Requirements: Mathematics

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Natural Sciences Division

For more than 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 through 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

and statistics have 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, statistics 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.

Mathematics and statistics are engaging fields, rich in beauty, with powerful applications to

other subjects. Thus we strive to ensure that Kenyon students encounter and learn to solve

problems using a number of contrasting but complementary mathematical perspectives:

continuous and discrete, algebraic and geometric, deterministic and stochastic, theoretical

and applied. In our courses we stress mathematical and statistical thinking and

communication skills. And in courses where it makes sense to incorporate technological

tools, our students learn to solve problems using computer algebra systems, statistical

packages and computer programming languages.

New Students

Those students interested only in an introduction to mathematics or statistics or a course to satisfy

a distribution requirement may select from MATH 105, 111, 128, STAT 106, 116 and SCMP 118.

Students wanting to continue the study of mathematics beyond one year, either by pursuing a

major or minor in mathematics or 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 STAT 106 and 206 or SCMP 118. A few well-prepared students may take

MATH 222 or 224 in their first year. Please see the department chair for further information.

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. STAT 106 is an

introduction to statistics, which focuses on quantitative reasoning skills and the analysis of data.

SCMP 118 introduces students to computer programming.

To facilitate proper placement of students in calculus courses, the department offers placement

tests that help students decide which level of calculus course is appropriate for them. 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. Students who have Advanced

Placement credit for STAT 106 should consider enrolling in STAT 206 or 216.

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The ready availability of powerful computers has made the computer one of the primary tools of

the mathematician and absolutely indispensable for the statistician. Students will be expected to

use appropriate computer software in many of the mathematics and statistics courses. However, no

prior experience with the software packages or programming is expected, except in advanced

courses that presuppose earlier courses in which use of the software or programming was taught.

Students Graduating in 2020-2022

Use the major requirements found in the archived course catalog.

REQUIREMENTS FOR THE MAJORS

There are three different areas of emphasis within the mathematics major: classical mathematics,

applied mathematics and statistics. Regardless of one's concentration, all math majors are required

to complete the same eight core courses.

Core Requirements

A student must have credit for the following core courses:

Three semesters of calculus (MATH 111, 112, 213 or the equivalent)

 One semester of statistics (STAT 106 or the equivalent)

 One semester of computer programming (SCMP 118, MATH 138 or PHYS 270)

 MATH 222 Foundations

 MATH 224 Linear Algebra

 MATH 480 Senior Seminar in Mathematics

Beyond the core requirements, there are three other types of requirements: the "area of focus"

requirement, the "depth" requirement and the "breadth" requirement. It is the "area of focus"

requirement that determines a student's emphasis within the math major.

Area of Focus Requirement

Every math major is required to take (at least) three courses from a single column in the table given

below. Additionally, at least one of those courses must be at the 300 level. (Note: special topics

courses may also count toward a major's area of focus, even though they are not listed in the table;

the department chair will sign off on such courses when appropriate.)

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Category I

Category II

A. Algebraic

B. Continuous/

analytic

C. Discrete/

combinatorial

D. Computation/

modeling/applied

E. Statistical/

data science

MATH 335

Abstract Algebra I

MATH 341

Real Analysis I

MATH 336

Probability

MATH 347

Mathematical

Modeling

STAT 206

Data Analysis

MATH 435

Abstract Algebra

MATH 441

Real Analysis II

MATH 236

Random Structures

Math 358

Mathematical Biology

STAT 436

Mathematical

Stats

MATH 327

Number Theory

MATH 360

Topology

Math 328

Coding Theory

SCMP 218

Data Structures

STAT 416

Linear

Regression

MATH 328

Coding Theory

MATH 230

Geometry

MATH 327

Number Theory

MATH 333 Applied

Differential Equations

STAT 216

Nonparametrics

Math 322

Mathematical

Logic

MATH 352

Complex

Functions

MATH 227

Combinatorics

MATH 324

Applied Linear

Algebra

MATH 336

Probability

MATH 368

Design and Analysis

of Algorithms

Additionally, (only) one of the following courses offered outside of the Department of Mathematics

and Statistics may be counted towards Column D:

 ECON 357 Economics with Calculus

 ECON 375 Advanced Econometrics

 PHYS 340 Classical Mechanics

 PHYS 350 Electricity and Magnetism

 PHYS 360 Quantum Mechanics

The math major's choice of column determines both the area of emphasis and the area of focus

within the mathematics major.

1. Classical Mathematics

To earn a math major with an emphasis in classical mathematics, the student must choose an area

of focus residing within Category 1 in the above table. So, for example, a math major taking three

courses from the first column would be a math major with an emphasis in classical mathematics

and a focus on algebra.

2. Applied Mathematics

To earn a math major with an emphasis in applied mathematics, the student must take three

courses residing in column D. Applied mathematics will also be the area of focus for this student.

3. Statistics

To earn a math major with an emphasis in statistics, the student must take three courses from

column E. Statistics will also be the area of focus for this student.

Depth Requirement

Majors are expected to attain a depth of study within mathematics. To this end, every major must

take at least two courses at or above the 300 level. At least one of these 300- or 400-level courses

must reside within the major's area of focus.

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Breadth Requirement

Majors are also expected to attain a breadth of knowledge spanning pure and applied mathematics

and statistics. Hence every major must take at least two different columns that are not the area of

focus. (These courses must not also be listed within the area of focus.) Additionally, every major

must take at least one course from Category I and one course from Category II.

For instance, a student that is pursuing a mathematics major with an emphasis in classical

mathematics and a continuous/analytic focus must choose a course from each of two columns

besides column B, and at least one of these columns must reside in Category II. Neither of these two

additional courses can be Probability (MATH 336) because the course resides in the student's area

of focus.

To summarize, a student earning a major in mathematics will take (or have credit for) at least 13

courses: eight core courses (including the Senior Seminar), three courses in an area of focus and

two additional courses outside the area of focus and spanning Categories I and II. Students with IB

or AP credit can place out of some of the introductory core coursework, and this will decrease the

number of required courses to a number less than 13.

SENIOR CAPSTONE

The "Senior Capstone" 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. While seniors will be studying their topics

individually, all must be enrolled in the Senior Seminar during this fall semester as it will provide

the structure and a timeline for completing the paper. Juniors are encouraged to begin thinking

about possible topics before they leave for the summer. Students are required to take the Major

Field Test in Mathematics produced by the Educational Testing Service. Evaluation of the "Senior

Capstone" is based on the student's performance on the paper and the standardized exam. Detailed

information on the "Senior Capstone" is available on the mathematics department website.

SUGGESTIONS FOR MAJORING IN MATHEMATICS

Students wishing to keep open the option of a major in mathematics and statistics typically begin

with the study of calculus and normally complete the calculus sequence, MATH 222 and either

SCMP 118 or STAT 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 and statistics major

should consult with a member of the mathematics and statistics department 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 and 335, since these courses are required for certification in most states, including Ohio.

HONORS IN MATHEMATICS

To be eligible to enroll in the "Mathematics Honors Seminar" by the end of junior year, students

must have completed the following:

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 One of the following year-long sequences: MATH 335/435, MATH 336/STAT 416, MATH

336/STAT 436 or MATH 341/441

 Have earned an overall Kenyon GPA of at least 3.33

 A GPA in Kenyon mathematics and statistics courses of at least 3.6

 The student also must have, in the estimation of the mathematics and statistics faculty, a

reasonable expectation of fulfilling the requirements to earn honors (listed below)

To earn honors in mathematics, a student must:

 Complete two of the year-long sequences: MATH 335/435, MATH 336/STAT 416, MATH

336/STAT 436 or MATH 341/441

 Complete at least six, half (0.5) unit courses in mathematics and statistics numbered 300 or

above

 Pass the "Senior Capstone" in the fall semester

 Pass the Mathematics Honors Seminar MATH 498 or the Statistics Honors Seminar STAT

498

 Present the results of independent work in MATH 498 or STAT 498 to a committee

consisting of an outside examiner and members of the mathematics and statistics

department

 Successfully complete an examination written by an outside examiner covering material

from MATH 498 and previous mathematics or statistics courses

 Maintain an overall Kenyon GPA of at least 3.33

 Maintain a GPA in mathematics courses of at least 3.6

Based on performance in all of the above-mentioned areas, the department (in consultation with

the outside examiner) can elect to award Honors, High Honors, or Highest Honors, or not to award

honors at all.

REQUIREMENTS FOR THE MINOR

There are two minors in mathematics and statistics. Each minor deals with core material of a part

of the discipline and each reflects the logically structured nature of the subject through a pattern of

prerequisites. A minor consists of satisfactory completion of the following courses:

Mathematics

The calculus sequence MATH 111, 112, 213 or the equivalent

Four other courses offered by the Department of Mathematics and Statistics. SCMP 118 and/or

SCMP 218 may also be used toward this four-course requirement. Of these four other courses,

students may count at most one at the 100 level.

Statistics

 STAT 106 or an equivalent introductory statistics course

 STAT 206

Three courses from the following:

 STAT 216

 MATH 236

 MATH 258

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 MATH 336

 STAT 416

 STAT 436

Students may count at most one statistics course from another department. ECON 205 or 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, Maple and R.

Deviations from the list of approved minor courses must be approved by the mathematics

department. Students considering a minor in mathematics or statistics are urged to speak with a

member of the department about the selection of courses.

TRANSFER CREDIT

Transfer credit from other institutions, and the applicability of this credit to the major or minor,

must be approved by the department chair.

CROSS LISTED COURSES

The following course is cross-listed in biology and will satisfy the natural science requirement:

MATH 258 Mathematical Biology

Courses in Mathematics

MATH 100 First-Year Seminar in Mathematics

Credit: 0.25

The first-year seminar in mathematics provides an introduction to the rich and diverse nature of

mathematics. Topics covered will vary from one semester to the next (depending on faculty

expertise) but will typically span algebra and number theory, dynamical systems, probability and

statistics, discrete mathematics, topology, geometry, logic, analysis and applied math. The course

includes guest lectures from professors at Kenyon, a panel discussion with upper-class math majors

and opportunities to learn about summer experiences and careers in mathematics. The course goals

are threefold: 1) to provide an overview of modern mathematics, which, while not exhaustive, will

expose students to some exciting open questions and research problems in mathematics; 2) to

introduce students to some of the mathematical research being done at Kenyon and; 3) to answer

whatever questions students might have during their first semester here, while exposing them to

useful resources and opportunities that are helpful in launching a meaningful college experience.

Open to first-year students. Prerequisite or corequisite: MATH 112 (or equivalent) and

concurrently enrolled in another MATH, STAT or SCMP course or permission of instructor. Offered

every fall semester.

MATH 105 SURPRISES AT INFINITY

Credit: 0.5 QR

Our intuitions about sets, numbers, shapes and logic all break down in the realm of the infinite.

Seemingly 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 and

how a finite shape can have an infinite perimeter. This very likely will be quite different from any

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mathematics course you have ever taken. This course focuses on ideas and reasoning rather than

algebraic manipulation, though some algebraic work will be required to clarify big ideas. The class

will be a mixture of lecture and discussion, based on selected readings. Students can expect essay

tests, frequent homework and writing assignments. No prerequisite. Generally offered every other

year.

MATH 111 CALCULUS I

Credit: 0.5 QR

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, also will 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 0.5 units of credit for calculus may not receive credit for MATH

111. Prerequisite: solid grounding in algebra, trigonometry and elementary functions. Offered

every semester.

MATH 112 CALCULUS II

Credit: 0.5 QR

The second in a three-semester calculus sequence, this course has two primary foci. The first is

integration, including techniques of integration, numerical methods and applications of integration.

This study leads into the analysis of differential equations by separation of variables, Euler's

method and slope fields. The second focus is the notion of convergence, as manifested in improper

integrals, sequences and series, particularly Taylor series. Prerequisite: MATH 111 or AP score of 4

or 5 on Calculus AB exam or an AB subscore of 4 or 5 on the Calculus BC exam or permission of

instructor. Offered every semester.

MATH 128 HISTORY OF MATHEMATICS IN THE ISLAMIC WORLD

Credit: 0.5 QR

This course examines an important and interesting part of the history of mathematics and, more

generally, the intellectual history of humankind: the history of mathematics in the Islamic world.

Some of the most fundamental notions in modern mathematics have their roots here, such as the

modern number system, the fields of algebra and trigonometry, and the concept of algorithm,

among others. In addition to studying specific contributions of medieval Muslim mathematicians in

the areas of arithmetic, algebra, geometry and trigonometry in some detail, we will examine the

context in which Islamic science and mathematics arose, and the role of religion in this

development. The rise of Islamic science and its interactions with other cultures (e.g., Greek, Indian

and Renaissance Europe) tell us much about larger issues in the humanities. Thus, this course has

both a substantial mathematical component (60-65 percent) and a significant history and social

science component (35-40 percent), bringing together three disciplines: mathematics, history and

religion. The course counts toward the Islamic Civilization and Cultures Concentration.

Prerequisite: solid knowledge in algebra and geometry.

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MATH 138 INTRODUCTION TO COMPUTER SCIENCE

Credit: 0.75 QR

This course is an introduction to the intellectual scope of computer science and to the art of

computer programming. This entry-level course is for students of all majors, including those with

and without previous programming experience. We teach the Python programming language to

introduce programming concepts. The course covers topics in abstraction, algorithms and program

design, basic data structures, security, networking, privacy and history. Web technologies including

HTML, CSS and Javascript are examined. Offered every semester.

MATH 213 CALCULUS III

Credit: 0.5 QR

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 also will be covered as time permits: vector fields, line

integrals, flux integrals, curl and divergence. Prerequisite: MATH 112 or a score of 4 or 5 on the BC

Calculus AP exam or permission of instructor. Offered every semester.

MATH 222 FOUNDATIONS

Credit: 0.5 QR

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. The course will

emphasize helping students read, write and understand mathematical reasoning. Students will be

actively engaged in creative work in mathematics. Students interested in majoring in mathematics

should take this course no later than the spring semester of their sophomore year. Advanced first-

year students interested in mathematics are encouraged to consider taking this course in their first

year. Students wanting to do so should contact a member of the mathematics faculty. Prerequisite:

MATH 213 or permission of instructor. Offered every spring semester.

MATH 224 LINEAR ALGEBRA

Credit: 0.5 QR

This course will focus on the study of vector spaces and linear functions between vector spaces.

Ideas from linear algebra are highly useful in many areas of higher-level mathematics. Moreover,

linear algebra has many applications to both the natural and social sciences, with examples arising

often in fields such as computer science, physics, chemistry, biology and economics. In this course,

we will use a computer algebra system, such as Maple or Matlab, to investigate important concepts

and applications. Topics to be covered include methods for solving linear systems of equations,

subspaces, matrices, eigenvalues and eigenvectors, linear transformations, orthogonality and

diagonalization. Applications will be included throughout the course. Prerequisite: MATH 213.

Generally offered three out of four semesters.

MATH 227 COMBINATORICS

Credit: 0.5 QR

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Combinatorics 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 algebras and

combinatorial methods of counting, including combinations and permutations, partitions,

generating functions, recurring relations, the principle of inclusion and exclusion, and the Stirling

and Catalan numbers. This course will cover a selection of these topics. Combinatorial mathematics

has applications in a wide variety of nonmathematical areas, including computer science (both in

algorithms and in hardware design), chemistry, sociology, government and urban planning; this

course may be especially appropriate for students interested in the mathematics related to one of

these fields. Prerequisite: MATH 112 or a score or 4 or 5 on the BC Calculus AP exam or permission

of instructor. Offered every other year.

MATH 230 EUCLIDEAN AND NON-EUCLIDEAN GEOMETRY

Credit: 0.5 QR

The "Elements" of Euclid, written over 2,000 ago, is a stunning achievement. The "Elements" and

the non-Euclidean geometries discovered by Bolyai and Lobachevsky in the 19th century form 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 also will

have the opportunity to learn about non-Euclidean geometries. Prerequisite: MATH 222 or

permission of instructor. Offered occasionally.

MATH 231 MATHEMATICAL PROBLEM SOLVING

Credit: 0.25

Looking at a problem in a creative way and seeking out different methods toward solving it are

essential skills in mathematics and elsewhere. In this course, students will build their problem-

solving intuition and skills by working on challenging and fun mathematical problems. Common

problem-solving techniques in mathematics will be covered in each class meeting, followed by

collaboration and group discussions, which will be the central part of the course. The course will

culminate with the Putnam exam on the first Saturday in December. Interested students who have a

conflict with that date should contact the instructor. Prerequisite: MATH 112 or a score of 4 or 5 on

the BC Calculus exam or permission of instructor.

MATH 236 RANDOM STRUCTURES

Credit: 0.5 QR

This course will explore the theory, structure, applications and interesting consequences when

probability is introduced to mathematical objects. Some of the core topics will be random graphs,

random walks and Markov processes, as well as randomness applied to sets, permutations,

polynomials, functions, integer partitions and codes. Previous study of all of these mathematical

objects is not a prerequisite, as essential background will be covered during the course. In addition

to studying the random structures themselves, a concurrent focus of the course will be the

development of mathematical tools to analyze them, such as combinatorial concepts, indicator

variables, generating functions, discrete distributions, laws of large numbers, asymptotic theory

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and computer simulation. Prerequisite: MATH 112 or a score of 4 or 5 on the BC Calculus AP exam

or permission of instructor. Generally offered every other year.

MATH 258 MATHEMATICAL BIOLOGY

Credit: 0.5 QR

In biological sciences, mathematical models are becoming increasingly important as tools for

turning biological assumptions into quantitative predictions. In this course, students will learn how

to fashion and use these tools to explore questions ranging across the biological sciences. We will

survey a variety of dynamic modeling techniques, including both discrete and continuous

approaches. Biological applications may include population dynamics, molecular evolution,

ecosystem stability, epidemic spread, nerve impulses, sex allocation and cellular transport

processes. The course is appropriate both for math majors interested in biological applications and

for biology majors who want the mathematical tools necessary to address complex, contemporary

questions. As science is becoming an increasingly collaborative effort, biology and math majors will

be encouraged to work together on many aspects of the course. Coursework will include homework

problem-solving exercises and short computational projects. Final independent projects will

require the development and extension of an existing biological model selected from the primary

literature. This course will build on (but not be limited by) an introductory-level knowledge base in

both math and biology. Interested biology and math majors lacking a prerequisite are encouraged

to consult with the instructor. Prerequisite: STAT 106 or MATH 111 or 112 (or any math or

statistics AP credit of 4 or 5) and either BIOL 115 or 116. Offered every other year.

MATH 322 MATHEMATICAL LOGIC

Credit: 0.5

This course is a mathematical examination of the formal language most common in mathematics:

predicate calculus. We will examine various definitions of meaning and proof for this language and

will consider its strengths and inadequacies. We will develop some elementary computability

theory en route to rigorous proofs of Godel's Incompleteness Theorems. Concepts from model logic,

model theory and other advanced topics will be discussed as time permits. Prerequisite: MATH 222

or PHIL 201 or permission of instructor. Offered occasionally.

MATH 324 LINEAR ALGEBRA II

Credit: 0.5 QR

This course builds on the concepts that arise in MATH 224. Topics will vary and will likely include

some of the following: abstract vector spaces, inner product spaces, linear mappings and canonical

forms, linear models, linear codes, the singular value decomposition, wavelets. Prerequisite: MATH

224. Offered every other year.

MATH 327 NUMBER THEORY SEMINAR

Credit: 0.5 QR

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. Rich with problems

that are easy to state but fiendishly difficult to solve, the subject continues to fascinate professionals

and amateurs alike. In this course, we will get a glimpse at both the old and the new. In the first

two-thirds of the semester, we will study topics from classical number theory, focusing primarily on

divisibility, congruences, arithmetic functions, sums of squares and the distribution of primes. In

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the final weeks we will explore some of the current questions and applications of number theory.

We will study the famous RSA cryptosystem, and students will read and present some current

(carefully chosen) research papers. Prerequisite: MATH 222. Offered every other year.

MATH 328 CODING THEORY AND CRYPTOGRAPHY

Credit: 0.5 QR

Coding theory, or the theory of error-correcting codes, and cryptography are two recent

applications of algebra and discrete mathematics to information and communications systems. The

goals of this course are to introduce students to these subjects and to understand some of the basic

mathematical tools used. While coding theory is concerned with the reliability of communication,

the main problem of cryptography is the security and privacy of communication. Applications of

coding theory range from enabling the clear transmission of pictures from distant planets to quality

of sound in compact discs. Cryptography is a key technology in electronic security systems. Topics

likely to be covered include basics of block coding, encoding and decoding, linear codes, perfect

codes, cyclic codes, BCH and Reed-Solomon codes, and classical and public-key cryptography. Other

topics may be included depending on the availability of time and the background and interests of

the students. Other than some basic linear algebra, the necessary mathematical background (mostly

abstract algebra) will be covered within the course. Prerequisite: MATH 224 or permission of

instructor. Offered every other year.

MATH 333 APPLIED DIFFERENTIAL EQUATIONS

Credit: 0.5 QR

Differential equations arise naturally to model dynamical systems such as often occur in physics,

biology, chemistry 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: MATH 224 or PHYS 245 or permission of

instructor. Offered every other year.

MATH 335 ABSTRACT ALGEBRA I

Credit: 0.5 QR

Abstract algebra is the study of algebraic structures that describe common properties and patterns

exhibited by seemingly disparate mathematical objects. The phrase "abstract algebra" refers to the

fact that some of these structures are generalizations of the material from high school algebra

relating to algebraic equations and their methods of solution. In Abstract Algebra I, we focus

entirely on group theory. A group is an algebraic structure that allows one to describe symmetry in

a rigorous way. The theory has many applications in physics and chemistry. Since mathematical

objects exhibit pattern and symmetry as well, group theory is an essential tool for the

mathematician. Furthermore, group theory is the starting point in defining many other more

elaborate algebraic structures including rings, fields and vector spaces. In this course, we will cover

the basics of groups, including the classification of finitely generated abelian groups, factor groups,

the three isomorphism theorems and group actions. The course culminates in a study of Sylow

theory. Throughout the semester there will be an emphasis on examples, many of them coming

from calculus, linear algebra, discrete math and elementary number theory. There also will be a

couple of projects illustrating how a formal algebraic structure can empower one to tackle

seemingly difficult questions about concrete objects (e.g., the Rubik's cube or the card game SET).

Finally, there will be a heavy emphasis on the reading and writing of mathematical proofs. Junior

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standing is recommended. Prerequisite: MATH 222 or permission of instructor. Offered every other

fall.

MATH 336 PROBABILITY

Credit: 0.5 QR

This course provides a calculus-based 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. Offered every fall.

MATH 341 REAL ANALYSIS I

Credit: 0.5 QR

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. "Analysis" is the branch of mathematics that deals with limiting processes. Thus the

concept of distance also will be a major theme of the course. In the context of a general metric space

(a space in which we can measure distances), we will consider open and closed sets, limits of

sequences, limits of functions, continuity, completeness compactness and connectedness. Other

topics may be included if time permits. Junior standing is recommended. Prerequisite: MATH 213

and 222. Offered every other fall

MATH 347 MATHEMATICAL MODELS

Credit: 0.5 QR

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.

Prerequisite: STAT 106 and MATH 224 or 258 or permission of instructor. Offered every other year.

MATH 348 SOFTWARE AND SYSTEM DESIGN

Credit: 0.5

A study of software design project that requires planning, analysis, design, implementation, testing

and maintenance. Different methods of planning, definition, requirements analysis and cost

estimation are considered. A central component of the course is a semester long team project which

engages a team of three to five students in the analysis, design, implementation and documentation

of a significant applied project. The goal of this team project is for the students to engage with the

material as they work to solve a real-world problem. These projects are real needs of organizations

in the surrounding community (including Gambier, Knox county and, at times, beyond).

Prerequisite: MATH 138, SCMP 118, 218 or 318.

MATH 352 COMPLEX FUNCTIONS

Credit: 0.5 QR

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 exists.

Subsequently, we will study the amazing results that come from making such a seemingly innocent

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assumption. Differentiability for functions of one complex variable turns out to be a very different

thing from differentiability in functions of one real variable. Topics covered will include analyticity

and the Cauchy-Riemann equations, complex integration, Cauchy's Theorem and its consequences,

connections to power series, and the Residue Theorem and its applications. Prerequisite: MATH

224. Offered every other year.

MATH 360 TOPOLOGY

Credit: 0.5 QR

Topology is an area of mathematics concerned with properties of geometric objects that remain the

same when the objects are "continuously deformed." Three of these key properties 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. Engineering, chemistry and physics are among the subjects that find

topology useful. The course will touch on selected topics that are used in applications. Prerequisite:

MATH 222 or permission of instructor. Generally offered every two to three years.

MATH 368 DESIGN AND ANALYSIS ALGORITHMS

Credit: 0.5

This course introduces students to the analysis and design of computer algorithms. Upon

completion of this course, students will be able to do the following: 1) analyze the asymptotic

performance of algorithms; 2) demonstrate a familiarity with major algorithms and data structures;

3) apply important algorithmic design paradigms and methods of analysis, and; 4) synthesize

efficient algorithms in common engineering design situations. Prerequisite: MATH 222 and SCMP

118 or PHYS 270 or equivalent.

MATH 435 ABSTRACT ALGEBRA II

Credit: 0.5 QR

This course picks up where MATH 335 ends, focusing primarily on rings and fields. Serving as a

good generalization of the structure and properties exhibited by the integers, a ring is an algebraic

structure consisting of a set together with two operations — addition and multiplication. If a ring

has the additional property that division is well-defined, one gets a field. Fields provide a useful

generalization of many familiar number systems: the rational numbers, the real numbers and the

complex numbers. Topics to be covered include polynomial rings; ideals; homomorphisms and ring

quotients; Euclidean domains, principal ideal domains, unique factorization domains; the Gaussian

integers; factorization techniques and irreducibility criteria. The final block of the semester will

serve as an introduction to field theory, covering algebraic field extensions, symbolic adjunction of

roots; construction with ruler and compass; and finite fields. Throughout the semester there will be

an emphasis on examples, many of them coming from calculus, linear algebra, discrete math and

elementary number theory. There also will be a heavy emphasis on the reading and writing of

mathematical proofs. Prerequisite: MATH 335. Offered every other spring.

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MATH 441 REAL ANALYSIS II

Credit: 0.5 QR

This course follows MATH 341. Topics will include a study differentiation and (Riemann)

integration of functions of one variable, sequences and series of functions, power series and their

properties, iteration and fixed points. Other topics may be included as time permits. For example: a

discussion of Newton's method or other numerical techniques; differentiation and integration of

functions of several variables; spaces of continuous functions; the implicit function theorem; and

everywhere continuous, nowhere differentiable functions. Prerequisite: MATH 341. Offered every

other spring.

MATH 493 INDIVIDUAL STUDY

Credit: 0.25-0.5

Individual study is a privilege reserved for students who want to pursue a course of reading or

complete a research project on a topic not regularly offered in the curriculum. It is intended to

supplement, not take the place of, coursework. Individual study cannot be used to fulfill

requirements for the major. Individual studies will earn 0.25–0.50 units of credit. To qualify, a

student must identify a member of the mathematics department willing to direct the project. The

professor, in consultation with the student, will create a tentative syllabus (including a list of

readings and/or problems, goals and tasks) and describe in some detail the methods of assessment

(e.g., problem sets to be submitted for evaluation biweekly; a 20-page research paper submitted at

the course's end, with rough drafts due at given intervals, and so on). The department expects the

student to meet regularly with his or her instructor for at least one hour per week. All standard

enrollment/registration deadlines for regular college courses apply. Because students must enroll

for individual studies by the end of the seventh class day of each semester, they should begin

discussion of the proposed individual study preferably the semester before, so that there is time to

devise the proposal and seek departmental approval before the registrar's deadline. Permission of

instructor and department chair required. No prerequisite.

MATH 498 SENIOR HONORS

Credit: 0.25 QR

This course will consist largely of an independent project in which students read several sources to

learn about a mathematical topic that complements material studied in other courses, usually an

already completed depth sequence. This study will culminate in an expository paper and a public or

semi-public presentation before an audience consisting of at least several members of the

mathematics faculty as well as an outside examiner. Permission of department chair required.

Prerequisite: Senior standing and at least one "depth sequence" completed.

Courses in Statistics

STAT 106 Elements of Statistics

Credit: 0.5 QR

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. The course concludes with inference regarding correlation, linear regression,

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chi-square tests for two-way tables and one-way ANOVA. Statistical software will be used

throughout the course, and students will be engaged in a wide variety of hands-on projects. No

prerequisite. Offered every semester.

STAT 116 STATISTICS IN SPORTS

Credit: 0.5 QR

Appropriate applications of statistical methods have changed the way some Major League Baseball

teams manage the game. (See "Moneyball: The Art of Winning an Unfair Game.") Statistics are used

in other sports to evaluate the performance of individual players or teams. The focus of this course

will be on the proper application of statistical models in sports. Students will use appropriate

methods to examine interesting questions such as: Are there unusual patterns in the performance

statistics of "steroid sluggers" such as Barry Bonds and Mark McGwire or pitchers such as Roger

Clemens? Other possible topics include the impact of a penalty kick in soccer, of home field

advantage in football, of technological improvements in golf or cycling, and of training methods in

marathon running. Although the sport and question of interest will change, the focus on proper

applications of appropriate statistical methods will remain the same. Students will analyze data and

present their results to the class. Oral and written reports will be expected. No prerequisite. Offered

every other year.

STAT 206 DATA ANALYSIS

Credit: 0.5 QR

This course focuses on choosing, fitting, assessing and using statistical models. Simple linear

regression, multiple regression, analysis of variance, general linear models, logistic regression and

discrete data analysis will provide the foundation for the course. Classical interference methods

that rely on the normality of the error terms will be thoroughly discussed, and general approaches

for dealing with data where such conditions are not met will be provided. For example,

distribution-free techniques and computer-intensive methods, such as bootstrapping and

permutation tests, will be presented. Students will use statistical software throughout the course to

write and present statistical reports. The culminating project will be a complete data analysis

report for a real problem chosen by the student. The MATH 106–206 sequence provides a thorough

foundation for statistical work in economics, psychology, biology, political science and many other

fields. Prerequisite: STAT 106 or 116 or a score of 4 or 5 on the Statistics AP exam. Offered every

semester.

STAT 216 NONPARAMETRIC STATISTICS

Credit: 0.5 QR

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 also will 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. Prerequisite: STAT 106, 116 or a score of 4 or 5 on the Statistics AP exam

or permission of instructor. Offered every other year.

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STAT 416 LINEAR REGRESSION MODELS

Credit: 0.5 QR

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 also may be studied if time permits.

Prerequisite: STAT 106 and MATH 224. Offered every other spring.

STAT 436 MATHEMATICAL STATISTICS

Credit: 0.5 QR

This course follows MATH 336 and introduces the mathematical theory of statistics. Topics include

sampling distributions, order statistics, point estimation, maximum likelihood estimation, methods

for comparing estimators, interval estimation, moment generating functions, bivariate

transformations, likelihood ratio tests and hypothesis testing. Computer simulations will

accompany and corroborate many of the theoretical results. Course methods often will be applied

to real data sets. Prerequisite: MATH 336. Offered every other spring.

STAT 493 INDIVIDUAL STUDY