent exploration in mathematics. Students will work under the close supervision of a faculty member on three areas of interest. Topics of study will be chosen by the student. As a culmination of the course, each student will write a proposal describing his or her plan of study for senior honors. Prerequisite: permission of department. Real Analysis II This is an analysis course with variable content, depending on the needs and interests of the students. Prerequisite: MATH 341. Individual Study This course enables students to study a topic of special interest under the direction of a member of the department. Prerequisites: permission of instructor and department chair. Senior Honors The content of this course is variable and adapted to the needs of senior candidates for honors in mathematics. Prerequisite: permission of department. Additional courses available another year include the following:Surprises at Infinity Our intuitions about sets, numbers, shapes, and logic all break down in the realm of the infinite. The paradoxical facts about infinity are the subject of this course. We will discuss what infinity is, how it has been viewed through history, why some infinities are bigger than others, how a finite shape can have an infinite perimeter, and why some mathematical statements can be neither proved nor disproved. This will very likely be quite different from any mathematics course you have ever taken. Surprises at Infinity focuses on ideas and reasoning rather than algebraic manipulation; a calculator will be entirely useless. The class will be a mixture of lecture and discussion, based on selected readings. You can expect essay tests and frequent writing assignments. No prerequisites. Introduction to Number Theory Part of the appeal of number theory, the study of the properties of the system of whole numbers, is the lure of the unknown: even a beginner can understand problems that the greatest mathematicians in history have been unable to solve. In this course, we will probably not solve them either, but we will learn what they are. We will also learn about such topics as primes and prime factorization, perfect numbers, arithmetic modulon, Diophantine equations, “Fermat’s Last Theorem,” and possibly continued fractions or quadratic number fields. The only prerequisites are a good understanding of high-school algebra and an interest in learning mathematics for its own sake. Prospective majors and students who plan to take only one or two math courses in college are equally welcome. Enrollent limited to first- and second-year students. Modeling Biological Growth and Form This course will explore various areas of mathematics involved in modeling the growth and form of biological organisms and populations. In particular, we will ask such questions as: How do you model the growth of a population of animals? How can you model the growth of a tree? How do sunflowers and seashells grow? How do mathematicians quantify symmetry? The course will be a “hands-on” course and will make extensive use of the graphical capabilities of the computer software package Maple. The course will not involve significant amounts of symbolic manipulation. Rather, assignments will usually involve readings, papers, and computer projects. The course will rely on ideas from a wide range of mathematical fields, including geometry, linear algebra, mathematical modeling, and computer graphics. Prerequisites: Precalculus or permission of the instructor. Enrollment limited. Nonparametric Statistics This course will focus on nonparametric and distribution-free statistical procedures. These procedures will rely heavily on counting and ranking techniques. In the one and two sample settings, the sign, signed-rank, and Mann-Whitney-Wilcoxon procedures will be discussed. Correlation and one-way analysis of variance techniques will also be investigated. A variety of special topics will be used to wrap up the course, including bootstrapping, censored data, contingency tables, and the two-way layout. The primary emphasis will be on data analysis and the intuitive nature of nonparametric statistics. Illustrations will be from real data sets, and students will be asked to locate an interesting data set and prepare a report detailing an appropriate nonparametric analysis. Prerequisites: MATH 106 or permission of instructor. Methods of Discrete
Mathematics Discrete mathematics is, broadly speaking, the study of finite sets and finite mathematical structures. A great many mathematical topics are included in this description, including graph theory, combinatorial designs, partially ordered sets, networks, lattices and Boolean algebra, and combinatorial methods of counting, including combinations and permutations, partitions, generating functions, the principle of inclusion and exclusion, and the Stirling and Catalan numbers. This course will cover a selection of these topics. Discrete mathematics has applications in a wide variety of non-mathematical areas, including computer science (both in algorithms and hardware design), chemistry, sociology, government, and urban planning, and this course may be especially appropriate for students interested in the mathematics related to one of these fields. Prerequisite: Math 107 or Math 222 or permission of instructor. Euclidean and Non-Euclidean Geometry The Elements of Euclid, written over two thousand years ago, is a stunning achievement. The Elements and the non-Euclidean geometries discovered by Bolyai and Lobachevsky in the nineteenth century formed the basis of modern geometry. From this start, our view of what constitutes geometry has grown considerably. This is due in part to many new theorems that have been proved in Euclidean and non-Euclidean geometry but also to the many ways in which geometry and other branches of mathematics have come to influence one another over time. Geometric ideas have widespread use in analysis, linear algebra, differential equations, topology, graph theory, and computer science, to name just a few areas. These fields, in turn, affect the way that geometers think about their subject. Students in MATH 230 will consider Euclidean geometry from an advanced standpoint, but will also have the opportunity to learn about several non-Euclidean geometries such as (possibly) the Poincare plane, geometries relevant to special relativity, or the geometries of Bolyai and Lobachevsky. In addition, the course may take up topics in differential geometry, topology, vector space geometry, mechanics, or other areas, depending on the interests of the students and the instructor. Prerequisite: MATH 222 or permission of instructor. Vector Analysis Physical and natural phenomena depend on a complex array of factors, and to analyze these factors requires the understanding of geometry in two and three (or more) dimensions. This course will continue the study of multivariable calculus begun in MATH 213. Topics of study will include vector fields, line and surface integrals, potential functions, classical vector analysis, and Fourier Series. Computer labs will be incorporated throughout the course, and physical applications will be plentiful. Prerequisite: MATH 213. Numerical Analysis This course presents a study of the major topics of classical numerical analysis. These include the solution of nonlinear equations, interpolation and approximation, numerical integration, matrices and systems of linear equations, and the solution of differential equations. The course requires extensive use of the computer. Prerequisites: MATH 118 and MATH 213 or permission of department chair. Dynamical Systems The theory of dynamical systems is the study of the behavior of physical or mathematical systems that change over time according to specific rules. Dynamical systems have applications to many areas of science and social science-research, including models of population growth and decline, interspecies relationships, traffic-flow problems, battles, river meanders, weather patterns, heartbeat rates, chemical reactions, and financial markets. In this course we will study both discrete and continuous time models, presenting the two approaches in a unified manner. Upon completion of the course, students should comprehend the basic concepts and recent developments in the field of dynamical systems, including the stability theory of equilibria and the theory of transitions to chaos. Students will develop the ability to analyze simple nonlinear discrete and continuous dynamical systems and to chart parameter regions of stability, periodicity, and chaos. Further, students will gain an appreciation for the power as well as the limitations of dynamical systems theory and chaos when applied to realistic systems such as ecologies and financial markets. Rather than taking a formal theorem-proof style, the course will be taught in a manner that stresses the geometry, intuition, and appreciation of dynamical systems. Computer technology will be used extensively to perform simulations and experiments. Prerequisite: MATH 111. Co-requisite: MATH 112. Linear Algebra II This course deepens the studies begun in MATH 224. Topics will vary depending on the needs and interests of the students. However, the topics are likely to include some of the following: abstract vector spaces, linear mappings and canonical forms, linear models and eigen vector analysis, inner product spaces. Prerequisite: MATH 224. Complex Functions QR MATH 352 (1/2 unit) The course starts with an introduction to the complex numbers and the complex plane. Next students are asked to consider what it might mean to say that a complex function is differentiable (or analytic, as it is called in this context). For a complex function that takes a complex number z to f(z), it is easy to write down (and make sense of) the statement that f is analytic at z if
Abstract Algebra II This course picks up where MATH 335 ends. In Math 435, however, the focus is on using the tools considered in Abstract Algebra I. Mathematicians and scientists apply the fundamental algebraic notions of group, ring, and field to a wide variety of mathematical areas and scientific disciplines; in MATH 435, the student explores these applications. The structure will be that of a topics course, the focus being on classical problems that can be solved (and historically were solved) using algebraic structures as tools. Topics that may be considered include insolvability of a quintic polynomial, the factoring of polynomials (just as in high school, but over arbitrary rings rather than the real numbers), the classification of finite simple groups (something proven very recently), special cases of “Fermat’s Last Theorem,” Eisenstein’s criterion for irreducibility, the beautiful subject of Galois theory, and more. The class may borrow knowledge from subjects including linear algebra, number theory, complex numbers, calculus, and computer programming, though all one needs to know about these subjects will be covered in class. Prerequisite: MATH 335. Mathematical Statistics QR MATH 436 (1/2 unit) This course follows MATH 336 and introduces the mathematical theory of statistics. Topics include sampling distributions, point estimation, maximum likelihood estimation, methods for comparing estimators, interval estimation, likelihood ratio tests, and hypothesis testing. These methods will be applied to real data sets. Prerequisite: Math 336.
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symbol, which designates those courses particularly appropriate for first-year students or upperclass students new to the mathematics curriculum. To facilitate proper placement of entering students, the department administers a calculus readiness exam and a calculus placement exam during Orientation. This and other entrance information is used during the orientation period to give students advice about course selection in mathematics. We encourage all students who do not have advanced placement credit to take the placement exam that is appropriate for them.
